The Biomechanical Foundation of Clinical Orthodontics 0867156511, 9780867156515

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The Biomechanical Foundation of Clinical Orthodontics

The Biomechanical Foundation of Clinical Orthodontics Charles J. Burstone, dds, ms Professor Emeritus Division of Orthodontics School of Dental Medicine University of Connecticut Farmington, Connecticut

Kwangchul Choy, dds, ms, phd Clinical Professor Department of Orthodontics College of Dentistry Yonsei University Seoul, Korea

Quintessence Publishing Co, Inc Chicago, Berlin, Tokyo, London, Paris, Milan, Barcelona, Istanbul, Moscow, New Delhi, Prague, São Paulo, Seoul, and Warsaw

Library of Congress Cataloging-in-Publication Data Burstone, Charles J., 1928- , author. The biomechanical foundation of clinical orthodontics / Charles J. Burstone and Kwangchul Choy. p. ; cm. Includes bibliographical references and index. ISBN 978-0-86715-651-5 I. Choy, Kwangchul, author. II. Title. [DNLM: 1. Biomechanical Phenomena. 2. Orthodontic Appliances. 3. Malocclusion--therapy. 4. Orthodontic Appliance Design. WU 426] RK521 617.6'43--dc23 2014043660

© 2015 Quintessence Publishing Co, Inc Quintessence Publishing Co, Inc 4350 Chandler Drive Hanover Park, IL 60133 www.quintpub.com 5 4 3 2 1 All rights reserved. This book or any part thereof may not be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, or otherwise, without prior written permission of the publisher. Editor: Leah Huffman Design: Ted Pereda Production: Sue Robinson Printed in China

Contents Preface  vii In Memoriam   ix Contributors  x A Color Code Convention for Forces   xi

Part I. The Basics and Single-Force Appliances  1

1  Why We Need Biomechanics  3 2  Concurrent Force Systems  11 3  Nonconcurrent Force Systems and Forces on a Free Body  25 4  Headgear  39 5  The Creative Use of Maxillomandibular Elastics  63 6  Single Forces and Deep Bite Correction by Intrusion  89 7  Deep Bite Correction by Posterior Extrusion  117 8  Equilibrium  135 Part II. The Biomechanics of Tooth Movement  155

9  The Biomechanics of Altering Tooth Position   157 10  3D Concepts in Tooth Movement   193



Rodrigo F. Viecilli

11  Orthodontic Anchorage   199



Rodrigo F. Viecilli

12  Stress, Strain, and the Biologic Response   209



Rodrigo F. Viecilli

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Part III. Advanced Appliance Therapy  227

13  Lingual Arches   229 14  Extraction Therapies and Space Closure   275 15  Forces from Wires and Brackets   323 16  Statically Determinate Appliances and Creative Mechanics   369



Giorgio Fiorelli and Paola Merlo

17  Biomechanics and Treatment of Dentofacial Deformity   389

Mithran Goonewardene and Brent Allan

18  The Biomechanics of Miniscrews   433

Kee-Joon Lee and Young Chel Park

Part IV. Advanced Mechanics of Materials  451

19  The Role of Friction in Orthodontic Appliances   453 20  Properties and Structures of Orthodontic Wire Materials   477



A. Jon Goldberg and Charles J. Burstone

21  How to Select an Archwire   491 Part V. Appendices  503 Hints for Developing Useful Force Diagrams   505 Glossary  511 Solutions to Problems   515

Index  563

vi

Preface

Historically, the mainstay of orthodontic treatment has been the appliance. Orthodontists have been trained to fabricate and use appliances and sequences of appliance shapes called techniques. However, appliances are only the instrument to produce force systems, which are the basis of tooth position and bone modification. And yet a thorough understanding of scientific biomechanics has not been a central part of orthodontic training and practice. Both undergraduate and graduate courses in most dental schools lack sound courses in mechanics and physics. What makes this problem worse is that there are few textbooks that describe biomechanics in a way that is suitable for the clinician. The authors hope this text will fill this void. This book was motivated by the request of orthodontists at all levels—from graduate students to experienced clinicians—to learn, understand, and apply scientific orthodontics and, in particular, efficiently manipulate forces in their everyday practices. This is particularly relevant at this time, when orthodontics is undergoing a wide expansion in scope. Twenty-first-century orthodontics has introduced substantial changes in the goals and procedures: bone modification by orthognathic surgery and distraction osteogenesis, airway considerations, temporary anchorage devices, plates and implants, brackets with controlled ligation forces, new wire materials, and nonbracket systems such as aligners. No longer can clinicians depend entirely on their technical skills in the fabrication and selection of appliance hardware to adequately treat their patients. The establishment of treatment goals and the force systems to achieve them has become the paramount characteristic of contemporary orthodontics. Different orthodontic audiences can benefit in special ways from a force-driven approach to treatment. The clinician is aided in the selection of appliances, creative appliance design, and treatment simulation. Simulation is the most valuable because it allows the clinician to plan different strategies using force systems and then select the best. It enables more predictive appliance shapes that approach optimal forces. Unlike an older approach of trying

out new procedures directly in the mouth, it is also cost-effective. Particularly in orthodontics, clinical evaluation requires long-term observation. With sound theory, many appliances can be evaluated so that long-term studies or trials can be avoided. While commercial orthodontic companies may not initially welcome clinical orthodontists who are knowledgeable in biomechanics, it is to their advantage when new important products are introduced to be able to discuss the innovations with scientifically trained clinicians. Researchers in orthodontic physics and material science also need this background. Biologic research at all levels also needs to control force variables. Studies on experimental animals where forces or stresses are delivered must control the force system to have valid results. Many times biologists do not understand the forces in their research and, hence, erroneous or insignificant results are obtained. Because most orthodontists do not have a strong background in physics and mathematics, the goal of this book is simplicity and accuracy in developing a scientific foundation for orthodontic treatment. In an orderly, step-by-step approach, important concepts are developed from chapter to chapter, with most chapters building on the previous one. From the most elementary to the most advanced concepts, examples from orthodontic appliances are used to demonstrate the biomechanical principles; thus, the book reads like an orthodontic text and not a physics treatise. Yet the principles, solutions, and terminology are scientifically rigorous and accurate. The biomechanics described in the book are ideal for teachers and students. The simplest way to teach clinical orthodontics is to describe the force systems that are used. Clear force diagrams are far better than vague descriptions. The teaching of the past, such as “I make a tip-back bend here” or “I put a reverse curve of Spee in the arch” is obviously lacking. What is the best way to learn biomechanics? The simplest approach is to carefully read each chapter and to understand the fundamental principles. Then solve each of the problems at the end of the chapter. It will be quickly apparent if one genuinely vii

understands the material. Over time, introduce biomechanics into your practice. When undesirable side effects are observed, use what has been learned to explain the problem. How could the side effect be avoided with an altered force system and appliance? Critically listening to lectures and reading articles can also be good training for developing a high level of biomechanical competence. One learns to bond a bracket quickly, but development of creative-thinking skills using biomechanics will take time. It was the intent of the authors to write a basic book on orthodontic biomechanics that would be simple and readable. Clear diagrams and clinical cases throughout ensure that it is neither dull nor pedantic. Our philosophy is that the creative thinking involved in manipulating forces and appliance design should be fun.

Note on the metric system The authors have adopted the metric system as their unit system of choice. However, the long shadow of American orthodontics has influenced the terminology in this book. Because the United States is the only major country not to fully adopt the metric system and is a major contributor to the literature, some units used throughout the book are not metric. Tradition and familiarity require some inconsistencies: inches are used for wire and bracket slot sizes, and a nonstandard unit—the “gram force”—is the force unit. It is our hope that the specialty of

Sadly for us, after finishing this book, a giant fell. Most of the contents of this book are based on Dr Burstone’s energetic and rigorous research for more than 200 research articles. The format of this book was adapted from the lectures on biomechanics that we gave at the University of Connecticut and Yonsei University for many years. Over the last 3 years, my work with Dr Burstone to convert those lectures and ideas into this book was one of the most challenging, most exciting, and the happiest moments in my life. As one of his students, an old friend, and a colleague, I have to confess that all of the concepts in this book are his. In the beginning, Force was created with the Big Bang. Fifteen billion years later, Newton discovered the Law of Force in the universe. How­ever, the

viii

orthodontics will adhere fully to the International System of Units in the future; therefore, future editions of this book will most likely use only metric units.

Acknowledgments This book would not have been possible without the input of many graduate students and colleagues. One of the authors (CJB) has been teaching graduate students for over 62 years. Long-term teaching has guided us both in how to most effectively present material and where most difficulties lie in acquiring biomechanical skills in a group of biologically trained orthodontists. This book could not have been developed in this manner without their intriguing questions and interaction. Special thanks are given to the staff at Quintessence Publishing for their valuable contribution in the development of this book: Lisa Bywaters, Director of Publications; Sue Robinson, Production Manager, Book Division; and particularly Leah Huffman, our editor, who worked so hard on a difficult book combining biology, physics, and clinical practice complicated by specialized dental and physics terminologies and equations. Dr Choy wishes to acknowledge the help he received from his wife Annie and his daughter Christa in the preparation of the manuscript. He is also grateful to his student Dr Sung Jin Kim for taking the time out of his busy schedule to review the questions and answers.

knowledge of how to control orthodontic force remained an occult practice that was only revealed through years of orthodontic apprenticeship. It was Dr Burstone who uncovered the magic and found the principles governing this treatment method that was once thought to be mysterious. There is no doubt that the Law of Orthodontic Force was his discovery. I would like to share Dr Burstone’s words from his last lecture with me on February 11, 2015, in Seoul: “Don’t believe blindly in experience, but believe in theory, and think creatively.” My father shaped my body; you shaped my thoughts. Charles, our dearest friend, may you rest in peace. Kwangchul Choy

In Memoriam

Dr Charles J. Burstone (1928–2015)

Dr Charles J. Burstone, orthodontist, educator, researcher, and friend to many, passed away February 11, 2015, of an apparent heart attack in Seoul, Korea. He died doing what he loved to do and in a place where he loved to be. Dr Burstone is well known for the development of the field of scientific biomechanics. He was a master teacher in orthodontics who could bridge the gap between understanding key engineering concepts and applying them to clinical practice. He made biomechanics understandable by showing how to use simple engineering principles to solve most orthodontic problems. He developed the Segmented Arch Technique through the use of sound engineering principles. Dr Burstone was unwavering in his enthusiasm for student learning and was dedicated to ensuring clinical excellence in his students. When I was a student, I can remember reviewing a patient’s treatment plan and him asking me, “What do you want to do with the lower incisors and why?” He emphasized the importance of having clear, specific, and defensible treatment objectives and then designing mechanical plans that would achieve those treatment objectives, step by step.

Over his lifetime, Dr Burstone trained hundreds of orthodontists, first at Indiana University and then later at the University of Connecticut. He served as Department Chairman while at each institution. He was a recipient of many awards and honors and remained active in organized dentistry throughout his life, serving in many positions and lecturing around the world. Dr Burstone also had a deep connection to Korea. He served there during the Korean War, and his photographs and movies from this period depicted everyday Korean life in a time of conflict. The National Folk Museum in Seoul developed an exhibit around his images entitled “Korea, 1952,” and his images were also used in a Korean documentary about the Korean War. He was devoted to Korea, and it is indeed fitting that his last lecture was delivered in Seoul. He truly loved his profession and was a beloved mentor and colleague to many. He leaves the worldwide orthodontic community to mourn his passing. Michael R. Marcotte, dds, msd Bristol, Conneticut

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Contributors Brent Allan, BDS, MDSc

A. Jon Goldberg, PhD

Head of Oral and Maxillofacial Surgery St John of God Hospital, Subiaco Perth, Australia

Professor Department of Reconstructive Sciences Institute for Regenerative Engineering University of Connecticut Farmington, Connecticut

Consultant in Oral and Maxillofacial Surgery Oral Health Centre School of Dentistry The University of Western Australia Nedlands, Australia

Charles J. Burstone, DDS, MS Professor Emeritus Division of Orthodontics School of Dental Medicine University of Connecticut Farmington, Connecticut

Kwangchul Choy, DDS, MS, PhD Clinical Professor Department of Orthodontics College of Dentistry Yonsei University Seoul, Korea

Giorgio Fiorelli, MD, DDS Professor Department of Orthodontics University of Siena Siena, Italy Private Practice Arezzo, Italy

Mithran Goonewardene, BDSc, MMedSc Program Director, Orthodontics
 School of Dentistry The University of Western Australia
 Nedlands, Australia

Kee-Joon Lee, DDS, MS, PhD Professor Department of Orthodontics College of Dentistry Yonsei University Seoul, Korea

Paola Merlo, DDS Private Practice Lugano, Switzerland

Young Chel Park, DDS, MS, PhD Professor Department of Orthodontics College of Dentistry Yonsei University Seoul, Korea

Rodrigo F. Viecilli, DDS, PhD Associate Professor Center for Dental Research Department of Orthodontics Loma Linda University School of Dentistry Loma Linda, California x

A Color Code Convention for Forces This book has several force illustrations that are used for different applications; there are activation and deactivation forces, equivalent forces, and resultant and component forces. To make it easier for the reader to understand the logical development of important concepts, a color code convention is utilized in this book.

Solid straight arrows and solid curved arrows represent forces and moments, respectively. Red arrows are forces that act on the teeth. Newton’s Third Law tells us that there are equal and opposite forces acting on the wire or an appliance.

Gray arrows denote unknown or incorrect forces.

Forces acting on a wire are drawn in blue. In special situations, forces can act both on a wire and on the teeth; in this book, therefore, depending on the point of view, the function being considered determines the color of the arrow.

Body motion including tooth motion is shown by a dotted straight or curved arrow. Motion arrows that describe linear and angular displacement are purposely different so that they are not confused with forces or moments.

Equivalent forces such as a force and a couple or components are identified with yellow arrows.

The diagrams for the “Problems” in each chapter and their solutions at the end of the book are kept simple, so the standard code above is not used. Problem figures for emphasis show known and unknown forces as green arrows. Solutions are shown in red arrows. Equilibrium diagrams (forces acting on a wire, for example) can show force arrows in blue in the solution section.

In situations where multiple forces must be shown, other colors may be utilized. xi

PART

I The Basics and Single-Force Appliances

CHAPTER

1 Why We Need Biomechanics “We build too many walls and not enough bridges.”

OVERVIEW



— Isaac Newton

Dentofacial changes are primarily achieved by the orthodontist applying forces to teeth, the periodontium, and bone. Hence, the scientific basis of orthodontics is physics and Newtonian mechanics applied to a biologic system. The modern clinician can no longer practice or learn orthodontics as a trade or a technique. He or she must understand forces and how to manipulate them to optimize active tooth movement and anchorage. Communication with fellow clinicians and other colleagues in other fields requires a common scientific terminology and not a narrow “jargon.” There is no such thing as a unique “orthodontics physics” divorced from the rest of the scientific community. New appliances and treatment modalities will need a sound biomechanical foundation for their development and most efficient use.

3

1 Why We Need Biomechanics Every profession has its trade tools. The carpenter uses a hammer and a saw. The medical doctor may prescribe medication and is therefore a student of proper drug selection and dosage. Traditionally, the orthodontist is identified with brackets, wires, and other appliances. Such hardware is only a means to an end point: tooth alignment, bone remodeling, and growth modification. The orthodontist achieves these goals by manipulating forces. This force control within dentofacial orthopedics is analogous to the doctor’s dosages. An “orthodontic dosage” includes such quantities as force magnitude, force direction, point of force application (moment-to-force ratios), and force continuity. Historically, because the end point for treatment is the proper force system, one might expect the development and usage of orthodontic appliances to be based on concepts and principles from physics and engineering. On the contrary, however, most appliances have been developed empirically and by trial and error. For that reason, treatment may not be efficient. Many times undesirable side effects are produced. If appliances “work,” at a basic minimum the forces must be correct, which is independent of the appliance, wires, or brackets. Conversely, when bad things happen, there is a good possibility that the force system is incorrect. These empirically developed appliances rarely discuss or consider forces. Forces are not measured or included in the treatment plan. How is it possible to use such mechanisms for individualized treatment? The answer is that they are shape driven rather than force driven. Different shapes and configurations are taught and used to produce the desired tooth movement. This approach is not unreasonable because controlled shapes can lead to defined wire deflections that relate to the produced forces. Unfortunately, there is so much anatomical variation among different patients that using a standard shape for a bracket or a wire or even modifying that shape will not always produce the desired results predictably. An example of a shape-driven orthodontic appliance is what E. H. Angle called the ideal arch. In a typical application of this ideal arch, an archwire is formed with a shape so that if crooked teeth (brackets) are tied into the arch, the deflected wire will return to its original shape and will correctly align the teeth. Today, wires have been improved to deflect greater distances without permanent deformation, and brackets may have compensations to correct anatomical variation in crown morphology. The principle is the same as Angle’s ideal arch, but this approach is now called straight wire. Straightwire appliances can efficiently align teeth but can 4

also lead to adverse effects in other situations. The final tooth alignment may be correct, but the occlusal plane may be canted or the arch widths disturbed. Intermediate secondary malocclusions can also occur. An understanding of biomechanical principles can improve orthodontic treatment even with shape-driven appliances by identifying possible undesirable side effects before any hardware is placed. Aligners also use the shape-driven principle of an ideal shape. All orthodontic treatment modalities, including different brackets, wires, and techniques, can be improved by applying sound biomechanics, yet much of clinical orthodontics today is delivered without consideration of forces or force systems. This suggests that many clinicians believe that a fundamental knowledge and application of biomechanics has little relevance for daily patient care.

Scientific Biomechanics There are many principles and definitions used in physics that are universally accepted by the scien­ tific community. At one extreme, there is classical physics—concepts developed by giants like Newton, Galileo, and Hooke. There are also other scientific disciplines, such as quantum mechanics. What the authors find disturbing is the hubris of what they call pseudo-biomechanics—new physical principles developed by orthodontists that are separate and at odds with classical mechanics. Orthodontists’ journal articles and lecture presentations are filled with figures and calculations that do not follow the principles of classical mechanics. Orthodontists may be intelligent, but we should not think we can compete with the likes of Newton. There is another major advantage in adopting scientific or classical mechanics. The methodology, terminology, and guiding principles allow us to communicate with our scientific colleagues and set the stage for collaborative research. Imprecise words can confuse. We speak of “power arms,” but power has a different meaning to an engineer than it does to a politician or a clinician. Force diagrams in orthodontic journals are difficult to decipher and may not be in equilibrium. The concepts, symbols, and terminology presented in this book are not trade jargon but will be widely recognized in all scientific disciplines. Note that the theme of this book is orthodontic biomechanics. The “bio” implies the union of biologic concepts with scientific mechanics principles. Let us now consider some specific reasons why the modern orthodontist needs a solid background in

Selecting or Designing a New Appliance biomechanics and the practical ways in which this background will enhance treatment efficacy.

Optimization of Tooth Movement and Anchorage The application of correct forces and moments is necessary for full control during tooth movement, influencing the rates of movement, potential tissue damage, and pain response. Furthermore, different axes of rotation are required that are determined by moment-to-force ratios applied at the bracket. For example, if an incisor is to be tipped lingually around an axis of rotation near the center of the root, a lingual force is applied at the bracket. If the axis of rotation is at the incisor apex, the force system must change. A lingual force and lingual root torque with a proper ratio must then be applied. These biomechanical principles are relevant to all orthodontic therapy and appliances—headgears, functional appliances, sliding mechanics, loops, continuous arches, segments, and maxillomandibular elastics (also sometimes referred to as intermaxillary elastics). The hardware is only the means to produce the desired force system. Equally important as active tooth movement is the control over other teeth so that they do not exhibit undesirable movements. This is usually referred to as anchorage and depends in part on optimally combining and selecting forces. Some orthodontists might think that anchorage is determined by factors independent of forces. For instance, the idea that more teeth means greater anchorage is very limiting. Working with forces can be more effective in enhancing anchorage, such as in pitting tipping movement against translation. All archwires produce multiple effects. Many of these effects are undesirable, which should also be considered anchorage loss. In a sense, a new malocclusion is created, resulting in an increased treatment time. Let us assume that translation of teeth could be accomplished at the rate of 1 mm per month. In a typical orthodontic patient, rarely does tooth movement exceed 5 mm. Not considering any waiting for growth, total treatment time should be no longer than 5 months. So why is treatment longer? Usually, more time is required to correct side effects. The use

of temporary anchorage devices (TADs) may eliminate side effects. As will be shown in chapter 18, a good biomechanical understanding is required to successfully use TADs; otherwise, adverse effects can still occur.

Selecting or Designing a New Appliance New appliances and variations of older existing appliances are continually presented in journal articles or at meetings. What is the best way to evaluate these appliances? One approach is to try them in your clinical practice. This evaluation will be quite limited because there is a lot of variation in a small sample of malocclusions. Moreover, it is timeconsuming and unfair to the patient. Because treatment is so long term, it may take many years to arrive at a conclusion on the efficacy of a new appliance. A better approach would be an evaluation based on sound and fundamental biomechanical principles. Drawing some force diagrams is much easier than protracted treatment. This is particularly valid when considering that most new appliances and techniques do not stand the test of time. Orthodontists have always been very creative. Not all great research has come from university research laboratories. Whether in their own offices or on typodonts in the lab, clinicians have made significant achievements in bracket design, various wire configurations, and treatment sequences (techniques). It is much more efficient to work with a pencil and a sheet of paper (or a computer) than it is to go through the demanding trial and error approach. The best appliances of the future will require rigorous engineering and sound biomechanical methodology. Let us assume for now that we have selected the best appliance for our individual patient. There are still many variables that require a sound biomechanical decision. For example, what size wire should we use? A 0.014-inch nickel-titanium (Ni-Ti) superelastic wire is not the same as a 0.014-inch Nitinol wire. The choice between a 0.016- and a 0.018-inch stainless steel (SS) archwire is significant. The larger wire gives almost twice the force.

5

1 Why We Need Biomechanics

Research and Evaluation of Treatment Results The clinician can be surprised at the progress of a patient. When the patient arrives for an appointment, mysterious changes are sometimes observed. Why is there now an open bite or a new reverse articulation (also referred to as crossbite), or why is the malocclusion not improving? These unexpected events may be attributed to biologic variation. Or it may be the wrong appliance (or manufacturer). In reality, most of the clinical problems that develop can be explained by deviation from sound biomechanical principles. Thus, an understanding of applied biomechanics allows the orthodontist to determine both why a puzzling and problematic treatment change occurred and also what to do to correct it. Sometimes the force system is almost totally incorrect; other times, a small alteration of the force system can produce a dramatic improvement. The prediction of treatment outcomes requires precise control and understanding of the applied force system as well as the usual cephalometric and statistical techniques. Good clinical research must control all of the known variables if the efficacy of one appliance is to be compared to another. Let us consider a study that is designed to compare the different outcomes between a functional appliance and an occipital headgear. It is insufficient to simply specify headgear or even occipital headgear. Headgears can significantly vary not only in force magnitude but also in direction and point of force application. It is little wonder that some research studies lead to ambiguous and confusing conclusions. A biomechanical approach to clinical studies opens up new avenues for research to help predict patient outcomes. The relationship between forces and tooth movement and orthopedics requires more thorough investigation. Relationships to be studied include force magnitudes, force constancy, momentto-force ratios at the bracket, and stress-strain in bone and the periodontal ligament. Force systems and “dosage” determine not only tooth or bone displacement with its accompanying remodeling; unwanted pathologic changes involving tissue destruction can also occur. Root resorption, alveolar bone loss, and pain are common undesirable events during treatment. Some histologic and molecular studies suggest a relationship between force or stress and tissue destruction. Although other variables may be involved, a promising direction for research is between stress-strain 6

and the mechanisms of unwanted tissue changes. To control pain and deleterious tissue destruction, it is likely that future research will validate that “dosage” does count.

How Scientific Terminology Helps As previously discussed, orthodontic appliances work by the delivery of force systems. In this book, the methods and terminology of the field of physics are adopted. Tooth movement is only part of a subset of a broader field of physics. This allows orthodontic scientists and clinicians to communicate with the full scientific community outside of dentistry, setting the stage for collaborative research. Many of the specialized orthodontic terms produce a jargon that is imprecise and certainly unintelligible to individuals in other disciplines. The orthodontist speaks of “torque.” Sometimes it means a moment (eg, the force system). At other times, however, it means tooth inclination (eg, “the maxillary incisor needs more torque”). Imprecision leads to faulty appliance use, which will be discussed later. A universal biomechanical and scientific language is the simplest way to describe an appliance and how it works. It not only allows for efficient communication with other disciplines for joint research but also offers the best way to teach clinical orthodontics to residents or other students. The old approach was primarily to teach appliance fabrication. Treating patients was just following a technique. An adjustment was how you shaped an arch: “Watch how I make a tip-back bend, and duplicate it.” Emphasis was on shape, and therefore we can call it shape-driven orthodontics. The biomechanical approach emphasizes principles and force systems. This approach, force-driven orthodontics, is the theme of this book. With clear terminology and sound scientific principles, the learner can better understand how to fabricate and use any appliance or configuration. It shortens the time and confusion in teaching students. It is said that a number of years of experience is required to complete the education of an orthodontist. Some say as many as 10 years. Why? It is the time needed to make and learn from your mistakes. If the student understands the biomechanical basis of an appliance, many common mistakes will never be made. It is not only the beginning student who benefits from sound biomechanical teaching. As new appliances are developed, the experienced orthodontist can better learn the “hows” and “whys” so that the

Advantages of Biomechanical Knowledge

Fig 1-1 Jacques Carelman painting of a pitcher. Although the pitcher looks reasonable, it will not actually pour coffee, much like some orthodontic appliances seem reasonable but do not actually work.

learning interval is shortened. More important, fewer errors will be made. Lectures at meetings will be shorter and easier to understand.

Knowledge Transfer Among Appliances The orthodontist may feel comfortable treating with a given appliance because routine treatment has become satisfactory and predictable. However, if he or she wants to change appliances (eg, moving from facial to lingual orthodontics), the mechanics may not be the same. When lingual orthodontic appliances were introduced a few years ago, some orthodontists were troubled that their mechanics (wire configurations and elastics) did not do the same on the lingual that they did on the buccal. Biomechanical principles that determine the equivalent force system on the lingual are simple to apply. Clinicians could have saved much “learning time” spent doing trial and error experimentation. A few simple calculations covered in chapter 3 could have helped the clinician avoid any aggravation.

Advantages of Biomechanical Knowledge Historically, there have been many exaggerated claims made by clinicians and orthodontic companies about the superiority of appliances or tech-

Fig 1-2  A wine bottle in a curved wine rack. Although it would seem that the bottle would fall over, it is in a state of static equilibrium so that it does not move. Similarly, some orthodontic principles that seem illogical are actually quite effective because they are based on sound biomechanics.

niques. Hyperbole is used with such terms as controlled, hyper, biologic, and frictionless. Journals and orthodontic associations are now doing a better job of monitoring possible conflicts of interest. The best defense against unwanted salesmanship is to stay vigilant and always apply scientific biomechanics. What may look possible becomes clearly impossible when the underlying principles are understood. The pitcher in Jacques Carelman’s painting looks reasonable, but it will not pour coffee (Fig 1-1). On the other hand, a sound biomechanical background can make possible what appears impossible. A filled wine bottle is placed in a curved wine rack. The rack is not glued to the table, so one might think that the bottle will tip over, but it does not (Fig 1-2). As will be discussed later, the bottle is in static equilibrium and, hence, the impossible becomes possible. Figure 1-3a shows an auxiliary root spring on an edgewise arch designed to move the maxillary incisor roots to the lingual. Is this possible or impossible? If the spring is bent to push lingually on the crown, the edgewise wire will twist to produce labial root torque (Figs 1-3b). This is an example of an impossible appliance. Placing the auxiliary on a round wire makes the mechanism possible (Fig 1-3c). The many advantages of a biomechanical knowledge for the clinician, including better and more efficient treatment, have been mentioned here. But what about for the patient? Obviously, one benefit is better and shorter treatment. Another significant advantage is the elimination of undesirable side effects. Side effects might require more patient cooperation. To correct problems, new elastics, headgear, surgery, or TADs may be prescribed. With better me7

1 Why We Need Biomechanics Fig 1-3  (a) An auxiliary root spring on an edgewise arch designed to move the maxillary incisor roots to the lingual. (b) If the spring is bent to push lingually on the crown, the edgewise wire will twist to produce labial root torque, making this appliance impossible. (c) Placing the auxiliary on a round wire makes the mechanism possible.

a

b

chanics, such anchorage loss would not have happened. It is not fair to ask our patients to cover up our mistakes with added treatment time or added therapy requiring considerable appliance wear, such as headgear. The future of the profession will be determined by how well we train our residents. Currently, not all graduate students are being trained in scientific biomechanics in any depth. Ideally, when a student graduates from a program, an understanding of biomechanics should be second nature. Otherwise, he or she will not be able to apply it clinically. Lectures and problem-solving sessions are very useful; however, biomechanical principles must be applied during chairside treatment. Carefully supervised patients and knowledgeable faculty are the key ingredients to teaching biomechanics. Conventional wisdom in orthodontics has emphasized the appliance. Graduate students and orthodontists were taught to fabricate appliances or make bends or adjustments in these appliances. Perhaps some lip service was given to biomechanics or biology, but basically the clinician was a fabricator and user of appliances. Treatment procedures were organized into a technique sequence. This empirical approach to clinical practice led to the development of different schools of thought, sometimes identified with the name of a leading clinician. Shape8

c

driven orthodontics (where forces are not considered) is usually a standard sequence or cookbook approach that does not adequately consider the individual variation among patients. The new wisdom is not appliance oriented. It involves a thinking process in which the clinician identifies treatment goals, establishes a sequence of treatment, and then develops the force systems needed for reaching those goals. Only after the force systems have been carefully established are the appliances selected to obtain those force systems. This is quite a contrast to the older process in which the orthodontist considered only wire shape, bracket formulas, tying mechanisms, friction, play, etc, without any consideration whatsoever of the forces produced. It is easy for the clinician to harbor negative feelings about orthodontic biomechanics. Some may believe that treatment mechanics are only common sense and that intuition and everyday knowledge are sufficient. Others may regard biomechanics as too sophisticated, demanding, and complicated for daily practice. Indeed, many of us became dentists and orthodontists because, as students, we disliked mathematics and physics and preferred the biologic disciplines. Fortunately, the physics used in orthodontics is not complicated, and many simple principles and concepts can be broadly and practically ap-

Advantages of Biomechanical Knowledge plied. Orthodontics is not nuclear physics. Scientific biomechanical thinking is actually easier than vague and disorderly thought processes and simplifies our overall treatment. The genius of pioneers such as Newton is that their principles are anything but common sense. Aristotle reasoned that if a heavy weight and a light weight were dropped from the same height, the heavy weight would hit the ground first. This seems like common sense. Galileo, on the other hand, thought that both weights would hit the ground at the same time. He supposedly dropped two different weights from the Leaning Tower of Pisa to prove his point. Many common-sense ideas are false. Common sense would tell you that the earth is flat and that the sun rotates around the earth, and yet the earth is round, and it rotates around the sun. As will be shown in this text, many of our conventional and accepted orthodontic ideas from the past are invalid. There are many textbooks and articles that describe techniques involving different types of brackets, sequences of wire change, and slot formulas, much like a recipe in a cookbook. Many malocclusions might be successfully treated following such cookbook procedures. However, surprises can occur as unpredicted problems develop during treatment. One or more recipes will not always work because malocclusions vary so much. Therefore, the clinician must seek sound biomechanical principles rather than a technique to correct the problem. Thus, bioengineering is needed not only for the challenging situation but also for the routine patient who may show an unexpected response to an appliance. Even if we typically treat by a certain technique, we must have biomechanical knowledge and skill in reserve, which will be required when unfortunate surprises strike. If that knowledge is not readily available because we do not continually apply it, we limit our ability to get out of trouble. By way of analogy: One of the authors recently tried to do some simple plumbing. When the house became flooded,

an experienced plumber was called, and his backup knowledge and expertise solved the problem. Unfortunately, when the orthodontist gets into trouble, he or she traditionally does not seek the advice of others, leading to either a poorer result or a lengthier treatment time. What about the “easy” case we may routinely treat successfully? It could be argued that applying creative biomechanics could also improve our treatment result or allow us to treat more efficiently. We might treat a Class II patient without extraction with some leveling arches and Class II elastics. A certain technique might work, negating any biomechanical thinking. However, the end point might be different than our treatment goals. Perhaps the mandibular incisors are undesirably flared or the occlusal plane angle steepened too much. The goals and quality of treatment can vary so much that it is difficult to define what a routine or “easy” case entails. It takes a very knowledgeable orthodontist to identify what an “easy” case really is. Technical competence is developed by fabricating and inserting appliances, but understanding principles involves thinking. Admittedly, technical skill is important in daily practice. But performing techniques without understanding the fundamental principles behind them is risky. At the same time, principles without technique lack depth. This book therefore explains the “hows” and “whys” of orthodontic treatment, which are inseparable. Orthodontic biomechanics is not just a theoretical subject for academics and graduate students. It is the core of clinical practice; orthodontists are biophysicists in that daily bread-and-butter orthodontics is the creative application of forces. The 21st century will be characterized by a major shift from shape-driven orthodontic techniques to a biomechanical approach to treatment, and with this shift will come rapid advancements in treatment and concepts.

9

CHAPTER

2 Concurrent Force Systems

“Goodbye, old friend. May the Force be with you.”

OVERVIEW

­— Obi-Wan Kenobi, From Star Wars Episode IV: A New Hope

The branch of physics dealing with forces is called mechanics. The most relevant Laws of Newton are the First and Third Laws. Many orthodontic questions and their solutions can be considered equilibrium situations, so Newton’s Second Law, which relates forces to bodies that accelerate, is less important. The division of mechanics describing bodies in equilibrium is called statics and for bodies that accelerate, kinetics. The simplest force system is force acting on a point; it is fully defined by force magnitude, force direction, and sense. Manipulating a force system includes adding a number of forces to obtain a resultant or breaking up a resultant into separate components. Forces are vectors that must be added geometrically and cannot be added algebraically. Simple orthodontic appliances that act on a point are maxillomandibular elastics (also known as intermaxillary elastics), finger springs, and cantilevers.

11

2 Concurrent Force Systems

a

Deactivated

100 g

b

a

100 g

Activated

Fig 2-1  Simple coil spring demonstrating Newton’s First Law. (a) Deactivated. (b) Activated. The spring is in equilibrium in both a and b.

Fig 2-2  (a and b) Newton’s Third Law. Equal and opposite forces act at the canine. FA (blue arrow), activation force on the coil spring; FD (red arrow), deactivation force on the canine.

Medical doctors may use thousands of medicines to treat their patients, but orthodontists use only one treatment modality: force. No matter what kinds of wires, springs, and brackets or appliances used, the hardware serves only as an intermediate tool to deliver a force or a series of forces. With proper force positioning and dosage, all kinds of tooth movement can be achieved. Therefore, knowledge of force is essential for understanding tooth movement. Because the word force has different meanings in common language and in physics, important definitions and concepts required for the application of force analysis to the field of orthodontics are developed in this chapter.

periodontium are examples of the first law. A simple orthodontic appliance component, a coil spring, is shown in Fig 2-1. The deactivated spring in Fig 2-1a is at rest; there are no forces acting on it. (This freebody diagram purposely ignores gravity and other nonrelevant forces.) The application of two forces in Fig 2-1b extends the spring, which can now be placed in the mouth between the anterior and posterior teeth for space closure. The forces are equal (100 g) and opposite, allowing the spring to remain in equilibrium. The spring deforms but does not accelerate, demonstrating the First Law. Newton’s Second Law (law of acceleration) states that when force is applied to an object, it accelerates proportional to the amount of force applied. The famous Newtonian formula is

The Field of Mechanics Mechanics is the field of physics dealing with the study of forces. Mechanics can be subcategorized into statics, kinetics, and material science. Statics deals with a force on a body with constant velocity, including a state of rest. Kinetics deals with a force on a body with acceleration. Finally, material science deals with the effect of forces on materials. The classic laws explaining the relationship between force and bodies were presented by Newton in 1686. Newton’s First Law (law of inertia) describes bodies at rest or bodies with uniform velocity (no acceleration): An object at rest tends to stay at rest, and an object in motion tends to stay in motion with the same velocity and in the same direction unless acted upon by an unbalanced force. This is the most important law for orthodontics, because it is the basis of all equilibrium applications. Activated appliances and restrained teeth within the bone and 12

b



F = ma

where m is mass, a is acceleration, and F is force. This formula defines the nature of force—an ability to accelerate an object. One would think that Newton’s Second Law would have important applications in orthodontics. Are teeth not moving? Although they move, they are not accelerating. Teeth are restrained objects and hence are bodies in equilibrium and at rest. Imagine a simple model in which a tooth is suspended by coil springs on all sides. Similar to the spring in Fig 2-1, the tooth is still in equilibrium after a force is applied to the crown. Therefore, this book does not cover applications in the field of kinetics. Newton’s Third Law (law of action and reaction) states that for every action there is always an equal and opposite reaction (ie, for every force there is an equal and opposite force). The commonly used ex-

Characteristics of a Force ample of this law is a rifle shot where the bullet feels the force and one’s shoulder feels the equal and opposite force. In Fig 2-2a, a coil spring is activated by a mesial force to allow its placement on the canine hook. Because this force (FA) produces an elongation of the elastic, it is called the activation force (Fig 2-2b). This force extends the elastic during the act of placement by the orthodontist, and later the canine hook maintains the mesial activation force, holding the elastic in place. At the canine hook, one observes the two equal and opposite forces of Newton’s Third Law (see Fig 2-2b). The blue force (FA) is the activation force (the force on the appliance), and the red force (FD) is the equal and opposite force on the tooth or the hook. This equal and opposite force (red arrow) is called the deactivation force and is in the direction of the tooth movement. In other words, the hook pulls on the elastic, and the elastic pushes on the hook. These action and reaction forces occur at the hook. In this example, it is also true that the canine and the molar feel equal and opposite forces, but this is not an expression of Newton’s Third Law. Why? The elastic is in equilibrium; hence, the forces on the elastic are equal and opposite. The explanation lies in Newton’s First Law, which covers equilibrium on bodies at rest (the elastic is not accelerating). Newton's Third Law is properly used when both activation and deactivation forces are showing on the canine (see Fig 2-2b). This chapter introduces how to manipulate or handle orthodontic forces. First, concurrent forces (ie, forces acting on a point) are considered. In the next chapter, this will be developed further to consider forces in three dimensions on a body.

Characteristics of a Force A force has three attributes: magnitude, direction and sense, and point of force application. Figure 2-3 shows three forces acting on a point (red dot), a hook on the maxillary arch. Because the hook defines the point of force application, only force magnitude and direction require further description. From where do the forces originate? Their source could be maxillomandibular elastics* or intra-arch elastics. Forces are vector quantities that cannot be added algebraically but are rather added geometrically. Note that the elastics have different angles to each other, representing different lines of force application and denoting their vector properties.

Fig 2-3  Elastic forces acting at a point. The hook (red dot) defines the point of force application. Different force magnitudes are represented by the length of the red arrows. The direction of the forces can be measured by the force angle to the occlusal plane.

Force magnitude The force magnitude is given in grams (g). The force magnitudes in Fig 2-3 are represented by arrows; the length of the arrow is proportional to the magnitude of the force. Note that the 150-g maxillomandibular elastic arrow is three times as long as the 50-g vertical elastic arrow and half the length of the 300-g intra-arch elastic arrow. Why are grams the unit of force in this example? This unit is technically incorrect, as shown below. Historically in America, ounces were used, and spring measuring scales were calibrated in ounces. More universal metric force gauges then became available, and the units were in grams. Generally, scales used by the layman for measuring body weight can be calibrated in pounds or kilograms. For the physicist, these are not force (weight) units but rather units for measuring mass. So let us briefly consider the relationship between mass and force. Again, the classic Newtonian formula is force equals mass times acceleration (F = m × a). The force is the product of mass (kilograms) and acceleration (m/s2). The unit of this magnitude of force is therefore kg•m/s2, and 1 kg•m/s2 equals 1 Newton (N). The terms gram weight and kilogram weight are therefore incorrect. Traditionally, orthodontists use the gram as the unit of force. In the strict sense as explained above, this is incorrect because grams are a unit of mass and not force. For example, gravity (a force) at sea level attracts a 100-g mass (the amount of material). The calculated acceleration of gravity is 9.8 m/s2. Let us now calculate how much force is acting on the 100-g mass at sea level using Newton’s Second Law. F=m×a F = 100 g × 9.8 m/s2 F = 0.98 kg•m/s2 = 0.98 N = 98 cN

*Traditionally, orthodontists have used the term intermaxillary elastics to denote elastics placed between the maxillary and mandibular arches, because both jaws used to be referred to as maxillae. However, because the mandible is no longer considered a maxilla, the term intermaxillary makes no sense, hence the new term: maxillomandibular elastics.

13

2 Concurrent Force Systems

a

b

Fig 2-4  (a) Force depends on gravitational acceleration, so people weigh less on the moon than on earth. (b) However, the same activation of an orthodontic appliance would produce the same force on the moon as on earth.

Scientifically, a centi-Newton (cN) is the correct unit for force, but in this book, grams will be used as the unit of force because of its tradition in orthodontics; perhaps this unit will be easier to understand for the clinician. However, the authors recommend that scientific publications and presentations use Newtons or centi-Newtons as the unit of choice. For purposes of practical conversion, 1 g equals 1 cN.* Perhaps in the not-so-distant future, an orthodontist might have a satellite office on the moon. If we attach a 100-g mass to a force gauge, as shown in Fig 2-4a, the measured gravitational force will be about 1 N on the earth but only 0.16 N on the moon. This is why people can jump with less effort on the moon, because they actually weigh less there. Let us now use this same force gauge to measure the force from an orthodontic appliance (Fig 2-4b). This type of force gauge uses a calibrated spring and has nothing to do with gravity. A spring gauge is based on Hooke’s law, where force is proportional to wire deflection. If the same appliance is used on the moon as on earth, there would be no difference in the forces, provided the activation is the same (see Fig 2-4b). Our imaginary orthodontist could therefore use the same appliances and activations used on earth, provided that there were no biologic differences required in outer space.

Force direction and sense Force also has sense and direction. The direction of the force is defined by its line of action. This direction is referred to as the sense. The arrows shown in Fig 2-3 demonstrate direction, sense, and the line of *More accurately, 1 g equals 0.98 cN, and 1 cN equals 1.02 g.

14

action of three elastics. The origin of each arrow is the point of force application (hook, red dot), the line (of action) indicates direction, and the arrowhead indicates the sense. The direction of the force in Fig 2-5 is demonstrated by the dotted line, and the arrowhead shows the sense. The two red forces have the same direction but different senses. To specify the direction of a force vector, a proper coordinate system is required; the direction of the force can be represented by the angle between a given axis of the coordinate system and line of action. There are several coordinate systems, but rectilinear Cartesian coordinates are most frequently used. Figure 2-6a shows the three axes of a Cartesian coordinate system and a sign convention in three dimensions. In this book, two-dimensional diagrams, such as those in Fig 2-6b, are used for simplicity’s sake. Any coordinate system and sign convention is acceptable, provided that it is clearly specified. The orientation of a coordinate system can be set arbitrarily, depending on the problem to be studied. In an orthodontic analysis, frequently used axes include the occlusal plane, the Frankfort horizontal, the midsagittal plane, and the long axis of a tooth. The direction of an orthodontic force is specified in accordance with a given established coordinate axis. For example, in Fig 2-7, a crisscross elastic (red arrow) is applied at 90 degrees to the mandibular right first molar relative to the midsagittal plane. What is the best coordinate system to evaluate the molar movement? Of the three shown (dotted lines), the authors would most likely select the system based on the mesiodistal or buccolingual axes of the tooth. Resolving the force into rectilinear

Characteristics of a Force

n Li e of tio ac n

Point of application Sense

Fig 2-5  Force sense and direction. The line of action (dotted line) demonstrates the direction, and the arrowheads denote the sense of the force.

a

b

Fig 2-6  Cartesian coordinates in three dimensions. (a) Three mutually perpendicular axes with a sign convention specified on each axis. (b) Two-dimensional diagrams with the same coordinates as those shown in a. For simplicity’s sake, most diagrams in this text show only two dimensions.

Fig 2-7  A crisscross elastic is attached at the buccal of the mandibular right molar. A coordinate system is selected that gives the information that is most useful. Here the mesiodistal crown axis system was selected. The elastic force (red arrow) has both lingual and mesial components of the force (yellow arrows).

15

2 Concurrent Force Systems

Fig 2-8 Law of transmissibility of force. The effect is the same whether the force is applied at the mesial or the distal of a canine as long as the force is along the same line of action (dotted line).

Fig 2-9  A force from an occipital headgear (FR , red arrow) can be resolved graphically into two rectilinear horizontal (Fx) and vertical (Fy) components (yellow arrows).

Manipulating Forces Components

Fig 2-10  The same force from an occipital headgear (FR) shown in Fig 2-9 is resolved into Fx and Fy components mathematically using simple trigonometric functions.

components tells us that there are both mesial and lingual forces (yellow arrows). It has been argued at some orthodontic meetings that there are advantages to canine retraction achieved by either pushing from the mesial or pulling from the distal. As observed in Fig 2-8, however, there is no difference in the line of action if the force is applied at the mesial or the distal. A force acting anywhere along this line of action has the same effect. In other words, a force can be moved along its line of action without changing its effect. This principle is called the law of transmissibility of force. The appliance may differ with either an open or closed coil spring, but if the force is along the same line of action, the response should be the same (assuming no other variables). A locomotion engine can either push or pull a train car with the same effect.

16

It is convenient to resolve a force into rectilinear components (ie, two forces at 90 degrees to each other). Another clinical way to look at direction is to ask how much force is parallel to the occlusal plane and how much is vertically directed. If distances are accurately drawn to represent the forces, the solution can be obtained graphically. A force from an occipital headgear is shown in Fig 2-9. The direction is clearly shown as 30 degrees to the occlusal plane. Note that the headgear force (FR, red) can be achieved by mentally walking from the hook (application point) at 30 degrees upward and backward. However, we can resolve this force into rectilinear components graphically by drawing two perpendicular lines: Force X (Fx) and Force Y (Fy ), with Fx parallel and Fy perpendicular to the occlusal plane. Now let us take our imaginary walk using these lines. Starting at the hook, we walk along the occlusal plane (Fx) to the right and then walk upward at 90 degrees to the occlusal plane (Fy ). This route may take longer, but we still end up at the apex of the original red arrow. Forces are vectors, so we can establish components using geometric addition. If measured, the two component force lengths (yellow) tell us the magnitude and sense of the vertical and horizontal rectilinear components of the original force. Although the Fy component is depicted at the arrowhead of Fx for analysis, Fy acts at the hook. During clinical visits, many times a diagram may be good enough to evaluate the rectilinear com-

Manipulating Forces

Fig 2-11  The two forces (red arrows) applied at the canine bracket can be added arithmetically to give a resultant (yellow arrow) because they act along the same line of action.

Fig 2-12  Two elastics (red arrows) applied at the canine hook. The resultant, FR (yellow arrow), is determined using the parallelogram method. The arrow (FR) connects the origin at the hook with the opposite corner of the parallelogram.

ponents of a force (graphic method). However, we might prefer to use an analytical method, employing some simple trigonometry. Figure 2-10 is the same diagram as Fig 2-9, where the angle of the headgear force can be any angle (θ). Fx and Fy can be determined using the following trigonometric relationships: Fy = FR sin θ Fx = FR cos θ

In Fig 2-12, two forces from maxillomandibular elastics (F1 = 300 g, F2 = 100 g) are applied to the canine hook. The magnitudes of each force are the same as those in Fig 2-11, but they lie on different lines of action. What is the magnitude of the resultant? If you said 400 g, the arithmetic total, the answer is incorrect. Forces are vectors and must be added geometrically. Forces F1 and F2 do not lie along the same line of action. The addition must be done graphically. Lines parallel to F1 and F2 are constructed, forming a parallelogram. A diagonal line (FR, yellow arrow) is drawn from the force origin (the hook) to the opposite corner of the constructed parallelogram. This line represents the vector sum of F1 and F2 and is the resultant force. The length of the diagonal line proportionally represents the force magnitude, and the angle to any plane represents the sense and direction of the resultant. Note that the length of FR (resultant) is not the arithmetic sum of the lengths of F1 and F2, and its direction is different than the applied individual elastics. The clinician might be advised to place a single elastic (the resultant) for simplicity rather than two, because the action on the arch will be the same.

Resultants Often clinical situations require that we add a number of forces. The canine shown in Fig 2-11 has two forces acting at the bracket from two chain elastics (red arrows). Because both elastics act along the same line of action, we can find the sum of forces by simple arithmetic (addition), remembering that a force vector has a sense (direction), and therefore sign (+ or –) must be considered.

(–100 g) + (+300 g) = +200 g

The principle that forces along the same line of action can be simply added together is important for orthodontists. The two elastics on the canine in Fig 2-11 produce a total of +200 g (yellow arrow). This sum of all the forces is called the resultant.

17

2 Concurrent Force Systems Perhaps a more useful and universal graphic method for force addition is the enclosed polygon method. Figure 2-13 shows the same component forces acting on the hook as shown in Fig 2-12. Sequential forces will be geometrically added instead of forming a parallelogram. Starting with F1, an arrow is drawn downward and to the distal. At the arrowhead of F1, F2 is drawn, keeping its angle and magnitude the same as the original F2 in Fig 2-12. Connecting the origin (the hook) with the arrowhead of the new F2 gives the resultant. In other words, we can walk the short way (yellow arrow resultant) or take the long way following the arrows of the F1 and F2 components (red arrows), ending up in the same place. The closed polygon method is particularly useful if more than two components are to be added. Four noncollinear forces are to be added in Fig 2-14. Each force is laid out in sequence, arrowhead to tail. The resultant force (FR, yellow) is a line connecting the origin hook and the final component (F4) arrowhead. Graphic methods for finding a resultant are very practical for the clinician. Most of the time, they are accurate enough for patient care; more important, they do not require complicated calculations. During chairside treatment, we are able to visualize the forces and come to correct conclusions in our “mind’s eye” visualization of force vectors and overall geometry. Nevertheless, an actual diagram is most helpful as a starting place for manipulating forces, either by graphic or analytical methods.

Analytical method for determining a resultant Instead of the graphic method, resultants can be calculated by using trigonometric functions and the Pythagorean theorem. Figure 2-15a shows two forces (red arrows) acting on a hook mesial to the canine. F1 is a long Class II elastic, and F2 is a short and more vertical Class II elastic.

Step 1: Resolve all forces into components using a common coordinate system. In order to add forces, common lines of action can be obtained by resolving F1 and F2 into x and y components. Figure 2-15a shows the forces F1 and F2 resolved into rectilinear components relative to an occlusal plane coordinate system. Fx is the horizontal component of force F, and Fy is the vertical component of force F. Using trigonometry, 18

Fx = F cos θ Fy = F sin θ

Step 2: Add all x forces and y forces. All forces on the x-axis are added. All forces on the y-axis are added (Fig 2-15b).

Fx1 + Fx2 = Fx Fy1 + Fy2 = Fy

Step 3: Draw a new right triangle using the summed Fx and Fy values.

A new right triangle is drawn based on Fx (the sum of Fx1 and Fx2) and Fy (the sum of Fy1 and Fy2) (Fig 2-15c).

Step 4: Calculate the magnitude and direction of the resultant. The magnitude of the resultant is calculated using the Pythagorean theorem.



FR =

FRx2 + FRy2

And the tangent function is used to calculate the direction (angle). FR tan θ = y FRx Below are some actual calculations using this method. Let us suppose that F1 = 300 g and F2 = 100 g, with the direction specified in Fig 2-15a.

Step 1: Find the components of each force. Fx1 = F1 cos θ1 = 300 g × cos 30° = 300 g × 0.87 = 261 g Fy1 = F1 sin θ1 = 300 g × sin 30° = 300 g × 0.5 = 150 g Fx2 = F2 cos θ2 = 100 g × cos 60° = 100 g × 0.5 = 50 g Fy2 = F2 sin θ2 = 100 g × sin 60° = 100 g × 0.87 = 87 g

Step 2: Add each component. FRx = Fx1 + Fx2 = 261 g + 50 g = 311 g FRy = Fy1 + Fy2 = 150 g + 87 g = 237 g

Step 3: Now we have the x and y coordinates of a resultant, and we can draw a new right triangle. Step 4: Find the magnitude and direction of the resultant. FR =

FRx2 + FRy2 = 3112 + 2372 = 391 (g)

FR 237 tan θ = y = = 0.76 FRx 311 Therefore, θ = 37.3°.

Clinical Applications

Fig 2-13 Enclosed polygon method for adding forces graphically. Starting at the hook, each force is laid out tail to arrow, maintaining the magnitude, direction, and sense (red arrows). Connecting the origin at the hook and the end point gives the resultant (yellow arrow).

a

b

Fig 2-14  The enclosed polygon method is useful, especially when there are more than two components of force. FR (yellow arrow) is the vector sum of all four components (red arrows).

c

Fig 2-15  Analytical method for determining a resultant. (a) Resolve all forces into rectilinear components (yellow arrows). (b) Add all x and y forces. (c) Construct a new right triangle with the summed Fx and Fy (yellow arrows). The hypotenuse (red arrow) is the resultant (FR). The magnitude and angle are found using the Pythagorean theorem and the tangent of θ.

Clinical Applications This chapter has discussed important concepts relating to a force or a group of forces acting on a point. A force on a point was selected in one plane because it offers a simple introduction to force manipulation. The same principles will operate with forces on a body in two or three dimensions. The major difference is the location of the point of force application, which will be considered in the next chapter. However, the clinician will be confronted with many challenges that will involve forces on a single point

only, so let us now consider some of these clinical applications. Forces resolved into their rectilinear components are always useful in planning the force system for proper treatment. For example, we may want to know how large the distal force component is in comparison to the occlusal (vertical) component using the occlusal plane as our coordinate system. Another clinical application is the simplification of the orthodontic appliance. In Fig 2-16a, two maxillomandibular elastics are used, a Class II elastic and a vertical elastic. These elastics could be replaced with a single elastic, the resultant (yellow arrow) in Fig

19

2 Concurrent Force Systems

a

b

Fig 2-16  (a) A Class II elastic is applied. A vertical elastic is also used to close an open bite. (b) By using the enclosed polygon method, the two elastics can be replaced with one elastic (yellow arrow), which is simpler for both the orthodontist and the patient.

a

b

Fig 2-17  Intrusive force from a cantilever (vertical red arrow) and distal force from an elastic chain (horizontal red arrow) produce a resultant force (yellow arrow) acting parallel to the long axis of the incisors. (a) Deactivated spring. (b) Activated spring.

2-16b. The replacement makes it easier for the patient and is therefore more likely to ensure patient compliance. Conversely, sometimes it is better to use two or more elastics that will produce the same effect as a single elastic, because sometimes the direction of the force needs to be changed slightly. For example, the objective in Fig 2-17 is to deliver an intrusive force parallel to the long axis of the incisors. Two forces are used: (1) an intrusive force from an intrusive cantilever attached to the first molar auxiliary tube and (2) a chain elastic producing a distal force. Note that the resultant force (yellow arrow) is parallel to the mean of the root long axis. Moreover, multiple forces can replace a single force when a single force cannot be placed clinically because of anatomical limitations (eg, during canine retraction, three or more forces are applied at the bracket instead of one on the root). Figure 2-18 shows an elastic chain engaged between brackets and a transpalatal arch. What would be the resultant force acting on the maxillary right second premolar and canine? Suppose the tension of the elastic is uniform; we could easily imagine a parallelogram and find the resultant graphically. The resultants (yellow arrows) on the premolar and canine are in the right directions to correct the malocclusion. 20

Suppose we want to apply lingual force on the canine. Figure 2-19a shows a single force directly acting on a canine using an auxiliary spring soldered to a passive lingual arch. What if there is no lingual arch present, and yet we need to apply a lingual force? The single lingual force can be resolved along the arch into two components (Fig 2-19b). Two simple elastics (component forces are yellow arrows) will produce the same effect on the canine as the auxiliary spring in Fig 2-19a. Anchorage, of course, will be different. Note that components are not always rectilinear. Figure 2-20a seems to show the force system acting on the molar using a temporary anchorage device (TAD) and an elastic chain (gray arrows). But this diagram is incorrect. One might think that an intrusive force would be present because the elastic is wrapped above the entire crown. However, only a buccal force (red arrow) is produced from the elastic connecting the molar-bonded hook and the TAD (Fig 2-20b). Figure 2-20c demonstrates that although the chain elastic between the two hooks on the molar is stretched, part of the elastic produces no force on the molar.

Clinical Applications

a

b

Fig 2-18  (a and b) One can easily imagine a parallelogram or enclosed polygon and estimate the magnitude and direction of the resultants (yellow arrows) graphically. The predicted direction of tooth movement is correct.

a

b

Fig 2-19  (a) A single force from a cantilever attached to a lingual arch gives a lingual force to the canine. (b) If no lingual arch is present, two components (yellow arrows) from an elastic chain could deliver a similar force.

a

b

c

Fig 2-20  An elastic chain is attached from a buccal TAD to the molar. (a) The gray force does not exist. (b) Only the intrusive buccal force (red arrow) is produced. (c) The elastic stretched between the two buttons on the molar delivers no vertical force to the molar because both forces (red arrows) cancel to zero.

21

2 Concurrent Force Systems

Summary

Fiorelli G, Melsen B. Biomechanics in Orthodontics 4. Arezzo, Italy: Libra Ortodonzia, 2013.

This chapter developed the key principles and methods for manipulating forces acting on a point. In most of orthodontic treatment, the clinician must plan for multiple point applications on three-dimensional bodies. The next chapter considers forces acting on more than one point in two and three dimensions—nonconcurrent forces. The principles and methods will be the same as for concurrent forces. Determining the point or points of force application will require consideration of an additional physical quantity—the moment.

Gottlieb EL, Burstone CJ. JCO interviews Dr. Charles J. Burstone on orthodontic force control. J Clin Orthod 1981;15:266–278.  alliday D, Resnick R, Walker J. Vectors. In: Fundamentals of H Physics, ed 8. New York: Wiley, 2008:38–115. Isaacson RJ, Burstone CJ. Malocclusions and Bioengineering: A Paper for the Workshop on the Relevance of Bioengineering to Dentistry [DHEW publication no. (NIH) 771198m, 2042]. Washington, DC: Government Printing Office, 1977.  oenig HA, Vanderby R, Solonche DJ, Burstone CJ. Force sysK tems from orthodontic appliances: An analytical and experimental comparison. J Biomech Engineering 1980;102:294–300.  elsen B, Fotis V, Burstone CJ. Biomechanical principles in orM thodontics. I [in Italian]. Mondo Ortod 1985;10(4):61–73.

Recommended Reading Burstone CJ. Application of bioengineering to clinical orthodontics. In: Graber LW, Vanarsdall RL Jr, Vig KWL (eds). Orthodontics: Current Principles and Techniques, ed 5. Philadelphia: Elsevier Mosby, 2011:345–380.  urstone CJ. Biomechanical rationale of orthodontic therapy. B In: Melsen B (ed). Current Controversies in Orthodontics. Berlin: Quintessence, 1991:131–146. Burstone CJ. Malocclusion: New directions for research and therapy. J Am Dent Assoc 1973;87:1044–1047. Burstone CJ. The biomechanics of tooth movement. In: Kraus B, Riedel R (eds). Vistas in Orthodontics. Philadelphia: Lea and Febiger, 1962:197–213.

22

 anda R, Burstone CJ. JCO interviews Charles J. Burstone. II: N Biomechanics. J Clin Orthod 2007;41:139–147.  anda R, Kuhlberg A. Principles of biomechanics. In: Nanda R N (ed). Biomechanics in Clinical Orthodontics. Philadelphia: WB Saunders, 1996:1–22. Nikolai RJ. Introduction to analysis of orthodontic force. In: Bioengineering: Analysis of Orthodontic Mechanics. Philadelphia: Lea and Febiger, 1985:24–70. S mith RJ, Burstone CJ. Mechanics of tooth movement. Am J Orthod 1984;85:294–307.

PROBLEMS

1. Compare A, B, C, and D. Is there a difference? The force is acting on a very rigid, nondeformable wire in B and C, and the force is applied on a very flexible wire in D.

2. A 300-g force of headgear and a 100-g force of intra-arch elastic act on the first molar. Find the resultant.

Activated Deactivated shape shape

3. A headgear (300 g) and a Class II elastic (100 g) act at a hook on the archwire. Find the resultant.

4. Find the resultant of the forces from the lingual arch and the crisscross elastic.

5. Resolve the 100-g force into two components parallel and perpendicular to the long axis of the tooth graphically and analytically when the angle is (a) 60 degrees, (b) 45 degrees, and (c) 110 degrees.

a

b

c

23

2 Concurrent Force Systems

6. Resolve the 150-g force from a Class II elastic into two components parallel and perpendicular to the occlusal plane when the angle is (a) 20 degrees and (b) 45 degrees.

a

7. Resolve the 100-g crisscross elastic force attached at the buccal tube of the first molar into buccolingual and mesiodistal components.

9. A Class II elastic and a headgear are simultaneously applied. The direction and magnitude of the elastic are kept constant. The resultant force must lie along the archwire axis. Find the angle when the headgear force is (a) 200 g, (b) 600 g, and (c) 1,000 g.

24

b

8. Find the resultant of a 400-g headgear force and a 200-g Class II elastic force.

CHAPTER

3 Nonconcurrent Force Systems and Forces on a Free Body “Everything should be made as simple as possible, but not simpler.”

OVERVIEW



— Albert Einstein

Teeth, segments, and arches are three-dimensional, and all appliance forces may not act on a single point. Nonconcurrent forces and their manipulation are described in this chapter. The principle of vector addition or resolution is the same as with forces on a point. One new parameter must be found: the point of force application. The concepts of moment and moment of force are introduced in this chapter. The point of force application of a resultant can be found by summing all separate moments around an arbitrary point; the distance from that arbitrary point to the resultant gives an identical moment. The useful concept of equivalence is also introduced. Components and resultants are equivalent because their action is the same. Any force can be replaced with a force and a couple that is equivalent. Force and couple equivalents at the center of resistance of a tooth or an arch or at a bracket are a powerful tool for understanding and predicting tooth movement.

25

3 Nonconcurrent Force Systems and Forces on a Free Body

Fig 3-1  If forces are parallel, they can be added algebraically to establish a resultant.

Fig 3-2  Nonparallel forces are resolved into components. Parallel components can then be added, and the magnitude and direction of the resultant can be determined.

Fig 3-3 The point of force application of a resultant on a body must be determined among many gray arrows. The moment must be considered in order to determine the correct point of force application.

Fig 3-4  Resultants for angled elastics in Fig 3-3. The correct point of force application as in Fig 3-3 is determined by considering moments.

In chapter 2, we considered forces acting at one point and learned how to resolve a force into components and find a resultant. As mentioned in that chapter, however, most orthodontic treatment involves forces that act on anatomical structures in three dimensions. This chapter considers such three-dimensional (3D) force systems (eg, more than one force acting at different points on a full dental arch).

clinical situations. This can be problematic if major asymmetries influence the location of the center of resistance (CR) of teeth (ie, if the CR varies from one plane to another). A simpler approach allows us to study one plane at a time. Thus, in Fig 3-1, both forces are projected on the xy plane (also called the z plane). Our analysis for now is limited to just one plane. Both forces have the same direction (angle) and sense but lie on different lines of action. Because they are parallel to each other, they can be added algebraically, just like multiple forces on a point with a common line of action.

Determining the Magnitude and Direction of the Resultant Figure 3-1 shows a lateral view of a maxillary arch with two vertical elastics applied at different points. Let us find a resultant—a single elastic that will do the same thing. A clarification is required before we start. For simplicity, in this text we will look at separate perpendicular projections while analyzing 3D 26

F1 + F2 = FR 100 g + 100 g = 200 g FR = 200 g The resultant is therefore 200 g.

Moments and Couples

Fig 3-5  Downward force on the wrench tightens the bolt. Force (F) times perpendicular distance (D) equals the moment (M), which is 1,000 gmm in this case. The moment is more specifically a moment of force.

Fig 3-6  Equal and opposite forces not along the same line of action produce a pure moment or a couple. The forces cancel each other out so that the force magnitude is 0 g and the moment is –1,000 gmm.

Determining the magnitude of angled component forces uses the same method presented in chapter 2. If the forces are not parallel, they can be resolved into two rectilinear components (Fig 3-2). Graphically or analytically, the resultant and its direction (angle) can then be calculated using the Pythagorean theorem and trigonometric functions. With multiple forces on a single point, the point of force application is given. However, on a 3D body, this point of force application must be determined, requiring an additional calculation, as shown in Fig 3-3. The gray lines are the resultant. The magnitude is 200 g, and the direction is parallel to the y-axis. But where is the point of force application? Any of the gray lines is a possibility. The same is true if the resultant is angled as in Fig 3-4. Any of the lines could be the point of force application. Finding the correct point of force application involves an understanding and calculation of moment.

M=F×D

Moments and Couples What is moment? Moment is a measure of the tendency of a body to rotate around a point or axis perpendicular to any given plane. For example, let us use a wrench to tighten a bolt in Fig 3-5. The 100 g is applied perpendicular to and 10 mm away from the bolt (known as the perpendicular distance). The tendency to rotate, or the moment (M), is calculated by multiplying the force (F) times the perpendicular distance (D) to the bolt:

This quantity M is more specifically called the moment of force, because it originates from the force on the wrench. The unit of measurement is the gram-millimeter (gmm) or centi-Newton-millimeter (cN-mm), and it is represented by the curved arrow in Fig 3-5. If the force is doubled or if the distance is increased to 20 mm, the moment increases to 2,000 gmm. This moment is positive because of the direction (clockwise). This is based on our sign convention, in which moments are positive (+) when they are in a clockwise direction (sense) and negative (–) when they are in a counterclockwise direction (sense) in this book. The 10-mm distance is called the moment arm. We all know from experience that it is not the force alone but the moment that determines the ease of tightening the bolt. This moment (ie, bolt tightening) is useful and readily visualized, but many of the moments used in this book are imaginary and only used for calculation. Thus, a broader definition of moment of force is the force times a perpendicular distance to any arbitrary or imaginary point. This concept is used to determine the location of the point of force application for a resultant of nonconcurrent forces. Before locating the resultant, let us consider a special type of moment where there is no force. Note that the screw in Fig 3-6 has two forces applied. The forces are parallel but have opposite senses. If we add them, the resultant force is 0 g. The screw feels no force. The moment, however, is one force times the perpendicular distance to the other force. 27

3 Nonconcurrent Force Systems and Forces on a Free Body

Fig 3-7  A couple is a free vector. The moment effect is therefore the same no matter where the couple is placed. The moment magnitude can be the same if both force magnitude and distance are varied.

Fig 3-8  The forces of a couple are not always shown. A curved arrow with proper magnitude and direction are sufficient. Sometimes a double-headed arrow is used to represent a couple. Imagine that the right hand has grasped any rotation axis with four fingers pointing in the direction of the desired moment; then the thumb is in the direction of the double-headed arrow.

M = 50 g × 20 mm = 1,000 gmm

they can be located anywhere on a free-body diagram. Figure 3-9 shows a couple acting on a canine. Two equal and opposite forces not along the same line of action deliver a pure moment of –1,200 gmm (300 g × 4 mm) (Fig 3-9a). The moment can be correctly represented as a curved arrow anywhere on or off of the tooth (Fig 3-9b). The tooth movement will always be the same because a couple rotates a tooth around its CR (purple circle). In Fig 3-10, a bar with three brackets is bonded to the labial of a canine. Does it make any difference whether the couple is applied at A, B, or C? No. Because the canine will spin around its CR (purple circle); a couple is a free vector not dependent on location of application. Erroneously, some orthodontists have believed that a couple causes a tooth to rotate between the two wings of a bracket. Three different views are shown in Fig 3-11. Traditional orthodontic jargon has described the actions as rotation (Fig 3-11a), tipping (Fig 3-11b), and torque (Fig 3-11c). This terminology is confusing and certainly not in line with the rest of the scientific and engineering specialties. First, all teeth rotate around their CRs. Rotation occurs in all views, not just from the occlusal. Second, the force systems are identical in their application of couples. It is simpler to classify all the force systems as couples and recognize that these couples will produce rotation of the teeth in all three planes of space. “Tip-back bends” and “torque” both involve moments. Incidentally, an archwire into the brackets would deliver a more complicated force system than that shown in Fig

This special moment has the tendency to produce pure rotation and can be called either a pure moment or a couple. With a moment of force, a body (eg, a tooth) will feel both a force and a moment. With a couple, however, only the moment is felt. A unique property of a couple is that it acts as a free vector. This means that it does not make any difference where it is applied; the effect will be the same. If we move the forces off-center as in Fig 3-7, the effect on the screw is the same. The magnitude of the forces has been changed, but the effect is the same because the moment is identical to that in Fig 3-6. M = 100 g × 10 mm = 1,000 gmm Because couples are free vectors and the force magnitudes and the distances between them can be varied, it may not be necessary to show the individual forces. For that reason, a curved arrow is used, provided the magnitude of the moment is given with the correct direction. Sometimes a double-headed arrow is used to represent a couple (Fig 3-8). The point of force application is critical in under­ standing appliance design and also the biomechanics of tooth movement. During all planning and diagram drawing, a force must be located exactly where it will act. Chapter 9 emphasizes that different types of tooth movement are produced by changing the position of the line of action of the force(s). By contrast, however, because couples are free vectors, 28

Moments and Couples

a

b

Fig 3-9  A couple will rotate a canine around its CR (purple circle). (a) Forces acting at bracket wings. Moment arm refers to the distance between the two bracket wings. (b) A couple of –1,200 gmm is shown as a curved arrow.

a

b

Fig 3-10  An imaginary appliance with three brackets on a rigid bar attached to the canine. It does not make any difference where the couple is applied. The tooth will rotate around its CR.

c

Fig 3-11  Orthodontic jargon for the shown force systems: rotation (a), tipping (b), and torque (c). These are all more simply described as a couple. All produce rotation around the CR.

3-11, usually including both forces and moments. Note that in Fig 3-11a, a couple to rotate the molar counterclockwise will have the same effect on the buccal as that applied on the lingual with a lingual appliance. Figure 3-12 illustrates a patient with a severe midline discrepancy (Fig 3-12a). However, in the posteroanterior cephalometric radiograph, there is no apical base discrepancy (Fig 3-12b). The maxillary incisors are tipped to the left side, and the mandibular incisors are tipped to the right. Correct bracket placement will produce couples at each tooth in a favorable direction to correct the midline. The couples on each tooth produced rotation of each tooth around its CR, not the bracket. After leveling, the roots became parallel to each other by rotation around each CR, with the crown and root apex moving in opposite directions (Fig 3-12c). The midline

was corrected using the couples acting at the brackets without requiring any lateral forces (Fig 3-12d). Figure 3-13 illustrates a patient with a posterior crossbite on the right side at the second molars (Fig 3-13a). The crossbite was mostly due to linguoversion of the mandibular second molar (Fig 3-13b). A piece of 0.016 × 0.022–inch nickel-titanium wire was twisted and inserted at the tubes of the mandibular second molar and second premolar brackets, bypassing the first molar tube, so that it produced equal and opposite couples, as depicted by red arrows (Fig 3-13c). Even though the couple is acting at the tube or bracket, the rotation still occurs around each CR. The second molar rotated counterclockwise, and the second premolar rotated clockwise. After the second molar was uprighted, the second premolar was extracted.

29

3 Nonconcurrent Force Systems and Forces on a Free Body

a

b

c

d

Fig 3-12  (a) A patient with a severe midline discrepancy (dotted lines). (b) In the posteroanterior cephalometric radiograph, there is no apical base discrepancy. Correct bracket placement will produce couples at each tooth. (c) The couples on each tooth produce rotation of each tooth around its CR. (d) The midline was corrected using the couple acting at the brackets without requiring any lateral force.

a

b

c

Fig 3-13  (a) A patient with a posterior crossbite on the right side at the second molars. (b) The crossbite was mostly due to linguoversion of the mandibular second molar. (c) A piece of 0.016 × 0.022–inch nickel-titanium wire was twisted and inserted at the tubes of the second molar and second premolar brackets, bypassing the first molar tube, so that it produced an equal and opposite couple, as depicted by red arrows.

30

Determining the Point of Force Application of the Resultant

states: The sum of the moments around any arbitrary point of the component forces equals the moment of the resultant force around that point (Fig 3-14a). Therefore,

Let us now go back to the determination of the resultant in Fig 3-3. The principle of vector addition was used to find the force magnitude of 200 g. The only unknown is the position of the line of the force (Fig 3-14). Here a moment formula is used that



∑M* = Rd

∑M* d= R

Determining the Point of Force Application of the Resultant

a

b

c

d

Fig 3-14  (a) To find the location of a resultant, an arbitrary point is selected (red dot). The sum of the component moments around that point equals the resultant magnitude times the distance to the same point. (b) The point of force application of the resultant is found at 15 mm from the arbitrary point (red dot). The moments from the red and yellow forces are equal (–3,000 gmm around the red dot). (c) Selection of a point at which to sum the moments is based on convenience. Any point is valid. Even a point on the Taj Mahal or the Eiffel Tower will give the correct answer as long as the distances are known, but they are not convenient. (d) The resultant can only be placed 15 mm anterior to the arbitrary point for the sense of the moment (–) to be correct. The red arrow is correct, while the gray arrow is incorrect.

where M is the moment measured in respect to any point (*), R is the resultant magnitude, and d is the location of the point of application of the resultant measured to the same point. First an arbitrary point is chosen. Because any point will do, a convenient one is selected that is located at the 100-g force in the molar region (red dot). The sum of all of the moments around that point is calculated (Fig 3-14b). 100 g × 30 mm = 3,000 gmm (counterclockwise) Note that no moment is produced by the molar vertical elastic because the perpendicular distance to the selected point is 0. Finally, dividing the sum of the moments by the resultant magnitude gives the distance to the selected point:

The resultant is located 15 mm anterior to the molar elastic. The two individual component elastics (red arrows) or one resultant elastic (yellow arrow) delivers the same moment to the selected point. Any point can be selected on or off the body under consideration to sum the moments, and the answer will be the same (Fig 3-14c). Even a point at the Taj Mahal or the Eiffel Tower will work if the distances are known. We select a point for convenience—here at the molar force to simplify the calculation. Only one last ambiguity remains. Is the 15-mm distance anterior to or posterior to the selected molar point? This distance must be anterior to the point for the direction of the moment (counterclockwise) to be correct (Fig 3-14d). The gray arrow is therefore incorrect.

3,000 gmm = 15 mm 200 g 31

3 Nonconcurrent Force Systems and Forces on a Free Body

Fig 3-15  Equivalence. Force systems that are equivalent produce the same effects. Examples include resultants and components.

Fig 3-16 Any force can be replaced with a force and a couple. A single force applied at the root (red arrow) could produce root movement (rotation around the incisal edge, blue dot). A force and couple (yellow arrows) at the bracket will do the same thing.

Equivalence of Forces

with a center of rotation on the crown (blue dot). But it is not practical to place a force so far apically; the appliance might have to be inserted by way of the nose. Let us replace the force at the bracket. The equivalent force is still 200 g (yellow arrow) based on the first formula. The bracket is picked as a convenient point to sum the moments. The original 200-g force times the 12-mm perpendicular distance gives a +2,400-gmm moment. The new replacement force has no moment (200 g × 0 mm = 0 gmm). For equivalence, a couple of +2,400 gmm must be added at the bracket (yellow curved arrow). The red or yellow force systems do the same—the teeth will not notice any difference. Replacing a force with a couple is also useful in predicting tooth movement. A 200-g distal force is applied at the mesial of the canine bracket (Fig 3-17). A replaced equivalent force acting at the CR of a tooth produces translatory movement, and a couple produces rotation around the CR (see also chapter 9). If we replace the original 200-g force (red arrow) at the bracket, the magnitude of the replacement force (yellow arrow) at the CR (purple circle) is the same: 200 g. This CR is merely a convenient point at which to sum the moments. The original force (red arrow) moment is 200 g × 7 mm = 1,400 gmm (counterclockwise). The replacement moment from the force (yellow arrow) is zero because there is no distance between the force and the point of application, so a couple of –1,400 gmm must be add-

Forces that are interchangeable are called equivalent. In other words, the effect on teeth or arches is the same. If the exchange rate is considered, an equivalence can be established between the Euro and the American dollar. This form of equivalence can change often and may not always reflect buying power. Newtonian equivalence, on the other hand, is more precise and always produces the same effect, ignoring any local effects where the forces are applied. Two force systems are equivalent if the sums of their forces are equal and if the sums of their moments around any arbitrary point are also equal (Fig 3-15). ∑F1 = ∑F2 ∑M1* = ∑M2* We have already considered some examples of equivalence. Resultants are equivalent to their component forces. A single force can be replaced by a number of forces or vice versa. The clinician can choose the best equivalence for effective and convenient treatment. But perhaps the most useful equivalence is the replacement of a force with a force and a couple. Let us consider the incisor in Fig 3-16. If a 200-g lingual force (red arrow) is placed on the root, one might expect the root to move lingually 32

Equivalence of Forces

Fig 3-17  Any force can be replaced with a force and a couple. If the replacement force and moment act at the CR, useful information on how the tooth will move is obtained. The distal force at the bracket causes distal-in or counterclockwise rotation of the canine.

Fig 3-18  A distal force and a moment translate the canine from the occlusal view (red arrows). If the same thing is to be accomplished using a lingual orthodontic appliance, a distal force of 200 g and a –400-gmm moment are required (sum of moments shown by yellow curved arrows). Note that moment direction on the lingual is opposite to the moment applied on the labial.

ed at the CR. The force system at the CR allows us to predict that the canine will translate distally and rotate in a counterclockwise direction. But are the force systems different between labial and lingual orthodontics? Equivalence can minimize any surprises when working from the lingual. Let us consider canine retraction as shown in Fig 3-18. A satisfactory force system (red arrow) from the labial is 200 g with a 1,400-gmm clockwise moment (200 g × 7 mm = 1,400 gmm). We want to replace the labial force system at the lingual bracket, which is 2 mm lingual to the CR. The force (yellow arrow) is the same: 200 g. The moments are conveniently summed around the lingual bracket: 200 g × 9 mm = –1,800 gmm [counterclockwise]. The 1,400-gmm red couple is moved to the lingual bracket. Why can this be done? It is a free vector. The total moment acting at the lingual bracket is therefore only –400 gmm (–1,800 gmm + 1,400 gmm = –400 gmm), acting in a counterclockwise direction. The force system working from the lingual is actually simpler than the labial system. Less moment is needed, thereby preventing undesirable tooth rotation, and that moment is in the opposite direction. A two-dimensional (2D) plastic model of two teeth was fabricated (Fig 3-19). Springs were attached to the teeth to simulate the periodontal ligament. A chain elastic connects the two teeth (Fig 3-19a) and is angled so that the right tooth tips and the left tooth translates. This is one method for anchorage

control to pit translation against tipping. However, it is impossible to connect teeth in this manner clinically because of tissue location. It is possible, on the other hand, to place an equivalent force system at the bracket, which is accomplished by a T-loop (Fig 3-19b). A more detailed explanation of loop shape for this purpose is discussed later. Figures 3-19c and 3-19d show that the original force system from the elastic (red arrows) is replaced by a force and a couple (yellow arrows) at the bracket of each tooth. The anchorage tooth has the greater moment. Equivalence allows us to practically make an appliance that will work using brackets on the crowns of the teeth. Figure 3-20a is the initial frontal view of a patient with a midline deviation. The patient is an extraction case where the anterior segment needs translation to the left side. A single force acting through the CR (purple circle in Fig 3-20b) is very difficult to achieve due to anatomical limitations. The resultant oblique single force is therefore replaced with vertical and horizontal force components. Note that the horizontal force component can be moved along its line of action so that the resultant can be determined as a force acting on a point at the patient’s right. The anterior segment translated to the left side without tipping (Fig 3-20c). The law of transmissibility and the equivalence principle allow us to practically make an appliance that will work using brackets at crown level.

33

3 Nonconcurrent Force Systems and Forces on a Free Body

a

b

c

d

Fig 3-19  2D plastic model with teeth supported by springs. (a) An elastic is stretched at an angle between the roots, causing the right tooth to tip and left tooth to translate. (b) An activated T-loop achieves similar movement with forces at the bracket. (c) Same elastic as in a, showing the forces. (d) Equivalent force systems: the original force from the elastic (red arrows) and the replacement at the bracket with a T-loop (yellow arrows).

a

b

34

Fig 3-20  (a) Initial frontal view of a patient with a midline deviation. (b) A single force acting through the CR (purple circle) is very difficult due to anatomical limitations. The oblique single force resultant (yellow arrow) is replaced with vertical and horizontal force components (red arrows). The horizontal component of force acting at the patient’s left can be moved along its line of action to intersect with the right vertical force to simply solve for the resultant, since all forces act then at a point. (c) The anterior segment translated to the left side without tipping.

c

2D Projections of 3D Force Systems

a

b

Fig 3-21  (a) 2D projection (xy plane) of a 3D body. The 100-g elastic is on the patient’s right side, and the 20-g elastic is on the left side. The yellow arrow shows the 120-g resultant. (b) 2D projection (yz plane) of a 3D body. Note the location of the resultant (yellow arrow).

a

b

Fig 3-22  (a) The resultant is replaced with a force and a couple (yellow arrows) at the CR in the xy plane, which will extrude the maxillary arch and steepen the occlusal plane. (b) The resultant is replaced with a force and a couple (yellow arrows) at the CR in the yz plane, which will extrude the maxillary arch and make it cant downward on the patient’s right side.

2D Projections of 3D Force Systems Describing a force system acting in a 3D space and how it influences tooth movement is not a trivial pursuit. Chapter 10 considers 3D tooth movement and questions such as what is the meaning of the CR for asymmetric teeth or groups of teeth. For now, however, we will handle 3D forces by projecting them onto three mutually perpendicular planes: x, y, and z. For simplicity, we will place two vertical elastics. In Fig 3-21, the 100-g force acts between the right lateral incisor and the canine on the labial arch. The other 20-g force acts at the left tube of a lingual arch. Figure 3-21a shows the xy plane (or the z plane), and Fig 3-21b shows the yz plane (x plane). The original component forces are the red arrows.

In each separate plane, a resultant is calculated using the principles explained in this chapter. The yellow arrow gives the resultant force in each plane (120-g magnitude, direction, and point of force application). The xz plane (occlusal view) is not shown because the forces are vertical and no force is seen from this view. The force direction here is completely vertical and parallel to simplify the discussion. The general solution, of course, allows the handling of many forces that are angled and not parallel with each other. This is one way to evaluate forces in three dimensions. The clinician can compress all forces in a given plane to a 2D projection and evaluate what happens in each plane. In Fig 3-21, what will happen in each plane? To answer that question (Fig 3-22), the resultant force (red arrow) must be replaced at the CR with a force and a couple (yellow arrows). 35

3 Nonconcurrent Force Systems and Forces on a Free Body Fig 3-23  3D view of the force system.

This applies once again the concept of equivalence. From a lateral view (Fig 3-22a), it can be seen that the cant of the occlusal plane will steepen (increase) and the maxillary arch will extrude. From the frontal view (Fig 3-22b), the resultant force off-center will extrude the arch and cant the occlusal plane, extruding the right side more than the left. Figure 3-23 depicts the 3D rendering of the forces. There are two components to the calculations involved in three dimensions projected into two dimensions. The first part is defining a resultant (adding forces). That part is pure Newton and is readily done. The second part involves a concept—the center of resistance—which involves physics, biology, and unproven assumptions. For that reason, the present discussion should only be considered an introduction to the relationship of tooth movement to forces, particularly in three dimensions.

36

Much of the basic knowledge has now been presented about the scientific handling of forces from orthodontic appliances. The following chapters con­ sider the simplest of all orthodontic appliances: appliances that deliver single forces. These appliances are statically determinate, which means that a single force measurement can completely define the force system. Simple, however, does not mean inferior. In many cases, these appliances may be the most useful, practical, efficient, and predictable of all appliances.

Recommended Reading See the recommended reading in chapter 2.

PROBLEMS

1.  A 100-g force is applied at the labial. Give the force system at the lingual bracket that will do the same.

2.  A 100-g force and a –400-gmm moment are applied at the labial. Give the force system at the lingual bracket that will produce the same effect.

3.  A single force through the CR will translate the canine distally. We can use a force through the CR; however, this is not always practical. Find the two forces at points A and B that will have the same effect.

4.  We want to translate the canine from this view. Find the two forces at points A and B that will do the same. This is the same problem as #3 except that the CR is further lingually. Find the equivalent forces at points A and B.

5.  Replace the 200-g elastic with two forces at FA and FB.

6.  Find the resultant of the two vertical elastics.

37

3 Nonconcurrent Force Systems and Forces on a Free Body

7.  Find the resultant of the two Class II elastics.

8.  For a given patient, 100 g of force and a 1,000-gmm moment at bracket A produce the desired movement. If bracket B is used, give the force system for the same effect.

9.  A wire extension is used at point A for canine retraction. Find the equivalent force system for point B.

10.  Compare the effect on the molar of the 200-g force if placed at the buccal (a) or lingual (b).

a

b

11.  A tip-back spring is attached to the molar. The activation force on the spring is 100 g. Compare the molar movement in a, b, and c.

a

b

12.  Compare the effect on the mandibular incisor of an 80-g force placed at the labial bracket (A) or on the lingual attachment (B). The circle is the CR of the incisor.

38

c

CHAPTER

4 Headgear

“A perfection of means and a confusion of aims seems to be our main problem.”

OVERVIEW



— Albert Einstein

The effect of a headgear on a tooth or a full arch is usually easy to calculate. First the center of resistance (CR) of a tooth, segment, or arch is determined. Deformation of the inner bow can be ignored unless relevant. A line of action of the force through the CR produces translation in the direction of the force. A line of action away from the CR produces both translation and rotation around the CR. Older terminology and descriptions of headgear usage, such as cervical or occipital with a short or long outer bow, are too complicated and may not be predictive. The clinician must first select a proper line of action of the headgear force and design the headgear to deliver that force. With a correct line of force, teeth can be translated, crowns tipped, or roots moved distally. With a correct line of force, the occlusal plane cant can also be maintained, increased, or decreased, thus increasing or decreasing the vertical overlap (also known as overbite).

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4 Headgear Fig 4-1  Inner and outer bow headgear. The inner bow is inserted into the molar tube, and a cervical elastic strap is connected at the hook of the outer bow. The red arrow is the force from the cervical elastic strap.

The headgear is the ideal appliance with which to start a discussion of appliance design. Headgear is a statically determinate appliance, which means that a single measurement with a force gauge can provide the clinician with most of the information required. Headgear classification is usually determined by anatomical geometry, such as where it is attached to the head (cervical or occipital), the direction of the pull (high or low), and outer bow length or position. These classifications and the rules of thumb based on them have led to serious errors in the use of headgear and have certainly complicated the explanation of how headgears work. In this chapter, emphasis is placed not on the geometry and shape but rather the force system of the headgear. Two different designs, based on the manner of attachment to the teeth, are presented in this chapter. One type has an inner archwire (bow) that attaches to a tube on the molar and an outer bow where elastic force is placed (Fig 4-1). This inner and outer bow headgear was developed by Oppenheim and taught at the University of Illinois to his student Kloehn, who helped popularize it in America. This design is referred to as the Oppenheim headgear, although historically similar headgears have been developed by others. It is commonly called the inner and outer bow headgear. The second design presented in this chapter is the J-hook headgear, in which separate right and left outer bows attach at a hook near the canines on the arch. Because the mode of force delivery is different for each of these two designs, they are considered separately in this chapter. First let us consider the inner and outer bow headgear and its force system in three dimensions from the lateral, occlusal, and frontal views.

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Inner and Outer Bow Headgear from the Lateral View Figure 4-1 shows a cervical headgear activated by an elastic strap. The inner bow of the headgear is inserted into a tube on the molar. For this analysis, we will assume that there is no play between the inner bow and the molar tube. In special conditions in which the inner bow is round, as from the frontal view, we will consider play because it is relevant. The relationship between the tube and the tooth is considered rigid, but the bows of the headgear are allowed to deform and deflect. The applied force (red arrow) is the force acting at the attachment hook on the outer bow. One could calculate the force acting at the molar tube on the molar band, but this is unnecessary because play normally does not exist in a second-order direction (tip) between the inner bow and the tube. The cervical elastic neck strap supplies the force. Figure 4-2a is the equilibrium diagram of the neck strap. Blue forces are equal and opposite and sum to zero (Newton’s First Law). The forces are reversed in Fig 4-2b (red force) and show the forces on the neck and the outer bow (Newton’s Third Law). This force and its line of action will determine the tooth movement. One can measure the force with a force gauge, and the line of action is in the direction of the stretched elastic. Both inner and outer bows can be bent elastically during force application. At point A in Fig 4-3, the inner bow is placed into the molar tubes, and the appliance is passive. But at point B, the strap

Inner and Outer Bow Headgear from the Lateral View

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Fig 4-2  The neck strap is in equilibrium (Newton’s First Law). (a) The forces that stretch the elastic strap (blue arrows) sum to zero. (b) The force on the outer bow (red arrow) is equal and opposite to the force at the hook stretching the elastic (blue arrow). This is an example of Newton’s Third Law.

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Fig 4-3 The effect of the headgear is determined by its activation position. (A) The headgear is placed in the molar tubes, but the appliance and the cervical strap are not yet engaged. This is the passive position. (B) The strap is now placed, and the bows deform elastically to their loaded shape and position. This is the activated position from which the force system effects are determined.

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Fig 4-4  The force system from an occipital headgear. (a) The force from an occipital headgear is shown as a red arrow (1). It is replaced with an equivalent force system at the CR by a force (2) and a couple (3) (yellow arrows). The force translates the molar’s CR apically and distally. The couple rotates the tooth in a counterclockwise direction around its CR. (b) The combined movement of the molar (2). The root moves apically and distally as the tooth inclination changes.

is placed and the bows deform elastically to their loaded shape and position. It is from this shape that the line of action of the force is determined by observing the direction of stretch of the elastic. The new position of the outer bow hook is the point of force application from which all calculations start. To analyze the effect from any headgear, first the line of action of the force must be determined. In Fig 4-4a, the line of action from an occipital headgear is shown. The force (red arrow in 1) is directed upward and backward, anterior to the center of resistance (CR) (purple circle in 1). The headgear force is then replaced at the CR with a force and a couple (yellow arrows in 2 and 3). This is based on the concept of equivalence (see chapter 3). The yellow force produces translation along its line of action at the CR, and the yellow couple will rotate the tooth around its CR (crown forward and root

back). We assume that the displacements will be in the same direction as the force and moments at the CR and that all displacements will be proportional to the forces and moments acting at the CR. Chapter 9 will discuss this concept in greater detail. For now, these simple assumptions are good enough for all clinical purposes. In our current analysis, we also assume that the headgear and the molar tube are rigidly connected and that there is no significant play between the molar tube and the inner bow of the headgear. Tipping from this play, as seen from the lateral view, is ignored. The molar in Fig 4-4a has two component movements based on the force and couple at the CR: translation of the molar upward and backward (2) and rotation of the molar (3). The combined movement is shown in Fig 4-4b (2). The headgear would cause the crown to tip mesially, the root to move apically

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4 Headgear

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Fig 4-5  A typical cervical headgear (design 1). (a) The line of action of the force (red arrow) from the neck strap is downward and backward and lies inferior to the CR of the molar. The equivalent force system at the CR is a downward and backward force and a large clockwise couple (yellow arrows). (b) The molar erupts and tips distally. The high moment-to-force ratio at the CR produces significant tipping.

and distally, and the CR to move upward and backward. This method does not tell us how much molar rotation occurs in comparison with the translation at the CR. This question will be discussed later. For now, we have at least established the general tooth movement of a given molar to a headgear force.

Typical Headgear Designs Because there are an infinite number of possible clinical situations, let us analyze typical cases from the lateral aspect.

Design 1: Typical cervical headgear In Fig 4-5a, the line of action of the force (red arrow) from the neck strap is downward and backward and lies inferior to the CR of the molar. To predict the tooth movement, we replace the red force with an equivalent force system at the CR: a force and a couple. The replacement force at the CR is 300 g (straight yellow arrow). If we sum the moments around the CR, we find that the replacement couple has a magnitude of +3,000 gmm (curved yellow arrow). 300 g (red force) × 10 mm = +3,000 gmm 300 g (yellow force) × 0 mm = 0 gmm For equivalence, a couple (curved yellow arrow) of +3,000 gmm must be added at the CR. Note that the same answer will be obtained even if the moments

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were summed around a different point, such as the hook of the outer bow. The enlargement in Fig 4-5b shows the expected molar movement. The CR moves downward and backward (from the force shown by the yellow arrow). The couple (curved yellow arrow) is clockwise and will rotate the molar around its CR, moving the crown distally. When we combine the two component movements, it is apparent that most of the movement is distal tipping of the molar. (Note that the amount of translation is exaggerated in the figure.) This much tipping occurs because the line of action of the force is far away from the molar’s CR. Relative to the occlusal plane, the molar moves distally and also extrudes.

Design 2: Low cervical headgear The design of the typical cervical headgear (design 1) has the disadvantage that an extrusive force is placed on the molar. But is it possible to use a neck strap and not produce extrusion? Yes. Figure 4-6a shows a line of action parallel to the occlusal plane. Because the perpendicular distance from the applied force (red arrow) is greater than that in design 1, force magnitude is reduced to 150 g to deliver the same amount of moment. What is the effect of the force on the molar? Again, we replace the 150-g applied force with a force and a couple at the CR. At the CR, the force is 150 g and the couple is +3,000 gmm. The molar translates distally along a line parallel to the occlusal plane (Fig 4-6b). The crown tips distally, spinning around the CR.

Typical Headgear Designs

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Fig 4-6  A low cervical headgear (design 2). (a) The force (red arrow) is placed much lower than in design 1 so that the force is parallel to the occlusal plane. An equivalent force system at the CR produces the same amount of moment (3,000 gmm) with half the amount of force (150 g) as in Fig 4-5 because the length of the moment arm is doubled (20 mm). (b) Because the moment is so large relative to the force, the molar primarily tips distally around the CR. Note that there is no extrusive force component relative to the occlusal plane.

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Fig 4-7  A cervical headgear for translation (design 3). (a) The line of action of the force (red arrow) passes through the CR of the molar. It is possible for a cervical headgear to produce distal translation; however, an extrusive direction is required. (b) The molar translates downward and backward.

Note the differences in the force systems between design 1 and design 2. Less translation in comparison to rotation occurs proportionally in design 2. The moment-to-force ratio at the CR is 10:1 in design 1 and 20:1 in design 2. In addition, design 2 gives better vertical control by not causing eruption of the molar. What are the indications for a low cervical headgear? A maxillary first molar may be tipped forward because of early loss of primary molars or extraction of a premolar. If tipping is mainly needed around the CR, the force system of the low cervical headgear is ideal. A high moment-to-force ratio at the CR will quickly tip the molar posteriorly. A line of action closer to the CR will produce less tipping and hence would be slower and less efficient. With this design, less force is used at the neck strap; other-

wise, the moment would be too large for the tooth movement and produce discomfort for the patient. If the line of action of the force is placed too far apically, the root will move distally (see Fig 4-9). Design 2 would be an ideal design to correct this problem.

Design 3: Cervical headgear for translation It is possible for a cervical headgear to produce translation of a maxillary molar. Note that the line of action of the required force must go through the CR (Fig 4-7). The 300-g force from the cervical strap passes through the CR of the tooth (red arrow), and therefore the tooth translates without any rotation. The direction of the translation is both downward

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4 Headgear

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Fig 4-8  An occipital headgear for intrusion and tipping of a molar distally (design 4). (a) An upward and distal force (red arrow) from an occipital harness is applied distal to the CR. The equivalent force system at the CR is 300 g of force accompanied by a 3,000-gmm clockwise moment (yellow arrows). (b) The molar’s CR moves upward and backward, and the molar tips backward.

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Fig 4-9  An occipital headgear for moving the molar root distally (design 5). (a) The upward and backward force (red arrow) is placed anterior to the CR. The equivalent force system (yellow arrows) at the CR is in an upward and backward direction accompanied by a 3,000-gmm counterclockwise moment. (b) The molar root moves backward and upward. The vertical dimension is controlled by the intrusive component of the force.

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and distal. The distal force could be useful for Class II correction, but the extrusion of the molar could be problematic if the mandible rotates downward and backward. However, in some patients, occlusal forces may prevent molar extrusion (small mandibular plane angles); in growing patients, growth can allow for molar eruption.

shows an upward and distal 300-g force (red arrow) from an occipital harness applied distal to the CR of a maxillary molar. The tooth feels 300 g of force accompanied by a 3,000-gmm clockwise moment at the CR. The CR of the molar will move distally and apically, and at the same time the molar will tip back (Fig 4-8b).

Design 4: Occipital headgear for tipping a molar distally

Design 5: Occipital headgear for moving the molar root distally

Many different lines of action from a cervical headgear are capable of tipping the molar distally. At best, this type of headgear produces no occlusal erupting force. If better vertical control is required or an open bite is present, it may be necessary to create an intrusive force on the molar. Figure 4-8a

During the correction of a Class II malocclusion, the first molar may inadvertently tip back; in this situation, distal root movement to correct the axial inclination is required. How is this accomplished without erupting or intruding the molar? The headgear design is similar to design 4, except now the line of

Typical Headgear Designs

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Fig 4-10  A headgear for molar translation parallel to the occlusal plane (design 6). (a) The force (red arrow) passes through the CR parallel to the occlusal plane. (b) The tooth translates along the occlusal plane. Note that there is no extrusive or intrusive component.

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Fig 4-11  (a to d) Many points of force application allow for versatility. The type of tooth movement is determined only by the relationship between the CR (purple circle) and the force (red arrow).

action is placed anterior to the CR (Fig 4-9). The 300-g force is applied upward and backward anterior to the CR by the elastic (red arrow). The equivalent force system (yellow arrows) at the CR is 300 g in an upward and backward direction, accompanied by a –3,000-gmm moment. Note that the moment is counterclockwise, which will tip the root back and the crown forward. Figure 4-9b shows the predicted tooth movement from the headgear. The CR will move backward and upward. At the same time, the axial inclination will be corrected by the couple acting at the CR.

Design 6: Headgear for molar translation along the occlusal plane

plane. Figure 4-10 shows design 6, a headgear that is capable of translating a molar distally without changing its vertical position. A connecting bar between the occipital harness and the cervical strap bypasses the positioning limitations imposed by the ear. Another method is the use of a separate occipital harness and cervical strap individually connected to the outer bow of the headgear. In this case, the resultant force must go through the CR and lie parallel to the occlusal plane. This type of headgear more uniformly distributes force to the teeth because they are not tipping. For that reason, it may be appropriate to use heavier forces. A full range of headgear force directions and points of force application are shown in Fig 4-11. Any method of headgear classification and proper headgear use depend on an accurate definition of the force system.

Perhaps one of the most useful force systems is a distal force through the CR parallel to the occlusal

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4 Headgear Fig 4-12  A headgear force applied at the CR (purple circle) of a full arch can produce translation in many directions.

Fig 4-13 Forces (red arrows) with different directions produce a clockwise moment of the maxillary arch.

Fig 4-14 Forces (red arrows) with different directions produce a counterclockwise moment of the maxillary arch.

Headgears Acting on a Full Arch

moment, steepening the occlusal plane (Fig 4-13). Other forces could produce a counterclockwise moment, rotating the occlusal plane upward in the anterior region (Fig 4-14). Similar appropriate headgear designs as discussed for molar movement can be selected to deliver the correct force system based on treatment goals for the full arch. As before, the key to success is the determination of a proper line of action for the force. Headgear is commonly used for nonextraction Class II treatment. Figures 4-15 and 4-16 demonstrate the extremes of clinical response in two patients. The patient in Fig 4-15 had a full-cusp Class II malocclusion with minimum growth during treatment. The headgear pull to the entire arch was directed through the CR and parallel to the occlusal plane (Fig 4-15a). The occlusion is shown before treatment (Figs 4-15b to 4-15d), after treatment (Figs 4-15e to 4-15g), and 2 years after retention (Figs 4-15h to 4-15j). As seen on the superimposition, maxillary distal translatory movement of the molars occurred during treatment (Figs 4-15k and 4-15l). Some uprighting of the maxillary incisors occurred along with considerable translation of the incisors’ CRs.

Headgear forces can be applied not only to the first molar but also to the full arch. For our analysis, we will assume that all teeth are rigidly connected. A round wire into the anterior teeth will allow tipping of the incisors and would not accurately reflect the anticipated changes. If more flexible wires are used, the wire can bend between brackets, producing secondary tooth-totooth effects (see chapter 15). Nevertheless, considering a full arch as a unit is very useful in anticipating treatment effects even if complete rigidity is not present. The location of the CR of an entire arch can be estimated to lie between the roots of the two premolars (purple circle in Fig 4-12). Therefore, a headgear force applied at the CR of the full arch can produce translation in many directions. A force away from the CR produces a force and a couple. Many lines of action from these potential forces can translate the CR and produce rotation around the CR of the entire arch. Some forces could produce a clockwise

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Headgears Acting on a Full Arch Fig 4-15  (a) The combination headgear. The resultant force (yellow arrow) from the occipital and cervical headgears passes through the CR of the full arch parallel to the occlusal plane. (b to d) Pretreatment intraoral casts of a patient with a Class II malocclusion. (e to g) Intraoral casts after headgear treatment. (h to j) Intraoral casts 2 years postretention. (k) Lateral superimposition of the maxilla, mandible, and cranial base before (black) and after (red) headgear treatment. The molar has translated distally in this nongrowing patient. (l) Occlusal superimposition of tooth positions before (black) and after (red) headgear treatment.

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4 Headgear

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Fig 4-16  (a to c) Pretreatment intraoral casts of a patient with a Class II malocclusion. (d to f) Intraoral casts after headgear treatment. (g to i) Intraoral casts 2 years postretention. (j) Cranial superimposition of tooth positions before (black) and after (red) headgear treatment. The headgear prevented the maxillary molar from coming forward while the mandible grew normally. (k) Maxillary and mandibular superimpositions before (black) and after (red) headgear treatment. The maxillary molar did not translate distally, yet the root of the molar moved distally from intraoral mechanics.

By contrast, the patient in Fig 4-16 had almost no distal crown movement of the maxillary first molar. The root of the molar moved distally by intraoral mechanics. The maxillary incisors tipped to the palatal during closure of the space between them. The headgear force was similar to that used for the

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patient in Fig 4-15, but here the headgear served a different function. It held the maxillary arch while the mandible grew normally to correct the Class II malocclusion. This included maintaining the cant of the occlusal plane and the expected vertical height during growth.

How to Design a Headgear

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Fig 4-17  Designing a headgear. (a) Step 1: Determine the location of the CR (purple circle). (b) Step 2: Establish the treatment goal. We want the arch to translate upward and backward and to rotate in a counterclockwise direction (white dotted arrows). (c) Step 3: Replace the force and the couple at the CR (yellow arrows) with a single equivalent force. The red arrow is correct, and the gray arrow is incorrect. (d) Step 4: Adjust the harness and headgear so that the location of the two hooks (red circles) lies along the line of action of the force (dotted line). (e and f) The length of the outer bow (e, short; f, long) does not affect the force system because they share the same line of action (dotted line). The shape of the outer bow also does not alter the force system.

How to Design a Headgear Much of the orthodontic literature has described the fabrication and use of headgear based primarily on the shape, length, and position of the outer bow: Outer bows are to be short or long, high or low, and up or down. This approach often can lead to improper designs and inaccuracies. Outer bows can be cut short when not required. Moreover, these shape-based descriptions make a simple design problem complicated. Headgear design should instead be based on establishing a desired line of action of the desired force. The first step clinically is to determine the CR of the tooth, segment, or full arch under consideration. We will use a full arch for our example (Fig 4-17a). The center of resistance is the geometric center of the roots through which a force produces translation. A more thorough discussion is covered in chapter 9. The CR is not a single point, so it is usually represented in diagrams by a large circle. Although there can be other confounding factors in three-dimensional space, we can practically establish a general area (large circle) for the CR. Even if the estimate of the CR is somewhat incorrect, its position remains constant on a tooth or group of teeth as treatment progresses. This allows for estimation of the force direction for CR translation. The second step is to establish our treatment goal for both translation of the arch and rotation around the CR (Fig 4-17b). Note that we want the arch to translate upward and backward and to rotate in a counterclockwise direction (ie, incisor moves superiorly).

The third step is to replace the force and the couple (yellow arrows) at the CR with a single equivalent force (red arrow) (Figs 4-17c and 4-17d). The force must be parallel to the force at the CR and must lie apical to the CR to produce the correct moment for equivalence. The red arrow is correct, and the gray arrow is incorrect (see Fig 4-17c). The correct red arrow in Fig 4-17d represents the line of action and its sense of the headgear force. If a greater moment is required at the CR, the force is moved further away according to the laws of equivalence. The ultimate design and fabrication of the headgear is now simple. The line of action of the force is established (dotted line). There are two hooks, one from the headgear harness and one from the outer bow (red circles). Both will lie along the line of action. The elastic is also stretched along the line of action of the force. The outer bow can be longer or shorter, and there is no difference in the effect of the headgear on the maxillary arch provided the line of action of the force is the same (Figs 4-17e and 4-17f). The design shown in Fig 4-17f may look ridiculous, but the line of force is the same as that in Fig 4-17e, so the effect is the same no matter how the outer bow is shaped. Some fine-tuning of the inner and outer bows may be required to compensate for any bending during placement of the elastic. It should be remembered that the relationship between the elastic and the CR is determined after the elastic is placed and the bows have elastically deformed. In short, the simple secret to headgear design is to establish a line of action of the force and design the headgear around it. Note that in Fig 4-18 the three outer bows vary in length and position, and yet the

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4 Headgear Fig 4-18  Three outer bows vary in length and position, yet the effect is the same because the single force produced by each bow (red arrow) shares the same line of action (dotted line).

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Fig 4-19  (a) Altering the cant of the maxillary occlusal plane with a cervical-pull headgear. To understand the effect on the maxillary arch, the headgear force (red arrow) is replaced with an equivalent force system at the CR (yellow arrows). The maxillary arch extrudes relative to the occlusal plane and rotates around the CR in a clockwise direction. (b) The direction of rotation of the maxillary arch. This rotation compensates for the extrusive component of the cervical headgear. The overall effect is to steepen the maxillary occlusal plane and reduce the open bite.

Fig 4-20  Altering the cant of the occlusal plane with an occipitalpull headgear. The headgear force (red arrow) is replaced with the equivalent force system at the CR (yellow arrows). The maxillary arch CR intrudes relative to the occlusal plane and rotates around the CR in a clockwise direction. This type of headgear reduces the vertical dimension and rotates the mandible forward more effectively than the cervical headgear shown in Fig 4-19.

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Clinical Monitoring and Corrective Action effect will be the same because the line of action is constant. So here we see that a force-driven rather than a shape-driven approach is more effective and that shape must be subservient to force. Headgear descriptions also must be more specific. For example, “high-pull occipital” is vague and describes many different lines of force and effects. A better description might be “45 degrees to the occlusal plane (upward and backward) and anterior to the CR.” This description is based on the force system.

Altering the Cant of the Occlusal Plane with the Headgear A Class II open bite patient with nonparallel occlusal planes is shown in Fig 4-19. The goal is to move the maxillary arch distally to correct the Class II molar relationship and to steepen the maxillary occlusal plane (clockwise rotation of the arch) so that it parallels the cant of the mandibular occlusal plane. Ideally, the vertical facial height should be maintained or reduced. The maxillary and mandibular occlusal planes must be parallel in order to achieve normal vertical overlap (also referred to as overbite). Could a cervical headgear be used (Fig 4-19a)? The equivalent force system at the CR produces two effects: The maxillary arch extrudes relative to the occlusal plane and rotates around the CR in a correct clockwise direction (yellow arrows). The extrusion is undesirable because it might increase the vertical dimension. This may not be the case if sufficient rotation around the CR occurs because of the equivalent moment at the CR rotating the maxillary arch downward in the front and upward in the back. Note that in Fig 4-19b, rotation of the maxillary arch at the CR compensates for any downward and backward translation of the arch. Therefore, the cervical headgear line of force may be a reasonable choice. We could not have necessarily assumed undesirable extrusion just because the pull was downward. The relationship of the maxillary incisor to the lip will also improve as the maxillary occlusal plane steepens, because the original incisor position is superior to stomion.

If a reduction in vertical dimension and forward rotation of the mandible are the goal, the line of force is changed to upward and backward posterior to the CR (Fig 4-20). The further posterior the force, the greater is the moment relative to the force that angles the maxillary arch cant to fit the mandibular arch. Sometimes it is advantageous to add extra wire to the outer bow to achieve this added rotation effect. On the other hand, moving the force closer to the CR (more anteriorly) decreases the moment and reduces the occlusal plane cant.

Clinical Monitoring and Corrective Action The headgear offers many possibilities if we first define the proper line of action for our headgear force system. Unfortunately, sometimes the clinical response can vary, requiring corrective procedures. Perhaps difficulty in identifying a CR or a deformed bow could be the cause. Understanding the same principles that guided us during design can also help during troubleshooting. The goal for the patient shown in Fig 4-21 was to translate the first molar distally, primarily parallel to the occlusal plane. The direction of the line of force (red arrow) was correct but lay too far apically (Fig 4-21a). Because of the equivalent force system (yellow arrows) at the CR, the root moved distally and the crown came forward slightly (Figs 4-21b and 4-21c). To correct the problem, the force was moved occlusal to the crown (Fig 4-21d), and the final result was satisfactory (Fig 4-21e). The inclination correction primarily required rotating the molar around its CR. One could argue that the pull in Fig 4-21f might be more efficient with reduced force magnitude because the outer bow is so far from the CR that the force system approaches a couple. This design is ideal when no further translation is needed during correction.

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4 Headgear

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d Fig 4-21  (a) A headgear was applied to an end-to-end Class II malocclusion. Note that the line of action of the force is too far apical to the CR of the molar. (b) Pretreatment cephalometric radiograph. (c) Cephalometric radiograph after initial headgear treatment. Note that the root moved distally and the crown came forward slightly. (d) The principle of equivalence is used to correct the undesired tooth movement. The force was moved occlusal to the CR. The headgear force (red arrow) is replaced with an equivalent force system at the CR (yellow arrows). This force system will move the root mesially and the crown distally. (e) Cephalometric radiograph after headgear treatment. The molar axial inclination has been corrected. (f) A low cervical headgear may be more efficient if no further translation of the CR is needed and correction of the axial inclination is mainly by rotation around the CR. The headgear force (red arrow) is replaced with an equivalent force system at the CR (yellow arrows), showing the high moment-to-force ratio at the CR.

Inner and Outer Bow Headgear from the Occlusal View

Fig 4-22  Occlusal view of a symmetric headgear. The headgear forces (red arrows) are replaced with a resultant (yellow arrow) at the CR. Assuming that the inner bow of the headgear connects both molars rigidly, the two molars are considered a single rigid body, and the CR of the two molars lies at the midpoint (purple circle). No rotation moment exists around the CR, and the molars equally translate to the distal.

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The occlusal view is important if a different molar distalization is needed on either side. It is possible to deliver a greater force on one side. The reasoning and the solution for how this is done require the application of equilibrium laws. Equilibrium diagrams are described in detail later in this book (see chapter 8); therefore, we will adopt a simpler approach to understand asymmetric headgear applications. A symmetric headgear is shown in Fig 4-22. Let us assume that the inner bow of the headgear is joined (glued) into each buccal tube. This bow connects both molars rigidly. The two molars are considered a single rigid body, and the CR of the two molars lies at the midpoint between them (purple circle). The red arrows on the hook of the outer bow are the forces from the neck strap or elastics from the

Inner and Outer Bow Headgear from the Occlusal View

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Fig 4-23  An asymmetric headgear. (a) The forces from the headgear (red arrows) are replaced with a resultant (yellow arrow), which is off-center. (b) The off-center force (red arrow) is replaced with an equivalent force system at the CR (yellow arrows). Note that it comprises an oblique force and a counterclockwise couple. The right molar will move more distally than the left molar.

Fig 4-24  Clinical application of an asymmetric headgear. (a) The maxillary right molar needs more distal movement than the left molar, and both molars need to move to the left side for axial inclination decompensation. The resultant from the asymmetric headgear forces is replaced with an equivalent force system (yellow arrows) at the CR, showing a desirable moment and a lateral force to the left. (b) The lateral force for decompensation makes the unilateral reverse articulation worse before the surgery.

a

harness. Its resultant is the yellow arrow. As the resultant force passes through the CR in the occlusal view, it will translate the CR. Because there is no moment produced around the CR, no rotation occurs. Symmetric headgear will always have equal forces on each side. Even if we apply different force magnitudes to each side of the headgear, the neck strap will instantly slide to find a new state of equilibrium with equal and opposite forces. Therefore, maintaining different magnitudes of force on each side of the outer bow in a symmetric headgear is impossible unless the neck strap is glued to the neck. Figure 4-23 shows an asymmetric headgear. The length of the outer bow is asymmetric with the left side, which is longer than the right side. The neck strap is still in equilibrium but now lies off-center to the patient’s right. This is the position that the neck strap will move to as the patient functionally turns his or her head. The force magnitudes applied to the outer bow end hooks are equal (red arrows). The oblique resultant of the individual red arrows (yel-

b

low arrow) is off-center toward the left molar (closer to the longer outer bow side in Fig 4-23a). If we replace this single force with an equivalent force system at the CR of the molars (Fig 4-23b), it consists of an oblique distal force with a counterclockwise couple (yellow arrows). Therefore, the left molar will move more in a distal direction than the left molar because of this counterclockwise rotation. The single force at the CR (yellow arrow) has not only a posterior component of force but also a lateral component that moves both molars to the right side. This lateral component of the force is usually considered an undesirable side effect. Figure 4-24 shows a clinical application of asymmetric headgear in which the right molar needs more distal movement and both molars need to move to the left side for buccolingual axial inclination decompensation. The force system is the same as that in Fig 4-23 except that it is a submental vertical view. The molars feel asymmetric distal forces and, at the same time, a lateral (horizontal) compo-

53

4 Headgear

a

54

b

Fig 4-25  Vertically placed headgear tubes on the molars allow freedom of rotation from a bilateral distal force (red arrows). The equivalent force system at the CR (yellow arrows) shows that the molars will move distally and rotate with their mesial sides outward.

Fig 4-26 (a and b) A patient with severe mesial-in rotation on the maxillary right first molar only. A headgear tube was welded on the right side, oriented vertically. Because the vertical tube allowed y-axis rotation, the maxillary right first molar not only moved distally but also rotated mesial out, distal in.

nent of force. For this clinical situation, the lateral force is desirable to decompensate the buccoversion of the maxillary right molar area for orthognathic surgery and is not considered a side effect. Clinical cases should be carefully selected with a complete understanding of the possible limitations of asymmetric headgear force systems. Narrowing of the inner bow width can occur with the asymmetric design. Along with lateral force, this can correct unilateral reverse articulation in properly selected patients. In Fig 4-25, vertical tubes (red circles) are placed to increase the freedom to rotate around the y-axis. Play between the molar tube and the inner bow allows for symmetric rotation of molars. A round inner bow bent at 90 degrees is inserted into the vertical tube so that the molars are free to rotate in the occlusal view. This might be indicated if the first molars are badly rotated. The forces from the headgear are the red distal arrows. As each molar is free to rotate individually, the two molars cannot be considered a single rigid body anymore. How will the molars move? Let us replace the red distal forces with an equivalent force system at the CR of each molar (yellow arrows). The molar now feels both a distal force at the CR and a mesial-out, distal-in rotation couple. Clinically, when the molar has tipped forward for any reason, such as following the early loss of primary molars or extraction of the permanent premolar, the mesial usually rotates inward and the distal outward. In this situation, vertically placed headgear tubes on rotated molars could help

not only for the distal movement but also to correct the rotation. Vertical tubes allow for easier placement of the headgear. Figure 4-26 shows a patient who presented with severe mesial-in rotation of the maxillary right first molar only. Unparalleled horizontal tubes on each molar make it difficult to insert the rigid inner bow, so a headgear tube was welded on the right side, oriented vertically. Because the vertical tube allowed y-axis rotation, the maxillary right first molar not only moved distally but also rotated mesially outward. Here, asymmetric tooth movement was accomplished with a symmetric outer bow shape.

Inner and Outer Bow Headgear from the Frontal View By varying the direction of the headgear force superiorly, there is not only a horizontal force component but also vertical components of force. Figure 4-27a shows the frontal projection of such a force system from design 5 (see Fig 4-9). In the analysis of the force system from the frontal view, the teeth and the headgear cannot be considered as a rigid body because the inner bow and molar tube are round, so individual rotation of each tooth around the tube is possible. The vertical component of the headgear force (red arrow) is replaced with an equivalent force system at the CR of each molar (yellow arrows). In Fig 4-27b, equivalence tells us that the molar will

J-Hook Headgear

a

b

c

Fig 4-27  (a) Frontal view of an occipital headgear force system. The round wire of the inner bow allows freedom of rotation around the x-axis of the molars. Any intrusive component of the headgear (red arrows) could cause buccal tipping. Note the equivalent force system (yellow arrows) at the CR. (b) The molar intrudes, the crown tips buccally, and the root moves lingually. (c) If a passive transpalatal arch is placed, individual molar tipping is not allowed, and intrusive forces (red arrows) can be replaced with a resultant (yellow arrow) at the midpoint between the molars (purple circle).

Fig 4-28  J-hook headgear. The eyelet of the bow is inserted into a hook on the archwire and can deliver only a single force at a fixed point (the hook). No moment is produced.

Fig 4-29 Possible force systems from the J-hook headgear. The red arrows show the possible directions of force at the fixed point of force application (yellow circle).

intrude, the crown will tip buccally, and the root will move lingually. The root will tip more to the palatal side if the inner bow is rigid enough to prevent buccal tipping of the crown of the molar. If a rigid, passive transpalatal arch is engaged as shown in Fig 4-27c, individual molar tipping is not allowed, and the two molars can be considered a single rigid body in which the CR lies at the midpoint (purple circle). The resultant (yellow arrow) of the two headgear forces (red arrows) passes through the CR; therefore, the molar will not tip but will translate in an intrusive direction.

J-Hook Headgear Figure 4-28 shows another type of headgear called a J-hook headgear. It is shaped like a J and has an

Fig 4-30 The gray arrows show the forces that are not possible because the lines of action do not pass through the fixed point of application (green hook).

eyelet at the anterior ends of separate right and left bows, which are inserted into right and left hooks (green) on the archwire. There is no inner bow connecting the left- and right-side outer bows of the headgear. Unlike the inner and outer bow headgear, the force application point is limited to the archwire hook. Only a single force can be applied there because the connection at the anterior eyelet and hook allows freedom of rotation, eliminating any moments. Figure 4-29 shows the possible force systems from the J-hook headgear. Red arrows show the possible directions of forces with a fixed force application point—the green hook (yellow circle). Only magnitude and direction can be varied in this type of headgear, so the types of tooth movement are limited. For example, forces from the gray arrows shown in Fig 4-30 are not possible because their lines of action do not pass through the hook.

55

4 Headgear

a

b

e

10 y 2 mo

f

c

10 y 3 mo

10 y 2 mo

g

d

12 y 10 mo

12 y 10 mo

Fig 4-31  Protraction headgear acting on a full arch. (a) The elastic is engaged between the hook (yellow circle in c) of the labiolingual appliance and the hook of the face mask. (b to d) Intraoral view before (b), during (c), and after (d) headgear treatment. (e) The force system from the protraction headgear. A single force (red arrow) acts on the hook of the labiolingual appliance, and its line of action (dotted line) passes through the CR of the maxillary arch (purple circle). For analysis of orthopedic effects, the equivalent force system on the CR of the maxilla (purple circle) is shown with yellow arrows. (f) Cephalometric radiograph prior to protraction headgear treatment. (g) Cephalometric radiograph after protraction headgear treatment. (h) Superimposition before (black) and after (red) protraction headgear treatment. A significant amount of forward translation of the maxillary arch with a little possible forward orthopedic effect on the maxilla is shown.

h

Protraction Headgear Figure 4-31 shows a protraction (or reverse) headgear on a full maxillary arch. It was designed to apply an extraoral anterior force on a nonextraction patient. The elastics are stretched from the face mask to the hooks on the labiolingual appliance (Fig 4-31c). Figures 4-31b and 4-31d show the pretreatment and posttreatment intraoral photos, re-

56

spectively. This protraction headgear mechanism is similar to that of a J-hook headgear in that only a single force is applied on each side and the force application point is fixed at the hook (yellow circle in Fig 4-31c). Figure 4-31e shows the force system from the protraction headgear. Assuming that the labiolingual appliance rigidly connects all of the teeth in the maxillary arch, the CR of the maxillary arch lies between the roots of the premolars (purple circle).

Protraction Headgear The force from the protraction headgear (red arrow) was designed so that the line of action passes through the CR of the maxillary dentition and the direction of force is forward and downward (dotted line). Therefore, the maxillary dentition will translate forward and downward along the line of action. In the analysis of the orthopedic effect, the force from the protraction headgear was replaced with an equivalent force system at the CR of the maxilla (purple circle). The yellow arrows show the equivalent force system acting on it. The maxilla will not only translate forward and downward but will also rotate in a counterclockwise direction. Figures 4-30f to 4-30h show the before and after cephalometric radiographs and superimposition tracings. The superimposition tracings show that the Class III molar relationship and anterior reverse articulation were corrected mainly by forward translation of the maxillary arch, accompanied by a possible slight orthopedic forward movement of the maxilla. There is debate about how much of an orthopedic effect is achieved with protraction headgear. Typically, the mandible can rotate downward and backward. Maxillary anterior translation may be limited. Also, the CR of the maxilla is not exactly known. Most of the correction with a protraction headgear appears to be dental rather than skeletal; however, more research is needed.

A protraction headgear on the first molar only was used in a Class III extraction patient to help translate the maxillary first molar forward rather than to translate the entire arch (Fig 4-32). An extension bar was rigidly connected to the molar tube (Figs 4-32a and 4-32c), and the force (red arrow) was applied at the hook (yellow circle) so that the line of action of the force passed through the CR of the first molar (dotted line). A passive transpalatal arch rigidly connected both molars to prevent individual molar rotation (Fig 4-32e). Figures 4-32g and 4-32h show the cephalometric radiographs before and after protraction headgear treatment along with the superimposition tracings (Fig 4-32i). The Class III molar relationship and anterior reverse articulation were corrected by protraction of the maxillary molar and retraction of the mandibular anterior teeth. Note that there were no significant orthopedic effects. A common error is to position the line of action of the protraction force at the level of the brackets, somewhat parallel to the occlusal plane shown in Fig 4-33. Because the force is not acting through the CR, a moment is produced that will reduce the occlusal plane angle, resulting in an open bite.

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4 Headgear

a

14 y 11 mo

b

e

g

i

58

14 y 11 mo

c

15 y 2 mo

d

19 y 3 mo

f

14 y 11 mo

h

19 y 3 mo

Fig 4-32 Protraction headgear acting on a molar. (a) An extension bar is placed at the buccal tube, and the force (red arrow) acts on the hook (yellow circle) of the extension bar. The line of action (dotted line) passes through the CR (purple circle) of the molar. (b) Intraoral view before treatment. (c) The extension bar in place with a hook for an elastic (yellow circle). (d) Intraoral view after headgear treatment. (e) A passive transpalatal arch is placed so that individual rotation of the molar is prevented. (f) The force system from the protraction headgear. The line of action (dotted line) of the protraction headgear force (red arrow) passes through the CR (purple circle) of the molar. (g) Cephalometric radiograph before protraction headgear treatment. (h) Cephalometric radiograph after protraction headgear treatment. (i) Superimposition before (black) and after (red) protraction headgear treatment. The Class III molar relationship and the anterior reverse articulation were corrected by protraction of the maxillary molar and retraction of the mandibular incisors. Note in the cranial base superimposition that the amount of orthopedic effect is insignificant.

Recommended Reading Fig 4-33  An improper line of action of elastic force (red arrow) would produce an anterior open bite because of undesirable rotation caused by a couple (replaced force system at the CR, yellow arrows). Ideally, force should be directed through the CR of the maxillary arch to prevent unwanted rotation of the arch.

Recommended Reading

Jacobson A. A key to the understanding of extraoral forces. Am J Orthod 1979;75:361–386.

Badell MC. An evaluation of extraoral combined high-pull traction and cervical traction to the maxilla. Am J Orthod 1976;69:431–466.

Kloehn SJ. An appraisal of the results of treatment of Class II malocclusions with extraoral forces. In: Kraus BS, Reidel RA (eds). Vistas in Orthodontics. Philadelphia: Lea & Febiger, 1962:227–258.

Baldini G. Unilateral headgear: Lateral forces as unavoidable side effects. Am J Orthod 1980;77:333–340.

Kuhn RJ. Control of anterior vertical dimension and proper selection of extraoral anchorage. Angle Orthod 1968;38:340–349.

Baldini G, Haack DC, Weinstein S. Bilateral buccolingual forces produced by extraoral traction. Angle Orthod 1981;51: 301–318.

Melsen B, Enemark H. Effect of cervical anchorage studied by the implant method. Eur J Orthod 2007;29:i102–i106.

Barton JJ. High-pull headgear versus cervical traction: A cephalometric comparison. Am J Orthod 1972;62:517–539. Drenker EW. Unilateral cervical traction with a Kloehn extraoral mechanism. Angle Orthod 1959;29:201–205. Gould E. Mechanical principles in extraoral anchorage. Am J Orthod 1957;43:319–333. Güray E, Orhan M. “En masse” retraction of maxillary anterior teeth with anterior headgear. Am J Orthod Dentofacial Orthop 1997;112:473–479. Haack DC, Weinstein S. The mechanics of centric and eccentric cervical traction. Am J Orthod 1958;44:345–357. Hershey HG, Houghton CW, Burstone CJ. Unilateral facebows: A theoretical and laboratory analysis. Am J Orthod 1981;79:229–249. Hubbard GW, Nanda RS, Currier GF. A cephalometric evaluation of nonextraction cervical headgear treatment in Class II malocclusion. Angle Orthod 1994;64:359–370.

Nikolai RJ. Bioengineering: Analysis of Orthodontic Mechanics. Philadelphia: Lea & Febiger, 1985:322–371. Perez CA, de Alba JA, Caputo AA, Chaconas SJ. Canine retraction with J hook headgear. Am J Orthod 1980;78:538–547. Tanne K, Hiraga J, Kakiuchi K, Yamagata Y, Sakuda M. Biomechanical effect of anteriorly directed extraoral forces on the craniofacial complex: A study using the FEM. Am J Orthod Dentofacial Orthop 1989;95:200–207. Tanne K, Matsubara S, Sakuda M. Stress distributions in the maxillary complex from orthopedic headgear forces. Angle Orthod 1993;63:111–118. van Steenbergen E, Burstone CJ, Prahl-Andersen B, Aartman HA. The role of a high pull headgear in counteracting side effects from intrusion of the maxillary anterior segment. Angle Orthod 2004;74:480–486. Worms FW, Isaacson RJ, Speidel TM. A concept and classification of centers of rotation and extraoral force systems. Angle Orthod 1973;43:384–401.

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PROBLEMS

1. Compare forces FA, FB, and FC. Do length and position of the outer bow make any difference?

2. The outer bow is the same as in problem 1, but the direction of pull is different. Compare forces FA, FB, and FC. Do length and position of the outer bow make any difference?

3–8. R  eplace the 500-g headgear force with an equivalent force system at the CR. Describe how the molar will move.

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3

4

5

6

7

8

Problems

9. Replace the combination headgear forces with a single equivalent force. How will the molar move? Show the line of action of the resultant force. If ϴ1 and ϴ2 are equal and ϴ1 = 20 degrees, then what is the magnitude of the resultant?

10. Two identical headgears are used on a Class II patient. What is the effect of moving the tube 4 mm gingivally? Compare FA and FB.

11. A J-hook headgear is used. Draw the force vector at the hook to translate the maxillary dentition. Is it possible to translate the maxillary dentition parallel to the occlusal plane?

12–13. Replace the headgear force at the CR. Describe the movement of the maxillary arch.

12

13

61

CHAPTER

5 The Creative Use of Maxillo­ mandibular Elastics “Truths are easy to understand once they are discovered; the point is to discover them.”

OVERVIEW



— Galileo Galilei

Maxillomandibular elastics (or intermaxillary elastics) are commonly used because of their simplicity; however, a lack of understanding of their force system can lead to many serious problems. Elastics are usually classified by the direction of the force (eg, Class II or Class III elastics). Sometimes force magnitude is considered, but point of force application is left out. Therefore, many different types of Class II elastics can be applied. There are short or long elastics. Short elastics can be placed anteriorly or posteriorly to produce asynchronous occlusal plane cant effects. Proper use of maxillomandibular elastics requires consideration of the attachment point of the elastic (line of action of the force) in respect to the center of resistance of each arch. In this way, vertical dimension, occlusal plane cants, and vertical overlap can be controlled and corrected if necessary. Often too many elastics are used when a single resultant elastic at the correct location would work better. However, sometimes more than a single elastic is needed when the attachment point is not directly accessible. Vertical elastics to cover up intra-arch mechanical side effects are rarely the best solution. All maxillomandibular elastics and their actions should be analyzed in three dimensions.

63

5 The Creative Use of Maxillo­mandibular Elastics Elastomeric rings, more commonly called elastics, are routine in orthodontic treatment and are some of the simplest appliances used. Unfortunately, how­ ever, the mode of operation and the proper selection of maxillomandibular elastics (also referred to as intermaxillary elastics) are not well known or practiced clinically. Different types of elastics are used either within the same arch (intra-arch elastics) or between arches (maxillomandibular elastics). One orthodontic sequence is to accomplish intra-arch alignment early and then later coordinate the arches that do not fit either because of side effects during intra-arch alignment or because of remaining original discrepancies. Commonly, maxillomandibular elastics are needed to correct problems introduced during early states of treatment. Maxillomandibular elastics can also be used directly to enhance anchorage or correct intra-arch or maxillomandibular problems. Because there are an infinite number of possible maxillomandibular applications (directions and points of force application), for simplicity in this chapter, our discussion will be limited to different types of bilateral Class II elastics, vertical elastics, and transverse or crisscross elastics, followed by asymmetric elastics such as unilateral Class II elastics and combined Class II/Class III elastics. This approach should provide a sufficient explanation that can be extrapolated to all maxillomandibular applications so as to minimize undesirable effects. Because of the infinite number of clinical situations that surround treatment planning with maxillomandibular elastics, some assumptions are made. Let us assume that a rigid arch is placed in all of the brackets, allowing no rotation or play in all planes. The theme of this chapter is to show the effects of single forces to the maxillary and mandibular arches (as rigid units without play), so play is only mentioned briefly. The centers of resistance (CRs) are best estimates, recognizing variation and three-dimensional difficulties. These are our boundary conditions for studying the effects of maxillomandibular elastics; other conditions, such as the use of round wires allowing rotation of the incisors or wire elastic deformation beyond our boundary, may still be partially valid. Treating both arches as rigid bodies simplifies our analysis so that we may arrive at core concepts without losing ourselves in the details. Although the effect of elastics on full rigid arches is the main consideration, the effects on segments and nonrigid wires are also discussed briefly. Maxillomandibular elastics represent a simple appliance that is not precise. As the patient moves the jaw and alters the vertical dimension, the forc64

es change. Variation in elastic size and degradation caused by the fluids in the mouth add to this variability. For standardization, we will maintain the vertical dimension near the mandibular rest position, because typically elastics are not worn during chewing of food. To develop the concepts in this chapter, Class II elastics are discussed initially, with the understanding that similar principles can be used for other force directions. Not all Class II elastics are the same, and different types of Class II elastics may produce radically different effects. Part of the problem is the traditional classification that is based on force direction: Class II, Class III, vertical, crisscross, etc. Direction is insufficient to describe a force system. Lacking in particular is the point of force application. Therefore, a Class II elastic is evaluated from three separate views: the lateral view parallel to the midsagittal plane, the frontal view, and the occlusal view. The coordinate system used is the occlusal plane.

What Does a Class II Elastic Do? If a Class II elastic is placed between the maxillary canine and the mandibular second molar (Fig 5-1), the point of force application is at the canine and molar hooks, and the force acts along its line of action (green elastic). To predict arch movement, we replace the applied force (red arrow) with an equivalent force system (yellow arrows) at the CR (purple circle) of each arch. Figures 5-1a and 5-1b are identical except that the force system is depicted in the maxillary arch in Fig 5-1a and in the mandibular arch in Fig 5-1b. Note that the perpendicular distances from the CR of the maxillary and mandibular arches to the elastic (D1, D2) are about the same, and hence the equivalent moments (curved yellow arrows) are the same. So how will the arches move? From the equivalent force system at the CR, the maxillary arch moves downward and backward and the mandibular arch moves upward and forward. Simultaneously, the large moments cause rotation of both arches around their CRs (Fig 5-1c). This analysis is similar to how the headgear was studied in chapter 4, except now two arches are being considered. In short, to understand what a maxillomandibular elastic will do, the elastic force is replaced with an equivalent force system at the CR of each arch by a force and a couple. Two movements are noted: (1) the direction and magnitude of the translation by the force and (2) the direction and magnitude of rotation around the CR. Figure 5-1 is an example of a long Class II elastic, where the distances from

Synchronous Class II Elastics

a

b

c

d

Fig 5-1  Long Class II elastic. (a and b) A single force from the elastic (red arrow) is replaced with an equivalent force system (yellow arrows) at the CRs of the maxillary and mandibular arches. (c) Both arches rotate synchronously (dotted curved arrows) in the same clockwise direction because of the same magnitude and direction of the moments (D1 = D2). (d) Lateral superimposition of cephalometric radiographs before (black) and after (red) long-term use of Class II maxillomandibular elastics in an extraction case using round wire. Both the maxillary and mandibular arches rotated in a clockwise direction, with extrusion of the maxillary anterior teeth and mandibular posterior teeth.

the elastic to the maxillary and mandibular CRs are the same, thereby causing the moments at the CRs to be equal and rotation around each CR to be the same. In other words, the occlusal plane of both the maxillary and mandibular arches will rotate clockwise (steepen) by about the same amount. If the occlusal planes rotate by the same amount, the rotation is called synchronous. Ignoring other factors, the vertical overlap (also known as overbite) will be maintained; otherwise, factors such as tooth extrusion and rotation of the mandible could influence the amount of vertical overlap.

Synchronous Class II Elastics The green elastic shown in Fig 5-2 is also synchronous, because the CR is an equal distance from the force in both arches. The point of attachment is

between the distal of the maxillary first premolar bracket and the mesial of the mandibular second premolar bracket. Because the distances (D1, D2) are equal, it can be described as a synchronous short Class II elastic. Note that the length of the red arrow is the same as that in Fig 5-1 because the force magnitude is the same. How does the predicted movement of a short elastic compare to that of the long elastic? The direction is different, with the short elastic having a greater vertical component of force in comparison with the horizontal force. Perhaps of more significance are the equivalent moments at the CR (yellow arrows), which are much smaller than those with the long elastic. Treatment goals can vary in respect to the level and cant of the treated occlusal plane. Do we want to extrude teeth and increase the vertical dimension by rotating the mandible downward and backward? Do we want to maintain the original cant 65

5 The Creative Use of Maxillo­mandibular Elastics Fig 5-2  Short Class II elastic. A single force from the elastic (red arrows) is replaced with an equivalent force system (yellow arrows) at the CR of each arch. It is also synchronous because the CR is an equal distance from the force in each arch (D1 = D2). The moment is lower and the vertical component of force is greater than that with the long Class II elastic.

or increase its angle? The choice of a long or short synchronous elastic is important in achieving the patient-specific treatment objective. The short elastic will not steepen the occlusal plane very much. But what about the increased vertical force? We cannot assume that this will always be deleterious. Patients can grow vertically, and the eruption of teeth can be masked with the increased vertical dimension. Some patients with flatter mandibular planes are highly resistant to an increase in vertical dimension from the extrusive component of the elastic. Nongrowing adult patents usually show little increase in vertical dimension following treatment. Thus, the effect of a Class II elastic is dependent not only on the elastic force but also on the occlusal forces and function. Traditionally, the orthodontist has evaluated the Class II elastic primarily by its direction. It was assumed that the more horizontal the force, the better. In most applications, direction may be modified some but not significantly; of greater importance are the equivalent moments operating at the CRs of the arches. These moments are determined by the points of force application of the elastic and the lines of action of the forces. In some Class II patients, the goal is to minimize the increase in vertical dimension. To determine whether a short or long elastic is preferable in this situation, the effect of large moments on the posterior end of the arches should be considered. The long elastic (see Fig 5-1b) that produces a large moment around the CR will extrude the mandibular second molar (see Fig 5-1c). Long-term use of Class II elastics and round wire led to clockwise rotation of both arches and extrusion of the maxillary anterior and mandibular posterior teeth in the patient shown in Fig 5-1d, producing excessive exposure of the maxillary anterior teeth with relaxed lips. For every millimeter of molar eruption, the mandible could rotate, significantly reducing the vertical 66

overlap in the incisor region by 2 to 3 mm. Because it is the moment that rotates the arch, it could be argued that the short elastic (with the smaller moments) is more conservative in preserving vertical dimension even though the vertical component of force is greater (see Fig 5-2). However, the horizontal component of force, which contributes to the displacement of the CR, is much less. More clinical research is needed to study the effect of varying the elastic force position and magnitude on the control of vertical dimension. The effects of short-term use of Class II elastics are very different from changes after considerable growth, which may mask the initial tooth movement. This chapter only describes the immediate force systems and their effects on the teeth.

Asynchronous Class II Elastics Let us now move the short elastic anteriorly so that its line of action passes through the CR of the mandibular arch (Figs 5-3a and 5-3b). The translatory effects at the CR are the same as in Fig 5-2; however, the moment in respect to the CR will be different for each arch. A large moment (yellow curved arrow) at the CR of the maxillary arch steepens the maxillary occlusal plane (rotates it clockwise). Because the line of action of the force on the mandibular arch goes through the CR of that arch, no moment and hence no rotation are produced in the mandibular arch. The arch rotations are asynchronous, with only a maxillary arch rotation leading to a lack of occlusal plane parallelism and an increase in vertical overlap (Fig 5-3c). If we move the same short elastic posteriorly so that the force (red arrow) now goes through the CR of the maxillary arch, the opposite effect will be observed (Fig 5-4a and 5-4b). Replacing the force system at each CR, it is observed that the maxillary

Asynchronous Class II Elastics

a

b

Fig 5-3  Short Class II elastic placed anteriorly. (a and b) A single force from the elastic (red arrow) is replaced with an equivalent force system (yellow arrows) at the CR of the maxillary arch. (c) The moment in respect to the CR will be different for each arch; therefore, only the maxillary arch rotates asynchronously.

c

arch will not rotate and the mandibular arch occlusal plane will steepen (rotate clockwise), leading to a reduction in vertical overlap (Fig 5-4c). Finally, let us move the same short elastic further forward so that the line of action is between the distal of the maxillary central incisors and the distal of the mandibular lateral incisors (Fig 5-5). This is called an anterior vertical elastic, but the distal translatory effect on the CR from the distal (horizontal) component of force is the same as the previous short elastics shown in Fig 5-2 to 5-4, except that the rotation tendency will be different; the maxillary arch will steepen, and the mandibular arch will flatten, leading to an increase in the deep bite. Note that the moment at the CR is greater on the maxillary arch than on the mandibular arch because D1 is greater than D2. The various Class II elastics discussed so far are representative but not fully inclusive of all possibilities.

Changing the magnitude and direction of the force is important. Of even greater significance is the positioning (point of force application) of the elastic force. It is the point of force application that will determine the amount and direction of arch rotation around the CR and the occlusal plane cant effect. Let us now consider three clinical examples of how an equivalence analysis can help us to select the best possible elastic pull. The patient shown in Fig 5-6 has nonparallel maxillary and mandibular occlusal planes, a severe anterior open bite, and a Class II malocclusion. Patients with this type of malocclusion are most likely candidates for surgery and not just Class II elastic therapy. However, let us assume that some Class II maxillomandibular elastics are needed. What would be the best line of action for the force? A short elastic placed at the maxillary canine bracket attached to the mandibular first premolar bracket would produce a large moment 67

5 The Creative Use of Maxillo­mandibular Elastics

b

a

b Fig 5-4  Short Class II elastic placed posteriorly. (a and b) A single force from the elastic (red arrow) is replaced with an equivalent force system (yellow arrows) at the CR of the mandibular arch. (c) The moment in respect to the CR will be different for each arch; therefore, only the mandibular arch rotates asynchronously.

c

Fig 5-5 Anterior vertical elastic. The maxillary and mandibular arches will rotate in opposite directions, leading to an increase in vertical overlap. Most of the rotation will occur in the maxillary arch (D1 > D2).

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Fig 5-6  Short Class II elastic force (red arrow) placed anteriorly in a Class II open bite case. An equivalent force system at the CR of the maxillary arch (yellow arrows) indicates that a large moment is only produced in the maxillary arch, closing the open bite and reducing the Class II malocclusion. The cant of the mandibular plane of occlusion will not change.

Asynchronous Class II Elastics

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Fig 5-7  Short Class III elastic placed anteriorly in a Class III open bite case. (a) The single force (red arrow) is through the CR of the maxillary arch. (b) An equivalent force system at the CR of the mandibular arch (yellow arrows) has a large moment, closing the open bite and reducing the Class III malocclusion. The cant of the maxillary plane of occlusion will not change.

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Fig 5-8  Short Class II elastic placed posteriorly in a Class II deep bite case. (a) The single force (red arrow) is through the CR of the maxillary arch. (b) An equivalent force system at the CR of the mandibular arch (yellow arrows) produces a large moment, opening the bite and reducing the vertical overlap. The cant of the maxillary occlusal plane will not change.

at the CR of the maxillary arch, which would steepen the maxillary occlusal plane, close the open bite, and reduce the Class II malocclusion. The cant of the mandibular occlusal plane would not change. In a similar manner, the best placement of the line of force for a Class III elastic for the Class III open bite shown in Fig 5-7a would be through the CR of the maxillary arch. Asynchronous rotation around the CR occurs in the mandibular arch (Fig 5-7b), and no rotation is produced in the maxillary arch. These two open bite cases are shown as exaggerations to better explain proper maxillomandibular elastic use and not to suggest that maxillomandibular elastics are the best treatment modality. Sometimes a patient with deep bite will still have excessive vertical overlap after leveling. A careful

appraisal will show two perfectly leveled arches that are not parallel and anteriorly converging (Fig 5-8). If Class II elastics are needed, one must be careful not to increase the vertical overlap. The best elastic use is a short elastic placed as far distal as possible on the second molar with a line of action through the maxillary CR. Note that the maxillary arch does not rotate and the mandibular arch steepens to better parallel the cant of the maxillary occlusal plane. A bite block placed anteriorly is helpful to disengage the posterior teeth during this corrective stage. It is best to avoid this undesirable leveling side effect with good mechanics during initial alignment, because correction may be difficult.

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a Fig 5-9  Frontal view of the long Class II elastic shown in Fig 5-1. The replaced equivalent force system (yellow arrows) at the CR shows that the mandibular second molar (terminal molar) will move in a superior direction; at the same time, the moment produced by the elastic at the CR will tip the molar crown lingually.

Fig 5-10  Posterior crisscross elastic in proximal view. (a) Force magnitude and direction can vary depending on jaw opening and hook placement. (b) For simplicity, we will assume the line of action to be an arbitrary line connecting the points at the brackets or hooks where the elastic is attached.

Nonrigid Arches with Third-Order Play So far this discussion of Class II elastics has outlined many of the important principles of usage and has not necessarily presented elastics as optimal treatment for the correction of a specific Class II malocclusion. There are other modalities, such as headgear, functional appliances, temporary anchorage devices [TADs], and loop springs, for that purpose. It has been assumed that the wires are rigid and that no play exists between the wire and the bracket. If round wires are used, one might expect a similar response; however, incisors could change their axial inclinations. There would be less change in the cants of the occlusal planes with full rigidity of wires and teeth. Furthermore, with a round wire without third-order control, maxillary molars can tip to the buccal, and mandibular molars can tip to the lingual, resulting in a crossbite (also known as a scissor bite). Viewed from the frontal plane, the long Class II elastic shown in Fig 5-1 produces a vertical component of force away from the CR of the mandibular arch. The mandibular second molar (terminal molar) will move in a superior direction; at the same time, the moment produced by the elastic at the CR will tip the molar crown lingually (Fig 5-9). Nevertheless, the determination of the effects of a maxillomandibular elastic in a rigid, no-play system is usually a good starting place in deciding proper maxillomandibular elastic use. In this chapter, this boundary condition is applied to demonstrate the principle of equivalence. 70

b

Lateral or Crisscross Elastics Crisscross elastics are commonly placed on individual molars, on an entire posterior segment, or on a full arch. Let us consider the effect of crisscross elastic placement on a full arch, assuming again a rigid archwire without any play. This may or may not be a good idea in clinical situations, but it will serve as the basis for evaluating lateral elastic force. A buccolingual crisscross elastic is shown in Fig 5-10. What is the line of action of the force? The answer is complicated, because the elastic contacts the molar’s occlusal anatomy (Fig 5-10a). For simplicity, we will assume the line of action to be an arbitrary line connecting the points (bracket or hooks) where the elastic is attached (Fig 5-10b). Some clinicians might place a maxillary and mandibular passive continuous arch in the malocclusion presented in Fig 5-11, which shows a crisscross elastic inserted only on the right side. This may not be correct, but let us analyze the force system. The equivalent force system (yellow arrows) is calculated to the CRs (purple circle) of the maxillary and mandibular arches. A large moment is produced in the maxillary arch that cants the occlusal plane downward on the right side and upward on the left side. The opposing force (yellow arrow) on the mandibular arch produces a negligible moment to the CR of that arch and, hence, no rotation occurs around the CR (Fig 5-11a). A large moment is produced in the maxillary arch that cants the occlusal plane occlusally on the right side and apically on the left side (Fig 5-11b). If this elastic is used, an open bite will be produced on the patient’s left side. It is common for a unilateral elas-

Lateral or Crisscross Elastics Fig 5-11  Unilateral posterior crisscross elastic in a continuous arch. (a) The forces from the crisscross elastic (red arrows) are replaced with equivalent force systems at the CRs (yellow arrows). (b) Asynchronous occlusal plane effects causing an open bite on the left side are anticipated.

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Fig 5-12  (a) To balance the moment created by the unilateral posterior crisscross elastic in Fig 5-11, a vertical elastic force (red arrow) is applied on the left side. The vertical elastic’s equivalent force system at the CR (yellow arrows) is equal and opposite to the moment from the crisscross elastic on the right side. (b) The yellow arrows are the resultant from the two elastics. (c) The resultant force is replaced with the equivalent force system at each CR (yellow arrows).

tic to be placed on a continuous arch to correct a posterior unilateral crossbite. Unfortunately, this can produce an asynchronous effect with an open bite and dual cants of the occlusal plane as seen from the frontal aspect. The open bite on the left side could be reduced by placing a vertical elastic so that a clockwise moment acts at the maxillary CR and a counterclockwise moment acts at the mandibular CR (Fig 5-12a). The vertical elastic must be placed in the identical anteroposterior position as the crisscross elastic. The resultant forces produced at the maxillary and mandibular arches are depicted in yellow arrows in Fig 5-12b. Once again, resultant single forces are replaced with an equivalent force system at the CRs of the maxillary and mandibular arches (Fig 5-12c), showing that both arches will rotate in a counterclockwise direction and no open bite will be produced on the left side. Another alternative to avoid the asynchronous effect in Fig 5-11 is to use two bilateral crisscross

elastics (Fig 5-13a). In this application, the resultant (yellow arrow) of the two forces (red arrows) of the crisscross elastics lies an equal distance from the maxillary and mandibular CRs (D1 = D2). Therefore, once again resultants are replaced with equivalent force systems on the CRs of the maxillary and mandibular arches (Fig 5-13b). Figure 5-13b also shows the same moments acting on each arch. The maxillary arch will translate downward to the left and synchronously rotate around the CR in a counterclockwise direction. The mandibular arch will translate upward to the right and synchronously rotate around the CR in a counterclockwise direction. Both the maxillary and the mandibular occlusal planes will cant in the same direction so that no lateral open bite will be produced. The difference between Fig 5-12 and Fig 5-13 is the direction. Figure 5-12 shows short maxillomandibular elastics with a more vertical component of force, while Fig 5-13 shows long maxillomandibular elastics with a more horizontal component of force in the frontal view. 71

5 The Creative Use of Maxillo­mandibular Elastics

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Fig 5-13  Bilateral crisscross elastics. (a) The resultant (yellow arrows) of the two forces of the crisscross elastics (red arrows) lies an equal distance (D1 = D2) from the maxillary and mandibular CRs. (b) The equivalent force systems at the CRs of each arch have synchronous moments (yellow arrows), rotating the maxillary and mandibular arches equally so that no lateral open bite will be produced.

a Fig 5-14 Changing the point of force application of a single force (red arrows) from crisscross elastics in a continuous arch. Arch rotation (occlusal view) will be produced unless the force passes through the CR. A, clockwise rotation; B, translation; C, counterclockwise rotation produced with various positions of the force.

Fig 5-15  The location of a crisscross elastic force (red arrow) and arch rotation (occlusal view). The equivalent force system at the CR is represented by yellow arrows. (a) An anterior crisscross elastic rotates the arch in a clockwise direction. (b) An elastic placed at the first molar rotates the arch in a counterclockwise direction.

The application of a unilateral elastic to correct a malocclusion with a unilateral crossbite may not be the best mechanics according to this analysis. Better alternatives include asymmetric lingual arches, crisscross elastics on only the offending segment, and TAD applications (see chapter 18). The use of a crisscross elastic on a continuous arch is not trivial because many side effects can occur. This requires us to look also from the occlusal view (Fig 5-14). If the force is in alignment with the CR, the arch will only translate to the patient’s left side. If the force is anterior or posterior to the CR, both translation and rotation around the CR will be observed. Figure 5-15a shows an anterior crisscross elastic rotating an arch in a clockwise direction; in Fig 5-15b, the elastic is placed at the first molar and the maxillary arch rotates in a counterclockwise direction. An anteriorly placed crisscross elastic is sometimes used to correct a midline discrepancy and an 72

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asymmetry in buccal occlusion. But is this an effective method with minimal side effects? Figure 5-16 shows the equivalent force system (yellow arrows) at the maxillary and mandibular CRs. From the frontal view, the force is away from the CR, tipping the maxillary arch to the left and the mandibular arch to the right. This could aid in the midline correction; however, the lateral tipping and the canting of the maxillary and mandibular occlusal planes (counterclockwise) are undesirable. The canting of the occlusal planes would be particularly unesthetic from the frontal view (Fig 5-17). What about the moment acting in the maxillary occlusal view? Is this moment useful (see Fig 5-15a)? In theory, this should help correct maxillary midline movement to the left and Class II correction on the maxillary left side. The couple (yellow arrows) will produce rotation of the arch en masse—both roots and crowns move distally on the left side. This type of tooth translation is very slow, and it is not

Lateral or Crisscross Elastics

Fig 5-16  Anterior crisscross elastic (red arrows). Equivalent force systems at the maxillary and mandibular CRs are represented by yellow arrows. The force system would aid in midline correction; however, the canting from the moments of the maxillary and mandibular occlusal planes is unavoidable.

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Fig 5-17  Predicted treatment result from an anterior synchronous crisscross elastic. The canting of the frontal occlusal planes would be particularly unesthetic.

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Fig 5-18  Anterior crisscross elastic (red arrows) placed off-center. Equivalent force systems (yellow arrows) at the CRs show a maxillary occlusal plane that rotates very little (a) and not at all (b) because D1 is very small. Therefore, the cant of the maxillary occlusal plane will be maintained. An open bite may occur on the right side because of the counterclockwise rotation of the mandibular arch due to the large moment at the mandibular CR (large D2).

a practical way to correct an asymmetry because of side effects if the force is acting at bracket level. Furthermore, note that if actual translation and rotation of the full arch occurs, arch harmony is lost, and a reverse articulation and crossbite are created (Fig 5-15b). Note the changes in the canine and molar regions; even rotation alone around the CR can create crossbite and reverse articulation. What can occur more rapidly is a mandibular shift to the patient’s right side. In that situation, the correction may only be temporary, unless the original crossbite and reverse articulation are a functional shift. The moment around the CR from the occlusal view can also produce an iatrogenic crossbite and reverse articulation as a side effect. Note the changes in the molar region after the rotation (see Fig 5-15a).

Unlike Figs 5-16 and 5-17, Fig 5-18 shows an off-center anterior crisscross elastic producing an asynchronous occlusal plane effect. In addition to the equal and opposite translatory movements, only the mandibular occlusal plane rotates in a counterclockwise direction, producing an open bite on the right side. If not carefully planned, placement of anterior crisscross elastics can lead to many undesirable side effects. Anterior crisscross elastics placed for midline discrepancy correction may cause many side effects in three dimensions. However, sometimes these side effects may be desirable; once we completely understand the force system, we can use these side effects to our advantage in carefully selected cases.

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Fig 5-19  A patient with a midline discrepancy and occlusal plane canting. (a) Initial facial view. The maxillary and mandibular occlusal planes cant downward on the left side. (b) Initial intraoral frontal view. The maxillary and mandibular midline discrepancy is shown. (c) An anterior crisscross elastic is placed. (d) Intraoral view after 17 months of treatment. (e) Facial view after treatment. Not only was the midline corrected, but the abnormal canting of the occlusal plane was also reduced.

Figure 5-19 shows a patient with a midline discrepancy whose maxillary and mandibular occlusal planes canted downward on the left side from the frontal view. The treatment objective was to correct the midline and cant the occlusal planes synchronously in a counterclockwise direction. An anterior crisscross elastic was placed (Fig 5-19c), and the force system was exactly the same as in Fig 5-16. Within 5 months, significant correction had occurred. The forces and moments produced by the elastic simultaneously corrected the midline discrepancy and the occlusal plane cant. It was demonstrated that a functional shift repositioned the mandible back to centric relation. Little tooth movement or growth was evident during this period of time. Figure 5-19d shows that the midline discrepancy was resolved. The occlusal plane canting correction was not enough; however, both arches rotated in a corrective direction (Fig 5-19e). 74

So what did the anterior crisscross elastic do? It helped eliminate an occlusal interference so that the mandible was allowed to reposition to centric relation. In this patient, narrowing of the mandibular arch or expansion of the maxillary arch with an increase of buccal horizontal overlap (also referred to as overjet) might also have eliminated the mandibular shift. The rationale for use of the crisscross elastic was the added effect of altering the cant of the frontal plane of occlusion and helping to correct any lateral dental asymmetry that might have been present. Figure 5-20a shows a patient with a midline discrepancy accompanied by a unilateral buccal reverse articulation on the right side. Particularly in some young patients, it is difficult to guide a mandible into centric relation to diagnose a functional shift or a true asymmetry. As a therapeutic treatment (diagnosis with treatment), rapid palatal expansion was

Lateral or Crisscross Elastics

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Fig 5-20  A mixed dentition patient with a midline discrepancy and unilateral buccal reverse articulation. (a) Initial intraoral view. The midline discrepancy and buccal reverse articulation on the right side are shown. (b) A rapid palatal expander was used in the maxillary arch. (c) Intraoral view after rapid palatal expansion treatment. The midline discrepancy was reduced. The midline correction was the result of a mandibular shift back to centric relation. With proper diagnosis, it is better to remove occlusal interferences by planned tooth movement than by blind use of an anterior crisscross elastic.

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Fig 5-21  An adult patient with a midline discrepancy and unilateral buccal reverse articulation. (a) Initial intraoral view. The mandibular midline is off to the right side with a buccal reverse articulation on the right side. (b) A unilateral constriction lingual arch was placed. (c) Intraoral view after unilateral constriction. (d) Frontal view after the buccal reverse articulation was resolved. The midline discrepancy is resolved. (e) Frontal view after retraction of the mandibular anterior teeth using symmetric mechanics. Midline correction was produced by correcting a mandibular shift. Proper diagnosis and specific tooth movement to eliminate occlusal interferences offered a better treatment modality than an anterior crisscross elastic.

performed on the maxillary arch (Fig 5-20b). Note that the midline discrepancy was reduced because a functional shift was resolved (Fig 5-20c). Blindly using an anterior crisscross elastic to treat this patient for midline correction would not make any sense; rather, treatment was specifically aimed at reducing the occlusal interferences of the posterior teeth in reverse articulation. Figure 5-21a shows an adult patient with a midline discrepancy and a buccal reverse articulation on the right side. The midline discrepancy was due to a functional shift of the mandible. An asymmetrically activated lingual arch was inserted on the

mandibular molars for unilateral constriction of the mandibular right first molar (Figs 5-21b and 5-21c). Figure 5-21d shows the frontal view after the buccal reverse articulation was resolved, and Fig 5-21e shows the frontal view after retraction of the mandibular anterior teeth. This case demonstrates an important principle: Do not automatically place a crisscross elastic if there is a midline discrepancy, but rather diagnose the problem first. If there is a mandibular shift, select the best mechanics to move the offending tooth or teeth; do not select an asymmetric elastic unless indicated for other sound reasons.

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5 The Creative Use of Maxillo­mandibular Elastics

Fig 5-22  An untreated skeletal asymmetry case with the chin and midline deviated to the right side. The teeth have naturally compensated for the given skeletal asymmetry.

Fig 5-23  Various locations of vertical elastics. Canting of the occlusal plane will be produced unless the force is passing through the CR. A, counterclockwise rotation; B, no rotation; C, clockwise rotation.

Fig 5-24  Vertical elastic placed off-center. The equivalent force system at the CR produces an open bite on the left side.

Although Class II/Class III elastics or anterior crisscross elastics are not usually effective in correcting the buccal occlusion in skeletal asymmetric subdivision patients (unless they help to unlock a mandibular shift), an anterior or posterior crisscross elastic might be helpful in the following type of asymmetry. It is common for compensations for a skeletal asymmetry to occur in the axial inclination of anterior teeth and the cant of the occlusal plane. Figure 5-22 shows a skeletal asymmetry with the chin and midline to the patient’s right. Note that the teeth have compensated for this asymmetry—the maxillary right canines are tipped to the buccal, and the mandibular right canines are tipped to the left side. The occlusal plane has canted downward on the left side. This is nature’s way of improving the occlusion. There are some patients for whom this natural compensation has not occurred or for whom more compensation is indicated; here a laterally directed force to the full arch or individual segments via maxillomandibular 76

elastics may be indicated. In these applications, the principles outlined in this chapter for general maxillomandibular elastic use should be most helpful in planning a special force system.

Vertical Elastics Vertical elastics are often used together with an archwire to augment vertical alignment. However, one must be careful in using vertical elastics because they may incorrectly cant an occlusal plane. We have seen how Class II elastics can cant an occlusal plane. The same can occur with vertical elastics in the lateral view (Fig 5-23). A force near the CR will translate the maxillary arch vertically; forces away from the CR will translate and rotate the maxillary arch. In a similar manner, the off-center vertical force in Fig 5-24 could produce an unwanted open bite on the left side. Vertical elastics should be only used af-

Subdivision Elastics

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Fig 5-25  Unilateral Class II elastic. (a) Occlusal view of the maxillary arch showing an equivalent force system at the CR (yellow arrows). The maxillary arch tends to rotate around the CR and can produce a buccal reverse articulation. (b) Frontal view showing an equivalent force system at the CR (yellow arrows), producing an open bite on the left side. The mandibular arch rotates more because D2 > D1.

ter careful study including their rotational effect in three dimensions.

Subdivision Elastics When there is a difference in occlusion between the left and right sides, such as Class II on one side and Class I on the opposite side, unilateral Class II elastics on a continuous arch are commonly used to attempt to correct the asymmetry. However, this can be problematic for a number of reasons. Figure 5-25a shows that the elastic produces a moment in the occlusal view of the maxillary arch that tends to rotate the entire arch around the CR. Although the Class II molar relationship on the right side may sometimes be improved with a unilateral Class II elastic, correction is usually disappointing. Why? This couple (curved yellow arrow) seen from the occlusal view would have to rotate the dental arch en masse in each arch. This would take a very long time; therefore, the usual successes are mostly the result of initial mandibular shifts. Unilateral elastics appear to correct asymmetric occlusions but later relapse because the correction involved a shift into a temporary pseudo-centric occlusion. Therefore, this en masse movement is not typically recommended. Rather, individual tooth move­ment around the arch is the proper goal, or, more often, extraction of teeth is indicated. In addition, a unilateral Class II elastic can produce other side effects. The lateral forces can lead to a reverse articulation on the right side and an open bite and increased maxillary molar overjet (cross-

bite) on the left side (Fig 5-25b). If we look from the frontal view, the unilateral Class II elastic force on the patient’s right side (red arrow) will translate the maxillary arch occlusally to the right, and the mandibular arch will translate occlusally to the left. The yellow arrows are the equivalent force system at the CR. In a similar manner to a vertical elastic (see Fig 5-24), moments are produced at both CRs in opposite directions that will cant both the maxillary and the mandibular occlusal planes, producing a lateral open bite on the left side. Unlike with the unilateral vertical elastic, however, the moment to the mandibular arch CR is considerably greater than that of the maxillary arch CR because the perpendicular distance is larger in the mandibular arch (D2 > D1). Can this side effect be avoided when a unilateral Class II elastic is used? Many orthodontists will first try a supplemental vertical elastic on the left side to close the open bite (Fig 5-26a). As depicted, the magnitude of the left vertical elastic force (red arrows) is set to balance the maxillary and mandibular moments from the right Class II elastic. The replaced equivalent force system of the left vertical elastic at the CR is shown in purple arrows (Fig 5-26b). After all moments are summed to their respective CRs from both the Class II elastic and the vertical elastic, the problem still remains. If the vertical elastic force is carefully calibrated, no lateral open bite will be observed; however, the cant of the occlusal plane will move downward on the left side, though this canting could be negligible because the magnitude of the moment is so low. If one can accept the slight canting of the occlusal plane side effect, this meth77

5 The Creative Use of Maxillo­mandibular Elastics

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Fig 5-26  (a) Unilateral Class II elastic with a vertical elastic on the opposite side. (b) The magnitude of the moment at the CR from the left vertical elastic force (purple arrows) is set to balance the moment (equal and opposite) from the right Class II elastic (yellow arrows). It may prevent the rotation of the maxillary arch, but the mandibular moment still exists. (c) Unilateral Class II elastic with a posterior vertical elastic. The horizontal component of force from the Class II elastic produces an unwanted lateral reverse articulation. (d) Unilateral Class II elastic with a posterior crisscross elastic on the opposite side. (e) The posterior crisscross elastic on the maxillary left side with a downward lingual direction could produce a synchronous occlusal plane change without an open bite. An equivalent force system is replaced at the CR (Class II elastic, yellow arrows; crisscross elastic, purple arrows). (f) Unilateral Class II elastic with a posterior crisscross elastic. The added component of force from the crisscross elastic might produce a greater crossbite. (g) A unilateral Class II elastic with a crisscross elastic from the maxillary palatal to the mandibular buccal. (h) There will be no occlusal plane rotation in the frontal view, but extrusive forces are increased. (i) Occlusal view of the force system of the unilateral Class II elastic and crisscross elastic. There is no lateral component of force. The moment will probably be ineffective to rotate the entire arch, but it operates in the correct direction.

od could be desirable; unfortunately, balancing the moments might be hard to do clinically. In short, the vertical elastic might help close the lateral open bite, but it is not an optimal solution. The anteroposterior placement of the vertical elastic is also very critical. It must be placed at the anteroposterior CR; otherwise, the action of the Class II elastic will be changed. For example, if the 78

left vertical elastic is placed distal to the anteroposterior CR, an asynchronous movement occurs, seen from the lateral view. From this view, an anterior open bite will be produced. The maxillary arch will flatten, and the mandibular arch will steepen. If this effect is understood by the clinician, this can be either desirable or undesirable, depending on the circumstances. Care must be taken to always think in

Subdivision Elastics Fig 5-27  (a) Class II elastic on the right side and Class III elastic on the left side. (b) The resultant forces (yellow arrows) show that both arches will rotate synchronously in a clockwise direction.

Fig 5-28  Class II elastic on the right side and Class III elastic on the left side (occlusal view of the maxillary arch). (a) The purple arrows are the replaced equivalent force system of the Class III elastic on the left side, and the yellow arrows are the replaced equivalent force system of the Class II elastic on the right side. (b) The final resultant of each force system at the CR is depicted in yellow arrows. It shows not only a couple but also a lateral and posterior force.

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three dimensions. However, a lateral component of force still exists (Fig 5-26c). Another solution for the right unilateral Class II elastic situation (see Fig 5-25b) is to use a posterior crisscross elastic on the left side at the anteroposterior CR of the arch (Fig 5-26d). The replaced equivalent force system of the left crisscross elastic at the CR is depicted with a purple arrow (Figs 5-26e and 5-26f). Note that with the proper angle and magnitude of the left crisscross elastic, the sum of all CR moments acting on the maxillary arch and the mandibular arch are equal and in the same direction; hence, synchronous clockwise movement will occur without producing the open bite. In addition, Fig 5-26f shows an added lateral component of force from the crisscross elastic (purple arrow), which might produce an additional unwanted side effect—a lateral crossbite. Perhaps the best solution for the lateral open bite problem associated with a unilateral Class II elastic is to change the direction of the crisscross elastic (Fig 5-26g). This will prevent occlusal plane rotation from the frontal view (Fig 5-26h). However, there is an increase in the vertical forces on each arch, which is unwanted. The occlusal view moment will probably be ineffective, but it operates in the correct direction. The replaced equivalent force system at the CR is depicted with yellow (right) and purple (left) arrows in Fig 5-26i.

Another strategy for a subdivision patient is to use a Class II elastic on the right side and a Class III elastic on the left side (Fig 5-27). From the frontal view, all of the elastic forces to both arches will produce moments at the CR that will rotate the maxillary and mandibular arches clockwise (yellow arrow in Fig 5-27b is the resultant of the two red elastic forces). The net lateral component of force to the right on the maxillary arch and the canting of both maxillary and mandibular occlusal planes are usually undesirable side effects. In the occlusal view of the maxillary arch (Fig 5-28a), the individual single force from the Class II elastic on the right side (red arrow) is replaced with a yellow equivalent force and moment at the CR; the Class III elastic on the left side (red arrow) is replaced at the CR by a purple force and moment. The resultant force system from both elastics is given in Fig 5-28b. It is clear that not only anteroposterior forces but also a counterclockwise moment and a significant lateral component of force are present (Class II/Class III elastics do not produce only a couple). Just like with a unilateral Class II elastic, the value of the CR moment in the occlusal view from Class II/Class III elastics can be questioned as an efficient way to correct an asymmetric occlusion. In the lateral view, it can be seen that Class II/ Class III elastics produce large extrusive forces with minimal change of the canting of the occlusal plane (Fig 5-29). 79

5 The Creative Use of Maxillo­mandibular Elastics Fig 5-29  Class II elastic on the right side and Class III elastic on the left side (lateral view). In this special case, the resultants (yellow arrows) are acting at the maxillary and mandibular CRs, and no change in the occlusal plane cants is anticipated.

Fig 5-30  Posterior crisscross elastic on a rigid continuous arch without play. The equivalent force system at the CR (yellow arrows) shows lateralocclusal translation and rotation around the CR of the full arch. This type of movement has little contribution to the crossbite correction.

Segmental Elastics It can be advantageous to apply a maxillomandibular elastic to a buccal segment rather than a full arch for crossbite correction. Let us consider, for example, a unilateral crossbite on the right side. If crisscross elastics are applied to the full arch, two effects will be observed in the frontal view of the maxillary arch: (1) lateral and occlusal translation and (2) rotation around the CR of the full arch (Fig 5-30). The translation force could be useful; however, let us concentrate on the moment at the CR. The moment will cause the maxillary arch to rotate around the CR. Note that the maxillary right molar moves occlusally and the maxillary left molar moves apically, with little contribution to the crossbite correction. Let us compare this with the crisscross elastic to the right buccal segment alone (Fig 5-31). For anchorage purposes, a continuous archwire is placed in the mandibular arch. The maxillary buccal segment will also translate laterally and downward. For 80

Fig 5-31  A posterior crisscross elastic on the right buccal segment only. An equivalent force system at the CR is depicted in yellow arrows. The moment on the segment at the CR produces more lingual crown movement than in Fig 5-30.

the same crisscross elastic force, more crossbite correction by translation and rotation around the CR is observed. The most important difference between a crisscross elastic on a segment rather than on a full arch is the effect of the moment at the CR. Note in Fig 5-31 that the moment on the segment alone tips the maxillary molar around its CR and produces more lingual crown movement for crossbite correction. Segmental crossbite correction therefore minimizes the vertical response and emphasizes the desired horizontal effects. The full arch is less sensitive to tipping. Detailed explanation can be found in chapter 9. It may be argued that a segment might extrude more from the vertical component of the elastic force; however, occlusal force may minimize this effect. Just like with the use of maxillomandibular elastics on a full arch, three-dimensional thinking is required to ensure full control in achieving segmental crossbite correction. We are required to look from the occlusal view as well as from the lateral and frontal views.

Elastic Redundancy Fig 5-32  Various locations of forces and predicted tooth movements from posterior crisscross elastics on the buccal segment (occlusal view). (a) Force placed at the CR. (b) Force placed anterior to the CR. (c) Force placed posterior to the CR.

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Fig 5-33 (a and b) Bent wire lingual hook using a hinge cap lingual bracket.

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c

Fig 5-34  (a) Extended lingual hook. (b and c) The hook can be slid posteriorly or anteriorly Fig 5-35 Multiple maxillomandibular elasto vary the location of the elastic force. tics. It would be difficult to get the patient’s compliance to wear so many elastics. All elastics can be replaced with an equivalent single elastic.

A buccal force on the maxillary segment is placed at the CR if translation is required in a buccal direction (Fig 5-32a). If the force is placed anterior to the CR, the canine end of the segment will move more than the terminal molar (Fig 5-32b); if the force is placed posterior to the CR, the terminal molar will be displaced the most (Fig 5-32c). Crisscross elastics may require additional hooks on the lingual of the teeth. An alternative is the use of a lingual attachment, such as a hinge cap lingual bracket, whereby a wire is placed that acts as a hook (Fig 5-33). A short, segmental lingual wire with a hook can also be placed in the hinge cap bracket and can be slid forward or backward to alter the anteroposterior position of the elastic force (Fig 5-34).

Elastic Redundancy Orthodontic patients are sometimes seen wearing many elastics in a complicated manner. It is therefore desirable to simplify the use of maxillomandibular elastics, both for the orthodontist, so he or she can better understand the force system, and for the patient, so he or she can more easily insert them. Multiple elastics (Fig 5-35) can challenge patient compliance. It would be simpler to use only one elastic that acts as the resultant of the many elastic forces. The best approach is to first determine what type of arch movement (translation and rotation around the CR) is required and then to establish the line of action of the elastic to achieve that goal. Our 81

5 The Creative Use of Maxillo­mandibular Elastics

a

b

Fig 5-36  Posterior woven up-and-down elastic. (a) The up-and-down elastic is placed on a patient with a posterior lateral open bite. Is this the correct treatment? (b) The required force system for closing the posterior open bite would be a single force (red arrows) as far posteriorly as possible.

Fig 5-37  Anterior up-and-down elastic. The up-and-down elastic is poorly designed because a single elastic placed as far anteriorly as possible (green elastic) would close the anterior open bite most efficiently.

patients will appreciate the fewer elastics that must be placed and reciprocate with better compliance. When many elastics are used, there is a good probability that the force system has not been carefully reasoned and that treatment errors will develop. On the other hand, many times a single elastic is not practical. We could use a single synchronous anterior crisscross elastic in the incisor region, but a crisscross elastic placed further back in the molar region requires two elastics on each side to avoid tongue impingement. Another common error is the use of an up-anddown elastic woven between many teeth to close an open bite. The patient in Fig 5-36a has a lateral open bite that developed during the use of a Herbst-type appliance. The open bite exists because the maxillary and mandibular planes of occlusion are not parallel—they diverge at the posterior. Which force system is the best to make parallel the occlusal planes with a vertical elastic? In order to obtain the largest moment for parallelism, the force should be placed as far posterior to the CRs as possible. The simple elastic in Fig 5-36b achieves that objective. The added elastic material in Fig 5-36a actually reduces the 82

efficiency because some of the vertical force is anterior to the maxillary and mandibular CRs. The woven up-and-down elastic in Fig 5-37 is also poorly designed, because the goal is to place the elastic force anterior to the CRs of the maxillary and mandibular arches. A single elastic on each side as far forward as possible would be the simplest and optimal placement. A vertical elastic at the midline would give the largest moment to make parallel the two occlusal planes and close the open bite. An elastic placed at the midline would have a high momentto-force ratio at the CR; therefore, one might even expect some molar intrusion.

Common Side Effects with Maxillomandibular Elastics Vertical elastics are commonly used to augment the forces from alignment archwires. It may be true that the leveling of a high canine bilaterally could flatten the cant of the maxillary plane, and this might be desirable. However, a unilateral high canine af-

Using Class II Elastics and Headgear Simultaneously

Fig 5-38 Anterior vertical elastics used to enhance canine extrusion; however, the force of the elastics is anterior to the maxillary and mandibular CRs, so these elastics will increase the deep bite.

ter leveling may show a frontal cant to the plane of occlusion. In this situation, a unilateral vertical elastic can overcompensate for the occlusal plane canting. The vertical (Class II) elastic in Fig 5-38 was used unilaterally to enhance the canine extrusion and to close the lateral open bite. Because the force of the elastic is anterior to the maxillary and mandibular CRs, an adverse side effect would be an increase in the deep bite of a patient who had too much vertical overlap initially. Indiscriminate vertical elastic use from elongated box shapes can present problems other than loss of simplicity. The box elastic shown in Fig 5-39 can irritate the gingiva with its long horizontal portion of the elastic. The horizontal portion of the elastic produces unnecessary mesiodistal forces that could potentially rotate the attached teeth, particularly if wire engagement is not complete.

Using Class II Elastics and Headgear Simultaneously Let us compare two modalities of Class II treatment. A headgear with a force through the CR of the maxillary arch and parallel to the occlusal plane could give good correction with excellent vertical control, including the cant of the occlusal plane. However,

Fig 5-39  Elongated box-shaped vertical elastic (green elastic). Its long horizontal portion could irritate the gingiva, and the horizontal portion of the elastic force produces unnecessary mesiodistal forces that could potentially rotate the attached teeth. This elastic could be replaced with a simpler form (purple elastic).

good patient compliance may be lacking. Class II elastics may be better accepted by the patient, but the occlusal plane could steepen. One approach is to combine both the headgear and the Class II elastics in the following way. A short Class II elastic is anteriorly placed so that the cant of the mandibular occlusal plane is maintained (Fig 5-40a). The maxillomandibular elastic is worn at all times so that continuous force is applied. This elastic will produce a large moment at the maxillary CR in a clockwise direction (curved yellow arrow). A headgear is used to negate this moment (Fig 5-40b). Many different lines of action are possible from both cervical and occipital headgears, provided that the moment from the headgear is counterclockwise to the maxillary CR (purple arrows in Fig 5-40b). Most likely, the headgear moment to the maxillary CR does not completely balance the elastic force moment instantaneously. Because the headgear is only worn parttime, the headgear moment must be greater than the elastic moment. This requires careful patient monitoring at each appointment. The overall resultant force (averaged over time) acting on the maxillary arch, if an occipital headgear is used, is shown as a yellow arrow in Fig 5-40c. The upward and backward translation of the maxillary arch should occur if this combination approach is used with minimal headgear wear.

83

5 The Creative Use of Maxillo­mandibular Elastics

a

b Fig 5-40  Combination of Class II elastics and headgear. (a) An anterior, short Class II elastic force (red arrows) can be replaced with an equivalent force system (yellow arrows) at the CR of the maxillary arch, which would rotate the maxillary arch clockwise. (b) A headgear force is applied to negate this moment. Any line of action of the depicted headgear forces (purple arrows) is possible. (c) The overall resultant force (yellow arrow) acting on the CR of the maxillary arch from both the Class II elastic force (red arrow) and the occipital headgear force (purple arrow).

c

Recommended Reading

84

Adams CD, Meikle MC, Norwick KW, Turpin DL. Dentofacial remodeling produced by intermaxillary forces in Macaca mulatta. Arch Oral Biol 1972;17:1519–1535.

Kim KH, Chung CH, Choy K, Lee JS, Vanarsdall RL. Effects of prestretching on force degradation of synthetic elastomeric chains. Am J Orthod Dentofacial Orthop 2005;128:477–482.

Dermaut LR, Beerden L. The effects of Class II elastic force on a dry skull measured by holographic interferometry. Am J Orthod 1981;79:296–304.

Kuster R, Ingervall B, Bürgin W. Laboratory and intra-oral tests of the degradation of elastic chains. Eur J Orthod 1986;8:202–208.

Hanes RA. Bony profile changes resulting from cervical traction compared with those resulting from intermaxillary elastics. Am J Orthod 1959;45:353–364.

Reddy P, Kharbanda OP, Duggal R, Parkash H. Skeletal and dental changes with nonextraction Begg mechanotherapy in patients with Class II division 1 malocclusion. Am J Orthod Dentofacial Orthop 2000;118:641–648.

PROBLEMS 1.  Replace the two vertical maxillomandibular elastics with a single elastic.

2.  Replace the two anterior crisscross elastics with a single elastic in a and b.

a

b

3.  Two vertical elastics are placed at the maxillary arch bilaterally between the canine and the first premolar. (a) Frontal view. (b) Occlusal view. Find a resultant single force that is equivalent. Is it possible to place this resultant on the archwire?

4. Two vertical elastics are placed at the maxillary arch asymmetrically. (a) Lateral view. (b) Occlusal view. Replace these elastics with an equivalent single elastic. Do you have to know the location of the CR to solve this problem?

a a

b

b

85

5 The Creative Use of Maxillo­mandibular Elastics

5–7. A unilateral Class II elastic is attached to the maxillary arch (lateral view). Replace this elastic with a force system at the CR. (Note that problems 5 to 7 are the same elastic from different views.)

5

7

6

8. Maxillary and mandibular rigid archwires are placed during finishing. A 100-g off-center force (green dot) is needed to make the maxillary and mandibular occlusal planes parallel. (From the frontal view, the maxillary left side must move downward; from the lateral view, the anterior open bite must be closed.) You are only allowed to place two equivalent vertical elastics attached to the maxillary archwire. Where are they located and what is their magnitude?

9–10. Replace the Class II maxillomandibular elastic with a force system at the maxillary and mandibular CRs. Discuss occlusal plane synchrony and asynchrony.

9

10

86

Problems

11–12. Replace the anterior crisscross elastic with a force system at the maxillary and mandibular CRs. Discuss occlusal plane synchrony and asynchrony.

13. The treatment objective is to close the anterior open bite and correct maxillary anterior protrusion by moving the maxillary arch backward and rotating it clockwise. Design the proper maxillomandibular elastic and briefly explain your design.

11

12

14. The maxillary midline is off-center, and the maxillary occlusal plane is canted upward on the right side. Design the proper maxillomandibular elastic for correction and briefly explain your design (consider the frontal view only).

15. The maxillary and mandibular midlines are off, and both the maxillary and the mandibular occlusal planes have a cant. Design the proper maxillomandibular elastic for correction and briefly explain your design (consider the frontal view only).

87

CHAPTER

6 Single Forces and Deep Bite Correction by Intrusion “Carve the peg by looking at the hole.”

OVERVIEW



— Korean proverb

Deep bite (or excessive vertical overlap) is best described as a symptom. Many variables are involved in this type of malocclusion, including a small vertical dimension, extruded incisors, or posterior teeth in infraocclusion. Hence, many different modalities are required for correction. This chapter describes the principles, biomechanics, and appliances for intruding incisors. In the past, intrusion was not considered a possibility because, with heavier forces, posterior extrusion was mainly observed. The force system needed for incisor intrusion requires proper force magnitude and force constancy (a low force-deflection rate). Ideally, force direction should be somewhat parallel to the long axis of the tooth. With flared incisors, the three-piece intrusion arch has the advantage that the force is delivered more posteriorly. Initial leveling with a continuous arch can be nonproductive if intrusion is required. Anchorage considerations include using very small force magnitudes, moving the posterior segment center of resistance (CR) as far anteriorly as possible, and controlling the large moment on the posterior segment.

89

6 Single Forces and Deep Bite Correction by Intrusion

Fig 6-1 Deep bite correction by posterior eruption. Class II elastics and round archwires were used, and the mandible was rotated backward.

Fig 6-2  Deep bite correction by genuine incisor intrusion. An intrusion arch was used, and no backward mandibular rotation occurred.

A deep bite is a common characteristic of many malocclusions, particularly Class II patients. Although continuous arches and many complicated force systems can be involved in deep bite correction, single forces by cantilevers or elastics can be most efficient in applying an optimal force system for either anterior intrusion or posterior extrusion. Because deep bite is a symptom, there are many causes and hence many solutions for correcting it. Sometimes incisor intrusion is indicated, and in other patients the extrusion of posterior teeth is required. Differential diagnosis must be followed by differential treatment mechanics. Some important indications for anterior intrusion include excessive maxillary incisor exposure, large vertical dimension of the lower face, facial convexity, a Class II skeletal discrepancy, little anticipated mandibular growth, normal or small interocclusal space, periodontal considerations, existing tooth alignment, and desirable occlusal plane cant. The variability in treatment goals must also differentiate between the amount of intrusion of the maxillary and the mandibular incisors. Figure 6-1 is an anterior cranial base superposition of a patient treated with Class II elastics and round wires. The deep bite was corrected by eruption of the mandibular posterior teeth, which produced downward and backward rotation of the mandible. Undesirably, the lower face vertical dimension is too great, and the total facial convexity has worsened. The incisor is vertically exposed too much in respect to the upper lip. Without further growth, mandibular position may not be stable, which will lead to counterclockwise jaw rotation and recurrence of 90

the deep bite. This result can best be described as “opening the bite.” This is not a desirable goal. Furthermore, as the mandible rotated backward during treatment, the Class II malocclusion became worse and hence harder to treat. Compare these results with the Class II patient treated by incisor intrusion in Fig 6-2, where the lower face vertical dimension has been controlled. The chin has moved forward as a result of the horizontal mandibular growth, and none of this growth has been lost by backward mandibular rotation. This chapter is devoted to the mechanics for successful and efficient incisor and canine intrusion. Chapter 7 describes methods of posterior segment extrusion. Both anchorage control and occlusal plane considerations are discussed.

Can Teeth Be Intruded? In the past, it was believed that intrusion of teeth would lead to undesirable sequelae such as tooth devitalization and loss of alveolar attachment. It is now established that light continuous forces can intrude incisors without jeopardizing the gingival attachment and can perhaps in some instances improve the attachment configuration. Early cephalometric studies showed little incisor intrusion because the heavy forces used primarily erupted the posterior teeth. Root resorption can be associated with incisor intrusion, but this can be minimized by reducing the intrusive forces. It has been shown that as the forces from intrusion increase, the rate of tooth movement does not increase, but the amount of

Continuous Intrusion Arch Fig 6-3  Intrusion means the apical movement of the CR of a tooth or group of teeth, not the incisal edge of the tooth. (a) True intrusion of incisors. The CR has moved apically. (b) Pseudointrusion. The incisor edge has tipped superiorly, but the CR has not intruded.

a

a

b

b

c

Fig 6-4  Continuous intrusion arch mechanism. (a) Deactivated continuous intrusion arch. The intrusion arch (yellow) is inserted into auxiliary tubes on the two first molars and is ready for activation. A rigid wire is placed in the buccal segment (teal). In the anterior segment (gray), leveling can be carried out at the same time as intrusion; therefore, the material and the size of the wire may vary. (b) Activated continuous intrusion arch. The intrusion spring is activated and tied to the anterior segment. (c) A passive lingual arch (green) is inserted to prevent adverse side effects from an intrusion arch.

root resorption does increase. Control of the vertical dimension or incisor intrusion in the maxillary arch is often attempted using a high-pull J-hook headgear. This may be contraindicated because heavy forces are directly and intermittently placed on the maxillary incisors. Confusion often surrounds the term intrusion. In this text, it has a strict meaning: the apical movement of the CR of a tooth or groups of teeth. In Fig 6-3a, the CR exhibits true intrusion. In Fig 6-3b, the deep bite is corrected by tipping of the crown. The CR has not moved. This is an example of deep bite correction without intrusion, or pseudointrusion. There are two basic mechanisms for incisor intrusion: the continuous intrusion arch and the threepiece intrusion arch. Both are based on the cantilever principle of applying single forces without a moment in the incisor region. The selection of each mechanism is dependent on the treatment goals required for the individual patient’s malocclusion.

Continuous Intrusion Arch The classic continuous intrusion arch (spring) mechanism is shown in Fig 6-4. A relatively rigid wire (teal) is placed in the right and left buccal segments (usually at least 0.018 × 0.025–inch stainless steel [SS] wire). Anterior segment leveling can be carried out at the same time as intrusion; therefore, a sequence of wires from flexible to more rigid can be placed in the anterior segment (gray). A lingual or transpalatal arch is present to control widths and maintain overall occlusal symmetry. The two buccal segments and the lingual arch form the posterior anchorage unit. The active intrusion arch (yellow) is inserted into auxiliary tubes on the two first molars and is attached anteriorly. The anterior point of attachment can be in the midline or even distal to the incisors (discussed later). The intrusion arch is a rectangular titanium-molybdenum alloy (TMA) or SS wire, rectangular so that the wire will not rotate in 91

6 Single Forces and Deep Bite Correction by Intrusion Fig 6-5  A rigid continuous incisor “bypass arch” (teal). This archwire provides similar anchorage control to a passive lingual arch.

a

b

Fig 6-6  Intrusion of all four maxillary incisors. (a) Pretreatment. (b) Posttreatment.

the molar auxiliary tube for greater force accuracy and reproducibility. Its overall configuration shape, including location of the bend, material of the wire (TMA or SS), and its cross section, which can vary from 0.016 × 0.022 to 0.018 × 0.025 inch, is not critical because the intrusive force is measured with a force gauge. The intrusion arch is bent apically (see Fig 6-4a) and activated occlusally by a ligature tie to the anterior segment (see Fig 6-4b). Note the occlusal step in the intrusion arch to prevent the arch from contacting the canine bracket after activation. The lingual arch is necessary because the arch width and arch form may be difficult to maintain with the intrusion arch alone due to the highly flexible wires and large activations for intrusion. A lingual arch gives the security of positive control while preventing side effects from an intrusion arch (see Fig 6-4c). An alternative method is the use of a continuous arch with a stepped bypass around the teeth requiring intrusion (Fig 6-5). The rigid continuous arch gives the anchorage control, and the intrusion arch that delivers the forces is the same as that shown in Fig 6-4. Figure 6-6 shows intrusion of the four maxillary incisors. A rope ligature tie is applied in the midline during activation. Note that as the deep bite correction has occurred, the position of the maxillary inci92

sor in respect to the maxillary canine has changed (see Fig 6-6b). This positional change does not occur in bite opening where the mandible is hinged open.

Global Characteristics of the Intrusion Force System The overall intrusion force system is depicted in Fig 6-7. The deactivation intrusive force (red arrow) acting on the incisors is produced by the intrusion arch (yellow). The activation force (blue arrow) is the equal and opposite force (Newton’s Third Law). The blue force is also the force acting on the posterior segments and can be replaced with the yellow equivalent force system at the posterior CR. If the posterior teeth move during anchorage loss, they will erupt, and the posterior occlusal plane will steepen (see Fig 6-7a). Let us look in more detail at the incisors, where the intrusive force acts anterior to the CR (Fig 6-7b). If we replace the red intrusive force with an equivalent force system at the CR of the incisors, the yellow moment shows that the incisors would tip labially. One method of preventing the incisors from flaring is to position the force further posteriorly so that it acts through the CR of

Global Characteristics of the Intrusion Force System

a

b

Fig 6-7  The force system of a continuous intrusion arch. (a) The anterior blue arrow is the force to activate the arch (activation force). An equal and opposite force (red arrow) is acting on the tooth as the arch deactivates (deactivation force). The posterior yellow arrows are the equivalent force system of the blue force acting at the posterior CR. The posterior segment will erupt, and the posterior occlusal plane will steepen. (b) The force system of an intrusion arch at the anterior segment. The force (red arrow) from the intrusion arch is replaced with an equivalent force system (yellow arrows) at the anterior CR. The incisor will not only intrude but also tip to the labial.

Fig 6-8  A distally directed ligature tie changes the line of action (yellow arrows) through the CR, preventing undesirable labial flaring.

Fig 6-9  Measuring the intrusive force with a force gauge. An intrusion arch is basically a free-end cantilever, so the accurate force system can be determined simply with a force gauge and a ruler.

the incisors (discussed later). If we want to keep the point of force application forward, it will be necessary to tie the intrusion arch with a distally directed tie. Note that the combination of an intrusive force and a distal force produces a resultant that is directed upward and backward (yellow arrows) in Fig 6-8. If properly applied, the resultant force will act through the CR of the incisors, and no undesirable flaring will occur. The continuous intrusion arch is basically a freeend cantilever. It can be tied anteriorly by a single tie or with a single force; hence, the force system is easy to understand and measure. By comparison, an archwire tied into multiple edgewise anterior brack-

ets is much more complicated (forces and moments) and unpredictable. Figure 6-9 shows the direct measurement of an intrusion arch with a force gauge. The advantage of a cantilever is that accurate force systems can be measured and predictably placed in the patient. Only a force gauge and a ruler are required—not elaborate force-sensing equipment. The key to successful intrusion involves particularly good force control: force magnitude, force constancy, a point contact, point of force application, and force direction. In addition, one should take advantage of segmental leveling and understand special anchorage considerations. Let us consider these points separately. 93

6 Single Forces and Deep Bite Correction by Intrusion

Force (g)

Fig 6-10  Recommended average intrusive forces measured at the midline. Note that very light forces are used for intrusion.

Central incisors

All four incisors Canine and incisors

Optimal force magnitude Early experiments on monkeys by Dellinger1 established the importance of controlling force magnitude for incisor intrusion. No advantage is gained by increasing the magnitude of force; higher force levels only increase root resorption and potential anchorage loss without increasing the rate of tooth movement. Figure 6-10 provides some recommended force averages. These values represent both studies and clinical experience.2 Overall, lighter forces are used for intrusion today than were used 50 years ago. Variation in the applied force is recommended based on root size and other biologic considerations. Force values are also given for intrusion of all six anterior teeth. This is usually not recommended unless a temporary anchorage device (TAD) is used, because the heavier forces needed for more teeth will disturb posterior anchorage. With the continuous intrusion arch, the force can be measured with a force gauge in the mouth or simulated outside of the mouth. Standardized tipback bends are not recommended for the individual patient because the length, width, and cross section of the intrusion arch can be quite variable.

Force constancy An important characteristic of any orthodontic appliance is the constancy of the force delivered during tooth movement. Figure 6-11 shows a cantilever by which a 100-g force is applied at the free end. The free end deflects 10 mm before it comes to rest. The force-deflection (F/∆) rate measures the constancy of the force. 94

F/∆ =

100 g = 10 g/mm 10 mm

This F/∆ rate denotes that for every millimeter of activation within the elastic range of the wire, 10 g of force is produced. For example, 5 mm of activation will produce 50 g; 7 mm will give 70 g. If there is a linear relationship between force and deflection, this is referred to as Hooke’s law—a law of physics stating that the force required to extend or compress a spring (or archwire in this case) by a certain distance is proportional to that distance. Not all orthodontic appliance components show a linear relationship between force and deflection, but many are close to linear, so the concept of F/∆ rate, even if it is only an approximation, is very useful. The F/∆ rate also tells us about the constancy of the force as the teeth move. Let us activate the cantilever to a full 10 mm. The attached tooth will feel 100 g of force initially. The tooth moves 1 mm; force magnitude drops 10 g as given by the F/∆ rate. After 5 mm of movement, the force is reduced to 50 g. Figure 6-12 compares an appliance with a high F/∆ rate with an appliance with a low F/∆ rate. Based on Hooke’s law, the greatest force is produced after the initial activation and reduces during deactivation. Three force zones are depicted: red, excessive; green, optimal; and blue, suboptimal. Let us follow the appliance with the low F/∆ rate (blue line). The force magnitude is more constant over a wider range of deactivation (blue arrow); furthermore, the forces remain longer in the optimum green range. On the other hand, the appliance with the high F/∆ rate (red line) abruptly changes force level during deactivation and requires a larger-than-needed deflection

Global Characteristics of the Intrusion Force System

Fig 6-11  The force-deflection rate (F/∆) of a cantilever. If the rate is linear, it follows Hooke’s law.

Fig 6-12  High versus low F/∆ rate appliances. The slope of the line represents the F/∆ rate (red line, high; blue line, low). The force zones are depicted in colors: red, excessive; green, optimal; blue, suboptimal. The horizontal blue arrows represent the range of action of each appliance in the optimal force zone.

Fig 6-13  F/∆ rate of a 0.016-inch SS continuous full archwire. To obtain 80 g of intrusive force, 0.05 mm of activation is required, which is practically unrealistic.

that produces excessive force (red range) to be practical. The appliance with the low F/∆ rate delivers a more constant optimal force and also requires fewer activations because it works over a larger distance (horizontal blue arrows). Leveling archwires even of so-called light wire have high F/∆ rates. For example, consider a 0.016inch SS wire used to level two incisors in a Class II, division 2 patient (Fig 6-13). Stiffness can vary depending on many factors, including bracket width and interbracket distance, but these numbers are representative. Suppose that we want to deliver 80 g of force in the midline. The F/∆ is 1,600 g/mm. How much activation is required? Only 0.05 mm. This small of an activation is not realistic. No orthodontist can see or activate 0.05 mm. Even if a correct 0.05mm activation is achieved, once the tooth moves 0.05 mm, the force will drop to zero. The high F/∆ wire would require frequent reactivations. Because

of this, typically with appliances with high F/∆ rates, no effort is made to deliver optimal force levels. The tooth is initially jolted with excessive force, and a healing process is allowed between appointments. Typical F/∆ values of a continuous intrusion arch are given in Fig 6-14. The F/∆ rate is 8 g/mm. Five mm of activation is required to obtain 40 g of force, a reasonable magnitude for two maxillary incisors. The low F/∆ rate has several advantages. Accuracy is enhanced in that an error of 1 mm in activation leads to only an 8-g force error. The force is also relatively constant, changing only 8 g for every 1 mm of intrusion. The large deflection required also allows little reactivation and longer intervals between appointments. Note that an intrusion of 3 mm or more could occur within an optimal magnitude force zone (green zone). Sometimes there is confusion about how much to activate or deflect an orthodontic appliance. In the 95

6 Single Forces and Deep Bite Correction by Intrusion

Fig 6-14  F/∆ rate for a 0.017 × 0.025–inch TMA continuous intrusion archwire. The force zones are depicted in colors: red, excessive; green, optimal; blue, suboptimal. The low F/∆ rate intrusion archwire (blue wire) is activated in the green zone. Note that the F/∆ rate is dramatically reduced due to large interbracket distance.

Fig 6-15  The amount of activation of the intrusion spring is determined by the amount of force required; therefore, it varies with F/∆, which is dependent on shape, material, and cross section of the wire. Suppose F/∆ = 8 g/mm; then 7.5 mm of activation is required for 60 g of intrusive force.

Fig 6-16  The intrusion archwire is tied to the anterior segment, but it is not placed in the bracket slot. Therefore, it does not produce side effects from unpredictable forces and/or moments.

traditional shape-driven concept, deflection was dictated by how far a tooth should move. For example, 2 mm of intrusion is required; therefore, the activation is 2 mm. In a force-driven appliance, however, activation is determined by the required force level. For example, in the intrusion arch shown in Fig 6-15, 7.5 mm of activation was required for 60 g of force. The amount of intrusion required has nothing to do with the desired force level delivered, much like the distance we are driving on a highway has nothing to do with the speed we drive. The force-driven appliance uses the amount of deflection to control force magnitude.

Force application at a point The intrusion arch is not placed in the brackets of the incisors so as to avoid common side effects from extraneous forces and moments acting between the 96

incisor brackets (Fig 6-16). A separate wire is placed in the anterior segment that can be either active or passive. A separate ligature tie is placed from the intrusion arch to the anterior segment. This tie delivers a single force (the force acts at a point) or a series of single forces if more than one tie is used. The intrusion arch is tied incisally or gingivally (Fig 6-17) to the incisor brackets to avoid gingival impingement. If an intrusion arch is placed into the incisor brackets, second- and third-order side effects are commonly produced. The second-order tip effect is shown in Fig 6-18. If an archwire is placed in the molar auxiliary tubes and deflected by a force in the incisor region, a curvature is produced (see Fig 6-18a). This curvature will cause the roots of the incisors to displace to the mesial (see Fig 6-18b). The patient in Fig 6-19 had good mesiodistal incisor axial inclinations at the start of treatment (see Fig 6-19a). After

Global Characteristics of the Intrusion Force System

a

b

Fig 6-17  (a) A separate wire is placed in the four-tooth segment, and the intrusion arch is tied labially. (b) A wire is placed in the two-tooth segment, and the intrusion arch is tied gingivally.

a

b

Fig 6-18  Second-order side effects from the intrusion arch placed into the bracket slot. (a) The intrusion arch (yellow) is activated with the force (blue arrow), and a curvature is produced (teal). (b) The curvature will produce convergence of the incisor roots.

a

b

Fig 6-19  An intrusion arch placed into the incisor brackets. (a) Before treatment. (b) After treatment. Note the secondorder side effects of root convergence; furthermore, there is little intrusion.

the intrusion arch was placed in the incisor brackets, the roots moved to the mesial (see Fig 6-19b). Furthermore, little intrusion occurred. If the intrusion arch is not placed in the brackets, this side effect produced by any secondary curvature in the intrusion arch can be avoided. Even more troubling can be third-order side effects. Any torque placed in the incisor region will alter the intrusive force. The green wire in Fig 6-20a

is passive with no active vertical force. The incisor position of the wire is twisted (inset) so that lingual root torque is produced during insertion. In order to place the wire into the incisor brackets, a clockwise activation moment (curved blue arrow) is required to twist the wire so that it is parallel to the bracket. Note that not only does this moment cause twist of the incisor portion of the wire, but also the entire wire bends elastically to the occlusal; thus, the plac97

6 Single Forces and Deep Bite Correction by Intrusion

a

b

Fig 6-20  Third-order side effects from an intrusion arch placed into the bracket slot. (a) Placement of a twist in the wire (crown labial, root lingual direction) produced a vertical extrusive force. (b) The opposite direction of twist in the wire (crown lingual, root labial direction) produced an additional vertical intrusive force. Additional vertical intrusive force may jeopardize the posterior anchorage unit because of the increased moment.

a

b

Fig 6-21  Simultaneous intrusion and leveling. (a) Before leveling. A 0.016 × 0.022–inch braided SS archwire was placed in the anterior segment for leveling. (b) After leveling. A 0.017 × 0.025–inch SS archwire was placed for en masse intrusion.

ing of a localized twist in the wire (incisor torque or third-order moment) produced a vertical extrusive force. The clinical significance of torque producing an extrusive force is obvious. During intrusion, we carefully measure the magnitude of the intrusive force. If we purposely or accidentally place any lingual root torque in the intrusion arch, this can reduce or completely overwhelm the intrusive force. The incisors may actually extrude. A similar effect can occur if labial root twist is placed in the incisor portion of the wire (Fig 6-20b). In this case, an activation torque (moment) is needed to twist the anterior part of the intrusion arch into the incisor brackets in a counterclockwise direction (blue arrow). The intrusion arch deflects apically and hence increases the intrusive force. But is this acceptable since the added vertical force is in the correct direction? Unfortunately, any increase in intrusive force can tax the anchorage. The increased intrusive force produces an increase in extrusive force on the posterior teeth and, even more significant, a moment on the posterior teeth, potentially steepening the occlusal plane. Therefore, because incisor intrusion requires low force magnitudes that must be carefully measured, one should avoid plac98

ing the intrusion arch into the incisor brackets due to the deleterious effects of torque. Is it possible to use round wire for an intrusion arch to avoid the side effect of vertical forces associated with torque? Most anterior segments of at least four incisors exhibit some form of a curvature in the occlusal view, so third-order anterior moments can be created even with round wire. In a Class II, division 2 case, however, a round wire might be satisfactorily inserted into the two central incisor brackets with minimal side effects from the moment. Another advantage of not placing the intrusion arch directly into the incisor brackets is the convenience and efficiency of simultaneously aligning and intruding the incisors. One can begin with flexible, large deflection wires and sequentially replace them with stiffer wires in the incisor segment. This avoids the need for a separate leveling phase, and the same intrusion mechanism is left in place over a long time frame. Only the incisor segment requires changing (Fig 6-21). The primary reason for not inserting the intrusion arch into the incisor brackets is the predictability of the force system. A single force can be measured with a force gauge, and with that information the

Changing the Point of Force Application Fig 6-22 Changing the point of force application, frontal view. Intrusive force is applied off-center for frontal canting correction.

force system is statically determinate. Statically determinate means that the law of static equilibrium is sufficient to solve the given problem. It may be theoretically true that small deviations from this condition can occur if the anterior segment is active or if the intrusion arch twists to deliver torque to the molars; however, most of the time these deviations are not relevant and can be practically ignored.

Changing the Point of Force Application If a continuous arch applies an intrusive force on an incisor bracket, the force is most likely anterior to the incisor’s CR so that the incisor may flare or the root apex may move lingually. It is often desirable during intrusion to move the force further posteriorly. From the frontal view, the intrusion force can be applied at the midline or off-center if canting of the frontal occlusal plane is required (Fig 6-22). Let us consider three incisors with identical and typical axial inclinations; an intrusive force is applied to each incisor at 90 degrees to the occlusal plane (Fig 6-23). If the force is applied at the incisor bracket (see Fig 6-23a), an equivalent force system at the CR tells us that the incisor will intrude and the moment will cause the crown to flare and the root to move lingually. This intrusive force is not completely efficient for two reasons. Relative to the long axis of the tooth, there are two components (purple arrows)—a labial force and an intrusive force—and thus some of the force is lost in an unwanted labial direction. Furthermore, the moment at the CR that would flare the incisor requires the arch to be tied back. Although difficult to be exactly delivered, the ensuing lingual root movement could be desirable for many Class II, division 2 patients. It might also be useful in some extraction Class II, division 1 patients

to help minimize lingual crown inclinations after space closure. In Fig 6-23b, the intrusive force is applied posterior to the incisor bracket with a line of action through the CR. This is possible by extending the wire distally on the anterior segment, forming the posterior extension. Typically the force is applied near the center of the canine crown. (If the canine has erupted, the posterior extension is stepped apically to avoid the canine.) Intrusion can be efficient because no moment side effect in respect to the CR is produced. The force direction may not be ideal if parallel intrusion along the long axis of the incisor is intended. The intrusive force in Fig 6-23c is applied lingual to the incisor’s CR. Here the moment at the CR is in a direction to tip the incisors lingually. This can be most useful in the treatment of flared incisors, where intrusive forces are required. In Class II, division 2 patients, the force can be successfully applied labially at the bracket (Fig 6-24). The direction of the intrusive force, almost parallel to the long axis of the tooth and the line of action, is very near to the CR—the most favorable intrusion configuration for teeth with parabola-shaped roots. The direction of the CR moment will flare the crown and improve the incisor inclination. If lingual brackets are used, intrusion would still be efficient, but the desirable flaring moment effect would be lost. The flared mandibular incisor in Fig 6-25 is problematic if the force is applied labially at the bracket (see Fig 6-25a). A very large moment is present that will flare the incisor more. The intrusive force is best applied distally so that the line of force is passing through the incisor’s CR (see Fig 6-25b). It is common to find mandibular arches with excessive curves of Spee that require more extrusion of posterior teeth than incisor intrusion. Even with these patients, one must control the reciprocal vertical forces on the incisors to avoid unwanted incisor flaring.

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6 Single Forces and Deep Bite Correction by Intrusion

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Fig 6-23  (a to c) Changing the point of force application, lateral view. The applied intrusive force (red arrow) is replaced with an equivalent force system at the CR (yellow arrows). Labial or lingual tipping of the incisor is produced unless the force is through the CR.

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100

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Fig 6-24  Intrusive force applied at the maxillary central incisor in a Class II, division 2 case. The intrusive force (red arrow) at the bracket is helpful. The replaced force system at the CR (yellow arrows) shows both a favorable intrusion force and a moment, improving the tooth inclination.

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Fig 6-25  Intrusive force applied to labially flared mandibular incisors. (a) Force applied at the bracket causes a large moment that will flare the incisors more labially. (b) The intrusive force is best applied distally so that the line of force passes through the incisor’s CR.

Fig 6-26  The effect of tying back the archwire. (a) Tying back the archwire may prevent labial movement of the incisor bracket; however, the incisor still changes its axial inclination. (b) Even if the force is through the CR, undesirable lingual root movement is inevitable.

Can this incisor flaring be avoided by tying back the archwire? Unfortunately, this could produce lingual movement of the root apex, and the root is already too far lingual. In Fig 6-26a, the tying back of the archwire may prevent labial movement of the incisor bracket; however, the incisor still changes its inclination by lingual movement of the apex. Moreover, the CR moment is so great that it is difficult to tie back the archwire sufficiently to prevent actual labial flaring of the incisor bracket. Even if the distal force is exactly correct as in Fig 6-26b, with the force through the CR, undesirable lingual root movement still occurs. Generally, teeth that are flared are not efficiently intruded because the force is not primarily directed along the long axis of the incisor and

a great horizontal distance from the CR is present. For that reason, in major intrusion cases the force is redirected parallel to the long axis of the teeth. This concept is discussed later in this chapter. The best solution for intruding a protrusive incisor is to position the force through the CR or any position posterior to the incisor bracket by tying bilaterally the continuous intrusion arch on the posterior extension of the anterior segment. Another possibility is the use of two separate intrusion arches (placed bilaterally). This mechanism, called the three-piece intrusion arch, is more versatile in treating asymmetries, handling space opening and closing requirements, and changing force direction.

Three-Piece Intrusion Arch Fig 6-27  Components of a three-piece intrusion arch.

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Fig 6-28  Shape of the three-piece intrusion arch. (a) Shape when deactivated. (b) Shape when activated. (c) The hook on the mesial end allows free sliding of either the posterior or anterior segment.

Fig 6-29 The force system of the threepiece intrusion arch. (a) Deactivation force system from the arch (red arrows). (b) Replaced equivalent force system at the CR (yellow arrows). The intrusive force is easily applied at the CR by using a distal extension.

a

Three-Piece Intrusion Arch Figure 6-27 shows the three-piece intrusion arch. It consists of (1) right and left posterior segments, (2) right and left intrusion springs, and (3) an anterior segment with a posterior extension. In addition, a lingual or transpalatal arch is inserted for arch width and symmetry control. The deactivated and activated shapes of the three-piece intrusion arch are shown in Figs 6-28a and 6-28b. Separate intrusion springs allow for independent intrusion activations on each side of the arch. Note the hook on the mesial end of the intrusion spring, which allows free sliding of either the posterior or anterior segment

b

to open or close space (Fig 6-28c). For example, if posterior teeth are allowed to tip back, the hook would slide distally on the posterior extension of the anterior segment. The force system acting on the teeth of the three-piece mechanism is shown in Fig 6-29 (activated spring). Deactivation forces and moments (red arrows) are acting on the teeth at the hook and the molar tube. The equivalent force systems at the anterior and posterior CRs are shown in yellow. Because the force acts through the CR of the incisor segment, the incisors will exhibit translatory intrusion. No flaring will occur. The posterior segment will extrude, and the posterior occlusal plane will tend to steepen. Nevertheless, anchorage control can be very good. 101

6 Single Forces and Deep Bite Correction by Intrusion

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Fig 6-30  Anchorage control in a three-piece intrusion arch. (a) Shape when deactivated. Force is applied to activate the arch (blue arrow). (b) Shape when activated. There is less reciprocal posterior moment due to the decreased moment arm (30 mm) compared with a continuous intrusion arch (50 mm).

Consider the forces to activate the spring to intrude four maxillary incisors in Figs 6-30a (deactivated intrusion spring) and 6-30b (activated intrusion spring). For a total of 60 g, 30 g are applied on each side. The deactivation force (red) to the buccal segments is also 30 g, which produces a 900-gmm moment (30 g × 30 mm) at the posterior CR. The primary concern in anchorage loss is the moment steepening the plane of occlusion. Here the moment is smaller because of the light 30-g intrusive force and also because of the shortened distance from the hook to the posterior CR. (The horizontal distance from the incisor bracket to the posterior CR is greater with the continuous intrusion arch.) Added to the promising anchorage potential are five posterior teeth on each side. (Including a second molar can be helpful.) Note that the posterior extension has been stepped apically to avoid the canine bracket. The lingual arch contributes to the overall anchorage control because it prevents individual movements of the posterior segments; if any tooth movement does occur, an en masse movement of all posterior teeth would be required.

Altering Force Direction During intrusion, it can be desirable to alter the direction of the force. Intrusion of single-rooted teeth is most efficiently accomplished with forces parallel to the long axis of the tooth. The intrusion spring in Fig 6-31 acts at 90 degrees to the occlusal plane. To change its direction so that it is parallel to the long axis of the incisors, a distally directed chain elastic can be added. Not much force is needed. If the intrusive force from the intrusion spring is 30 g, the force can be similar—not hundreds of grams. A force from a chain elastic that undergoes degradation due to 102

the fluids in the mouth is not very accurate, and a metal spring might be more predictable (Fig 6-32), but the simplicity and comfort of the chain elastic, if compensated for and carefully observed, allow for its use. Another method to redirect the force is to use a simple cantilever (Fig 6-33a). The angle of the ligature tie in Fig 6-33b denotes the force direction. To facilitate the use of a cantilever for this application, the position of the helix can be varied so that the direction of the intrusive force can be kept relatively constant (Fig 6-34). The force direction may also be altered by bending the posterior extension on the anterior segment at a suitable angle (Fig 6-35). This can only work if there is minimal friction between the hook and the wire extension (Fig 6-36a). Imagine the hook soldered or glued to the wire; then the force direction (red arrow) does not change (Fig 6-36b). If friction is low, the horizontal force component along the wire is lost, and the force can be redirected. Of course, some friction must be present, so the force redirection will be unpredictable. A common clinical challenge is to have a line of action parallel to the long axis of an incisor and through its CR. Can this be accomplished if the point of force application is at the incisor bracket? We can add a horizontal force component to the vertical intrusive force shown in red in Fig 6-37a. The resultant force (yellow arrow) acts through the CR, but its direction is not parallel to the long axis (dotted black line) of the incisor. In order to have the proper direction, the point of force application must be changed to lie posterior to the lateral bracket on the posterior extension of the anterior segment (Fig 6-37b). This is one application of a three-piece intrusion arch.

Altering Force Direction

Fig 6-31  Altering the force direction in a three-piece intrusion arch using an elastic chain. A distally directed chain elastic can be added to redirect the intrusive force parallel to the long axis of the incisors.

a

Fig 6-32  Altering the force direction in a three-piece intrusion arch using a metal coil spring. A force from a metal coil spring might be more predictable.

b

Fig 6-33  Altering the force direction in a three-piece intrusion arch by changing the angle of the ligature tie. (a) Deactivated shape. (b) Activated shape. The length of the cantilever arm is adjusted so that the angle of the ligature tie coincides with the intended force direction.

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Fig 6-34  Altering the force direction in a three-piece intrusion arch by changing the position of the helix. (a) Deactivated shape. (b) Activated shape. The position of the helix can be varied so that the direction of the intrusive force can be kept relatively constant.

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6 Single Forces and Deep Bite Correction by Intrusion

a Fig 6-35  Altering the force direction in a three-piece intrusion arch by bending the distal extension. The intrusive force acts perpendicular to the distal extension, provided that there is no friction between the hook and the distal extension.

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b

Fig 6-36  (a) With minimal friction, no horizontal force component (yellow dotted arrow) is produced. (b) With high friction, the direction will remain perpendicular to the occlusal plane.

Fig 6-37  Control of the line of action in each type of intrusion arch. (a) In a continuous intrusion arch, if the intrusive force is applied at the bracket, the resultant force can be through the CR by adding a horizontal force component. However, its line of action (dotted black line) is not parallel to the long axis of the incisor. (b) In a three-piece intrusion arch, the point of force application is moved posterior to the lateral bracket on the extension so that the resultant is not only through the CR but also parallel to the long axis of the incisor.

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c

Thus, force control involves not only force magnitude and force constancy but also point of force application (line of action) and direction. Many other possibilities are summarized in Fig 6-38. All forces are parallel to the long axis of the incisor except Fig 6-38b. Parts a and d have opposite CR moment effects, while parts b and c produce pure intrusional translation without a moment (rotation). Translation along the long axis of the tooth is only possible in c by changing the force angle and point of force 104

d

Fig 6-38 (a to d) Various possibilities of intrusive force application replaced at the CR. All forces are parallel to the long axis of the incisor except b. Parts a and d have opposite CR moment effects, and parts b and c have pure intrusional translation without a moment. Translation along the long axis of the tooth is only possible in c by changing the force angle so that its line of action goes through the CR; b also translates, but the force is no longer parallel to the long axis of the tooth.

application so that its line of action goes through the CR; b also shows translation, but the force is no longer parallel to the long axis of the tooth. The importance of force direction and point of force application is demonstrated in Fig 6-39. The young adult patient shows considerable bone loss, extrusion, and flaring of the maxillary right central incisor associated with localized periodontitis (Figs 6-39a and 6-39b). Leveling with a continuous edgewise archwire is limited because the forces are too

Altering Force Direction

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Fig 6-39  The patient shows bone loss, extrusion, and flaring of the maxillary right central incisor. (a) Frontal view before treatment. (b) Occlusal view before treatment. (c) The force system was designed so that the resultant force (yellow arrow) was through the incisor’s frontal CR. (d) Lateral view. The resultant is made parallel to the long axis of the tooth by adding a distal force. (e) Frontal view after treatment. (f) Occlusal view after treatment. (g) Pre- and posttreatment radiographs. Note the improved bone level and attachment. The tooth is still functioning in the mouth after 30 years.

great and unknown. Also, it would be difficult to direct the force along the long axis of the incisor. Therefore, a three-piece intrusion arch with right and left chain elastics were used to redirect the force along the long axis of the tooth. At the start, 10 g was applied at each side. From the frontal view, this was incorrect because the CR of the single right central incisor is off-center to the right side. In the second adjustment, the force system was corrected to reflect this asymmetry. The ideal and correct force system is given in Fig 6-39c, with 13.3 g on the right side and 6.7 g on the left side. In addition, about 15 g of force in a distal direction was applied bilaterally

on the anterior segment. From the lateral view, note that the combination of intrusive force and distal force redirects the resultant so that it lies parallel to the long axis of the incisor (Fig 6-39d). Thus, in three dimensions, the resultant force (approximately 20 g) acted through the incisor’s CR from the frontal view and lay slightly posterior to the CR from the lateral view (see Fig 6-39d). After intrusion, the tooth was stabilized with a continuous arch (Figs 6-39e and 6-39f). Note the improved bone architecture and attachment (Fig 6-39g). After treatment, no pocket was evident during periodontal probing. The tooth is still functioning in the mouth after 30 years. 105

6 Single Forces and Deep Bite Correction by Intrusion

a

b

c

d Fig 6-40  The patient has a large interlabial gap and 9 mm of incisor showing at rest. A continuous intrusion arch was used. (a and b) Before treatment. (c and d) After treatment. (e) Superimposition shows the dramatic intrusion of the maxillary incisor.

e

The patient in Fig 6-40 had a large interlabial gap and a maxillary incisor that was positioned 9 mm below the upper lip at rest (Figs 6-40a and 6-40b). The maxillary and mandibular premolars were extracted. To correct the vertical incisor exposure, a continuous intrusion arch was used. The mandibular arch curve of Spee was maintained, and a curve of Spee was built into the maxillary arch to fit the mandibular curvature (Figs 6-40c and 6-40d). Note the dramatic intrusion of the maxillary incisor (Figs 6-40e). Unlike hinging open a mandible during deep bite correction, intrusion is a slow movement. One should not expect over 1 mm of movement per month. In this patient, no growth was present, so the deep bite correction was accomplished mainly by maxillary incisor intrusion. The force system of simultaneous intrusion and retraction of four maxillary incisors with the threepiece intrusion arch is shown in Fig 6-41. This is par106

ticularly appropriate if a flared maxillary incisor is present (Figs 6-42a to 6-42c). A force lingual to the CR produces an intrusive force at the CR and a moment that tips the incisor lingually. The patient in Fig 6-42 was treated with a three-piece intrusion arch with a distal force from a chain elastic. Initially the two incisors were retracted, and this was followed by retraction of the four maxillary incisors as a unit (Figs 6-42d and 6-42e). During the en masse movement, the point of force application was moved posteriorly to about the position of the center of the canine (Fig 6-42f). Note that considerable retraction of the maxillary incisor occurred with minimal anchorage loss (Figs 6-42g to 6-42k). This is to be expected because the only distal forces on the anterior region were approximately 20 g per side. This force was used not for direct retraction but rather to redirect the force of the intrusion arch so that the resultant was parallel to the long axis of the incisors.

Altering Force Direction Fig 6-41  The force system of simultaneous intrusion and retraction with the three-piece intrusion arch. The yellow arrows show the replaced force system at the CR.

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Fig 6-42  The patient has a flared maxillary incisor. (a) Lateral cephalometric radiograph before treatment. (b and c) Frontal and lateral intraoral views before treatment. (d and e) A three-piece intrusion arch was used to produce intrusive force and a moment that tipped the incisors lingually. (f) Occlusal view after intrusion and retraction. (g to i) Frontal and lateral intraoral views after treatment. (j and k) Superimposition of lateral tracings and occlusogram. Note that considerable retraction of the maxillary incisor occurred with minimal anchorage loss.

107

6 Single Forces and Deep Bite Correction by Intrusion Fig 6-43  Leveling of a Class II, division 2 case with a continuous full archwire. (a) Before leveling. (b) After leveling. Note that little intrusion has occurred. The lateral incisors erupted, and the posterior plane of occlusion was steepened.

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Fig 6-44  (a to d) The two central incisors are intruded to the level of the lateral incisors first.

Fig 6-45  Leveling of a Class II, division 1 case. (a) In many cases, four maxillary incisors can be intruded as a unit to the level of the canine. (b) Mandibular arches can display an excessive curve of Spee, with the four incisors stepped as a unit occlusally. This step can be used advantageously if intrusion is required.

Avoiding Initial Leveling Arches It is common practice to use high-deflection and lower-force wires to align and level at the beginning of treatment. This may not be a good idea if genuine intrusion of incisors is a treatment goal. If a Class II, 108

Fig 6-46  Deep bite may not be apparent with flared incisors. Note after lingual tipping that the CR must be moved apically (∆).

division 2 malocclusion is leveled, for example, with a nickel-titanium wire, little intrusion occurs (Fig 6-43). The lateral incisors will erupt, and the posterior plane of occlusion will steepen because of the large moment at the posterior CR from the incisor intrusive force. It is better to take advantage of the original anatomical geometries in tooth arrange-

Avoiding Initial Leveling Arches

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Fig 6-47  Patient with severely flared incisors. (a) The vertical overlap looks normal before treatment. (b) A three-piece intrusion arch was used for retraction of the maxillary incisors and leveling of the mandibular arch. (c) After treatment. (d) Visual treatment objectives. Note that significant intrusion of the maxillary incisors was anticipated. (e and f) Lateral cephalometric radiographs before and after treatment. The three-piece intrusion arch achieved the planned treatment objectives.

d

e

ment. Thus, in the Class II, division 2 malocclusion, the two central incisors are intruded to the level of the lateral incisors (Fig 6-44). Similarly, in many Class II, division 1 patients, the four maxillary incisors can be intruded as a unit to the level of the canine (Fig 6-45a). Some mandibular arches can display an excessive curve of Spee with four incisors stepped as a unit occlusally. This step can be used advantageously if intrusion is required (Fig 6-45b). Flared incisors can give the illusion that no vertical discrepancy exists. In Fig 6-46, a straight wire from the posterior teeth lines up with the incisors (dotted line at the brackets). Note that when the incisor is tipped lingually to a correct axial inclination, the CR moves apically (∆). Early recognition of the potential step between the incisors and the canine allow for the most efficient treatment. This is not necessarily force redirection along the flared tooth’s long axis. If space is available, the incisors first can be retract-

f

ed by tipping. Later, with the better inclination, the incisor can be intruded. Because intrusion of the incisors may be necessary in many patients, it is obvious that the original malocclusion alignment is most important. The patient in Fig 6-47 has normal vertical overlap (or overbite) at the start of treatment but potential deep bite after retraction of the maxillary incisors by tipping (Fig 6-47a). Intrusion and retraction of the maxillary anterior segment are simultaneously accomplished by the three-piece mechanism (Figs 6-47b to 6-47d). Some incisor intrusion occurred in the mandibular arch as the posterior curve of Spee was leveled. Note the significant intrusion of the maxillary incisors in Figs 6-47e and 6-47f. The anatomical tooth arrangements with their steps between segments discussed here for intrusion may also be suitable for resolution with extrusive mechanics (see chapter 7). 109

6 Single Forces and Deep Bite Correction by Intrusion

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b

Fig 6-48  Special anchorage control. (a) The reciprocal force system in the posterior segment shows extrusive force and tip-back moment. (b) Occlusal chewing forces may help negate the vertical extrusive force, but the tip-back moment remains.

Fig 6-49  The effect of number of teeth on occlusal plane cant in the buccal segment. If a posterior segment has only a first molar in comparison with a canine to first molar, anchorage is poor, and the first molar will tip back. SD, standard deviation.

Special Anchorage Considerations Historically, the major limitation in achieving genuine incisor intrusion has been anchorage control. In response to intrusive forces, extrusive forces operate on the posterior teeth. Note in Fig 6-48 that, in addition to the extrusive force, a large moment at the CR steepens the occlusal plane of the posterior segment; it is this moment that is the most important aspect of intrusion anchorage control. Occlusal chewing forces may help negate the vertical extrusive force, but the tip-back moment remains. Sound anchorage principles such as increasing the number of teeth in the buccal segment, increasing the rigidity of the posterior segment wire, and placing a lingual arch are beneficial. If a posterior segment has only a first molar in comparison with a canine to first molar, anchorage is poor, and a steepening of the posterior occlusal plane will occur (Fig 6-49). One should look beyond the number and root 110

size of the posterior teeth. Consider the distance between the anterior point of attachment and the posterior CR. Figure 6-50a shows a flared incisor. A good strategy might be to retract the incisors (even partially) before intrusion (Fig 6-50b). This reduces the perpendicular distance to the CR (L1 > L2) and the unwanted steepening moment to the posterior teeth. Adding teeth to the buccal segments also influences the position of the posterior CR. Adding a canine moves the CR forward, and adding a second molar moves it posteriorly. Although the canine root mass is smaller, its addition to the posterior segment may be more significant for anchorage than addition of the second molar (Fig 6-51). The steepening moment is reduced by shortening the distance to the posterior CR. It is the same principle as moving the attachment hook posteriorly with a three-piece intrusion arch, which has already been discussed. The use of a headgear can negate the steepening occlusal moment from any intrusion arch. A number of different directions (purple arrows) are possible,

Special Anchorage Considerations

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Fig 6-50  The effect of length of the intrusion arch. (a) A flared incisor. It is better to retract the incisors even partially before intrusion. (b) Retracting the incisor reduced the length of the intrusion arch (L1 > L2) and reduced the unwanted steepening moment to the posterior teeth.

a

b

Fig 6-51  The location of the posterior CR in accordance with the addition of teeth. (a) Adding a canine to the anchorage unit moves the CR anteriorly. (b) Adding second molars to the anchorage unit moves the CR posteriorly.

Fig 6-52  The use of a headgear with an intrusion arch. A number of different directions (purple arrows) can negate undesirable tip-back moments (curved red arrow) from the intrusion arch. The headgear forces are much larger; therefore, minimal headgear wear is sufficient.

as shown in Fig 6-52, with both occipital and cervical headgears. Because intrusion forces are relatively small and headgear forces are much larger, minimal headgear wear is required. Usually little or no headgear is required to back up anchorage if the mechanics described in this chapter are used (Fig 6-53). The use of implants and TADs can certainly simplify anchorage considerations for incisor intrusion (see chapter 18). However, the improved anchorage should not be used as an excuse to employ excessive intrusive forces. Figure 6-54 shows a patient in whom the first premolars were used as anchorage to intrude all the maxillary incisors. Note the genuine intrusion of incisors relative to the canine as

a reference. Later the first premolar was extracted; therefore, this tooth served like an implant for the incisor intrusion stage of treatment. Sometimes a high-pull headgear anterior to the CR of the maxillary arch (or anterior to the CR of the posterior segment to back up the anchorage) is used to intrude incisors and to reduce the cant of the occlusal plane of the maxillary arch. A high-pull J-hook headgear anterior to the CR is commonly used to intrude incisors along with a full archwire (Fig 6-55a). However, the direct application of intermittent, fluctuating high-level forces in the incisor region may lead to root resorption. Because the maxillary archwire may not be sufficiently rigid, it is not a 111

6 Single Forces and Deep Bite Correction by Intrusion Fig 6-53 The effect of headgear on occlusal plane cant. Usually little or no headgear is required to back up anchorage if intrusion force levels are kept low. SD, standard deviation.

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b

good idea to place heavy forces in the incisor region that could lead to root resorption. If a headgear is used, it is preferable to use an inner bow inserted into the molar tube for improved force control and distribution of forces away from the incisors (Fig 6-55b) than the J-hook headgear. It is better to use the headgear only to help control the posterior segments during incisor intrusion, with the headgear force on the posterior teeth and not the incisors. Typically, patients that require both incisor and canine intrusion are considered too challenging for intrusion of six anterior teeth by dental anchorage 112

Fig 6-54 The use of a tooth to be extracted as an anchorage unit. (a) Before intrusion. The first premolars were used as anchorage to intrude the four maxillary incisors. (b) After intrusion. Note the genuine intrusion of the incisors relative to the canine as a reference. Later the first premolar was extracted.

Fig 6-55  (a) A high-pull J-hook headgear anterior to the CR is commonly used to intrude incisors along with a full archwire. The direct application of intermittent fluctuating high-level forces in the incisor region may lead to root resorption. (b) Headgears with inner and outer bows better distribute force to the posterior teeth.

alone. These patients may require orthognathic surgery or the use of plates and TADs. If canine intrusion is needed, it might be best to gain full control with a continuous edgewise arch stepped around the canine (a canine bypass). A separate cantilever can be employed for the canine intrusion (Fig 6-56). Another possibility is a bypass arch with a rectangular loop welded to vertically align the canine (Fig 6-57). The loop can also be used to simultaneously rotate the canine or change mesiodistal axial inclination.

Special Anchorage Considerations Fig 6-56  Separate canine intrusion using a cantilever. A continuous archwire bypassing the canine provides the best anchorage for separate canine intrusion.

Fig 6-57  Separate canine intrusion using a loop. A rectangular loop welded to a bypass archwire is used to simultaneously rotate the canine or change mesiodistal axial inclinations during intrusion. (a) Before treatment. (b) After treatment.

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Fig 6-58  The use of a reciprocal tip-back moment. (a) Wire is not inserted into the full posterior segment so that the continuous intrusion arch produces a tip-back moment on the first molar only. The incisors will intrude, and reciprocally the molar will tip back, helping to correct the Class II malocclusion. (b) A three-piece intrusion arch can be more efficient because it allows an unrestrained molar tip-back.

The last anchorage consideration is to accept and practically use the reciprocal tip-back moment from the intrusion arch. Wire is not inserted into the entire posterior segment, so that the continuous intrusion arch shown in Fig 6-58a produces a tipback moment on the first molar only. The incisors will intrude, and reciprocally the molar will tip back,

helping to correct a Class II malocclusion. A threepiece intrusion arch can be more efficient because it allows an unrestrained molar tip-back effect (Fig 6-58b). Tip-back mechanics from cantilevers and three-piece arches are further discussed in the next chapter on posterior extrusion mechanics.

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6 Single Forces and Deep Bite Correction by Intrusion

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Fig 6-59  A patient with deep bite and a Class II malocclusion. Note not only that the incisors are intruded but also that the Class II malocclusion is improved as the reciprocal moment tips back the posterior teeth.

The Class II malocclusion in Fig 6-59 can be improved using a continuous intrusion arch because the reciprocal moment steepens and tips back the posterior teeth. If posterior segments are tipped forward, one can take advantage of this inclination problem during incisor intrusion for acceptable molar tip-back.

Burstone CJ, Marcotte MR. Problem Solving in Orthodontics: Goal-Oriented Treatment Strategies. Chicago: Quintessence, 2000.

References

Shroff B, Lindauer SJ, Burstone CJ, Leiss JB. Segmented approach to simultaneous intrusion and space closure: Biomechanics of the three-piece base arch appliance. Am J Orthod Dentofacial Orthop 1995;107:136–143.

1. Dellinger E. A histologic and cephalometric investigation of premolar intrusion in the Macaca speciosa monkey. Am J Orthod 1967;53:325–355. 2. van Steenbergen E, Burstone CJ, Prahl-Andersen B, Aartman IH. The influence of force magnitude on intrusion of the maxillary segment. Angle Orthod 2005;75:723–729.

Recommended Reading Burstone CJ. Applications of bioengineering to clinical orthodontics. In: Graber TM, Vanarsdall RL (eds). Orthodontics: Current Principles and Techniques, ed 4. Philadelphia: Mosby, 2005:293–330. Burstone CJ. Biomechanics of deep overbite correction. Semin Orthod 2001;7:26–33. Burstone CJ. Deep overbite correction by intrusion. Am J Orthod 1977;72:1–22.

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Choy K, Pae EK, Kim KH, Park YC, Burstone CJ. Controlled space closure with a statically determinate retraction system. Angle Orthod 2002;72:191–198. Romeo DA, Burstone CJ. Tip-back mechanics. Am J Orthod 1977;72:414–421.

Shroff B, Yoon WM, Lindauer SJ, Burstone CJ. Simultaneous intrusion and retraction using a three-piece base arch. Angle Orthod 1997;67:455–462. van Steenbergen E, Burstone CJ, Prahl-Andersen B, Aartman IHA. The role of a high pull headgear in counteracting side effects from intrusion of the maxillary anterior segment. Angle Orthod 2004;74:480–486. Vanden Bulcke M, Sachdeva R, Burstone CJ. The center of resistance of anterior teeth during intrusion using the laser reflection technique and holographic interferometry. Am J Orthod 1986;90:211–219. Vanden Bulcke M, Sachdeva R, Burstone CJ. Location of the center of resistance of anterior teeth during retraction using the laser reflection technique. Am J Orthod 1987;90:375–384.

PROBLEMS For problems 1 through 7, a 60-g force (30 g per side) is placed in the middle of a four-tooth anterior segment by a continuous intrusion arch. 1. Replace the force system at the CR of the anterior and posterior segments.

2. The incisors are flared. Replace the force system at the CR of the anterior and posterior segments.

3. The posterior CR has moved anteriorly because the canine was included in the anchorage unit. Replace the force system at the CR of the anterior and posterior segments.

4. The posterior CR has moved further anteriorly because the maxillary second molar was excluded from the anchorage unit. Replace the force system at the CR of the anterior and posterior segments.

5. Find the magnitude and sense of the headgear force (FHG) on the given line of action to keep the occlusal plane from canting. Assume an exact balance.

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6. Replace the force system at the CR of the anterior and posterior segments.

7. Compare the anchorage using only one molar with problem 1.

8. A 30-g force is placed on the distal extension of the anterior segment (on each side) by a three-piece intrusion arch. Replace the force system at the CR of the anterior and posterior segments.

9. A 30-g force is placed on the distal extension of the anterior segment (on each side) by a three-piece intrusion arch. Compare the effect of FA, FB, and FC. Replace the force system at the CR of the anterior and posterior segments.

10. A continuous intrusion arch is tied off-center to the left. What effect does it have on the cant of the occlusal plane of the incisor and the posterior segment in the frontal view? Posterior segments are rigidly connected by a lingual arch. Replace the force through the CR for the answer.

11. A continuous intrusion arch is tied off-center to the left as in problem 10, except the posterior segments are not connected with a lingual arch. Will the movement of the posterior and anterior segments be different? Replace the force through each CR for the answer.

12. An intrusive force is directed parallel to the long axes of the flared incisors by a three-piece intrusion arch. A coil spring (FH) and intrusion arch (FV) are the components of the 30-g intrusive force. Find the magnitude of FH and Fv. What will happen to the incisors and the posterior segment? 116

CHAPTER

7 Deep Bite Correction by Posterior Extrusion “Respond intelligently even to unintelligent treatment.”

— Lao Tzu

“The more perfect a thing is, the more susceptible to good and bad treatment it is.”

OVERVIEW

— Dante Alighieri

Most patients with deep bite (excessive vertical overlap) require some extrusion of posterior teeth. Along with facial growth, this can lead to stable deep bite correction. Archwires with an intrusive force to the incisors can produce posterior extrusion that is a combination of rotation and translation. In the mandibular arch, translatory extrusion and reduction in the occlusal plane angle are described as Type I extrusion. Type II extrusion is produced by anterior arch disengagement with posterior vertical elastics. Combining Type I arch mechanics with an extrusive headgear offers an additional method for parallel posterior extrusion. Leveling arches may lead to posterior extrusion, but control is usually lacking. The placement of exaggerated curves of Spee in the maxillary arch or reverse curves of Spee in the mandibular arch produces unwanted moments. The completely flat arch is an orthodontic construct that is not necessarily valid biologically.

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b

Fig 7-1  Type I posterior extrusion. An intrusive force to the anterior teeth produces a tip-back moment and an extrusive force on the posterior segments.

Fig 7-2  Type II posterior extrusion. A single force at the center of resistance (purple circle) produces parallel extrusion of the posterior segment without rotation.

b a

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Fig 7-3  Force system from straight-wire leveling. The posterior teeth tip back and extrude (a), and an anterior moment displaces the canine root to the mesial (b).

Not every patient with deep bite requires intrusion of the incisors. Because intrusion requires demanding mechanics, it is desirable to identify the situations where posterior extrusive mechanics can be employed. Patients who exhibit good mandibular growth or have a large interocclusal space may be successfully corrected by increasing the vertical dimension. Let us consider three treatment possibilities by extrusion: (1) intra-arch tooth movement where initial alignment is poor; (2) fully aligned arches where respective occlusal planes converge anteriorly; and (3) synchronous occlusal plane changes associated with Class II or Class III maxillomandibular elastics.

Type I Posterior Extrusion Some patients exhibit a mesially tipped posterior segment that requires a combination of extrusion and rotation (Fig 7-1). An intrusive force to the canine or anterior teeth produces a tip-back moment on the posterior segments. This combination of ro118

tation and extrusion is classified as a Type I posterior extrusion for deep bite correction. It is the easiest extrusion to produce during leveling because intrusive forces to the canines or the incisors will create a rotational moment (tip-back) on the posterior segment (see chapter 6). Other patients require extrusion of the occlusal plane of the posterior segment without changing its cant. In Fig 7-2, parallel extrusion (Type II extrusion) is shown. The mechanics of Type II extrusion require special methods to eliminate the undesired tip-back moments. Type I extrusion is most commonly seen in the mandibular arch and is associated with an exaggerated curve of Spee. Although a straight wire can create a desirable moment on the posterior segment, anteriorly a straight wire could displace the canine roots to the mesial (Fig 7-3). Sometimes a reverse curve of Spee is placed in a continuous mandibular archwire. This might help alleviate some of this canine root displacement; however, it is too imprecise to solve the problem and may create other unwanted effects.

Type I Posterior Extrusion

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Fig 7-4  Three-piece tip-back (or intrusion) arch. (a) The force system rotates and extrudes the posterior segment. (b) A larger intrusive force than that used for intrusion is applied to all six anterior teeth so that intrusion is kept to a minimum.

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Fig 7-5  Tip-back of posterior teeth. (a) When indicated, a posterior stabilizing archwire is not inserted, and a tip-back spring is engaged on the first molar only. (b) The premolars will tip individually, not en masse, due to the transseptal fibers or a ligated ligature wire. Space (∆) distal to the canine can be gained without any distal force.

Fig 7-6  The tip-back is accomplished without flaring of the mandibular incisors by placing the intrusive force distal to the CR of the anterior segment (red arrow). The replaced equivalent force system (yellow arrows) at the CR indicates that the anterior segment will intrude and tip slightly to the lingual.

The three-piece intrusion arch discussed in chapter 6 can be used to extrude and tip back (rotate) the posterior segment (Fig 7-4). The appliance and force system look similar to the mechanism for intrusion, but there are significant differences. During intrusion, the forces are distributed to the incisors only. For extrusion, on the other hand, the intrusive force is applied to all six anterior teeth. Also, larger forces are used—perhaps 100 g per side instead of 30 to 40 g per side. Emphasis is placed on delivering a large enough moment to efficiently tip back and extrude the posterior segment. Figure 7-4a shows a –2,000 gram-millimeters (gmm) tip-back moment. Note that the intrusive force acts at the center of resistance (CR) of the anterior segment to prevent incisor flaring. If indicated, space can be effectively

regained by individual tip-back of posterior teeth in selected cases without any distal force. The premolars will spontaneously tip distally due to transseptal fibers or ligature wire tying together the posterior teeth. The hook on the tip-back spring, which can slide distally on the anterior segment, allows for distal movement of the posterior segment (Fig 7-5). Thus, space can be opened to alleviate moderate crowding in the mandibular arch. Unlike most continuous archwires, this is accomplished without flaring of the mandibular incisors by placing the intrusive force distal to the CR of the anterior teeth (Fig 7-6). This further ensures that no incisor flaring will occur and allows a longer range of distal sliding on the distal extension of the anterior segment. A lingual arch or a transpalatal arch (TPA) can be use119

7 Deep Bite Correction by Posterior Extrusion

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Fig 7-7  The components of a three-piece tip-back (intrusion) arch. (a) Posterior segment (red rectangle), anterior segment (green rectangle), and tip-back extrusion spring (yellow rectangle). (b) Left and right posterior segments are stabilized by a passive lingual arch (red rectangle in the center). (c) Deactivated shape. (d) Activated shape.

Fig 7-8  The continuous tip-back spring is indicated if distal movement of the posterior teeth is not required. Anterior force position variation can still be achieved.

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Fig 7-9  Varying the anterior point of force application. Intrusive force acting at the CR (a), anterior to the CR (b), and distal to the CR (c). The position of the force influences not only the flaring tendency of the incisors but also the rotation of the posterior segments—incisor intrusive forces applied further forward produce more posterior rotation.

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Type II Posterior Extrusion Fig 7-10  (a) The deep bite makes it difficult to bond brackets. (b) A fixed bite plate was inserted (not shown) in the maxillary arch temporarily, and a continuous tipback arch was placed to extrude and rotate the posterior teeth.

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Fig 7-11 A mandibular bite plate opens the bite and creates occlusal space for the posterior teeth to extrude.

ful to maintain arch form, arch width, and molar buccolingual axial inclinations. The three-piece tip-back (Type I extrusion) arch (or intrusion arch) consists of (1) a posterior segment (right and left segments connected by a lingual arch), (2) an anterior segment, and (3) right and left tip-back and extrusion springs (Fig 7-7). The design can be simplified if distal movement of the posterior teeth is not required by using a continuous tip-back spring (without sliding hooks) from the right to the left molar auxiliary tubes (Fig 7-8). The point of force application to the incisors can also be moved closer to the CR or further anteriorly to the incisor brackets (Fig 7-9). The position of the force influences not only the flaring tendency of the incisors but also the rotation of the posterior segments—incisor intrusive forces applied further forward produce more posterior rotation relative to the extrusive posterior translation. It might be difficult to level a mandibular arch using wires and brackets in a patient with deep bite, because occlusal forces can shear off the mandibular incisor brackets. Figure 7-10 shows a Type I continuous intrusion arch placed in the auxiliary molar tubes. An intrusive force (150 g) was placed at the incisor midline, causing the posterior teeth to extrude and rotate. A fixed bite plate, a temporary appliance, was inserted in the maxillary arch to prevent the occlusion from debonding the mandibular anterior brackets. The bite plate was removed once the mandibular arch was leveled by extrusion (after approximately 10 weeks).

Type II Posterior Extrusion It is much easier to extrude and rotate a posterior segment during intra-arch alignment than to extrude it in a parallel manner, maintaining the posterior plane of occlusion. The mechanics of Type II (parallel) extrusion require vertical forces, and intra-arch moments are to be avoided. Undesirable arch moments can come from a step apically to the incisors, mandibular curves of Spee, and almost any type of incisor intrusion arch. The appliance for Type II extrusion relies on disengaging the posterior teeth with an anterior bite plate. Posterior segments are now free to spontaneously erupt or more likely be extruded using vertical elastics. The mandibular bite plane opens the bite and creates occlusal space for the posterior teeth to extrude (Fig 7-11). It is usually preferable not to have any continuous archwires in place so that the separate maxillary and mandibular posterior segments can be extruded as a unit. A bite plate in the maxillary arch attached to a precision palatal horseshoe arch (Fig 7-12) or attachments bonded to the palatal surface of the maxillary incisors (Fig 7-13) can also be employed for the disengagement. Bite plates are not comfortable and not easily worn by many patients; for that reason, bonded bite plates are preferable. When the bite plate is in place, the posterior teeth are separated. Depending on the malocclusion, they can be joined together as a unit (Fig 7-14) or han121

7 Deep Bite Correction by Posterior Extrusion Fig 7-12  (a and b) A maxillary bite plate is easily bonded to a palatal horseshoe arch. Fabrication can be either directly in the mouth or a laboratory procedure.

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Fig 7-13  (a to c) Bonded palatal attachments can be used to disengage the posterior teeth.

Fig 7-14 Posterior teeth are separated by a bite plate and extruded en masse using a stabilizing archwire. The resultant forces (red arrows) from the elastic pass distal to the CR for extrusion and rotation.

dled as individual teeth (Fig 7-15). Vertical elastics are applied through the CR of a segment if parallel eruption is required. The yellow elastic will flatten the maxillary and steepen the mandibular posterior occlusal plane cant (see Fig 7-14). Sometimes, the direction of the elastics will be slightly Class II or Class III or not exactly at the CR; hence, care should be taken to minimize any problems by carefully evaluating the progress of the tooth movement. In summary, for deep bite correction via Type II extrusion, undesirable moments are avoided by avoiding mechanics from the archwire and relying directly on vertical forces from vertical maxillomandibular elastics. Undesirable side effects are possible with these mechanics. The elastic force is buccal to the CR, which 122

Fig 7-15 Posterior teeth are extruded individually without a stabilizing archwire when individual movement is required for maximal intercuspation and alignment.

means that the posterior teeth could tip toward the lingual. For the short period of time in which the elastics are used, no major problem should be observed; for full control of arch width, lingual arches can be used. Figure 7-16a shows a deep bite in which the discrepancy is in both arches. Note the step relationship between the canine and the first premolar brackets. The posterior teeth are disengaged by a bite plate (pink in Fig 7-16b). Space is now present between the maxillary and mandibular posterior teeth. The resultant of the box elastic (green) is close to the CR of both posterior segments. The posterior segments now erupt without changing the occlusal plane cants (Fig 7-16c).

Type II Posterior Extrusion

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Fig 7-16  (a) A deep bite that requires Type II extrusion of posterior teeth in both arches. (b) The posterior teeth are disengaged by a bite plate (pink), and a rectangular elastic (green) is placed. The resultant forces (red arrows) are close to the CR of both posterior segments. (c) The posterior segments are extruded without changing their occlusal plane cants. (d) The discrepancy is in the maxillary arch only, and an FRC ribbon is used for full mandibular anchorage. (e and f) The FRC can be used before full bracketing to correct the deep bite in the early stages of treatment. (g) After deep bite correction, connecting wire must be used to maintain the extrusion. (h and i) The discrepancy is in the mandibular arch, and the same principle of placing an FRC ribbon is applied.

It could be advantageous to use a fiber-reinforced composite (FRC) ribbon instead of a stabilizing archwire in the anchorage arch (Fig 7-16d). Here the discrepancy is in the maxillary arch, with the step between the canine and the first premolar. Only the maxillary teeth are bracketed. For some patients,

the deep bite should be corrected at the initial stage of treatment. One approach is to place the FRC ribbon before brackets and vertical elastics are placed (Figs 7-16e to 7-16i). The FRCs can then be removed and brackets placed for the continuation of treat­ment. 123

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Fig 7-17  The leveling of a deep bite by a continuous archwire. (a) Red arrows show the forces on the teeth at the brackets. (b) The replaced equivalent force system at each CR. Note the large moment acting on the posterior segment because the canine extrusive force has a large perpendicular distance to the posterior CR. (c) The occlusal plane will steepen as a result of posterior rotation and extrusion accompanied by possible intrusion of the anterior segment.

Fig 7-18  Deep bite leveled by a continuous archwire. Note the steep maxillary occlusal plane and flat mandibular occlusal plane (white lines). The anterior vertical overlap (or overbite) was little improved (red circle).

If an intrusive force is placed on the incisors with a continuous archwire (Fig 7-17a), the response of the posterior teeth is both extrusion and rotation. Rotation is inevitable because an extrusive force has a large perpendicular distance to the posterior CR. Therefore, the replaced equivalent force system at the posterior CR shows a large moment (Fig 7-17b). As a result, the occlusal plane will be steepened by posterior rotation and extrusion accompanied by possible intrusion of the anterior segment (Fig 7-17c). It still may be advantageous to use a wire connecting the anterior and posterior teeth so that some incisor intrusion will occur even if the objective is primarily posterior extrusion. At a minimum, it is de124

Fig 7-19 Parallel eruption of posterior teeth can be accomplished by adding an extrusive force (Fβ) distal to the CR of the posterior segment. As L1 is limited, Fβ has to be problematically large to balance the moment produced anteriorly by an intrusive force (Fα).

sirable to prevent the incisors from erupting further. But how can we prevent the undesirable side effects of maxillary occlusal plane steepening and mandibular occlusal plane flattening (Fig 7-18)? Parallel eruption of posterior teeth can be accomplished by adding an extrusive force (Fβ ) distal to the CR of the posterior segment (Fig 7-19). In order for the resultant force to go through the CR (required for posterior translation), the moments from both anterior and posterior forces must sum to zero, measured around the CR. The use of a vertical elastic is problematic because the force (Fβ) should be larger due to the limited length of the moment arm (L1 in Fig 7-19). One of the solutions to this problem could

Type II Posterior Extrusion

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Fig 7-20  A cervical headgear with a force as far back as possible and mainly directed downward can be a very efficient way to balance the steepening moment from the anterior force. (a) A resultant force (yellow arrow) passes through the CR. (b) The moments of the replaced force systems of the headgear (yellow arrow) and the tip-back spring (purple arrows) at the CR cancel each other out. (c) The resultant force on the posterior segment (red arrow) acting on the CR.

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be a cervical headgear (Fig 7-20) with a force as far back as possible and mainly directed downward; this can be a very efficient way to balance the steepening moment from the anterior force. Of course, the force system from the headgear is not absolutely balanced with the intrusive force to the incisors, because the intrusion arch acts continually and the headgear is worn only part-time. Note in Figs 7-20b and 7-20c that an exact balance for parallel posterior extrusion requires a line of action though the maxillary arch CR that is downward and backward.

This is not an instantaneous balance but a balance over time, requiring careful monitoring of the patient’s progress. The patient in Fig 7-21 was treated with a continuous intrusion arch and a cervical headgear. The headgear force was directed downward and backward and produced a large moment to balance the intrusive force. The patient had favorable mandibular growth related to his puberty growth spurt, which accounted for most of the Class II malocclusion and deep bite correction (see Fig 7-21i).

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Fig 7-21  A patient with deep bite treated with a continuous intrusion arch and a cervical headgear. (a to d) Before treatment. (e to h) After treatment. (i) The maxillary posterior teeth were significantly extruded, and the occlusal plane cant was maintained. Favorable mandibular growth aided the correction of the Class II malocclusion and deep bite.

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Curves and Reverse Curves of Spee A method that is commonly suggested for deep bite correction is the placement of an exaggerated reverse curve of Spee in the mandibular archwire or an exaggerated curve of Spee in the maxillary arch126

wire. However, both diagnostically and biomechanically, this does not make much sense. Let us start by considering the occlusal plane or occlusal curvature goals of orthodontic treatment. Naturally, the occlusal surfaces do not line up in a plane; some curve of Spee is present. As shown in Fig 7-22, little curvature may be evident anterior to the first molars (green line). Curvature distal to the first molar (orange line) can be marked because of

Curves and Reverse Curves of Spee

Fig 7-22  The natural occlusal plane shows the curve of Spee. Little curvature is present anterior to the first molars (green line). The curvature distal to the first molar (orange line) can be marked because of the axial inclinations of the second molars as they erupt.

Fig 7-23  Leveling the curve of Spee could make a Class II malocclusion worse as the maxillary second molars move forward and the mandibular second molars move backward due to the leveling moments (green teeth).

Fig 7-24  Misconception of the mandibular curve of Spee in which the teeth are analogous to spokes on a wagon wheel.

the immature axial inclinations of the second molars as they erupt. Later in development, the maxillary second molar crowns can exhibit a similar distoaxial inclination. This posterior curvature is natural and should not be leveled. Leveling moments could make a Class II malocclusion worse as the maxillary second molars move forward and the mandibular second molars move backward (Fig 7-23). Why then do some orthodontists believe that a finished dental arch should have teeth aligned along a flat occlusal plane? Perhaps it is an inability to easily correct deep bites or the desire to have a wire slide through a molar tube during sliding mechanics. There are other sound biomechanical methods to solve deep bite and space closure problems while still allowing a normal occlusal plane curvature. Another misconception relates to patients with an excessive mandibular curve of Spee. It is assumed that the brackets are aligned along a curvature with roots diverging from the front to the back of the arch. It has been suggested that the teeth are anal-

ogous to spokes on a wagon wheel (Fig 7-24). In the typical deep bite patient, however, axial inclinations of the teeth may be relatively normal because the deep bite is caused by vertical position and not tooth angulation. The patient in Fig 7-25 has a deep bite, but the tooth angulations are relatively normal; the mandibular incisor roots are not labially positioned through the labial plate of the bone as in the “wagon wheel” model. What is required is primarily vertical tooth movement either by extrusion or intrusion with little change of axial inclination. As discussed previously, any posterior eruption should be by translation (Type II extrusion). Because a curvature in a wire produces moments, a reverse curve of Spee is not indicated for the mandibular arch in this type of malocclusion. The effects of placing a continuous archwire with a reverse curve of Spee was studied by Kojima and Fukui1 using a numerical analysis (Fig 7-26). Brackets were aligned in a straight line, and a reverse curve of Spee was incorporated into the mandibular arch. 127

7 Deep Bite Correction by Posterior Extrusion Fig 7-25  A patient showing a severe curve of Spee and deep bite. (a and b) Incisor angulations are relatively normal. (c and d) The incisors have overerupted.

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Fig 7-26  The effects of a reverse curve of Spee in the mandibular arch. Note that no vertical change occurred on individual teeth but that the roots of all teeth converged toward the center of the arch. (Reprinted from Kojima and Fukui1 with per­mission.)

Note that no vertical change occurred on individual teeth while the roots of all teeth converged toward the center of the arch. This is to be expected because uniform curves that are part of a circle basically deliver moments, not vertical forces. The vertical overlap (or overbite) might improve, but favorable axial inclinations are sacrificed. Let us follow what would happen if an archwire with an exaggerated reverse curve of Spee was placed in a mandibular arch (Fig 7-27a), contributing to a deep bite. Unlike the patient in Fig 7-25, there is an axial inclination problem that needs correction. A first premolar was extracted, and the anterior and posterior segments tipped during space closure. Even here, where moments are needed, a reverse curve of Spee arch is a mistake. The tooth (bracket) discrepancy is only between the canine and the second premolar. To the mesial of the canine and distal of the second premolar, all brackets 128

are in good alignment (Fig 7-27b). Now let us do a tooth by tooth analysis, starting with the problem area—canine to second premolar. A straight wire between these two brackets (Class VI geometry; see chapter 15) will produce equal and opposite couples (Fig 7-27c). This would be appropriate to move roots together, provided the arch is tied back or these teeth are tied together. Now let us add a curvature to the entire archwire, placing it only between these two teeth (Fig 7-27d). Again, this is not a problem because the desirable moments moving the roots together are augmented. A smooth curvature or a segment of a circle produces equal and opposite couples. The difficulty arises when a curvature is placed along the entire arch in all the brackets. Note in Fig 7-27e that the curvature produces a moment in the wrong direction, moving the root of the second premolar distally. It can also be seen that any curvature

Curves and Reverse Curves of Spee

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Fig 7-27  The force system on the mandibular teeth from an archwire with a reverse curve of Spee. (a) After extraction of the first premolar, the anterior and posterior segments tipped into the extraction site, causing an axial inclination problem. An archwire with a severe curve of Spee was placed. (b) There is an intersegmental discrepancy only between the canine and the premolar. (c) If a straight wire is placed between these two brackets, the Class VI geometry will produce favorable equal and opposite couples. (d) A curvature between these two brackets will augment the favorable force system. (e) The same force system is produced between the premolar and the first molar. Note that the moment in the wrong direction moves the root of the second premolar distally. (f) Similarly, the force system between the first and second molars will produce unwanted distal root movement of the first molar. If all moments are added, the first molar will feel no moment because moments from the adjacent teeth (second premolar and second molar) cancel each other out. The second molar will feel an unwanted tip-back moment.

between the first and second molars (Fig 7-27f) will produce unwanted tooth movement. If all moments are added, the first molar will feel no moment because moments from the adjacent teeth (second premolar and second molar) cancel each other out. The second molar will feel an unwanted tip-back moment. Thus, even in a patient requiring axial inclination change, using wire moments can be incorrect if delivered from a continuous archwire with a reverse curve of Spee. Therefore, if a continuous archwire is to be used for this malocclusion, the reverse curvature should only be placed between the canine and the second premolar. In an extraction case in which loss of control has led to tipping, a continuous archwire (either straight or augmented with a localized curvature) can be used to correct the problem. This mechanism has a small range of action because the wire activation

is over a small interbracket distance between the canine and the second premolar. A segmental approach can lead to a superior force system allowing for a simple root spring. Figures 7-28a and 7-28b show the required force system to correct the axial inclinations and to eliminate the excessive curve of Spee. The arch is divided into two segments: an anterior segment and a posterior segment. A 0.018 × 0.025–inch titanium-molybdenum alloy wire with a curvature (a reverse curve of Spee) connects the auxiliary tubes on the first molar and on the canine (Figs 7-28c and 7-28d). The interbracket distance has been markedly increased from 5 mm to 14 mm, decreasing the force-deflection rate and producing a more constant moment magnitude that acts over a longer range. The good axial inclinations within each segment are maintained as an efficient en masse movement is achieved. 129

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Fig 7-28  Segmental approach to the malocclusion in Fig 7-27 with separate, passive anterior and posterior segments. (a) The correct force system from a root spring (red arrows). (b) The response to the force system. The gray shadow image is the malocclusion before root spring placement. (c) Deactivated shape of a root spring. The curvature delivers the correct required force system continuously. Blue arrows show the activation force system to activate the spring. (d) Activated shape. Red arrows show the deactivation force system acting on the tooth segments.

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b

The patient in Fig 7-29 had excessive vertical overlap during the finishing phase of orthodontics, and it was decided to correct this overlap by placing a localized exaggerated curve of Spee in the maxillary arch. This resulted in the roots of the canine and the first premolar contacting. This localized curvature arch produced the wrong force system for a vertical displacement discrepancy mainly caused by poor incisor bracket height. The correct force system requires mainly vertical forces. It has been commonly said that leveling a curve of Spee requires added arch length. In fact, a general rule of thumb is that for every millimeter of depth of curve of Spee, one millimeter of added arch length is needed. However, this oft-repeated idea is incorrect for many reasons. First, there are different types of excessive curves. The tooth arrangement with normal axial inclinations (Fig 7-30a) does not 130

Fig 7-29 (a) A patient showing excessive vertical overlap during finishing. (b) A localized exaggerated curve of Spee was placed in the maxillary arch, which resulted in the roots of the canine and the premolar contacting.

require more space in the arch because the teeth and brackets need to move only vertically. The confusion regarding added length arose because formerly orthodontists would take a flexible brass wire and contour to the occlusal surface of the dental cast. When the wire was flattened out, it became longer. Unfortunately, this flattening and lengthening has nothing to do with the amount of space required. We can see this same effect if a continuous archwire is placed into the brackets of an arch with an excessive curve (Fig 7-30b). The inserted teal archwire is shorter than the extended flat archwire. This is an appliance problem that can lead to flaring of the incisors and possible space opening but tells us nothing about the space needed. If the wire is free to slide distally, this lengthening effect disappears. Some added arch length may be required for tipped teeth with abnormal axial inclinations. The

Curves and Reverse Curves of Spee

a

b

Fig 7-30  Leveling the curve of Spee. (a) Normal axial inclinations do not require added space in the arch. (b) The misconception that added space is required came from the fact that there is a difference (Δ) between the projected lengths of a curved wire and a straight wire. This is one limitation of leveling with a straight continuous archwire.

Fig 7-31 Tipped posterior teeth with abnormal axial inclinations may require added space, but the amount may not be significant.

a

b

Fig 7-32 Many curves of Spee appear similar (green curve), but tooth position and the required biomechanics are different for each situation. (a) Parallel eruption with no moments. (b) Mandibular posterior segment tip-back. (c) Root movement with equal and opposite moments between the canine and the first premolar.

c

posterior segments in Fig 7-31 are tipped mesially and hence require more space. But even with this type of occlusal curvature, the diagram shows that little added space is needed for small angular changes. In short, the need for added arch length to compensate for an excessive curve of Spee has been exaggerated.

Some curves of Spee may be too large, and reduction is indicated. The biomechanics will differ because the original anatomical basis (tooth position) varies widely (Fig 7-32). Note that parts a, b, and c have similar occlusal curvatures but structurally are very different if one looks at the axial inclinations of the teeth. 131

7 Deep Bite Correction by Posterior Extrusion

Reference 1. Kojima Y, Fukui H. A numerical simulation of tooth movement by wire bending. Am J Orthod Dentofacial Orthop 2006;130:452–459.

Recommended Reading Andrews FL. The six keys to normal occlusion. Am J Orthod 1972;62:296–309. Baldridge DW. Leveling the curve of Spee: Its effect on the mandibular arch length. J Pract Orthod 1969;3:26–41.

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Braun S, Hnat WP, Johnson BE. The curve of Spee revisited. Am J Orthod Dentofacial Orthop 1996;110:206–210. Germane N, Staggers JA, Rubenstein L, Revere JT. Arch length considerations due to the curve of Spee: A mathematical model. Am J Orthod Dentofacial Orthop 1992;102:251–255. Roberts WW III, Chacker FM, Burstone CJ. A segmental approach to mandibular molar uprighting. Am J Orthod 1982;81:177–184. Romeo DA, Burstone CJ. Tip-back mechanics. Am J Orthod 1977;72:414–421. Woods M. A reassessment of space requirements for lower arch leveling. J Clin Orthod 1986;20:770–778.

PROBLEMS

1. A three-piece tip-back (intrusion) arch is used to upright and extrude the posterior segments. A moment of –2,000 gmm is needed per side on the posterior CR. How much force should be applied anteriorly between the central and lateral incisors?

2. A three-piece tip-back (intrusion) arch is used to upright and extrude the posterior segments. A moment of –2,000 gmm is needed per side on the posterior CR, and force is applied on a distal extension. How much force is applied anteriorly? Compare the extrusive force on the posterior segment between problem 1 and problem 2.

3. To upright the second molar, –2,000 gmm is required. How much vertical force (FA, FB) is applied in a and b? Which is better? Discuss.

a

b

4. The three-piece tip-back arch is activated with a 50-g force. What happens to the anterior segment in a ? What happens to the anterior segment in b ? What happens to the posterior segment in each?

a

b

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7 Deep Bite Correction by Posterior Extrusion

134

5. An incisor bite plate bonded to the mandibular incisors disengages the posterior occlusion. A vertical elastic is used to erupt the maxillary posterior segment by translation (Type II extrusion). Are there side effects on the mandibular arch? If so, what is their effect on the deep bite correction?

6. The maxillary arch right posterior segment requires Type II extrusion with unilateral vertical elastics. The mandibular arch was stabilized by a full continuous archwire for rigid anchorage. Replace the force system on the maxillary and mandibular CRs. Discuss the desirable and undesirable effects.

7. The three-piece tip-back spring on the mandibular arch and a Class II elastic are placed. Two anterior forces (100 g) are measured. Find the equivalent force systems at the CR of the maxillary arch, the mandibular posterior segment, and the mandibular anterior segment.

8. A uniform reverse curve of Spee is placed in the mandibular arch. The measured moment values between each of the two brackets are given sequentially (eg, first molar and second molar). Moments vary because of different angulations and interbracket distances. Add the total moments for the canine, second premolar, first molar, and second molar. Ignore any extraneous vertical and horizontal forces and the moments anterior to the canine. Are the moments uniform on each tooth? Discuss the effects.

CHAPTER

8 Equilibrium “When you light a candle, you also cast a shadow.”

— Ursula K. Le Guin

“Physics depends on a universe infinitely centered on an equals sign.”

— Mark Z. Danielewski

“You may hate gravity, but gravity doesn’t care.” — Clayton Christensen “Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.”

OVERVIEW



— Isaac Newton, First Law (translated from Principa Mathematica)

The important concept of equilibrium is based on Newton’s First Law. In a body at rest or at uniform velocity, the sum of all the forces and moments is zero. When an archwire is inserted into all of the brackets with activation forces, the wire is elastically deformed yet is in equilibrium. The archwire does not accelerate or exert any force to move the patient. Therefore, the activation force diagram is in equilibrium. An equilibrium diagram and the laws of equilibrium are very useful in solving for unknowns in an orthodontic appliance. It aids in the selection of the best appliance or adjustment. Clinicians always have treatment goals for their therapy, and unless the archwire or appliance can be placed in equilibrium, the goals are not possible. Free-body (deactivation force) diagrams depicting forces that act on the teeth can be obtained from the equilibrium diagram using Newton’s Third Law by reversing force and moment direction. The equilibrium equations are not only useful for the understanding of appliances but are also applicable to understanding the biomechanics of tooth movement and physiologic stresses in the temporomandibular joint.

135

8 Equilibrium

a

b

Fig 8-1  Newton’s First Law. (a) An elastic is hooked on the mandibular second molar and is ready to be activated. The orthodontist brings it forward with a force of 100 g. (b) Once the elastic is engaged, it remains stretched and keeps the state of rest in equilibrium.

Equilibrium Principle The most important concept from physics that can be applied to orthodontics is the equilibrium principle. It is based on Newton’s First Law, which states that a body remains at rest or in motion with a constant velocity unless acted upon by an external force. The overall mechanism that delivers orthodontic forces to teeth and bones is a spring, and this energystoring device is the active component of all orthodontic appliances. The spring, which can be fabricated from many materials, including metal, rubber, or polymer, stores the mechanical energy charged by the orthodontist during activation and releases it slowly. The orthodontic spring comes in many different applications and shapes: archwires, coil springs, elastics, aligners, and loops, among others. It may be surprising that orthodontic appliances are in equilibrium, because they are required to deliver forces to move teeth. Equilibrium means that no resultant forces are acting on the orthodontic appliance. Let us consider a simple appliance, an elastic. If the elastic is hooked on the mandibular second molar (Fig 8-1a), we can bring it forward with an applied force of 100 g (Fig 8-1b). The two blue forces are responsible for stretching the elastic, and they are considered activation forces because they are applied to the appliance to activate it. The activated (stretched) elastic is also in equilibrium, because the sum of all forces is zero (100 g + [–100 g] = 0 g). Clinically, we know that the elastic is in equilibrium; it is not accelerating. Most important, it is not exerting an unbalanced resultant force on the patient. It does not push the patient up to the ceiling, out the front door, or out of the window. Once it is recog136

Fig 8-2 Newton’s Third Law. Sisyphus pushes the rock up (action, blue arrow) and the rock pushes down on Sisyphus (reaction, red arrow) with equal and opposite forces to each other.

nized that the appliance is in equilibrium (not pushing, pulling, or rotating the patient), the laws of equilibrium can help us solve for unknown forces. In Fig 8-1b, the force on the right (100 g) was measured with a force gauge. We do not need to measure the left force because the elastic has to be in equilibrium; therefore, the left force must be equal in magnitude and opposite in direction (–100 g). The activated elastic demonstrates Newton’s First Law of a body at rest or at uniform velocity. Figure 8-2, on the other hand, demonstrates Newton’s Third Law: For every action (or force), there is an equal and opposite reaction (or force). In this figure, Sisyphus pushes the rock up (action, blue arrow), and the rock pushes down on Sisyphus (reaction, red arrow). Now let us take a closer look at the canine hook (Fig 8-3). Two forces (blue arrow and red arrow) are present: The hook holds the stretched elastic forward; the blue activation force deforms the appliance, and the red deactivation force acts to move the teeth during deactivation of the spring. Newton tells us that there are always two equal and opposite forces when bodies interact: the activation force and the reactive deactivation force. It is these forces and their direction (red arrow) that will move the teeth (see Fig 8-3). Note that Fig 8-4 is the same as Fig 8-1b except that the force directions are reversed. A diagram showing only the force system and objects of interest, such as forces from the spring and the teeth (excluding forces from mastication, gravity, etc), is called a force (or free-body) diagram. Because the activated appliance is always in equilibrium, its force diagram should always be in equilibrium. Figure 8-1b is the activation force diagram of the elastic; conceptually, it correctly depicts the elastic (the appliance) in a state of rest. Figure 8-4, however,

Equilibrium Principle

Fig 8-3  Two equal and opposite forces from an appliance and the tooth are action and reaction. Clinically the force required for activation of the appliance is called activation force, and the force on the tooth as the appliance deactivates is called deactivation force.

a

Fig 8-4  Deactivation force diagram showing separate unbalanced deactivation forces on both arches. This is not an equilibrium diagram, although all of the forces sum to zero.

b

Fig 8-5  Two equilibrium diagrams. (a) The spring (yellow rectangle) with blue activation forces. (b) The tooth (yellow) with a red deactivation force and stresses (forces) in the periodontium. Compression-side and vertical components of the stresses in the PDL are not depicted because they also sum to zero.

shows separate unbalanced forces on the maxillary and mandibular arches. It is an incomplete force diagram in that it does not show all forces (eg, stresses at the periodontal ligament [PDL]) acting on the object (ie, the teeth). However, all of the forces sum to zero, so we can call it a deactivation force diagram, remembering that the appliance is the only entity in equilibrium, not the teeth as so represented. Figure 8-5 shows two valid force diagrams in equilibrium. Figure 8-5a demonstrates that the appliance is in equilibrium from the activation forces. Figure 8-5b shows the deactivation forces, and the restraining forces (stresses) from the periodontium are also in equilibrium. Thus, teeth as well as appliances can have suitable and correct equilibrium diagrams if the PDL support is considered. In Fig 8-6, an open coil spring is shown pushing both canines distally. Because only distal forces acting on the canine are depicted in the figure, it might imply that using an open coil spring between the canines instead of a

Fig 8-6 An open coil spring is pushing both canines distally. No matter how noble and sophisticated an appliance may be, it cannot overcome the laws of physics, so a single distal force is not possible.

closed coil spring between the canine and the first molar can eliminate anchorage loss. However, that implication is wrong. No matter how noble and sophisticated an appliance is, it cannot overcome the laws of physics; a single distal force is not possible. In our simple example of a 100-g maxillomandibular elastic attached to the maxillary arch, what is the force on the mandibular arch at the molar? The answer comes from applying both Newton’s First and Third Laws. Step 1 is to place the appliance in equilibrium (see Fig 8-1b). Because the measured force to the right is 100 g, the unknown force to the left must also be 100 g in magnitude (First Law). Step 2 is to reverse the direction of the forces determined from the appliance activation force diagram (see Fig 8-4). This gives us the deactivation forces acting on the teeth (the dental arches); hence, the conclusion is that the elastic gives equal and opposite forces to the dental arches. Some orthodontists have believed that this conclusion comes from Newton’s Third Law, 137

8 Equilibrium

Fig 8-7  The equilibrium diagram of the activated intrusion arch (solid teal). The sum of forces and moments is zero. The deactivated arch is shown in transparent teal.

a

138

b

Fig 8-8  The deactivation force diagram shows that the forces are acting on the tooth. The direction of the deactivation force system (red) is the reverse of the activation force system shown in Fig 8-7 (based on Newton’s Third Law).

c

Fig 8-9  The force system acting on the molar can be depicted in a number of ways: A force and a couple at the center of the bracket (a), a downward force and two single forces producing a couple at the edge of the bracket (b), or two forces with the larger force at the mesial (c).

Fig 8-10  Anchorage assumed to be equal and opposite. However, this is a misuse of Newton’s Third Law.

but that is incorrect. It actually comes from the equilibrium application of the First Law. Although the Third Law is involved in reversing force direction, no calculation is involved in Step 2 because there is always an equal and opposite force between bodies. Some orthodontists have also incorrectly assumed that it is Newton’s Third Law that explains anchorage. For example, this assumption would state that during space closure, an appliance (eg, a loop) delivers equal and opposite force systems to the anterior and posterior segments. Newton’s Third Law seems to support this supposition, but it is not a correct application. The First Law, on the other hand, provides the correct answer that some forces may be equal while the force system can differ between the anterior and posterior teeth. An elastic delivers forces only; let us now consider an intrusion arch where both forces and moments are present. Figure 8-7 shows the equilibrium diagram with the forces and moments on the intrusion

arch given in blue; the activated arch is depicted in teal, and the deactivated arch shape is depicted in transparent teal. A downward activation force is present anteriorly, while an upward force and a counterclockwise couple are present at the posterior end of the arch. The activated intrusion arch is in equilibrium because the sum of the forces (100 g + [–100 g]) is zero. The sum of the moments, measured from the molar tube, is also zero (100 g × 35 mm = +3,500 gmm and a molar couple of –3,500 gmm). In Fig 8-8, the forces are reversed in red to show the deactivation forces that are in the direction of the tooth movement. The incisor brackets feel intrusive force, and the molar bracket feels an extrusive force and clockwise moment that tips the crown backward and the root forward. The forces acting on the molar can be depicted in a number of ways (Fig 8-9): (1) a force and a couple at the center of the bracket (see Fig 8-9a); (2) a downward force and two single forces producing a couple at the edge of the bracket

Basic Concepts and Formulas of Equilibrium

a

b

Fig 8-11  Equilibrium tells us that the sum of all forces and moments should be zero. Any unbalanced force (a) or moment (b) is impossible on the archwire.

(see Fig 8-9b); or (3) two forces with the larger force at the mesial (see Fig 8-9c). They are all equivalent. Figure 8-7 is the force diagram showing the intrusion arch in equilibrium. If a 100-g force is measured in the incisor region, the activation force and moment on the molar bracket need not be measured. Here Newton’s First Law is used to calculate any unknowns. Figure 8-8 reverses the direction of all forces and moments based on the Third Law to show the forces acting on the teeth; no calculations are needed. It is the no-brainer part of the analysis. Clinicians are usually more interested in this deactivation force diagram—forces acting on the teeth (see Fig 8-8)—than the activation force diagram from which it was derived. Although all forces and moments sum to zero, the deactivation force diagram does not show any real object in equilibrium and should not conceptually be called an equilibrium diagram. Misinterpretations of deactivation force diagrams are common; the forces and moments in these diagrams are independent and act separately at the molar tube and at the incisor bracket. These unbalanced forces are what move the teeth. It is an incomplete equilibrium diagram because the teeth are in equilibrium by the deactivation force and the stresses in the PDLs, which are not depicted. By contrast, in the true equilibrium diagram the activation forces and moments act on the entire arch and not as independent entities. Also, note that the incisor force system of the active teeth is equal and opposite to the force system of the anchorage molar tooth (Fig 8-10). The incisors feel an intrusive force, and the molar brackets feel both an extrusive force and a moment (see Fig 8-10). However, this example demonstrates and emphasizes the principle of equilibrium for discussing

anchorage in orthodontics, which is based on Newton’s First Law and not his Third Law.

Basic Concepts and Formulas of Equilibrium The formulas used in equilibrium calculation are very simple. Because the appliance is at rest and not accelerating, the following information is known: 1.

∑F = 0

2.

∑M = 0

where F is force and M is moment. Imagine that we are inserting an archwire into all of the brackets of a malocclusion and we must apply forces to elastically deform the wire or the PDL. When the patient leaves the office, there is no force acting on the archwire. The equilibrium formula (1) tells us that any unbalanced force is impossible on the arch (Fig 8-11a). The balanced forces are possible because they sum to zero. The same is true with any moments (Fig 8-11b). One cannot deliver only a moment from an arch to rotate a molar (see Fig 8-11b). Balanced moments produced by a couple and a force or a couple alone are possible to meet equilibrium requirements. Thus, in Fig 8-11, the unbalanced force system to the archwire is impossible— no amount of wire bending or bracket placement can make it happen. A simple equilibrium diagram can keep us out of trouble by quickly identifying what is impossible from a given archwire.

139

8 Equilibrium Fig 8-12 No matter how diverse and complicated the forces on the teeth may be, the archwire is always in equilibrium. (a) Deactivation forces (red arrows) act independently on each tooth. (b) The activation force system (blue arrows) acts on the entire archwire. Moments and forces sum to zero. All forces and moments are artistic approximations.

a

b

No matter how diverse, complicated, and independent the forces on the teeth may be, the archwire or the appliance is always in equilibrium. In Fig 8-12a, a flexible nickel-titanium (Ni-Ti) wire is placed into all of the brackets, which are badly misaligned. Many forces and moments acting on the teeth (red arrows) can be produced at every bracket, which are artistically depicted in the diagram. The teeth move in response to these deactivation forces, and yet the forces that activate the archwire (Fig 8-12b) are in equilibrium—the sum of the forces and moments on the wire is zero. The force system from this straight wire does not always deliver desired tooth movement, but at least we can be assured that the archwire and the bracket geometry will produce a force system that is in equilibrium. It is better for us to do the thinking than to let the appliance do it. The primary value of the equilibrium principle is that it gives us a powerful tool to solve for unknown forces, so that we can better design our appliances. The tool is called boundary conditions, which provide simple equations to solve for unknown forces. The deactivation force diagram from the intrusion arch shown in Fig 8-8 is very easy to determine. We can measure the force on the arch at the incisor and replace it with an equivalent force system at the molar bracket or some posterior center of resistance (CR). An activation force diagram of the intrusion arch will give the same result but is not necessary. A similar deep bite scenario is depicted in Fig 8-13a. Let us measure the activation moments at both the incisors (500 gmm) and the molar bracket (–3,500 gmm). All of the other forces do not need measurement and can now be calculated by boundary conditions. 140



1.

∑M = 0 500 gmm + (–3,500 gmm) + M = 0 M = 3,000 gmm

M will come from vertical forces. There are no more couples at the incisor and molar because all moments have been measured there. Any horizontal forces are ignored. FB × 35 mm = 3,000 gmm FB at molar = +86 g

2.

∑F = 0 FA at incisor = –86 g

The full answer is given in Fig 8-13b, the activation force diagram in equilibrium. To obtain the forces on the teeth, all forces and moments have their direction reversed (Fig 8-13c). The principle of equilibrium applies not only to rigid bodies but also to nonrigid deformed bodies and even bodies with a constant velocity. An activation force is applied to a coil spring (Fig 8-14a). Although it changes shape, it is continually in equilibrium with the activation forces. The spring in its original shape before activation is the passive shape, and the blue arrows are the activation force system and the red arrows the deactivation force system (Fig 8-14b). Movement can occur between points A and B in Fig 8-14c, but still the net force is zero, and no acceleration is occurring. Other examples of objects in equilibrium are a squeezed balloon (where distances between points change) and a flexible seesaw that changes its shape.

Solving Problems Using Equilibrium

a

b

Fig 8-13 The equilibrium principle provides boundary conditions to solve for unknown forces. (a) The activation moments at both the incisors (500 gmm) and the molar bracket (–3,500 gmm) are measured, and FA and FB are unknowns. (b) The spring is in static equilibrium by vertical forces of 86 g. The vertical forces do not need to be measured. (c) The deactivation force system acting on the tooth (red arrows). The direction of the force system is reversed from the equilibrium diagram (b).

c

a

b

c

Fig 8-14  The object of equilibrium does not need to be a rigid body. (a) An activation force is applied to a flexible coil spring in equilibrium. (b) The green spring (top) in its passive shape before activation. The blue arrows are the activation force system (middle), and the red arrows are the deactivation force system (bottom). (c) The shape of the coil spring is continually changing as it moves from point A to point B, but the spring is always in equilibrium with the activation forces.

Solving Problems Using Equilibrium Let us now follow the steps in applying the equilibrium principle to clinical situations. We decide to

use a 2 × 4 appliance from the first molars to the incisors (Fig 8-15). We would like to place a 40-g intrusive force on the incisors. To prevent flaring of the incisors, labial root torque of 300 gmm is also applied. The question is: What will happen to the first molar?

141

8 Equilibrium

Fig 8-15  The red force system is planned on the incisor segment for intrusion without flaring.

a

Fig 8-16  The equilibrium diagram for the activated intrusion arch shown in Fig 8-15. The direction of the known force system on the incisor is reversed in blue.

Step 1

Step 3

Figure 8-15 shows the deactivation forces acting on the teeth, from which we must determine all other forces. In this step, we start the force diagram (Fig 8-16) by reversing the direction of the known forces so they now become activation forces—forces acting on the wire (Newton’s Third Law). All activation forces are in blue.

Apply formula 2.

Step 2 Apply formula 1.

∑F = 0 Add any forces at the molar that will be required for equilibrium. Because there is a downward force acting at the incisors, place an upward arrow at the molar (Fig 8-17a). The diagram is now in equilibrium from the forces except for the magnitude of the molar force. According to formula 1, the force magnitude is –40 g (Fig 8-17b). 142

Fig 8-17  The vertical upward force at the molar is added to maintain the equilibrium of force (a). It is calculated as 40 g (b).

b

∑M = 0 In order for equilibrium to exist, the sum of the moments around any point must equal zero. Let us sum the moments around the red point on the incisors in Fig 8-18a. This point is selected as convenient because it simplifies calculation, eliminating the force at the incisors. The equation holds true, however, for any arbitrary point. 40 g × 30 mm + (–300 gmm) + Mmolar = 0 Mmolar = –900 gmm The moment acting on the arch in the molar region is a counterclockwise 900 gmm (Fig 8-18b). The equilibrium diagram is now complete, and the vertical forces balance. Couples on the wire at the molar and at the incisors balance with the couple produced by the 40-g vertical forces.

Solving Problems Using Equilibrium Fig 8-18  (a) A moment is necessary at the molar for the appliance to be in equilibrium. (b) It is calculated as –900 gmm. The intrusion arch is now in equilibrium with balanced forces and moments.

a

b

Fig 8-19 We are interested in the force system acting on the tooth during deactivation of the arch. The direction of the force system is reversed in red. The molar will extrude, the crown will move distally, and the root will move forward.

Step 4 Of course, clinically we want to know the force acting on the teeth. As explained before, this is a simple step in which all forces and moments on the force diagram are reversed. The red arrows in Fig 8-19 denote the deactivation force system on the teeth. Now we can answer the question as to what will happen to the molar. The molar will extrude, the crown will move distally, and the root will move forward. The forces and moments are low, so anchorage is relatively good for incisor intrusion. Anchorage could be augmented by adding more teeth to the posterior segment. The force diagram could be more complicated (eg, more horizontal forces), but we have kept it simple to develop the equilibrium principle and how it is applied. To better define molar movement, the force at the molar tube should be replaced with an equivalent force system at the CR. The CR is so close to the center of the tube that this step can be ignored in this case.

Important considerations Because Fig 8-19 shows the forces on the teeth, the question is asked: Should we use this diagram only without first putting the archwire (the appliance) in equilibrium? This can be erroneous unless the orthodontist understands what the diagram means.

All of the red forces and moments sum to zero, but each force and moment acts separately on individual teeth (the molar and two incisors). The blue force equilibrium diagram in Fig 8-18b has the forces acting on the entire archwire. Sometimes beginning students look at the deactivation force diagram (see Fig 8-19) and think that the vertical forces will flatten the occlusal plane of the whole arch or that the moments will steepen the occlusal plane. But this is not correct. The +900-gmm moment and 40-g force act only on the molar, primarily tipping the molar crown distally. The same is true of the incisors, which will primarily intrude. The blue forces act on the entire archwire, but the red forces are independent and do not interact. The second error in starting with a deactivation force diagram of the unbalanced forces on individual teeth is improper force and moment direction. Some forces may be erroneously placed from other appliance components, or maybe activation forces are used instead of deactivation forces. It is wise to first put the appliance in equilibrium to avoid mistakes and to allow full understanding and only then reverse force direction to obtain the forces on the teeth. Later, with experience, one can use only the deactivation force diagram if fully understood. It is certainly true that for one wire, activation and deactivation forces are equal and opposite. Each appliance component should be separately studied with an equilibrium diagram if unknowns are present. 143

8 Equilibrium

a

b

c

d Fig 8-20  (a) The red force system is planned on the incisor segment for intrusion and lingual root movement. (b) The activated arch in the equilibrium diagram. The direction of the known force system is reversed in blue. (c) The equilibrium of forces is used to solve for the unknown forces at the molar. (d) The equilibrium of moments is used to solve for the unknown moment at the molar. An arbitrary red dot is selected for calculation of moment. Now the activation force diagram of the spring can be completed with all known forces and moments. (e) All forces and moments are reversed in direction to produce the deactivation force diagram of the forces (red arrows) acting on the teeth.

e

Working example In another patient with deep bite, the maxillary incisors are in linguoversion (Fig 8-20a). We decide to apply a 40-g intrusion force with a 100-g tie-back force. To improve the incisor axial inclinations, a moment (torque) of 1,200 gmm is placed in the archwire. Again we ask the question: What is the reciprocal force system on the molar? First, we start to compose the equilibrium diagram (Fig 8-20b), which shows the known forces and moments on the archwire. The blue activation forces are the same as in Fig 8-20a, except their direction has been reversed. In Fig 8-20c, based on the formula ΣF = 0, the horizontal and vertical forces (blue arrows) are placed 144

on the wire at the molar tube. The vertical upward force is 40 g, and the distal force is 100 g. The formula ∑M = 0 is then used to calculate the unknown moment at the molar. Arbitrarily, a convenient point is selected at the incisor brackets (red dot), and all moments are summed in respect to this point (see Fig 8-20c). 40 g × 30 mm + (–100 g × 3 mm) + 1,200 gmm + Mmolar = 0 Mmolar = –2,100 gmm The equilibrium diagram is now complete (Fig 8-20d); all forces and moments are reversed in direction to show the forces (red arrows) acting on the teeth (Fig 8-20e). However, although all the forces in

Equilibrium and Equivalence

a

b

Fig 8-21  (a) A 2,100-gmm moment will excessively tip the molar back. (b) The deactivation force diagram tells us that eliminating the 40-g intrusive force will give a 900-gmm moment at the molar.

Fig 8-20e sum to zero, this is not an equilibrium diagram because unbalanced independent forces are acting on the teeth. If we want to place this force system on the incisors, there exists a major anchorage problem on the molar. A 2,100-gmm moment could tip back the molar (Fig 8-21a). With the aid of the activation force diagram (forces on the wire) and the deactivation force diagram (forces on the teeth), we can now do some creative thinking to improve the force system. Maybe 40 g is not needed to intrude the incisors. Let us eliminate the intrusive force. 100 g × 3 mm – 1,200 gmm + Mmolar = 0 Mmolar = 900 gmm Now the moment on the molar becomes 900 gmm (Fig 8-21b). This is much better. Perhaps anchorage can be reinforced by including more teeth or by using a headgear if necessary. Another possibility is to start the incisor root movement at a later stage of treatment. Notice that the tie-back horizontal forces (100 g) produce a counterclockwise rotation on the equilibrium diagram. If we can increase these forces, the molar will feel a smaller moment. This is a possibility, but the incisor crown will retract further to the lingual, which might not be an acceptable treatment objective. Therefore, the tie-back solution to the large molar moment is limited. We can try out many possibilities and choose the best force system; however, unless our diagram is in equilibrium, it will not be possible. The thinking sequence in Figs 8-20 and 8-21 can be helpful even without calculating the actual numbers. It tells us, for example, that if we elect not to measure the intrusive force and the lingual root torque, it is possible to obtain much higher magnitudes, leading to significant anchorage loss.

Summary The use of a deactivation force diagram with the direction reversal on the teeth gives the clinician a powerful tool for treatment planning. The clinician can try out many different strategies to optimize the treatment. Trial and error on a patient using an appliance is time-consuming and could lead to permanent harm. Trial and error on a diagram, however, is not only friendly to both the orthodontist and the patient but is also more versatile. The equilibrium principle has been used to determine unknown forces acting from an orthodontic appliance when the forces are known at the bracket positions. Another step is necessary, however, when the forces and moments at a tube or bracket are replaced at a relevant CR. In the examples given so far, the molar CR (purple circle) and molar tube center were close enough that replacement was not necessary (see Fig 8-21).

Equilibrium and Equivalence In this text, two important principles of physics have been applied to clinical orthodontics: equivalence and equilibrium. In this section, both formulations are needed. Figure 8-22a shows an intrusion arch with a separate anterior segment and a solitary intrusive force of 100 g. For anchorage, an entire buccal segment from canine to second molar is used. We want to know what will happen to the posterior segment and how good its anchorage will be. Figure 8-22b is the activation force diagram showing a downward force to activate the arch. Applying the ΣF = 0 formula, an upward force of 100 g is placed on the arch at the center of the molar tube (Fig 8-22c). Switching to the ΣM = 0 formula, the moment at 145

8 Equilibrium

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d Fig 8-22  (a) An intrusion arch with an intrusive force of 100 g is placed. The entire buccal segment is used for anchorage. (b) The activation force system on the tooth, reversing the direction of the intrusive force. (c) The equilibrium of forces is used to calculate the upward force of 100 g at the molar. (d) The equilibrium of moments is used to calculate the counterclockwise moment of 3,000 gmm. The equilibrium diagram is completed with the complete force system. (e) The reversed force system shows the deactivation force system acting on the brackets.

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Fig 8-23  (a) The movement of the posterior segment is predicted with a replaced equivalent force system (yellow arrows) at the CR of the posterior segment. (b) The direct replacement of anterior force by equivalence, not using the equilibrium principle, gives the same result.

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Fig 8-24 A patient whose first premolar was extracted presents with a high mesially tipped canine. The goal is to tip the canine distally. Note how the CR of the canine will move only slightly distally with a marked occlusal displacement (gray shadow).

Fig 8-25  A simple vertical loop spring is planned to bring the canine downward into occlusion. The force system (red arrows) that we want acts on the canine. The moment-to-force ratio of 10 mm was placed at the molar for reasonable anchorage. Is this force system possible?

the molar tube on the arch is calculated to be –3,000 gmm (Fig 8-22d). In this relatively simple situation, force direction is reversed from the equilibrium diagram to give the forces and moments acting on the molar. Our answer is the 100-g extrusive force at the center of the molar tube and a 3,000-gmm clockwise moment (red arrows in Fig 8-22e). To predict posterior segment movement, it is necessary to replace this force system at the CR of the posterior segment (Fig 8-23a). Here the equivalence equations are applied. At the posterior segment CR, there is a 100-g occlusal force on the teeth with a moment of 2,200 gmm (yellow arrows are the replaced equivalent force system of the red arrows). In the chapters on equivalence (see chapter 3) and deep bite correction (see chapter 6), we solved similar problems without using the principle of equilibrium. We directly replaced the 100-g downward force at the incisors with a 100-g extrusive force at the CR and a moment of +2,200 gmm (100 g × 22 mm = +2,200 gmm) (Figure 8-23b). The answer, of course, has to be the same, even if only the principle of equivalence is applied in this special case. Therefore, in a statically determinate force system using mechanisms like an intrusion arch in which all forces are known on the incisors, it is not necessary to develop an equilibrium diagram.

basis for appliance selection and use. Let us consider a few examples by which we can try out different treatment modes with simple diagrams rather than subjecting the patient to trial and error procedures. A patient whose first premolar will be extracted presents with a high mesially tipped canine (Fig 8-24). The goal is to tip the canine distally. Note how the CR of the canine will move only slightly distally with a marked occlusal displacement. To eliminate any friction considerations, let us use a simple vertical loop to bring the canine down. Figure 8-25 shows the force system acting on the tooth that we want, including horizontal forces of 200 g to close the space (any moments from the vertical loop are ignored for simplicity). To prevent the molar from tipping forward, we will apply a 2,000-gmm moment at the molar tube. With a 10-mm distance to the molar CR, this might give a 10-mm moment-toforce ratio and hence reasonable molar anchorage. Is this a possible force system for the patient? Let us set up a few valid examples of force diagrams in equilibrium to answer this question. All forces and moments have their direction reversed in Fig 8-26. The sum of the forces is zero, but the sum of the moments is not zero because there is an unbalanced –2,000-gmm moment acting at the wire; therefore, this force system is not possible. We decide to apply two vertical forces (a couple) at the canine and the molar to put the loop into equilibrium (Fig 8-27a). The deactivation force diagram (Fig 8-27b) with force directions reversed is now valid and can be used to discuss the loop appliance and its desirability. The force system on the molar is reasonable. The resultant forces are depicted as yellow arrows in Fig

Equilibrium and Creative Biomechanics Equilibrium and other principles from physics allow both simulation of treatment stages and a scientific

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8 Equilibrium Fig 8-26  The activation force system on the spring with a reversed direction of force. The diagram is not in equilibrium, because there is an unbalanced 2,000-gmm moment acting at the molar.

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8-27c. There should be sufficient moment to keep the molars from tipping forward from the red mesial force on the molar. The canine is more problematic. The desired movement of the CR is downward and backward, but the intrusive component of force on the canine is inadequate. This is not a good outcome to have because the canine is already in infraocclusion. Also, the line of action of the force is moved closer to the CR to produce less effective mesial root displacement. The canine will probably tip around a center of rotation near the apex level, and the apex is too far to the distal already (note the predicted canine position after movement by this force system [in green] in Fig 8-27c). There is no reason to place 148

b Fig 8-27  (a) The spring in the diagram can be put in equilibrium by adding two vertical forces (100 g). (b) The deactivation force diagram for simulation and evaluation. The forces acting on the teeth are shown, reversing the force directions from a. (c) The horizontal and vertical forces (red arrows) are replaced with single forces (yellow arrows). The forces on the molar are reasonable, with sufficient moment to keep the molars from tipping forward. The yellow resultant force on the canine is slightly intrusive, and the line of action of force is closer to the CR, which is not desirable. This simulation procedure avoids a lengthy clinical error in treatment. Note that the canine intrudes instead of extruding.

this appliance in the patient because we know the force system is undesirable. Now let us work with the red forces on the teeth from the deactivation force diagram in Fig 8-27b; practically, it is not necessary to go back to the original equilibrium diagram of the forces on the loop. We will try out different deactivation force systems from the loop to see if we can improve the outcome. Perhaps if the horizontal magnitude of the force is reduced to 100 g, less molar moment and hence less vertical force will be needed. However, an intrusive force is still present on the canine, and the line of action of the resultant forces remains the same, so this is not a substantial improvement.

Equilibrium and Creative Biomechanics

Fig 8-28 The deactivation force diagram of forces on the teeth after placing a couple at the canine. This is also unsatisfactory because the canine may translate or even flare.

Fig 8-29 The deactivation force diagram of forces on the teeth after an extrusive force is placed at the canine. This is desirable for the canine, but the favorable moment is lost on the molar, and it will tip forward.

Another possibility is to remove the vertical forces and to place a counterclockwise couple (crown forward and root backward) on the canine. But alas, this is also unsatisfactory because the canine may translate or even flare—a movement opposite to what we want (Fig 8-28). Let us consider placing an extrusive force on the canine. This is better for the canine, but our deactivation force diagram tells us that the favorable moment is lost on the molar, and it will tip forward (Fig 8-29). In short, because we cannot change the laws of physics, there is no good solution if we want to use the vertical loop appliance. It is better to find this out by an equilibrium analysis than by any unexpected side effect observed after the appliance is inserted into the patient’s mouth. In fact, any configuration of a single-wire appli­ ance will not have the proper solution for this problem. A straight-wire arch with a coil spring or a chain elastic with a tip-back bend to preserve posterior anchorage will also intrude the canine or at least prevent its eruption; even a straight leveling wire will erupt the canine but significantly intrude the incisors, leading to an anterior open bite and tipping the molar forward as a major loss of anchorage occurs (Fig 8-30). A better solution to obtaining desirable and consistent forces is the use of dual wires for this highly displaced canine case. The molar and the incisors are connected by a rigid bypass wire to stabilize the arch (Fig 8-31a). A separate elastic or spring is attached to the canine, originating from the buccal hook of the molar. The spring is in equilibrium (Fig

Fig 8-30  A straight leveling wire will erupt the canine but significantly intrude the incisors, leading to an anterior open bite and tipping the molar forward as a major loss of anchorage occurs. A single-wire appliance has inherent limitations to obtaining a desirable force system in this malocclusion.

8-31b). The 200-g spring has a downward and backward pull on the canine (Fig 8-31c). It is far enough from the canine CR that the canine crown will move distally, and the root will come forward. As an independent force member, the spring force magnitude can be varied. If desired, the spring can be attached to a hook on the lingual of the canine to minimize canine rotation. Unlike the single-wire solution, the retraction force has an extrusive component of force (Fig 8-31d). The rigid bypass wire can be replaced with fiber-reinforced composite for better stabilization. The mesial force from the spring attached at the molar tube acts on the entire bypass arch (Fig 8-31e). Because the bypass arch is rigid, any anchorage loss would be seen primarily as a flattening of the occlusal plane. With a light force, this might be adequate to maintain posterior anchorage. The flattening of the entire occlusal plane is helpful if the incisors require significant intrusion, in which case an intrusion arch can be placed instead of a bypass archwire (Fig 8-32a). A 100-g intrusion force acting over 20 mm can produce a 2,000-gmm moment on the molar (Fig 8-32b). That moment could be adequate with the 200-g mesial force (Fig 8-32c) to translate the molar forward and thus serve as excellent anchorage. Note how the dual wires allow good control over the force system, force magnitudes, force direction, and points of force application, which is not always possible with a single wire. At the same time, there is the beneficial effect of reducing deep bite if indicated.

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Fig 8-31  Dual-wire system for a superiorly displaced canine. (a) The molar and the incisors are connected by a rigid bypass wire to stabilize the large anchorage unit. (b) A separate elastic or spring is attached to the canine and a hook at the molar. The spring is in equilibrium by activation forces (blue arrows). (c) The reversed deactivation force system on the canine (red arrows) shows that the 200-g force is far enough from the canine CR that the canine will easily tip. (d) The replacement components of the resultant force (yellow arrows) on the canine shows that the CR will move occlusally and distally, which is desirable. (e) The replaced equivalent force system (yellow arrows) at the CR of the total anchorage segment by the entire bypass arch. With a rigid bypass arch and light force, the posterior anchorage should be maintained.

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Fig 8-32 (a) An intrusion arch can be placed for needed intrusion of incisors. (b) Deactivation force system. A 100-g intrusion force acting over 20 mm can produce a 2,000-gmm moment on the molar. (c) Combined force system for the spring and intrusion arch. A 2,000-gmm moment could be adequate with the 200-g mesial force to translate the posterior segment and serve as excellent anchorage.

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Equilibrium and Creative Biomechanics

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Fig 8-33  (a) A patient with a unilateral constriction on the right side. What is the best way to deliver greater force on the right side with a lingual arch in the mandibular arch? (b) The lingual arch with a step-out bend at the right side. This appliance is not feasible because the lingual arch is not in equilibrium. (c) The lingual arch stiffness is reduced on the left side, but this appliance is still not feasible. (d) No matter how cleverly an appliance is fabricated, the forces must be equal and opposite. We cannot change the laws of physics. (e) The deactivation force system acting on the teeth (red arrows) shows that buccal crown torque is placed on the right molar. The side effects (vertical movements) are small due to the large distance between the vertical forces at each molar. (f) The activation force diagram in equilibrium, with forces reversed in blue. (g) The mandibular right molar primarily tips to the buccal. Some bilateral buccal expansion is needed.

Important note to remember In applying equilibrium, we have gone back and forth between the activation force diagram and the deactivation force diagram of forces acting on the appliance and the teeth. The deactivation force diagram can certainly be used to plan orthodontic strategies as long as it is understood that, conceptually, it is the appliance—not the teeth—that is in equilibrium.

Common misconceptions A patient has a unilateral constriction on the right side. The treatment goal is to expand the right side of the mandibular arch more than the left side (Fig 8-33a). What is the best way to deliver greater force on the right side with a lingual arch in the mandibular arch? Do we step the arch unilaterally as in Fig 8-33b? Or do we bilaterally expand and yet reduce

g

the stiffness on the left side unilaterally with a loop to reduce the force (Fig 8-33c)? This is actually a trick question. Equilibrium requires that the ∑F = 0; therefore, no matter how cleverly an appliance is fabricated, the forces must be equal and opposite (Fig 8-33d). Is it possible to do differential tooth movement using a lingual arch? Yes, because instead of horizontal forces, differential moments are applied on each tooth. The red arrows in Fig 8-33e show the forces acting on the teeth. Buccal crown torque is placed on the right molar. The side effects of vertical forces are small due to the large distance between the brackets at each molar. Figure 8-33f (blue arrows) demonstrates the valid equilibrium diagram, which has a moment on the right molar along with vertical forces on the arch at each molar. The mandibular right molar primarily tips to the buccal (Fig 8-33g). Asymmetric applications of lingual arches will be discussed in more detail in chapter 13. 151

8 Equilibrium Fig 8-34  An example of a confusing free-body diagram. Diagrams should be clearly labeled as to whether the shown forces act on the appliance or the teeth. Here some forces act on the teeth and some are reacting from the archwire insertion. Forces from the headgear and elastics can have different effects if the archwire is elastically deformed or acts as a rigid body. The shown elastic acts on the archwire but is not a reactive force from archwire insertion.

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Recommended Reading

Some therapy attempts the impossible. Equilibrium theory lets us know what is scientifically possible. Free-body diagrams published in journals or presented at meetings are helpful in presenting force systems of new ideas. The object in question—the appliance, not the teeth (unless they are in equilibrium with their periodontal support)—should always be in equilibrium. Diagrams should be clearly labeled as to whether the shown forces act on the appliance or the teeth. Free-body diagrams like Fig 8-34 should be avoided, where some forces act on the teeth and some act on the archwire without explanation. Moreover, the wire (its depicted force system) is not in equilibrium. This chapter has applied the equilibrium concept to appliance design. Other applications are equally useful and valid. The biomechanics of tooth movement or orthopedics can start with placing the teeth or bones into a state of equilibrium. Functional movements of the mandible and stresses in the temporomandibular joint are studied or modeled based on equilibrium. Not confined to orthodontic appliances, the equilibrium concept is universal and has broad applications.

Burstone CJ. The biomechanics of tooth movement. In: Kraus B, Riedel R (eds). Vistas in Orthodontics. Philadelphia: Lea and Febiger, 1962:197–213. Burstone CJ, Koenig HA. Force systems from an ideal arch. Am J Orthod 1974;65:270–289. Halliday D, Resnick R, Walker J. Fundamentals of Physics, ed 8. Hoboken, NJ: Wiley, 2008. Smith RJ, Burstone CJ. Mechanics of tooth movement. Am J Orthod 1984;85:294–307.

PROBLEMS The problems below use the concept of equilibrium. First make an equilibrium diagram in which all given forces are acting on whatever is in equilibrium—an arch or appliance, a tooth or a mandible. You must check the force direction carefully in the diagrams below. For example, if the given direction is for deactivation force, you must reverse the direction on your diagram. In short, first get the correct results from your activation equilibrium diagram; then reverse the force direction if forces on the teeth are required.

1. A 30-g force is applied anteriorly on each side (60 g at the center) from an intrusion arch. Find all forces and moments acting at the molar tube.

2. A 200-g force is applied from a lingual arch on the left molar. Find all forces and moments acting on the lingual tubes. (a) Assume that right and left couples are equal. (b) Assume that no couple exists on the right side. The direction of force and moments must be determined.

3. A 2,000-gmm couple is applied from a lingual arch to rotate the left molar clockwise. No other couples or horizontal forces are applied. Find all other forces acting on the molar tubes.

4. A vertical loop placed off-center for space closure produces unequal couples. Find all forces acting on the canine and premolar. Given forces and moments are on the teeth.

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5. A straight wire is used to level the canine and premolar brackets. Solve for unknown forces and moments. Ignore horizontal forces, and assume that couples on each bracket are equal. What are the undesirable side effects?

6. A lingual arch is used to tip the right molar to the buccal by an applied couple. Ignore horizontal forces; no couple is used on the left molar. Find all forces acting on both molars. What are the undesirable side effects?

7. A root spring acts to move the canine root to the distal. Give the forces and moments acting on the canine and molar bracket.

8. The mandible is in equilibrium as the subject clenches his teeth with 1,000 g of muscle force. Determine the force at the condyle. This is one of many simple temporomandibular joint models. Some believe the condyle is not a stress-bearing region.

9. Give all resultant forces and moments acting at the CR of the posterior segment from all appliances—headgear, loop arch, and maxillomandibular elastics. Ignore the vertical discrepancy of the tubes of molar attachment. Rectilinear components for the resultant are satisfactory.

10. The Herbst appliance, activated by the muscle force, is in equilibrium. Give all forces and moments acting at the CR of each molar.

PART

II The Biomechanics of Tooth Movement

CHAPTER

9 The Biomechanics of Altering Tooth Position “And yet…it moves.”

— Galileo Galilei

“Prediction is very difficult, especially if it’s about the future.”

— Niels Bohr

“It is tough to make predictions, especially about the future.”

— Yogi Berra

“Few things are harder to put up with than the annoyance of a good example.”

OVERVIEW



— Mark Twain

This chapter considers the correct force systems and optimal force magnitudes to produce different types of tooth movement. Before the required forces are established, an accurate method of describing tooth movement is necessary. Concepts such as an axis (center) of rotation or a “screw axis” are applicable. Primary tooth movement is produced by a couple or the line of action of force acting through the center of resistance (CR). Derived tooth movement is produced by a combination of primary movements where the line of force is away from the CR. In theory, all axes of rotation can be produced with a single force whose line of action lies on or off of the tooth. The clinician must estimate where that force is and may replace it with a moment and a force at the bracket (moment-to-force [M/F] ratio). Force systems change over time following tooth displacement; this consideration is part of an optimal force system. Stress and strain in the periodontal ligament and bone are more relevant than force alone in determining the force level to use clinically.

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Fig 9-1  A force acting through the CM of a free body causes linear acceleration by translation. If the force is moved away from the CM, the body will undergo a combination of linear and angular acceleration. Each CM point in a given plane projected at 90 degrees forms an axis, and all axes—no matter the plane—will intersect at a point.

Fig 9-2  The block is held in place by springs attached to supports. It moves but does not accelerate. The force acting on a point called the CR will cause the rectangular block to move by translation. Each CR point projected at 90 degrees forms an axis, but all red axes may not intersect at a point. Therefore, because of unknowns, the CR is usually depicted as a circle (in 2D) or a sphere (in 3D) rather than a point.

The core of orthodontic treatment involves tooth movement, bone displacement, growth, and overall remodeling by force systems. This chapter discusses the relationship between the force system and the change in tooth position. What is an optimal force system for tooth movement? To answer that question, a number of factors must be considered. What level of force magnitude should be delivered? Should the force be intermittent, constant, or pulsating? Will the force move the tooth to the target position correctly? Here the concern is the quantitative explanation of tooth-displacement patterns—how to move a tooth from position A to position B and what force system is required for this movement. How is the force system different if an incisor is to be tipped lingually, translated lingually, or the crown held in place as the root is moved lingually?

block from Fig 9-1 as a restrained body held in place by attached springs (Fig 9-2). We push on it with a force, but it does not accelerate. The rectangular block suspended by springs can be considered a structure undergoing deformation. Force acting on a point called the CR will cause the rectangular block to move by translation. The CR is analogous to the CM in that it is a point at which translation occurs, but they have entirely different locations. For one thing, a CM is a point that can be found in three-dimensional (3D) space. Each point projected at 90 degrees forms an axis, and all axes—no matter which plane they are in—will intersect at a point. This is not necessarily true for a restrained body (note in Fig 9-2 that the red axes do not intersect at a point). The location of the CR of a tooth or a group of teeth is influenced by many factors. The direction of force may alter the location of the CR because the tooth is not morphologically symmetric or the PDL has asymmetric properties (anisotropy). Biologic changes during tooth movement over time can alter root length and periodontal support along with PDL properties. Therefore, the CR (or, in 3D, an axis of resistance) may change somewhat during treatment. The tooth with a single-point CR in three dimensions is a special case with an ideal isotropic PDL. Fortunately, great asymmetry and variations are not the rule, so practically the concept of CR is both useful and valid. It is still better to think of the CR of a tooth not as a point but rather as a circle (a sphere in three dimensions). Although chapter 10 discusses

Free and Restrained Bodies Imagine a square body floating in space. Force acting through its center of mass (CM) will produce linear acceleration (Fig 9-1). If the force is moved away from the CM, the body will undergo a combination of linear and angular acceleration. A pure moment or couple produces only angular acceleration around the CM. The tooth is not a free body. It is a body restrained by the periodontal ligament (PDL) and other periodontal and bony structures. Now imagine the 158

Methods to Describe Change of Tooth Position

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Fig 9-3  A simplified tooth model with a spring support. (a) Passive state. (b) A force is applied at the CR that causes the tooth to translate. The tooth moves, yet it is still in equilibrium as one spring extends and the other compresses. (c) A realistic tooth model. The applied force is balanced by the sum of all the compressive, tensile (small red arrows), and shear stresses in the PDL.

these axes of resistance and rotation in three dimensions in greater detail, this chapter introduces and develops these concepts in two dimensions. In chapter 8, the concept of equilibrium was applied to the archwire and the orthodontic appliance, but the tooth is also in a state of equilibrium. In that chapter, a two-dimensional (2D) model of a tooth with attached elastics demonstrated the relationship between forces and tooth displacement. Here, for further simplification, a model of a canine with springs attached at the root is shown (Fig 9-3a). A force is applied at the CR of the tooth that causes the tooth to translate (Fig 9-3b). The tooth moves, and yet it is still in equilibrium as one spring extends and the other compresses. The equilibrium diagram (Fig 9-3c) shows the tooth in equilibrium—the applied force is balanced by the sum of all the compressive and tensile stresses (small red arrows) in the PDL. One should not confuse the appliance equilibrium diagram with the tooth equilibrium diagram; they are two distinct systems.

Methods to Describe Change of Tooth Position Before we can accurately relate forces to tooth movement, it is necessary to have an accurate method to describe a change in tooth position. For example, to say that the central incisor should tip lingually is too vague. An incisor could tip around an axis at its apex or at the center of the root, and each would require an entirely different force system. Other qualitative descriptions such as “flaring,” “aligning,” and “opening the bite” are too imprecise to serve as a basis for establishing a proper force

system. There are many possible and valid methods to describe tooth placement. Let us consider some of the advantages of each method.

Method 1 The vertical incisor in Fig 9-4a is to be moved lingually and intruded, with its axial inclination corrected. So far the description is qualitative and too vague. Method 1 is to describe translation and rotation around the CR of the incisor. The purple circle is the CR, and a dotted line is connected to the CR before and after the planned movement (Fig 9-4b). This is its translatory path, and its x and y coordinates can therefore be plotted. The incisor not only translates but also must rotate around its CR (Fig 9-4c) to reach it final desired position. The total rotation angle and the translation magnitude and direction define the tooth movement (Figs 9-4d and 9-4e). The rotation occurs around an axis at 90 degrees to the plane in which the force acts. There are advantages and disadvantages to this method. The advantage is the direct relationship between applied forces and the CR (a force acting at the CR translates a tooth, and a couple rotates a tooth around the CR). The disadvantage is that the position of the CR is calculated and is not a real anatomical point. Particularly with asymmetric teeth and in three dimensions, an actual point may not exist. However, any estimated CR point can be copied accurately from an initial position to a final position tracing of a tooth and reliably gives the direction of the desired force. The line of action of an applied force is not necessarily identical to the CR path. Any small deviation that is not clinically relevant will be ignored.

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Method 2 The most common method of describing tooth movement is based on bracket position change (method 2). In a sense, a local coordinate system is built into the bracket. If the bracket is correctly bonded to the tooth, in addition to its occlusogingival level, three rotational axes are established (Fig 9-5a). E. H. Angle classified these axes as first, second, and third order. Currently, clinicians speak of “torque,” “rotation,” and “tip.” To avoid confusion, this chapter describes rotation around an x-axis (Fig 9-5b), a y-axis (Fig 9-5c), and a z-axis (Fig 9-5d). Terms like torque used to describe tooth movement are wrong. The term torque will be used in this chapter only to describe the force system consisting of a couple or a pure moment. Inclination angles or change of axial inclination should not be called “torque.” Z-axis rotation around the bracket changes the mesiodistal axial inclination, x-axis rotation changes the labiolingual (faciolingual) inclination, and y-axis rotation rotates the tooth (see Fig 9-5). Various slot angles known as prescriptions are built into the bracket. Imagine a bag filled with many loose brackets; one could describe it as a bag of individual coordinate systems. In

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c Fig 9-4 Description of tooth movement (method 1). (a) Translation of the CR and rotation around the CR. The vertical incisor is to be moved lingually and intruded, with its axial inclination corrected. (b) The purple circles indicate the CR, while the dotted line is the CR path before and after the planned movement. (c) The incisor incrementally rotates around its CR (dotted arrow). (d) Final incisor position after rotation around the CR. (e) The total rotation angle (ϴ), the translation magnitude (D), and the force directions (dotted arrows) define the tooth movement.

three dimensions, three translations and three rotations of any bracket (tooth) are possible. The potential to move in a given direction is quantified as the degrees of freedom, and a tooth in space has six degrees of freedom for full control. An edgewise appliance using round wire may have only five degrees of freedom. Tooth displacement at the bracket can be described in two ways. The change of the coordinate system from position 1 to position 2 can be based on the coordinate system of the initial tooth (bracket) position, or a global coordinate system away from the individual tooth under consideration can be used. Let us consider method 2, where the only coordinate system is on the bracket-tooth. In Fig 9-6, an incisor is moved forward and intruded; displacement is exaggerated for demonstration purposes. Because only translation occurs, the sequence is not important for describing tooth movement. Three paths are possible (red, blue, and green arrows), and all have the same end point (Fig 9-6a). The clinician may look at the bracket path from start to finish to determine the direction (line of action) of the force. This is incorrect except in the special case of tooth translation in which the CR and bracket move

Methods to Describe Change of Tooth Position

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Fig 9-5  Description of tooth movement using change in bracket position (method 2). (a) A local individual coordinate system for each tooth is built into the bracket with three rotational axes. (b) X-axis rotation: third-order rotation, or torque. (c) Y-axis rotation: first-order rotation, or rotation. (d) Z-axis rotation: second-order rotation, or tipping.

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Fig 9-6  (a and b) An incisor is moved forward and intruded. Because only translation occurs, the sequence of movement is not important. Three paths are possible (red, blue, and green arrows), but all have the same end point. The clinician typically looks at the bracket to determine the desired tooth movement and to evaluate the result (a). The motion path of the bracket and the line of action of the force that is required for that movement are identical only if the tooth translates. The paths of the CR, the bracket, and the force direction are parallel (b).

parallel to each other. In Fig 9-6b, a line connecting the CR of position 1 and position 2 is parallel to the bracket path. Note that the three paths in Fig 9-6a are correct. The clinician, however, must select the most advantageous path to get there. This method of describing tooth movement is simple only for tooth translation; however, it becomes more complicated if we must also rotate the bracket around any axis or multiple axes, because the sequence of the rotations leads to different end points for tooth position. The use of angles to describe movement is nothing new. The famous physicist Euler described in detail the proper manipulation of descriptive angles— Euler angles. The interesting thing about using angles for describing motion is that they are not cumulative (ie, they cannot be added irrespective of sequence). For example, a bank account is cumulative.

$100 + $50 – $100 = $50 is the same as $50 – $100 + $100 = $50 The total balance will be the same regardless of the sequence of deposits and withdrawals. It is the same for the translated bracket in Fig 9-6; the sequence makes no difference. The incisor can first be moved to the labial and then intruded, or it can first be intruded and then moved to the labial; either way, it will still end up in the same position. But this is not true for angles—angles are noncumulative. Consider the lingually tipped incisor in Fig 9-7. The coordinate system is drawn at the bracket. Let us try two different paths. In Fig 9-7a, the incisor will first be translated. After translation, the tooth is shown 161

9 The Biomechanics of Altering Tooth Position

a

b

Fig 9-7 Rotation and translation using the individual coordinate system at each bracket. (a) The incisor will first be translated (white tooth) and then rotated to its final position (green tooth). (b) The sequence is changed: The incisor bracket is first rotated (white tooth) and then translated (green tooth). The Euler angles used for describing motion are not cumulative. The end result depends on the sequence.

a

b

in white. Now the tooth is rotated around the x-axis at the bracket, and its final position is shown in green. In Fig 9-7b, the sequence is changed: First the incisor is rotated around the x-axis and then translated. Remember that the coordinate system is fixed on the tooth. Note that the green tooth in Fig 9-7b ends up in an entirely different position than in Fig 9-7a. This example is in 2D, and adding additional rotations on other planes will only compound the sequence problem. In short, any three given bracket angles are insufficient to describe tooth movement to an end point if the bracket itself is used as the only coordinate system. Sequence must be given. On the other hand, if an additional outside reference or global coordinate system is used, three axis angles measured to the common reference can be sufficient. A modification of method 2 uses a global coordinate system 162

a

b

Fig 9-8  Rotation and translation at each bracket measured to a global coordinate system. It makes no difference whether the incisor is translated first and then rotated (a) or rotated and then translated (b).

Fig 9-9  The end result of x-axis displacement and z-axis rotation depends on the sequence. (a) Displacement and then rotation. (b) Rotation and then displacement. The final position of the tooth (green tooth) is sequence dependent, and the movements are noncumulative.

based on the occlusal plane (Fig 9-8). Now it makes no difference if the incisor is translated first and then rotated around the x-axis (Fig 9-8a) or rotated around the x-axis first and then translated (Fig 9-8b). Figure 9-9 shows the proposed central incisor tooth movement in 2D: an x-axis displacement to the left and a z-axis rotation. Without a global reference coordinate, the end point is indeterminate because it depends on the sequence. It is important not only to have a meaningful bracket prescription and correct positioning of the bracket on the tooth but also to know the relationship between the bracket coordinate system and the global coordinate system. A straight wire or series of wires between the two brackets could eventually align the teeth. The sequence of alignment is determined by the biomechanics of the wire-bracket system and the biologic response and is not dictated by any arbitrary se-

Methods to Describe Change of Tooth Position Fig 9-10 (a) When a couple is applied at the bracket, the tooth rotates around the CR. (b) To rotate around the bracket, a lingual force and a couple must be applied. (c) Some clinicians may believe that a twisted wire at the bracket produces a force system that rotates the tooth around the bracket if a couple is applied, but that is incorrect. (d) Incorrect bracket path for a couple.

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Fig 9-11  A single force on the crown will tip an incisor to the labial with a center of rotation near the CR. (a) Correct force system. (b) Observing only the change in bracket position (bracket path) does not directly give the correct force system. Note the wrong direction of an intrusive force with an unnecessary moment.

quence of translations and rotations. Furthermore, the final occlusal plane cant may not be parallel to any original global occlusal plane reference. The bracket position in method 2 has the advantage of supplying a definitive landmark during treatment, provided that it is not changed. This method is easy for the clinician to understand and use because it relates well to the appliance (eg, “I must step the tooth up; therefore, I must bend the wire with an upward step”). The global reference plane should also be kept constant. This may be difficult if most teeth are misaligned. Unfortunately, the relationship between the change in tooth position and the force system to produce such a change according to bracket position is complicated. A good example is a twist placed in an archwire, where the force system in an ideal situation could be a couple (Fig 9-10a). When a couple is applied at the bracket, some clinicians believe that the tooth will tend to rotate around the bracket. The rotation of the tooth around the bracket also requires a force; hence, pos-

Fig 9-12 An incisor is moved lingually and intruded, and its axial inclination is changed. A force along the path of the CR (green arrow) and rotation around the CR give good information about the tooth movement and the force system. A line connecting the brackets before and after movement (red arrow) is not the line of action of the force.

sible anchorage loss and an unexpected final tooth position may be expected (Fig 9-10b). This fallacy is often based on looking at the bracket position before and after and assuming that any translation means a force in that direction, while any rotation implies a couple (Figs 9-10c and 9-10d). The force system in Fig 9-11a is correct; a single force on the crown will tip an incisor to the labial with a center of rotation (CRot) near the CR. Observing only the change in bracket position (Fig 9-11b) gives the wrong conclusion: an intrusive force with the wrong direction and an unnecessary moment. In Fig 9-12, an incisor is moved lingually and intruded, and its axial inclination is changed. Following the path (green arrow) of the CR and rotation around the CR gives good information about the tooth movement and relates directly to the force system. Note that a line connecting the brackets before and after (red arrow) is not the line of action of the force. The bracket path parallels the CR path only in the special case of translation. 163

9 The Biomechanics of Altering Tooth Position

a

b

a

c

d

b

Fig 9-14  The occlusogingival level of the bracket path does not change in a and b. It parallels the CR path only during pure translation (a).

Let us consider a bracket with the latest and best slot prescription so that if the teeth were aligned on a straight wire with a correct occlusal plane cant, a perfectly treated occlusion would be observed. Here is the problem: If we start with a malocclusion, which rotations should we do first (x-, y-, or z-axis rotation)? Because the angles of the axes are sequence-sensitive or a good reference plane could be lacking, the best sequence is not obvious. With identical bracket angle prescriptions, the final result may vary depending on the sequence of tooth movement. There are an infinite number of possibilities; it is not determined by a prescription of angles alone. The orthodontist should decide on the best path for a tooth to move from its initial position to its final position at the end of treatment. Perhaps one stage of movement is the best, not a series of independent rotations. The bracket and a straight wire based on the forces delivered by an appliance may eventually align teeth, but the sequence may not be the best for the particular situation, and extraneous tooth movement may occur in the interim. 164

Fig 9-13 The goal is to tip the maxillary incisor lingually and to maintain the same occlusogingival level of the bracket. (a and d) Tipping the incisor lingually by a single force displaces the root forward using round wire (clockwise rotation from a to b). (c) In the next stage, the CR must be intruded and moved distally with rotation around the CR to correct the faciolingual axial inclinations (counterclockwise rotation). (d) The most direct movement with clockwise rotation. Note that changes in the bracket path do not follow changes in the location of the CR.

Fig 9-15 A mandibular molar tipped to the lingual. A single force from a crisscross elastic could directly bring the molar into good alignment in one stage of movement.

In another example, the goal is to tip the maxillary incisor lingually and to maintain the same occlusogingival level of the bracket. One possibility (Fig 9-13) is to tip the incisor lingually by a single force, displacing the root forward using round wire (Figs 9-13a and 9-13b). No intrusive force is applied, so the position of the CR does not change much. In the next stage (Fig 9-13c), the CR must be intruded and moved distally to correct the faciolingual axial inclinations. The most direct movement is depicted in Fig 9-13d. Note that the path of the bracket deviates widely from the CR path. Figure 9-14 shows another possibility: lingual translation of the bracket followed by x-axis rotation around the bracket. In both Figs 9-13 and 9-14, following the path of the CR and rotation around the CR is more useful than planning mechanics only from the change in bracket position. Building in bracket angulations in three dimensions is certainly helpful during finishing, when it is simpler to have a straight wire hold and finely tune the occlusion, but it is fallacious to think that it can correctly dictate the sequence of

Methods to Describe Change of Tooth Position

a

b

c

Fig 9-16  (a) If an archwire is used, a round wire is adequate because no moments are needed. (b) Torque is not indicated if rotation around the CR is desired because a buccal force is more efficient. (c) If a CRot at the apex is wanted, the direction of torque is reversed, accompanied with a single force at the bracket. The rotation direction of the tooth (dotted arrow) is the same with each force system.

Roll (x) axis Yaw (y) axis Pitch (z) axis

Fig 9-17 Pilots control aircraft considering 3D rotations: roll (x), yaw (y), and pitch (z). They cannot blindly use a series of rotation commands unrelated to an outside coordinate system. A series of harmonized maneuvers such as banking to the right by roll followed by a slight nose-up pitch is needed for a smooth turning to the right while maintaining altitude. The sequence of control is very important. Simply controlling the rudder by yaw only will crash the plane. Can we expect a straight wire to automatically give a harmonized maneuver?

tooth movement or provide the correct force system required during treatment. Once again, the orthodontist must do the thinking, not the appliance. Method 2 of describing tooth movement can therefore lead to an inefficient or even wrong path of tooth movement; moreover, moment and force direction are not directly related to bracket angle and translation changes. For example, consider a malocclusion in which a mandibular molar is tipped to the lingual. A single force from a crisscross elastic could directly bring the molar into good alignment in one stage of movement (Fig 9-15). If an archwire is used, a round wire is appropriate because no moments are needed (Fig 9-16a). Torque is not indicated if rotation around the CR is desired because a buccal force is more efficient (Fig 9-16b). If root apices are to be maintained, the reverse direction of torque accompanied with a single force at the bracket is needed (Fig 9-16c). Note that some types of tooth movement require torque, while others do not. It is very easy to place improper torque after

simply looking at bracket angles without understanding the forces and moments involved. Orthodontists are not the only professionals that use Euler angles with strange terminology (tip, torque, and rotation). Pilots control planes considering 3D rotations (Fig 9-17): roll (x), yaw (y), and pitch (z). They cannot blindly use a series of rotation commands unrelated to an outside coordinate system. A series of harmonized maneuvers such as banking to the right by roll followed by a slight nose-up pitch is needed for a smooth turning to the right while maintaining the altitude. The sequence of control is very important. Simply controlling the rudder by yaw only will crash the plane.

Method 3 Method 3 of describing tooth displacement is the establishment of a center of rotation (CRot) in two dimensions or an axis of rotation in three dimensions. The procedure is as follows: 165

9 The Biomechanics of Altering Tooth Position

a

Fig 9-18  Description of tooth movement using a CRot (method 3). (a) Identify an arbitrary landmark (green dot) on the tooth. (b) Connect the two points before and after tooth movement. (c) Bisect the line and drop a perpendicular line from it. (d) Place a second arbitrary landmark (red dot) on the tooth before and after movement. (e) Connect the two second landmarks (red dots) and drop a perpendicular line from its midpoint. (f) The CRot is the intersection of the two perpendicular lines, marked as a blue dot. (g) By definition, every point on the tooth or any extension from the tooth rotates around the CRot. (h) The beginning and terminal tooth positions can be correct while the intermediate tooth positions (gray tooth) can differ, and the movement of the tooth does not necessarily follow the path of the arc.

b

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1. Identify an arbitrary landmark on the tooth. It is at the apex (green dot) in this case (Fig 9-18a). 2. Place the landmark on the tooth before and after movement. 3. Connect the two points (Fig 9-18b). 4. Bisect the line and drop a perpendicular line from it (Fig 9-18c). 5. Place a second arbitrary landmark (incisal tip, red dot) on the tooth before and after movement (Fig 9-18d). Theoretically the first and second landmarks can be placed anywhere; however, it is recommended to place the first (green dots) and second (red dots) landmarks as far from each other as possible to enhance the accuracy of intersection. 166

6. Connect the two second landmarks (red dots) and drop a perpendicular line from its midpoint (Fig 9-18e). The CRot is the intersection of the two perpendicular lines marked as a blue dot (Fig 9-18f). By definition, every point on the tooth or any extension off the tooth (such as an attached bracket or lever) is rotating around this center of concentric circles (Fig 9-18g). Note that the sequence of all the intermediate tooth displacements follows an arc, while tooth move­ ment between any two positions is a straight line. In reality, the beginning and terminal tooth positions can be correct while the intermediate tooth positions can differ, and the movement of the tooth does not

Primary Tooth Movement Fig 9-19  Universal description of tooth movement in 3D. “Screw movement” along the axis of rotation. Note that it is a left-handed screw, which means that a counterclockwise rotation will advance the screw.

necessarily follow the path of the arc (Fig 9-18h). For this reason, strictly speaking, the CRot is called an instantaneous center of rotation. There are a number of limitations to the CRot concept. First, it is two-dimensional. To describe tooth movement in 3D, an axis of rotation is used. A perpendicular line to a 2D plane can form an axis of rotation. The tooth can rotate around the axis in 3D space. An axis of rotation does not have to lie inside the tooth; for example, an axis of rotation could be any place away from the tooth (x, y, and z), with any angle to a tooth’s usual coordinate system. This, of course, makes it difficult for the clinician to visualize tooth movements and relate them to an appliance. It may be easier for the orthodontist to use perpendicular projections that are more familiar and more intuitive to visualize. Although it may seem that an axis of rotation in 3D space could define most kinds of tooth movement, there are special situations that are not covered. Nägerl et al have suggested a more universal approach using the concept of a screw movement1 (Fig 9-19). The screw movement incorporates an axis of rotation with simultaneous translation along the axis. The screw movement along the axis of rotation can describe any movement in 3D space. The axis of rotation is also instantaneous, describing before and after positions only and not the path of the intermediary tooth movement. For simplicity and because 3D orthodontics with 3D biomechanics is in its infancy, this book uses 2D representation of CRot for most of its discussion. Chapter 10 considers 3D displacements and their biomechanics.

Force Systems and Tooth Movement The following sections discuss the relationship between force systems and the pattern of tooth displacement. These sections are based on research using theoretical analysis, numerical techniques like finite element analysis, experimental studies on humans and animals, and direct measurement using transducer, laser reflection, and holographic interferometry. Studies range from macroscopic cephalometric measurement to microscopic wavelength of light.

Primary Tooth Movement The two primary tooth movements are translation and rotation around the CR. By definition, in an ideal model, a force with a line of action acting through the CR (red arrow in Fig 9-20) produces translation (dotted arrow in Fig 9-20). Every part of the tooth moves parallel with a force vector. In many orthodontic texts, this movement is called bodily movement. A pure moment or a couple will rotate a tooth around its CR (Fig 9-21). For simplicity, it is assumed that the tooth displacement (dotted arrow in Fig 9-20) is parallel to the applied force (red arrow). This is probably a good estimate; however, recent research shows that this is not exactly true in special situations.2 167

9 The Biomechanics of Altering Tooth Position

1h 3

Fig 9-20  A force with a line of action acting through the CR (red arrow) produces translation (dotted arrow). Every part of the tooth moves parallel with a force vector. In many orthodontic texts, this movement is called bodily movement.

Fig 9-21  A pure moment (couple) will rotate a tooth around its CR.

Fig 9-22 Research shows that the CR of a single symmetric, parabola-shaped root (3D) is positioned approximately one-third of the distance from the alveolar crest to the apex, measured from the alveolar crest.3

Force = 50 g

a

Position of force, mm (distance from alveolar crest)

Rotation

b

Fig 9-23  (a) The centroid (CM) of a paraboloid of revolution is also located at a one-third distance, whose calculation is independent from most CR studies. (b) Shell theory, which may better model the PDL as a sum of several thin 2D shells (centroid at two-fifths the distance each, gray dots) covering the paraboloid of revolution, also gives one-third the distance (red dot in a).

Research shows that the CR of a single symmetric, parabola-shaped root (3D) is positioned approximately one-third of the distance from the alveolar crest to the apex, measured from the alveolar crest (Fig 9-22). It is interesting to note that the centroid (CM) of a paraboloid of revolution is also located at a one-third distance, whose calculation is independent from most CR studies (Fig 9-23a). Shell theory (Fig 9-23b), which may better model the PDL as a sum of several thin 2D shells (centroid at two-fifths the distance each, gray dots) covering the paraboloid of revolution, also gives one-third the distance (red dot in Fig 9-23a). It has been estimated that the CR of a molar is near its trifurcation or bifurcation4 (Fig 9-24). 168

Fig 9-24  The CR on a mandibular molar is near its bifurcation. (Reprinted from Burstone et al4 with permission.)

The location of the CR is relatively independent of the applied force magnitude if the PDL strain is small (Fig 9-25). However, if the magnitude of the force is large enough, not only the PDL but also the alveolar bone or even the tooth itself will undergo nonlinear deformation; hence, the location of the CR can change with force magnitude. This chapter considers the CR and CRot independent of force magnitude and only considers the tooth displacement in the PDL space, not in the surrounding bone. This simplifies the understanding and presentation of the biomechanics of tooth movement. Future research is needed to better define the relationship between force or couple magnitude and the CRot. Note that in a study in which the couple magnitude

Primary Tooth Movement

Fig 9-25  The location of the CR is relatively independent of the applied force magnitude if the PDL strain is small.

Fig 9-27  Clinical implications of the CR in three situations. (a) A typical given M/F ratio of 10 mm at the bracket will be equivalent to a force acting through the CR (D1). (b) With significant root resorption, a lower M/F ratio is required (D2). (c) The adult patient with significant alveolar bone loss requires the highest M/F ratio to translate the tooth lingually (D3).

was increased, the CRot moved horizontally; ie, the tooth extruded more as the couple increased (Fig 9-26). This study did not differentiate tooth movement produced by PDL strain from that produced by alveolar bone deformation.2 More research is still needed to locate the CR of all teeth in all planes, of groups of teeth (segments), and of full arches. This knowledge is needed even without considering the complications of a CR in 3D space. It should also be recognized that significant variation exists among patients with differing tooth morphologies, periodontium, and periodontal changes during treatment. Consider the clinical implications of the three central incisors shown in Fig 9-27. In Fig 9-27a, an arbitrary but typical M/F ratio of 10 mm at the bracket will be equivalent to a force acting through the CR. In Fig 9-27b, the amount of root resorption means that a lower M/F ratio is required. The adult patient in Fig 9-27c, with much alveolar bone loss, requires the highest M/F ratio to translate the tooth lingually. Mesiodistal sliding mechanics would create the

Fig 9-26 When the couple magnitude is increased, the CRot moves horizontally; ie, the tooth is extruded more as the couple is increased.

a

b

c

highest friction in part c. As mentioned previously, a CR should not be considered a point but rather a circle in 2D because of variations. Even with these limitations, a CR is a useful concept for practical clinical application. A pure moment or a couple rotates a tooth around its CR. A couple applied at the various positions of the incisor bracket in Fig 9-28 produces rotation around the CR and not—as some orthodontists may believe—around the bracket. If couples of the same magnitude are applied at different points on the tooth or even on extensions away from the tooth, the action is the same. Rotation around the bracket requires a couple and a force. Couples are free vectors, and unlike forces, their point of force application does not change how a tooth will move. Note also in Fig 9-29 that the directions of the two equal and opposite forces comprising a couple make no difference as long as the magnitudes of the couple moments are the same. The visualization of a couple as a free vector rotating a tooth around its CR may be difficult, so instead 169

9 The Biomechanics of Altering Tooth Position

Fig 9-28  A couple applied at various bracket positions on an incisor; however, all the actions are the same.

a

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d

let us calculate the effect of identical couples at a tooth. In Fig 9-30a, two equal and opposite forces of 100 g are applied to a canine. The moment is equal to one force times the perpendicular distance to the other force (Fig 9-30b). 100 g × 10 mm = +1,000 gmm Now let us calculate the moment in respect to the CR.

170

Fig 9-29  The directions of the two equal and opposite forces that comprise a couple make no difference in the force system. All incisors will rotate around the CR.

Fig 9-30 (a to d) The visualization of a couple as a free vector rotating a tooth around its CR. See text for explanation.

Adding the two moments gives +1,000 gmm, which is the same answer we got when we multiplied the force times the perpendicular distance to the other force. Now let us move the couple occlusal to the crown (Fig 9-31a). The moment at the CR (Figs 9-31b and 9-31c) of the upper force (left red arrow) is 100 g × 22 mm = +2,200 gmm [clockwise]

100 g × 5 mm = +500 gmm (Fig 9-30c)

The moment of the lower force (right red arrow) at the CR (Figs 9-31d and 9-31e) is

100 g × 5 mm = +500 gmm (Fig 9-30d)

100 g × 12 mm = –1,200 gmm [counterclockwise]

Derived Tooth Movement

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Fig 9-31  (a to f) The couple is moved occlusal to the crown; however, the CR feels the same moment from the occlusally placed couple. See text for explanation.

Adding the two moments from the occlusal and apical forces still gives the same answer: +1,000 gmm (Fig 9-31f). The CR feels the same moment from the occlusally placed couple as from the more apically placed couple because couples are free vectors. A pure moment or a couple delivers no force to a tooth. The sum of all forces is zero. It may seem confusing to the clinician that a tooth will move without a resultant force. The special case of a couple produces rotation around the CR by a moment alone.

Derived Tooth Movement When the line of action of a force is away from the CR, the displacement is called derived tooth movement. Various lingual forces in Fig 9-32 have lines of action occlusal and apical to the CR, and each of them produces a different axis of rotation. The forces occlusal to the CR produce varying degrees of lingual tipping (clockwise rotation), and the forces apical to the CR produce lingual root movement (counterclockwise rotation). Let us select one of the forces at the level of the alveolar crest (Fig 9-33) that will tend to tip the incisor lingually (rotate it clock-

wise) around an axis at the apex. Because the force is not at 90 degrees to the long axis of the tooth, a small intrusive component is created that will be ignored for now. The lingual force can be replaced with an equivalent lingual force and a couple at the CR (green and purple arrows in Fig 9-33b). The tooth will translate from the force, and the couple will rotate the tooth around its CR. In this particular case, the balance of each primary displacement produces the CRot at the apex. In Fig 9-33b, the translatory tooth movement is shown in the green transparent tooth and the rotation around the CR in the purple transparent tooth. The starting point in determining what force system is needed for an orthodontic appliance is to locate the single force that will produce the desired CRot. Any desired CRot can be accomplished with a single force either on or away from the tooth; the challenge is to find the line of action of force application. The exception, of course, is rotation around the CR where a couple is required. To help in determining force position, a “stick” diagram is most useful. Let us use the example of lingual crown tipping around a CRot at the apex (blue dot in Fig 9-34). The tooth is represented as a blue stick with a purple circle as the CR (see Fig 9-34). A force (red arrow) is ap171

9 The Biomechanics of Altering Tooth Position

a Fig 9-32  Various lingual forces have lines of action occlusal and apical to the CR, and each of them produces a derived tooth movement.

b

Fig 9-33  (a) A force at the level of the alveolar crest will tend to tip the incisor lingually (rotate it clockwise) around an axis at the apex. (b) The lingual force can be replaced with an equivalent force and a couple at the CR (green and purple arrows). The tooth will translate (green tooth) from the force, and the couple will rotate the tooth around its CR (purple tooth). The balance of each primary displacement produces the CRot at the apex. Fig 9-34 Stick diagram of controlled tipping. (a) A force (red arrow) is applied at the alveolar crest region that typically tips an incisor around the apex. The equivalent force is replaced at the CR with a force (green arrow) and a couple (purple arrow). (b) The stick (tooth) is translated to the lingual from the green force component. (c) The clockwise moment (purple arrow) rotates the stick around its CR so that the CRot is at the apex (blue dot).

a

a

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plied at the alveolar crest region that typically tips an incisor around the apex. The force is replaced at the CR with a force (green arrow) and a couple (purple arrow). In Fig 9-34b, the stick (tooth) is translated to the lingual from the green force component. Finally, after the translation (Fig 9-34c), the clockwise purple moment rotates the stick (tooth) around its CR so that the CRot is at the apex (blue dot). In this example, we knew the correct position of the red force to produce rotation around the apex, based on research5; so our analysis only serves to explain 172

d

Fig 9-35 Stick diagram of root movement. (a) Suppose we did not know where to position the force to rotate an incisor around its incisal edge. (b) It is readily seen that the CR moves along the line of action of applied force. Therefore, the direction of the force must be lingual (green arrow). (c) The stick must be rotated so that the CRot is at the incisal tip. A counterclockwise couple can achieve this (purple arrow). (d) Only a force apical to the CR (red arrow) can achieve this result; thus, the location of the force must be apical to the CR.

why the CRot could be at the apex. But suppose we did not know where to position the force to tip an incisor around the apex. Here the stick diagram is useful. If we draw the before and after tooth positions, it is readily seen that the CR moves along the line of action of applied force. This tells us that the direction of the force must be lingual (see Fig 9-34b). The remaining question is the location of the force. The stick must be rotated so that the CRot is at the apex. A clockwise couple can achieve this (see Fig 9-34c). On which side of the CR could a force pro-

Derived Tooth Movement

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Fig 9-36  The effect of changing force position in the simple geometry of horizontal forces at 90 degrees to the long axis of a tooth. The position of force will be moved apically from the crown sequentially. (a) The force acts at the level of a typical bracket, and the CRot (blue dot) is about a millimeter or so apical to the CR (purple circle). (b) The force is moved apically. The tooth tips, with the root still moving in the opposite direction as the crown, but less so than in part a. (c) The force is placed at the alveolar crest. The CRot is at the apex. No part of the tooth is moving in the opposite direction as the applied force. (d) The force is slightly occlusal to the CR. Both the crown and the root apex move in the same direction. The CRot moves off the tooth. (e) The force acting through the CR produces translation, where the CRot is located at infinity. (f) The force is placed slightly apical to the CR. The CRot comes back from infinity and is occlusal to the crown. (g) The force is placed more apically. Root movement with a CRot at the incisal edge could occur. (h) Moving the force further apically places the CRot near but occlusal to the CR, and the tooth movement begins to approach the movement of a couple.

duce a clockwise couple? Only a force occlusal to the CR can achieve this result; thus, the location of the force must be occlusal to the CR, and its exact location must be determined by additional calculation. Let us now consider another example (Fig 9-35) of an incisor that requires root movement (a CRot around the incisal edge, blue dot). Unlike Fig 9-34, all that is known at the start of our determination is the force direction. In Fig 9-35a, the CR moves lingually; therefore, a lingual green force at the CR is required (Fig 9-35b). What is the direction of the couple to rotate the stick (tooth) so that the CRot is at the incisal edge? The correct direction is counterclockwise (Fig 9-35c). So where does the single force go? It is positioned somewhere apically to the CR (red force in Fig 9-35d). Additional research tells us that it is approximately 2 to 4 mm apical to the CR. But what is the force system at the bracket, where an appliance delivers the force that produces the incisor root movement? That is the easy part: an equivalent force system to the single force on the

root. A more detailed discussion is given later in this chapter. Figure 9-36 describes the effect of changing force position in the simple geometry of horizontal forces at 90 degrees to the long axis of a tooth. The graphics are general in nature and do not necessarily reflect accurate positions. Within a moderate range, force magnitude does not seem to influence the CRot. The old idea that heavy forces tip teeth more than light forces is certainly incorrect. In Fig 9-36a, the force acts at the level of a typical bracket, and the CRot (blue dot) is about a millimeter or so apical to the CR (purple circle). The force is far enough from the CR that the effect from the moment overwhelms the effect of the force so that tooth movement is similar to that produced by a couple. Now let us sequentially move the force apically (Fig 9-36b). The tooth tips, with the root still moving in the opposite direction as the crown, but less so than in part a. If the force is placed further apically at the alveolar crest (Fig 9-36c), the CRot 173

9 The Biomechanics of Altering Tooth Position Fig 9-37 Stick diagrams can be used in the occlusal view. A molar is to be moved around a CRot near the mesial contact area. (a) A force (red arrow) placed offcenter to the distal will translate the stick buccally and rotate it around its CR in a counterclockwise direction. (b) The CRot could be at the mesial contact area, depending on how far distal the force is placed.

a

b

is at the apex. The CRot moves off the tooth in Fig 9-36d, where the force is slightly occlusal to the CR; both the crown and the root apex move in the same direction. Force acting through the CR produces translation, where the CRot is located at infinity (Fig 9-36e). Moving the force slightly apical to the CR (Fig 9-36f) places the CRot occlusal to the crown tip. The CRot comes back from infinity and will begin to approach the CR, starting at infinity occlusally as the force is moved further apically. In Fig 9-36g, root movement with a CRot at the incisal edge could occur. Moving the force further apically places the CRot near but occlusal to the CR, and the tooth movement begins to approach the movement produced by a couple (Fig 9-36h). Note in Figs 9-36a and 9-36h that when the force is far away from the CR, there is a part of the tooth that moves in the opposite direction to the applied force. In other words, the tooth movement becomes similar to pure rotation by a couple. The same “stick” diagrams as in Figs 9-34 and 9-35 can be used in the occlusal view. A molar is to be moved around a CRot near the mesial contact area (Fig 9-37). A force (red arrow) placed off-center to the distal (Fig 9-37a) will translate the stick (tooth) buccally and rotate it around its CR in a counterclockwise direction. The CRot (blue dot in Fig 9-37b) could be at the mesial contact area depending on how far distal the force is placed. The further distally the force is placed, the closer the CRot will be to the CR. Eight physical 2D models demonstrate the effect of force position and magnitude in Fig 9-38. The teeth are suspended by a series of elastics simulating the PDL. The blue line is drawn on the transparent tooth and the red line on the background so that they coincide at rest. As the tooth is displaced by a force (red arrows), the amount of displacement and rotation is seen by the gap and the angle between 174

the two lines. The location of intersection (blue circle) of the blue and red lines represents a CRot such as that in Figs 9-34 and 9-35. The first thing to notice is that the CR moves parallel to the applied force, no matter where it is applied. In some cases, as in Figs 9-38a and 9-38b, some part of the tooth may move in the opposite direction as the force, but the CR never does. As the force position moves downward from the bracket to the apex (Figs 9-38a to 9-38d), the intersection (blue circle) moves as explained by the curves in Fig 9-43, which are discussed later in this chapter. Note in Figs 9-38a and 9-38b how the force magnitude does not affect the location of intersection (CRot, blue circles) and that only the amount of rotation is increased as the force magnitude is increased. Also note the amount of tooth displacement of the teeth (white arrow) in Figs 9-38a and 9-38f. These two force systems show a similar amount of tooth displacement at the crown, yet the amount of force used is totally different. Based on the elongation of the chain elastic, very light force is used in Fig 9-38a and very heavy force is used in Fig 9-38f; however, the maximum amount of stress the PDL feels would be the same. In other words, very light force can induce very high stress in uncontrolled tipping (see Fig 9-38b). Even though the force magnitude is very low, repeated light faciolingual forces at the crown by the muscle of the cheek and the tongue induce physiologic mobility of uncontrolled tipping. As a result, the width of the PDL is most narrow near the middle of the root and widens coronally and apically from the middle. Empirically, clinicians have learned that the most efficient way (the least force used) of extracting a tooth is repeated uncontrolled tipping, which can induce very high stress on the PDL and bone. Also note that the location of the CRot changes abruptly near the CR. In summary, in derived tooth movement (1) the CR moves parallel to the applied force; (2) a single force with a correct line of action can produce any

Derived Tooth Movement Fig 9-38 Eight physical 2D models demonstrate the effect of force position and magnitude. The teeth are suspended by a series of elastics simulating the PDL. The blue line is drawn on the transparent tooth, and the red line is drawn on the background so that they co­ incide at rest. As the tooth is displaced by a force (red arrows), the amount of displacement and rotation is seen by the gap and the angle between the two lines. The location of the intersection of the blue and red lines (blue circle area) is like a CRot in the stick diagram. (a and b) Uncontrolled tipping. (c and d) Controlled tipping. (e and f) Translation. (g and h) Root movement. Comparing a and b, c and d, e and f, and g and h, the position is the same but the magnitude is greater at the right (b, d, f, and h). The force magnitude does not affect the location of the CRot. Note that the amounts of displacement of the bracket in a and f (white arrows) are similar even though the applied force magnitude is different.

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b

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required center or axis of rotation; (3) the CRot is independent of force magnitude; and (4) the CRot changes abruptly when the force is near the CR. Even the exception—rotation around the CR—can be obtained by a force applied away from the CR, and that distance does not have to be great. Figure 9-39 reminds us that there are infinite force positions in 3D and that they do not necessarily have a line of action through the tooth itself.

Fig 9-39 There are infinite force positions and corresponding axes of rotation in 3D.

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9 The Biomechanics of Altering Tooth Position Fig 9-40 (a) A force (red arrow) apical to the CR that could produce root movement with a CRot around the incisal edge. (b) The yellow force system is equivalent to the single red force.

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b

Force Systems at the Bracket An obvious question is: what is the force system that an appliance must place at the bracket to produce the required CRot? It is nice to know that a force placed 2 mm apical to the CR could produce root movement around the incisal edge, but it may not be possible to place the force so far apically. The use of wire extensions (lever arms) sometimes allows a force to be placed apically on the root, but there are limitations because of gingival impingement. The most common approach is to place the forces at a bracket. It is therefore necessary to replace the correct single force with an equivalent force system at the bracket. Figure 9-40a shows a force (red arrow) apical to the CR that could produce root movement with a CRot around the incisal edge. The formulas for equivalence discussed in chapter 3 can now be used to replace the 100 g on the root with an equivalent force system at the bracket (Fig 9-40b).

∑ F1 = ∑ F2 Therefore, 100 grams must be applied at the bracket.

∑ M1 = ∑ M2 A point on the bracket (red dot) is selected to sum the moments. A couple of –1,200 gmm must be placed at the bracket (curved yellow arrow). The yellow force system is equivalent to the single red force. Other force characteristics, such as the force-deflection (F/∆) rate and moment-deflection (M/ϴ) rate, may change during deactivation of the spring so that an equivalent M/F ratio may not be constantly maintained. If a ratio is made between the moment (couple) and the force, the force system at the bracket is defined. Here the ratio is 12:1. The CRot of a tooth is determined by the M/F ratio and is 176

mainly independent of force magnitude. If the M/F ratio is used, the point of force application must be given. The unit of measurement for the M/F ratio is millimeters, and the M/F ratio simply denotes how many millimeters away from the bracket a single force must be placed. One might think that if a lot of effort or work is required to insert a wire into the bracket, then the tooth will feel excessive stress in the PDL, which could be harmful and could lead to undesirable side effects like root resorption. However, this is not always true. One should not confuse the effort or work that is required to insert an appliance with the stress the PDL will feel. For example, consider a walrus with a very long canine (Fig 9-41). Arbitrarily, let us apply 100 g at the CR (Fig 9-41a). We will replace this force system at two different levels for comparison: (1) at bracket A, which is 10 mm away from the CR, and (2) at bracket B, which is 60 mm away from the CR at the tip of the tooth (Fig 9-41b). Because bracket A is 10 mm from the CR, a force of 100 g and a moment of –1,000 gmm is needed at the bracket (yellow arrows in Fig 9-41c). The distance from the CR to bracket B is 60 mm, so a moment of –6,000 gmm is needed at that bracket along with the 100 g force (yellow arrows in Fig 9-41d). What does the orthodontist feel when he or she inserts the wire into each bracket position? If a bracket existed at the CR, the 100 g would feel light and the wire would be easy to insert because no moment would be necessary. If the orthodontist inserts the wire at bracket A, more effort would be needed because of the 100-g force and torque of –1,000 gmm. Yet the tooth itself will feel the same as if the appliance delivered the 100 g through the CR. If the orthodontist inserts the wire at bracket B, he or she will need to be very strong to place it because of the –6,000gmm torque. However, even though it would require hard work to activate the appliance, the force system acting on the tooth at bracket B would have

Force Systems at the Bracket

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Fig 9-41  A walrus with a very long canine demonstrates how the effort required to place a wire does not affect the force system acting on the tooth. (a) A force of 100 g is applied at the CR. (b) Bracket A is placed 10 mm away from the CR, and bracket B is placed 60 mm away from the CR at the tip of the tooth. (c) The replaced equivalent force system at bracket A (yellow arrows). (d) The replaced equivalent force system at bracket B (yellow arrows). All force systems are equivalent, and hence the tooth feels the same stress.

Fig 9-42 LVDT with infinite resolution is used to detect and trace the micromovement of the maxillary canine and the silicone periodontium model under various loading conditions.

no different effect on the incisor than that 100 g applied through the CR. With this general concept in mind, let us investigate further about the location of the CR and CRot based on some experiments. Because the amount of displacement or rotation of the tooth within the PDL space is very small, sophisticated technology like laser hologram or a linear variable displacement transducer (LVDT) with infinite resolution are required to detect and trace the movement of the tooth under various loading conditions. An experiment in which an in vitro maxillary canine was placed in a silicone periodontium model (Fig 9-42) shows the normal sequential change in the CRot as

a horizontal force is moved vertically. When the M/F ratio at the bracket (position of a single force measured from the bracket) is plotted against the CRot, a typical hyperbolic curve is produced (Fig 9-43). The horizontal axis gives the force position (a) from the CR. The blue vertical axis gives the M/F ratio at the bracket; 0 mm is a horizontal force at the bracket, and positive values are apical and negative (–) values occlusal to the bracket. The black vertical axis gives the position (b) of the CRot in respect to the CR (inset of Fig 9-43). The absolute numbers of the CRot position are less important than the general shape of the graph, because there can be much variation in tooth morphology and PDL constitutive behavior. 177

9 The Biomechanics of Altering Tooth Position Fig 9-43  The CRots are plotted on the force position from the CR (a) versus the distance between the CRot and the CR (b). CRots pro­duce a typical hyperbolic curve. Each M/F ratio (arrows) corresponds with a CRot (its same-colored dot). The CRot abruptly changes from the incisal tip to the apex near the CR (gray area).

Let us start with a primary tooth movement—the application of a pure moment or a couple in Fig 9-43. At a large occlusal distance from the bracket (gray arrow, M/F ratio = –12 mm), the CRot (gray dot) approaches the CR. For all practical purposes, force so far from the CR acts like a couple because the effect from the moment overwhelms the effect of the force. Let us now move the force apically to the level of the bracket (blue arrow, M/F ratio = 0 mm). The CRot is still very close to the CR (blue dot), perhaps a few millimeters further apically. The clinician should be aware that either a couple or a force at the bracket will produce about the same effect. Next the force is moved apically to a level at approximately the alveolar crest (green arrow, M/F ratio = 8 mm). Here the CRot is at the apex (green dot). If the force level is moved to the CR, translation occurs, and the CRot diverges at infinity. When the force is moved slightly apical to the CR (red arrow, M/F ratio = 14 mm), root movement is observed with a CRot around the incisal edge. The M/F ratios are the moments and forces needed at the bracket to effect the different CRots. The numbers also denote how many millimeters the force position must be changed to have the desired effect. The shape of the curve shows us that great accuracy is required in a narrow range of points of force application (gray area between the green and red arrows) for CRots around commonly required points on a tooth from the in178

cisal edge to the apex of the root, where important CRots such as controlled tipping, translation, and root movement occur. On the other hand, tipping movements around the CR are easily obtained with many forces, provided that they are at a distance from the CR. The M/F ratios represent both the position of a single force (measured from the bracket) and the actual couple and force that must be applied at the bracket. During anterior retraction, most clinicians prefer to think of the force system that must be applied at the bracket. An increasing number of orthodontists are beginning to use extensions and require a sound biomechanical protocol for determining the position of the single force away from the bracket. Let us look in greater detail at the force system at the bracket based on the data and concepts described in Fig 9-43 from the clinical point of view. A round archwire with a force from an elastic or a spring delivers only a force (Fig 9-44a). The crown moves lingually and the root labially with a CRot slightly apical to the CR. This is referred to as simple tipping or uncontrolled tipping. To avoid movement of the apex in the opposite direction, perhaps to achieve more desirable axial inclinations, the CRot must be placed near the apex; this requires both a lingual force (straight red arrow) and a counterclockwise moment (curved red arrow in Fig 9-44b). The direction of the moment is commonly called lingual

Force Systems at the Bracket

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a

b

d

e

Fig 9-44  The force system at the bracket and CRot. (a) A single force at the bracket results in uncontrolled tipping. (b) Controlled tipping. (c) Translation. (d) Root movement. (e) The CRots described in parts a to d are given. Each CRot corresponds to a force of the same color.

root torque. If the incisor requires lingual translation and the force remains the same, the magnitude of the moment must be increased (Fig 9-44c). Finally, for root movement (rotation around an axis at the incisal edge) when the force is kept constant, the moment must be further increased (Fig 9-44d). Note that controlled tipping (rotation around the apex), lingual translation, and root movement all require moments or torques in the same direction. This may not be intuitive because in parts b and d the teeth are rotating in opposite directions. The positions of the single forces to produce the CRots described in parts a to d are given in Fig 9-44e. Each CRot and force corresponds to the same color. As the hyperbola-shaped tooth-movement graph demonstrated (see Fig 9-43), most of the important rotation centers are found in a narrow range of M/F ratios. This narrow range of force positions (gray area in Fig 9-45) is the “critical zone”; note that CRots from infinity to the incisal edge are possible with single

Fig 9-45 The important rotation centers from the apex to the incisal tip are found in a narrow range of M/F ratios. This narrow range of force positions is the “critical zone” (shaded area).

forces acting in this zone (see Fig 9-45). By contrast, rotating a tooth around the CR or close to the CR is not challenging because many force positions outside the gray zone can achieve that result. Nägerl et al developed a theory of proportionality (Fig 9-46) by which the distance from the applied force to the CR (a) multiplied by the distance from the CR to the CRot (b) gives a constant (σ2).1 The value of σ2 may vary in accordance with the direction of force; however, this formula is valid three-dimensionally: a × b = σ2 σ2 is a measure of the variance of the support of the tooth by the periodontium and is called the center of rotation constant of a tooth at a given direction of force. Schematically visualized PDL support in accordance with varying σ2 values is depicted in Fig 9-47. The higher the sigma (σ), the wider is the “critical zone” (Fig 9-47a). A high σ2 makes the pre179

9 The Biomechanics of Altering Tooth Position Fig 9-46 Nägerl et al developed a theory of proportionality by which the distance from the applied force to the CR (a) multiplied by the distance from the CR to the CRot (b) gives a constant (σ2) known as the center of rotation constant. σ2 is a measure of the variance of the support of the tooth by the periodontium.1

a

b Fig 9-47  (a) A high σ2 value provides a wider critical zone. (b) A low σ2 value provides a narrower critical zone. (c) When σ2 = 0, no matter where the force is placed (except at the CR), rotation around the CR will occur.

c

dictability of achieving any CRot much easier, and a low σ2 means a greater sensitivity for locating a force to acheive an exact CRot (Fig 9-47b). Theoretically, if σ2 = 0 (Fig 9-47c), no matter where the force is placed (except at the CR), rotation around the CR would occur (red arrows in Fig 9-47c). The translation is theoretically possible but extremely difficult to achieve practically (orange arrow in Fig 9-47c). Just a small amount of deviation from the CR 180

or a force placed anywhere except at the actual CR results in pure rotation. It is beyond the scope of this book to describe σ2 in detail; however, longer roots obviously have greater σ2 value. Also, wider roots and teeth splinted together as a unit have greater σ2 value. σ2 is determined by root morphology and periodontal behavior and not just by root length.

A Couple or Single Force at the Bracket for Rotation Near the CR

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Fig 9-48  A Class II, division 2 nonextraction case. (a) The maxillary central incisors are to be flared and the arch length increased. Either labial force at the bracket (b) or lingual root torque (c) can achieve that goal.

A Couple or Single Force at the Bracket for Rotation Near the CR Twist in an orthodontic wire produces a couple or torque. A couple anywhere or a single force at the level of a bracket produces about the same CRot; the tooth tips, with the crown moving in one direction and the root apex moving in the opposite direction. Some orthodontists historically were taught that torque in a wire or “torque” in the bracket will spin a tooth around the bracket. To spin a tooth around a bracket, however, both a moment (couple or torque) and a force are required. A force tips a tooth around an axis about 1 to 2 mm below its CR, and torque tips it around the CR. Clinically, it is impossible to tell the difference in how the tooth moves between the two approaches. Which is the most efficient when this type of tipping movement is indicated? Because the end result from two completely different force systems is similar, the choice depends on the feasibility of clinical application and the best possible equilibrium diagram. Let us consider a few examples in which a force is the better choice over torque placement. Chapter 13 shows some cases in which a couple is a better choice over a single force. Figure 9-48a shows a Class II, division 2 nonextraction case in which the maxillary central incisors are to be flared and the arch length increased. The CRot needed at the incisor is near the CR; the crown comes forward and the root moves back. Either labial force at the bracket (Fig 9-48b) or lingual root torque (Fig 9-48c) can achieve that goal. But which is more efficient? The better choice is the labial force, which allows simultaneous leveling with a round

flexible archwire and is very simple. A labial force can be introduced at the beginning of treatment. To deliver torque, on the other hand, a full-size edgewise wire is needed, and much tooth alignment must be accomplished before a rectangular archwire with a twist can be placed. Friction is more of a problem with a twisted archwire because the wire must slide anteriorly to allow for flaring. Moreover, inserting a twisted edgewise wire could cause the bracket bond to fail or the bracket to fracture if it is ceramic. The single force is also more favorable because the slightly more apical CRot allows less labial movement of the apex. It is common to finish treatment of a Class II patient before the second molars fully erupt. Figure 9-49 shows a frontal view of second molars. The mandibular molars are in a correct position, but the maxillary molars are erupting too far to the buccal. Spinning the molar around an axis near the CR will reduce the buccal overjet (horizontal overlap) and improve its axial inclination. One possibility is to bond tubes on the second molars and place lingual crown torque in a full-size edgewise wire (Fig 9-49a). Theoretically, such a force system is excellent, but this is not a good choice practically for many reasons. The localized tensile stress at the bonding surface is very high, so that the newly bonded molar tubes may not reach sufficient bond strength to withstand the torque. Practically it is difficult to place a full-size wire with torque through the first and second molar attachments. Some leveling may first be required. Also, it is difficult to deliver a constant couple because unpredictable forces are associated with deflection of an edgewise wire. As the molar tips lingually, the geometry of the wire to the bracket changes, and pure torque no longer pre­ sents itself. Overall indeterminacy increases. 181

9 The Biomechanics of Altering Tooth Position

a

b

Fig 9-49  The maxillary molars are erupting to the buccal side. Rotating the molar around an axis near the CR by a couple (a) or applying a single lingual force to the crown (b) will reduce the buccal overjet. A lingual force (b) is the best choice. Placing torque may require preliminary leveling. Pure continuous torque is difficult to practically achieve because as the molar moves lingually, the force system changes and is indeterminate.

Fig 9-50 In a simple anterior reverse articulation, a round wire appliance delivering a single force (red arrow) without any torque would flare the maxillary incisors while correcting their axial inclination. Note that the CR only moves horizontally.

A single lingual force applied to the crown is much simpler and is accomplished with a light round archwire or a finger spring (Fig 9-49b). The force system is much more predictable because no moment is associated, yet the end result will be similar. Many times torque is selected for the wrong reason; “torque” is often incorrectly used to mean axial inclination rather than a couple force system. “The molar has improper torque” does not make sense semantically. Sophisticated appliances like the edgewise arch were developed to simultaneously deliver forces and moments, and it is sometimes thought that more primitive appliances with less control and hence fewer degrees of freedom always produce compromised results. But this is not necessarily the case because not all treatment requires six degrees of freedom. For example, in a simple anterior reverse articulation (also sometimes referred to as crossbite), a round wire appliance (or even a tongue depressor as a lever) could flare the maxillary incisors while correcting their axial inclination (Fig 9-50). No “torque” is necessary. There are many examples of a single-force appliance (one degree of freedom) being inserted and 182

beneficial tooth movement occurring in all planes of space. This involves both translation and rotation around all axes (x, y, and z) through the CR or another relevant point. Even removing an archwire from the tube or bracket to eliminate unwanted moments from the wire can further enhance this simple “single force only” appliance therapy if indicated. This text also discusses the use of wire extensions (cantilever arms), by which single forces can be placed near or at the CR or away from the CR to produce a different CRot. This is a separate application of the concept of applying a simple force without a couple to the tooth’s crown; if properly positioned and feasible, a single force can produce any needed CRot.

When a Force and a Couple Are Required for Tipping It has already been discussed how a single force at the bracket can produce satisfactory tipping if the goal is to allow the root to move in the opposite

When a Force and a Couple Are Required for Tipping direction. In most malocclusions requiring significant retraction of incisors, it is preferable to prevent the root apex from being displaced forward. Figure 9-51a compares tipping an incisor lingually around the CR (green) with tipping it around the apex (yellow). The end point for both is a normal incisor axial inclination. Tipping around the apex, however, allows for more retraction and hence is indicated for extreme overjets and for extraction cases. Maintaining the position of the apex is called controlled tipping. No part of the tooth is displaced in the opposite direction to the applied force. As previously discussed, a moment and a force are required. For simplicity, the force direction has been ignored in this chapter up to now. The M/F ratios of the graph in Fig 9-43 refer to forces at 90 degrees to the long axis of the tooth. Figure 9-51b shows an incisor tipping around the apex (blue dot). Note that both a lingual force and a moment (lingual root torque) are required. The CR moves occlusally and lingually. The direction of the force is an average reflecting the displacement direction of the CR (dotted arrow). The CR must move in the same direction as the force (with some exceptions and considerations that have been discussed and are here disregarded). Many orthodontists describe the tooth movement path at the bracket, as discussed earlier in this chapter. Note in Fig 9-51c how the bracket moves downward and backward and rotates in a clockwise direction. Some orthodontists erroneously think that the necessary force system is found in this path (gray arrows). The correct force system to produce the desired CRot, however, is shown in red. Chapter 13 discusses the relationship between different bracket positions and the force system produced; this is an entirely different question, and even here we must relate our answers back to the CR of a tooth. The clinician cannot reason that any angle deviation means a corresponding couple and any x, y, z discrepancy gives the correct direction or amount of force. Looking at the change in bracket position in this way is letting the bracket do the thinking. Connecting the CR at position 1 with the desired position 2 gives the average direction of the force; the angle of rotation around the CR gives the direction and is related to the magnitude of the required moment. It is true that we may not accurately know the position of the CR as properly defined; however, we can accurately copy any CR point from the first to the next tooth tracing. Even if there is some error, an estimate of the CR is better than a bracket landmark, which is only indirectly influenced by a force system because it is far from the CR. The closer our estimate is to the CR, the more reliable is the interpretation of force

and moment direction, because we are looking at primary displacements. In short, the change in position of the bracket should be called the bracket path and is not directly related to the force system of that path (except in the rare case of pure translation). The path of the CR (translation and rotation) is directly related to the direction and magnitude of the moment and force needed for that CR path. In some patients, an increase in the vertical overlap (also known as overbite) is undesirable; for these patients, a more demanding approach is necessary (Fig 9-51d). The root is not displaced forward, and the CR must translate upward and backward. The direction of force is now upward and backward, and the moment is in the direction of lingual root torque. The correct force system is shown in red. If the bracket does the thinking instead of the orthodontist, not only is the torque placed in the wrong direction, but the force is also placed in the wrong direction (gray arrows). If a straight-wire appliance is used in this case, the force system is also incorrect. In Fig 9-52, a recent experimental study using LVDT shows that the displacement vector of the tooth (straight dotted arrow) does not necessarily parallel the applied force vector (red and blue arrows).2 The small red and blue circles are the CRots under varying horizontal forces (red and blue arrows), which are perpendicular to the anatomical long axis of the canine. The varying CRots have a linear asymptote (dotted black line) with an angle of 11 degrees to the long axis of the tooth. The CR of the tooth moves perpendicular to this line along the straight dotted arrow. The black dotted line formed by varying CRots is called a functional axis of the tooth under given applied forces. A little horizontal dispersion of the CRots near the CR is due to minor extrusive vertical displacement by a couple, as explained earlier in Fig 9-26. The angulation of the anatomical long axis to a given force direction, the morphology and curvature of the root, and the anisotropic property of the PDL may affect the angle of the functional axis. More research is indicated to investigate the relationship between force direction and tooth displacement; the best and most practical assumption for now is that the CR moves parallel to the line of action of the force. We should expect that, with new knowledge, some modification of this principle may be required. It is often said that a “light” force producing uncontrolled tipping can also create an optimal rate of tooth movement. This is correct in that the tooth movement measured at the crown of a tooth can be rapid. Look at the maxillary superimposition in Fig 9-53, in which case the incisors were retracted using 183

9 The Biomechanics of Altering Tooth Position

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Fig 9-51  In most malocclusions requiring significant retraction of incisors, it is preferable to prevent the root apex from being displaced forward. (a) Compare the uncontrolled tipping (green tooth) and controlled tipping (yellow tooth). The end point for both is a normal incisor axial inclination. Controlled tipping allows more retraction. (b) Both a lingual force and a moment are required (red arrows). A force and a moment are indicated for extreme overjets and for extraction cases. The CR moves in the same direction as the force (dotted arrow). (c) The bracket moves downward and backward and rotates in a clockwise direction (dotted arrows). Some clinicians erroneously think that the required force system is in the same direction as the bracket displacement (gray force system). (d) A patient showing an increase in deep bite. The root is not displaced forward, and the CR must translate upward and backward. The correct force system is shown with red arrows. The displacement of the bracket (gray arrows) is not the correct force system.

Fig 9-52  The functional axis of a tooth. The blue and red dots are the CRots under vertically varying horizontal forces (red and blue arrows). The blue and red colors indicate clockwise and counter­ clockwise rotation, respectively. The black dotted line is a functional axis of the tooth, an asymptote of the CRots. The CR moves perpendicularly to the functional axis, as shown by the straight dotted arrow. The red forces rotate the tooth counterclockwise, and the blue forces rotate the tooth clockwise (curved dotted arrows).

184

a round wire and an elastic. As predicted, the CRot is close to the CR (Fig 9-53a). Let us look at the compression side of the root, which determines the rate of tooth movement. The stress at the compression side is depicted in small red arrows in Fig 9-53b; the same analysis is valid for the tension side. The goal is to move the incisor lingually, and progress is measured by the amount of bone resorption on the lingual surface of the root. The only useful resorption is a small blue area of bone lingually at the alveolar crest. The bone resorption is in the wrong direction on the labial root surface (green area). Why does tooth movement appear so rapid? It is partly caused by the tipping; the angle created by the tipping gives an optical illusion as it augments the amount of tooth displacement at the incisal edge. Note that the CR has moved little. Based on the bone remodeling, little of the modification is useful in reaching the final tooth position. More importantly, even with very light force, localized stress in the PDL may be excessively high (see Figs 9-38a and 9-38b). Numerical analysis tells us that the localized high stress in uncontrolled tipping is five times greater than the uniform stress in translation produced by the same magnitude of force.6,7 The radiograph and autopsy material in Figs 9-54a and 9-54b show a patient whose orthodontic treatment tipped the incisors lingually; the root apices became very prominent. It is not proven that the root resorption was caused by this tooth movement, but possible root resorption (Fig 9-55) could occur

When a Force and a Couple Are Required for Tipping Fig 9-53 The incisors were retracted by uncontrolled tipping using a round wire and an elastic. (a) As predicted, the CRot is close to the CR. (b) The stress at the compression side is depicted by small red arrows. Note that the maximum stress occurs at the apex and alveolar crest. The only useful resorption is a small blue area of bone lingually at the alveolar crest. Bone resorption is in the wrong direction on the labial root surface (green area).

a

a

b

b

Fig 9-54  Radiograph (a) and autopsy model (b) showing posterior roots protruding through bone.

because of higher stresses at the apex and the large amount of root displacement through the cortical plate of bone (or subsequent to moving the root lingually). Figure 9-56 shows a series of lateral cephalometric radiographs from a case in which an incisor was moved lingually by uncontrolled tipping followed by root movement (see Figs 9-44a and 9-44d). The apices of the incisor went through so-called “round tripping,” moving forward and backward, which is undesirable. The patient showed relatively normal axial inclinations at the start of treatment (Fig 9-56a); uncontrolled tipping occurred, and the apex apparently penetrated the labial cortical plate of the bone (arrow in Fig 9-56b). Later, root movement was performed to bring the apices to the lingual, and new bone formation is seen at the labial side (Fig 9-56c). Uncontrolled tipping by a single force at the crown is biomechanically very easy, but it occurs too quickly for the clinician to notice its side effects. The force system required for root movement (CRot

Fig 9-55  Note the possible root resorption due to high stress at the apex.

at the crown) may look simple because it can be performed by a single force apical to the CR or its equivalent force system at the bracket; however, it needs a sophisticated clinical appliance to avoid the adverse side effects of anchorage loss. Figure 9-57 shows a diastema between the central incisors. But what is the best way to treat it? A straight wire placed in the brackets would produce equal and opposite couples (Fig 9-57a). The direction of the moments will tip the crowns together. Although the force system seems correct, pure rotation around the CR may not be manifested because of the initially high frictional forces in a distal direction on the incisors. Figure 9-57b, on the other hand, is better because a force without a couple is used without a wire. Using button hooks on the lingual of the teeth will further reduce the tendency for incisor rotation from the occlusal view. Ideally, the CRot should be closer to the apex for better inclination and stability. A force with a moment from a wire segment with a curvature could produce the 185

9 The Biomechanics of Altering Tooth Position

a

b

c

Fig 9-56  The incisors moved lingually by uncontrolled tipping followed by root movement. The apices of the incisors went through socalled “round tripping,” which is undesirable. (a) The patient showed relatively normal axial inclinations at the start of treatment. (b) The apex penetrated the labial cortical plate of the bone (arrow). (c) Root movement was performed to bring the roots back, and new bone formation is seen at the labial side (arrow).

a

b

c

Fig 9-57  Methods of closing a diastema between the central incisors. (a) A straight wire placed into the brackets would produce equal and opposite couples to tip the crowns together. Although the force system seems correct, pure rotation around the CR may not be manifested because of high friction. (b) A simple elastic without a wire along with button hooks on the lingual of the teeth would be better than a straight wire. The CRot would be closer to the apex for better inclination and stability. Rotation of the teeth will be less because D1 is smaller than D2 in the lateral view. (c) A force with a moment from a wire segment with a curvature could produce the required equal and opposite couples needed for controlled tipping.

required equal and opposite couples (whose direction is crowns apart, roots together) needed for controlled tipping (Fig 9-57c). However, because friction is unpredictable, force system control is very difficult. Frictionless mechanics with loops delivering both forces and moments may provide a more predictable force system. Note that the proper moment direction is opposite of the direction given by a straight wire with a normal prescription.

Characteristics of an Optimal Force System What is an optimal force? Of course, it depends on treatment goals. It includes the traditional aspects of a force vector: magnitude, direction, and point of application. This chapter has considered mainly the point of force application (ie, M/F ratios). Most 186

evidence suggests that the M/F ratio at the bracket determines the CRot and that force magnitude is not a major factor in determining the CR.8 What is the best force magnitude to use with any given M/F ratio? The goal is usually defined as rapid tooth movement, minimal pain response, minimal tissue damage (root resorption and alveolar bone loss), and minimal anchorage loss. On a clinical level, the rate of tooth movement has been studied and related to force magnitude. A typical graph is shown in Fig 9-58. There may be three distinct phases of response over time—the initial phase, the lag phase, and the post lag phase. In the initial phase, very rapid tooth displacement is observed instantly. It is due to the nonbiologic, purely mechanical deformation of the PDL. This mechanical displacement is also known as physiologic mobility. The stress induced in the PDL during this initial phase initiates the bone-remodeling cascade. This bone remodeling requires time, hence the lag phase. High stresses

Characteristics of an Optimal Force System

Apicogingival level H G

Initial

F E D C B –8 –6 –4 –2

0

2

4

6

Post lag

8

A H G

Lag

F E D C

Distance from the alveolar crest (mm) 13.3 13.0

H G

10.5

F

8.0 5.0

E D C

2.5

B

0.0

A

6.5

Labial

Distal

Mesial

Lingual

B –6 –4 –2

0

2

4

6

A

Stress (g/mm2)

Fig 9-58 The rate of tooth movement versus time. The graph shows three distinct phases: initial, lag, and post lag.

Fig 9-59  A 3D stress diagram of a tooth can be a starting point for understanding the subsequent biologic changes that occur. It shows that for different CRots, if the applied force is kept constant, maximum stress levels can vary significantly. Translation produces the lowest stress levels because the stress-strain distribution is more uniform (green line).

producing necrosis also increase the lag phase. The post lag phase involves a biologic response to bone remodeling: resorption and apposition. Most significant is the biologic displacement in the post lag phase. With lower stresses, sometimes no lag phase is observed. More force does not necessarily produce faster tooth movement. In this text, some relative force magnitudes for different types of tooth movement are given based on the maximum stress level in the PDL. This may be state of the art, but a better approach is needed. It is the stress, not the force, that is distributed to the cementum, PDL, and bone; the biologic response is to these stresses. Therefore, more emphasis should be put on the stress and strain in the PDL rather than the absolute magnitude of force applied to the tooth. Therefore, the following chapters focus on the relationship between applied force on a tooth and the distributed forces (stress) and deformations (strain) in the PDL and the alveolar bone. These discussions will provide a more basic understanding of optimal force levels for tooth movement and anchorage control, both physically and biologically. A 3D stress diagram of a tooth can be a starting point in understanding the subsequent biologic changes that occur (Fig 9-59). It shows that for different CRots, if the applied force is kept constant, maximum stress levels can significantly vary. Translation produces the lowest stress levels because

the stress-strain distribution is more uniform (green line in Fig 9-59). Figure 9-60 is a “working hypothesis” graph that shows the relationship between compressive stress in the PDL and the rate of bone resorption. The dimensions and slopes are conceptual; however, future studies of the graph’s concepts at stress-strain and molecular levels will be important. The graph shows that with no added stress, no bone resorption occurs. The stress is increased to a magnitude where bone resorption will start to occur (threshold). The idea of threshold force has been suggested to explain a rationale for anchorage control. How low is low enough? There is an interesting classical study by Weinstein.9 He showed that 2 g of force can initiate tipping movements of a tooth and concluded that the threshold, if it exists, is less than 2 g. Future studies of tooth translation with lower stresses might demonstrate a threshold more definitively. As the stress is increased, there is a proportional increase in the rate of bone resorption (optimal stress). Any further increase in stress does not increase the rate of bone resorption. Higher stress levels leading to undesirable tissue changes reduce the rate of bone resorption (excessive). Our general understanding of the response of a tooth to an applied force tends to fit the graph, but more research is needed. Histologic studies (Fig 9-61) on experimental animals show that heavy force can collapse 187

9 The Biomechanics of Altering Tooth Position Fig 9-60 The hypothetical relationship between compressive stress in the PDL and the rate of bone resorption.

Heavy force

Heavy force

the capillary arteries and block the blood flow in the PDL, leading to aseptic necrosis and hyalinization (arrows).8 Necrotic areas that need to be removed can temporarily slow down the rate of tooth movement, although rapid undermining resorption can follow. Frontal bone resorption without hyalinization is observed with lighter forces. Any definition of an optimal force must include force continuity. Some appliances store energy and deliver force over a long time period, while others are relatively intermittent in nature. Currently continuous forces are in vogue, using supposedly biologic magnitudes of force or stress. Rapid palatal expansion is an example of an intermittent force application using a screw. How is force continuity measured? One measure is change in force per unit time (g/day). The most common method is to relate the change of force to the appliance deflection or the change in tooth position (g/mm). In other chapters, the force constancy of orthodontic appliances has 188

Light force

Fig 9-61 Microscopic view of the compression side in an ex­ peri­mental animal. Aseptic ne­ crosis and hyalinization (arrows) are shown with heavy force. Frontal bone resorption without hyalinization is observed with light force.

been described in terms of force deflection—the F/∆ rate describes the change in force as a tooth moves, as the deflection of the appliance is reduced by deactivation. However, not all appliance components follow Hooke’s law, and they may not have a linear relation between force and deflection. Let us consider a situation in which a premolar is far to the lingual (Fig 9-62a). If an archwire is inserted that has a high F/∆ rate, too much force is exerted at the initial activation, and the tooth is exposed to the excessive force zone (red). As the tooth continues to move, the optimal and suboptimal force zones are passed through in rapid sequence; the tooth is within a very limited range of the optimal force zone (narrow green zone in Fig 9-62b). Therefore, the total force level is not optimal. By contrast, a wire with a low F/∆ rate (ie, not a straight wire) that is overbent or uses an overformed loop could provide a wide range in the optimal force zone (green zone in Fig 9-63) during the entire move-

Characteristics of an Optimal Force System

a

b

Fig 9-62  A wire with a high F/∆ rate is used to move a premolar located to the lingual. (a) Too much force is exerted at the initial activation. (b) The tooth is within a very limited range of the optimal force zone (narrow green zone).

a

b

Fig 9-63  A wire with a low F/∆ rate is used to move a premolar located to the lingual. (a and b) The wire is overbent to provide a wide range in the optimal force zone (wide green zone) during the entire movement process to the buccal.

ment process to the buccal. For more details about F/∆ rates, see chapter 6). Not only can forces change as an appliance deactivates, but other parameters such as M/F ratios can also change (see chapter 14). The influence of dynamic forces, such as rapidly changing forces (both high and low frequencies), may offer an interesting possibility for optimizing orthodontic force delivery. Box 9-1 lists several concepts that can help to define the force continuity of an orthodontic appliance. Hopefully, future research will focus on orthodontic force system optimization.

Box 9-1  Force continuity expressions Force change with displacement • F/∆ rate • M/F ratio change with deflection (ϴ) Force change with time • Intermittent or continuous • Hours/day (active force time) • Displacement/time change • Force/time change • F/∆ change/time • Force × time Force times displacement • Energy • Work • Power Pulsating forces

189

9 The Biomechanics of Altering Tooth Position

References

Christiansen RL, Burstone CJ. Centers of rotation within the periodontal space. Am J Orthod 1969;55:353–369.

1. Nägerl H, Burstone CJ, Becher B, Messenburg DK. Center of rotation with transverse forces: An experimental study. Am J Orthod Dentofacial Orthop 1991;99:337–345. 2. Choy K, Kim KH, Burstone CJ. Initial changes of centres of rotation of the anterior segment in response to horizontal forces. Eur J Orthod 2006;28:471–474. 3.  Burstone CJ. The biomechanics of tooth movement. In: Kraus B, Riedel R (eds). Vistas in Orthodontics. Philadelphia: Lea and Febiger, 1962:197–213. 4. Burstone CJ, Pryputniewicz RJ, Weeks R. Center of resistance of the human mandibular first molars [abstract]. J Dent Res 1981;60:515. 5. Tanne K, Koenig HA, Burstone CJ. Moment to force ratios and the center of rotation. Am J Orthod Dentofacial Orthop 1988;94:426–431. 6. Tanne K, Koenig HA, Burstone CJ, Sakuda M. Effect of moment to force ratios on stress patterns and levels in the PDL. J Osaka Univ Dent Sch 1989;29:9–16. 7. Choy KC, Pae EK, Park YC, Kim KH, Burstone CJ. Effect of root and bone morphology on the stress distribution in the periodontal ligament. Am J Orthod Dentofacial Orthop 2000; 116:98–105. 8. Burstone CJ. Application of bioengineering to clinical orthodontics. In: Graber LW, Vanarsdall RL, Vig KWL (eds). Orthodontics: Current Principles and Techniques, ed 5. Philadelphia: Elsevier Mosby, 2012:345–380. 9. Weinstein S. Minimal forces in tooth movement. Am J Or­ thod 1967;53:881–903.

Coolidge ED. The thickness of the human periodontal membrane. J Am Dent Assoc 1937;24:1260–1270.

Dermaut L, Kleutghen J, Clerck H. Experimental determination of the center of resistance of the upper first molar in a macerated, dry human skull submitted to horizontal headgear traction. Am J Orthod Dentofacial Orthop 1986;90:29–36. Geramy A. Alveolar bone resorption and the center of resistance modification. Am J Orthod Dentofacial Orthop 2000; 117:399–405. Nikolai RJ. On optimum orthodontic force theory as applied to canine retraction. Am J Orthod 1975;68:290–302. Nikolai RJ. Periodontal ligament reaction and displacements of a maxillary central incisor loading. J Biomech 1974;7:93–99. Smith R, Burstone CJ. Mechanics of tooth movement. Am J Orthod 1984;85:294–299. Soenen PL, Dermaut LR, Verbeeck RMH. Initial tooth displacement in vivo as a predictor of long-term displacement. Eur J Orthod 1999;21:405–411. Steyn CL, Verwoerd WS, Merwe EJ, Fourie OL. Calculation of the position of the axis of rotation when single-rooted teeth are orthodontically tipped. Br J Orthod 1978;5:153–156.

Recommended Reading

Synge JL. The tightness of teeth, considered as a problem concerning the equilibrium of a thin incompressible elastic membrane. Phil Trans R Soc Lond 1933;231:435–470.

Burstone CJ. The biophysics of bone remodeling during orthodontics—Optimal force consideration. In: Norton LA, Burstone CJ (eds). Biology of Tooth Movement. Boca Raton, FL: CRC Press, 1989:321–333.

Tanne K, Nagataki T, Inoue Y, Sakuda M, Burstone CJ. Patterns of initial tooth displacements associated with various root length and alveolar bone height. Am J Orthod Dentofacial Orthop 1991;100:66–71.

Burstone CJ, Every TW, Pryputniewicz RJ. Holographic measurement of incisor extrusion. Am J Orthod 1982;82:1–9.

Vanden Bulcke MM, Burstone CJ, Sachdeva RC, Dermaut LR. Location of the center of resistance for anterior teeth during retraction using the laser reflection technique. Am J Orthod Dentofacial Orthop 1987;91:375–384.

Burstone CJ, Pryputniewicz RJ. Holographic determination of center of rotation produced by orthodontic forces. Am J Orthod 1980;77:396–409. Burstone CJ, Pryputniewicz RJ, Bowley WW. Holographic measurement of tooth mobility in three dimensions. J Periodontal Res 1978;13:283–294. Choy KC, Kim KH, Park YC, Han JY. An experimental study on the stress distribution in the periodontal ligament. Korean J Orthod 2001;31:15–24.

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Dathe H, Nägerl H, Kebein-Meesenburg D. A caveat concerning center of resistance. J Dent Biomech 2013;4:1758736013499770.

Vanden Bulcke MM, Dermaut LR, Sachdeva RC, Burstone CJ. The center of resistance of anterior teeth during intrusion using the laser reflection technique and holographic interferometry. Am J Orthod Dentofacial Orthop 1986;90:211–219. Yettram AL, Wright KWJ, Houston WJB. Center of rotation of a maxillary central incisor under orthodontic loading. Br J Orthod 1977;4:23–27.

PROBLEMS

1. The perpendicular distances between the single force and the bracket are given. The black circle is the CR. To replace each of the forces with an equivalent force system at the bracket, give the M/F ratio required at the bracket. Draw the correct direction of the force system at the bracket, a dot at the approximate CRot, and the direction of rotation with a dotted curved arrow around the CRot in a to e.

a

b

c

d

e

2. Give the approximate force system at the lingual bracket for the buccally inclined maxillary left molar to rotate around the green dot in a to c. Denote the correct direction of moments and forces. The dark gray tooth outline is before the movement, and the light gray outline is after the movement.

a

b

c

191

CHAPTER

10 3D Concepts in Tooth Movement Rodrigo F. Viecilli

“Seek simplicity, but distrust it. We think in generalities, but we live in detail.”

OVERVIEW



— Alfred N. Whitehead

The scientific understanding of physical tooth-movement references evolved over time, shifting from a two-dimensional (2D) model to a three-dimensional (3D) model. However, the classic 2D concepts of tooth movement may not always work in 3D. This chapter discusses the evolution of concepts of tooth movement and outlines the differences among the concepts of fulcrum, pivot, center of mass and centroid, center and axis of rotation, and center and axis of resistance.

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10 3D Concepts in Tooth Movement

Fig 10-1  A primitive appliance to control crown and root tipping, invented by Calvin Case in 1916. (From A Practical Treatise on the Technics and Principles of Dental Orthopedia.)

Fig 10-2  A primitive experimental model of tooth movement depicted by Calvin Case in 1921, using a wood stick to determine the “fulcrum,” or center of rotation. (From A Practical Treatise on the Technics and Principles of Dental Orthopedia.)

The center of resistance of a tooth was originally conceptualized as an analogy to the center of mass of a free body. Initially idealized in two dimensions, it could be determined by the iterative trial application of a force until translation was obtained or by the application of a couple. In 2D, the center of resistance coincides with the center of rotation when a couple is applied, because the resultant force is zero. The center of resistance, as the center of mass, does not translate in the absence of a resultant force; hence, it coincides with the center of rotation only when a couple is applied. This chapter discusses the evolution of the idea of center of resistance, shifting from a simplified 2D model to a generalized 3D understanding of the biomechanics of tooth movement.

Such a gap in science and the abundant orthodontic literature with little physical rigor often led to confusion among the concepts of center of rotation, center of resistance, fulcrum, and pivot. Part of the problem was that fulcrum and pivot were historically mentioned by early orthodontists in an attempt to primitively describe the biomechanics of tooth movement.

Origins of a Tooth-Movement Reference In 1916, Calvin Case used individualized tooth movement appliances to record lines of action of the forces with the intention of controlling the tipping tendencies of the teeth (Fig 10-1). In so doing, he revealed some understanding of where the toothmovement reference should be to obtain predominant crown or root movement. However, it took over 40 years for this idea to evolve into the concept of center of resistance, developed by Charles Burstone and James Baldwin at Indiana University in the 1950s.

194

Fulcrum versus pivot In 1921, Calvin Case included a figure in his textbook to try to explain how a tooth would move in response to a force; he named what we know today as the center of rotation a fulcrum (Fig 10-2). Physically, a fulcrum is defined as the support of a lever, and a pivot is the point around which the lever pivots (rotates). When a force is applied to a supported lever, it typically pivots around the fulcrum, so the terms are often used interchangeably. However, if the total load applied to the lever is a couple (ie, net force of zero), then it would rotate around its center of mass, so the pivot (center of rotation) could technically be different from the fulcrum. Because these concepts are more applicable to simplified lever mechanics, they are not ideal to describe tooth movement.

Center of rotation versus axis of rotation Geometrically, a 2D body rotates around a center of rotation, and a 3D body rotates around an axis of

Scientific Development of the Concept of Center of Resistance

B

Midpoint

Midpoint

Centroid A

Midpoint

C

Fig 10-3  The centroid of a triangle (barycenter) can be determined by the intersection of the median lines.

Fig 10-4  The centroid of the area under a parabola represents the projected resistance of the root or PDL and is located at 40% of the height, closer to the base.

rotation. In a 2D orthogonal projection, the body will appear to rotate around a point. In 2D, a center of rotation is sufficient to describe tooth movement from position A to position B. In 3D, however, the only comprehensive way to describe the movement of a body is through screw axis theory, which involves more refined mathematics compared with center of rotation descriptions. Chasles theorem states that in Euclidean 3D space, any movement of any object can be described by rotation around an axis and translation along this same axis. For a rigorous 3D theory of tooth movement, this is the only method available to describe any type of movement. However, in orthodontics, because our tooth movement planning is often simplified, we typically think in terms of 2D projections of the 3D teeth, and thus projection points of the axes of rotation (centers of rotation) have been sufficient for clinical applications. On the other hand, describing tooth movement as a single 3D movement instead of movement combinations is technically more appropriate because it prevents ambiguous situations related to the order of movements, as combining different rotations in different orders in 3D can lead to a different final position.

volume of resistance, which are explained later in the chapter.

Origins of the concept of center of resistance The term center of resistance was first used by Leo­ nardo da Vinci in his 1505 book, Codex on the Flight of Birds. The concept of center of resistance in orthodontics originated from the scientific discussions between Charles Burstone and James Baldwin, who were responsible for establishing biomechanics in orthodontics as a science at Indiana University. Burstone and his research group were involved in the evolution and formalization of all 2D and 3D scientific models of a tooth-movement reference, which culminated in the concepts of axes of resistance and

Scientific Development of the Concept of Center of Resistance As with any scientific model, the models for a reference to tooth movement have been refined over time. The following sections describe the rationale for each model and how it was established.

Centroid, barycenter (center of gravity), and center of mass The centroid is the average geometric center of an object (Fig 10-3). In free homogenous bodies, the center of mass (center of gravity or barycenter) and centroid are located at the same point. The inertia or resistance to movement of a volumetric body is the same in all directions, and hence a point is sufficient to describe this resistance. Because the process of or­ thodontic mechanotransduction (ie, where and how mechanical stimuli translate into tooth movement) was incompletely understood, the first mathematical models of center of resistance utilized centroids of a 2D root or periodontal ligament (PDL) projections to represent resistance to tooth movement.

2D projection model The first model was based on the centroid of an approximate 2D projection of a root or PDL, with the rationale that the greatest resistance to tooth movement would be represented by these. For instance, the model of a single-rooted parabolic root or PDL projection can be appreciated in Fig 10-4. With this model, the centroid that represents the center of re195

10 3D Concepts in Tooth Movement Fig 10-5 Determination of the centroid of a paraboloid of revolution. (Reprinted from Burstone and Pryputniewicz1 with permission.)

Centroid of a thin section Centroid of a paraboloid of revolution Thin section i

y

×

(1

/3

i

y

)H

×

(2

/5

)H

Locus of centroids of thin section

sistance is located at 40% of the length of the root, closer to the alveolar ridge than the apex.

3D symmetric model With this model, the entire root, represented by a paraboloid of revolution (constructed by rotation of the area of a parabolic section along its long axis) is considered the element of resistance to tooth movement1 (Fig 10-5). The centroid is located at 33% of the long axis, within the root half that is closer to the alveolar ridge. It could be argued that if 3D PDL resistance should be modeled, perhaps the centroid of a surface area (or a thin volumetric shell) of a paraboloid of revolution could be a more reasonable model for the center of resistance. This model has never been published, but the author calculated it to be at 34% for a surface, which is clinically insignificantly different from the original volumetric model published by Burstone and Pryputniewicz.1 It is also worth noting that because the paraboloid of revolution is a symmetric entity, if we assume PDL resistance is uniform, this allows for the resistance to be represented as a point, like the center of mass. 196

3D asymmetric models and the axes of resistance Recently, the reference model for tooth movement has been revisited in light of recent research findings with regard to orthodontic mechanotransduction. Because stress measures the internal resistance (force per infinitesimal area), it could be an ideal scientific measure of “resistance” of the PDL to instantaneous tooth displacement. However, if the clinical purpose of the concept of center of resistance is to predict future tooth movement after bone modeling takes place, perhaps the deformations of bone and tooth should not necessarily be accounted for in the model of PDL stress, because these are mostly recoverable. We have shown that the PDL stresses are not uniform in naturally shaped teeth. This is indeed very logical, because tooth PDLs do not have axisymmetric morphology, and the PDL is histologically heterogeneous and anisotropic (ie, material properties vary in magnitude depending on direction of the load). Furthermore, because the stress-strain curve of the PDL is nonlinear, the PDL may become stiff-

Scientific Development of the Concept of Center of Resistance

z y z

y

y x

a

x

x

z

b

c

Fig 10-6  Axes of rotation determined by three perpendicular couples in the buccal (a), mesial (b), and apical (c) directions.

y'

y' x'

x'

z'

z

z'

x

z'

y'

y

a

x'

y x

b

z

c

Fig 10-7  (a to c) 3D locations of the axes of resistance. In each field of view, the correct reference for translation is the intersection of the two axes that can be seen as lines.

er in one direction than another if resistance due to morphology is different in different directions. The biologic reactions to PDL stress vary with stress thresholds and, over time, add even more potential differences in references to tooth movement in different directions. In 1991, Nägerl et al demonstrated large differences in the positions of the centers of resistance for each direction of tooth movement.2 In 2009, the author confirmed this finding, noting that axes of resistance could be more adequate references in 3D because a couple causes a rotation around an axis that could then be used as a reference for translation.3 In 2010, it was shown in dog teeth that the centers of resistance were statistically different in different directions.4 The author recently used finite element analysis in a model of a maxillary first molar to determine possible differences in locations of axes of resistance solely due to lack of PDL axisymmetry.5 It was found

that the 3D axes of resistance indeed do not intersect at a 3D point; in the maxillary first molar study, the axes of resistance missed each other by a maximum of 0.6 mm. Hence, it is reasonable to believe that, considering all possible causes of differences in the PDL stress fields for each movement direction, the center of resistance as a point does not exist as a realistic physical entity. If there are three axes of resistance for each possible orthogonal direction of movement (Fig 10-6), how do we know which of the three should be used as references for translation in each direction? Well, the reference for translation in the direction perpendicular to the field of view is always at the intersection of the two axes that can be seen as lines on that view (Fig 10-7). Another question that arises is whether the axes of resistance and rotation are the same when a couple is applied. However, this question is revealed to be meaningless because a combination of two axes 197

10 3D Concepts in Tooth Movement Fig 10-8 The projection of the z-axis of rotation (obtained by a buccally oriented couple) is different than the projected center of resistance for translation in the buccal direction.

y' Center of resistance for translation along z

x' z' Center of rotation for a moment parallel to z

y x

must be used to determine a 2D projected point used as a translation reference (so that the body does not rotate in any direction). So there is at least one axis of rotation that is never coincident with this point. This can be better understood in Fig 10-8. It is important to note that in segments of teeth, the morphologic asymmetries that lead to nonco­ incident axes of resistance will build up, and distances between the axes will likely increase. This is a subject for future studies.

3D volume of resistance Dathe and Nägerl formalized the mathematics for a 3D volumetric center of resistance,6 which perhaps could be thought of as all possible locations of intersections of axes of resistance for different tooth translation directions. Clinically, one can think that there is a 3D volume of resistance (a volumetric center, not a point), with a certain level of uncertainty that is dependent both on morphology and other sources of asymmetric behavior discussed earlier.

Practical limitations of the concepts of center of resistance and axes of resistance It is important to note that the axes of resistance can vary their positions during tooth movement because 198

the PDL, root, and bone are subject to constant biologic modeling. Hence, the clinician should not see the position of the axes as a static feature. Clinically, if using a system for controlled tooth movement, we should start with our best guess based on scientific literature and then continually correct the appliance and force system, depending on the movement that was achieved with the previous configuration and the desired future outcome.

References 1. Burstone CJ, Pryputniewicz RJ. Holographic determination of centers of rotation produced by orthodontic forces. Am J Orthod 1980;77:396–409. 2. Nägerl H, Burstone CJ, Becker B, Kubein-Messenburg D. Centers of rotation with transverse forces: An experimental study. Am J Orthod Dentofacial Orthop 1991;99:337–345. 3. Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE. Three-dimensional mechanical environment of orthodontic tooth movement and root resorption. Am J Orthod Dentofacial Orthop 2008;133:791.e11–791.e26. 4. Meyer BN, Chen J, Katona TR. Does the center of resistance depend on the direction of tooth movement? Am J Orthod Dentofacial Orthop 2010;137:354–361. 5. Viecilli RF, Budiman A, Burstone CJ. Axes of resistance for tooth movement: Does the center of resistance exist in 3-dimensional space? Am J Orthod Dentofacial Orthop 2013;143:163–172. 6. Dathe H, Nägerl H, Kubein-Meesenburg D. A caveat concerning center of resistance. J Dent Biomech 2013;4: 1758736013499770.

CHAPTER

11 Orthodontic Anchorage Rodrigo F. Viecilli

“He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast.”

— Leonardo da Vinci

“Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry.”

OVERVIEW



— Richard P. Feynman

This chapter explains the biomechanical basis for orthodontic anchorage. The intensity of the biologic response relates to mechanical stimulus, and this stimulus, when combined with the biologic environment, leads to the clinical perception of anchorage value. Certain appliances have the potential to change the degrees of freedom for tooth movement and selectively enhance anchorage potential. These appliances and the scientific rationale for typical clinical strategies to improve anchorage are discussed in the chapter.

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Definition and Clinical Perception of Anchorage In its broadest sense, orthodontic anchorage is resistance to tooth movement. Hence, the anchorage value of an appliance or dentoalveolar complex relates to its capacity to resist movement. In intraoral anchorage, the clinical perception of anchorage directly relates to the difference in relative speed of tooth movement between units. The following section discusses the variables that could potentially influence the anchorage value of dentoalveolar units.

Rationale for Anchorage from a Basic Science Perspective The speed of tooth movement is the result of interaction among many intertwined basic scientific variables. The effects of many of these variables have not yet been quantified, but some have recently begun to be better understood.

Mechanical variables Periodontal ligament stress and the total tooth load at the axis of resistance As described in detail in chapter 12, orthodontic tooth movement is initiated when stress is applied to the periodontal ligament (PDL). The number of resorbing osteoclasts is initially directly proportional to the third principal (“most compressive”) stress. If the stress is low enough for the tissue to remain viable, direct bone resorption occurs, which can quickly result in tooth movement as the PDL space naturally widens after bone resorption. If the stress is too high, hyalinization can occur, which can delay tooth movement because the necrotic tissue has to be removed after undermining resorption. Depending on the magnitude of the intraoral load, it is possible for teeth with less support to undergo delayed tooth movement following excessive stresses, especially when compared with larger teeth in which necrosis did not occur. Direct bone resorption occurs in the large tooth so that it can start moving more rapidly, at least initially. Larger teeth naturally have more PDL support, and hence the stress magnitudes in the PDL are smaller than those in little teeth when the same

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force is applied. Consequently, the anchorage value of larger teeth is also greater. Everything else being equal, smaller stresses should attract fewer osteoclasts so that tooth movement is slower. This is the main reason a molar has more anchorage value than an incisor. It is important to note that different types of movements lead to different patterns of stress fields in the PDL, and hence the anchorage value of a tooth will also depend on the type of movement desired. Translation typically has more anchorage value than controlled tipping, which in turn has more value than uncontrolled tipping. For instance, in translation, the compressive stresses can be onethird of those for uncontrolled tipping for the same tooth with the same force applied to the bracket because the total load acting on the tooth is reduced. The total load acting on a tooth is the equivalent force system at the axis of resistance. The total load for a tooth being tipped with 100 cN at the bracket is 100 cN (force) + 100 cN × d (moment), where d is the distance to the axis of resistance. With translation, the load is smaller because the total moment is zero, and a countermoment is applied to the bracket to cancel out the moment of the tipping force. This explains why, in tipping, larger peak stresses affect the PDL.

Biologic variables Biologic variables are intrinsic factors that can vary locally (from tooth to tooth) or between individuals. The two main categories are the inflammatory response and the bone quantity and quality.

Inflammatory response For a given equal mechanical stimulation, there can be variations in the intensity of inflammatory response from one periodontal or bone site to another, which can affect the attraction of osteoclasts and the speed of tooth movement. These differences can affect the anchorage value of a tooth from individual to individual as well as within the same individual. One cause of this type of intra-individual variation in response is vascularity. Decreased vascularity in one dentoalveolar site compared with another provides less opportunity for cellular recruitment and may promote ischemia and necrosis, which can delay tooth movement. Furthermore, between different individuals, there can be differences in the genetic profile that lead to differences in the performance of biologic mediators such as prostaglan-

Anchorage Values According to PDL Stress Fig 11-1  Loads applied to the teeth (shown here in only one direction) used to calculate the anchorage values. The force was fixed at 8 cN and the moment at 50 cNmm.

dins, cytokines, leukotrienes, and growth factors. These can translate in different levels of inflammation and thus influence anchorage potential.

Bone quantity and quality Density. If the trabecular bone in the alveolus surrounding the tooth has decreased bone volume fraction, it follows that there is increased porosity in the bone. Hence, osteoclasts need to resorb less bone to result in space for tooth movement. Decreased bone density may also facilitate the work of osteoclasts. Thickness of cortical bone, trabecular bone volume fraction, and bone density vary among dentoalveolar sites and individuals, possibly affecting the speed of tooth movement and anchorage value.

Bone remodeling rate or turnover. High bone turnover, or a quick cycle of renewal, means that large numbers of osteoclasts are rapidly resorbing bone and osteoblasts are rapidly reforming it. It is a natural process of bone repair. The larger the number of cells already at work performing bone remodeling, the easier it may be for bone modeling associated with tooth movement to occur. Hence, the anchorage value of a dentoalveolar site will decrease in rapidly remodeling bone. The rate of turnover may change depending on the jaw, dentoalveolar site, or individual. It is also important to remember that bone morphology and turnover directly relate to overall bone metabolism, including nutritional deficiencies; abnormalities of the kidney, gut, or parathyroid function; or local pathologies. These factors can all alter the anchorage potential

of dentoalveolar sites in different individuals or at different periods for the same individual. Clinical application of injury to the bone can cause a decrease in bone volume and density and an increase in bone turnover due to increased inflammation. Different methods to apply bone injury and increase tooth movement have been used over the last 100 years. The concept is nothing new, but originally the effect was explained only by a reduction in the volume of cortical bone, so procedures were more aggressive and involved dramatic decortications. The effects of frequent injuries to the bone in the alveolar bone level and root resorption still have not been fully explored.

Anchorage Values According to PDL Stress The author has conducted finite element analyses to calculate the load ratios necessary to promote equal PDL stresses in teeth of average size and ideal occlusion (except third molars) for different types of movement that occur during the orthodontic alignment phase.1 At this point, a linear model was used that is applicable to small loads. The methodology consisted of the application of the same load (force, 8 cN; moment, 50 cNmm) for different movements (intrusion/extrusion force at the bracket, buccal/ lingual crown tipping at the bracket, distal/mesial crown tipping couple, and couples with the vector perpendicular to the occlusal plane) to all teeth. The simulated loads are shown in Fig 11-1. The PDLs were 201

11 Orthodontic Anchorage

Mandibular central = –3.7 kPa

Maxillary canine = –2 kPa

Stress normalized to MC = 1

Stress normalized to MC = 0.54

then divided into thirds longitudinally (so that one of the regions contained the third with the highest stress) and in fourths transversally (so that one of the regions contained the fourth with the highest stress). The intersection of these two regions determined the region of analysis where stresses were averaged. The ratio between the stresses determined the anchorage values for each tooth. An example calculation is shown in Fig 11-2.

Clinical Intraoral Anchorage Strategies Number of teeth and segment units The easiest strategy to enhance the anchorage of a unit is to add more teeth to the unit. This expands the overall support of the unit, decreasing the peak stresses and, hence, the number of osteoclasts. What is the clinical gain in posterior anchorage of adding second molars to a maxillary posterior anchorage unit that contains the second premolar and first molar, if space is closing against the incisors and canine? The anchorage ratio of anterior to posterior is nearly 1:1 without the second molar, which means we could expect the space to close 50% by each unit. By adding the second molar, this ratio changes to 1.6:1, which means that in a 7.8-mm space closure, we could expect 4.8 mm of closure from anterior movement and 3 mm from posterior movement (saving roughly 2 mm of anchorage loss). 202

Fig 11-2  Example of third principal stress analysis for labial crown tipping, comparing standardized PDL regions for the maxillary canine and the mandibular incisor. The same force (8 cN) causes stresses on the maxillary canine that are 54% of the stresses on the mandibular incisor. Hence, the anchorage value of the maxillary canine for this particular movement is 1.85 times that of the mandibular incisor. MC, mandibular central incisor.

Differential moments to attain differential stress (anchorage) The second major strategy to reinforce anchorage is to apply differential moments by achieving a system in equilibrium for the two units. To understand this, let us examine the example in Fig 11-3. In order to tip one of the molars with the intent of correcting a reverse articulation (also known as a crossbite), the reactive system is planned so that the contralateral molar translates. As explained earlier, one can expect stresses at least three times lower for translation compared with tipping, because during tipping the total load acting on the tooth includes the force and a moment. In translation, the total load acting on the tooth consists only of the applied force. Vertical forces in the system are equal and opposite and tend to maintain the stress differential achieved by the application of differential moments. This strategy is also used effectively for space closure (Fig 11-4) and has been considered effective in clinical studies. In this section, it is also relevant to discuss whether the inclination of a tooth promoted by a distal crown tipping bend affects its anchorage potential. Consider a molar that is vertical compared with a molar that has its crown angulated distally 10 degrees. If a tipping force is applied to the vertical molar at the tube, the force system at the axis of resistance will be the tipping force plus a moment of 10 × d, where d is the distance to the axis of resistance (see chapter 3). In the molar that is angulated distally, the distance to the axis of resistance is reduced. The cosine function of 10 can be used to calculate the perpendicular distance, which is 98.4% less (see Fig 2-10). Hence, there is only a 1.6% reduction in the distance

Clinical Intraoral Anchorage Strategies Fig 11-3  Differential moment load system set to correct a reverse articulation due to a lingually tipped maxillary molar. The intent of the force system is to apply a translation load to the patient’s right anchorage molar and a tipping load to the left molar, which can be achieved by activation of a transpalatal arch. The total moment at the CR from the red horizontal force (F) acting on the anchorage tooth plus the applied orange couple (M) from the vertical forces (F1) is zero, while the tipped tooth will suffer large corrective tipping moments from both F1 and F. This will allow the PDL of the reactive tooth to be under significantly less stress, thus reinforcing its anchorage.

M = F1d

F1 F1 d F

F

M/F 10 If d = 40 mm and 2,000 gmm is required on the patient’s right molar, the vertical forces are 50 g.

Fig 11-4  Example of T-loop mechanics utilized to close spaces in a 16-year-old hyperdivergent patient with 12 mm of horizontal overlap (also known as overjet) (a). (b) The T-loop has classic Burstone preactivation curvatures and is activated at 4.5 mm to deliver a force of approximately 250 cN. It was displaced posteriorly 3 mm to achieve an initial momentto-force ratio of 5 mm in the anterior teeth and 8 mm on the posterior teeth. (c) As the anterior unit tipped, V-bends were added to the anterior portion of the loop to maintain the desirable force system on the anterior unit. Note that intrusion of the mandibular incisors and canines had to be performed to normalize the vertical overlap (also known as overbite) and avoid anterior interferences when space closure was completed. (d) After root movement of the anterior unit, realignment was performed. No extraoral, elastic, or implant anchorage was used to close spaces in this case. Differential moments to achieve differential stress is a powerful anchorage strategy with a sound biomechanical basis.

b

c

to the axis of resistance, which means the moment of the force will be 1.6% smaller. This is a negligible effect with no clinical consequence. To decrease the moment acting on the molar by 30%, it would have to be angled back at least 45 degrees, which is clinically impractical. Hence, it is possible to change the anchorage potential of a tooth by changing its angulation. However, minor angulation changes of less than 10 degrees, as historically proposed in orthodontic techniques and prescriptions, are essentially useless as anchorage-enhancement strategies. Active application of moments during space closure can better enhance anchorage because they have a proven scientific rationale, as explained earlier.

a

d

It is interesting to speculate as to whether the anchorage gain effect originally observed by Tweed in his original technique occurred because bends were applied during space closure (and not before, as later proposed). As we have shown, applying tipback moments during space closure does not have the same effect as bends that are applied before space closure and tip the tooth before space closure. Some advantage could be gained if the bends are performed during movement because differential moments are being delivered, hence decreasing the total load at the axis of resistance.

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a

b

c

d

Increasing the force at the active unit or decreasing the force at the reactive unit to obtain a moment-to-force differential It is also possible to increase the movement of the active unit by adding more force to the active unit or by canceling some of the force on the reactive unit, with the force and moment magnitudes planned so that one unit translates and the other tips. Intraorally, this can be accomplished by the use of maxillomandibular elastics or fixed functional appliances (such as Forsus, Herbst, and others). An example of this strategy is shown in Fig 11-5.

Occlusal interlocking and interferences The third major strategy to reinforce intraoral anchorage is based on occlusal interlocking. Orthodontists have long noticed that it is often more difficult to move teeth or close spaces in patients with strong masticatory muscle patterns. Although little data is available on this subject, it is logical that a patient that maintains occlusal interlocking and pressure on the surfaces of the teeth will add mechanical resistance to tooth movement if the target tooth interlocks with opposing teeth. If a patient maintains occlusal interlocking most of the time, the 204

Fig 11-5 Stages of space closure using a custom calibration 8 × 16–mm T-loop made of 0.017 × 0.025–inch titaniummolybdenum alloy (TMA) wire. (a) The T-loop initial load system delivers a moment-toforce (M/F) ratio of roughly 6 mm to each unit and a force of 300 cN at 8 mm of activation. After approximately 4 mm of space closure, when a Class I canine relationship was achieved, the M/F ratio delivered by the loop to each unit dropped to roughly 10 mm. (b) At that point, a maxillomandibular Class III elastic was added. (c) A calibrated α-β root spring made of 0.019 × 0.025–inch TMA wire was added to deliver an M/F of 12 mm to the posterior unit and 10 mm to the anterior unit. (d) The activations of this spring resulted in a force of roughly 250 cN at the ligature connecting the anterior and posterior segments.

loads acting on the tooth will also be distributed to the opposing dentition, thus lowering the stresses acting in the PDL. Although this is a natural occurrence, orthodontists have tried to enhance occlusal interlocking by adding acrylic occlusion rims that are adapted to the occlusion on both arches and promote enhanced occlusal interlocking to the target teeth to preserve anchorage. Even in situations where occlusal interlocking is intermittent, the constant occlusal loading may disturb the pattern of stresses in the PDL that would promote an organized cellular response to achieve the intended movement. Hence, occlusal loading is an unpredictable factor in tooth movement patterns and speed, especially in posterior teeth. Lack of horizontal overlap or excessive vertical overlap can also lead to more difficulty controlling tooth movement in the anterior region (Fig 11-6). Occlusal interferences, especially in patients with strong muscle patterns, are a critical factor to consider when planning orthodontic mechanics.

Soft tissue loads and growthrelated changes The effects of soft tissue loading on the teeth cannot be ignored. Parafunction of the perioral tissues can have dramatic effects in tooth position. The effect of parafunctional habits or invading soft tissue space must be considered in the anchorage plan be-

Degrees of Freedom and the Biomechanical Basis of Intraoral Anchorage Devices Fig 11-6  (a) A Burstone three-piece intrusion arch used to intrude the mandibular incisors along their long axis and eliminate the anterior interference prior to space closure to correct the Class II canine relationship. After enough horizontal overlap and minor vertical overlap were achieved, the minor spaces were closed using a 0.017 × 0.025–inch TMA base arch with a tip-back bend to enhance anchorage and distal-pull activation with cinch-back. Closing spaces with an anterior interference could have caused a Class II molar relationship or distal displacement of the condyle. Evaluating occlusal interferences is a critical part of a sound anchorage plan.

a

cause these tissues are able to produce loads on the teeth. For example, a patient with a forward tongue posture will probably increase the apparent anchorage value of the anterior unit, because the loads applied by the tongue will cancel out some of the appliance-generated force on the anterior teeth. Growth displacement of bones (differential growth of the mandible) can also change the apparent anchorage value of teeth. Clinically, it may appear that maxillary posterior teeth have more anchorage because the mandible is not a stable reference during the peak growth period.

Degrees of Freedom and the Biomechanical Basis of Intraoral Anchorage Devices When an anchorage unit is being designed, it is sometimes convenient to connect teeth in opposite sides of the arch. This is often done with appliances such as the transpalatal arch (TPA), lingual arch, Nance arch, and horseshoe arch. The horseshoe arch is similar to a lingual arch but is used in the maxillary arch. Besides connecting teeth to establish a new anchorage unit with more value, these appliances change the way tooth movement occurs on these teeth. This section discusses how these types of appliances can alter the degrees of freedom for tooth movement and how this can affect or assist in treatment outcomes. There are six degrees of freedom in 3D, consisting of three translation and three ro-

b

tation components. Each of them can be affected in different ways depending on the appliance chosen. A TPA connects first or second maxillary molars by means of a stiff wire (typically 0.036-inch stainless steel for stabilizing purposes). Because the wire is very stiff, it modifies the movement of the molars in the six degrees of freedom as follows: 1. Rotation of the teeth perpendicular to the occlusal plane: This is useful in cases where the orthodontist wants to control molar rotation (eg, during space closure). Another use is to use the position of the molar tube as a guide to determine the shape of the maxillary arch during alignment. The molars are unable to rotate independently but can rotate as a unit. 2. Buccolingual translation: Both molars are partially constrained to translate lingually as determined by the stiffness of the TPA. When used in the orthodontist’s favor, this also helps in the control of arch form. For a molar to translate lingually, the contralateral molar needs to translate buccally. This adds some level of stabilization to the arch form and width when an archwire is used to align adjacent teeth that are rotated, such as premolars and second molars. 3. Occluso-apical translation: The presence of the palate and the tongue are possible constraints to tooth movement in this direction. Addition of acrylic to the center of the TPA could theoretically enhance this, but the effect on extrusion control is unpredictable due to the intermittent and variable nature of tongue forces. In any case, the molars must translate together in this degree of freedom. 205

11 Orthodontic Anchorage

a

b

Fig 11-7  (a) Diagram demonstrating the forces acting on the wires connecting the teeth. If the miniscrew is stable, it is able to stabilize the wire by generating a load that is equal and opposite to the one applied to the posterior teeth. (b) Stainless steel ligature wire tied to the first molar in an attempt to stabilize the posterior segment. This mechanical design will actually force the posterior segment to rotate around the miniscrew. A tension load will develop on the ligature wire so that the resultant force system (hypothetical blue force) matches the constraint; ie, the axis of rotation has to be located at the miniscrew. The blue force will cause a moment around the axis (center) of resistance. This mechanical configuration will cause flattening of the occlusal plane and ultimately opening of the occlusion (open bite).

4. Lingual or buccal crown rotation: Most TPAs have connection wire terminals with tooth inclination control (eg, the original Burstone or Atkinson [universal] types). Hence, the teeth cannot have independent changes in inclination; ie, for a molar crown to rotate lingually, the contralateral crown would have to rotate buccally. This can also help to align the inclinations of all teeth after insertion of the rectangular alignment archwires, using the molar as a reference. Naturally, this is only true if alignment is performed with a TPA when the molars are already ideally positioned. Other TPA designs such as the Burstone Precision TPA system can fit a round wire at the hinge cap bracket and keep the teeth free to incline. This also allows for use of a rectangular wire that is round only on one side, which can be useful for planning certain movements. 5. Mesiodistal translation: Both molars are constrained to translate together. This means that the molars can still move in this direction but must do it together. If space exists in front of only one of the molars, then there could be some value in prevention of molar mesial drift. Otherwise, there are no restrictions. 6. Distal or mesial crown rotation: Both molars are constrained to rotate together. The same rationale for mesiodistal translation applies here. For some time, it was suggested that a TPA could enhance anteroposterior (mesiodistal) anchorage 206

because it would force the molars to move into cortical bone when they would normally follow the normal line of the arch. Studies have shown that the TPA does not seem to add any anchorage value to prevent mesial drift.2 The conclusion of the clinical studies is logical, because any kind of mesial molar movement, with or without a TPA, would involve cortical modeling because the alveolar pro­ cess is always thicker at the molars. However, some anchorage enhancement probably exists if space is present only mesial to one of the molars, because the contralateral tooth would be constrained by its adjacent tooth. Another method of modifying the degrees of freedom for movement of teeth is the use of orthodontic miniscrews. It is important to note that complete restriction of tooth movement in all directions is achieved only if the miniscrew is solidly connected to a tooth by means of a stiff, rigid material—the classic indirect anchorage method. For instance, inserting a rectangular wire in the miniscrew slot and bonding it to a tooth results in solid anchorage that depends only on the stability of the miniscrew (Fig 11-7a). On the other hand, it has often been proposed that stainless steel ligatures be used in an attempt to obtain indirect anchorage; ie, a tooth is connected to a miniscrew by ligating it to the bracket (Fig 11-7b). This strategy will not result in solid anchorage; rather, it will add a constraint that will modify tooth movement and perhaps lead to dramatic clinical side effects if not carefully planned.

Recommended Reading

References 1. Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE. Orthodontic mechanotransduction and the role of the P2X7 receptor. Am J Orthod Dentofacial Orthop 2009;135: 694.e1–694.e16. 2. Zablocki HL, McNamara JA Jr, Franchi L, Baccetti T. Effect of the transpalatal arch during extraction treatment. Am J Orthod Dentofacial Orthop 2008;133:852–860.

Recommended Reading Burstone CJ. The segmented arch approach to space closure. Am J Orthod 1982;82:361–378. Hart A, Taft L, Greenberg SN. The effectiveness of differential moments in establishing and maintaining anchorage. Am J Orthod Dentofacial Orthop 1992;102:434–442.

Kawarizadeh A, Bourauel C, Zhang D, Gotz W, Jager A. Correlation of stress and strain profiles and the distribution of osteoclastic cells induced by orthodontic loading in rat. Eur J Oral Sci 2004;112:140–147. Viecilli RF. Self-corrective T-loop for differential space closure. Am J Orthod Dentofacial Orthop 2006;129:48–53. Viecilli RF, Kar-Kuri MH, Varriale J, Budiman A, Janal M. Effects of initial stresses and time on orthodontic external root resorption. J Dent Res 2013;92:346–351. Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE. Three-dimensional mechanical environment of orthodontic tooth movement and root resorption. Am J Orthod Dentofacial Orthop 2008;133:791.e11–791.e26. Xia Z, Chen J, Jiang F, Li S, Viecilli R, Liu S. Clinical changes in the load system of segmental T-loops for canine retraction. Am J Orthod Dentofacial Orthop 2013;144:548–556.

Burstone CJ, Koenig HA. Optimizing anterior and canine retraction. Am J Orthod 1976;70:1–19.

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CHAPTER

12 Stress, Strain, and the Biologic Response Rodrigo F. Viecilli “Don’t get involved in partial problems, but always take flight to where there is a free view over the whole single great problem, even if this view is still not a clear one.”

— Ludwig Wittgenstein

“An ancestor of mine maintained that if you eliminate the impossible, whatever remains, however improbable, must be the truth.”

OVERVIEW



— Spock, from Star Trek

The mechanical terminology of force, pressure, stress, strain, and other terms are often confused in orthodontics. However, evidence can help to clarify the science behind which type of mechanical stimulus initiates an orthodontic response. This chapter explains the various mechanical terms related to orthodontic tooth movement and focuses on the mechanical effects of loads on biologic structures, discussing how each structure may contribute to an orthodontic response.

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12 Stress, Strain, and the Biologic Response tooth movement phases, the speed of tooth movement, periodontal ligament (PDL) necrosis, bone modeling, and root resorption are often questionable in light of the deficiencies pointed out above. The critical issue is that appropriate physical concepts and mathematics must be used to quantify the mechanical stimulus in the tissue environment; for meaningful conclusions to be drawn from experimental models, these entities need to be properly understood by both scientists and clinicians. In this chapter, the interrelationships among mechanical stimuli and orthodontic responses are clarified under the light of classic mechanics of materials and recent experimental findings. Fig 12-1  External load and a traction-bound vector. F, an internal reaction force associated with Δs, an infinitesimal area or datum plane.

Force, Load, Stress, and Strain Force

Significant confusion permeates the orthodontic literature in the use of the concept of force. At the root of this problem lies a misunderstanding of the differences among the concepts of force, load (forces and moments), and stress and strain. The problem is so widespread that, after many decades of orthodontic research, a systematic review published in 2003 found no significant evidence to support ideal or optimum orthodontic forces to be used clinically.1 Most of the understanding of the relationship between mechanical stimulus and orthodontic response comes from animal models, but smaller animals in which standard clinical armamentaria are used to apply orthodontic loads are typically exposed to stresses in their alveolar structures with orders of magnitude that are higher than would be applied in humans.2,3 Root asymmetries, multiple roots, and uncontrolled or unavoidable loading offsets produce complex and unintuitive stress patterns due to simultaneous displacements and rotations in all three planes that are difficult to quantify without proper methodology.4 Hence, histologic results can be puzzling if the sectioning planes are not oriented according to specific stress and loading patterns.5 Moreover, there are differences in remodeling cycles and inherent morphometric bone characteristics of animal species. These issues have often been neglected in clinical and basic science studies, so knowledge gaps still exist in almost every quantitative aspect of the relationship between mechanical stimulation and orthodontic response as well as its clinical implications. Despite recent advancements, previous conclusions with regard to the timing of 210

A force is a vectorial (first-order tensor) quantity representing the physical action of one body on another that leads to instantaneous acceleration. The vectorial quality of force limits its applicability to a point and the control of only two pieces of information: direction and magnitude. Mathematically, a force is described by a 3 × 1 matrix. Forces alone are sufficient to study the behavior of material points that do not deform.

Load A load is a combination of forces. That it, a couple of forces producing a pure moment is a load. Multiple forces acting in the same direction on different points on the same body are a load. Orthodontic force systems, which often combine couples and forces, can simply be called loads. Because loads are forces acting on different points, they are still limited to vectorial qualities (ie, a point of force application).

Stress Stress represents the combined action of forces (vectors) in a body (divided into small datum [given] planes or areas). In other words, if we imagine that a body is divided into these small pieces, we can establish that, after a load is applied, each small piece of the body will be associated with a share of the reaction that will balance the external load applied. Mathematically, these “shares” can be conceptually developed as “bound vectors” (nonmovable vectors associated with a location) because now that there is

Force, Load, Stress, and Strain Fig 12-2  (a) A couple load applied on the surface of the body leads to shear stress. (b) Compressive load leading to compressive stress in the vertical direction. Note that the body expands in the perpendicular direction according to Poisson’s ratio. The initial shape is the red dotted line, while the final shape is the solid black line.

a

Fig 12-3 Mathematic tensor representations of stress. The matrix on the far right discriminates the normal stresses (σ) along the diagonal, with shear stresses (Ƭ) located across the diagonal.

σ=

b

σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33

an additional piece of information to be controlled (the datum plane orientation), an unbound vector is mathematically no longer sufficient to describe it (Fig 12-1). A second-order tensor must be used, which is just a larger matrix to accommodate more information. The internal reaction forces, while acting at an arbitrary angle with a small datum plane inside the body, cause either traction (shear), compression, tension, or a combination of these. At each infinitesimal datum plane, there will be a specific stress component that is related to the orientation of the plane that was chosen as a reference. Because of the way it is mathematically defined, with an infinitesimal calculus base that allows dealing with continuous surfaces, stress is mathematically a better concept to relate to bodily deformations than forces alone. The fundamental difference between force and stress is that force is a simple vector while stress is always a vector defined with respect to an area (datum plane), also called a second-order tensor, that can be used for proper physical and mathematic treatment of deformable continuous entities (bodies).

Types of stress Bodies can deform by shear (resisted by the shear modulus, a material property) or tension and com-

σxx σxy σxz σyx σyy σyz σzx σzy σzz

τxy τxz τyx σy τyz τzx τzy σz σx

pression (resisted by the bulk modulus, which is related to the modulus of elasticity and Poisson’s ratio). A shear stress is created by a force acting within or parallel to the datum plane, while a normal stress is created by a force acting perpendicular to the plane. Shear stresses (Fig 12-2a) create a change in the shape of the body without a change in volume and are defined in number pairs that are equal (xy = yx). Normal stresses (Fig 12-2b) create a change in volume and are defined as a single value (x or y).

Stress tensor The stress tensor is defined by the stress components obtained using the datum planes that form a coordinate system as a reference. The datum planes can form an area (two-dimensional [2D]) or volume (three-dimensional [3D]) as small as can be imagined, and hence stress is a point property; however, unlike with force, this point has an infinitesimal size defined. In a single plane (2D), two equal shear stresses (xy and yx) are defined as well as two unique normal stresses (x and y). Hence, the tensor quantity is defined by a 2 × 2 matrix. In 3D, three pairs of shear stresses (xy/yx, xz/zx, yz/zy) and three normal stresses (x, y, z) exist, forming a 3 × 3 matrix (Fig 12-3). Therefore, although the stress tensor has nine components, only six are independent. 211

12 Stress, Strain, and the Biologic Response

a

b

Fig 12-4  (a) General state of stress. σ, normal stresses; Ƭ, shear stresses. (b) Principal stresses in the principal (x, y, and z) directions.

Equivalent stresses As explained earlier, the definition of the stress components depends on the establishment of arbitrary datum planes, but the tensor itself is always the same physical quantity. The logical consequence of this is that although there is a single phenomenon on the body being loaded, there are multiple algebraic representations possible depending on the reference planes. It is important to keep in mind that this property is shared by forces; ie, a force can be described in different ways depending on the coordinate system used to define its components, but this does not change the physical nature of the force.

magnitudes, not their directions (the algebraically largest number is the first and so on). The magnitudes and directions of the principal stresses are invariant; thus, they are a simple way to present stress field results.

Pressure (scalar) Pressure is typically considered a special case of stress that can be defined as a scalar (force divided by area, F/A); ie, only one piece of information is sufficient to describe it. For instance, when applying a load to a gas or fluid inside a container, there is no shear, and stresses in all directions are equal. In that case, the stress state can be defined by a single number.

Principal stresses In the midst of all mathematically equivalent stress tensors, there is a particular coordinate system– body orientation in which all the shear stresses become zero. The axes of the coordinate system in which this occurs are called principal axes. Hence, any stress state can be defined by principal stresses that are only tensile or compressive but in one specific orientation. In 2D, any state can be defined by two normal components and in 3D by three normal components (Fig 12-4). The obtained maximum, intermediate, and minimum principal stresses at a point are referred to as first (S1), second (S2), and third (S3) stresses, respectively, all perpendicular to each other. These principal stresses are numbered by convention solely on the basis of their algebraic 212

Hydrostatic pressure (stress) tensor The hydrostatic pressure tensor is the part of the stress tensor that causes only a change in the volume of the body. It can be determined by averaging the three principal stresses after the stress matrix is transformed to the principal axes.

Deviatoric stress tensor This tensor is responsible for the change in shape of the body. It can be obtained by subtracting the hydrostatic tensor from the original stress tensor.

Force, Load, Stress, and Strain Fig 12-5  General 3D representation of Hooke’s law, where the “spring constant” is related to the compliance matrix. ε, strain; σ, stress; Ƭ, shear; S, stiffness factors.

Fig 12-6  Compliance matrix of a completely isotropic material (E, elastic modulus; v, Poisson’s ratio). This matrix represents the 3D material properties of a material.

Strain Strain is the technical term for a change in the spatial location of points of a body with respect to their original position. Mathematically, it is defined by the relative contributions of the displacement vector of an arbitrary point in each spatial direction. For example, in its simplest form, in one dimension, it is defined as the relative change in length of a linear body with uniform stress. All stress quantities previously defined have an equivalent strain quantity of the same name, because stress and strain are mathematically related.

shear modulus in response to shear strain. In a linear (one-dimensional) homogeneous stress state, stress equals the elastic modulus times the strain. In more dimensions, the stress in one direction affects the strain in the other directions (Poisson’s ratio). For this reason, Hooke’s law establishes the complete relationships between stresses and strains by using what is called a compliance matrix, which is a fourth-order tensor (or its inverse, a stiffness matrix), as shown in Fig 12-5. In this matrix, the S terms are related to each of the three material properties described earlier, depending on the degree of iso­ tropy and linearity of the material.

Hooke’s law

Isotropic and anisotropic materials

Stress and strain components are related to each other mathematically by Hooke’s law. In order to relate the stress and strain tensors in a general 3D case, three constants that are related to the material properties must be utilized: Poisson’s ratio, the elastic modulus (also called the Young modulus), and the shear modulus. Poisson’s ratio is the negative ratio of transverse to axial strain when a body is compressed. The elastic modulus represents the stiffness of the material in response to linear strain and the

In an isotropic material, the material properties are the same independent of the direction of the load. A good example is steel. An orthotropic material is a material that has different properties in axial, circumferential, and radial directions. A good example is a wooden trunk. An anisotropic material has different properties in every direction that require many mechanical tests to describe. In the case of a completely isotropic material, the compliance matrix assumes the simple form shown in Fig 12-6. 213

12 Stress, Strain, and the Biologic Response

Linearity of material properties Linearity typically measures whether the elastic modulus is constant (ie, whether it changes depending on the level of strain). Steel is an example of a linear material because it has a constant elastic modulus; ie, if we plot the stress-strain curve during load application, the graph is a line before plastic deformation occurs. Rubber is an example of a nonlinear material because it gets stiffer with strain, hence the graph is not a line. There are many types of nonlinear behaviors, and an appropriate mathematic model is typically developed to describe each. The mathematic formulation of Hooke’s law becomes progressively more complicated with complex nonlinearity and anisotropic behaviors.

Viscoelasticity Simple elasticity is found in materials such as steel. When a load is applied, elastic deformation occurs immediately. If the elastic limit is not surpassed, the material returns to its original shape immediately. Viscous materials resist the application of strain over time. They behave somewhere in between a liquid and a solid. Materials considered viscoelastic have mixed viscous and elastic properties. The dominating property can depend both on the strain magnitude and the rate at which it is applied.

Von Mises stress The von Mises stress criterion is often cited in biologic studies. However, it is questionable to apply von Mises stress to PDL, bone, and tooth. It is independent of the hydrostatic stress component of the tensor and hence is better applied to the yield of ductile materials such as metals. Results of von Mises stresses in the dentoalveolar structures do not seem to have any obvious relationship with known biologic responses.

Material Properties of the Dentoalveolar Complex This section discusses the role of dentoalveolar material properties in stress and strain analysis for orthodontic tooth movement. A common challenge for engineers in the analysis of behavior of bodies or structures is the choice of how to model the material properties of a substance to solve a specific problem. Some materials, like 214

steel, are relatively simple and can often be modeled linearly and isotropically in their elastic range. Others, like concrete, composite materials, wood, and even some alloys commonly used in orthodontics like nickel-titanium (Ni-Ti), require more comprehensive testing for the full understanding of their properties, and thus the modeling mathematics are more complex. With organic materials, like enamel, dentin, bone, and PDL, real mechanical properties are complex as well. Because of the nature of tissue formation and apposition, their internal structure, and the presence of fluids and solids, these materials have viscoelastic, nonlinear, and anisotropic characteristics to different extents. Perhaps the best and most general statement about modeling structures and the necessary amount of detail that needs to be put into a material model is an old engineering saying: “All models are wrong, but some of them work.” In other words, the decision as to how much detail is necessary to model a material is often dependent on what problem needs to be analyzed, what answer is being sought, or what hypothesis is being tested. These types of decisions require some experience and expertise, just like in any scientific field that utilizes a particular experimental model to study a particular subject. Hence, the judgment of whether a material model is appropriate to describe a material is dependent on the context of the problem. For instance, if an engineer is modeling a large and very flexible beam supported by reinforced concrete to find out how the beam behaves, the relative difference in stiffness of the materials demands that the beam material properties be modeled with much more detail than the concrete, which could be modeled as a general linear isotropic material even though, in reality, it is nonlinear and anisotropic. In other cases, experimental difficulties arise, and the engineer may have to (1) assume some of the properties of a specific material, (2) compare simpler loading scenarios in the analytical or numeric model to simple experimental results to verify if the material simplifications are plausible, and then (3) estimate an error. This is called model validation. It is known that enamel, bone, and dentin have anisotropic properties. Bone and dentin also demonstrate significant viscoelasticity. However, in the context of modeling tooth movement, there is a difference of four to five orders of magnitude in the properties of these stiffer materials and the PDL.6 For this reason, most authors who construct numeric (finite element) models decide to model the mineralized structures with simple isotropic and linear material properties, because it is the PDL behavior

Material Properties of the Dentoalveolar Complex that mostly controls the movement of the tooth. The approximate elastic moduli of dentin and enamel are quite well known, and sometimes authors decide to model these two materials as a single material because the differences in tooth movement analysis results are negligible. On the other hand, the stiffness of certain bone areas and the PDL may be very similar because bone material properties vary depending on the degree of mineralization and type of bone. For this reason, many authors often assign the bone heterogeneous material properties.7 For instance, when an animal computed tomography (CT) scan is used, it is possible to make the material properties proportional to the gray values that are representative of bone mineral density. Whether this is going to be significant or not in the model results depends again on what problem is being analyzed. The PDL is the most important structure when modeling tooth movement.8,9 If one wants to look at stresses in the PDL with ultimate accuracy and within microns of resolution, the blood vessels, PDL fibers, and cells/intercellular matrix must be modeled. The first difficulty involved in discovering the exact material properties of each of these structures is to isolate them and test them in a practical way, but their properties are all mathematically complex. The second difficulty is to properly determine their spatial position, because they are soft tissues that are difficult to image in 3D altogether with other mineralized structures. Because of these difficulties, experimentally validated material models of the PDL model it as a single material to describe the behavior of these structures altogether. The goal is to find a way to mathematically model the PDL behavior as a single material and at the same time make sure the experimental results match the results of the numeric model (eg, tooth movement or the expected biologic reaction in the PDL, bone, or root). The anisotropy of the PDL is an obvious reality because of the previously explained irregularity of the presence of the PDL components. Although attempts have been made, it is difficult to establish a testing protocol that would encompass material properties in all necessary directions of PDL loading and completely describe PDL behavior. To further complicate matters, the PDL, like rubber, also pre­sents a stiffening behavior with strain and is therefore nonlinear. In rats and pigs, a bilinear isotropic material model with a strain threshold was previously validated by matching simple tipping tooth movement with significant accuracy.6,10 This means that a linear PDL model may be reasonably accurate for smaller loads that do not reach that threshold. Viscoelasticity is

another material property that is often discussed. Because biologic phenomena typically occur after a relative steady state is reached in PDL strain, the viscoelastic behavior has been de-emphasized by most authors, and studies of the phenomenon are less common. In summary, there is no absolute correct model of the PDL. Some less detailed models may work more efficiently than highly detailed models, depending on the hypothesis being tested, by sacrificing material behavior that is not relevant. For analysis of biologic responses, it appears that an isotropic nonlinear PDL model could be sufficient to approximate PDL stresses, as the expected tooth movement can correspond very closely to reality. So there is evidence that either linear or nonlinear models could work to describe biologic responses, depending on load magnitude (and the critical strain threshold). The most common numeric method used to simulate orthodontic tooth movement and calculate mechanical quantities based on the material properties of the dentoalveolar structures is the finite element method. The finite element method attempts to calculate mechanical quantities at any point in the body using two fundamental strategies: (1) discretization of the body into smaller and relatively regular pieces (elements) that together better approximate the shape and deformation behavior of the body, and (2) combining mathematic treatments of static equations for the smaller elements to solve the problem for all bodies. For example, a curved surface could be approximated by small planar elements that have their shape defined linearly (straight plane) or nonlinearly (a quadratic element), or a curved volume could be approximated by small volumetric entities (linearly or nonlinearly defined tetrahedrons or bricks). The density of the element mesh must be enough to bring the solution of the problem to an acceptable error. This is usually performed by conducting convergence analysis; ie, the element mesh density is increased until the desired solution no longer varies significantly within the mechanical context of the problem. Typically, quadratic elements better approximate the initial and deformed shape of organic entities and tend to produce convergence faster than linear elements. It is often discussed among investigators whether stresses or strains are a better mechanical measurement to relate to a biologic response.11 There are many points that need to be considered when discussing this matter. First, it is a problem of mechanotransduction (ie, which type of mechanical stimulus generates which type of biologic response). For instance, failure criteria of viscoelastic materials are 215

12 Stress, Strain, and the Biologic Response

Developmental and physiologic genetic signaling

Fig 12-7 Diagram of contemporary orthodontic biomechanics. Environmental effects

Dentoalveolar morphology, physiology, and material properties Orthodontic appliance Orthodontic force system

Functional loads

Finite element analysis

Mechanical environment Mechanotransduction/ adaptive genetic signaling

Environmental effects

Increased modeling and remodeling (tooth, PDL, and bone) Tooth movement

experimentally related to stress. Hence, if one would think about crushing or destruction of tissue, perhaps stress could be a better measurement. Piezoelectricity is also related to stress, so if there was any contribution of that in the tissue response, stress is a better measurement to relate to it. One could think that the strain, which represents the actual cellular deformation (which would require a different stress in hard or soft tissue) could be a parameter that better relates to the response over time. Choosing strain would then make sense to analyze different cells in tissues of different material properties. However, it has been shown that the material properties of the tissue change with time.12 For example, cells change their configuration and function to maintain a constant stress at different parts of their structure, due to increases in actin production, which reinforce the cytoskeleton within cells to maintain a stress threshold homeostasis. In that case, stress could be a better parameter to relate to the response because it is independent of the mechanical property (depending only on force and area). On the other hand, the measured stress–biologic response relationship would be restricted to the specific tissue in question. 216

In spite of these difficulties, the controversy of stress versus strain is probably of secondary importance11 if we think that stress and strain fields for tooth movement have subtle differences and are numeric approximations. When stimulus-response relationships are established, picking maximum or minimum stress or strain seems to have little effect on the results of the ultimate goals, which are to establish an approximate level of relative load equivalence between animal and human models and relationships between clinical orthodontic loads and the expected tissue or molecular response.

Orthodontic Mechanotransduction The fundamental biologic responses attributed to orthodontic loads are bone formation, bone resorption, and iatrogenic external root resorption. In the classic view of orthodontic tooth movement, bone formation is associated, somewhat vaguely, with a tension side and resorption with a compression side.13,14 Other theories have attempted to link tooth movement to flexion of the alveolar wall.15–17

Orthodontic Mechanotransduction

a

b

Fig 12-8  (a) Computer-aided design model showing internal morphology. (b) Finite element model and coordinate system.

The tension-compression theory is the most popular, and it is ubiquitous in orthodontic research. Orthopedics proposes several mechanisms of mechanotransduction attributed to bone cells, but applicability to orthodontics is unclear, and there is no analogy to root resorption. Although clarification has been attempted—eg, parametric analyses and partial descriptions—knowledge gaps abound at virtually all steps of the cascade shown in Fig 12-7. Because the dentoalveolar assembly consists of complex shapes of three materials (tooth, bone, and PDL), closed-form analytical calculations to characterize the mechanical environment are impossible. Useful results are obtainable with experimental and computer-based numeric models, such as finite element analysis (FEA). Intricate objects such as the dentoalveolar complex should be modeled in three dimensions. Even loads initially applied to volumetric bodies in one dimension can lead to stresses in three dimensions, depending on the supporting constraints. If a cube is uniformly compressed in the y direction, it is free to expand in the x and z directions (Poisson’s effect), and there will be stress only in the y direction, although there is strain in all directions. If the cube’s expansion is constrained in the x or z directions, there will be compressive stresses in all directions because the constraints will also load the body in response to Poisson’s effect. These two scenarios differing on constraint are an example of how strain and stress fields can have subtle differences. It also explains why stresses of smaller magnitude develop in the PDL in directions other than that of the force. Therefore, it is inadequate to model the mechanical environment of the PDL as a simple pressure scalar, as is still often done in the orthodontic literature.

The mechanical environment for tooth movement and root resorption This section summarizes the mechanical environment in the dental structures that are subject to known orthodontic loads. The author used FEA to calculate the stresses and deformations produced during distal translation or tipping of a stylized maxillary right canine into a first premolar extraction site (Fig 12-8). In addition to the typically considered principal stress magnitudes, emphasis was placed on their directions. With this approach, two critical ambiguities in the orthodontic tensioncompression concept were examined. First, the specific tissue in which the pertinent tensioncompression acts is sometimes not explicitly stated. It is occasionally unclear whether the mechanical environment under consideration is acting in bone, root, or PDL. Second, attention is generally not paid to the direction of the tension-compression when interpreting stress fields. Nonetheless, there have been attempts to correlate biologic activity with changes in the mechanical environment as characterized by the six stress components,3 2D principal stresses,18 or hydrostatic stresses.19 Because of the previously mentioned limitations, the results are inconclusive. In the literature, the mechanical environment is typically depicted with color-coded stress magnitude gradients superimposed on the structure or line graphs, showing stress or strain magnitudes along specific paths that, until recently, showed no regard for their directions.20

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12 Stress, Strain, and the Biologic Response

0

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5 Distance from apex (mm)

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Tipping 25 30 –0.0008

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Fig 12-9  (a) Exaggerated (10 times) displaced positions of the root. CRot, center of rotation; CR, center of resistance. (b) Deformed root shapes. (c) Exaggerated (5,000 times) total tooth displacement in translation. The scale is in millimeters.

For descriptive purposes in this chapter, a cylindric coordinate system nomenclature affords a relatively precise and unambiguous way to discuss the mechanical environment in the alveolus, even if it is not entirely applicable to the tapered root or socket. In a cylindric coordinate system, the three mutually perpendicular directions are longitudinal, radial, and circumferential (hoop). The longitudinal direction is in the apicogingival direction. The radial direction could be represented by the spokes of a wheel, and the tangential (circumferential) direction would be the tangent to the wheel perimeter. Thus, on the mesial and distal sides of the tooth, the circumferential direction is in the buccopalatal direction, and the radial direction corresponds to the mesiodistal direction. On the buccal and palatal sides of the tooth, the circumferential and radial directions are the mesiodistal and buccopalatal directions, respectively. The first matter to understand is that the stress fields on the root, PDL, and bone are completely different and that the stresses with the largest absolute magnitudes in each one occur in different locations in the apicocoronal direction. For instance, 218

the highest tensile and compressive stresses on the tooth root during tipping are near the middle of the root, and in the PDL they are near the apex and cervical region. Let us examine in detail the case of simple distal crown uncontrolled tipping of a canine in order to understand what happens with compressive stresses on the tooth root. There are three principal stresses acting at the root surface. One is the most intuitive, and it is the compression that would be naturally expected near the cervical region in the direction the force points and in the apical region in the opposite direction (as one would expect from the tension-compression theory). Another is generated by the bending of the root within the socket, caused mostly by the force at the crown and the load reaction along the root, within the alveolus. This bending generates a pattern of root deformation as shown in Fig 12-9, which is very similar (but in the opposite direction) in translation and tipping. Looking at that pattern of deformation and the shape of the root, it is somewhat intuitive to understand that the highest compressive stress will be at the distal side, slightly apical to the center of the root and in the

Orthodontic Mechanotransduction

a

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Fig 12-10  Principal stresses associated with tooth tipping in the root: (a) distal-side S3 (S1 is almost zero); (b) mesial-side S1 (S3 is almost zero); (c) distal-side S2; (d) mesial-side S2.

longitudinal direction, and is caused predominantly by constraint in the radial direction. The remaining stress is negligibly small and is the one generated by the natural circumferential constraint of the socket (see earlier explanation about Poisson’s ratio). Note that the most compression here (ie, third principal stress) is the longitudinally directed stress (Fig 12-10) and that known locations of increased root resorption (cervical and apical regions), for instance, do not match with this stress distribution. Analysis of the stresses in tension also lead to the same conclusion. So if we make the reasonable assumption that the stress of highest magnitude will affect the response more than the others, it seems unlikely that root stresses and strains (and, consequently, cells) play a major role in the mechanotransduction of root resorption. The second entity of interest we will consider for the same loading scenario is the bone surface, and it is the most complex-shaped entity of the three. Therefore, it has the most complex deformation pattern and, furthermore, the most complex stress

distribution pattern. There are many nuances on the bone stress field, including simultaneous tension and compression in different directions at the same location, but the most interesting aspect is that, in the distal cervical side and the mesial apical side of the bone surface adjacent to the PDL, the largest absolute stress on the bone surface is compressive (Fig 12-11). In the cervical region, it is in the circumferential direction, and in the apical region it is radial. The magnitude of compressive stresses would fall into the “bone formation” category in Frost’s mechanostat (a theory that relates bone modeling with strain). However, biologically, it has been shown multiple times that this side of the bone is resorbed during tooth movement. Logically, this suggests that the mechanism that dominates tooth movement is independent of Frost’s mechanostat—ie, that the bone is not the main mediator of orthodontic mechanotransduction. By the process of logical elimination, the PDL should be the only candidate. Indeed, if we observe the pattern of PDL stresses, it is always consistent with the expected biologic re219

12 Stress, Strain, and the Biologic Response

a

b

Fig 12-11  3D direction plots of principal stresses associated with tooth tipping in bone: (a) distal side (canine socket is on the right); (b) mesial side (canine socket is on the left).

a

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Fig 12-12  Principal stresses associated with tooth tipping and their components on the distal side of the PDL: (a) S1; (b) S2; (c) S3.

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sponses of bone and root resorption. At the distal side, the PDL is compressed in all three directions simultaneously, and the locations where we expect the bone to resorb are perfectly proportioned to the principal stress compression patterns (as well as for tension and formation). The most important thing to acknowledge about the PDL is that the stresses have different magnitudes in different directions. 220

The largest absolute stress is in the radial direction, followed by the other two directions that are approximately one-third of the former (Fig 12-12). These data point out that the principal stresses in the PDL, particularly the first and third, are good candidates to study orthodontic mechanotransduction.9

Orthodontic Mechanotransduction

Bone remodeling, modeling, and root resorption Once the PDL maximum (first) and minimum (third) principal stresses were determined as candidates for bone formation and resorption, respectively, the problem was to demonstrate that they are indeed related to tissue and cellular response. To solve this problem, an animal model that properly mimics the mechanical environment present in human tooth movement needed to be developed.4,7,21 Logically, from the differences between force, load, and stress explained earlier, this means that a force level that is tuned to the morphology of the animal tooth must be used. The influence of tooth size on the loadstress relationship is not trivial, and it cannot be derived with simple calculations such as force divided by overall area. Tooth morphology and the type of movement are also important complicating factors because they result in a great range of stresses and distributions. If a study aims to obtain a biomechanical animal model that is clinically relevant, it is first necessary to perform a finite element stress analysis on a human tooth subjected to a clinically realistic orthodontic load system to ascertain the associated stresses in its structures. Then, with a finite element model of the animal tooth in question, the orthodontic load magnitude necessary for the animal model to approximate a similar stress magnitude range in the human tooth can be determined. Thus, rather than matching orthodontic load levels, physical matching must be done on the calculated stress levels and directions. To match stress directions, the complex shapes of rodent teeth require histologic sectioning planes that consider the 3D nature of stresses and strains. It is incorrect to assume that compression and tension areas in the PDL are always along the line of action of the force because, typically, with forces with a 3D offset to the center of resistance, tooth displacements and rotations may occur in all three planes of space. Therefore, the directions of compression and tension must also be calculated in order to obtain 2D sections where maximum tension and compression occur in the different sides of the PDL. For the purposes of the author’s studies, sectioning planes in a rodent tooth root were used to approximate the stress distributions associated with a single-rooted tooth (ie, a canine) tipping in one plane. If such a match is found, the other mouse roots become irrelevant because the model is stress

based, and specific morphologic features are merely circumstantial. There are also biologic complications involved in controlling stress and strain in orthodontic experiments. Most studies are performed on animals with diverse genetics and highly variable tooth morphology. This results in highly variable biologic responses that obscure the interpretation of variability into purely biologic (eg, inflammatory) and mechanical (stress differences due to morphology) factors. In the author’s experiments, mice and rats were used because, among other advantages, they share 99% of their genes with humans; thus, they are good animal models for studies of human biology. Mice from an inbred strain share 99.99% of their genetic material. By using a single inbred strain of mice or rats, the role of genetic variation in the dispersion of data is practically eliminated. Because inbred mice have the same tooth and bone morphology, they can be modeled by a single finite element model. By using stress calculations from a finite element model of a human canine subjected to an orthodontic load, it is possible to calculate the orthodontic load that would cause similar peak stress levels in mice or rats. Furthermore, with the rodent finite element model representing all rodents within a specific inbred strain, it is possible to localize histologic planes in which the stress magnitudes and directions approximate those of the human PDL compressive and tensile stress zones and map histologic sections to the finite element data. A 3-cN force in mice and a 10-cN force in rats are approximately enough to model the largest compression found in a human canine being tipped with 1.2 N of force applied at the bracket. All three loads generate absolute values of compression of around 20 kPa at the highest compressive region of the smallest root. Because the chosen root is significantly smaller than the others, the axis of rotation of the tooth will be apical to this root’s apex. Hence, the smallest root reveals a gradient of compression in the PDL where it is possible to analyze the tissue and cell response with decreasing stress, minimizing the need to apply different loads. The first hypothesis tested in the animal model was whether the PDL compression would affect the bone away from the PDL interface. Using microcomputed tomography (microCT) imaging techniques, a sample 3D sphere image of bone was taken in between the roots of mice, and the bone morphometric parameters were compared. When orthodontic force was applied, there was roughly a 40% decrease in bone volume, with a 30% decrease in over-

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12 Stress, Strain, and the Biologic Response

20 kPa

10 kPa 250 µm

all bone density away from the PDL-bone interface. Many other parameters of bone quality also decreased. This suggests that the inflammatory mediators generated by PDL compression spread widely enough to start general bone remodeling in a relatively large area around the tooth socket. That is, independent of local effects, tooth movement is largely facilitated by a regional acceleratory phenomenon (which increases bone remodeling) generated by PDL inflammation. After the PDL is compressed, if the stress is large enough, tissue necrosis may occur. This necrosis can occur through different mechanisms, such as direct crushing of cells, interruption of blood flow, or induction by inflammatory factors. The relationship of these factors with stress is not static; ie, time can also contribute to necrosis. Another form of cell death that can be induced by stress is apoptosis, but it tends to be silent physiologically, perhaps only accelerating tissue remodeling. If compression is sufficiently large, cells will not be able to resorb tissue from within the PDL space and will therefore need to remove tissue (root and bone) around the compressed area to reach the necrotic tissue. In the mouse model, the author showed that the PDL tissue can undergo necrosis with stresses over approximately 10 kPa after 10 days of force application (see Fig 12-10). Because the mouse study was limited in time (to 10 days), a second study in rats was later performed where the force was actively maintained for 3, 15, 21, or 30 days. It was confirmed that, near areas of necrosis, the number of resorbing cells (macrophages, osteoclasts, and odontoclasts) and root resorption were significantly larger than in non-necrotic areas even after 30 days. The most important finding was that no necrosis occurred in the PDL in a region where the largest absolute compressive stress (prin222

Fig 12-13 Visual relationships between initial third principal stresses (PDL), necrosis, indirect bone resorption, root resorption, and direct bone resorption in a mouse model after 10 days of force application. Note that root resorption occurs mostly near the area of significant hyalinization. The figure suggests that stresses above 10 kPa tend to lead to necrosis, while stresses below that number tend to promote direct resorption.

cipal) was between 7.8 and 9.9 kPa. The bone in this region of the PDL underwent direct resorption, and necrotic cells were never observed within it in any time period. Moreover, in this necrosis-free region, root resorption was not significantly different from the controls at any time during the experiment. When the compressed tissue does not die, it is possible for osteoclasts to remove bone directly from within the compressed PDL zone, and it was observed that root resorption in these circumstances is minimal. This occurred in animal models when the third principal stress was kept below approximately 10 kPa (Fig 12-13). For an average canine-tipping finite element model, keeping a tipping force under an average of 40 cN should prevent PDL necrosis. For translation, this value would be around 120 cN (roughly three times more). For controlled tipping, the value would be somewhat in between these two. Note that these are purely frictionless forces, and in the context of orthodontic treatment these values could change depending on the frictional mechanism used to move the tooth. The relationship between stresses and biologic response is complicated, especially when time is built into the problem (Figs 12-14 to 12-16). For instance, in the compressive side of the PDL, the number of resorbing cells is proportional to the third principal stress after 3 days. However, at 15 days, the cells are actively resorbing necrotic tissue, and things start to change (see Fig 12-14). Although resorption of the bone would imply less root support and a higher stress concentration, the number of resorbing cells decreased after 15 days and continued to decrease afterward up to 30 days. The most likely explanation is that anti-inflammatory mediators are released after phagocytosis (as a signal for resolution of inflammation) or after necrotic tissue is resorbed. The stresses will change as the bone resorbs, increasing

Orthodontic Mechanotransduction

a

OERR (× 1,000 mm3)

r

30 d Cervical middle

Time (d)

Cervical

3d

b

b

r

b

3rd principal stress

r

b

Apical

Apical

b

r

c

Fig 12-14  Orthodontic external root resorption (OERR) and PDL necrosis over time. (a) Illustration of the average third principal stress on the PDL of the area of interest and the corresponding representative microCT sections. The tooth is represented in white, and the red areas are resorption cavities, illustrating the method of resorption quantification on each of 20 sections per specimen. Resorption was measured volumetrically in three average stress zones over time. (b) Relationship between OERR and third principal stresses in different time groups. (c) Acellular or pyknotic area of necrosis (hyalinization, indicated by triangular arrows) in the PDL as classically determined in the hematoxylineosin (H&E) stain (r, root; b, bone). No hyalinization was ever detected in the apical (low-stress) region at any time point after force application in any of the H&E sections examined.

underneath the bone and decreasing outside of it. In spite of this, the biologic effect (mediators) likely dominates, and the number of resorbing cells decreases. This continued to happen up to 30 days, when the cycle started over after the original necrotic tissue had been resorbed and a new necrotic area was formed, likely leading to a new “spike” in

pro-inflammatory mediators. Basically, although initially there is a proportional relationship between stresses and resorptive activity, there is no constant relationship between the number of cells and resorbing activity throughout time. Although this was the first quantitative description of the relationship between the mechanical 223

12 Stress, Strain, and the Biologic Response

3d

15 d

21 d

30 d

Apical

Middle

Cervical

Control

a

15 d

30 d

b Fig 12-15  Histologic features and events in the root resorption cycle. (a) Representative photographs of the TRAP+ (small triangular arrows) cell populations and events in different root regions over time. The panels are focused in different areas of each region. Note the necrosis, evidenced by tissue degeneration or pyknotic nuclei near the alveolar crest at 3 days. Active orthodontic external root resorption (OERR) occurs near the necrotic area at 15 days. At 21 days, past resorption can be observed on the root, and necrosis is seen under the bony spicule. At 30 days, almost half of the alveolar crest has been resorbed, and a new stress concentration zone develops a new resorption cycle in a more apical location of the high-stress (cervical) zone. (b) Whole-root view of two highly active OERR time points due to the processing of necrotic tissue from highly stressed PDL zones. At 15 days, it is possible to visualize a pattern of OERR that is proportional to initial PDL stress; ie, more OERR occurs near the cervical region. At 30 days, the alveolar crest is partly resorbed, new necrosis occurs more apically, and OERR increases near the middle root zone. b, bone; r, root; p, PDL; s, bony spicule; oc, osteoclast; m, macrophage; odc, odontoclast.

environment and the cellular response over time during tooth movement,7 these findings cannot be generalized. The timing of these reactions may change depending on the animal species, rate of bone remodeling, occlusal load, and genetic profile. A time-based description of tooth movement phe224

nomena is probably largely based on specific characteristics of the experimental model and the mechanical environment. That is, whereas in this rat model the root resorption cycle seems to restart between 21 and 30 days (see Fig 12-15), it could be different in humans or even another rat strain.

References

Fig 12-16  Changes in overall TRAP+ cell population in the different root regions (asterisk denotes significant differences of P < .05). These results suggest that the population is directly related to initial stress, but it then changes, likely due to the change in mechanical environment (stress relief) and release of anti-inflammatory mediators after necrotic and mineralized tissue resorption.

Conclusion

References

The unveiling of the relationships between tissue response and mechanical stimulus locally in the PDL facilitates the understanding of the molecular mechanisms of mechanotransduction. Past work has indicated a mechanism of how different individual responses to cellular damage and necrosis may affect orthodontic responses.4,21 A comprehensive review of molecular aspects of mechanotransduction in tooth movement is beyond the scope of this chapter but is available elsewhere.22 Much has been said about individual variation and how orthodontic tooth movement, speed, or root resorption greatly varies among individuals, but the problem has seldom been approached properly. Future research in the biomechanics of tooth movement should determine whether individual variation affects the relationship between different stresses and cellular damage or only the inflammatory response associated with it. Furthermore, the quantification of mechanical thresholds for necrosis and root resorption in a genetically diverse population together with the clarification of the role of occlusal forces are fundamental problems to be solved. This will allow the growth of an evidence base for orthodontic load dosage supported by solid basic science.

1. Ren Y, Maltha JC, Kuijpers-Jagtman AM. Optimum force magnitude for orthodontic tooth movement: A systematic literature review. Angle Orthod 2003;73:86–92. 2. Kawarizadeh A, Bourauel C, Gotz W, Jager A. Numerical study of tension and strain distribution around rat molars [in German]. Biomed Tech (Berl) 2003;48:90–96. 3. Kawarizadeh A, Bourauel C, Zhang D, Gotz W, Jager A. Correlation of stress and strain profiles and the distribution of osteoclastic cells induced by orthodontic loading in rat. Eur J Oral Sci 2004;112:140–147. 4. Viecilli R, Katona T, Chen J, Roberts E, Hartsfield J Jr. Comparison of dentoalveolar morphology in WT and P2X7R KO mice for the development of biomechanical orthodontic models. Anat Rec (Hoboken) 2009;292:292–298. 5. Verna C, Zaffe D, Siciliani G. Histomorphometric study of bone reactions during orthodontic tooth movement in rats. Bone 1999;24:371–379. 6. Kawarizadeh A, Bourauel C, Jager A. Experimental and numerical determination of initial tooth mobility and material properties of the periodontal ligament in rat molar specimens. Eur J Orthod 2003;25:569–578. 7. Viecilli RF, Kar-Kuri MH, Varriale J, Budiman A, Janal M. Effects of initial stresses and time on orthodontic external root resorption. J Dent Res 2013;92:346–351. 8. Viecilli RF, Budiman A, Burstone CJ. Axes of resistance for tooth movement: Does the center of resistance exist in 3-dimensional space? Am J Orthod Dentofacial Orthop 2013;143:163–172. 9. Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE. Three-dimensional mechanical environment of orthodontic tooth movement and root resorption. Am J Orthod Dentofacial Orthop 2008;133:791.e11–791.e26. 10. Ziegler A, Keilig L, Kawarizadeh A, Jager A, Bourauel C. Numerical simulation of the biomechanical behaviour of multi-rooted teeth. Eur J Orthod 2005;27:333–339. 11. Humphrey JD. Stress, strain, and mechanotransduction in cells. J Biomech Eng 2001;123:638–641. 12. Taber LA, Humphrey JD. Stress-modulated growth, residual stress, and vascular heterogeneity. J Biomech Eng 2001; 123:528–535.

225

12 Stress, Strain, and the Biologic Response 13. Krishnan V, Davidovitch Z. Cellular, molecular, and tissue-­ level reactions to orthodontic force. Am J Orthod Dentofacial Orthop 2006;129:469.e1–469.e32. 14. Reitan K. The initial tissue reaction incident to orthodontic tooth movement as related to the influence of function: An experimental histologic study on animal and human material. Acta Odontol Scand Suppl 1951;6:1–240. 15. Baumrind S. A reconsideration of the propriety of the “pressure-tension” hypothesis. Am J Orthod 1969;55:12– 22. 16. Epker BN, Frost HM. Correlation of bone resorption and formation with the physical behavior of loaded bone. J Dent Res 1965;44:33–41. 17. Melsen B. Tissue reaction to orthodontic tooth movement—A new paradigm. Eur J Orthod 2001;23:671–681. 18. Katona TR, Paydar NH, Akay HU, Roberts WE. Stress analysis of bone modeling response to rat molar orthodontics. J Biomech 1995;28:27–38. 19. Dorow C, Sander FG. Development of a model for the simulation of orthodontic load on lower first premolars using the finite element method. J Orofac Orthop 2005;66:208– 218.

226

20. Cattaneo PM, Dalstra M, Melsen B. The finite element method: A tool to study orthodontic tooth movement. J Dent Res 2005;84:428–433. 21. Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE. Orthodontic mechanotransduction and the role of the P2X7 receptor. Am J Orthod Dentofacial Orthop 2009;135: 694.e1–694.e16. 22. Krishnan V, Viecilli R, Davidovitch Z. Cellular and molecular biology behind orthodontic tooth movement. In: Krishnan V, Davidovitch Z (eds). Biological Mechanisms of Tooth Movement. Oxford: Wiley-Blackwell, 2014.

Recommended Reading Beer FP, Johnston ER, DeWolf JT. Mechanics of Materials, ed 3. New York: McGraw-Hill, 2002.

PART

III Advanced Appliance Therapy

CHAPTER

13 Lingual Arches “Necessity ... the mother of invention.”

— Plato



“Almost all new ideas have a certain aspect of foolishness when they are first produced.”

— Alfred North Whitehead

“Any man may easily do harm, but not every man can do good to another.”

— Plato

“New opinions are always suspected, and usually opposed, without any other reason but because they are not already common.”

OVERVIEW



— John Locke

Precision lingual arches can be quite versatile if used independently or in combination with a facial archwire. Many malocclusions have discrepancies that are best solved by bilateral mechanics rather than the use of adjacent teeth as in a continuous arch. The lingual arch that connects only two attachments bilaterally is a simple system for understanding bracket-wire interactions. New designs allow for insertion of a horseshoe or transpalatal lingual arch in the maxilla. The traditional “ideal” arch shape may not be the correct shape if defined by a correct force system. Active applications unique from the lingual include unilateral tip-back and unilateral and bilateral molar rotation. This chapter describes in detail how to shape a lingual arch to produce a desired force system.

229

13 Lingual Arches

a

b

Fig 13-1  Two lingual arch designs connecting first molars. (a) Lingual transpalatal arch (TPA). (b) Horseshoe lingual arch.

A lingual arch can refer to many different things. Lingual archwires can be placed in multiple brackets on the lingual surfaces of the crowns. This chapter considers only lingual arches that connect two teeth across the arch, usually at the first molar (Fig 13-1). Lingual arches can be used for passive applications to preserve tooth position or for active applications to move the teeth. Passive applications include space maintenance, anchorage re­ inforcement to minimize side effects, and as a base structure for attaching auxiliary springs. Active applications include molar rotation, arch width expansion and constriction both symmetrically and asymmetrically, and unilateral tip-back.

Limitations of a Labial Appliance The lingual arch can be used by itself or inserted to complement a labial appliance (Fig 13-2). An additional lingual arch is sometimes necessary because the labial archwire has two major limitations: adjacent tooth anchorage considerations and posterior width instability. The arch has been used in architecture for thousands of years because structurally it is very stable and can resist vertical loads. It is found in many cathedrals, bridges, and triumphal arches (eg, the Gateway Arch in St Louis). However, the arch as a structure is very unstable laterally at its free ends, so it requires strong support at these ends. The same is true in a dental archwire. Even the stiffest full-size 0.022 × 0.028–inch stainless steel labial archwire may have a very low force-deflection (F/Δ) rate at the free end if loaded with a lateral force (Fig 13-3). Therefore, loss of terminal molar width can be frequently observed after application of Class II and Class III 230

elastics, from headgear forces, and during interarch alignment because the wires may have been subjected to some lateral components of force. Consider the use of a low-stiffness nickel-titanium (Ni-Ti) wire that is straight without any arch form. If placed from terminal molar to terminal molar, it may effectively align the teeth; however, the low stiffness of the wire probably will not change the arch form. Related to width stability is molar buccolingual axial inclination control. An edgewise arch fully engaged in the molar tube or bracket in theory might actively control or passively keep the buccolingual molar inclination; in practice, however, the wire “play” allows for molar inclination and potential width changes as a result of vertical or horizontal components of force from an elastic or a headgear. A more significant limitation of a labial archwire is its inherent anchorage selection, where adjacent teeth determine the anchorage and force system produced. In Fig 13-4, first molars are positioned bilaterally and symmetrically to the buccal. A labial archwire uses second molars and second premolars as anchorage (red forces), most likely leading to side effects on the teeth adjacent to the molars (Fig 13-4a). A lingual arch, however, works across the arch to utilize reciprocal anchorage (Fig 13-4b). Many useful possibilities exist for applying crossarch anchorage selection either with symmetric or asymmetric force systems. The labial arch gives only limited options for using adjacent teeth for anchorage to move molars. Molar anchorage for posterior tooth movement opens up more useful possibilities for sound mechanics. The cross-arch distance between two molars is one of the largest interbracket distances available in the oral cavity. Increased interbracket distance provides many advantages, such as low F/Δ rate, increased

Attachments

Fig 13-2  The lingual arch can be used alone, or it can be inserted to complement a labial appliance, because any labial appliance has inherent limitations.

a

Fig 13-3  Even a full-size 0.022 × 0.028–inch stainless steel labial archwire may have a very low F/Δ rate if loaded with a lateral force at its free end.

b

Fig 13-4  (a) A labial archwire uses second molars and second premolars as anchorage for moving a first molar, most likely leading to side effects on adjacent teeth. (b) A lingual arch working across the arch can utilize reciprocal anchorage without any side effects.

range of action, increased moment arms, and ease of evaluating wire-bracket geometry. Because of this, the lingual arch can be one of the simplest fixed appliances where wires are inserted into brackets. Methods of fabrication, insertion, and removal of lingual arches are found elsewhere,1,2 while this chapter places emphasis on the biomechanics of the lingual arch.

Attachments For passive applications, the lingual arch can be securely soldered to a band; however, attachments for removable wires allow for frequent active and passive adjustment changes when indicated. A folded 0.036-inch (0.9-mm) stainless steel wire fits snugly into a lingual sheath in Fig 13-5. There can still be some play, and the sheath commonly deforms, changing shape so that full control with six degrees of freedom is lacking. Therefore, a robust and more

precision fit lingual bracket (Fig 13-6) or a hinge cap bracket (Fig 13-7) is preferable. In a labial archwire, there is always a little play needed between the bracket and the wire, even with a full-size wire, because sliding mechanics is commonly required. By contrast, a 0.032 × 0.032–inch square wire accurately fits in the slot of a precision lingual bracket or hinge cap bracket so that full three-dimensional control with six degrees of freedom is assured. In very special applications, a 0.032-inch round wire is used to allow some rotational movement around the x-axis of the bracket, eliminating one degree of freedom to remove unnecessary torque. Lingual precision brackets are preangulated (–12 degrees for maxillary teeth and +6 degrees for mandibular teeth) for ease of use (Fig 13-8). Orienting the bracket slot parallel to the occlusal plane at the end of treatment simplifies any twisting of the lingual arch. If buccolingual axial inclinations are initially favorable, flat wires can be easily inserted with minimal adjustment devoid of torque. 231

13 Lingual Arches

Fig 13-5 A folded 0.036-inch (0.9-mm) stainless steel wire snugly fits into a lingual sheath; however, there can still be some play so that full control with six degrees of freedom is lacking. Torque can easily deform the sheaths.

a

a

Fig 13-6  A precision fit lingual bracket uses an O-ring or metal ligature ties.

b

b

Lingual Arch Configurations In this chapter, all lingual appliances connecting just two brackets across the arch are called lingual arches. Typically, it is first molars that are connected; however, second molars can also be connected, or even canine to canine bars can comprise a lingual arch. In the maxillary arch, two designs are basic: the transpalatal arch (TPA) and the horseshoe arch. Although a TPA is usually inserted from the mesial of the hinge cap (Fig 13-9a), it sometimes is desirable to insert it from the distal (Fig 13-9b). This can avoid impingement if a torus palatinus or a lingually positioned second premolar is present. Placing a TPA 232

Fig 13-7 The precision lingual hinge cap bracket. A 0.032 × 0.032–inch square wire accurately fits in the slot so that full three-dimensional control with six degrees of freedom is assured. (a) Cap opened. (b) Cap closed.

Fig 13-8 Lingual precision brackets are pre­angulated in the third order for ease of use. (a) Maxillary arch, –12 degrees. (b) Mandibular arch, +6 degrees.

further distally can also influence the force system to produce association (described later in the chapter). The maxillary horseshoe arch has the advantage of simplicity and ease of fabrication because minimal palatal contouring is needed (Fig 13-10). Because wire orientation is at 90 degrees to a TPA, the force system is uniquely suited to special types of tooth movement, which are discussed later in this chapter. Because of the tongue, mandibular lingual arches must have the horseshoe configuration. Two types are commonly used. The high mandibular lingual arch (Fig 13-11a) touches the incisor cingulum and is used for space maintenance or for added incisor anchorage. It can also be used to prevent the mandibular incisors from tipping lingually in extraction ther-

Lingual Arch Configurations

a

b

Fig 13-9  A maxillary TPA. It is usually inserted from the mesial of the bracket (a); however, sometimes it is desirable to insert it from the distal (b).

a

Fig 13-10 The maxillary horseshoe lingual arch has the advantage of simplicity and ease of fabrication. Because the wire orientation is at 90 degrees to a TPA, the force system is uniquely suited to special types of tooth movement.

b

Fig 13-11  (a) A high mandibular lingual arch touches the incisor cingulum and is used for space maintenance or for added incisor anchorage. (b) The low mandibular lingual arch is placed below the tongue and does not touch the mandibular incisors. It is used passively to prevent side effects or actively for reverse articulation, arch width control, molar rotation, and molar tip-back applications.

apy. The low mandibular lingual arch (Fig 13-11b) is placed below the tongue and does not touch the mandibular incisors. It is more universal in its applications, including control of posterior width, molar buccolingual axial inclinations, reverse articulation mechanics, and as a base for finger springs. The low mandibular lingual arch should be fabricated as far apical as possible so that the tongue will not exert any vertical or forward force on it. Its low position has the added advantage of a smooth curvature that is easy to fabricate and fit because contouring around irregular teeth is not required (Fig 13-12).

Wire size and material The F/Δ rate of the lingual arch can be varied by altering the overall configuration of the arch, the wire cross section (size and shape of the lingual arch), and the material. When a high F/Δ rate is

Fig 13-12 The low mandibular lingual arch is placed below the tongue so that its low position has the added advantage of a smooth curvature that is easy to fabricate and fit because contouring around irregular teeth is not required.

needed for a passive application, full-size 0.032 × 0.032–inch stainless steel wire is used. For active applications, 0.032 × 0.032–inch beta-titanium alloy is better because the modulus of elasticity is only 0.42 that of stainless steel; thus, the force magnitude is 0.42 times that of stainless steel for an identical appliance, and the range of action is twice that of stainless steel. If undesirable torque is to be avoided, 0.032-inch stainless steel or betatitanium round wires are used. Table 13-1 summarizes the relative stiffness of different lingual archwires by shape and dimension of different wire cross sections and materials. For simplicity, the relative stiffness of a 0.036-inch round stainless steel wire is denoted as a base value of 1.0. Note that a full range of wire stiffness and forces can be obtained (with and without third-order torque) using a 0.032 × 0.032–inch bracket; these wires have a precision fit with minimal play. 233

13 Lingual Arches

Table 13-1  Relative stiffness of lingual archwires Material

Wire size (inch)

Relative stiffness*

Stainless steel

0.036

1.0

Stainless steel

0.032 × 0.032

1.06

Stainless steel

0.032

0.62

Beta-titanium

0.032 × 0.032

0.45

Beta-titanium

0.032

0.26

*The relative stiffness of a 0.036-inch round stainless steel wire is denoted as a base value of 1.0.

a

b

c

d

Fig 13-13 One of the simplest applications of a lingual arch is as a space maintainer. The molar is prevented from tipping forward by the arch contact on the cingulum of the mandibular incisor.

Fig 13-14  An extraction case treated without a lingual arch. (a) Before space closure. The anterior force from the space closure spring acts buccal to the CR. The replaced force system at each CR is shown with yellow arrows. (b) After space closure. The mandibular buccal segments displaced not only mesially but also rotated mesial in after space closure (dotted line). (c) The force system at the mandibular buccal segment is replaced by lateral and anterior components of the forces. (d) If a rigid passive lingual arch is placed (teal wire), reciprocal equal and opposite forces and moments will cancel each other out and will be effectively avoided; however, the anterior component of force (yellow arrows) still exists.

Passive applications An important function of a passive lingual arch is to stabilize the posterior teeth together as a unit and to maintain arch width and form. Full-size stainless steel wire is used. One of the simplest applications would be a space maintainer where the anterior arc (the apex of the lingual arch) touches the lingual surface of the incisors (Fig 13-13). A lingual arch can also prevent side effects during space closure. Figure 13-14 shows an extraction case treated without a lingual arch. Only the force systems on the mandibular posterior segments are depicted. The anterior force from the space closure spring acts 234

buccal to the center of resistance (CR, red arrow in Fig 13-14a); the replaced force system at each CR is shown in yellow. As a result, the mandibular buccal segments were not only displaced mesially but also rotated mesial in after space closure (Fig 13-14b). In Fig 13-14c, the force system at the mandibular buccal segment is once again replaced by lateral and anterior components of the forces. If a rigid passive lingual arch is placed (teal wire in Fig 13-14d), the major side effects of displacement and rotation can be eliminated. Reciprocal equal and opposite lateral forces and moments cancel each other out and will be effectively avoided; however, the anterior component of force (yellow arrows in Fig 13-14d) still

Lingual Arch Configurations Fig 13-15 A localized unilateral buccal crossbite on the maxillary left second molar was treated with a flexible Ni-Ti wire. (a and b) Before treatment. (c) After level­ ing, good tooth-to-tooth alignment is seen from the occlusal view. (d) However, the lateral view shows that alignment was produced in part by buccal movement of the entire left posterior segment.

Fig 13-16  (a and b) A passive lingual arch can serve as a rigid base structure for the attachment of auxiliary springs or elastics. Unique lines of force can be achieved working from the lingual.

a

b

c

d

a

b

exists. No matter how rigid the wire is, this anterior component of force is not avoided. It cannot overcome the laws of physics. A localized unilateral crossbite on the maxillary left second molar (Figs 13-15a and 13-15b) was corrected with a flexible labial Ni-Ti wire. After leveling, good tooth-to tooth alignment is seen from the occlusal view (Fig 13-15c); however, the lateral view (Fig 13-15d) shows that alignment was produced in part by anchorage loss with buccal movement of the entire left posterior segment. Thus, a simple localized malocclusion became a new generalized crossbite of many teeth that may be more difficult to treat. A passive maxillary lingual arch (TPA or horseshoe arch from first molar to first molar), if placed before level­ ing, would have prevented this side effect. A passive lingual arch can serve as a rigid base structure for the attachment of auxiliary springs or elastics (Fig 13-16). Leveling and alignment performed with only a labial archwire may produce unwanted side effects in some patients. The lingual arch offers many creative possibilities that can con-

tinually be modified during treatment without depending on adjacent teeth for anchorage. Hooks, lever arms, elastomers, and metal springs are simple to design and fabricate. In Fig 13-17, a finger spring with a helix soldered to a mandibular lingual arch was used for incisor alignment. Anchorage was provided by the bilaterally rigidly connected first molars. A labial wire might have been satisfactory, but it uses adjacent teeth for anchorage that could lead to side effects. Sometimes the lingual arch can be placed before incisor brackets are placed. A delay in bracket placement due to occlusal interference, esthetic reasons, or improved biomechanics can eliminate an unnecessary side effect of facial bracket-wire alignment. In a patient with a bilateral maxillary second molar buccal crossbite (Fig 13-18a), the only forces needed for correction are bilateral single forces on the second molars (red arrows in Fig 13-18b). A single elastic or a coil spring placed bilaterally on the second molars would be the simplest and the best appliance mechanically; however, it would be uncomfortable 235

13 Lingual Arches

a

b

c

Fig 13-17  (a) A finger spring with a helix soldered to a mandibular lingual arch was used for incisor alignment. Because the anchorage was the bilaterally rigidly connected first molars, not the adjacent teeth, no side effects were seen during leveling. (b) Before. (c) After.

a

b

c

d

for the patient. Instead, elastics are placed on the right and left extensions soldered to a passive lingual arch that delivers the same force system as the single coil spring but is much more comfortable for the patient (Figs 13-18c and 13-18d). Without the lingual arch, any labial alignment archwire would have expanded the posterior end of the arch and produced a tapered arch form (see Fig 13-15). The patient in Fig 13-19 required extraction of the maxillary first molars instead of the premolars because there was localized enamel hypoplasia on both first molars (Fig 13-19a). Before extraction, provisional crowns were placed on the first molars. Two separate, passive TPAs were placed anteriorly and posteriorly. Hooks were placed on the TPAs near the level of the center of resistance (CR) (Figs 13-19b and 13-19c).The anterior segment was a rigid unit including right and left premolars. Initially the ante236

Fig 13-18 (a and b) In a patient with a bilateral maxillary second molar buccal crossbite, only bilateral single forces are needed for correction (red arrows) of the second molars. Therefore, a single elastic or a coil spring placed bilaterally on the second molars would be the simplest and the best appliance mechanically; however, it would be uncomfortable for the patient. (c) Instead, elastics are placed on right and left extensions soldered to a passive lingual arch that delivers (more comfortably) the same force system in the occlusal view. (d) After bilateral constriction.

rior teeth were not bracketed. The posterior segment was also a rigid unit; it included only second molars. Note that the posterior TPA provided additional space for the elastics (see Fig 13-19c). Because the elastic forces were applied at the CR of both anterior and posterior segments, space closure was primarily translation. Also note that the mesial movement of the second molars provided the space for the third molars to erupt (Fig 13-19d). After space closure, coordination of the anterior segment width and second molar width is required. The biomechanics of this case are discussed in more detail in chapter 14. With two passive lingual arches, the line of force can be placed obliquely so that differential space closure is achieved (Figs 13-20a to 13-20d). It is seen from the lateral cephalometric radiograph of this patient that the line of force (Ni-Ti coil spring) passes through the CR of the posterior segment (Fig 13-20e).

Lingual Arch Configurations Fig 13-19 The patient required maxillary arch extraction. (a) The maxillary first molars were extracted instead of the premolars because there was localized enamel hypoplasia on both first molars. (b) Two separate, passive TPAs were placed anteriorly and posteriorly. Hooks were placed on the TPAs near the level of the CRs. (c) Because the elastic forces were applied at the CR of both anterior and posterior segments, space closure was primarily translation. Note that the mesial movement of the second molar provided the space for the third molars to erupt (d).

a Fig 13-20 Use of two maxillary lingual arches for space closure. (a) Before treatment. (b) The line of force was placed obliquely so that differential space closure (tipping versus translation) could be achieved. (c) After space closure, the incisors were aligned. (d) After debonding. (e) The lateral cephalometric radiograph shows that the line of force (Ni-Ti coil spring) passes through the CR of the posterior segment and lies occlusal to the CR of the canine. (f) Various lines of action from an elastic or spring are possible between the anterior and posterior segments. It is even possible to place a resultant force (yellow arrows) away from the hooks by using two elastics on a side. (g) Two TPAs with dual elastics on each side.

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The line of force lies occlusal to the CR of the canine, so the canine will tip distally. Note the various lines of actions that are possible from an elastic or spring between the anterior and posterior segments for various types of tooth movements (Fig 13-20f). Also

note that it is possible to place a force off the hook (yellow arrow in Fig 13-20f) using two elastics (Fig 13-20g). The yellow resultant force is equivalent to the red forces (bilaterally) of the two separate elastics at the hooks. 237

13 Lingual Arches

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Fig 13-21  The passive lingual arch can serve as effective rigid anchorage for asymmetric tooth movement. (a) This patient has a unilateral buccal crossbite on the maxillary right second molar. (b) Before treatment. The deactivation force diagram shows a lingual force on the second molar in red and the reciprocal force system on the anchorage unit at the CR. (c) After treatment. The anchor teeth connected by a passive TPA remained unchanged after treatment. (d) Before treatment. The same principle was applied in the mandibular arch using a buccal force from a cantilever inserted in the first molar bracket on the buccal side of the second molar. (e) After treatment. Note that the mandibular left second premolar was moved buccally with a loop anchored by the two molars connected by the passive lingual arch. (f to h) After debonding, the maxillary and mandibular arch coordination was good, which suggests minimal side effects with this approach.

But can a passive lingual arch also be used in cases requiring unilateral asymmetric tooth movement? We have already seen how placement of a passive lingual arch before leveling can prevent side effects. The passive application of a lingual arch can serve as effective rigid anchorage for asymmetric tooth movement. The patient in Fig 13-21 had a unilateral buccal crossbite on the maxillary right second molar (Fig 13-21a). Before a leveling arch was placed, a passive TPA with an auxiliary cantilever spring welded to the maxillary right second molar was placed. The rigidly connected first molars act as one unit, with its CR midway between the CRs of the first molars. The deactivation force diagram in Fig 13-21b shows a lingual force on the second molar and the reciprocal force system on the anchorage unit at the CR. Note that the anchor teeth acted on by the reciprocal force system remained unchanged (Fig 238

13-21c). The same principle was applied in the mandibular arch using a buccal force at the mandibular right second molar (Figs 13-21d and 13-21e) from a cantilever inserted in the first molar bracket on the buccal side of the tooth. If the second molar is also rotated mesial in, the hook should be placed as far mesial as possible; a vertical loop at the distal of the first molar tube can be used to lower the F/∆ rate. Also note that the mandibular left second premolar was moved buccally with a loop anchored by the two molars connected by the passive lingual arch. After treatment, the maxillary and mandibular arch coordination is good, which also suggests minimal side effects with this approach (Figs 13-21f to 13-21h). In order to rotate the second premolar distal in, either a distal force on the buccal or a mesial force on the lingual can be used (Fig 13-22). If a mesial force is needed along with the moment, the lingual

Lingual Arch Configurations

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Fig 13-22  (a) In order to rotate the second premolar distal in and move it mesially, a mesial force (red arrow) from a lingual arch can be used. The yellow arrows are the equivalent force system at the CR. A chain elastic on the buccal would have an inconsistent force system with the force in the wrong direction. (b) During canine retraction, placing part or all of the distal force on the canine lingual hook can solve or reduce unfavorable canine rotation. (c) The lingual arch prevents molar or posterior segment rotation.

Fig 13-23 (a and b) An elastic from a hook on the lingual surface of the tooth attached to a lingual arch can produce the desired moment to rotate the molar, which is not easy with buccal wire.

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arch is ideal for attaching an elastic (Fig 13-22a) because bilateral molars are used as anchorage, not the anterior teeth. The lingual elastic along with a buccal elastic are needed to produce a pure moment or a couple. During canine retraction, canines tend to rotate distal in because the CR is lingual to the bracket. Placing part or all of the distal force on a lingual button on the canine can solve this problem (Figs 13-22b and 13-22c). The labial archwire can also be used to deliver an antirotation moment to the canine; the disadvantage is that this approach adds friction to the system. Leveling of a buccally erupted second molar has an inherent side effect of mesial-in rotation in continuous arch alignment. This common problem is discussed in detail in chapter 15. A buccal straight wire undesirably expands the intermolar width at the first molars. Sometimes after treatment has been completed successfully in the anterior region, second molars erupt rotated. In such cases, an elastic from a hook on the lingual surface of the tooth attached to a lingual arch can produce the desired moment to rotate the molar (Fig 13-23). The lingual elastic also produces a mesial force at the CR (yellow equivalent force system in Fig 13-23b). If the second molar and first molar contact areas collide, the force system on the second molar approaches a couple.

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The lingual elastic in Fig 13-24 is ideal to rotate the second premolar mesial in and to move the premolar distally to close a small space; the lingual arch offers good anchorage to prevent the first molars from rotating mesial out. Unlike a buccal wire that is depended on primarily for alignment, friction is small. If the molar also needs mesial-out rotation unilaterally, the lingual arch is placed after the molar rotation is corrected. An equal and opposite couple at each CR can be produced by a single elastic only (Fig 13-25a). The maxillary left first molar is rotated mesial in, and the second premolar is rotated mesial out. A single force is applied at the lingual side by an elastic without a buccal wire. A force at the lingual (red arrows) is replaced with a force at the CR and a couple (yellow arrows). These equal and opposite forces near the contact area cancel each other out because the teeth are already in contact (dotted line). Only the couples that rotate the teeth in the desired direction—molar mesial out and premolar mesial in—remain (Fig 13-25b and 13-25c). After the molar is sufficiently rotated, a symmetric passive lingual arch can be inserted and the premolar rotation continued if needed. If only the second premolar is rotated, the elastic can still be used; only now the passive lingual arch must be in place.

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13 Lingual Arches Fig 13-24  The lingual elastic is ideal to rotate the second premolar mesial in and move it distally to close a small space. The lingual arch offers good anchorage to prevent first molars from rotating mesial out.

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Fig 13-25  (a) The molar is rotated mesial in, and the second premolar is rotated mesial out. A single force was applied at the lingual side by an elastic without any archwire. (b) The single force at the lingual (red arrow) is replaced with a yellow force and a couple at the CR. The couple rotates the teeth in the desired direction. (c) Final alignment. Adding an archwire on the buccal during rotation would only increase the friction, and a passive lingual arch prevents desirable rotation of the first molar.

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Fig 13-26  Auxiliary springs attached to a lingual arch. (a) Spring for palatal traction of the maxillary second premolar. (b) Springs for extruding impacted canines. (c) Spring for moving a canine labially. A reverse articulation prevents bracket placement on the labial.

A spring is attached to the passive lingual arch for palatal traction of the maxillary second premolar (Fig 13-26a). Other passive lingual arch applications include springs to erupt impacted canines (Fig 13-26b) and teeth in reverse articulation where occlusal interference will debond brackets at the labial surface (Fig 13-26c). Major tooth movement via finger springs from a lingual arch can be considered an efficient adjunct to labial wires for initial tooth alignment and leveling with or without the labial brackets. Waiting to 240

bond buccal brackets on individual teeth until after initial major tooth movement is completed can be the most efficient mechanics to avoid unnecessary tooth movement by inadvertent leveling errors using a continuous archwire.

Active applications The lingual arch passive shape can be modified so that when it is inserted into the molar attachments, a force system can be produced. This appliance can

Forces Produced from the Shape-Driven Lingual Arch

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Fig 13-27  An infinitely rigid ideal arch would reduce side effects. (a) Only lingual forces (red arrows) are delivered to the molars, which will translate lingually. (b) The equilibrium diagram shows buccal forces (blue arrows) and no moments acting on the lingual arch. The amount of tooth displacement within the PDL space will be so small that it is exaggerated in the figure; adjustments must be made by small increments.

be very useful in delivering both symmetric and asymmetric force systems. In addition, because only two attachments are involved, a lingual arch provides a simple model with which to understand force systems from an orthodontic appliance. In chapter 15, two-bracket systems are studied using straight wires. This chapter analyzes more complicated configurations in three dimensions.

Forces Produced from the Shape-Driven Lingual Arch An entire chapter could be written on the use of active lingual arches without mentioning forces at all, which would emphasize the “shape” of different lingual arch applications. However, the major focus of this book is to delineate both the correct and incorrect force systems produced by any orthodontic appliance. The shape-driven appliance usually uses an “ideal arch shape,” where the brackets at the start of treatment are imagined to move out to the passive shape or final shape of the arch. E. H. Angle called this predetermined form the ideal archwire. In some shapes today, it is referred to as straight wire or, more specifically, preformed archwire. Using an ideal archwire in a labial appliance is a typical shape-driven method where the wire is elastically bent and placed into malaligned brackets. As the wire deactivates to the original preformed and deactivated ideal shape, it will hopefully bring the teeth into ideal positions. The same approach is commonly used with lingual arches. The force system from the ideal-shaped arch depends on the rigidity of the arch. For better understanding, the wire without any stress (or bending

moment) is depicted in green, and the wire under stress is depicted in orange. Let us imagine an infinitely rigid arch (F/Δ = ∞) in Fig 13-27a. Because the arch does not deflect during tying of the molars, tooth movement is limited to deformation of the periodontal ligament (PDL) support (ie, initial mechanical displacement and subsequent biologic response). Only lingual forces (red arrows) are applied to the molars, which translate lingually from the occlusal view. The equilibrium diagram (Fig 13-27b) shows buccal forces (blue arrows) and no moments at the lingual arch. The amount of displacement of the teeth will be so small within the PDL that it is exaggerated in the figure. Of course, the high rigidity of the lingual arch will require frequent adjustments and, hence, lacks efficiency. Let us now, by contrast, fabricate a low-rigidity arch (Fig 13-28a), where the preformed ideal shape (green shape) is identical. To place the highly flexible arch into the two molars (orange shape), attachments require both buccal forces and moments (blue arrows). Why are additional moments necessary? As the lingual arch is expanded to fit into the brackets by buccal forces only, the green U shape becomes an orange V shape (Fig 13-28b) that necessitates moments for full insertion (Fig 13-28c). Unlike the rigid lingual arch in Fig 13-27, the flexible lingual arch undergoes a complicated elastic deformation of its shape that introduces moments on each molar. Each first molar in Fig 13-28c initially receives both a desired lingual force as well as unwanted moments rotating the molars mesial in. The flexible lingual arch has the advantage of a greater range of activation, requiring fewer adjustments. In fact, its shape can be made narrower than the desired width to assure more efficient force levels as the molars approach their final widths. The shape can still be an exagger241

13 Lingual Arches

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Fig 13-28  (a) A flexible ideal arch where the performed ideal shape (green) is identical to Fig 13-27. To place the highly flexible arch into the two molar attachments (activated shape, orange) requires both buccal forces and couples (blue arrows). (b) As the lingual arch is expanded to place it into the brackets using a force only, its U shape becomes a V shape that necessitates couples for full insertion. (c) Deactivation force system on the teeth. Each first molar initially receives a desired lingual force but also unwanted moments rotating the molars mesial in.

ated ideal arch if the arms are made parallel to the molar brackets; however, undesirable side effects are produced by the elastic archwire deformation. Thus, the ideal arch shape works best for relatively rigid wires where the F/Δ rate is high and the displacement of the tooth is confined to PDL deformation at each activation.

Force system changes during deactivation of the ideal shape Suppose the intermolar width at the first molars (Fig 13-29a) is to be expanded (Fig 13-29b), and we select the green ideal arch shape (Fig 13-29c). The shape of the arch is determined without considering the force system. Clinically, the desired molar position is determined first (blue teeth in Fig 13-29c), and then the wire is fabricated passive to that position (green wire in Fig 13-29c). Then the lingual arch is elastically deformed by applying the necessary force system during activation to place it into the initial bracket positions (orange wire in Fig 13-29c). The activated lingual arch exerts a force system on the molars, and as the teeth move to the final desired positions, the lingual arch eventually deactivates to its preformed ideal shape. This common treatment procedure seems conceptually logical and very easy to understand and apply; however, it has inherent limitations and some disadvantages as discussed previously. Unwanted forces or moments can be produced, and the force system can also change during deactivation of this so-called ideal shape approach. First, let us consider the relative magnitude of the force system only. The magnitude of the force system of the activated lingual arch is initially 100% (orange wire in Fig 13-29c), whereas the final preformed and deactivated ideal shape (green wire in Fig 13-29c) would exert no force system at all (0%). If the magnitude of the initial force system was set 242

for an optimal range, the tooth will move rapidly initially. After the tooth passes through the optimal zone and as force continues to reduce, reaching a suboptimal force zone, the movement would slow down; the molars may not reach their final target position because force magnitude approaches zero at the final stage of deactivation. More importantly, the initial force system may not be correct. The mandibular left first molar is shown in an enlarged view in Fig 13-29d. When a constriction force (blue arrow) is applied at the free end to insert the lingual arch (activation force), the deactivated arch (green wire) will elastically deform to the orange shape. When a lingual force is applied for insertion, the free ends of the arch cut across the bracket at an angle (see Fig 13-29d). To place the orange wire into the molar bracket, not only a clockwise moment but also a greater magnitude of lingual force are needed (Fig 13-29e). Note the difference in the orange shape between Fig 13-29d and 13-29e. A greater magnitude of lingual force is required because clockwise moments tend to further expand lingual arches. In this appliance configuration, the forces and moments are associated and do not act independently. (The principles of association and dissociation are discussed later in this chapter). The blue arrows are the activation force system that the clinician applies to the lingual arch to activate it during insertion. The deactivation force system acting on the molars, depicted in Fig 13-29f, is equal and opposite to the activation force system. In this example, the mesial-out moments on the molars are unnecessary and undesirable side effects. This is similar to the example shown in Fig 13-28. The changes in the force system over time are shown in the enlarged view of the mandibular left first molar in Fig 13-29g. The large unwanted moments in Fig 13-29f could overwhelm the expansion force, and for the first month or so, the molars would rotate mesial out

Force-Driven Method

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Fig 13-29 The shape-driven method. (a) The narrow intermolar width needs expansion. (b) The target positions of the molars are in blue. (c) The selected ideal arch shape is in green. The shape is determined without considering the force system. The lingual arch is elastically deformed by applying the necessary force and moments during activation to place it into the initial bracket positions. (d) If a constriction force (blue arrow) is applied at the free end to insert the lingual arch, the preformed shape (green) will elastically deform to the orange shape. The left free end of the lingual arch cuts across the bracket with an angle. (e) To place the orange archwire into the molar bracket, not only is a clockwise moment needed but also a greater magnitude of lingual force. (f) The deactivation force system (red) acting on the molars is equal and opposite to the activation force system required for insertion. (g) During the course of deactivation (from orange to green wire), the molar rotates counterclockwise (red dotted arrow) and then clockwise (purple dotted arrow), which is unnecessary. Initial rapid tooth movement occurs in the optimal force zone, which includes significant side effects. Moreover, the tooth movement is very slow in the suboptimal zone to reach the target position.

and not demonstrate the desired parallel expansion. Not only does the force reduce over time as the lingual arch deactivates, but the moment-toforce (M/F) ratio at the molar lingual attachment continually changes. Note that unnecessary rotation is produced when an ideal lingual arch shape is used to expand the mandibular first molars (see Fig 13-29g). Because of the wire shape changes during deactivation and subsequent changes in bracket geometry, the centers of rotation are continually being modified. Initial rapid mesial-out rotation (red tracing and red dotted curved arrow) and small lingual translations (green tracing) are followed by very slow mesial-in rotation (purple tracing and purple dotted arrow). As the molar approaches its final deactivated position (purple tracing), it may rotate in a reverse direction to correct the rotation side ef-

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fect; however, this may take a relatively long time because the suboptimal zone moment and force values are dissipated. It is certainly better to directly move teeth to their final positions without the side effects associated this “round trip” ride.

Force-Driven Method It is apparent that the ideal arch may be the desired shape when teeth have moved to the predetermined fully deactivated shape; however, the force system to get there may not always be ideal with this shape, as discussed previously. A better approach is to design the shape of a lingual arch to produce the desired force system. This is called a force-driven appliance. The tooth reacts to the ap243

13 Lingual Arches plied force system; it does not matter what kind of material, cross section, or configuration of the lingual arches is used. Therefore, in the force-driven method, deciding the force system takes priority over establishing the final tooth position. Considering any given lingual arch configuration (horseshoe or TPA), the force system changes during deactivation. Both force magnitude and M/F ratios can be changed. In other words, the force system may not always be correct throughout the full range of deactivation. Therefore, certain principles must be applied. First, the initial force system, which is set to an optimal force magnitude range and a correct M/F ratio, must be correct. Second, because the magnitude of the forces and moments after appliance insertion decrease as the teeth move, their optimal levels need to be maintained as much as possible, particularly during the terminal phase of molar movement. This is accomplished by lowering the F/∆ rate and producing a range of activation that is larger than the required tooth movement. This allows for the delivery of more constant and optimally lighter forces. The accurate shape of a lingual arch to deliver a specific force system can be obtained using computers applying beam theory and iterative methods; however, the clinician can easily apply these principles chairside to achieve a close approximation of the appropriate shape. The clinical procedure follows. The first step is to determine the desired force system. An equilibrium diagram is useful to assure that a valid force system exists. See, for example, the mandibular first molar expansion (translation from the occlusal view) with a single buccal force on both molars in Fig 13-30a. In the second step, a passive shape of the arch for the original molar position is fabricated and contoured with minimal clearance between the soft tissues for maximum comfort (green wire in Fig 13-30b). The third step is a simulation shaping of the lingual arch by the deactivation force system. Here the clinician performs a loading on the passive shape, applying the predetermined deactivation force system (forces on the teeth) from the first step. In our example, the loading is two simple equal and opposite forces in the direction of the desired molar movement (red arrows in Fig 13-30c). Note that as bilateral expansive forces are applied, the lingual arch gets wider at the free end and assumes a less tapered V shape. This is the simulated shape (orange wire in Fig 13-30c) and the general shape needed in the lingual arch (green wire in Fig 13-30d). In the fourth step, the wire is permanently bent to the deactivated shape until it becomes identical 244

to the simulated shape. To determine the amount of activation for the given shape, a force gauge can be used when single forces alone are needed (Fig 13-30e). Grid paper could also be helpful to record the amount of activation in any given simulated shape (Fig 13-30f). The simulated shape is observed as the lingual arch is elastically deformed; it is therefore necessary to permanently deform the lingual arch to the simulated shape by increasing the load. However, increasing the load only to bend the wire to the correct deactivated shape is not enough. Arches are more resistant to permanent deformation if they are bent further than the simulation shape and then bent back to the established shape (Bauschinger effect). The fifth step is the trial activation, where the shape is checked in the mouth for correctness before final insertion. Activation forces (blue arrow) are applied on the deactivated lingual arch (Fig 13-30g), and if the lingual arch is correctly fabricated, it will fit easily in the molar attachments with a single force only. If any moment is necessary to engage the lingual arch, the shape needs further adjustment. Note the difference in shape of the ideal arch shape (see Fig 13-29d, orange wire) and the force-driven shape (see Fig 13-30g, orange wire), where the free ends cross the molar brackets at an angle with a single force. After placement (orange activated shape in Fig 13-30g), the activated shape is identical with the original passive shape (green shape in Figs 13-30b and 13-30c) in the force-driven method. Once it is placed and the clinician’s hand is released, it exerts the forces on the tooth as it deactivates (Fig 13-30h). If the lingual arch is carefully contoured for comfort in its passive shape, it should maintain these patientfriendly contours after insertion. Fabricating a comfortable passive lingual arch that does not impinge on tissues is time well spent using a force-driven lingual shape. Because it is force driven and based on the original lingual shape, the deactivated shape is usually more comfortable than the ideal arch shape, where forces during insertion are ignored. Computer iteration methods using beam theory follow the same steps described above. If any deviation from the activated shape and passive shape is detected, a little modification to the deactivated shape is added, and the cycle is repeated until the correct shape is obtained. The force-driven lingual arch is more efficient than the shape-driven one. It delivers the correct force system initially within the optimal force level zone, where the most significant tooth movement occurs. The tooth will move directly to the target position without any unnecessary wiggling or side effects. It

Force-Driven Method

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Fig 13-30 The force-driven method. (a) The desired force system is first established. An equilibrium diagram is useful to assure its validity. (b) A passive shape (green) to the original molar position is fabricated. (c) Simulation is performed with the deactivation force system. (d) The deactivated shape, which is identical to the simulated shape, is the final shape needed before placement. (e) To determine the amount of activation for the given shape, a force gauge can be used. (f) Grid paper could also be helpful to record the amount of activation. (g) The activated shape is identical to the passive shape. (h) Once the archwire is placed and the clinician’s hand is released, the initial forces on the molar are correct.

is comfortable for the patient because the activated shape bypasses problematic anatomical structures. Let us track the displacement of the tooth in the force-driven method. The tooth will displace quickly to the target position (teal tooth or wire in Fig 13-31), which lies in the optimal force zone (green area). This is a desirable feature because tooth movement will not slow down as the final target position is reached, and no side effect correction is required. If the force-driven bilateral expansion lingual arch remains in place for a sufficiently long time after the target position is reached, it will enter the subopti-

mal force zone (yellow area in Fig 13-31). Even if the tooth movement in the suboptimal zone is very slow, the molars may pass beyond their target positions, and molar position may follow the shape of the wire, which creates molar expansion and distal-out rotation (green tooth or wire in Fig 13-31). Therefore, when the tooth reaches the planned desired position, the lingual arch is removed and reformed to a passive shape. Unlike the shape-driven ideal arch, the deactivated force-driven shape has little clinical relevance; the lingual arch is removed or made passive before the fully deactivated shape is reached. 245

13 Lingual Arches Fig 13-31  Force magnitude zones with the force-driven method. Initially the force system is correct and the magnitude the greatest. The optimal zone assures that efficient force levels are maintained when the molar approaches its target position (teal tooth or wire). If the force-driven bilateral expansion lingual arch remains in place for a sufficiently long time after the target position is reached, it will enter the suboptimal force zone (yellow area). The lingual arch is removed when the molar reaches the target position.

It is easy to determine the deactivated shape of the lingual arch in the shape-driven method; however, the force system may or may not be correct. The amount of linear (parallel) and angular bends are determined by the final position of the molar’s bracket, ignoring the force system. In the force-­ driven method, on the other hand, the deactivated shape is determined by the force system, not the final lingual bracket position. Universally, the correct shape is formed by applying the desired deactivation force system to any passive arch, called simulation. This arch shape is first simulated and then deformed permanently into that simulated shape. The amount of activation, both linear and angular, may be dependent on the overall configuration of the lingual arch, its dimensions, cross-sectional shape, and material. Therefore, we will limit our suggestions of specified activations in millimeters or degrees and rather emphasize where and how to make bends or twists in the wire to deliver relatively correct force systems. Force system simulation provides the correct shape; however, understanding where and how the wire is bent or twisted is more important not only to fabricate the correct shape but also to modify the force system when necessary. Remember, even if the force system and the shape are correct, the response of teeth can vary, and the force system always needs monitoring and modification. The principle of correct shape based on the required forces is obtained by an understanding of beam theory. Further discussion of beam theory is beyond the scope of this chapter, but it is important for the clinician to note that the amount of bending or twist is proportional to the bending moment or torsional moment (torque) at a given section of a wire. The sections of a wire that have the highest stresses are called the critical sections. At these sections, most of the bending or torsion occurs during 246

the simulation procedure. More detailed analytical and computer evaluations can be useful and require further discussion.3

Symmetric Applications Bilateral expansion Bilateral (symmetric) molar and posterior segment arch width modification requires reciprocal anchorage and hence is a logical application of the lingual arch. First a passive lingual arch is fabricated (Fig 13-32). A flexible typodont is used to demonstrate the movement of the teeth using the lingual arch. The resin teeth are embedded in an elastic material so that the forces displace the teeth, demonstrating the resulting tooth movement. After the passive lingual arch is placed in the typodont, red dots are marked on the mesial and distal sides of the first molar (see Fig 13-32). Along with the red dots on the adjacent teeth that form a line, one can easily visualize the movement of the first molar after an active arch is placed. Let us consider a clinical situation where mandibular molars require expansion. Our thinking must be in three dimensions. From the frontal view, tipping around the root center can be allowed. From the occlusal view, translation is the defined movement. Bilateral expansion requires two single forces acting buccally at the lingual bracket. This required deactivation force system is applied to the passive lingual arch, and the simulated correct deactivation shape is formed. Figure 13-33a is the deactivated shape, which is identical to the simulated shape for bilateral expansion. The amount of activation is determined during the simulation step by the de-

Symmetric Applications Fig 13-32 A passive lingual arch is first fabricated for every active application. Red dots in alignment are marked on the mesial and distal sides of the first molar to visualize the movement of the first molar on the elastic typodont.

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Fig 13-33  Bilateral expansion by the force-driven method. (a) Simulated and deactivated shape. Note that the wire and brackets are not parallel (dotted lines). (b) After an active arch is placed. The two red dots on the first molar moved equally to the buccal. (c) If controlled tipping or translation should occur, a couple is added by torsion of the square wire. Double-headed arrows represent couples (buccal root torque) using the right-hand rule of thumb (see Fig 3-8).

activation force system. Note that distal free ends are not parallel with the brackets (dotted lines in Fig 13-33a). Next, before final placement, constriction force is applied by squeezing both free ends with single forces as a trial activation in the mouth, and the fit is checked. Modification can be made to alter the magnitude or to ensure passivity. Figure 13-33b shows that the first molar moves to the buccal without noticeable rotation after placement of an active lingual arch. The red dots on the first molar indicate equal mesial and distal contact area displacement. If a single force is required, a round wire is preferable (0.032-inch beta-titanium) so that the clinician does not have to adjust a rectangular wire for third-order passivity to eliminate unnecessary moments (torque) from the frontal view. Note that Fig 13-33b shows tooth movement within an optimal force zone, as the lingual arch is not fully deactivated. The appliance is force driven with the initial force system correct, with only a buccal force and no moment from the occlusal view. The arch is made passive when alignment is reached. A rectangular wire can be useful if the center of rotation is to be moved from a centered position on the root to the root apex. In addition to the shape previously described, a twist is placed along the pos-

terior of the arch to deliver equal and opposite couples in a buccal root torque direction. This is not a localized twist but a shape that simulates loading the lingual arch with couples (double-headed red arrow in Fig 13-33c). If an ideal arch was used with posterior free-end arms parallel to the brackets for expansion (Fig 13-34a), an incorrect force would be produced. Couples would rotate the molars mesial out during the delivery of the initial force system (Fig 13-34b). Initially, the undesired rotation would be seen more than the expected widening of the intermolar width. Also note that if the wire deforms unexpectedly, it may impinge on the soft tissues.

Bilateral constriction The same principles and sequence are used for bilateral constriction using a force-driven appliance. The deactivated shape is determined by simulation (ie, applying the deactivation constrictive force to the passive shape). Note that the free end of the lingual arch and the brackets are not parallel as in the ideal shape (Fig 13-35a). After insertion, only forces (no moments) act on both first molars, and red dots indicate that the molars do not rotate (Fig 13-35b). 247

13 Lingual Arches Fig 13-34 Bilateral expansion by the shape-driven method (ideal shape). (a) The free ends are fabricated parallel to the brackets (dotted lines). (b) Moments (mesial-out) are associated with buccal force in the initial force system.

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Fig 13-35  Bilateral constriction by the force-­ driven method. (a) Simulated and deactivated shape. Note that the wire and brackets are not parallel (dotted lines). (b) After insertion, only translation (no rotation) is observed in the occlusal view.

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Estimating bending and torsional moments at arbitrary sections along a wire The simulation procedure will correctly determine the deactivation shape of an arch, but fabrication of the correct shape also requires technique; nevertheless, understanding the principle of how a wire undergoes deformation during simulation greatly helps during fabrication of the correct deactivation shape. Figure 13-36a depicts a straight-wire cantilever with a deactivation force (red arrow) applied at the free end. This could represent an intrusion arch to the mandibular incisors with the terminal tube at the mandibular first molar and only a point contact at the incisor brackets. Note that the forces that act on the wire to deform it are in the direction of the forces that act on the teeth for intrusion. In response to an intrusively directed force, the cantilever will bend downward elastically, assuming the simulated shape. This is the correct deactivated shape for intrusion, when the wire is permanently deformed to that shape (Fig 13-36b). During activation (see Fig 13-36b), the force direction (blue arrows) and the bending of the cantilever are in the opposite direction to the last bend we used to permanently deform the wire. It is recommended to overbend past the simulation shape and then to bend back to the final deactivated shape. If this is done, activation to place the wire in the mouth is in the same direction as the last bends used 248

to permanently deform the wire. The deactivated shape (green wire) in Figs 13-36b and 13-36c is identical, but Fig 13-36c is more resistant to permanent deformation (Bauschinger effect). Intuitively, we know that the general shape will be a downward curvature from the terminal molar bracket forward. The question is: Where should we bend the wire, and how much curvature should we place? Let us make an imaginary cut with a knife at point B of the wire (Fig 13-36d). If the cut was real, the small element in Fig 13-36d (bottom) would fall off; however, it does not because stresses (forces) hold the wire together. Therefore, the element is in a state of equilibrium. The surface of the element where we imagined the cut is called a section. When we shape a wire or an arch, permanent deformation occurs at every imaginary section along the wire. Let us look at the element (point B) again. Stress and strain will occur in three dimensions at each section. The red vertical force perpendicular to the long axis of the wire is the shear force. The moment acting at the section, needed for equilibrium, is the bending moment. Horizontal axial stresses (pure tension and compression), which are not shown in this figure, act parallel to the long axis of the wire. If the wire is twisted (not in this example), torsional moments (torque) can operate around the long axis of the wire. The torsional moment along the long axis does not change the overall configuration of the wire. When we shape the appliance, the F/Δ rate is low mainly in the bending and torsional mode; therefore, we are primarily interested in the bend-

Symmetric Applications

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Fig 13-36  The procedure to obtain the correct shape for a downward single force at the free end of a cantilever spring. (a) Passive shape and simulated shape. The red arrow at the free end is in the direction of the deactivation force system that we want. The cantilever is in equilibrium. (b) The wire is permanently deformed to the deactivated shape (green), which is identical to the simulated shape. The activation force system (blue arrows) is applied. The activated shape (orange) becomes identical to the passive shape. (c) To increase the range of action, the wire is overbent (purple shape) and then bent back to the goal-simulated shape. Note that the final bends are in the direction of the activation force system (blue arrows in b). (d) An imaginary cut (section) at point B will make an element that is also in equilibrium. (e) An arbitrary section can be made to find the bending moment at a given section (red curved arrows). The curvature of the deactivated shape of the wire at each section is proportional to the bending moment. More bending occurs near point A, where the bending moment is the greatest, and there is no bending at point C.

ing and torsional moments. Figure 13-36e shows the magnitude of the bending moments at arbitrary sections along the wire. It also informs us where and how much to bend the wire. The magnitude of curvature at each section is proportional to that of the bending moment. At point C, no element or bending moment exists; hence, no bend is placed there. Compare sections at points B and A. The element at section A is the longest and, hence, the bending moment is the largest. The amount of the bend (or curvature) must be proportionally increased at point A. Where a section has the highest bending moment, it is called a critical section. Critical refers to its sensitivity to wire failure, such as permanent deformation or fracture. So how is the straight-wire cantilever bent to produce the force-driven shape for incisor intrusion? The wire is curved downward gradually, increasing the bend magnitude as the pli-

er is moved distally (see Fig 13-36e). The simulation approach (see Fig 13-36a) described previously gives the same result. Push down on a cantilever anteriorly, and you will see it deform increasingly more to the distal. Understanding simulation and the principle of where to bend a wire greatly enhances the proper use of a force-driven appliance. Two basic loading conditions in bending a wire are forces and couples. Let us now apply a couple to the free end of a cantilever and see how the wire will bend (Fig 13-37). Note that at each section, no shear force is present, and only an equal and opposite bending moment is required to keep all elements in equilibrium. What is the force-driven shape to deliver only a couple at the free end? Unlike the single force example in Fig 13-36, all sections require identical bending moments. This requires equal bending all along the wire in a downward direction. In other 249

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Fig 13-37  A straight-wire cantilever with a couple applied at the free end. At all arbitrary sections, the bending moment is the same. Therefore, the curvature is uniform, a segment of a circle.

words, placing any straight wire in equilibrium with only equal and opposite couples produces a wire curvature that is a segment of a circle. The curved wires in Figs 13-36e and 13-37 may look very similar, but the curvatures and their positions are very different; thus, the force system is totally different in each. Figure 13-38 shows a lingual arch used to narrow intermolar widths by a lingual force alone. An imaginary section is made at the apex of the arch (dotted line). The orange element is in equilibrium, and the activation force system is depicted by blue arrows. Note the highest bending moment at this section. This is observed during the simulation procedure and is also anticipated with the bending moment diagram. During the simulation of bilateral constriction, single forces at the free ends are applied, and because the largest perpendicular distance from these single forces is near the apex, most bending occurs there. This region is a high-stress critical section where overbending is advised to minimize permanent deformation. If lingual arches are overbent and then reformed, residual stresses are operating in the correct direction to minimize permanent deformation. Therefore, desirable residual stresses should not be removed (Bauschinger effect). It is not strange to find that a wishbone is always broken near the apex, which is the critical section.

Association and dissociation of moment and force The phenomenon of wire association and dissociation can be demonstrated with an ideal arch. The lingual arch in Fig 13-39 is shaped with arms parallel 250

Fig 13-38  An imaginary section is made at the apex of the arch (dotted line), which is the critical section in bilateral constriction by a single force at the free end. The orange element is in equilibrium by the activation force system (blue arrows).

to the brackets and wide; we have already learned that both a buccal force and a mesial-out moment are produced; furthermore, the wider the parallel arms, the greater the increase in both buccal forces and mesial-out moments. With this horseshoe configuration, an association exists between the force and the moment (buccal force and mesial-out moment). Now let us angle the free arms (Fig 13-40) so that they cross the bracket in a direction of mesial out with the width not changed. Not only mesial-out moments but also an expansion force are produced. If width or angle is modified, an association always exists. Narrowing the width produces a moment directed mesial in; the association is the same but reversed. Let us suppose that two identical bilateral maxillary expansion horseshoe arches are placed for bilateral expansion using an ideal shape, one inserted from the anterior (Fig 13-41a) and the other from the posterior (Fig 13-41b). Typical mesial insertion will produce expansion and a mesial-out rotation (see Fig 13-41a). Although it would be practically impossible to place, the distally inserted lingual arch with its apex directed posteriorly will produce expansion and a mesial-in rotation, which is opposite in direction from the mesial insertion (see Fig 13-41b). These differences in the force system despite the same shape at the bracket are not accidental; they reflect how wires bend in respect to any applied forces and demonstrate the association of force and moment. Because the arches in Figs 13-41a and 13-41b are mirror images, with the expansion the angles at which the archwires cross the bracket during activation are equal and opposite in the anteriorly or posteriorly inserted lingual arch.

Symmetric Applications Fig 13-39  Bilateral expansion in the shapedriven method. Parallel buccal wire (a) is accompanied by molar mesial-out moments (b).

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Fig 13-40  Bilateral rotation in the shapedriven method. Mesial-out angular bends at molars (a) are accompanied by buccal forces (b).

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Fig 13-41  Bilateral expansion using an ideal shape inserted from the anterior (a) and from the posterior (b) will generate moments in opposite directions in the occlusal view. With a vertically placed TPA, no moment will be produced with parallel expansion (c). (a and b) Associated designs for expansion. (c) Dissociated design for expansion.

Let us now make a TPA, where the amount of wire anterior and posterior to the attachment is about the same (Fig 13-41c). It should not surprise us that the correct expansive force only is produced by parallel expansion, only without any free-end angulation. In other words, because of its unique wire configuration in 3D, the TPA dissociates the force from the moment. Increasing the amount of expansion does not change or produce a moment, and increasing a first-order angle does not change the force. Interestingly, the force-driven shape and the ideal arch shape are the same from the occlusal view. During the force-driven simulation with forces only, the free ends expand with a relatively constant angle parallel to the passive shape. The linear parallel displacement in expansion takes place mostly by the bending of the apex of the TPA. Dissociation is only present from the occlusal view in this example; associations with a TPA between expansion and

x-axis moments (third-order rotation) from the rear view are discussed later (see Fig 13-49). If a roundwire TPA is used, pure expansion or constriction (occlusal view translation) can be delivered with single forces; here, the force-driven shape is the same as the shape-driven shape, with the exception that it is exaggerated so that a larger range of activation maintains the optimal force zone. The dissociation phenomenon is demonstrated on the elastic typodont in Fig 13-42. From the occlusal view, an ideal arch shape is used. The freeend arms are parallel to the brackets (Figs 13-42a and 13-42b). After insertion (Fig 13-42c), only translation occurs because a force acting at the CR and no moments are produced. The teeth translate to the buccal as shown by the red dots (see Fig 13-42c). In this application, the ideal shape gives the correct force system. Also, the M/F ratio of 0 mm is constant throughout the entire range of action. It is desir251

13 Lingual Arches

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Fig 13-42  Dissociated type of application with a TPA. (a) Passive shape. (b) Simulated (deactivated) shape. Note that the wire and brackets are parallel (dotted lines). (c) Once it is inserted, only force (no moment) is produced in the occlusal view.

able to use a round wire with small cross section and lower modulus of elasticity such as 0.032-inch beta-titanium to lower the magnitude of forces and to deliver it more constantly. If so, the deactivation shape should be wider than the required amount of translation. When ideal width is achieved, the TPA is made passive.

Bilateral rotation Many patients require molar rotation (rotation around the y-axis). Individual molars can be rotated, or the overall arch form can be modified by rotation of a segment. The shapes of horseshoe and TPA lingual arches to produce equal and opposite couples for rotation are very different. First, let us consider a horseshoe lingual arch that can be used in either the maxillary or mandibular arch. To rotate the molars bilaterally (mesial out in this case), a simulation is performed on the passive shape using equal and opposite couples (Fig 13-43). The deactivated shape (or simulated shape) of the lingual arch is depicted in Fig 13-43a. In the simulation process, every section of the lingual arch feels the same bending moment; therefore, the entire lingual arch is bent evenly from one end to the other. This bilateral rotation lingual arch looks similar to the bilateral constriction arch (see Fig 13-35). Both have a shape with a curvature that may suggest a distal-in molar rotation effect; however, this shape with its added uniform curvature only exerts bilateral mesial-out moments (Fig 13-43b). This force-driven shape, which looks nothing like an ideal shape, only rotated the molars mesial out. In Fig 13-43b, a little width change is seen, which suggests that it has passed the optimal zone. The lingual arch should have been removed and reshaped for passivity when the tooth target position was reached. A round 0.032-inch beta-titanium wire is a good choice for this type of tooth movement. The deactivated shape should be narrower, produced by applying the simu252

lation couples. By using a less stiff wire, the moments can be lower and act more constantly, and the range of the optimal force zone is increased. Orthodontists have learned to use less stiff wires and lower forces on the buccal for alignment; unfortunately, there is a tradition of fabricating rigid high-force lingual arches. Bilateral molar rotation with a TPA requires an entirely different deactivated shape (Fig 13-44). The free ends of the TPA cut across the centers of the lingual brackets, forming equal angles bilaterally. Note that the deactivation force-driven shape is similar to the shape-driven shape. When dissociation is present, a configuration like an ideal arch should work. Wire-bracket angles produce moments, and linear parallel displacements produce forces. This is the special case where a straight wire or an ideal arch works according to the older predictions. First, the passive TPA is fabricated (Fig 13-44a), and simulation of the force system is performed (Fig 13-44b). The simulation-determined deactivated shape should be overlaid on the molar brackets and the geometry checked. Arch width should be maintained at the bracket center (see Fig 13-44b). Equal angles of the TPA arms in respect to the slot should be present on both sides. If the angle is hard to evaluate with this method, only one side of the TPA can be placed, and the distance (Δ) of the other free end should be measured to the contralateral molar bracket (Fig 13-44c). This is repeated on the other side (Fig 13-44d). The distances (Δ) should be the same on both sides. Note that the molars have rotated and maintained their original widths after placement (Fig 13-44e). When a mesial-out rotation is simulated on the passive TPA, most of the wire deformation occurs at the green part of the TPA (Fig 13-45), which is torsion. Torsion of the vertical arms of the TPA does not change the arch width. By contrast, any bending in the horseshoe lingual arch changes both width and rotation.

Symmetric Applications Fig 13-43 Bilateral rotation with a mandibular lingual arch. (a) Simulated (deactivated) shape. Every section of the wire needs an even amount of bending. (b) Once it is inserted, only moments are generated initially.

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Fig 13-44  Bilateral rotation with a maxillary TPA. (a) Passive shape. (b) Simulated (deactivated) shape. With this lingual arch (TPA) design, the force-driven and shape-driven shapes look similar in the occlusal view. Horizontal arm–bracket crossover angles on the right and left should be the same. (c and d) If the angles are too small to compare, only one side is placed, and displacement (Δ) of the contralateral side is measured and repeated on the other side sequentially with the same Δ. (e) The molars have rotated and maintained their original widths after placement.

a Fig 13-45  Most of the elastic deformation of a TPA during bilateral rotation occurs at the marked green part of the arch by torsion.

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Fig 13-46  (a) Both molars are severely rotated mesial in so that a buccal wire cannot be placed initially. (b) Molars have been rotated independent of all other teeth by the TPA.

Both molars in Fig 13-46a are severely rotated mesial in so that a buccal wire cannot be placed initially. Alignment using brackets on the facial surface can lead to side effects, particularly arch expansion. Reciprocal mechanics across the arch solves the prob-

lem more efficiently; later, brackets can be placed on the facial surfaces to finish the maxillary alignment. Molars have been rotated independent of all other teeth (Fig 13-46b). This has the advantage of pure rotation around the CR of the molars. 253

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The mesial contact area of the molar moves distally (Δ in Fig 13-47a) because the molar rotates around the CR, which lies lingual to the central groove, even though there is no distal movement of the CR (Fig 13-47b). This distalization is helpful for Class II mal­occlusion correction. A facial continuous archwire with molar rotation moments when tied back will not allow the buccal tube to move distally, and hence this movement is inhibited. The CR on the maxillary first molar may lie lingually because of the large lingual root, the lingual divergence of the lingual root, and the buccal inclination of the molar (see Fig 13-47b).

Associated versus dissociated type of application In the maxillary arch, a TPA has a clear advantage over a horseshoe arch if pure rotation is required. Because force and moment are dissociated, the selection of a proper shape is simplified. This is also true for either linear parallel expansion or constriction in the occlusal view. Bend laterally in a parallel direction, and you get a force only; bend an angle, and you get a moment only. This is not true of the horseshoe arch, where the association of force and moment can require complicated linear displacements and curvatures for either pure translation or pure rotation of a molar. Thus, for pure translation or pure rotation, a dissociated mechanism is easier to use. Many times, we may want to have both a couple to rotate a molar mesial out and a lateral force to increase arch width in the occlusal view. The horseshoe arch from the mesial can give a favorable moment (mesial out) and force direction if the wire is formed parallel to the bracket (see Fig 13-41a). We call this a consistent configuration. If the horseshoe had the arch apex placed posteriorly (see Fig 254

Fig 13-47  (a) The mesial contact area of the molar moves distally because the molar rotates around the CR, which lies lingual to the central groove even though there is no distal movement of the CR. (b) The CR on the maxillary first molar may lie lingually because of the large lingual root, the lingual divergence of the lingual root, and the buccal inclination of the molar.

13-41b), the moment to the molars would be mesial in, the opposite direction with the same ideal shape. Because the direction is wrong, this is an inconsistent configuration. In Fig 13-48, the required force system is shown with red arrows. Two arches are shown: a TPA (Fig 13-48a) and a horseshoe arch (Fig 13-48b). The TPA is the better choice for this situation because dissociation occurs in the occlusal view. Dissociated moment-force appliances are the most efficient because the magnitude of force and moments are controlled independently and the correct shape (correct force system) is the easiest to determine and maintain throughout the full range of tooth movement. The horseshoe arch, which has a consistent configuration to give both force and moment in the correct direction with an ideal arch shape, gives at least the correct direction of force and moment; however, it may not give the correct M/F ratio to deliver the desired balance of force and moment. Force and moment cannot be controlled independently because they are associated. Therefore, calibration becomes difficult. The greater the distance to the apex from the bracket, the greater will be the M/F ratio. It may seem that the applications for a TPA with dissociation is simple and easy to apply—that is, until we think in three dimensions. Let us consider two first molars viewed from the posterior (Fig 13-49a). We prefer to tip the molars to the buccal with a center of rotation at the apex of the molar roots. This requires both a buccal force and a couple in the direction of buccal root torque (red arrows). If the buccal root torque is increased, translation could occur. Let us first look at an ideal arch configuration using a TPA, where the green arch is fabricated so that its arms are parallel and buccal to the molar brackets (Fig 13-49b). Now we apply a blue activation force to constrict and insert the TPA into the brackets; notice

Symmetric Applications Fig 13-48  If pure rotation or expansion is required in the maxillary arch, a TPA (a) has an advantage over a horseshoe arch (b) in the occlusal view because the TPA exhibits dissociation of force and moment.

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Fig 13-49  Force and moment association with a TPA from the rear view. (a) We want to tip the molars to the buccal with a center of rotation at the apex of the molar roots. The required deactivation force system is shown with red arrows. (b) The TPA was fabricated into an ideal shape (green) with arms parallel and buccal to the molar brackets without considering the force system. (c) We now apply e a blue activation force to constrict the TPA. Notice that the wire arms develop an angle (third-order) with the bracket. (d) Not only a lingual force but also a palatal root torque are required for placement. (e) Equal and opposite deactivation moments with forces are a correctly produced association. It is a consistent configuration because the direction of the force and moment is correct; however, we are not sure of the exact M/F ratio. If more twist were added to increase the buccal root torque, the lingual arch width would increase; hence, the buccal force would be increased. Exact control of the M/F ratio is difficult due to this association.

that not only a force (Fig 13-49c) but also a moment with increased force are required for engagement (Fig 13-49d). Equal and opposite deactivation moments with forces that are consistent are correctly produced (Fig 13-49e); although we are not sure of the exact M/F ratio, only the direction is required to identify this configuration as associated using an ideal arch for analysis. The arm parallel to the buccal expansion shape of the lingual arch increases buccal force and buccal root torque. The M/F ratio depends on many parameters such as the vertical height of the arch; if controlled tipping with a center of rotation at the apex should occur with this lingual arch, it would be just sheer luck, although the force and couple directions are valid. The association of force and moment is also shown in Fig 13-50. Let us make a passive TPA arch and maintain the arch width. The wire at its insertion area is twisted (torsion) in the direction of buccal

root torque (Fig 13-50a). When the wire is elastically twisted by trial activation (blue curved arrows in Fig 13-50b), the arch will expand. The so-called twist at the bracket produces a torque (couple) and an associated buccal force (Fig 13-50c). Thus, placing a twist (torsional angle) in an arch again produces an association of a buccal force and buccal root torque. The lingual arch shape to produce a force and couple is not trivial. We could simulate the shape by applying the deactivation force system; however, we are applying both a moment and a force, which is not easy to do or accurate enough. If the curvature to produce buccal root torque is placed in the wrong place, buccal forces or the torques will be incorrectly altered. Theoretically, both TPA and horseshoe-type lingual arches can deliver identical and correct force systems initially (Fig 13-51). Practically in this example, the better shape for applying a buccal force and buccal root torque is the horseshoe arch rather than 255

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b Fig 13-50  Effect of TPA twist on arch width. (a) The archwire before insertion has the same width as the brackets. (b) When the wire is elastically twisted during a trial activation (blue curved arrows), the arch will expand. (c) Thus, placing a torsional angle in an arch produces an associated buccal force and buccal root torque.

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Fig 13-51 Both TPA (a) and horseshoe (b) lingual arches can theoretically deliver identical and correct force systems initially. Practically, the better shape for applying a buccal force and buccal root torque is the horseshoe arch because its design dissociates the expansion or constriction force from the torque. Fine-tuning can be accomplished because increasing the torque does not change the arch width.

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Fig 13-52 A horseshoe lingual arch has been used to expand the maxillary posterior segments by tipping at the root apices. Expansion forces and moments (buccal root torque) were applied to the molars. In this type of application, the force and moment are dissociated; manipulation is more user-friendly because force and moment are controlled independently.

the TPA. With the horseshoe arch, forces are dissociated from moments; applying the couples does not change the arch width. The TPA, on the other hand (see Figs 13-49 and 13-50), behaves exactly the same as a vertical U-loop retraction spring, which is considered in detail in chapter 14. Because the height and interbracket distance (arch width) of a TPA may vary individually, it is difficult to standardize the shape, and hence forces and moments are difficult to predict. Adding more root-moving moments by twisting the free end of the arch induces unpredictable alteration of associated horizontal forces (Fig 13-51a). In the horseshoe type of application (Fig 13-51b), forces and moments are working independently (dissociated) so that it is easier to modify the force system. If the molar tips when we need 256

translation, force can be decreased and the amount of moment maintained, and/or the moment can be increased by twisting the free end without changing the force magnitude (arch width). Dissociated mechanisms, if available, should be the appliance of choice because they reduce the indeterminacy of the force system so that tooth movement becomes more predictable. The horseshoe lingual arch (Fig 13-52) has been selected to expand the maxillary posterior segments. The goal is to tip the molars around a center of rotation at their apices. Force application requires both a buccal force and buccal root torque on both sides. The horseshoe design should be kept simple; note that the entire arch is in one plane and not stepped apically into the palate. If off-plane, any moments

Asymmetric Applications Fig 13-53  A common anchorage approach is to pit a larger segment with more teeth against a segment with fewer teeth; however, this rarely works.

can be associated with a width change. An increase in the M/F ratio at the bracket can produce translation; this type of activation should be done carefully because the buccal plate of bone is thin. In younger patients, it is possible that the midpalatal suture will open during expansion if the force system approaches translation in the frontal view and tipping is minimized.

Asymmetric Applications Unilateral expansion or constriction We have learned from the laws of equilibrium that the right and left horizontal forces acting on each molar cannot have different force magnitudes when unilateral expansion is required. A common asymmetric approach is to pit a larger segment with more teeth against a segment with fewer teeth (Fig 13-53). In practice, this rarely works because it is difficult to control force magnitude so that the anchorage side is in a subthreshold range. The approach described here uses the lingual arch to produce differential moments between the right and left sides to achieve unilateral expansion or constriction. Two options for delivering differential moments are possible. The first method is to apply bilateral expansive single forces obliquely so that the line of action passes through the CR on one side and is near the crown level on the other side (Fig 13-54). The deactivation force diagram (Fig 13-54a) shows the line of action of forces in respect to both CRs. This is the most

appropriate diagram to understand the principles that underpin the mechanics. The equivalent force system replaced at the left molar brackets is given with yellow arrows in Fig 13-54b. The anchor tooth on the left side translates laterally and intrudes. The more uniform stress distribution in the PDL prevents anchorage loss. The active molar on the right has a high stress distribution because the single force at the bracket produce a large moment, tipping the molar to the buccal; therefore, it tips rapidly buccally. The undesirable side effect is the extrusion of the right molar; however, the vertical component of force can be small, and occlusal forces can minimize the molar eruption. The deactivation force diagrams are based on an equilibrium diagram of the appliance; hence, all forces and moments must sum to zero. For brevity, the equilibrium diagrams of the appliance have been omitted. Because the appliance is in equilibrium, all forces at the bracket add to zero, and hence the equivalent forces at the CR also add to zero. The CR position of the molars can vary individually measured from the molar brackets, such as variation of the root length, shape, inclination, and occlusogingival placement of the bracket; therefore, consideration must be given to the replacement force system at the brackets, where the lingual arch adjustment occurs. Fine-tuning a lingual arch activation based on tooth inclination and morphology (altered equivalent force system) is required in achieving success. For that reason, it is necessary to make a plan based on the CR position first and later figure what is needed at the bracket. The following protocol outlines how these principles can be transferred to daily practice.

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Fig 13-54  Asymmetric expansion by equal and opposite forces with a moment on the anchorage side (method 1). (a) The line of action passes through the CR on the anchorage (left) side and is near the crown level on the mandibular lingual crossbite (right) side. (b) The equivalent force system replaced at the molar brackets (yellow arrows). The right side will tip, and the left side will translate. (c) The arbitrary 100-g expansion force requires 800 gmm for translation on the left side. (d) Simulation is performed in two steps: horizontal activation for horizontal force followed by vertical activation for moment. Therefore, the force is resolved into vertical and horizontal components (purple arrows). (e) The horizontal force component on the molars is used to simulate the shape using a force gauge, and the deactivated shape is produced. (f) Trial activation. A force gauge is applied in the direction of the blue arrow for fine-tuning. (g) The vertical force component on the molars is used to simulate the deactivated shape needed for the moment. (h) Trial activation. A force gauge is applied in the direction of the blue arrow. A 34-g vertical force is needed to produce an 800-gmm moment.

Unilateral expansion by force (method 1) Step 1: Establish a valid force system A good diagram can be helpful where a valid force system is designed (see Fig 13-54a). The red diagonal single force acting at the CR of the left molar is 258

replaced at the molar bracket with yellow arrows (see Fig 13-54b). Let us arbitrarily use 100 g of expansion force. The measured interbracket distance is 23.5 mm, and the distance to the CR is 8 mm; thus, a counterclockwise 800-gmm moment is needed on the anchor molar (Fig 13-54c). Shape simulation is performed horizontally and vertically (independently of each other); therefore, the force is resolved into vertical and horizontal components (Fig 13-54d).

Asymmetric Applications Fig 13-55 During simulation, torsion occurs at the green regions (a and b), and bending is produced at the red region (c). (d) The right-side free end of the lingual arch after simulation has a second-order (tip) angulation to the bracket.

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Step 2: Perform a horizontal simulation The horizontal force simulation is performed. The deactivated lingual arch shape is determined by simulation as described previously. It is more complicated with both a force and a moment delivered on one tooth, but vertical and horizontal forces and moments are dissociated in this type, so forming can be done in sequential stages. The passive lingual arch is first horizontally expanded for simulation (Fig 13-54e). A total of 94 g can be measured with a force gauge. An accurate horizontal deactivated shape that is the same as the simulated shape is fabricated, and the final deactivated shape is confirmed by a trial activation (Fig 13-54f).

Step 3: Perform a vertical simulation A vertical force simulation is performed (Fig 13-54g). Because the passive lingual arch width is 23.5 mm, 34 g of occlusal vertical force is needed. If the force is maintained at 90 degrees to the occlusal surface of the lingual archwire at the free end of the right tooth, the left restrained end will undergo torsion. The vertical force for torque is measured using a plier to stabilize the lingual arch on the left side and a force gauge on the right side. The right free end is pushed occlusally with a force gauge. The measured vertical force times the horizontal distance to the opposing bracket is the moment (see Fig 13-54g). A grid paper is helpful to measure and record the amount of deflection (see Fig 13-30f). Final-

ly, shape is confirmed by trial activation (Fig 13-54h). Force-driven arches can be readily fabricated using the simulation principle. But unlike with an ideal arch, where one learns to copy a shape, an understanding of biomechanics is necessary. The correct deactivation shape has been determined by simulation in steps 2 and 3; however, more details of where the bends and twists are placed are further described here. The passive lingual arch undergoes both torsion and bending in reaching its simulated shape in three dimensions. Most of the torsion occurs between the left lingual bracket and the apex of the lingual arch (green region in Figs 13-55a and 13-55b), which is the most distant region from the single force. Slight bending occurs at the red region of the lingual arch (Fig 13-55c). The deactivated shape is fabricated using torsion and bending at these critical sections. The right-side free end of the lingual arch after simulation has an angle with the bracket (Fig 13-55d) that looks like it might tip the right molar forward, but this does not happen.

Step 4: Perform a trial activation After the deactivation shape is fully completed, a trial activation is performed to check both the force system and comfort (ie, the archwire is inserted into one or more brackets as a check). Pushing the free end of the right side apically to the level of the right lingual bracket with a single force should show no first-, second-, or third-order angle (potential moment) because the wire changes its angle during 259

13 Lingual Arches

Fig 13-56 After insertion of the lingual arch in the elastic typodont, the red dots show little tooth movement on the left anchorage side. The right side shows buccal movement without any rotation.

activation; if the deactivated shape is correct, it is ready for final insertion. If not, a little adjustment is made until the right-side free end fits without applying torque. Grinding off the rectangular edge of the right-side free end could also ensure freedom from third-order torque. Figure 13-56 shows the effect on the elastic typo­ dont of the activated unilateral expansion lingual arch using an oblique single force (a force and a couple on the anchor bracket). The red dots show little tooth movement on the left anchorage side. The right side shows buccal movement without any rotation. Vertical side effects are minimized by occlusal forces and the large mechanical advantage of a large transpalatal distance between the CRs of the first molars.

Unilateral expansion by a couple (method 2) The second method of unilateral crossbite correction is to directly apply a couple to the molar to be expanded. The valid force system at the CR is presented in Fig 13-57a, and its equivalent force system at the bracket is shown in Fig 13-57b with yellow arrows. In this second method, the couple for tipping the problematic side is produced by vertical (not horizontal) forces. The deactivation force diagram in Fig 13-57a has no horizontal expansion forces and uses a couple instead on the right side. The red arrows are the forces acting through the molar CRs. The right molar will tip to the buccal around an axis near the CR of the right root. Anchorage is afforded by the intrusive-extrusive resistance of the molars. As in the first method, the distance between the CRs is large and, hence, the vertical forces are relatively small. A replaced equivalent force system on the lingual brackets is shown in Fig 13-57b. It is somewhat complicated by the need to have two couples in opposite directions and different magnitudes on the 260

right and left molars. Although not perfect, the force system in Fig 13-57c is simpler and gives a close approximation with only a couple on the tooth in the problematic side. Because the vertical forces do not act through the CR, side effects are produced; however, moments produced by these forces in respect to the CR are in a favorable direction to help correct the asymmetry. They move the molars to the right. For convenience, the force system of Fig 13-57c, with one couple only, is usually selected. As the right molar is tipped to the patient’s right side, the left molar will be carried to the right. To hold its position, a compensatory expansion may be placed in the lingual arch. Other than the compensatory few millimeters of expansion, the deactivation shape is simple to make because only deformation from the couple on the active molar (along with the vertical forces) is needed. The lingual arch is held with a plier on the couple side, and a vertical force is applied on the other side at 90 degrees to the occlusal surface of the wire to simulate the shape for the desired force system. During the simulation process, torsion and bending occur. The anchorage-side free end will lie apical to the bracket. A force gauge attached to the free end can measure the force by pushing down on the archwire arm until the required level of force is reached. After the correct deactivated shape is fabricated, the required force system should be obtained when the arm is pushed up to the bracket level. In this example (Fig 13-58), 1,000 gmm is the moment wanted for correction; therefore, a 33-g vertical force is needed to insert the wire into the left anchor bracket (33 g × 30 mm = ~1,000 gmm). Figure 13-58a shows all the forces on the teeth (deactivation forces) after insertion. If all arrows are reversed, it would be the activation force diagram that correctly shows the validity of the force system (Fig 13-58b). This simple simulation approach (applying the deactivation force system and observing the shape change) determines the deactivated shape where both torsion (green region in Fig 13-59a) and some

Asymmetric Applications

a

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Fig 13-57  Asymmetric expansion by the couple on the affected side (method 2). (a) The force system at the CR. The couple moves the right molar to the buccal. Vertical forces act as anchorage. (b) Equivalent force system at the brackets (yellow arrows). (c) No couple at the bracket on the anchorage side is a simpler force system that approximates the force system in b.

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Fig 13-58  (a) The deactivation force system (red) is used to simulate the correct shape. The vertical force is calculated to deliver the required moment. (b) Trial activation. The activation force system is applied using a force gauge in the direction of the blue arrow for fine-tuning of the shape.

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Fig 13-59  In method 2, during simulation, torsion is produced at the green region (a), and bending occurs at the red region (b).

bending (red region in Fig 13-59b) are needed. The effect of the vertical force is negligible; therefore, little lingual tipping would occur by the intrusive force acting on the left molar in Fig 13-58a. Should lingual tipping of the left molar occur, it could be compensated by adding proper moments to counter the effect. Some prefer to round the archwire on the noncouple side to assure that no moment is delivered from the archwire on the “good” side. Figure 13-60 shows the typodont result of applying the unilateral couple of method 2. The right side moved buccally, and the left side was maintained. Note that there is no difference between method

Fig 13-60 The typodont result of applying the unilateral couple of method 2. The right side moved buccally, and the left side was maintained. Note that there is no apparent difference between method 1 (see Fig 13-56) and method 2.

1 (see Fig 13-56) and method 2 (see Fig 13-60), and little difference would be expected clinically. The choice between the first and second method depends on clinical feasibility. Generally, using a couple (second method) gives a more constant force system in that the center of rotation does not change much as the lingual arch deactivates. In other words, the sensitivity of tooth movement is reduced in method 2. Method 1, on the other hand, is clinically very difficult because translation itself is very sensitive to force position. In method 2, the procedure of measuring the moment requires only placing the archwire in one bracket and having the other be 261

13 Lingual Arches

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Fig 13-61  (a and b) A patient with a severely lingually tipped mandibular left second molar that needs unilateral expansion. The mandibular left first molar is a pontic. Unilateral expansion by a couple (method 2) to the left second molar was the method of choice. (c) Note that the left second molar uprighted and moved buccally (green rectangle) while the anchorage of the left first molar has been preserved.

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displaced vertically, either gingivally or occlusally. If the archwire is displaced occlusally, it is far easier to place. This is another factor to consider when selecting between method 1 and method 2. Figure 13-61 shows a patient with a severely lingually tipped mandibular left second molar that needs unilateral expansion. Note that the left second molar is so severely tipped lingually that a pontic with an occlusal rest on the first molar was used (Fig 13-61a). Unilateral expansion was achieved by adding a couple to the problem tooth. Note that the left second molar uprighted and moved buccally (green rectangle in Figs 13-61b and 13-61c). There was no expansion force but only buccal crown torque for the asymmetric movement; the right second molar maintained its original position. If adjacent teeth had been used for anchorage on the left side with a facial continuous arch, it would have been more difficult to achieve success and could have potentially caused many problematic side effects. The patient in Fig 13-62 with a unilateral reverse articulation needs unilateral constriction of the 262

Fig 13-62  (a and b) A patient with a unilateral reverse articulation needed unilateral constriction of the mandibular right first molar. A couple only on the affected side (method 2) was used for the unilateral constriction. (c) A passive lingual arch with comfort bends. (d) The deactivation force system and shape produced by simulation. (e) Unilateral constriction of the mandibular right molar was achieved with no apparent side effects.

e

mandibular right first molar. A couple was added only on the affected side (method 2) for unilateral constriction. A passive lingual arch was fabricated for comfort with minimal clearance between all anatomical structures (see Fig 13-62c). The simulation or the correct deactivated shape before placement into the right-side bracket is shown in Fig 13-62d. The red force system in Fig 13-62d is the deactivation force system that was used to simulate the archwire to study its shape. Unilateral constriction of the mandibular right molar was achieved with no apparent side effects (see Fig 13-62e). Figure 13-63 is a frontal view of a TPA for unilateral constriction where the maxillary right side is to be narrowed. The red arrows are the force system during simulation and are also the forces acting on the teeth after full insertion (Fig 13-63a). The simulation from the posterior view shows that the deactivated shape is created by most bending occurring near the critical sections (red region in Fig 13-63b) that lie close to the affected molar bracket. Cyclic bending and sharp bends or twists should be

Asymmetric Applications

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Fig 13-63  A TPA for unilateral constriction of the maxillary right side using method 2. (a) The deactivation force system and simulated shape. (b) The deactivated shape is created by most bending occurring near the critical sections (red region). (c) The maxillary right side has narrowed, and anchorage on the left side has been preserved.

Fig 13-64  Unilateral expansion by method 1. (a) The patient had a unilateral reverse articulation on the right side. (b) The maxillary right side was expanded unilaterally. (c) The lateral force simulation shape. (d) The vertical force and moment simulation shape.

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avoided in this high-stress region (especially near the 90-degree bend near the lingual bracket) because they can lead to fatigue fracture. As before, the desired force and moment were calculated, and the force was measured with a force gauge. Trial activation was performed to bring the free end to the level of the left molar bracket, and the original width was maintained. Width should be checked because there can be an association between the force and the moment in a TPA. Note that there is no torsion required in this simulation. In Fig 13-63c, the response after insertion into the typodont can be seen. The maxillary right side has narrowed, and anchorage on the left side has been preserved. The patient in Fig 13-64 had a unilateral reverse articulation without a mandibular shift. Correction involved both maxillary and mandibular lingual arches (Figs 13-64a and 13-64b). The force system of method 1 (adding a couple to the anchorage side) was de-

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livered by a maxillary horseshoe arch. The maxillary force-driven shape was accomplished in two separate steps. Step 1 was the expansion, where the typical free ends diverge (Fig 13-64c). A unilateral bendtwist was placed on the anchorage left side, and a small compensating second-order bend was placed on the right side as step 2. A force gauge was used to measure both the horizontal force and the left couple magnitude (vertical forces). The original passive shape and the simulated (deactivated) shape (Fig 13-64d) were recorded on graph paper for reference. For both method 1 and method 2 for unilateral reverse articulation correction, there are a number of advantages in selecting the horseshoe lingual arch over the TPA. The force is dissociated from the moment, and the clinician can use interactive mechanics and thus continually modify the appliance to fine-tune the result. Horseshoe arches are easi263

13 Lingual Arches

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er to make than TPAs where contouring for comfort is demanding. Nevertheless, the force systems in methods 1 and 2 with the horseshoe arch or TPA configuration are valid choices by the orthodontist.

Unilateral rotation A lingual arch can be very useful if an asymmetric (unilateral) rotation needs correction. The mandibular right molar shown in Fig 13-65 requires a mesial-out rotation. The forces acting on the teeth (brackets) are shown in Fig 13-65a. This force diagram is based on the equilibrium diagram, which is identical except that the force and moment directions are reversed; equilibrium demonstrates that the solution is valid. The deactivation force system was applied to deform the arch, giving the simulated shape (see Fig 13-65a). Note that to create this 264

Fig 13-65  Unilateral rotation with a mandibular lingual arch. The mandibular right molar requires a mesial-out rotation. (a) Simulated shape. Most bending is done near the right attachment (red region). (b) After insertion, the right molar has rotated mesial out. No side effects are evident.

Fig 13-66  Unilateral rotation with a maxillary TPA. It is an efficient design because of dissociation in the occlusal view. (a) Torsion (green region) and bending (red region) during simulation. (b) The right molar is corrected. Anchorage control is good because the distal force on the left side is small.

Fig 13-67  (a) The patient has a maxillary right first molar rotated mesial in. (b) Unilateral rotation was accomplished with no side effects.

shape, most bending is done near the right attachment (red region), where the critical section is located. After insertion (Fig 13-65b), the right molar has rotated mesial out. No side effects are evident. According to the free-body force diagram, a distal force is present on the anchorage side and a mesial force on the active right molar; however, these mesiodistal forces are small because of the large distance across the arch. This is another example of a favorable mechanical advantage of a desired large moment associated with unwanted small magnitude forces. Similar mechanics are used in the maxillary arch, where a TPA is an efficient design because of dissociation in the occlusal view. The force system on the teeth (red arrows) and the deactivation shape are shown in Fig 13-66a. Most twisting is done near the right attachment (green region), and supplementa-

Asymmetric Applications Fig 13-68 The maxillary right posterior segment requires rotation in a counterclockwise direction. (a) Before treatment. A modified active horseshoe-shaped maxillary lingual arch was fabricated. The configuration has a free end with a single-force contact at the mesial of the left canine. The deactivated shape is shown in green. (b) The unilateral rotation was corrected without any noticeable side effects. (c and d) Before treatment. This patient also needed reverse articulation correction of the right canine with a midline discrepancy. From the buccal tube of the right molar, a labial cantilever was extended to the maxillary right incisor segment, exerting a lateral force. The deactivated shape is shown in green. The direction of the moment at the molar (red arrows in d) is opposite to what is required; however, the magnitude of the moment from the TPA overwhelmed the moment from the labial cantilever. (e and f) After treatment. The midline and reverse articulation were also corrected.

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ry bending (red region) is performed to ensure passivity on the left side. Distributing torsion along the vertical arms can minimize permanent deformation of the TPA. Figure 13-66b shows the right molar corrected with good anchorage control after the TPA was inserted. As in Fig 13-65, the effect of the mesiodistal forces is negligible. The patient in Fig 13-67 has a maxillary right first molar rotated mesial in. A straight wire on the buccal would tend to expand the molar buccally; furthermore, if a facial arch is tied back, the buccal tube on the molar will be prevented from moving distally (Fig 13-67a). The efficient rotation correction is shown in Fig 13-67b; side effects are minimized. Along with the correction of the rotation, no other side effects like distal movement are detected on the left side. The maxillary right posterior segment (Fig 13-68a) requires rotation in a counterclockwise direction. A modified active horseshoe-type maxillary lingual arch was placed for unilateral rotation of the maxillary right buccal segment. The configuration has a free end with a single-force contact at the mesi-

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al of the left canine. Because it is a cantilever, the force magnitude measured by a force gauge is sufficient to make the force system statically determinate. The deactivated shape is depicted in green (see Fig 13-68a). Shape is not as critical with a cantilever as long as magnitude, direction, and point of force application are correct. The unilateral rotation was corrected without noticeable side effects from the forces (Fig 13-68b). In the frontal view, this patient also needs correction of the reverse articulation of the right canine and midline discrepancy. From the buccal tube of the right molar, a labial cantilever was extended to the maxillary right incisor segment, exerting a lateral force (Figs 13-68c and 13-68d). The direction of the moment at the molar is opposite to what we need; however, the magnitude of the moment from the TPA overwhelms the one from the labial cantilever. The midline and reverse articulation were also corrected after the treatment (Figs 13-68e and 13-68f). Here one of the largest interbracket distances inside the oral cavity was used; therefore, the force side effects were kept to a minimum.

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13 Lingual Arches

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Another example of a unilateral rotation with a cantilever of the maxillary second premolar is found in Fig 13-69. The premolar underwent a rotation of over 100 degrees (dotted arrows); however, the side effects are hardly seen. When a single force is applied to an anchor tooth, there are two choices; one is full bracket engagement, and the other is a free end with a ligature tie. Full engagement may be more secure in maintaining the appliance, but the cantilever is the simpler option for the delivery of the correct force system. The cantilever does not require an equilibrium diagram for checking the validity of the force system; the applied force diagram (based on the force gauge reading) and its equivalent force system through an appropriate CR (yellow arrows in Fig 13-69e) should be sufficient. Also, the shaping is not sensitive. As long as the force is correct, any shape will work. Still, any fine-tuning of a cantilever involves considering possible changes in force direction over time and appliance comfort. 266

c

Fig 13-69  A cantilever from the maxillary second premolar is used to correct its rotation. The premolar underwent a rotation of over 100 degrees (dotted arrow); however, the side effects are hardly seen (a to d). The cantilever does not require an equilibrium diagram for checking the validity of the force system; the applied force and its equivalent force system through an appropriate CR should be sufficient (e).

Fig 13-70  Fallacy of moving right and left molars distally in two steps. (a) First, the right molar is moved distally with a single force by shaping a TPA. (b) Then the force system is reversed, and the left molar is moved distally. This figure is actually the same as Fig 13-44 (bilateral rotation) except that the force is applied sequentially in two steps.

It could be suggested that to accomplish bilateral distal movement of the molars on a Class II patient, an asymmetric activation of the TPA could be used. The suggested concept is first to move the right molar distally with a single force by shaping a TPA as shown in Fig 13-70a. The opposite anchorage molar feels a mesial force and a moment that is mesial out on the molar. After the right molar has moved distally, the force system is reversed, and the left molar is moved distally (Fig 13-70b). Is it possible to produce bilateral distal molar movement by alternating sides with this force system? Unfortunately, the anchorage molar feels very high shearing stress because the mesial-out moment on the molar is so large. The mechanical advantage is to rotate the anchorage molar rather than move the opposite side distally. The reason is the same as explained for the delivery of the moment to accomplish unilateral rotation; the large interbracket distance favors the moment rather than the force. Figure 13-70 is actu-

Asymmetric Applications Fig 13-71  (a and b) A patient in whom the asymmetric force system from Fig 13-70a was used. Note that the left molar received a very large moment while the right molar received a small distal force, so little movement is observed on the right side.

a

ally the same as the bilateral rotation in Fig 13-44, but the shape was applied sequentially in two steps. Figure 13-71 shows a patient in whom the force system from Fig 13-70a was used. Notice the effect. The right molar did not appreciably move distally. The left molar radically rotated because of the large mesial-out moment. Alternating sides only will produce bilateral molar rotation and no distalization of the CR. This could help correct a Class II malocclusion for the reasons explained in Fig 13-47; however, the same result can be more simply achieved by applying equal and opposite rotation couples to the first molars in one step.

Unilateral tip-back and tip-forward mechanics Typically, equal and opposite single forces at the facial side of a continuous arch are considered when the distal movement of a posterior segment or molar is indicated. This is problematic unilaterally if the anterior teeth are used as anchorage because incisors can flare and a midline discrepancy can occur along with other side effects. Can a lingual arch offer other possibilities? Let us first look from the occlusal view. It would be easy to think of equal and opposite single forces (right acting distally) on the lingual brackets to generate unilateral tip-back (right) and tip-forward (left); however, a valid force diagram in equilibrium cannot be constructed with these forces alone (Fig 13-72a). This is impossible. However, couples added to the same plane of occlusion are a possible equilibrium force system. Figure 13-72b is a valid force diagram that is in equilibrium after the addition of equal-magnitude counterclockwise rotational couples at both molars. If a lingual arch were inserted, more buccal segment rotation (right side mesial in

b

and left side mesial out) than tip-forward and tipback from the mesiodistal forces would be observed. Both would probably occur, so this lingual arch force system may be valid for very specific malocclusions requiring such movements. Another possibility is to place the couple on one side only (Fig 13-72c) if only that side (molar) requires rotation. Sometimes, rotating a molar or a posterior segment can give a favorable distal force to distalize the affected side; however, this is usually unlikely, as shown in Fig 13-71. A better approach is needed. An important and unique force system that a lingual arch can provide is equal and opposite couples that act in the sagittal plane (Fig 13-72d). The effects from a single force at the crown and those from couples are indistinguishable clinically. Also, this force system is not available in a full facial continuous arch where tip-back or tip-forward of posterior teeth is pitted against the anterior segment. The starting shape is always passive, and then equal and opposite couple loading is simulated. A mandibular horseshoe lingual arch has been simulated by applying equal and opposite couples: tipback moments on the right side and tip-forward moments on the left side (Fig 13-73a). The lingual arch is twisted at the apex (green region in Figs 13-73a and 13-73b) and bent with a gentle curvature bilaterally (red region). Note the smooth curvature of the arch, which is not surprising because couples deform a straight wire into a segment of a circle. Once the adjacent tooth is removed from the typodont, the tooth displacement is more evident after insertion of the activated lingual arch. Note that space has opened on the right side mesial to the first molar as it has tipped back and the left molar has tipped forward (Fig 13-73c). This action is independent from the anterior segment because equilibrium is created across the arch bilaterally. Basically, there is no difference in deactivation shape 267

13 Lingual Arches

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Fig 13-72  Producing unilateral distal movement with a lingual arch. (a) An invalid deactivation force diagram. This is not possible because the archwire would not be in equilibrium. (b) One possible valid deactivation force diagram with equal-magnitude counterclockwise rotational couples at both molars; however, this rotation is rarely indicated. (c) Another valid deactivation force diagram with the moment on one side only; this is also rarely indicated, ie, that a molar or buccal segment needs rotation in this direction on one side only. In b and c, moments are large, and the distal force is too small to be effective. (d) A valid deactivation force diagram with equal and opposite couples acting in the sagittal plane; this is the most practical force system and does not use a force for distalization.

c

Fig 13-73  Unilateral tip-back and tip-forward with a mandibular lingual arch. (a) The simulated (deactivated) shape. Torsion (green region) and bending (red region) occur along the arch (a and b). (b and c) Unilateral tip-back and tip-forward occurs on the typodont without forces.

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of the TPA in the maxillary arch (Fig 13-74) where equal and opposite tip-forward and tip-back moments are generated simultaneously, governed by the law of equilibrium. Let us see the force system developed in a unilateral tip-back and tip-forward lingual arch if it were fabricated in an ideal shape in three dimensions. Figure 13-75a shows the passive shape of a TPA, as268

Fig 13-74  (a and b) Unilateral tip-back and tip-forward with a maxillary TPA. The deactivated shape of the TPA is basically the same as the horseshoe arch except that it is turned at 90 degrees. The left side is the tip-back side.

suming that the molar lingual attachments are parallel with each other. In Fig 13-75b, second-order bends were placed for unilateral tip-back and tip-forward based on the ideal shape method. This ideal shape generates a very complicated force system three dimensionally. When activated by equal and opposite couples on both sides for the free ends to be parallel to each other, the free ends displace

Asymmetric Applications

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Fig 13-75 Why the ideal arch shape for unilateral tip-back and tip-forward mechanics is incorrect. (a) Passive shape. (b) Second-­ order bends next to the brackets were placed for unilateral tip-back and tip-forward based on the ideal shape concept. (c and d) When equal and opposite couples were applied on both sides of the free ends to make them parallel to each other, the tip-back arm moved anteriorly (Δ). Therefore, the tip-back side feels both a tip-back moment and an anterior force. Instead of the molar crown moving distally, the root could come forward. (e) The mesiodistal forces can produce undesirable molar rotations from the occlusal view.

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Fig 13-76  The force-driven shape (a to c) produces only equal and opposite couples (d) without any side effects.

anteroposteriorly (Figs 13-75c and 13-75d). This requires additional mesiodistal forces for engagement into the brackets. This is not desirable because the force is acting in the opposite direction to what we want. The molar mesial root movement can occur on the tip-back side (Fig 13-75e). Furthermore, rotational side effects on the molars in the occlusal view will be produced (see Fig 13-75e). This is also the force system produced by a TPA with parallel arms from the lateral view if right and left lingual molar brackets are not parallel (one side molar bracket is tipped back and the other side molar bracket is tipped forward). In short, the ideal arch shape for this application should not be used because of very complicated

three-dimensional side effects. If the TPA in Fig 13-75 is turned at 90 degrees, it becomes the horseshoe lingual arch, and its improper shape gives the same side effects. Therefore, the force-driven shape (Figs 13-76a to 13-76c) is mandatory in unilateral tip-back and tip-forward applications to produce equal and opposite couples only (Fig 13-76d). Another application of a lingual arch with equal and opposite couples is to equalize right and left posterior occlusal planes, acting on the entire buccal segment rather than on the molar alone. What if we want tip back a molar on one side only? Would it be possible? The anchorage side will tip forward from the lingual arch’s equal and opposite couple; a 269

13 Lingual Arches

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Fig 13-77  (a to c) The patient has an asymmetric molar relationship: Class I on the right side and Class II on the left side. (d) A unilateral tip-back/tip-forward active TPA was placed. (e and f) On the labial, two symmetric tip-back cantilever springs were inserted bilaterally on the anterior segment. Note the correction of the Class II molar relationship on the left side.

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method must be available to prevent that from happening. In reality, there are two possible approaches for handling the anchorage side effect. The first method is to counter the tip-forward moment on the anchorage side by a labial appliance. The patient in Fig 13-77 has an asymmetric molar relationship: Class I on the right side and Class II on the left side. The treatment plan was to tip the left 270

Fig 13-78 (a and b) An asymmetric tipback/tip-forward active TPA was placed in a patient with a Class II malocclusion on the right side. (c and d) Right and left tipback cantilevers were added on the labial side. No buccal archwire but only a ligature wire was placed in the premolars and canine to allow them to drift distally individually on the right side and to maintain the plane of occlusion of posterior teeth.

molar back using a tip-back/tip-forward TPA. Figures 13-77c and 13-77d show the maxillary arch before and after TPA treatment. It is seen that only the left first molar has tipped back. On the labial, two symmetric tip-back cantilever springs were inserted bilaterally (see Figs 13-77e and 13-77f). The sum of the force systems from both appliances are the following: The tip-forward lingual arch moment on

Summary Fig 13-79  (a and b) A patient needed unilateral second molar tip-back on the maxillary left side. (c) A tip-back moment was applied to the left molar. (d) All other teeth were rigidly joined to comprise an anchorage unit. Space opened with minimal side effects. A couple, not a distal force, was used for the distalization of the left second molar.

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the right side and the cantilever spring sum to zero, whereas the lingual arch tip-back moment is doubled on the left side from the cantilever activation. In a similar patient, an asymmetric tip-back/tipforward arch was placed in a patient with a Class II malocclusion on the right side (Figs 13-78a and 13-78b). Right and left tip-back cantilevers were added on the labial to enhance the right-side moment and to prevent the left posterior segment from tipping forward. No buccal archwire but only a ligature wire was placed in the premolars and canine to allow them to drift distally individually on the right side and to maintain the plane of occlusion of the posterior teeth (Figs 13-78c and 13-78d). A segment of wire connected at the posterior segment, including the first molar, would undesirably cant the right-side plane of occlusion as the first molar tipped back. The second method to prevent the tip-forward side from tipping forward is to connect the anchorage side to an arch segment joining all the teeth on that side or, even better, connecting more teeth around the arch. The patient in Fig 13-79 requires unilateral tip-back on the maxillary left side (Figs 13-79a and 13-79b). The required force system is depicted in Fig 13-79c. The tip-forward moment on the right side is prevented because all the teeth are rigidly joined together as an anchorage unit (Fig 13-79d).

d

Summary In this chapter, force systems from lingual arches of different configurations and applications have been described. Because there are only two brackets involved, the lingual arch offers an excellent opportunity to analyze simple appliance equilibrium in three dimensions. Emphasis has been placed not only on arch fabrication techniques but also on developing the principles of correct force delivery. Force-driven appliances have shapes that may look unusual and nothing like an ideal arch. Absolute force values in three dimensions were determined using beam theory. The ideal arch shape is easy to visualize and simple to make, but the result is unpredictable; the force system is complicated, and many side effects may occur. The starting point of all appliance design is to establish the desired force system and to make sure it is valid; the lingual arch must be in equilibrium. A simple equilibrium force diagram is not wasted time. Appliance selection, specific design considerations, and fabrication follow in later chapters.

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References 1. Burstone CJ, Hanley KJ. Modern Edgewise Mechanics and the Segmented Arch Technique. Glendora, CA: Ormco, 1986. 2. Burstone CJ. Precision lingual arches. Active applications. J Clin Orthod 1989;23:101–109. 3. Burstone CJ, Koenig HA. Precision adjustment of the transpalatal lingual arch: Computer arch form predetermination. Am J Orthod 1981;79:115–133.

Recommended Reading Burstone CJ, Koenig HA. Force systems from an ideal arch. Am J Orthod 1974;65:270–289. Burstone CJ, Manhartsberger C. Precision lingual arches. Passive applications. J Clin Orthod 1988;22:444–451. DeFranco JC, Koenig HA, Burstone CJ. Three-dimensional large displacement analysis of orthodontic appliances. J Biomech 1976;9:793–801. Drenker EW. Forces and torques associated with second order bends. Am J Orthod 1956;42:766–773. Koenig HA, Burstone CJ. Analysis of generalized curved beams for orthodontic applications. J Biomech 1974;7:429–435. Koenig HA, Burstone CJ. Force systems from an ideal arch— Large deflection considerations. Angle Orthod 1989;59:11–16.

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PROBLEMS The following problems may require the principle of equilibrium for their solution. The first step should be an equilibrium diagram with forces and moment on the appliance. Solve for unknowns on the appliance. Then the direction of the force system is reversed for the forces on the teeth. 1. The maxillary right second molar is erupted buccally with mesial-in rotation. A passive lingual arch with a rigid wire extension was used to pull the second molar palatally. Find the resultant force system at the CRs of the second molar and the anchorage unit (two first molars).

2. The situation is the same as in problem 1, but a flexible wire was used for the extension instead of a rigid one. How does it affect the force system?

3. The mandibular molars require bilateral expansion, and the deactivation force on the left molar is given (200 g). What would be the force magnitude required at the right molar? It is still not in equilibrium, so find the moments for the following conditions:

4. The maxillary right first molar requires unilateral expansion. The maxillary TPA was fabricated with angulation (buccal crown twist) of the right side arm only based on an ideal shape-driven shape. Draw the directions of the force system. What side effect is anticipated at the anchorage tooth? Ignore the horizontal forces.

a. Add a moment on the right molar only for a valid deactivation force diagram. b. Add a moment on the left molar only for a valid deactivation force diagram. c. Equally distribute the moments at both molars for a valid deactivation force diagram.

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5. The lingual arch arms (deactivated) were designed parallel to the brackets for bilateral constriction. Give the relative force system at the bracket and molar CR (direction only is required, not magnitude) acting at each molar. Assume that the force at the bracket passes through the CR.

6. The situation is the same as in problem 5, but the posterior segment is connected by rigid wire forming right and left rigid units. Replace the force system at the CR of each posterior segment. Describe the difference with problem 5.

7. A 1,000-gmm tip-back moment is delivered to the maxillary left first molar. What is the valid force system acting on the maxillary right molar if no other forces exist?

8. The situation is a the same as in problem 7 except that the maxillary right molar is already located distally. What is the valid force system acting on the maxillary right molar if no other forces exist?

9. Bends were placed next to the molar brackets (tip-back on the left and tip-forward on the right) based on an ideal arch–shaped TPA. The ideal shape gives an unwanted mesial force on the left molar. Calculate all other forces and moments on the teeth. What are the side effects if this TPA arch shape is used?

10. A patient has a maxillary palatal reverse articulation on the right side and a buccal crossbite on the left side with poor axial inclinations. a. A lingual arch is used to correct the reverse articulation. Draw a deactivation force diagram (only direction of forces, not magnitude, required). b. What type of lingual arch is preferred? Why?

CHAPTER

14 Extraction Therapies and Space Closure “It is necessary for him who would reach his journey’s end to sometimes go round about.”

— Persian proverb

“The Tao that can be described is not the true Tao.”

— Lao Tzu

“A short cut is always a long road. A short cut is often a wrong cut.”

OVERVIEW



— Danish proverb

Extraction cases require a solid understanding of biomechanics whether or not sliding or friction-free appliances are used. The ratio of anterior retraction to posterior protraction is primarily determined by the force system. Group A (anchorage cases) mechanics may also use an increased number of teeth, rigid segments, and undisturbed anchorage. Little evidence suggests that separate canine retraction is more conservative of anchorage than en masse space closure. In both sliding mechanics and friction-free loops, the moment-toforce (M/F) ratios at the bracket are continually changing. Distinct phases are tipping, translation, and root movement. Cantilevers (gingival extensions) for space closure are statically determinate and deliver more constant M/F ratios at the bracket, minimizing different stages of retraction. The scientific basis for spring design for a space closure loop is discussed in this chapter. Variables include loop height, apical added wire placement, and interbracket distance. The roles of the activation moment and the residual moment are explained. Temporary anchorage devices also require good biomechanics because side effects can occur.

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a

b

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Fig 14-1  Three categories of differential space closure. (a) Group A mechanics. Most of the extraction space is closed by retraction of anterior teeth. (b) Group B mechanics. The extraction space is closed by equal attraction of both anterior and posterior teeth. (c) Group C mechanics. Most of the extraction space is closed by protraction of posterior teeth.

Many appliances and techniques have been introduced for closing space in extraction patients. Emphasis has been placed on wires, brackets, named techniques, and specially designed appliances. If the clinician is to select the best methods and optimally use any mechanism, less consideration should be given to the hardware and more to sound biomechanical principles applied to space closure. The movement of a tooth is solely a response to the periodontal ligament (PDL) stress produced by the appliance. The PDL does not have a preference for the material, size, or shape of the wire; type of the bracket; or kind of configuration used. This chapter discusses the biomechanical principles governing both retraction springs and sliding mechanics that are used for extraction space closure. The force system developed during space closure can be very sophisticated due to factors such as friction, threedimensional (3D) effects, cross-section shape and dimension of the wire, bracket width, loop shape, and modulus of elasticity, among others. However, the discussion here is kept as simple as possible. The objective of this chapter is not to advocate any specific appliance or technique but rather to introduce important principles so that the clinician can creatively design, select, and modify his or her appliance to fully control the force system during space closure.

Differential Space Closure Extraction therapy is frequently necessary in orthodontic treatment of patients with severe crowing or protruding anterior teeth (with or without crowding). Once extraction is decided upon, the anteroposterior position of the incisors must be established, and only then can the optimal force system be determined. For example, if the treatment goal is to retract a canine and maintain the anteroposte276

rior position of the molar, differential space closure is required. Differential space closure is divided into three categories, depending on how proportionally anterior and posterior segments contribute to the space closure. In group A mechanics, most of the extraction space is closed by retraction of anterior teeth (Fig 14-1a). In group B mechanics, the extraction space is closed by somewhat equal attraction of both anterior and posterior teeth (Fig 14-1b). In group C mechanics, most of the extraction space is closed by protraction of posterior teeth (Fig 14-1c). Obviously, differential space closure with group A and group C mechanics is more challenging than with group B mechanics.

Strategies for Maintaining Posterior Molar Position: Group A Mechanics The stress developed in the PDL by the orthodontic appliance is the initiator of tooth movement. Even the insertion of the separation elastic ring for banding the molar could create a large stress on the molar, causing mesial movement by initiating an orthodontic tooth movement cascade. Therefore, the strategies for maintaining posterior anchorage are concentrated on keeping the stress in the PDL of the anchorage unit as low as possible so as not to initiate orthodontic tooth movement. The simplest method of reinforcing the anchorage is to increase the number of teeth in the anchorage unit. However, this method is very limited unless stress is evenly distributed among the roots. In addition to the number of teeth included in the anchorage unit, the posterior teeth can be connected rigidly so that individual movements are not allowed. Passive lingual arches provide cross-arch stabiliza-

Strategies for Maintaining Posterior Molar Position: Group A Mechanics Fig 14-2 The posterior anchorage unit. (a) Rigid wires are inserted into the posterior segments. A transpalatal arch is used for cross-arch stabilization. (b) Rigidly connected posterior teeth with minimum play form the posterior anchorage unit.

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b

a

b

Fig 14-3  Bonded FRC segment. FRC provides better passivity and stability of the segment than a wire does.

Fig 14-4  A patient with incisor protrusion. (a) The maxillary anterior segment and the mandibular buccal segments are connected by an FRC. (b) Space is closed. Note that the mandibular anterior teeth have been retracted with minimal anchorage loss and that the good intercuspation was not disturbed.

tion, and the insertion of full-size wires may prevent play between the wire and brackets so that the stress is more evenly distributed to the whole unit (Fig 14-2). Rigidly joined posterior teeth are called a posterior anchorage unit. Starting space closure with undisturbed posterior anchorage may be an important factor in group A mechanics. Technically, insertion of “straight” fullsize wires into the posterior brackets may require a leveling process because most sliding techniques require rigid archwires for space closure. The leveling process may be avoided by using completely passive wires, but this is very difficult to achieve because both wire bending and torsion may be required to ensure passivity. Even if the wire may look passive, very small wire deflections could create instantaneous heavy forces, leading to high stress on the anchorage unit. These stresses added to the stress from the space closure mechanism can lead to anchorage loss. A bonded fiber-reinforced composite (FRC) segment is a good alternative to a passive heavy wire for an anchorage unit because passivity is ensured

(Fig 14-3). Figure 14-4 shows the mandibular buccal segment connected by an FRC so that the stress is evenly distributed and initial extraneous heavy leveling forces between posterior teeth are avoided. Note that the mandibular anterior teeth have been retracted with minimal anchorage loss. Also, the good intercuspation is not disturbed, which is another contributing factor to reinforcing the anchorage. Temporary anchorage devices (TADs) are used as anchorage for space closure in the maxilla (see Fig 14-4a). The use of TADs is described in chapter 18. Supplemental forces from headgears, maxillomandibular elastics (also known as intermaxillary elastics), and TADs are additional possibilities for consideration in group A mechanics. Perhaps the most important factor in differential space closure is the application of a force system that provides differential stress on each unit so that the anchorage unit (posterior teeth) translates and the active unit (anterior teeth or canine) tips. Figure 14-5 shows a physical model of tooth displacement occurring within the PDL. The red line is drawn on the background, and the green line is drawn on the 277

14 Extraction Therapies and Space Closure

a

b

Fig 14-5  A physical model of teeth showing differential space closure. (a) An elastic is placed at an angle from the left tooth to the right tooth. (b) The T-loop acts identically to the angled elastic.

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b

c

d

Fig 14-6  A physical model of teeth showing the effect of location of the force application point and magnitude of force. (a) A light force is applied near the alveolar crest. (b) A heavier force is applied at the CR. Note that the crown movement at the bracket is about the same. (c and d) The same amounts of force are applied at different locations. Note that less crown tooth movement is observed from the force approaching the CR.

transparent teeth that are suspended by a series of elastics around the root, simulating the principal fibers of the PDL. In Fig 14-5a, an elastic is placed at an angle from the left tooth (β, posterior or anchorage tooth) to the right tooth (α, anterior or canine tooth). Compare the red and green lines. The red force on the anchorage tooth acting through its center of resistance (CR) translates the tooth. By contrast, the right anterior tooth with the red force 278

applied at the alveolar crest tips around a point at its apex. Translation produces a more uniform stress distribution in the PDL and, hence, preserves anchorage. One should also notice with the angled elastic an intrusive force on the anterior unit. Is it possible to run an elastic in this manner clinically? The answer is no, of course, because of anatomical limitations. But we can place an equivalent force system at the brackets with the T-loop shown in Fig

En Masse Versus Separate Canine Retraction

a

b

Fig 14-7 Two basic types of space-closing mechanics. (a) Sliding (or friction) mechanics. (b) Loop (or frictionless) mechanics. In sliding mechanics, the tooth feels less force than that applied as a result of friction. In loop mechanics, there is no loss of applied force due to friction.

14-5b that acts identically to the angled elastic (see discussion of equivalence in chapter 3). Let us compare the effect of force placement level on the amount of crown tooth movement using the same model. A light force is applied near the alveolar crest (Fig 14-6a), and a heavier force is applied at the CR (Fig 14-6b). Note that the elastic chain is stretched more in Fig 14-6b. Crown movement at the bracket is about the same. Why? Because forces away from the CR produce higher stresses and strain. The sum of all strain, clinically, we call tooth movement. If we use the same model again and keep the forces identical, more crown tooth movement will be observed from the force occlusal to the CR (Figs 14-6c and 14-6d). This is to be expected because higher PDL strains are present. Thus, differential tooth movement can be produced by changing the point of force application. This model represents the initial stage of tooth movement (mechanical displacement) and relates only to the stress-strain pattern. The biologic displacement occurs later. But is it possible to achieve differential tooth movement with the same force? Yes. Because tooth morphology and tooth number can influence the stress. As demonstrated by the above model, even if the resultant forces on the anterior and posterior segments are equal and opposite, differential tooth movement can be produced if the force is angled relative to the occlusal plane. Other equilibrium situations are possible where the resultant force system has equal and opposite forces and unequal moments. (The forces do not have to be on the same line of action. The elastic used in this model is a special case.) Sometimes the phrase differential force is used to describe the differences between anterior and posterior tooth movement. Without explanation, this concept can be confusing. If only one appliance is used, the force magnitudes to the anterior and posterior teeth must be the same (equilibrium principle). However, because we normally apply our forces

at the brackets, a differential M/F ratio system can exist. Forces may be equal and opposite, but moments can be different at the anterior and posterior brackets; hence, differential M/F ratios can be expected.

En Masse Versus Separate Canine Retraction In cases of severe crowding, separate canine retraction is necessary for gaining space for incisor alignment. Classically, anterior teeth were retracted in two stages. It was believed that separate canine retraction followed by four-incisor retraction would preserve the posterior anchorage because lighter forces could be used at each stage. Perhaps this could work if sufficiently low magnitudes of force were used; however, most clinicians use about the same forces for separate canine retraction as for en masse space closure. Clinical studies show that there is no difference in anchorage loss between en masse and two-stage retraction.1 Therefore, there is no reason to retract the anterior segment in two stages unless it is indicated for special situations such as anterior crowding, flared incisors, extruded or high canines, and midline discrepancies. Two-stage retraction is more complicated, unesthetic due to transient spacing between the canine and lateral incisor, has a longer treatment time, and is more likely to lead to iatrogenic side effects like incisor extrusion, especially during sliding mechanics.

Intra-arch space-closing mechanics There are two basic types of space-closing mechanics: sliding (or friction) mechanics (Fig 14-7a) and loop (or frictionless) mechanics (Fig 14-7b). In sliding mechanics, the force is applied by an elastic or spring, and the bracket slides along the guiding archwire. 279

14 Extraction Therapies and Space Closure

a

b

There is always friction between the bracket and the archwire so that the tooth feels less force than the force applied by the elastic or coil spring. The guiding wire provides moments required for prevention of tipping and rotation (see Fig 14-7a). In loop mechanics, there is no guiding wire, and the spring provides both force and moment so that there is no loss of applied force due to friction; because friction is usually unknown, results may be more predictable (see Fig 14-7b) with loop mechanics than with sliding mechanics. On the other hand, sliding mechanics may provide better control over all tooth movement because the archwire serves as a definitive guide. Placing differential moments is easier with frictionless loop design and allows for better delivery of group A and group C specialized mechanics.

Continuous Versus Segmented Arches In applying sliding space closure mechanics, the archwire can be continuous from the posterior tooth (second or first molar) on one side to the posterior tooth on the opposite side, and the shape of the wire cross section or material used is usually unchanged (Fig 14-8a). A loop can be incorporated in a continuous archwire for frictionless mechanics (Fig 14-8b). However, with a continuous archwire many possible interactions can occur between each tooth (bracket); therefore, activations can become complicated and indeterminate. Furthermore, the small interbracket distances limit the accuracy of both linear and angular activations. Figure 14-2 shows a possible segmented archwire where the maxillary arch is divided into an anterior segment and two posterior segments. If the right and left posterior segments are connected by a passive lingual arch (transpalatal arch [TPA]) and rigid stabilizing archwires are inserted, one could consid280

Fig 14-8  Continuous archwire. (a) The archwire is relatively straight for sliding mechanics. (b) A loop can be incorporated for loop mechanics.

er the maxillary arch as composed of only two teeth, a multirooted anterior tooth and a multirooted posterior tooth. These two segments can be connected by two springs of varying cross section and material. By reducing an arch into two segments, the orthodontist needs only place a single activation per side between the auxiliary tubes on the canine and the molar. The use of only these two attachments per side also increases the intertube distance (distance between auxiliary tubes), allowing for larger activations and greater accuracy of activation.

Friction (Sliding) Mechanics Tooth movement from the facial view typically follows four phases during sliding mechanics (Fig 14-9). In three dimensions, it is more complicated because from the occlusal view rotational moments must be considered. This is described in more detail in chapter 19 (on friction). In this chapter, only the force system in the lateral view is considered. During phase I, after the tooth is leveled, a distal force is applied. Uncontrolled tipping can occur because of the play between the bracket and the wire. In phase I there is no friction. With wide brackets, little phase I tipping occurs because play is small. In comparison, narrow brackets used in the Begg technique allow for considerable tipping without friction. In phase II, more tipping is observed; friction is now increasingly present as the tipped bracket deflects the wire more and more. As a reaction from the deflected wire, two vertical forces (couple) act on the bracket slot, fighting the tipping tendency. Forces from these moments produce normal forces, which cause the friction. When the tooth tips sufficiently to deflect the wire enough to create higher moments, the tooth translates (phase III). Distal tooth movement or space closure can stop as the root-uprighting moments increase or the distal force lessens. Root correction now occurs (phase IV).

Friction (Sliding) Mechanics Fig 14-9  The facial view of four phases in sliding mechanics (right to left). A force is applied distally from right to left (red arrows). Phase I: Uncontrolled tipping with play. The tooth freely tips without friction until the bracket slot and wire touch each other. Phase II: Controlled tipping. More tipping is observed, and the wire starts to deflect. The deflected wire exerts two equal and opposite normal forces (couple), and friction increases. Phase III: Translation. If the moment from the deflected wire is high enough, translation occurs. Phase IV: Root movement. The tooth movement can stop as the rootuprighting moments increase or the distal force lessens.

a

b

Fig 14-10 The side effects of sliding mechanics with an inadequately stiff guide wire. (a) Before retraction of the canine. (b) After retraction of the canine. (c) Iatrogenic curve of Spee. Note that the curve of Spee was developed in the mandibular arch. As the canine tipped distally, the wire deflected occlusally in the incisor region and gingivally in the premolar region (phase II tipping). The anterior vertical overlap was increased (dotted lines show the occlusal plane before retraction and the axis of the canine).

c

One problem with sliding mechanics is that friction is unpredictable and, hence, the delivered force system is unpredictable. The usual treatment goal is to translate adjacent teeth at the extraction site during space closure. Unfortunately, the highest level of friction will occur during translatory tooth movement. In fact, frictional forces acting in the opposite direction to the applied force may be large enough to stop the tooth movement completely. This phenomenon is referred to as appliance ankylosis. In phase IV, the tooth uprights without space closure; the orthodontist then reactivates the spring or chain elastic, and the tipping phases can start again. With each repeating phase, the tooth will wiggle back and forth until space is closed. In other words, the center of rotation keeps changing during space closure, which biologically is not the most direct way to stimulate the PDL. Along with a varying M/F ratio at the bracket, the absolute magnitude of the force is fluctuating due to the variable amount of friction

in accordance with each phase. Therefore, even if the magnitude of applied force is constant and not excessive, the stress in the PDL might be too high in the uncontrolled tipping phase or too low in the translation phase. Space closure such as canine retraction with sliding mechanics may appear to be pure translation from the start to the end; however, a complicated series of phases normally occurs. Much friction can be eliminated by avoiding translatory movement and allowing primarily tipping. This has the disadvantages of requiring later root movement (axial inclination correction) and potentially leading to undesirable side effects. If the guiding continuous wire is not stiff enough, a canine might tip, resulting in increased vertical overlap (also known as overbite). Figure 14-10 shows this effect. The canine was retracted by an elastic force along a 0.016-inch stainless steel wire. As the tooth tipped distally, the wire deflected occlusally 281

14 Extraction Therapies and Space Closure

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b

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Fig 14-11  Cantilever intrusion spring in a continuous archwire. (a) Before retraction. (b) During retraction. (c) After retraction. The incisor eruption side effect is prevented by an additional cantilever intrusion force during canine retraction. Also, it provides additional tip-back moment at the molar for better anchorage. Note that the amount of anterior vertical overlap is not changed.

Fig 14-12  A second overlay continuous archwire with an intrusion force anterior to the canine prevents incisor eruption during canine retraction.

in the incisor region. Note that a curve of Spee was developed in the mandibular arch with an increase in vertical overlap as the mandibular incisors erupted and premolars intruded. To compensate for these side effects, a so-called compensating reverse curve may be incorporated in the archwire; however, it will make the force system more unpredictable and many times incorrect. Some might place a V-bend (or, better, a curvature) between the canine and the second premolar. This might prevent the canine from tipping but increases the sliding friction. The high friction can impede retraction. Larger–cross section wires can also minimize this side effect, but again the frictional forces are increased. Another approach to prevent incisor eruption during sliding canine retraction mechanics is the use of a separate cantilever that bypasses the canine with an intrusive force anterior to the canine (Fig 14-11). Or a second continuous intrusion arch can be inserted into an auxiliary tube on the molar and tied to the anterior segment (Fig 14-12). If the anterior segment is retracted en masse, the sliding occurs within the posterior segment; therefore, posterior teeth should be leveled prior to sliding (see Fig 14-7a). As mentioned previously, this leveling of posterior segments may initiate anchorage loss during tooth movement and may not be desirable in group A mechanics. 282

Because the applied force system is unknown to the operator because of the friction, and because the interbracket distance between the canine and second premolar bracket is small, differential space closure via the application of differential M/F ratios is very limited. Hence, most orthodontists only accomplish group B mechanics when using sliding mechanics. Differential space closure (group A and group C mechanics) may require additional appliances including headgears, maxillomandibular elastics, and TADs, among others.

Frictionless (Loop) Mechanics For simplicity, let us consider the force system using sliding mechanics only from the facial view (Fig 14-13a). What is the role of the guide wire? The archwire provides the required amount of moment to control the tooth movement. If the canine is translating (phase III movement), the correct moment for the needed M/F ratio at the bracket comes from the wire. If wire extensions (Fig 14-13b) are attached to the auxiliary tubes on the molar and canine so that the line of action of the force passes through each CR, the canine and molar also translate. Here no guide wire is necessary.

Frictionless (Loop) Mechanics

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b

Fig 14-13  Separate canine retraction. (a) Facial view of the force system in phase III (translation) of sliding mechanics. The wire provides required moments. (b) Sliding mechanics with an extension hook. The line of action of the force passes through each CR. There is no moment from the guide wire; therefore, a guide wire may not be necessary.

Fig 14-14 Separate canine retraction by frictionless mechanics. The spring delivers required forces and moments for controlled movement of the canine without any guide wire (facial view).

Fig 14-15  Separate canine retrac­ tion using frictionless mechanics. (a) Force system in the facial view of the spring. (b) Because the force is applied buccal to the CR, an occlusal antirotation moment is also required in the occlusal view. Note that all 3D forces are incorporated in the spring.

a

In frictionless mechanics, there is no guide wire, and specially designed springs may be used. The spring provides the required M/F ratios in three dimensions. There is no loss of applied force due to friction, providing greater predictability and versatility. Typically, with a properly shaped loop or frictionless spring, space closure will go through the same three phases of closure seen with sliding mechanics: tipping followed by translation and then root movement. The difference in a properly designed appliance is a greater activation range and a more constant force level and M/F ratio, leading to a more constant center of rotation (less wiggling).

b

Differential moments between anterior and posterior segments become easy and practical. During separate canine retraction with a T-loop spring, the facial view force system (Fig 14-14) delivers forces and moments for controlled movement of the canine without any guiding wire. Because the force is applied buccal to the CR, an occlusal antirotation moment is also required (Fig 14-15). These 3D effects are discussed in detail later in this chapter. The appliances used in frictionless mechanics can be either statically determinate or statically indeterminate (Fig 14-16).

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14 Extraction Therapies and Space Closure Fig 14-16  Two types of frictionless mechanics. (a) Statically indeterminate system. The point of force application is at the bracket. (b) Statically determinate system. The point of force application is on the root of the tooth. The red force on the left is acting at the CR of the tooth. Note that the indeterminate and determinate systems deliver equivalent force systems.

a

b

Statically Determinate Space Closure Appliances A statically determinate appliance or spring means that the law of statics (equilibrium) is sufficient to solve for all unknown forces and moments acting on the wire and the teeth. Unless we know all the forces and moments, we cannot predict the result. A common statically determinate appliance is the cantilever intrusion spring, where the force is measured at the incisors with a force gauge. All of the appliances discussed previously for space closure are indeterminate, even if the forces are measured or known. If sliding mechanics are used, the many unknowns include friction and the moments acting along the arch. A frictionless spring (loop) requires the determination of both forces and moments at their anterior and posterior ends. Thus, even if forces are known, the system is indeterminate unless the moments are also determined. We can make these springs determinate by calibrating springs in which the material and dimensions are kept constant either experimentally or theoretically. Another approach is to design a space closure device that is so simple that it delivers only a force and no moments at either end. Such an appliance is statically determinate if the force is measured at one end. A simple rubber band, coil spring, or elastic fits that description. Of course, a spring placed parallel to the occlusal plane at the level of the brackets would tip teeth during space closure. If the point of force application is moved from the bracket to a point apically in line with the CR, translation will occur. This system has both control and determinacy using equal and opposite forces without requiring moments. From the occlusal view, a rigid anterior 284

segment and posterior segments connected with a TPA allow rotational control without the use of an archwire. Figure 14-16b shows differential space closure using an oblique single force that is applied at the root below the clinical crown of the tooth. Applying the force near the CR of the tooth is limited due to anatomical considerations, such as a shallow vestibule or a buccal frenum. The point of force application may freely slide along the line of action of the force by the law of transmissibility. If we move the elastic forces closer to the right tooth along the line of action of the forces, the forces have the same relationship to the CR (same effect), but the more occlusal position of the force will impinge less on the mucobuccal fold. Figure 14-17 shows a cantilever spring made of rectangular 0.017 × 0.025–inch beta-titanium wire with a helix used for retracting the anterior segment. Note that the point of force application of the activated spring is located anteriorly as much as possible so that the spring is not extended too far apically, yet the line of force (black dotted line) passes near or apical to the CR. Predicted types of tooth movement by this spring are controlled tipping of the anterior segment and translation or slight mesial root movement of the posterior segment based on the imaginary visualized line of action of the force. The dotted arc is the estimated deactivation path of the hook of the spring. The deactivation path of the spring produces a relatively constant line of action during space closure. Figure 14-18 shows an extraction case treated with the cantilever spring (statically determinate retraction system) described above. Controlled tipping of the anterior segment followed by root movement of the anterior segment occurred as predicted. There was no translation phase, so greater magnitude of force was not required. Note that two elastics are

Statically Determinate Space Closure Appliances Fig 14-17 Statically determinate retraction system made of 0.017 × 0.025–inch rectangular beta-titanium wire. The transparent shadow image shows the shape of the activated spring. The dotted line is the initial line of action. The dotted curved arrow is the imaginary deactivation curve of the hook of the spring, which shows that the line of action remains relatively unchanged within the initial effective range of tooth movement.

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Fig 14-18  A uniarch extraction case treated with the statically determinate retraction system. (a) Before treatment. (b) Controlled tipping of the anterior segment. (c) Root movement of the anterior segment. (d) After treatment. Note that there is no translation phase. (e and f) Cephalometric radiographs before and after treatment. (g) Superimposition of cephalometric tracings.

g

used to redirect the line of force in Fig 14-18c. This is discussed in detail later in the chapter in the root movement section. The cephalometric radiographs

and superimposition show that the treatment goals were achieved (see Figs 14-18e to 14-18g).

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14 Extraction Therapies and Space Closure There are anatomical limitations to placing a force far enough apically on the facial side to produce translation. Better possibilities are available on the lingual. If the palatal vault is sufficiently high, the line of action can be placed near or apical to the CR. The patient shown in Fig 13-19 (see chapter 13) had localized enamel hypoplasia on the first molars only. Therefore, the left and right first molars were extracted, and two passive TPAs were placed anteriorly and posteriorly with hooks near the estimated occlusogingival location of the CR. Anterior and posterior segments were translated into the extraction space, and mesial movement of the second molars allowed the space for the third molars to erupt. The patient shown in Fig 13-20 shows differential space closure using two TPAs. It is theoretically possible but practically difficult for translation to deliver a line of action of force at the CR. The location of the CR is compounded by many factors, such as root length and shape, alveolar bone level, and the position of the tooth itself. Furthermore, even if the exact location of the CR is identified, placing the force slightly occlusal or apical to the CR can produce significant second-order rotation (tipping or root movement). Therefore, it is recommended that the lingual hook be placed sufficiently apical to the estimated CR and that additional force be applied at the brackets of the crown so that the resultant force can be easily modified. If the tooth tips too much, rather than relocating the hook further apically by fabricating a new TPA, the magnitude of force of the elastic at the crown is reduced, or the apical elastic force is increased. If the tooth undergoes root movement, the magnitude of force of the elastic at the crown is increased or the apical elastic force is reduced. This is another example of equivalence—using many forces rather than a single force for better clinical ease and control. The statically determinate spring, which delivers force at the desired point of force application so that no moments are required, has another advantage over both frictionless springs occlusally positioned at brackets and sliding mechanics. In occlusally positioned space closure mechanisms with well-designed springs, the change in M/F ratios is not abrupt but very gradual; hence, the center of rotation is relatively constant. With an elastic or spring without moments, the M/F ratio at the CR is a constant that is independent of the force magnitude (see Fig 14-6). Therefore, the “wiggling” side effect is reduced to a minimum. Statically determinate space closure utilizes a single force without moments. Statically indeterminate mechanisms ideally achieve the same result with an 286

equivalent force system using moments and forces at the level of the brackets. The result can be the same, but it is simpler for the orthodontist to visualize a single force and an imaginary line of action. Force determination or calibration is usually simple, requiring only a force gauge. Clinically or experimentally statically indeterminate mechanisms involve sophisticated theory or equipment.

Statically Indeterminate Spring Design If both ends of a spring are inserted in the bracket or tube by activations involving forces and moments, it becomes statically indeterminate. Because the number of unknowns increases, the laws governing static equilibrium are not sufficient to determine all unknown forces and moments. A simple linear force gauge is not adequate to measure the force from the appliance because all moments should be measured at the same time. One solution is to measure the force system in the laboratory with sophisticated sensors; test results may be presented by graphs or data tables. When buying a car, we do not need to test every aspect of its function ourselves. Its technical performance has already been tested in the factory, and the results are presented in the user’s manual. The orthodontic loop or spring with test results is called a calibrated spring and is similar to an automobile specification. The calibrated test results provide the amount of force and moment at any given activation, the moment differential between the anterior and posterior ends, and the maximum amount of activation allowed within the elastic limit to avoid permanent deformation. Many loops and springs have been suggested for space closure characteristics, making it hard to choose between them without a scientific basis for evaluation (Fig 14-19). With the exception of a few examples, most loops have not been fully or even partially tested in the laboratory or in the clinic; furthermore, the shape of the loop usually lacks a sound biomechanical basis. The important variables that should be considered in designing a space closure spring are discussed here. The advantage of a statically indeterminate spring is that the point of force application lies at the occlusal plane level of the bracket, yet the replaced equivalent single force is placed far apical to the bracket on the root. No matter what type of spring is used, some properties are required to deliver the optimal force sys-

Stress Distribution in a Spring Wire Fig 14-19  (a and b) Various shapes of loop springs (top row, deactivated shape; bottom row, activated shape). Most of them are not precalibrated in the laboratory and produce unknown force systems and unpredictable results.

a

b

tem and ensure operator convenience. The force system that can move the teeth as quickly as possible to the target position without any adverse side effects to the tooth itself and the periodontium is considered the optimal orthodontic force system. The exact value of the optimal level of stress in the periodontium is not known; however, continuous light force is thought to be the most efficient for producing reasonable rates of tooth movement, minimizing anchorage loss, and reducing possible discomfort and tissue damage. More importantly, an optimal force system should control the center of rotation to move the tooth to its correct position. Therefore, the force magnitude, force/deflection (F/∆) rate, and M/F ratio of the spring are very important characteristics or specifications for space closure springs. Physical properties such as cross section, shape, and length of the wire along with material properties (modulus of elasticity and yield strength) determine the force system of the spring. In this chapter, emphasis is placed on the shape properties of the spring and how they relate to the F/∆ rate and M/F ratio. Shape properties, unlike material properties, are fully under the clinician’s control in designing a biomechanically efficient loop or spring. Knowing where and how to bend and place additional wire is very important, but first we must investigate what happens inside a wire during activation of an orthodontic appliance.

Stress Distribution in a Spring Wire Figure 14-20a shows a segment of straight wire. If forces are applied at each end, the wire will elongate elastically; however, the F/∆ rate is tremendously high so that the amount of elongation (∆L1) is not detectable with a naked eye. Every clinician knows that a wire is very stiff in axial tension and would make a useless spring. Adding just a few bends to the wire by making a loop can dramatically reduce the F/∆ rate so that it elongates much more (∆L2) under the same amount of applied force (Fig 14-20b). The F/∆ rate is reduced because the wire bends and undergoes a nonuniform stress distribution instead of a uniform stress distribution with a pure axial load. When a wire bends, the length of the upper longitudinal surface of the wire (point A in Figs 14-21a and 14-21b) is decreased by compression while the length of the lower surface (point C) is increased by tension. Somewhere near the center of the wire, where the length is unchanged and the stress is zero, is the neutral axis (green line at point B in Figs 14-21a and 14-21b). Let us define the direction of bending (upper compression, lower tension) as positive and the other direction (upper tension, lower compression) as negative.

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14 Extraction Therapies and Space Closure

a

b

Fig 14-20  The role of a loop in a wire. (a) A straight wire has a tremendously high F/∆ rate in tension. (b) Even a simple vertical loop dramatically reduces the F/∆ rate (∆1