Selman's the Fundamentals of Imaging Physics and Radiobiology [10 ed.] 0398093180, 9780398093181

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SELMAN’S THE FUNDAMENTALS OF IMAGING PHYSICS AND RADIOBIOLOGY

Copy of Roentgen’s radiograph of his wife’s “hand with rings,” made soon after his discovery of x-rays in November 1895 in Wurzburg, Germany. (See Glasser O. Dr. W. C. Roentgen, 2nd ed., Charles C Thomas, 1958, Springfield, p. 39.)

TENTH EDITION

SELMAN’S THE FUNDAMENTALS OF

IMAGING PHYSICS AND

RADIOBIOLOGY By

VICTOR WHITE, PH.D., RT (R) Radiologic Technology and Sonography Program Director and Instructor Montana State University Billings, Billings, MT Radiographer, Advanced Care Hospital, Billings, MT Radiography Lecturer, RiverStone Health, Billings, MT Exercise Science Instructor, Yellowstone Christian College, Billings, MT

C H A R L E S C T H O M A S • P U B L I S H E R • LT D. Springfield • Illinois • U.S.A.

Published and Distributed Throughout the World by CHARLES C THOMAS • PUBLISHER 2600 South First Street Springfield, Illinois 62794-9265 This book is protected by copyright. No part of it may be reproduced in any manner without written permission from the publisher. All rights reserved. 2020 by CHARLES C THOMAS • PUBLISHER, LTD

©

ISBN 978-0-398-09318-1 (hard) ISBN 978-0-398-09319-8 (ebook) First Edition, 1954 Second Edition, 1957 Third Edition, 1961 Fourth Edition, 1965 Fifth Edition, 1972 Sixth Edition, 1977 Seventh Edition, 1985 Eighth Edition, 1994 Ninth Edition, 2000 Tenth Edition, 2020 Library of Congress Catalog Card Number: 2019045612 (print) 2019045613 (ebook) With THOMAS BOOKS careful attention is given to all details of manufacturing and design. It is the Publisher’s desire to present books that are satisfactory as to their physical qualities and artistic possibilities and appropriate for their particular use. THOMAS BOOKS will be true to those laws of quality that assure a good name and good will. Printed in the United States of America PM-S-2 Library of Congress Cataloging-in-Publication Data Names: White, Victor (Victor N.), author. Title: Selman’s the fundamentals of imaging physics and radiobiology / by Victor White. Other titles: Fundamentals of imaging physics and radiobiology Description: Tenth edition. | Springfield, Illinois : Charles C Thomas, Publisher, Ltd., 2020. | Preceded by The fundamentals of imaging physics and radiobiology / by Joseph Selman. 9th ed. 2000. | Includes bibliographical references and index. Identifiers: LCCN 2019045612 (print) | LCCN 2019045613 ( ebook) ISBN 9780398093181 (hardback) | ISBN 9780398093198 (ebook) Subjects: MESH: Health Physics | Diagnostic Imaging--methods | Diagnostic Imaging--instrumentation | Radiation, Ionizing Classification: LCC RC78.7.D53 (print) | LCC RC78.7.D53 (ebook) | NLM WN 110 | DDC 616.07/54--dc23 LC record available at https://lccn.loc.gov/2019045612 LC ebook record available at https://lccn.loc.gov/2019045613

Dedicated first, and foremost, to all my medical imaging students: past, present, and future. Also dedicated to my father, Robert. G. White and high school physics instructor, Larry Armstrong; great mentors and retired teachers, formerly of Taylorville High School in Taylorville, IL, and to Steven B. Dowd, Ed.D, my former Radiography Program Director. Also dedicated to Dan Bar, D.O., Michael Rafati, M.D., Sharon Hall, M.D., Hector Stella, M.D., and Ben Roberts, MFA, RT (R)(CT)(T), some of the best clinicians and academicians that I have had the pleasure of working with over the years. Also dedicated to friends and colleagues Craig Reed, Andrea Decker, Jenna Andujar, BAS, RT (R), and Melanie Schmidt, BAS, RT (R), who all offered useful advice, helped with contracts, and/or provided illustrations regarding completion of this textbook. Finally, dedicated to Max Lee Newlin, RT (R), a great person, radiographer, and student mentor who left us much too soon.

PREFACE

A

number of significant changes have been made in this tenth edition. Color photographs and new illustrations have been provided for a number of existing chapters and for the new chapters in this book. Revisions and updates have been completed for Chapters 1 through 28, whereas Chapters 29 to 33 are all new. The E = MC2 formula has been added to Chapter 3. An updated Periodic Table of the Elements has been added to Chapter 4. A photograph of a modern rotating anode x-ray tube, a diagram of the photodisintegration effect, and the half-life formula have been added to Chapter 12. A new photograph of a rotating anode has been added to Chapter 13, along with some heat unit problems and a diagram of a new type of x-ray tube being developed. In Chapter 14, a new photograph of a modern x-ray control panel has been added. In Chapter 15, a table showing the relationship of screen speed number and screen speed types has been added. In Chapter 20, a new photograph of a Vidicon tube and diagrams of fluoroscopy units and C-Arms have been added. In Chapter 21, a xeromammogram image has been added. In Chapter 22, new mammography, breast US, and breast MRI images, along with photographs of modern mammography equipment and digital mammography quality assurance items have been added. In Chapter 23, information regarding medical and nonmedical computer languages and data have been added. Chapter 24 is significantly revised. Chapter 25 has been significantly revised and updated with new formulas, new diagrams, and CT images. In Chapter 26, images concerning types of nuclear reactors that produce power and radionuclides, diagrams of different types of radiation detectors, and PET diagrams are provided. In Chapter 27, new tables regarding the Acute Radiation Syndrome (ARS) and cancer model data have been provided. In Chapter 28, updated tables about annual dose equivalency, organ system radiation exposure, a photograph of a Luxel® OSL dosimeter, diagrams and data regarding various radiation detectors and current radiobiology practice have been added. Chapter 29, Bone Densitometry, is a completely new chapter. Chapter 30, Magnetic Resonance Imaging, is now its own thoroughly revised chapter. Chapter 31, Ultrasound, is now its own thoroughly revised chapter. Chapter 32, Fusion Imaging, is a completely new chapter. Chapter 33, Molecular Imaging, is a completely new chapter. In addition, an increased focus has been placed on computed radiography and digital imaging (Indirect and Direct Digital Radiography) to adapt to current technology and ARRT (R) Board exam topics. Chapters related to film processing, the darkroom, and film-screen technology have been maintained because this technology is still used in rural and other facilities worldwide and is still included on the ARRT State Exam for Limited X-Ray Machine Operators (LXMOs). vii

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Additional questions have been added to the end of most chapters, and the answers or locations in the textbook where the answers can be found for all questions are now included in this edition. I would like to thank Radiology Administrators Michael Wright, Michael Klein, Stuart Schwab, Michelle Walker, and Chief MRI Technologist Terri Camp at Billings Clinic in Billings, MT, for allowing me to take photographs of their MRI Department and related MRI equipment. I would also like to thank St. Vincent’s Healthcare Radiology Department Administrator Aubrey Brennemann for allowing me to take photographs of ultrasound machines, CT units, and Mammography units at St. Vincent’s Healthcare in Billings, MT. I would especially like to thank Jenna Andujar, BAS, RT (R), for her expert help and assistance in obtaining photographs of the DXA unit, DXA equipment, and a Luxel® OSL Dosimeter from St. Vincent’s Healthcare. Finally, I would like to express my sincere gratitude to Michael Thomas, president of Charles C Thomas, Publisher, Ltd., for his great patience and understanding, and his most competent staff, particularly Michael Fagg, Cindy Marcy, and Sharon Moorman, for their superb help with my illustrations and the publication of this tenth edition of Selman’s The Fundamentals of Imaging Physics and Radiobiology. Victor N. White, PhD, RT (R)

INTRODUCTION

T

his, the tenth edition of Selman’s The Fundamentals of Imaging Physics and Radiobiology, is the continuation of a seminal work in radiation physics and radiation biology first published by Joseph Selman, MD, in 1954 by Charles C Thomas, Publisher, Ltd., Springfield, IL. A brief history about the originating author of this book is now in order. Dr. Joseph Selman was born August 25, 1915, in Albany, New York, and passed away on October 31, 2010, in Plano, Texas. Dr. Selman was a widely respected physician and an author of many medical textbooks. He was also one of the founders of the East Texas Cancer Center in Tyler, Texas, in 1951. Editions of this book start with the inaugural edition in 1954. The second edition was published in 1957, the third edition in 1961, the fourth edition in 1965, the fifth edition in 1972, the sixth edition in 1977, the seventh edition in 1985, the eighth edition in 1994, and the ninth edition in 2000. This coauthor happened to use the sixth edition of this book while a radiologic technology student in the St. John’s Hospital/Lincoln Land Community College Radiologic Technology Program in Springfield, IL, graduating in 1985. In this tenth edition, published in 2020, the overall style of Dr. Selman is still present, but, with any revision, the style of the current author is also present. In essence, my raison d’être in revising this book was to better reflect current radiology practice and to honor the work of Dr. Selman. Topics discussed in this textbook deal with the physics of x-radiation, the biological interaction of radiation with matter, and all aspects of imaging equipment and technology commonly found in the modern radiology department. The chapter on computed tomography (CT) has been heavily revised and updated. Protective measures regarding radiation safety and radiation hazards for workers and patients are thoroughly discussed, and new chapters on dual energy x-ray absorptiometry (DXA), magnetic resonance imaging (MRI), ultrasound (US), fusion, and molecular imaging have been added. This book will be very helpful to students about to take the ARRT (R) registry examination but is not a registry review book per se. This book also serves as a good overview of radiologic imaging physics for radiographers and other medical professionals. V.N.W.

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CONTENTS

 Page Preface............................................................................................................................. vii Introduction...................................................................................................................... ix Chapter 1.  SIMPLIFIED MATHEMATICS..................................................................... 3 Arithmetic......................................................................................................... 3 Fractions....................................................................................................... 3 Percent......................................................................................................... 4 Decimal Fractions........................................................................................ 4 Significant Figures........................................................................................ 5 Algebra.............................................................................................................. 6 Ratio and Proportion........................................................................................ 8 Plane Geometry................................................................................................ 9 Similar Triangles.......................................................................................... 10 Graphs and Charts............................................................................................ 11 Large and Small Numbers................................................................................ 12 Logarithms........................................................................................................ 13 Questions and Problems................................................................................... 14 2.  PHYSICS AND THE UNITS OF MEASUREMENT.................................. 16 Standard Units.................................................................................................. 17 Fundamental Units........................................................................................... 17 Derived Units.................................................................................................... 18 Manipulation of Units...................................................................................... 20 Prefixes Applied to SI Units.............................................................................. 20 Questions and Problems................................................................................... 21 3.  THE PHYSICAL CONCEPT OF ENERGY................................................. 22 Force.................................................................................................................. 22 Work and Energy.............................................................................................. 22 Law of Conservation of Energy....................................................................... 23 Radiation as Waves and Particles...................................................................... 25 Fission................................................................................................................ 25 Fusion................................................................................................................ 25 Questions and Problems................................................................................... 26 xi

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4.  THE STRUCTURE OF MATTER................................................................ 27 Subdivisions of Matter...................................................................................... 27 Atomic Structure: The Electrical Nature of Matter......................................... 30 Atomic Number................................................................................................ 32 Mass Number.................................................................................................... 32 Isotopes and Nuclides....................................................................................... 32 Isobars, Isotones, and Isomers.......................................................................... 33 The Periodic Table............................................................................................ 33 Chemical Behavior............................................................................................ 34 Ionization.......................................................................................................... 36 Questions and Problems................................................................................... 38 5. ELECTROSTATICS....................................................................................... 39 Definition.......................................................................................................... 39 Electrification.................................................................................................... 39 Methods of Electrification................................................................................ 39 Laws of Electrostatics........................................................................................ 41 Electroscope...................................................................................................... 43 Static Discharge................................................................................................ 44 Questions and Problems................................................................................... 45 6.  ELECTRODYNAMICS: ELECTRIC CURRENT....................................... 46 Definition.......................................................................................................... 46 The Nature of an Electric Current................................................................... 46 Sources of Electric Current.............................................................................. 47 Factors in an Electric Circuit............................................................................ 47 Ohm’s Law........................................................................................................ 50 Cells and Batteries............................................................................................. 50 Components of Elementary Electric Circuits................................................... 51 Series and Parallel Circuits............................................................................... 52 Electric Capacitor (Condenser)......................................................................... 56 The Work and Power of a Direct Current........................................................ 57 Questions and Problems................................................................................... 57 7. MAGNETISM................................................................................................. 59 Definition.......................................................................................................... 59 Classification of Magnets.................................................................................. 59 Laws of Magnetism........................................................................................... 60 Nature of Magnetism........................................................................................ 60 Magnetic Fields................................................................................................. 61 Characteristics of Lines of Force...................................................................... 62 Magnetic Induction (Magnetization)................................................................ 63

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Magnetic Permeability and Retentivity............................................................ 63 Magnetic Classification of Matter.................................................................... 64 Earth’s Magnetism............................................................................................ 64 Questions and Problems................................................................................... 65 8. ELECTROMAGNETISM............................................................................... 66 Definition.......................................................................................................... 66 Electromagnetic Phenomena............................................................................ 66 The Electromagnet........................................................................................... 67 Electromagnetic Induction................................................................................ 68 Direction of Induced Electron Current............................................................ 69 Self-Induction.................................................................................................... 70 Mutual Induction.............................................................................................. 71 Questions and Problems................................................................................... 71 9.  ELECTRIC GENERATORS AND MOTORS............................................. 72 Electric Generator............................................................................................. 72 Definition..................................................................................................... 72 Essential Features......................................................................................... 72 Simple Electric Generator........................................................................... 72 Properties of Alternating currents............................................................... 75 Direct Current Generator............................................................................ 76 Advantages of Alternating Current............................................................. 77 Electric Motor................................................................................................... 77 Definition..................................................................................................... 77 Principle....................................................................................................... 77 Simple Electric Motor.................................................................................. 78 Types of Electric Motors............................................................................. 78 Current-Measuring Devices......................................................................... 80 Questions and Problems................................................................................... 81 10. PRODUCTION AND CONTROL OF HIGH VOLTAGE: REGULATION OF CURRENT IN THE X-RAY TUBE............................ 82 Transformer...................................................................................................... 82 Principle............................................................................................................ 82 Efficiency and Power Losses.............................................................................. 86 Control of High Voltage................................................................................... 86 Autotransformer................................................................................................ 87 Control of Filament Current and Tube Current.............................................. 88 Choke Coil................................................................................................... 88 Rheostat....................................................................................................... 88 High-Frequency Control of Current and Voltage............................................ 89

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Questions and Problems................................................................................... 90 11. RECTIFICATION AND RECTIFIERS........................................................ 92 Definition.......................................................................................................... 92 Methods of Rectifying an Alternating Current................................................ 93 Rectifiers........................................................................................................... 98 Rectifier Failure................................................................................................. 103 Questions and Problems................................................................................... 105 12. X-RAYS: PRODUCTION AND PROPERTIES............................................ 106 How X-Rays Were Discovered......................................................................... 106 Nature of X-Rays............................................................................................. 107 Source of X-Rays in Radiology: The X-Ray Tube.......................................... 109 Details of X-Ray Production............................................................................ 111 Electron Interactions with Target Atoms: X-Ray Production.......................... 111 Target Material................................................................................................. 113 Efficiency of X-Ray Production....................................................................... 113 Properties of X-Rays........................................................................................ 114 Specifications of the Physical Characteristics of an X-Ray Beam................... 114 X-Ray Exposure (Quantity).............................................................................. 114 X-Ray Quality.................................................................................................. 116 “Hard” and “Soft” X-Rays............................................................................... 120 The Interactions of Ionizing Radiation and Matter......................................... 121 Relationship of Energy Mass and the Speed of Light................................. 125 Relative Importance of Various Types of Interaction...................................... 126 Detection of Ionizing Radiation....................................................................... 127 Summary of Radiation Units........................................................................... 127 Exposure: Coulombs per Kilogram or the Roentgen....................................... 128 Exposure...................................................................................................... 128 Kerma.......................................................................................................... 128 Absorbed Dose: The Gray (Gy)................................................................... 128 RAD, REM, and Sievert.............................................................................. 128 Tissue Weighting Factors and Radiation Weighting Factors............................ 129 Tissue Weighting Factors............................................................................. 129 Radiation Weighting Factors....................................................................... 129 Equivalent Dose (EqD)................................................................................ 129 Effective Dose (EfD)..................................................................................... 130 Collective Effective Dose (SE)...................................................................... 130 Standard Unit of Radioactivity................................................................... 130 Activity Measurement.................................................................................. 130 Modification of Kilovoltage X-Ray Beams by Filters...................................... 130 Questions and Problems................................................................................... 132

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13. X-RAY TUBES................................................................................................ 133 Thermionic Diode Tubes.................................................................................. 133 Radiographic Tubes.......................................................................................... 134 Glass Envelope............................................................................................. 134 Cathode........................................................................................................ 134 Anode........................................................................................................... 135 Space Charge Compensation........................................................................... 140 Factors Governing Tube Life............................................................................ 141 New Developments in X-Ray Tube Technology.............................................. 147 Questions and Problems................................................................................... 148 14. X-RAY CIRCUITS.......................................................................................... 149 Source of Electric Power................................................................................... 149 The Main Single Phase X-Ray Circuits........................................................... 150 Completed Wiring Diagram............................................................................. 157 Basic X-Ray Control Panel or Console............................................................ 158 Three-Phase Generation of X-Rays................................................................. 160 High-Frequency Generation of X-Rays........................................................... 164 Falling-Load Generator.................................................................................... 165 Special Mobile X-Ray Equipment.................................................................... 166 Battery-Powered Mobile X-Ray Units......................................................... 166 Capacitor (Condenser)-Discharge Mobile X-Ray Units................................... 167 Questions and Problems................................................................................... 170 15. X-RAY FILMS, FILM HOLDERS, AND INTENSIFYING SCREENS...... 171 Composition of X-Ray Film............................................................................. 171 Types of Films................................................................................................... 173 Practical Suggestions in Handling Unexposed Film......................................... 173 Film Exposure Holders................................................................................ 174 Intensifying Screens.......................................................................................... 175 Questions and Problems................................................................................... 185 16. THE DARKROOM......................................................................................... 186 Introduction...................................................................................................... 186 Location of the Darkroom................................................................................ 186 Building Essentials............................................................................................. 187 Entrance............................................................................................................ 187 Size............................................................................................................... 188 Ventilation.................................................................................................... 188 Lighting........................................................................................................ 188 Film Storage Bin.......................................................................................... 189 Questions and Problems................................................................................... 190

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17. CHEMISTRY OF RADIOGRAPHY AND FILM PROCESSING.............. 191 Introduction...................................................................................................... 191 Radiographic Photography............................................................................... 191 Radiographic Chemistry................................................................................... 192 Manual Processing............................................................................................ 194 Film Fog, Stains, and Artifacts.......................................................................... 195 Automatic Processing........................................................................................ 196 Summary of Processor Care............................................................................. 200 Silver Recovery from Fixing Solutions.............................................................. 200 Federal Environmental Protection Agency Rules and X-Ray Film Chemistry.... 200 Questions and Problems................................................................................... 201 18. RADIOGRAPHIC QUALITY........................................................................ 202 Blur .................................................................................................................. 202 Geometric or Focal Spot Blur........................................................................... 203 Focal Spot Evaluation.................................................................................. 205 Motion Blur....................................................................................................... 207 Screen Blur........................................................................................................ 207 Object Blur.................................................................................................. 208 Radiographic Density....................................................................................... 208 Contrast............................................................................................................ 212 Radiographic Contrast..................................................................................... 213 Subject Contrast................................................................................................ 213 Film Contrast.................................................................................................... 216 Distortion.......................................................................................................... 217 Direct Magnification or Enlargement Radiography......................................... 220 Modulation Transfer Function.......................................................................... 223 Questions and Problems................................................................................... 224 19. DEVICES FOR IMPROVING RADIOGRAPHIC QUALITY.................... 226 Scattered Radiation.......................................................................................... 226 Removal of Scattered Radiation by a Grid...................................................... 228 Principle of the Radiographic Grid.................................................................. 228 Efficiency of Grids............................................................................................ 229 Types of Grids.................................................................................................. 231 Precautions in the Use of Focused Grids.......................................................... 234 Practical Application of Grids.......................................................................... 236 Removal of Scattered Radiation by an Air Gap............................................... 239 Reduction of Scattered Radiation by Limitation of the Primary Beam.......... 240 Other Methods of Beam Limitation................................................................. 245 Moving Slit Radiography............................................................................. 245 The Anode Heel Effect—Anode Cutoff ........................................................... 246

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Compensating Filters........................................................................................ 247 Summary of Radiographic Exposure............................................................... 248 Questions and Problems................................................................................... 248 20. FLUOROSCOPY............................................................................................. 250 The Human Eye............................................................................................... 250 Fluoroscopic Image Intensification................................................................... 251 Magnification in Image Intensifier.................................................................... 254 Multiple-Field Intensifiers................................................................................. 255 Viewing the Fluoroscopic Image...................................................................... 255 Optical Lens System.................................................................................... 255 Television Viewing System........................................................................... 256 Video Cameras............................................................................................ 256 Television Monitor....................................................................................... 257 Charge-Coupled Device (CCD) TV Camera................................................... 259 Quality of the TV Image.................................................................................. 259 Recording the Fluoroscopic Image................................................................... 260 Recording the Video Image Magnetic Tape and Disc........................................... 260 Laser Discs................................................................................................... 261 Digital Fluoroscopy...................................................................................... 262 Digital Fluoroscopy Imaging Process........................................................... 262 Patient Radiation Protection during Fluoroscopy....................................... 263 C-Arms: Best Orientation for Radiation Protection.................................... 264 Questions and Problems................................................................................... 265 21. VANISHING EQUIPMENT........................................................................... 266 Stereoscopic Radiography................................................................................ 266 Tomography...................................................................................................... 269 Xeroradiography (Xerography)........................................................................ 275 X-Ray Film and Film Processors................................................................. 276 Questions and Problems................................................................................... 277 22. MAMMOGRAPHY......................................................................................... 278 Digital Mammography..................................................................................... 280 Breast Sonography....................................................................................... 281 How Is a Breast Sonogram Performed?....................................................... 282 Advantages of Breast Ultrasound..................................................................... 282 MRI of the Breast............................................................................................. 282 Patient Positioning for a Breast MRI Study................................................. 282 Breast MRI Limitations and Risks............................................................... 283 Quality Standards in Mammography............................................................... 284 Qualifications of Radiologic Technologist (Mammographer).......................... 285

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Equipment................................................................................................... 285 Quality Assurance........................................................................................ 285 Quality Assurance Records.......................................................................... 286 Radiologic Technologist Responsibilities..................................................... 286 Film Screen Mammography Quality Control Tests......................................... 286 Daily............................................................................................................. 286 Weekly.......................................................................................................... 287 Monthly....................................................................................................... 288 Quarterly...................................................................................................... 288 Semiannual.................................................................................................. 288 Mobile Mammography Units........................................................................... 288 Digital Mammography Quality Control...................................................... 288 FDA Alternative Standards for Digital Mammography QA/QC.................... 289 Questions and Problems................................................................................... 290 23. BASIC COMPUTER SCIENCE.................................................................... 291 Introduction...................................................................................................... 291 History.............................................................................................................. 291 Data vs. Information.................................................................................... 292 Computer Operations....................................................................................... 293 Computer Components.................................................................................... 293 Computer Language......................................................................................... 299 Binary Number System..................................................................................... 299 Summary of Applications in Radiology........................................................... 301 Questions and Problems................................................................................... 301 24. DIGITAL X-RAY IMAGING.......................................................................... 302 Introduction: Digital Imaging........................................................................... 302 Digital Radiography: Direct Capture.......................................................... 302 Digital Radiography: Indirect Capture....................................................... 302 Digital Radiography: Indirect Detectors..................................................... 303 Computed Radiography (CR)..................................................................... 304 CR Equipment Design................................................................................ 306 Computed Radiography Stimuable Phosphor Read-Out Process............... 306 Hounsfield Units and Digital Imaging......................................................... 306 Attenuation Coefficients, Voxel, Pixel, and Matrix...................................... 307 Matrix.......................................................................................................... 308 Exposure Index............................................................................................ 308 Dose Creep.................................................................................................. 309 Post Processing............................................................................................. 309 Proper Technique......................................................................................... 311 HIS, RIS, and PACS.................................................................................... 311

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Digital QA.................................................................................................... 312 Monitor QA Tests........................................................................................ 312 Digital Radiography Image Quality Items/Tests........................................ 312 Digital Fluoroscopy........................................................................................... 313 Future Prospects........................................................................................... 317 Digital Imaging Summary........................................................................... 317 Conclusion........................................................................................................ 319 Questions and Problems................................................................................... 319 25. COMPUTED TOMOGRAPHY.................................................................... 320 Conventional CT Scanning.............................................................................. 320 CT Detectors.................................................................................................... 323 CT Subject Contrast.................................................................................... 325 Hounsfield Equation.................................................................................... 327 Linear Attenuation Coefficient ( µ)............................................................... 328 CT Reconstruction Processes....................................................................... 328 Backprojection............................................................................................. 328 Fourier Reconstruction................................................................................ 329 Clinical Utility of CT Reconstruction......................................................... 330 Spiral (Helical) CT Scanning....................................................................... 332 CT Tubes..................................................................................................... 333 Pitch............................................................................................................. 334 Single Section Pitch..................................................................................... 334 Multiple Section Pitch.................................................................................. 334 Multislice CT............................................................................................... 335 Dose Issues................................................................................................... 335 CT Dose Methods............................................................................................. 337 Questions and Problems................................................................................... 337 26. RADIOACTIVITY AND DIAGNOSTIC NUCLEAR MEDICINE............ 338 Introduction...................................................................................................... 338 Natural Radioactivity........................................................................................ 339 Unstable Atoms............................................................................................ 339 Radioactive Series........................................................................................ 339 Radium............................................................................................................. 339 Introduction................................................................................................. 339 Properties..................................................................................................... 340 Types of Radiation........................................................................................... 340 Radioactive Decay............................................................................................ 341 Decay Constant................................................................................................. 341 Half-Life............................................................................................................ 342 Average Life...................................................................................................... 342

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Radon................................................................................................................ 343 Radioactive Equilibrium................................................................................... 343 Artificial Radioactivity...................................................................................... 343 Isotopes............................................................................................................. 344 Artificial Radionuclides..................................................................................... 344 Nuclear Reactor................................................................................................ 345 Nuclear Transformations.................................................................................. 348 Properties of Artificial Radionuclides............................................................... 348 Radioactive Decay............................................................................................ 349 Application of Radionuclides in Medicine....................................................... 351 Radionuclide Instrumentation.......................................................................... 351 Sources of Error in Counting........................................................................... 354 Efficiency and Sensitivity of Counters.............................................................. 356 Geometric Factors in Counting........................................................................ 357 Methods of Counting....................................................................................... 359 Important Medical Radionuclides.................................................................... 359 Examples of the Use of Radionuclides in Medical Diagnosis.......................... 360 Imaging with Radionuclides: The Gamma Camera........................................ 362 Collimators........................................................................................................ 363 Gamma Camera Collimators........................................................................... 364 CRYSTAL.................................................................................................... 364 Crystal-Photomultiplier Complex..................................................................... 365 Single-Photon Emission Computed Tomography (SPECT)............................. 365 Available Radionuclide Imaging....................................................................... 366 Positron Emission Tomography (PET).............................................................. 368 PET Scanner Design.................................................................................... 368 PET Radionuclides...................................................................................... 369 PET Physics................................................................................................. 369 Correction Factors....................................................................................... 369 Calibration of Radiopharmaceuticals and Gamma Cameras.......................... 370 Radiopharmaceuticals................................................................................. 370 Cyclotrons and Nuclear Medicine............................................................... 370 Questions and Problems................................................................................... 371 27. RADIOBIOLOGY........................................................................................... 373 Definition.......................................................................................................... 373 History.............................................................................................................. 373 The Physical Basis of Radiobiology................................................................. 374 Radiobiologic Effects........................................................................................ 376 Structure and Function of Cells....................................................................... 376 Nucleus.............................................................................................................. 376 Cytoplasm......................................................................................................... 377 Cell Reproduction............................................................................................. 378

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Mitosis............................................................................................................... 378 Meiosis.............................................................................................................. 380 Structure of DNA............................................................................................. 381 Functions of DNA............................................................................................ 383 The Radiobiologic Lesion................................................................................. 384 Modes of Action of Ionizing Radiation........................................................... 384 Cellular Response to Radiation........................................................................ 386 Nuclear Damage............................................................................................... 386 Cytoplasmic Damage........................................................................................ 387 Cellular Radiosensitivity................................................................................... 387 Modifying Factors in Radiosensitivity............................................................... 389 Acute Whole-Body Radiation Syndromes........................................................ 390 Explanation of Acute Whole-Body Radiation Syndromes............................... 392 Dose–Response Models.................................................................................... 393 Sigmoid Dose–Response Curve........................................................................ 394 Linear Dose–Response Curve........................................................................... 394 Linear-Quadratic Dose–Response Curve......................................................... 395 Injurious Effects of Radiation on Normal Tissues........................................... 396 Early Effects: Limited Areas of the Body.......................................................... 396 Late Effects........................................................................................................ 397 Late Somatic Effects in High-Dose Region...................................................... 397 Dose Response Models and Cancer Initiation.................................................. 398 One-Hit Model............................................................................................ 398 Probit Model................................................................................................ 398 Weibull Model.............................................................................................. 398 Multistage Model......................................................................................... 399 Late Somatic Effects in Low-Dose Region........................................................ 400 Risk Estimates for Genetic Damage................................................................. 402 Radiation Injury to Embryo and Fetus............................................................. 403 Questions and Problems................................................................................... 405 28. PROTECTION IN RADIOLOGY—HEALTH PHYSICS.......................... 406 Introduction...................................................................................................... 406 Background Radiation...................................................................................... 407 BERT........................................................................................................... 407 Dose Limits, Stochastic and Non-Stochastic Effects......................................... 409 Risk Assessment................................................................................................ 409 Hazard Identification........................................................................................ 409 Dose–Response Assessment.............................................................................. 410 Exposure Assessment........................................................................................ 410 Risk Characterization.................................................................................. 410 Risk Communication................................................................................... 410 Derivation of Unit for Risk Assessment............................................................ 410

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Equivalent Dose................................................................................................ 411 Effective Dose.............................................................................................. 412 Numerical Dose Equivalent Limits................................................................... 415 Occupational................................................................................................ 415 Fertile or Pregnant Radiation Workers........................................................ 416 Nonoccupational (General Public) Limit..................................................... 416 ALARA Concept.............................................................................................. 417 Personnel Protection from Exposure to X-Rays............................................... 417 Protective Measures.......................................................................................... 420 Protective Barriers in Radiography and Fluoroscopy....................................... 420 Working Conditions.......................................................................................... 422 Acceptance (Compliance) Testing..................................................................... 422 Protection Surveys............................................................................................. 423 Protection of the Patient in Diagnostic Radiology...................................... 425 Dose Reduction in Radiography....................................................................... 427 Protection in Mammography............................................................................ 429 Computed Tomography Scanning................................................................... 429 CT Dose Calculations.................................................................................. 430 Computed Tomography Dose Index (CTDI) for Noncontiguous Slices   and CTDI for Helical Scans.................................................................... 430 Dose–Length Products and Effective Dose.................................................. 431 Risk Estimates.............................................................................................. 431 CT Noise........................................................................................................... 431 Patient Protection in Fluoroscopy..................................................................... 432 Protection from Electric Shock......................................................................... 434 Protection in Nuclear Medicine........................................................................ 434 Radiation Monitoring....................................................................................... 436 Gas-Filled Detectors.......................................................................................... 437 Detector Efficiency....................................................................................... 439 Unsafe Radiation Practices in Nuclear Medicine............................................. 439 Questions and Problems................................................................................... 439 29. BONE DENSITOMETRY.............................................................................. 441 Overview of Bone Densitometry...................................................................... 441 Definition of Osteoporosis................................................................................ 441 General Risk Factors for Osteoporosis......................................................... 442 Prevalence of Osteoporosis.................................................................................. 442 Detection of Osteoporosis................................................................................ 443 Osteoporosis Detection Technologies............................................................... 443 Quantitative Ultrasound (QUS)................................................................... 444 QUS Scanning Protocol and Machine Characteristics............................... 445 Physics of QUS............................................................................................ 445 QUS Advantages......................................................................................... 445

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Dual Energy X-Ray Absorptiometry (DXA)............................................... 445 DXA Scanning Protocol.............................................................................. 446 DXA Physics................................................................................................ 446 DXA Advantages......................................................................................... 447 Comparison of QUS T Scores and DXA T Scores.................................... 448 Quantitative Computed Tomography (QCT)............................................. 449 QCT Scanning Protocol.............................................................................. 449 QCT Physics................................................................................................ 450 QCT Advantages and Disadvantages.......................................................... 450 Quality Assurance/Quality Control............................................................ 450 Responsibility for Bone Densitometry Unit QA/QC.................................. 451 Use of T Scores in Osteoporosis Analysis................................................... 451 Z Scores and Osteoporosis Screening.......................................................... 452 Bone Mineral Density and Bone Mass........................................................ 453 Treating Osteoporosis.................................................................................. 453 Nonclinical Methods for Treating Osteoporosis............................................... 454 The Importance of Vitamins Regarding Bone Health................................ 455 Exercise and Osteoporosis Prevention......................................................... 456 Conclusion........................................................................................................ 456 Questions and Problems................................................................................... 456 30. MAGNETIC RESONANCE IMAGING........................................................ 457 Nuclear Magnetic Resonance........................................................................... 457 Angular Momentum Theory....................................................................... 459 Resonance......................................................................................................... 461 Importance of Relaxation to Clinical MR Imaging......................................... 464 From NMR to MRI..................................................................................... 465 Magnetic Resonance Imaging........................................................................... 466 Pulse Sequences and Pulse Flip Angles........................................................ 466 Production of a Magnetic Resonance Image.............................................. 466 Noise in MRI.................................................................................................... 473 MR Image Scanning Artifacts.......................................................................... 474 Chemical Shift Artifacts............................................................................... 474 Wraparound Artifact.................................................................................... 474 Material Artifacts......................................................................................... 475 MRI Equipment............................................................................................... 475 Surface Coils................................................................................................ 477 Typical MRI Unit........................................................................................ 478 Hazards of MRI............................................................................................... 480 Human Exposure Considerations................................................................ 480 Functional MRI as a Screening Tool........................................................... 484 Functional MRI and Stroke Assessment...................................................... 485 Perfusion MRI Imaging............................................................................... 485

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The Fundamentals of Imaging Physics and Radiobiology

Fast FLAIR Brain-Imaging Techniques....................................................... 486 Diffusion-Weighted Brain Imaging.............................................................. 486 Scanning Pulse Sequences for DWI............................................................. 488 Diffusion Weighted MRI Image Appearance.............................................. 488 Diffusion-Weighted Imaging Pulse Sequences............................................. 489 Diffusion-Weighted Brain Mapping............................................................ 489 Conclusion........................................................................................................ 490 Questions and Problems................................................................................... 491 31. ULTRASOUND IMAGING........................................................................... 492 Nature of Sound............................................................................................... 492 Properties of Ultrasound.................................................................................. 493 Ultrasound Process............................................................................................ 494 Production of Ultrasound............................................................................ 494 Types of Piezoelectric Crystals.................................................................... 495 Image Construction..................................................................................... 496 Ultrasound Beam Characteristics................................................................ 496 Echo Reception of Ultrasound.................................................................... 498 Behavior of Ultrasound in Matter............................................................... 498 Different Types of Reflection...................................................................... 500 Specular Reflection...................................................................................... 500 Scattering..................................................................................................... 500 Ultrasound Image Displays............................................................................... 501 Common Ultrasound Procedures..................................................................... 503 Features of an Ultrasound Image: Presentation.......................................... 504 Features of an Ultrasound Image: Depth.................................................... 504 Typical Appearance of Normal Tissue....................................................... 504 Enhancement............................................................................................... 504 Anisotropy.................................................................................................... 504 Frame Rate.................................................................................................. 506 Ultrasound Quality Assurance Factors............................................................. 506 Artifacts........................................................................................................ 506 Potential Biohazards of Ultrasound.................................................................. 508 Questions and Problems................................................................................... 508 32. FUSION IMAGING........................................................................................ 509 Definition and Purpose of Fusion Imaging...................................................... 509 Technical Aspects of Fusion Imaging.......................................................... 509 History of Fusion Imaging........................................................................... 509 Fusion Imaging Advantages......................................................................... 509 Fusion-imaging Algorithms/Processes......................................................... 510 Classification of Fusion-imaging Algorithms.................................................... 510 Low-level Fusion Techniques (Pixel level).................................................... 510

Contents

xxv

Spatial Transform Domain and Frequency Domain Transform................. 511 Image Registration Overview...................................................................... 512 Data Registration......................................................................................... 512 Structure Mapping....................................................................................... 512 Pixel-level Techniques: Transform Domain Multiresolution....................... 513 Networking Requirements........................................................................... 513 Fusion-imaging Hardware........................................................................... 513 Modalities of Image Fusion......................................................................... 514 Nuclear Medicine and Fusion Imaging............................................................ 514 Radiation Therapy and Fusion Imaging.......................................................... 515 Overall Advantages of Fusion Imaging by Modality........................................ 515 Technologist Training in Fusion Imaging......................................................... 516 Summary........................................................................................................... 516 Questions and Problems................................................................................... 516 33. MOLECULAR IMAGING............................................................................. 517 Molecular Imaging............................................................................................ 517 Increased Digitization of Diagnostic and Therapeutic Imaging Data........ 517 Human Genome.......................................................................................... 518 Increased Technological Ability to Analyze Human Genes and Tissue   at the Molecular Level............................................................................. 518 Genes........................................................................................................... 522 Chromosomes.............................................................................................. 523 Mitosis and Meiosis...................................................................................... 524 Disease and Pathology...................................................................................... 524 Molecular Imaging, Biomarkers, and Monoclonal Antibodies.................... 524 Molecular Imaging and Monoclonal Antibodies......................................... 524 Imaging Probes............................................................................................ 525 Molecular Probes......................................................................................... 525 Nuclear Medicine Imaging Probes.............................................................. 526 Magnetic Resonance Imaging Probes.......................................................... 527 Ultrasound Imaging Probe.......................................................................... 527 Optical Imaging........................................................................................... 528 Molecular Imaging Strategies........................................................................... 529 Direct Imaging............................................................................................. 529 Conclusion........................................................................................................ 530 Questions and Problems................................................................................... 530 Appendix–Answers to Problems............................................................................................ 531 Bibliography..................................................................................................................... 541 Index............................................................................................................................... 551

SELMAN’S THE FUNDAMENTALS OF IMAGING PHYSICS AND RADIOBIOLOGY

Chapter 1 SIMPLIFIED MATHEMATICS Learning Objectives After completing this chapter, the reader will be able to: • Define mathematics and the differences between arithmetic, algebra, and plane geometry. • Describe ratio and proportion. • Describe differences between charts and graphs.

• Describe differences between small and large numbers. • Describe logarithms. • Answer questions and solve the problems at the end of the chapter.

A

algebra, ratio and proportion, and plane geometry. Such a review should be beneficial in at least two ways. First, it should make it easier to understand the basic principles and concepts of radiologic physics. Second, it should aid in the solution of such everyday problems as conversion of radiographic techniques, interpretation of tube rating charts, determination of radiographic magnification, and many others that may arise from time to time. The discussion will be subdivided as follows: (1) arithmetic, (2) algebra, (3) ratio and proportion, (4) geometry, (5) graphs and charts, (6) large and small numbers and (7) logarithms. Only fundamental principles will be included in this chapter.

LL OF THE PHYSICAL SCIENCES  have in common  a firm basis in mathematics. This is no less true of radiologic physics, an important branch of the physical sciences. Clearly, then, in approaching a course in radiologic physics you, as a student radiographer, graduate student, or radiology resident, should find your path smoothed by an adequate background in the appropriate areas of mathematics. We shall assume here that you have had at least basic exposure to mathematics, although this may vary widely from place to place. However, realizing that much of this material may have become hazy with time, we shall review the simple but necessary aspects of arithmetic,

ARITHMETIC Arithmetic is calculation or problem-solving by means of definite numbers. We shall assume that you are familiar with addition, subtraction, multiplication, and division and shall therefore omit these operations.

quantity below the line is called the denominator; it indicates the number of equal parts into which the unit is divided. The quantity above the line is the numerator; it indicates the number of equal parts taken. Thus, if a pie were divided into three equal parts, the denominator would be 3; and if two of these parts were taken, the numerator would be 2, so, the two segments would represent 2 3 of the pie. Fractions represent the division of one quantity by another. This extends the concept

Fractions In arithmetic, a fraction may be defined as one or more equal parts of a unit. For example, 12, 13, and 2 5 are fractions. The 3

4

The Fundamentals of Imaging Physics and Radiobiology 4 3 4 × 3 12 × = = 5 10 5 × 10 50 of fractions to expressions in which the numerator is larger than the denominator, as in the fraction 5 2. The resulting fraction can be reduced by If the numerator is smaller than the denomidividing the numerator and the denominator by nator, as 3 5, we have a proper fraction. If the the same number, in this case, 2: numerator is larger than the denominator, as 5 , we have an improper fraction, because 12 2 6 3 ÷ = 5 ÷ 3 = 1 2 3, which is really an integer plus a 50 2 25 fraction. which cannot be further simplified. In adding fractions, all of which have the Note that when the numerator and the denom­ same denominator, add all the numerators first inator are both multiplied or divided by the same and then place the sum over the denominator: number, the value of the fraction does not change. For instance, 2 3 6 5 2 + 3 + 6 + 5 16 + + + = = 7 7 7 7 7 7 3 2 6 × = 16 2 5 2 10 =2 7 7 3 3 is the same as × 1 = Subtraction of fractions having identical 5 5 denominators follows the same rule: When two fractions are to be divided, as 4 5 ÷ 3 7, the fraction that is to be divided is the dividend 6 4 6−4 2 − = = and the fraction that does the dividing is called the 7 7 7 7 divisor. In this case, 4 5 is the dividend and 3 7 the If fractions are added or subtracted, and the divisor. The rule is to invert the divisor (called “takdenominators are different, then the least ing the reciprocal”) and multiply the dividend by it: common denominator must be found. This is the smallest number which is exactly divisible 4 3 ÷ by all the denominators. Thus, 5 7

1 2 3 + − =? 2 3 4

The smallest number which is divided exactly by each denominator is 12. Place 12 in the denominator of a new fraction:

 12

Divide the denominator of each of the fractions in the old equation into 12, and then multiply the answer by the numerator of that fraction; each result is then placed in the numerator of the new fraction:

6+8−9 5 = 12 12

Multiplying fractions means taking their product. To multiply fractions, take the product of the numerators and place it over the product of the denominators,



4 7 28 13 × = =1 5 3 15 15 Percent

A special type of fraction, percent, is represented by the sign % to indicate that the number standing with it is to be divided by 100. Thus, 95% = 95/100. We do not use percentages directly in mathematical computations but first convert them to fractions or decimals. For instance, 150 × 40% is changed to 150 × 40/l00 or 150 × 2/5 or 150 × 0.40. All these expressions equal 60. Decimal Fractions Our common method of representing numbers as multiples of ten is embodied in the



Simplified Mathematics

decimal system. A decimal fraction has as its denominator 10, or 10 raised to some power such as 100, 1000, 10,000, etc. The denominator is symbolized by a dot in a certain position. For example, the decimal 0.2 = 2/10; 0.02 = 2/100; 0.002 = 2/1000, etc. Decimals can be multiplied or divided, but care must be taken to place the decimal point in the proper position:    2.24   × 1.25   1120   448  224 2.8000 First, add the total number of digits to the right of the decimal points in the numbers being multiplied, which in this case turns out to be four. Then point off four places from the right in the answer to determine the correct position of the decimal point. The decimal system is used everywhere in science and in the vast majority of countries in daily life. Significant Figures The precision (reproducibility of results) of any type of measurement is limited by the design of the measuring instrument. For example, a scale calibrated in grams as shown in Figure 1.1 allows an estimate to the nearest tenth of a gram. Thus, the scale in Figure 1.1 reads 8.4 g. The last figure, 0.4, is estimated and is the last significant figure— that is, it is the last meaningful digit. Obviously, no greater precision is possible with this particular instrument. To improve precision, the scale would have to show a greater number of subdivisions.























Figure 1.1.  With this calibrated scale, we can estimate to the nearest tenth. Thus, the position of the pointer indicates 8.4 units, the 0.4 being the last significant figure.

Significant figures are used in various mathematical operations. For example, in addition:

5

Item 1 Item 2 Item 3

  98.26 g   1.350 g 260.1 g 359.710 g

Notice that three digits appear after the decimal point in the answer. But in item (3), there is only one digit, 1, after the decimal point; beyond this, the digits are unknown. Therefore, the digits after the 7 in the answer imply more than is known, since the answer can be no more precise than the least precise item being added. In this case, the answer should be properly stated as 359.7. In addition and subtraction, the answer can have no more significant figures after the decimal point than the item with the least number of significant figures after its decimal point. A different situation exists in multiplication and division. Here, the total number of significant figures in the answer equals that in the items having the least total number of significant figures. For example, in   25.23 cm   × 1.21 cm    2523   5046 2523 30.5283  cm2 1.21 cm has fewer significant figures—three in all—so the answer should have three significant figures and be read as 30.5 (dropping the 0.0283). In general, to round off significant figures, observe the following rule: if the digit following the last significant figure is equal to or greater than 5, the last significant figure is increased by 1; if less than 5, it is unchanged. The rule is applied in the following examples:

45.157 is rounded to 45.16 45.155 is rounded to 45.16 45.153 is rounded to 45.15

where the answer is to be expressed in four significant figures.

6

The Fundamentals of Imaging Physics and Radiobiology

ALGEBRA The word algebra, derived from the Arabic language, connotes that branch of mathematics which deals with the relationship of quantities usually represented by letters of the alphabet— Roman, Greek, or Hebrew. Operations. Mathematical operations with letter symbols are the same as with numerals, since both are symbolic representations of numbers which, in themselves, are abstract concepts. For example, the concept “four” may be represented by 4, 22, 2 × 2, 2 + 2, or 3 + 1; or by the letter x if the value of x is specifically designated to represent “four.” In algebra, just as in arithmetic, the fundamental operations include addition, subtraction, multiplication, and division; and there are fractions, proportions, and equations. Algebra provides a method of finding an unknown quantity when the relationship of certain known quantities is specified. Algebraic operations are indicated by the same symbols as in arithmetic: + (plus) add − (minus) subtract × (times) multiply ÷ (divided by) divide = equals To indicate addition in algebra, use the general expression

x + y (1)

The symbols x and y, called variables, may represent any number or quantity. Thus, if x = 4 and y = 7, then, substituting these values in equation (1),

4 + 7 = 11

Similarly, to indicate subtraction in algebra, use the general expression x − y If x = 9 and y = 5, then 9 − 5 = 4 Notice that algebraic symbols may represent whole numbers, fractions, zero, and negative numbers among others. Negative numbers are those whose value is less than zero and are

designated as − x. In algebraic terms, add a positive and a negative number as follows: x + − y If x = 8 and − y = − 3, then

8+−3

is the same as 8 − 3 = 5 The + sign is omitted in the designation of positive numbers, being reserved to indicate the operation of addition. On the other hand, to subtract a negative quantity from a positive one x − − y If x = 4 and − y = − 6, then 4 − − 6 is the same as

4 + 6 = 10

Multiplication in algebra follows the same rules as in arithmetic. However, in the multiplication of letter symbols the × sign is omitted, x × y being written as xy. If x = 3 and y = 5, then substituting in the expression xy,

3 × 5 = 15

Division in algebra is customarily expressed as a fraction. Thus, x ÷ y is written as x/y. If x = 3 and y = 5, then

3÷5 =

3

5

When two negative quantities are multiplied or divided, the answer is positive. Thus, − x × − y = xy and − x/− y = x/y. When a positive and a negative quantity are multiplied or divided, the answer is negative; thus, x × − y = − xy, and x ÷ − y = − x/y. In solving an algebraic expression consisting of a collection of terms we must perform the indicated multiplication and division first



Simplified Mathematics

and then carry out the indicated addition and subtraction. An example will clarify this:

7

weight must be added to the other end in order to keep the board level.

ab + c/d − f = ? Suppose a = 2, b = 3, c = 4, d = 8, and f = 5. Substituting in the preceding expression,

2×3+

D

D

4 −5 = ? 8

Performing multiplication and division first,

4 6 + −5 = ? 8 Then, performing addition and subtraction,



4 4 1 6 −5 =1 =1 8 8 2

A parenthesis inclosing a group of terms indicates that all of the terms inside the parenthesis are to be multiplied by the term outside the parenthesis. This is simplified by performing all the indicated operations in correct sequence—inside the parenthesis first—and then multiplying the result by the quantity outside the parenthesis. For example,

6 (8 − 4 + 3 × 2) = 6 (8 − 4 + 6) = 6 × 10 = 60

Equations. The simpler algebraic equations can be solved without difficulty by the application of basic rules. You can easily verify these rules by substituting numerals. In the equation

a + b = c + d (2)

a + b is called the left side and c + d the right side. Each letter is called a term. If any quantity is added to one side of the equation, the same quantity must be added to the other side in order for this to remain an equation. Similarly, if any quantity is subtracted from one side, the same quantity must be subtracted from the other side. To simplify the concept of the equation, we may picture it as a see-saw as in Figure 1.2. If persons of equal weight are placed at each end, the board will remain horizontal—the equation is balanced. If a second person is now added to one end of the see-saw, a person of similar

ED

DE

Figure 1.2.  Analogy of algebraic equation to a see-saw.

Return again to the simple equation: a + b = c + d in which each term is a variable. If any three of the variables are known, the fourth can be found. Suppose, a is 3, b is 4, c is 1, and d is unknown. Substituting these values in the equation,

3+4=1+d

How can d be found? Simply rearrange the equation so that d is alone, that is, the only term in its side. In this case, subtract 1 from both sides of the equation. However, a mathematical short cut can be used: a term may be transposed from one side of an equation to the other side, provided it is given the opposite sign. Following this rule, 1 becomes a minus 1 when moved to the left side. Thus, 3+4−1=d 6=d or, d=6

Usually, in solving algebraic equations, rearrange the terms before substituting their numerical values. In the equation a + b = c + d to find d, transpose c to the left side and change its sign. a + b − c = d       or, d=a+b− c (Reversing both sides of an equation does not alter its equality.)

8

The Fundamentals of Imaging Physics and Radiobiology

In algebraic equations in which terms are multiplied or divided, analogous rules apply. For example, equation x = y/z

and move x into the denominator of the right side as a multiplier, and z = y/x (3)



may be solved for y by multiplying both sides by z,     or,

xz = y

xz = yz / z xz = y

The above rule can be readily tested. Suppose that y is 12 and x is 3. Substituting in equation (3), z = 12 3 z= 4 4 = 12 3 (4)



y = xz



The same result may be obtained by moving z from the denominator of the right side to the numerator of the left side. Thus, we have the short-cut rule for cross-multiplication: if the denominator of one side of an equation is moved, it multiplies the numerator of the other side and, conversely, if the numerator of one side is moved, it multiplies the denominator of the other side. Suppose that in the equation x = y/z, x and y are known, then z is solved as follows: move z into the numerator of the opposite side as a multiplier,

If we wish to move 3, we must place it in the numerator of the left side of equations (4), 4 × 3 = 12



(4a)

Note that numerical equations (4) and (4a) balance. Now, referring again to equation (4), suppose we wish to move 12. We must place it in the denominator of the left side 4



12

=

1

3

Again, it is evident that the equation balances.

RATIO AND PROPORTION A ratio is a fixed relationship between two quantities, simply indicating how many times larger or smaller one quantity is relative to another. It has essentially the same meaning as a fraction. One symbol that expresses a ratio is the colon (:). Thus, a:b is read “a is to b.” Or, 1:2 is read “1 is to 2.” In modern mathematics, ratios are usually represented as fractions: a:b is the same as a/b 1:2 is the same as 12 As noted above, the fraction 12 indicates that the numerator is 12 as large as the denominator. Similarly, 2 3 indicates that the numerator is 2 3 as large as the denominator. The meaning of ratio is important to the technologist because it underlies the concept of proportion, defined as an expression showing that two given ratios are equal. Thus, we may have an algebraic proportion,



a/b = c/d (5)

which is read “a is to b as c is to d.” The same idea can also be represented numerically. For example,

3

6

= 48

If any three terms of a proportion are known, the fourth may easily be determined. Suppose in proportion (5) a is 2, b is 4, d is 8, and c is to be found. Then,

2

4

=

c

8

Moving 8 to the numerator of the left side (cross multiplying),

2×8 =c 4



c = 16 4 = 4

There are three general types of proportions that pertain to radiography.



Simplified Mathematics

1. Direct proportion. Here, one quantity maintains the same ratio to another quantity as the latter changes. For example, the statement “a is proportional to b” means that if b is doubled, a is automatically doubled, if b is tripled, a is tripled, etc. In order to represent this mathematically, let us assume that the quantity a1 exists when b1 exists, and that if b1 is changed to b2, then a1 becomes a2. Thus, a1/b1 = a2/b2

a a1 = 2 1/ b1 1/ b2

a2 = 8/8 =1 Thus, a is halved when b is doubled. 3. Inverse square proportion (law:) The inverse square law, taken from the work of Sir Isaac Newton, is as follows: I i ( D2 ) = (7) I 2 ( D1 )2 2

I1 = Initial intensity of radiation I2= Second intensity of radiation D1 = Initial distance D2 = Second distance 1. The radiation intensity is 240 mSv at 72” SID. What would the radiation intensity be at 40” SID?



Cross-multiplying,

2/8 = a2/4





The numbers below the letters are called “subscripts” and have no significance except to label a2 as being different from a1. Such a direct proportion is solved by the method described above. 2. Inverse proportion. Here, one quantity varies in the opposite direction from another quantity as the latter changes. Thus, the statement “a is inversely proportional to b” means that if b is doubled a is halved, if b is tripled, a is divided by 3, etc. Such a proportion is set up as follows:



a1 a2 = (6) b2 b1

A numerical example should help clarify this. Suppose that when a1 is 2, b1 is 4; if a is inversely proportional to b, what will a1 become if b1 is changed to 8? In this case, a2 is the unknown and b2 is 8. Substituting in equation (6),

9

2 240 ( 40) = 2 = X (72)

240 1600 = = X 5184 1600 X = 240 × 5184 = 1600 X = 1,244,160 =

1,244,160 = 1600 X = 776 mSv X=

Due to the inverse square law, the radiation intensity reaching the image receptor (IR) increases as the tube gets closer to the IR (i.e., 40” is closer to the IR than 72”).

PLANE GEOMETRY Plane geometry is that branch of mathematics which deals with figures lying entirely in one plane; that is, on a flat surface. Some of the elementary rules of plane geometry will now be listed. 1. A straight line (in classical geometry) is the shortest distance between two points. It has only one dimension—length.

2. A rectangle is a plane figure composed of four straight lines meeting at right angles. Its opposite sides are equal. The sum of the lengths of the four sides is called the perimeter. The area of a rectangle is the product of two adjacent sides. Thus, in Figure 1.3 the perimeter is the sum a + b + a + b = 2a + 2b. The area is a × b or ab.

10

The Fundamentals of Imaging Physics and Radiobiology D

%

$

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Similar Triangles

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Of great importance in radiology, and closely related to ratio and proportion, is the proportionality of similar triangles. Such triangles have an identical shape, although they differ in size. In similar triangles, the corresponding sides are directly proportional and the corresponding angles are equal. Thus, similar triangles represent the geometric equivalent of direct proportion. Figure 1.7 shows two similar triangles. The corresponding sides (those opposite the equal corresponding angles) are proportional, so that

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$

%

Figure 1.3.  Rectangle. Perimeter = a+ b + a  + b = 2a + 2b

Figure 1.4.  Square. Perimeter = 4a. Area = a × a = a2

3. A square is a special rectangle in which all four sides are equal, as in Figure 1.4. 4. A triangle is a figure made up of three straight lines connecting three points that are not in the same line and lie in the same plane. The perimeter of a triangle is the sum of the three sides. The area of a triangle is one half the base times the altitude (the altitude is the perpendicular distance from the apex to the base). This is shown in Figure 1.5.

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%

5$

',

86

+

',$0(7(5 &( 17

(5

&

Figure 1.5.  Triangle. Perimeter = a + b + c. Area = ½ ch. 

Figure 1.6.  Circle. Circumference = p × diameter = pd. Area = p × radius2 = pr2.

5. A circle is a closed curved line which is everywhere at an equal distance from one point called the center, lying in the same plane as the center. A straight line from the center to any point on the circle is the radius. A straight line passing through the center and meeting the circle at two points is the diameter, which is obviously equal to twice the radius. The length of the circle is its circumference, obtained by multiplying the diameter by a constant, n (pi). Pi always equals about 3.14. The area enclosed by a circle equals n times the square of the radius. Figure 1.6 shows these relationships.

a/A = b/B, or a/A = c/C, or b/B = c/C \

\

D [

$

E

]

F

[

% ]

&

Figure 1.7.  Similar triangles. The corresponding angles are equal, that is, x = x, y = y, z = z. The corresponding sides are proportional; thus, a/A = b/B = c/C.

A line drawn across any triangle parallel to its base produces two similar triangles, one partly superimposed on the other, as shown in Figure 1.8. Line BE has been drawn parallel to the base CD. Then triangle ABE is similar to triangle ACD and their corresponding sides are proportional. This may be simplified by separating the two triangles as in Figure 1.7. $

%

&

(

'

Figure 1.8.  Similar triangles. BE is parallel to CD. Triangle ABE is similar to triangle ACD. Therefore, AB/AC = AE/AD = BE/CD.



Simplified Mathematics

A thorough comprehension of similar triangles is essential to the understanding of photographic and radiographic projection. Suppose an object is placed in a beam of light originating at a point source and allowed to cast a shadow or image on a surface (see Figure 1.9). The image has to be larger than the object. Knowing the distances of the object and image, respectively, from the light source, as well as the object size, we can determine the image size from the similarity of triangles ABC and ADE. We do this by setting up the following proportion:

also applies in radiography, as will be shown in Chapter 18.

$

%

image size image distance = object size object distance Substituting the known values in the equation, we can predict the size of the image by solving the equation. Note that if any three of the values are known, the fourth can be readily obtained from the same equation. This principle

11

3RLQW6RXUFH RI/LJKW

2%-(&7

&

)

6+$'2:²,0$*(

'

*

(

Figure 1.9.  Projection of the shadow image of an object by a point source of light.



DE/BC = AG/AF.

GRAPHS AND CHARTS For practical purposes, graphs and charts may be regarded as diagrams representing the relationship of two quantities, one of which depends on the other. The dependent factor is called the dependent variable. The factor which changes independently is called the independent variable. When data are accumulated, showing how a dependent variable changes with a change in the independent variable, they can either be compiled in a table, or represented by a graph. However, tables do not give intermediate values as do graphs. The construction and interpretation of a graph may be exemplified as follows: the optimum developing time for x-ray film increases as the temperature of the developing solution decreases (not inversely proportional, however). Table 1.1 reproduces the data for a developer in manual processing. To chart this information on graph paper, first plot the temperature (independent variable) along the horizontal axis of the

graph. Then plot the developing time (dependent variable) on the vertical axis, as shown in Figure 1.10. In mathematics, the dependent variable is said to be a function of the independent variable. To graph the tabulated data, take the first temperature listed in the table and locate it on the horizontal axis. Trace vertically upward from this point to the horizontal line corresponding to the correct developing time, and mark the intersection with a dot. Repeat this for all the values in the table, and then draw a line that best fits the dots. This constitutes a line graph or curve.

Table 1.1 Time–Temperature Development Data. Temperature Developing Time in Minutes °C

°F

17

62

4 12

18.5

65

3 12

12

The Fundamentals of Imaging Physics and Radiobiology

Temperature Developing Time in Minutes °F

20

68

3

21

70

2 34

22

72

2 12

24

75

2

Having constructed a graph, how do we read it? Suppose that we wish to determine the correct developing time for a temperature of 18.5°C (65°F). Locate 18.5°C (65°F) on the horizontal axis and trace vertically upward to the intersection with the curve; from this point of intersection trace horizontally to the vertical axis on the left, where the correct developing time can be read. This is shown in Figure 1.10, where the tracing lines are formed by broken lines with arrows. The horizontal tracing line meets the vertical axis at approximately 3 12 min, which is the correct developing time at this temperature. Thus, a properly constructed graph gives the developing times for temperature readings that did not appear in the original table, a process called interpolation.

'(9(/23,1*7,0(,10,1

°C

 



7(03(5$785(,1ƒ&     

























7(03(5$785(,1ƒ)

Figure 1.10.  Method of using a time–temperature development curve in manual processing. Select the correct temperature on the horizontal axis, in this case 18.5°C (65°F). Follow vertically, as shown by the arrows, to the curve. Then follow horizontally to the time axis where the correct developing time in this case is seen to be 3 12 min. Tally to the vertical axis on the left, where the correct developing time can be read. This is shown in the figure, where the tracing lines are formed by broken lines with arrows.

LARGE AND SMALL NUMBERS Because ordinary notation is too cumbersome to express the extremely large numbers often encountered in physics, we resort to the exponential system in powers of ten. For example, the number 103 means 10 × 10 × 10. In the quantity 103, 10 is the base and 3 the exponent and is read as “ten to the third power.” In general, 101 = 10 102 = 100 103 = 1000, etc. Note that the exponent of 10 indicates the number of zeros required to express the number in ordinary notation. Now apply this principle: The velocity of light in air or vacuum is 300,000,000 meters per second (m/s). Place a decimal point after 3,

following which there are eight zeroes. Therefore, this number can be simply expressed as 3.0 × 108 m/s and read as “three point oh times ten to the eighth.” A more complicated number such as 4,310,000,000,000,000 may be simplified in the same way by placing a decimal point after the first digit, 4, and counting the number of places to the right, in this case 15. The corresponding number in the powers-of-ten system is 4.31 × 1015. Suppose, instead, we are to convert 6.3 × 106 to ordinary notation. Count six places to the right of the decimal point, filling out the places with zeroes. Since the first place is already occupied by a digit, 3, five zeroes must follow. Therefore, the number is 6,800,000. In general, any number followed by 106 is expressed in millions.



Simplified Mathematics

13

The same system can be applied to extremely small numbers by the use of negative exponents. This is based on

multiply the digits and make the exponent of 10 in the answer equal to the algebraic sum of the exponents of 10 in the multipliers. Thus:

10− 1 = 1/10 = 0.1 10− 2 = 1/100 = 0.01 10− 3 = 1/1000 = 0.001, etc.

4 × 103 × 2 × 102 = 4 × 2 × 10(3 + 2) = 8 × 105 3 × 106 × 2 × 10− 4 = 3 × 2 × 10(6 − 4) = 6 × 102

From this you can see that a negative exponent indicates the number of decimal places to the left of a digit, including the digit. Thus, 0.004 would be expressed as 4 × 10− 3. To obtain this result, count the number of places to the right of the decimal point, including the first digit,

To divide numbers in this system, we divide the digits and derive the exponent of 10 in the answer by subtracting algebraically the exponent in the divisor from the exponent in the dividend:

6 × 105 6 × 10( 5 −3 ) = = 3 × 102 2 2 × 103



9 × 107 9 × 10(7 + 2 ) = = 3 × 109 3 3 × 102

0.004 The counted number of places (three in this instance) then becomes the negative exponent of 10; thus, 4 × 10− 3. A more complicated number such as 0.00000000000372 would be simply expressed as 3.72 × 10− 12, read as “three point seven two times ten to the minus twelfth.” The simplified system of notation facilitates the multiplication and division of large and small numbers. To multiply such numbers, we

Note in the above examples that when a power number is moved from the denominator of a fraction to the numerator, or vice versa, we simply change the sign of the exponent. Thus,

8 8 × 105 = −5 3 × 10 3

LOGARITHMS In several areas of imaging, we shall have to use quantities known as logarithms. For our purposes, only the basic aspects of logarithms will be explained. In fact, the preceding section included powers-of-ten; these powers are logarithms “to the base 10,” also known as common logs. As an example of the use of logs, consider the intensity of sound. It so happens that the human ear, in comparing a sound 100 times louder than another, perceives it as being only twice as loud; 100 = 102, so the exponent is the “twice.” If the sound is 1000 times the first, the ear perceives it as 3 times louder; 1000 = 103, so now 3 represents the “three times.” It turns out that 2 is the logarithm of 100, stated as log 100 = 2; similarly, log 1000 = 3. In other words, logs are simply the exponent to which 10 must be raised to equal a given number. Note that 10 is the base of the

system of common logs. (There is another system—the Naperian—with a different base.) Conversely, the antilogarithm (antilog) of 2 is 100. The following examples will help summarize simple logs: 100 = 1 101 = 10 102 = 100 103 = 1000

log 1 = 0 log 10 = 1 log 100 = 2 log 1000 = 3

antilog 0 = 1 antilog 1 = 10 antilog 2 = 100 antilog 3 = 1000

You can see that in each case, for example, log 100 = 2 because 10 has to be raised to the second power to equal 100. The same applies to negative exponents: 10− 1 = 0.1

log 10− 1 = − 1 antilog − 1 = 0.1

10− 2 = 0.01

log 10− 2 = − 2 antilog − 2 = 0.01

10− 3 = 0.001 log 10− 3 = − 3 antilog − 3 = 0.001

14

The Fundamentals of Imaging Physics and Radiobiology

You can see that in each instance, for example, log 0.01 = − 2 because 10 has to be raised to the − 2 power to equal 0.01. Why does 100 = 1? Arrange exponents in decreasing order:

Special tables of logarithms are available to find the logs to the base 10 (i.e., common logs) for numbers that are not integer powers of ten; for example, let us multiply 43.61 × 25.23 by using logs.

...5 4 3 2 1 0 − 1 − 2 − 3 − 4 − 5...

log 43.61=1.6396

Obviously, each exponent is one less than that preceding, as we go from left to right. What lies between 1 and − 1? This must be zero because 1 − 1 = 0, and − 1 − (l) also = 0. Thus, 100 = 1. The same applies to all real numbers except the number 0. Logarithms have proved extremely useful in multiplication and division, especially of large and small numbers, because to multiply powers of 10, one simply adds the exponents, and to divide powers of 10, one subtracts exponents. In a chain multiplication with a mixture of positive and negative exponents, all to the base 10, one simply adds the exponents algebraically and uses the resulting exponent as a power of 10, as in the preceding section.

log 25.23=1.4017

3.0413total valueof exponent Applying this new exponent to 10,



103.0413 = antilog of 3.0413 = 1100

Actually multiplying the original numbers 43.61 × 25.23 = 1100.2. If a 5-place log table had been used, the answer would have shown the .2; this example has been given simply to show how logarithms work. Manual calculations with the aid of log tables can be done, but the use of calculators or computers are most commonly used to solve these types of problems.

QUESTIONS AND PROBLEMS   1. Reduce the following fractions: (a) 4 8 (b) 912 (c) 10150 (d) 4 5   2. Solve the following problems: (a) 3 5 + 2 5 + 4 5 = (b) 2 3 + 3 4 + 3 7 = (c) 12 + 2 3 + 4 5 =   3. Mr. Jones has 100 bushels of potatoes and sells 75% of this crop. How many bushels does he have left?   4. Divide as indicated: (a) 4 5 ÷ 3 5   (b) 4 9 ÷ 716 (c) 8 9 ÷ 3 5  5. If a = b/c, what is b in terms of a and c? What is c in terms of a and b?  6. Solve the following equations for the unknown term: (a) x + 4 = 7 − 3 + 8 (b) 6 + 3 − 1 = 4 + 1 − a (c) x 3 = 7 9

 7. Solve the following proportions for the unknown quantity: (a) a 7 = 2 21   (b) 3 9 = x 15 (c) 4 6 = 7 y   (d) 3 5 = 9 x   8. The diameter of an x-ray beam is directly proportional to the distance from the tube target. If the diameter of the beam at a 20-in. distance is 10 cm, what will the diameter be at 40-in. distance?   9. The exposure of a radiograph is directly proportional to the time of exposure. What will happen to the exposure if the time is tripled? 10. What is the area of a circle having a diameter of 4 cm? 11. What is the area of a rectangular plot of ground measuring 20 ft on one side and 30 ft on the adjacent side? l2. Convert 3.6 × 106 to the ordinary number system.



Simplified Mathematics

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Change 424,000 to the exponential system. What is log 105 to base 10 (common logs)? State the antilog of 4. (a) Multiply 105 × 106 in the powers of ten system. (b) State the log of the answer. (a) Divide 108 by 105. (b) State the log of the answer. What is antilog (4 + 5)? Find the log of 1000 × 10,000. If the distance of the image receptor (IR) from the x-ray tube is 40 in., and the x-ray intensity is 100 mR, what will the x-ray intensity be when the x-ray tube is moved to 72 in.? If the distance of the image receptor from the x-ray tube is 44 in., and the x-ray intensity is 25 mR, what will the new x-ray intensity be when the tube is moved to 40 in.? When reading a graph, which is the Y axis and which is the X axis?

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Chapter 2 PHYSICS AND THE UNITS OF MEASUREMENT Learning Objectives After completing this chapter, the reader will be able to: • Describe standard units. • Describe fundamental units. • Describe derivative units.

• Understand prefixes and SI units. • Be able to manipulate units. • Answer questions and solve the problems at the end of the chapter.

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In fact, it is ultimately an important contributor even to the biologic sciences. What, then, is physics? It may be defined as that branch of science that deals with matter and energy, and their relation to each other. It includes mechanics, heat, light, sound, electricity, and magnetism, and the fundamental structure and properties of matter. For our purpose, we shall be interested in those aspects of physics pertaining to the origin, nature, and behavior of x-rays and related types of radiation, that is, radiologic physics. Thus far, we have used the word science rather freely without definition. Science is organized and classified knowledge. Natural processes

  HYSICS, AS AN EXACT SCIENCE, requires a precise vocabulary in which each term has a clear and definite meaning. Not only does this simplify the learning process, but it also facilitates the organization of concepts and their accurate communication to others. In order to appreciate the position of physics within the framework of science, we may use the term natural science to include the systematic study of the universe and its contents. Figure 2.1 shows the subdivision of natural science into its major categories. Here you can readily see that while physics is listed as one of the physical sciences, it actually underlies all of them.

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