Finite Element Method and Medical Imaging Techniques in Bone Biomechanics 9781786305183, 1786305186, 9781119681618, 1119681618, 9781119681625, 1119681626

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

Mathematical and Mechanical Engineering Set coordinated by Abdelkhalak El Hami

Volume 8

Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

Rabeb Ben Kahla Abdelwahed Barkaoui Tarek Merzouki

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2019 The rights of Rabeb Ben Kahla, Abdelwahed Barkaoui and Tarek Merzouki to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019948686 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-518-3

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1. Main Medical Imaging Techniques . . . . . . . . . . . . . . .

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1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. X-ray imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Definition of X-rays . . . . . . . . . . . . . . . . . . . . . . 1.2.2. X-ray instrumentation and generation . . . . . . . . . . . . 1.2.3. Applications of X-ray imaging . . . . . . . . . . . . . . . . 1.2.4. Advantages and disadvantages of X-ray imaging . . . . . 1.3. Computed tomography . . . . . . . . . . . . . . . . . . . . . . . 1.3.1. Description of the technique . . . . . . . . . . . . . . . . . 1.3.2. Development of computed tomography . . . . . . . . . . . 1.3.3. Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5. Advantages and disadvantages of computed tomography 1.4. Magnetic resonance imaging . . . . . . . . . . . . . . . . . . . 1.4.1. Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2. Generation of the resonance effect . . . . . . . . . . . . . . 1.4.3. Relaxation and contrast . . . . . . . . . . . . . . . . . . . . 1.4.4. Applications of magnetic resonance imaging . . . . . . . 1.4.5. Advantages and disadvantages of magnetic resonance imaging . . . . . . . . . . . . . . . . . . . . . 1.5. Ultrasound imaging . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1. Definition of ultrasound . . . . . . . . . . . . . . . . . . . . 1.5.2. Development of ultrasound imaging . . . . . . . . . . . . . 1.5.3. Generation of ultrasound . . . . . . . . . . . . . . . . . . . 1.5.4. Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.5. Applications of ultrasound techniques . . . . . . . . . . .

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1.5.6. Advantages and disadvantages of ultrasound imaging . . . . . . . . 1.6. Comparison between the different medical imaging techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Medical Image Analysis and Processing . . . . . . . . . . .

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2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2. Image compression . . . . . . . . . . . . . . . . . 2.3. Image restoration . . . . . . . . . . . . . . . . . . 2.4. Image enhancement . . . . . . . . . . . . . . . . . 2.4.1. Window and level . . . . . . . . . . . . . . . 2.4.2. Gamma correction . . . . . . . . . . . . . . . 2.4.3. Histogram equalization . . . . . . . . . . . . 2.4.4. Image subtraction . . . . . . . . . . . . . . . . 2.4.5. Spatial filtering . . . . . . . . . . . . . . . . . 2.5. Image analysis . . . . . . . . . . . . . . . . . . . . 2.5.1. Texture features . . . . . . . . . . . . . . . . . 2.5.2. Edges and boundaries . . . . . . . . . . . . . 2.5.3. Shape and structure . . . . . . . . . . . . . . . 2.6. Image segmentation . . . . . . . . . . . . . . . . . 2.6.1. Simple methods of image segmentation . . 2.6.2. Active contour segmentation . . . . . . . . . 2.6.3. Variational methods . . . . . . . . . . . . . . 2.6.4. Level set methods . . . . . . . . . . . . . . . . 2.6.5. Active shape and active appearance models 2.6.6. Graph cut segmentation . . . . . . . . . . . . 2.6.7. Atlas-based segmentation . . . . . . . . . . . 2.6.8. Deformable model-based segmentation . . . 2.6.9. Energy minimization-based segmentation . 2.6.10. Learning-based segmentation . . . . . . . . 2.6.11. Other approaches . . . . . . . . . . . . . . . 2.7. Image registration . . . . . . . . . . . . . . . . . . 2.7.1. Dimensionality . . . . . . . . . . . . . . . . . 2.7.2. Nature of the registration basis . . . . . . . . 2.7.3. Nature of the transformation . . . . . . . . . 2.7.4. Transformation domain . . . . . . . . . . . . 2.7.5. Interaction . . . . . . . . . . . . . . . . . . . . 2.7.6. Optimization procedure . . . . . . . . . . . . 2.7.7. Modalities involved . . . . . . . . . . . . . . 2.7.8. Subject . . . . . . . . . . . . . . . . . . . . . . 2.7.9. Object . . . . . . . . . . . . . . . . . . . . . . . 2.8. Image fusion . . . . . . . . . . . . . . . . . . . . .

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49 49 50 50 51 51 52 52 52 53 53 55 57 58 58 60 61 62 62 63 63 65 65 65 66 66 67 68 69 70 71 72 72 73 74 74

Contents

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74 75 75 75 76 76 76

Chapter 3. Recent Methods of Constructing Finite Element Models Based on Medical Images . . . . . . . . . . . . .

79

2.8.1. Pixel fusion methods . . . . . . . . . . . . . . . . . . . . 2.8.2. Subspace methods . . . . . . . . . . . . . . . . . . . . . 2.8.3. Multi-scale methods . . . . . . . . . . . . . . . . . . . . 2.8.4. Ensemble learning techniques . . . . . . . . . . . . . . 2.8.5. Simultaneous truth and performance level estimation 2.9. Image understanding . . . . . . . . . . . . . . . . . . . . . 2.10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1. Introduction . . . . . . . . . . . . . . . . . . 3.2. X-ray-based finite element models . . . . 3.3. CT-based finite element models . . . . . . 3.4. MRI-based finite element models . . . . . 3.5. Ultrasound-based finite element models . 3.6. Conclusion . . . . . . . . . . . . . . . . . .

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Chapter 4. Main Bone Sites Modeled Using the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.1. Introduction . . . . . . . . . . . . 4.2. FE modeling of the calcaneus . 4.3. FE modeling of phalanges . . . 4.4. FE modeling of the metatarsal . 4.5. FE modeling of the tibia . . . . 4.6. FE modeling of the knee . . . . 4.7. FE modeling of the femur . . . 4.8. FE modeling of the vertebrae . 4.9. FE modeling of the humerus . . 4.10. FE modeling of the elbow . . 4.11. FE modeling of the ulna. . . . 4.12. FE modeling of the wrist . . . 4.13. Conclusion . . . . . . . . . . .

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125 125 127 129 131 137 140 143 147 149 149 150 152

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

Introduction

Investigation of the inside of the human body can be carried out directly, through surgical intervention, or indirectly, using, for example, autonomous encapsulated cameras or endoscopes. The latter mainly consists of flexible or rigid optical tubes, taking images at the distal end while passing into the body through an orifice or surgical opening and transmitting these images to the proximal end. However, these two investigation methods are invasive techniques, causing potential damage or injuries. The need for less invasive methods to visualize the body anatomy and internal composition has led to the development of various medical imaging techniques. These techniques have enjoyed a spectacular expansion due to the significant contributions from computer science, engineering, medical physics, applied mathematics, biology and chemistry. In general, a medical imaging process involves the use of an energy source moving through the body to create images of its internal anatomy or of the region of interest, and to provide relevant information on the composition and functioning of different biological structures. X-ray imaging, computed tomography (CT), magnetic resonance imaging (MRI), as well as ultrasound (US) imaging are the most commonly used medical imaging techniques. Each has its advantages and weaknesses, as well as its specific indications and application areas. Furthermore, medical images obtained using these techniques may subsequently be used to reconstruct two-dimensional (2D) or three-dimensional (3D) models of the target organs or structures. To do this, several numerical methods may be used, such as the finite element (FE) method, which is one of the most widely used. The FE method involves the

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

construction of models according to a three-step planning process, consisting of pre-mesh, mesh and post-mesh. During the pre-mesh planning, medical images are segmented and registered, surfaces are smoothed and the regions of interest are extracted, using specific scientific software. The meshes are then generated using several approaches, based on different algorithms depending on the element size and type (triangular, tetrahedral, hexahedral, brick, etc.). Creating tetrahedral elements for 3D volume meshing is relatively easy, as the construction of the raw elements requires the connection of several vertices. From a computational perspective, the brick elements are more accurate, but their construction requires complicated computer-assisted design surface topology, in addition to other efforts. After the preliminary mesh generation, verifying element quality is important to provide a strong solution for the FE model. Precise material property assignment must be monitored to ensure the rigidity required for tissue meshes. However, segmentation is considered as the most important and difficult step in processing and analyzing medical images, because addressing the intensity inhomogeneity in the regions to segment is challenging, which influences the outcome of the entire analysis. More specifically, the intensity inhomogeneity usually results from flaws in the image acquisition process, while the background represents the basic anatomical structure of many medical images. The segmentation process consists of extracting the region of interest using an automatic or semi-automatic process, and various segmentation methods have been used in several medical applications. Digital models based on data from medical images have recently been widely used in biomechanics. In particular, digital bone models are largely used in biomechanical engineering to better understand bone behavior and the fracturing process by evaluating the fracture risk based on image diagnosis. Digital bone fracture modeling is not an easy task to perform, since bone is a heterogeneous composite material and its modeling depends mainly on its mechanical properties. It should be noted that accurate predictions are tightly linked to realistic characterization of the material behavior. Digital models can be obtained from volumetric images using two main categories of procedure, namely the voxel-based method and the geometry-based method, both of which are based on the FE method. Developed in mechanical engineering, the FE analysis was first introduced in orthopedic biomechanics in the early 1970s, in order to evaluate the stresses to which human bones are subjected. Since then, this analysis method has been widely used to study bone mechanics.

Introduction

xi

In this book, Chapter 1 covers the physical principles and applications, as well as the advantages and limitations, of the four most common medical imaging techniques used in bone diagnosis and investigation. Chapter 2 covers the main steps of processing medical images generated using these four techniques. Chapter 3 is a review of recent studies on the FE model of reconstruction of bone structure using medical images, and finally Chapter 4 is a review of recent studies using the FE method to model and analyze bone behavior at several sites on the human skeleton.

1 Main Medical Imaging Techniques

1.1. Introduction The field of medical imaging has experienced revolutionary progress, with improved accuracy and reduced invasiveness. Medical imaging techniques make it possible to better understand human behavior and are essentially composed of an energy-emitting source able to penetrate the human body. During its penetration, this energy can be absorbed or attenuated at different levels depending on the tissue density and the penetrated atomic number. This process generates signals that can be detected by special detectors specific to the energy source. Mathematical models are then used to manipulate these signals in order to create medical images. An imaging modality is a specific imaging technique or system used to visualize the inside of the body. According to the used energy source, several modalities can be distinguished. Diagnostic radiology is based on the use of the electromagnetic spectrum beyond the visible light region, such as X-rays used in mammography and computed tomography, whereas magnetic resonance imaging is based on the use of radiofrequencies and ultrasound imaging is based on the use of mechanical energy in the form of high-frequency sound waves. During the acquisition of a medical image, the acquisition conditions and the technical quality of the image are the two determinant factors of its diagnostic utility. In general, the image quality involves compromises; it improves when the dose of X-rays increases in radiography and computed tomography, when the image acquisition time increases in MRI and when

Finite Element Method and Medical Imaging Techniques in Bone Biomechanics, First Edition. Rabeb Ben Kahla; Abdelwahed Barkaoui and Tarek Merzouki. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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the power levels increase in ultrasound imaging. However, the safety and the comfort of the patient are crucial parameters that should be taken into account during the acquisition process, and an excessive radiation dose should not be applied in pursuit of a perfect image. Indeed, the quality of the image and the safety of the patient must be balanced. This chapter briefly addresses the four medical imaging modalities that are most commonly used to create FE models of different parts of the human skeleton: X-ray imaging, computed tomography (CT), magnetic resonance imaging (MRI) and ultrasound imaging. 1.2. X-ray imaging X-ray imaging is the foremost technique that is used to perform medical imaging. The X-rays used in radiography were discovered in 1895 by physicist Wilhelm Roentgen, who created the first radiographic images of the human anatomy (Figure 1.1) (Bushberg and Boone 2011). Hence, X-ray imaging has gone on to define the radiology field, leading to the emergence of radiologists and specialists in interpreting medical images (Bushberg and Boone 2011). 1.2.1. Definition of X-rays X-rays fall within the electromagnetic rays (Figure 1.2) that transport radiating energies through space by waves and photons. These rays can be represented by a photon or a wave model and can be categorized according to their energy, frequency or wavelength (Berger et al. 2018): =

[1.1]

where denotes the wave propagation speed, i.e. the speed of light. The photon energy ( ) is directly linked to or to as follows (Berger et al. 2018): =

=

[1.2]

Main Medical Imaging Techniques

where is Planck’s constant ( of light ( , ×

, ).

×

) and

3

is the speed

By passing through different materials, X-rays lose a certain amount of energy according to the absorption behavior of each material, which describes the basic principle of traditional X-ray radiography that measures the amount of lost energy. The contrast in the image comes from the difference in the energy amount lost between the different materials. The absorbed energy amount also depends on the dose released during acquisition (Berger et al. 2018).

Figure 1.1. This famous image represents the oldest existing human X-ray image. It was taken on December 22, 1895 by Roentgen, and shows the hand of his wife. This X-ray, clearly showing the bones of the hand as well as two rings on one of the fingers, represents the beginning of diagnostic radiology. Within several months, Roentgen was able to determine the physical properties underpinning X-rays and published, on December 28, 1895, his conclusions in a preliminary report entitled “On a New Kind of Rays” in the Proceedings of the Physico-Medical Society of Würzburg. On January 23, 1896, an English translation was published in the scientific journal Nature, and was quickly followed by the news of the discovery spreading around the world. Thanks to the medical applications of this kind of ray, radiological imaging has made rapid progress to become an essential part of medical care. The letter “X”, assigned by Roentgen, indicates the unknown nature of these rays, hence the name “X-ray” (Gunderman 2012)

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

Figure 1.2. Frequencies and wavelengths of the different types of electromagnetic rays. X-rays are spread across a wavelength ranging from 0.01 to 10 , corresponding to an energy level of 100 –100 (Berger et al. 2018)

1.2.2. X-ray instrumentation and generation The X-ray tube is a glass vacuum tube with a cathode made of a tungsten filament and an anode that can be made of tungsten, molybdenum or rhodium (Figure 1.3). By heating the cathode filament, electrons are produced and released into the tube (Reilly 2019). Owing to the significant difference in potential between the anode and the cathode, the produced electrons are accelerated and hit the target element of the anode at a high speed. The incident electrons are deviated because of the magnetic field of the atoms composing the anode material, with a decrease in their speed, thereby generating X-rays. The direction of these rays is defined by the anode tilt angle (Berger et al. 2018). The interaction between the high-energy electrons and the target element produces heat with a small amount of X-rays. The vacuum conditions make it possible to avoid any interaction between the electrons and gas molecules before reaching the anode. For the majority of diagnostic procedures using X-rays, the applied tension ranges from 40 to 150 kV, but it is lower for mammography (36–40 kV) (Delbeke and Segall 2011). The anode is therefore often made of tungsten due to its high melting point and high atomic number, which may increase the interaction rate and, as a result, the production of X-rays. In mammography, the anode can also be made of molybdenum or rhodium (Reilly 2019).

Main Medical Imaging Techniques

5

X-rays can be produced according to two different processes (Figure 1.4). An initial process corresponds to a relatively strong interaction between the electron emitted by the cathode and another electron from the inner layer of the anode atom, which results in a complete elimination of the electron from the inner layer, with its place taken by an orbital electron from the outer layer of the target atom. The transition from the outer layer to the inner layer is accompanied by the emission of X-rays with energy equal to the difference in the binding energies of the two involved orbital electrons. Characteristic radiation produces a discrete spectrum of typical peaks. A second process corresponds to an interaction between the electron emitted by the cathode and the nucleus of the anode target atom, generating a deviation and a decrease in the speed of this incident electron. This loss of kinetic energy is at the root of the production of Bremsstrahlung X-rays (“bremsen” in German means “to brake”), producing a continuous spectrum (Berger et al. 2018).

Figure 1.3. An X-ray tube containing a cathode filament that, when heated, releases electrons accelerated because of a difference in the potential of 100 towards an anode generally made of tungsten. The heat generated during this process can be cleared by rotating the anode. X-rays are produced as a result of the interaction of incident electrons with orbital electrons from the target tungsten atom nuclei (Reilly 2019)

All low-energy photons are, in fact, absorbed by the human body as they do not have enough energy to pass through the body, and thus they never reach the detector. These photons thereby significantly increase the absorbed radiation dose during image acquisition without actually increasing its quality. As a result, a thin aluminum plate is placed between the X-ray

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

source and the human body, which acts as an X-ray filter (different from the mathematical filters used in image processing) (Berger et al. 2018). This aluminum filter is often completed by a second filter made of copper. The low-energy electrons are known as soft radiation, high-energy ones are called hard radiation, and the process of eliminating soft radiation is called beam hardening. The area of the human body to be irradiated is delimited by a collimator, and the dispersion photons are absorbed by a collimating scatter grid. The latter can be, for example, made of lead. It allows the passage of photons with a low incidence angle and blocks those with a high incidence angle. In pediatrics, the dispersion grid is not always used because of the limited dispersion in small children (Suetens 2009).

Figure 1.4. X-ray spectrum from a tungsten tube showing the continuous spectrum produced by Bremsstrahlung and the peaks produced by the characteristic radiation (Berger et al. 2018)

The X-rays reaching the detector make it possible to create medical images of the inside of the body for the area they have passed through. The detector may be (1) a cassette containing a storage plate, (2) an image intensifier coupled with a camera, (3) a screen–film combination, in which a film is sandwiched between two screens, (4) a flat panel active matrix detector, or (5) a dual layer detector (Suetens 2009).

Main Medical Imaging Techniques

7

Figure 1.5 shows a complete radiographic imaging system comprising: (1) an X-ray source, (2) an aluminum filter to remove low-energy photons and thus increase the average energy of the photon beam, a process called beam hardening, (3) a collimator limiting the area to be irradiated, (4) a patient attenuating the X-ray beam and causing dispersion, (5) a collimating scatter grid to absorb the dispersion photons, and (6) a detector (Suetens 2009). In the radiographic image, tissues with high X-ray absorption, such as bones, are represented by light structures.

Figure 1.5. Schematic representation of the radiographic imaging system (Suetens 2009)

1.2.3. Applications of X-ray imaging X-ray imaging is used in several applications. The most common applications are listed below. 1.2.3.1. Conventional radiography Conventional radiography refers to the process of creating 2D images using X-ray projection, by measuring the attenuation of those rays as they pass through the body. Therefore, this technique is mainly used to examine fractures and changes in the skeletal system due to the high attenuation coefficient of bones compared to that of the surrounding tissue, which

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

provides a good contrast (Figure 1.6) and an accurate detection, as well as a distinct classification of different fractures (Berger et al. 2018). Moreover, radiography is frequently used to accurately measure bone mineral density, making it possible to assess and monitor the risk of osteoporosis (Shi et al. 2016), and is considered as the primary screening method for detecting bone tumors and tumor-like lesions (Morrison et al. 2013). In addition, this imaging modality is able to provide information on lesion location, internal matrix, margins and associated periosteal response, and thus make it possible to guide differential diagnosis (Wu and Hochman 2012). Using radiography, a single image or a small number of images can be acquired for a specific view.

Figure 1.6. Radiographic images showing (left) fractures in the forearm and (right) metal plates used to fix the ulna and the radius (Berger et al. 2018). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

1.2.3.2. Dual energy X-ray absorptiometry Dual-energy X-ray absorptiometry (DXA or DEXA), also known as bone densitometry, is an imaging modality subtracting two images obtained using different X-ray spectra (Flower 2012). It is based on two concepts: (1) the passage of two X-ray beams or an X-ray beam of two different energies through the body or the region of interest, and (2) the attenuation of the calculated beam or beams, since the emitted photon number is known (Ahmad et al. 2014). The attenuation (absorption or dispersion) of the X-ray beam varies with the energy intensity, as well as with the human tissue

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thickness and density. It decreases as photon energy increases and increases as tissue thickness and density increase (Ceniccola et al. 2018). For example, the attenuation is more important in bone tissue than in soft tissue (Guglielmi et al. 2016). In fact, this modality makes it possible to estimate the value of the attenuation coefficient related to two different energy levels. The value of is constant for bones and fats, but variable for soft tissues in accordance with their composition (the higher the percentage of fat in the soft tissue, the lower the values of ) (Ceniccola et al. 2018). Assuming that the body is subdivided into three groups: bone, lean tissue and adipose tissue, DXA provides measurements of the mineral content, lean mass and fat mass of the body or the region of interest, without directly measuring these three components (Kendler et al. 2013; Ahmad et al. 2014; Bazzocchi and Diano 2014). The amount of fat mass and lean mass is inferred on the basis of the ratio of fat mass to lean mass at the boneless neighboring pixels, thereby assuming that the amount of fat at the bone level is similar to that at the adjacent boneless tissues (Bazzocchi et al. 2014). In accordance with the amount of radiation reaching the detector and the two different energies used simultaneously, a gray value with two different attenuation values is assigned to each pixel. In the case of bone, both attenuation values provide information on bone mineral content that, when divided by the bone surface, make it possible to determine bone mineral density (Lucas et al. 2017). The attenuation itself can be subdivided owing to photoelectric absorption and photon dispersion. These two components have different energy dependence, because of which a photoelectric component and a dispersion component can be extracted from the two initial images. Similarly, the difference in the amounts of photoelectric dispersion and absorption between different materials makes it possible to create images of the thicknesses of two selected materials. By combining these thicknesses, the attenuation maps recorded in the two original images may be reproduced. Therefore, an appropriate selection of these materials makes it possible to create images showing or eliminating any type of tissue, as desired (Figure 1.7). For example, images of bone structure and soft tissue can be separately viewed by selecting the bone and the tissue (with a nominal composition) as base materials (Flower 2012).

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Figure 1.7. Image of a DXA examination (Ceniccola et al. 2018). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

DXA was originally developed to measure bone mineral density and is frequently used to diagnose osteoporosis. This modality is currently the reference technique in body composition analysis at the molecular level, which is recognized for the accuracy of its measurements for a specific region of interest or for the whole body (Marinangeli and Kassis 2013; Guglielmi et al. 2016). With the latest generation of densitometers, body composition can be assessed using a single whole body scanner, which consequently reduces the duration of radiation exposure and acquisition time. The obtained images are comparable to the radiological images and provide accurate data (Bazzocchi et al. 2014; Ceniccola et al. 2018). 1.2.3.3. Fluoroscopy Fluoroscopy is a transmission projection imaging modality, making it possible to periodically obtain X-ray images at a certain frequency, which is equivalent to a real-time radiography (Bushberg and Boone 2011). The frame rate depends on the acquisition speed of the detection system, which may reach almost 30 frames per second. Image intensifiers and FPD screens are potential detection systems (Berger et al. 2018). Fluoroscopy is used in minimally invasive therapeutic procedures, requiring real-time image feedback to guide the used tools, such as endoscopes or catheters, without direct visual contact with the intervention

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area (Figure 1.8). It makes it possible to visualize the gastrointestinal tract and vessels using contrast agents, and can provide radiographic films of anatomical movement, such as those of the esophagus or heart (Bushberg and Boone 2011; Berger et al. 2018).

Figure 1.8. Image of a fluoroscopy sequence while introducing two pacemaker electrodes (left). Typical clinical configuration of a minimally invasive surgical procedure guided by a fluoroscopy sequence using a freely positioned device, with the X-ray source placed under the patient and the FPD right above him (right) (Berger et al. 2018)

1.2.3.4. Digital subtraction angiography Owing to the poor contrast of X-ray imaging between the vessels and the surrounding tissue, digital subtraction angiography (DSA) is used to visualize the gastrointestinal tract and blood vessels using a contrast agent that can be whether swallowed or injected into the bloodstream, in order to obtain information about the size, shape, flow or light of the target area. Barium and iodine are two typical contrast agents, with the first being used for gastrointestinal examinations and the second for intravascular examinations (Flower 2012; Berger et al. 2018). Figure 1.9 shows the stages of angiography obtained by acquiring an image of the region of interest without a contrast agent, and then by acquiring another image during the transit of the contrast agent. By subtracting the image showing the surrounding tissue from the image showing the tissue and the vessels, the attenuation caused by the contrast agent can then be measured and the resulting angiogram clearly shows the blood vessels in the region of interest. Two further examples of DSA are shown in Figure 1.10.

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Figure 1.9. Creation of a DSA, starting by creating an image of the hand without any injected contrast agent, known as a mask image (left), followed by the creation of a fill image of the same hand after injecting a contrast agent into the vascular system (center). The subtraction of the two images provides the angiogram (right), which only shows the contributions of the contrast agent and, thereby, visualizes the vessels in the hand (Berger et al. 2018)

Temporal and spatial filters can be used on all images to improve the signal-to-noise ratio (SNR) in the subtracted image, and corrections can be made to adjust for the movements of the patient (Flower 2012).

Figure 1.10. Two examples of DSA. The left angiogram refers to the upper mesenteric artery feeding the intestines, with an anterior–posterior projection from the upper abdomen; the catheter is introduced into the aorta line. The right angiogram refers to one of the arteries feeding a hemangioma of the left trapezius of the shoulder, with an anterior–posterior projection of the left lung and shoulder; the catheter is introduced into the left subclavian artery (Flower 2012)

1.2.3.5. Mammography Mammography is an imaging modality specifically used to image the breast using X-ray projection, with energies much lower than those used in general radiography, making it possible to accentuate the contrast in the

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breast in order to detect breast abnormalities, such as calcifications or masses. The used systems and detectors are therefore designed to particularly produce high quality images based on low X-ray doses. Mammography can be either (1) diagnostic, facilitating the diagnoses of women with breast symptoms, such as lumpiness, or (2) screening, making it possible to detect breast cancer in asymptomatic women (Figure 1.11A). Digital mammography enables computer-assisted detection, with which some systems provide a rotational movement of approximately 7 deg to 40 deg of the X-ray tube (and in some cases of the detector) around the breast, thereby producing a tomosynthesis (Figure 1.11B), with images parallel to the detector plane and capable of reducing the overlap of the anatomy below and above the plane in focus (Bushberg and Boone 2011).

Figure 1.11. (A) Digital mammogram showing the skin line of the breast, the fat tissue and a possible cancerous mass, indicated by the arrow. In projection mammography, tissue superposition at different depths can mask the malignancy characteristics or cause artifacts mimicking tumors. (B) Tomogram synthesized at mid-depth. By reducing the anatomy underlying and overlying the tomosynthesis, the suspicious mass in the breast is clearly represented with a spiculated appearance, indicative of cancer. Owing to its high sensitivity, very high benefit/risk ratio and low cost, X-ray mammography is currently the procedure of choice for screening and early detection of breast cancer (Bushberg and Boone 2011)

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1.2.4. Advantages and disadvantages of X-ray imaging Table 1.1 summarizes the advantages and disadvantages of X-ray imaging. Advantages

Disadvantages

Fast execution Inadvisable during pregnancy Non-invasive technique Largely ineffective in distinguishing between different fat types (visceral, intramuscular Low cost and subcutaneous) Low radiation levels Difficult to carry out for sick people or very Safe in repeated measurements small children who are unable to keep their Effective characterization of bone lesions back extended and primary bone tumors Operation requires specific technical skills, as well as experience Table 1.1. Advantages and disadvantages of X-ray imaging (Ceniccola et al. 2018)

1.3. Computed tomography Computed tomography (CT) is also an imaging technique that uses X-rays, based on the same physical principles as the X-ray imaging technique mentioned above: X-rays pass through the body or the region of the body to be scanned, and are attenuated according to the physical properties of the tissues through which they pass. Unlike “simple” X-ray techniques, CT makes it possible to create tomographic images, where “tomography” refers to an image (graphic) of a slice (tomo). One of the first scanners was composed of a single X-ray emitter with a single in-line detector, capable of rotating around the body or the region being scanned. The individual, who is in the tube containing the transmitter and detector, moves one centimeter or more through the tube after obtaining each slice in order to acquire the next slice. This process is repeated until scanning the whole region of interest, enabling the acquisition of a series of CT scans of that region. The number of transmitters and detectors has changed with the progressive development of this imaging technique, making it possible to obtain a 3D reconstruction and create cross-sectional views of the body or the region of interest. The use of CT dates back to the early 1970s and represents the first computer-assisted medical imaging modality, due to which the rate of exploratory surgery has been significantly reduced (Bushberg and Boone 2011; Miller et al. 2014).

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1.3.1. Description of the technique Fundamentally, a scanner measures the attenuation of a well-collimated X-ray beam along a line between the emission source and the detector. The body or region of interest is analyzed by the source–detector assembly, carrying out a linear translation movement. By rotating the X-ray tube around the body and combining X-rays from different angles, CT scanners produce a 3D replication of the body or region of interest. A numerical value, called CT value, is assigned to each voxel of the generated image, making it possible to measure the attenuation of body tissues in Hounsfield units (HU), using a standard scale for a linear attenuation coefficient and defining water as and air as (Prado and Heymsfield 2014; Fosbøl and Zerahn 2015; Yip et al. 2015; Lucas et al. 2017; Ceniccola et al. 2018). The CT value is expressed as follows (Vogl et al. 2016): =

[1.3]

is the linear where is the linear attenuation coefficient and attenuation coefficient of water. Using the HU scales, body tissues and organs can be determined by quantifying the muscle mass ( =− to + ), the subcutaneous and intramuscular fat tissue ( =− to − ), as well as the visceral fat tissue ( =− to − ). For each image, the cross-sectional area of the different tissues (Figure 1.12) can be determined manually or using a software (Ceniccola et al. 2018).

Figure 1.12. Cross-section of the abdominal segment showing the third lumbar vertebra with different colored tissues: skeletal muscle in red, visceral adipose tissue in yellow, subcutaneous adipose tissue in blue and intramuscular adipose tissue in green (Ceniccola et al. 2018). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

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1.3.2. Development of computed tomography The first CT system was built in 1971 by Sir Godfrey Newbold Hounsfield and Allan McLeod Cormack, and was used to carry out the first clinical scan (Figure 1.13) in the same year. This seminal invention enabled them to win the Nobel Prize in Medicine in 1979. In 1990, the helical scanning system was introduced by Willi Kalender and his colleagues, and was named for its helical trajectory during image acquisition. The first scanners were characterized by a relatively long data acquisition time, with the reconstruction of a single 2D slice, a slow quantification and a low spatial resolution. In 2002, the quantification depth, the spatial resolution and the rotation speed of the system were considerably improved, with the ability to obtain up to slices in parallel. In recent years, spatial and temporal resolutions have been continually improved, and the integration of two X-ray emission sources into a single scanner in 2005 was an important step. The use of these two sources with different voltages, equivalent to a double energy scan, can considerably increase the acquisition rate, as well as the slice number acquired at the same time. Hence, the field of view in the axial direction has also increased with voxel sizes less than one millimeter. It has therefore become possible to scan an entire organ with a single rotation, which significantly reduced movement artifacts. Today, CT is one of the most important medical imaging technologies, providing advantageous and fascinating views of the human body (Figure 1.14) (Flower 2012; Maier et al. 2018).

Figure 1.13. First clinical CT scan obtained in October 1971 at Atkinson Morley hospital in London (Maier et al. 2018)

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Figure 1.14. Volume-rendered CT scan of the head (Maier et al. 2018). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

1.3.3. Instrumentation Seven scanner generations have been developed over time to improve overall performance, mainly for the duration of data acquisition. 1.3.3.1. First-generation scanners Although first-generation (1G) scanners are no longer manufactured for medical imaging, their geometry is still useful for understanding the theoretical ideas behind image reconstruction. 1G scanners have a single finely collimated source, emitting a pencil-shaped X-ray beam, with a single detector, all of which moves in linear and circular motions around the body, performing a “translation–rotation” movement (Gupta et al. 2013; Prince and Links 2015). Initially, the source–detector assembly takes a linear step around the individual, allowing parallel projections to be measured, one sample at a time. After each projection, it rotates at a certain angle to make the next projection (Figure 1.15). The presence of a single detector allows easy calibration with relatively low costs. Owing to 2D collimation at the source, as well as with the detector, 1G scanners are characterized by a higher dispersion rejection than any other generation. The acquisition time is very slow, even for acquiring relatively low resolution images (Flower 2012).

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Figure 1.15. Geometry of a 1G scanner (Prince and Links 2015)

1.3.3.2. Second-generation scanners Second-generation (2G) scanners are also no longer used in medical imaging. They consist of a single source and a network of detectors arranged along a line or in a circle (Figure 1.16). The source and detectors also move in a “translation–rotation” movement in the same way as in 1G scanners, but the X-ray beam is fan-shaped, maintaining the energy in the section while having distributed it over the detector network. The central detectors thereby detect the projection in a similar manner to detectors in 1G scanners, but the additional detectors allow further projections to be obtained at different angles (Prince and Links 2015).

Figure 1.16. Geometry of a 2G scanner (Prince and Links 2015)

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1.3.3.3. Third-generation scanners Third-generation (3G) scanners are systems that rotate continuously by means of a sliding slip ring for power supply and data collection. This generation of scanners has of a single X-ray emission source and a network of detectors, with a sufficiently wide fan beam to cover a complete cross-section of the patient (Figure 1.17). The source–detector unit performs only synchronized “rotation–rotation” movements without the need for linear scanning (Aichinger et al. 2011; Adam et al. 2014). 1.3.3.4. Fourth-generation scanners Fourth-generation (4G) scanners have a single rotating X-ray source and a set of fixed detectors arranged in a ring. X-rays are emitted in a fan beam wide enough to cover the cross-section of the patient’s body. Each detector therefore receives energy from the source moving around the patient, with small gaps separating the detectors and allowing X-rays to escape (Figure 1.18). This generation of scanners is characterized by more efficient detection than 3G scanners; however, the image quality of both generations is comparable owing to the likelihood of the detectors to scatter, since collimation cannot be used with 4G scanners. The image display is almost instant due to the high rotational speed of the X-ray emission source; however, the design has been abandoned owing to a number of problems, namely the extent of dispersion artifacts (Flohr 2013; Prince and Links 2015).

Figure 1.17. Geometry of a 3G scanner (Prince and Links 2015)

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Figure 1.18. Geometry of a 4G scanner (Prince and Links 2015)

1.3.3.5. Fifth-generation scanners Fifth-generation (5G) scanners, also known as “electron beam computed tomography” (EBCT), are based on the use of an accelerated electron beam in a funnel-shaped vacuum tunnel. Deflection coils, organized in a semi-circular configuration and placed beneath the patient, deflect these electrons electromagnetically onto targets corresponding to four tungsten anode bands surrounding the individual’s body. X-rays are thereby generated in a fan-shaped beam, captured by a semi-circular detector located around the patient (Prince and Links 2015; Vogl et al. 2016). EBCT scanners are characterized by the very short acquisition time needed for the thicknesses of the individual layers, virtually eliminating movement artifacts. However, the image generated is of a poor quality owing to the loud noise and improving it requires longer acquisition times, with an increase in the radiation dose (Vogl et al. 2016). 1.3.3.6. Sixth-generation scanners Sixth-generation (6G) scanners generally refer to helical scanners (Figure 1.19), although most scanners are nowadays able to perform a helical scan. Based on the same arrangement of the source–detector unit as that of 3G and 4G scanners, helical scanners consist of moving the patient table

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perpendicular to the source–detector plane, while maintaining the rotation of the X-ray emission source. This creates a helical movement around the body, making it possible to acquire a continuous volume of data during a single breath of the patient. As a result, the acquisition rate is considerably improved and the artifacts are significantly reduced. This acquisition technique provides a retrospective reconstruction of several superimposed slices, thereby improving small lesion visualization and allowing the creation of a very detailed 3D CT angiogram (Klein et al. 2018).

Figure 1.19. Geometry of a 6G scanner or helical scanner (Aichinger et al. 2011)

1.3.3.7. Seventh-generation scanners Seventh-generation (7G) scanners emerged following the advent of multidetector CT (MDCT) scanners. As their name indicates, these scanners use several axial rows of parallel detectors that collect a thick fan-shaped X-ray cone beam (Figure 1.20). Because of this conical shape and the typical enlargement of the region of interest located in the center of the field of view, the detector rows are approximately twice as wide as the minimum available cross-section thickness. This geometry makes it possible to simultaneously collect a large number of one-dimensional projections (Adam et al. 2014; Prince and Links 2015). MDCT scanners are characterized by a short sampling period with a reduction in movement artifacts, which can be very useful for children, seriously ill patients and victims of accidents. Longer examination sections may be used to trace the

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vessels. Isotropic imaging can be carried out with finer collimation, which is particularly advantageous for visualizing the musculoskeletal system and temporal bone with multi-plane reconstructions (Vogl et al. 2016).

Figure 1.20. Geometry of a 7G scanner or MDCT scanner (Adam et al. 2014)

The first four generations of CT scanners are characterized by the use of a single X-ray source, as it is expensive, large and must be constantly calibrated. In addition, rotating a large object (source and/or detectors) limits the overall imaging speed. However, new data processing procedures and methods must be developed in view of the emergence of helical and MDCT scanners. 1.3.4. Applications 1.3.4.1. Computed tomography fluoroscopy CT fluoroscopy is a real-time imaging technique that considerably improves the capacity and speed to carry out image-guided percutaneous interventions, while applying a moderate radiation dose. Images are quickly reconstructed, with the scanning table displaced to observe anatomy and lesions, as well as catheter and needle placement in real time. This imaging technique is frequently used to guide operations through the body, in addition to drainage and biopsy procedures. It is particularly used to guide

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the placement and movement of needles or catheters in moving organs, such as the abdomen or chest (Klein et al. 2018). 1.3.4.2. Perfusion computed tomography Perfusion CT imaging is based on the use of contrast attenuation curves in tissues, as well as in their afferent and efferent vessels, allowing the measurement of blood volume, flow and transit in tissues, in addition to blood vessel leakage. A rapid intravenous bolus injection is performed, and the data from the same region of interest is obtained in succession. Hence, perfusion CT represents a multi-phase imaging modality, with a reduction in the radiation dose during each phase to avoid excessive exposure to radiation. It is particularly used for diagnosing strokes (Adam et al. 2014) (Figure 1.21), and is increasingly used in oncology to examine the anti-angiogenic treatment effects at the early stage of clinical trials (García-Figueiras et al. 2013). Two main methods of image analysis have been described: (1) the first, more robust method is based on contrast slope improvement, requiring high injection rates but not able to provide absolute measurements, and (2) the second method is more susceptible to noise and inconsistencies in data, but able to provide absolute values based on Fourier deconvolution methods (Adam et al. 2014). 1.3.4.3. Dual-energy computed tomography Dual-energy, or double-source, CT is an imaging technique that uses two sources and two detectors of X-rays, in order to determine the behavior and composition of the target tissue at different radiation energies. Differences in soft tissue and fat tissue, as well as contrast agents, at different energy levels, increase lesion visibility and characterization. Data acquisition time can be half the time needed when using conventional MDCT, which significantly improves the ability to visualize certain organs, such as the heart, without using potentially dangerous heart rate depressant substances. Dual-energy CT can also be used to determine urinary stone chemical composition and to select appropriate medical or surgical treatment (Klein et al. 2018). Moreover, it can help to accurately and sensitively distinguish hemorrhage from iodinated contrast agent staining that may occur after an endovascular thrombectomy in cases of infarction (Figure 1.22) (Gupta et al. 2010).

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Figure 1.21. Perfusion CT showing an infarct of the left middle cerebral artery territory. The mean transit time map (A) shows an area of delayed mean transit time in the posterior part of the left middle cerebral artery territory. The cerebral blood flow map (B) shows a larger area of cerebral blood flow revealing ischemic and infarcted tissues. The cerebral blood volume map (C) shows a small area of reduced cerebral blood volume in the left parietal convexity corresponding to core infarct. The area of incompatibility between the reduced cerebral blood flow and cerebral blood volume regions is a potentially salvageable ischemic penumbra (Adam et al. 2014). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

Figure 1.22. Dual-energy CT showing (left) a single-energy axial CT scan, (middle) a corresponding iodine superimposition image indicating hyperattenuation at the left frontal lobe and caudate nucleus (arrows), and (right) a virtual non-contrast image indicating the absence of hyperattenuation (arrow) and the presence of contrast staining rather than hemorrhage (Peh 2014)

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1.3.4.4. Movement analysis The increasing width of the scanner detectors allows the evaluation of motion effects, due to the possibility of acquiring a large amount of data from the region of interest, either continuously or with very short time intervals. This technique is particularly used to assess tarsal or carpal joint instability, as well as to perform functional examinations of arterial compression syndromes (Adam et al. 2014). 1.3.5. Advantages and disadvantages of computed tomography Table 1.2 summarizes the advantages and disadvantages of CT imaging technique. Advantages

Disadvantages

Cross-section view Tissue contrast High-resolution image Rapid acquisition High quantitative and qualitative precision Possibility of determining tissue quality

Contrast allergy Relatively high cost High exposure to radiation Image analysis requiring technical skills Convenience sampling

Table 1.2. Advantages and disadvantages of CT imaging techniques (Ceniccola et al. 2018)

1.4. Magnetic resonance imaging Magnetic resonance imaging (MRI) is currently a major technique in the field of medical imaging, using magnetic fields about 10–60 thousand times stronger than the magnetic field of Earth (Bushberg and Boone 2011). This imaging technique is based on the principle of nuclear magnetic resonance (NMR), which enables imaging the nuclei of atoms, such as hydrogen, carbon, nitrogen, sodium and phosphorus, located inside the body (Miller et al. 2014). In most clinical cases, the used MRI focuses on imaging hydrogen nuclei (1H), since hydrogen is the most abundant atom in the human body, providing a relatively large magnetic moment. Indeed, the number of hydrogen atoms in the body exceeds 10 (Maier et al. 2018) and a typical MRI voxel contains about 10 (Haidekker 2013). NMR was first

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reported in 1946 and its use as an in vivo imaging technique, now known as MRI, dates back to the early 1970s. Conventional NMR was first used to study the chemical content of material. The development of modern MRI machines has since allowed the acquisition of a detailed depiction of the human body and the various contrasting soft tissues, representing a very important step forward in the field of medical imaging in the 20th Century (Miller et al. 2014; Maier et al. 2018). 1.4.1. Instrumentation The magnetic resonance system mainly consists of two equipment groups. The first is the control center, which is composed of a “host” computer with its graphical user interface. The associated electronic components and power amplifiers are generally located in an adjacent room and connected to the second equipment group. The latter is housed in the machine in which the patient is located (Figure 1.23), and includes the components generating and receiving the magnetic resonance signal, comprising a set of main magnetic coils, three gradient coils, compensation coils, as well as an integrated radiofrequency (RF) transmitter coil (Ridgway 2010). In order to use RF electromagnetic waves, or radio waves, the room containing this second equipment group must block potential electromagnetic noise sources and isolate them from its own RF. The magnet and its associated coils are therefore enclosed in a special examination copper-lined room, forming what the physics community calls a Faraday cage.

Figure 1.23. Schematic representation of the relative positions of the different magnetic coils composing the magnetic resonance machine (Currie et al. 2013)

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To obtain MRIs, the individual is placed in the magnetic field and subjected to RF pulses generated by the surrounding coils. These pulses are absorbed by positively charged hydrogen atoms of the patient (protons) and re-emitted to be detected by the coils surrounding the body. Absorption and re-emission are separated by a time period characteristic of each tissue. By slightly modifying the applied external magnetic field intensity according to the subject position, the resonance frequency of the proton of interest varies. The frequency and phase of the re-emitted RF pulses determine the position of each signal emitted from the body. The images produced by MRI represent slices of the body or region of interest, in which each point of the image depends on the micro-magnetic properties of the corresponding tissue (Bushberg and Boone 2011). 1.4.2. Generation of the resonance effect The hydrogen atom nucleus has an intrinsic property called spin, corresponding to the rotation of its unique proton around an axis, which provides a magnetic effect (Figure 1.24). In the absence of an external magnetic field, the axes of the hydrogen nuclei are randomly oriented, i.e. the spin distribution will be random and the net macroscopic magnetic moment is null (Figure 1.25). When applying a magnetic field, , the water in the body becomes polarized, and the spins undergo a torque that aligns , in the same them in the same direction as the external magnetic field, way as a compass needle gradually aligns itself in the direction of this field. This alignment briefly activates an RF electromagnetic field, which in turn generates the rotation of the axes around the magnetic field direction, in the same way that the tilt of a spinning-top rotates around the gravity direction. It should be recalled that an RF wave corresponds to the variation of a magnetic field over time (Feeman 2010; Miller et al. 2014; Maier et al. 2018). RF waves are emitted at the same oscillation frequency as the hydrogen atoms; they disappear when their axes of rotation reach a stable position and are emitted again when these atoms are re-unbalanced (Maier et al. 2018).

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Figure 1.24. Representation of the hydrogen atom nucleus rotating (spinning) as it begins to have a magnetic effect (Maier et al. 2018)

The spin orientation may be parallel, i.e. in the same direction as the applied magnetic field, or antiparallel, i.e. in the opposite direction. Because the parallel orientation requires a lower energy level, more spins are oriented parallel than antiparallel to the magnetic field, and the difference is called , since the parallel and antiparallel spin excess ∆ . In an external field spins cancel each other out, the spin excess ∆ determines the net magnetic moment (Haidekker 2013), corresponding to the sum of all spin is always directions. By convention, the external magnetic field considered to be oriented along the z-axis, and thus the net magnetic moment is also parallel to the z-axis. The net magnetic moment is thereby generally called longitudinal magnetization (Haidekker 2013).

Figure 1.25. Hydrogen atoms schematically represented by their axes of rotation: (a) randomly distributed in the absence of an external magnetic field, (b) partially , owing to random aligned in the same direction as the applied magnetic field interactions between the nuclei, and (c) with the accumulated magnetization of all the spins around (Maier et al. 2018)

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Moreover, the alignment of the axes of rotation of the hydrogen atom nuclei in relation to the direction of the magnetic field is partial, owing to the random interactions between the nuclei (Figure 1.25). The spins obey the following movement equation (Haidekker 2013): =

×

[1.4]

where is the gyromagnetic ratio. Therefore, determines the angular displacement and precession rate of the magnetic moment . It is important to highlight the inconsistency in the literature and between the different specialties regarding the definition and units of . Some authors call it the magnetogyric ratio and describe the gyromagnetic ratio as the reciprocal. Others treat it as a synonym for the Landé g-factor. The gyromagnetic ratio is sometimes expressed in , causing a difference factor of , or , which is ambiguous (Flower 2012). At the beginning of the precession, the axis of rotation rotates around the according to a certain angle. This angle direction of the magnetic field are decreases over time, until the axis of rotation and the direction of aligned. This alignment is represented by the net magnetization vector . Due to the accumulation of hydrogen nuclei inside the body in the presence , the net magnetization is proportional to the of a strong magnetic field intensity of this field (Maier et al. 2018). The precession frequency (also known as the Larmor frequency) of the nuclear spin depends on the intensity of the applied magnetic field and the gyromagnetic ratio (Maier et al. 2018): =

[1.5]

⁄ For hydrogen, the gyromagnetic ratio is approximately . and the precession frequency in a magnet of . is approximately . (Haidekker 2013). It should be noted that precession and rotation should not be confused (nucleus precession around the magnetic field direction and nucleus rotation around its own axis). In the following, only the net magnetization will be considered.

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When is excited, i.e. moved from its initial equilibrium position, RF waves are emitted from the body. The direction of changes when a to is applied, with RF waves weaker orthogonal magnetic field coming from a coil at the same resonant frequency as . The magnetic field must, in fact, be orthogonal to and its direction must be aligned with the direction of the present angle of rotation of (Maier et al. 2018). In order to better understand this phenomenon, a turntable is considered to be moving around the direction of the magnetic field (Figure 1.26). The point of application of is assumed to be on this turntable. Being based on a fixed inertial reference point, not belonging to the turntable, the movement of follows a sort of spiral, since it has the same direction as the (Figure 1.26). If the reference point is fixed on the magnetic field turntable, the direction of becomes constant and the excitation induced can be differentiated, but the precession movement cannot be by observed. By eliminating , gradually returns to its equilibrium position through a process called relaxation, during which the RF waves emitted by the body are used to generate MRIs (Maier et al. 2018). 1.4.3. Relaxation and contrast Relaxation refers to the return of net magnetization to its equilibrium state before excitation by an RF pulse. The rate of relaxation depends on the intensity of the applied magnetic field, as well as the tissues, owing to the limited interactions between protons in the case of large molecules or dense tissues characterized by an impeded movement of water molecules. The rate of relaxation therefore differs from one type of tissue to another. For instance, the water relaxation rate differs from the fat relaxation rate, hence the difference between the amounts of the received signal for each of them during the relaxation time. This is at the root of the tissue contrast in MRIs (Maier et al. 2018). Considering the microscopic origins of relaxation phenomena, NMR relaxation parameters are extremely sensitive indicators of the physical and chemical microenvironment, including nuclei of interest (Flower 2012).

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Figure 1.26. Excitation and precession of the magnetization vector . This figure was reconstructed from Maier et al. (2018). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

A mathematical description of relaxation was first proposed by (Bloembergen et al. 1948). Other papers were subsequently published, providing a detailed description of relaxation physical principles and quantum mechanics (Abragam and Abragam 1961; Abragam 1983; Ernst et al. 1987; McConnell 1987; Levitt 2001; McConnell 2009). This section will provide a summary of these principles. By considering a system of coordinates (o, x, y, z) and supposing that is directed along the z-axis, each of the magnetization vectors can be divided into two components, a longitudinal M and a transverse M , such as 0. During M = M + M . Therefore, before relaxation, M = 0 and M excitation, an RF pulse moves the vector M tip towards the transverse plane, so that M = 0 and M is maximized. By deactivating the RF pulse, the spins regain their equilibrium state following, in general, two spin relaxation mechanisms: (1) a longitudinal relaxation, during which M tends to return to its initial intensity; and (2) a transverse relaxation, during which the intensity of M tends towards 0. Hence, the intensity of M varies according to trajectory time, as shown in Figure 1.27 (Maier et al. 2018). Longitudinal relaxation is often called T or “spin–lattice relaxation”, where the term “lattice” implies a “thermal bath” of a well-defined temperature, to which each of the spin systems is independently coupled. T results from the loss of proton energy in the ambient atmosphere (Flower 2012). The recuperation of M

is actually expressed as M (t) = M (1 − e ). The relaxation T

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

describes the necessary time constant for M to recuperate 1 −

(

63%)

from its initial magnetization M (Maier et al. 2018). It corresponds to the energy transfer between the spin system and the network, modifying the magnetization longitudinal component (component z) (Flower 2012).

Figure 1.27. Precession and relaxation of the white substance according to the two reference frames. Relaxation is represented by the blue trajectory, and the magnetization vector (vector in green) turns towards the plane (vector in red). When relaxation begins, the length of the red vector ( ) exponentially decreases ) with a speed defined by . At the same time, the length of the green vector ( increases at a speed defined by . Moreover, the magnetization vector turns around the direction of the magnetic field (direction of the z-axis) with the Larmor (Maier et al. 2018). For a color version of frequency dependent on the intensity of the figure, see www.iste.co.uk/benkahla/finite.zip

The transverse relaxation is also called T or “spin–spin relaxation”, leading to two processes: the indirect exchanges using the network and the small differences between the precession frequencies between spins (Flower 2012), which result in a loss of magnetization perpendicularly to the outer field B (Miller et al. 2014). In fact, M is described by M (t) = M e . T corresponds to the time needed for M to decrease following excitation to a value of ( 37%) of its initial value M (Maier et al. 2018), describing the loss of magnetization in relation to the xy plane. This relaxation time depends on the tissue chemical structure and results in a slight modification of the nucleus effective field B , and therefore of the Larmor frequency. T is always lower than T , and, for the majority of tissues, it is much faster than T (Flower 2012). The tight relationship between the relaxations T and T and the physical properties of the tissues of the region of interest create a tissue contrast.

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Therefore, measuring the magnetic resonance signal during relaxation provides a contrast, resulting in images with different gray levels, whose intensities reflect the corresponding tissue densities. Changing the imaging parameters can lead to image “weighting”, making it possible to reflect one type of relaxation more than another (Miller et al. 2014). For a pulse sequence, two parameters are used to determine the way the resulting image will be weighted: (1) the echo time (ET) corresponds to the time period between the emission of an RF pulse and the peak of the echo signal; and (2) the repetition time (RT) corresponds to the time period between two successive RF pulses. Three main types of contrasts can therefore be identified: (1) the T weighting of a short ET sequence as well as a short RT sequence, (2) the T weighting of a long ET sequence as well as a long RT sequence, and (3) the proton density (PD) weighting of a short ET sequence and a long RT sequence. For example, liquids are generally characterized by long T and T and fats by short T and T (Miller et al. 2014). Consequently, a T weighted image shows liquids in dark and fats in light, while a T weighted image shows liquids in light and fats in dark (Figure 1.28). The contrast or weighting selection in the MRI to be generated is of major importance, representing a determining factor of the subsequent medical application and therefore of its diagnostic importance (Maier et al. 2018). The majority of clinical exams are carried out with the images having T weighting and T weighting. Moreover, tissue abnormalities can be better defined by introducing a magnetic resonance contrast agent. Gadolinium is the most commonly used contrast agent. It is a paramagnetic agent that modifies the local magnetic field and considerably reduces the relaxation of the neighboring water protons, which locally increases the signal time on a T weighted image (Miller et al. 2014). 1.4.4. Applications of magnetic resonance imaging A number of MRI techniques have been developed over the last two decades, allowing its use in clinical trials for purposes of monitoring and assessment (Yu 2011).

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

1.4.4.1. Diffusion-weighted imaging In water, molecules in continuous random motion, known as Brownian motion or Wiener process, induce isotropic diffusion that becomes anisotropic in the presence of structural components, such as cell membranes. This random molecular motion in water can be quantitatively measured using diffusion MRI. In basic diffusion-weighted imaging, the apparent diffusion coefficient is usually used to quantify water diffusion. When diffusion is restricted, for example in the case of ischemia, the apparent diffusion coefficient decreases and the signal on the diffusionweighted imaging becomes clearer. This imaging modality is widely used in several therapeutic areas and is considered as an evaluation criterion in clinical trials (Miller et al. 2014).

Figure 1.28. Difference between the (a) and (b) weighted images of brain contrast (Scheef and Träber 2012). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

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1.4.4.2. Magnetic resonance spectroscopy Magnetic resonance spectroscopy is an imaging modality that can provide metabolic information from tissues in vivo, using the relationship between the resonance frequencies of nuclei and their chemical environment: a phenomenon known as chemical shift. Based on the anatomical reference images, MRS signals can be located on the volumes of interest of the targeted tissues. A magnetic resonance spectroscopy spectrum can be obtained by converting the signal acquired in the time domain into one in the frequency domain. For each spectrum peak, the resonance frequency position is determined based on the chemical shift. The resulting spectrum can be quantified either in standard signal strength or in absolute value. Magnetic resonance spectroscopy is widely used for diagnosing and assessing response to treatment, as well as for neurological and hepatic studies. For example, the decrease in signal to a certain value reflects the decrease in a neural marker called NAA, indicating a neural loss or dysfunction (Miller et al. 2014). 1.4.4.3. Functional magnetic resonance imaging Based on the fact that a region of the brain is more active when oxygen consumption is higher, functional MRI (fMRI) consists of measuring the degree of brain activity in relation to the level of oxygen consumed in each brain region using conventional MRI equipment. Indeed, the magnetic field generated by an MRI machine is able to detect the different concentrations of oxygenated and deoxygenated hemoglobin in the blood of the brain regions of interest. A color spectrum gradient is then produced, allowing the most active regions to be distinguished from the less active ones. The fMRI also makes it possible to determine the most active region of an individual’s brain when speaking or moving, or by using a variety of specific stimuli. This field has been the subject of several studies, leading to a significant evolution in the understanding of how the brain works during complex cognitive processes, such as deception (Mohamed et al. 2006).

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

1.4.5. Advantages and disadvantages of magnetic resonance imaging The MRI technique also has its advantages and disadvantages, which are summarized in Table 1.3. Advantages

Disadvantages

Relatively high cost Possibility of providing slices according Relatively limited availability to the axial, sagittal and coronal planes Longer than average length of examination Detailed description of the contrast in Limited sensitivity to detecting calcification soft tissues and periosteal reaction High sensitivity to fats and to small Contraindications in the presence of changes affecting them ferromagnetic aneurysm forceps, metal Non-invasive, non-ionizing and foreign bodies in orbits, pacemakers, non-contrast iodinated technique cochlear implants, insulin pumps and other No side effects for children or pregnant electronic implants, as well as in the case of women severe renal failure or dialysis Multi-plane and multi-parametric Frequent appearance of artifacts during examination relatively long imaging sequences Table 1.3. Advantages and disadvantages of MRI (Ceniccola et al. 2018)

1.5. Ultrasound imaging Acoustic waves can be classified into three categories: (1) infrasound, in the subsonic range, with a frequency of less than 16 Hz, (2) acoustic sound, in the audible range that can be detected by the human auditory system, with a frequency between 16 Hz and 20 kHz, and (3) ultrasound, with a frequency exceeding 20 kHz. The latter category is used in several applications ranging from measuring distances at sea to medical applications. 1.5.1. Definition of ultrasound Ultrasound refers to sound waves with frequencies exceeding 20 kHz (Figure 1.29). This form of mechanical energy can be used to create medical images of different body parts, including muscles, blood vessels and soft tissues, due to its balanced penetration and interaction with the body. Ultrasound is reflected and refracted at the interface level separating media with different acoustic refractive indices (Flower 2012; Cikes et al. 2017).

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Figure 1.29. Acoustic wave spectrum (Mikla and Mikla 2013). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

As with all other types of wave, ultrasound is mainly characterized by the frequency , the speed , the wavelength and the intensity . Typically, ultrasound intensity used for diagnosis varies from 1 to 10 mW m−2. It should be recalled that the wavelength is linked to the frequency and the speed of sound according to the fundamental wave equation: =

[1.6]

Owing to the significant attenuation of high frequencies in tissues, the different body parts are examined using different frequencies ranging from 3 to 5 MHz for abdominal areas, from 5 to 10 MHz for superficial parts and small sizes, and from 10 to 30 MHz for eyes and skin (Mikla and Mikla 2013). 1.5.2. Development of ultrasound imaging The development of ultrasonography dates back to the 21st Century BCE, owing to the study carried out by the Roman architect and engineer Vitruvius on the acoustics of theaters, in which interference, echoes and reverberation were discussed. The application of ultrasonography in medicine began with the first description of the influence of sound waves through liquids, in 1927, by Robert Williams Wood and Alfred Lee Loomis. In the 1930s, Karl Theodore Dussik and his brother began studying the human brain using sonograms (Figure 1.30) (Christian et al. 2014). Today, ultrasound is frequently used in medicine for diagnostic purposes, at frequencies typically ranging from 2 to 40 MHz. However, these sound waves are now undergoing a development, making it possible to use them in therapeutic applications (Maier et al. 2018).

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

1.5.3. Generation of ultrasound Mechanical pressure can be converted into an electrical voltage (piezoelectric effect) using piezoelectric crystals or materials. This electrical voltage can be measured with two electrodes (Figure 1.31). Conversely, applying electrical energy to a piezoelectric crystal (indirect or reciprocal piezoelectric effect) causes the mechanical distortion of the crystal, leading to its vibration and to the generation of sound waves. This is the basic principle of ultrasound generation from transducers (Mikla and Mikla 2013; Sehmbi and Perlas 2015).

Figure 1.30. Karl Dussik and his ultrasound apparatus in 1946 (Christian et al. 2014)

Figure 1.31. Piezoelectric effect (Maier et al. 2018)

Typically, barium titanate (BaTiO3) and lead zirconate titanate (PZT) are used as piezoelectric materials (Maier et al. 2018).

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1.5.4. Transducers An ultrasonic transducer plays the role of both a generator and a detector of ultrasonic waves. It converts mechanical energy into electrical energy, and vice versa. As they penetrate, ultrasound echoes are generated by reflection and diffusion, following their contact with tissues and body fluids. Using the characteristics of these echoes and depending on the type of the used transducer, 1D, 2D or 3D images of the regions of interest can be reconstructed and visualized (Maier et al. 2018). 1.5.4.1. Conventional construction (single-element transducers) The main components of a single-element transducer are shown in Figure 1.32.

Figure 1.32. Typical components of a conventional single-element transducer (Flower 2012)

The piezoelectric element is cut and generally shaped from a piezoelectric ceramic (usually lead zirconate titanate, PZT) or plastic (polyvinylidene difluoride, PVDF). Electrodes, made of silver, are deposited on both the front and back faces, with a permanent polarization of the element across its thickness. Therefore, applying a voltage across these electrodes induces a proportional change in thickness, and applying a pressure across the two faces induces a difference in potential between the electrodes. Focusing can be applied by several methods: a shaped concave ceramic, a lens plus a plane ceramic disk, a bowl plus a defocusing lens, as well as overlapping beams from two elements (Flower 2012). To overcome acoustic mismatching between the element and the tissue, a matching layer is applied, thereby allowing the increase in the energy transfer efficiency. However, this is only applicable in the case of continuous waves, where the ideal layer thickness and impedance are respectively (Flower 2012):

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

= =(

[1.7] ×

)

[1.8]

It should be mentioned that PVDF elements do not require a matching layer. A backing medium is usually used to provide mechanical support. However, it should be minimal if maximum efficiency is desired. In addition, single-element transducers are also composed of a casing that should be electrically screened and acoustically decoupled from the element, in order to prevent dynamic range reduction caused by electrical interference or acoustic ringing. Electrical tuning is frequently used to filter out the element’s low frequency radial mode vibrations, as well as to manipulate the electrical amplitude parameters, in order to provide the best compromise between resolution and sensitivity. The transducer element capacitance is expressed as follows (Flower 2012): =



where is the element dielectric constant and

[1.9] is the element area.

Although the same basic design aspects of the conventional construction are maintained, modern scanning equipment is based on using multipleelement transducers rather than single-element ones (Flower 2012). 1.5.4.2. Multiple-element transducers As their name indicates, multiple-element transducers involve more than one element to enable the use of continuous waves, where separate transmitting and receiving elements are required. The general principle is based on simultaneously exciting a group of tiny elements (piezoelectric crystals), making it possible to produce a plane wavefront emerging from the aperture formed by the spatial extent of these elements. Then, the excitation of a different, but overlapping, group of elements induces the translation of the sound beam from position 1 to position 2 (Figure 1.33a). Focusing or steering the sound beam requires exciting each element via a variable delay. Ultrasound systems, operating based on this technology, are known as “phased-array” systems, creating a sector-shaped image by phase steering the sound beam (Flower 2012).

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Receive beam formation consists of (1) selecting an array element group that will form the desired acoustic receiving aperture, (2) applying a variable delay to the signal received on each element and (3) summing the signals over all the elements in the group. Therefore, a single receive beam line is formed. Owing to the delays on transmission, the synthetized beam is steered and focused to a given angle and distance (Figure 1.33b and 1.33c). Then, a receiver is generated with a maximum sensitivity to echoes arriving from that focal distance and steering direction. Indeed, during the time that a complete sequence of echoes takes to return, following a single transmitted sound pulse, the focusing delays may be continuously adjusted, in order to maximize the sensitivity, and thus the focusing of the system receiving directivity pattern on the position of each echo when arriving from each depth. This is known as swept or dynamic focusing approach, which is only applied to the received signal. Modifications in the transmitted beam focal properties can only be made over successive sound pulses, by applying a different delay combination on each pulse (Flower 2012).

Figure 1.33. Schematic 2D representation of the principles of beam forming (a–c), focusing (c), lateral scanning (a) and steering (b), using arrays of elemental sound sources (Flower 2012)

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

1.5.5. Applications of ultrasound techniques Given the very broad use of ultrasound imaging, this brief overview only focuses on presenting its main applications. 1.5.5.1. Grayscale imaging The use of this imaging modality in grayscale is very widespread. 1.5.5.1.1. Brain In adults, ultrasound is not able to visualize the brain anatomy, because of the attenuation and refraction properties of the skull, although several brain disorders, such as Parkinson’s disease, have recently been depicted. However, ultrasound imaging can be used for intraoperative guidance in neurosurgery carried out on adults, although less frequently than MRI and CT (Flower 2012). In newborns, the brain can be visualized due to the anterior fontanelle (Figure 1.34), allowing several conditions to be diagnosed during the first months of life (Suetens 2009).

Figure 1.34. Ultrasound imaging showing a normal skull (left) and fluid-filled cavities on both sides following intraventricular hemorrhage (right) (Suetens 2009)

1.5.5.1.2. Eye and orbital The small size of ocular and eye socket (or orbital) structures requires high resolution without deep penetration. Therefore, equipment has been specifically designed for ophthalmic use (Figure 1.35), with frequencies typically ranging from 13 to 20 MHz, but reaching up to 50 MHz for some applications, such as biometry. This imaging modality makes it possible to measure the eye axial length and to accurately determine the reflective

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interface location. In addition, information on eye movement and topography can be obtained (Flower 2012). Transducers of 20 MHz may be used to image the retina (Figure 1.36). This type of wave provides limited penetration depth because of their high attenuation coefficient. This loss of acoustic energy is however negligible for surface tissues, such as the retina (Suetens 2009).

Figure 1.35. Ophthalmic ultrasound showing (left) a normal eye and (right) a collar button-shaped malignant melanoma in the shape of a “collar” (solid arrow) with a detachment of the overlying retina (dotted arrows) (Flower 2012)

Figure 1.36. Ultrasound image of an eye showing a malignant melanoma at the top (arrow) and a localized detachment of the retina at the bottom (arrow head), using a 10 MHz (left) and 20 MHz (right) scan (Suetens 2009)

1.5.5.1.3. Neck Soft superficial tissues, such as the thyroid (Figure 1.37), lymph nodes and salivary glands, can be easily imaged using ultrasound (Suetens 2009).

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

In the thyroid, functional studies of radionuclides can be supplemented by structural information obtained using ultrasound, which mainly contributes by differentiating solid and cystic nodules (Flower 2012).

Figure 1.37. Thyroid ultrasound showing a slight bilateral enlargement (arrows), suggesting a hormonal imbalance or an inflammatory disease (Suetens 2009)

1.5.5.1.4. Fetus and gynecology For pregnant women, ultrasound imaging has become a normal routine, enabling the acquisition of information on the fetus (Figure 1.38), the uterus and the placenta (Suetens 2009). It has also become the preferred method for studying the female pelvis. Gynecological examinations can be performed with transabdominal techniques using a full bladder as an acoustic window, or with transvaginal techniques that can be combined with sonohysterography, where a catheter is passed through the cervix to the uterine cavity by gently injecting a sterile saline solution, thus providing an excellent delimitation of the endometrial cavity and facilitating the diagnosis of structural and morphological anomalies (Flower 2012). 1.5.5.2. Doppler ultrasound Doppler ultrasound is based on a phenomenon familiar to train enthusiasts: by standing beside railroad tracks when a train is whistling while moving rapidly, the pitch of the whistle increases as the train approaches and becomes lower as the train passes by the observer and speeds off into the distance. The variation in the whistle pitch results from the Doppler effect, which consists of an apparent change in the sound frequency. The same phenomenon occurs at ultrasound frequencies, and the Doppler effect is used to determine the direction and measure the speed of blood flow. For a

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grayscale image of the region of interest, the blood flow is displayed in red in one direction and in blue in the other direction (Figure 1.39) (Bushberg and Boone 2011).

Figure 1.38. Transverse ultrasound of a fetus abdomen, showing a fluid-filled and dilated left kidney collection system (arrows) caused by the kidney outlet obstruction (Suetens 2009)

Figure 1.39. Doppler color flow imaging used for vascular assessment. The color flow can be obtained by many ultrasound systems. This figure shows an internal carotid artery superimposed on the grayscale image, showing an aneurysm in the left internal carotid artery (Bushberg and Boone 2011). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

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1.5.5.3. Echocardiography Echocardiography is used to evaluate cardiac movements and structures by inserting a probe into the esophagus (transesophageal echocardiography (Figure 1.40) via the surface of the chest wall (transthoracic echocardiography). A beam, or varying ultrasound beams, produces an image or video showing the heart wall, vascular system and valves, as well as the movements of these different structures during blood pumping. Pathologies, such as fluid accumulation around the heart and valve regurgitation, can therefore be determined. Furthermore, a 3D echocardiography can be performed using a set of multiple transducers with an advanced processing system that allows a 3D visualization of the heart in several planes (Miller et al. 2014).

Figure 1.40. Transesophageal echocardiography showing an atrial septal defect (Suetens 2009)

1.5.5.4. Bone ultrasonometry Bone ultrasound was developed in the 1980s to evaluate osteoporosis. Unlike the ultrasound imaging modalities described above, this modality is based on a transmission at a nominal frequency of 1 MHz or lower, producing a sufficient measurement signal with attenuation correlated with bone quantity and quality. Most systems are able to assess the calcaneus and measure broadband ultrasound attenuation (BUA) from 0.2 to 0.6 MHz or the speed of sound (SOS) through the bone. However, the accuracy of this modality is lower than that of DXA; hence, bone ultrasonometry is acceptable for clinical trials (Miller et al. 2014).

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1.5.6. Advantages and disadvantages of ultrasound imaging Similar to all other medical imaging techniques, ultrasound imaging has advantages and disadvantages, which are summarized in Table 1.4. Advantages

Disadvantages

Ability to provide real-time information No required special infrastructure Wide availability Portability Low cost Non-invasive and non-ionizing technique Ability to evaluate changes in longitudinal muscle

Agreement and clinical protocol paucity on their use Technique limited by situations of excessive edema Little contribution to the assessment of bone tumors Inability to entirely visualize intraosseous lesions Inability to penetrate adult cortical bone

Table 1.4. Advantages and disadvantages of ultrasound imaging technique (Ceniccola et al. 2018)

1.6. Comparison techniques

between

the

different

medical

imaging

Medical imaging techniques can be compared in terms of three major concepts. The first concept is image quality, which can be represented by (1) contrast, referring to the difference in brightness or darkness of the image between a region of interest and its background; (2) spatial resolution, referring to the spatial extent of small objects in the image; and (3) noise, referring to the accuracy with which the signal is received. The second concept is the system availability, which can be represented by its cost and its ability to provide real-time information. The third concept is safety, involving the effects of ionizing radiation and heat on the body (Kasban et al. 2015). Table 1.5 summarizes and compares the different medical imaging techniques detailed above.

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Image quality

Availability of the system

Spatial resolution

Good contrast

Cost

Radiography 1 mm

Soft tissues and fluid

CT

0.5 mm

MRI Ultrasound

Imaging technique

Real-time information

Safety Ionizing radiation

Heating effect

Average No

Yes

Low

Hard tissues and fluid

High

No

Yes

Low

0.5 mm

Hard tissues and fluid

High

No

No

Average

1 mm

Soft tissues

Low

Yes

No

Negligible

Table 1.5. Comparison between medical imaging techniques (Kasban et al. 2015)

1.7. Conclusion Medical imaging has been broadly developed to be a particularly effective and necessary technique for clinical care among medical professionals. The multiple existing modalities have offered numerous advantages spurring the use of medical imaging as an eligibility requirement, as well as a biomarker in clinical trials for several endpoints for efficacy and safety. Nevertheless, each technique has its advantages and its limitations and counter-indications. Therefore, none of them can be reliable individually in all medical applications. While this chapter addresses the basic physics of the main medical imaging techniques, the next chapter focuses on the major steps and methods used in processing the generated medical images, in order to create digital models of the different organs and tissues in the human body.

2 Medical Image Analysis and Processing

2.1. Introduction The data obtained from medical images, using various imaging techniques, have contributed to the transformation of radiological practices on the basis of specific computer-based approaches, making it possible to better understand and analyze these data. Medical image processing has undergone a dramatic expansion in recent decades and has become an interdisciplinary field of research, drawing on skills in engineering, computer science, applied mathematics, physics, statics, biology and medicine. In several clinical settings, automation and reliability associated with computer use have significantly contributed to the accuracy and speed of clinical investigations. Consequently, computer-assisted diagnostic treatment is now an important part of clinical routines. However, the development of new advanced technologies and the diversity of medical imaging modalities have created new challenges, such as the procedures for processing and analyzing a large volume of images that can be applied to provide high-quality information, in order to demonstrate significant pathological or atypical differences, even if they are often minimal. This chapter does not cover all stages of medical image processing and analysis, but highlights the major methods and steps of these processes. 2.2. Image compression Improved gray-level and spatial resolution of medical images, along with color information and 3D imaging, significantly increase the size of the

Finite Element Method and Medical Imaging Techniques in Bone Biomechanics, First Edition. Rabeb Ben Kahla; Abdelwahed Barkaoui and Tarek Merzouki. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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image to be stored, thus having an impact on data transmission rates or the network “bandwidth”. Image compression therefore saves storage and transmission by depicting images with a minimum number of bytes. Thus, two main categories of techniques can be used: (1) the first category is called unlossy, in which the compressed image is an exact reproduction of the original one, and (2) the second category is called lossy, in which some information is lost when the image is compressed. The file size for medical images is generally reduced by a factor of 2 using the unlossy technique and by a factor of 20 using the lossy technique (Flower 2012). 2.3. Image restoration Imaging techniques and digitization devices induce image intensity or geometry distortion effects. The elimination of these effects requires modeling with analytic expressions or calibration experiments, in addition to approximations or numerical techniques in the case of more complex effects. Image restoration refers to the correction of these effects, which may not be necessary for successful image processing. However, restoration avoids the use of customized techniques for particular properties of the image (Flower 2012). 2.4. Image enhancement Image enhancement refers to the transformation of the original generated image into a more suitable one for further processing. For visual inspection, modifying the image contrast is often beneficial, making the dark pixels darker and the bright pixels brighter. Since medical images are poorly illuminated, many structures in the image are not clearly visible. Thus, the contrast can be enhanced to highlight and accentuate these structures (Chaira 2015). Image intensity values are therefore scaled in a meaningful way. For this purpose, the simplest way is to apply a function to the intensity values (Bernecker 2018): ( , )=

( , )

[2.1]

Images are initially enhanced and then processed, which is often referred to as the pre-processing stage: the images are filtered to eliminate any noise present in them (Chaira 2015).

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51

2.4.1. Window and level Window and level functions are commonly used to semi-automatically adjust the display, which is required when the image has gray values significantly greater than 8 bits. The gray values in CT scans have known physical properties, which allow material interpretation. Figure 2.1 shows this effect, where the image on the left displays Hounsfield units (HU) ranging from −1,000 to 1,000, which covers bone, contrast agent, soft tissue and air, whereas the image on the right, corresponding to the exact same slice, displays HU ranging from 200 to 500, which only covers cortical bone and contrast agent in the heart. It should be noted that the image was obtained using a C-arm CT system, generating images with a much lower quality than conventional CT (Bernecker 2018).

Figure 2.1. Window and level effect on a slice obtained using a C-arm system, showing an animal experiment. The image on the left displays −1,000 to 1,000 HU, covering bone, contrast material, soft tissue and air, while the image on the right displays 200 to 500 HU, only covering contrast material and cortical bone (Bernecker 2018)

2.4.2. Gamma correction A power law is commonly used for the function, i.e. ( ) = ∙ , where denotes a constant that can be used to normalize the resulting intensities. This type of contrast enhancement is called gamma correction, which adapts to the way humans perceive images. The gamma correction is performed in the same way for all pixel locations, thereby corresponding to a

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global transformation. Other types of functions can also be used for , such as log (Bernecker 2018). 2.4.3. Histogram equalization Histogram equalization is another different approach for enhancing the image display. The intensity values in an image are often limited to a small range of possible values (i.e. a narrow peak in the histogram). Therefore, histogram equalization is used to equally distribute all the intensity values in the image (Bernecker 2018). This equal distribution corresponds to a linear ( ) = ∙ , where denotes the slope CFD, expressed as depending on the image pixel number (Bernecker 2018). 2.4.4. Image subtraction Images obtained before and after the administration of the contrast agent can be subtracted, in order to remove a distracting background and reveal the structures of interest. Image subtraction is mainly prevalent in X-ray angiography and pathology enhancement in neuroanatomical magnetic resonance. It should be noted that this technique is based on assuming a perfect alignment of the source image and the target image, in addition to remaining constant non-enhancing tissue voxel values (Flower 2012). 2.4.5. Spatial filtering The output of a spatial filtering operation at the voxel is related to its neighborhood. The implementation is performed through convolution with single or multiple masks (Flower 2012): ∗[

, ]=∑

∑

[ , ]∙ [ − , − ]

where [ , ] denotes a mask defined over an operation is expressed as: ∗

=

∗

×

[2.2] neighborhood. This [2.3]

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53

Several convolution masks can be used for image analysis, but only a few are of significance in the context of enhancement. The mean of the region is determined by assigning to each [ , ]. This makes it possible to × smooth the data and eliminate the noise. Increasing and results in increasing both the smoothing amount and the processing time. The median is also useful when the image contains especially large or small isolated voxel values, and the mode is more suited when the image contains only a few values, such as segmented images (Flower 2012). 2.5. Image analysis Each image scene is composed of features such as textures, lines and edges, the quantification of which forms the basis of image processing systems. Predefined region features can be directly used to perform serial measurements, but they are more often used to segment the image objects by classifying them into the tissues that the voxel represents. The same features can be calculated using several methods, and the choice of the appropriate method depends on the properties of image acquisition (Flower 2012). 2.5.1. Texture features Textures qualitatively refer to well-known properties. They are quantified based on the local neighborhood of the voxel, making it possible to carry out the segmentation process, particularly in the case of indistinct borders between objects in a scene (Flower 2012). 2.5.1.1. Characteristics of a grayscale histogram Voxel values represent the simplest texture descriptors. Objects embedded in a complex background can be revealed using the histogram shape of the voxel’s local neighborhood. For instance, a bimodal histogram may indicate the presence of a “high density” object. Setting a threshold value at the minima between the two modes makes it possible to only retain the object voxels (Davies and Dance 1990).

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

Local gray-level histograms, such as the central moments and the absolute central moments ∗ , can help extract other measurements (Flower 2012):

∗

=∑

( − ̅)

=∑

|( − ̅ )|

( ) ( )

[2.4] [2.5]

where and respectively denote the minimum and maximum voxel values in the neighborhood, and ( ) denotes the distribution of voxel probability, i.e. the number of voxels having the value divided by the total is the variance, voxel number in the region. Then, ∗ is the dispersion, − 3 is the kurtosis, as shown in Figure 2.2 (Flower is the skewness and 2012). 2.5.1.2. Edge density The texture roughness can be estimated based on the number of edges or ridges per unit area. In the processing scheme of a moving neighborhood, edge direction and magnitude properties induce larger-scale structure measurements in the scene (Flower 2012). 2.5.1.3. Fractal dimension The term “fractal form” refers to the shape and structure of natural formations, such as living tissues, which cannot be generated from simple formulae characterized by a few sizes, such as the length of edges or radii. Instead, a fractal form is described by recursive, scale-independent relationships. Unlike Euclidean shapes having an integer dimension, a fractal form has a continuous fractal dimension, D, with an upper dimensionality limit of the space in which it is embedded (Flower 2012). The following expression can be used to approximately measure the fractal dimension (Mandelbrot 1983): =− where

( )

denotes a line measured at scale and ( ) is its apparent length.

[2.6]

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Figure 2.2. Examples of grayscale histogram textures: (a) dispersion, (b) variance, (c) asymmetry and (d) kurtosis (Flower 2012)

2.5.2. Edges and boundaries To test first-order spatial derivative large values, first-order operators are used, where edge-detection algorithms are implemented using two and . Sobel and orthogonal convolution masks or spatial filters: Prewitt are common voxel edge-detection spatial masks (Figure 2.3), generating almost identical output. Two intermediate images, and , are used for the implementation, which are generated by mask convolution with the original image, (Flower 2012): =

×

[2.7]

=

×

[2.8]

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At each voxel, the edge gradient magnitude, , and direction obtained using the following formulae (Flower 2012): =

(

= tan

) +(

)

are then

[2.9] [2.10]

Second-order derivatives are estimated using second-order operators, such as the Laplacian, which are useful in the case of smooth- and long-range edges (Figure 2.3). However, these operators are characterized by a higher sensitivity to noise than the first-order operators (Flower 2012). Here, a single mask is convolved with the image and the identified zero-crossing points.

Figure 2.3. Examples of edge detection: (a) Sobel filter magnitude, (b) Sobel filter angle, (c) Prewitt filter magnitude, (d) Prewitt filter angle and (e) Laplacian secondorder filter. The curved edge of images (b) and (d) is caused by a slight nonlinearity of the edge of the imaging coil, which is a common feature of the hardware and is only revealed when the background is non-zero (Flower 2012)

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2.5.3. Shape and structure Segmentation allows the generation of binary images used to derive shape and structure measurements. These features describe the properties of localized objects and not the voxels that form them (Flower 2012). 2.5.3.1. Geometric features Geometric features represent effective gross morphology descriptors. Perimeter and area, as well as minimum and maximum radii from the center of mass to the boundary are well-known and easily calculated properties. Compactness is also an important parameter and takes a minimum value of 1 for a circle (Flower 2012). 2.5.3.2. Object moments In a binary image, the central moments of an object, follows (Flower 2012): ,

=∑ ∑ [ − ] [ − ]

, ∈

, are defined as [2.11]

where [ , ] denotes the center of mass of . Using this double moment sequence, only can be determined. The moments of inertia ( + = 2) are combined to calculate the angle, , of the axis of the least moment of inertia (Flower 2012): = 0,5 tan

, ,

,

[2.12]

The smallest rectangle enclosing the object, , is the bounding rectangle, which is aligned with the angle . The coordinates are rotated + and ∗ = − + ) on each point of the (∗= ∗ ∗ boundary and the minimum ( and ) and maximum ( ∗ and ∗ ) ∗ ∗ ∗ values are recorded. The rectangle lengths are − and − ∗ , and the ratio of the sides provides a further value. A comprehensive and succinct representation can be formed based on the features derived from moments, and used to manipulate and compare objects (Flower 2012).

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2.5.3.3. Object skeleton The object skeleton is a compact representation of the object form. It is composed of lines or curves which allow the reconstruction of the original object. Several features can be determined from the skeleton, such as the skeleton total length, the line segment number and the node number, in addition to more complex measures obtained based on the graph theory. The skeleton can be obtained using several methods, including morphological processing and distance transform. The distance transform is used in the case of a binary mask image, with a zero value for the background and an infinite or suitably large voxel value for the boundary of the object. The distance transform may be calculated using an algorithm with two sequentially applied “half masks”, one operating from the bottom to the top of the raster image, while the other operating in the opposite direction. Morphological processing is based on the application of two operations on the object in a binary image: the first corresponds to the structure element and the second corresponds to the element translation to the selected location. These two basic operations are singularly used to contract or expand the object boundary. They can also be used in combination for various operations (Flower 2012). 2.6. Image segmentation Segmenting a medical image consists of partitioning this image into several regions and connecting its features (and therefore voxels) to form objects. These objects are then identified as specific organs or tissues. Two segmentation approaches are generally used: (1) a bottom-up processing approach, aggregating features by classification in order to form regions, and (2) a top-down processing approach, requiring target object specification by locating, scaling and distorting these objects, in order to better correspond them to local features (Flower 2012). 2.6.1. Simple methods of image segmentation The simplest segmentation method is thresholding, in which the grayscale image is converted to a binary image based on thresholds. It is therefore

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important to choose an appropriate threshold value (Zhou 2015). An example of segmentation by thresholding using an MRI is shown in Figure 2.4.

Figure 2.4. Original MRI of slices 62 and 90 with user-defined initial pixels (top) and segmentation results using functions T1 and T2 (bottom) (Mabrouk et al. 2019)

Figure 2.5. MRI before (a) and after (b) the application of K-means clustering (Ng et al. 2006). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

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Clustering is often based on the use of the K-means algorithm, in order to assign a voxel or pixel to one of the K cluster labels, to which its distance is minimal. Using all current pixels that belong to the cluster, the centers of the latter are recomputed until convergence (Zhou 2015). Figure 2.5 shows an MRI before and after the use of K-means clustering. Region growing functions are under the assumption that, within one region, the neighboring voxels/pixels share similar values. Starting from a group of seed voxels/pixels, the regions undergo an iterative growth, by merging new neighboring unallocated voxels/pixels into the region if the unallocated voxel/pixel is sufficiently close to those that belong to the region (Zhou 2015). 2.6.2. Active contour segmentation The active contour model, often called the snake, is a dynamic structure that was formally introduced in 1987 by Kass and Witkin (Kass et al. 1988), and used as a segmentation method to find a parameterized curve ( ) that ( ): minimizes the cost function ℰ ℰ

( )= − ∇

( )

+

( )| ( )| +

( )|

( )|

[2.13]

where ∇ denotes the gradient operator, denotes the image, denotes the potential amplitude controlling parameter, and ( ) and ( ) respectively denote the curve tension and rigidity controlling parameters (S. K. Zhou 2015). The snake model implicitly assumes that the curve is defined by the edge, owing to gradient operator use. The force exerted on the snake is computed by minimizing the gradient descent. This force is defined as the negative of the energy field gradient, due to which the curve evolves. The gradient vector flow snake model (Xu and Prince 1998) and the geodesic active contour (Caselles et al. 1997) are two major variants of the active contour model. Figure 2.6 shows the results of an enhanced active contour model.

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Figure 2.6. Development process from the initial contour curve (the first column) to the final contour curve (the last column), from left to right, using an enhanced active contour model (Zhao et al. 2018). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

2.6.3. Variational methods The variational method introduced by Mumford–Shah (Mumford and Shah 1989) allows the study of the problem of minimal partition, by finding and , which minimize the cost function a curve and two constants ℰ ( ) (Zhou 2015): ℰ

( )=

ℒ( )

| ( , )−

|

+

| ( , )−

|

+ [2.14]

where Ω and Ω are respectively the interior and exterior regions, in relation to the curve , and are the piecewise constants for both of these

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regions, and ℒ( ) is the curve length. This method assumes homogeneous regions (Zhou 2015). 2.6.4. Level set methods Image segmentation using a level set method is performed based on a level set function , also called a level curve function, which implicitly represents the shape, while considering its boundary as the zero level set of the function . The use of only one function allows the segmentation of only two regions. Using the level set model (Osher and Sethian 1988), tractable numerical computations involving curves and surfaces on a fixed Cartesian grid can be performed, and following shapes with changed topology can be easily monitored. A model was proposed by Chan and Vese (2001) to combine variational methods and level sets into a single framework, resulting in an active contour evolution, without explicitly depending on the image edges (Zhou 2015). 2.6.5. Active shape and active appearance models Active shape models (Cootes et al. 1995) and active appearance models (Cootes et al. 2001) are two of the most frequently used methods in model-based segmentations, in which an offline learning and an online fitting of a model to an unseen image is performed. In active shape models, a shape model is learned, using a point-based representation, and fitted by finding a line for every point, making it possible to deform and thus provide the best matching between the shape and the image evidence. Then, the deformed shape is constrained to meet the learned statistical shape model (Zhou 2015). Furthermore, active appearance models include the image shape and appearance in the statistical model. A linear generative model is used to jointly characterize the shape and the appearance (Zhou 2015): = ̅ +

[2.15]

=

[2.16]

+

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where ̅ is the mean shape, is the mean appearance in a normalized patch, and is the blending coefficient vector shared by both the appearance and the shape. Fitting the parameters of the active appearance models requires the use of an analysis-by-synthesis approach, where the deviation from the assessed appearance is minimized and parameterized by the active appearance models and the target image. This optimization is dictated by the difference between the target image and the current appearance estimate (Zhou 2015). 2.6.6. Graph cut segmentation A graph = ( , ), including a set of vertices or nodes, in addition to a set of edges or links, each two vertices being linked by an edge, is used to perform a graph-based segmentation, in which the grid points of the image are usually considered as nodes in a graph, and edges are used to connect the neighboring voxels/pixels. Hence, the segmentation process becomes a graph cut problem, labeling the nodes with different labels and splitting the graph into several subgraphs. A binary labeling function = | ∈ is defined to label all the pixels in the image as 0 or 1. From a mathematical perspective, the graph cut problem finds the optimal binary function that can minimize the energy function expressed as (Zhou 2015): ( )=∑

∈

(

) + ∑(

, )∈

,

(

,

)

[2.17]

( ) denote the unary term data that determines the assignment where cost of the pixel to a label , and , denotes the pairwise interaction function that allows the assignment of neighboring pixels having similar properties to the same label (Zhou 2015). 2.6.7. Atlas-based segmentation The specific case of brain atlas deformation is known as atlas-based segmentation, allowing the creation of a new individualized brain atlas. This segmentation method is based on the use of a pre-existing reference MRI, in which the structures of interest have been previously segmented (labeled image). A non-rigid registration is then performed between the individual specific MRI and the reference image (Cuadra 2003). The resulting transformation codes a voxel-by-voxel correspondence between the two

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MRIs, which can be applied to the reference MRI in order to seek the individual structures of interest (Figure 2.7) (Zhou 2015). Based on this image segmentation method, several works have been carried out, such as the automatic organ segmentation performed using a probabilistic abdomen atlas, based on non-contrast CT scans of the abdomen (Park et al. 2003), the multiple abdominal organ segmentation performed by combining an atlas with fuzzy connectedness (Zhou and Bai 2007), as well as the gallbladder, vena cava and liver automatic segmentation using CT scans and constructing hierarchical, multi-organ statistical atlases (Okada et al. 2008). Moreover, a statistical atlas was used to label anatomical structures in magnetic resonance fast view scans (Fenchel et al. 2008), the missing organ problem was addressed in multi-organ segmentation by including a detection module for explicitly missing organs in an atlas-guided segmentation solution (Suzuki et al. 2012), and the atlas approach was further extended to create hierarchically weighted atlases for a specific subject, allowing multi-organ segmentation based on abdominal CT scans (Wolz et al. 2012).

Figure 2.7. Atlas-based segmentation process (Cuadra 2003). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

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2.6.8. Deformable model-based segmentation Deformable models make it possible to adapt a curve in order to maximize its overlap with the real contour of an object of interest in the image to be analyzed. The term “deformable model” refers to surfaces or curves defined in the image domain and deformed under the influence of (1) “internal” forces related to specific curve features, imposing regularity constraints while maintaining model smoothness during deformation, and (2) “external” forces corresponding to the regions surrounding the curve and defined so that the model is attracted to an object or other features of interest in the image (Mesejo et al. 2016). Deformable models can be subdivided into two major categories: (1) parametric/explicit deformable models, which explicitly represent surfaces and curves in their parametric forms during deformation, thereby allowing direct interaction with the model and rapid implementation in real time through compact representation, and (2) geometric/implicit deformable models based on the curve evolution theory (Malladi et al. 1994; Adalsteinsson and Sethian 1995; Sethian 1999), which implicitly represent surfaces and curves as a set of levels with a higher dimension scalar function and make it possible to naturally manage topological changes (Mesejo et al. 2016). In this context, several approaches have been developed, including the 3D deformable models presented by Yang et al. (2004) and the fully automatic method presented by Costa et al. (2007). 2.6.9. Energy minimization-based segmentation This image segmentation method can be performed using (1) an energy function that features the competition between neighboring shape models, in order to incorporate prior shape information and deformable model interactions, while avoiding overlapping (Yan et al. 2005), or (2) an energy minimization method that allows the extraction of 12 organs using non-contrast 3D CT scans of the abdomen (Shimizu et al. 2007). The energy function takes into account the uniformity of gray values, as well as the hierarchy and exclusiveness between organs (Zhou and Xu 2016). 2.6.10. Learning-based segmentation Computer vision learning refers to the development of statistical methods based on an initial database, in order to perform tasks that are too complex to

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solve using more traditional equations and algorithms (Julian 2012). Learning-based marginal space learning models have been used to automatically detect organs from CT scans (Kohlberger et al. 2011). The learned boundary detectors are leveraged to segment several organs, and the level set optimization is invoked to refine organ segmentation. Moreover, a similar learning-based approach has been developed, but followed by an information theory-based boundary deformation, making it possible to perform the final segmentation of the rectum, the bladder and the prostate from CT scans (Lu et al. 2012). 2.6.11. Other approaches A 4D graph was constructed and the graph cut algorithm was used to allow a simultaneous segmentation of four abdominal organs, based on contrast-enhanced CT scans (Linguraru et al. 2010). In contrast, graphical deformable models were integrated to carry out multi-organ segmentation in a collaborative manner (Uzunbaş et al. 2013). In addition, a more accurate organ segmentation was achieved by combining the segmentation process with the registration process (Lu et al. 2010). 2.7. Image registration Image registration is an iterative process that makes it possible to find the geometrical transformation that aligns or maps an object with a coordinate to an object with a different coordinate. The aim is to determine a corresponding functional or anatomical position in at least two images, in order to obtain as much information as possible. The registration process is defined as the unambiguous correspondence between the coordinates of one space and those of another so that the points of two spaces, referring to the same anatomical point, are mapped to each other. The main purpose of image registration is to detect changes in the same types of images. The source image is transformed into a target image, the measures of similarity between them are computed, and the resulting image is generated after fulfilling the measures of similarity. The process is carried out several times, in order to optimize the transformation parameters in the case of a non-perfect alignment of the similarity measure between the source image and the target image. 2D and 3D source and target image mapping can be expressed as follows (Alam et al. 2018):

Medical Image Analysis and Processing

( ,

)=

( ,

,

)=

( , ) ( , , )

67

[2.18] [2.19]

where denotes a fixed target image, is a moving source image, g is an intensity mapping function, is a special transformation. , , and , , respectively denote the source image and the target image coordinates used for 2D image transformation, whereas , and , respectively denote the source image and the target image coordinates used for 3D image transformation (Alam et al. 2018). The image registration algorithm generally includes four steps, on which depend the registration outcome: (1) feature detection, (2) feature matching, (3) transform model assessment and (4) resampling and transformation (Alam and Rahman 2016). Over time, several registration methods have been developed, which can be categorized according to nine criteria (Mani and Arivazhagan 2013). 2.7.1. Dimensionality The dimensionality criterion can be spatial or temporal. 2.7.1.1. Spatial dimensions The registration process consists of computing a transformation between the image coordinate systems or between an image and a physical space. Spatial dimensions thus refer to the number of space geometric dimensions, which can be (Mani and Arivazhagan 2013): – 2D-to-2D, which is used to align 2D slices by performing one rotation and two orthogonal translations, with the possibility of correcting the changes in scale; – 3D-to-3D, which is used to accurately register tomographic datasets, or to register a single tomography image to a spatially defined information, by performing 3D rigid registration that consists of three translations and three rotations, assuming that the organ of interest behaves like a rigid body and that the spatial relationship between the internal organs of the body has not altered or distorted;

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– 2D-to-3D, which is mainly used in image-guided surgery, where the registration process allows the establishment of the correspondence between 2D projection images and 3D volumes, when the position of one or more slices must be established relatively to a volume. 2.7.1.2. Registration of time series Image registration obtained during different time periods makes it possible to monitor the evolution of a given disease and evaluate the response to a given treatment, thereby providing a tool for studying dynamic processes, including physiological or metabolic processes (Mani and Arivazhagan 2013). 2.7.2. Nature of the registration basis Depending on the nature of the registration basis, the methods used can be either extrinsic, when they are based on foreign objects introduced into the imaged area, or intrinsic, when they are based on image information produced by the patient himself/herself (Mani and Arivazhagan 2013). Extrinsic registration methods consist of attaching artificial objects to the patient. These objects must be clearly visible and precisely detectable in all the imaging modalities. Extrinsic registration is computationally efficient and easily automated, without requiring complex optimization algorithms. However, patient-specific image information is not included, and the registration nature is generally restricted to rigid transformations. In addition, images with low spatial content require additional spatial information to carry out the registration process (Mani and Arivazhagan 2013). Intrinsic registration methods consist of using information provided by the patient, such as image voxel intensities, binary divided structures or apparent prominent landmarks. The intrinsic methods can be subdivided into the following: (1) landmark-based methods, which consist of identifying salient elements in one image, such as curves, surfaces and point landmarks, and matching them with their corresponding elements in the other image, thereby ensuring the biological validity of mapping; (2) segmentation-based methods, which use either rigid models, where the surfaces extracted from both the source and target images represent the input to the registration process, with a relatively easy segmentation, or deformable models, where the surfaces or curves extracted from one image make it possible to fit the

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other image through an elastic deformation, which complicates the method in terms of cost function regularization; (3) voxel property-based methods, which consist of matching the intensity patterns in each of the two images using mathematical or statistical criteria, where the transformation is adjusted by measuring the similarity of the intensities of the input images, but these methods do not take into account the spatial dependency of the pixels, which results in a failure of the measures against the distortion corruption of the spatially varying intensity when the registration of the two images is required; and (4) hybrid-based methods, which combines intensity and geometric features in order to produce more robust methods capable of establishing more accurate correspondences in the case of a more complex registration (El-Gamal et al. 2016). 2.7.3. Nature of the transformation The transformation may also be classified according to its nature (Figures 2.8 and 2.9). 2.7.3.1. Rigid Rigid object images can be sufficiently registered using only translations and rotations, for example in the case of bone or brain image registration when neither the skull nor the dura is opened. In addition, rigid registration makes it possible to approximately align images showing slight changes in the intensity of the shape of the object, occurring in the time series of fMRIs or in successive histological sections. The global rigid transformation is a popular registration technique, because a good approximation is provided by the rigid body constraint. In addition, few parameters need to be determined and most of the techniques do not allow more complex transformations (Mani and Arivazhagan 2013). 2.7.3.2. Affine Affine transformation provides line parallelism preservation, without preserving line lengths or angles. This technique also extends the degrees of freedom of rigid transformation with a scaling factor and additionally a shearing in each image dimension (Mani and Arivazhagan 2013). It can be used to rectify calibration errors in the voxel dimension (Viola and Wells III 1997; Jenkinson and Smith 2001), as well as to measure the relative scaling error between scans (Holden et al. 1999).

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2.7.3.3. Projective Projective transformation is used in the case of a tilted scene, making it possible to maintain straight lines, but parallel lines converge towards vanishing points. Indeed, this technique has no real physical basis, except in the case of a 2D/3D registration. In the case of a fully elastic transformation with an inadequate behavior or a large number of parameters to be solved, projective transformation can be used as a “constrained-elastic” transformation (Mani and Arivazhagan 2013). 2.7.3.4. Curved Based on computer vision, several algorithms have been adapted and used over time. For example, these algorithms make it possible to match the crest based on a combination of geometric hashing and the Hough transform, match experimental data of 2D images with their corresponding images superimposed on a curved surface within a volumetric image, or perform inter-modality matching of sectional data with a volumetric image of homologous objects (Mani and Arivazhagan 2013). 2.7.3.5. Non-rigid Today, the most interesting registration works are carried out based on non-rigid techniques, for applications including tissue deformations, modeling and anatomical structure variability. Non-rigid registration remains a challenging research subject, because it requires smoothness and a large number of degrees of freedom in the deformation process. Several algorithms have been developed to perform nonlinear registration of medical images. However, a large computation time is required, which is a major disadvantage in many clinical applications. Therefore, further research is needed to improve accuracy, increase speed and enhance the evaluation of the registered results (Mani and Arivazhagan 2013). 2.7.4. Transformation domain The transformation of image coordinates can be global or local. The former concerns the entire image, with mapping function parameters being valid for the whole image. The latter concerns a small part of the image, with local mapping function parameters being only valid for a small region or patch around the selected control point location (Mani and Arivazhagan 2013).

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Figure 2.8. Examples of rigid transforms: (a) reference image, (b) similarity transform, (c) affine transform and (d) projective transform of the target image (Khalifa et al. 2011)

Figure 2.9. Example of local (non-rigid) deformation (Duy 2009)

2.7.5. Interaction This concerns the registration algorithm control by human experts. Three interaction levels can be distinguished: (1) the first level corresponds to interactive algorithms, in which a specific software is used to provide an initial transformation parameter estimate, making it possible to carry out the registration process, (2) the second level corresponds to semi-automatic algorithms, where interaction can be achieved by initializing the algorithm parameters based on data segmentation, or by steering the algorithm towards the desired solution based on adjustments made throughout the process according to the visual evaluation of the alignment, and (3) the third level corresponds to automatic algorithms which operate without any interaction (Mani and Arivazhagan 2013).

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In some methods, the user interaction narrows down the search space, rejects the mismatch and accelerates the optimization process, yet the lack of interaction level quantification or control can complicate the validation process with further human interaction. Extrinsic methods are often used with automatic algorithms, due to the visibility and ease of detection of the used markers. However, the user has to determine an initial location or provide a seed point when required. In contrast, anatomical landmark and segmentation-based methods are considered semi-automatic as the process must be initialized by the user (Mani and Arivazhagan 2013; El-Gamal et al. 2016). 2.7.6. Optimization procedure Optimization techniques are used to optimize the transformation parameters needed for image alignment. For example, transformation parameters can be quickly and reliably determined using good optimization algorithms. In non-rigid registration applications, the difficulty of designing or choosing an optimizer increases with the flexibility (or non-rigidity) of the transformation model because of the increase in the number of parameters necessary to describe the model. Thus, the time required for the optimizer to determine the parameters increases and, because of the local minima problem, the probability of choosing a set of parameters, providing a good image match but not the best increases. The transformation parameters can be explicitly determined from the available data or searched by finding an optimal cost function defined on the parameter space. Due to this cost function, the similarity between the source and target images can be determined based on a certain transformation. In the case of monomodal registrations, the use of objective functions decreases the complexity of the problem, due to the linear relationship between the two images, as well as the simplicity of the similarity metric. Moreover, the cost function includes explicit regularization terms for smoothness and diffeomorphic constraints to preserve topology (Mani and Arivazhagan 2013). 2.7.7. Modalities involved Depending on the involved modality, the registration process can be subdivided into four tasks: (1) a monomodal task, which is applied between images obtained with the same medical modality, (2) a multimodal task,

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which is applied between images obtained with different modalities, (3) a modality-to-model task, which helps in tissue morphology through gathering statistics, and (4) a model-to-modality task, which is often applied in intraoperative registration techniques. For the third and fourth tasks, only one image is involved and the other registration input corresponds to the model or even the patient. Monomodal and multimodal procedures are the two most well-known and frequently used tasks. Monomodal tasks assist in applications dealing with subtraction imaging, verifying interventions, monitoring growth and comparing rest–stress, while multimodal tasks assist in a large number of applications generally used for diagnosis and mainly performed as anatomical–anatomical and functional–anatomical processes. The difference between these two processes is that anatomical–anatomical registration makes it possible to combine images showing different tissue morphology sides, while functional–anatomical registration makes it possible to link tissue metabolism and its associated spatial location, while respecting anatomical structures. In addition, the multimodal task plays an important role in many medical applications, such as image analysis, object recognition and 3D image reconstruction. Despite the difficulty of the multimodal procedure, caused by the use of images of different modalities which present a very high degree of dissimilarity, the resulting registered images provide physiological and anatomical information that can facilitate clinical diagnosis and treatment (El-Gamal et al. 2016). 2.7.8. Subject The patient from whom the medical images are obtained forms another criterion for classifying medical image registration methods. The latter can therefore be (1) intra-subject, when all the involved images are obtained from a single patient, i.e. from identical modalities, (2) inter-subject, when the images are obtained from different patients, or (3) atlas, when one image is obtained from a single patient and the other image is constructed from a database of image information obtained from the imaging of several subjects (Mani and Arivazhagan 2013). The registration of an image of a patient on an image of a normal subject is known as atlas registration. Intra-subject registration allows the detection of structure intensity or shape changes. It is frequently used in diagnosis and surgery, but is mainly used to align serial brain MRIs. In contrast, inter-subject registration is often used to determine size, shape and grosser topology changes. In turn, atlas

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

registration makes it possible to obtain statistics about a particular structure, shape and size and therefore to identify anomalous structures, which can be used to transfer the segmentations from one image to another (El-Gamal et al. 2016). Medical image registration can be carried out using the neural network method, genetic algorithms or fuzzy ensembles. 2.7.9. Object In the medical image registration process, the object corresponds to the involved anatomy part. Objects can be classified into several categories: limb, head, thorax, abdomen, vertebrae and spine, each of which can be further divided into subcategories (see Mani and Arivazhagan 2013 for more detail). 2.8. Image fusion The image fusion process consists of combining information from different images to form a single image or combining some of their features into a single image, in order to optimize its diagnostic value. The resulting image offers a more accurate description of the source image, and better fits the human visual system or further image analysis procedures. Image fusion is used to generate additional clinical information that cannot be revealed when using separate images, in addition to reducing storage costs by only recording the fused image instead of recording all images gathered from different sources. Several methods have been suggested to perform the fusion process. 2.8.1. Pixel fusion methods These methods are based on the use of simple pixel-to-pixel operations involving sophisticated operators, such as the Markov random field, but also simple arithmetic operators, such as addition, subtraction, multiplication and division, as well as a maximum, minimum, median and rank. However, these methods have certain limitations, including the image contrast reduction, but the obtained results are satisfactory in some cases, such as input images with high overall brightness and contrast (El-Gamal et al. 2016).

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2.8.2. Subspace methods Subspace methods represent a set of statistical techniques used to eliminate the correlation between input images. In other words, a high-dimensional input image is projected onto a lower-dimensional space or subspace in order to (1) facilitate understanding of the intrinsic structure of input data, (2) ensure better generalization resulting from the image projection into smaller dimensions, (3) speed up the image processing with lower-dimensional spaces and less memory, and (4) construct a model from data representation in the lower dimensions (El-Gamal et al. 2016). Well-known examples of subspace methods are principal component analysis, independent component analysis, linear discriminant analysis, canonical correlation analysis and non-negative matrix factorization (Mitchell 2010). 2.8.3. Multi-scale methods Multi-scale or multi-resolution image analysis consists of transforming each input image ( ) in order to represent it in a multi-scale manner, ( ) ( ) ( ) , ,…, , using a set of techniques such as those used in the i.e. discrete wavelet transform, the undecimated discrete wavelet transform and the dual-tree complex wavelet transform. To show this graphically, the decomposed sequences of the images can be arranged in a pyramid where the bottom contains an image identical to the input image . Then, at the are reconstructed by applying a low pass following levels, the images filtering, along with subsampling of the image − (El-Gamal et al. 2016). 2.8.4. Ensemble learning techniques Ensemble learning aims to build accurate predictors or classifiers by assembling weak predictors or classifiers. In the image fusion context, ensemble learning corresponds to the fusion of images , ∈ ∗ , , … , , all derived from the same base image . This makes it possible to substantially improve the fused image quality. The simplest way to generate images is to apply different transformations to the base image ∗ . To provide an effective ensemble learning, the images must be

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independent and must highlight different characteristics in et al. 2016).

∗

(El-Gamal

After the demonstration, the spatial and temporal alignment of the images , feature images or decision maps from the base image ∗ , and the pixel-based fusion operators, such as trimmed or arithmetic mean, can be and the feature images. Other methods, such as a applied on the images majority vote, can be used for the decision maps (El-Gamal et al. 2016). 2.8.5. Simultaneous truth and performance level estimation The simultaneous truth and performance level estimation (STAPLE) algorithm is the category in which the expectation–maximization algorithm is used as the basis for image fusion in the case of a large number of segmented images (Mitchell 2010). Indeed, the expectation–maximization algorithm makes it possible to evaluate the maximum probability of an underlying distribution of a given set of incomplete data on a powerful iterative basis. Based on the STAPLE algorithm, the expectation– maximization algorithm makes it possible to obtain an iterative estimation of individual segmentation quality. The latter is then taken into account when computing the final segmentation by weighing the decisions made by a more reliable segmentation algorithm than those made by a less reliable algorithm (El-Gamal et al. 2016). 2.9. Image understanding Image understanding refers to the interpretation of the elements identified within the image. It allows the construction of a statement about the scene, by integrating the co-registered images, the image and object features, and the “world knowledge”. Decisions can then be made following this stage (Flower 2012). 2.10. Conclusion Digital image data have transformed radiological practice by enabling the introduction of computer-based approaches, and particularly image processing, to assist (or, as some even claim, to replace) the radiologist. The automation and reliability accompanying the use of computers is recognized

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as beneficial in a number of clinical settings with a particular potential for health screening programs, where a large number of cases need to be accurately and rapidly investigated. This also occurs in epidemiological imaging studies. To demonstrate significant atypical or pathological differences (which are often minimal), statistical power lies in the use of large cohorts. Therefore, automation with image processing becomes attractive in terms of reliability and ease of processing.

3 Recent Methods of Constructing Finite Element Models Based on Medical Images

3.1. Introduction In recent decades, finite element (FE) models have been widely used in the fields of biomechanics and engineering in regenerative medicine. The FE modeling of human bones has provided a potential tool for parametric and reproducible evaluation with a wide variety of results. These models can be reconstructed from medical images, mainly using one of the imaging techniques presented in Chapter 1 and based, overall, on the analysis and processing procedures presented in Chapter 2. This chapter presents a brief review of some recent work that has reconstructed FE bone models based on medical images. 3.2. X-ray-based finite element models In a study carried out by Luo et al. (2011), a patient-specific 2D FE model was developed using femur DXA images, in order to assign material properties and apply stress/strain conditions. The FE mesh was generated using the outermost contour of the DXA image. The following empirical functions were used to correlate bone mechanical properties (elasticity modulus E, normal yield strength σ and shear yield strength τ ) with volumetric bone mineral density (vBMD), ρ : = 2838 ×

.

Finite Element Method and Medical Imaging Techniques in Bone Biomechanics, First Edition. Rabeb Ben Kahla; Abdelwahed Barkaoui and Tarek Merzouki. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

[3.1]

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

= 26.9 ×

.

[3.2]

= 56.3 ×

.

[3.3]

where areal BMD (aBMD), ρ was obtained based on the femur DXA image. A standard critical loading profile (SCLP), describing all the main static forces that could cause a hip fracture in an osteoporotic individual, was defined to mimic a sideways fall. The loading/constraint profile (Figure 3.1) was adopted, assuming that the impact lateral force was proportional to the body weight and that the distance between the top of the femoral head and the constrained cross-section was proportional to the total length of the femur. The process of building the FE model for this study (Figure 3.2) can be summarized as follows: (i) extraction of the outermost contour of the proximal femur from the DXA image, (ii) generation of a 2D FE mesh from the contour, (iii) assignment of material properties based on the above empirical functions and the aBMD extracted from the DXA image, (iv) application of the SCLP and (v) determination of 2D strain/stress fields using FE analysis. The nearest-nodes FE method was used to deal with the distorted FEs.

Figure 3.1. Loading/constraint conditions used to assess the risk of femur fracture (Luo et al. 2011). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

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Figure 3.2. Process for constructing a DXA-based FE model: (a) contour of the proximal femur extracted from a DXA image; (b) 2D FE mesh generated from the contour; (c) assigned material property (Young’s modulus); (d) application of load/stress conditions; (e) distribution of deformation (Luo et al. 2011). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

In one study, L. Yang et al. (2014) carried out a linear-elastic plane stress FE analysis based on DXA scans. Pixel-by-pixel BMD maps obtained from these images were used to analyze the hip structure and identify the proximal femur of cadaveric trabecular samples using an image processing algorithm that combined edge detection and thresholding, followed by manual addition or removal. Each femur was considered to be a plate with a thickness t specific to the patient, such as t = 3,5πW⁄16 , where W is the average thickness of the middle third cross-sections of the femoral neck on the BMD map. The aBMD was converted into the vBMD, i.e. vDMO = aDMO⁄t . In turn, the vBMD was converted into apparent = vDMO⁄(1,14 × 0,598), from which the material density (ρ ), i.e. ρ properties were derived using the following empirical equations: =

,

15010 ,

6850

if

≤ 0,280 ⁄

if

> 0,280 ⁄

=

85,5

,

if

≤ 0,355 ⁄

38,5

,

if

> 0,355 ⁄

=

50,1

,

if

≤ 0,355 ⁄

22,6

,

if

> 0,355 ⁄

)

(

[3.4]

)

(

[3.5]

(

) [3.6]

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In order to take into account any side-artifact errors that occur during biomechanical cadaveric specimen testing and to determine the relationship between material properties and bone density, the former were multiplied by a factor of 1.28. A fall on the greater trochanter was simulated by subjecting it to a peak impact force as a function of body height and weight. The distal femoral shaft displacement of the femoral stem and the medial displacement of the femoral head were prevented (Figure 3.3). A linear-elastic analysis was performed without taking into account the post-yield behavior of the proximal femur. The estimated femoral strength was defined as the onset impact force causing von Mises stress in a contiguous area of 25 mm , within an anatomical region bounded proximally by the subcapital line and distally by a transverse line passing through the lesser trochanter distal end. This made it possible to exceed an apparent yield stress corresponding to the average yield stress in compression and tension. The ratio of von Mises stress to apparent yield stress was determined to generate a stress ratio map, after which a contiguous area of at least 25 mm containing the maximum stress ratio was identified. Noting , the minimum stress ratio in this area, the estimated femoral strength was determined by scaling the peak impact force of 1⁄β , since the FE analysis was linear-elastic and the stress was proportional to the applied force. This approach provided a successful definition of the fracture.

Figure 3.3. Loading conditions of the proximal femur used to simulate a sideways fall by subjecting the greater trochanter to impact forces, fixing the distal end and restraining the femoral head in the vertical direction. The distribution of the elasticity modulus is illustrated by the intensity of the DXA scan (Yang et al. 2014)

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In a study by L. Yang et al. (2018), an elastic-linear FE analysis was performed based on DXA images of a cadaveric trabecular specimen. A pixel-by-pixel BMD map was extracted from each image using a special program. These maps were used to segment the proximal femur and construct an FE model by converting each pixel into a brick element with four nodes at each corner. The material mechanical properties were assigned to each pixel: the proximal femur was considered to be a plate with a constant thickness t, i.e. t = 3,5πW⁄16, with W being the average thickness of the middle third cross-section of the femoral neck on the BMD map (Figure 3.4). The cross-sectional areas and moments of inertia were chosen to be as close as possible to the rectangular and anatomical circular cross-sections of the plate. The aBMD, ρ , was converted into vBMD, i.e. = ρ ⁄(1,14 × 0,598). ρ = ρ ⁄t, and then into apparent density, i.e. ρ The apparent density was used to calculate the elasticity modulus E based on the following empirical equations: =

,

15010 6850

,

if if

≤ 0,280 ⁄ > 0,280 ⁄

[3.7]

Side-artifact errors that occur during biomechanical specimen testing were taken into account by multiplying the material properties mentioned above by a factor of 1.28. Subsequently, the relationship between material properties and bone density was estimated by assigning the Poisson’s ratio to be 0.35 and simulating a lateral fall by applying a force of 500 N to the greater trochanter. A linear-elastic analysis was performed without taking into account the post-yield behavior, given the linearly elastic behavior of the proximal femur until failure. The femoral strength was defined based on the principal compressive yield strain ε : =

−

+

[3.8]

where ε and ε are the element normal strain along the and directions, and γ is the element shear strain, which is the deformation of elements by shear strength.

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For the femur trabecular bone, a compressive yield strain ε of 1.04% for each was used to calculate the strain ratio between ε and ε element. Following the identification of a contiguous area of 9 mm with the highest strain ratio, where femur fracture may occur or initiate, the femoral strength was calculated by dividing the applied force by the minimum strain ratio in that area.

Figure 3.4. The proximal femur plate model with a constant thickness (Yang et al. 2018)

A methodology for the automatic design of hexahedral FE meshes of the pelvis was developed by Fougeron et al. (2018), based on a 3D reconstruction of a patient-specific FE model using biplanar X-rays. EOS biplanar radiographs were used to derive images and reconstruct the pelvis in 3D, which distinguished the male pelvis from the female pelvis due to their anatomical differences. Generic FE meshes were designed based on generic anatomical atlases of the pelvic bones in both males and females. Non-uniform rational B-spline surfaces were generated over triangular surface meshes using Geomagic Studio, while closing the surface mesh at the entrance of the sacrum canal because of the complex shape of the sacrum. Ansys Mechanical APDL was used to virtually partition the pelvic volume into cubic interior volumes, based on the surface patches that were meshed with linear hexahedral elements. Using a dual kriging algorithm, the geometric deformation of the source mesh to the target mesh was computed to obtain a personalized mesh. Particular attention was paid to the sacral promontory, the acetabulum and the ischial tuberosities

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where higher control point density was assigned, and the patient-specific hexahedral mesh of the pelvis was defined by applying the kriging transformation (Figure 3.5). Excessive mesh spatial distortions were corrected by regularizing the generic mesh sizes, using an optimization method that minimized the errors identified using four quality metrics: parallel deviation, aspect ratio, Jacobian ratio, maximum angle and wrapping factor. A comparison was made between the personalized surface mesh and the initial 3D reconstruction surface mesh, and the point-to-surface distances were computed in high curvature zones, contact zones and the remaining zones.

Figure 3.5. Patient-specific mesh design: (a) EOS images and semi-automatic 3D reconstruction, (b) 3D reconstruction of the pelvic bone, (c) generic FE mesh, (d) kriging of the FE mesh obtained using the personalized 3D model (Fougeron et al. 2018). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

A study by Varga et al. (2015) evaluated and quantified the 3D structure of the lacuno–canalicular network of human cortical bone tissue, using synchrotron X-ray phase nano-tomography (SR-PNT). These geometric details were then incorporated into the FE models, in order to study local deformations of the extracellular matrix and osteocytes and to assess their relationship with the morphology of the lacuno–canalicular network. SRPNT imaging was carried out by first obtaining an overview image of each

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sample containing the whole cross-section of the cylindrical femur sample (Figure 3.6(d)). The region of interest was selected from this image, and the spatial distribution of the refractive index (δ) was reconstructed using filtered back projection (Figure 3.6(e)). The volumes of interest with individual osteocyte lacunae were selected from the youngest tissue domain of each nano-tomography image. The mass density and lamellar organization of all the regions in the image were used to estimate the relative age of local tissues. Each volume of interest was defined as a cuboid image region comprising the perilacunar bone matrix of a given thickness and an osteocyte lacuna at the center (Figure 3.7(a)).

Figure 3.6. Preparation and imaging of samples. (a) Location of the diaphyseal section within the proximal femur. (b) Cortical bone section in red and sample selected for imaging in blue. (c) Result of the final preparation of the sample of approximately 500 μm diameter milled down on the one end of the cuboid, and location of the overview image slice of sample C represented by the red rectangle. (d) The overview image slice of sample C, with a white scale bar of 100 μm, and the region of the final nano-tomography image indicated by the red circle. (e) Region of the final nano-tomography image, with a white scale bar of 20 μm, containing the outer osteon lamellae on the right side and two more mineralized regions on the left side, with each region being separated by thin cement lines with distinct grayscale intensities. The black areas of the image correspond to the pores of the osteocytes (Varga et al. 2015). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

In order to quantify the morphology of the lacuno–canalicular system and distinguish it from the mineralized matrix, the images were resampled and segmented. A connectivity enhancement algorithm was applied to avoid artificial local disconnections of the canaliculi caused by image artifacts, and the segmented image was then cleaned to retain only the largest pore containing the lacuna and the connected canaliculi.

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The refractive index value of each voxel was converted to units of mass density (ρ). The grayscale values were shifted by a global offset value, determined for each volume of interest individually as the mean density value within the central domain of the lacuna and gray values of all voxels were corrected accordingly. The median of the extracellular matrix mass density ( ρ ) was then determined from the volume-of-interest domain, without taking pores into account. A morphological opening operation was used to separate the canaliculi from the lacuna, followed by a morphology quantification of the two pore compartments (Figure 3.7(a), (b)).

Figure 3.7. Segmentation of the lacuno–canalicular network and generation of the FE geometry. (a) Sub-volume of the raw image corresponding to the volume of interest, represented with two perpendicular slices. (b) Mesh of the lacuno– canalicular network boundary, showing the lacuna in yellow and the canaliculi in red. (c) Mesh of the cell boundary extrapolated from the lacuno–canalicular network geometry, showing the cell body in green and the dendrites in turquoise. (d) Complete mesh of the volume of interest, including the cell, and pericellular tissue surrounding the cell body shown in yellow and the dendrites shown in red, while dividing the extracellular bone matrix into two parts, one corresponding to the direct vicinity of the lacuno–canalicular network shown in blue and the other corresponding to the part further away from the pore network shown in purple. A sub-volume of the extracellular and pericellular matrix was made transparent. The FE model boundary conditions are also shown in (d) and correspond to a prescribed displacement on the upper surface, the total constraint on the lower surface and the symmetry on the sides (Varga et al. 2015). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

The properties of the lacuna (e.g. volume, surface, aspect ratio, main orientation with respect to the long bone axis) and the canaliculi (e.g. volume, surface, volume normalized by the bone volume, diameter, spacing, number of canaliculi per lacuna, percentile canalicular lacuna interface). Three areas were considered in the FE models: (i) the extracellular matrix defined as the volume-of-interest domain

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without considering the lacuno-canalicular network, (ii) the osteocyte with the cell body and dendrites, and (iii) the pericellular matrix filling the space between the extracellular matrix and the cell. These domains were meshed together with linear tetrahedral elements using the 3D Delaunay triangulation with the CGAL library. The size of the elements was adjusted in accordance with the size of the local features, in order to minimize the element number, optimize their quality and minimize the loss of accuracy (Figure 3.7(d)). Homogeneous, isotropic and linear elastic properties were assigned to all materials. Young’s modulus ( E ) and Poisson’s ratio ( ) were respectively fixed at 16 GPa and 0.38 for the extracellular matrix, at 4.47 kPa and 0.3 for the cell and at 40 kPa and 0.4 for the pericellular matrix. The constrained uniaxial compression, caused by the displacement of each volume of interest, was simulated (Figure 3.7(d)). The proximal surface of the volume of interest was then subjected to a prescribed displacement along the femur axis, while the distal surface was fully constrained and the sides were symmetrically constrained. The loading direction effect was assessed by loading the volume of interest in three perpendicular directions, and three simulations were performed by applying the load along one of the three lacunar eigendirections, while symmetrically constraining the other sides of the volume of interest (Figure 3.8). Numerical simulations were performed using a standard Abaqus solver. The main deformations were assessed individually in the three domains of the FE model and studied as a function of distance from the lacunar boundary. In addition, the portion of the dendrite volume with high strain near the lacuna was quantified. A sub-model analysis was carried out for each volume of interest, in order to determine the influence of the size of the FEs on the results, and a cuboid sub-volume of the volume of interest was meshed with a constrained, much finer element size. The conditions at the displacement limits on the six sides of the sub-volume were determined from the corresponding displacement results of the total volume-of-interest model. A qualitative and quantitative comparison of the principal strain was performed between the sub-model and the corresponding part of the entire model for the extracellular matrix and the osteocyte. A MATLAB® algorithm was used to automatically carry out the various stages of image processing, meshing and post-processing.

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Figure 3.8. Boundary conditions used for the FE analyses, allowing the study of the loading direction influence on the strains surrounding the osteocytes. Application of the constrained uniaxial compression along the directions corresponding to the (a) longest, (b) middle and (c) shortest eigenvalue of the ellipsoid representing the approximate lacunar shape. The side opposite the loaded surface was fully constrained and the neighboring sides were symmetrically constrained (Varga et al. 2015)

3.3. CT-based finite element models The influence of the CT direction was numerically analyzed by Marques et al. (2014) based on the FE method. This study was carried out using a cadaveric adult mandible without teeth and soft tissue. CT scans were acquired in two different directions and used to build three FE models using the Scan IPw software: (i) the first model corresponds to CT scans acquired perpendicular to the -axis (Figures 3.9 and 3.10), (ii) the second model corresponds to CT scans acquired perpendicular to the -axis (Figures 3.9 and 3.10) and (iii) the third model corresponds to half of the second scanned model. Bone was considered to be linear elastic, and material properties were assigned based on the CT scan grayscale. In order to estimate the density of mandible models, the pixel scale value was linearly related to BMD as follows: = 0.0012 ×

+ 0.264

[3.9]

Young’s modulus was determined using the following formula: = 9.482 ×

[3.10]

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

For the three FE models, the meshing was carried out using Scan Few software with tetrahedral linear elements, the smoothing tool was applied with a recursive Gaussian filter, the numerical simulations were performed using the MSC MarcTM software and the obtained results for the three models were then compared. The boundary conditions were applied in such a way as to mimic the mandible behavior and the involved muscle forces, using six forces on each side of the mandible and taking three constraints into account. The condyles were fixed on the and axes and constraints were applied to the incisive tooth in the and directions.

Figure 3.9. Cadaveric mandible and the corresponding CT scans shown in two different directions (Marques et al. 2014)

Figure 3.10. FE model and boundary conditions (Marques et al. 2014)

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CT scans were used by Osawa et al. (2014) to analyze hip joint cartilage using the FE method and to accurately examine the stress distribution, taking into account articular cartilage inhomogeneity. 3D scans were used to extract the bone shape, and cartilage geometry was reproduced based on the bone surface. Figure 3.11 shows the protocol for creating a geometric cartilage model. The ZedView software was used to segment and visualize bone and cartilage in 3D. The region of interest surrounding the joint was selected and the contrast was adjusted to clearly show the cartilage geometry. The masked areas of the coxal bone, the cartilage and the femur were selected from the region-of-interest images and segmented by only extracting the masked areas maintaining continuity between the CT scans. The articular cartilage mask was then modified to fill the gap between the acetabular fossa and the femoral head. The 3D reconstruction of the different components of the FE model is shown in Figure 3.12. The reproduced cartilage geometric model was then discretized into eight-noded brick FEs. The laminar structure of the cartilage was expressed by dividing it into multiple layers, and a single bone element layer was provided for the subchondral bone layer of the femur or the acetabular fossa. The cartilage layers on the femur and the acetabular fossa were discretized into five layers, and inhomogeneous material properties were represented for each of these layers. Cartilage was modeled as a transversely isotropic composite material, with the transverse plane parallel to the articular surface and the bone–cartilage interface. The inhomogeneity and anisotropy of the cartilage microstructure were taken into account in the mechanical model used, allowing the assessment of the mechanical property distribution as a function of cartilage depth. The influence of histological characteristics on this depth was assessed by considering two patterns of the distribution of elastic properties on the cartilage cross-section: (i) an inhomogeneous distribution, for which the cartilage was considered to be a transversely isotropic elastic body, with the transverse plane being parallel to the articular surface and inhomogeneous material properties varying with depth, and (ii) a homogeneous distribution, considering the middle layer material property across the whole depth, for which the cartilage was considered to be an isotropic elastic material, with Young’s modulus of 0.45 MPa and Poisson’s ratio of 0.27 for the cartilage and Young’s modulus of 2000 MPa and Poisson’s ratio of 0.3 for the bone. The lower surface of the subchondral bone of the femoral side was fully constrained and a compressive load

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was applied to the surface of the coxal bone side (Figure 3.13), taking into account the conditions of the one-leg stance. This hip joint compression was simulated and analyzed using Abaqus software.

Figure 3.11. Protocol of CT-based FE modeling of the hip cartilage (Osawa et al. 2014)

Figure 3.12. 3D CT scan showing the femur, the coxal bone and the cartilage. (a) Horizontal view of the masked components. (b) Multi-valued image of each component. (c) Frontal 3D view of these components (Osawa et al. 2014). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

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Figure 3.13. Multilayered FE model of the joint cartilage and the boundary conditions for FE analysis (Osawa et al. 2014)

Using pre- and post-operative CT scans, Yamako et al. (2014) evaluated the load transfer model from the implant to the femur by performing an FE analysis and using an interface stress between the bone and the stem. The generation of the FE model (Figure 3.14) was carried out by constructing a pre-operative femur surface model and obtaining the post-operative femur and the rough surface of the implanted stem. The 3D construction of the surface models was carried out using semi-automatic segmentation with Wiener noise filtering and Canny edge detection. The ICP (Iterative Closest Point) algorithm was used with the leastsquare method to transform the post-operative femur coordinates to fit the pre-operative model. The rough stem model was also transformed. A computer-assisted design was used to construct a precise stem model. The latter was matched with the rough stem model that was transformed using the ICP algorithm (Figure 3.14(A)). A mesh with four-noded tetrahedral elements was generated for the proximal femurs, and coincidental nodes were generated for the bone–stem interface. Then, a customized FE model was obtained, showing the precise bone geometry, the actual stem position and BMD data (Figure 3.14(B)). Linearly elastic isotropic material properties were attributed to the stem and the bone. Young’s modulus E = 110 GPa and Poisson’s ratio = 0.3 were assigned to the stem, and Young’s modulus E = 6850 ρ . and the same Poisson’s ratio = 0.3 were assigned to the bone. Finite sliding face-to-face contact elements were used to model, and

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a penalty formulation was used to solve the contact interactions. The femur distal end was fully constrained, and static loads (representing the hip contact load with a set of simplified muscle forces) were applied to the FE models (Figure 3.15) in order to simulate the peak load when climbing stairs. Load transfer from the stem to the femur was studied by determining the stress at the bone–stem interface and dividing the interface area into three regions: distal, middle and proximal coating. The mean interface stress was determined for each region. The bone remodeling stimulus was evaluated by calculating the strain energy density, and FE models were analyzed using the Abaqus software. The scapular complex was numerically analyzed by Pomwenger et al. (2014) using a CT-based FE model. In order to carry out patient-specific modeling, a 3D FE model was established to analyze the effect of incorporating a CT-based BMD distribution on a glenoid implant and its primary fixation, instead of incorporating a uniform density distribution. The CT scans of the human shoulder complex were obtained and then segmented to separate the scapula bony structures from the soft tissue. Thresholding and region-growing algorithms were applied to the scans using Mimics software, in order to perform an initial segmentation, and manual refinement was required to carry out adequate segmentation of the trabecular and cortical bone, owing to the very thin sections of the cortical bone within the scapula, as well as to the low resolution of the CT scans. Then, the computed 3D FE model was imported into SolidWorks, in order to place the implant and perform the virtual surgery. The bone cement used for the glenoid component primary fixation was constructed, in addition to a humeral head component acting as a force transmitter on the implant surface. Further processing, such as bone density integration and boundary condition configuration, was then applied to the final 3D model (Figure 3.16) using the Ansys software. Equations linking bone density to Hounsfield units (HU) from the CT data and to Young’s modulus were used to take into account the bone density distribution in custom-made FE models. A homogeneously layered shell was defined as a compacta with a Young’s modulus of 17,500 MPa, in order to ensure cortical stability. A calibration obtained using linear regression was applied to provide a linear relation between bone density and HU: = 1.1187 ∙ 10

∙ HU

[3.11]

Recent Methods of Constructing Finite Element Models Based on Medical Images

95

Young’s modulus was then obtained using the bone density calculated by only addressing the open and closed trabecular bone cell structures: = 1049.45 ∙ = 3000 ∙

for

≤ 0.35 g⁄cm

for 0.35 ≤

≤ 1.8 g⁄cm

Figure 3.14. Patient-specific FE model. (A) Readjustment procedures. (B) BMD distribution (Yamako et al. 2014). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

[3.12] [3.13]

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Finite Element Method and Medical Imaging Techniques in Bone Biomechanics

Figure 3.15. Boundary conditions applied to the patient-specific FE models in the case of total hip arthroplasty (Yamako et al. 2014)

The maximum density value ρ = 1.8 g⁄cm was considered for a cortical bone and ρ = 0 g⁄cm in the absence of bone. The scapula was attached to the insertion points of the deltoideus, rhomboideus, trapezius, and serratus anterior and inferior muscles. The model was also fixed at the contact to the thorax, which was represented by a point near the angulus inferior, on the scapula dorsal side. The glenohumeral force measurement obtained by an instrumented implant was used instead of the forces obtained by numerical shoulder models. The force data were analyzed for 90° flexion and 90° abduction. The humeral head translation was initiated, and the force application point on the implant surface was eccentrically changed. A coefficient of friction μ = 0.07 was defined between the implant surface and the humeral head, and μ = 0.6 was defined at the interface between the bone and cement. Young’s modulus and Poisson’s ratio were respectively set at E = 500 MPa and μ = 0.4 for the implant, E = 2000 MPa and μ = 0.4 for the bone cement, and E = 230.000 MPa and μ = 0.4 for the humeral head component. A mesh with quadratic tetrahedral elements was generated for the FE model using the Ansys meshing size control method, in order to obtain finer mesh resolution at important areas such as the glenoid and the implant (Figure 3.17).

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Figure 3.16. Final 3D model showing a human scapula, a keeled implant, bone cement and a humeral head prosthesis used to apply the glenohumeral forces (Pomwenger et al. 2014)

Figure 3.17. (A) Glenoidal region FE mesh and (B) keeled streamlined implant (Pomwenger et al. 2014)

The stresses applied on the mandibular implants, abutments and bone surrounding the mandible were studied by Hussein (2015) using a 3D FE analysis based on CT scans. A 3D FE model of the mandibular arch, corresponding to an edentulous individual with a moderately resorbed residual ridge, was reconstructed from the CT scans, and four endosseous implants were created with their abutments and modeled using CAD design,

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followed by the design of the mandibular prosthesis. The CT scans were imported into Mimics software to create the 3D model of the edentulous mandible. SolidWorks software was used to design implants, abutments and prostheses in the form of a hybrid fixed–detachable prosthesis, including both the base and the teeth. Two anterior implants were placed bilaterally in the anterior area, and two angled ones (20°) were placed in the premolar simulating area (All-on-4). The models were then all imported into the Ansys software and used to perform the FE analysis. All components were considered to be homogeneous, linear and isotropic materials, by setting Young’s modulus and Poisson’s ratio at 110 GPa and 0.3 respectively for implants and abutments, 20 GPa and 0.3 for cortical bone, 2 GPa and 0.4 for trabecular bone and 200 GPa and 0.3 for prostheses. Meshing was carried out with tetrahedral elements (Figure 3.18). The nodes located at the condyle joint surface and the masticatory muscle attachment regions were all fully constrained. Boundary conditions were applied to simulate complete osseointegration. An oblique load was applied from the buccal to the lingual direction of the first lower molar.

Figure 3.18. (A) FE model assembly meshing with the different components. (B) Cross-section at the molar areas with 3D elements (Hussein 2015)

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A study by Pauchard et al. (2016) developed and released a highly automated tool to segment bone from CT scans. Its use in the retrospective prediction of proximal femur strength was then demonstrated using a sample cohort. The obtained results were compared to those of a manual segmentation. Proximal femur CT scans for normal, osteopenic and osteoporotic individuals were obtained. Manual image segmentation was carried out using an interactive method, combining region growth and manual correction, and the MITK-GEM analysis software was used to implement and integrate custom plugins. A 3D femur scan was processed in 2D axial slices from the proximal to the distal end, and the segmentation resulted in the creation of a voxel mask, defining the voxels belonging to the femur. This mask was converted to a closed surface using a Gaussian filter, followed by marching cubes, which provided a smooth surface with custom plugins integrated into MITK-GEM. The interactive segmentation method proposed in this study was based on graph cut, using the Boykov–Jolly algorithm: hard constraints established by the user were added to classify the image voxels to belong to either the object or the background, and the relationship between the neighboring voxels was encoded. This interactive segmentation was then implemented in C++ and integrated as an MITKGEM plugin. Similar to manual segmentation, the resulting 3D voxel mask was converted into a closed surface using a Gaussian filter, followed by a marching cube that provided a smooth surface with custom plugins integrated into MITK-GEM. The meshing of the smoothed segmentation surface was performed using ICEM CFD. In each pixel, the ash density (ρ ) was linked to the calibrated bone mineral content (mgHA) and the apparent density (ρ ) using the following equations: .

= =

[3.14]

.

[3.15]

.

Young’s modulus (E) and the ultimate stress (σ ) were determined as a function of ρ : = 6850(

)

.

[3.16]

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= 68 ∙ ( )

.

∙

[3.17]

with a constraint rate (ε): =

.

[3.18]

The trabecular bone yield stress (σ ) was defined as: =

.

[3.19]

A tension compression asymmetry ratio of 0.7 was implemented based on the Drucker–Prager yield criterion. At its ultimate stress, the strain was set at 2%, beyond which the response was considered perfectly plastic. Yield strain was defined by a 0.2% offset rule, and the proportional limit was defined as 80% of the yield stress. Poisson’s ratio was set at 0.3 . The numerical simulation boundary conditions were applied to mimic those that occurred during a sideways fall on the hip, aligning the FE models in accordance with their scanning position on the medial–lateral -axis of the image plane, defined as the impact direction. The greater trochanter region was supported in the drop direction and a free sliding was performed in the plane (Figure 3.19). In-house MATLAB® scripts were used to automate the FE model construction, and Ansys was used to solve the FE equations.

Figure 3.19. Coordinate system and boundary condition definition. The femur head and was subjected to a load applied at a distance ∆ set as 4% of H. The dimensions correspond to the distances between the most medial node of the femoral head and the most lateral node of the greater trochanter in the and directions respectively (left). Material property assignment sample (right). The curves show the stress–strain relationship in compression. A tension–compression asymmetry ratio of 0.7 was considered (Pauchard et al. 2016)

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Jackman et al. (2016) evaluated the accuracy of FE analysis based on quantitative computed tomography (QCT) scans, in order to predict vertebral failure patterns. In this study, the QCT scans were used to reconstruct FE models of the entire vertebral body of T8 with the posterior elements. Each voxel was converted into a hexahedral element. Based on the average local BMD (ρ ) for each element, the material was considered as linear elastic with uniaxial yield strength (ρ ). Young’s modulus along the axial direction (E ) and ρ were computed using the following formula: = −34.7 + 3.230

[3.20]

= −0.75 + 0.0249

[3.21]

Young’s modulus was measured in MPa, and a value of 0.1 kPa was attributed to it when it had a negative value, corresponding to the elements . For E and σ , a scaling was carried out with a factor with a low ρ of 1.28 to account for side-artifact errors. Two yield criteria were used: (i) a crushable foam plasticity model requiring isotropic elastic properties and (ii) the von Mises criterion requiring transversely isotropic elastic properties. The post-yield behavior was perfectly plastic for both criteria. The T8 FE models were subjected to four different boundary conditions (Figure 3.20). The boundary condition corresponded to displacement measurements at the superior and inferior endplates, providing a better prediction accuracy assessment using FE analysis, but without being clinically available. The second boundary condition corresponded to an idealized condition, as it represented the loading between rigid plates for flexion, by applying a uniform angular displacement at the superior endplate combined with a uniform compressive displacement, and for compression, by applying a uniform compressive displacement. The third and fourth boundary conditions, known as “generic intervertebral disks” and “specific intervertebral disks” respectively were created using average pressure for moderately healthy, degenerated and all intervertebral disks in both flexion and compression. An interpolation of the mid-sagittal and mid-coronal profiles was then carried out in the transverse plane of intervertebral disks, producing an applied load distributed over the superior endplate. This applied load was uniformly scaled to represent the total applied force fitting the force developed across the endplate in the case of the first boundary condition. For samples representing very small displacements at the inferior endplate of the T8 vertebrae, this endplate was

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thus fixed in the second, third and fourth boundary conditions. For the remaining samples representing failure at the inferior endplate and very small displacements at the superior endplate, the loading was applied to the inferior endplate, while fixing the superior one.

Figure 3.20. Load distribution across the superior endplate for boundary conditions in flexion (right) and in compression (left) for the first boundary condition (A+E), the second boundary condition (B+F), generic intervertebral disks (C+G) and specific intervertebral disks (D+H) (Jackman et al. 2016). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

A study by Chen et al. (2017) developed the vertebral strength measurement using a QCT-based FE analysis. The association between this vertebral strength and the incident vertebral fracture was determined. The basic CT scans were used to perform the FE analyses and volumetric bone measurements using the VirtuOst software. Analyses of the L3 vertebra were carried out using the abdominal CT series, with bone segmentation and conversion of voxel intensity values to BMD using a hydroxyapatite phantom. This was followed by resampling the bone volume into isotropic voxels and converting each voxel into a hexahedral FE to which material properties were assigned based on empirical relationships with the BMD. Uniform axial compression was simulated by applying the compressive force through a virtual layer of bone cement, defining the vertebral force as the compressive force at 2% strain. The trabecular and integral BMD were determined based on vertebral body bone mineral content, using the same VirtuOst software. The integral vBMD was defined as the average density over the entire vertebral body, with the bone mineral content being the total mineral mass of the entire vertebral body and the trabecular BMD being the average density of an ellipsoidal volume in the trabecular compartment in

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the middle of 10 mm of the vertebral volume. An aBMD of the L3 vertebrae was measured using a semi-automated algorithm, creating a customized program in MATLAB® that allows the circumvention of the vertebral body with the posterior elements and defining the surfaces in 3D. These surfaces were projected onto a single plane, and an anteroposterior measurement of the L3 vBMD was made based on the mean calibrated vBMD attenuation values, in addition to the known pixel size (Figure 3.21). The strength thresholds for fragile bones in the spine were set at 6,500 N for males and 4,500 N for females, corresponding to the thresholds for trabecular vBMD in osteoporotic individuals. The measurements were limited to L3 and adjusted to an equivalent average L1/L2 by a simple multiplication using a scale factor of 0.89 in order to allow these measurements to be compared to those of the previously established L1/L2 thresholds. For osteoporotic bones, a proposed diagnostic threshold for trabecular vBMD was used to compare the FE analysis sensitivity and the vBMD determined from the CT scans.

Figure 3.21. (a) Profiled CT scan with posterior elements. (b) CT-based (Chen et al. 2017)

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Figure 3.22. (a) Reconstruction diagram of the FE model. (b) The loading direction is indicated by the red arrows making an angle with the femur axis in the frontal plane and an angle with the femoral neck axis in the horizontal plane (Kawabata et al. 2017). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

A study by Kawabata et al. (2017) developed CT-based femur FE models for both the right and left proximal femurs, by calibrating HU with a water phantom. These models were used to assess the predicted fracture loads and their locations in patients with lytic femoral lesions. The predicted fracture loads were analyzed, and the risk factors involved in pathological fractures were determined. CT data from the Mechanical Finder software were used to create 3D FE models of healthy and pathological femurs. The mesh was generated with tetrahedral elements for trabecular and inner cortical bone and three-nodal shell elements for the outer surface of the cortical bone (Figure 3.22). Material properties were assigned and numerical simulations were carried out to simulate the experimental setup of loading femurs until fracture for determining bone stiffness. The loading was applied according to an angle γ with respect to the femoral long axis along the frontal plane

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and δ with respect to the femoral neck axis along the horizontal plane (Figure 3.22(b)). Yield stress and Young’s modulus were calculated for each tetrahedral element and Poisson’s ratio of 0.4 was set for each element. When the principal strain exceeded 3,000 microstrains, Young’s modulus was considered to be zero, a point corresponding to the cortical shell element track. The applied virtual load was incrementally increased until the occurrence of shell crack, with the fracture load being defined as the virtual load that caused the first shell crack. Fracture lines were simulated for both the healthy and the pathological femurs (Figure 3.23) and bone lesions were calculated as empty cavities with Young’s modulus equal to zero due to differences in the material characteristics of each lesion. Cortical bone thickness and lesion size were measured using a Synapse Vincent volume analyzer. Cortex thickness was measured at six different axial planes perpendicular to the neck or shaft axis and in eight different directions on each plane.

Figure 3.23. Reconstruction image showing fracture loads and fracture initiation points for healthy and pathological femurs (Kawabata et al. 2017)

The distribution of loading stress in the femoral shaft was assessed by Oh et al. (2017), using an FE model reconstructed from the CT scans of patients with complete and incomplete atypical femoral fractures, categorized according to the location of the lesion. In addition to femurs with atypical fractures, femoral morphology and FE analysis were also studied for control femurs corresponding to outpatients with thigh pain without a history of lower limb fracture, as well as patients with spine disorders or osteoarthritis without atypical femoral fractures. The lateral buckling angle and the anterior radius of curvature of the femoral stem were determined. Using fluoroscope femoral radiographs, the side with the most intense pain in patients without internal fixation of both femurs and the contralateral side

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in patients with internal fixation of one femur were assessed. A digital planning system was used to mechanically measure the femoral bowing degree (Figure 3.24). The femoral shaft bowing angle was measured by aligning the center of the medullary cavity at three different points on an anteroposterior view in the lying position; the angle between the neck and femoral shaft was measured on the same anteroposterior view; and the radius of anterior curvature of the femoral diaphysis was measured by aligning the medullary cavity center at three different points on a lateral view in the lateral decubitus, with overlapping of the distal femur medial and lateral condyles. 3D nonlinear FE models of femurs on the same side were reconstructed from CT DICOM soft tissue data using the Mechanical Finder software and were used to precisely assess the femoral shaft (Figure 3.25(A)). The 3D femur model mesh was generated with tetrahedral solid elements, using triangular shell elements for the outer surface of the cortical bone. Young’s modulus and Poisson’s ratio were determined using the Keyak model. The load was applied along an axis from the femoral head center to the distal condyle center (Figure 3.25(B)), based on individual body weights while constraining the distal condyle. The maximum principal stress distribution was then evaluated, the tensile stress location characteristics were compared, and the correlation between femoral morphology and the largest maximum principal strain ratio was calculated.

Figure 3.24. Radiographic measurement of femoral bowing, with the center of the medullary cavity being aligned with the anteroposterior and lateral views: (A) lateral bowing angle and (B) anterior radius of curvature (Oh et al. 2017)

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Figure 3.25. CT-based nonlinear FE model. (A) Full-length FE models were created based on CT DICOM data, with tetrahedral elements and triangular shell elements. (B) The load axis and the constrained distal condyle (Oh et al. 2017). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

An FE model was reconstructed by Miura et al. (2017), based on the CT scans of fresh frozen cadaveric proximal femurs corresponding to individuals who did not undergo hip surgery. The FE model was created using the Mechanical Finder software, by generating a mesh with linear tetrahedral elements for the trabecular bone and the inner parts of the cortex, and triangular shell elements for the outer cortex. The CT value of each element was defined as the average of the voxels contained in an element. A pilot study was conducted using CT DICOM data to determine the equations that could be reproduced to calculate the yield stress and Young’s modulus for a proximal femur with the selected mesh size. The yield stress and Young’s modulus for the element shell were calculated while assuming that the CT value of this element is equal to 1000 HU. The tensile yield stress was considered to be 0.8 times the compressive yield stress. Poisson’s ratio of 0.3 was established for each element. The FE model was sloped at 20° in the coronal plane to the shaft axis, fixing its distal end with a resin box and placing a resin cap on the femoral head (Figure 3.26) to reproduce the experimental conditions of the mechanical tests carried out in this study. This resin cap was subjected to a uniformly distributed uniaxial compressive load, recording the reaction force and the degree of displacement of each point, and thereby obtaining the force–displacement curve. Based on this curve, mechanical stiffness was calculated as between

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20% and 80% of the maximum fracture load, and the predicted fracture load was defined as the load when the stiffness decreased by more than 20% of the estimated values.

Figure 3.26. FE analysis of the proximal femur plate model (e), sloped at 20% to the axis of the femoral shaft in the coronal plane and fixed at its distal end by placing a resin cap on the femoral head (f), applying a uniaxial compressive load (g) until predicted fracture (h) (Miura et al. 2017). For a color version of the figure, see www.iste.co.uk/benkahla/finite.zip

An FE analysis was carried out by Jalil et al. (2017) to calculate the stress distribution on the lumbar vertebrae and analyze the effect of two different biomaterials used for the construction of a lumbar interbody fusion cage, namely polyether ether ketone and polylactic acid. In addition, an FE model was built without any cages, and an implanted posterior instrumentation was used as a control model. This study was conducted on the fourth and fifth lumbar vertebrae, called L4 and L5 respectively. The CT scans of these two L4-L5, vertebrae were used to reconstruct the corresponding FE model using the Mechanical Finder software. The functions of this software and the regions of interest extracted based on the bone edges made it possible to obtain the anatomical structure of L4-L5, which were defined as the trabecular bone core enclosed by the cortical bone. The mesh was generated using two types of elements: solid tetrahedral elements for trabecular bone, facet joint cartilage and intervertebral disks, and linear shell triangular elements for cortical bone. For each tetrahedral element, bone density was defined as the average number of HU within the element. Table 3.1 presents the values of the Young’s modulus, yield stress and Poisson’s ratio for the bone model, and Figure 3.27 shows the solid

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and mesh model of L4-L5 attached to a later instrumentation. A standard surgical procedure for lumbar interbody fusion was simulated by trimming the facet joints to perform posterior pedicle screw fixation. Two rods and two screws were considered for the subsequent instrumentation of the screw rods. Several parts of the intervertebral disks, including the annulus fibrous and nucleus pulposus, were subtracted to allow bilateral insertion of the cages. The same tetrahedral elements were used to mesh all cages and components of the subsequent instrumentation. Boundary and load conditions were applied to simulate those occurring during normal physiological activities, using two compressive and four rotational loads, namely extension, flexion, axial rotation and lateral bending, and the load was applied to the superior surface of L4 while fully constraining the inferior surface of L5 (Figure 3.28). The spinal bone stress behavior was evaluated for the cage and non-cage models, taking both construction biomaterials into account and applying different loading conditions. Bone fracture analysis at L4-L5 was also evaluated to determine the distribution of fracture risk, and the maximum von Mises stress distribution was estimated to understand the effect of the two biomaterials used, as well as the different loading conditions on the lumbar interbody fusion cage. Young’s modulus,

(MPa)

=0

= , .

= =

0