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Three dimensional finite element model for lesion correspondence in breast imaging
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by Yan Qiu.
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[Tampa, Fla.] :
University of South Florida,
2003.
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Thesis (M.S.C.S.)University of South Florida, 2003.
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Includes bibliographical references.
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Text (Electronic thesis) in PDF format.
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ABSTRACT: Predicting breast tissue deformation is of great significance in several medical applications such as surgery, biopsy and imaging. In breast surgery, surgeons are often concerned with a specific portion of the breast, e.g., tumor, which must be located accurately beforehand. Also clinically it is important for combining the information provided by images from several modalities or at different times, for the planning and guidance of interventions. Multimodality imaging of the breast obtained by mammography, MRI and PET is thought to be best achieved through some form of data fusion technique. However, images taken by these various techniques are often obtained under entirely different tissue configurations, compression, orientation or body position. In these cases some form of spatial transformation of image data from one geometry to another is required such that the tissues are represented in an equivalent configuration. We constructed the 3D biomechanical models for this purpose using Finite Element Methods (FEM). The models were based on phantom and patient MRIs and could be used to model the interrelation between different types of tissue by applying displacements of forces and to register multimodality medical images.
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Coadviser: Goldgof, Dmitry
Coadviser: Li,
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finite element model.
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registration.
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Dissertations, Academic
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x Computer Science
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u http://digital.lib.usf.edu/?e14.192
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Three Dimensional Finite Element Model fo r Lesion Correspondence in Breast Imaging by Yan Qiu A thesis submitted in partial fulfillment of the requirements for the degree of Masters of Science in Computer Science Department of Computer Science and Engineering College of Engineering University of South Florida CoMajor Professor: Dm itry Goldgof, Ph.D. CoMajor Professor: Lihua Li, Ph.D. Dmitry Goldgof, Ph.D. Lihua Li, Ph.D. Sudeep Sarkar, Ph.D. Date of Approval: November 11, 2003 Keywords: finite element model, mammogr am, registration, correspondence, MRI Copyright 2003, Yan Qiu
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Acknowledgments I gratefully acknowledge the help of Dr. Dmitry Goldgof a nd Dr. Lihua Li, who have supervised my thesis work and guided me through all steps of research and writing. I am grateful to Dr. Sudeep Sarkar for his valu able guidance and feedback. I also thank Sorin Anton and Yong Zhang for thei r comments and suggestions. I have received invaluable support fr om my wife, Yu Huang, who has provided help and encouragement through my graduate studies.
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i Table of Contents List of Tables................................................................................................................. ..........iii List of Figures................................................................................................................ ..........iv ABSTRACT....................................................................................................................... ......vi Chapter 1 Introduction......................................................................................................... .....1 1.1 Overview................................................................................................................... ...1 1.2 Previous Work and Pilot study.....................................................................................3 1.3 Elastic Theory............................................................................................................. ..8 Chapter 2 Theoreti cal Background........................................................................................12 2.1 Breast Imaging............................................................................................................1 2 2.2 Finite Element Model.................................................................................................13 2.3 Generic Model............................................................................................................14 2.4 Breast Model Compression Simulation.....................................................................15 2.5 Material Pr operties.....................................................................................................17 2.6 Meshing and Di scretization.......................................................................................19 2.7 Registration............................................................................................................... ..22 Chapter 3 Algorithm and Phantom Study..............................................................................26 3.1. Algorithms................................................................................................................ .26 3.1.1 Algorithm Based on Mini mum Euclidean Distance........................................26 3.1.2 Algorithm for Cu rve Predic tion........................................................................27 3.2 Modeling................................................................................................................... ..28 3.2.1 Image Acquisition a nd Data Extraction...........................................................28 3.2.2 Determine Mate rial Property............................................................................29 3.2.3 Meshing.............................................................................................................30 3.2.4 Boundary Condition..........................................................................................32 3.3 Compression simulation results...........................................................................33 3.4 Validation................................................................................................................. ..34 3.4.1 Validation Based on Mini mum Euclidean Distance........................................34 3.4.2. Validation for Pr edicted Curve........................................................................39 Chapter 4 Patient Study........................................................................................................ ..42 4.1 Image Acquisition and Data Extraction.....................................................................42 4.2 Patient Generic Model................................................................................................43 4.3 Validation Based on Minimum Euclidean Distance.................................................44 4.4 Validation for Predicted Curve..................................................................................45 Chapter 5 C onclusion........................................................................................................... ..48
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ii References..................................................................................................................... ..........51
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iii List of Tables Table 1. Error Comparison for Differ ent Models for Y oungÂ’s Modulus.............................19 Table 2. Number of N odes and El ements..............................................................................30 Table 3. Compression Values Used in Xray Imaging..........................................................33 Table 4. Process Illustra tion Using Node 4719....................................................................36 Table 5. Minimum Euc lidean Distance..................................................................................38 Table 6. Minimum Euclidean Distance Ex periment Results for Patient Data......................45
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iv List of Figures Figure 1. Compression of Ma mmogram in Two Views........................................................12 Figure 2. Phantom Finite Elem ent Model Illustration...........................................................13 Figure 3. Phantom Finite Element Generic Model Illustration.............................................14 Figure 4. Compression in the CC View to Matc h Mammogram..........................................15 Figure 5. Finding Correspondent Lesi on Elements in Two Views.......................................16 Figure 6. Tetrahedral Elements and Hexahedral Elements Â– ANSYS..................................21 Figure 7. Phantom Imag e Segmentation................................................................................29 Figure 8. Model before and after meshing.............................................................................31 Figure 9. Finite Element Mode l Elements and Nodes...........................................................32 Figure 10. Finite Element M odel Elements and Nodes.........................................................33 Figure 11. Phantom CC view compression simulation.........................................................34 Figure 12. Phantom ML View Compression Simulation......................................................34 Figure 13. Validation Using Mi nimum Euclidean Distance.................................................35 Figure 14. Nodes Registered to Feature Point.......................................................................35 Figure 15. Simulated Calcifica tion with Predicted Curve.....................................................37 Figure 16. Validation fo r Predicted Curve.............................................................................39 Figure 17. Predicted Lesion Pos ition in Phantom ML View................................................40 Figure 18. Patient MR Images................................................................................................42
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v Figure 19. Segmented MRI Images af ter Morphological Operation....................................42 Figure 20. Generic Model Scaling Acco rding to CC View and ML View..........................43 Figure 21. Patient Data with Visi ble Lesion in Both Views.................................................45 Figure 22. Predicted Lesion Positio n in Patient Data ML View...........................................46
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vi THREE DIMENSIONAL FINITE ELE MENT MODEL FOR LESION CORRESPONDENCE IN BREAST IMAGING Yan Qiu ABSTRACT Predicting breast tissue deform ation is of great significance in several medical applications such as surgery, biopsy and imag ing. In breast surgery, surgeons are often concerned with a specific portion of the br east, e.g., tumor, which must be located accurately beforehand. Also clinically it is important for combining the information provided by images from seve ral modalities or at different times, for the planning and guidance of interventions. Multimodality im aging of the breast obtained by mammography, MRI and PET is thought to be be st achieved through some form of data fusion technique. However, images taken by th ese various techniques are often obtained under entirely different tissue configurations, co mpression, orientation or body position. In these cases some form of spatial transforma tion of image data from one geometry to another is required such that the tissues are represented in an equivalent configuration. We constructed the 3D biomechanical models for this purpose using Finite Element Methods (FEM). The models were based on pha ntom and patient MRIs and could be used to model the interrelation between different ty pes of tissue by applying displacements of forces and to register mu ltimodality medical images.
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1 Chapter 1 Introduction 1.1 Overview Breast cancer is the second leading cause of cancer death for all women (after lung cancer), and the leading overall cause of cancer death in women between the ages of 40 and 59. In 2002, 257,800 new cases of breast cancer will be diag nosed, and 39,60 0 women will die from the disease. The ri sk of developing breast cancer seems to depend on several factors including age, personal or family histor y of breast cancer, parity, age at first birth, hormonal replacement, etc. However, over 70% of cancer cases are women with no identifiable risk factors. Ea rly diagnosis is very important for proper treatment and cure and this has led many countries including US, to develop regular screening programs that are primarily based on mammograp hy and physical examination. Mammography is the main scr eening tool for breast cancer with a sensitivity of about 85% and specificity up to 25%. Despite their proven effectiveness, both screening tools entail significant variability and there are few attempts todate to standardize either one or correlate mammographic to physical examination findings. Techniques that improve the accuracy of mammography or phys ical breast examination or both are still highly desirable and could bene fit breast cancer diagnose. Furthermore, methodologies that yield a 3D representation of the breast with accurate volume and lesion location are
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2 expected to offer a unique tool for accura te and consistent fo llowup, for correlation between findings from different screening pr ocedures, as well as correlation of serial examinations (annual or serial exams). Several approaches have been investigated until now to improve the diagnostic accuracy of mammography includi ng computeraided detecti on and diagnosis, computer displays, new direct digital mammography systems, and, recently, digital 3D mammography or tomosynthesis. Excluding to mosynthesis research, all other studies todate have been focused on the reconstruction of stereotactic biopsy images, the use of liquid crystal display glasses to create a 3D pe rception of the breast from monitors, and the combination of the 2D mamm ographic views using warping techniques to establish a degree of correlation and stereo representation. We have implemented several published methods from the latter class of methodologies in an effort to correlate the mammographic views and generate a 3D simu lation of the compression. Screening mammograms usually consist of a craniocaudal (CC) and mediolateral oblique (ML) view of each breast. Breast xra ys show areas of fatty and glandular tissue, pectoral muscle (if the view is ML), skin boundary, nipple and the nonbreast region. Due to variation in compression and physical cha nges of the breast, consecutive mammograms of the same patient are difficult to fully corre late from one examinatio n to the next and the expert reader may identify only general sim ilarities. Similarly, du e to differences in compression geometry and lack of common, refe rence points or fixed landmarks other than the nipple, onetoone correspondence betw een the mammographic views is nearly impossible, and wellknown stereo imaging algo rithms widely used in stereo navigation, such as stereovision or passive rang ing, cannot be applied to mammography.
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3 The goal of this thesis is to construct solid model for human breast, and design solid modelbased methods to estimate nonrigid registration and nonrigid motion attributes, that is, to predict physical deformation and displacements of the breast and perform a nonrigid registration between two different Xray view s. This is useful for surgical procedures and diagnoses purposes. A 3D finite element model of the br east was constructed based on the MR slice images. Once we had the FE M model, a compression similar to one performed during mammography data acqui sition, was simulated and a registration between the projected image of 3D volum e and Xray images was performed. The commercial software ANSYS was used to compute the FEM of the breast solid model. 1.2 Previous Work and Pilot study There are many cases where the rigid body motion paradigm is inadequate. For instance most of the biological objects are flexible and articulated. In order to describe these types of deformations one must mode l the physics by which these objects deform. Underlying equations describing rigid motion and methods for solving them has been studied thoroughly [35][49]. In [47] and [48] the author s investigated the influence of different tissue elasticity values, Poisson ratios, boundary conditions, finite element so lvers and mesh resolutions on a data set. MR images were acquired before and after breast compression. Images were aligned using a 3D nonrigid registration al gorithm [37]. Biomechanical model of the breast was constructed using FEM. Two geometri cal models were used in this study with different geometric resolutions. For the firs t, coarser geometric model, the segmented images were blurred by a Gaussian kernel a nd resampled to an 8mm voxel size, and after
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4 the triangulation of the surface, the number of elements was decimated by 120 iterations. After meshing, these led to a model that had 51072 nodes and 34873 elements for fat and 4484 surface shells for skin. The second model was less coarse and resulted in 102102 nodes and 72756 elements, and 4950 surface shells. Fi ve material models were used in this study: breast assumed linear and homogenous, 1K Pa Young modulus was assigned to the mixture of glandular and fatty tissues; an additional YoungÂ’s modulus of 88KPa was chosen for the skin; in addition to previous material mo del a 10KPa Young modulus was used to model separately the glandular tissue; nonlinear material model [2], exponential curves were used to describe the stressstrain curves, Young modulus of fat increases linearly with stra in from zero up to the value for glandular tissue; nonlinear model proposed in [38], inst ead of exponential curves to describe stressstrain curves, quadratic and third order polynomials were used for fatty and glandular tissue respectively. Four sets of boundary conditions were used: all surface nodes were constrained to the corresponding displacements obtained from 3D nonrigid registration; a subset of the previous nodes was used, which is the posterior, medial and lateral side of the breast; the nodes on the wall chest boundary were assigned to have zero displacement and no boundary conditions to other nodes;
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5 besides the previous conditions, the nodes on the medial side of the breast were assigned to have zero displacement. In ANSYS both the Â“Frontal SolverÂ” and Â“Sparse DirectÂ” solvers are direct elimination solvers and are recommende d when robustness is required. The Â“Preconditioned Conjugate Grad ientÂ” is recommende d for large solid models. All three solvers were used on different models. The co nclusions of this test were that boundary conditions and the value of Poisson ratio have a much larger effect on the performance of the FEM model than using different tissue pr operties, although models with accurate boundary conditions seemed not to be much influenced by Poisson ratio. The mesh resolution and choice of finite element solver had almost no effect on the performance of the FEM of the breast. In [16] six similarity measures are co mpared for use in intensity based 2D3D image registration. The accuracy of each re gistration method is compared to a gold standard which was calculated using a featurebased algorithm. Simila rity measures were used to register a fluoroscopic image a nd a digitally reconstructed radiograph. The similarity measures used are: normalized cross correlation; entropy of the difference of the im age, the entropy measures operate on the difference image obtained by subtracting the scaled DRR image from fluoroscopy image; mutual information or relative entropy, it has been found very effective in 3D3D multimodality image registration gradient correlation, the cross correlation is computed on the gradient images obtained using Sobel kernel;
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6 pattern intensity also operates on diff erence image like entropy, but count the number of patterns in differ ence image that should tend to zero for a perfect registration gradient difference involves the difference of the gradient images The study concluded that the least accurate similarity measure for this experiment was mutual information. Gradie nt correlation has been shown to be sensitive to thin line structures. On images that contained both soft tissue and a stent cross correlation, entropy and mutual information failed often, and gradie nt correlation was the most accurate. Pattern intensity, gradient correlation and gradient difference were affected very little by the presence of the soft tissue. In [14] three 3D2D regist ration modalities were addresse d for CT scan images and Xray images. Here is presented the framework for finding a geometric transformation between a 3D image and a 2D radiography 3D2D curve registration problem, 3D3D surface rigid registration by using passive ster eo to reconstruct a 3D surface, and 3D2D surface registration by using silhouettes for fi nding the transformation for blood vessels. In order to find an initial estimate of the pr ojective transformation, the authors used the bitangentline properties of the curves. For 3D3D surface rigid registration, a passive stereo system was used and re sulted in a dense description of the surface using points and normals. For 3D2D projective surface regist ration, the authors used a combinatorial approach to find the initial estimate based on the property: Â“If a point M on a 3D surface S is such that its projection m=Proj(M) lies on the occluding contour c, then the normal vector N to S at point M is eq ual to the normal vector n of th e plane P defined by (m,O,t), where t is the tangent vector to the occluding contour at point mÂ”.
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7 In [42] and [43] nonrigid registration methods were e xplored and in [9] a novel validation method was proposed. The method is based on finite element modeling of the breast in order to simulate plausible breast deformations. The model similar to [31] used the value of the Young modulus 1KPa for fatty tissue, 1 6.5KPa for the carcinoma and 88KPa for the skin. The registration was perfor med for the real compress breast image and the simulated image of the breast using FEM. The average error was about 1mm, with the minimum values as low as 0.08mm in tumorous tissue and 0.15 mm in the overall tissue. In [43] the FEM method was used for validati on for the registration technique using multiresolution with freeform deformations ( FFD), based on multilevel Bsplines and nonuniform control point distribution. In [2] the authors developed a procedur e to predict the displacement by plate compression of the breast that takes less than ha lf an hour, making it clinically practical. In this study, they used the FE M of the deformable breast for guiding breast biopsy with MR imaging and registration between different breas t MR data sets from the same patient, obtained at different times and at different compressions. Volume elements used in this finite element model of the breast were: hexa hedral trilinear isoparametric elements to model the breast tissue and threenode triangular isoparametric elements to model the skin. All elements assumed to have nonlinear elastic material properties, isotropic, homogenous and incompressible. For each element after each deformation increment, the stiffness value of every element is updated to model th e nonlinear behavior of material. The Young modulus for the fat tissue was assumed to ha ve a quadratic behavior with respect to stiffness in order to try to take into account the Cooper ligaments. Only 8 MRI slices were stacked in order to obtain the 3D breast vol ume for the compression experiment and 58 for
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8 registration experiment. The results showed that the performance of the model was modestly affected by the material properties, but the shape and size of the patient breast, and the boundary properties between breast and plates have a great influence. Sorin Anton [1] described Finite Element Model using different material property and boundary condition in his thesis. During my graduate work I further used Finite Element Model for phantom and patient study. 1.3 Elastic Theory Nonrigid motion has been continuously studi ed in the last years and a variety of approaches had been presente d, but until now no one paradigm can be applicable to all types of nonrigid motions. Active contours were used to track motion. Active c ontours are a minimization problem and have been largely used. The gene ral concept and one of its applications are presented in [27]. An insightful view of th e underlying mathematics and a new algorithm for detecting objects which do not have boundarie s defined by gradient is presented in [9]. In [19] the authors used the active contours to track the endocardial and epicardial borders of the left ventricle. In [51] the authors us ed active contours to track feature points between two images. In [15] is proposed a new method to incorporate prior shape information into geometric active contours for c ontour detection. Unlike traditio nal active contours (snakes) that are represented as parameterized curves in a Lagrangian form ation, geometric active contours are represented as level sets of two dimensional distance functions which evolve according to an Eulerian formation. Active c ontours enforce constraints on smoothness and the amount of deformation, but they cannot be used to constrain the types of deformation
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9 valid for a particular object class. To overco me this problem aprior i constraints must be enforced on the types of allowa ble deformations [41] [12]. Finite element models incl ude material properties of the object and an underlying geometry model and can accurately predict displacements and motion based on applied forces, or recover the loads given nodal displ acements. In [28] aut hors used FEM model to compute elastic properties of the scars relative to the surrounding area. In [51] the authors used FEM to recover elastic properties of the skin by incrementally modifying the material properties until the model matches the image of the deformed object. In [51] the authors used finite element model to compute intermediate images given images at two time instances and their corresponding features. In [52] the authors used FEM based algorithm for accurate nonrigid motion tracking. They us ed the difference between the actual behavior from the motion images and predicted behavior of the object in order to refine the model, and unknown parameters ar e recovered during the search for the best finite element model that approximates the nonrigi d movement of a given object. The dynamics of the elastic body is gove rned by the following system of partial differential equations: xf u x z w x y v x u t u 2 2 2 2 2 2 2 0 (1) yf v y z w y v y x u t v 2 2 2 2 2 2 2 0 (2) zf w z w z y v z x u t w 2 2 2 2 2 2 2 0 (3) where: ( u,v,w ) represents the displacement vector in Cartesian coordinates
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10 E representsYoungÂ’s modulus represents PoissonÂ’s ratio fi represents force field are Lame consta nts, computed with the formulas: 1 2 E (4) 2 1 1 E (5) For small deformations The YoungÂ’s modulu s can be considered constant and the deformation elastic. However in medical imaging large deformations are desirable to maximize the signal to noise ratio. At large de formations biological ti ssues will have more or less strain hardening, depending on tissue property. In our case for breast compression which undergoes significant deformation we can assume YoungÂ’s modulus constant and equal to an average value of the initial and fina l state. In order to de scribe the deformation in response to an external solicitation, a tissue can be considered as isotropic and linear continuous elastic medium. In this case the relation betwee n strain and stress can be expressed in tensor notation: nn ij ij ij 2 (6) where ij represents the symmetric stress tensor ij represents the symmetric strain tensor ij represents the Kroneker delta defined as:
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11 j i if j i ifij 0 1 and 3 1 3 1 ij ij nn and for i=j 33 22 11 nn (7) In order to have a solution and to be uni que for the system of partial differential equations we must have some boundary conditions (the init ial conditions for the partial differential equations). The mechanical boundary conditions are given by [20]: 00 i i i j iju u F n (8) where: jn is the jth component of the unit normal vector at the body surface iF is the force per unit area at the surface acting in direction xi 0 iu is the initial surface displacement is a variation symbol
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12 Chapter 2 Theoretical Background 2.1 Breast Imaging In Xray imaging breast compression is necessary to flatten the breast so that the maximum amount of tissue can be imaged, and to reduce xray scatter which leads to image degradation. Usually two or more Xrays are taken on different angles, with different compression values for each angle of view. To match the features from one view to the feature in another view is not a trivial task and usually is done by a trained physician. (a)Compression of Mammogram in CC Vi ew (b)Mammogram in CC View (c) Compression of mammogram in ML view (d) Mammog ram in ML view Figure 1. Compression of Mammogram in Two Views
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13 2.2 Finite Element Model The finite element model approach is based on the underlying geometry of the object and on the material properties of the ob ject. Using a system of partial differential equations to predict the moveme nt of each node, shape analysis of the constitutive elements in each state, material properties and a set of border conditions to insure the convergence of the solution, FEM can predict with high accur acy the final state of the object, or any intermediate state. (a) Digital image of phantom (b ) Phantom model after meshing Figure 2. Phantom Finite Element Model Illustration Biomechanical models of the breast are being developed for a wide range of applications including image alignment tasks to improve diagnosis and therapy monitoring, imaging related studies of the biomechanical properties of lesions, and image guided interventions. From the need to register featur es in different views of the same organ, or the need to know what displace ment will have a particular tissue under external controlled applied deformations emerged the necessity to have a 3D representation of the object and to be able to predict deformations and/or strain distributions inside the object during external deformations. The necessity for a nonrigid registration between two different
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14 views inspired the first part of this study. The modeling concepts also hold for the second part of the study, but the goal is different. 2.3 Generic Model Figure 3. Phantom Finite Element Generic Model Illustration In our experiments using patient data, ge neric model was used in the case when no MRI from the patient is available. The first step is to build a generic breast model based on well segmented MR images for one breast and apply this generic model for registration between Xray images for differe nt breast. For this method to work, the generic model must be scaled first and afte r that attempt the registration. The generic model was first compressed and projected in CC and ML views. The x and y scaling factors were obtained by sca ling the projected CC view al ong x and y axis to match the (a) CC View Xray (b) Generic Mode l Scaled According to CC View (c) ML View Xray (d) Generic Mode l Scaled According to ML View
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15 Xray image. These scaling factors were th en taken into the consideration when we simulated the breast compressi on. Using these x, y scaling factors we recomputed the generic model and use it for registration be tween the two Xray views. The results for matching the scaled model with the ML Xray view are presented as follows: 2.4 Breast Model Compression Simulation In order to be able to match a feature from the Xray CC view to Xray ML view, first the breast model is compressed along the z direction with a displacement value identical to Xray compression. To solve the system of partial differential equations, a sparse method solver and a preconditioned conjugate gradient (PCG) with full NewtonRaphson iterative method were used. The PCG method performs better for solid model which has a large number of elements. After computing the predicted displacement of the nodes, the 3D compressed breast model was pr ojected on XY plane perpendicular to the compression direction, resulting a 2D image that should match the Xray CC view. The next step involved in experiment was to register the projected image and the CC Xray image, matching the feature pixels from CC Xray image to projec ted node positions. (a) Uncompressed Model (b) Compressed Model Simulating CC view Mammogram Figure 4. Compression in the CC Vi ew to Match Mammogram
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16 After the registration we back project the f eature point into the 3D breast model and record the elements that correspond to that feature in Xray CC vi ew. These corresponding elements are aligned on a line in the compressed model and on a curve in the decompressed volume model. (a) Finding correspondent lesion elements in compressed model (b) Finding correspondent lesion elements in uncompressed model (c) Correspondent lesion elements in uncompressed model (d) Finding Correspondent Lesion Elements In ML View Figure 5. Finding Correspondent Lesi on Elements in Two Views The uncompressed breast model was rotated ar ound the Oy axis with an angle equal to the angle of the Xray ML view. Again the predicted node displacements were computed using the same solver. After model prediction, the breast m odel was projected again on a plane perpendicular to the compression dir ection and the new 2D projected image was registered against the ML Xray image. The elem ents recorded in the CC projection lie on a
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17 curve in the 2D projected image and the feat ure corresponding to the same feature in the Xray CC view is also found on this curve. Figure. 5. shows the process simulating mammogram compression in the ML view and predicting lesion position in ML view. 2.5 Material Properties The first step toward FEM is to define the material properties for the solid model constructed in previous steps. The mode ling of biomechanical tissue has gained considerable interest in a range of clinical and research app lications. According to literature in the domain the breast is considered to be made of biological soft tissues, which are known to be incompressible. The female breast is essentially composed of four structures: lobules or glands, milk ducts, fat and connectiv e tissue. The breast tissues are joined to the overlying skin by fibrous strands called Cooper Â’s ligaments [48]. Most biologic tissues have both a viscous and an elastic response to external deforma tions. Because we are interested only in slow deformations the res ponse of the tissue can be considered entirely due to elastic forces [2]. A ll tissues in the breast can be considered isotropic, homogenous and incompressible with nonlinear elastic pr operties for large deformations. The Young modulus represents how much a material will de form when a load is applied, and Poisson ratio express how much a materi al will shrink in one direc tion when is stretched in a perpendicular direction, an incompressible material will c onserve the volume so same volume stretched must be shrinked. Since brea st tissues exhibit a nonlinear behavior for large deformations the YoungÂ’s modulus can be considered as a function of strain for each tissue:
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18 n nm n n n ne b E (9) where: n is the nominal stress for tissue type n, n is the nominal strain for tissue type n, nb and nm are fit parameters determined experimentally for tissue type n [2]. For glandular tissue type 2/ 100 15 m N bgland and 0 10 glandm, for fatty tissue type 2/ 460 4 m N bfat and 4 7 fatm [2]. For skin YoungÂ’s mo dulus can be considered given by the followi ng formula [2]: i skina E 3 1 to i where the values for YoungÂ’ modulu s depend on strain as following: 1 68 0 / 10 57 1 68 0 54 0 / 10 89 2 54 0 0 / 10 43 32 8 2 7 2 6 skin skin skin im N m N m N a The fitted YoungÂ’s modulus versus strain polynomials are as follows [38]: 0049 0 0024 0 5197 02 fatE (10) 0121 0 6969 0 7667 11 8889 1232 3 glandE (11) According to experiments and comparisons presented in [48], the improvement in error registration for the overall model provi ded by the previous defined equations for YoungÂ’s modulus over the consta nt value for YoungÂ’s modulus is not very important as presented in Table 1, where EL represents error of transformation with respect to landmarks.
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19 The conclusion of the study in [48] is th at inaccurate assumptions about the surface displacement vectors have a much larger ef fect on the performance of the FEM breast models than using a model with different tissue properties. Table 1. Error Comparison for Diffe rent Models for YoungÂ’s Modulus PCG Solver Frontal Solver Sparse Direct Solver EL [mm] EL [mm] EL [mm] No Model mean std maxmea n std ma x mean std max 1 Constant YoungÂ’s modulus [1KP] 2.12 1.01 3.462.12 1.013.4 6 2.12 1.0 1 3.45 2 Additional skin [88KP] 2.22 1.21 3.772.22 1.213.7 6 2.22 1.2 1 3.77 3 Constant YoungÂ’s modulus [10KP] 2.49 1.01 3.992.49 1.003.9 9 2.49 1.0 1 3.99 4 YoungÂ’s modulus modeled (35) 2.17 0.98 3.382.17 0.983.3 8 2.17 0.9 8 3.38 5 YoungÂ’s modulus modeled (36) (37) 2.53 0.85 3.862.53 0.853.8 6 2.53 0.8 5 3.86 Max. difference 0.41 0.36 0.530.41 0.360.5 3 0.41 0.3 6 0.54 2.6 Meshing and Discretization Mesh generation is one of the key compone nts in device simulation. A quality mesh not only is necessary for obtaini ng good simulation results, but also has a significant impact on the computation time and efficient usage of computer resources. A me sh is a partition of geometric region into a set of nonoverlapping subregions. Each subregion is called an element and is characterized by its points (a lso called vertices or nodes), edges and faces. Mesh elements are simply connected, convex polyhedrons. The mesh process is based on the divideandconquer principle and invol ves dividing the volume in many small nonoverlapping entities called elements for which th e equations (22)(24) can be easily defined and computed. Usually these elements are te trahedrons or hexahedrons for 3D volume
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20 meshing. These element types an d their properties are presented in Fig. Generally meshes with hexahedral elements are superior to ones with tetrahedral elements in terms of convergence, stability of solution in nonlinear systems and accuracy. Many researchers had conducted studies to measure the elasticity parameters of soft tissues [42] [2] [46] [44] [47]. According to these studies the average YoungÂ’ modulus value for fatty tissue is 1KPa, for skin is 88KP a, for glandular tissue is 10KPa. The overall YoungÂ’s modulus is considered to lie in the range 5KPa15KPa for the entire breast modeled as a linear, continuous, incompressible, isotropic and homoge nous tissue. Since we are interested in constructing a generi c model we choose an initial value for YoungÂ’s modulus of 10KPa. Since the breast is cons idered to be an incompressible tissue, theoretically volume is pres erved for a PoissonÂ’s ratio equal to 0.5. However high PoissonÂ’s ratio can lead to instabilities in FE model, a value between 0.4900.495 is generally accepted as a computational stab le and minimum disp lacement error for PoissonÂ’s ratio. After model calibration we used a PoissonÂ’s ratio value of 0.490. To mesh the solid model of the breast we us ed a feature first type of mesh based on Delaunay principle. The features that comp osed initial mesh poin ts were the sampled border points. The mesh was composed of tetr ahedral elements with 10 nodes (each side has an additional node in the middle to m odel the deformations more accurately) The resulted meshed volume is presented in Figure 14. This is the finite element of the breast to which the deformatio ns will be applied. For a 10 pi xels sample interval in the original image slice, 52 slices were stacked to construct the volume and an element size of 8 units, the meshing resulted in 13225 nodes and 8744 elements, with 2 bad shape elements. Because of the complex geometry of the breast solid model and the size of the
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21 element resulted a small percent of bad shaped elements. This could be alleviated by further reducing the element size but this w ould lead to large numb er of elements and nodes and a long computational time. This va lue for element size is acceptable regarding computational time, the percent of bad shaped elements and final result error. Further reduction of the element size would have no effe ct on the final result. Table 2 presents the number of nodes and elements of the finite element model as a function of element size for sampling interval along the z axis equal to 3. As it can be seen, th e number of nodes and elements grows exponentially and so does the computational time. Table 3 presents the number of nodes and elements has the finite element model as a f unction of element size for sampling interval along the z axis equal to 1. Increasing the sample interval on the z axis results in a smoothing of the breast mode l, value of three was used in our experiments because it was the lowest value for which th e FEM was stable and number of bad shape elements minimum resulti ng in a robust solution. (a)Tetrahedral elements 10 nodes quadratic behavior (b)Hexahedral elements 8 nodes large deformation Figure 6. Tetrahedral Elements a nd Hexahedral Elements Â– ANSYS
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22 2.7 Registration In order to identify the elements that corre spond to a selected feature in CC image or to identify the locus of th e candidate feature in the ML image, a 2D rigid registration must be accomplished between the Xra y image and the projected 3D volume. Because of the characteristics of both images only rigid image registration is considered, which means two images can be matched through the linear transform. The registration involves th e following basic steps: featurebased registration; the 3D compressed model must be pr ojected on a plane perpendicular to the direction of compression; the Xray image must be scaled such th at one pixel in both images represent the same distance; a geometrical transformatio n must be found that will ma p one image into another. bilinear interpolation applied to the tr ansformed image because in the transformed image not all the pixels will have a co rrespondent in the original image. The transformation is composed of th ree components: rotation, scaling and translation, which can be modeled as follows: 0 0 ' cos sin sin cos Y X Y X S Y Xt t t t (12) To find the parameters of the transformation first the eigenvalues and eigenvectors will be computed for the auto correlation matrix for both imag es. The autocorrelation matrix is:
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23 ) ( 2 ) ( ) ( ) ( 2 x y I x y I x y I x y Iy xy xy x M (13) Image registration is based on the alignment of the eigenvectors for (46) corresponding to two largest eigenvalues of two images. From above, we get the rotation angle.     cos* * *t b t bV V V V 2cos 1 sin (14) The equation (47) doesnÂ’t give us the di rection of rotation, thus we have to determine the direction according to the sign of eigenvectors. } { 0 0 ) . tan( 2 0 0 ) . tan( 0 0 ) . tan( 0 0 ) . tan( t b j x V y V if x V y V a x V y V if x V y V a x V y V if x V y V a x V y Vif x V y V aj (15) Thus the rotation angle is t b The translation is estimated according to the mass center of two images. The mass centers of two images ) (y x are defined according to the following equation: ) 0 ) ( ( ) 0 ) ( (0 ) ( 0 ) ( j i I count y j i I count xj i I y j i I x and thus t y b y t x b xY X 0 0 (16)
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24 The similarity measure is based on the pair wise pixel comparison of the base image and aligned image. The co mparison function is as follows: ) ( ) ( 0 ) ( ) ( 1 ) (' 'j i I j i I if j i I j i I if j i dt b t b (17) Then the similarity measure can be defined as: 2 ) 0 ) ( ( ) 0 ) ( ( ) ( ) ( [%]' j i I area j i I area j i d I I f simt b N i M j t b (18) The objective of the image registration is to minimize the similarity measure. Therefore, the rigid image registration can be present as an optimization model: ) ( min' t bI I f 0 0 'cos sin sin cos .Y X I I stt t (19) From the discussion above, the algori thm can be outlined in three steps: Step 1: Initialize the system. Step 2: While ( ) (1n nf f abs) do Translate the image Rotate the image Dilate the image and erode the imag e to remove the holes generated by the transform. (or perform a bilinear interpolation) Calculate the similarity )) ( ) ( ('n I n I f ft b n Step 3: Output the transform and the result image
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25 Because the MR images imaged more of the breast than Xray images, the projected model is larger than the correspon ding Xray image. For a correct registration process, the size of the project ed model has to be reduced to the mammogram size.
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26 Chapter 3 Algorithm and Phantom Study The following investigations utilized Triple Modality Biopsy Phantom. Experiments were designed to provide quantitative answers to the following questions that are often raised in twoview mammography: 1. If suspicious features are found in two views, to what degree we can determine that they actually correspond to the same tumor? Quantitative answers were computed based on the distance between 3D curves generated from backprojection and de formation modeling, which pr oved the accuracy of the proposed method using breast phan tom with ground trut h of abnormalities. 2. If a feature is found only in the first view, what is its possible position in the second view? Suspicious area was computed in the second view th rough a series of projection and breast deformation modeling. 3.1. Algorithms 3.1.1 Algorithm Based on Minimum Euclidean Distance Following algorithm is implem ented for validation purpose: 1. Suspicious features are found in one mammogram (for example, ML view). 2. Compress 3D finite element breast mode l with recorded compression data in ML view.
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27 3. Each feature found in ML view will be backprojected into compressed breast model to create a straight line. Backp rojection is accomplished through simple raytracing of the feature using known camera configuration. 4. Adaptively remesh the model elements that are adjacent to the straight lines generated in step 3. 5. Label the elements through wh ich the straight lines pass. 6. Decompress breast model to its original shape by moving plates outwards, and the straight line will be de formed into a curve. As a result, each feature in ML view will have a corresponding 3D cu rve in the uncompressed breast model. 7. Repeat steps 16 for each f eature found in CC view. 8. A feature in ML view will be paired w ith all features in CC view. For each pair, the distance between their 3D curves gene rated at step 6 will be computed. Two curves will be evenly divided into segmen ts by key points (including two end points of the curve). The di stance between two curves w ill then be computed as the sum of Euclidean distances between all key points on the curves. Each feature pair will then be ranked based on the computed distance of their 3D curves. Finally, the feature pair with the highest rank (smallest distance) will be considered as a match. In other words, they are most likely related to the same breast abnormality. 9. Repeat step 8 for each feature in ML view. 3.1.2 Algorithm for Curve Prediction 1. A suspicious feature is found in ML vi ew, but not in CC view (or vice versa).
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28 2. Compress 3D finite elem ent model with recorded compression data in ML view. 3. Backproject ML feature into 3D model to gene rate a straight line. 4. Adaptively remesh elements that ar e adjacent to the straight line. 5. Label the elements through wh ich the straight line passes. 6. Decompress breast model to its original shape by moving plates outwards, and the straight line will be deformed into a curve. 7. Compress model with recorded CC data The curve of ML feature is further deformed. 8. Project the curve onto CC view film. The projected curve in CC view indicates the most suspicious area in which a s econd reading is strongly recommended to see whether a featured was overlooked. 3.2 Modeling 3.2.1 Image Acquisition and Data Extraction The phantom data consist of a set of pa rallel twodimensional MR images of the phantom. The distance between slices is 2. 5mm and the voxel size for the MRI is 1.41x1.41x2.50mm. The extracted contour set is loaded to ANSYS FEM software package where it would then be meshed into tetrahedral structural elements.
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29 (a) Phantom MRI slice 20 (b) Phantom MRI slice 30 (c) Phantom MRI slice 37 (d) Phantom MRI slice 20 segmented (e) Phantom MRI slice 30 segmented (f) Phantom MRI slice 37 segmented Figure 7. Phantom Image Segmentation 3.2.2 Determine Material Property In order to simulate breast deformations, we constructed and tested several models, each combined with different sets of simulation parameters: 1. Inhomogenous nonlinear model 2. Homogenous nonlinear model 3. Homogenous linear model For example, for homogenous linear model, these tissues have been modeled for sake of simplicity as fatty tissue. The Young's modulus were set to 1 kPa for the fatty
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30 tissue, and to 16.5 kPa for the carcinoma, using publis hed values [49]. For nearincompressibility of the tissue, the Poisson's ratio was set to 0.495. 3.2.3 Meshing Voxeland surfaceoriented meshing meth ods have been used to generate the meshed FEM model. The phantom volume was th en meshed into isoparametric tetrahedral structural solids (elements). The elements cons ist of four corner no des and an additional node in the middle of each edge. Each node has three associated degrees of freedom (DOF) which define translation into the nodal x, yand zdirections. Each element has a quadratic displacement behavior, and provides nonlinear material properties as well as consistent tangent stiffness for large strain applicatio ns. The skin was modeled by adding shell elements consisting of eight no des onto the surface of the fa tty tissue. Fig. 2. shows renderings of the FEM models. Table 2. Number of Nodes and Elements Element size Number of nodes Number of elements 20 2836 1668 15 4010 2424 10 7899 5068 8 13225 8744 5 52982 36824 The values used in our experiments were 3 for z axis sampling and 10 size of the element. Since in breast model compression we do not have any information about the force applied but only about the displacemen t used, the boundary conditions used were Dirichelet conditions.
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31 Voxeland surfaceoriented meshing meth ods have been used to generate the meshed FEM model. The phantom volume was th en meshed into isoparametric tetrahedral structural solids (elements). The elements cons ist of four corner no des and an additional node in the middle of each edge. Each node has three associated degrees of freedom (DOF) which define translation into th e nodal x, yand zdirections Each element has a quadratic displacement behavior, and provides nonlinear material properties as well as consistent tangent stiffness for large strain applicatio ns. The skin was modeled by adding shell elements consisting of eight nod es onto the surface of the fatty tissue. Figure 8. shows the renderings of the FEM models. (a) Breast Volume Model (b) Breast Model after Meshing Figure 8. Model before and after meshing The mesh was composed of tetrahedral el ements with 10 nodes (each side has an additional node in the middle to model the de formations more accurately. The elements have a quadratic displacement behavior. The resulted meshed volume is presented in the following figure. This is the finite element of the breast to which the deformations will be applied. For a 10 pixels sample interval in th e original image slice, 52 slices were stacked to construct the volume and an element size of 8 units, the meshing resulted in 4762
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32 nodes and 2952 elements. The number of nodes a nd elements grows ex ponentially and so does the computa tional time. (a) FEM Breast Element Model (b) FEM Breast Model Nodes Figure 9. Finite Element Model Elements and Nodes 3.2.4 Boundary Condition During Xray imaging, force is applied through two plates that moves towards each other to compress the breast. This is a dynam ic contact problem that must be simulated numerically. We will approximate breast defo rmation during compression by incremental stepwise simulation. The underl ying assumption is that the motion of plate is slow enough so that breast deformation in each step can be described by a static equilibrium equation. More importantly, the mesh topology will not be too distorted to aff ect the displacement prediction. In clinical practice, the final co mpression magnitude is recorded, but the force exerted on plates is rarely measured. So, we will specify Dirichlet condition (displacement) on plates. To avoid sliding mo vement between plates and breast we assume that once in contact with plates, the node will move only in the direction of compression. We also assign zero displacement to the node s that lie on the ribs (chest wall). The advantage of this modeling scheme is that it can be applied to both compression and decompression, simply
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33 by changing directions of pl ate movement. The boundary co ndition imposed to our model is that the contact between the breast and the plates is a rough contact without any sliding, and the plates are restricted to move only along one axis. (a) Friction for left plate and model = 0.9 (b) Friction for righ t plate and model = 0.9 Figure 10. Finite Element Model Elements and Nodes 3.3 Compression simulation results To match the features betw een two Xray views, simila r compressions at exactly the same angle are applied to the finite elem ent model of the breast in order to achieve similar displacements and to perform nonri gid registration between Xray views. The compression data recorded from the Xray machine is as follows: Table 3. Compression Values Used in Xray Imaging Xray view Compression angle [degrees] Compression displacement [mm] RML 90 60 RCC 0 59 LML 90 71 LCC 0 60
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34 The results below (Figure 11. and Figure 12.) show the compressed model in CC view and ML view when th e same displacement was added to the FEM model. (a) Phantom CC View Xray (b) Phanto m model CC view after compression simulation Figure 11. Phantom CC view compression simulation (a) Phantom ML View Xray (b) Phanto m model CC view after compression simulation Figure 12. Phantom ML Vi ew Compression Simulation 3.4 Validation 3.4.1 Validation Based on Minimum Euclidean Distance To better understand the algorithm we impl emented, it is impo rtant to remember that no matter the model is compressed in which directio n, when we uncompress it, the correspondent feature points in each view have always the sa me original position. The following procedure is repeatedly used in our experiment: we trace the original position
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35 first, then we combine it with the displacement to determ ine the feature pointÂ’s final position in the correspondent view, Figure 13. Validation Using Minimum Euclidean Distance (a) Original CC View (b) Project ion from Model (c) Registration Figure 14. Nodes Registered to Feature Point
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36 We used a small region to represent feature point 1. Only 2952 elements were used in the model to improve the computing speed. Thus, the nodeÂ’s coordinate may not correspond to each single point. A correspo ndent point is actually a small correspondent region in our experiment. Table 4. Process Illustration Using Node 4719 N ode N umber CCX After Compression CCY After Compression 4719 79.463 132.515 Displacement in X Direction Displacement in Y Direction Displacement in Z Direction 4719 0.56224 0.98684 38.42 CCX Before Compression CCY Before Compression CCZ Before Compression 4719 80.02483 133.5017 98.22318 Displacement in X Direction Displacement in Y Direction Displacement in Z Direction 4719 3.2633 18.199 8.0645 MLX Before Compression MLY Before Compression MLZ Before Compression 4719 80.02483 133.5017 98.22318 MLX After Compression MLY After Compression MLZ After Compression 4719 83.28813 151.7007 106.2877 MLX After Compression MLZ After Compression 4719 83.28813 106.2877 FEM was computed for different displacem ent values applied to breast model, each resulting in physically plausible disp lacements at each node. The method can be further developed to predict the displacement for each tissue inside the breast and for ductile tissues which have nonli near elastic and anisotropic behavior. In this study we considered only two Xray views for matchi ng because so it is today the screening mammography process. Because of this the f eature point in the second view must be searched along a line, which represents the locu s of the feature point back projected from
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37 the first view. Simulated calcification lies on the dotted curve. Ambiguity still exists since two calcifications are on that curve. (a) Feature point 1 in CC View (b) Feature Point 1 in CC View Figure 15. Simulated Calcification with Predicted Curve Xray mammograms of the phantom were us ed to evaluate the homogenous linear tissue model and the mammography compre ssion setup. Using FE M model with the algorithm based on Minimum Euclidean Dist ance, we achieved the coordinates for the same lesion in the CC and ML view when the phantom was uncompressed. We calculated the Euclidean distance of all the nodes listed, generated from differen t views and found the minimum. The two nodes corr esponding to the minimum Eu clidean distance representing the lesion position in 3D coordi nate system. Since each lesion position in CC view and ML correspond to a curve when we uncompress the model to the original state, the two curves would cross each other at one point, which is th e lesionÂ’s exact positio n when the model is not compressed. In perfect situation, the smallest distan ce between the curves in the uncompressed model should be zero. Table4 summarizes th e accuracy of compre ssion simulation using 1Â’ 1Â’Â’ 1 1
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38 FEM deformation model. Further validation was done using MRI data. The computed coordinates from Finite Element Model for f eature points were matched with coordinates calculated from MRI volume set. Distance betw een a feature point and its prediction: 2.6mm 0.8. Comparing with the distan ce between feature point s: 25.9 mm Â– 80 mm, this result showed that the Finite El ement Model can reasonably predict lesion correspondence. Table 5. Minimum Euclidean Distance Simulated Mass and Calcification Minimum Euclidean Distance (mm) 1 0.816 2 1.134 3 0.229 4 1.191 5 0.432 6 0.824 7 0.719 8 1.364 9 0.619 Using generic model, we further tested on pa tient data with visi ble lesion in both CC and ML view. The minimu m Euclidean distances achieved for different from patient data set were larger comparing with phant om study. The accuracy could be improved when models generated from patient MRIs were used.
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39 3.4.2. Validation for Predicted Curve Figure 16. Validation for Predicted Curve In cases when a suspicious feature is f ound in one view but not in the other view, we use FEM model to predict area in which a second reading is strongly recommended to see whether a feat ure was overlooked. We tested on several cases in which lesions were visible in both views using FEM model. The results are shown as in Figur e 17. The shape of the final predicted area depends on the region defined fo r the lesion. In the cases when the whole region of lesion was used, the resulted predicted area covered the lesion in the other view. The detailed procedure is listed below:The uncom pressed breast model is compressed along the y direction with a displacement value id entical to Xray comp ression. After model prediction, the breast model was projected again on a plane perpendicular to the compression direction and the ne w 2D projected image was re gistered against the ML X
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40 ray image. The elements recorded in the CC projection now lie on a curve in the 2D projected image and the featur e corresponding to the same f eature in the Xray CC view is also found on this curve. In order to be able to match a feature from the Xray CC view to Xray ML view, first the breast model is compressed along the z direction with a displacement value identical to Xray compression. To solve the system of pa rtial differential equations, a sparse method solver and a preconditioned c onjugate gradient (PCG) with full NewtonRaphson iterative method were used. The PC G method performs better for solid model which has a large number of elements. After computing the predicted displacement of the nodes, the 3D compressed breast model was projected on XY plane perpendicular to the compression direction, resulting a 2D imag e that should match the Xray CC view. The next step involved in experiment was to re gister the projected image and the CC Xray image, matching the feature pixels from CC Xray image to projec ted node positions. (a) Simulated Lesion in Phantom CC View (b) Lesion Predictio n in Phantom ML View Using Finite Element Model Figure 17. Predicted Lesion Po sition in Phantom ML View After the registration we back project the feature point into th e 3D breast model and record the elements th at correspond to that featur e in Xray CC view. These
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41 corresponding elements are aligned on a line in the compressed model and on a curve in the decompressed volume model. Combining feature pointsÂ’ c oordinates from MRI data, we further calculated the distance between the feature point and pr edicted curve. The minimum Euclidean distance is less than 2.1mm.
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42 Chapter 4 Patient Study 4.1 Image Acquisition and Data Extraction The patient data consist of a set of para llel twodimensional MR images of patient breasts. The distance between slices is 2. 5mm and the voxel size for the MRI is 1.41x1.41x2.50mm. The extracted contour set is loaded to ANSYS FEM software package where it would then be meshed into tetrahedral structural elements. (a) Slice 28 Subject 577 (b) Slice 34 Subject 580 (c) Slice3 Subject 577 Figure 18. Patient MR Images (a) Slice 28 Subject 577 (b) Slice 34 Subject 580 (c) Slice3 Subject 577 Figure 19. Segmented MRI Images after Morphological Operation
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43 4.2 Patient Generic Model As similarity measures the m ean and standard deviation of the Euclidian distance of corresponding landmarks were us ed. Landmarks were defined at prominent regions, which could be identified by physicians. In the pilot study of phantom and patient data, it was shown that a homogeneous linear elastic FEM model can best simulate the deformation of a female breast as applied during Xray mammography. Our goal is to generate a generic mo del which after scaling according to the mammogram data could be used to find corres pondence in mammograms even with no MR volume. Figure 20. Generic Model Scaling A ccording to CC View and ML View (a) CC View Xray (b) Generic Mode l Scaled According to CC View (c) ML View Xray (d) Generic Mode l Scaled Accordi ng to ML View
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44 4.3 Validation Based on Minimum Euclidean Distance Xray mammograms of patients were used to evaluate the homogenous linear tissue model and the mammog raphy compression setup. Us ing FEM model with the algorithm listed in Chapter 3 Phantom Study, we achieved th e coordinates for the same lesion in the CC and ML view when the ph antom was uncompressed We calculated the Euclidean distance of all the nodes listed, ge nerated from different views and found the minimum. The two nodes corr esponding to the minimum Eu clidean distance representing the lesion position in 3D coordi nate system. Since each lesion position in CC view and ML correspond to a curve when we uncompress the model to the original state, the two curves would cross each other at one point, which is the lesionÂ’s exact positi on when the model is not compressed. In perfect situation, the smallest di stance between the curves in the uncompressed model should be zero. Tabl e 6 summarizes the accuracy of compression simulation using FEM deformation model. Using generic model, we further tested on pa tient data with visi ble lesion in both CC and ML view. The minimu m Euclidean distances achieved for different from patient data set were larger comparing with phantom study. The accuracy could be improved when models generated from patient MRIs were used.
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45 (a) Feature Point in CC View (b) Feature Point in ML View Figure 21. Patient Data with Visible Lesion in Both Views The minimum Euclidean distances achieved for different fr om patient data set were larger comparing with phantom study. Th e accuracy could be im proved when models generated from patient MRIs were used. Table 6. Minimum Euclidean Distance Experiment Results for Patient Data Patient data with visible lesion in CC and ML view Minimum Euclidean Distance (mm) 1 1.374 2 2.016 3 2.349 4 2.542 4.4 Validation for Predicted Curve In cases when a suspicious f eature is found in one view but no in the other view, we use FEM model to predict area in which a second reading is strongly recommended to see whether a feature was overlooked. We tested on several cases in which lesion s were visible in bo th views using FEM model. The results are shown as in Fig. 22. The shape of the final predicted area depends on the region defined for the lesion. In the cases when the whole region of lesion was used, the resulted predicted area covered the lesion in the other view.
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46 (a) Lesion in CC view (b) Lesi on prediction in ML view Figure 22. Predicted Lesion Positi on in A Different View The detail procedure is listed as follo ws. The uncompressed breast model is compressed along the y direction w ith a displacement value identical to Xray compression. After model prediction, the breast model was pr ojected again on a plane perpendicular to the compression direction and the new 2D proj ected image was registered against the ML Xray image. The elements recorded in the CC projection now lie on a curve in the 2D projected image and the feature corresponding to the same feature in the Xray CC view is also found on this curve. In order to be able to match a feature fr om the Xray CC view to Xray ML view, first the breast model is co mpressed along the z direction with a displacement value identical to Xray compression. To solve the system of partial differential equations, a sparse method solver and a preconditioned co njugate gradient (PCG) with full NewtonRaphson iterative method were used. The PC G method performs be tter for solid model which has a large number of elements. After computing the predicted displacement of the nodes, the 3D compressed breast model was pr ojected on XY plane perpendicular to the compression direction, resulting a 2D image that should match the Xray CC view. The next step involved in experiment was to re gister the projected image and the CC Xray image, matching the feature pi xels from CC Xray image to projected node positions.
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47 After the registration we back project the f eature point into the 3D breast model and record the elements that correspond to that feature in Xray CC vi ew. These corresponding elements are aligned on a line in the compressed model and on a curve in the decompressed volume model.
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48 Chapter 5 Conclusion Although biomechanical model has been u sed in breast cancer related studies [2, 21, 22, 25], to our knowledge, this is the first attempt of using a sophis ticated finite element model to improve breast cancer detection with two mammographic vi ews. Because finite element model has a unique capability of predicting nonrig id deformation accurately, a significant improvement in twoview ma mmographic reading can be expected. To relieve the computational burden of finite element modeli ng, adaptive meshing technique will be used to redu ce the element number. An in cremental approach is also devised to simulate br east deformation through a series of static equilibrium computation. We design two schemes of using 3D breast model to improve twoview mammographic interpretation. One schem e is for cases where both MRI and mammography are available and subjectspecifi c model will be built from MRIs. In case that MR images are not available, we will u tilizes a generic breast model to facilitate twoview mammography interpretation. This generi c modelbased approach is especially useful for situations where expensive MR im aging is not an affordable option. In this work we presented a FEM for the br east, which can be su ccessfully used in nonrigid registration. The exact values for material propertie s are not critical for this purpose. FEM was computed for different disp lacement values applied to breast model,
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49 each resulting in physically plausible disp lacements at each node. The method can be further developed to predict the displacement for each tissue inside the breast and for ductile tissues which have nonli near elastic and anisotropic behavior. In this study we considered only two Xray views for matchi ng because so it is today the screening mammography process. Because of this the f eature point in the second view must be searched along a line, which represents the lo cus of the feature point back projected from the first view. In the future work more da tasets from Lifetime Screening Center, Tampa FL will be combined into our work, which will be used to achieve higher accuracy of lesion correspondence analysis in breast imaging usi ng finite element model. Xray mammograms of the breast phantom a nd patient were used to evaluate the performance of the FE model and the de scribed algorithm. We computed model prediction error as compared to featur e size and distance of image features. In experiments of compression simulati on using FEM deformation model when suspicious area is visible in 2 views, the sm allest distance between curves for 9 feature points we tested is 0.6mm. Ideally the sma llest distance between the curves in the uncompressed model should be zero. Compared with the distance betw een feature points, 25.9mm to 80mm, this result showed that the Finite Element Model can predict lesion correspondence. Validation was also performe d using MRI data. The computed lesion coordinates were compared with coordinates calculated from MRI volume set. Distance between a feature point and its prediction is 2.6mm based on 9 f eature points in the phantom. To validate the algorithm for cases when suspicious area is visible only in one view, a feature points visible in both views were selected. First the predicted position is
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50 computed (the deformed projected curve), then the minimum Euclidean distance between the real feature position and its prediction is calculated as an indi cator for accuracy. The average error is less than 2.1mm. Additional experiments were performed on patient breast data. 3D breast model was scaled to fit mammographi c projections (just global 3D scaling). The feature points were manually identified on three sets of two view mammograms. The minimum 3D distances for projected curves were comput ed to validate the model and the algorithm. The average minimum distance were below 2.7 mm, somewhat larger than in the phantom study but still very promising. In conclusion, our initial experiment s have shown that we can construct sufficiently details 3D FEM model to establis h correspondences of f eatures identified in two mammographic views. The proposed algo rithm needs to be further tested and validated on larger data set. Further optimization of element sizes and meshing strategies are needed for improved accuracy.
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