PATIENT- SPECIFIC COMPUTATIONAL BIOMECHANICS OF THE BRAIN

Transcription

1 PATIENT- SPECIFIC COMPUTATIONAL BIOMECHANICS OF THE BRAIN Johnny (Yang) Zhang M.Sc. B.Eng. This thesis is presented for the degree of Doctor of Philosophy of The University of Western Australia Intelligent Systems for Medicine Laboratory School of Mechanical and Chemical Engineering The University of Western Australia 2013

2 FOR MY PARENTS PROF. HONG LI AND YUEFEI ZHANG MY WIFE JINGYA CUI FOR ENCOURAGING, SUPPORTING AND ALLOWING ME TO INDULGE IN MY INTEREST

3 Abstract In this research, I studied a method to generate patient-specific biomechanical models without segmentation and meshing by using an alternative framework to commonly used Finite Element method. To utilise computational biomechanics of the brain in practical (clinical) applications, such as computer-aided surgery planning, we need a framework that can 1) generate the patient-specific computational grid efficiently; 2) handle unknown invivo patient-specific material properties; and 3) produce meaningful results from a fully nonlinear simulation in a short time without supercomputers. I created a Fuzzy Mesh-Free Total Lagrangian Explicit Dynamic (FMTLED) framework that met these requirements. The framework utilised a Mesh-Free method so that 1) The biomechanical model worked directly on an unstructured cloud of points that did not form elements; 2) It performed numerical integration over a nonconforming background grid, 3) It assigned material properties to integration points based upon fuzzy tissue classification; and 4) The results were weakly sensitive to the patient-specific mechanical properties (the model could have much less stringent requirements for tissue classifications). I showcased three two-dimensional models and one three-dimensional comprehensive model using FMTLED, based on patient-specific datasets. All studied cases had high resolution pre-operative T1 and T2 Magnetic Resonance Imaging (MRI) taken before the operation and lower resolution intra-operative MRI right during the operation, after craniotomy (skull is opened). I used the intra-operative MRI for validation purpose. 1

4 I compared the FMTLED computation results against 1) intra-operative MRI and 2) results from validated Finite Element models using both contour projection and Hausdorff distance method. The comparisons showed excellent agreement between the FMTLED results, intra-operative MRI (the ground truth) and the Finite Element results. It demonstrated that the FMTLED models, for all practical purposes, produced equivalent accuracy to Finite Element models. Considering the reduced workload during model generation, this research made the FMTLED framework an appealing option for computational biomechanics in clinical applications. 2

5 Preface The market for scientific computations in medicine would be as large as in engineering by Russell H. Taylor Johns Hopkins University Over the years during my PhD candidature, I had witnessed the growing interest in the field of patient-specific modelling for human body-related problems. I expected personalised treatments utilising computational biomechanics becoming available to sufferers all over the world in the near future. This was partly resulting from the increasing demand and interest for more accurate therapeutic technologies. While more essentially, the availability of computing power and more advanced tools and algorithms to carry out such studies had been emerging over the last decades. It now allowed simulations to be run almost anywhere from either parallel clusters, or even personal computers (possibly with GPU parallel acceleration). Patient-specific modelling had great potential to study several aspects of the human body related illnesses and improve clinical outcomes and efficiency of health care delivery, which were otherwise not possible. There were a lot of ups and downs to be one of the explorers venture into the unknown field of patient-specific biomechanical modelling. I was on my mettle, with 3

6 my dedication for scientific research, and my supervisors inducements. I hoisted the sail, heading towards the adventure of my PhD 4

7 Table of Contents: Abstract... 1 Preface... 3 Table of Contents:... 5 List of Figures... 8 Acknowledgements Statement of Candidate Contribution Chapter 1 Introduction Background Motivation of Research Thesis Outline Chapter 2 Literature Review Computational Biomechanics of the Brain for Computer-Assisted Neurosurgery Patient-Specific Modelling Mesh-Free Methods Fuzzy Tissue Classification of MRI

8 Chapter 3 Mesh-Free Computational Grid Influence Domain Shape Functions and Construction of Approximation Choice of Weight Functions Algorithm Overview Chapter 4 Material Properties based on Fuzzy Tissue Classification Chapter 5 Solution Methods without Accurate Segmentation and Material Properties Total Lagrangian Explicit Dynamics Dynamic Relaxation Displacement Loading Brain-Skull Interaction Solution Algorithm Verification Chapter 6 Computation of Brain Deformation with 2D and 3D Patient-Specific Datasets D Cases D Case Chapter 7 Discussion and Conclusions

9 References:

10 List of Figures Figure 1 3D image of brain tumour (shown as green) and ventricle (shown as blue) was projected on the patient as an aid to surgical planning and navigation using Computer-Integrated Surgery (CIS) systems, Courtesy of Surgical Planning Laboratory, Harvard University Figure 2 Comparison of the brain surface determined from images acquired preoperatively with the one determined intra-operatively from the images acquired after craniotomy, superior view. a) Preoperative surface is semitransparent; b) Intra-operative surface is semi-transparent. Deformation of brain surface due to craniotomy is clearly visible. Surfaces were determined from the images provided by Department of Surgery, Brigham and Women s Hospital (Harvard Medical School, Boston, Massachusetts, USA). Image adapted from (Wittek et al., 2007) Figure 3 Comparison of the brain surface determined from images acquired preoperatively with the one determined intra-operatively from the images acquired after craniotomy, inferior view. a) Preoperative surface is semitransparent. Notice lateral deformation of left parietal lobe surface (shift to the right); b) Intra-operative surface is semi-transparent. Deformation of brain surface due to craniotomy on both craniotomy side and the side opposite to craniotomy is clearly visible. Surfaces were determined from the images provided by Department of Surgery, Brigham and Women s Hospital (Harvard Medical School, Boston, Massachusetts, USA). Image adapted from (Wittek et al., 2007)

11 Figure 4 Work Flow to Use the Patient-Specific Computational Biomechanical Model for Neurosurgery. 1 - Neuroimage; 2 - Segmented Neuroimage; 3 - Surface Model Generated from Segmented Neuroimage; 4 - Patient-specific Brain Mesh; 5 - Computational Result Visualisation Figure 5 Comparison between conventional Finite Element biomechanical modelling (left column) and the proposed fuzzy Mesh-Free modelling (right column) Figure 6 The commonly used supports of node I Figure 7 Schematic representation of Mesh-Free discretisation facilitating arbitrary distribution of nodes Figure 8 Workflow of Mesh-Free Total Lagrangian Explicit Dynamic Algorithm Figure 9 Schematic representations of background integration cells that do not conform to the intracranial geometry Figure 10 Comparison between an FE model constructed using segmentation and a fuzzy Mesh-Free model based on fuzzy tissue classification. (a) T2 MRI of the brain, including the tumour and ventricles - notice that no clear boundaries can be easily defined, especially for the tumour; (b) Finite Element model of ventricles generated from segmentation; (c) Finite Element model of the tumour generated from segmentation; (d) the fuzzy membership function for ventricle indicated at each pixel by the level of blue; (e) the fuzzy membership function for tumour indicated at each pixel by the level of red; (f) fuzzy Mesh-Free model of ventricle and (g) fuzzy Mesh-Free model of tumour. Green dots represent nodes while grey grids define the background integration cells. Material 9

12 properties are assigned directly to integration points based on the fuzzy classification results Figure 11 Three cases selected for 2D computation with T2 pre-operative MRI (left column) and intra-operative MRI showing craniotomy area (right column). The MRIs are provided by Computational Radiology Laboratory (CRL), Harvard Medical School Figure 12 Models generated from three different datasets. Left column: Finite Element Models, with parenchyma, tumour (red) and ventricle (blue) modelled separately. Right column: Fuzzy Mesh-Free Model without explicitly separating the tumour and ventricles from the brain; the fuzzy membership functions are indicated by the level of red for tumour and blue for ventricles; nodes are shown as green dots Figure 13 Evaluation of accuracy for the three cases. The colour shows the difference of the simulation results (computed deformation field) over the whole brain [mm] Figure 14 Patient-specific MRI data provided by Computational Radiology Laboratory (CRL), Harvard Medical School. Pre-operative T2 weighted MRI (left) shows the brain contains a large anterior tumour close to the parenchyma surface. Intra-operative MRI (right) was taken right after the craniotomy (skull was opened) Figure 15 Main anatomical components (parenchyma, ventricle and tumour) as surfaces generated from the segmented pre-operative MRIs

13 Figure 16 Geometry discretisation of the patient-specific problem domain with Mesh- Free nodes represented with cloud of blue points Figure 17 Geometry discretisation of the patient-specific problem domain with uniformly distributed, non-conforming integration grid (only one slice is shown) Figure 18 3D Fuzzy Tissue Classification shown in 4 slices. The fuzzy membership functions for tumour and ventricles are indicated correspondingly by the level of red for tumour and blue for ventricles Figure 19: Locations of sections shown in Figure 21. The sections are chosen at 5mm intervals while cutting across the regions of interest (tumour and ventricle) Figure 20 Four axial Intra-operative MRI sections with Fuzzy MTLED deformed contours of tumour and ventricle surfaces (green lines). For reference Finite Element results for ventricle and tumour are shown in red. Notice that most of the red lines are inline with the green contours which mean great consistency. Figure 19 shows the locations of the sections displayed here. The sections are chosen at 5mm intervals while cutting across the regions of interest (tumour and ventricle) Figure 21: Four sagittal Intra-operative MRI sections with Fuzzy MTLED deformed contours of tumour and ventricle surfaces (green lines). For reference Finite Element results are shown in red. Notice that most of the red lines are inline with the green contours which mean great consistency. Figure 19 shows the locations of the sections. The sections are chosen at 5mm intervals while cutting across the regions of interest (tumour and ventricle)

14 Figure 22: Differences (in mm) between Fuzzy MTLED predicted surfaces and Finite Element predicted surfaces of tumour and ventricle. The colours represent the difference d(a,b) described in Equation (4) to the nearest point on the control surface (the same surface deformed by validated Finite Element algorithm). Most of the surfaces are shown blue which represent good matching between the two results. The top image shows the Fuzzy MTLED predicted ventricle surface. The bottom image shows the Fuzzy MTLED predicted tumour surface. The 95% Hausdorff distance for ventricle and tumour surface is respectively mm and mm

15 Acknowledgements I must express my sincere gratitude for the following: W/Prof. Karol Miller, primary advisor of my research, for his supervision; mentoring in my work and life; encouragement when I was feeling frustrated and timely advice during all the time of the candidature. Also I extremely enjoyed the daily conversation during coffee time and of course, the ping pong games. Prof. Adam Wittek, my co-advisor, for his encyclopaedic knowledge of the research literature and patience to sit through hours of discussion with me, explaining, reasoning and correcting my ideas as well as pointing me to the right direction, especially during the first year of my candidature when Karol was away. Dr. Grand Joldes my co-advisor, and Ashley Horton my research colleague for their generous help. I couldn t finish the project without their solid mathematical background and brilliant programming skills. W/Prof. Yinong Liu, Head of School and Prof. Hong Yang, Professor Melinda Hodkiewicz Graduate Research Coordinator for their supporting during my candidature. My research colleagues Dr. Guiyong Zhang, Sachin Shrestha, Dr Gordon Wu, Dr Mu Zhang, Mr. Jingyang Li and Mr. Jiajie Ma etc. who have made the research journal more enjoyable. My grandparents, parents, my wife Jingya Cui and my newborn daughter for their patient support and indulgence. 13

16 Financial support of the Scholarship for International Research Fees (SIRF), Grants for Research Student Training (GRST), Graduate Research Student Travel Award and UWA Convocation Ken and Julie Michael Postgraduate Research Award is gratefully acknowledged. 14

17 Statement of Candidate s Contribution The original contribution and the main novelty of the thesis is the Fuzzy Mesh-free TLED framework. It is a new concept of the modelling process without segmentation and meshing. This concept is a step towards the idea of an image as a model which can substantially change the practise of patient-specific biomechanical modelling as well as other fields that depend on image data as input. The Fuzzy Mesh-free TLED framework development is followed by implementation, application and evaluation of the framework. Chapters 1 and 2 present the introduction, background and the literature review. Each of Chapters 3, 4, and 5 represents a major workflow component of the framework. The novelty element in these chapters is the careful selections of proper individual methods and implementation to fit in the framework. Chapter 3, 4 and 5 all have portions that have been published in peer-reviewed publications as listed below. The bibliographical details of the work with relative contribution of the candidate and other parties to the co-authored work are outlined below. Each author has given permission for this work to be included in the thesis. Zhang, J. Y. (70%), Joldes, G.R (15%), Wittek, A. (5%), & Miller, K. (10%). Patient-Specific Computational Biomechanics of the Brain without Segmentation and Meshing International Journal for Numerical Methods in Biomedical Engineering 29(2): Zhang, J. Y. (70%), Joldes, G.R. (9%), Wittek, A. (5%), Horton, A. (3%), Warfield, S. (3%), Miller, K (10%) (2012). Neuroimage as a Biomechanical Model: Toward New Computational Biomechanics of the Brain. In P. M. F. Nielsen, A. Wittek & K. Miller (Eds.), Computational Biomechanics for Medicine 15

18 Deformation and Flow (pp ): Springer New York. 16

19 Chapter 1 Introduction 1.1 Background Among all surgical operations, removing brain tumours is one of the most challenging surgeries. The operation has to be applied precisely over the patient-specific location of the anatomic / functional abnormality, according to the current (intraoperative) patient s anatomy. On one side, if the tumour cells are not completely removed after surgery, cancer can return. While on the other side, removing too much brain tissue can interfere with the patient s memory and other functions. Nakaji and Speltzer (Nakaji and Speltzer, 2004) listed the accurate localisation of the target as the first principle in modern surgical approaches. Moreover, from the cost point of view, brain surgery is an expensive procedure. There has been a broad international concern about the cost of meeting rising expectations for such procedures, particularly if the number of people requiring such procedures is large. Using improved machinery to help surgeons performing these procedures quickly and accurately with minimal side effects could significantly reduce the cost. The difficulties and the related high cost of brain surgery call for the need of a novel partnership between surgeons and machines to extend the ability to plan and carry our surgical interventions more accurately and efficiently. Computer-Integrated Surgery (CIS) systems, such as computer-aided surgical planning and image-guided intervention could help with surgical precision and patient survival, as shown in Figure 1. Neurosurgical planning for image-guided intervention is typically conducted using pre- * Portions of this chapter have been published in: Zhang, J. Y., G. R. Joldes, et al. (2013). "Patient-specific computational biomechanics of the brain without segmentation and meshing." International Journal for Numerical Methods in Biomedical Engineering 29(2):

20 operative radiographic images. The advantage of using pre-operative radiographic images is the high image quality. Therefore the abnormity can be identified more clearly. The surgeon then can plan more accurately. Figure 1 3D image of brain tumour (shown as green) and ventricle (shown as blue) was projected on the patient as an aid to surgical planning and navigation using Computer-Integrated Surgery (CIS) systems, Courtesy of Surgical Planning Laboratory, Harvard University. However due to the softness of brain tissue, for a number of common procedures, significant deformations can be expected. The deformations lead to severe misalignment between the positions of pathology determined from pre-operative images and their actual positions (Warfield et al., 2005, Roberts et al., 1998). A typical example is the craniotomy operation (i.e. opening of the skull) a common neurosurgical procedure that results in brain shift and tumour movement associated with the shift. As shown in Figure 2 and Figure 3 (Image adapted from (Wittek et al., 2007). Courtesy of K. Miller 18

21 and A. Wittek, Intelligent System for Medicine Lab, UWA), craniotomy can result in deformation of brain surface as large as 10 mm (Miga et al., 2003) (i.e. around 10% of distance between the left and right cortical landmarks). These significant intra-operative soft tissue deformations will compromise the accuracy and applicability of imageguided-surgery and pose one of the key challenges faced by high-quality neurosurgery. Figure 2 Comparison of the brain surface determined from images acquired preoperatively with the one determined intra-operatively from the images acquired after craniotomy, superior view. a) Preoperative surface is semi-transparent; b) Intraoperative surface is semi-transparent. Deformation of brain surface due to craniotomy is clearly visible. Surfaces were determined from the images provided by Department of Surgery, Brigham and Women s Hospital (Harvard Medical School, Boston, Massachusetts, USA). Image adapted from (Wittek et al., 2007). 19

22 Figure 3 Comparison of the brain surface determined from images acquired preoperatively with the one determined intra-operatively from the images acquired after craniotomy, inferior view. a) Preoperative surface is semi-transparent. Notice lateral deformation of left parietal lobe surface (shift to the right); b) Intra-operative surface is semi-transparent. Deformation of brain surface due to craniotomy on both craniotomy side and the side opposite to craniotomy is clearly visible. Surfaces were determined from the images provided by Department of Surgery, Brigham and Women s Hospital (Harvard Medical School, Boston, Massachusetts, USA). Image adapted from (Wittek et al., 2007). The intra-operative imaging would be the most straightforward solution to accurately determine the current position of a tumour and other pathologies during surgery. However the quality of intra-operative imaging suffers from the constraints of the operating room, e.g. intra-operative 3D scanner is incompatible with conventional instruments and surgical practice. They are also expensive in terms of capital and running cost (Steinmeier et al., 1998). Therefore, spatial resolution and contrast of intraoperative images are typically inferior to those of pre-operative ones (Warfield et al., 2005). This problem can, in principle, be solved by aligning (i.e. registering) the high quality pre-operative data to scans of the brain acquired intra-operatively to retain the 20

23 pre-operative image quality. To achieve accurate alignment, the brain deformation must be taken into account, which implies non-rigid registration. Predicting the intraoperative brain tissue deformations is recognised as a critical procedure in image-guided neurosurgery (Gering et al., 2001). In the past, non-rigid registration relied on purely image-based methods such as optical flow (Beauchemin and Barron, 1995, Dengler and Schmidt, 1988), mutual information-based similarity (Viola and Wells III, 1995), entropy-based alignment (Warfield et al., 2001), and block matching (Dengler and Schmidt, 1988). These methods did not take into account mechanical properties of anatomical structures depicted in the image. Therefore they might yield non-physical deformation fields and wrong correspondence between pre-operative and intra-operative images. A viable solution which has drawn particular interest recently is to use patientspecific biomechanical models together with sparse intraoperative information to realistically warp the pre-operative image to intra-operative configurations (Kyriacou et al., 1999, Edwards et al., 1998, Archip et al., 2007, Hu et al., 2007, Miga et al., 2000b, Miga et al., 2001, Miga et al., 2003, Skrinjar et al., 2002, Warfield et al., 2002). In most practical cases, the models utilise the Finite Element method to solve sets of partial differential equations of solid mechanics governing the behaviour of the analysed continuum. Then the models are loaded to reflect the intra-operative scene in order to predict the soft tissue deformation during surgery. To-date, real-time prediction of the brain deformation relies on linear Finite Element procedures in which the deformation is assumed to be infinitesimally small (the equations of solid mechanics are integrated over the undeformed preoperative brain geometry) and the brain tissue is treated as a continuum exhibiting linear stress-strain relationship (Archip et al., 2007, Clatz et al., 2005, Ferrant et al., 2001, Skrinjar et al., 21

24 2002, Warfield et al., 2002). It implies that geometry changes of the analysed continuum are negligible and equations of continuum mechanics can be solved over the initial geometry. However, the brain surface deformations due to craniotomy can exceed 20 mm as reported in (Roberts et al., 1998). These values are inconsistent with the infinitesimally small deformation assumption above. Therefore, in several studies (Hu et al., 2007, Wittek et al., 2009b, Wittek et al., 2010a, Xu and Nowinski, 2001) Finite Element models utilising geometrically non-linear (finite deformations) formulation of solid mechanics are applied. With improved non-linear deformation compensation that ensures plausibility of the predicted displacements, intra-operative procedures can benefit from a less invasive approach and more accurate guidance. All these benefits are achieved based on the assumption that the patient-specific biomechanical model can be generated without significant effort. This is, however, not the case. Over the last five years, the interest in developing patient-specific models for realistic computational simulation has gained more and more attention from research groups and government funding agencies (such as ARC, NHMRC, EPSRC and NIH). These individualised high-quality patient-specific biomechanical models have not yet been used in clinical practice because of the tedious manual steps in the workflow from medical images to model results (Neal and Kerckhoffs, 2010). The majority of such biomechanical models utilise the Finite Element method (FEM). The workflow to use Finite Element models for neurosurgery is illustrated in Figure 4. 22

25 Figure 4 Work Flow to Use the Patient-Specific Computational Biomechanical Model for Neurosurgery. 1 - Neuroimage; 2 - Segmented Neuroimage; 3 - Surface Model Generated from Segmented Neuroimage; 4 - Patient-specific Brain Mesh; 5 - Computational Result Visualisation Segmentation stands as the first bottleneck within the workflow. In the process of Finite Element based patient-specific modelling, surface models are required for each intracranial tissue type. Therefore segmentation is required to exclusively enforce each voxel a binary decision based on tissue types. Manual segmentation of high-resolution 3D volumetric image is a tedious and irreproducible task. It is impractical for processing large amount of data in clinical practice. Fully automatic and unsupervised methods, while having received significant attention in the literature, are still challenging (Balafar et al., 2010). In particular, the segmentation of the pathology (tumour) often requires intensive manual input to achieve good or even acceptable results. This is certainly confirmed by the experience in our laboratory (Wittek et al., 2010a). 23

26 Another necessary but tedious step in the development of the Finite Element model is the creation of a computational grid (or meshing). Nowadays, meshing of a regular geometry is a routine task. Varieties of mesh generation methods are available in the literature such as Delaunay meshing (Shewchuk, 2002) and Marching Cubes (Lorensen and Cline, 1987). However biological objects, especially pathological tissues, often have complicated geometry. Generating a high quality mesh for biological structures still presents considerable challenges. Many algorithms are now available for fast and accurate automatic tetrahedral mesh generation from biological objects, but not for automatic hexahedral mesh generation (Viceconti and Taddei, 2003). Unfortunately, linear tetrahedral element exhibits volumetric locking (Zienkiewicz et al., 1998), especially in case of soft tissues such as the brain, which are modelled as almost incompressible materials (Miller and Chinzei, 1997). Automatic hexahedral mesh generation for complex geometries remains a challenging problem. Even using IA-Mesh (Grosland et al., 2009) - one of the latest developments in hexahedral meshing for biomechanics, an experienced analyst has to manually adjust the mesh (Wittek et al., 2010a). So far, mesh generation for biological structures remains the most tedious and time-consuming step in the application of computational biomechanics. Therefore, to automate the modelling workflow as much as possible becomes the most critical challenge in patient-specific biomechanics. In this research the boundary condition does not represent a major difficulty compared with segmentation and meshing. This is because the modelling method used in this study considers brain shift due to craniotomy as a displacement zero traction problem. The model is loaded by the enforced motion of nodes at the brain surface in the craniotomy area. (Refer to Chapter 5.3 and 5.4 of the thesis). 24

27 1.2 Motivation of Research The aim of the research is to address the major obstacle that stands in the way of widespread use of computational mechanics in medicine, which is efficient generation of patient-specific computational model from medical images, with a focus on imageguided neurosurgery. The rationale is straightforward. In neurosurgical workflow, 3D images (e.g. magnetic resonance images) are acquired (Kikinis et al., 1996, Warfield et al., 2005, Warfield et al., 2008). Then in order for patient-specific biomechanical computations to be practical in clinical environment, computational grids must be obtained from these images automatically and rapidly. Currently this procedure involves image segmentation and meshing. As discussed above, both present themselves as formidable problems that are very difficult to automate. The present work attempts to address this issue by a combination of the below two approaches: I. By assigning material properties to integration points within the problem domain based on Mesh-Free method and fuzzy tissue classification of the image. II. By utilising robust framework and solution algorithms so that models are discretised by Mesh-Free computational grids and material properties are characterised by fuzzy/probabilistic membership functions obtained directly from the image. 25

28 These ideas will lead to a novel concept: Neuroimaging as a Biomechanical Model. Both image segmentation and meshing will be eliminated from the patient-specific model generation pipeline, as shown in Figure 5. Also due to the selection of modelling methodology that considers brain shift due to craniotomy as a displacement zero traction problem, the model was loaded by the enforced motion of nodes at the brain surface in the craniotomy area. Therefore the boundary condition does not represent a major difficulty compared with segmentation and meshing due to the selection of modelling method (refer to Section 5.3 and 5.4 of the thesis). 26

29 PRE-OPERATIVE IMAGING PREPROCESSING SEGMENTATION SURFACE GENERATION FUZZY TISSUE CLASSIFICATION COMPUTATIONAL NODES INSERTION MESHING MATERIAL PROPERTIES BOUNDARY CONDITION & LOADING FINITE ELEMENT MODEL MATERIAL PROPERTIES BOUNDARY CONDITION & LOADING FUZZY MESH-FREE MODEL Figure 5 Comparison between conventional Finite Element biomechanical modelling (left column) and the proposed fuzzy Mesh-Free modelling (right column). 27

30 1.3 Thesis Outline A review of relevant literature is presented in Chapter 2. This includes the review of modelling the brain for image-guided surgery and patient-specific biomechanical modelling. It also includes the review of the two key methodologies utilised in the current study, which are Mesh-Free method and fuzzy tissue classification. Chapter 3 describes the design of Mesh-Free computational framework that fits the current study. This includes the selection of proper influence domain; how to build the shape function; the choice of weight functions and an overview of the algorithm. Chapter 4 presents the assignment of material properties based on fuzzy tissue classification membership functions. The difference between segmentation and tissue classification is clarified. Chapter 5 describes the robust solution method for the fuzzy-mesh-free model even without accurate segmentation and mechanical properties of the living tissue. This includes the Total Lagrangian Explicit Dynamics algorithm, the use of dynamic relaxation method, loading method and the treatment of boundary condition. Chapter 6 investigates the computation of patient-specific brain deformation using the fuzzy Mesh-Free model, respectively in 2D slice model and 3D comprehensive model. Then I evaluate the fuzzy Mesh-Free simulation results against the Finite Element results computed using Abaqus TM (ABAQUS 6.9 User Documentation, Internet Manual, Simulia, Retrieved 10 September 2011) for three cases of craniotomy-induced brain shift. For 2D section cases, I compare the displacement fields within the whole problem domain. For the 3D simulation, in addition to section-wise accuracy 28

31 comparison, I utilise the 3D Hausdorff Distance for a more straightforward quantitative result. Chapter 7 presents the conclusions drawn from the research and also provides suggestions for future research into patient-specific biomechanical modelling. 29

32 Chapter 2 Literature Review 2.1 Computational Biomechanics of the Brain for Computer-Assisted Neurosurgery Computational mechanics has been proven to be a powerful and effective tool for understanding complex physical phenomena. It has enabled technological developments in almost every area of our lives. One of the greatest challenges for mechanists is to extend the success of computational mechanics to fields outside traditional engineering, in particular to biology, biomedical sciences, and medicine (Oden et al., 2003). The application of computational mechanics in neurosurgery has attracted more and more interest in recent years. (Roberts et al., 1998, Warfield et al., 2002, Bucholz et al., 2004, Warfield et al., 2005, Miller et al., 2011a, Wittek et al., 2010a). Biomechanics researchers often come with an engineering or physics background. For these new to the field of brain neurosurgery, it would be beneficial to read the introductory chapters in (Miller, 2011) on brain anatomy (Chapter 2) and brain imaging (Chapter 3). The information in these primers should be easily digestible with no medical background. It can save a considerable amount of time spent studying the specialized literature in these fields. During neurosurgery, the brain significantly deforms (Miller et al., 2011a). Despite the enormous complexity of the brain, many aspects of its response can be reasonably described in purely mechanical terms, such as displacements, strains and stresses. Therefore they can be analysed using established methods of continuum mechanics (Miller, 2011). 30

33 Edwards et al (Edwards et al., 1998) developed a simplified brain model based on a three-component system to study the brain deformation. Solid regions were constrained by the rigid-body transformation and fluid regions were unconstrained. A number of energy models for deformable tissues were compared. The model could be deformed with intraoperative data using a technique similar to active contours. The authors in (Kyriacou et al., 1999) presented a biomechanical model of the brain using a finite-element formulation. Emphasis was given to the modelling of the soft-tissue deformations induced by the growth of tumours and its application to the registration of anatomical atlases, with images from real patients. Different loading methods have been proposed in the literature. Paulsen et al (Paulsen et al., 1999) proposed a model using preoperative information with intraoperative data to construct and drive a 3D computational model. Then it estimated volumetric displacements in order to update the navigational image set. Miga (Miga et al., 2001) proposed a new approach to account for brain shift using computational methods driven by sparsely available operating room data for modelling tissue retraction and resection. Surgical procedures involving the retraction of tissue and the resection of a left front parietal tumour were simulated computationally. The simulations were used to update the preoperative image volume to represent the dynamic operating room environment. Miller and Wittek (Miller and Wittek, 2006, Wittek et al., 2010b) loaded the models through imposed displacements on the model surface. With such loadings prescribed as forced motion of boundaries, the unknown deformation field within the domain depended very weakly on the mechanical properties of the tissue. However most real-time prediction of the brain deformation relied on linear Finite Element procedures in which the deformation is assumed to be infinitesimally small (i.e. 31

34 the equations of solid mechanics were integrated over the un-deformed preoperative brain geometry) and the brain tissue was treated as a continuum exhibiting linear stress strain relationship. Skrinjar et al (Skrinjar et al., 2002) studied two different biomechanical-model-based approach for brain shift compensation: a damped spring mass model and a model based on linear continuum mechanics. Both models were guided by limited intraoperative (exposed brain) surface data, with the aim to recover the deformation in the full volume. Warfield et al (Warfield et al., 2002, Warfield et al., 2000a) developed an algorithm to create enhanced visualizations of tumour and critical brain structures by aligning preoperatively acquired image data with intraoperative images of the patient s brain during surgery. They demonstrated that the implementation was sufficiently fast to capture volumetric brain deformation during three neurosurgeries. The volumetric deformation was inferred through a biomechanical simulation with boundary conditions established via surface matching. Ferrant et al presented an algorithm for the non-rigid registration of 3D Magnetic Resonance (MR) intraoperative image sequences showing brain shift (Ferrant et al., 2001). The algorithm tracked key surfaces (cortical surface and the lateral ventricles) in the image sequence using an active surface algorithm. The volumetric deformation field is then inferred from the displacements at the boundary surfaces using a biomechanical finite element model of these objects. Clatz et al proposed a new model to simulate the 3D growth of glioblastomas multiforma (GBM, the most common and most aggressive malignant primary brain tumour in humans (Clatz et al., 2005). In the work, they used linear Finite Element method (FEM) to simulate the invasion of the glioblastomas multiforma in the brain parenchyma and its mechanical interaction with the invaded structures (mass effect). As discussed before, the brain surface deformations due to craniotomy can exceed 20 mm (Roberts et al., 1998) and tend to be above 10 mm for around 30% of patients (Maurer et al., 1998). These values are inconsistent with the infinitesimally 32

35 small deformation assumption (The assumptions implied that geometry changes of the analysed continuum were negligible and equations of continuum mechanics could be solved over the initial geometry). Therefore, in several studies Finite Element models utilising geometrically non-linear (i.e. finite deformations) formulation of solid mechanics have been applied to compute deformation field within the brain for neuroimage registration. Xu and Novinski et al (Xu and Nowinski, 2001) proposed a novel non-rigid registration method for registration of the Talairach-Tournoux brain atlas with MRI images and the Schaltenbrand-Wahren brain atlas. A non-linear Finite Element method was used to find the local deformation. Finite element equations were governed by constraints in the form of displacements derived from the correspondence relationship between extracted feature points. The advantage of the method is the use of existing brain data atlas so that the degree of freedom of the system is reduced. However for pathological data, use of atlas-based registration is limited and re-meshing is required. Hu et al (Hu et al., 2007) developed a template-based algorithm to build a non-linear 3D patient-specific FE brain model using shell element for membranes and brick element for the brain. To demonstrate the potential of Finite Element modelling in surgical planning, intraoperative brain shift was predicted for two additional head orientations. Two patient-specific Finite Element models were constructed. The resulting models had the same mesh quality with the template model. One of the two Finite Element models was selected to validate the predicted brain shift against data acquired on intraoperative MR imaging. The brain shift predicted using the model was greater (0.5 ~2.3 mm) than that observed intra-operatively. Also, pathological tumour tissue was not modelled separately. Despite facilitating accurate predictions of the brain deformations, the non-linear biomechanical models have been, so far, of little practical importance as the algorithms used in such models led to computation times greatly exceeding the real-time 33

36 constraints of neurosurgery. For instance, Wittek et al (Wittek et al., 2009b) reported the computation time of over 1700 s on a standard personal computer (3 GHz dual-core processor) using a model with around 50,000 degrees of freedom implemented in the commercial non-linear Finite Element solver LS-DYNA. Recently our group (ISML) developed and implemented Total Lagrangian Dynamic Explicit (TLED), a specialised non-linear Finite Element algorithms and solvers for real-time computation of soft tissue deformation (Miller et al., 2011a, Miller et al., 2010a, Miller, 2011, Joldes et al., 2009d, Wittek et al., 2010a, Miller et al., 2007). In Total Lagrangian formulation, stresses and strains are measured with respect to the original configuration. This choice allows for pre-computing of most spatial derivatives before the commencement of the time-stepping procedure. By using explicit time integration, the need for iterative equation solving is eliminated during the timestepping procedure. Miller, Wittek and Joldes et al. used patient-specific biomechanical models together with sparse intra-operative information to realistically warp the preoperative image to intra-operative configurations, i.e. using a computational model that reflected the intra-operative scene to predict the soft tissue deformation during surgery. Verification of the numerical accuracy and numerical performance of these algorithms were reported in the literature (Miller et al., 2007, Joldes et al., 2009d, Joldes et al., 2009b, Wittek et al., 2010a, Joldes et al., 2010a). The algorithm s accuracy for large strain non-linear elastic behaviour was validated using a reputable commercial software ABAQUS. The average number of floating-point operations per element per time step was 35% lower than for the similar implementation of the algorithm based on updated Lagrangian formulation. Joldes et al. implemented the algorithms on a graphics processing unit (GPU) using the new NVIDIA Compute Unified Device Architecture (CUDA). (Joldes et al., 2010b) It led to more than 20 times increase in the computation 34

37 speed. This made possible the use of more elements / integration points to better represent the geometry. 35

38 2.2 Patient-Specific Modelling Patient-specific modelling is the development of computational models that are individualised based on patient-specific data. It has great potential to improve diagnosis and optimise clinical treatment by predicting outcomes of therapies and surgical interventions. Most current medical diagnostic practices lead to rough estimates of outcomes for a particular treatment plan. The treatments and their estimated outcomes are usually based on the results of clinical trials (Neal and Kerckhoffs, 2010). However these results might not apply directly to individual patients as they are based on averages (Kent and Hayward, 2007). As an alternative, patient-specific modelling can be used to tailor treatment to optimise for the individual s therapy. With patient-specific modelling, a prior knowledge of post-operative conditions can be provided before the therapy and help the clinicians to make the best decision. Therefore patient-specific modelling has great potential to study and research several aspects of human-related illness, which are otherwise not possible. For example, authors in (Rajagopal et al., 2008) presented a modelling framework for creating patient-specific biomechanical models of the breast to capture realistic breast shape and internal deformations across MRIs. Avril et al. presented the patient-specific numerical model of leg compression in the treatment of venous deficiency (Avril et al., 2012). They used patient-specific material properties of both the compression stocking and biological soft tissues inside the leg to account for the inter-individual variability of tissue material property. The patient specific modelling approach is currently being put into practice to assist in managing a wide range of different medical conditions, such as in orthopaedics (Cohen et al., 2003, Hunziker et al., 2006, Woo et al., 2006, Chao et al., 2004, Delay et al., 2001), cardiology (Aguado-Sierra et al., 2011, Sermesant et al., 2006, Taylor and 36

39 Figueroa, 2009) and neurology (Cusick and Yoganandan, 2002, Keshner and Kenyon, 2004, Petit et al., 2004). Aside from the mainstream research areas discussed above, patient-specific modelling has also been employed to model other anatomical entities, such as lung (Sundaresan et al., 2009, Shi et al., 2006), larynx (Li et al., 2008) and kidney (Zhou et al., 2008). Over the last few years, the interest in developing patient-specific model for realistic computational simulation has gained more and more attention from government and other funding agencies. In 2006, the European Commission funded the action entitled STEP: Structuring the Euro Physiome ( which led to the famous Virtual Physiological Human (VPH) project ( The Virtual Physiological Human now forms a core target of the 7th Framework Programme ( of the European Commission, and aims to support the development of patient-specific computer models and their application in personalised and predictive healthcare. In November 2007, the National Institute of Health (NIH) posted a funding opportunity announcement (PAR ) regarding patient-specific modelling. One of the major goals of this grant was to stimulate the design of realistic computational models to make predictions about clinical outcomes. As a direct result of the American Recovery and Reinvestment Act of 2009, the National Institute of Biomedical Imaging and Bioengineering announced a challenge grant Towards the Virtual Patient, with the same goal as the prior NIH announcement. More recently, Engineering and Physical Sciences Research Council (EPSRC), UK funded a Patient Specific Modelling Network ( project until 2012 to improve collaboration from different disciplines and address the difficulties faced by patient-specific modelling community. 37

40 Despite the growing attention and amount of work in this field, studies of patient-specific modelling for large patient populations are still very limited. Patientspecific modelling has not yet become a standard of care in clinical practice. The evaluation of the predictive capability of these models has not yet been performed on a large scale (Antiga et al., 2008). The likely cause of the paucity of studies is the fact that the process from image acquisition to simulation results is still based on several operator dependent steps. It involves image segmentation, meshing. Both of them are still open problems. This makes the use of image-based patient-specific modelling for large-scale studies non-trivial and encumbered by time-consuming operator-dependent tasks. They also constitute a potential source of error. The influence is difficult to control and quantify. Therefore again, to automate the modelling workflow as much as possible becomes the most critical challenge in patient-specific biomechanics. To further justify the importance of patient-specific modelling I have to clearly distinguish whether the intended application is generic or patient-specific. For example, biomechanical model of the brain has two distinct applications: one is neurosurgical simulation for surgical training and skill assessment; the other is neuroimage registration for image-guided-surgery (Miller, 2011). The goal of surgical simulation research is to model and simulate deformable materials for applications requiring realtime visual and haptic interactions. These feedbacks require computing the deformation field and the interaction force between the surgical tool and the tissue at frequencies of at least 500 Hz (DiMaio and Salcudean, 2005). Also when a simulator is intended for surgical training, a generic model developed form average organ geometry and material properties can be used while patient-specific model is not required. This application is not the focus of the current study. 38

41 This thesis focuses on the other application - modelling the brain for imageguided surgery. In this application, while it does not require interaction forces (accurate stress and reaction forces are not the focus). Instead I am only interested in the finalstate displacement of the area of interest (such as tumour). For practical operation planning and computer-aided surgery, a generic model without patient-specific information (such as the position of the tumour) used in surgical training simulation is of little use. 39

42 2.3 Mesh-Free Methods Numerical computation of patient-specific soft tissue deformation is typically based on Finite Element Analysis (FEA) (Bathe, 1996, Cotin et al., 1999, Székely et al., 2000, Picinbono et al., 2003, Luboz et al., 2005, Wittek et al., 2007, Miller et al., 2007). The results from these finite element experiments were promising. They demonstrated that a high level of precision could be achieved in real-time simulations of surgical procedures using nonlinear (both geometric and material) biomechanical models. However there are limitations. Due to mesh-based interpolation, the accuracy of the finite element calculation relies heavily on the element mesh that discretises the geometry. Distorted or low quality meshes lead to high errors. Hence, it is desirable to use only good-quality hexahedral elements. When the geometry is highly irregular, an experienced analyst is required to manually create such a mesh, which consumes valuable human time. It is not guaranteed to be feasible in limited time for complex three-dimensional geometries. The good hexahedral mesh used in (Wittek et al., 2007) took more than 2 months to generate by an experienced analyst. This is a major bottleneck in the efficient generation of patient-specific models for real-time simulation of surgical procedures. Mesh-Free methods were developed with the objective of eliminating part of the difficulties associated with the reliance on a mesh to construct the approximation. In Mesh-Free methods, the approximation (corresponding to interpolation in FE methods) is built from nodes only. The advent of the Mesh-Free idea dated back from 1977, with Monaghan and Gingold (Gingold and Monaghan, 1977) and Lucy (Lucy, 1977) developing a Lagrangian method based on the Kernel Estimates method to model astrophysical phenomena without boundaries. This method, named Smoothed Particle Hydrodynamics (SPH), was a particle method based on the idea of replacing the fluid 40

43 by a set of moving particles and transforming the governing partial differential equations into the kernel estimates integrals (Liu, 2003). Despite the success of the SPH in modelling astrophysics phenomena, it was only after the 90 s that the SPH was applied to others classes of problems. This consequently exposed some inherent problems of the method, such as interpolation consistency (Swegle et al., 1995, Belytschko et al., 2000, Xiao and Belytschko, 2005) and difficulty in enforcing essential boundary conditions (Belytschko et al., 1996). They were consequences of limited usage of SPH in modelling bounded problems, since originally the SPH was formulated to model open problems, such as exploding stars and dust clouds. Over the past years, many improvements were incorporated into SPH in an attempt to restore the consistency and consequently the accuracy of the method (Belytschko et al., 1996, Bonet and Kulasegaram, 2000, Bonet and Lok, 1999, Dilts, 2000, Dilts, 1999, Johnson et al., 2000, Johnson and Beissel, 1996, Libersky et al., 1997, Vila, 1999). Several techniques have been proposed to solve the problem of imposing essential boundary condition in mesh-free methods. They mainly fall into two categories. One is the Lagrange multiplier method. The other one is the penalty method. For the Lagrange multiplier method, there are several choices for the approximation space for the Lagrange multipliers (e.g. finite element interpolation on the boundary, mesh-free approximations on this boundary and the point collocation method which uses the Dirac delta function). One drawback of the Lagrange multiplier method is the introduction of additional unknowns to the problem. In addition, due to the introduction of zero terms on the diagonal of the system of equation matrix, the matrix is no longer positive definite. The main advantage of the penalty method compared with the Lagrange multiplier approach is that no additional unknowns are required. However, the conditioning of the matrix much depends on the choice of the penalty parameter. 41

44 What is more, in the penalty method, the constraints are only satisfied approximately. For detailed description, one can refer to (Nguyen et al., 2008). Since then the SPH method had been successfully applied to a wide range of problems such as free surface, impact, magneto-hydrodynamics, explosion phenomena, heat conduction and many other computational mechanics problems (Bonet and Kulasegaram, 2000, Monaghan, 1988, Monaghan, 1982, Libersky et al., 1993). Also in 1995 Liu proposed the Reproducing Kernel Particle method (RKPM) (Wing Kam et al., 1995) in an attempt to construct a procedure to correct the lack of consistency in the SPH method. The RKPM has been successfully used in multiscale techniques, vibration analysis, fluid dynamics and many other applications. Later, an extension of RKPM based on moving least square approximation resulted in the Moving Least Square Reproducing Kernel Particle method (Liu et al., 1997, Li and Liu, 1996). The first Mesh-Free method based on the Galerkin technique was only introduced over a decade after Monaghan and Gingold first published the SPH method. The Diffuse Element Method (DEM) was introduced by Nayroles and Touzot in 1991 (Nayroles et al., 1992). The idea behind the DEM was to replace the FEM interpolation within an element by the Moving Least Square (MLS) local interpolation (Breitkopf et al., 2004). In 1994 Belytschko and colleagues introduced the Element-Free Galerkin Method (EFG) (Belytschko et al., 1994), an extended version of Nayroles s method. The Element-Free Galerkin method was one of the first Mesh-Free methods based on a global weak form. It was found to be more accurate than the Diffuse Element method, although the improvements implemented in the method increased its computational costs (Belytschko et al., 1994). EFG is one of the most popular Mesh- Free methods and its application has been extended to different classes of problems such as fracture and crack propagation, wave propagation, acoustics and fluid flow. 42

45 Many authors state that the use of a background cell to perform the numerical integration eliminates the Mesh-Free characteristic of the EFG, therefore the method is not truly Mesh-Free (Rabczuk and Belytschko, 2005, Wen et al., 2006, Li and Liu, 2002, Atluri and Shen, 2002b). However, in the context of using Mesh-Free method for neurosurgical simulation, the numerical integration cells don t have to comply with the problem geometry, therefore can be easily generated (Zhang et al., 2012b, Miller et al., 2012, Horton et al., 2010). In contrast to RKPM and the EFG method that use intrinsic basis, other methods were developed that use an extrinsic basis and the partition of unity concept (Romero and Armero, 2002, BabuŠKa and Melenk, 1997). This extrinsic basis was initially used to increase the approximation order similar to a p-refinement as, e.g. Duarte and Oden introduced in 1995 the H-p Cloud method which is a Mesh-Free method based on h and p enrichment of the approximation functions (Duarte and Oden, 1996, Liszka et al., 1996). Melenk and Babuška (Melenk and Babuška, 1996) pointed out the similarities of Mesh-Free and Finite Element methods and developed the so-called partition of unity Finite Element method (PUFEM). Another class of Mesh-Free methods are methods that are based on local weak forms where the integration is carried out in local subdomains. The most popular method is the Mesh-Free local Petrov Galerkin (MLPG) method (Atluri and Zhu, 1998, Atluri and Zhu, 2000, Atluri and Shen, 2002a). Atluri (Atluri and Shen, 2002b) introduced the notion truly Mesh-Free since no construction of a background mesh is needed for integration purposes which eliminates the need of the background cell and, consequently, performs the numerical integration in a meshfree sense. The MLPG uses the Petrov-Galerkin method in an attempt to simplify the integrand of the weak form. The method was originally formulated using the MLS technique and after Atluri 43

46 extended the MLPG formulation to other Mesh-Free approximation techniques. The freedom of choice for the test function in the Petrov-Galerkin method gives rise to different MLPG schemes (Atluri and Zhu, 2000). The MLPG and its different schemes have been applied to a wide range of problems such as Euler-Bernoulli Beam Problems, solid mechanics, vibration analysis for solids, transient heat conduction, amongst many others. The main difference of the MLPG method to methods such as EFG or RKPM is that local weak forms are generated on overlapping subdomains rather than using global weak forms. Radial basis functions (RBF) were first applied to solve partial differential equations in 1991 (Kansa, 1990a, Kansa, 1990b). The direct Collocation procedure used by Kansa is relatively simple to implement, however it results in an asymmetric system of equations due to the mix of governing equations and boundary conditions. Moreover, the use of Multiquadric RBF results in global approximation, which leads to a system of equations that is characterised by a dense stiffness matrix. The RBF Hermite- Collocation was proposed as an attempt to avoid the asymmetric system of equations. Both globally and compactly supported radial basis functions have been used to solve PDEs and results have shown that the global RBF yielded better accuracy. However the compactly supported stiffness matrix is sparse, while the global RBF results in a dense matrix that may become very expensive to solve in the case of a large number of collocation points. Another class of Mesh-Free method is Point Interpolation method (PIM) which uses the Polynomial Interpolation technique to construct the approximation. It was introduced by Liu in 2001 as an alternative to the Moving Least Square method (Liu and Gu, 2001). The PIM, originally based on the Galerkin method, suffers from the singularity of the interpolation matrix and also from the fact that it does not guarantee 44

47 the continuity of the approximation function. Several approaches have been investigated by Liu in an attempt to overcome these problems (Liu, 2003) such as Local Radial Point Interpolation methods (LRPIM). The LRPIM has been applied to solid mechanics, fluid flow problems and others(liu and Gu, 2001). Overall, among most Mesh-Free methods, they share the common advantages such as (i) large deformation can be handled more robustly, (ii) natural solution with moving discontinuities such as crack propagation, (iii) no mesh alignment sensitivity, (iv) higher-order continuous shape functions, (v) non-local interpolation character and (vi) h-adaptivity is simpler to incorporate in Mesh-Free Methods than in mesh-based methods. Beside these advantages, Mesh-Free Methods are not without disadvantages. The Mesh-Free methods shape functions are rational functions. They requires highorder integration scheme to be correctly computed. The treatment of essential boundary conditions is not as straightforward as in mesh-based methods since they do not satisfy the Kronecker delta property. In general, the computational cost of Mesh-Free Methods is higher than that of FEM (Flyer et al., Kim et al., 2002, Chen and Wang, 2000). To avoid some difficulties inherent in Mesh-Free Methods, Mesh-Free Methods were coupled successfully to Finite Element methods (Belytschko et al., 1995, Huerta and Fernández-Méndez, 2000, Berger et al., 2008). Meanwhile, hybrid methods are available that exploit the advantages of Mesh-Free methods and Finite Element method (Idelsohn et al., 2004, Liu et al., 2004, Huerta et al., 2004, Rabczuk et al., 2006), e.g. the shape functions fulfil the Kronecker delta property while simultaneously exploiting the smoothness and higher-order continuity of Mesh-Free shape functions. In the context of computational biomechanics, Horton et al. and Miller et al. firstly apply Mesh-Free method in brain modelling (Horton et al., 2006, Horton et al., 2007, Horton et al., 2010, Miller et al., 2012). They used a modified element-free 45

48 Galerkin method combined with Total Lagrangian Explicit Dynamics (Miller et al., 2007) to deal with fully nonlinear (both geometric and material) formulations and large (finite) deformations of tissues. In (Miller et al., 2012), Miller et al. showcased the Mesh-Free Total Lagrangian Explicit Dynamics Method (MTLED) and demonstrated the methods ability to fulfil all of the important requirements for surgical simulation (e.g. 1. Fully nonlinear (both geometric and material) formulations to deal with large (finite) deformations of tissue; 2. Producing meaningful results in a short time on consumer hardware; and 3. Not requiring significant manual work while discretising the problem domain). However to generate the model, segmentation of the image into distinct tissue classes is still required in order to assign material properties to the integration points. In the current work, I further improved the method by abandoning image segmentation, and using fuzzy tissue classification to reduce the difficulties in the model generation process. 46

49 2.4 Fuzzy Tissue Classification of MRI Tissue classification plays a crucial role in many medical imaging applications, including image-based modelling and computer-integrated surgery. However, classification of voxels exclusively into distinct classes is a difficult task because of the common artefacts in the medical images such as noise and partial volume effects (Pham and Prince, 1999a). Partial Volume effect represents a blurring of intensity across boundaries where multiple tissues are present in a single voxel. It is very common in medical images. Particularly for 3D MRI data, the resolution is not isotropic, and in many cases is quite poor along one axis of the image (Pham, 2000). To compensate for these artefacts, the most common approach is to allow regions or classes to overlap, called soft or fuzzy classification methods. There has recently been growing interest in development of soft or fuzzy classification methods (Bezdek et al., 1993, Udupa and Samarasekera, 1996, Brandt et al., 1994, Yang and Fei, 2011, Lin et al., 2012, Chang et al., 2012, Caldairou et al., 2011). In fuzzy tissue classifications, voxels may be classified into multiple classes with a varying degree of membership. The membership thus gives an indication of where noise and partial volume averaging have occurred in the image. In addition to partial volume effect, another major difficulty that is specific to the classification of MRI is the intensity inhomogeneity artefact (Condon et al., 1987, Simmons et al., 1994). This artefact is typically caused by non-uniformities in the RF field during acquisition. It causes spatial changes in tissue statistics, i.e., mean and variance of the same tissue class vary over the image domain. This artefact can significantly degrade the performance of classification that assumes the intensity value of a tissue class constant over the image (McVeigh et al., 1986, Wicks et al., 1993). Although improvements in scanner technology such as nonparametric, multichannel 47

50 methods have reduced this artefact, inhomogeneities remain a problem, particularly in images acquired by using surface coils (Ji et al., 2012, Tsang et al., 2013). The removal of the spatial intensity inhomogeneity from MRI is difficult because the inhomogeneities could change with different MRI acquisition parameters from patient to patient, and from slice to slice. Therefore, the correction of intensity inhomogeneities is usually required for every new image. Standard fuzzy classification algorithms cannot effectively compensate for intensity inhomogeneities either. Many approaches had been proposed in the literature for performing tissue classification in the presence of intensity inhomogeneity artefacts. Most of them assumed that the shading in MRI was well modelled by the product of the original image and a smooth slowly varying gain field (Wells Iii et al., 1996, Pham and Prince, 1999b, Van Leemput et al., 1999, Zhang et al., 2001, Ahmed et al., 2002, Liew and Yan, 2003, Li et al., 2009a, Li et al., 2009b, Sikka et al., 2009, Ji et al., 2010, Ji et al., 2011). Among them, many were based on Fuzzy C-Mean method (Pham and Prince, 1999b, Ahmed et al., 2002, Liew and Yan, 2003, Li et al., 2009a, Li et al., 2009b, Sikka et al., 2009, Ji et al., 2010, Ji et al., 2011). Typically, corrupted images were segmented using either a two-step approach or a one-step segmentation algorithm that could simultaneously classify the voxels and compensate for the shading effect. In the two-step approach, the image was first corrected to remove intensity inhomogeneities. A standard segmentation algorithm that assumed no inhomogeneity followed this correction. Numerous methods had been proposed in the literature to perform the correction step. Dawant et al. (Dawant et al., 1993) used manually selected reference points in the image to guide the construction of a spline correction surface. The performance of this method depended substantially on the labelling of the reference points. Considerable user interactions were usually required to obtain good correction 48

51 results. Later, authors in (Gilles et al., 1996) proposed an automatic and iterative B- spline fitting algorithm for the intensity inhomogeneity correction of breast MRI. The application of this algorithm was restricted to MRI with a single dominant tissue class, such as breast MRI so that was not quite suitable for brain MRI (typically at least three dominant tissue classes). Another polynomial surface fitting method (Brechbuhler et al., 1996) was proposed based on the assumption that the number of tissue classes, the true means, and standard deviations of all the tissue classes in the image were given. Unfortunately, the required statistical information was usually not available. Therefore more and more researchers focused on the one-step method. It could simultaneously compensate for the shading effect and segment the image. These methods offered the advantage of being able to interleave bias field correction and segmentation in an iterative process. Also they could use intermediate information from the iteration and benefit from each other. Pham and Prince (Pham and Prince, 1999b) proposed an adaptive FUZZY C- MEAN (AFUZZY C-MEAN) algorithm, which incorporated a spatial penalty term into the objective function (In optimisation problem, the objective function is the equation to be optimised given certain constraints and with variables that need to be minimised or maximised) to enable the estimated membership functions to be spatially smoothed. Ahmed (Ahmed et al., 2002) added a neighbourhood averaging term to the objective function, and thus developed the bias-corrected FUZZY C-MEAN (BCFCM) algorithm. Liew and Yan (Liew and Yan, 2003) used a B-spline surface to model the bias field and incorporated the spatial continuity constraints into fuzzy clustering algorithms. Li (Li et al., 2009a) proposed an energy-minimization approach to the coherent local intensity clustering (An integrated objective function with energy on a bias field, membership functions of the tissues, and the parameters that approximate the 49

52 true signal from the corresponding tissues), with the aim of achieving tissue classification and bias field correction simultaneously. Ji Z, etc (Ji et al., 2010, Ji et al., 2011) incorporated the global information into the coherent local intensity clustering model to enhance its robustness to the involved control parameters. Although it improved the segmentation accuracy, it also dramatically increased the computational complexity. Also various kernel techniques had been used to improve the performance of Fuzzy C-Mean approaches (Chen and Zhang, 2004, Liao et al., 2008, Yang and Tsai, 2008). Chen and Zhang (Chen and Zhang, 2004) replaced the original Euclidean distance with a kernel-induced distance and supplemented the objective function with a spatial penalty term, which modelled the spatial continuity compensation. Yang and Tsai (Yang and Tsai, 2008) proposed an adaptive Gaussian-kernel-based FUZZY C- MEAN (GKFCM) algorithm with the spatial bias correction. Liao (Liao et al., 2008) developed a spatially constrained fast kernel FUZZY C-MEAN (SFKFCM) clustering algorithm to improve the computational efficiency. However, the clustering performed in a kernel space was generally very time consuming. criteria: In the current study, the fuzzy tissue classification method requires the following 1) 3D the algorithm must be able to do clustering in 3D space; 2) Inhomogeneity Compensation the algorithm must have inhomogeneity compensation capabilities; 3) Input Parameters the algorithm should not have too many input parameters to adjust the performance, otherwise the output will be more subjective depending on the input parameters; 4) Robust the algorithm must be able to produce results for general MRIs; 50

53 Based on the above selection criteria, the fuzzy tissue classification used in this study was developed based on a bias-corrected FUZZY C-MEAN (BCFCM) algorithm developed by (Ahmed et al., 2002). The BCFCM algorithm was formulated by modifying the objective function of the standard fuzzy c-means algorithm to compensate for inhomogeneities and to allow the labelling of a voxel to be influenced by the labels in its immediate neighbourhood. Since the computation speed for tissue classification is not a big concern in this prototype stage, the code was implemented in Matlab. 51

54 Chapter 3 Mesh-Free Computational Grid In the past few years, there has been growing interest in Mesh-Free methods for solving partial differential equations numerically, especially in engineering community. There are several key concepts in Mesh-Free method, including influence domain, shape function and weight function. 3.1 Influence Domain The influence domain for a node is the bounded region S Ωthat contains all the points whose shape functions (see Section 3.3) are influenced by that node. The most commonly used supports are circular support and rectangular support, as shown in Figure 6. Figure 6 The commonly used supports of node I I use the simplest form of influence domain, which is a sphere with radius r. The value of r is based on factors such as the size of the problem domain Ω, the density of nodes and the required accuracy. All points within this radius are considered to be influenced by the centre node, see Figure 7. The support domain for a point includes all * Portions of this chapter have been published in: Zhang, J. Y., G. R. Joldes, et al. (2013). "Patient-specific computational biomechanics of the brain without segmentation and meshing." International Journal for Numerical Methods in Biomedical Engineering 29(2):

55 the nodes that have that point in their influence domains. These nodes will be used for shape functions computation at that point. The construction of influence domains is of critical importance since it is the basis for computing shape functions and their derivatives. If influence domains are too large, too many nodes will be included in the support domain. The shape function will lose its local compact support characteristics. It will influence the approximation accuracy. Also due to computational complexity requirements, a large influence domain will normally increase the computation burden. At the same time, if influence domains are too small and the support domain does not have enough nodes, the shape function computation will become singular. In (Horton et al., 2010) the authors fixed the number of nodes per integration point by modifying the support domains for each integration point. The benefit of that approach was the control over the number of nodes involved in the computation of shape functions, so that the possibility of using either too many or too few nodes per integration point was eliminated. However, due to different size of support domains, the shape functions might not be continuous in the problem domain. It might introduce problems when calculating shape function derivatives or calculating displacements for a point that was not an integration point and could influence the convergence of the solution. In this work, I eliminate the restrictions on the number of nodes per integration point, to ensure the continuity of the shape functions. 53

56 Figure 7 Schematic representation of Mesh-Free discretisation facilitating arbitrary distribution of nodes 3.2 Shape Functions and Construction of Approximation Mesh-Free methods construct approximations only in terms of nodes (see Section 3.1). How to build the approximation function is the most significant feature of the method. There are a number of ways to construct shape functions proposed in the literature, such as kernel methods, moving least square methods (MLS) and partition of unit methods. All these methods share many features of the same framework. For example, in most cases, MLS methods are identical to kernel methods when the parent kernel is identical to the weight function and the same consistent basis is used. In other words, a consistent discrete kernel approximation must be identical to the related moving least square approximation. Also any MLS may serve as a partition of 54

57 unity(belytschko et al., 1994). Therefore in this study, moving least squares shape functions are used due to their simplicity and robustness. These shape functions were initially developed by (Lancaster, 1981) and used in Mesh-Free methods in the Diffuse Element Method (Nayroles et al., 1992). For any point x Ω, the scalar valued field variable ux ( ) is approximated with h u ( x ) by the inner product of a vector of the polynomial basis p(x) and a vector of the coefficients a(x) as: u h (x) = m! j=1 p j (x)a j (x) =p T (x)a(x) (1) T Where m is the number of terms in the polynomial basis functions p ( x ), and axis ( ) an m-vector of unknown shape function coefficients that depend on spatial coordinates x. Examples of the commonly used bases are the polynomial bases: p'(x; 1, 1)={1, x} p'(x; 1, 2)={1, x, y} p'(x; 2, 2)={1, x, y, x 2, y 2, xy} (2) p'(x; 3, 2)={1, x, y, x 2, y 2,xy,x 2 y, xy 2 } p'(x; 3, 3)={1, x, y, z, x 2, y 2,z 2, xy, yz, xz, x 3, y 3,z 3, x 2 y, xy 2, y 2 z, yz 2,z 2 x,zx 2, xyz} where p (x; k, n) are the k th order n-dimensional polynomial basis functions of x. The number of coefficients m depends on the order of polynomial and dimensions. In the current research, first order basis function is utilised. To calculate the coefficient vector ax ( ), a functional J representing the h difference between the local approximation u ( x ) and the function u( x ) at the n nodes is created from the weighted residuals. 55

58 n h 2 ( ) = [( ( i) - i) w( i)] i= 1 J x u x u x x (3) Where n is the number of nodes in the influence domain and w( x x ) is the weight function based on the Euclidean distance between the nodes xi and the centre point x to make distinction between nodes that are very close to x and those are only just inside the influence domain. i For any x, ax ( ) is chosen to minimise the functional J by considering J = 0 a (4) which leads to the following set of linear equations for a(x) A(x)a(x) = B(x)U where U T = {U 1,U 2,,U n } n! A = W i (x)p(x I )p T (x I ) i=1 B = {W 1 (x)p(x 1 ),W 2 (x)p(x 2 ), W n (x)p(x n ),} (5) The vector of coefficients a(x) can be written in matrix form: a(x) = A(x)!1 B(x)u (6) A(x)is an m x m matrix, known as the weighted moment matrix. It is defined as n A(x) = " w(x! x i ) p(x i )p T (x i ) (7) i Matrix A has to be non-singular for every sampling point x in the domain to make the equation solvable. 56

59 B(x) is an m x n matrix, defined as B(x) =! B 1 B 2! B " n # $ (8) B i = w(x! x i )p(x i ) (9) u is an n-vector. It contains the value of the field variable at each node in the support domain. u =! u 1 u 2! u " n T # $ (10) axis ( ) substituted back into equation so that approximant can be expressed as u h (x) = n m "" i n " = # i (x)u i i=1 j p j (x)(a(x)!1 B(x))u i (11) which takes the form of an inner product between vectors of shape functions and nodal values u. As in the Finite Element method, φi ( x) is the shape functions for the ith node in the support domain. The shape function and its derivative can be obtained as!(x) = p'(x)a "1 (x)b(x)!,i = p',i A "1 B + p' A "1,i B + p' A "1 B,i (12) A "1,i = "A "1 A,i A "1 It should be noted that the approximation over the whole problem domain is no longer a polynomial based on the shape function (Belytschko et al., 1996). However if u(x) is a polynomial, it can be reproduced exactly by any function included in the basis 57

60 space. If the weight function W I (x) and its first k th derivatives are continuous, then the shape function Φ(x) and its first k th derivatives will be continuous. Same as other numerical methods, Mesh-Free method must converge. This means that the numerical solution obtained by the Mesh-Free method must approach the exact solution when the nodal spacing approaches zero. For any numerical method to converge, it must be stable and consistent. Stability is associated with the quadrature of the Galerkin form and the character of the Galerkin procedure. Consistency inherently arises from the character of the approximation. The minimum requirements for consistency depend on the order of the partial differential equations to be solved so that the approximation using these shape functions is capable to exactly reproduce the function (in all the elements or cells that forms the entire problem domain). For partial differential equations of order 2k, solution by a Galerkin method requires consistency of order k, i.e., a constant field for the k th derivatives must be represented exactly as the discretization parameter h tends to zero. For a second order PDE, this implies that consistency is satisfied if constant first derivative can be represented exactly. Consistency conditions are closely related to completeness and reproducing conditions. An approximation is complete if it provides a basis that can produce the function with an arbitrary order of accuracy. Any approximation that can exactly reproduce linear polynomials can reproduce any smooth function and its first derivative with arbitrary accuracy as the approximation is refined, and approximation that has linear consistency also has linear completeness (Belytschko et al., 1996). Reproducing condition refers to the ability to reproduce a function if the nodal values are set by the function. Therefore, the ability to reproduce n th order polynomials is equivalent to n th order consistency. The order of the consistency of an approximation is also called the 58

61 order of the polynomial that can be exactly represented, and consistency conditions are often expressed in terms of the order of the polynomial that can be exactly represented. I use low-order monomials as basis functions, but it is possible to use more advanced functions in the basis to deal with singularities that may arise in certain problems. It should be noted that in MTLED all shape functions and their derivatives are calculated entirely in the pre-processing stage. 3.3 Choice of Weight Functions The weight functions W(x) play an important role in the performance of the Mesh-Free methods. They are used in all varieties of Mesh-Free methods. The weight functions determine the continuity of the shape function and therefore the final approximation. If the weight function W(x) and its first k th derivatives are continuous, then the shape function Φ(x) and its first k th derivatives will be continuous. Also by choosing some special weight functions, the shape function can shift between the approximation and the interpolation properties. For example, the standard least-squares interpolant is obtained if the weight function is chosen to be constant over the entire domain. However, all the unknowns are then fully coupled. By choosing the weight function to have a large domain of influence, the approximation behaves like a polynomial of higher order than p(x). Limiting the weight function to be nonzero over a small subdomain results in a sparse system of equations. The standard Finite Element formulation will be obtained if the weight function is chosen to be constant over each subdomain or element (Babuška et al., 2003). Most Mesh-Free weight functions are bell-shaped so that they are positive and a unique solution a(x) in Equation (6) is guaranteed. They are monotonic decreasing 59

62 functions with respect to the distance to the centre of influence domain. Therefore they are nonzero in the domain of support and zero outside of the support domain. The following is a list of commonly used weight functions. They are functions of the distance between the node and the integration points, i.e., W I (x) = W ( x! x I ) (13) Exponential: # % W (s I ) = $ &% e!(s I /a) 2, for s I " 1 0, for s I > 1 (14) or $ & W (s I ) = % & ' e!(s I /c) 2! e!(" I /c) 2 1! e!(" I /c) 2 for s I # " I 0, for s I > " I (15) where ρ is the radius of the influence domain. Cubic Spline: # % % W (s I ) = $ % % &% 2 3! 4s 3 I + 4s I for s I " ! 4s + 4s I I! 4 3 s 3 I for 1 2 <s " 1 I 0, for s I > 1 (16) Quartic Spline 60

63 # % W (s I ) = $ &% 1! 6s I 2 + 8s I 3! 3s I 4 for s I " 1 0, for s I > 1 (17) where s I = x! x i 2 r (18) In this study, the quartic spline weight function is used for its simplicity and its second order of continuity. The derivative of the weight function is!w I (x)!x where = dw (s I ) ds I = % ' & ('!s I!x ("12s I + 24s 2 I "12s 3 I )r I / # I R I, if s I $ 1 0, if s I > 1 (19) r I = X " X I R I = r I The quartic spline weight function also satisfies the following conditions so that first- and second-order derivatives of the weight function and the final approximations are continuous. W (0) = 1, W (1) = 0 dw = 0, ds I s I =0 dw = 0 ds I s I =1 (20) d 2 W = 0 ds 2 I s I =1 61

64 3.4 Algorithm Overview Figure 8 shows an overview of the Mesh-Free Total Lagrangian Explicit Dynamic Algorithm (MTLED) algorithm used in the current work (Horton et al., 2010, Miller et al., 2012). Figure 8 Workflow of Mesh-Free Total Lagrangian Explicit Dynamic Algorithm In the pre-processing stage of MTLED, it loads the simulation geometry in the form of two lists: node locations and integration point locations. Before influence domains (discussed in Section 3.1) are defined and shape functions (discussed in Section 3.2) are created, nodes for field variable approximation and the integration points for calculating local nodal force are independent particles in the problem domain 62

65 with no connections to each other. Then it loads the material properties for each integration point (see Section 4). To create the shape functions and calculate the shape function derivatives, it loops through all the integration points to find the local nodes within the support domain (see Section 3.1) and calculates the shape functions (see Section 3.2) and the shape function derivatives. It saves all the pre-calculated shape function derivatives. Therefore in the solving stage, I can directly use them to speed up the algorithm. As the nodes are where the masses are concentrated and displacements are calculated, I loop through all the nodes and associate each node with proportionally allocated mass. In the solving stage, it solves the problem for each incremental time step after all global nodal displacements are initialised. Within each time step, it loops through the integration point list. At the same time it gets the local nodes in the influence domain and their associated shape function derivatives from the pre-processing results. Then it calculates the local deformation gradient, the strain-displacement matrix, second Piola- Kirchoff stress and finally the local nodal reaction forces. For nodal reaction force spatial integration, I use a uniform background grid that does not need to conform to the domain geometry. Numerical integration is performed in each cell using a single-point Gaussian quadrature. Since it does not require the background cell to conform to the intracranial geometry, its generation can be performed automatically (see Figure 9). Previous studies on computing the brain responses with the similar approach resulted in very small absolute errors (0.85mm), which is smaller than the voxel of the MRI (Horton et al., 2006, Horton et al., 2007). In this chapter I present the detail design of the Mesh-Free computational framework that fits the current study for efficient model generation. It includes the selection of proper influence domain; how to build the shape function; the choice of 63

66 weight functions; and an overview of the algorithm. However it introduces another question: how to assign material properties to the model without segmentation and meshing. Figure 9 Schematic representations of background integration cells that do not conform to the intracranial geometry 64

67 Chapter 4 Material Properties based on Fuzzy Tissue Classification In the modelling process, material properties (Young s modulus and Poisson s Ratio in the current study) need to be assigned to each integration point based on information from the medical image. Traditional hard segmentation result contains less information than the original image as it converts intensity values to discrete label maps. Each tissue class is assigned with uniform material properties based on the label map, as shown in Figure 10 (b, c). On the other hand, fuzzy tissue classification result includes the image intensity information in the fuzzy membership functions - in Figure 10 (d, e). In the present study I use fuzzy tissue classification instead of hard segmentation in the modelling process. Before further discussion, I would like to clarify the difference between segmentation and tissue classification. Although interchangeable usage is often seen in the literature, segmentation and tissue classification do have differences in terms of procedures and outcomes. Segmentation partitions an image into non-overlapping and connected regions with discretely defined boundaries, while tissue classification assigns each pixel (voxel) to a number of different tissue types (Pham, 2000). Note that the requirement for the regions to be non-overlapping is removed so that the classification can be fuzzy (or soft). Therefore it produces fuzzy membership functions for each tissue class rather than discrete label maps. This leads to an easier and more flexible modelling process. * Portions of this chapter have been published in: Zhang, J. Y., G. R. Joldes, et al. (2013). "Patient-specific computational biomechanics of the brain without segmentation and meshing." International Journal for Numerical Methods in Biomedical Engineering 29(2):

68 Figure 10 Comparison between an FE model constructed using segmentation and a fuzzy Mesh-Free model based on fuzzy tissue classification. (a) T2 MRI of the brain, including the tumour and ventricles - notice that no clear boundaries can be easily defined, especially for the tumour; (b) Finite Element model of ventricles generated from segmentation; (c) Finite Element model of the tumour generated from segmentation; (d) the fuzzy membership function for ventricle indicated at each pixel by the level of blue; (e) the fuzzy membership function for tumour indicated at each pixel by the level of red; (f) fuzzy Mesh-Free model of ventricle and (g) fuzzy Mesh-Free model of tumour. Green dots represent nodes while grey grids define the background integration cells. Material properties are assigned directly to integration points based on the fuzzy classification results. The use of fuzzy tissue classification makes a substantial difference in the modelling workflow. As shown in in Figure 10 (a), malignant brain tumour can spread within the brain and spine. Lacking distinct borders, tumour is normally very difficult to segment. 66

69 There are extensive literatures on tissue classification of brain images (Lin et al., 2012, Weisenfeld and Warfield, 2011, Wells et al., 1996). To cope with extreme and variable anatomies due to pathologies, I utilised Fuzzy C-Mean (FCM) clustering (Pham and Prince, 1999a) for fuzzy tissue classification. FUZZY C-MEAN clusters similar intensity data by computing the membership function at each voxel for a specified number of classes C and minimising the objective function J!"#. J FCM = " C " u jk I j #V k j!! k=1 (21) C 0 $ u jk $ 1, " u = 1 jk k=1 I j k where Ω is the image domain; j is the location index; k is the class index; and V represents the degree of similarity between the intensity I at that location j and the centroid of its class V k. Notice that the fuzzy membership u jk will form a partition of unity for different classes. A high membership value means that the data value at that location is close to the centroid for that particular class. Our Mesh-Free model contains three tissue classes (C = 3): healthy tissue, ventricles and tumour (as the same properties for white and grey matter are used without introducing noticeable errors in computed displacements (Wittek et al., 2010a). I developed a semi-automatic subroutine based on a bias-corrected FUZZY C- MEAN (BCFCM) algorithm (Ahmed et al., 2002). The BCFCM algorithm was formulated by modifying the objective function of the standard fuzzy c-means algorithm to compensate for inhomogeneities and to allow the labelling of a voxel to be influenced by the labels in its immediate neighbourhood. Verification using simulated MRI data was performed in (Ahmed et al., 2002). The results showed that intensity variations across patients, scans, and equipment changes were accommodated in the estimated bias 67

70 field without the need for manual intervention. The fuzzy tissue classification subroutine used in this study converges within 2 minutes. Continuous material properties MP j at each location j are interpolated based on these membership functions from FUZZY C-MEAN as follows C MP j =! u " MP (22) jk k k=1 Despite continuing efforts (Sinkus et al., 2005, Turgay et al., 2006), commonly accepted non-invasive methods for determining patient-specific constitutive properties of the brain have not been developed yet. In the context of brain biomechanics for nonrigid registration, constitutive models of brain tissue vary from simple linear-elastic models (Warfield et al., 2000b) to Ogden hyperelastic models (Wittek et al., 2007) and bi-phasic models based on consolidation theory (Miga et al., 2000a). However the modelling approach used in this study ensures that the calculated brain deformation depends very weakly on the constitutive model and mechanical properties of the brain tissues. Therefore, following (Wittek et al., 2010a, Joldes et al., 2009d, Joldes et al., 2009b), I use the hyper-elastic Neo-Hookean model. The literature indicated that hyperelastic models well captured the behaviour of brain tissues undergoing large deformation (Miller, 2011). The Neo-Hookean model is chosen here for its simplicity, which is essential for large-scale real-time simulation. Abaqus: I use the same form of Neo-Hookean strain energy potential as the one in U = µ 0 2 (I 1! 3) + k 0 2 (J!1)2 (23) 68

71 where µ is the initial shear modulus, k 0 0 is the initial bulk modulus, I 1 is the first deviatoric strain invariant, and J is the volumetric change. Based on the experimental data (Miller et al., 2000) and prior modelling experience (Joldes et al., 2009a), the Young s modulus is set to 3000 Pa for the brain parenchyma tissue and two times larger for the tumour. There are strong experimental evidences (Sahay et al., 1992, Miller, 2011) that the brain tissue is (almost) incompressible. Therefore I assign a Poisson s ratio of 0.49 for parenchyma and tumour which has been widely used in the literature (Miller et al., 2011b, Jin et al., 2011, Horton et al., 2010, Joldes et al., 2009d, Wittek et al., 2007). Following (Wittek et al., 2007, Wittek et al., 2010a) the ventricles are set to be very soft and compressible I set Young s modulus to 10 Pa and Poisson s ratio to 0.1 (to account for the possible leakage of cerebrospinal fluid from ventricles during surgery). In the fuzzy Mesh-Free TLED model material properties (Young s modulus and Poisson s ratio) are assigned directly to each Gauss (integration) point. More complicated assignment schemes can be defined (such as averaging of properties over the integration cell), but in this work I use the nearest-neighbour approach for its simplicity. 69

72 Chapter 5 Solution Methods without Accurate Segmentation and Material Properties 5.1 Total Lagrangian Explicit Dynamics The Fuzzy Mesh-Free TLED method is based on Total Lagrangian formulation. This means that all the calculations are referring to the initial configuration of the analysed continuum. The decisive advantage of this formulation is that large amount of computation (such as all derivatives with respect to spatial coordinates) can be carried out during the pre-processing stage. As indicated in (Miller et al., 2007), the Total Lagrangian formulation significantly reduced the computation time compared with the Updated Lagrangian formulation used in vast majority of commercial Finite Element solvers (such as LS-DYNA, Abaqus). Also the material law implementation was much simpler since hyper-elastic models could be easily described using the deformation gradient. Following (Miller et al., 2007, Bathe, 1996), the nodal forces are computed as: t F = 0! t 0 X t 0 S 0 BdV 0 (24) V 0 where 0 t X is the deformation gradient at time t, 0 t S is the second Piola-Kirchoff stress and 0 B is the matrix of shape function derivatives (Horton et al., 2010). The solver applies explicit integration in time domain using the central difference method. This allows very straightforward treatment of nonlinearities, as no * Portions of this chapter have been published in: Zhang, J. Y., G. R. Joldes, et al. (2013). "Patient-specific computational biomechanics of the brain without segmentation and meshing." International Journal for Numerical Methods in Biomedical Engineering 29(2):

73 iterations are required during each time step. The time stepping scheme for solving the equation of motion can be expressed as: t+!t U =!t 2 M "1 t F + 2 t U " t"!t U (25) 0 where t Uis the displacement calculated at time t, F is the reaction force, and M is the constant lumped mass matrix. The mass associated with an integration point is distributed equally to all nodes found in the support domain of that particular integration point. Explicit time integration is only conditionally stable and therefore requires careful selections of the time step to keep stable. I use an estimate of the stable time step derived specifically for the mass lumping algorithm in Mesh-Free methods, as described in (Joldes et al., 2012). For central difference integration used in this study, the stable critical time step can be obtained from the maximum frequency of free vibration. It is equal to the smallest characteristic length Le of an element in the mesh divided by the dilatational wave speed c. In this study, the characteristic length Le is the smallest distance between integration points. The advantage of using a lumped mass matrix is that the system of Equations (25) can be decoupled and the solution is computed separately for each degree of freedom. Therefore, during the whole solving process, the global stiffness matrix for the entire model is not required. There is no need to solve the whole system of equations. The computation time can be reduced by an order of magnitude compared with implicit integration (Wittek et al., 2007). 5.2 Dynamic Relaxation Dynamic relaxation is an explicit method that can be used for computing the steady state solution for a discretised continuum mechanics problem. It can be used for 71

74 finding the deformed state for a discretised continuum mechanics problem. The method relies on the introduction of an artificial mass dependent damping term in the equation of motion, which attenuates the oscillations in the transient response and increases the convergence towards the steady state solution. The dynamic relaxation method is especially attractive for highly nonlinear problems (including both geometric and material nonlinearities) solved using the finite element method. Because of its explicit nature there is no need for solving large systems of equations. All quantities can be treated as vectors, reducing the implementation complexity and the memory requirements. Although the number of iterations to obtain convergence may be quite large, the computation cost for each iteration is very low, which makes it a very efficient solution method for nonlinear problems. I use dynamic relaxation (Joldes et al., 2009b) in the solution algorithm to allow for fast and accurate convergence to the deformed state. Several different cases are used to compare the convergence speed in (Joldes et al., 2009b). The convergence is more than 10 times faster than algorithms without dynamic relaxation In dynamic relaxation, a mass proportional damping component is added to the equation of motion so that Equation (25) becomes t+!t U = t U + "( t U # t#!t U) + $M #1 t F 0 " = 2 # c!t 2 + c!t $ = 2!t c!t (26) where c is the damping coefficient. 72

75 In the relaxation stage, the integration time step Δt is kept constant, while the damping coefficient c and lumped mass matrix M are initiated following (Joldes et al., 2011) and automatically adjusted to maximise the convergence rate and improve the computational efficiency without compromising the solution convergence (Joldes et al., 2011). During dynamic relaxation it uses a termination criteria based on an estimate of the maximum absolute error in the solution instead of residual forces or energy, as commonly used in Finite Element software. As explained in (Joldes et al., 2009b), dynamic relaxation attenuated the high frequency oscillations more so that the calculated displacement would converge towards the real solution at a frequency close to the smallest oscillation frequency. The absolute error vector would also converge towards the eigenvector corresponding to the lowest eigenvalue of the equation of motion. Therefore, the solution error decreased at each step by: U n+1! U * " #(U n! U * ) (27) where ρ is the estimated convergence rate and * U is the steady state solution. The maximum absolute error can be approximated by applying the infinity norm to equation (27): U n+1! U * " # $( Un+1! U * " + Un+1! U n " ) => U n+1! U * " # $ 1! $ Un+1! U n " (28) During dynamic relaxation, if the estimated absolute error (given by the right hand side of eq. 28) is smaller than a configured threshold for a consecutive number of steps, I consider the tolerance is satisfied and the simulation is terminated. In the 73

76 current study, following (Joldes et al., 2011), the termination threshold is set to 0.1mm and the number of consecutive steps is set to Displacement Loading The exact physiological mechanisms behind the craniotomy-induced brain shift are hotly disputed among neurosurgical community and require further studies (Aarabi et al., 2006). There is still lots of work to be done to rigorously characterise the response of brain tissue under mechanical loading (Bilston, 2011). Furthermore, there are always uncertainties in patient-specific tissue properties. To reduce the effects of such uncertainties, in the present study, craniotomyinduced brain shift is modelled as a displacement zero traction problem (Miller, 2011, Miller et al., 2010a). As the basis for the framework, the insensitivity of the predicted deformation on mechanical properties has been quantitatively verified in (Wittek et al., 2009a, Miller et al., 2010b, Miller and Lu, 2013b) when the deformation problem is formulated as the pure displacement and displacement-zero traction problem. The model is loaded by enforced displacement of the nodes on the brain surface exposed by craniotomy. Such information can be measured using a variety of techniques, such as StealStation neuro-navigation system by Medtronic and Image Guided Surgery Platform by BrainLab. In this study, as a prototype research, the displacements for loading are derived from the distances between corresponding points on the pre-operative and intra-operative cortical surfaces. The correspondence between these surfaces is determined by applying a B-spline registration algorithm to their curvature maps. More details can be found in (Joldes et al., 2009c). 5.4 Brain-Skull Interaction 74

77 I model the interaction between the skull and the brain as a finite sliding, frictionless contact. As the skull is orders of magnitude stiffer than the brain tissue, I assume the skull to be rigid. A contact interface is defined between the rigid skull and the brain surface. The contact formulation described in (Joldes et al., 2009d, Joldes et al., 2008) is used, which detects and prevents the penetration of the skull by any brain nodes while allows for frictionless sliding. The basic contact algorithm is implemented in two stages. In the preprocessing stage: 1 - Study skull surface and create lists with additional edges and faces to check for each node; 2 - Pre-compute all dimensions related to the skull surface that are needed in the local search stage (such as normal directions, lengths, bisectors, etc.); speed. 3 - Distribute skull nodes into subgroups to increase the calculation In the computation stage: At the end of each time step, for each node P to detect penetration 1 - Identify the subgroup containing the node P and search for the closest skull node C in that subgroup and all the surrounding subgroups; 2 - Find the closest point on the skull surface, R, by searching the master edges and faces that contain C and the additional master edges and faces related to node C; 75

78 3 - Check for penetration, using the normal to the skull surface in R; 4 - If penetration is detected, move the node P to the point R. Such modelling, although simple, was used in (Wittek et al., 2010a, Joldes et al., 2011) for computing brain deformations during brain shift due to its reliability and low computational cost. The computational examples in (Joldes et al., 2008) proved the accuracy and the computational efficiency of the selected method. For a model having more than degrees of freedom, a complete simulation can be done in less than a minute on a standard personal computer. 5.5 Solution Algorithm Verification The verification of the solution algorithm was conducted in (Horton et al., 2010). In the verification, a fully nonlinear Neo-Hookean material model (a cylinder of height 0.1m and radius 0.05 m) was employed. Numerical experiments were performed and the results were compared with those obtained with established Finite Element code. The verification showed negligible forces and displacement differences (less than 5%) between the Finite Element results and the Mesh-Free results. These results showed the solution algorithm in the current work to be working and useful. 76

79 Chapter 6 Computation of Brain Deformation with 2D and 3D Patient-Specific Datasets Three 2D cases and one 3D case were used in this study to validate the framework (refer to Chapter 6.1 for 2D cases and Chapter 6.2 for 3D case) D Cases The three cases of 2D anatomical data and boundary conditions used in the presented work are extracted from high resolution pre-operative MRI taken before the brain surgery and intra-operative MRI during the surgery at Department of Surgery, Brigham and Women s Hospital (Harvard Medical School, Boston, Massachusetts, USA). The three 2D case image represent different situations that may occur during neurosurgery as characterised by tumour located in different parts of the brain: posteriorly (for case 1), laterally (for case 2) and anteriorly (for case 3), as shown in Figure 11. * Portions of this chapter have been published in: Zhang, J. Y., G. R. Joldes, et al. (2013). "Patient-specific computational biomechanics of the brain without segmentation and meshing." International Journal for Numerical Methods in Biomedical Engineering 29(2): Zhang, J. Y., G. Joldes, et al. (2012). Neuroimage as a Biomechanical Model: Toward New Computational Biomechanics of the Brain. Computational Biomechanics for Medicine Deformation and Flow P. M. F. Nielsen, A. Wittek and K. Miller (Eds.), Springer New York:

80 78

81 Figure 11 Three cases selected for 2D computation with T2 pre-operative MRI (left column) and intra-operative MRI showing craniotomy area (right column). The MRIs are provided by Computational Radiology Laboratory (CRL), Harvard Medical School. The fuzzy Mesh-Free models for the three cases are shown in Figure 12 (right column). The intensity levels of red and blue in the image correspond to the membership functions for tumour and ventricles from the fuzzy tissue classification. Nodes are represented by green dots in the domain. I also generate Finite Element meshes from the three cases. The Finite Element models (as shown in Figure 12, left column) are constructed from the segmented preoperative MRIs with standard linear quadrilateral plane strain elements. Each element belongs to only one tissue type: parenchyma (white), tumour (red) or ventricle (blue). To allow deformations to propagate between the parenchyma, tumour and ventricles, common nodes are used between different tissues. This ties the surfaces of different tissues together without needing to define any extra internal boundary conditions. The meshes are generated using Hypermesh TM (a high-performance commercial Finite Element mesh generator by Altair, Ltd. of Troy, MI, USA) with local manual adjustments. H-refinement of mesh density is used to obtain a converged control solution in finite element model. I use approximately the same number of integration cells (corresponding to the number of elements used in Abaqus) in fuzzy mesh-free model. Comparing the fuzzy Mesh-Free model (right column) with the Finite Element model (left column), the MRI voxel intensity information is lost after segmentation and mesh, but is included in the fuzzy Mesh-Free model. Uncertainties about tissue classes 79

82 are utilised in the numerical simulation. However, unlike creation of Finite Element meshes, generation of the fuzzy Mesh-Free models is a trivial exercise. Table 1 shows the modelling details for the three cases. Maximum brain shift after craniotomy is shown as Maximum boundary loading in the last column. Table 1 Modelling Details Case Nodes (FEM) Elements (FEM) Nodes (Mesh-Free) Integration Points (Mesh- Free) Maximum Boundary Loading [mm] 1 (Top) (Middle) (Bottom) I compare the fuzzy Mesh-Free simulation results against the Finite Element results computed using Abaqus TM (ABAQUS 6.9 User Documentation, Internet Manual, Simulia, Retrieved 10 September 2011). For both simulations, I use the same constitutive material laws (refer to Chapter 4 for detailed constitutive equations and parameters), loading and boundary conditions (refer to Chapter 5.3 for loading method and boundary conditions). 80

83 81

84 Figure 12 Models generated from three different datasets. Left column: Finite Element Models, with parenchyma, tumour (red) and ventricle (blue) modelled separately. Right column: Fuzzy Mesh-Free Model without explicitly separating the tumour and ventricles from the brain; the fuzzy membership functions are indicated by the level of red for tumour and blue for ventricles; nodes are shown as green dots. The differences in the resulting deformation fields are shown in Figure 13. For each node in the Finite Element model, the displacements are compared to the displacements at the corresponding position in the fuzzy Mesh-Free model using the Moving Least Square approximation scheme. The maximum and average differences for the three cases are shown in Table 2. As shown in Figure 13, most of the larger localised errors are located near ventricles. This is because in FEM, ventricle is modelled as fully compressible material with very small Poisson s Ratio. In fuzzy- Meshfree method, some ventricle will have large Poisson s ratio near the boundary with the brain tissue as the brain tissue is modelled as fully incompressible material with Poisson s Ratio near 0.5. As the accuracy of manual neurosurgery is not better than 1mm (Bucholz et al., 2004) and the voxel size in high-quality pre-operative MRI is usually of a similar magnitude, therefore the differences (maximum absolute errors for the three cases are respectively 0.57mm, 0.38mm and 0.54mm as shown in Table 2) is, for practical purposes, negligible and the proposed Mesh-Free models can produce equivalent accuracy to the Finite Element models. 82

85 83

86 Figure 13 Evaluation of accuracy for the three cases. The colour shows the difference of the simulation results (computed deformation field) over the whole brain [mm]. Table 2 Numerical details of comparison for the three cases Case Maximum Boundary Loading [mm] Average Absolute Difference [mm] Maximum Absolute Difference [mm] 1 (Top) (Middle) (Bottom) D Case The 3D study I present here is based on MRI data provided by the Computational Radiology Laboratory (CRL) at Harvard Medical School. The 3D image dataset was selected based on the size and the location of the tumour (a large anterior tumour close to the parenchyma surface, see figure 14 left. The surgical procedure involved removing a large section of the skull immediately over the tumour, after which a significant retraction of the cortical surface was observed near the craniotomy, as shown in Figure 14 right.). Also this case was investigated in our previous studies (Joldes et al., 2009a, Joldes et al., 2009b, Wittek et al., 2007 and Wittek et al., 2009) 84

87 using finite element simulation. The previously obtained results are used in this study for validation purpose in Chapter 6.2. Figure 14 Patient-specific MRI data provided by Computational Radiology Laboratory (CRL), Harvard Medical School. Pre-operative T2 weighted MRI (left) shows the brain contains a large anterior tumour close to the parenchyma surface. Intraoperative MRI (right) was taken right after the craniotomy (skull was opened). Figure 15 shows the main anatomical components (parenchyma, ventricle and tumour) as surfaces generated from the segmented pre-operative MRIs in (Wittek et al., 2010a). 85

88 Figure 15 Main anatomical components (parenchyma, ventricle and tumour) as surfaces generated from the segmented pre-operative MRIs. Following (Horton et al., 2010, Zhang et al., 2012a, Zhang et al., 2013, Miller et al., 2012), for field variable approximation I consider the overall brain as a single domain so that the parenchyma, tumour and ventricle volumes are filled with 31,326 nodes with average spacing of 3.5mm as shown in Figure 16. For spatial integration, I use a uniform grid of 53,672 integration points with 3 mm spacing, as shown in Figure 17. Mesh density convergence study was done for FEM analysis. Approximately the same number of integration cells (corresponding to the number of elements used in Abaqus) is used in fuzzy mesh-free model. Numerical integration is performed using the single point Gaussian quadrature in each regular cell of the grid. Such a uniform background grid does not need to conform to the domain geometry so that it is very easy to build (simply generate an isometric grid and delete the cells with centroid outside the problem domain). This approach can introduce small volumetric errors 86

89 (maximally half of a volume of a hexahedron intersected by a domain boundary). However it has been proved to be too small to have any significant effect on the results as shown in (Horton et al., 2010, Zhang et al., 2013). Figure 16 Geometry discretisation of the patient-specific problem domain with Mesh-Free nodes represented with cloud of blue points. 87

90 Figure 17 Geometry discretisation of the patient-specific problem domain with uniformly distributed, non-conforming integration grid (only one slice is shown). I assign material properties (Young s modulus and Poisson s Ratio) to each integration point based on fuzzy tissue classification results as shown in Figure 18. I develop a simple yet efficient semi-automatic fuzzy tissue classification method based on FUZZY C-MEAN clustering. More detail about the FUZZY C-MEAN clustering can be found in Chapter 4. It is by no means the only or the best approach to build the Fuzzy Mesh-Free model. As I mentioned about the less strict requirement for accurate segmentation, any working classification methods utilising fuzzy membership functions such as (Weisenfeld and Warfield, 2011, Wells et al., 1996), will fit into our Fuzzy Mesh-Free framework. Figure 18 also shows local misclassification of cerebrospinal fluid (CSF). The misclassification is due to similar MRI intensity for tumour and cerebrospinal fluid (CSF). In T1 MRI they both appear darker than normal brain tissue. 88

91 In T2 MRI they both appear lighter than normal brain tissue. To show the robustness of the framework, especially for tumour, the results in the present study is actually quite coarse without any further refinement as shown in Figure

92 Figure 18 3D Fuzzy Tissue Classification shown in 4 slices. The fuzzy membership functions for tumour and ventricles are indicated correspondingly by the level of red for tumour and blue for ventricles. Despite continuing efforts (Turgay et al., 2006, Sinkus et al., 2005), one of the obstacles standing before the computational biomechanics community is the difficulty in obtaining accurate patient-specific tissue properties. The ability to directly use fuzzy tissue material properties, even with possible local misclassifications in our Fuzzy Mesh-Free framework is based on our reformulation of computational mechanics problems as a displacement-zero traction problem (Miller et al., 2010a) so that the results are weakly sensitive to the variation in patient-specific mechanical properties of simulated continua (described in Chapter 5). Therefore I could have much less stringent requirements for accurate segmentation of different tissue classes. In the field of image-guided surgery where the current, intra-operative configuration of the soft organ is of critical importance, I load the model by prescribing displacements on the portion of the exposed parenchyma surface. Following (Zhang et al., 2013, Miller et al., 2012, Horton et al., 2010) I apply Total Lagrangian formulation so that all the calculations are referring to the initial configuration of the analysed continuum. As discussed before, the advantage of this formulation is that large amount of computation, such as all derivatives with respect to spatial coordinates, can be carried out during the pre-processing stage. Also the material law implementation is much simpler because Neo-Hookean hyper-elastic model used in the current study can be easily described using the deformation gradient. I apply explicit integration in time domain using the central difference method. This allows very straightforward treatment of nonlinearities because no iterations are required during each time step. Explicit time integration is only conditionally stable and therefore 90

93 requires careful selections of maximum stable time step that can be used. I use dynamic relaxation in the explicit solution algorithm to allow for very fast and accurate convergence to the deformed state. During dynamic relaxation I utilise a termination criteria based on an estimate of the maximum absolute error in the solution as described in (Joldes et al., 2009b). For brain skull interaction I model the contact as a finite sliding, frictionless contact (Joldes et al., 2009d). The current research focuses on the steady state solution after craniotomy, but before the surgical cutting or puncturing. In this context of image-guided surgery, the only variable of interest is the final steady state solution of the deformation field. It is used to warp pre-operative MRI so that they correspond to the current, intra-operative configuration of the brain (on the other hand, accurate computation of stresses is unnecessary). To evaluate the accuracy of the 3D deformation field generated by the Fuzzy MTLED model, I compare the results with intra-operative MRI by imposing the predicted surface contours onto intra-operative MRI sections. I also compare our results with those Finite Element results, which was presented and validated in (Wittek et al., 2010a). Due to the fuzzy feature (no surfaces defined within the model) and the complexities of 3D model, I utilise the major component surfaces (in this study: tumour and ventricle) from segmented preoperative MRI in (Wittek et al., 2010a) to assist with the visualization. Notice that the use of surface is not necessary for the modelling and simulation in the current study, but purely for better visualization and evaluation purpose. Cross Sections on Intra-operative MRI In this study, intra-operative MRI taken right after craniotomy is used as ground truth data to validate the computation results. Below shows four sections, each taken at 5mm intervals through regions of the brain with both tumour and ventricle 91

94 from axial and sagittal orientation. Figure 19 shows the locations of the sections. At each section, I impose tumour and ventricle surfaces deformed by FMTLED onto intraoperative MRI (Figure 21). For reference, I also impose the Finite Element result from (Wittek et al., 2010a) onto the corresponding slice. Figure 19: Locations of sections shown in Figure 21. The sections are chosen at 5mm intervals while cutting across the regions of interest (tumour and ventricle). 92

95 Figure 20 Four axial Intra-operative MRI sections with Fuzzy MTLED deformed contours of tumour and ventricle surfaces (green lines). For reference Finite Element results for ventricle and tumour are shown in red. Notice that most of the red lines are inline with the green contours which mean great consistency. Figure 19 shows 93

96 the locations of the sections displayed here. The sections are chosen at 5mm intervals while cutting across the regions of interest (tumour and ventricle). Figure 21: Four sagittal Intra-operative MRI sections with Fuzzy MTLED deformed contours of tumour and ventricle surfaces (green lines). For reference Finite Element results are shown in red. Notice that most of the red lines are inline with the green contours which mean great consistency. Figure 19 shows the locations of the sections. The sections are chosen at 5mm intervals while cutting across the regions of interest (tumour and ventricle). Figure 21 shows negligible difference (less than the size of the MRI voxel) between the Fuzzy MTLED results and the Finite Element results. As the tumour is very difficult to identify in the intra-operative MRI while the ventricle is much easier to identify, the ventricle contours show very good alignment with the image. All 94

97 comparisons show that the results obtained using the Fuzzy MTLED method are as useful as those obtained with the Finite Element method. Hausdorff Distance To get an objective quantitative evaluation, following (Archip et al., 2007, Oguro et al., 2010, Wittek et al., 2010a), I utilise the 95% Hausdorff distance to measure the differences between two results. The Hausdorff distance H (A, B) (Hausdorff, 1957, Fedorov et al., 2008) between set A and set B is denoted as: H (A, B) = max(h(a, B),h(B, A)) h(a, B) = max a!a {d(a, B)} d(a, B) = min b!b a " b (29) where h(a,b) is the maximum Euclidean distance from any of the points in set A to set B, and d(ambo) is the Euclidean distance from point a to the nearest point b in set B. The percentile Hausdorff distance is used for removing outlier comparison pairs (Fedorov et al., 2008). In this study set A is the surface of the Fuzzy MTLED deformed ventricles and tumour and set B is the surface of the Finite Element deformed ventricles and tumour. Figure 22 shows the distance d(a,b) described in Equation (29) on a colour scale. The values are projected onto the surface deformed by Fuzzy MTLED. The majority of the distances (shown blue) are within 0.5 mm. The calculated 95% Hausdorff distance for ventricle surfaces is mm. For the tumour surfaces, it is mm. Since the level of accuracy in image guided surgery is 1mm (Bourgeois et al., 1999), it shows that the results from the current study are accurate enough to be useful in clinical situations. 95

98 Max: mm Max: mm Figure 22: Differences (in mm) between Fuzzy MTLED predicted surfaces and Finite Element predicted surfaces of tumour and ventricle. The colours represent the difference d(a,b) described in Equation (4) to the nearest point on the control surface (the same surface deformed by validated Finite Element algorithm). Most of the surfaces are shown blue which represent good matching between the two results. The top image shows the Fuzzy MTLED predicted ventricle surface. The bottom image 96