The Pennsylvania State University The Graduate School College of Engineering NEURO-MECHANICAL MODELING OF HYDROCEPHALUS:

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1 The Pennsylvania State University The Graduate School College of Engineering NEURO-MECHANICAL MODELING OF HYDROCEPHALUS: STEPS TOWARD A FINITE ELEMENT APPROACH A Thesis in Engineering Mechanics by Jason Fritz c 2010 Jason Fritz Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science December 2010

2 The thesis of Jason Fritz was reviewed and approved by the following: Corina Drapaca Assistant Professor of Engineering Science and Mechanics Thesis Advisor Francesco Costanzo Professor of Engineering Science and Mechanics Clifford Lissenden Professor of Engineering Science and Mechanics Judith A. Todd P. B. Breneman Department Head Chair Signatures are on file in the Graduate School.

3 Abstract Hydrocephalus is a condition which occurs when an excessive accumulation of cerebrospinal fluid in the brain causes enlargement of the ventricular cavities. Although known instances of hydrocephalus date back to the time of Hippocrates, the process by which hydrocephalus develops is still not well understood. In order to better understand the mechanisms of hydrocephalus we need to design novel mathematical and computational models capable of correctly predicting the brain response to mechanical loading induced by hydrocephalus, which in turn will help design superior diagnostic and treatment protocols. In this thesis, we propose novel numerical simulations for hyrdocephalus using medical images and available finite element software. In our first model, we studied the changes in the brain geometry due to the ventricular deformations caused by the pulsatile ventricular pressure. We used the real geometry of the brain taken from MR images and we assumed that the brain parenchyma behaves as a nonlinear hyperelastic solid. Our finite element simulations show that this model is able to correctly predict the large displacements seen in hydrocephalic brains and that once a pressure threshold of 1960Pa has been reached, any further pressure increase due to the natural pulsations of the brain will induce hydrocephalus. We also formulate the first neuro-mechanical model of the brain that will couple the electrochemical and mechanical properties of the brain. We assume that the brain tissue is a charged hydrated soft tissue made of a solid phase, an interstitial fluid phase and an ion phase with two monovalent ion species. This model is able to predict the onset of normal pressure hydrocephalus due to a change in the ionic concentrations of the ventricular cerebrospinal fluid in the absence of an elevated intracranial pressure. This opens the possibility of treating hyrdocephalus noninvasively through the use of pharmaceutical drugs. Finally, we developed a process for taking medical images and turning them into meshes that could be imported into finite element software. This process allowed us to create geometrically accurate meshes, therefore increasing the accuracy of our simulations. Using these meshes, we perform elastic models simulating an endoscopic third ventriculostomy surgery and brain tumor growth, using the deal.ii finite element library. iii

4 Table of Contents List of Figures List of Tables Acknowledgments vi vii viii Chapter 1 Introduction Hydrocephalus Modeling Approaches Outline Chapter 2 General Constitutive Theory Kinematics of Finite Deformations Principles of Classical Continuum Mechanics Hyperelastic Constitutive Law Isotropic Hyperelastic Materials Incompressibility The Mooney-Rivlin Material Chapter 3 The Hyperelastic Model Mooney-Rivlin Model Material Properties Geometry Boundary Conditions Results Finite Element Simulation of Hydrocephalus Chapter 4 The Triphasic Mechano-Electrochemical Model The Triphasic Model Numerical Simulation of NPH iv

5 4.2.1 Geometry and Boundary Conditions Material Parameters Results Conclusion Chapter 5 Geometrically Accurate Meshes of the Brain Creating meshes using Simpleware Using Cubit Using the mesh converter in deal.ii Importing mesh into the deal.ii Library Chapter 6 Elastic Models Using deal.ii Library Finite Element Simulations Using the deal.ii Library Endoscopic Third Ventriculostomy Tumor Growth Chapter 7 Conclusions Achievements Future Work Appendix A Example C++ source code which reads in a mesh and outputs a picture of the mesh 36 Appendix B Example C++ source code of elasticity model 38 Appendix C Nontechnical Abstract 50 Bibliography 51 v

6 List of Figures 1.1 Circulation of the cerebrospinal fluid (light blue) [1] A normal brain vs. a hydrocephalic brain. Ventricles are indicated by red arrow Diagram of a CSF shunt, which is used to reverse the dilation of the ventricles [2] Reference configuration (left), deformed configuration (right) Stress vs. Strain Plot, of the brain parenchyma, created by Abaqus material evaluate tool (a) One slice of a MR image. (b) Segmentation of the MR image (b) 3mm thick mesh of the brain parenchyma, created from the segmention of the MR image Boundary conditions of the model Plot of the pulsation forcing function a(t) given by equation (3.4) Displacements for a Ventricular Pressure of 1,013Pa Displacements for a Ventricular Pressure of 2,549Pa Displacements for a Ventricular Pressure of 5,886Pa Mises Stress for a Ventricular Pressure of 2,549Pa (left) and 5,886Pa (right) Structure of a charged hydrated soft tissue Schematic representation of the 1D brain Ventricular wall displacement showing an initial swelling followed by shrinking The ventricular wall displacement for c F 0 = 0.2, 0.4, and The ventricular wall displacement for ϕ F 0 = 0.25, 0.5, and Process of importing MR images into Simpleware Screen capture of using the flood fill tool to segment the MR images Examples of exported meshes from Simpleware STL Import Options window Screen capture of Cubit with a preview of the element size Screen capture of adding a surface mesh to a block Screen capture of exporting a mesh into an Abaqus.inp file Example output from the FEM Mesh conversion tool A mesh representing an accurate geometry of the brain, which was created in Cubit and then imported into the deal.ii library Endoscopic third ventriculostomy initial solution Solution after four steps of grid refinement Tumor growth initial solution Solution after four steps of grid refinement vi

7 List of Tables 3.1 Mooney-Rivlin material parameters for different strain rates vii

8 Acknowledgments I want to thank Dr. Corina Drapaca for her support, advice, and motivation. I also want to thank my family, friends, and fellow students for giving me the inspiration and encouragement to finish my degree. viii

9 Dedication This work is dedicated to my father. I would not be who I am today if it were not for his continual support and motivation. ix

10 Chapter 1 Introduction 1.1 Hydrocephalus Hydrocephalus is a clinical condition characterized by abnormalities in the cerebrospinal fluid (CSF) circulation of the brain (Figure 1.1), resulting in ventricular dilation (Figure 1.2b). Normally, there is a delicate balance between the rate of formation and of absorption of CSF, the entire volume being absorbed and replaced once every twelve to twenty-four hours [3]. The CSF forms within the lateral ventricles of the brain, circulates through the ventricles and within the subarachnoid space surrounding the brain, and drains into the venous blood by passing through the arachnoid villi located in the dura mater above the brain [4]. The driving forces to convey the CSF out of the ventricular system are: the pressure gradients between the different parts of the ventricular system, the subarachnoid space, and the venous sinuses [5]. These pressure gradients are created by the continuous CSF secretion and are enhanced by the arterial pulsations of the brain with each heartbeat. Figure 1.1. Circulation of the cerebrospinal fluid (light blue) [1].

11 2 (a) MR image (left [6]) and schematic (right [2]) of a normal brain. (b) MR image (left [6]) and schematic (right [2]) of a hydrocephalic brain. Figure 1.2. A normal brain vs. a hydrocephalic brain. Ventricles are indicated by red arrow. Although hydrocephalus cases were regularly described by Hippocrates ( B.C.) and Galen ( A.D.), the first detailed description of a treatment was given in the tenth century by Abulkassim Al Zahrami [7]. Still another six centuries had to pass for the first clear description of hydrocephalus to be given by Vesalius ( ), and only recently has its treatment started to be more successful [8]. Currently, it is believed that hydrocephalus may be caused by increased CSF production, by obstruction of CSF circulation or of the venous outflow system [5], or due to genetic factors [9]. One possible, age-related, classification of hydrocephalus is into infantile and normal pressure (NPH) hydrocephalus (Figure 1.2). In the infantile hydrocephalus, the intracranial pressure is raised, and, as the CSF accumulates in the ventricles, the brain tissue compresses, and both the ventricles and the skull, enlarge. On the other hand, NPH is predominantly found in adults over 60 years of age, and it is characterized by a normal intraventricular pressure. Unlike the infantile type, NPH is hard to diagnose since many conditions affecting older individuals can mimic the symptom profile of NPH, including Parkinson s disease, Alzheimer s disease, metabolic and psychiatric disorders, endocrine dysfunction, infections, trauma, vascular and neurodegenerative disorders, and incontinence from urinary tract disorders [10]. In most of the cases, the cause of NPH is unknown.

12 3 The incidence of infantile hydrocephalus in the U.S. is approximately 1-3 per 1,000 births, while worldwide one birth in every 2,000 results in hydrocephalus [11]. In 2000, there were 27,870 patients treated for NPH and more than 8,000 new cases diagnosed [12]. Recent estimates of NPH incidence range from 50,000 to 375,000 people in the United States, with the higher figure more likely to be correct [13]. According to the U.S. Census Bureau, in 2002, there were nearly 60 million people age 55 or older living in the United States. Average life expectancy was approximately 77 years in 2001, according to the National Center for Health Statistics, Centers for Disease Control and Prevention [14]. Since average life expectancy is expected to continue to increase, the number of diagnosed cases of NPH and the associated treatment costs will continue to grow, as well. The efforts in treatment have been principally through CSF flow diversion. Within limits, the dilation of the ventricles can be reversed by a surgical placement of a shunt in the brain to drain excess CSF into the abdomen where it can be absorbed (Figure 1.3). The extent of improvement after neurosurgical shunt procedures varies greatly: 45%-65% of patients respond positively [15 17], while morbidity is about 40%-50% [15, 17, 18]. Therefore, there is an urgent need for a proper selection of patients who may benefit by a shunt operation. An alternative to the shunt procedure is performing an endoscopic third ventriculostomy surgery, which opens a hole inside the brain to re-establish flow of the CSF. However, despite the efforts of neurosurgeons and great advances in technology, the two treatment options display no statistically significant difference in the efficacy for treating hydrocephalus because the process by which hydrocephalus develops is still poorly understood. Figure 1.3. Diagram of a CSF shunt, which is used to reverse the dilation of the ventricles [2].

13 4 1.2 Modeling Approaches In order to better understand the pathophysiology of hydrocephalus, appropriate mathematical and computational models are required. So far, there have been two approaches to modeling the biomechanics of the hydrocephalic brain parenchyma. In the first approach the brain is modeled as a linear elastic or viscoelastic material [19 24]. In the second approach, the brain parenchyma is assumed to be made of a porous, linearly elastic solid with Newtonian fluid filled pores [25 30]. Both linear viscoelastic and poroelastic models are based on the assumption of small strain theory, which means that they are capable of predicting only small deformations. To correctly model the large deformations seen in hydrocephalus, a nonlinear material law is required. The first quasi-linear viscoelastic model of the brain parenchyma was proposed in [31]. However, all the above mentioned mechanical models suffer from the assumption that the brain s geometry is either a cylinder or a sphere. In addition, with the exception of the models presented in [30] and [24], none of the other models takes into account the intracranial pulsations due to the heartbeats and, thus, fails to address the possible role of the pulsations in the development of hydrocephalus [32, 33]. All of the previously stated modeling approaches are also of no use in trying to model normal pressure hydrocephalus, where the deformation occurs without the existence of an increased intracranial pressure. In order to design better diagnostic and treatment protocols for NPH, we need to develop realistic biomechanical models of the brain for the numerical simulation of NPH. Most of the models presented in the engineering literature on NPH are based on the hydrodynamics of cerebro-spinal fluid (CSF) which tends to accumulate in the brain ventricular system during the development of NPH (in a healthy brain, the CSF circulates continuously between the ventricles, the site of CSF production, and the subarachnoid space, the site of CSF absorption) [34,35]. All these models are based on the bulk flow theory: the driving force of the CSF bulk flow is the CSF pressure at the production site being slightly in excess of the pressure at the absorption site. In this theory, the enlargement of the ventricles during the development of hydrocephalus is due to an increased intracranial pressure. However, NPH is incompatible with the bulk flow theory since in the case of NPH the ventricles dilate without an increase of the CSF pressure [36]. Recently, Levine [37] postulated that there exists an abnormal but very small gradient of static pressure across the cerebral mantle that should be sufficient to produce the ventricular dilatation of NPH. Although this is an attractive theory, it has very limited medical applicability since there are no instruments sensitive enough to measure such small abnormal gradients. Finally, none of the published biomechanical models of NPH incorporates any relevant clinical information about abnormal electro-chemical processes taking place during the development of NPH [38, 39].

14 5 1.3 Outline In Chapter 2, we provide a short review of the continuum mechanics framework, including the kinematics of a deformable body, imcompressibility, and the constitutive laws of a hyperelastic material, specifically a Mooney-Rivlin material. In Chapter 3, we present the modeling approach that we used to model the deformation seen in hydrocephalus through the use of the finite element package Abaqus. We assume that the brain parenchyma is made of a nonlinear hyperelastic material of a Mooney-Rivlin type. We also use the real geometry of the brain as it is shown in medical images taken from [6] to analyze the mechanical behavior of the brain. We will examine the effect of the ventricle pulsations and how these pulsations affect the development of hydrocephalus. Finally, we will show that there exists a pressure threshold after which any increase in pressure due to pulsations will induce hydrocephalus In Chapter 4, we discuss the need for a triphasic mechano-electrochemical theory in order to accurately model normal pressure hyrdocephalus. We formulate the first neuro-mechanical model of the brain that couples the electro-chemical and mechanical properties of the brain. We assume that the brain tissue is a charged hydrated soft tissue made of a solid phase, an interstitial fluid phase and an ion phase with two monovalent ion species. We will use the proposed model to study the onset of NPH due to a change in the ionic concentrations of the ventricular CSF in the absence of an elevated intracranial pressure. In Chapter 5, we show the process we designed to create geometrically accurate meshes of the brain. A stack of MR images of a normal brain were imported in the software called Simpleware, and then segmented to obtain the geometry of just the white and gray matter. This geometry could then be exported as an Abaqus part file and then imported straight into Abaqus. If further mesh editing was required, the part file could be imported into a mesh generation and geometry preparation software called Cubit. In Chapter 6, we present the results of a elastic models representing the theoretical displacements seen in an endoscopic third ventriculostomy surgery performed in some cases of hydrocephalus and also the displacements caused my the growth of a tumor. These models were computed using the C++ deal.ii finite element library. In Chapter 7, we discuss our achievements and future work.

15 Chapter 2 General Constitutive Theory 2.1 Kinematics of Finite Deformations In the theory of continuum mechanics a body s reference or undeformed configuration is denoted by β r, with its boundary denoted by β r [40]. Points in β r are labeled by their position vectors X relative to an origin. When the body is deformed from β r, it occupies a new configuration called the current or deformed configuration of the body. This configuration is denoted by β with its corresponding boundary β (Figure 2.1). The deformation from the reference to the deformed configuration is represented by the mapping χ : β r β, such that: x = χ(x, t), (2.1) where x is the position vector of the point X in β. The mapping χ is called the deformation from β r to β, and is assumed to be a smooth one-to-one map of every material point of the body. Figure 2.1. Reference configuration (left), deformed configuration (right).

16 7 The deformation gradient, denoted by F, is defined as: F = x X, (2.2) such that: 0 < J det F <. (2.3) Equation (2.3) says that no region of an object of positive, finite volume is deformed into one of zero or infinite volume. Every material body whose deformations can be represented by equations (2.1)-(2.3) is called a continuum. The deformation gradient has the unique polar decompositions: F = RU = VR, (2.4) where R is the proper orthogonal tensor which characterizes the local rigid body rotation, and the positive definite, symmetric tensors U and V, called the right and left stretch tensors, respectively, describe local deformations. Decomposition (2.4) of the deformation gradient F can be seen as a pure stretch U at X followed by a rigid body rotation R, or as the same rigid body rotation followed by a pure stretch V at x. The stretch tensors may be written in the following spectral forms: 3 3 U = λ i u (i) u (i), V = λ i v (i) v (i) (2.5) i=1 i=1 where v (i) = Ru (i), i {1, 2, 3}, λ i are the principal stretches, u (i) are the unit eigenvectors of U (the Lagrangian principal axes), v (i) those of V (the Eulerian principal axes), and denotes the tensor product. It can be shown that J = λ 1 λ 2 λ 3. Because U and V are in general tedious to compute, it is customary to use the following tensors: C = F T F = U 2, B = FF T = V 2 (2.6) The positive definite, symmetric tensors C and B are called the right and left Cauchy-Green deformation tensors, respectively. 2.2 Principles of Classical Continuum Mechanics There are two kinds of forces that act on any part P β of a continuum body: a contact force t n per unit area of the boundary P of P, and a body force b per unit volume of P [40]. The total force f(p, t) and the total torque τ(p, t) acting on the part P are related to the momentum and the moment of momentum of the material points of β in a deformed configuration in accordance

17 8 with Euler s laws of motion: and τ(p, t) = f(p, t) = P P t n da + b dv = d v dm (2.7) P dt P x t n da + x b dv = d x v dm. (2.8) P dt P Let ρ r and ρ be the mass densities in β r and β, respectively. In the classical theory of continuum mechanics, the mass of a body is conserved during deformation, which means that the following conservation law is true: ρ r = Jρ (2.9) Application of the first law (2.7) to an arbitrary tetrahedral element leads to the proof of existence of the second order Cauchy s stress tensor T: t n = Tn (2.10) where n is the outward unit normal to P in the deformed configuration. By substituting equation (2.10) into (2.7) and using the divergence theorem, we obtain the Cauchy s first law of motion: div T + b = ρ a (2.11) where a is the acceleration of a particle. Equation (2.11) corresponds to Newton s second law of motion in continuum mechanics. The second law (2.8), along with equations (2.10) and (2.11) restricts the Cauchy stress, T, to the space of symmetric tensors: T = T T. (2.12) The Cauchy stress characterizes the contact force distribution t n in the deformed configuration per unit area in the deformed configuration. In solid mechanics, the Cauchy stress is an inconvenient measurement, because the deformed configuration is generally not known before the deformation has occurred. Therefore the first Piola-Kirchoff stress tensor T R, is introduced to define the contact force distribution t N = T R N in the deformed configuration per unit reference area in the undeformed configuration, where N is the outward unit normal to P in the undeformed configuration, such that P t n da = P R t N da. The balance laws in the reference configuration are: Div T R + b R = ρ R a (2.13) T R F T = FT T R (2.14) So far, the deformations of a continuum and the forces that produce them have been presented without any mention of any specific material properties the body may have. However, the constitutive nature of the material dictates its response to applied forces. In mechanics, the

18 9 intrinsic relationship between the deformation F, the rate of deformation Ḟ, and the stress T or T R is described by an equation known as a constitutive equation. In the following section the principle of mechanical energy will be used to derive the constitutive equation of hyperelastic solids. 2.3 Hyperelastic Constitutive Law By definition there exists a strain energy function: Σ (X, t) = Σ (F (X, t) X), (2.15) such that the constitutive equation for a hyperelastic solid is: T R = 2.4 Isotropic Hyperelastic Materials Σ (F) F. (2.16) From the spectral theorem the principle invariants of tensors C and B are the same for every deformation F. Thus, for example, the principal invariants of B are defined by: I 1 = tr (B) = λ λ λ 3 I 2 = 1 [ 2 I1 tr ( B 2)] = λ 2 2 λ λ 2 3 λ λ λ 2 2 I 3 = det B = λ 2 1 λ λ 3 (2.17) (2.18) (2.19) A hyperelastic material is isotropic if and only if its strain energy function (2.15) is a function of the principal invariants (2.17) (2.19). One common representation of the general constitutive equation for an isotropic solid is: T = α 0 I + α 1 B + α 2 B 2, (2.20) or, by use of the Cayley-Hamilton theorem: T = β 0 I + β 1 B + β 1 B 1. (2.21) The scalar coefficients: α i = α i (I 1, I 2, I 3 ) (2.22) β j = β j (I 1, I 2, I 3 ) (2.23) where i = 0, 1, 2 and j = 0, 1, 1, are called the material or elastic response functions. These are

19 10 given in terms of the strain energy function by: β 0 = α 0 I 2 α 2 = 2 [ ] Σ Σ I 2 + I 3 I3 I 2 I 3 (2.24) 2.5 Incompressibility β 1 = α 1 I 1 α 2 = 2 I3 Σ I 1 (2.25) β 1 = I 3 α 2 = 2 I 3 Σ I 2 (2.26) Incompressibility, inextensibility, and rigidity are important examples of internal material constraints. An incompressible material may undergo only isochoric (or volume preserving) deformations. Every deformation of an incompressible material is subject to the internal material constraint: γ(f) = J 1 = 0 (2.27) This equation must hold for all deformations of an incompressible body. The Cauchy stress of an incompressible material is determined by F only to within a stress p I, where p = p(x) is a Lagrange multiplier to be determined by the boundary conditions. The total Cauchy stress T of an incompressible elastic material, which need not be hyperelastic, is thus given by: T = p I + T E (F) (2.28) The extra stress T E (F) represents the elastic constitutive response of the material. If the incompressible material is hyperelastic and isotropic, the invariant strain energy function is only dependent on the first and second principal invariants: Σ = Σ (I 1, I 2 ) (2.29) Therefore, the constitutive equation for an incompressible, isotropic hyperelastic material is given by: T = p I + α 1 B + α 2 B 2, (2.30) or, equivalently, T = p I + β 1 B + β 1 B 1. (2.31) The scalar field p differs in (2.30) and (2.31), and the material response coefficients: α i = α i (I 1, I 2 ), β j = β j (I 1, I 2 ) (2.32)

20 11 with i = 1, 2 and j = 1, 1 are given by: β 1 = α 1 + I 1 α 2 = 2 Σ I 1, β 1 = α 2 = 2 Σ I 2 (2.33) 2.6 The Mooney-Rivlin Material Natural rubber, synthetic elastomers, and biological tissue are examples of materials that have been modeled as incompressible, isotropic hyperelastic materials [40]. One constitutive model used to describe these materials is called a Mooney-Rivlin material. The Mooney-Rivlin material is such that β 1 = α and β 1 = β are constants in (2.33) is the most general theoretical model for which both response functions are constant. Therefore the general strain energy function of the Mooney-Rivlin material is given by: Σ(I 1, I 2 ) = α 2 (I 1 3) + β 2 (I 2 3) (2.34) This is the first constitutive model for hydrocephalus that we will analyze in the next chapter.

21 Chapter 3 The Hyperelastic Model 3.1 Mooney-Rivlin Model Material Properties We assume that the brain parenchyma behaves as an incompressible, isotropic hyperelastic material of the Mooney-Rivlin type. The Mooney-Rivlin strain energy function implemented in Abaqus for incompressible materials has the following general form [41]: U = C 10 (Ī1 3) + C 01 (Ī2 3), (3.1) where U denotes the strain energy per unit of reference volume, C 10 and C 01 are material parameters, Ī1 and Ī2 are the first and second deviatoric strain invariants defined as: Ī 1 = λ λ λ 2 3 (3.2) Ī 2 = λ 1 + λ 2 + λ 3 (3.3) where the deviatoric stretches λ i = J 1/3 λ i ; J is the total volume ratio; and λ i are the principal stretches. Strain Rate [mm/s] C 10 [P a] C 01 [P a] Table 3.1. Mooney-Rivlin material parameters for different strain rates

22 13 The constants C 10 and C 01 were taken from [31] and can be seen in Table 3.1. These parameters were obtained from fitting the model to mechanical experiments perform on human brain tissue in vitro [42]. The material parameters for the 5.08 mm/s strain rate were used in the model because hydrocephalus is a slow process which develops over an extended period of time. Therefore the slower strain rate material parameters were chosen in the model of the brain parenchyma. Figure 3.1. Stress vs. Strain plot, of the brain parenchyma, created by Abaqus material evaluate tool. This plot corresponds to the Mooney-Rivlin material parameters, in Table 3.1, for a strain rate of 5.08m/s Geometry The geometry of the brain parenchyma plays an important role in the brain s response to mechanical loading. As we can see in Figure 3.2a, the wavy nature of the cortical surface of the brain makes it difficult to create a realistic geometry of the brain directly in Abaqus. To insure an accurate geometry, a stack of MRI images were obtained from [6], in which image segmentation (Figure 3.2b) could be performed to create a data set importable into Abaqus (Figure 3.2c). Two software packages called ScanIP T M and +ScanFE T M, from Simpleware Ltd. [43], were used to segment the brain parenchyma and generate a data set readable in Abaqus (This process is shown in Chapter 5. For simplicity, we chose to model just a thin slice of the brain. The model includes a 3mm thick slice, which was the minimum allowed by the image segmentation software. In order to avoid any problems with the creation of the brain parenchyma mesh, we used a recursive Gaussian filter to smooth out the cortical surface and then manually created the vertical slots within the image segmentation software. This mesh kept the main geometric features of the brain and allowed the analysis in Abaqus to be done without any issues.

23 14 Figure 3.2. (a) One slice of a MR image. (b) Segmentation of the MR image (b) 3mm thick mesh of the brain parenchyma, created from the segmention of the MR image Boundary Conditions We will focus on the development of hydrocephalus in young adults (this is not normal pressure hydrocephalus). The outer cortical surface (the red in Figure 3.3) was constrained to prevent any outward displacement. This constraint is due to the rigidity of the adult skull. The vertical slots (the purple in Figure 3.3) were free to move and were also allowed to contact each other. The top and bottom surfaces (normal to the page) of the brain slice were also restrained from any displacement normal to the surface. This was done to mimic a plane strain model. A pulsatile pressure (Figure 3.4) was applied to the outer surface of the ventricular walls (the blue in Figure 3.3) to imitate the pulsations of brain due to the heartbeats [44]: a(t) = α(1.3 + sin(ωt π 2 ) 1 2 cos(2ωt π )) (3.4) 2 In our simulations we used the following parameters: α = 2500Pa, ω = 1Hz. Figure 3.3. of the model. Boundary conditions Figure 3.4. Plot of the pulsation forcing function a(t) given by equation (3.4).

24 Results In Figures 3.5, 3.6, and 3.7, we show contour plots of the horizontal and vertical displacements at different ventricular pressures. The normal intracranial pressure in adults varies from approximately 780Pa to 1170Pa. Intracranial pressure values above 1960Pa in adults suggest the presence of hydrocephalus [45]. The size of the lateral ventricles shows a 10-20% change during the cardiac cycle as measured from medical images [45]. One of the base horizontal distances in the middle of the left ventricle is 1.16cm. During the cardiac cycle, this distance has a 1.16mm -2.32mm change. In our model, when a ventricular pressure of 1,013Pa is applied, to mimic the normal pressure within the brain, the ventricle horizontally expands 1.5mm, which is between the above mentioned normal limits. With a pressure of 2,549Pa the horizontal expansion is 3.86mm and for 5,886Pa the horizontal expansion is 5.04mm. This is a clear indication of the presence of hydrocephalus for pressures above 1960Pa. Contour plots of the Mises stresses at different ventricular pressures are shown in Figure 3.8. Figure 3.5. Horizontal displacements (limits: red = mm and blue = mm) (left) and vertical displacements(limits: red = mm and blue = mm) (right) with a ventricular pressure of 1,013Pa.

25 16 Figure 3.6. Horizontal displacements (limits: red = mm and blue = -2.67mm) (left) and vertical displacements(limits: red = mm and blue = mm) (right) with a ventricular pressure of 2,549Pa. Figure 3.7. Horizontal displacements (limits: red = mm and blue = mm) (left) and vertical displacements(limits: red = mm and blue = mm) (right) with a ventricular pressure of 5,886Pa.

26 17 Figure 3.8. Mises Stress with a ventricular pressure of 2,549Pa (left) and with a ventricular pressure of 5,886Pa(right) (max stress = Pa). (limits: red = Pa and blue = 0Pa). 3.3 Finite Element Simulation of Hydrocephalus We modeled the brain parenchyma as an incompressible, isotropic, Mooney-Rivlin hyperelastic solid and used the real geometry of the brain as seen in medical images to analyze the response of the brain to pulsatile ventricular pressure. We have shown that for pressure values above 1960Pa the model correctly predicted large displacements of the ventricles of about 3-5mm. This means that once a pressure threshold of 1960Pa has been reached, any further pressure increase due to the pulsations will induce hydrocephalus. This is in agreement with clinical observations reported in [45].

27 Chapter 4 The Triphasic Mechano-Electrochemical Model 4.1 The Triphasic Model The triphasic mechano-electrochemical theory has been applied, so far, to model the mechanics of the articular cartilage [46 49]. However the theory can be applied to any charged hydrated soft tissue made of an intrinsically incompressible, porous-permeable, charged solid phase; an intrinsically incompressible, interstitial fluid phase; and an ion phase with two monovalent ion species anion ( ) and cation (+). Although the theory has been extended to include multiple species of ions [47], for simplicity we will consider the 1:1 electrolyte case first. In addition, there exist positively and negatively charged groups on the solid phase called fixed charges since they are much less mobile than the freely mobile ions dissolved in the fluid phase. The solid phase and the ion phase are electrically charged, while the fluid phase and the tissue as a whole are electrically neutral. A schematic picture of the structure of the triphasic soft tissue is shown in Figure 4.1. Figure 4.1. Structure of a charged hydrated soft tissue.

28 19 The constitutive equations of the thriphasic soft tissue with infinitesimal deformations are [46,49]: σ = pi + λ s tr (ɛ) I + 2µ s ɛ (4.1) µ f = µ f 0 + p RT Φ(c+ + c ) ρ f (4.2) µ a = µ a 0 + RT M a ln (γ a c a ) + z af c Ψ M a, a = +, (4.3) where equation (4.1) is Hooks law for the linear elastic phase, and equations (4.2), (4.3) are the constitutive equations for the fluid phase and the ion phase. We have denoted by p the fluid pressure, σ the stress tensor in the elastic solid, ɛ the strain tensor in the elastic solid, λ s, µ s the Lame coefficients which depend on solid volume fraction and ion concentrations c a, a = +,, R is the universal gas constant, T is the absolute temperature, µ f is the chemical potential of the fluid phase with µ f 0 the reference chemical potential, Φ the osmotic coefficient, ρf the true mass density of the fluid, Ψ the electric potential, γ a the activity potential coefficients, µ a the electro-chemical potential of the ion species a with µ 0 a its reference electro-chemical potential, z a the valence of ion species a including sign, M a the molar weight of the a ionic species, and F c is Faraday constant. The governing equations consist of the equilibrium equation for the mixture, and the continuity equations of the mixture and of the ions, which combined with the electroneutrality condition give [49]: (λ s tr (ɛ) I + 2µ s ɛ) ( RT ɛ f + RT Φc k) = 0 (4.4) ν s RT α (ϕ f ɛ f + ϕf c + ɛ + ɛ+ + ϕf c ) ɛ ɛ = 0 (4.5) [ RT ( ϕ α ϕf c F ɛ f f c + D + ɛ + + RT α [( ϕ f c D ɛ ϕ f (c + ) 2 ɛ + + RT α RT α ϕ f (c ) 2 ɛ ϕ f c + c ɛ + RT α ) ɛ + ] + ϕ f c + c ) ] ɛ = 0 (4.6) ɛ ( ϕ f c k) t [( ϕ f c D [( ϕ f c + D + = ɛ + RT α ɛ + ϕ f (c ) 2 ɛ + RT α ϕ f (c + ) 2 ɛ + + RT ϕ f c + c α ɛ + ϕ f c + c ) ] ɛ + RT α ɛ ) ɛ + ] + ( ϕ f c k ν s RT ) α ϕf c k ɛ f (4.7) In equations (4.4) - (4.7) we denoted ) by c k = c + +c, α = ϕ f /k with k the hydraulic permeability and porosity ϕ f = ϕ f 0 (1 + ϕ f 0 tr (ɛ), ν s the velocity of the solid phase, and D +, D the diffusivity coefficients of the two ion species. The modified electrochemical potential functions

29 20 are defined as: ɛ f = p RT, ɛ+ = γ + c + exp ( ) ( ) Fc Ψ, ɛ = γ c Fc Ψ exp RT RT (4.8) In addition, the fixed charged density (FCD) is by definition c F = c + c 4.2 Numerical Simulation of NPH Geometry and Boundary Conditions In this section we investigate the onset of NPH due to a change in the ionic concentrations of the ventricular CSF in the absence of an elevated intracranial pressure. For simplicity, we consider the linearized one-dimensional (1D) case. Initially, a brain tissue sample made of white matter only is at equilibrium with the intraventricular CSF (external bathing solution) with concentration c 0 of monovalent ions. Since there is no existent clinical literature on what ions in the ventricular CSF might cause NPH, we assume for example that the two monovalent ions are Na + and Cl, since they have the largest concentrations in the ventricular CSF [50]. We assume that the tissue of brain with NPH has length h and is confined by the rigid, impermeable skull which does not allow for lateral movement of the tissue (Figure 4.2). The boundary conditions at the bottom of the specimen are that the solid displacement and all the electrochemical fluxes are zero, while the boundary conditions at the top are that the stress and the electrochemical potentials are continuous across this boundary. Initially, the ionic concentration in the ventricular CSF decreases from c 0 to c 1 linearly. If the total ionic concentration in the brain tissue is zero, then the brain tissue will swell. However, if we assume that the brain tissue with NPH varies linearly from c 0 at the top to zero at the bottom, then the brain tissue will shrink. Figure 4.2. Schematic representation of the 1D brain.

30 Material Parameters The parameters used in our simulations are: h = 1mm, λ s +2µ s = kpa [32], D + = 0.5x10 9 m 2 /s, D = 0.8x10 9 m 2 /s, α = 0.7x10 15 Ns/m 4, T = 298 K, ϕ f 0 = 0.75, Φ = 1, cf 0 = 0.2 meq/ml (most of these parameters are not known for the brain so we have taken them from [49]). The initial concentration of ions in the ventricular CSF has a normal value of c 0 = mol/l [50]. It decreased linearly to the abnormal value of c 1 = mol/l within 1500 s. The displacement of the ventricular wall is shown in Figure Results Figure 4.3. Ventricular wall displacement showing an initial swelling followed by shrinking. In Figure 4.4 we show the influence of the initial fixed charged density c F 0 on the displacement of the ventricular wall, while in Figure 4.5 we show how this displacement varies with the initial porosity ϕ F 0. We notice that as c F 0 increases the swelling and the displacement decrease, and as decreases the swelling and the displacement increase. ϕ F Conclusion We have proposed the first neuro-mechanical model of the brain that links the electro-chemical and mechanical properties of the brain. Using the thriphasic theory, we assumed that the brain is a charged hydrated soft tissue made of a solid phase, an interstitial fluid phase and an ion phase with two monovalent ion species. We have shown that the proposed model can predict the shrinkage of the brain tissue seen in NPH patients due to a change in the ionic concentrations of the ventricular CSF and in the absence of an elevated intracranial pressure. This model opens the possibility of treating hyrdocephalus noninvasively through the use of pharmacutical drugs.

31 22 Figure 4.4. The ventricular wall displacement for c F 0 = 0.2 (straight line), 0.4 (dashed line) and 0.8 (dotted line). Figure 4.5. The ventricular wall displacement for ϕ F 0 = 0.25 (straight line), 0.5 (dashed line) and 0.75 (dotted line).

32 Chapter 5 Geometrically Accurate Meshes of the Brain A multi-step process was used to create the anatomically correct meshes of the brain. This process started with obtaining a stack of MR images with a raw byte (unsigned) file format from the BrainWeb: Simulated Brain Database [6]. Then the images were segmented, using Simpleware software, to obtain just the gray and white matter of the brain parenchyma. A finite element mesh was then applied to the segmented part. That mesh was then imported into the finite element package for modeling. Below, I will describe the process and parameters that were used in creating the meshes. 5.1 Creating meshes using Simpleware The first step was to import the MR images into Simpleware. To import the images, there are some values that need to be know about the stack images like size of pixels, number of slices, and the spacing between slices (See Figure 5.1a). These values are provided by the BrainWeb website [6]. Therefore the required values are Size-X = 181, Size-Y = 217, Number of Slices = 181, and the spacing for all three dimensions is 1mm. After inputing the correct values, the images that are too be segmented need to be choosen. For our model, we only modeled a thin slice (3mm) of the brain parenchyma, so images were picked because it was near the vertical center of the brain and they were the best images for segmenting the ventricles (See Figure 5.1b).

33 24 (a) Parameters for importing. (b) Choosing the MR images. Figure 5.1. Process of importing MR images into Simpleware. Once the images were imported, then they could be segmented using multiple tools. To segment out the white and gray matter, we used the floodfill tool to segment the image based on grayscale intensity values of the image. Figure 5.2 shows the brain parenchyma segmented by using a lower intensity value of 96 and an upper intensity value of 185. To edit the segmentation, not based on intensity values, the paint tool provides a pointer that can be used to add to or erase parts of the segmentation. During the segmentation procress, the CSF in the ventricles and the CSF between the skull and brain parenchyma were also segmented in order to create surface pair interactions that could be used in Abaqus to define contact, load, or displacement boundary conditions. Figure 5.2. Screen capture of using the flood fill tool to segment the MR images.

34 25 After all the segmentation is complete, then a mesh can be created. For a model in Abaqus (Chapter 3) the mesh can be exported as an.inp file, which can be directly imported into Abaqus as a orphan meshed part. For a model using the deal.ii library, a different approach is required. The deal.ii library is a C++ program library targeted at the computational solution of partial differential equations using adaptive finite elements. The library enables rapid development of modern finite element codes, while taking care of the details of grid handling and refinement, handling of degrees of freedom, input of meshes and output of results in graphics formats. It has support for several space dimensions at once, so programs can be written independent of the space dimension without unreasonable penalties on run-time and memory consumption. Other aspects of the library include adaptive meshes and a wide array of tools classes often used in finite element programs [51]. The library is only capable of using either quadrilateral elements or tetrahedral elements, not both in the same mesh. In Simpleware, if quadrilateral elements are selected to mesh a complicated geometry, Simpleware has trouble with using just quadrilateral elements to get an accurate mesh. So it includes tetrahedral elements at the trouble spots on the geometry. Therefore, to create a mesh usable in the library, an external meshing program called Cubit is required to create the single element type meshes. To allow Cubit the ability to create these meshes, the segmented part in Simpleware must be exported as an.stl file. An STL file describes a raw unstructured triangulated surface by the unit normal and vertices (ordered by the righthand rule) of the triangles using a three-dimensional Cartesian coordinate system. Figure 5.3a shows an example of a mesh representing an accurate geometry of the brain and Figure 5.3b shows how Simpleware is capable of creating larger meshes of complex geometries. (a) A mesh created from a few MR images. (b) A mesh created using the full stack of MR images. Figure 5.3. Examples of exported meshes from Simpleware.

35 Using Cubit The STL mesh uses many triangles to describe the surface of a part, but for our purpose we needed to combine these triangles into larger surfaces that are useful for our purposes. When importing the file, there is a feature angle option (seen in Figure 5.4) which specifies the angle at which surfaces will be split by a curve or where curves will be split by a vertex. So for examples, 180 degrees will generate a surface for every element face, while 0 degrees will define a single, unbroken surface from the shell of the mesh. The default angle is 135 degrees, but for our purposes it needed to be changed to 150 degrees to created a single top and bottom surface. The top or bottom surface could then be remeshed to create a single 2D cross section of the brain parenchyma. Figure 5.4. STL Import Options window. With the geometry imported into Cubit, the surface could then be meshed. Each surface in Cubit is label with an ID number which is used to select it in the program. After selecting a surface then it has to be seeded by going to the Command Panel, selecting the Mode as Meshing, Entity as Surface, and then Action as Intervals (these buttons are pushed in Figure 5.5). The Interval setting can be set as Auto, where the program decides what the element size should be, or Constant Size, where the user can define the size of the elements. For our purposes a constant size of 5 provided a sufficient mesh size. Each red dot in Figure 5.5) shows where a vertex of an element will be. Once the surface is seeded, then it can be meshed by clicking the Mesh button to mesh the surface using the default mesh parameters. Before the mesh can be exported, it needs to be added to a block by going to the Command Panel, selecting the Mode as Materials and BCs, then Entity as Blocks (these buttons are pushed in Figure 5.6). After clicking the Surface radio button, the meshed surface can be selected using its ID number, then added to a block by clicking apply. Now the block containing the meshed surface can be exported to an output file.

36 27 Figure 5.5. Screen capture of Cubit with a preview of the element size. Figure 5.6. Screen capture of adding a surface mesh to a block. The mesh can be exported by going to the Command Panel, selecting the Mode as Analysis Setup, then the Operation as Export Mesh. The Abaqus option should already be selected in the drop box, so then the output file location can be defined by clicking on the Filename button. Then the Block ID number needs to be entered into the Block ID(s) text box (since only one block was created, the ID number should be 1). Before clicking the Apply button, the Export Using Cubit IDs should be unchecked before exporting because it will make the node and element numbering start at 1.

37 28 Figure 5.7. Screen capture of exporting a mesh into an Abaqus.inp file. 5.3 Using the mesh converter in deal.ii The file that is created using Cubit is not yet ready to be used with the deal.ii library because it is not formatted correctly. Currently, it is a.inp file but needs to be converted to the.ucd file format. To do this, there is a mesh conversion tool (Figure 5.8) in the deal.ii library located in the deal.ii/contrib/mesh conversion folder. There is a readme file that helps with using the conversion program. Before using the program it has to be complied by using the make command in a terminal window. Then a.ucd file can be created from an Abaqus.inp file by using the following general command:./convert mesh <dimension> <ABAQUS input file type> /path/to/input file.inp /path/to/output file.ucd For our purposes <dimension> = 2, and <ABAQUS input file type> = 1. Figure 5.8. Example output from the FEM Mesh conversion tool.

38 Importing mesh into the deal.ii Library Now that the mesh 5.3a is in the right file format, it can be imported to use with the deal.ii library. We used the step-5 tutorial C++ program (Reading a grid from a Disk) in the deal.ii library [51] as a starting point (See Appendix A for my C++ program). We removed all the code pertaining to doing computations and edited the code to just read in the.ucd file and then export the mesh as a.eps file. This was done to make sure that the mesh could be imported correctly without any errors like an incorrect numbering of the nodes or elements. The.eps output file can be seen in Figure 5.9 and that the complex geometry was meshed with only quadrilateral elements. Figure 5.9. A mesh representing an accurate geometry of the brain, which was created in Cubit and then imported into the deal.ii library.

39 Chapter 6 Elastic Models Using deal.ii Library With the ability to create meshes from MR images and then import them to use with the deal.ii library, we needed to test if we could perform elastic computations on the meshes. The models in this chapter were created by editing the C++ code for the step-8 tutorial program (Systems of PDE. Elasticity) with the deal.ii library [51]. See Appendix B for an example of the source code used to run these models. In this tutorial, the elastic equations are solved. They are an extension to Laplace s equation with a vector-valued solution that describes the displacement in each space direction of a rigid body which is subject to a force. Of course, the force is also vector-valued, meaning that in each point it has a direction and an absolute value. The elastic equations are the following: j (c ijkl k u i ) = f i, i = 1... d (6.1) where the values c ijkl are the stiffness coefficients and will usually depend on the space coordinates. In many cases, one knows that the material under consideration is isotropic, in which case by introduction of the two coefficients λ and µ the coefficient tensor reduces to: c ijkl = λ ij kl + µ ( ik jl + il jk ) (6.2) The elastic equations can then be rewritten in much simpler a form: λ ( u) ( µ ) u µ ( u) T = f (6.3) 6.1 Finite Element Simulations Using the deal.ii Library In this section, we show finite element simulations using the deal.ii library for modeling the mechanical behavior of the brain due to endoscopic third ventriculostomy surgery (subsection 6.1.1) and the presence of a tumor (subsection 6.1.2). In both simulations the λ and µ values

40 31 were both set to 1, for the purpose of demonstration. Also the brain is a rigid body, meaning that the outer boundaries are fixed and unable to move. In these models, the mesh is locally refined in multiple steps, using an error estimator class which estimates the energy error with respect to the Laplace operator. The refinement then results in nodes on the interfaces of cells which belong to one side, but are unbalanced on the other. The common term for these is hanging nodes, and can be seen in both Figure 6.2a and Figure 6.4a. This error estimator, although developed for Laplace s equation has proven to be a suitable tool to generate locally refined meshes for a wide range of equations, not restricted to elliptic problems. Although it will create non-optimal meshes for other equations, it is often a good way to quickly produce meshes that are well adapted to the features of solutions, such as regions of great variation or discontinuities [51] Endoscopic Third Ventriculostomy In an endoscopic third ventriculostomy surgery a small hole would be made between the ventricles to aid in the removal of excess CSF from the brain. This process was mimicked by using a constant (unit) force in the x-direction located in a little circle of radius 2, around the point (90, 115), and y-force of this circle was set to zero. It can be seen in Figure 6.1b that the displacement solution is course and inaccurate, but after locally refining the grid four times near the location of the applied force, the displacement solution becomes smoothed out (see Figure 6.2b). (a) Initial mesh. (b) x-direction displacement. Figure 6.1. Endoscopic third ventriculostomy initial solution.

41 32 (a) Mesh after four local refinements. (b) x-direction displacement. Figure 6.2. Solution after four steps of grid refinement Tumor Growth In the case of tumor growth, otherwise known as neoplasia, which is the uncontrolled division of cells, the mass of a tumor will increase in size. In a confined space such as the intracranial cavity, this quickly becomes problematic because the mass invades the space of the brain pushing it aside, leading to compression of the brain tissue, increased intracranial pressure, and destruction of brain parenchyma [52]. To mimic this growth we used a constant (unit) force in the x-direction located in a circle of radius 7, around the point (125, 60), and y-force of this circle was also set to zero. As with the previous model the inital solution (Figure 6.3b) is course, but after the grid refinement, the solution becomes smoothed out (see Figure 6.4b).

42 33 (a) Initial mesh. (b) x-direction displacement. Figure 6.3. Tumor growth initial solution. (a) Mesh after four local refinements. (b) x-direction displacement. Figure 6.4. Solution after four steps of grid refinement.