The mechanical response of the midbrain to indentation

Transcription

1 Technische Universiteit Eindhoven Mechanical Engineering Section, Mechanics of Materials Group, Materials Technology Eindhoven, September 27, 2009 The mechanical response of the midbrain to indentation Author: Thiam Wai Chua ( ) Document number: MT Supervisors: dr. ir. J. A. W. van Dommelen ir. R. J. H. Cloots

2 Abstract This study was to investigate the mechanical behavior of midbrain tissue undergoing indentation in three directions: anterior-posterior, inferior-superior and lateral-medial directions. Midbrain tissue sample was prepared from porcine midbrain and its material parameter values were obtained from the studied of X.G. Ning et al. [4] and derived from M. Hrapko et al. [6]. Indentation is used because the mechanical response of midbrain tissue can be considered as homogeneous at the length scale (less than 2 mm) of the indenter. Midbrain tissue samples are indented with different indentation speeds and the force relaxation after a step indent is measured as well. The indentation tests shown that the reaction force level and stiffness of the midbrain is lower in inferiorsuperior direction than in the other two directions. It may suggest that the axonal fibers cannot sustain compression loading and may buckle. Midbrain was characterized as an anisotropic hyperelastic incompressible material and a finite element model of the indentation experiment was made using the Holzapfel-Gasser-Ogden model with axonal fibers that were aligned along the inferior-superior direction. Finally, the mechanical response obtained from indentation tests are compared with results from the finite element model of the indentation experiment. Keywords: midbrain tissue, compression, indentation, anisotropy, hyperelastic, Holzapfel- Gasser-Ogden model, finite element analysis.

3 Contents 1 Introduction 2 2 Experiment Midbrain tissue sample preparation Experiment setup Indentation procedure Model formulation and finite element models Transversely isotropic representation of axonal fibers Anisotropic hyperelastic formulation Finite element model Results and discussions 14 5 Conclusions 19 Appendices 21 Bibliography 24 1

4 Chapter 1 Introduction In the Netherlands, there were 791 deaths (2007) and 16,750 injuries requiring hospitalisation (2006) due to traffic accidents although the number of the road injuries has dropped by over 10% and the number of road fatalies has decreased by 30% in ten years time. These traffic accidents cost the society about 9 billion euros every year [1]. Therefore, extensive scientific researches have to be carried out in order to decrease the number of deaths and injuries caused by the traffic accidents in the future. The human nervous system, especially the brain, can be considered as the utmost important part in the human body and traumatic brain injury (TBI) is still a main factor for a large number of fatalities. Thus, continuing research and the development of brain finite element model for determining its mechanical behavior is indispensable in order to design and improve the head protective measures. Soft biological tissue such as brain tissue can be modeled as inhomogeneous, anisotropic, nonlinear hyperelastic materials. In the midbrain region (Figure 1.1), bundles of highly oriented axonal fibers (or neural tracts) are embedded in matrix and provide mechanical strength. The axonal fibers within the midbrain are highly oriented as the midbrain connects the brain to the spinal cord. In the Holzapfel-Gasser-Ogden model [3], the important fundamental hypothesis is that axonal fibers cannot support any compressive stress and would buckle under compressive load. In addition, brain tissue is a complex material to model as test results from different sources vary greatly due to differences in testing conditions, sample preparation, loading conditions, and loading rates [5,7]. The objective of this study is to investigate the anisotropy of midbrain tissue and the ability of the Holzapfel-Gasser-Ogden model to described it. Indentation has been used because midbrain tissue can be considered as a local measurement and homogeneous at this scale. Samples with three different directions of axonal fibers with respect to the loading direction have been prepared for indentation experiments as illustrated in Figure 1.2. A numerical model of the midbrain sample and indenter for the indentation experiment is made. The midbrain sample is modeled using Abaqus software/standard Version (HKS Inc. of Rhode Island, USA) with the anisotropic hyperelastic Holzapfel- Gasser-Ogden model [2,3]. First, the experimental methods for indentation of the midbrain samples that are used to examine the anisotropic hyperelastic mechanical behavior of the midbrain will be discussed. Next, the model formulation of the anisotropic hyperelastic Holzapfel-Gasser- Ogden model and the numerical model of midbrain sample indentation will be described. Finally, results from the experiments and numerical simulations will be compared and discussed. 2

5 CHAPTER 1. INTRODUCTION 3 Figure 1.1: Brain regions: 1. posterior, 2. superior, 3. thalamus, 4. midbrain, 5. anterior [5]. Figure 1.2: Schematic diagram of indentation directions with respect to axonal orientations.

6 Chapter 2 Experiment An experimental setup for an indentations on midbrain tissue samples is used. Twenty four midbrain tissue samples were prepared from nine post-mortem porcine midbrains and all indentations were performed within six hours post-mortem. Two indentation protocols were used. The first protocol was used to determine the minimum duration for full recovery of the sample between each indentation. The second protocol was used to characterize the mechanical properties of midbrain tissue for further analysis. 2.1 Midbrain tissue sample preparation Approximately rectangular midbrain samples (approximately, mm) were prepared by using a Leica VT1000S Vibrating blade microtome. The midbrain samples were prepared from fresh halves of porcine brains from 6-month-old pigs obtained from a local slaughterhouse. Considering the factors of availability and for minimization of postmortem time at testing, porcine brain tissue was chosen as a substitute for human brain tissue. Fresh halves of porcine brains were placed in Phosphate Buffered Saline (PBS) solution during transportation and during the samples preparation process, to prevent dehydration and swelling as well as to slow-down degradation of the porcine brain tissue. The Dura-mater layer was removed from each porcine brain before the preparation process and all tests were conducted within 6 hours post-mortem, as suggested by Garo et al. [11]. 2.2 Experiment setup The experimental indentation setup previously used to investigate brain tissue interregional mechanical properties variations [5], is used for this experiment as well. The indentation experimental setup is depicted in Figure 2.1. A spherical indenter with 2 mm diameter is attached to a leaf spring (depicted as F, leaf spring force) with spring stiffness of 1320 N/m. The leaf spring is controlled by a motor to move the leaf spring wit a speed of 0.05 µm to 1500 µm vertically (shown as symmetrical line in Figure 2.1). The force required for indentation was measured with a resolution of 15 µn and a sample frequency of 40 Hz in order to obtain sufficient data points for post-processing purpose. The sample was placed on dish with a diameter of 40 mm and a height variation of mm. The surface of the dish and and indenter were covered by a hydrophobic coating (Sigmacoate, Sigma) to prevent the adhesion of midbrain sample to dish surface and indenter surface. 4

7 CHAPTER 2. EXPERIMENT 5 Figure 2.1: Schematic diagram of indentation experimental setup with F denoting the leaf spring force. The indentation setup was placed on an air-spring suspension table to reduce force fluctuations and all measurements were performed at room temperature at the Laboratory of Cell and Tissue Engineering of the Eindhoven University of Technology. 2.3 Indentation procedure Two indentation protocols were used as shown in Figure 2.2. First, the approximate height of sample (excluding thickness of fluid layer presents on the surface) was determined. An approximate sample height was determined from the transition from tensile force (due to surface tension when indenter made contact with fluid layer) to a compression force. Next, for all subsequent indentations of first and second protocols, an indentation depth of 10% of the sample height (excluding fluid layer thickness) was chosen [5]. The first protocol (Figure 2.2(a)) consists of three subsequent indentations at a constant speed of 0.1 mm/s. Each indentation starts without a contact between indenter and sample surface. The sample tends to adhere to the indenter and the dish due to a fluid layer present on the surface of the sample. Therefore, the indenter was raised such that the indenter was fully detached from the sample before each recovery period. The duration required for full viscoelastic recovery of the sample between each indentation is determined such that repeated indentations showed a reproducible result. The second protocol (Figure 2.2(b)) was used to characterize the mechanical properties of midbrain tissue for further analysis. In addition, the indentation procedure of the second protocol provides additional information compared with the first protocol and the sample height is determined from the transition from tensile force to compressive force during initial indentation at 0.1 mm/s in the second protocol indentation procedure [5]. The second protocol consists of five indentations at a speed of 0.1 mm/s, 0.32 mm/s, 1.0 mm/s and 0.1 mm/s for the first, second, third and fourth indentation, respectively,

8 CHAPTER 2. EXPERIMENT 6 Protocol 1 Indentor vertical position versus time indenter vertical position arbitrary sample height Protocol 2 Indenter vertical position versus time indenter vertical position arbitrary sample height Indentor vertical position (mm) Indenter vertical position (mm) Time (s) Time (s) (a) protocol 1 (b) protocol 2 Figure 2.2: Indentation protocols: (a) protocol 1 and (b) protocol 2. The solid line indicates the vertical position of indenter and the dashed line indicates the surface of midbrain sample. Table 2.1: Number of midbrain samples and tests in three different directions Midbrain sample directions Number of samples Number of tests inferior-superior 7 18 lateral-medial 8 18 anterior-posterior 9 20 followed by a stress relaxation test. In the stress relaxation test, the sample was indented at a speed of 1.0 mm/s and held for 25 seconds at an indentation depth of 10% of the sample height. Midbrain samples with three different directions (inferior-superior, lateral-medial and anterior-posterior directions) with respect to loading axis (shown as symmetrical line in Figure 2.1) from nine porcine brains were prepared and subjected to the second protocol. Indentations were performed on the samples with the axonal fibers aligned perpendicular and parallel to the loading axis. The indentation test sequence was applied to two or three different locations for each sample. The number of midbrain samples and tests for different directions are tabulated in Table 2.1.

9 Chapter 3 Model formulation and finite element models In soft biological tissue, there is a correlation between its internal microscopic structure and its macroscopic mechanical properties. Therefore, constitutive equations are important for the numerical analysis of the soft biological tissues. The axonal fibers and matrix are the key ingredients in the structure of midbrain. Experimental results in this report (chapter 4) show that the axonal fibers can be characterized as anisotropy (or transversely isotropy) material. These axonal fibers form the main connection between the brain and the spinal cord and they are responsible for transfering the mechanical stimuli (signal) from the brain to the spinal cord and vice versa. As a result, the mechanical properties and behavior of the midbrain are highly dependent on the orientation (or dispersion) of axonal fibers. Therefore, continuum models that do not account for the dispersion of axonal fibers are not able to capture accurately the mechanical behavior of the midbrain [3]. Gasser et al. have developed a hyperelastic arterial layers model which is able to represent the dispersion of the collagen fiber orientation [3]. Arterial tissue is composed of three layers with highly uniaxial collagen fibers. The model incorporates an additional scalar structure parameter, k, that characterizes the dispersed fiber orientation [3]. Therefore, the structural continuum framework of the Holzapfel-Gasser-Ogden model for arterial tissue is efficient to describe the anisotropic hyperelastic mechanical behavior of midbrain tissue and this model is used in the finite element model of the indentation experiment in Abaqus software/standard Version Transversely isotropic representation of axonal fibers A constitutive model with transversely isotropic hyperelastic mechanical behavior of a family of collagen fibers for arterial wall has been developed by Gasser et al. [3]. Therefore, this model is a good departure point for the anisotropic hyperelastic mechanical behavior of midbrain tissue which consists bundles of highly oriented axonal fibers embedded in matrix. The continuum representation of distributed fibers is discussed in more detail in Appendix A. It is assumed that there is only one family of fiber bundles axons and these axonal fibers are embedded in an isotropic incompressible matrix. These axonal fibers 7

10 CHAPTER 3. MODEL FORMULATION AND FINITE ELEMENT MODELS 8 are distributed uniaxially in inferior-superior direction, let n o be the fiber direction unit vector in the reference configuration. For the sake of simplification, the preferred fiber direction vector n o is aligned with the Cartesian basis vector e 3, i.e. the orientation density function is independent of Eulerian angle Φ as defined in Figure 1 from Appendix A. Therefore, an orientation density function ρ( M(Θ, Φ)) becomes ρ( M(Θ)). The coefficients α 12, α 23 and α 13 in Equation (5) from Appendix A is vanish. The remaining coefficients α 11, α 22 and α 33 in Equation (5) from Appendix A, are given by α 11 = α 22 = κ, α 33 = 1 2κ, κ = 1 4 π 0 ρ( M(Θ))sin 3 ΘdΘ, (3.1) where the notation κ has been introduced which represents the fiber distribution and described the degree of anisotropy. Consequently, the generalized second order structure tensor H can be written in compact form H = κi + (1 3κ) n o n o, (3.2) where I is the identity tensor. Hence, H depends only on single dispersion structural parameter κ which κ [0, 1/3]. κ = 0, describes the full alignment of axonal fibers and κ = 1/3 describes the isotropic distribution of axonal fibers. Gasser et al. [3] have assumed that the embedded fibers are distributed according to a transversely isotropic and π-periodic von Mises distribution in order to determine the range of values for κ [0, 1/3]. 3.2 Anisotropic hyperelastic formulation The continuum representation of the axonal fiber orientation derived in section 3.1 forms the foundation for an anisotropic hyperelastic formulation. In order to derive the anisotropic hyperelastic strain energy potential W for the midbrain, it is assumed that it can be separated into an isotropic strain energy potential of the matrix, W m and an anisotropic strain energy potential of axonal fibers, W f. Therefore, the anisotropic strain energy potential function is N W( C,H i ) = W m ( C) + W f i ( C,H i ( n oi,κ)), (3.3) i=1 where the general second order structural tensor H i ( n oi,κ) is defined according to equation (3.2), N is the number of fiber families and C is the isochoric part of the right Cauchy-Green strain tensor C. According to Ning et al. [4], matrix material has been modeled as an incompressible isotropic neo-hookean model, i.e. W m = C 10 (Ī1 3) + 1 D 1 (J 1) 2, (3.4) where Ī1 = tr( C) denotes the first invariant of C, J is the volume ratio, C 10 and D 1 are neo-hookean coefficients. These neo-hookean coefficients can be related to the initial shear modulus G o and the bulk modulus K o as follows

11 CHAPTER 3. MODEL FORMULATION AND FINITE ELEMENT MODELS 9 C 10 = G o 2, D 1 = 2 K o (3.5) The additional contribution of the anisotropic strain energy potential for the i th family of axonal fibers is W f i ( C,H i ) = k 1 N ( e k ) 2Ē2 i 1, (3.6) 2k 2 i=1 Ē i = H i : C 1, (3.7) H i = κi + (1 3κ) n oi n oi, (3.8) where k 1 > 0 is a stress-like parameter to quantify the axonal fibers tensional mechanical strength and k 2 > 0 is a dimensionless parameter. The important fundamental hypothesis of the Holzapfel-Gasser-Ogden model is that the axonal fibers cannot support any compression stress and would buckle under compressive load, i.e. fibers contribute only their mechanical strength during tension [3]. In order to represent this important assumption, it is assumed that the anisotropic part (1 3κ) n oi n oi contributes to H i and Ēi from Equation 3.8 and 3.7, respectively, only if the strain in the direction of n oi is positive, i.e. C: noi n oi > 1. Therefore, Ēi becomes 0 when Ēi is negative, where are Macaulay brackets. General form of Ēi can be derived by substituting the Equation 3.8 into Equation 3.7 as introduced by Gasser et al. [3], Ē i = κ(ī1 3) + (1 3κ)(Ī4i 1), (3.9) to show the differences between Equation (3.6) and the anisotropic contribution of the strain energy function as introduced in Holzapfel & Gasser (2001) and Holzapfel et al. (2000) and Ī4i = C: n oi n oi. For completeness, the strain energy potential function of the isotropic matrix and the contribution from the anisotropic axonal fiber reinforcements is given by W = C 10 (Ī1 3 ) + 1 D 1 ( J lnj ) + k 1 2k 2 N i=1 ( e k ) 2 Ēi 2 1. (3.10) The corresponding Cauchy stress tensor consisting of hydrostatic (denoted as superscript h ) and deviatoric (denoted as superscript d ) parts is expressed as The hydrostatic and deviatoric parts are given by σ = σ h + σ d. (3.11) ( ) ( K σ h J 2 ) 1 = I, 2 J (3.12) σ d = 2 C N 10 J + k 1 e k 2 Ēi 2 Ēi ( κ B d + Ī4i(1 3κ)( n i n i ) d), (3.13) i=1 where B d is the deviatoric part of the Finger tensor B.

12 CHAPTER 3. MODEL FORMULATION AND FINITE ELEMENT MODELS Finite element model Three-dimensional finite element (FE) models of the indentation experiment were created using Abaqus software/standard Version Due to symmetry planes of the indenter and the midbrain sample, only a quarter of the indenter and the midbrain sample was modeled. The indenter was modeled as a discrete rigid shell in order to reduce the memory space consumed by the central processing unit (compared to discrete rigid solid indenter model). The mesh of the indenter has 1875 discrete rigid elements of type R3D4 (4-node 3-D bilinear rigid quadrilateral element) as shown in Figure 3.1(b). The midbrain sample was modeled as a deformable solid body. The mesh of the midbrain sample has solid elements in the directions of length width height, respectively, of type C3D8R (8-node linear brick element with reduced integration and hourglass control). The meshing of midbrain sample model is biased in order to obtain fine elements in the area close to the indenter for accuracy, as depicted in Figure 3.1(b). The dimensions of midbrain sample and indenter finite element models are depicted in Figure 3.1(a). Degrees of freedom for translation in Y-direction at each node of the bottom plane were constrained to represent the slip boundary between the midbrain tissue sample and the dish surface. The translation in Z-direction at each node in the XY-plane were constrained to represent a symmetry boundary condition. Also a symmetry boundary condition was applied in the YZ-plane which the transition in X-direction were constrained. All nodes were allowed to move in every transitional direction on three remaining planes (top plane, outer XY-plane and outer YZ-plane). Initially, the tip of the indenter model (master surface) had contact with the top surface of the midbrain model (slave surface) to prevent sudden changes in contact conditions, which lead to severe discontinuity iterations [12], and the symmetrical line of indenter model and midbrain sample model were aligned with each other (Figure 3.1). Next, in order for the indenter to indent into the midbrain sample, the reference point of the indenter model was prescribed a displacement of 0.2 mm (10% of the height of the midbrain sample model) in minus Y-direction. The values of displacement, force, pressure and stress in the midbrain sample model were recorded in every increment. The Holzapfel-Gasser-Ogden model as discussed in Section 3.1 and 3.2, [2,3] was used for the midbrain sample. Parallel (Y-direction) and perpendicular (X- or Z-direction) directions of axonal fibers with respect to the indenter loading axis (symmetrical line in Figure 3.1(a)) were defined. The mechanical behavior of two different midbrain tissue sample models with different axonal fiber directions were simulated and investigated. Material parameters 1 for midbrain region as tabulated in Table 3.1 were used in the Holzapfel-Gasser-Ogden model. Material parameters for the brainstem [4] shown in Table 3.1 cannot be used for the midbrain tissue indentation analysis because an error occured due to a large difference in order of magnitude between material parameters. In addition, the shear modulus calculated by Ning et al. [4] is relatively low compared with the shear modulus calculated from Hrapko et al. [6]. Therefore, the shear modulus G 0 and axonal fibers reinforcement strength k 1 for midbrain model indentation analysis have to be increased more than the values used in brainstem shearing test [4] (but less than the values in corona radiata shearing test [6]). The mechanical behavior of the midbrain, brainstem and corona 1 Units in Table 3.1 have been converted to Newton (N) and millimeter (mm) for finite element analysis in Abaqus software/standard Version in order to minimize the difference in order of magnitude between material parameters.

13 CHAPTER 3. MODEL FORMULATION AND FINITE ELEMENT MODELS 11 Table 3.1: The material parameters of the midbrain tissue sample model and brain tissue from other resources [4,6] Brain regions G 0 [Pa] C 10 [Pa] K 0 [GPa] D 1 [1/GPa] k 1 [Pa] k 2 [ ] κ[ ] midbrain brainstem [4] corona radiata [6] radiata is comparable because these regions consist of bundles of highly uniaxial axonal fibers embedded in matrix material. Two equal three-dimensional finite element models 2 (length width height dimensions were equal to the midbrain model shown in Figure 3.1) for shearing finite element analysis were modeled. Following steps are the procedure to increase the values of shear modulus G 0 and axonal fibers reinforcement strength k 1 for midbrain model indentation analysis. 1. Model the corona radiata FE model 2. Analyze the corona radiata FE model in simple shear deformation using the material parameters 3 in Table Extract the results of deviatoric stress σ d and Cauchy stress contour from corona radiata FE model analysis 4. Model the brainstem FE model 5. Analyze the brainstem FE model in simple shear deformation using the material parameters in Table Extract the results of deviatoric stress σ d and Cauchy stress contour from brainstem FE model analysis 7. Compare the results of σ d and Cauchy stress contour from corona radiata and brainstem FE models. 8. Increase or decrease the values of G 0 and k 1 in the ratio of G 0 /k 1 (12.7 / 121.2) and repeat the step 5 to step 7 if the σ d and Cauchy stress contour are not identical 9. Utilize the current values of shear modulus G 0 and axonal fibers reinforcement strength k 1 for midbrain FE model indentation analysis 2 FE models of corona radiata and brainstem are modeled and compared isotropically because the value of axonal fibers reinforcement strength k 1 for corona radiata is not determined in [6] 3 G 0 =478.0 Pa is not explicitly mentioned in [6], but this value is calculated from [6]

14 CHAPTER 3. MODEL FORMULATION AND FINITE ELEMENT MODELS 12 (a) Y RP Z X Y Z X (b) Figure 3.1: Three-dimensional finite element model of a quarter of midbrain tissue sample: (a) midbrain sample and indenter dimensions (Length=10.0 mm, Width=10.0 mm, Height=2.0 mm, Radius=1.0 mm), and (b) 3-D midbrain finite element model.

15 CHAPTER 3. MODEL FORMULATION AND FINITE ELEMENT MODELS 13 Figure 3.2: Algorithm of determination of the G 0 and k 1 values for the midbrain tissue using the material parameters of brainstem and corona radiata in Table 3.1

16 Chapter 4 Results and discussions The mechanical behaviors of 6-month-old porcine midbrain in three different directions (Figure 1.2) were investigated using indentation and compared with a finite element model using Abaqus software/standard Version Force levels at an indentation depth of 0.1 mm for each direction with mean and standard deviation are shown in Figure 4.1. Every data point represents one indentation force. From Figure 4.1, the mean of force level and standard deviation for the anteriorposterior direction and the lateral-medial direction are comparable may suggest that the axonal fibers are aligned perpendicular to loading axis in anterior-posterior direction and lateral-medial direction. However, the force level (mean and standard deviation) of axonal fibers aligned parallel to loading axis in inferior-superior direction is lower compared with the other two directions. It may due to axonal fibers cannot support any compression and would buckle in compressive load as discussed in Section 3.1 and 3.2. Therefore, the force response of axonal fibers defined in X- (anterior-posterior) and Z-directions (lateralmedial) in midbrain finite element model are equal. Henceforth, axonal fibers defined in anterior-posterior and lateral-medial directions in midbrain model are equivalent. Three average force-indentation depth curves of three directions were obtained for two or three locations on each sample. The experimental averaged force response with standard deviation bandwidth for three directions of midbrain obtained from indentation tests and numerical force-indentation depth curve obtained from finite element analysis are given in Figure 4.2. As discussed previously in Chapter 3, assumption of Holzapfel- Gasser-Ogden model, the axonal fibers parallel to loading axis (inferior-superior direction) do not provide mechanical strength and would buckle. In addition, mechanical strength is only dominated by matrix in this direction. As a result, the average response of midbrain from experimental and numerical indentations is considerably stiffer in anterior-posterior and lateral-medial directions than in inferior-superior direction. In the experiment indentation test, the force level ratio at 0.10 mm indentation depth are , and for lateral-medial/anterior-posterior, lateral-medial/inferior-superior and anteriorposterior/inferior-superior directions, respectively. On the other hand, force level ratio at 0.10 mm indentation depth for indentation analysis in anterior-posterior/inferior-superior directions (or lateral-medial/inferior-superior) is and the force level ratio in lateralmedial/anterior-posterior direction is The average responses and standard deviations of lateral-medial and anterior-posterior directions are comparable because the average responses of midbrain in both directions are considerably contributed by axonal fibers and matrix. Moreover, the average response in inferior-superior direction is lower than other two directions which indicated 14

17 CHAPTER 4. RESULTS AND DISCUSSIONS 15 Force level at 0.1mm for indentation of different directions for 0.1 mm/s Force [mn] Indentation of different directions (1: anterior posterior, 2: inferior superior, 3: lateral medial) Figure 4.1: Force levels with mean and standard deviation at indentation of 0.1 mm for three directions (1: anterior-posterior, 2: inferior-superior, 3: lateral-medial) on midbrain at speed of 0.1 mm/s. Each data point represents one indentation force Force vs indentation depth anterior posterior direction inferior superior direction laterial medial direction anterior posterior direction (FE model) inferior superior direction (FE model) Force [mn] Indentation depth [mm] Figure 4.2: Averaged force-indentation depth response with standard deviation obtained from indentation test and force-indentation depth response from finite element analysis.

18 CHAPTER 4. RESULTS AND DISCUSSIONS 16 the midbrain average response in inferior-superior direction is may only dominated by matrix, and assumed that the axonal fibers do not contribute mechanical strength in this direction. The standard deviation bandwidth of the midbrain in three directions in this studies are relatively large compared to the standard deviation bandwidth of the midbrain in lateral-medial direction from a previous study, [5], which maybe due to uncertainty variation of midbrain microstructure from every porcine brain and experimental conditions, [5, 7] or the indentations on midbrain samples conducted in this study are too close to the midbrain sample edge (approximately 6.0 mm from the midbrain sample edge to symmetrical line of indenter as shown in Figure 2.1). In addition, from Figure 4.2, there are differences of the force level between anterior-posterior/inferior-superior directions (or lateral-medial/inferior-superior directions) obtained from indentation experiments and the force level (anterior-posterior/inferior-superior direction) obtained from midbrain sample finite element analysis. These differences of the force level between experiments and finite element analysis maybe due to the uncertainty variation of midbrain micrstructure from every porcine brain and the unknown degree of dispersion of the axonal fibers with respected to the anterior-posterior, inferior-superior and lateral-medial directions in midbrain samples but axonal fibers were defined and aligned parallel to inferior-superior direction in midbrain model. A two-sample hypothesis t-test of distributions with equal means, unknown and unequal variances, against the alternative that the means are unequal. And, a two-sample F-test of equal normal distributions with same variance, against the alternative that equal normal distributions with different variances. Both two-sample hypothesis tests were performed to examine the results obtained from different indentation directions for midbrain tissue samples at the default significance level of 5% [9]. The results of these analyses are summarized in Table 4.1. From the t-test table, P-values of and were found for anterior-posterior/inferior-superior and lateral-medial/inferiorsuperior directions, respectively, indicating that a significant difference between mean stiffness of anterior-posterior or lateral-medial with inferior-superior exists. In contrast, there is no significant difference (P-value = ) of stiffness between anterior-posterior and lateral-medial directions in the midbrain. In addition, there are no significant differences between the spread of the midbrain in three directions in an F-test (Table 4.1) at the default significance level of 5% which indicate the variances between three directions in midbrain are relatively similar. Table 4.1: Statistical comparison of different indentation directions for midbrain tissue sample: P-values for t-test and F-test t-test anteriorposterior inferiorsuperior lateralmedial F-test anteriorposterior inferiorsuperior lateralmedial anteriorposterioposterior anterior inferiorsuperior inferiorsuperior The midbrain stiffness (force/indentation depth) at 0.1 mm of indentation depth for three indentation speeds was determined. Figure 4.3 shows the dependence of midbrain stiffness on speed of indentation. From Figure 4.3, it can be seen that the midbrain stiffness in three directions increases with increasing speed but the slope of stiffness decreases with increasing speed of indentation. Figure 4.3 also shows that the midbrain

19 CHAPTER 4. RESULTS AND DISCUSSIONS 17 stiffness in inferior-superior direction is lower than in the other two directions maybe due to the mechanical strength only resulted from the matrix in inferior-superior direction as discussed in Section 3.1 and 3.2. In addition, the estimated spread increases with increasing indentation speed. Figure 4.3 is also comparable to stiffness versus indentation speed plot in a previous study, [5] Stiffness versus indentation speed at 0.1mm indentation Midbrain anterior posterior direction MIdbrain inferior superior direction Midbrain lateral medial direction 3.5 Stiffness [N/m] Indentation speed [mm/s] Figure 4.3: Midbrain indentation stiffness at 0.1 mm versus three indentation speeds (0.1 mm/s, 0.32 mm/s and 1.0 mm/s). The error bars indicate the estimated spread between indentation speeds. Figure 4.4 shows the average normalized relaxation force (force/maximum force) and relaxation stiffness (force/maximum indentation depth) versus time for porcine midbrain tissue during the relaxation indentation step at the end of protocol 2 in three directions. The mechanical response of brain tissue consists of elastic and viscoelastic parts. The plateau values of the relaxation curves result from the elastic part, whereas the height of the peak response and relaxation response during the indentation step applied are determined by the elastic and viscoelastic parts. From Figure 4.4(a), it can be observed that the average normalized force relaxation after the peak value in anterior-posterior and lateral-medial directions is slower than in the inferior-superior direction because viscoelastic relaxation in anterior-posterior and lateral-medial directions results from axonal fibers and matrix but viscoelastic relaxation in inferior-superior direction is only influenced by the matrix. Figure 4.4(b) shows similar conclusion from Figure 4.4(a) because viscoelastic relaxation stiffness in inferior-superior direction is less stiff than another directions. The viscoelastic contribution only from the matrix (without axonal fibers) in inferior-superior direction makes the stiffness less than another two directions (contribution from axonal fibers and matrix). The difference of average relaxation stiffness in anterior-posterior and lateral-medial directions may be due to variation of average stiffness from various midbrain tissue samples.

20 CHAPTER 4. RESULTS AND DISCUSSIONS 18 Average normalized relaxation force vs time 1 anterior posterior direction inferior superior direction laterial medial direction Normalised force [N/m] Time [s] (a) Average relaxation stiffness vs time anterior posterior direction inferior superior direction laterial medial direction 6 Stiffness [N/m] Time [s] (b) Figure 4.4: (a) Average normalized relaxation force, and (b) average relaxation stiffness versus time for porcine midbrain tissue in three directions.

21 Chapter 5 Conclusions The objective of current research was to investigate the mechanical behavior of midbrain tissue in three directions (anterior-posterior, lateral-medial and inferior-superior directions) undergoing indentation deformation. Compression is one of the deformations associated with directional loading of brain during collision, which may lead to traumatic brain injury (TBI). In the current study, mechanical behavior of a 6-month-old porcine midbrain undergoing compression deformation was obtained using the indentation technique because the mechanical behavior of midbrain tissue can be considered to be homogeneous at the length scale of the indenter [5]. Midbrain tissue samples were indented in two protocols with three different indentation speeds (0.1 mm/s, 0.32 mm/s and 1.0 mm/s). The mechanical response obtained from indentation experiment test is compared with finite element analysis of midbrain tissue sample model. A quarter of the midbrain tissue sample was modeled as an anisotropic, hyperelastic incompressible material and a quarter of the indenter was modeled as a rigid body. The midbrain tissue sample model is represented as an axonal fiber reinforced material where axonal fibers are aligned along inferior-superior direction. The midbrain tissue sample model was modeled using Holzapfel-Gasser-Ogden model where axonal fibers were described using an anisotropic strain energy potential function and the matrix material using a neo-hookean model. The axonal fibers were modeled with an initial modulus of Pa and the initial modulus of that midbrain matrix was Pa. The current study indicates that the midbrain tissue sustained less compressive force level in inferior-superior direction compared with compressive force level in anteriorposterior and lateral-medial levels. This may suggest that the axonal fibers do not contribute in compressive stress and would buckle in this direction compare with another two directions. The current experiment result also shows that the stiffness of midbrain in inferior-superior direction is lower than in another directions. The mean response of midbrain tissue in inferior-superior direction is lower than in anterior-posterior and lateral-medial directions because this may due to only matrix contributes to the midbrain tissue mechanical strength in inferior-superior direction, i.e. matrix and axonal fibers are contributed to the midbrain tissue mechanical strength in anterior-posterior and lateral-medial directions. The force-indentation depth curves of the midbrain tissue in anterior-posterior (or lateral-medial) and inferior-superior directions obtained from indentation experiments are relatively different from the results obtained from finite element analysis may due to uncertainty variation of midbrain structure and an unknown degree of dispersion axonal fibers from every porcine midbrain tissue sample with respect 19

22 CHAPTER 5. CONCLUSIONS 20 to anterior-posterior, lateral-medial and inferior-superior directions but the axonal fibers were defined aligned parallel to inferior-superior direction in the midbrain model. There was no statically significant difference of mean force level between anteriorposterior and lateral-medial directions based on statistical analysis. However, there was a significant difference between anterior-posterior (or lateral-medial) and inferior-superior directions. In addition, the relaxation response of a step indent was measured and showed that midbrain in inferior-superior direction is less stiff than another two directions. Microstructurally, many factors that contribute to midbrain compression stiffness should be considered such as fiber orientation, fiber bundle buckling stability, sample dimensions, undulation of axonal fibers and fiber-matrix bond strength [4]. In nature, most biological tissues are anisotropic and viscoelastic. Therefore, these factors can be taken into account in midbrain tissue modeling for improving midbrain tissue finite element analysis results.

23 Appendices 21

24 A. Continuum representation of distributed fiber orientations The objective of this appendix is to present the directional data of distributed fibers into a continuum model, presented by Gasser et al. [3]. An orientation density function ρ( M) is introduced, which characterizes the distribution of fibers in the reference configuration Ω o with respect to the fiber direction vector M. The vector M is an arbitrary unit vector ( M = 1) located in three-dimensional Eulerian space and vector M can be represented in terms of two Eulerian angles Θ [0,π] and Φ [0, 2π] M(Θ, Φ) = sinθcosφ e 1 + sinθsinφ e 2 + cosθ e 3, (1) where { e 1, e 2, e 3 } denote the axes of a rectangular Cartesian coordinates system as illustrated in Figure Figure 1: Arbitrary unit direction vector M in terms of Eulerian angles Θ [0,π] and Φ [0, 2π] in three-dimensional Cartesian coordinate system { e 1, e 2, e 3 } [3]. The orientation density function ρ( M) can be written in terms of two Eulerian angles (Θ and Φ), ρ( M(Θ, Φ))sinΘdΘdΦ to represent the number of fibers with orientations in the range [(Θ, Θ + dθ), (Φ, Φ + dφ)]. In addition, it is assumed that ρ( M) is normalized. 1 ρ( M(Θ, 4π Φ))sinΘdΘdΦ = 1. (2) Φ Θ Next, the generalized second order structure tensor, H which represents the fiber distribution, is defined by H = 1 ρ( M(Θ, 4π Φ)) M(Θ, Φ) M(Θ, Φ)sinΘdΘdΦ (3) Φ Θ By using Equation (1), the generalized second order structure tensor (3) can be written in compact form H = α ij e i e j, where i,j = 1, 2 and 3, (4)

25 23 where the coefficients α ij = α ji are defined as α 11 = 1 ρ( M)sin 4π 3 Θcos 2 ΦdΘdΦ Φ Θ α 22 = 1 ρ( M)sin 4π 3 Θsin 2 ΦdΘdΦ Φ Θ α 33 = 1 ρ( M)cos 4π 2 ΘsinΦdΘdΦ Φ Θ α 12 = 1 ρ( M)sin 4π 3 ΘsinΦcosΦdΘdΦ Φ Θ α 23 = 1 ρ( M)sin 4π 2 ΘcosΦsinΦdΘdΦ Φ Θ α 13 = 1 ρ( M)sin 4π 3 ΘcosΦcosΦdΘdΦ (5) Φ Θ Therefore, the generalized second order structure tensor H is a measurement of the fiber distribution can be computed once the density distribution function ρ( M) is given.

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