A biphasic hyperelastic model for the analysis of fluid and mass transport in brain. tissue. José Jaime García

Transcription

1 A biphasic hyperelastic model for the analysis of fluid and mass transport in brain tissue José Jaime García Escuela de Ingeniería Civil y Geomática Universidad del Valle, Cali, Colombia Joshua H. Smith Department of Mechanical Engineering Lafayette College Easton, PA 18042, USA Corresponding author institution and address: José Jaime García Universidad del Valle Calle 13 Carrera 100 Edificio 350 Cali Colombia South America Tel: (572) Fax: (572) josejgar@univalle.edu.co 1

2 Abstract A biphasic hyperelastic finite element model is proposed for the description of the mechanical behavior of brain tissue. The model takes into account finite deformations through an Ogden-type hyperelastic compressible function and a hydraulic conductivity dependent on deformation. The biphasic equations, implemented here for spherical symmetry using an updated Lagrangian algorithm, yielded radial coordinates and fluid velocities that were used with the convective-diffusive equation in order to predict mass transport in the brain. Results of the model were equal to those of a closed-form solution under infinitesimal deformations, however, for a wide range of material parameters, the model predicted important increments in the infusion sphere, reductions of the fluid velocities, and changes in the species content distribution. In addition, high localized deformation and stresses were obtained at the infusion sphere. Differences with the infinitesimal solution may be mainly attributed to geometrical nonlinearities related to the increment of the infusion sphere and not to material nonlinearities. Keywords Convection enhanced delivery; infusion; hydrated tissues. 2

3 Nomenclature Variable or constant Definition C C a D E e J M p r v i v f v s W α k Tissue concentration of a chemical species Aqueous concentration of a chemical species Effective diffusion coefficient Young s modulus Volume dilatation Determinant of the deformation gradient tensor Nonlinear parameter for hydraulic conductivity Pore pressure (interstitial fluid pressure) Radial coordinate Interstitial fluid velocity Bulk fluid velocity Velocity of the solid phase Energy function Parameter of the energy function φ f Volume fraction of the fluid phase Volume fraction of the fluid phase at zero strain κ κ 0 λ r Hydraulic conductivity Hydraulic conductivity at zero strain Radial stretch ratio λ θ Circumferential stretch ratio 3

4 µ k Parameter of the energy function µ ' Parameter of the energy function Radial effective stress Circumferential effective stress ν Poisson s ratio Introduction Various biological tissues such as brain and articular cartilage are composed of a solid phase and a fluid, where the fluid plays an important role in the physiological activities and the mechanical response of the tissue. 23 Mathematical models that have been proposed to simulate the mechanical response of biological hydrated tissues are important tools to understand the etiology of diseases, to develop biomedical devices, or to test protocols to treat disorders. In particular, efforts to develop models for brain tissue have been motivated by the need to simulate hydrocephalus, head trauma, neurosurgical procedures and protocols to deliver therapeutic drugs into the brain. Early mathematical models considered the brain as a single-phase, linear, viscoelastic material, 6,19 and more recent models have considered single-phase, nonlinear, viscoelastic formulations. 5,11,18,20,21,26,27 Poroelastic or biphasic models have been used to study hydrocephalus 12,30,32,36 and the fluid transport and tissue deformation that occurs during positive pressure infusion, also called convection enhanced delivery. 2,3,31 Each of the biphasic studies, with the exception of the recent study by Dutta-Roy et al., 12 is limited by the assumption of linear elasticity of the solid phase. Initial attempts at modeling the mass transport that occurs during convection enhanced delivery were 4

5 conducted under the even more limiting assumption that the tissue is rigid during the infusion, 22,28,29 although this was relaxed by Netti et al. 24 and Chen and Sarntinoranont, 8 who also assume linear elasticity of the solid phase. Nonlinear stress-strain curves under finite deformations have been documented for brain tissue under tension 13,21 and compression. 20 Whether or not this nonlinear behavior is relevant under all clinical conditions is still focus of discussion, with, for example, Taylor and Miller 33 arguing that a linear constitutive model is sufficient for modeling hydrocephalus but Clarke and Meyer 10 recently stating that a nonlinear formulation is needed. The nonlinear variation of hydraulic conductivity with strain has also been taken into account and this effect has been deemed to play an important role in both the mechanics of the tissue and the associated fluid transport. 32,36 One of the primary goals of modeling convection enhanced delivery is the prediction of the distribution of the infused therapeutic agent. Previous models of infusion into the brain and spinal cord have generally either considered (1) anatomically realistic geometry but assumed no tissue deformation during the infusion, 16,17,28,29 or (2) the concomitant tissue deformation and drug transport but for a simplified, spherical geometry. 8,22,24 To the best of our knowledge, no model has been proposed to consider simultaneously nonlinear stress-strain curves under finite deformation, nonlinear variation of hydraulic conductivity, and convective-diffusive transport of the infused agent. We present a spherically symmetric model of the concomitant fluid and mass transport that occurs during the large deformation of brain tissue as a result of convection enhanced delivery. 5

6 Whereas commercial finite element solvers have recently been used to study infusion 8 and hydrocephalus, 12 no package supports modeling the nonlinear behavior of brain tissue coupled with the transport of the infused agent. The spherical model we present, implemented in a custom-written code, is used to study the physics that occurs during a constant pressure infusion and to determine the importance of using a nonlinear material model for brain tissue and of including a deformation-dependent hydraulic conductivity in such simulations. Knowing the significance of these effects is important for the development of more advanced, three-dimensional computational models that may ultimately aid in devising clinical infusion protocols for the treatment of diseases of brain tissue. Description of the mathematical model Biphasic model Fluid flow and strain of a hydrated biological tissue considering spherical symmetry and finite deformations can be described in terms of the mixed solid velocity pressure formulation 1 by the following equations: (1). (2) The variables and are the effective stresses in radial and tangential direction, respectively, v s is the velocity of the solid phase, p is the pore pressure (interstitial fluid pressure) and the coefficient κ is the hydraulic conductivity. Considering the spherical symmetry all functions only change with the radial coordinate r, which represents the 6

7 current position of a material element. Eq. (1) is a statement of force equilibrium and Eq. (2) is a statement of fluid mass continuity that neglects any transvascular fluid exchange. The solid phase was represented by a hyperelastic isotropic energy function W as W N = k= 1 µ k α α µ ' k k ( λr + 2λθ 3 α k ln( J )) + ( J 1) α 2 k 2 (3) where α k, and λ r and λ θ are, respectively, the radial and circumferential stretch ratios, µ k, µ ' are material parameters and J is the determinant of the deformation gradient tensor. 25 The material parameters can be adjusted to yield the Young s modulus E and Poisson s ratio ν at zero strain as follows N E = ( 1+ ν ) α µ (4) k = 1 k k ν = 2µ ' + µ ' N k = 1 α k µ k. (5) The incompressible form of this energy function allows reproducing the nonlinear behavior of brain tissue under finite deformations, as shown by Miller and Chinzei 21 and Franceschini et al. 13 The variation used here allows using Poisson s ratios different than 0.5, which is consistent with the experimental behavior of brain tissue. 34 Note that material parameters α k can be adjusted to describe the shape of the stress-strain curve. To be consistent with other analyses of brain tissue, 8,32,36 the variation of hydraulic conductivity with strain was assumed to depend on tissue dilatation as, (6) 7

8 where κ 0 is the hydraulic conductivity at zero strain, M is a non-dimensional parameter that controls the variation of hydraulic conductivity, and e is the volume dilatation, that can be related to the stretch ratios as. (7) Eqs. (1) and (2) were solved using well known procedures of finite elements and an updated Lagrangian scheme, 4 where the validation of the code was accomplished comparing numerical results with closed-form solutions reported in the literature. 14 The Galerkin method was used with linear interpolation for both the shape functions and the weighting functions. After solution of Eqs. (1) and (2), the fraction of the fluid phase φ f was calculated using φ f =1 (1 φ f0 ) /J, (8) where φ f0 is the initial fluid fraction. Darcy s law was then used to calculate the radial bulk velocity v f and interstitial fluid velocity v i as v f p = viφ f = κ + φ f vs, (9) r where v s is the velocity of the solid phase. Mass transport model The convective-diffusive mass transport equation considering a coordinate system fixed in space is described by C+ C v i ( D C) = 0, (10) 8

9 where C is the material derivative of the concentration C of the chemical species per unit volume of tissue (in contrast to interstitial concentration, which is per unit volume of the interstitial fluid within the tissue), v i is the interstitial fluid velocity vector, and D is the effective diffusion coefficient as measured in the tissue. 8,22,28.29,35 In this statement of mass conservation of the species, we assume that transport is limited to the interstitial space, and furthermore, that no vascular uptake, cellular absorption, or metabolism occur. Lastly, since convective effects dominate diffusive effects, for simplicity we assume that the effective diffusion coefficient D does not change as the tissue deforms. Both the transient fluid velocity and deformation of the tissue from the biphasic analysis were considered in the transport model. Thus, at each time step the finite element mesh was not fixed in space, as assumed in Eq. (10), but updated according to the deformation of the tissue. To take this effect into account in the finite element solution, the material derivative of C in Eq. (10) was replaced in terms of the convected velocity (the difference between the velocity of the fluid and the velocity of the solid phase) as 4 C C = + ( vi v s ) C. t (11) Therefore, in this formulation the transport equation becomes C + ( v v s ) C + C vi ( D C) = 0 t i, (12) which is represented by the following equation for a spherically symmetric geometry: C t + (v i v s ) C r + 1 r 2 [ r r2 v i ]C 1 r 2 r r2 D C r = 0. (13) Eq. (13) was solved in space with the Galerkin method using linear interpolation functions and the time derivative of C was approximated with the backward difference. 9

10 Usually concentration distributions are presented normalized against the concentration in the interstitial space, that is C (φ f C a ) where C a is the aqueous concentration. Normalized in this way, the concentration represents the amount of the therapeutic agent per unit volume of the interstitial fluid, and hence will be referred to as the normalized interstitial concentration. While not often shown in previous literature on infusion, the concentration distributions can also be normalized against the aqueous concentration, that is C C a. Normalized in this way, the concentration represents the amount of the therapeutic agent per unit volume of brain tissue, and hence will be referred to as the normalized bulk concentration. While the former method is a more true liquid concentration, this latter definition is more comparable to quantifying the distribution of radiolabeled agents in vivo using MR or SPECT since these imaging techniques measure intensity per voxel. Geometry, boundary conditions, and mesh Consistent with the geometry used by Chen and Sarntinoranont, 8 an infusion cavity of 0.18 mm initial radius was considered in the analyses. The domain was represented using 120 elements with a length of 0.02 mm for the first element and a geometric increment of Each element represents a spherical domain comprised between the two nodes. Other analyses undertaken with a mesh composed of 240 elements showed results within the 2% of those obtained with the current mesh, so results obtain using the mesh with 120 elements were considered to be mesh independent. In the coupled biphasic and mass transport analyses, a small region of normalized interstitial concentration 2% greater than 1 was obtained for the more compliant 10

11 material. Since normalized values in excess to 1 are not physically admissible, analyses with meshes of 240 and 360 elements were performed and showed convergence of those values to 1. Hence, concentration curves are shown for the mesh of 360 elements. The 120 and 360 element meshes resulted in an outer radius of the spherical domain greater than 20 mm, large enough to be consistent with the biphasic boundary conditions at infinity, which were zero traction and zero pore pressure. Pore pressure was prescribed at the inner sphere and a traction equal to the pore pressure, which implies a radial effective stress equal to zero at the inner boundary, in agreement with other analyses of infusion. 8 A time step of 5 s was used. At the inner boundary the interstitial concentration is known. Therefore, as a boundary condition for the mass transport equation, the concentration C per unit volume of the tissue was made equal to the aqueous concentration C a multiplied by the fluid fraction calculated from the biphasic analysis, that is, φ f (r i,t)c a where r i is the current radius of the infusion sphere. Material properties A value of 0.35 was used for the Poisson s ratio, in agreement with the analyses by Chen and Sarntinoranont. 8 Based on this Poisson s ratio, a baseline Young s modulus of 421 Pa was obtained from the equilibrium shear modulus (156 Pa) that can be calculated from results reported by Miller and Chinzei

12 Two sets of nonlinear elastic parameters were used in this study using one term in the energy function, Eq. (3). In the first set, α1 = 4. 7 from the experimental study of Miller and Chinzei. 21 The other energy function parameters were calculated using Eqs. (4) and (5) as µ 1 = Pa and µ '= Pa. In the second nonlinear set, α 1 = 1, which allowed for reproducing a more linear behavior of the stress-strain curve (Figure 2). For this set, the other energy function parameters were calculated as µ = Pa and µ '= Pa. For the initial hydraulic conductivity κ 0, a range between 0.1 mm 4 N -1 s -1 used by Chen and Sarntinoranont 8 and 4 mm 4 N -1 s -1 measured by Cheng and Bilston 9 was used in the analyses. To our knowledge, the nondimensional parameter M governing the variation of hydraulic conductivity with deformation has not been measured in brain tissue, but has been assumed to be in the range of 0 to 5 in previous studies of brain tissue. 8,32,36 Because these studies were limited to infinitesimal deformations, and therefore smaller values of the dilatation, we consider values of M in the range of 0 to 2. To be consistent with Chen and Sarntinoranont 8 a diffusion coefficient D equal to mm 2 s -1. A summary of material parameters used in this study is presented in Table 1. Results of the finite element implementation were compared with analytic solutions developed by Basser 3 for the pore pressure and the fluid velocity and developed by Chen and Sarntinoranont 8 for the radial displacement. For this validation, we used a Young s modulus of 4210 Pa, tenfold the value calculated using the results of Miller and Chinzei, 21 to guarantee infinitesimal deformations and a value for M of 0, so that the hydraulic conductivity was constant with deformation. 12

13 Results At steady state, good correlations were observed between the closed form linear solution and the nonlinear finite element curves at an infusion pressure of 100 Pa (Figure 3), with differences of maximum values of fluid velocity and radial displacement of 5.4% and 0.7%, respectively, using a Young s modulus of 4210 Pa and M = 0 to guarantee infinitesimal deformations. The difference between maximum fluid velocities increased to 10.8% for an infusion pressure of 500 Pa, while the radial displacement from the nonlinear analyses was 14.6% higher than that of the linear closed-form solution (Figure 3). This indicates that even for an artificially stiffer material, finite strains at 500 Pa occur and the nonlinear analyses diverge from the linear solutions. For the Young s modulus of 421 Pa, even at a low infusion pressure of 200 Pa there were important variations between the closed-form linear solution and the nonlinear analyses with constant hydraulic conductivity (Figure 4), with differences of 63.7% and -52.3% for the maximum values of fluid velocity and radial displacement with respect to the results obtained for the parameter set with α 1 = 1. While there was only a small difference of 3.7% between maximum values of fluid velocity for the analyses with the two sets of material parameters (α 1 = -4.7 vs. α 1 = 1) at this infusion pressure, there was a moderate difference of 11.6% between the maximum values of the radial displacement. These results suggest that the differences between the nonlinear analyses and the linear solution are mainly due to geometric nonlinearities. Based on this, the following results are only presented for the parameter set with α 1 =

14 Allowing the hydraulic conductivity to vary with deformation with M = 1, variations in the initial hydraulic conductivity κ 0 caused important differences in the fluid velocity distributions but did not significantly alter the maximum radial displacement or the pore pressure curves (Figure 5). Changes in the parameter M, which governs the variation of hydraulic conductivity with deformation, caused important differences in all distributions (Figure 6). Mainly, the increment of M caused a reduction in the slope of the pore pressure curve, an increase in the maximum fluid velocity, and a change in the form of the displacement curve that was monotonic for M = 0 but had local minimum for M = 2. For the baseline material properties (E = 421 Pa), local circumferential and radial stretch ratios of approximately 6 and 0.34, respectively, were obtained at the infusion cavity along with significant increases in the fluid fraction (Figure 7). For the baseline Young s modulus of 421 Pa, maximum radial displacement, fluid velocity and flow rate exhibited a significant nonlinear behavior with respect to the infusion pressure. Specifically, the displacement and flow rate increased exponentially while the fluid velocity depicted a bell shaped curve (Figure 8). On the other hand, for the Young s modulus of 4210 Pa, all three functions displayed an approximately linear behavior with respect to the infusion pressure (Figure 8). Important differences in the normalized interstitial concentration were obtained from the analyses using different values of the Young s modulus and hydraulic conductivity (Figure 9). These distributions correspond to infusion volumes of 4.85 ml and 28.8 ml at 30 s and 1200 s, respectively, for the more compliant material and to volumes of 0.18 ml and 1.56 ml at 30 s and 1200 s, respectively, for the stiffer material. At 1200 s, the content of infused agent (30.29) for E = 421 Pa was almost twenty-fold that (1.59) for E 14

15 = 4210 Pa using a hydraulic conductivity κ 0 = 1 mm 4 N -1 s -1, while for κ 0 = 0.1 mm 4 N -1 s -1 the content (5.41) for E = 421 Pa was twenty seven-fold that (0.20) obtained with E = 4210 Pa. Since the flow rate was greater for E = 421 Pa than for E = 4210 Pa and the infusion cavity had increased in radius more substantially, the corresponding concentration distributions showed increased penetration of the infused agent. There were striking differences among the normalized bulk concentration distributions using different values of the Young s modulus and hydraulic conductivity (Figure 10). As the tissue deformed, the size of the infusion sphere increased, and within the infusion sphere the bulk concentration (per unit total volume) was equal to the aqueous concentration. At the interface of the infusion sphere with the tissue, there was a sudden drop in the bulk concentration since the infused agent can only be transported into the fluid portions of the tissue. This drop was greatly pronounced in the stiffer material (E = 4210 Pa) since the fluid fraction φ f reached only moderate increments from the baseline value of 0.2 to values between 0.25 and 0.3, in contrast with the minor drop for the more compliant material (E = 421 Pa) due to the important increase of the fluid fraction in the vicinity of the infusion sphere to values between 0.8 and Discussion Under the framework of biphasic theory, a mathematical model was proposed here for infusion into brain tissue using a hyperelastic constitutive equation for the solid phase that can describe nonlinear stress-strain curves under finite deformations documented in experimental studies. 9,13 In addition, a hydraulic conductivity dependent on deformation can be considered, which has been regarded as an important factor influencing both the 15

16 elastic response and associated fluid flow. 8,32,36 Deformations and fluid velocity fields from the biphasic analyses can then be used to predict mass transport in the brain, which may be useful to accurately evaluate the influence of various parameters in the outcome of drug delivery protocols into the brain. Because commercial finite element solvers do not currently allow for the inclusion of all the desired effects in the modeling of infusion, a custom-written finite element code was implemented with the assumption of spherically symmetry. This allowed for an efficient implementation of the nonlinear iterative scheme by using two-node elements that represent spherical domains. The spherical model was used to study the physics that occurs during infusion and to test the importance of each of the nonlinear effects in such simulations. The numeric solution converged to the linear under infinitesimal deformations, e.g., for lower infusion pressures and higher Young s modulus, maximum radial displacements were of the order of microns and differences between maximum radial displacement obtained with the closed form linear solution and the nonlinear finite element analyses were less than 0.7%. However, an increase of infusion pressure yielded maximum radial displacements of the order of 8% of the infusion radius, and the difference between the radial displacements from both analyses increased to 15%, which indicates the important contribution of the nonlinear effects in the analyses of drug infusion even considering a relatively high elastic modulus (4210 Pa). Higher displacements from the nonlinear model with respect to those from the linear solution may also be predicted with the infinitesimal solution by comparing displacements obtained with infusion spheres of different radius. On the other hand, the similarities between numerical results 16

17 obtained with two different sets of nonlinear parameters indicate that the nonlinear effects are mainly due to geometrical changes of the infusion cavity taken into account in the finite element procedure and not to the material nonlinearities. This can be explained since maximum values of strain are highly localized at the infusion cavity, affecting only a small percentage of the material (Figure 7). A similar conclusion about the importance of material nonlinearities was recently obtained by Wittek et al., 37 who studied the influence of different constitutive formulations in predicting brain deformation during craniotomy-induced shift. Different from the results reported by Chen and Sarntinoranont, 8 this analysis predicts important changes of fluid velocity, pore pressure, and displacement distributions with variations of the parameter M, which defines the change of hydraulic conductivity with deformation. The reason is that strains calculated in this study are higher than those of Chen and Sarntinoranont 8 due to the consideration of smaller elastic moduli and the updated geometry at each time step. Reductions of the slope of pore pressure with increments of parameter M near the infusion cavity may be mainly attributed to the corresponding growth of the hydraulic conductivity, due to the high increase of fluid content at the infusion cavity (fourfold the baseline value 0.2). The reduction of pore pressure slope was greatly counteracted by the rise in hydraulic conductivity and therefore, relative fluid velocity greatly increased with M, since it is equal to the product of pressure slope with the hydraulic conductivity. As a consequence, an important increase of fluid flow was also observed under these conditions, which implied a higher volume of drug distribution (Figure 9). 17

18 Differences in the shape of the curves of fluid velocity versus infusion pressure for different values of the Young s modulus can be explained since there are two mechanisms responsible for fluid velocity changes with variations of the infusion pressure (Figure 8b). First, considering a constant radius of the infusion sphere, increments of infusion pressure tend to raise the slope of the pressure curve, which also increases fluid velocity. On the other hand, increases of the infusion cavity due to deformations tend to reduce the slope of the pressure curve and, consequently, the fluid velocity. Therefore, at lower strains, the first mechanism dominates and this is why the relationship between fluid velocity and infusion pressure is approximately linear for a higher Young s modulus. At higher deformations the second mechanism may dominate since the growth of the infusion sphere becomes important at higher infusion pressures and this explains the bell shaped form of the curve when a lower Young s modulus was considered. However, since the flow rate depends on the square of the radius of the infusion sphere, an increment of the infusion sphere implies an increase in the flow rate even though the fluid velocity may be lower (Figure 8c). The larger penetration of drug into the tissue observed for the more compliant material may be mainly attributed to the greater flow rate, which is a consequence of the important increment of the radius of the infusion sphere (fivefold the radius of initial sphere), in contrast with the moderate increment of the infusion sphere (~10%) obtained for the stiffer material. Differences in the shape of the normalized bulk concentration curves between the stiffer and compliant materials are due to the great changes that the fluid fraction near the infusion sphere (approximately four to fivefold the reference value of 0.2) in the more compliant material, results that have been documented experimentally in hydrocephalic cats. 15 Considering that the Young s modulus of

19 Pa is more representative of the properties of the brain tissue, it can be concluded that nonlinear geometric effects represented in a larger infusion cavity should be taken into account for a more accurate simulation of mass transport. While the purpose of this work was to study the physics of constant pressure infusions, the primary limitation of this work is the use of a simplified spherical geometry and associated boundary conditions. For example, including the interaction between the brain and skull would require a more sophisticated boundary condition than the zero traction currently applied to the outside of the spherical domain, such as that proposed recently by Wittek et al. 38 Other limitations of this study include the use of some material parameters that have been extracted from the literature and have not been measured directly (i.e. Poisson s ratio) and others that may not necessarily apply to the model, such as the parameter M in the formula for the nonlinear variation of hydraulic conductivity with strain. In addition, transvascular flow may be an important effect that should be included in future studies. The isotropic and homogeneous model proposed here is only a crude approximation of the microstructure of the brain. Other experimental and analytical studies have to be undertaken to quantify anisotropies and heterogeneities of brain and include these effects for the description of the mechanical response of the tissue. Acknowledgments J. J. García and J. H. Smith thank the Universidad del Valle and Lafayette College for giving them the time to undertake this study. Furthermore, the authors thank Dr. Raghu Raghavan for discussions regarding measuring concentration distributions in vivo. 19

20 References 1. Almeida, E.S. and R. L. Spilker. Mixed and penalty finite element models for the nonlinear behavior of biphasic soft tissues in finite deformations: Part I Alternate formulations. Comput. Methods Biomech. Biomed. Eng. 1:25 46, Barry, S.I. and G. K. Aldis. Flow-induced deformation from pressurized cavities in absorbing porous tissues. Bull. Math. Biol. 54: , Basser, P. J. Interstitial pressure, volume, and flow during infusion into brain tissue. Microvascular Research 44:143 65, Belytschko, T., W. K. Liu, and B. Moran. Nonlinear Finite Element for Continua and Structures, Chichester: John Wiley & Sons, Ltd., Bilston, L. E., Z. Liu, and N. Phan-Thien. Large strain behaviour of brain tissue in shear: Some experimental data and differential constitutive model. Biorheology 38:335 45, Bilston, L.E., Z. Liu, and N. Phan-Thien. Linear viscoelastic properties of bovine brain tissue in shear. Biorheology 34: , Bruehlmeier, M., U. Roelcke, P. Blauenstein, J. Missimer, P. A. Schubiger, J. T. Locher, R. Pellikka, and S. M. Ametamey. Measurement of the extracellular space in 20

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23 22. Morrison, P. F., D. W. Laske, H. Bobo, E. H. Oldfield, and R. L. Dedrick. Highflow microinfusion: Tissue penetration and pharmacodynamics. Am. J. Physiol. 266:R292 R305, Mow, V. C., W. Zhu, and A. Ratcliffe. Structure and function of articular cartilage and meniscus. In: Basic Orthopaedic Biomechanics, edited by V. C. Mow, W. C. Hayes. New York: Raven Press, Netti, P. A., F. Travascio, and R. K. Jain. Coupled Macromolecular Transport and Gel Mechanics: Poroviscoelastic Approach. AIChE J. 49: , Ogden, R. W. Non-Linear Elastic Deformations. Mineola, New York: Dover Publications, Inc., Prange, M. T. and S. S. Margulies. Regional, directional, and age-dependent properties of the brain undergoing large deformation. J. Biomech. Eng. 124: , Prange, M. T., D. F. Meaney, and S. S. Margulies. Defining brain mechanical properties: Effects of region, direction, and species. Stapp Car Crash Journal 44: ,

24 28. Sarntinoranont, M., R. K. Banerjee, R. R. Lonser, and P. F. Morrison. A computational model of direct interstitial infusion of macromolecules into the spinal cord. Ann. Biomed. Eng. 31: , Sarntinoranont, M., X. Chen, J. Zhao, and T. H. Mareci. Computational model of interstitial transport in the spinal cord using diffusion tensor imaging. Ann. Biomed. Eng. 34: , Smillie, A., I. Sobey, and Z. Molnar. A hydroelastic model of hydrocephalus, Journal of Fluid Mechanics 539: , Smith, J. H. and J. A. C. Humphrey. Interstitial transport and transvascular fluid exchange during infusion into brain and tumor tissue. Microvascular Research 73:58 73, Sobey, I. and B. Wirth. Effect of non-linear permeability in a spherically symmetric model of hydrocephalus, Math. Med. Biol. 23: , Taylor, Z. and K. Miller. Reassessment of brain elasticity for analysis of biomechanisms of hydrocephalus, J. Biomech. 37: , Tenti, G., J. M. Drake, and S. Sivaloganathan. Brain biomechanics: mathematical modeling of hydrocephalus. Neurol. Res. 22:19 24,

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26 Table 1. Summary of material parameters values used in this study. Parameter Range References Young s modulus, E Pa 21 Poisson s ratio, ν 0.35 Strain energy function exponent, α 1-4.7; Initial hydraulic conductivity, κ mm 4 N ,9 s Nonlinear parameter, M 0 2 Porosity, φ f 0.2 8,32,36 7 Diffusivity, D mm 2 s

27 Figure 1. Schematic diagram of a spherically symmetric model of the brain and infusion site. Figure 2. Stress-strain curves under uniaxial stress obtained for the two sets of nonlinear parameters. Since the same Young s modulus and Poisson s ratio were used, the two curves have the same slope at zero strain, i.e., stretch ratio equal to one. 27

28 (a) Pore pressure (b) Fluid velocity (c) Radial displacement Figure 3. Steady state pore pressure, fluid velocity, and radial displacement distributions obtained for the closed form solution with E = 4210 Pa and the nonlinear strain energy function at two infusion pressures. 28

29 (a) Pore pressure (b) Fluid velocity (c) Radial displacement Figure 4. Steady state pore pressure, fluid velocity, and radial displacement distributions obtained for the closed-form linear solution with E = 421 Pa and the nonlinear strain energy function with two sets of material parameters at an infusion pressure of 200 Pa. 29

30 (a) Pore pressure (b) Fluid velocity (c) Radial displacement Figure 5. Steady state pore pressure, fluid velocity, and radial displacement distributions for different values of hydraulic conductivity κ 0 for α 1 = -4.7 and M = 1 at an infusion pressure of 500 Pa. Note that the pore pressure distributions are nearly coincident for κ 0 = 0.1 mm 4 N -1 s -1 and 1.0 mm 4 N -1 s -1, and that the radial displacement distributions are nearly coincident for κ 0 = 1 mm 4 N -1 s -1 and 4 mm 4 N -1 s

31 (a) Pore pressure (b) Fluid velocity (c) Radial displacement Figure 6. Steady state pore pressure, fluid velocity, and radial displacement distributions for different values of nonlinear hydraulic conductivity parameter M for α 1 = -4.7 and κ 0 = 0.1 mm 4 N -1 s -1 at infusion pressure of 500 Pa. 31

32 Figure 7. Steady state fluid fraction, radial stretch, and circumferential stretch distributions displaying significant maxima localized at the infusion site, for α 1 = -4.7 and κ 0 = 0.1 mm 4 N -1 s -1 at an infusion pressure of 500 Pa. 32

33 (a) Radial displacement (b) Fluid velocity (c) Flow rate Figure 8. Steady state maximum radial displacement, fluid velocity, and flow rate as a function of infusion pressure for the nonlinear strain energy function with α 1 = -4.7, κ 0 = 0.1 mm 4 N -1 s -1, M = 1, and two values of the Young s modulus. 33

34 (a) 30 seconds (b) 1200 seconds Figure 9. Normalized interstitial concentration distributions at (a) 30 s and (b) 1200 s for the nonlinear strain energy function with α 1 = -4.7 and M = 1 using two values of the Young s modulus E and two values of the hydraulic conductivity κ 0 at an infusion pressure of 500 Pa. (a) 30 seconds (b) 1200 seconds Figure 10. Normalized bulk concentration distributions at (a) 30 s and (b) 1200 s for the nonlinear strain energy function with α 1 = -4.7 and M = 1 using two values of the Young s modulus E and two values of the hydraulic conductivity κ 0 at an infusion pressure of 500 Pa. 34