Nonlinear poroplastic model of ventricular dilation in hydrocephalus

Transcription

1 J Neurosurg 109: , 2008 Nonlinear poroplastic model of ventricular dilation in hydrocephalus Laboratory investigation SHAHAN MOMJIAN, M.D., 1 AND DENIS BICHSEL, PH.D. 2 1 Department of Clinical Neurosciences/Service of Neurosurgery, Geneva University Hospitals and University of Geneva; and 2 Geneva School of Engineering, Geneva, Switzerland Object. The mechanism of ventricular dilation in normal-pressure hydrocephalus remains unclear. Numerical finiteelement (FE) models of hydrocephalus have been developed to investigate the biomechanics of ventricular enlargement. However, previous linear poroelastic models have failed to reproduce the relatively larger dilation of the horns of the lateral ventricles. In this paper the authors instead elaborated on a nonlinear poroplastic FE model of the brain parenchyma and studied the influence of the introduction of these potentially more realistic mechanical behaviors on the prediction of the ventricular shape. Methods. In the proposed model the elasticity modulus varies as a function of the distension of the porous matrix, and the internal mechanical stresses are relaxed after each iteration, thereby simulating the probable plastic behavior of the brain tissue. The initial geometry used to build the model was extracted from CT scans of patients developing hydrocephalus, and the results of the simulations using this model were compared with the real evolution of the ventricular size and shape in the patients. Results. The authors model predicted correctly the magnitude and shape of the ventricular dilation in real cases of acute and chronic hydrocephalus. In particular, the dilation of the frontal and occipital horns was much more realistic. Conclusions. This finding suggests that the nonlinear and plastic mechanical behaviors implemented in the present numerical model probably occur in reality. Moreover, the availability of such a valid FE model, whose mechanical parameters approach real mechanical properties of the brain tissue, might be useful in the further modeling of ventricular dilation at a normal pressure. (DOI: /JNS/2008/109/7/0100) KEY WORDS biomechanical study finite-element analysis hydrocephalus mathematical model normal-pressure hydrocephalus poroplasticity T HE pathogenesis of ventricular dilation in NPH remains obscure. The initial hypothesis considered an increased resistance to CSF absorption with a slight pressure elevation and pressure waves, 22 but it has also been posited that a change in the mechanical properties of the brain parenchyma could favor or even cause ventricular dilation at a normal pressure. 12 Hakim et al. 5 proposed that the brain could be considered as a porous viscoelastic material. This concept has been taken over and formalized by Nagashima et al. 10 based on the theory of poroelasticity that studies the deformation of a porous elastic material. If the geometry of the object is complex like that brought about by the ventricular shape, the mechanics of the deformation of such a material has to be computed using FE analysis. 13 So far, several studies Abbreviations used in this paper: CSF = cerebrospinal fluid; FE = finite-element; NPH = normal-pressure hydrocephalus. have simulated ventricular dilation in 2D brain slices with the aid of FE analysis. 10,13,14,21 However, these FE models yielded an almost uniform ventricular dilation, whereas in reality the horns of the ventricles appear to enlarge more than the bodies of the lateral ventricles; the elongation of the horns was clearly not reproduced by these models. This discrepancy could be explained by the fact that these FE models implemented uniform linear properties for the poroelastic matrix. It is known, however, that the material properties of the brain are nonlinear and also dependent on the stress strain rate and time, 21 but most authors have argued that a linear stress strain relationship is sufficient to model deformation in hydrocephalus. A recent model introduced a nonlinearity of the fluid permeability in the pores, 18,23,24 but this also did not resolve this very small dilation of the horns. We instead hypothesized that the elastic modulus of the porous matrix might behave nonlinearly as a function of the stretching of the pores of that matrix. 100 J. Neurosurg. / Volume 109 / July 2008

2 Nonlinear poroplastic model of ventricular dilation Moreover, after successful shunt treatment for NPH, the ventricular volume decreases only moderately, 1 which suggests that a plastic deformation of the brain has occurred and that plasticity should also be taken into account in the modeling of ventricular dilation. 2 In the present study we have therefore implemented a nonlinear elastic modulus in a 2D poroplastic FE model of hydrocephalus and analyzed the influence of the introduction of these potentially more realistic mechanical behaviors on the prediction of the shape of the ventricular dilation, by quantitatively comparing the results of the simulations with the natural evolution of real cases of human hydrocephalus. Brain Geometry Methods We used the real anatomy of the brain surface and ventricles in 2 patients in whom hydrocephalus was developing to construct 2D geometrical forms that served as input for the simulations. For each patient, 2 CT scans were acquired. In the first case, chronic hydrocephalus developed 3 months after subarachnoid hemorrhage. The first CT scan was obtained when the ventricles were still normal in size and the second when hydrocephalus had occurred. In the second case, the patient presented with acute hydrocephalus due to intraventricular hemorrhage. The first scan was obtained at this time and the second after 3 days of external ventricular drainage with normalization of the ventricular size. The boundaries of the ventricles and the brain surface extracted from the CT scans showing normal-size ventricles defined the poroelastic domain in which the differential equations have to be solved to obtain the distortion of the brain under hydrocephalus. To test the validity of the model, the deformation of the ventricles obtained by calculation was compared with the real ventricular contours extracted from the scans with dilated ventricles. Brain Modeling The important technical terms of the mechanics involved in the present model have been defined in a glossary (see Appendix 1). In its normal state (no hydrocephalus) the brain may be modeled as a poroelastic material composed of an elastic porous matrix saturated by an interstitial fluid in the pores. 13 The elastic material properties of the brain are not experimentally well defined, but measurements suggested that the stiffness value of the normal brain tissue is in the range from 10 to 100 kpa. 4 In chronic hydrocephalus, indirect evidence from histopathological findings in experimental models and from CSF compensatory parameters (pressure volume index) measured in patients 20 suggests that brain tissue elasticity is reduced. A recent experimental study demonstrated a nonsignificant difference between gray and white matter elasticity in the spinal cord; 11 initial isotropic and homogeneous material properties were therefore also assumed in the entire brain. The nonlinearity that we have implemented in the model is to make the Young modulus of the brain parenchyma dependent on the local parenchymal void ratio or fluid concentration, which reflects the dilation of the pores of the matrix. Namely, the local Young modulus is a decreasing function of the void ratio, which is very similar to what J. Neurosurg. / Volume 109 / July 2008 happens in a sponge. The nonlinearity chosen in this model allows the Young modulus (E) to vary between E min and E max in an exponential way E(e) = E 0 a e. The constant a is defined by E(0.01) = E min = E 0 /10 and E max = 20E 0, E 0 = 10 4 Pa. These values were chosen in accordance with numerical experimentations, which have shown a void ratio varying between at most and As no experimental data for E min and E max are available, the chosen values are arbitrary; however, these values could vary significantly without important alterations in the results of the calculations. Only the number of iterations depended strongly on these parameters. The time evolution of the mechanical properties has been modeled so as to simulate not only nonlinear elasticity but also plasticity, which implies a relaxation of stress and pressure with time. In practice, each time iteration has been divided into 2 steps. The first step is devoted to solving the static poroelastic problem, which defines the new geometric boundaries (deformed ventricles) and to calculating the void ratio of the deformed parenchyma. The second step consists of updating the Young modulus (nonlinearity of elasticity) and relaxing the internal stress and pressure (plasticity). A detailed presentation of the poroelastic equations involved in the present model is found in Appendix 2. 3 Boundary Conditions The skull is treated as a nondeformable medium, which implies that the brain surface does not enlarge; in other words, the elastic medium is fixed at the brain surface (Dirichlet boundary condition). In the brain, the ventricles are considered to be cavities whose walls deform under the pressure of CSF (Neumann boundary condition). The fluid pressure is 0 at the brain surface boundary and the constant fluid pressure in the ventricles was set at 500 Pa (3.8 mm Hg). This value was chosen to be as low as possible while giving an acceptable calculation time. Calculation Methods We used the MATLAB (The MathWorks, Inc.) and COMSOL Multiphysics (COMSOL AB) programs. The image processing (extraction of the brain surface and ventricle contours from the CT scans) and the construction of the objects used in COMSOL to define the geometry have been done with MATLAB and the COMSOL library. The numeric resolution is done with COMSOL in which the needed functions are defined, the boundary conditions are set, a mesh is created, and the partial differential equations are solved in each time iteration using the FE methods provided by the software. As mentioned earlier, each time iteration is divided into 2 steps. First, the time-independent partial differential equations describing the elastic and fluid properties are solved. Second, the Young modulus is updated according to the newly obtained void ratio, and the pressure and stress are relaxed. This process is repeated 50 times. After each time iteration, a picture of the deformed brain is saved, allowing the calculated deformations of the ventricles to be visualized. Furthermore, to objectively evaluate the predictive capacity of our model, we defined a quality factor as follows. Let S lo be the surface of the dilated ventricles, S so be the surface of the normal-size ventricles 101

3 S. Momjian and D. Bichsel obtained from the CT images, and S lc be the surface of the dilated ventricles obtained from the calculations; S im = S lo S so is the surface difference between the dilated and the normal-size ventricles in the real case; and S cal = S lc S so is the surface difference between the dilated ventricles and the normal-size ventricles in the simulated case. The quality factor (Q) of the deformation is defined as follows: 1 A(S im S cal ) Q = A(S lo ) A(S lc ) 2 A(S so ) where A is the area of the domain and is the symbol for the symmetric difference. At the beginning of the calculations, Q = 0 since A(S cal ) = 0. In the best possible case, S cal = S im, the symmetric difference is 0, and the quality factor is 1. In every other case, the quality factor varies between 0 and 1. Results The real and simulated evolution of a case of chronic hydrocephalus is represented in Fig. 1. Figure 1a shows the initial normal-sized ventricles; this image is used to set the boundaries of the ventricles and brain surface at the beginning of the simulation. The same slice of the brain after the onset of hydrocephalus is shown in Fig. 1b. The ventricle boundaries are extracted from this image to be compared with the result of the calculations. Both initial and dilated real boundaries are shown on all other images to allow a visual comparison. The brain displacement obtained after the last iteration is shown in Fig. 1c. The calculated deformed ventricles are very similar to the real deformed ventricles, particularly for the right ventricle. The left ventricle, whose horns are longer in the real case, is not as well rendered. The void ratio (Fig. 1d) is much higher near and around the horns of the ventricles. The inhomogeneous Young modulus obtained after the last iteration is depicted in Fig. 1e, and the pressure distribution in the solid matrix is shown in Fig. 1f. As described in Methods, Fig. 1d and e show that the Young modulus is a decreasing function of the void ratio. The real and simulated evolution of a second case of hydrocephalus is presented in Fig. 2. In this case the hydrocephalus was acute, and the patient presented with dilated ventricles (Fig. 2b). However, the CT image obtained after external ventricular drainage and normalization of the ventricular size was used as input for the simulation (Fig. 2a). The same descriptions and comments for Fig. 1c f apply to Fig. 2c f. The quality factors, 68% in the first case and 56% in the second, provide a good agreement between the results of the simulation predicting the ventricular size and shape and the observed real data. Discussion The model of hydrocephalus created for the present study implemented a nonlinearity of the Young modulus as a function of the ongoing deformation of the porous matrix as well as a mechanical plasticity of the brain parenchyma modeled as a relaxation of mechanical stresses with time. This greatly improved the prediction of the magnitude and shape of the ventricular dilation in real cases of acute and chronic hydrocephalus. In particular, the dilation of the frontal and occipital horns was much more realistic. The morphological superiority of the present nonlinear poroplastic model over a simple linear poroelastic model is shown in Fig. 3. To our knowledge, this is the first FE model of hydrocephalus taking into account the possible nonlinearity of the Young modulus and the probable mechanical plasticity of the brain. In addition, this is the first study to quantitatively measure the prediction of the model when applied to real cases of developing hydrocephalus. In fact, previous studies have estimated the validity of the model they implemented only qualitatively by noticing a visual agreement with the general clinical observation. 10 Here, we quantitatively established the validity of the model by comparing, in the same patient, the deformation pattern predicted by simulation with the actual brain deformation after hydrocephalus had occurred. The close matching between the prediction of the simulation and the evolution of the ventricles in the patients suggests that hydrocephalic brain tissue sustains a nonlinear plastic change in its elastic modulus. The generation of anatomically realistic and precise 3D FE models is expected soon as computing power becomes readily available. However, the results that will be obtained using such models may be misleading if the baseline mechanical properties of the brain as well as its behavior under stress are not also as realistic as possible. The knowledge of this information in vivo, however, is largely lacking, 6,9 which partly is because of the unknown relationship between microscopic living structures and macroscopic mechanical parameters. 18 In vivo animal experiments are difficult to set up, and the necessary mechanical parameters must therefore be determined through in vitro experiments and/or simulation. Simulation with validation with observational data offers a valid indirect method to characterize these unknown mechanical properties and behaviors to build valid and predictive models. Nonlinear Models of Brain Material So far, the biomechanical models of brain parenchyma were conceived as being made of a homogeneous and isotropic material with uniform elastic properties, which is a very simplistic assumption for a living tissue. As advocated by previous authors, 18 models with nonlinear brain material properties must be developed to be able to correctly reproduce the evolution of the body and horns of the ventricles. However, this intention has not been confirmed by previous researchers, probably because of the limited computing power readily available at that time and because of the difficulty of choosing which mechanical parameters could have a nonlinear behavior. Recently, an initial FE model implementing a nonlinear property was presented. 18,23,24 The fluid permeability in the pores was the parameter chosen to be nonlinear and varied nonlinearly as a function of the tissue strain. This refinement, however, did not change the pattern of ventricular enlargement with a persisting uniform dilation. Moreover, the latest model, 24 albeit 3D, used artificial geometry to represent the ventri- 102 J. Neurosurg. / Volume 109 / July 2008

4 Nonlinear poroplastic model of ventricular dilation FIG. 1. Real and simulated evolutions obtained in a case of chronic hydrocephalus. Computed tomography scans showing the initial (a) and dilated (b) real ventricles. Computer-generated images showing color-coding representing displacement (c), void ratio (d), Young modulus (e), and pressure in the solid matrix (f) after the last calculation iteration. The inner red line denotes the real initial ventricular contour; the white area represents the calculated final ventricular contour that can be compared with the real dilated ventricles (outer red line). The values on the x and y axes are in meters. cles; the ventricular shape was therefore not realistic and had only some characteristic features of the actual ventricles. Findings from previous FE studies and observations of real cases of hydrocephalus can help to point out a possible mechanical parameter behaving nonlinearly. Regional differences in stress concentration have been found in previous FE models of hydrocephalus after ventricular dilation. 10,13,21 Overall, expansive stresses have been demonstrated around the frontal and occipital horns, whereas compressive stresses have been shown along the body of the J. Neurosurg. / Volume 109 / July 2008 lateral ventricles. 13 These regional differences in mechanical stress could explain the preferential location of pathophysiological alterations observed in hydrocephalus in humans. An expansive stress around the frontal and occipital horns could lead to local cavitations in the solid component of the brain tissue and to disruptions of the ependyma and periventricular white matter, and hence readily explain the classic pattern of periventricular CSF edema observed in these regions. 13 Moreover, the changes of pore size and of fluid content may induce alterations in the mechanical properties of the porous matrix. The mechanical effect 103

5 S. Momjian and D. Bichsel FIG. 2. Real and simulated evolution of a case of acute hydrocephalus. Computed tomography scans showing the initial (a) and dilated (b) real ventricles. Computer-generated images showing color-coding of displacement (c), void ratio (d), Young modulus (e), and pressure in the solid matrix (f) after the last calculation iteration. The inner red line denotes the real initial ventricular contour, and the white area represents the calculated final ventricular contour that can be compared with the real dilated ventricles (outer red line). brought about by distension of the pores as well as a possible deleterious metabolic effect of such a stretching may cause a loss of elasticity. Experimentation found that the elastic response of brain tissue had a nonlinear behavior with an increase of the modulus with increasing strain, 8 and that the edematous brain was relatively immutable and stiff. 7 However, these conclusions were based on a very short time evolution and, on the contrary, a sufficiently prolonged mechanical stress may alter mechanically and biologically the structural properties of the brain tissue, 12 which could lead to a decrease in its elasticity. We therefore chose to introduce into our model a dependence of the elastic modulus on the size of the pores (void ratio), which resulted in a larger dilation of the horns and markedly improved the prediction of ventricular dilation in real cases of hydrocephalus. Plasticity of the Brain After a large and prolonged deformation, plastic changes of the brain structure may occur. The living cerebral microstructure may reorganize itself to reduce the sustained 104 J. Neurosurg. / Volume 109 / July 2008

6 Nonlinear poroplastic model of ventricular dilation FIG. 3. Computer-generated images of ventricular dilation obtained using a simple linear poroelastic model (a) and the present nonlinear poroplastic model (b), with the same boundary conditions and ventricular pressure. The quality factor is much better in the latter model (0.68 versus 0.22). stress. This means that when the ventricular pressure diminishes, the brain does not completely recover its original configuration. In fact, after successful shunt treatment for chronic hydrocephalus, the ventricular volume decreases only moderately, 1 which suggests that a plastic change of the brain structure has occurred. The nonlinear poroelastic model has therefore to be sophisticated by implementing a plastic behavior of the solid matrix. The realistic agreement between the prediction of ventricular size by simulation by using the present poroplastic model and the evolution of ventricular volume in reality suggests that mechanical plasticity is a key factor to explain the large ventricular dilation observed in longstanding hydrocephalus. This could be particularly important in the pathogenesis of NPH because the reduction of internal stresses may allow an ongoing deformation to occur at a relatively low pressure. In our model, the ventricular dilation has been induced by establishing a small pressure gradient between the ventricles and the subarachnoid spaces, which may appear unrealistic given that such a gradient has not been reproducibly found, either in obstructive or in communicative hydrocephalus. 19 However, Peña et al. 14 showed that ventricular dilation could also occur when imposing a normal fluid pressure in the ventricles and in the subarachnoid spaces but a slightly lower fluid pressure inside the cerebral mantle. This lower fluid pressure inside the parenchyma is supposed to exist to generate a transependymal resorption of CSF by the parenchyma, with the brain acting as a CSF sink. 14 As such an alternative CSF pathway is probably present in NPH, this latter way of placing the pressure gradient in the model would have probably been conceptually closer to the real situation of NPH where the pressure is normal. However, from a mechanical standpoint, the results in terms of ventricular size and shape are expected to be the same whatever the configuration of the pressure gradient as long as its magnitude is preserved. As the investigation focused mainly on the influence of the mechanical parameters on the resultant ventricular shape, we opted for a pressure gradient configuration that could be more easily implemented with the available FE modeling software packages. J. Neurosurg. / Volume 109 / July 2008 The aforementioned reasoning and notably the fact that the same results can be contemplated with normal ventricular and subarachnoid spaces pressures also mean that our FE model is applicable to the study of NPH. Conclusions The nonlinear poroplastic FE model of the brain used to simulate the formation of hydrocephalus correctly predicted the size and shape of the ventricular dilation. Such a model may serve as a tool to further investigate the disturbing mechanisms enabling a ventricular dilation to occur at a normal pressure. Moreover, as suggested by Wirth and Sobey, 24 a valid and accurate numeric model of ventricular dilation could also serve to simulate the recovery toward the original and undistorted brain and ventricle configuration from the dilated and deformed ventricular shape in a given hydrocephalic patient. Such a reverse process could predict the evolution in time of the return of the ventricles and brain to a normal configuration after treatment. The knowledge of the expected treatment effect could be used as a theoretical end point to tailor and optimize treatment by adjusting the settings of a programmable valve. Appendix 1 The elastic modulus or modulus of elasticity or Young modulus (E) is the mathematical description of the tendency of a material to be distorted elastically (that is, nonpermanently) when a load is applied to it. Mathematically, the definition of the Young modulus is E = stress/strain, where the stress is the force per unit area and the strain is the ratio of the distortion caused by the stress to the original state of the object. Mechanical plasticity is the propensity of a material to undergo a permanent distortion under load. In other words, the distortion persists even when the load has returned to zero. A medium is isotropic when its properties are the same in all the 105

7 S. Momjian and D. Bichsel directions, and a medium is homogeneous when its properties are the same everywhere in the object. The void ratio of a porous medium is defined as the ratio of the volume of void to the volume of solid. The fluid permeability is the ability of a fluid to move through a medium. The permeability depends on the nature of the fluid and on the porosity of the medium. Finite-element analysis is a very efficient method to numerically solve differential equations. Appendix 2 In each time iteration, we have to solve partial differential equations to obtain the distributions of stress, strain, displacement, pressure, and void ratio. The first differential equation is related to the stress tensor of the elastic media and is as follows: ij = p x j where is the stress tensor and p is the pressure due to the fluid. The second differential equation, the Biot equation, describes the fluid pressure and is as follows: (K ij p) =0 x j in the static approximation where K ij is the hydraulic permeability that depends on the void ratio. In the present model, the permeability is assumed to be isotropic and the Biot equation simplifies to the following: i (K p) =0 The elastic properties of the brain are described by the constitutive equations of the elastic medium: ij = kk ij 2 ij = e ij 2 ij, e = kk = u k x k where and are the Lamé constants, which are related to the Young modulus E and the Poisson ratio as follows: Ev E = and = (Eq. 5) (1 v)(1 v) 2(1 v) and e is the void ratio. The strain tensor is defined as follows: ij = 1 u i u j (Eq. 6) 2 x j where u is the displacement vector. On merging Equations 6 and 4 and deriving, we get 2 u i ( ) e = p x 2 j (Eq. 1) (Eq. 2) (Eq. 3) (Eq. 4) (Eq. 7) Equations 7 and 2 with the boundary conditions must be solved to get the displacement of the brain under ventricular dilation. The boundary conditions associated with Equation 7 are as follows: u k = 0 at the brain surface (the brain surface does not enlarge), u k is free at the ventricular walls; ij = p ij on the surface of the ventricles, p v is the uniform hydrostatic pressure in the ventricles. The boundary conditions associated with Equation 2 are as follows: p = 0 at the brain surface, and p = p at the ventricular walls. References 1. Anderson RC, Grant JJ, de la Paz R, Frucht S, Goodman RR: Volumetric measurements in the detection of reduced ventricular volume in patients with normal-pressure hydrocephalus whose clinical condition improved after ventriculoperitoneal shunt placement. 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J Biomech 3: , Miller K, Chinzei K, Orssengo G, Bednarz P: Mechanical properties of brain tissue in-vivo: experiment and computer simulation. J Biomech 33: , Nagashima T, Tamaki N, Matsumoto S, Horwitz B, Seguchi Y: Biomechanics of hydrocephalus: a new theoretical model. Neurosurgery 21: , Ozawa H, Matsumoto T, Ohashi T, Sato M, Kokubun S: Comparison of spinal cord gray matter and white matter softness: measurement by pipette aspiration method. J Neurosurg 95: , Pang D, Altschuler E: Low-pressure hydrocephalic state and viscoelastic alterations in the brain. Neurosurgery 35: , Peña A, Bolton MD, Whitehouse H, Pickard JD: Effects of brain ventricular shape on periventricular biomechanics: a finite-element analysis. Neurosurgery 45: , Peña A, Harris NG, Bolton MD, Czosnyka M, Pickard JD: Communicating hydrocephalus: the biomechanics of progressive ventricular enlargement revisited. Acta Neurochir Suppl 81:59 63, Rubin RC, Hochwald GM, Tiell M, Epstein F, Ghatak N, Wisniewski H: Hydrocephalus: III. Reconstitution of the cerebral cortical mantle following ventricular shunting. Surg Neurol 5: , Rubin RC, Hochwald GM, Tiell M, Liwnicz BH: Hydrocephalus: II. Cell number and size, and myelin content of the pre-shunted cerebral cortical mantle. Surg Neurol 5: , Rubin RC, Hochwald GM, Tiell M, Mizutani H, Ghatak N: Hydrocephalus: I. Histological and ultrastructural changes in the preshunted cortical mantle. Surg Neurol 5: , Sobey I, Wirth B: Effect of non-linear permeability in a spherically symmetric model of hydrocephalus. Math Med Biol 23: , Stephensen H, Tisell M, Wikkelso C: There is no transmantle pressure gradient in communicating or noncommunicating hydrocephalus. Neurosurgery 50: , J. Neurosurg. / Volume 109 / July 2008

8 Nonlinear poroplastic model of ventricular dilation 20. Tans JT, Poortvliet DC: Reduction of ventricular size after shunting for normal pressure hydrocephalus related to CSF dynamics before shunting. J Neurol Neurosurg Psychiatry 51: , Taylor Z, Miller K: Reassessment of brain elasticity for analysis of biomechanisms of hydrocephalus. J Biomech 37: , Vanneste JA: Three decades of normal pressure hydrocephalus: are we wiser now? J Neurol Neurosurg Psychiatry 57: , Wirth B: A Mathematical Model for Hydrocephalus. Oxford: Oxford University, Wirth B, Sobey I: An axisymmetric and fully 3D poroelastic model for the evolution of hydrocephalus. Math Med Biol 23: , 2006 Manuscript submitted June 18, Accepted October 29, The Swiss National Science Foundation provided a grant to Dr. Momjian (Grant No. K-32K ). Address correspondence to: Shahan Momjian, M.D., Service of Neurosurgery, Geneva University Hospitals, Rue Micheli-du- Crest, 24, 1211 Genève 14, Switzerland. Shahan.Momjian@ hcuge.ch. J. Neurosurg. / Volume 109 / July