Problem Formulation

12.2.1 FSI Problem

LetO2r^d,d¼2, 3, be a fixed control volume containing a solid bodybwhose reference configuration is denoted byB and whose current configuration at timetis denoted byBt. The fluid occupies the regionO∖Bt. Let the position of any material point ofbinBbe denoted bysand letx_Q(t) be the position of any pointQinOat timet. The motion ofbis denoted by,

Fig. 12.1 A T1-weighted transversal magnetic resonance (MR) image of a human brain showing ventricular walls in healthy as well as in hydrocephalic brains (Adapted from [3]

that uses the MR simulator developed in [4–7])

80 S. Roy et al.

z:B!Bt and x¼zðs;tÞ; s2B;x2O;t2½0;TÞ;T>0: (12.1) The displacement of the bodyB, denoted byw, is one of the primary unknowns in the FSI problem and is defined as

wðs;tÞ ¼zð Þ s;t s; s2B;t2½0;TÞ: (12.2) The velocity field, denoted byu(x,t), is also one of the primary unknowns of the FSI problem and the spatial description of the (material) velocity field is defined as

u xð Þ ¼;t @zð Þs;t

@t

s¼z¹ð Þx;t: (12.3)

From Eqs. (12.2) and (12.3) we can readily see that, _

wðs;tÞ ¼u x;ð Þjt _{x¼zð Þs;t}; (12.4)

where the dot denotes the material derivative ofwwith respect to timet.

The local form of the balance of mass and the balance of momenta for x∈O∖(∂O\∂Bt) and 8t ∈ (0,T) can be represented as follows,

r_ þrdivu¼0; (12.5)

divTþrb¼ru_; (12.6)

T¼T^T; (12.7)

wherer¼r(x,t) is the spatial description of the density,T¼T(x,t) is the Cauchy stress,b denotes the external force density per unit mass acting on the system, div denotes the divergence of a field with respect toxand the superscript T denotes the transpose operator.

At the interface between the solid and the fluid phases, i.e.8x∈O\∂Bt, the following conditions must be satisfied 8t∈(0,T):

1. Continuity of motion, i.e.,u(x⁺)¼u(x) 2. Continuity of traction, i.e.,T(x⁺,t)n¼T(x,t)n

wherex⁺ andx tend tox∈∂Btfrom outside and inside ofBt, respectively. If∂ODand∂ONdenote parts of the boundary where the Dirichlet boundary condition and the Neumann boundary condition, respectively, are specified then we must have u xð Þ ¼;t ugð Þ;x;t x2@OD and Tð Þx;t m xð Þ ¼;t tgð Þ;x;t x2@ON; (12.8) wheremdenotes the outward unit normal to∂Oandu_gandtgdenote the prescribed velocity and traction, respectively. We next assume constitutive models for the fluid and the immersed solid and then present the variational formulation of the problem.

12.2.2 Constitutive Behavior

Since the CSF is a water-like fluid, we assume it to be an incompressible, linearly viscous Newtonian fluid that has a uniform density denoted by rf. Incompressibility requires that r_f ¼0 and the constitutive function for such a fluid is given by Gurtin [19],

T^_f ¼ pIþ2fD (12.9)

12 An Immersed Finite Element Method Approach for Brain Biomechanics 81

wherep represents the hydrostatic pressure in the fluid;Iis the identity tensor;f>0 represents the dynamic viscosity coefficient of the fluid;D¼¹₂LþL^T

is the stretching tensor andL¼graduis the velocity gradient tensor.

Understanding the constitutive behavior of the brain parenchyma is still an active field of research. Under slow strain loading conditions that typically occur during hydrocephalus Taylor and Miller [20] have shown that the brain parenchyma can be modeled as an incompressible, quasi-linear viscous material (i.e. the rheological model comprises a linear viscous element in parallel with a hyperelastic element). The Cauchy stress for such a material is of the form

T^s¼ pIþT^^v_sþ^T^e_s; (12.10)

wherepis a Lagrange multiplier,T^^v_sdenotes the viscous part of the response andT^^e_sdenotes the elastic part of the response.

In fact, the viscous nature of the solid is assumed to be similar to that of the fluid:

T^^v_s¼2sD; (12.11)

whereZsdenotes the viscosity coefficient of the solid and in particular, we assume thatZs¼Zf¼Z. We have assumed that the hyperelastic contribution is of the NeoHookean type and hence the elastic part of the Cauchy stress response takes the form

^T^e_s¼J¹mðFF^TIÞ; (12.12)

where F¼I + Grad wwith Grad denoting the gradient of a field with respect tos,Jis the determinant of F andmis the shear modulus of the solid. The first Piola-Kirchhoff stress from Eq. (12.12) is given as

P^^e_s ¼mF^TF^T

: (12.13)

12.2.3 Variational Formulation of the FSI Problem

The FSI problem involves the primary unknownsu(x,t),p(x,t) andw(s,t). We seek the solution to this FSI problem in a weak sense and for Eqs. (12.5)–(12.8) we select the functional spaces as follows:

u2v¼H¹_Dð ÞO ^d:¼ u2L²ð ÞO^djgradu2L²ð ÞO^dd;uj_@OD ¼ug

; (12.14a)

p2q:¼L²ð Þ;O (12.14b)

w2y¼H¹ð ÞB ^d:¼ w2L²ð ÞO ^djGradw2L²ð ÞO^dd

; (12.14c)

whered¼2 or 3 and denotes the dimension of the Euclidean point space for the problem. The space containing the test functions for the velocity field is subspace ofvand is defined as

v0¼H¹₀ð ÞO^d :¼ v2L²ð ÞO ^djgradv2L²ð ÞO ^dd;vj_@OD ¼0

: (12.15)

We can reformulate Eq. (12.6) in a variational setting as follows:

ð

Orfðu_bÞ vdv ð

Opdivvdvþ ð

O

T^^v_f : gradvdv ð

@ON

tgvda þ

ð

B

½P^^e_sF^Tðs;tÞ:gradvðxÞ_x¼zðs;tÞdV¼0 8v2v0; (12.16)

82 S. Roy et al.

where, we have also made use of the constitutive models (Eqs.12.9–12.13), the boundary conditions (Eq.12.8) and the interface conditions. Note that the top line Eq. (16) represents the traditional components of the momentum balance equations of the Navier-Stokes equations whereas the first term in the second line of Eq. (12.16) represents the contribution arising from the elasticity of the immersed solid. We reformulate Eq. (12.5) as follows:

ð

Oqdivudv¼0 8q2q: (12.17)

We would also like to enforce Eq. (12.3) in a weak sense and do so in the following form:

FB

ð

B

_

w sð Þ ;t u xð Þj;t x¼zð Þs;t

y sð ÞdV¼0 8y2y; (12.18)

whereFBis a constant and we have assumed it to be 1. 0 for this study.

12.2.4 Discretization

We use the finite element method for discretizing the spatial domain of the problem. We consider two independent triangulations for our problem:Ohfor OandBhforB, comprising quadrilateral cellsK_O andKB, respectively, in 2D. In accordance with the Galerkin finite element method, we choose finite dimensional subspacesvhv,qhqdefined on Ohandyhydefined onBhas follows,

vh:¼ uh2vjuhjK 2pVð Þ;K K2Oh

span vⁱ_h NV

i¼1; (12.19a)

qh:¼ p_h2qjp_hjK 2pQð Þ;K K2Oh

span qⁱ_h N_Q

i¼1; (12.19b)

yh :¼ w_h2yjw_hjK 2pYð Þ;K K2Bh

span yⁱ_h N_Y

i¼1; (12.19d)

wherepVð ÞK ,pQð ÞK andpYð ÞK are polynomial spaces of degreerV,rQandrYrespectively on the cells ofKandNV,NQand NYare the dimensions of the finite-dimensional subspaces. Specifically, we choosevhandqhin a manner that is consistent with the Brezzi-Babuska inf-sup condition for the Navier-Stokes component of the problem.

The spatially discretized version of Eqs. (12.16)–(12.18) form a set of differential algebraic equations which we discretize temporally using the finite difference method and integrate using the implicit Euler method.