Biomedical Engineering Reference
In-Depth Information
In the above equations,
γ
ξR
is the effective fluid weight and
K
ξ
γ
ξR
K
Sξ
/μ
ξR
is
the Darcy permeability, involving the effective dynamic fluid viscosities
μ
ξR
and the
intrinsic permeabilities
K
Sξ
. Since the permeabilities are related to the deformation
of the solid skeleton, deformation-dependent intrinsic permeabilities (e.g., Markert,
2007
) are included via
=
n
ξ
n
ξ
0
S
κ
n
0
S
n
S
K
Sξ
0
S
K
Sξ
=
.
(19.13)
Therein,
K
Sξ
0
S
are the permeability tensors of the undeformed solid reference config-
uration, which need to be equipped with material parameters describing the initial
(anisotropic) intrinsic permeability of the tissue perfusion by the interstitial fluid
(
K
SI
0
S
) and by the blood plasma (
K
SB
≥
0 governs the non-linear
deformation-dependent behavior. A possibility for a patient-specific determination
of the coefficients of the permeability tensors and of the effective drug diffusion
tensor
D
D
is described in Sect.
19.2.2
.
Finally, there is a general need to express the volume fractions of all constituents
either by primary variables and initial conditions or by constitutive equations. In
case of an incompressible solid constituent with an initial volume fraction of
n
0
S
in
the solid reference configuration,
n
S
0
S
). The exponent
κ
n
0
S
(
det
F
S
)
−
1
holds and simplifies to
n
S
=
=
n
0
S
(
1
div
u
S
)
in case of small-strain theories. Thus, as a result of (
19.1
),
n
B
and
n
I
cannot be specified by primary variables individually but only as a sum:
n
B
−
n
I
+
=
n
S
. Thus, additional constitutive information is needed. To solve this problem,
it is assumed in the present study that the blood-vessel system is stable and inherent.
Furthermore, since the blood plasma is assumed to incompressible, this leads to the
assumption that
n
B
1
−
n
0
S
is constant. In particular, it is assumed that
n
B
n
0
S
=
=
=
0
.
05, cf. Table
19.1
.
19.2.2 Inhomogeneous and Anisotropic Perfusion Parameters
Regarding the micro-structural composition of the nervous brain tissue, one can as-
sume that perfusion in grey matter (cell bodies) is isotropic. In contrast, white-matter
perfusion is anisotropic due to a preferred flow direction in the ECS along the axonal
fibers. The micro-structural information of the white-matter tracts can be provided
by Diffusion Tensor Imaging (DTI), cf. Basser et al. (
1994
). The outstanding feature
of DTI is the possibility to determine the diffusion tensor
D
awd
of water molecules
in living biological tissue. The symmetric, positive definite apparent water-diffusion
tensor
D
awd
of each voxel of the DTI can be written as
⎡
⎤
⎡
⎤
γ
1
,
awd
D
11
D
12
D
13
0
0
D
awd
=
⎣
D
21
D
22
D
23
⎦
e
i
⊗
⎣
γ
2
,
awd
⎦
v
i
⊗
v
i
.
e
k
=
0
0
D
31
D
32
D
33
γ
3
,
awd
0
0
(19.14)
