Biomedical Engineering Reference
In-Depth Information
19.2.1.5 Constitutive Settings
The presented balance equations need to be completed with admissible constitutive
relations. An exploitation of the entropy inequality (Ehlers,
2002
,
2009
) yields re-
strictions and conditions for the formulation of constitutive equations such as the
principle of effective stresses, viz.,
T
ξ
T
S
T
E
−
n
S
p
I
T
ξ
n
ξ
p
ξR
I
,
=
and
=
E
−
(19.8)
where
I
denotes the second-order identity. The extra-stresses
T
ξ
E
of the pore liq-
uids are neglected due to dimensional reasons (e.g., Ehlers,
2002
)aswellasos-
motic pressure contributions. The partial stress of the solid skeleton contains the
pore pressure
p
=
(n
B
p
BR
+
n
I
p
IR
)/(
1
−
n
S
)
. Following this,
T
E
−
=
T
p
I
.
(19.9)
Therein,
T
E
is the effective stress governed by the solid deformation. Concerning
the CED problem, use is made of the small-strain (linear) elasticity concept. In ad-
dition, although the brain tissue exhibits an inhomogeneous and anisotropic nature
of the white-matter tracts, which strongly influences the pore-fluid and diffusion
properties, it is assumed that a standard linear elasticity law in the Hookean sense is
sufficient for the description of the brain deformation. Thus,
T
E
=
2
μ
S
ε
S
+
λ
S
(
ε
S
·
I
)
I
.
(19.10)
Therein,
ε
S
is the linear Green-Lagrangian strain, and
μ
S
and
λ
S
are the Lamé
constants.
In this contribution, the main attention is drawn to the flow of the administered
therapeutical including the drug. As a matter of fact, a slow infusion process with a
slight application dose as is applied within the CED causes only small deformations
in the solid skeleton (as is seen later in Fig.
19.7
). However, if tumor growth or other
diseases, i.e. hydrocephalus, have to be taken into consideration, the tissue model
can be extended to finite elasticity or to finite viscoelasticity (see, e.g., Taylor and
Miller,
2004
; Ehlers et al.,
2009
). In both cases, the inclusion of solid anisotropy is
possible.
Moreover, relations for the filter velocities
n
ξ
w
ξ
of the pore fluids and for the
drug diffusion
n
I
c
m
d
DI
need to be specified. Following the detailed constitutive
modeling process described in Acartürk (
2009
) and Ehlers (
2009
), appropriate as-
sumptions for the direct momentum production terms
p
D
have to be pos-
tulated and inserted into the respective partial momentum balances. In this regard,
Darcy-like filter laws for the pore liquids are obtained in terms of
p
ξ
and
ˆ
ˆ
K
ξ
γ
ξR
grad
p
ξR
ρ
ξR
g
n
ξ
w
ξ
=−
−
(19.11)
as well as a Fick-like diffusion law for the therapeutic agent:
n
I
c
m
d
DI
=−
D
D
grad
c
m
.
(19.12)
