Biomedical Engineering Reference
In-Depth Information
involving a force balance and conservation statement
G
12
G
u
-----------------
u
p
(
t
f
)
g
0
(15.1a)
---
1
p
-----
(
u
)
k
p
------
0
(15.1b)
t
t
where
G
is the shear modulus,
is Poisson's ratio,
u
is the displacement vector,
p
is the ratio of fluid volume extracted to volume
change of tissue under compression,
is the pore fluid pressure,
is the
amount of fluid which can be forced into tissue under constant volume,
and
k
is the hydraulic conductivity, 1
S
t
f
are the densities of tissue and surrounding fluid, respectively, and
g
is
the gravitational acceleration vector.
In Equation (15.1a), the first two terms are the classic PDE describing
mechanical equilibrium in a linearly elastic solid, i.e., the stress of the solid
matrix is linearly proportional to the strain. The third term reflects an applied
body force which is a function of interstitial pressure gradients arising from
hydraulic loading conditions, e.g., cerebrospinal fluid drainage, hyperos-
motic drugs, capillary transport, etc. The last term in Equation (15.1a)
approximates the effects of gravity on brain tissue. Here, a reduction in buoy-
ancy forces is modeled by changes in the density of the fluid surrounding tis-
sue which has become exposed to air (i.e., becomes the density of air).
In Equation (15.1b), the first term reflects the time rate of dilatational
changes of the solid matrix. The second term in the equation is a conservation
of mass statement. It contains (within the divergence operator) Darcy's law,
which linearly relates the movement of interstitial fluid to the pressure gra-
dient acting across the tissue, i.e.,
f
is the velocity of the
interstitial fluid. These first two terms taken together state that volumetric
changes correlate directly with the transport of fluid into and out of a control
volume. The last term in (15.1b) represents an accumulation of pressure
which allows compressibility of the interstitial fluid. This term may be used
in cases where the tissue is not completely saturated with fluid, i.e., small
voids of air or gas are present. Generally, the brain is considered to be a fully
saturated medium which sets
vk
p
where
v
to unity and eliminates the time rate of
change of pressure from (15.1b).
To first order, this characterization of brain tissue would seem to be a rea-
sonable starting point for modeling, especially under the acute loading condi-
tions associated with surgery and given that the mechanical properties for
brain tissue are not agreed to universally. As an aid to better understanding
the physical behavior associated with Equations (15.1a-b), Figure 15.1 shows
a fully saturated porous medium undergoing compression by a perforated
piston. In this example, the material is placed under compression
and drain-
age is allowed through the holes in the piston face. Figure 15.2 illustrates the
P
