Biomedical Engineering Reference
In-Depth Information
swelling due to edema, the deformation map
ϕ
can be obtained by solving the
mechanical BVP problem.
For real tumor cases,
T
r
and
D
r
are not known but are estimated via a statistical
approach as explained in Section 4.4. To simulate themass-effect of tumors starting
with normal brain images, approximations of these parameters must be used as
explained next.
4.3.1. Defining the relaxed configuration
Defining
κ
r
involves specifying the geometry of the brain and that of
T
r
(which corresponds to brain tissue that had died and is no longer present in
κ
t
or is infiltrated by tumor cells) and
D
r
(which corresponds to brain tissue that is
swollen by edema in
κ
t
). These regions are highly variable for different tumor
cases and types. In order to make the problem tractable, herein,
T
r
and
D
r
are
approximated with two concentric spheres. The center,
c
t
and radii
r
t
and
r
e
of
T
r
and
D
r
, respectively, are treated as model parameters.
It is worth noting that the shape of the final tumor depends on the generated
surrounding stresses and need not be spherical [79]. Furthermore, the goal of
the biomechanical model described here is, for a given definition of
T
r
and
D
r
,
to predict brain tissue deformation in a normal brain image. Having built such
a forward model for the deformation caused by tumors, the estimation of the
unknown model parameters, including
T
r
and
D
r
, for a real tumor case will be
dealt with in the context of inverse problem solving. This is explained in detail in
Section 4.4.
4.3.2. Tumor mass-effect and edema
To account for themass-effect of the bulk tumor, following thework ofWasser-
man and Acharya [81], the expansive force of the neoplasm is assumed to be gen-
erated by the proliferative tumor cells only. Accordingly, in model simulations,
brain tissue in region
T
r
is removed and a constant outward pressure
P
is applied
normal to the boundary of
T
r
.
P
is a model parameter that determines the mass
effect exerted by the bulk tumor, and therefore, to a large extent, the final tumor
size.
Depending on the type of tumor and its aggressiveness,
T
r
may be surrounded
with a peri-tumor edema region
D
r
. Since expansion due to edema occurs in white
matter (WM) only [83,84],
D
r
is restricted to WM tissues. Edema expansion in
WM is mostly perpendicular to the direction of the fibers. Here, no knowledge of
WM fiber orientation is assumed, and, therefore, an isotropic expansive strain
e
is applied to
D
r
by using analogy to thermal expansion. A thermal conductivity
value of zero for brain tissues prevents this expansion from spreading outside
D
r
.
Studies of brain edema that measured a volume expansion of 200 to 300% in WM
[83,84] imply
e
∈
[0
.
26
,
0
.
44]. For simulations starting with normal brain scans a
value of
e
=0
.
35 is adopted.
