Biomedical Engineering Reference
In-Depth Information
4.3.3. Brain tissue constitutive material models
Depending on the area of application, researchers studying the mechanics of
brain tissue have used a number of different material constitutive relationships. The
brain tissue behavior has been modeled as linear elastic [81, 85-89], hyperelastic
[79], hypervisco elastic [90-93], and poroelastic [94]. Given the time scale of the
tumor growth process, inertial effects can be ignored and the deformation of brain
tissues may be modeled as a quasi-static process. Furthermore, with such a large
time scale any viscoelastic effects of the brain tissues would have abated since
viscoelastic time constants of living soft tissues, such as the brain, do not exceed a
few hours [90, 92, 93]. Therefore, in this work, the viscous component of the brain
tissue's response will be ignored.
Clearly, the brain is heterogeneous, since it is composed of white and gray
matter, with a different molecular makeup. A difference in material properties
between white and gray matter has been reported in the literature [89, 93], although
the exact material parameter values vary widely. Directional property differences
were also reported for brain white matter [93]. Based on experiments on the
porcine brain, Miga et al. [95] observed that the use of a heterogeneous model
of the brain tissue did not significantly improve the accuracy of their brain-shift
model. Therefore, in simulations reported in this chapter, a homogeneous and
isotropic material model for the brain tissue is adopted. The simulations can easily
be adapted to deal with the heterogeneous anisotropic case if accurate material
property values and white matter orientation information are available and reliable.
Using the dataset described later in this chapter, experiments were performed
to compare the behavior of the linear material model and four hyperelastic mater-
ial models [79, 90, 92, 93] suggested for modeling brain tissue. All FE simulations
were performed using the commercial package ABAQUS [96], and nonlineari-
ties arising from large deformations were taken into account. The testing of the
poroelastic material model was deferred to future work since ABAQUS requires
the use of second-order finite elements with this material model, which is one or-
der higher than the one used in simulations reported here for the tested materials.
The stability of the different material models at strain levels encountered during
mass-effect simulations, the ability of the simulations to reach convergence, and
the error in predicting deformations caused by real tumors were the factors used
to select the most suitable material model to use for brain tissues. Based on these
experiments, which are described in [97], we adopted the isotropic and homo-
geneous material model proposed by Miller and Chinzei [92] while relaxing the
perfect incompressibility assumption and ignoring viscous effects. Under these
conditions, the strain energy density function of the material becomes [96]:
λ
α
3
−
3
+
J/J
th
−
1
2
,
=
2
µ
α
2
D
1
1
+
λ
α
2
+
λ
α
W
(3)
λ
i
J
−
1
/
3
λ
i
,
λ
i
,i
where
=
=1
,
2
,
3, are the principal material stretches,