Biomedical Engineering Reference
In-Depth Information
where M is a constant representing the motility of the cells. Further, tumor surface
tension is modeled through localized enhancement of the Cauchy stress tensor, i.e.
σ ST , on the tumor surface and is given by
γ
ρ c ( I
σ ST
=
− ˆ
n
⊗ ˆ
n ),
(21.16)
ρ c
where
n is the unit normal to the tumor surface and γ(
ˆ
|∇
|
) is the coefficient of
ρ c
surface tension, while γ(
) is defined to be non-zero only in the neighborhood
of the tumor boundary whose location is identified by the sharp gradient in cell
concentration.
|∇
|
21.3 Numerical Framework
The coupled system of PDE's (Eqs. ( 21.1 )-( 21.6 )) is solved using the Galerkin Fi-
nite Element Method and is integrated in time using the backward Euler scheme.
The time steps were adaptively chosen to ensure near quadratic convergence. The
code was based on the deal.ii (Bangerth et al., 2007 ) finite element library and heav-
ily utilized its hanging nodes based adaptive mesh refinement capability (Fig. 21.3 )
and its suite of iterative solvers. A monolithic scheme was adopted to concurrently
solve for the complete set of solution variables ( ρ c s n e o g and
u ). How-
ever, a matrix based monolithic scheme would require the computation of the Jaco-
bian of the global residual equation. In this work, the exact Jacobian was computed
using the Sacado ( 2011 ) automatic differentiation library which allows for efficient
run time computation of the variational derivative of the residual equations. 1
¯
21.4 Simulations
The chemomechanical framework presented above is used to simulate tumor
growth. Most of the growth related parameters, transport constants and elastic com-
pliance values were obtained from the literature. The results presented in this work
are primarily 2D simulations, however, a representative 3D simulation is also in-
cluded at the end of this section. The problem geometry and the initial tumor seed-
ing concentrations are depicted in Fig. 21.4 . The initial tumor seed is a uniform
spherical concentration of cells ( ρ c
250 fg / µm 3 ) and the surrounding matrix is
=
10 8 N / µm 2 , ν
modeled as an elastic soft material ( E
=
2 . 4
·
=
0 . 49). To model the
1 Automatic differentiation, also referred to as algorithmic differentiation, calculates derivatives of
functions up to any order to within machine precision by reducing complex functions to elementary
arithmetic operations and elementary functions by repeated application of the chain rule. It can
result in significant speedup of multiphysics implementations by computing the Jacobian of finite
element residuals. For variational problems, even greater ease of implementation is possible: only
the energy functional needs to be coded, and the system of residual equations and the Jacobian can
be computed by taking variational derivatives of the functional and residual equations, respectively.
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