Page 28 - Computational Modeling in Biomedical Engineering and Medical Physics

14 Computational Modeling in Biomedical Engineering and Medical Physics

Table 1.1 Partial differential operators and related mathematical models.
Principal part PDE type Canonical form obtained after Analytic problem
change of variables
2
b 2 ac , 0 Elliptic φ 1 φ 5 h ξ; ηð Þ 1 Dφ Laplace/Poisson
ξξ
ηη
2
b 2 ac 5 0 Parabolic φ 2 φ 5 h ξ; ηð Þ Helmholtz
ξξ
ηη
2
b 2 ac . 0 Hyperbolic φ 2 φ 5 h ξ; ηð Þ 1 Dφ Helmholtz
ξξ
ηη
Several common physical processes may lead to static and stationary diffusion mod-
els (elliptic PDEs, or Laplace and Poisson problem, in Chapters 4: Electrical activity of
the heart, and Chapter 6: Magnetic drug targeting), quasisteady and unsteady diffusion
models (parabolic PDEs or Helmholtz problems, in Chapters 5: Bioimpedance meth-
ods, and Chapter 8: Hyperthermia and ablation (Thermotherapy methods), propaga-
tion models (hyperbolic PDEs or Helmholtz problems, in Chapter 8: Hyperthermia
and ablation (Thermotherapy methods), steady and unsteady diffusion transport pro-
blems (diffusion convection PDEs, Navier Stokes problem, in Chapters 5,8).
Navier Stokes equations have a parabolic character because of the nonzero diffusion
term. However, depending on the specific situation, which from the physical point of
view means the rheological model of the blood, the flow rate, and the vessel size that
together define the Reynolds group, these equations are hyperbolic when they are
convection dominated, and parabolic when they are diffusion dominated. From the
mathematical point of view, the convection operator is hyperbolic, the diffusion oper-
ator is elliptic, and the time operator is parabolic.
Furthermore it is crucial to note that if the coefficients are variable then the PDE
type may vary locally, that is, elliptic, parabolic, or hyperbolic. If the PDE is nonlinear
then the type of problem to solve numerically may depend on the linearization tech-
nique that is utilized. These consequences may raise concerns in the selection of the
solver, as solvers are optimized for classes of problems described through algebraic sys-
tems of equations.

1.6 Numerical solutions to the mathematical models

Many numerical methods were proposed and successfully used to solve the PDEs pro-
duced by the mathematical models, and it is for the researcher to decide which of
them to use. Among them, FEM (Chapter 3: Computational Domains) has reached
the level of versatility and numerical accuracy where complex mathematical models
representing coupled physical processes and complex computational domains, such as
those constructed using CAD or image-based construction techniques, may be solved
successfully using available hardware and software resources. The mathematical models
are solved here (Chapters 4 8) using FEM in Galerkin formulation, which may be