Page 47 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 33

Apparently although the species field was solved considering the mass convection
process, Eq. (1.39), the image provided by the isoconcentration representation is less
relevant in depicting the mass transfer mechanism, which is fully unveiled by the mass
function field lines.

1.12 A roadmap to a well-posed, direct problem and its solution

In view of the complexity posed by multiphysics problems of medical engineering,
some strategy has to be envisaged. In general the evolution of a system of finite size,
under internal and external constrains, is presented by the laws of Physics. In a direct
problem, the system hence its boundary, structure, properties, internal sources, bound-
ary constraints, and initial state are known input data, and the concern is to find the
states that the system “visits” during its evolution. A practical path to tackle the prob-
lem is presented sequentially, but the steps to follow may be reiterated to define, in
the end, a consistent problem with a realistic, robust solution.
The keyword in the first initial stage of the solution is the qualitative analysis. It
may provide order of magnitude solutions and hints useful in reducing the complexity
of the problem while representing the outlining physics. A qualitative sketch of the
system and its boundary is drawn, its structure (materials, sources), the internal and
external constraints are presented, and the underlying physical laws are reviewed. The
physical (thermodynamic), mathematical, and numerical concepts of system and
boundary have to be considered. Then an order of magnitude analysis is performed
with the aim to find the characteristic space and time scales, and to contain, “bracket”
the solution. The outcome of this analysis is a well-defined system, and (hopefully) a
simpler; however, consistent physical model that accounts for the underlying mechan-
isms, couplings that need to be observed.
In medical physics problems, it is more likely that multidisciplinary or “multiphy-
sics” models with some irreducible, occurring couplings, for example, properties,
sources, body forces. Different time scales (diffusion, transport, propagation) of differ-
ent phenomena (flow heat transfer, mass transfer EMF) and dynamics for the EMF
may intervene, for example, stationary, harmonic, pulse, and PWM.
For example, as discussed earlier, in analyzing the heating and heat transfer processes in
MW hyperthermia and ablation the local EMF heat source may be introduced through the
r.m.s. value of the electric field strengthinthe SARterm(Chapter8:Magnetic
Stimulation). This assumption has physical grounds—thesamepower level—and avoids the
simultaneous integration of a dynamic EMF problem and the transient HT problem, which
happen at considerably different time scales. Moreover if arterial hemodynamic contributes
to the HT then the characteristic pulsating flow may be replaced with an equivalent per
mass flow rate stationary flow because the blood flow rate happens faster than the HT rate
within the surrounding medium.