Page 48 - Computational Modeling in Biomedical Engineering and Medical Physics

34 Computational Modeling in Biomedical Engineering and Medical Physics

Within the limits of the continuum media hypothesis, small to large space scales phe-
nomena may act simultaneously. For instance, the hemodynamic flow occurs through a
complex network of interconnected arterial, capillary, and venous vessel trees whose sizes
vary considerably, under different flow mechanisms, and with different local, rheological
characteristics for the blood. Homogenization techniques are then used to avoid the diffi-
culty of analyzing small and large, slow and fast flows altogether, while keeping the “big
picture” of the underlying physics: the small size vessels (slow flow) and the embedding
tissue are seen as an equivalent continuous medium that has their combined functionality.
For instance, in heat transfer analysis, the bioheat equation (Pennes, 1948) presents a per-
fused tissue as an equivalent, continuum medium that acts as a heat sink/source that mod-
els the convective contribution to the HT of capillaries and small size vessels.
In selecting the leading phenomenon in a model with coupled physics, a qualita-
tive power level analysis may be useful. For instance, in modeling the transcutaneous
electrical stimulation, the external power source exceeds that of the electrical activity
of the heart. It is then reasonable to neglect the latter, and model the body as a vol-
ume conductor to find the electrical current density distribution inside the thorax
(Chapter 4: Electrical Activity of the Heart).
Moreover an order of magnitude analysis of the physical model may provide the
arguments to simplify it, by neglecting the small and or slow (in the order of magni-
tude sense) terms, which provide negligibly small information while introducing, per-
haps, undesirable numerical complexities.
These are only some considerations. Other valuable conclusions that may be useful
in simplifying the numerical problem to solve may be drawn through the qualitative
analysis as will be see in the Chapters that follow.
Next the route marked by the following steps ought to be traveled mark the route
to solve the model:
1. Construct the system and its boundary, called the computational domain in what
follows. CAD tools, or medical image based construction techniques, or fractal
algorithms, or a combination of these may be used. When using CAD or fractal
geometry, the resulting computational domain has to represent realistically the
physical system under analysis.
2. Build the physical model:
a. list the laws of physics that describe the system state, evolution, the material
laws that apply and the internal sources;
b. define the boundary conditions, that is, the interactions with the surroundings
(heat transfer and, or mechanical work equivalents); and
c. define the initial conditions, that is, the initial state of the system.
3. Build the mathematical model out of the physical model. The set of equations that
make the physical model is reduce to a smaller set of higher order PDE(s) and/or
ODE(s).