Page 41 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 27

Summing up, a bioheat transfer problem requires an analysis that is specific to that
anatomic ROI—vascularization, tissue, heat source, and surrounding, boundary condi-
tions—before selecting one model or another. In general most existing bioheat models
reside in the theory of porous media specifically the heat and fluid flow in a fluid-
saturated porous medium. Therefore to distinguish between the heat flow by larger ves-
sels from smaller vessels, represented through participating, by diffusion and convection,
heat transfer continuum media with homogenized macroscopic properties, we use mod-
els of general heat transfer through hemodynamic conduction and convection for the
larger vessels, and Brinkmann-type model for the heat transfer through the participating
tissue and embedded smaller vasculature.
This approach enables connecting the two coexisting types of flows, larger vessel
flow and porous media flow, and is used in Chapter 6: Magnetic drug targeting,
Chapter 7: Magnetic drug targeting, and Chapter 8: Hyperthermia and ablation
(Thermotherapy methods).

1.10 The computational domain
The system of concern is usually outlined, drawn by a qualitative analysis aimed to
explain “what and why is working” has to be materialized, to embody its abstract con-
cept of physical domain to which the related physics and the associated interactions
apply. In the end, it becomes the computational domain or simply the “geometry,”
drawn with the pencil on the paper, properly sized and presented such as to enable
the quantitative analysis of the underlying physics—mathematical model, analytical, or
numerical solution.
For engineered systems, the contemporary natural representation of the computa-
tional domain is the CAD produced “geometry.” Its complexity and level of detail is
then a tradeoff between the required physical detail and the available resources—hard-
ware, software, etc. More recently, medical image based construction techniques fused
geometries (CAD and construction based) are used to provide patient-tailored computa-
tional domains. Chapter 3: Computational Domains, provides some details and examples
of the design techniques that are commonly utilized in this book.
Numerical modeling for medical engineering is about systems and anatomic struc-
tures. The sketch of the computational domain has then to be inspired and must rep-
resent the insights of the design and evolution in nature. Design phenomena are not
covered through the existing laws of physics though (Bejan, 2000). Instead scientists
propose empirical or theoretical skeletons to build upon: general models, fractal geom-
etry, network theories, chaos theory (Bergé et al., 1984), optimality statements (opti-
mum, maximum, minimum) (Bejan and Zane, 2012), allometric scaling rules (power
laws) (Darwin, 1845; Huxley, 1972; D’Arcy, 1992), and so on. All these methods are
descriptive rather than predictive.