Page 42 - Computational Modeling in Biomedical Engineering and Medical Physics

28 Computational Modeling in Biomedical Engineering and Medical Physics

Allometric laws, fractal geometry, and constructal law
The scaling method and the scaling rules were introduced and used here to provide
for order of magnitude, predictive relations that are useful in constructing consistent
and computationally efficient physical models. The scaling rules have much more
potential though as they reflect the underlying physical principles and the generic fea-
tures of the system: size, structure, and properties. In fact they may define the skeleton,
predict the quantities that are related to each other, for quantitative assertions to be
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found through experimental statistics. Along this path comes the allometry or the sta-
tistical shape analysis.
Allometry is also used in biology, where it studies the relationship of the body size
to its shape, anatomy, physiology, and finally behavior. Examples of allometric rules
α
are (Snell, 1982) body size mass law, in power law form, Y 5 Y 0 UM ,[Y is a depen-
dent parameter, Y 0 an integration constant, M the body mass, α the scaling exponent
(α . 0 positive allometry, α , 0 negative, allometry, α 5 1 isometry)], or in
log arithmic form, log Y 5 α log M 1 log Y 0 ;(Murray, 1926a,b) work uptake law for
transfer processes in mass conservative networks [the cube of vessels diameter at each
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generation is preserved, i.e., ΣD Bconstant, in animal circulatory systems, plant vas-
cular systems, ecosystems (e.g., forests), intracellular networks, etc., but also porous
materials, a.s.o.]; Huo Kassab model shows the Murray’s law exponent is equal to
7/3 (rather than 3) in coronary branching (Huo and Kassab, 2012; Kleiber, 1932) met-
b
abolic rates law for of mammals and birds, B ~ M [B is the metabolic rate, M is the
body mass, b is an allometric exponent (3/4 metabolic, 1/4 heart rate, 1/4 life span,
3/8 aorta/tree trunk diameters, 1/4 genome length, 3/4 population densities in for-
ests)], to name a few.
Allometric relationships between two measured quantities, e.g., mass and flow rate,
are often expressed as power law equations. However, it should be noted that the
existence per se of the power law does not indicate that the object is necessarily a
fractal.
The geometric and functional complexity of the tree structures in the human
body—hemodynamic system, respiratory tract, renal system, etc.—raises significant dif-
ficulties when passing from mathematical modeling to numerical simulation, and
Euclidean geometry that applies to smooth and regular shapes of integer geometric
dimension—zero for a point, one for a line, two for a plane, and three for a vol-
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ume—fails to resolve such problems. Fractal geometry was then introduced to
describe such “mathematical monsters” (Mandelbrot, 1975), self-similar objects that
have the same details in different scales, of noninteger geometric, fractal dimension
Order in Chaos (2013) and Bergé et al. (1984).

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Greek: allos means different and metrie means to measure.
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Latin: fract means broken.