Page 57 - Computational Modeling in Biomedical Engineering and Medical Physics

44 Computational Modeling in Biomedical Engineering and Medical Physics

A tree is not a net (mesh) and there are no loops in its structure, such that nodes
are connected through unique branches (unique paths), which render a certain tree
topologically equivalent with other similar trees. They may be the most difficult to
describe but when we see it we recognize it, and call it tree. Fractal geometry seems
dedicated to their representation as “mathematical monsters” (Mandelbrot, 1975,
2020; Falconer, 1990; Fractal, 2020).
Trees are seen everywhere and at all scales, in trees, plants, leaves, roots, lungs, vas-
1
cularized tissues, circulatory system, neuronal dendrites, bacterial colonies, electrical
discharges, cracks, hydrographic basins, deltas, and urban growths, etc. (Bejan, 2000a,
b, 2016; Bejan, Lorente, 2010). They are flow systems that connect a node, called
root, with an irrigated territory (an infinity of points) of finite size.
It may be inferred that if a single principle may explain all these forms (tree, round, slice)
then that principle acts everywhere. It may fill-in the gap between physics and biology,
between two ways of thinking, reasoning, two different perspectives of perceiving the sur-
rounding reality.
The natural macroscopic structure occurs in space and it evolves in time. In physi-
ology, it is known that the heartbeats and the respiratory frequency are related
(Petrescu et al., 2018), and specific to each animal. Their values decrease for increasing
bodies, and allometric relations (Chapter 1: Physical, Mathematical, and Numerical
Modeling) document such connections empirically. Yet again, which is that principle
that generates structure and shape in such a vast and diverse spectrum? Does it exist?
In the animate realm, the breathing frequencies, the pulse rate, the river morphol-
ogy, etc., their recurrent geometric or time particularities have been measured in detail
and correlated successfully (Murray, 1926a,b; Schmidt-Nielsen, 1972; Peters, 1983;
Huo and Kassab, 2012). However, these accurate and valuable correlations offer no
hint on what (if any) physical law they might epitomize.
More recently, arborescent macroscopic organizations have been quantified mathemat-
ically with fractal geometry methods, which propose iterative constructs, which mimic
shapes sized in abstract spaces of fractional order, but resembling natural tree structures,
when represented in the natural (2D or 3D) space representation (Mandelbrot, 1975,
2020; Falconer, 1990; Cheng and Huang, 2003; Order in Chaos, 2013; Uahabi and
Atounti, 2015). This approach is descriptive rather than predictive—it has no stemming
physics skeleton. However, quoting “future progress depends on establishing a basis much
more consistent on which the geometrical organization is deduced out of the mechanism
which is producing it” (Kadanoff, 1986; Bejan, 2000a,b).
The Constructal law answers precisely this desideratum and it spells out the principle
that explains and relates the system shape with its morphology dynamics (Bejan,
2000a,b). It is a completely deterministic principle that allows anticipating, predicting

1
dendron (Δενδρoν) means tree in Greek.