Page 64 - Computational Modeling in Biomedical Engineering and Medical Physics

Shape and structure morphing of systems with internal flows 51

Living trees
The constructal principle initially explained the shape of the systems that optimizes the
volume-to-point flow resistance and lays the theoretical bridge to explain several
known empirical allometric laws for living tree structures. In biology, where the
viewer usually contemplates a tree construct of higher order, the size concerns global

and external quantities such as the total volume, V i , the total mass, M i 5 ρV i 1 2 φ ,
i
2
the cross-sectional area of the tree’s root, S i 5 π=4 D . Then the total interior area
i
of all tubes inside the constructal, A i , are the outcome of the optimal constructal struc-
ture. In nondimensional form their definitions are
π π
M i S i A i
^
^
i
^ M i 5 5 2 1 2 φ ; S i 5 5 2 ð 2i25Þ=3 ; A i 5 5 a i ;
ρL 3 0 i ρL 2 0 λ 2 L 0 2 λ ð2:6Þ
where ρ is the tissue density, λ 5 L 0 /D 0 is the shape factor of the smallest tube, and i
is the construct order.
Figs. 2.3 and 2.4 (Morega and Proca, 2004) plots these quantities versus each other
using the points representing each level of assembly, i, marked at the upper part of the
curves, to show their origin. They synthesize the observations made from outside
(measurements performed by the biologist), that is, from the perspective of an

Figure 2.3 The relation between the cross-sectional area of the root and the total mass of the
construct.