Vascular Branching
171
(5.4.18)
333
ro
=
rl
+
r2
which is again known as Murray’s law or the cube law for bifurcation.
over the mother vessel lumen area, we have:
Defining an area ratio as the sum of the daughter vessel lumen areas
2
r,2
+r2
r0
4
=
2
For an equi-bifurcation, or that r1=r2, we have
(5.4.19)
(5.4.20)
Substitute, we have, for the area ratio:
A,
=
1.26
(5.4.21)
When the angle of branching is involved, with half angles of
branching of
0
and
cp,
the optimum rate of energy is obtained when
(5.4.22)
22
2
ro
=
r,
cos8
+
r2
cosq
Extensive data of vascular branches have been obtained by several
investigators for vascular branches in different vascular beds (e.g. Li et
al., 1984; Schmidt-Shoenbein, 1986;Kassab et al., 1993; Kassab and
Fung, 1995; Zamir, 2000). Data from pig coronary arteries by Kassab et
al. (1993) show that Murray’s law works very well in both control and
hypertension hearts.
A
modified cost function that includes a metabolic
constant
k,
and takes into consideration the wall thickness of the vessel,
h
p,
=
(,)Q2
8771
+
k(m22)
+
kJ2mhZ)
m
(5.4.23)
This does not appear to differ significantly from Murray’s formulation.
