170
Dynamics
of
the
Vascular
System
Thus, the optimal radius for a blood vessel when the minimum rate of
energy is required for steady flow is proportional to the 1/3 power of
flow,
Q.
5.4.2
Optimizing Branching Radii and
Angles
Now consider a bifurcation with a mother vessel with length
10
and radius
ro that branches into
two
daughter vessels with lengths and radii of 11, 12
and rl and r2, respectively, as shown in Fig. 5.4.2. It is assumed that the
vessels are lying in the same plane.
Fig.
5.4.2:
Schematic drawing
of
a bifurcating vascular branching junction.
A
mother
vessel branches into two daughter vessels. Lumen radius, length and bifurcation angles
are shown.
We have for the cost function that assumes optimum rate of energy
usage given by equation (5.4.1
5)
3k
2
2
p,
=
-(m0
zo
+
q2Z1
+
m;’)
The conservation of mass gives the equation of continuity of
Qo
=
Qi
+
Q2
(5.4.1
6)
(5.4.1
7)
which gives rise to the relation for the branching radii, from
(5.4.6):
