568
13 Modeling Biological Systems
This system of ten equations has an analytic solution, but the algebra is
complicated, and the solution was not published until 2012! 5 In our present case,
we are focused on finding the bifurcation angles. For this, we can use the argument
of Murray. 6 First, take the vessel radii to be optimal according to Eq. (13.8). Then
the cost function for the vessels in Fig. 13.5 will be proportional to the three-term
sum: f ∝
i r 2
i x i . Let P be the generalized Fermat Point. Now add a point P a
small distance δ along the line AP , and let θ AB be the angle between AP and P B,
and let θ AC be the angle between AP and P C. Since P minimized the cost function,
the variation in the cost for the total length as the sum of lengths from the new point
P to each vertex should vanish, to first order in δ. This reads
r
2
A δ = r
2
B δ cos θ AB + r
2
C δ cos θ AC .
(13.24)
Similarly, making variation in cost under infinitesimal extensions of the other two
line segments BP and CP will give
r
2
B δ = r
2
A δ cos θ AB − r
2
C δ cos(θ AB + θ AC )
(13.25)
and
r
2
C δ = r
2
A δ cos θ AB − r
2
B δ cos(θ AB + θ AC ) .
(13.26)
Solving for the cosine of the two angles gives
cos θ AB =
r 4
A + r 4
B − r 4
C
2r 2
A r 2
B
,
(13.27)
cos θ AC =
r 4
A + r 4
C − r 4
B
2r 2
A r 2
C
,
(13.28)
and the cosine of the angle between the vessels B and C coming from vessel A
becomes
cos(θ AB + θ AC ) =
r 4
A − r 4
B − r 4
C
2r 2
B r 2
C
.
(13.29)
With Eq. (13.8), this becomes
cos(θ BC ) =
(1 + (r C /r B ) 3 ) 4/3 − 1 − (r C /r B ) 4
2(r C /r B ) 2
.
(13.30)
5 Alexei Yu Uteshev, Analytic Solution for the Generalized Fermat-Torricelli Problem, Comp
Geometry 121:4, 318–331 (2014).
6 C.D. Murray, The Physiological Principle of Minimum Work Applied to the Angle of Branching
of Arteries, Jour Gen Physiol 9, 835–841 (1926).
13 Modeling Biological Systems
This system of ten equations has an analytic solution, but the algebra is
complicated, and the solution was not published until 2012! 5 In our present case,
we are focused on finding the bifurcation angles. For this, we can use the argument
of Murray. 6 First, take the vessel radii to be optimal according to Eq. (13.8). Then
the cost function for the vessels in Fig. 13.5 will be proportional to the three-term
sum: f ∝
i r 2
i x i . Let P be the generalized Fermat Point. Now add a point P a
small distance δ along the line AP , and let θ AB be the angle between AP and P B,
and let θ AC be the angle between AP and P C. Since P minimized the cost function,
the variation in the cost for the total length as the sum of lengths from the new point
P to each vertex should vanish, to first order in δ. This reads
r
2
A δ = r
2
B δ cos θ AB + r
2
C δ cos θ AC .
(13.24)
Similarly, making variation in cost under infinitesimal extensions of the other two
line segments BP and CP will give
r
2
B δ = r
2
A δ cos θ AB − r
2
C δ cos(θ AB + θ AC )
(13.25)
and
r
2
C δ = r
2
A δ cos θ AB − r
2
B δ cos(θ AB + θ AC ) .
(13.26)
Solving for the cosine of the two angles gives
cos θ AB =
r 4
A + r 4
B − r 4
C
2r 2
A r 2
B
,
(13.27)
cos θ AC =
r 4
A + r 4
C − r 4
B
2r 2
A r 2
C
,
(13.28)
and the cosine of the angle between the vessels B and C coming from vessel A
becomes
cos(θ AB + θ AC ) =
r 4
A − r 4
B − r 4
C
2r 2
B r 2
C
.
(13.29)
With Eq. (13.8), this becomes
cos(θ BC ) =
(1 + (r C /r B ) 3 ) 4/3 − 1 − (r C /r B ) 4
2(r C /r B ) 2
.
(13.30)
5 Alexei Yu Uteshev, Analytic Solution for the Generalized Fermat-Torricelli Problem, Comp
Geometry 121:4, 318–331 (2014).
6 C.D. Murray, The Physiological Principle of Minimum Work Applied to the Angle of Branching
of Arteries, Jour Gen Physiol 9, 835–841 (1926).