13.3 Optimality Conditions
569
Two Interesting Cases If the two arteries B and C have the same radii, then they
split off from A at an angle of arccos(2 2/3 − 1) ≈ 75 ◦ . If instead r C << r B , then
the angle between vessel A and B is well approximated by (2/3)(r C /r B ), and that
between A and C can be approximated by 90 ◦ − (1/
√
2)(r C /r B ) 2 . So a tiny artery
leaves the supply artery at an angle close to 90 ◦ , and the larger diameter exiting
artery leaves the supply artery at close to 0 ◦ .
Murray found that a comparison of the optimal angles predicted are in close
agreement with actual bifurcation angles measured in cat tissue. This shows that
the cost to supply blood to tissue has been optimized, and that a simple model
for minimizing of power needs predicts the radii and bifurcation angles of arteries.
Darwin would explain that individuals with less than optimal systems under given
environmental pressures have a lower chance of survival than those with optimal
systems. After a number of generations, the DNA coding for optimal operations
dominate the living population.
13.3.3 Optimized Aortic Diameter
The formula for the optimal radius for an artery, Eq. (13.8), works for most arteries,
but fails for the aorta, giving a value of 0.4 cm instead of about 1.5 cm. 7 In the flow
of blood from the heart, the possibility of turbulence in our aorta plays a significant
role. Turbulence produces a back pressure, forcing the heart to work harder. Laminar
flow will turn to turbulent flow in a straight pipe when the Reynolds number, R e =
ρvr/η exceed about 2100, but in the strongly curved upper aorta, turbulence starts
for R e above about 1000. The speed of flow, v, is inversely proportional to the
square of the aorta radius r: v = i/(πr 2 ), where the current i is the cardiac output.
With ρ/η ≈ 37 cm 2 /s, i ≈ 100 cm 3 /s, and R e ≈ 1000, the minimum radius to
have no turbulence is r ≥ 1.3 cm. As the speed in the aorta is actually varying, one
should consider the predicted value for the radius in surprisingly good agreement
with typical measured values.
13.3.4 The Pattern of Seeds on a Sunflower
The arrangement of seeds on a sunflower are in a spiral pattern which minimizes the
area taken up by the seeds. The minimization of space makes the number of spirals
on a given flower be a Fibonacci number. Alan Newell and Matt Pennybacker (2013)
have shown 8 that the growth of seeds on a sunflower can be predicted by a dynamical
7 See Rosen, ibid.
8 Alan C. Newell and Matthew Pennybacker, Fibonacci patterns: common or rare? Symposium on
Understanding Common Aspects of Extreme Events in Fluids, Procedia IUTAM 9, 86–109 (2013).
569
Two Interesting Cases If the two arteries B and C have the same radii, then they
split off from A at an angle of arccos(2 2/3 − 1) ≈ 75 ◦ . If instead r C << r B , then
the angle between vessel A and B is well approximated by (2/3)(r C /r B ), and that
between A and C can be approximated by 90 ◦ − (1/
√
2)(r C /r B ) 2 . So a tiny artery
leaves the supply artery at an angle close to 90 ◦ , and the larger diameter exiting
artery leaves the supply artery at close to 0 ◦ .
Murray found that a comparison of the optimal angles predicted are in close
agreement with actual bifurcation angles measured in cat tissue. This shows that
the cost to supply blood to tissue has been optimized, and that a simple model
for minimizing of power needs predicts the radii and bifurcation angles of arteries.
Darwin would explain that individuals with less than optimal systems under given
environmental pressures have a lower chance of survival than those with optimal
systems. After a number of generations, the DNA coding for optimal operations
dominate the living population.
13.3.3 Optimized Aortic Diameter
The formula for the optimal radius for an artery, Eq. (13.8), works for most arteries,
but fails for the aorta, giving a value of 0.4 cm instead of about 1.5 cm. 7 In the flow
of blood from the heart, the possibility of turbulence in our aorta plays a significant
role. Turbulence produces a back pressure, forcing the heart to work harder. Laminar
flow will turn to turbulent flow in a straight pipe when the Reynolds number, R e =
ρvr/η exceed about 2100, but in the strongly curved upper aorta, turbulence starts
for R e above about 1000. The speed of flow, v, is inversely proportional to the
square of the aorta radius r: v = i/(πr 2 ), where the current i is the cardiac output.
With ρ/η ≈ 37 cm 2 /s, i ≈ 100 cm 3 /s, and R e ≈ 1000, the minimum radius to
have no turbulence is r ≥ 1.3 cm. As the speed in the aorta is actually varying, one
should consider the predicted value for the radius in surprisingly good agreement
with typical measured values.
13.3.4 The Pattern of Seeds on a Sunflower
The arrangement of seeds on a sunflower are in a spiral pattern which minimizes the
area taken up by the seeds. The minimization of space makes the number of spirals
on a given flower be a Fibonacci number. Alan Newell and Matt Pennybacker (2013)
have shown 8 that the growth of seeds on a sunflower can be predicted by a dynamical
7 See Rosen, ibid.
8 Alan C. Newell and Matthew Pennybacker, Fibonacci patterns: common or rare? Symposium on
Understanding Common Aspects of Extreme Events in Fluids, Procedia IUTAM 9, 86–109 (2013).