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On deriving Murray's law from constrained minimization of flow resistance
1Georgian Court University, Lakewood, New Jersey, USA.
Journal of Theoretical Biology
|December 28, 2020
Summary
Sherman's analysis of Murray's law is flawed. Bifurcation (two daughter vessels) minimizes total flow resistance, explaining its prevalence in biological systems.
Area of Science:
- Physiology
- Biophysics
- Fluid Dynamics
Background:
- Murray's law relates parent and daughter vessel radii, derived from minimizing blood flow costs.
- Sherman's widely cited derivation claims minimal resistance for any number of daughter vessels satisfying Murray's law.
Purpose of the Study:
- To re-evaluate Sherman's analysis of Murray's law.
- To determine the optimal branching pattern for minimizing total flow resistance in vascular networks.
Main Methods:
- Mathematical analysis of fluid resistance in branching vessels.
- Comparison of vessel radii satisfying Murray's law with those yielding minimal resistance.
Main Results:
- Sherman's analysis is flawed; many radius sets satisfying Murray's law do not minimize resistance.
- Minimal total flow resistance increases with the number of daughter vessels (N) for N>=1.
- Bifurcation (N=2) achieves minimal total flow resistance.
Conclusions:
- Bifurcation is the optimal branching pattern for minimizing vascular resistance.
- This finding explains the biological prevalence of bifurcations over higher-order branchings.
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