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Pulsatile blood flow, shear force, energy dissipation and Murray's Law

BACKGROUND: Murray's Law states that, when a parent blood vessel branches into daughter vessels, the cube of the radius of the parent vessel is equal to the sum of the cubes of the radii of daughter blood vessels. Murray derived this law by defining a cost function that is the sum of the energy...

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Autores principales: Painter, Page R, Edén, Patrik, Bengtsson, Hans-Uno
Formato: Texto
Lenguaje:English
Publicado: BioMed Central 2006
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1590016/
https://www.ncbi.nlm.nih.gov/pubmed/16923189
http://dx.doi.org/10.1186/1742-4682-3-31
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author Painter, Page R
Edén, Patrik
Bengtsson, Hans-Uno
author_facet Painter, Page R
Edén, Patrik
Bengtsson, Hans-Uno
author_sort Painter, Page R
collection PubMed
description BACKGROUND: Murray's Law states that, when a parent blood vessel branches into daughter vessels, the cube of the radius of the parent vessel is equal to the sum of the cubes of the radii of daughter blood vessels. Murray derived this law by defining a cost function that is the sum of the energy cost of the blood in a vessel and the energy cost of pumping blood through the vessel. The cost is minimized when vessel radii are consistent with Murray's Law. This law has also been derived from the hypothesis that the shear force of moving blood on the inner walls of vessels is constant throughout the vascular system. However, this derivation, like Murray's earlier derivation, is based on the assumption of constant blood flow. METHODS: To determine the implications of the constant shear force hypothesis and to extend Murray's energy cost minimization to the pulsatile arterial system, a model of pulsatile flow in an elastic tube is analyzed. A new and exact solution for flow velocity, blood flow rate and shear force is derived. RESULTS: For medium and small arteries with pulsatile flow, Murray's energy minimization leads to Murray's Law. Furthermore, the hypothesis that the maximum shear force during the cycle of pulsatile flow is constant throughout the arterial system implies that Murray's Law is approximately true. The approximation is good for all but the largest vessels (aorta and its major branches) of the arterial system. CONCLUSION: A cellular mechanism that senses shear force at the inner wall of a blood vessel and triggers remodeling that increases the circumference of the wall when a shear force threshold is exceeded would result in the observed scaling of vessel radii described by Murray's Law.
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spelling pubmed-15900162006-10-05 Pulsatile blood flow, shear force, energy dissipation and Murray's Law Painter, Page R Edén, Patrik Bengtsson, Hans-Uno Theor Biol Med Model Research BACKGROUND: Murray's Law states that, when a parent blood vessel branches into daughter vessels, the cube of the radius of the parent vessel is equal to the sum of the cubes of the radii of daughter blood vessels. Murray derived this law by defining a cost function that is the sum of the energy cost of the blood in a vessel and the energy cost of pumping blood through the vessel. The cost is minimized when vessel radii are consistent with Murray's Law. This law has also been derived from the hypothesis that the shear force of moving blood on the inner walls of vessels is constant throughout the vascular system. However, this derivation, like Murray's earlier derivation, is based on the assumption of constant blood flow. METHODS: To determine the implications of the constant shear force hypothesis and to extend Murray's energy cost minimization to the pulsatile arterial system, a model of pulsatile flow in an elastic tube is analyzed. A new and exact solution for flow velocity, blood flow rate and shear force is derived. RESULTS: For medium and small arteries with pulsatile flow, Murray's energy minimization leads to Murray's Law. Furthermore, the hypothesis that the maximum shear force during the cycle of pulsatile flow is constant throughout the arterial system implies that Murray's Law is approximately true. The approximation is good for all but the largest vessels (aorta and its major branches) of the arterial system. CONCLUSION: A cellular mechanism that senses shear force at the inner wall of a blood vessel and triggers remodeling that increases the circumference of the wall when a shear force threshold is exceeded would result in the observed scaling of vessel radii described by Murray's Law. BioMed Central 2006-08-21 /pmc/articles/PMC1590016/ /pubmed/16923189 http://dx.doi.org/10.1186/1742-4682-3-31 Text en Copyright © 2006 Painter et al; licensee BioMed Central Ltd. http://creativecommons.org/licenses/by/2.0 This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( (http://creativecommons.org/licenses/by/2.0) ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
spellingShingle Research
Painter, Page R
Edén, Patrik
Bengtsson, Hans-Uno
Pulsatile blood flow, shear force, energy dissipation and Murray's Law
title Pulsatile blood flow, shear force, energy dissipation and Murray's Law
title_full Pulsatile blood flow, shear force, energy dissipation and Murray's Law
title_fullStr Pulsatile blood flow, shear force, energy dissipation and Murray's Law
title_full_unstemmed Pulsatile blood flow, shear force, energy dissipation and Murray's Law
title_short Pulsatile blood flow, shear force, energy dissipation and Murray's Law
title_sort pulsatile blood flow, shear force, energy dissipation and murray's law
topic Research
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1590016/
https://www.ncbi.nlm.nih.gov/pubmed/16923189
http://dx.doi.org/10.1186/1742-4682-3-31
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