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

Painter, Page R ; Edén, Patrik LU and Bengtsson, Hans-Uno LU (2006) In Theoretical Biology Medical Modelling 3(31).
Abstract
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Theoretical Biology Medical Modelling
volume
3
issue
31
publisher
BioMed Central (BMC)
external identifiers
  • wos:000208054800031
  • scopus:33749476025
  • pmid:16923189
ISSN
1742-4682
DOI
10.1186/1742-4682-3-31
language
English
LU publication?
yes
id
11319dfb-fedc-44fa-9de9-b2680edf939c (old id 168533)
alternative location
http://www.tbiomed.com/content/pdf/1742-4682-3-31.pdf
date added to LUP
2016-04-01 15:46:27
date last changed
2021-10-06 05:04:51
@article{11319dfb-fedc-44fa-9de9-b2680edf939c,
  abstract     = {Background<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
Methods<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
Results<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
Conclusion<br/><br>
<br/><br>
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.},
  author       = {Painter, Page R and Edén, Patrik and Bengtsson, Hans-Uno},
  issn         = {1742-4682},
  language     = {eng},
  number       = {31},
  publisher    = {BioMed Central (BMC)},
  series       = {Theoretical Biology Medical Modelling},
  title        = {Pulsatile blood flow, shear force, energy dissipation and Murray's Law},
  url          = {http://dx.doi.org/10.1186/1742-4682-3-31},
  doi          = {10.1186/1742-4682-3-31},
  volume       = {3},
  year         = {2006},
}