Pulsatile blood flow, shear force, energy dissipation and Murray's Law
(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:
https://lup.lub.lu.se/record/168533
 author
 Painter, Page R ; Edén, Patrik ^{LU} and Bengtsson, HansUno ^{LU}
 organization
 publishing date
 2006
 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
 17424682
 DOI
 10.1186/17424682331
 language
 English
 LU publication?
 yes
 id
 11319dfbfedc44fa9de9b2680edf939c (old id 168533)
 alternative location
 http://www.tbiomed.com/content/pdf/17424682331.pdf
 date added to LUP
 20160401 15:46:27
 date last changed
 20211006 05:04:51
@article{11319dfbfedc44fa9de9b2680edf939c, 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, HansUno}, issn = {17424682}, 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/10.1186/17424682331}, doi = {10.1186/17424682331}, volume = {3}, year = {2006}, }