A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function

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Abstract

Previously, by assuming a viscous dominated flow in the boundary layer and an inertia dominated flow in the vessel core, a velocity profile function for a 1D-wave propagation model was derived. Because the time dependent shape of the velocity profile in this boundary layer model depends on the size of the inviscid core and the boundary layer, and thus on the Womersley number, it differs along the arterial tree. In this study we evaluated a lumped model for a vessel segment in which the element configuration is based on physical phenomena described by the boundary layer model and for which all parameters have a physically based quantitative value dependent on the Womersley number. The proposed electrical analog consists of a Womersley number dependent resistor and an inductor arranged in parallel, representing the flow impedance in respectively the vessel core and the boundary layer, in series with a second resistor. After incorporating a capacitor representing the vessel compliance in this rigid tube model, the element configuration resembles the configuration of the four-element windkessel model. For arbitrary Womersley numbers the relative impedance of Womersley theory is approximated with high accuracy. In the limits for small and large Womersley numbers the relative impedances of the proposed lumped model correspond exactly to Womersley theory.
Original languageEnglish
Pages (from-to)27-40
JournalMathematical Biosciences and Engineering
Volume6
Issue number1
DOIs
Publication statusPublished - 2009

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Blood Pressure
Arteries
Velocity Profile
Blood Flow
Electric Impedance
arteries
blood flow
blood pressure
Blood
Boundary Layer
Boundary layers
Vessel
impedance
Physical Phenomena
Impedance
Compliance
Resistors
Configuration
Model
Dependent

Cite this

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title = "A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function",
abstract = "Previously, by assuming a viscous dominated flow in the boundary layer and an inertia dominated flow in the vessel core, a velocity profile function for a 1D-wave propagation model was derived. Because the time dependent shape of the velocity profile in this boundary layer model depends on the size of the inviscid core and the boundary layer, and thus on the Womersley number, it differs along the arterial tree. In this study we evaluated a lumped model for a vessel segment in which the element configuration is based on physical phenomena described by the boundary layer model and for which all parameters have a physically based quantitative value dependent on the Womersley number. The proposed electrical analog consists of a Womersley number dependent resistor and an inductor arranged in parallel, representing the flow impedance in respectively the vessel core and the boundary layer, in series with a second resistor. After incorporating a capacitor representing the vessel compliance in this rigid tube model, the element configuration resembles the configuration of the four-element windkessel model. For arbitrary Womersley numbers the relative impedance of Womersley theory is approximated with high accuracy. In the limits for small and large Womersley numbers the relative impedances of the proposed lumped model correspond exactly to Womersley theory.",
author = "W. Huberts and E.M.H. Bosboom and {Vosse, van de}, F.N.",
year = "2009",
doi = "10.3934/mbe.2009.6.27",
language = "English",
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pages = "27--40",
journal = "Mathematical Biosciences and Engineering",
issn = "1547-1063",
publisher = "Arizona State University",
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TY - JOUR

T1 - A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function

AU - Huberts, W.

AU - Bosboom, E.M.H.

AU - Vosse, van de, F.N.

PY - 2009

Y1 - 2009

N2 - Previously, by assuming a viscous dominated flow in the boundary layer and an inertia dominated flow in the vessel core, a velocity profile function for a 1D-wave propagation model was derived. Because the time dependent shape of the velocity profile in this boundary layer model depends on the size of the inviscid core and the boundary layer, and thus on the Womersley number, it differs along the arterial tree. In this study we evaluated a lumped model for a vessel segment in which the element configuration is based on physical phenomena described by the boundary layer model and for which all parameters have a physically based quantitative value dependent on the Womersley number. The proposed electrical analog consists of a Womersley number dependent resistor and an inductor arranged in parallel, representing the flow impedance in respectively the vessel core and the boundary layer, in series with a second resistor. After incorporating a capacitor representing the vessel compliance in this rigid tube model, the element configuration resembles the configuration of the four-element windkessel model. For arbitrary Womersley numbers the relative impedance of Womersley theory is approximated with high accuracy. In the limits for small and large Womersley numbers the relative impedances of the proposed lumped model correspond exactly to Womersley theory.

AB - Previously, by assuming a viscous dominated flow in the boundary layer and an inertia dominated flow in the vessel core, a velocity profile function for a 1D-wave propagation model was derived. Because the time dependent shape of the velocity profile in this boundary layer model depends on the size of the inviscid core and the boundary layer, and thus on the Womersley number, it differs along the arterial tree. In this study we evaluated a lumped model for a vessel segment in which the element configuration is based on physical phenomena described by the boundary layer model and for which all parameters have a physically based quantitative value dependent on the Womersley number. The proposed electrical analog consists of a Womersley number dependent resistor and an inductor arranged in parallel, representing the flow impedance in respectively the vessel core and the boundary layer, in series with a second resistor. After incorporating a capacitor representing the vessel compliance in this rigid tube model, the element configuration resembles the configuration of the four-element windkessel model. For arbitrary Womersley numbers the relative impedance of Womersley theory is approximated with high accuracy. In the limits for small and large Womersley numbers the relative impedances of the proposed lumped model correspond exactly to Womersley theory.

U2 - 10.3934/mbe.2009.6.27

DO - 10.3934/mbe.2009.6.27

M3 - Article

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SP - 27

EP - 40

JO - Mathematical Biosciences and Engineering

JF - Mathematical Biosciences and Engineering

SN - 1547-1063

IS - 1

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