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Lumped_Param_Circ_wReg

A closed loop circulatory model with varying elastance heart, pulmonary, coronary and systemic circulatory loops and actively regulating arterioles in the systemic circulation responding to pressure and shear stress on the vessel walls.

Model number: 0161

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Description

 This model is identical to the lumped parameter circulation model except for the
insertion of an arteriolar vessel compartment which varies resistance and compliance
actively based on the intraluminal pressure and vessel wall shear stress. The reason
for the incorporation of a vasoactive arteriolar compartment is that much of the
regulation of blood flow throughout the microvasculature can be attributed to regulatory
responses in the arterioles. Two known local regulatory responses are that to changes
in pressure (myogenic response) and that to changes in flow (shear-dependent response)
and both are characterized here. There are other local and global responses such as
conducted response, countercurrent flow response and sympathetic responses which are
not characterized in this model. The ultimate goal of incorporating a regulatory
response in a circulatory model is to describe the active response of the
microvasculature to changes in blood volume and systemic pressure which occur in
hemorrhagic injury.



Below we compare the beat to beat pressure response of the model with (top) and without (bottom) vasoactive response:
Figure: Where subcripts lvc, aopc, aodc, sap, sa, sc, sv and vcc stand for left ventricle, proximal aorta, distal aorta, systemic arteries, systemic arterioles, systemic capillaries, systemic veins and vena cava respectively. The pulse shape of the arterioles with vasoactive response is considerably damped because the vessels are instantaneously reacting to their local pressure and shear conditions. In actuality the local response can be divided into an active and passive component with two distinct time scales. The passive portion of the response has a time constant on the order of 1 second while the active vascular smooth muscle response of the vessel has a time constant on the order of 1 minute. Therefore if these two responses are separated and allowed to operate with their respective time characteristics, a more faithful representation of the beat to beat pulse pressue response will result. This model should still faithfully respond to a long time scale adjustment in systemic pressure.

Equations

In a lumped parameter model each compartment of the systemic circulation is described by the flow analog of an RC circuit similar to what is shown below. In the figure flow is F, pressure is P, total resistance is R, individual vessel diameter is D, compliance is C and total volume is V. In the vasoactive arteriolar compartment the resistance to flow, RSA, and vessel compliance, CSA, depend upon the average compartment pressure, PSAmid = (PSA - PSC)/2, and the wall shear stress, $\tau$SA. If we are given an initial blood volume and a typical resistance of all the arterioles in this compartment we are able to determine the total number of vessels in the compartment and their effective length explicitly. All the arterioles in the compartment are identical and the response to pressure and shear stress is determined from experimental data from Sun et al. (see reference below.)

A poster, presented at the Mathematical Biosciences Institute at Ohio State, describing the equations used to determine pressure, flow, total resistance to flow, individual arteriolar diameter, compliance and total arteriolar volume in the arteriolar compartment can be downloaded here (PDF file).

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

References


Brandenburg, Robert O. Cardiology: fundamentals and practice.
Yearbook medical publishers, 1987. p. 49.

Chung DC, Niranjan SC, Clark Jr. JW, Bidani A, Johnston WE,
Zwischenberger JB, Traber DL. A dynamic model of ventricular
interaction and pericardial influence.  Am J Phsiol Heart
Circ Physiol. 272: H2942-H2962, 1997.

Golden JF, Clark JW, Stevens PM. Mathematical Modeling of
Pulmonary Airway Dynamics.  IEEE Transactions on Biomedical
Engineering. 20(6): 397-404, 1973.

Heldt T, Shim EB, Kamm RD, Mark RG. Computational modeling of
cardiovascular response to orthostatic stress.  Journal of
Applied Physiology. 92: 1239-1254, 2002.

Lu K, Clark JW, Ghorbel FH, Ware DL, Bidani A.
A human cardiopulmonary system model applied to the analysis
of the Valsalva maneuver.  Am J Physiol Heart Circ Physiol.
281: H2661-H2679, 2001.

Rideout VC. Mathematical computer modeling of physiological
systems. Englewood Cliffs, NJ: Prentice Hall, 1991, 261 pp.

Sun Y, Beshara M, Lucariello RJ, Chiaramida SA.  A
comprehensive model for right-left heart interaction under the
influence of pericardium and baroreflex.  Am J Physiol Heart
Circ Physiol. 272: H1499-H1515, 1997.

Zinemanas D, Beyar R, Sideman S.  Relating mechanics, blood
flow and mass transport in the cardiac muscle.  Int. J. Heat
Mass Transfer.  37(suppl. 1) 191-205, 1994.



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Key Terms

Closed loop circulatory model, Vasoactive response, Pressure, Shear stress, Local regulation

Model History

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Posted by: BEJ

Acknowledgements

Please cite www.physiome.org in any publication for which this software is used and send an email with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.
Or send a copy to:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.