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.

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, R_SA, and vessel compliance, C_SA, depend upon the average compartment pressure, P_SAmid = (P_SA - P_SC)/2, and the wall shear stress, _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.

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.