Comp2ode is a model which describes the passage of a single solute through a simple organ. Although not as fully developed as the models used by the NSR for scientific research, the model does have some elements in common with these more complex models. Flow entering the organ passes through three regions in series: arteries, capillaries, and veins. The arteries and veins are modeled as vascular operators, while the capillary and surrounding tissues are modeled as two well-stirred compartments.
Vascular operators simulate the purely convective portions of the circulatory system by adding time delay and dispersion to an input function. The mean transit time for material passing through the vascular operator is given by , where V is the vessel volume and F is the flow. In the absence of any dispersion, any material entering the vessel at time t leaves the vessel at . This is the case in our first sample simulation. Load and run the parameter set vasop.par and bring up plot area 1 (accessed through the Results menu). The input function, C in , is set to a lagged-normal density function which simulates a brief injection of the solute, as would be used in a multiple-indicator dilution experiment. The characteristics of the input function can be set by choosing C in from the parameters menu of the XSIM interface. The output function C a is simply C in with an added time delay of 0.05 mL divided by 1 mL min -1 , or 3 seconds. You should be able to modify the transit time by appropriate changes in either V a or F. (Please see the NSR software demo page to learn how to run XSIM models over the internet.)
In real vessels, a number of factors including molecular diffusion, blood flow heterogeneity, radial variations in flow velocity, and red cell mixing can cause material entering the vessel to assume a range of possible transit times. The relative dispersion, RD, is defined as the standard deviation of transit times divided by the mean transit time: . This parameter controls the level of mixing the vessel induces. Select run config from the Model menu, click the "Activate" button for the inner loop, and rerun the model. You will now see three additional output curves representing three different levels of mixing as RD varies from 0.0 to 0.48. (See King, R.D., A. Deussen, G.M. Raymond, and J.B. Bassingthwaighte. A vascular transport operator. Am. J. Physiol. 265: H2196-H2208, 1993 for more information on vascular operators.)
The blood-tissue exchange region is modeled as an exchange between two well-stirred compartments representing the plasma and interstitial spaces. A pair of equations derived from the principle of conservation of mass describe concentrations in the plasma and interstitium.
The left hand side of the equation is the rate of change of material (the concentration times the volume) in the plasma compartment; the model assumes that V p is constant with time. The first term in the right hand side of this equation represents the convective transport of material into and out of the compartment, and the second term is the exchange flux with the interstitium.
The equation for plasma concentration is of similar form. Note that the terms on the right hand side are duplicated in the equation for plasma concentration with a sign change, ensuring conservation of mass is preserved.
Typical Parameter Values
To run a model simulation using the default parameters shown in table 1, load the parameter file comp2ode.par . Plot area 1 now shows the concentration vs. time curves at a number of locations in the organ, shown on the model's main window. The rate of molecular diffusion across the capillary wall is modeled as a concentration difference between compartments multiplied by a permeability-surface area product, PS. The higher the PS, the more rapid is the mixing between interstitium and plasma; increasing PS to 10 results in the interstitial and plasma concentration-time curves becoming almost identical as the interstitium and plasma essentially become a single compartment. Conversely, with low PS values (try 0.1) virtually no material is exchanged and the plasma just acts as a single compartment mixing space.
A little experimentation will show that it is the PS/F ratio rather than the absolute values of these parameters which sets the relative importance of the two terms in the expression for C p , and consequently determines the magnitude of the exchange flux. In FandPS.par , F and PS are both doubled in the second model loop. The volumes of the compartments are also doubled to keep the transit times the same. In this case, the two loops produce identical outputs.
More realistic models of transport in organs are available from NSR that include additional regions, such as parenchymal, endothelial, and red blood cells, multiple solutes, specialized transporters, sites for specific and non-specific binding, and reactions among different species. After experimenting with comp2ode you should be ready to pick up this added complexity.