Axially distributed 2-region capillary-tissue exchange operator and analogous 2 compartment model.
One-dimensional convection-permeataion-diffusion- reaction model consisting of two concentric cylinders separated by a membrane. The advecting plasma region with volume Vp has flow Fp, no consumption, and axial diffusion (disperion) Dp. Units are physiological per gram of tissue so that a single unit can model a homogeeously perfused organ. Radial diffusion is assumed instantaneous (short radial distances). Exchange into a second surrounding non-flowing region is passive with conductance, PS, the Permeability capillary Surface area product. This interstitial fluid region, isf, of volume Visf, like the capillary, is axially distributed, and the gradients axially are dissipated by a concentration-independent axial diffusion or dispersion. Radial diffusion within this space is considered instantaneous, and consumption, Gisf, is first order. This model is used in multicapillary models as one of a set of units in parallel.Sangren and Sheppard (1954) give the analytical solution for D = 0. VERIFICATION TEST: Change the input function from the LagNormal cuver to a 1 second pulse input of 1 mM from 3 to 4 seconds. Then set Dp =0. Leave the PSg unchanged, but increase Gisf to 1000 so that everything entering the ISF is consumed and backflux from ISF to plasma goes to zero. These are the condtions under which the Crone-Renkin expression for extraction becomes true: PSg/Fp = 1 - ln (1 - E), where E is the fractional extraction between entrance and exit, and represents a unidirectional flux. Or, restated: E = 1 - exp(-Psg/Fp. The spatial profile Cp(x) at the peak of the pulse has the envelope: Cp(x) = exp((-PSg/Fp)*x/L).. For a constant infusion input Cin, the profile is Cp(x) = Cin*exp((-PSg/Fp)*x/L). Plot this. It should fit the peaks of the pulses. Check the Text output for the graph to see how many digits accuracy are obtained. Test different solvers.
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.
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Sangren WC and Sheppard CW. A mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment. Bull Math Biophys 15: 387-394, 1953 (This gives an analytic solution for the two-region DISTRIBUTED model.) Goresky CA, Ziegler WH, and Bach GG. Capillary exchange modeling: Barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970. (This gives another derivation of the analytical form, and uses the model in both single and multicapillary models.) Bassingthwaighte JB. A concurrent flow model for extraction during transcapillary passage. Circ Res 35: 483-503, 1974. (This gives numerical solutions, which are faster than the analytic solutions, and imbeds the model in an organ with tissue volums conserved, and with arteries and veins.) Guller B, Yipintsoi T, Orvis AL, and Bassingthwaighte JB. Myocardial sodium extraction at varied coronary flows in the dog: Estimation of capillary permeability by residue and outflow detection. Circ Res 37: 359-378, 1975. (Application to sodium exchange in the heart.)
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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.