Simple Model of an axially distributed two-region capillary Blood-Tissue EXchange unit with consumption in interstitium
This capillary-tissue model consists of two "concentric" cylinders, a central capillary of volume Vcap with flow Fcap along the capilary of length L in which there is axial dispersion in the x-dimension. Radial diffusion is assumed instantaneous in both regions. The convective mean transit time is Vcap/Fcap = 3 seconds with the parameters above. The capillary length L is set at 1 mm, an approximation for cardiac capillaries. In the partial differential equation the boundary conditions at the capillary entrance, where x = x.min, are set so that diffusion in the upstream direction is balanced by the convection and no material is lost there from the capillary-tissue unit. Material is lost by convection into the outflow with concentration Cout. The conductance for exchange between the capillary and the ISF is purely passive, equivalent to diffusional exchange through the clefts between endothelial cells. Visf is the interstitial water space, in which the solubility is assumed to be the same as in the capillary plasma. Consumption, occuring only in the ISF, is by a first order reaction with rate constant Gisf. (Note that Fcap, PS, and Gisf all have the same units, those of flux per unit mass of tissue, ml/(g*min). The dispersion terms represent a combination of molecular diffusion and dispersion due to intravascular velocity profile, eddies, disturbed flow, branching, all causing some mixing. This model can be made into a stirred tank (compartmental) model by raising the dispersion coefficients to high values, e.g. 10^(-1) cm^2/sec, oblitering axial gradients to result in uniform concnetrations from end to end. Dcap = 1e-4 cm^2/sec is a good approximation for the observed physiological dispersion in capillaries.
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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W.C. Sangren and C.W. Sheppard. A mathematical derivation of the exchange of a labelled substance between a liquid flowing in a vessel and an external compartment. Bull Math BioPhys, 15, 387-394, 1953. C.A. Goresky, W.H. Ziegler, and G.G. Bach. Capillary exchange modeling: Barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970. J.B. Bassingthwaighte. A concurrent flow model for extraction during transcapillary passage. Circ Res 35:483-503, 1974. B. Guller, T. Yipintsoi, A.L. Orvis, and J.B. Bassingthwaighte. 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. C.P. Rose, C.A. Goresky, and G.G. Bach. The capillary and sarcolemmal barriers in the heart--an exploration of labelled water permeability. Circ Res 41: 515, 1977. J.B. Bassingthwaighte, C.Y. Wang, and I.S. Chan. Blood-tissue exchange via transport and transformation by endothelial cells. Circ. Res. 65:997-1020, 1989. Poulain CA, Finlayson BA, Bassingthwaighte JB.,Efficient numerical methods for nonlinear-facilitated transport and exchange in a blood-tissue exchange unit, Ann Biomed Eng. 1997 May-Jun;25(3):547-64.
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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.