Flow with axial dispersion through a one-region pipe of uniform cross-section. Statistics on inflow and outflow concentration curves */
Further reading: Distributed Blood Tissue Exchange Models Explained
Description
The partial differential equation models flow into, through and out of a pipe with plug flow and axial dispersion (diffusion) along the x-axis and instantaneous radial dispersion so that concentration is uniform across the cross-section at each x-position. Consumption,Gp, equivalent to loss by a first order reaction or loss by permeation is a uniform fraction per unit time along the pipe. (This can be modified by making G a function of concentration, Gp(Cp) or of position, Gp(x).) Flow is constant, as are all the other parameters.The boundary conditions are (1) At the inflow, the diffusion coefficient, Dp, cm^2/s, times the spatial gradient in concentration, dC/dx, balances the difference between the inflow concentration and the concentration Cp just inside; (2) At the outflow, the gradient dC/dx is set to zero, as if reflecting from an impermeable surface, so that mass is lost into the outflow only by flow, Cout = Cp(x=L,t). LIMITATIONS: This model cannot approximate Newtonian parabolic flow, where the response to a flow-proportiaonal cross-sectional pulse labeling at the inflow would give a sharp upstroke and peak at 1/2 the mean transit time and then, in the absence of axial dispersion, diminish in proportion to 1/t^2. See Gonzalez-Fernandez (1962) on this point.
Equations
Differential Equations
Left Boundary Conditions
Right Boundary Conditions
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Initial Conditions
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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. Gonzalez-Fernandez JM. Theory of the measurement of the dispersion of an indicator in indicator-dilution studies. Circ Res 10: 409-428, 1962. 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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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.