TranspMM.2sol2sided.Distrib2F

Model number: 0025

An axially distributed two region capillary-tissue exchange opreator with permeation across capillary walls via clefts (PS_g) and competition between two solutes for cell transmembrane transporter (PS_c) of two-sided Michaelis-Menten type.

Detailed Description

A one-dimensional convection-permeation-diffusion-reaction model consisting of two concentric cylinders separated by a membrane. The capillary plasma region, volume V_p, has flow F_p, first order consumption G_p, and axial diffusion (dispersion) D_p. Units are physiological per grame of tissue so that a single unit can model a homogenously perfused organ. Radial diffusion is assumed instantaneous (short radial distances).

The Interstitial Fluid Region, with volume V_ISF, is axially distributed. The gradients axially are dissipated by a concentration-independent axial diffusion. Radial diffusion within the ISF is considered instantaneous, and consumption, G_ISF, is first order.

Capillary-tissue exchange happens by way of two parallel routes.

1. PS_g: Passive exchange between plasma and surrounding non-flowing interstitial fluid is through interendothelial clefts. PS_g is Permeability Surface area product.

2. PS_c: Facilitated transport occurs via a transporter on the capillary membrane with PS_cmax as maximal conductance at low concentrations.

This model is used in multicapillary models as one of a set of units in parallel.

Relevant Equations

${ PS_{\text{\small{c}}} = \frac {PS_{\text{\small{cmax}}}}{1 + \frac {A_{\text{\small{p}}} + A_{\text{\small{ISF}}}}{K_{\text{\small{mA}}}} + \frac {B_{\text{\small{p}}} + B_{\text{\small{ISF}}}}{K_{\text{\small{mB}}}}}$

${ \frac {d}{dt}A_{\text{\small{p}}}(t,x) = - \frac {F_{\text{\small{p}}} L}{hV_{\text{\small{p}}} \frac {d}{dx}A_{\text{\small{p}}}} - \frac {(PS_{\text{\small{g}}} + PS_{\text{\small{c}}})(A_{\text{\small{p}}} - A_{\text{\small{ISF}}})}{hV_{\text{\small{p}}}} - \frac {G_{\text{\small{pA}}} A_{\text{\small{p}}}}{hV_{\text{\small{p}}}} + D_{\text{\small{pA}}} \frac {d^2 A_{\text{\small{p}}}}{dx^2}$

${ \frac {d}{dt}A_{\text{\small{ISF}}}(t,x) = \frac {(PS_{\text{\small{g}}} + PS_{\text{\small{c}}})(A_{\text{\small{p}}} - A_{\text{\small{ISF}}})}{hV_{\text{\small{ISF}}}} - \frac {G_{\text{\small{ISFA}}} A_{\text{\small{ISF}}}}{hV_{\text{\small{ISF}}}} + D_{\text{\small{ISFA}}} \frac {d^2 A_{\text{\small{ISF}}}}{dx^2}$

${ \frac {d}{dt}B_{\text{\small{p}}}(t,x) = - \frac {F_{\text{\small{p}}} L}{hV_{\text{\small{p}}} \frac {d}{dx}B_{\text{\small{p}}}} - \frac {(PS_{\text{\small{g}}} + PS_{\text{\small{c}}})(B_{\text{\small{p}}} - B_{\text{\small{ISF}}})}{hV_{\text{\small{p}}}} - \frac {G_{\text{\small{pB}}} B_{\text{\small{p}}}}{hV_{\text{\small{p}}}} + D_{\text{\small{pB}}} \frac {d^2 B_{\text{\small{p}}}}{dx^2}$

${ \frac {d}{dt}B_{\text{\small{ISF}}}(t,x) = \frac {(PS_{\text{\small{g}}} + PS_{\text{\small{c}}})(B_{\text{\small{p}}} - B_{\text{\small{ISF}}})}{hV_{\text{\small{ISF}}}} - \frac {G_{\text{\small{ISFB}}} B_{\text{\small{ISF}}}}{hV_{\text{\small{ISF}}}} + D_{\text{\small{ISFB}}} \frac {d^2 B_{\text{\small{ISF}}}}{dx^2}$

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References

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 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. The original Lagrangian sliding fluid element model with diffusion.)

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.)

Goresky CA. Hepatic membrane carrier transport processes: Their involvement in bilirubin uptake. In: Chemistry and Physiology of Bile Pigments. Washington, D.C.: Publishing House U.S. Government, 1977, p. 265-281.

Silverman M and Goresky CA. A unified kinetic hypothesis of carrier-mediated transport: Its applications. Biophys J 5: 487-509, 1965.

Dawson CA, Linehan JH, Rickaby DA, and Roerig DL. Inﬂuence of plasma protein on the inhibitory effects of indocyanine green and bromcresol green on pulmonary prostaglandin E1 extraction. Br J Pharmac 81: 449-455, 1984.

Related Models

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Key Terms

Transporter, Michaelis-Menten, capillary-tissue exchange, axial gradients, solute-solute competition, permeability surface area, BTEX, spatially distributed, convection diffusion reaction, axially distributed, gulosity.

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Copyright (C) 1999-2010 University of Washington. From the National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061. Academic use is unrestricted. Software may be copied so long as this copyright notice is included. This software was developed with support from NIH grant HL073598. Please cite this grant in any publication for which this software is used and send one reprint to the address given above.

Model development and archiving support at physiome.org provided by the following grants: 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.