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# TranspMM.2sided.Distrib2F

An axially distributed two region two-sided Michaelis-Menten transporter model, with permeation across the capillary wall via clefts (PSg) and cell transporters (PSc).

Model number: 0019

### Further reading:     Distributed Blood Tissue Exchange Models Explained Download PDF file.

 Run JSim model Java Applet: JSim Tutorial

## Description

 This is an axially distributed 2-region capillary-tissue exchange model with
permeation across the capillary wall via clefts (PSg) and  cell transporters (PSc).
The capillary plasma region has volume Vp, flow Fp, first order  consumption Gp,
and axial diffusion Dp. Units are physiological (i.e. per gram of tissue) so that
this can represent a homogeneously perfused organ. Radial diffusion is assumed
instantaneous (short radial distances).

This interstitial fluid region, isf, has volume Visf, first order consumption Gisf,
and axial diffusion Disf. Capillary-tissue exchange is modeled by two parallel routes:
1. PSg: Passive exchange between plasma and surrounding non-flowing interstitial
fluid is through interendothelial clefts. PSg is Permeability-Surface area product.
2. PSc: Facilitated transport occurs via a transporter on the capillary membrane
with PScmax as maximal conductance at low concentrations.
Transporter is modified from TranspMM.1sided.Distrib2F--facilitated transport can go either way.


## Equations

#### Two-sided Saturable Transporter Equation

$\large {\it PS_c} \left( t \right) ={\it PS_{cMAX}} \left( 1+{\frac {{\it C_p}}{{ \it K_{mc}}}}+{\frac {{\it C_{isf}}}{{\it K_{mc}}}} \right) ^{-1}$

#### Differential Equations

$\large \frac{\partial C_p}{\partial t} = \frac{-F_p \cdot L}{V_p} \cdot \frac{\partial C_p}{\partial x}- \frac{G_p}{V_p} \cdot C_p +\frac{PS_g+PS_c}{V_{p}}\cdot (C_{isf}-C_p) +D_p \cdot \frac{\partial^2 C_p}{\partial x^2}$
$\large \frac{\partial C_{isf}}{\partial t} = \frac{-G_{isf}}{V'_{isf}} \cdot C_{isf} + \frac{PS_g+PS_c}{V'_{isf}} \cdot (C_p-C_{isf}) +D_{isf} \cdot \frac{\partial^2 C_{isf}}{\partial x^2}$

#### Left Boundary Conditions

$\large -{\frac {{\it F_p}\cdot L \cdot \left( {\it C_p}-{\it C_{in}} \right) }{{\it V_p}}}+{D_{p} \cdot \it {\frac {\partial}{\partial x}}C_p=0$$\large {\it {\frac {\partial}{\partial x}}C_{isf}=0$ .

#### Right Boundary Conditions

$\large {\it {\frac {\partial }{\partial x}}C_p=0$$\large {\it {\frac {\partial }{\partial x}}C_{isf}=0$$\large {\it C_{out}={\it C_{p}$ .

#### Initial Conditions

$\large C_p=C_p0$ ,   $\large C_{isf}=C_{isf}0$   or
$\large C_p=C_p0(x)$ ,   $\large C_{isf}=C_{isf}0(x)$ .

The equations for this model may also 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.

## 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 embeds 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.



## Key Terms

Axially Distributed, two region, capillary-tissue exchange, facilitated transport, plasma, interstitial fluid region, radial diffusion, tutorial. Michaelis-Menten

## Model History

Get Model history in CVS.

## Acknowledgements

Please cite www.physiome.org in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.