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# BTEX50

Models a tissue cylinder consisting of five regions: plasma, interstitial fluid,endothelial cells, parenchymal cells, and mitochondria.

Model number: 0083

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## Description

These partial differential equations model a "tissue cylinder" consisting of five regions. The five regions are capillary plasma, p; endothelial cell, ec; interstitial fluid, isf; parenchymal cell, pc; and mitochondria, mito; and are separated by four barriers--the luminal or plasma surface and endothelial cell layer; the albuminal surface of the endothelial cell facing the interstitium; the membrane between the interstitial fluid and parenchymal cell; and the membrane between the parenchymal cell and the mitochondria. In addition, there is a diffusional path from plasma to ISF bypassing endothelial cells via intercellular clefts.

## Equations

#### Differential Equations

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

#### Left Boundary Conditions

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

### Right Boundary Conditions

$\large {\it {\frac {\partial }{\partial x}}C_p=0$ , $\large {\it {\frac {\partial }{\partial x}}C_ec=0$ , $\large {\it {\frac {\partial }{\partial x}}C_{isf}=0$ , $\large {\it {\frac {\partial }{\partial x}}C_{pc}=0$ , $\large {\it {\frac {\partial }{\partial x}}C_{mito}=0$ , $\large {\it C_{out}={\it C_{out}$ .

#### Initial Conditions

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

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

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.



## Related Models

Blood Tissue Exchange (BTEX) models

## Key Terms

BTEX50,PDE,convection,diffusion,permeation,reaction,distributed,capillary,plasma,isf,interstitial fluid,endothelial,parenchymal,cell,mitochondria