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# CTEX10

Models a capillary consisting of N compartments. Explores sensitivity analysis and optimization and details the Monte-Carlo GUI for robust estimates of parameters, confidence limits and covariance.

Model number: 0272

 Run JSim model Java Applet: JSim Tutorial

## Description

N compartments in series simulate a partial differential equation with convection and diffusion. The relative dispersion of N compartments in series is 1/sqrt(N). There is not an exact relationship between the diffusion coefficient and the relative dispersion in the partial differential equation for diffusion and convection in a bounded domain with inflow and outflow at the ends. This model explores sensitivity analysis and optimization in a guided exercise. Details for using the Monte-Carlo graphical user interface are included both from the main TABS, Top Left. The information is also repeated in the Notes (Run Time GUI, Bottom, Notes TAB).

## Equations

#### Ordinary Differential Equations

$\large {\frac {{\it V_p}}{N}} \cdot {\frac {d}{dt}}{\it C_{p,1}} \left( t \right) ={\it F_p}\cdot \left( {\it C_{in}} \left( t \right) - {\it C_{p,1}} \left( t \right) \right)$ ,   and for j=2..N:  $\large {\frac {{\it V_p}}{N}} \cdot {\frac {d}{dt}}{\it C_{p,j}} \left( t \right) ={\it F_p}\cdot \left( {\it C_{p,j-1}} \left( t \right) -{\it C_{p,j}} \left( t \right) \right)$ .

#### Initial Conditions

$\large {\it C_{p,j}} \left( 0 \right) ={\it C_{p,j}{0}}$   for j=1..N.

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

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Two Compartment Models:

N>2 Compartment Models:

Osmotic Exchange:

Pharmacology:

## Key Terms

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