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# Comp2Exchange

Models two compartments with a single substance passively exchanging between the two compartments.

Model number: 0245

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## Description

This is a model of two compartments, one with volume V1 and initial concentration C10, and the other with volume V2 and concentration C20. The two compartments exchange material at a rate PS. PS is the product of the permeability multiplying the surface area. The exchange is passive and equal in both directions in this model.

• Understanding the Comp2Exchange Model (PDF file).

## Equations

#### Ordinary Differential Equations

$\large {\frac {d}{dt}}{\it C1} \left( t \right) ={\frac {{\it PSg}\, \left( { \it C2} \left( t \right) -{\it C1} \left( t \right) \right) }{{\it V1 }}}$
$\large {\frac {d}{dt}}{\it C2} \left( t \right) ={\frac {{\it PSg}\, \left( { \it C1} \left( t \right) -{\it C2} \left( t \right) \right) }{{\it V2 }}}$

#### Initial Conditions

$\large {\it C1} \left( 0 \right) ={\it C10}$
$\large {\it C2} \left( 0 \right) ={\it C20}$

#### Analytic Solutions

$\large {\it analyticC1} \left( t \right) = \frac {\left( {\it C20}\,{\it V2}+{\it C10}\,{\it V1}+{\it V2}\, \left( {\it C10}-{\it C20} \right) {e^{-{ \frac {{\it PS}\, \left( {\it V1}+{\it V2} \right) t}{{\it V1}\,{\it V2}}}}} \right)} { \left( {\it V1}+{\it V2} \right) }$
$\large {\it analyticC2} \left( t \right) = \frac {\left( {\it C20}\,{\it V2}+{\it C10}\,{\it V1}-{\it V1}\, \left( {\it C10}-{\it C20} \right) {e^{-{ \frac {{\it PS}\, \left( {\it V1}+{\it V2} \right) t}{{\it V1}\,{\it V2}}}}} \right)} { \left( {\it V1}+{\it V2} \right) }$

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

 None.



## Related Models

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## Key Terms

Course, compartment, compartmental, tutorial, exchange, multiple compartments, flux, steady state