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# Comp6_Recirc

Six compartment model with flow, exchange, and recirculation.

Model number: 0118

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## Description

    System is composed of 6 compartments, five of them in series, with one
in parallel. The system is divided into three operators with recirculation
as diagrammed.  Exchange is allowed in only the first operator. The second
and third operators are each composed of two stirred tanks operators in series.
The input function, Cin, is positioned between the first and second
operators but could be positioned anywhere. A fractional clearance occurs
between the output of the third operator with the remainder recirculating
to the first operator.

This model can be used to explore the theory of John L.Stephenson,
Bull Math Biophys 10: 117-121, 1948.and Bull Math Biophys 22:1-17, 1960.



Figure: A plot of concentration as a function of time for each of the compartment outflows, where CC2 is the outflow conentration from the system before recirculating into compartment CA1. Concentrations are in mM.

Diagram:


CC2*(1-clear)
<----------<---------------------------<---------------<---------------<
Fp|    OPERATOR 1         Cin       OPERATOR 2             OPERATOR 3    |
|  _________________     |     ______   _______      _______  _______  ^
v  |Vp     CA1(t)  | CA1 v     |VB1  |  |VB2  |      |VC1  |  |VC2  |  |
Creturn -->|               |--->(+)--->| CB1 |->| CB2 |----->| CC1 |->| CC2 |->|
|      PSg      |           |_____|  |_____|      |_____|  |_____|
|       ^       |
--------|--------             FLOW = Fp ml/(g*min)  -->
|       v       |
|               |
|Visfp  CA2(t)  |
-----------------


## Equations

The equations for this model may 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.

Operator A, a single stirred tank with permeation into a nonflowing tank.

$\large \frac{dC_{A1}}{dt} = (\frac{F_p}{V_p}) \cdot (C_{return}-C_{A1}) - (\frac{PS_g}{V_p}) \cdot (C_{A1}-C_{A2})$

$\large \frac{dC_{A2}}{dt} = (\frac{PS_g}{V{isfp}}) \cdot (C_{A1}-C_{A2})$

Operator B, a pair of tanks in series with inputs from A1 and from Cin:

$\large \frac{dC_{B1}}{dt} = (\frac{F_p}{V_{B1}}) \cdot (C_{A1} + C_{in} -C_{B1})$

$\large \frac{dC_{B2}}{dt} = (\frac{F_p}{V_{B2}}) \cdot (C_{B1} -Ci_{B2})$

Operator C, same as B but no other input:

$\large \frac{dC_{C1}}{dt} = (\frac{F_p}{V_{C1}}) \cdot (C_{B2} -C_{C1})$

$\large \frac{dC_{C2}}{dt} = (\frac{F_p}{V_{C2}}) \cdot (C_{C1} -C_{C2})$

$\large C_{return} = (1.0-clear) \cdot C_{C2}$

Where Vp is the volume of the plasma space, PSg is the permeability surface area product, Visfp is the ISF volume fraction in 1 gram of tissue, clear is the fractional clearance, VB is the plasma volume fraction in 1 gram of B tissue and VC is the plasma volume fraction in 1 gram of C tissue.

## References

 John L.Stephenson, Bull Math Biophys 10: 117-121, 1948.

John L.Stephenson, Bull Math Biophys 22:1-17, 1960.



## Related Models

Single Compartment Models:

Two Compartment Models:

N>2 Compartment Models:

Osmotic Exchange:

Pharmacology:

## Key Terms

compartment, compartmental, flow, first order process, recirculating, tutorial, transit time, clearance, six compartment