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# CompNFlowDelay

Computes N compartment models in series, each with volume = Vp/N, and with an added delay.

Model number: 0250

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## Description

N well stirred tanks (i.e. compartmental models with flow) are connected in series. A delay is imposed on the result. The total volume is kept constant. The transit time and the variances of the input to the first tank and the output of the last tank after the delay are computed. From this information, the transit time, and the variance of N stirred tanks plus delay is computed.

## 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.

#### Analytic Solution for Relative Dispersion

$\large {\it RD_{analytic}}={\frac {{\it V_p}}{\sqrt {N} \left( {\it V_p}+{\it delay}\cdot {\it F_p} \right) }}$

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

Bassingthwaighte, J.B. Plasma indicator dispersion in
arteries of the human leg. Circ. Res. 19: 332-346, 1966.

## Related Models

Single Compartment Models:

Two Compartment Models:

N>2 Compartment Models:

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Pharmacology:

## Key Terms

RD, Relative Dispersion, Compartmental, Compartment, Stirred Tanks, delay, transit time, Tutorial