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# BTEX10_OscillatingFlow

Two single compartments use an oscillating flow through a pipe (BTEX10) to exchange material and volume.

Model number: 0308

 Run JSim model Java Applet: JSim Tutorial

## Description

This model consists of two well-stirred (instantaneously mixed) compartments connected by a pipe (BTEX10) through which they exchange material and volume by an oscillating flow. The solutions from BTEX10_OscillatingFlow are compared with Comp1_OscillatingFlow (2nd model in this project, where the BTEX10 pipe is replace by a one compartment pipe) by setting the diffusion cofficient to a large value (D=0.1 cm^2/sec).

## Equations

#### Algebraic Equations

$\large {F=A \cdot \sin \left( k \cdot t \right)}$
$\large {\it FlowFromLeft} = { {if} \left( F>0 \right)\ F \ else \ 0 }$
$\large {\it FlowFromRight} = { {if} \left( F<0 \right)\ F \ else \ 0 }$

#### Differential Equations

$\large {\frac {d {\it A_1}}{dt}}= { {if} \left( F>0 \right)\ -{\frac {F \cdot {\it A_1}}{{\it V_1}}} \ else \ -F \cdot {\it C_{outR}} }$
$\large {\frac {d {\it V_1}}{dt}}=-F$
$\large {\frac {d {\it A_2}}{dt}}= { {if} \left( F>0 \right)\ F \cdot {\it C_{outL}} \ else \ {\frac {F \cdot {\it A_2}}{{\it V_2}}} }$
$\large {\frac {d {\it V_2}}{dt}}=F$
$\large \frac{\partial C_m}{\partial t} = \frac{-F \cdot L}{V_m} \cdot \frac{\partial C_m}{\partial x} +D \cdot \frac{\partial^2 C_m}{\partial x^2}$

#### Left Boundary Conditions

$\large -{\frac {{\it FlowFromLeft}\cdot L \cdot \left( {\it C_m}-{\it C_{left}} \right) }{{\it V_m}}}+{D \cdot \it {\frac {\partial C_m}{\partial x}}=0$
$\large {\it C_{outR}}={\it C_m}$

#### Right Boundary Conditions

$\large -{\frac {{\it FlowFromRight}\cdot L \cdot \left( {\it C_m}-{\it C_{right}} \right) }{{\it V_m}}}+{D \cdot \it {\frac {\partial C_m}{\partial x}}=0$
$\large {\it C_{outL}}={\it C_m}$

#### Initial Conditions

$\large A_1=A_{1}0$ ,   $\large V_1=V_{1}0$ ,   $\large A_2=A_{2}0$ ,   $\large V_2=V_{2}0$ ,   and   $\large C_m=C_m0$   or   $\large C_m=C_m0(x)$

where F is the oscillating flow, A1 and A2 are the amounts of substrate in volumes V1 and V2 respectively, and Cm is the concentration in the pipe with volume Vm and length L.

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, F.P. Chinard, C. Crone, C.A. Goresky,
N.A. Lassen, R.S. Reneman, and K.L. Zierler.  Terminology for
mass transport and exchange.  Am. J. Physiol. 250 (Heart. Circ.
Physiol. 19): H539-H545, 1986.

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

BTEX10, Oscillating flow, reversing flow, PDE, advection, diffusion