BTEX10x2CircularFlow

Two single compartments use a circular flow through a two pipes (BTEX10s) to exchange material. Pipe 1 has flow to the right, pipe 2 has flow to the left.

Model number: 0309

Description

This model consists of two well stirred tanks (ODE's), connected by two smaller pipes (BTEX10 models (PDE's). The material circulates in a circular path. Pipe 1 is flow from the left, Pipe2 is flow from the right. The diffusion coefficient, D, is set to a high value for comparison with a corresponding version which uses compartmental models for the pipes. A second model is included, Compx2_CircularFlow comparing using compartmental models for the two pipes and a large value of the diffusion coefficient in the BTEX10 (partial differential equations) pipes. Model illustrates inflow boundary conditions and PDEs for inflow on both left and right boundaries.

Equations

Differential Equations

${\frac {d {\it A_1}}{dt}}= F \cdot {\it (C_{outR}- C_1)}$
${\frac {d {\it A_2}}{dt}}= F \cdot {\it (C_{outL}- C_2)}$
$\frac{\partial C_L}{\partial t} = \frac{-F \cdot L}{V_m} \cdot \frac{\partial C_L}{\partial x} +D \cdot \frac{\partial^2 C_L}{\partial x^2}$
$\frac{\partial C_R}{\partial t} = \frac{-(-F) \cdot L}{V_m} \cdot \frac{\partial C_R}{\partial x} +D \cdot \frac{\partial^2 C_R}{\partial x^2}$

Left Boundary Conditions

$-{\frac {{\it F}\cdot L \cdot \left( {\it C_L}-{\it C_{1}} \right) }{{\it V_m}}}+{D \cdot \it {\frac {\partial C_L}{\partial x}}=0$
$\frac{\partial C_R}{\partial x} =0$

${\it C_{outR}}={\it C_R}$

Right Boundary Conditions

$-{\frac {{\it -F}\cdot L \cdot \left( {\it C_R}-{\it C_{2}} \right) }{{\it V_m}}}+{D \cdot \it {\frac {\partial C_R}{\partial x}}=0$
$\frac{\partial C_L}{\partial x} =0$
${\it C_{outL}}={\it C_L}$

Initial Conditions

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

Additional Equations

${\it C_1}={\frac {{\it A_1}}{{\it V_1}}}$
${\it C_2}={\frac {{\it A_2}}{{\it V_2}}}$

where F is the flow (positive when from left to right and negative when from right to left), A₁ and A₂ are the amounts of substrate in volumes V₁ and V₂ respectively, and C_L (flow from left to right) and C_R (flow from right to left) are the concentrations in the pipes with volumes V_m and lengths 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.

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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, Compartmental, Circular flow, PDE, advection, diffusion

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Acknowledgements

Please cite www.physiome.org in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

[This page was last modified 22Sep10, 11:54 am.]

Model development and archiving support at physiome.org provided by the following grants: NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.