Model number

Flow with axial dispersion through a one-region pipe of uniform cross-section.

Further reading: Distributed Blood Tissue Exchange Models Explained


BTEX10 diagram

   The partial differential equation models flow into, through and out of 
 a pipe with plug flow and axial dispersion (diffusion) along the x-axis
 and instantaneous radial dispersion so that concentration is uniform across
 the cross-section at each x-position. Consumption,Gp, equivalent to loss by 
 a first order reaction or loss by permeation is a uniform fraction per
 unit time along the pipe. (This can be modified by making G a function of
 concentration, Gp(Cp) or of position, Gp(x).) Flow is constant, as are all
 the other parameters.The boundary conditions are 
   (1) At the inflow, the diffusion coefficient, Dp, cm^2/s, times the
 spatial gradient in concentration, dC/dx, balances the difference between
 the inflow concentration and the concentration Cp just inside; 
   (2) At the outflow, the gradient dC/dx is set to zero, as if reflecting
 from an impermeable surface, so that mass is lost into the outflow only
 by flow, Cout = Cp(x=L,t).

 LIMITATIONS: This model cannot approximate Newtonian parabolic flow, where
 the response to a flow-proportiaonal cross-sectional pulse labeling at the
 inflow would give a sharp upstroke and peak at 1/2 the mean transit time
 and then, in the absence of axial dispersion, diminish in proportion to
 1/t^2. See Gonzalez-Fernandez (1962) on this point. 



Differential Equations


Left Boundary Conditions


Right Boundary Conditions



Initial Conditions

e4 or


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.

Download JSim model project file

Help running a JSim model.

Download MATLAB model M-file

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,

 Gonzalez-Fernandez JM. Theory of the measurement of the dispersion of
 an indicator in indicator-dilution studies. Circ Res 10: 409-428, 1962.

 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, 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. 
Key terms
piston flow or plug flow

Please cite 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.

Model development and archiving support at provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, 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.