// MODEL NUMBER: 0079
// MODEL NAME: BTEX10
// SHORT DESCRIPTION: Flow with axial dispersion through a one-region pipe
// of uniform cross-section.
import nsrunit; unit conversion on;
math BTEX10 {
// INDEPENDENT VARIABLES
realDomain t sec ; t.min=0; t.max=20; t.delta=0.1; //TIME, seconds
realDomain x cm; real L=0.1 cm, Ngrid=61; x.min=0; x.max=L; x.ct=Ngrid;// SPACE
private x.min, x.max, x.ct;
// parameters labelled private do not appear on the run-time menu
/* PARAMETERS AND KEY TO NAMES
p = PLASMA */
real Fp = 1 ml/(g*min), // Plasma flow
// VOLUME
Vp = 0.05 ml/g, // Plasma Volume
// G is the consumption rate coefficient in each region (Gulosity)
Gp = 0.2 ml/(g*min), // Gulosity or consumption in plasma
// D is the axial diffusion coefficient
Dp = 1.0e-4 cm^2/sec; // Random Gaussian dispersion in plasma
// The hVolume protects against a zero divide if Vp is set to zero.
private real hVp =if(Vp>0) Vp else (1e-6 ml/g);
// SET INFLOWING CONCENTRATION
extern real Cin(t) mM;
// CONCENTRATION VARIABLES
real Cp(t,x) mM, // Concentration in plasma
Cout(t) mM; // Outflow Concentration from plasma region
// BOUNDARY CONDITIONS (Note total flux BC for inflowing region.)
when (x=x.min) { (-Fp*L/hVp)*(Cp-Cin)+Dp*Cp:x = 0; }
when (x=x.max) { Cp:x = 0; Cout = Cp;}
// INITIAL CONDITIONS
when (t=t.min) { Cp = 0; }
// PARTIAL DIFFERENTIAL EQUATION
Cp:t = - Fp*L/hVp*Cp:x - Gp/hVp*Cp + Dp*Cp:x:x;
} //END of MML program
/*
FIGURE:
Fp ________________________________________
Cin(t) ---> |Vp Cp(t)|---> Cout(t)
|Gp |
|Dp PLASMA|
_______________________________________|
|<----------------L------------------->|
|--> x
DETAILED DESCRIPTION:
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.
KEY WORDS: BTEX10,PDE,convection,diffusion,permeation,reaction,distributed,capillary,
plasma, piston flow or plug flow
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.
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.
REVISION HISTORY:
Revised by BEJ 04/14/09 to correct Boundary Conditions
Revised by JBB 11nov10 to add explanations and references
JSim SOFTWARE COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
JSim software was developed with support from NIH grants HL088516,
and HL073598. Please cite these grants in any publication for which
this software is used and send one reprint of published abstracts or
articles to the address given below. Academic use is unrestricted.
Software may be copied so long as this copyright notice is included.
Copyright (C) 1999-2011 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/