// MODEL NUMBER: 0080
// MODEL NAME: BTEX20
// SHORT DESCRIPTION: Models a tissue cylinder consisting of two regions: plasma,
// and interstitial fluid.
import nsrunit; unit conversion on;
math btex20_pde {
// INDEPENDENT VARIABLES
realDomain t sec ; t.min=0; t.max=30; t.delta=0.1;
realDomain x cm; real L=0.1 cm, Ngrid=61; x.min=0; x.max=L; x.ct=Ngrid;
private x.min, x.max, x.ct;
/* PARAMETERS AND KEY TO NAMES
p = PLASMA
isf = INTERSTITIAL FLUID REGION*/
real Fp = 1 ml/(g*min), // Plasma flow
// VOLUMES (a p after the region name implies "prime", a virtual volume)
Vp = 0.05 ml/g, // p
Visfp = 0.15 ml/g, // isf (Visf')
// PS means Permeability-surface area product between two regions
PSg = 1 ml/(g*min), // between p and isf
// G is the consumption rate coefficient in each region (Gulosity)
Gp = 0 ml/(g*min), // p
Gisf = 0 ml/(g*min), // isf
// D is the axial diffusion coefficient
Dp = 1.0e-5 cm^2/sec, // p
Disf = 1.0e-6 cm^2/sec; // isf
// The hVolumes protect against zero divides
private real hVp =if(Vp>0) Vp else (1e-6 ml/g);
private real hVisfp =if(Visfp>0) Visfp else (1e-6 ml/g);
// INFLOWING CONCENTRATION
extern real Cin(t) mM;
// CONCENTRATION VARIABLES
real Cp(t,x) mM, // p
Cisf(t,x) mM, // isf
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; Cisf:x = 0; }
when (x=x.max) { Cp:x = 0; Cisf:x = 0; Cout = Cp;}
// INITIAL CONDITIONS
when (t=t.min) { Cp = 0; Cisf = 0; }
// PARTIAL DIFFERENTIAL EQUATIONS
Cp:t = -Fp*L/hVp*Cp:x -Gp/hVp*Cp+ Dp*Cp:x:x
+ PSg/hVp*(Cisf-Cp);
Cisf:t = -Gisf/hVisfp*Cisf + Disf*Cisf:x:x
+ PSg/hVisfp*(Cp-Cisf) ;
}
/*
FIGURE:
Fp ________________________________________
Cin(t) ---> |Vp Cp(t)|---> Cout(t)
|Gp ^ |
|Dp | PLASMA|
___________PSg_________________________|
|Visfp | Cisf(t)|
|Gisf V INTERSTITIAL|
|Disf FLUID REGION|
________________________________________
|<----------------L------------------->|
|--> x
DETAILED DESCRIPTION:
These partial differential equations model a "tissue cylinder"
consisting of two regions. The two regions are capillary plasma,
p; and interstitial fluid, isf; there is a diffusional path from plasma
to the isf.
KEY WORDS:
BTEX20,PDE,convection,diffusion,permeation,reaction,distributed,capillary,
plasma,isf,interstitial fluid
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, 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
Boundary Conditions
Revised by GR 09/22/09
Added Statistics and Reformatted.
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) 1999-2009 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/