/* MODEL NUMBER: 0019
MODEL NAME: TranspMM.2sided.Distrib2F
SHORT Description: An axially distributed two region two-sided Michaelis-Menten transporter
model, with permeation across the capillary wall via clefts (PSg) and cell transporters (PSc).
*/
import nsrunit; unit conversion on;
math TranspMM2sidedDistrib2F {
// INDEPENDENT VARIABLES
realDomain t sec ; t.min=0; t.max=30; t.delta=0.05;
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; //Ngrid must be odd number
// PARAMETERS
real Fp = 1 ml/(g*min), // Plasma flow: Subscript p for plasma
Vp = 0.05 ml/g, // Plasma volume
Visf = 0.25 ml/g, // ISF volume of distribution (virtual)
PSg = 1.0 ml/(g*min), // Permeability-surface area product cleft between p and ISF
Gp = 0 ml/(g*min), // Plasma consumption rate for metabolite
Gisf = 100 ml/(g*min), // ISF consumption rate for metabolite
Dp = 1e-06 cm^2/sec, // Plasma axial diffusion coefficient
Disf = 1e-05 cm^2/sec, // ISF axial diffusion coefficient
PScmax= 1 ml/(g*min), // Max PSc across endo cells
Kmc = 1 mM; // Km, Michaelis const for transporter
// Using hVolumes protects against setting V = 0.
private real hVp = if(Vp>0) Vp else (1e-6 ml/g);
private real hVisf = if(Visf>0) Visf else (1e-6 ml/g);
// DEPENDENT VARIABLES
real PSc(t,x) ml/(g*min); // PS via MM transporter across endothelial cell
extern real Cin(t) mM;
real Cp(t,x) mM, Cisf(t,x) mM, Cout(t) mM; //Concentration plasma,
// isf and outflow
// BOUNDARY CONDITIONS FOR PDEs
when (x=x.min) { (-Fp*L/hVp)*(Cp-Cin)+Dp*Cp:x =0;} // Left Hand Total flux BC.
when (x=x.max) { Cp:x = 0; Cout = Cp; } // Right Hand (no flux)
when (x=x.min) { Cisf:x = 0;} // Left Hand (no flux)
when (x=x.max) { Cisf:x = 0;} // Right Hand (no flux)
// INITIAL CONDITIONS FOR PDEs:.
when (t=t.min) { Cp = if (x=x.min) Cin else 0; }
when (t=t.min) { Cisf = 0; }
// PARTIAL DIFFERENTIAL EQUATIONS
PSc = PScmax/(1 + Cp/Kmc + Cisf/Kmc);
Cp:t = -Fp*L/hVp*Cp:x + (PSg+PSc)*(Cisf-Cp)/hVp - Gp*Cp/hVp + Dp*Cp:x:x;
Cisf:t = (PSg+PSc)*(Cp-Cisf)/hVisf- Gisf*Cisf/hVisf + Disf*Cisf:x:x;
// ADDITIONAL CALCULATIONS
realState peakHtCin(t) mM;
when(t=t.min) peakHtCin=0;
event( t>t.min and peakHtCin |Vp Cp(t)|---> Cout(t)
|Gp ^ ^ |
|Dp | | PLASMA|
___________PSg_____PSc_________________|
|Visfp | | Cisf(t)|
|Gisf V V INTERSTITIAL|
|Disf FLUID REGION|
________________________________________
|<----------------L------------------->|
|--> x
Fp : Plasma Flow Rate, (ml/g)/min
Vp : Plasma Volume, ml/g
Visfp: Volumes of Distribution, ml/g
PSg: Permeability-surface area product exchange
coefficients, (ml/g)/min
Gp, Gisf: Consumption rates for metabolite, (ml/g)/min
Dp, Disf: Axial Diffusion Rate, cm^2/sec
Cin: Plasma metabolite inflow, mmol/ml
Cout: Plasma metabolite outflow, mmol/ml
Cp, Cisf: metabolite concentration, mmol/ml
DETAILED DESCRIPTION:
This is an axially distributed 2-region capillary-tissue exchange model with
permeation across the capillary wall via clefts (PSg) and cell transporters (PSc).
The capillary plasma region has volume Vp, flow Fp, first order consumption Gp,
and axial diffusion Dp. Units are physiological (i.e. per gram of tissue) so that
this can represent a homogeneously perfused organ. Radial diffusion is assumed
instantaneous (short radial distances).
This interstitial fluid region, isf, has volume Visf, first order consumption Gisf,
and axial diffusion Disf. Capillary-tissue exchange is modeled by two parallel routes:
1. PSg: Passive exchange between plasma and surrounding non-flowing interstitial
fluid is through interendothelial clefts. PSg is Permeability-Surface area product.
2. PSc: Facilitated transport occurs via a transporter on the capillary membrane
with PScmax as maximal conductance at low concentrations.
Transporter is modified from TranspMM.1sided.Distrib2F--facilitated transport can go either way.
SHORTCOMINGS/GENERAL COMMENTS:
KEY WORDS: Axially Distributed, two region, capillary-tissue exchange, facilitated transport,
plasma, interstitial fluid region, radial diffusion, tutorial. Michaelis-Menten
REFERENCES:
Sangren WC and Sheppard CW. A mathematical derivation of the
exchange of a labeled substance between a liquid flowing in a
vessel and an external compartment. Bull Math Biophys 15: 387-394, 1953
(This gives an analytic solution for the two-region model.)
Goresky CA, Ziegler WH, and Bach GG. Capillary exchange modeling:
Barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970.
(This gives another derivation of the analytical form, and uses the model in
both single and multicapillary models.
Bassingthwaighte JB. A concurrent flow model for extraction
during transcapillary passage. Circ Res 35: 483-503, 1974.
(This gives numerical solutions, which are faster than the analytic solutions,
and embeds the model in an organ with tissue volums conserved, and with arteries
and veins. The original Lagrangian sliding fluid element model with diffusion.)
Guller B, Yipintsoi T, Orvis AL, and Bassingthwaighte JB. 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.
(Application to sodium exchange in the heart.)
Goresky CA. Hepatic membrane carrier transport processes: Their involvement
in bilirubin uptake. In: Chemistry and Physiology of Bile Pigments.
Washington, D.C.: Publishing House U.S. Government, 1977, p. 265-281.
Silverman M and Goresky CA. A unified kinetic hypothesis of carrier-mediated
transport: Its applications. Biophys J 5: 487-509, 1965.
REVISION HISTORY:
Original Author : JBB Date: Nov/08
Revised by : BEJ Date: 16/Dec/08
Revision: 1) Update format of comments
Revised by : GMR Date 11/May/10
Revision: 2) Standard format
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2009 University of Washington. From the National Simulation Resource,
Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Academic use is unrestricted. Software may be copied so long as this copyright notice is included.
This software was developed with support from NIH grant HL073598.
Please cite this grant in any publication for which this software is used and send an email
with the citation and, if possible, a PDF file of the paper to: staff@bioeng.washington.edu.
*/