// MODEL NUMBER: 0164
// MODEL NAME: Lung_tumor_2region
// SHORT DESCRIPTION: 2-region capillary-tissue exch of iodinated contrast in solid tumor.
JSim v1.1
import nsrunit; unit conversion on;
math Lung_Tumor_2region {
// Diagram and References: At bottom:
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
real
Fp = 1 ml/(g*min), // Plasma flow, p for plasma
Vp = 0.05 ml/g, // Plasma volume
Visfp = 0.15 ml/g; // ISF volume of distribution
// Using hVolumes protects against zero divides
private real
hVp = if(Vp>0) Vp else (1e-6 ml/g),
hVisfp = if(Visfp>0) Visfp else (1e-6 ml/g);
real
PSg = 0.5 ml/(g*min), // Capillary Perm-Surface area product
Dp = 1e-05 cm^2/sec, // Plasma axial diffusion coefficient
Disf = 1e-05 cm^2/sec, // ISF axial diffusion coefficient
rho = 1.06 g/ml, // tissue density
scale = 5;
real
Cp(t,x) mmol/ml, // Concn in p
Cisf(t,x) mmol/ml, // Concn in isf
Cout(t) mmol/ml, // Concn in outflow
Q1(t) mmol/ml, // regional concn of contrast within ROI
Q1sc(t); // regional scaled concn of contrast
extern real
Cin(t) mmol/ml; // Cin is fitted to observed input in PA
// Boundary Conditions for PDEs:
when (x=x.max) { Dp*Cp:x = 0; Cout = Cp; } // Right Hand
when (x=x.min) { (-Fp*L/hVp)*(Cp-Cin)+Dp*Cp:x =0;} // Left Hand Total flux BC.
when (x=x.min) { Cisf:x = 0;}
when (x=x.max) { Cisf:x = 0;}
// Initial Conditions for PDEs:.
when (t=t.min) { Cp = if (x=x.min) Cin else 0; }
when (t=t.min) { Cisf = 0; }
// Partial differential equation, PDE
Cp:t = -Fp*L/hVp*Cp:x - PSg/hVp*(Cp-Cisf) + Dp*Cp:x:x;
Cisf:t = PSg/hVisfp*(Cp-Cisf) + Disf*Cisf:x:x;
// Calculating residue function:
when (t=t.min) Q1=0;
Q1:t = Fp * (Cin-Cout) * rho;
Q1sc = Q1*10^scale;
}
/*
FIGURE:
Fp ________________________________________
Cin(t) ---> |Vp Cp(t)|---> Cout(t)
|Gp ^ |
|Dp | PLASMA|
___________PSg_________________________|
|Visfp | Cisf(t)|
|Gisf 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:
One-dimensional convection-permeation-diffusion-
reaction model consisting of two concentric cylinders separated by a
membrane. The central plasma region of volume Vp has flow Fp, no
consumption, but has axial diffusion (disperion) Dp. Units are
physiological per gram of tissue so that a single unit can model a
homogeeously perfused region. Radial diffusion is assumed instantaneous
(short radial distances). Exchange into a second surrounding non-flowing
region is passive bidirectional with conductance, PS, the Permeability
capillary Surface area product. The interstitial fluid region, isf, of
volume Visfp (see Notes) is axially distributed, and the gradients axially
are dissipated by a concentration-independent axial diffusion or dispersion.
Radial diffusion within this space is considered instantaneous, and there
is no consumption. This model is sutiable for use in multicapillary models
as one of a set of units in parallel.
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: solid tumors, btex, capillary-tissue exchange, permeability, diffusion, convection
axial gradients, Combined Transport, Transport Physiology
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 imbeds the model in an organ with tissue volums conserved, and with arteries
and veins.)
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.)
REVISION HISTORY:
Original Author : JBB Date: 06/12/08
Revised by: BEJ Date:01aug11 : Update comment format
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2011 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@physiome.org.
*/