NOTES:
Upper panel of Cout potpage: Inflow and Outflow C(t)
The input function Cin(t) has been chosen to be a Lagged Normal Density Curve, selected by clicking on the circled ~ near the bottom of the INPUT list and setting fgen_1 as LagNormal instead of the default Pulse1. The parameters of this functiion are: tmean = its mean time; RD= its relative dispersion or standard deviation divided by tmean; skewn = skewness, its third moment of inertia, and frpeak is the height of the curve as a fration of its peak height at which teh function is terminated; Area = the area under the curve, with units in this case of mmol/ml as defined in the MML code "extern real Cin(t) mmol/ml.
1. Run loops changing PSg from 1 to zero. The second solution , with PSg=0, mimics a reference intravascular solute. In multiple indicagtor dilution experiments there are normally two tracers injected. The input fucntion (black curve Cin would be the same for both. Cout is the concentration-time curve of response observed at the outflow, The intravascular tracer (e.g. albumin) is represented by the dashed red line on the plot labelled Cout,while the continuus red curve is the permeant tracer.
2. Visfp = lambda*Visf, where lamda is the isf/plasma partition coefficient, that is the ratio of Cisf/Cp at equilibrium, and Visf is the anatomic voolume of the interstitial space.
Visfp > Visf if the "solubility" for a solute in isf is higher than in plasma. Normally for small solutes Visfp = Visf, but for large solutes, or those with the same charge as ISF matrix (mormally negatiely charged) then Visfp < Visf.
Lower panel of the plot page: This is set up by clicking on "View" at the top of the plot page, then "Reset #rows" and select "2".
3. Spatial profiles: These are plots of Cp(x,t) versus x, the position within the capillary, at a selected set of times. This is done by: View "plot 2" by selecting it on the button just uner "File" at the top of the plot page. Then to the right of the window in which the plot variable is to be entered, click on the down arrow and select "Cp(t,x)" and then in the window replace the "t" of "Cp(t,x)" with your selected time, e.g. 3.5 (seconds).
Add a curve to this plot or creat another plot and put Cisf(3.5,x) and othe curves of Cisf(t,x) in order to compare Cisf(t,x) and Cp(t,x). Do this for different permeabilities, by
putting PSg into the "Loops" and raising or lowering it a little. Adjust PSg to amke the ISF profiles closer to the Plasma profiles. In what direction does each move as PSg is raised ... lowered?
// 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.
*/
0.0 2.4 4.8 7.2 9.6 12.0 14.4 16.8
19.2 21.6
0.0 41.29 328.57 833.09 417.95 207.02 141.86 84.4
104.27 74.41