// MODEL NUMBER: 0191
// MODEL NAME: Sanshe
// SHORT DESCRIPTION: The Sangren-Sheppard model (Bull Math Biophys 15: 387-394, 1953) for the
// exchange of a labeled substance between a liquid flowing in a vessel and an external
// compartment. Similar to BTEX20 with no diffusion
import nsrunit; unit conversion on;
math sanshe {
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 Vp ml/g; // Plasma volume
real Visfp ml/g; // ISF volume of distribution (virtual)
real PSg ml/(g*min); // Permeability-surface area product between p and ISF
real Fp = 1 ml/(g*min); // Flow in plasma
real tcap = 3 sec; // transit time in capillary
real psf = 0.5; // ratio of PSg to Fp;
real gamma = 3; // ratio of Visfp to Vp;
Vp = Fp*tcap;
Visfp = gamma*Vp;
PSg = Fp*psf;
real Gp = 0 ml/(g*min), // Plasma consumption rate for metabolite
Gisf = 0 ml/(g*min); // ISF consumption rate for metabolite
// Using hVolumes protects 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);
extern real Cin(t) mmol/ml;
real Cp(t,x) mmol/ml, Cisf(t,x) mmol/ml, Cout(t) mmol/ml; //Concn p, isf, outflow
// Boundary Conditions for PDEs:
when (x=x.min) { Cp = Cin; }
when (x=x.max) { Cp:x = 0; Cout = Cp; }
// 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) - Gp/hVp*Cp ;
Cisf:t = PSg/hVisfp*(Cp-Cisf)- Gisf/hVisfp*Cisf ;
// STATISTICS from calculated moments:
// Cin is the model input, Cout is the model output
real S1_in(t) mM*sec, S2_in(t) mM*sec^2, S3_in(t) mM*sec^3;
real tbar_in(t) sec, RD_in(t) ;
when(t=t.min) {S1_in=0;S2_in=0;S3_in=0;} // Initial conditions for integrators
S1_in:t = Cin; // Integrating to get Area
S2_in:t = Cin*t; // Integrating to get mean time
S3_in:t = Cin*t*t; // Integrating to get variance
tbar_in = S2_in/S1_in; // tbar_in is mean time of input function
RD_in=sqrt((abs(S3_in/S2_in)/tbar_in)-1); // RD_in = relative dispersion of input
//
real S1_out(t) mM*sec, S2_out(t) mM*sec^2, S3_out(t) mM*sec^3;
real tbar_out(t) sec, RD_out(t);
when(t=t.min) {S1_out=0;S2_out=0;S3_out=0;}
S1_out:t=Cout;
S2_out:t=Cout*t;
S3_out:t=Cout*t*t;
tbar_out=S2_out/S1_out;
RD_out=sqrt((abs(S3_out/S2_out)/tbar_out)-1);
//Calculate moments of the operator from the input & output moments
real SD_op(t), tbar_op(t), RD_op(t);
SD_op= sqrt( (RD_out*tbar_out)^2 - (RD_in*tbar_in)^2 );
tbar_op=tbar_out-tbar_in; //Should be Vp/Fp when PSg = 0, else (Vp+Visfp)/Fp
RD_op=SD_op/tbar_op; // but these are only correct if S1_out =S1_in
}
/*
FIGURE:
Fp ________________________________________
Cin(t) ---> |Vp Cp(t)|---> Cout(t)
|Gp ^ |
| | PLASMA|
___________PSg_________________________|
|Visfp | Cisf(t)|
|Gisf V INTERSTITIAL|
| FLUID REGION|
________________________________________
|<----------------L------------------->|
|--> x
LEGEND:
Fp : Plasma Flow Rate, (ml/g)/min
tcap : capillary transit time, sec
psf : ratio of PSg/Fp, non-dimensional
gamma: ratio of Visfp/Vp, non-dimensional
Vp : Plasma Volume, ml/g
Visfp: Volume of Distribution, ml/g
PSg: Permeability-surface area product exchange
coefficients, (ml/g)/min
Gp, Gisf: Consumption rates for metabolite, (ml/g)/min
Cin: Plasma metabolite inflow, mmol/ml
Cout: Plasma metabolite outflow, mmol/ml
Cp, Cisf: metabolite concentration, mmol/ml
Vp=tcap*Fp
PSg=psf*Fp
Visfp=gamma*Vp
DETAILED DESCRIPTION:
The Sangren-Sheppard model may be seen as a varient of the btex20_pde
model. Instead of specifying Vp, Visfp, and PSg, three alternative
parameters are specified, (1) tcap, the capillary transit time, (2) gamma,
the ratio of Visfp to Vp, and (3) psf, the ratio of of PSg to Fp (flow).
Vp, Visfp, and PSg are then given by
Vp=Fp*tcap, Visfp=gamma*Vp, and PSg=psf*Fp. Once these parameters
are calculated, the model is the same as btex20_pde with out diffusion.
This is a model of a "tissue cylinder" consisting of a capillary
plasma region and an interstitial fluid (isf) region. The model is
multi-segmented in a single spatial dimension to solve the convection-
diffusion equation with exchange and consumption. It is modeled as a
differential equation coupled with an ordinary differential equation.
The convecting plasma region has volume Vp, flow Fp, first order
consumption coefficient Gp. The interstitial fluid (isf) region has volume
Visfp, first order consumption coefficient Gisf. The name of the
isf volume, Visfp, is derived from Visf' where the prime is rendered as p.
The isf volume is a volume of distribution which might or might not
correspond to a physical volume. It is referred to as a virtual volume.
PS is the permeability surface product of the membrane. It is also
known as PScap (PS for the CAPillary region) and PSg (the PS for the
Gap between endothelial cells which allows for the passive exchange of
material between the plasma and the isf. In this model, the PS is denoted
PSg. In other nomenclature it is denoted PSc where c stands for "cleft",
a synonym for gap.
Units are per gram of tissue (physiological) so that a single sanshe
unit can model a homogeneously perfused organ.
Instantaneous mixing is assumed in each individual segment in each
region. Passive exchange between plasma and isf regions has conductance
PSg.
This model can be adapted and expanded to represent multi-capillary
models in serial and parallel configurations.
Statistics:
tbar_in is the transit time of the input concentration.
tbar_out is the transit time of the output concentration.
tbar_op is the transit time of the BTEX20 model:
tbar_op=tbar_out-tbar-in.
SD_in is the standard deviation of the input concentration (spread in
time).
SD_out is the standard deviation of the output concentration.
SD_op is the standard deviation of the BTEX20 model:
SD_op= square root (SD_out*SD_out - SD_in*SD_in).
RD_in is the Relative Dispersion of the input concentration:
RD_in=SD_in/tbar_in.
RD_out is the Relative Dispersion of the output concentration:
RD_out=SD_out/tbar_out.
RD_op is the Relative Dispersion of the BTEX20 model:
RD_op=SD_op/tbar_op.
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: sanshe, btex20, two regions, blood tissue exchange, Sangren Sheppard, Publication,
PDE, Transport Physiology, convection, diffusion
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.)
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:
Original Author : garyr Date: 06/12/08
Revised by: BEJ Date:02nov11 : Update Format of comments
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.
*/