Model number

Axially-distributed 2-region capillary-tissue exchange operator with permeability varying along the capillary length.


 Diagram and References: At bottom:

realDomain t sec ; t.min=0; t.max=30;;
realDomain x cm; real L=0.1 cm, Ngrid=61; x.min=0; x.max=L; x.ct=Ngrid;    
Ngrid must be odd number

 Plasma flow: Subscript p for plasma
 Plasma volume  
 ISF volume of distribution (virtual)
 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);
 Plasma consumption rate for metabolite 
 ISF consumption rate for metabolite
 Plasma axial diffusion coefficient
 ISF axial diffusion coefficient

 Permeability-surface area product, PS, between p and ISF
 Average PS with linear change with x
 Pratio  = PS(x=L)/PS(x=0)
 PS = PS0 + x/L*(PSL-PS0)

 PS at x=0: PS0 =2*Pbar/(1+Pratio)
 PS at x=L: PSL =2*Pratio*Pbar/(1+Pratio)
     PS0 = 2*PSbar/(1+Pratio);      
     PSL = 2*Pratio*PSbar/(1+Pratio);

 Input function from function generator
 Reference intravascular
 Concn p, isf, outflow
 transcapillary extraction of permeant

 Boundary Conditions for PDEs:
when  (x=x.min) { Cp = Cin; Cref = Cin;}
when  (x=x.max) { Cp:x = 0; Cout = Cp; Cref:x = 0; Crefout = Cref;}
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; Cref = if (x=x.min) Cin else 0; }
when (t=t.min) { Cisf = 0; }

 Partial differential equation, PDE
Cref:t = -Fp*L/hVp*Cref:x;
Cp:t   = -Fp*L/hVp*Cp:x - PS/hVp*(Cp-Cisf)  - Gp/hVp*Cp         + Dp*Cp:x:x;
Cisf:t =                  PS/hVisfp*(Cp-Cisf)- Gisf/hVisfp*Cisf + Disf*Cisf:x:x;
 E(t) at outflow

 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) ;
 Initial conditions for integrators
 Integrating to get Area
 Integrating to get mean time
 Integrating to get variance
 tbar_in is mean time of input function
 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;}
Calculate mooments of the poperator 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 );
 Should be Vp/Fp when PS = 0, else (Vp+Visfp)/Fp
 but these are only correct if S1_out =S1_in

Fp          ________________________________________
Cin(t) ---> |Vp                               Cp(t)|---> Cout(t)
            |Gp         ^                          |              
            |Dp         |                    PLASMA|
            ___________PS _________________________|
            |Visfp      |                   Cisf(t)|
            |Gisf       V              INTERSTITIAL|
            |Disf                      FLUID REGION|
            |--> x
 Fp : Plasma Flow Rate, (ml/g)/min
 Vp : Plasma Volume,  ml/g 
 Visfp: Volumes of Distribution,  ml/g  
 PS: 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

 One-dimensional convection-permeataion-diffusion-
 reaction model consisting of two concentric cylinders separated by a 
 membrane. The central plasma region of volume Vp has flow Fp, first order 
 consumption, Gp, and axial diffusion (disperion) Dp. Units are physiological
 per gram of tissue so that a single unit can model a homogeeously perfused 
 organ. Radial diffusion is assumed instantaneous (short radial distances).
 Exchange into a second surrounding non-flowing region is passive with
 conductance, PS, the Permeability capillary Surface area product, which 
 is set to vary with position x/L along the capillary according to Pratio = PS(x=L)/PS(x=0). 
 This interstitial fluid region, isf, of volume Visf, like the capillary, 
 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 consumption, Gisf, is
 first order. A gradient in PS(x) along the capillary with Pratio  = PS(x=L)/PS(x=0) =
 about 2 tend to even out the gradients in Cisf.
 This model is used in multicapillary models as one of a set of units in parallel.

 A third PDE provides an intravascular reference indicator Cref(t,x), and against 
 which an instanteous calculation of the transcapillary extraction:
 E(t) = (Crefout-Cout)/Crefout  at the outflow.


The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

Download JSim model project file

Help running a JSim model.

   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.) Like this model, has gradients in PS along x/L. See Model 316 at

   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.)
Key terms
Blood-tissue exchange
gradient permeability

Please cite in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

Model development and archiving support at provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.