Fig 1: Inflow and Outflow: Default
All model parameters are set to their default values. The
inflow and outflow concentrations are plotted as a function
of time.
Fig 2: Effectively BTEX10,20,30,40, and 50: RunLoops
First Loop: Only plasma space is open.
Second Loop: Plasma and isf are open.
Third Loop: Plasma, isf, and pc are open. Note that the
peak heights of the outflow are the same, because
PSpc is the second barrier a molecule "sees."
Fourth Loop: plasma, isf, pc, and ec are open. Note that the
peak height is lower, than in second and third loop,
because a molecule can escape into either the isf or the
ec.
Fifth Loop: plasma, isf, pc, ec, and mito are open. There is
consumption in the mito region.
By setting different PS's to zero, the model can be reduced from
a five region model to four, three, two, or one regions.
Fig 3: Sensitivity d(log(Cout))/d(log(PS)) (RUN SENSITIVITY): SNS
Sensitivity can be calculated in two different ways:
Upper Panel: d(log(Cout))/d(log(PS)) = (Cout:PS)/(PS/Cout),
which is dimensionless, and
Lower Panel: d(Cout)/d(PS) = Cout:PS,
which is dimensional.
// MODEL NUMBER: 0083
// MODEL NAME: BTEX50
// SHORT DESCRITPION: Models a tissue cylinder consisting of five regions: plasma,
// interstitial fluid,endothelial cells, parenchymal cells, and mitochondria.
import nsrunit; unit conversion on;
math btex50_pde {
// INDEPENDENT VARIABLES
realDomain t sec ; t.min=0; t.max=30; t.delta=0.1;
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;
/* PARAMETERS AND KEY TO NAMES
p = PLASMA
isf = INTERSTITIAL FLUID REGION
ec = ENDOTHELIAL CELL
pc = PARENCHYMAL CELL
mito = MITOCHONDRIA */
real Fp = 1 ml/(g*min), // Plasma flow
// VOLUMES (a p after the region name implies "prime", a virtual volume)
Vp = 0.05 ml/g, // p
Visfp = 0.15 ml/g, // isf (Visf')
Vecp = 0.02 ml/g, // ec (Vec')
Vpcp = 0.6 ml/g, // pc (Vpc')
Vmitop = 0.01 ml/g, // mito (Vmito')
// PS means Permeability-surface area product between two regions
PSg = 1 ml/(g*min), // between p and isf
PSecl = 2 ml/(g*min), // between p and ec
PSeca = 2 ml/(g*min), // between ec and isf
PSpc = 3 ml/(g*min), // between isf and pc
PSmito = 1 ml/(g*min), // between pc and mito
// G is the consumption rate coefficient in each region (Gulosity)
Gp = 0 ml/(g*min), // p
Gisf = 0 ml/(g*min), // isf
Gec = 0.0 ml/(g*min), // ec
Gpc = 0.0 ml/(g*min), // pc
Gmito = 6.0 ml/(g*min), // mito
// D is the axial diffusion coefficient
Dp = 1.0e-5 cm^2/sec, // p
Disf = 1.0e-6 cm^2/sec, // isf
Dec = 1.0e-6 cm^2/sec, // ec
Dpc = 1.0e-6 cm^2/sec, // pc
Dmito = 1.0e-6 cm^2/sec; // mito
// The hVolumes protect 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);
private real hVecp =if(Vecp>0) Vecp else (1e-6 ml/g);
private real hVpcp =if(Vpcp>0) Vpcp else (1e-6 ml/g);
private real hVmitop =if(Vmitop>0) Vmitop else (1e-6 ml/g);
// INFLOWING CONCENTRATION
extern real Cin(t) mM;
// CONCENTRATION VARIABLES
real Cp(t,x) mM, // p
Cisf(t,x) mM, // isf
Cec(t,x) mM, // ec
Cpc(t,x) mM, // pc
Cmito(t,x) mM, // mito
Cout(t) mM; // Outflow Concentration from plasma region
// BOUNDARY CONDITIONS (Note total flux BC for inflowing region.)
when (x=x.min) { (-Fp*L/hVp)*(Cp-Cin)+Dp*Cp:x =0;
Cisf:x = 0; Cec:x = 0; Cpc:x = 0; Cmito:x = 0;}
when (x=x.max) { Cp:x = 0; Cisf:x = 0; Cec:x = 0; Cpc:x = 0; Cmito:x = 0; Cout = Cp;}
// INITIAL CONDITIONS
when (t=t.min) { Cp = 0; Cisf = 0; Cec = 0; Cpc = 0; Cmito = 0;}
// PARTIAL DIFFERENTIAL EQUATIONS
Cp:t = -Fp*L/hVp*Cp:x -Gp/hVp*Cp+ Dp*Cp:x:x
+ PSg/hVp*(Cisf-Cp) + PSecl/hVp*(Cec-Cp);
Cisf:t = -Gisf/hVisfp*Cisf + Disf*Cisf:x:x
+ PSg/hVisfp*(Cp-Cisf) + PSeca/hVisfp*(Cec-Cisf)
+ PSpc/hVisfp*(Cpc-Cisf);
Cec:t = -Gec/hVecp*Cec +Dec*Cec:x:x
+PSecl/hVecp*(Cp-Cec) +PSeca/hVecp*(Cisf-Cec);
Cpc:t = -Gpc/hVpcp*Cpc +Dpc*Cpc:x:x
+ PSpc/hVpcp*(Cisf-Cpc) + PSmito/hVpcp*(Cmito-Cpc);
Cmito:t = -Gmito/hVmitop*Cmito +Dmito*Cmito:x:x
+ PSmito/hVmitop*(Cpc-Cmito);
}
/*
FIGURE:
Fp ________________________________________
Cin(t) ---> |Vp Cp(t,x)|---> Cout(t)
|Gp ^ |
|Dp | PLASMA|
______PSecl_____ ^ _________________|
|Vecp | \ | \ Cec(t,x)|
|Gec V ^ \ PSg \ ENDOTHELIAL|
|Dec | \ | \ CELL|
__________PSeca____\ v \_____________|
|Visfp | Cisf(t,x)|
|Gisf V ^ INTERSTITIAL|
|Disf | FLUID REGION|
_______________________Pspc_____________
|Vpcp | Cpc(t,x)|
|Gpc ^ V PARENCHYMAL|
|Dpc | CELL|
| ----------PSmito----------------- |
| |Vmitop | Cmito(t)| |
| |Gmito V | |
| |Dmito MITOCHONDRIA| |
| --------------------------------- |
________________________________________
|<----------------L------------------->|
|--> x
DETAILED DESCRIPTION:
These partial differential equations model a "tissue cylinder"
consisting of five regions. The five regions are capillary plasma,
p; endothelial cell, ec; interstitial fluid, isf; parenchymal cell,
pc; and mitochondria, mito; and are separated by four barriers--the
luminal or plasma surface and endothelial cell layer; the albuminal
surface of the endothelial cell facing the interstitium; the membrane
between the interstitial fluid and parenchymal cell; and the membrane
between the parenchymal cell and the mitochondria. In addition,
there is a diffusional path from plasma to ISF bypassing endothelial
cells via intercellular clefts.
KEY WORDS: BTEX50,PDE,convection,diffusion,permeation,reaction,distributed,capillary
plasma,isf,interstitial fluid,endothelial,parenchymal,cell,mitochondria
REFERENCES:
W.C. Sangren and C.W. Sheppard. A mathematical derivation of the
exchange of a labelled substance between a liquid flowing in a
vessel and an external compartment. Bull Math BioPhys, 15, 387-394,
1953.
C.A. Goresky, W.H. Ziegler, and G.G. Bach. Capillary exchange modeling:
Barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970.
J.B. Bassingthwaighte. A concurrent flow model for extraction
during transcapillary passage. Circ Res 35:483-503, 1974.
B. Guller, T. Yipintsoi, A.L. Orvis, and J.B. Bassingthwaighte. 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.
C.P. Rose, C.A. Goresky, and G.G. Bach. The capillary and
sarcolemmal barriers in the heart--an exploration of labelled water
permeability. Circ Res 41: 515, 1977.
J.B. Bassingthwaighte, C.Y. Wang, and I.S. Chan. Blood-tissue
exchange via transport and transformation by endothelial cells.
Circ. Res. 65:997-1020, 1989.
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:
Revised by BEJ 04/14/09
Boundary Conditions
Revised by GR 09/22/09
Added Statistics and Reformatted.
JSim SOFTWARE COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
JSim software was developed with support from NIH grants HL088516,
and HL073598. Please cite these grants in any publication for which
this software is used and send one reprint of published abstracts or
articles to the address given below. Academic use is unrestricted.
Software may be copied so long as this copyright notice is included.
Copyright (C) 1999-2008 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/