Fig 1: Inflow, Outflow and ISF Concentrations: Default
Upper Plot displays the inflow and outflow concentrations
as well as the concentration in the ISF at the end of
the capillary.
Lower Plot shows snapshots at various times of the
concentration in the capillary as a function of distance
along the capillary. Lowest two curves are the snapshots of
the concentration in the ISF and parenchymal cell
as a function of distance.
Fig 2: Run Loops: LOOPparms
The model is run with four different sets of PSg and PSpc:
Run 1: PSg=0.5, PSpc=3 (standard run)
Run 2: PSg=5.0, PSpc=3 (shows effect of increasing just PSg. Peak
is shifted to later in time and smaller in magnitude. Backflux
increases magnitude of tail.
Run 3: PSg=30, PSpc=30. The PS's are very large and this essentially
becomes a single region model with volume =0.8 ml/gram of tissue.
Run 4: Psg=0, PSpc=0, Vp=0.8. Shows the correspondence between high
PS models with single region model with Vp=sum of the volumes of
all region.
Fig 3: Optimization: PreOpt
Load the PreOpt parameter set. Run the model.
Then Run the Optimizer.
Data was generated with PSg=1 ml/(g*min), PSpc=5 ml/(g*min),
Gpc = 0 ml/(g*min).
20% proportional noise was added to the output curve, e.g,
Cout*(1+0.4*(random()-0.5)) with t.delta=0.1, then
data was selected at 1 second intervals, beginning at t=4 seconds.
Running the optimizer yields PSg=1.179 ml/(g*min),
PSpc=5.197 ml/(g*min), and Gpc= 0.567.
Fig 4: Sensitivity Functions: SNSparms
The sensitivity of an output variable to a parameter is given as
d(Variable)/d(Parameter) = Variable:Parameter
A non-dimensionalsensitivity is given by
d(ln(Variable))/d(ln(Parameter)) = (Variable:Parameter)*(Parameter/Variable)
Sensitivity functions are calculated for d(Cout)/d(Psg), d(Cout)/d(PSpcp), and
d(Cout)/d(Gpc). Note that increasing PSg causes a decrease in peak Outflow.
Increasing PSg causes an increase in the height of the tail (11-30 seconds).
Increasing PSpc also increases the height of the tail after 16seconds.
After 25 seconds, Cout is approximately 50 times more sensitive to changes
in PSg than to changes in PSpc.
// MODEL NUMBER: 0081
// MODEL NAME: BTEX30
// SHORT DESCRIPTION: Models a tissue cylinder consisting of three
// regions: plasma, interstitial fluid, and parenchymal cells.
import nsrunit; unit conversion on;
math btex30_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
pc = PARENCHYMAL CELL */
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')
Vpcp = 0.6 ml/g, // pc (Vpc')
// PS means Permeability-surface area product between two regions
PSg = 1 ml/(g*min), // between p and isf
PSpc = 3 ml/(g*min), // between isf and pc
// G is the consumption rate coefficient in each region (Gulosity)
Gp = 0 ml/(g*min), // p
Gisf = 0 ml/(g*min), // isf
Gpc = 6.0 ml/(g*min), // pc
// D is the axial diffusion coefficient
Dp = 1.0e-5 cm^2/sec, // p
Disf = 1.0e-6 cm^2/sec, // isf
Dpc = 1.0e-6 cm^2/sec; // pc
// 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 hVpcp =if(Vpcp>0) Vpcp 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
Cpc(t,x) mM, // pc
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; Cpc:x = 0; }
when (x=x.max) { Cp:x = 0; Cisf:x = 0; Cpc:x = 0; Cout = Cp;}
// INITIAL CONDITIONS
when (t=t.min) { Cp = 0; Cisf = 0; Cpc = 0; }
// PARTIAL DIFFERENTIAL EQUATIONS
Cp:t = -Fp*L/hVp*Cp:x -Gp/hVp*Cp+ Dp*Cp:x:x
+ PSg/hVp*(Cisf-Cp);
Cisf:t = -Gisf/hVisfp*Cisf + Disf*Cisf:x:x
+ PSg/hVisfp*(Cp-Cisf) + PSpc/hVisfp*(Cpc-Cisf);
Cpc:t = -Gpc/hVpcp*Cpc +Dpc*Cpc:x:x
+ PSpc/hVpcp*(Cisf-Cpc);
}
/*
FIGURE:
Fp ________________________________________
Cin(t) ---> |Vp Cp(t)|---> Cout(t)
|Gp ^ |
|Dp | PLASMA|
___________PSg_________________________|
|Visfp | Cisf(t)|
|Gisf V ^ INTERSTITIAL|
|Disf | FLUID REGION|
_______________________Pspc_____________
|Vpcp | Cpc(t)|
|Gpc V PARENCHYMAL|
|Dpc CELL|
________________________________________
|<----------------L------------------->|
|--> x
DETAILED DESCRIPTION:
These partial differential equations model a "tissue cylinder"
consisting of three regions. The three regions are capillary plasma,p;
interstitial fluid, isf; and parenchymal cell, pc; and are separated by one
barrier-- the membrane between the interstitial fluid and parenchymal cell.
In addition, there is a diffusional path from plasma to the isf.
KEY WORDS:
BTEX30,PDE,convection,diffusion permeation,reaction,distributed,capillary,
plasma,isf,interstitial,fluid,parenchymal,cell
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.
REVISED JBB:08MAR16: Updated notes and parameter sets
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2016 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.
When citing JSim please use this reference: Butterworth E, Jardine BE, Raymond GM, Neal ML, Bassingthwaighte JB.
JSim, an open-source modeling system for data analysis [v3; ref status: indexed, http://f1000r.es/3n0]
F1000Research 2014, 2:288 (doi: 10.12688/f1000research.2-288.v3)
This software was developed with support from NIH grants HL088516 and HL073598, NIBIB grant BE08417,
the Cardiac Energy Grid HL199122 (PI: J.B. Bassingthwaighte), and the Virtual Physiological Rat program
GM094503 (PI: D.A.Beard). Please cite these grants 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.
*/
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