// MODEL NUMBER: 0146
// MODEL NAME: Crone63_BTEX
// SHORT DESCRIPTION: Models a tissue cylinder consisting of
// two regions: plasma, and interstitial fluid. Fit to Crone 1963 data.
import nsrunit; unit conversion on;
math Crone1963_BTEX {
// 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/BLOOD
isf = INTERSTITIAL FLUID REGION
*/
real Fp = 1 ml/(g*min), // blood flow
// VOLUMES (virtual volume)
Vp = 0.05 ml/g, // capillary blood
Visf = 0.15 ml/g, // isf (Visf')
OrgW = 1g, // Organ weight
DoseS = 1 umol, // Dose inj substrate
DoseR = 1 umol, // Dose inj vascular ref
// Capillary radius in dog, from Birks EK, 1994:
CapRadius = 3.4e-4 cm, // Cap radius in dog
CapArea = Vp*2/CapRadius,
// PS means Permeability-surface area product between two regions
PSg = 1 ml/(g*min), // between p and isf
// G is the consumption rate coefficient in each region (Gulosity)
Gp = 0 ml/(g*min), // p
Gisf = 0 ml/(g*min), // isf
// D is the axial diffusion coefficient
Dp = 1.0e-5 cm^2/sec, // p
Disf = 1.0e-6 cm^2/sec; // isf
// The hVolumes protect against zero divides
private real hVp =if(Vp>0) Vp else (1e-6 ml/g);
private real hVisf =if(Visf>0) Visf else (1e-6 ml/g);
// INFLOWING CONCENTRATION
extern real Cin(t) 1/sec;
real CinS(t) = DoseS*Cin(t)/(Fp*OrgW),
CinR(t) = DoseR*Cin(t)/(Fp*OrgW);
real CinSsum = sum(t=t.min to t.max, CinS);
real CinRsum = sum(t=t.min to t.max, CinR);
// CONCENTRATION VARIABLES
real Cp(t,x) mM, // p
Cisf(t,x) mM, // isf
Cout(t) mM, // Outflow Concentration from plasma region
Cp_ref(t,x) mM, // Plasma reference
Cpout_ref(t) mM; // Ref outflow concentration from plasma region
// Extraction:
real Extract(t) dimensionless;
Extract = if ((Cpout_ref >0.1) and (Cout*(DoseR/DoseS) |Vp Cp(t)|---> Cout(t)
|Gp ^ |
|Dp | PLASMA|
___________PSg_________________________|
|Visfp | Cisf(t)|
|Gisf V INTERSTITIAL|
|Disf FLUID REGION|
________________________________________
|<----------------L------------------->|
|--> x
DETAILED DESCRIPTION:
These partial differential equations model a "tissue cylinder"
consisting of two regions. The two regions are capillary plasma,
p, and interstitial fluid, isf. There is a diffusional path from plasma
to the isf. A consumption term is included but is set to zero here.
Model is fit to Crone 1963 data for sucrose uptake into
hind leg of a dog and inulin uptake into the kidney. This paper was
one of the first attempts at characterizing capilliary permeability
in various organs and regions of the body. Vascular reference is Evans' Blue Dye.
KEY WORDS:
BTEX20, PDE, convection, diffusion, permeation, reaction, distributed, capillary,
plasma, isf, interstitial fluid, Crone, Indicator dilution, sucrose,
vascular reference, data
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.
Christian Crone, The Permeability of Capillaries in Various Organs as
Deterimined by Use of the 'Indicator Diffusion' Method,
Acta physiol.scand. 58:292-305, 1963,
REVISION HISTORY:
Author BEJ 02/15/11
- Based on Model #0080: btex20
- Fit to Crone 1963 data
Revised by: BEJ Date:15may13 : Update optimization params
Copyright (C) 1999-2013 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.
*/