/* MODEL NUMBER: 0137
MODEL NAME: Renkin2region59
SHORT DESCRIPTION: Models uptake of tracer potassium by skeletal muscle during continuous
perfusion, flow constant or varied. Estimates the PS for the single composite barrier
between axially distributed plasma and the cell potassium pool. (derived from model 0080)
*/
import nsrunit; unit conversion on;
math Renkin59 {
// INDEPENDENT VARIABLES
realDomain t min; t.min=0; t.max=30; t.delta=0.1;
realDomain x cm; real L=0.1 cm, Ngridx =61; x.min=0; x.max=L; x.ct=Ngridx;
realDomain z cm; real Lz=1 cm, Ngridz =11; z.min=0; z.max=Lz; z.ct=Ngridz;
private x.min, x.max, x.ct, z.min, z.max, z.ct;
/* PARAMETERS AND KEY TO NAMES
p = capillary PLASMA
pc = parenchymal or muscle cell*/
// EXTERNAL INPUTS for governing flow and INFLOW TRACER CONCENTRATION
extern real Cin(t) mM; // Uses fgen1: Const. Infusion: click on circled squiggle
extern real Fratio(t) dimensionless; // Uses fgen2: Use pulse sequence for time varying flow
real Scale = 1 dimensionless,// with Flow fgen2 Ampl = 1, set scale = actual flow
Fp(t) ml/(g*min), // Plasma flow = Blood Flow*(1-Hct)
Hct = 0.51 ml/ml, // Hematocrit
// VOLUMES
Vpipe= 0.0036 ml/g, // cannula volume leading to artery, dispersive
Vp = 0.005 ml/g, // plasma
Vpc = 10.0 ml/g, // Virtual cell K+ space = cell vol x
// [K+]cell / [K+]plasma = 0.6*30 = 18 ml/g.
// The value 10.0 is less than expected for K+
// PS is Permeability-surface area product for "membrane" or barrier
// for exchange, i.e. endothelium, ISF, and cell membrane combined
// Consumption is not considered since there is none for K+
// Tracer decay could be considered: K-42 half life is 12.44 hours
PS(t) ml/(g*min), // PS between p and cell (composite membrane)
PS0 = 0.135 ml/(g*min), // PS at zero flow
A1 = 0.3 dimensionless, // coeff for PS(Flow) = PS0(A1 + A2*Flow)
A2 = 0.1 g*min/ml, // slope coeff for PS(Flow) - Flow dependence term of PS
thalfK= 12.44 hr, // half life for K42 is 12.44 hours, but data
// were corrected for isoptic decay
Mmuscle = 11.2 g, // Mass of perfused skeletal muscle
Flowinit= 1.37 ml/min, // initial blood flow (Fp(t) is changed by Fratio)
// D: axial dispersion coefficients in each region
Dz = 3e-2 cm^2/sec, // in pipe, dispersion is > molec diffusion
Dp = 8e-4 cm^2/sec, // in plasma, dispersion is > molec diffusion
Dpc = 2e-6 cm^2/sec; // parenchymal cell ... slow diffusion
// The hVolumes protect against zero divides
private real hVp =if(Vp>0) Vp else (1e-6 ml/g);
private real hVpc =if(Vpc>0) Vpc else (1e-6 ml/g);
private real hVpipe =if(Vpipe>0) Vpipe else (1e-6 ml/g);
// CONCENTRATION VARIABLES
real Cindec(t) mM, // tracer concn Cin corrected for decay
Cpipe(t,z) mM, // tracer concn in cannula to artery
Cpipeout(t) mM, // tracer concn out from cannula into artery
Cp(t,x) mM, // p for tracer k-42. Chemical [K]p = 4.58 mM
Cpc(t,x) mM, // pc for tracer K-42. [K]pc ~ 150 mM
Cout(t) mM, // Outflow Concentration from plasma region
Extract(t) dimensionless; // transoprgan A-V diff divided by C inflowing
// BOUNDARY CONDITIONS (Note total flux BC for inflowing region.)
when (z=z.min) { (-Fp*Lz/hVpipe)*(Cpipe-Cindec)+Dz*Cpipe:z = 0; }
when (z=z.max) { Cpipe:z = 0; Cpipeout = Cpipe;}
when (x=x.min) { (-Fp*L/hVp)*(Cp-Cpipeout)+Dp*Cp:x = 0; Cpc:x = 0; }
when (x=x.max) { Cp:x = 0; Cpc:x = 0; Cout = Cp;}
// INITIAL CONDITIONS
when (t=t.min) { Cpipe = 0; Cp = 0; Cpc = 0; }
// PARTIAL DIFFERENTIAL EQUATIONS
Cindec = Cin*exp(-0.69*t/thalfK);
Fp = Fratio*(Flowinit/Mmuscle)*(1-Hct);
PS = PS0*(A1 + A2*Fp);
Cpipe:t= -Fp*Lz/hVpipe*Cpipe:z + Dz*Cpipe:z:z;
Cp:t = -Fp*L/hVp*Cp:x + Dp*Cp:x:x + PS/hVp*(Cpc-Cp);
Cpc:t = Dpc*Cpc:x:x + PS/hVpc*(Cp-Cpc);
Extract= if (Cpipeout > 0) 1 - Cout/Cpipeout else 0;
real PStheor(t) ml/(g*min);
PStheor = -Fp*ln(1-Extract);
}
/*
FIGURE: Blood-tissue exchange in muscle capillaries
Fp _________ ______________________________
Cin(t) --->| Vpipe | |Vp Cp(t)|
| |-->-| ^ |-> Cout(t)
|PIPE, z | |Dp | PLASMA|
|________| |___________PS _______________|
|Vpc | Cpc(t)|
| V |
|Dpc PARENCHYMAL CELL|
|_____________________________|
|<------------ x/L ---------->|
0 1.0
DETAILED DESCRIPTION:
An entry spatial domain, z, accounts for delay in the cannula leading
to the arterial inflow. Model is an axially distributed pipe, BTEX10, with its
dispersion controlled by the dispersion coeffient, Dz.
These partial differential equations model a "tissue cylinder"
consisting of two regions, the capillary plasma, p, and parenchymal cell, pc.
The PS is that for a composite membrane with components in series (endothelium,
ISF, cell membrane). This is a tracer experiment so that chemical concentrations
do not change, and PS is not dependent on tracer concentration ir on the
mechanism of its conductance.
Isotopic decay is not the cause of the diminution in the inflow
concentration: Renkin states (1959a) that the data have all been
corrected for the decay. (The half life of K42 is 12.44 hours.)
The first data set from Renkin 1959a, Figure 1A, has flow 1.36 to 1.38 ml/min
for a 11.2 g muscle, 0.122 ml/(g*min). (Set this in "scale" with fgen2 Amp=1.)
Renkin's expt P26 was estimated to have PS = 4.8 ml / (min* 100 g) at a flow of
10 ml/(min*100g), while the solution modeled above with no PS flow dependence (A2=0)
estimated it to be 0.091 ml/(g*min) or 9.1 ml / (min* 100g) at a flow of
12.1 ml/(min*100g). Our calculated theoretical PS (PStheor) is 0.046 ml/(g*min)
or 4.6 ml/(min*100g). Renkins estimate was interpolated from the clearance of K+ at many flows.
Renkin's estimate does not explicitedly take into account back-diffusion of K+
back into the capillary but rather adds a correction to the outflow K42 counts
based on injection of tracer free blood afterwards and the outflow activity at that
point is considered due to back-diffusion.
When PS flow dependence is added and our solution fitted to variable
flow data (Renkin 1959a Figure 1B, expt P27, Load and run parameter set 'Fig1B')
the estimated PS is 0.033 ml/(g*min) or 3.3 ml/(min*100g) at a flow of
9.7 ml/(min*100g). Renkin's interpolated PS estimate was 6.4 ml/(min*100g) at
flow of 10 ml/(min*100g). Renkin's PS estimate corresponds to the Maximum PStheor
value from our model of 6.8 ml/(min*100g). If we turn off flow dependence (A2=0 and A1=1) we get
estimated PS of 3.2 ml/(min*100g) showing very little PS flow dependence at flows
modeled here (As the model stands, there is high correlation between A1, A2 and PS0).
The flow dependence coefficients A1 and A2 relate to the physiology in terms of
vascular perfusion. A1 relates to regions well perfused and should be set to 1 or zero
while A2 represents regions initially poorly perfused but as arterial pressure increases,
blood flow through the capillaries of poorly perfused regions increase more rapidly
then those regions already well perfused (Renkin 59b paper). The Renkin data presented
here shows little flow dependence (Set A2=0 and A1=1 and fit Fig1B data to confirm).
KEY WORDS:
BTEX20, PDE, convection, diffusion, permeation, reaction, distributed,capillary,
plasma, isf, interstitial fluid, Data
REFERENCES:
Renkin EM. Transport of potassium-42 from blood to tissue in isolated
mammalian skeletal muscles. Am J Physiol 197: 1205-1210, 1959a.
Renkin EM. Exchangeability of tissue potassium in skeletal muscle.
Am J Physiol 197: 1211-1215, 1959b.
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.
Crone C; The permeability of capillaries in various organs as determined by the
use of the `indicator diffusion' method. Acta Physiol Scand 58: 292-305, 1963.
Crone C; Facilitated transfer of glucose from blood into brain tissue. J Physiol
181: 103-113, 1965.
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.
Bohr C. U:ber die spezifische Ta:tigkeit der Lungen bei der respiratorischen
Gasaufnahme und ihr Verhalten zu der durch die Alveolarwand stattfindenden Gasdiffusion.
Skand Arch Physiol 22: 221-280, 1909.
Pappenheimer JR, Renkin EM, and Borrero LM. Filtration, diffusion and molecular
sieving through peripheral capillary membranes. A contribution to the pore theory
of capillary permeability. Am J Physiol 167: 13-46, 1951.
Conn HL Jr and Robertson JS. Kinetics of potassium transfer in the left ventricle of
the intact dog. Am J Physiol 181: 319-324, 1955.
Renkin EM. Effects of blood flow on diffusion kinetics in isolated, perfused
hindlegs of cats. A double circulation hypothesis. Am J Physiol 183: 125-136, 1955.
Crone C. Does `restricted diffusion' occur in muscle capillaries?.
Proc Soc Exp Biol Med 112: 453-455, 1963.
Kellen MR and Bassingthwaighte JB. An integrative model of coupled water and solute
exchange in the heart. Am J Physiol Heart Circ Physiol 285: H1303-H1316, 2003a.
Kellen MR and Bassingthwaighte JB. Transient transcapillary exchange of water driven
by osmotic forces in the heart. Am J Physiol Heart Circ Physiol 285: H1317-H1331, 2003b.
Renkin EM and Rosell S. Influence of sympathetic adrenergic vasoconstrictor nerves on
transport of diffusible solutes from blood to tissues in skeletal muscle. Acta Physiol Scand 54: 223-240, 1962.
Sheehan RM and Renkin EM. Capillary, interstitial, and cell membrane barriers to
blood-tissue transport of potassium and rubidium in mammalian skeletal muscle. Circ Res 30: 588-607, 1972.
Tancredi RG, Yipintsoi T, and Bassingthwaighte JB. Capillary and cell wall permeability
to potassium in isolated dog hearts. Am J Physiol 229: 537-544, 1975.
Yipintsoi T, Tancredi R, Richmond D, and Bassingthwaighte JB. Myocardial extractions of
sucrose, glucose, and potassium. In: Capillary Permeability (Alfred Benzon Symp. II),
edited by Crone C and Lassen NA. Copenhagen: Munksgaard, 1970, pp 153-156.
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 by: JBB Date:18oct11 : Applied 2 region BTEX model to Renkin data
JSim SOFTWARE COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
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.
*/