// MODEL NUMBER: 0205
/* MODEL NAME: Safford1978
SHORT DESCRIPTION: Calculates the bulk diffusion coefficient, Db, for water through a matrix
of cells surrounded by ECF, influenced by cell membrane permeability. This is contrasted with
results obtained from homogeneous sheet and dead-end pore models. From Safford et al. 1978 paper.
*/
import nsrunit;
unit uM = 1e-6 M; unit conversion on;
math saff1978 {
// Cell permeation model:
realDomain P cm/s; P.min=1e-6; P.max=1e-3; P.delta = 0.5e-6; // Sarcolemmal permeability to water
// PARAMETERS:
real alpha(P) 1/cm, E(P) cm, F(P) 1/cm, G(P) 1/cm, H(P) 1/cm, I(P) 1/cm, J(P) 1/cm,
K(P) dimensionless, n(P) dimensionless, Q(P) 1/cm, U(P) dimensionless,
W(P) dimensionless, X(P) dimensionless, Y(P) dimensionless,
Db(P) cm^2/s;
real V_chamber = 14.5 ml; // Volume of recipient and donar chambers
real l_avg = 0.14 cm; // Avg thickness of sheet
real At = 0.283 cm^2; // Total sheet area
real Vtiss = At*l_avg;
real f_water = 0.85 ml/ml; // Fractional water content - ml H2O per ml tissue
real Vwater = f_water*Vtiss;
real Ve; // Extracellular volume per ml tissue
real Vi; // cellular volume per ml tissue
real L = 12 um, // Edge length of end of a cardiac cell
Lzero = 2 um, // Thickness of extracellular space
Dw = 2.38e-5 cm^2/s, // diffusion coeff for water in water at 25C
Dsuc = 5.1e-6 cm^2/s, // diffusion coeff for sucrose in water at 25C
fe = 0.28, // fraction of free diffusion rate, e = extracellular
fi = 0.22, // fraction of free diffusion rate, i = intracellular
De = fe*Dw, // effective extracellular diffusion coeff
Di = fi*Dw; // effective intracellular diffusion coeff
real iVol = ((L^2/(L+Lzero)^2)*At*l_avg); // Cellular vol
Vi = iVol/Vtiss; // cellular volume ratio
Ve = (Vtiss -iVol)/Vtiss; // extracellular volume ratio
alpha = sqrt(2*P*(1/(L*Di) + 1/(Lzero*De))); // 1/cm
n = Di*alpha/P ; // dimensionless
W = (Lzero/L)*(De/Di); // dimensionless
X = W*(sinh(alpha*L) + n*cosh(alpha*L)); // dimensionless
Y = W*(cosh(alpha*L) + n*sinh(alpha*L)); // dimensionless
E = L + Di/P; // cm
F = (1-cosh(alpha*L))/L +P*(1+W)/Di; // 1/cm
G = sinh(alpha*L)/L + W*alpha; // 1/cm
H = (1+Y)/E +P*(1+W)/Di; // 1/cm
I = X/E - W*alpha; // 1/cm
U = (1/F)*(1/L + G*K); // dimensionless
J = G*H/F + I; // 1/cm
K = (1/J)* (1/E - H/(F*L)); // dimensionless
Q = alpha*(K*W - U*(1+W)/n); // 1/cm
// VARIABLES:
// bulk diff coeff for water, Eq 6:
Db = De/(Lzero/(L+Lzero)+1/(L*(Di/De)*(W*alpha*K-Q)-Lzero*(alpha*K+Q)));
real Dratio(P) = Db/Dw; // Ratio of bulk tissue diffusion to water diffusion
// ******************************************************************************
// Diffusion across an uneven sheet modeled using equations from Suenson 1974.
// Apply Db in the equations for diffusion from donar to receipient compartment (Dbulk is some value Db(P))
// This was not done in the Safford 1978 paper:
realDomain t min;t.min = 0;t.max = 200; t.delta = 0.1;
real M = 10 dimensionless; // Dimensionless, Total Number of terms in series approximation
realDomain m dimensionless; m.min =1;m.max = M; m.delta = 1;
real CR(t) dimensionless; // Ratio of tracer in receipient to donar compartment
// Bulk diffusion coeff is equivalent to D/lambda^2, where D is free-diffusion coeff in water
real DBulk = 0.003e-6 cm^2/sec; // Bulk diffusion coefficient calculated from above for a given P
real Dratio_water = DBulk/Dw; // Ratio of DBulk to Dw for diffusion value used to fit experimental data.
// *********************************************
// Multi-path (MP) parameters:
real N = 9 dimensionless; // Total number of paths
real l_rel_min dimensionless, l_rel_max dimensionless;
l_rel_min = 0.5; l_rel_max = 2; // Range of relative lengths
real l_width = if(N<2) (l_rel_max-l_rel_min)/N else (l_rel_max-l_rel_min)/(N-1); // check divide by zero
// relative paths used for weighting:
realDomain l_i dimensionless; l_i.min = l_rel_min;l_i.max=l_rel_max;l_i.delta = l_width;
real l_max cm;
l_max = l_avg*l_rel_max; // Max absolute path length
real l_min cm;
l_min = l_avg*l_rel_min; // Min abs path length
// Instead of actual length and areas from strips of tissue, use a distribution function to get relative lengths
// about l_avg and weights for each associated length:
extern real MPathDF(l_i) dimensionless; // Multi-path distribution function - returns weight for relative length
real wPathSum = sum(l_i=l_i.min to l_i.max, MPathDF *l_i.delta);
real weight(l_i) = MPathDF*l_i.delta;
// Put together the three terms for CR(t):
real thirdTermsum(t,m,l_i) = ((-1)^m/m^2) * exp((-DBulk*m^2*PI^2*t)/(l_i*l_avg)^2);
real thirdTermTotal(t,l_i) = sum(m=m.min to m.max, thirdTermsum);
real firstWeightedTerm(t) = (DBulk*At*t)*sum(l_i= l_i.min to l_i.max,weight(l_i)/(l_i*l_avg) );
real thirdWeightedTerm(t) = sum(l_i= l_i.min to l_i.max,(2*weight(l_i)*l_i*l_avg*At/(PI^2))*thirdTermTotal);
CR = (1/V_chamber)*(firstWeightedTerm - (l_avg*At)/(6) - thirdWeightedTerm);
} // end
/*
DETAILED DESCRIPTION:
Diffusion of water through a slab of uniform thickness. Tracer water at side 1 diffuses
through a matrix of cells evenly spaced throughout an extracellular space, ECF.
Cells are square beams,L by L, on a rectangular lattice, separated by Lzero.
Diffusion occurs through both ECF and cells in parallel. The cells have permeability P
on all surfaces allowing exchange between cells and ECF. (The cell shape (square,
hexagonal or cylindrical beams) has negligible effect.). Given fixed intracellular
and extracellular Ds, P dominates the effective intratissue effective bulk diffusion
coefficint Db.
Using P as an independent variable allows one to show the bulk D as a function
of P and to vary other parameters in the loops. Running the program from
P=P,min -1e-6 to P.max = 0.4 gives the plot in Figure 7 of Safford 1978 for which the
cell sizes and surface area matches that of cardiac tissue.
See also Figures 5 and 6 for variation in dimensions and Table II, p527.
This steady state Db was estimated experimentally by Safford from the slope of
dCr/dt in an experiment in which the tissue lies between Compartment 1
(stirred) with fixed concentration and tracer diffuses into compartment 2
whose concentration Cr(t) is initally zero. (A type of Barrer time-lag study.)
The two other models presented here (Sheet Diffusion and Dead-end Pore) were previously
presented in papers Suenson et al., 1974 and Safford et al., 1977. These models
predict higher hindrance of water in tissue (ratio of observed to free) compared to
that of sucrose due to physiologically unrealistic values for water permeability and
water space available for diffusion.
SHORTCOMINGS/GENERAL COMMENTS:
- There is instability in the solution at high P, approaching 1,
but it is reduced by enlarging Lzero. The cause is presumably stiffness
that occurs when exchange rates are high and the ECF volume small.
- Use Suensen eq w/ Db to get amt diffusing across sheet/slab from chamber 1 to chamber 2,
do not need dead-end pores as cell permeation model takes into account diffusion into and
around cells.
KEY WORDS: Diffusion, barrer, sheet diffusion, dead-end pore, DEP, publication, data, water,
sucrose, cell geometry, permeation, PMID722277
REFERENCES:
(primary): Safford RE, Bassingthwaighte EA, and Bassingthwaighte JB.
Diffusion of water in cat ventricular myocardium.
J Gen Physiol 72: 513-538, 1978.
BARRER, R. M. 1953. A new approach to gas flow in capillary systems. J. Phys. Chem.
57:35-40.
BASSINGTHWAIGHTE, J. B., and H. REUTER. 1972. Calcium movements and excitation-contraction
coupling in cardiac cells. In Electrical Phenomena in the Heart. W. C.
DeMello, editor. Academic Press, Inc., New York. 353-395.
BASSINGTHWAIGHTE, J. B., T. YIPINTSOI, and R. B. HARVEY. 1974. Microvasculature of
the dog left ventricular myocardium. Microvasc. Res. 7:229-249.
BERGER, W. K. 1972. Correlation between the ultrastructure and function of intercellular
contacts. In: Electrical Phenomena in the Heart. W. C. DeMello, editor. Academic
Press, Inc., New York. 63-88.
BIRD, R. B., W. E. STEWART, and E. N. LIGHTFOOT. 1960. Transport Phenomena. John Wiley & Sons, Inc., New York. 780 pp.
BLINKS, J . R. 1965. Influence of osmotic strength on cross-section and volume of isolated
single muscle fibres. J. Physiol. (London). 177:42-57.
BOYLE, P. J., and E. J. CONWAY. 1941. Potassium accumulation in muscle and associated
changes. J. Physiol. ( Lond. ). 100:1-63.
CRANK, J. 1956. The Mathematics of Diffusion. Oxford University Press, London. 347
PP.
GOODKNIGHT R. C., and I. FATT. 1961. The diffusion time-lag in porous media with
dead-end pore volume.J. Phys. Chem. 65:1709-1712.
PAGE, E., and R. S. BERNSTEIN. 1964. Cat heart muscle in vitro. V. Diffusion through a
sheet of right ventricle.J. Gen. Physiol. 47:1129-1140.
SAFFORD, R. E., and J. B. BASSINGTHWAIGHTE. 1977. Calcium diffusion in transient and
steady states in muscle. Biophys. J. 20:113-136.
SCHAFER, D. E., and J. A. JOHNSON. 1964. Permeability of mammalian heart capillaries
to sucrose and inulin. Am. J. Physiol. 206:985-991.
SUENSON, M., D. R. RICHMOND, and J. B. BASSINGTHWAIGHTE. 1974. Diffusion of
sucrose, sodium and water in ventricular myocardium. Am. J. Physiol. 227:1116-1123.
REVISION HISTORY:
Original Author : JBB Date: 04/dec/13
Revised by: BEJ Date:01jan14 : Added sheet and DEP model. added data from Safford '78 paper
Added notes and parameter sets.
Revised by: BEJ Date:02mar15: typo in comment fixed
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2015 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.
*/