Models in project file (Safford1978.proj):
1. 'Safford78': Calculates the bulk diffusion coefficient, Db, for water through a matrix
of cells surrounded by ECF, influenced by cell membrane permeability.
2. 'SheetDiffusion': Barrer type diffusion across a slab of varying thickness. From Suenson
et al. 1974.
3. 'Deadend_Pore': Parallel pathway, dead-end pore model (DEP) that accounts for sequestration
or binding of water within heart muscle sheet. From Safford and Bassingthwaighte, 1977
Parameter sets:
Use with model 'Deadend_Pore':
'DEP_fig2_H2O': DEP model fit to H2O diffusion data from Figure 2 of Safford 1978.
Note exchange rate constant (Ke), diffusion channel and DEP volume, and
the tissue diffusion coefficient for water (D_tiss). View results on
plotpage 'Fig2_H2O_plot'. Use ratio of tissue to free diffusion for
water and sucrose to compare between models and compare to values found in
Table 1 of Safford 1978 paper.
'DEP_fig2_suc': DEP model fit to sucrose diffusion data from Figure 2 of Safford 1978. Note
THe dead-end pore volume goes to zero showing that little sucrose is 'bound'
within the tissue, as expected. Use plot page 'Fig2_suc_plot'.
'DEP_fig3_H2O': DEP model fit to H2O diffusion data from Figure 3 of Safford 1978. Note
exchange rate constant (Ke), diffusion channel and DEP volume, and the tissue diffusion coefficient for water (D_tiss). View results on plotpage
'Fig3_H2O_plot' and compare to values found in Table 1 of Safford 1978 paper.
Use with model 'Safford78':
'fig2_H2O': Cell permeation model fit to Figure 2 of Safford 1978. Uses the sheet diffusion
equation but with Bulk diffusion term , Db(ulk), and the diffusional area, Ad,
is now just the total tissue area, At. The original paper did not explicitedly
fit this cell permeation model to the data but rather used it to just calculate
the diffusion coefficient for water in the tissue, Db, and confirmed the ratio
of Db to free diffusion coefficient is higher than that of sucrose.
'fig3_H2O': Cell permeation model fit to Figure 3 of Safford 1978. Uses the sheet diffusion
equation but with Bulk diffusion term , Db(ulk), and the diffusional area, Ad,
is now just the total tissue area, At. The original paper did not explicitedly
fit this cell permeation model to the data but rather used it to just calculate
the diffusion coefficient for water in the tissue, Db, and confirmed the ratio
of Db to free diffusion coefficient is higher than that of sucrose.
'fig5': Reproduce figure 3 of Safford 1978 paper. Load 'Safford78' model, confirm parameter set 'fig5' is loaded, and go to the 'Loops' tab and hit the 'Run' button. View results
in plot page 'Fig5_plot'. Simulation loops through values for cell size, L, and gap
space, Lzero. Adjust values for Lzero and L to get effect geometry has on bulk
diffussion as a function of cell permeability.
'fig7': Reproduce figure 7 of Safford 1978 paper. Use with plot page 'Fig7_DB.vs.P'. Shows
relationship between P and Db using most probable parameter values.
Use with model 'SheetDiffusion':
'sheet_fig2_H2O': Sheet diffusion model fit to H2O diffusion data from Figure 2 of
Safford 1978. Note diffusion channel volume, and the tissue diffusion
coefficient for water (Dtiss). View results on plotpage 'Fig2_H2O_plot'.
'sheet_fig2_suc': Sheet diffusion model fit to sucrose diffusion data from Figure 2 of
Safford 1978. Note diffusion channel volume, and the tissue diffusion
coefficient for sucrose (Dtiss). View results on plotpage 'Fig2_suc_plot'.
Plot pages:
'Fig7_DB.vs.P': Use with param set 'fig7'
'Fig2_suc_plot': Use with param sets 'sheet_fig2_suc, 'DEP_fig2_suc'
'Tiss_ThickDistrib': Shows weighted distribution of relative tissue thicknesses about
average tissue length, l_avg. Use with model 'Safford78'.
'Fig2_H2O_plot': Use with param sets 'fig2_H2O', 'sheet_fig2_H2O, 'DEP_fig2_H2O'
'Fig3_H2O_plot': Use with param sets 'fig3_H2O', 'DEP_fig3_H2O'
'Fig5_plot': Use with param set 'fig5'
// 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.
*/
10.0901 15.2476 19.9067 24.5631 29.7071 34.3634 39.5047 44.8871
49.7846 55.1724 60.3137 65.1923 69.8568 80.3832 84.785 90.6549
95.0567 100.198 106.071 110.237 116.099 120.755 125.891 155.98
166.019 185.582 190.236 195.607 200.272 210.286 220.073 224.946
229.874 234.755 245.048
0.00377595 0.00810068 0.013637 0.0197822 0.027151 0.0332962 0.0412738 0.0504721
0.0578379 0.0658185 0.0737961 0.0854236 0.0897423 0.106309 0.114278 0.1247
0.132668 0.140646 0.150459 0.15599 0.168238 0.174383 0.183578 0.235697
0.25104 0.286591 0.293345 0.304979 0.309298 0.33012 0.346678 0.359524
0.360192 0.371211 0.384731
Data from Figure 2 of:
Safford RE, Bassingthwaighte EA, and Bassingthwaighte JB.
Diffusion of water in cat ventricular myocardium.
J Gen Physiol 72: 513-538, 1978.
Abscissa is time, minutes.
Ordinate is percent relative concentration of THO (tritiated water)
in recepient compartment relative to donar compartment.
10.3611 20.4755 31.0706 36.2417 40.4134 46.0778 50.7524 55.9108
61.3191 66.4713 71.1174 77.5346 81.4312 86.3461 91.2705 96.3879
106.48 110.892 121.649 131.691 136.644 141.546 146.451 151.6
156.496 161.158 166.563 171.7 175.881 181.991 186.378 191.609
196.531 201.376 205.548 211.49 220.504 225.935 231.078 236.227
245.99 250.87 256.329 260.9 265.612 271.239 281.078 285.73
290.566 295.579 300.444
1.47644E-4 2.91774E-4 0.0028856 0.00479141 0.00912583 0.0110387 0.0135481 0.0178966
0.0216379 0.0272078 0.0352132 0.0346939 0.0445208 0.0482551 0.0501574 0.0624446
0.0668634 0.0724226 0.0915079 0.105697 0.102104 0.108281 0.113847 0.120027
0.127426 0.132378 0.13673 0.145353 0.147855 0.158935 0.16938 0.159683
0.162196 0.179365 0.183699 0.179509 0.201623 0.20109 0.208492 0.214672
0.234965 0.245416 0.239387 0.262048 0.25723 0.266471 0.272107 0.278891
0.297892 0.282696 0.296201
Data from Figure 2 of:
Safford RE, Bassingthwaighte EA, and Bassingthwaighte JB.
Diffusion of water in cat ventricular myocardium.
J Gen Physiol 72: 513-538, 1978.
Abscissa is time, minutes.
Ordinate is percent relative concentration of sucrose
in recepient compartment relative to donar compartment.
Models in project file (Safford1978.proj):
1. 'Safford78': Calculates the bulk diffusion coefficient, Db, for water through a matrix
of cells surrounded by ECF, influenced by cell membrane permeability.
2. 'SheetDiffusion': Barrer type diffusion across a slab of varying thickness. From Suenson
et al. 1974.
3. 'Deadend_Pore': Parallel pathway, dead-end pore model (DEP) that accounts for sequestration
or binding of water within heart muscle sheet. From Safford and Bassingthwaighte, 1977
Parameter sets:
Use with model 'Deadend_Pore':
'DEP_fig2_H2O': DEP model fit to H2O diffusion data from Figure 2 of Safford 1978.
Note exchange rate constant (Ke), diffusion channel and DEP volume, and
the tissue diffusion coefficient for water (D_tiss). View results on
plotpage 'Fig2_H2O_plot'. Use ratio of tissue to free diffusion for
water and sucrose to compare between models and compare to values found in
Table 1 of Safford 1978 paper.
'DEP_fig2_suc': DEP model fit to sucrose diffusion data from Figure 2 of Safford 1978. Note
THe dead-end pore volume goes to zero showing that little sucrose is 'bound'
within the tissue, as expected. Use plot page 'Fig2_suc_plot'.
'DEP_fig3_H2O': DEP model fit to H2O diffusion data from Figure 3 of Safford 1978. Note
exchange rate constant (Ke), diffusion channel and DEP volume, and the tissue diffusion coefficient for water (D_tiss). View results on plotpage
'Fig3_H2O_plot' and compare to values found in Table 1 of Safford 1978 paper.
Use with model 'Safford78':
'fig2_H2O': Cell permeation model fit to Figure 2 of Safford 1978. Uses the sheet diffusion
equation but with Bulk diffusion term , Db(ulk), and the diffusional area, Ad,
is now just the total tissue area, At. The original paper did not explicitedly
fit this cell permeation model to the data but rather used it to just calculate
the diffusion coefficient for water in the tissue, Db, and confirmed the ratio
of Db to free diffusion coefficient is higher than that of sucrose.
'fig3_H2O': Cell permeation model fit to Figure 3 of Safford 1978. Uses the sheet diffusion
equation but with Bulk diffusion term , Db(ulk), and the diffusional area, Ad,
is now just the total tissue area, At. The original paper did not explicitedly
fit this cell permeation model to the data but rather used it to just calculate
the diffusion coefficient for water in the tissue, Db, and confirmed the ratio
of Db to free diffusion coefficient is higher than that of sucrose.
'fig5': Reproduce figure 3 of Safford 1978 paper. Load 'Safford78' model, confirm parameter set 'fig5' is loaded, and go to the 'Loops' tab and hit the 'Run' button. View results
in plot page 'Fig5_plot'. Simulation loops through values for cell size, L, and gap
space, Lzero. Adjust values for Lzero and L to get effect geometry has on bulk
diffussion as a function of cell permeability.
'fig7': Reproduce figure 7 of Safford 1978 paper. Use with plot page 'Fig7_DB.vs.P'. Shows
relationship between P and Db using most probable parameter values.
Use with model 'SheetDiffusion':
'sheet_fig2_H2O': Sheet diffusion model fit to H2O diffusion data from Figure 2 of
Safford 1978. Note diffusion channel volume, and the tissue diffusion
coefficient for water (Dtiss). View results on plotpage 'Fig2_H2O_plot'.
'sheet_fig2_suc': Sheet diffusion model fit to sucrose diffusion data from Figure 2 of
Safford 1978. Note diffusion channel volume, and the tissue diffusion
coefficient for sucrose (Dtiss). View results on plotpage 'Fig2_suc_plot'.
Plot pages:
'Fig7_DB.vs.P': Use with param set 'fig7'
'Fig2_suc_plot': Use with param sets 'sheet_fig2_suc, 'DEP_fig2_suc'
'Tiss_ThickDistrib': Shows weighted distribution of relative tissue thicknesses about
average tissue length, l_avg. Use with model 'Safford78'.
'Fig2_H2O_plot': Use with param sets 'fig2_H2O', 'sheet_fig2_H2O, 'DEP_fig2_H2O'
'Fig3_H2O_plot': Use with param sets 'fig3_H2O', 'DEP_fig3_H2O'
'Fig5_plot': Use with param set 'fig5'
// SHORT DESCRIPTION: Diffusion across an uneven sheet (Suenson 1974).
// Part of Model Safford1978 ( model #0205 )
import nsrunit; unit uM = 1e-6 M; unit conversion on;
math sheetDiffusion {
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 Ad = 0.1 cm^2; // Area available for diffusion
real Aratio = Ad/At; // Ratio of diffusion area to total sheet area
real Vtiss = l_avg*At; // Tissue volume
real Vdiff = l_avg*Ad; // Volume available for diffusion
real Dsuc = 5.1e-6 cm^2/sec; // free sucrose diffusion coefficient in water at 25C
real DH2O = 2.38e-5 cm^2/sec; // free water diffusion coefficient at 25C
real Dfree = 1e-5 cm^2/sec; // Currently can be set to Dsuc or DH2O
real lambda = 1 dimensionless; // tortuosity factor
real Dtiss = Dfree/lambda^2; // Apparent diffusion coefficient in tissue
real Dratio = Dtiss/Dfree; // Ratio of tissue diffusion to free diffusion
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 CRsheet(t) dimensionless; // Ratio of tracer in receipient to donar compartment
// Multi-path (MP):
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;
real thirdTermsum(t,m,l_i) = ((-1)^m/m^2) * exp((-Dfree*m^2*PI^2*t)/((lambda^2)*(l_i*l_avg)^2));
real thirdTermTotal(t,l_i) = sum(m=m.min to m.max, thirdTermsum);
real firstWeightedTerm(t) = (Dfree*Ad*t/lambda^2)*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*Ad/(PI^2))*thirdTermTotal);
CRsheet = (1/V_chamber)*(firstWeightedTerm - (l_avg*Ad)/(6) - thirdWeightedTerm);
real CRsheetLin(t) = (1/V_chamber)*(firstWeightedTerm - (l_avg*Ad)/(6)); // Used to calc time lag of steady-state and its slope
} // 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.
*/
Models in project file (Safford1978.proj):
1. 'Safford78': Calculates the bulk diffusion coefficient, Db, for water through a matrix
of cells surrounded by ECF, influenced by cell membrane permeability.
2. 'SheetDiffusion': Barrer type diffusion across a slab of varying thickness. From Suenson
et al. 1974.
3. 'Deadend_Pore': Parallel pathway, dead-end pore model (DEP) that accounts for sequestration
or binding of water within heart muscle sheet. From Safford and Bassingthwaighte, 1977
Parameter sets:
Use with model 'Deadend_Pore':
'DEP_fig2_H2O': DEP model fit to H2O diffusion data from Figure 2 of Safford 1978.
Note exchange rate constant (Ke), diffusion channel and DEP volume, and
the tissue diffusion coefficient for water (D_tiss). View results on
plotpage 'Fig2_H2O_plot'. Use ratio of tissue to free diffusion for
water and sucrose to compare between models and compare to values found in
Table 1 of Safford 1978 paper.
'DEP_fig2_suc': DEP model fit to sucrose diffusion data from Figure 2 of Safford 1978. Note
THe dead-end pore volume goes to zero showing that little sucrose is 'bound'
within the tissue, as expected. Use plot page 'Fig2_suc_plot'.
'DEP_fig3_H2O': DEP model fit to H2O diffusion data from Figure 3 of Safford 1978. Note
exchange rate constant (Ke), diffusion channel and DEP volume, and the tissue diffusion coefficient for water (D_tiss). View results on plotpage
'Fig3_H2O_plot' and compare to values found in Table 1 of Safford 1978 paper.
Use with model 'Safford78':
'fig2_H2O': Cell permeation model fit to Figure 2 of Safford 1978. Uses the sheet diffusion
equation but with Bulk diffusion term , Db(ulk), and the diffusional area, Ad,
is now just the total tissue area, At. The original paper did not explicitedly
fit this cell permeation model to the data but rather used it to just calculate
the diffusion coefficient for water in the tissue, Db, and confirmed the ratio
of Db to free diffusion coefficient is higher than that of sucrose.
'fig3_H2O': Cell permeation model fit to Figure 3 of Safford 1978. Uses the sheet diffusion
equation but with Bulk diffusion term , Db(ulk), and the diffusional area, Ad,
is now just the total tissue area, At. The original paper did not explicitedly
fit this cell permeation model to the data but rather used it to just calculate
the diffusion coefficient for water in the tissue, Db, and confirmed the ratio
of Db to free diffusion coefficient is higher than that of sucrose.
'fig5': Reproduce figure 3 of Safford 1978 paper. Load 'Safford78' model, confirm parameter set 'fig5' is loaded, and go to the 'Loops' tab and hit the 'Run' button. View results
in plot page 'Fig5_plot'. Simulation loops through values for cell size, L, and gap
space, Lzero. Adjust values for Lzero and L to get effect geometry has on bulk
diffussion as a function of cell permeability.
'fig7': Reproduce figure 7 of Safford 1978 paper. Use with plot page 'Fig7_DB.vs.P'. Shows
relationship between P and Db using most probable parameter values.
Use with model 'SheetDiffusion':
'sheet_fig2_H2O': Sheet diffusion model fit to H2O diffusion data from Figure 2 of
Safford 1978. Note diffusion channel volume, and the tissue diffusion
coefficient for water (Dtiss). View results on plotpage 'Fig2_H2O_plot'.
'sheet_fig2_suc': Sheet diffusion model fit to sucrose diffusion data from Figure 2 of
Safford 1978. Note diffusion channel volume, and the tissue diffusion
coefficient for sucrose (Dtiss). View results on plotpage 'Fig2_suc_plot'.
Plot pages:
'Fig7_DB.vs.P': Use with param set 'fig7'
'Fig2_suc_plot': Use with param sets 'sheet_fig2_suc, 'DEP_fig2_suc'
'Tiss_ThickDistrib': Shows weighted distribution of relative tissue thicknesses about
average tissue length, l_avg. Use with model 'Safford78'.
'Fig2_H2O_plot': Use with param sets 'fig2_H2O', 'sheet_fig2_H2O, 'DEP_fig2_H2O'
'Fig3_H2O_plot': Use with param sets 'fig3_H2O', 'DEP_fig3_H2O'
'Fig5_plot': Use with param set 'fig5'
// SHORT DESCRIPTION: Parallel pathway, dead-end pore model (DEP) that accounts for sequestration or binding
// of water within heart muscle sheet. From Safford and Bassingthwaighte, 1977.
// Part of Model Safford1978 ( model #0205 )
import nsrunit; unit conversion on;
math deadEndPore {
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 V_cell = 14 ml; // Volume of recipient and donar chambers
real l_avg = 0.1 cm; // Avg thickness of sheet
real Atiss = 0.283 cm^2; // Exposed tissue area
real rho = 1.053 g/ml; // density of wet tissue (Suenson et al. 1974)
real f = 0.85 ml/g; // Fractional water content:g water per g tissue (water density assumed 1ml/g)
real Vw = Atiss*l_avg*rho*f; // Total tissue water volume
real Ad = 0.1 cm^2; // Area available for diffusion
real Aratio = Ad/Atiss;
real Vtiss = l_avg*Atiss;
real Vdiff ml; // Diffusion channel volume per ml of tissue
Vdiff = l_avg*Ad;
real beta(m) cm^-1;
beta = m* PI/l_avg; // Eq 8a
real Ke = 0.001 sec^-1; // exch rate const between D-E pore vol. and diff channel vol.
real Vdep = 0.1 ml; // dead-end pore water vol, equivalant sum of all binding sites
real Vdep_calc = Vw-Vdiff; // dead-end pore water vol calc if assumed all left over water space is Vdep
real VdepRatio = Vdep/Vdiff; // Ratio of dead-end pore to diffusion channel volumes
real D_tiss = 1.0e-5 cm^2/sec; // Tissue diffusion coefficent (D_free/lambda^2)
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:// Multi-path distribution function - returns weight for relative length
extern real MPathDF(l_i) dimensionless;
real wPathSum = sum(l_i=l_i.min to l_i.max, MPathDF *l_i.delta);
real weight(l_i) = MPathDF*l_i.delta;
real CR_DEP(t) dimensionless; // Relative conc of substrate in recipient to donar chamber
real betaMP(m,l_i) cm^-1;
betaMP = m* PI/(l_i*l_avg); // Eq 8a
real G_MP(m,l_i) sec^-1;
G_MP = Ke*(1 + Vdep/((l_i*l_avg)*weight(l_i)*Ad)) + D_tiss * betaMP^2; // Eq 8b
real S_posMP(m,l_i) sec ^-1;
S_posMP = -G_MP/2 + (1/2)*(G_MP^2 - 4*D_tiss*betaMP^2*Ke)^0.5; // Eq 8c
real S_negMP(m,l_i) = -G_MP/2 - (1/2)*(G_MP^2 - 4*D_tiss*betaMP^2*Ke)^0.5;
// Eq 8d is not needed for multi-path equations and algebraic simplification gives Eq 10.
real thirdSumMP_pos(t,m,l_i) = (m^2*((-1)^m)*exp(t*S_posMP))/((S_posMP^2)*(-l_i*l_avg)*(Ad*l_avg + Vdep*(Ke/(S_posMP +Ke))^2 ));
real thirdSumMP_neg(t,m,l_i) = (m^2*((-1)^m)*exp(t*S_negMP))/((S_negMP^2)*(-l_i*l_avg)*(Ad*l_avg + Vdep*(Ke/(S_negMP +Ke))^2 ));
real thirdTermMP_pos(t,l_i) = sum(m=m.min to m.max ,thirdSumMP_pos);
real thirdTermMP_neg(t,l_i) = sum(m=m.min to m.max ,thirdSumMP_neg);
// Put it all together (Eq 10):
real firstTermSum(t) = sum( l_i = l_i.min to l_i.max, (weight(l_i)/(l_i*l_avg))) ;
real thirdTermSum(t) = sum(l_i = l_i.min to l_i.max , (weight(l_i)/(l_i*l_avg)^2)* (thirdTermMP_pos(t,l_i) + thirdTermMP_neg(t,l_i)) );
CR_DEP(t) = (D_tiss*Ad*t/(V_cell)) *firstTermSum - (l_avg*Ad + Vdep)/(6*V_cell)
+ ((2*PI^2*D_tiss^2*(Ad^2)*l_avg)/V_cell) *thirdTermSum;
// Linear part of diffusion Eq 10:
real CR_DEPLinear(t) = (D_tiss*Ad*t/(V_cell)) *firstTermSum - (l_avg*Ad + Vdep)/(6*V_cell);
}
/*
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.
*/
2.78297 6.96246 9.74879 14.4855 19.7762 24.5063 29.7919 34.2399
38.1324 41.7446 47.0252 51.7519 56.4752 60.3626 65.3579 70.3616
75.0732 80.3454 85.0687 90.6246 95.6232 100.895 105.612 110.898
115.613 120.61 125.61 130.598 135.876 141.43 145.858 151.145
156.142 161.132 166.128 170.848 176.125 181.117 186.11 195.807
200.814 205.514 210.811 220.241 225.784 230.233 240.501 245.214
249.936 254.943 259.936 264.635 270.187 275.187 279.887 294.321
299.046 304.308 308.736 319.309 324.301 329.284 334.548 338.427
343.704 353.421 358.148
7.97488E-6 8.15723E-6 8.2788E-6 8.48546E-6 1.65697E-5 3.2483E-5 5.23473E-5 7.61017E-5
9.59052E-5 1.19623E-4 1.51267E-4 1.75034E-4 2.06654E-4 2.38237E-4 2.85576E-4 3.13281E-4
3.72388E-4 4.23665E-4 4.55285E-4 4.94795E-4 5.3428E-4 5.85558E-4 6.32884E-4 6.52748E-4
7.04002E-4 7.47414E-4 7.82972E-4 8.46018E-4 8.85515E-4 9.28951E-4 9.99826E-4 0.00101576
0.00105918 0.00111829 0.00116563 0.00120511 0.0012446 0.0012998 0.00135106 0.00148107
0.00150092 0.00158751 0.00157989 0.00168239 0.00175332 0.00177315 0.00187176 0.00192694
0.00196249 0.00198234 0.00203361 0.00212413 0.00217149 0.00220705 0.00229364 0.00242385
0.00245154 0.00252638 0.00259726 0.00263306 0.00268825 0.00276308 0.00283399 0.0028852
0.0029247 0.00300759 0.00303135