Model number


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.


 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.  

Safford et al. 1978 paper (Journal link)

fig 1

Safford et al., 1978


The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

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(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.
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

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.
Key terms
sheet diffusion
dead-end pore
cell geometry

Please cite in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

Model development and archiving support at provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.