Parallel pathway, dead-end pore model that accounts for sequestration or binding of calcium within heart muscle sheet. From Safford and Bassingthwaighte, 1977. Also contains an implementation of Suenson et al. 1974 diffusion model to validate new model with sucrose data.
Model number: 0202
|Run Model: ||    Help running a JSim model.|
(JSim model applet may take 10-20 seconds to load.)
ABSTRACT: Rates of diffusion through the extracellular space of thin sheets of myocardium from the right ventricular outflow tract of kittens were estimated at 23C for [45Ca(2+)] and an inert reference tracer, [14C]sucrose. The myocardial sheets were mounted in an Ussing chamber and equilibrated with Tyrode solution with varied calcium concentrations, Ca0. The tracers were added to one side and their concentrations on the other side measured at 5-15-min intervals for 6 h. The apparent tracer diffusion coefficient for sucrose was 1.11 0.06 x 10^-6 cm^2*s^-1 (mean t SEM, n = 74), 22% of the free diffusion coefficient; the lag time before reaching a steady state provided estimates of the intratissue volume of distribution or diffusion space of 0.41 +- 0.15 ml/ml tissue (n = 74), a value compatible with expectations for extracellular fluid space. Over the range of Ca0 from 0.02 to 9.0 mM, the intratissue apparent diffusion coefficient for Ca, Dca, averaged 1.65 0.10 x 10^-6cm^2*s^-1, n = 74, which is 21% of the free D0ca, and was not influenced by Ca0. Because transsarcolemmal Ca permeation is slow, Dca is the diffusion coefficient in the extracellular region. The paired ratios Dca/D. averaged 1.32 +- 0.05 (n = 67) for all levels of Ca0 but at physiologic or higher Ca0 averaged 1.45 +- 0.07 (n = 39), close to the ratio of free diffusion coefficients, 1.53. Equations distinguishing transient from steady state diffusion were fitted to the data, showing that the apparent distribution volume of "binding sites" external to the diffusion pathway diminished at higher Cao in a fashion suggesting that at least two different Ca(2+) binding sites were present. There are two diffusion models presented here: the dead-end pore model ('saff77_Binding') to account for Ca binding, and one based on Suenson et al. 1974 ('saff77_MPcrank', based on Crank 1956) that is used to model sucrose diffusion across the myocardium. The dead-end pore model contains the single and multi-path solutions using the same parameters so that it is easy to see the effect of heterogeneous tissue on diffusion across tissue.
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.
We welcome comments and feedback for this model. Please use the button below to send comments:
(Primary) Safford RE, Bassingthwaighte JB, Calcium diffusion in transient and steady states in muscle, Biophysical Journal, 20(1), Oct 1977, 113-136, ISSN 0006-3495 ALDRICH BI. The effects of the hyaluronic acid complex on the distribution of ions. Biochem J. 1958 Oct;70(2):236–244. Beeler GW, Jr, Reuter H. Membrane calcium current in ventricular myocardial fibres. J Physiol. 1970 Mar;207(1):191–209. Birks RI, Davey DF. Osmotic responses demonstrating the extracellular character of the sarcoplasmic reticulum. J Physiol. 1969 May;202(1):171–188. BLINKS JR. INFLUENCE OF OSMOTIC STRENGTH ON CROSS-SECTION AND VOLUME OF ISOLATED SINGLE MUSCLE FIBRES. J Physiol. 1965 Mar;177:42–57. Caillé JP, Hinke JA. Evidence for Na sequestration in muscle from Na diffusion measurements. Can J Physiol Pharmacol. 1972 Mar;50(3):228–237. Crank J. The Mathematics of Diffusion. Oxford: Clarendon Press, 1956. Crank J. The Mathematics of Diffusion, 2nd edition. Oxford: Clarendon Press, 1975. ENGEL MB, JOSEPH NR, CATCHPOLE HR. Homeostasis of connective tissues. I. Calcium-sodium equilibrium. AMA Arch Pathol. 1954 Jul;58(1):26–39. ENGEL MB, JOSEPH NR, LASKINDM, CATCHPOLE HR. Binding of anions by connective tissue: dermis and cartilage. Am J Physiol. 1961 Oct;201:621–627. Fawcett DW, McNutt NS. The ultrastructure of the cat myocardium. I. Ventricular papillary muscle. J Cell Biol. 1969 Jul;42(1):1–45. GERSH I, CATCHPOLE HR. The nature of ground substance of connective tissue. Perspect Biol Med. 1960;3:282–319. GINZBURG BZ, KATCHALSKY A. THE FRICTIONAL COEFFICIENTS OF THE FLOWS OF NON-ELECTROLYTES THROUGH ARTIFICIAL MEMBRANES. J Gen Physiol. 1963 Nov;47:403–418. Haljamäe H, Linde A, Amundson B. Comparative analyses of capsular fluid and interstitial fluid. Am J Physiol. 1974 Nov;227(5):1199–1205. HODGKIN AL, KEYNES RD. The mobility and diffusion coefficient of potassium in giant axons from Sepia. J Physiol. 1953 Mar;119(4):513–528. JOSEPH NR, ENGEL MB, CATCHPOLE HR. Homeostasis of connective tissues. II. Potassium-sodium equilibrium. AMA Arch Pathol. 1954 Jul;58(1):40–58. JOSEPH NR, CATCHPOLE HR, LASKIN DM, ENGEL MB. Titration curves of colloidal surface. II. Connective tissues. Arch Biochem Biophys. 1959 Sep;84:224–242. Kushmerick MJ, Podolsky RJ. Ionic mobility in muscle cells. Science. 1969 Dec 5;166(3910):1297–1298. Langer GA, Frank JS. Lanthanum in heart cell culture. Effect on calcium exchange correlated with its localization. J Cell Biol. 1972 Sep;54(3):441–455. Manery JF. Connective tissue electrolytes. Fed Proc. 1966 Nov-Dec;25(6):1799–1803. Martinez-Palomo A, Benitez D, Alanis J. Selective deposition of lanthanum in mammalian cardiac cell membranes. Ultrastructural and electrophysiological evidence. J Cell Biol. 1973 Jul;58(1):1–10. NIEDERGERKE R. The rate of action of calcium ions on the contraction of the heart. J Physiol. 1957 Oct 30;138(3):506–515. PAGE E, BERNSTEIN RS. CAT HEART MUSCLE IN VITRO. V. DIFFUSION THROUGH A SHEET OF RIGHT VENTRICLE. J Gen Physiol. 1964 Jul;47:1129–1140. Patlak CS, Fenstermacher JD. Measurements of dog blood-brain transfer constants by ventriculocisternal perfusion. Am J Physiol. 1975 Oct;229(4):877–884. Philpott CW, Goldstein MA. Sarcoplasmic reticulum of striated muscle: localization of potential calcium binding sites. Science. 1967 Feb 24;155(3765):1019–1021. Polimeni PI. Extracellular space and ionic distribution in rat ventricle. Am J Physiol. 1974 Sep;227(3):676–683. Reuter H, Seitz N. The dependence of calcium efflux from cardiac muscle on temperature and external ion composition. J Physiol. 1968 Mar;195(2):451–470. SCHAFER DE, JOHNSON JA. PERMEABILITY OF MAMMALIAN HEART CAPILLARIES TO SUCROSE AND INULIN. Am J Physiol. 1964 May;206:985–991. Shaw M. Interpretation of osmotic pressure in solutions of one and two nondiffusible components. Biophys J. 1976 Jan;16(1):43–57. Suenson M, Richmond DR, Bassingthwaighte JB. Diffusion of sucrose, sodium, and water in ventricular myocardium. Am J Physiol. 1974 Nov;227(5):1116–1123. Weidmann S. The diffusion of radiopotassium across intercalated disks of mammalian cardiac muscle. J Physiol. 1966 Nov;187(2):323–342. Weingart R. The permeability to tetraethylammonium ions of the surface membrane and the intercalated disks of sheep and calf myocardium. J Physiol. 1974 Aug;240(3):741–762. Wiederhielm CA, Fox JR, Lee DR. Ground substance mucopolysaccharides and plasma proteins: their role in capillary water balance. Am J Physiol. 1976 Apr;230(4):1121–1125.
- Diffusion Tutorial,
- 1-D Diffusion modeled as a partial differential equation,
- 1-D Diffusion with asymmetrical Consumption modeled as a partial differential equation,
- 1-D diffusion-advection equation with Robin boundary condition
- Random Walks of multiple particles in 1 dimension
- Random Walk of single particle in 2 dimensions
- Fractional Brownian Motion Walk in 2 dimensions
- Diffusion in a uniform slab
- Two Slab diffusion: Different diffusion coeffs in adjacent slabs require special boundary conditions
- Heat equation in two dimensions with Dirichlet boundary conditions
- Safford 1977 Dead end pore model for Calcium diffusion in muscle
- Safford 1978 Water diffusion in heart
- Suenson 1974 Diffusion in heart tissue, sucrose and water
- Facilitated diffusion through 2 regions
- Barrer Diffusion: Diffusion through 1-D slab with recipient chamber on right
tissue diffusion, plane sheet, binding, dead-end pore, ventricular myocardium, calcium, sucrose, tutorial, heterogeneous transport, Data, Publication, PMID901900, PMCID: PMC1473340
Model HistoryGet Model history in CVS.
Posted by: BEJ
Please cite www.physiome.org 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: email@example.com.
Or send a copy to:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
[This page was last modified 02Nov16, 2:41 pm.]
Model development and archiving support at physiome.org provided by the following grants: 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.