// MODEL NUMBER: 0279
// MODEL NAME: PoreTransport
// SHORT DESCRIPTION: Permeability and reflection coefficient for a hard spherical
// solute, radius = rs, traversing a cylindrical pore, radius = rpore.
// Hydrodynamic calculation from Bassingthwaighte 2006, corrected 2012.
import nsrunit; unit uM = 1e-6 M; unit conversion on;
math Pore {
realDomain alp dimensionless; alp.min = 0; alp.max = 1; alp.delta = 0.01; // alp = rs/rpore
//Codes Revised Equations for F and G for pores, but not slits.
// STATE VARIABLES
real Fjbb(alp) dimensionless, // JBB Eq 16 in Bassingthwaighte 2006
Gbean(alp) dimensionless, // Eq 6-120; Curry 5.51
Gjbb(alp) dimensionless, // JBB Eq 17
sigjbb(alp) dimensionless, // JBB Eq 18
rs(alp) nm, // solute radius, hard sphere
rpore = 7.9 nm, // pore radius
phi(alp) dimensionless, // partition coeff = vol fract of membrane accesible to solute
Prel(alp) dimensionless, // P/Pmax, predecessor to Eq 19 in Bassingthwaighte 2006
Prelcorr(alp) dimensionless,// P/Pmax, Eq 19 in Bassingthwaighte 2007
CorrG(alp) dimensionless; // Correction to substract from Gbean
// Essential Equations:
rs = alp*rpore;
phi = (1-alp)^2; // solute partition coeff = fraction of pore accessible to solute
Prel = phi*Fjbb; // Relative permeability due to exclusion
Gbean = (1-(2/3)*alp^2 -0.20217*alp^5) / (1 - 0.75851*alp^5); //Eq 6-120 Curry 5.51
CorrG = 0.0431*(1-(1-alp^10)); //Correction to Gbean at high alp
Gjbb = Gbean -CorrG; //Eq 17 in JBB 2006
Fjbb = (1-alp^2)^1.5*phi / (1 + 0.2*alp^2*(1-alp^2)^16); //Eq 16 in JBB 2006
sigjbb= (1-(1-(1-phi)^2)*Gjbb) + 2*alp^2*phi*Fjbb; //Eq 18 in JBB 2006
Prelcorr = phi * Fjbb *(1 + 9*alp^5.5 *(1-alp^5)^0.02); //Eq 19 in JBB 2006
// Variables useful for comparisons:
real Fcurry(alp) dimensionless, // F is Eq 6-119 of Curry 1984
sigBean(alp) dimensionless, // Curry Eq 6-118
sig549(alp) dimensionless; // Curry Eq 5-49
Fcurry = 1-2.10444*alp +2.08877*alp^3 -0.94813*alp^5 -1.372*alp^6
+3.87*alp^8 -4.19*alp^9; //Eq 6-119 or 5.17 Curry & by Faxen
sigBean= 1 - (2*phi - phi^2)*Gbean + (16/9)*alp^2*phi*Fcurry; //Eq 6-118 Curry =Eq 12JBB =Eq 5-49Curry
sig549= (1-(1-(1-phi)^2)*Gbean) + (16/9)*alp^2*phi*Fcurry; //Eq 5-49 Curry same as sigBean
// SLIT Equations and scalfn are recorded here for convenient reference only
// SLIT EQs from Bean and Curry, but not reworked with new hydrodynamic calculations, thus inaccurate
real sigslit(alp), Fslit(alp);
sigslit= 1-((2*alp^2/3)*(1-alp)*Fslit + (3/2)*(1-alp)*((2/3)*(1+alp)-(7/12)*alp^2));//Eq 6-121 Curry
Fslit = 1 - 1.004*alp + 0.418*alp^3 +2.10*alp^4 - 0.1696*alp^5; //Eq 6-115 Curry
// scalfn=1+k1*alp^k2*(k3-alp^k4)^k5; //Eq 15 in JBB 2006 generic scalar
// Parameters used in numeric solutions for scalfn, an arbitrary function to terminate Bean's F and G
// F converges to zero; G converges 10 0.5. The scalfn is used to find a correct function
real scalfn(alp) dimensionless,// scalfn is used to find corrections to Bean's F and G at high alp
k1 = 0.2 dimensionless, //used in scalfn, a generic scalar of 0 < alp < 1
k2 = 2 dimensionless, //used in scalfn, a generic scalar of 0 < alp < 1
k3 = 1 dimensionless, //used in scalfn, a generic scalar of 0 < alp < 1
k4 = 2 dimensionless, //used in scalfn, a generic scalar of 0 < alp < 1
k5 = 16 dimensionless; //used in scalfn, a generic scalar of 0 < alp < 1
scalfn = 1+0.2*alp^2*(1-alp^2)^16; //another example correction factor
} // End of Program
/*
// DETAILED DESCRIPTION:
//OBJECTIVE: The equations for transport of hydrophilic solutes through aqueous pores
// provide a fundamental basis for examining capillary-tissue exchange and water and
// solute flux through transmembrane channels, but the theory remains incomplete for
// ratios, alp, of sphere diameters to pore diameters greater than 0.4. Values for
// permeabilities, P, and reflection coefficients, sigma, from Lewellen [18], working
// with Lightfoot et al. [19], at alp = 0.5 and 0.95, were combined with earlier values
// for alp < 0.4, and the physically required alp < 1, to provide accurate expressions
// over the whole range of 0 < alp < 1. (alp =rs/rpore & stands for the greek alpha in the text.)
// METHODS: The "data" were the long-accepted theory for alp < 0.2 and the computational
// results from Lewellen and Lightfoot et al. on hard spheres (of 5 different alp's)
// moving by convection and diffusion through a tight cylindrical pore, accounting
// for molecular exclusion, viscous forces, pressure drop, torque and rotation of
// spheres off the center line (averaging across all accessible radial positions),
// and the asymptotic values at alp = 1.0. Coefficients for frictional hindrance to
// diffusion, F(alp), and drag, G(alp), and functions for sigma(alp) and P(alp), were
// represented by power law functions and the parameters optimized to give best
// fits to the combined "data".
// RESULTS: The reflection coefficient sigma = {1 - (1 - (1-PHI)^2)*G'(alp)} + 2*alp^2*PHI*F'(alp),
// and the reloative permeabiltiy P/Omax = PHI*F'(alp)*(1 + 9* alp^5.5 * (1 -alp^5)^(0.02)],
// where PHI is the partition coefficient or volume fraction of the pore available to solute.
// The new expression for the diffusive hindrance is
// F'(alp) = (1-alp^2)^(3/2)*PHI / (1 + 0.2 alp^2 *(1 - alp^2)^16), and for the drag factor is
// G'(alp) = (1-(2/3)*alp^2 - 0.20217*alp^5) / (1 - 0.75851*alp^5) - 0.0431*[1 - (1 - alp^10)].
// All of these converge monotonically to the correct limits at alp = 1.
// CONCLUSION: These are the first expressions providing hydrodynamically based estimates
// of SIGMA(alp) and P(alp) over 0 < alp < 1. They should be accurate to within 1 t0 2%.
KEY WORDS: porous transport, hydrodynamic molecular radius, cylindircal pore size,
capillary permeability, convection, diffusion, interstitial matrix,
molecular exclusion, osmotic water and solute fluxes, particle flow in fluids,
steric hindrance, Spiegler frictional coefficients, Kedem Katchalsky, Publication,
Data, Tutorial
REFERENCES:
Bassingthwaighte JB. A practical extension of hydrodynamic theory of porous
transport for hydrophilic solutes. Microcirculation 13: 111-118, 2006. (#591)
Bassingthwaighte JB. Erratum to: A practical extension of hydrodynamic theory
of porous transport for hydrophilic solutes. Microcirculation 19(7): 668, 2012. (#650)
-corrected Eq 12 to use +16/9 * rather than -16/9 *. and curve for Bean in Fig 3.
Apelblat A, Katzir-Katchalsky A, and Silberberg A. A mathematical analysis
of capillary-tissue fluid exchange. Biorheology 11: 1-49, 1974.
Bean CP. The physics of porous membranes: neutral pores. In: Membranes:
Macroscopic Systems and Models, edited by Einsenman G. New York: Dekker, 1972, p. 1-54
Biber TUL and Curran PF. Coupled solute fluxes in toad skin.
J Gen Physiol 51: 606-620, 1968.
Curry FE. Mechanics and thermodynamics of transcapillary exchange..
In: Handbook of Physiology, Sec. 2 The Cardiovascular System. Vol. 4, Microcirculation,
edited by Renkin EM and Michel CC. American Physiological Society: Bethesda, Maryland, 1984, p. 309-374.
Ginzburg BZ and Katchalsky A. The frictional coefficients of the flows of
non-electrolytes through artificial membranes. J Gen Physiol 47: 403-418, 1963.
Katchalsky A and Curran PF. Nonequilibrium Thermodynamics in Biophysics.
Cambridge, MA: Harvard University Press, 1965.
Kedem O and Katchalsky A. Thermodynamic analysis of the permeability of
biological membranes to non-electrolytes. Biochim Biophys Acta 27: 229-246, 1958.
Kedem O and Katchalsky A. A physical interpretation of the phenomenological
coefficients of membrane permeability. J Gen Physiol 45: 143-179, 1961.
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, 2003.
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, 2003.
Lewellen PC. Hydrodynamic Analysis of Microporous Mass Transport. Ph.D. Thesis,
University of Wisconsin-Madison (1982).
Lewellen PC, Bassingthwaighte JB, Lightfoot EN, and Stewart WE. Microporous
membrane transport. Microvasc Res 25: 245, 1983. (#195)
Lightfoot EB, Bassingthwaighte JB, Grabowski EF, Hydrodynamic models for diffusion in
microporous membranes. Ann Biomed Eng 4:78-90 (1976). (#130)
REVISION HISTORY:
Original Author : JBB July/05 Revised by JBB Dec/09 1. update Notes, eqs. 2. Add par sets and plots
Revised by BEJ 04Jan10 Update comments format
Revised by GMR 11Jul12 Added calc for Pcurry, fixed Fig 3, minor corrections to documentation.
Revised by JBB 28jan16 Added Refs to Bassingthwaighte 2012, Lewellen 1982, 1983; Lightfoot 1976.
Revised by JBB 28jan16 Reduced code to calculate reflection coeff and Perm vs rs/rpore (spherical)
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2016 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.
When citing JSim please use this reference: Butterworth E, Jardine BE, Raymond GM, Neal ML, Bassingthwaighte JB.
JSim, an open-source modeling system for data analysis [v3; ref status: indexed, http://f1000r.es/3n0]
F1000Research 2014, 2:288 (doi: 10.12688/f1000research.2-288.v3)
This software was developed with support from NIH grants HL088516 and HL073598, NIBIB grant BE08417,
the Cardiac Energy Grid HL199122 (PI: J.B. Bassingthwaighte), and the Virtual Physiological Rat program
GM094503 (PI: D.A.Beard). Please cite these grants 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.
*/