JSim v1.1
/*
// MODEL NUMBER: 0274
MODEL NAME: Osm.Uncoupled2
SHORT DESCRIPTION: Uncoupled, independent fluxes of water and of 2 solutes,
across a membrane separating 2 stirred tanks equipped with columns above each
to provide observable column heights as measures of their pressures.
*/
import nsrunit; unit conversion on;
math Osm.UncoupledB {realDomain t sec; t.min=0; t.max=60; t.delta=0.1;
//PARAMETERS:
real PermA = 0 cm/s, PermB = 0.1 cm/s, // Permeab for solutes 1 & 2
Am = 1 cm^2, // Surface area of membrane
Lp = 0.02 cm * s^(-1) * mmHg^(-1), // Hydraulic conductivity, Kf
p1zero= 0 mmHg, p2zero= 0 mmHg, // Initial pressures in V1 V2
V10 = 1 mL, V20 = 1 mL, // Initial volumes in V1 V2
A10 = 1 mM, A20 = 0 mM, B10 = 0 mM, B20 = 1 mM, // Initial concns at t=0.
Atot = A10*V10 + A20*V20,
Btot = B10*V10 + B20*V20; // Conservation is assumed for A and B to reduce ODEs
real TempC = 20 degK, // deg Centigrade
R = 0.062363 mmHg/(mM*degK), // Gas Constant in mmHg/(mM*degK)
T = 273.16 + TempC, // T degK
RT mmHg/mM, // calculate RT for TempC
RT = R*T, // RT = 19.3425*10^6 mmHg*cm^3*mol^(-1) at 37C
// RT = 19.3425 mmHg/mM at 37C
Base1 = 0.01 cm^2, // area of base of volume 1, to calc p1.
Base2 = 0.01 cm^2, // area of base of volume 2, to calc p2.
rho = 1.00 g/cm^3, // density -> cm H2O height to mmHg (rhoHg = 13.59508 g/ml)
grav = 980 cm*sec^(-2);// press p = rho*grav*height; height = VolChange/BaseArea
//VARIABLES:
real A1(t) mM, A2(t) mM, B1(t) mM, B2(t) mM, // concentrations
qA1(t) mole, qB1(t) mole,// qA2(t) mole, qB2(t) mole, // amts of A,B in V1
V1(t) mL, V2(t) mL, // volumes of V1 and V2
p1(t) mmHg, p2(t) mmHg; // Hydrostat press in V1 and V2
// INIT COND:
when (t=t.min) {V1 = V10; V2 = V20;
// A1 = A10; A2 = A20; B1 = B10; B2 = B20;
qA1=A10*V10; qB1=B10*V10; }
// ODEs: volume and solute flux equations: fluxes are from 1 to 2
V1:t = -Lp*Am*(p1-p2 - RT*(A1-A2 + B1-B2)); // water conductivity *area* pressure
V2:t = -V1:t; // assumes conservation of water volume
qA1:t = -Am*PermA*(A1-A2); // Quantity of A exchanging
qB1:t = -Am*PermB*(B1-B2); // Quantity of B exchanging
// qA2:t = - qA1:t;
// qB2:t = - qB1:t;
A1 = qA1/V1; B1 = qB1/V1; // Calculate Concns in V1
A2 = (Atot - qA1)/V2; B2 = (Btot - qB1)/V2; // Assumes Conservation
// A2 = qA2/V2; B2 = qB2/V2; // Calculate Concns in V2 independently to allow maass check
real Aerr(t) mole, Berr(t) mole;
Aerr = Atot- A1*V1-A2*V2; Berr= Btot- B1*V1 - B2*V2; // deviation from zero = Mass balance error
p1 = p1zero + rho*grav*(V1-V10)/(Base1); //p1 of height 1 h1 = (V1-V10)/(Base1)
p2 = p2zero + rho*grav*(V2-V20)/(Base2); //p2 of height 2 h2 = (V2-V20)/(Base2)
// Translate Lp = Kf into Permeability Pf:
real Pf cm/s, // water permeability of membrane
Vw = 18 ml/mol, // water partial molar volume
Pf = Lp*RT/Vw, // Pf = 1.015682*1e6* Lp at 20 degC; Pf is a diffusive permeability
// Pf uses activity driving forces, not pressure
RTVW mmHg, RTVW = RT/Vw; //1.0746*1e6 at 37C. Water activity = pressure*Vw/RT
real Y(t) ml; when (t=t.min) {Y = V10;} // Y(t):t uses Pf, not Lp
Y:t= -(Pf*Vw/RT)*Am*(p1-p2 - RT*(A1-A2 + B1-B2)); // Y(t) should= V1(t)
/* ANALYTICAL CHECK;
A simple check of arbitrary exactness is provided by a 1-compartmental equivalent.
Make V1 so large, e.g. 10000 ml,that A1 is changed negligibly when water flow into
or out of it from V2. With water fluxes, the column heights change just with
1 ml starting volumes.
*/
real tau1 s;
tau1 = Base1/(2*rho*grav*Lp*Am); // 2 is because what leaves V1 enters V2
real p_theor(t) mmHg, p_diff(t) mmHg;
p_theor = A1*RT*(1-exp(-t/tau1)); // when PermA = 0 and permB finite
p_diff = p1 - p2;
}
/* Lp = 0.02 cm/(s*mmHg), //Hydraulic conductivity = filtration coeff
// which translates to Pf,cm/s, by multiplying Lp by RT/Vw= 1.02*1e6 mmHg at 20 degC
// so Pf =2.031e4 cm/s.
// Mlekody, Moore Levitt JGP 81: 213, 1983 give Pf = 2e-2 cm/s at 23C in RBC.
.
Here Pf = 2.15*1e4 cm/s @37C. Note the the surface to volume ratio here is 1 cm^2 / 1 ml
whereas that for RBC is much higher/
MODEL VERIFICATION: Total Mass is conserved. See Aerr and Berr:
Check ro see if V1 + V2 is constant.
DETAILED DESCRIPTION:
This model is the same as Osm.Uncoupled1.proj except for using column heights
instead of elasticity. It also allows temperature changes.
Uncoupled, independent fluxes of water and of 2 solutes,
A and B, across a membrane separating 2 stirred tanks. Solute activities
are assumed unity so concentrations = thermodynamic activity. The model describes
a situation similar to that for the simplest expressions of Kedem and Katchalsky
(1958) but omits all interactions between solutes and between water and
any solute. One can think of the solutes passing though the membrane by
passive permeation with permeability coefficients PermA and Perm B, and
the water passing through aqueous pores with filtration coefficient or
hydraulic conductivity, Lp. The aqueous pores do not permit solute passage.
Lp is the same as the traditional filtration coefficient Kf. Lp translates
to a conventional permeability for water filtration, Pf cm/s,
Pf = Lp*RT/Vw
where RT = 19.347*10^6 mmHg*cm^3*mol^(-1) at 37C, Vw is the partial molar volume
of water, 18 ml/mol or the concentration of water in water is 55.55 M
The driving forces are the pressure difference for water flux and the
concentration for the solute fluxes. The pressure difference across the
membrane is the hydrostatic pressure difference minus the osmotic pressure
difference. The osmotic pressure is given by Van't Hoff's Eq:
p_osm = a.C.RT, where p_osm is the osmotic pressure, mmHg,
"a" is the activity coefficient, assumed in this model to equal unity,
C is concentration, M, and RT is the Gas Constant times Temperature Kelvin.
In this model the solute doesn't permeate the aqueous pore so there is
no consideration of a reflection coefficient, or rather it is assumed to be unity.
Thus solute concentration in the pore water is zero, and there is no solute advection..
The system is composed of two volumes of pressure-dependent size, yet
stirred instantaneously continually. The slope of the pressure/volume
relationship is linear and defined by the height of a column of fluid above
the rigid chambers The narrow columns of fluid have heights h1 and h2.
The pressure is rho*grav*h1 in chamber 1, where rho is fluid density,
grav is acceleration due to gravity;
The fluid in the columns is considered to be instantaneously mixed with that
in the chamber from which it rises. Fluid or volume flux, Jv, from side 1 to
side 2 causes a difference in the column heighta between the two sides by
Base*(h2-h1) = Jv, where Base = area of the base of the column, and the
pressure difference rises to rho* grav*(h2-h1) cm H2O, where rho is the fluid
density. g/ml, in the narrow colums..
The model OsmUncoupledA.proj uses an analogous linear chamber elastance, Elast mmHg/ml,
gives an equivalent measure for flexible chambers, assuming a linear relationship between
the pressure change and the volume change. (1 mmHg = 13.59 cm H2O.)
Notes: Situation 1:= Model default par. . PermA = 0, PermB > 0.1. See Notes.
Situation 2 = Model par2 PermB > 0.
SHORTCOMINGS/GENERAL COMMENTS:
ASSUMPTIONS:
1. Compartmental assumptions apply to the solutions on either side of the
membrane. These are: Instantaneously stirred tank. No concentration gradients.
No diffusion limitation for reactions.
KEY WORDS: Two ideal solutes, compartment, water and solute exchamge uncoupled,
passive transmembrane exchanes independent, tutorial
REFERENCES:
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.
Stein WD. The Movement of Molecules across Cell Membranes. New York: Academic Press, 1967.
Stein WD. Transport and Diffusion across Cell Membranes. Orlando, Florida:
Academic Press Inc., 1986.
REVISION HISTORY:
Original Author : JBB Date: 26/Dec/09
Revised by : BEJ Date: 29/Dec/09
Revision: Update format of comments.
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2009 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@bioeng.washington.edu.
*/