The first plot presents the permeability-surface product vs oxygen consumption
for co-current and counter-current configuration. It can be seen that for the
same consumption rate the surface area required for counter-current flow is always
less than the surface area required for co-current flows. However, the difference
is only relevant at high consumption rates typical for high performance athletes.
The second plot shows the relative difference in permeability-surface area product
vs oxygen consumption for the two configurations. In the third plot the
concentration profiles are shown for co-current and counter-current flows.
It can be easily seen why the counter-current configuration is more efficient.
In co-current configuration the concentration in blood and alveoli are approaching
each other. As the concentration curves cannot cross (second law of thermodynamics)
there is a limited amount of oxygen that can be removed from air. In contrast, for
counter-current flow concentration profiles decrease in opposite direction, thus
much more oxygen can be removed from air, i.e., concentration of oxygen in air can
drop almost to zero.
// MODEL NUMBER: 0129
// MODEL NAME: Co-vsCounter-CurrentExchange
// SHORT DESCRIPTION:
// Steady-state air-blood exchange for two geometric configurations.
JSim v1.1
import nsrunit;
unit conversion on;
math ExchangeArea {
//real j = 7 mmol/min; // physiological demand
real Fc = 5500.0 ml/min; // blood flow rate
real Fa1 = 1000.0 ml/s; // air flow rate (cocurrent)
real Fa2 ml/s; // air flow rate (countercurrent)
real Fa2 = -Fa1;
real Cc0 = 0.0 mmol/ml; // inlet concentration of gas in blood
real M02 = 32.0 g/mol;
real rho = 1.205 kg/m^3;
real Ca0 mmol/ml;
real Ca0 = rho/M02; // inlet concentration of gas in air
real Ca01 mmol/ml; // inlet concentration of gas in air (opposit end)
Ca01 = Ca0;
real jvn = 250.0 ml/min; // volumetric consumption of O2, Hlastala p. 16
real jn mmol/min;
jn = jvn*Ca0;
real F1 ml/s; // geometric mean of flows (cocurrent)
real F2 ml/s; // geometric mean of flows (countercurrent)
real B1 mmol/ml; // integration constant (cocurrent)
//real B2 mmol/ml; // integration constant (countercurrent)
F1 = Fc*Fa1/(Fc+Fa1);
F2 = Fc*Fa2/(Fc+Fa2);
B1 = Ca0-Cc0;
//B2 = (Ca01-Cc0)/F2/(1.0/Fc+1.0/Fa2*exp(-PS2/F2));
realDomain jv ml/min; jv.min = 25.0; jv.max = 2500; jv.delta = 25.0;
//realDomain j mmol/min; j.min = 1.0; j.max = 100.0; j.delta = 10.0;
real j(jv) mmol/min;
j = jv*Ca0;
real PS1(jv) ml/min; // permeability-surface product (cocurrent)
PS1 = -F1*ln(1.0-j/B1/F1);
PS1 >= 0.0;
real PS2(jv) ml/min; // permeability-surface product (countercurrent)
PS2 = ln(Fc*(j+Fa2*Ca01-Fa2*Cc0)/(Fa2*(-j+Ca01*Fc-Cc0*Fc)))*F2;
PS2 >= 0.0;
real PS1n ml/min;
PS1n = PS1(jvn);
real PS2n ml/min;
PS2n = PS2(jvn);
real dPS(jv) dimensionless;
dPS = abs(PS1-PS2)/(PS1+PS2);
real dPSn dimensionless;
dPSn = dPS(jvn);
realDomain x dimensionless; x.min = 0.0; x.max = 1.0; x.delta = 0.1;
real Cc1(x) mmol/ml;
Cc1 = (Ca0*Fa1+Cc0*Fc)/(Fa1+Fc)+Fa1*(Cc0-Ca0)*exp(-PS1n*(Fa1+Fc)*x/(Fa1*Fc))/(Fa1+Fc);
real Ca1(x) mmol/ml;
Ca1 = -((Cc0-Ca0)*exp(-PS1n*(Fa1+Fc)*x/(Fa1*Fc))*Fc/(Fa1+Fc)-(Ca0*Fa1+Cc0*Fc)/(Fa1+Fc));
real Cc2(x) mmol/ml;
Cc2 = (-Ca01*Fa2*exp(PS2n/Fc)+Cc0*exp(PS2n/Fa2)*Fc)/(exp(PS2n/Fa2)*Fc-Fa2*exp(PS2n/Fc))-
Fa2*exp(PS2n/Fc)*(-Ca01+Cc0)*exp((-Fa2+Fc)*PS2n*x/(Fc*Fa2))/(exp(PS2n/Fa2)*Fc-Fa2*exp(PS2n/Fc));
real Ca2(x) mmol/ml;
Ca2 = (-Fa2*exp(PS2n/Fc)*exp(PS2n/Fa2)*(-Ca01+Cc0)*exp(-PS2n*(Fa2+Fc)*x/(Fa2*Fc))*Fc/(Fc+Fa2*exp(PS2n/Fc)*exp(PS2n/Fa2))+
Fa2*(Ca01*Fa2*exp(PS2n/Fc)*exp(PS2n/Fa2)+Cc0*Fc)/(Fc+Fa2*exp(PS2n/Fc)*exp(PS2n/Fa2)))/Fa2;
}
/*
FIGURE:
LEGEND:
DETAILED DESCRIPTION:
Steady-state air blood exchange is presented for two different hypothetical geometric
configurations, i.e., for co-current and counter-current air-blood exchange.
Alveoli-capillary interconnection is modeled as two co-centric cylinders with air and
blood flowing in the same direction for co-current flow and in the opposite direction
in the counter-current flow.
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: co-current, counter-current, steady-state, Respiratory System,
Air-Blood gas exchange, alveolar exchange
REFERENCES:
REVISION HISTORY:
Original Author : AndyMat Date: 31/jul/07
Revised by: BEJ Date:13jun11 : Update comment format
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2011 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.
*/