Recirculation of O2 and CO2 between a 2-compartment lung and 2-compartment body.

Description

  This oversimplified whole body model captures the uptake of O2 in the
  lung, delivery to tissue, intratissue consumption and CO2 production
  with return of CO2 to the lung where it is lost into the expired air. 
  This lung/body exchange of O2 and CO2, while ridiculously simple, allows 
  one to explore the values for flows, PSs and Volumes, for metabolic rates, 
  and alveolar exchange that give rise to the values in a normally ambient 
  human. The alveolar exchanges are passiveand the tissue consumption is 
  first order Michaelis-Menten. The ratio of CO2 produced to O2 used in the 
  Respiratory Quotient, RQ.  Concentrations of O2 and CO2 in the blood are 
  expressed in the equivalent partial pressures assuming a Henry's Law type 
  of form (concO2 = alphaO2 * pO2) for conversion of partial pressures of 
  O2 and CO2 to the concentrations bound and dissolved in the blood, tissue
  and lung interstitial fluid (isf).    

Equations

The transport between and within each of the four regions can be represented by convective transport, e.g. F/V11*(O21-O11), diffusive transport between regions, e.g. PS1O/V11*(Cin-O12), and consumption/production terms, e.g. Vcytox/((Kcytox + O22)*V22)*O22, The equations are as follows:

// Input function
extern real Cin(t) mmHg; //Concentration in inflow stream.

//Initial conditions
when(t=t.min){  O11=0; O12=100; O21=0; O22=0; 
                C11=0; C12=0; C21=0; C22=60;  }

//ODEs
O11:t = F/V11*(O21-O11) + PS1O/V11*(O12-O11);
O12:t = 		  PS1O/V12*(O11-O12) + (PSalv/V12)*(Cin-O12);
O21:t = F/V21*(O11-O21) + PS2O/V21*(O22-O21);
O22:t =                   PS2O/V22*(O21-O22) - Vcytox/((Kcytox + O22)*V22)*O22;
C11:t = F/V11*(C21-C11) + PS1C/V11*(C12-C11);
C12:t =                   PS1C/V12*(C11-C12) - GCair*C12/V12;
C21:t = F/V21*(C11-C21) + PS2C/V21*(C22-C21);
C22:t =                   PS2C/V22*(C21-C22) + RQ*Vcytox/((Kcytox + O22)*V22)*O22;

Where F is blood flowrate, O11-O22 are partial pressures of O2 in each of the regions, C11-C22 are partial pressures of CO2 in each of the regions.
PSalv, PS1O and PS2O are the permeability-surface area product of O2 transport between lung and lung isf, lung isf and blood, and blood and tissue respectively, PS1C and PS2C are the permeability-surface area product of CO2 transport between lung isf and blood, and blood and tissue respectively.

V11-V22 are region volumes, the Vcytox/((Kcytox + O22)*V22)*O22 term is the O2 consumption in tissue, the RQ*Vcytox/((Kcytox + O22)*V22)*O22 is the CO2 production in the tissue and the GCair*C12/V12 term is the exhalation of CO2 from the lung.

Units for concentrations as mmHg: Partial pressures are "activities" equivalent to molar concentrations times an activity coefficient. Just as the p50 for the hemoglobin binding of oxygen can be expressed in mmHg or in mM, so also can the binding of O2 to cytochrome oxidase be expressed in either.