Disequilibrium Between Alveolar and End-Pulmonary-Capillary O₂ Tension in Altitude Hypoxia and Respiratory Disease: An Update of a Mathematical Model of Human Respiration at Altitude

Abstract

We have previously formulated and validated a mathematical model specifically designed to describe human respiratory behavior at altitude. In that model, we assumed equality of alveolar and end-pulmonary-capillary oxygen tensions. However, this equality may not hold true during rapid and prolonged changes to high altitudes producing severe hypoxia as can occur in aircraft cabin decompressions and in some respiratory diseases. We currently investigate this possibility by modifying our previous model to include the dynamics of oxygen exchange across the pulmonary capillary. The updated model was validated against limited experimental data on ventilation and gas tensions in various altitude-decompression scenarios. The updated model predicts that during rapid and sustained decompressions to high altitudes the disequilibrium of gas tensions between alveolar gas and capillary blood could be 10 Torr, or larger. Neglecting this effect underestimates the severity of a decompression and its potential to produce unconsciousness and subsequent brain damage. In light of these results, we also examined the effect of this disequilibrium on the diminished oxygen diffusion capacity that can occur in some respiratory diseases. We found that decreases in diffusion capacity which would have minimal effects at sea level produced significant disequilibrium of gas tensions and a large fall in hemoglobin oxygen saturation at a cabin altitude of 4000–8000 ft. As demonstrated, this new model could serve as an important tool to examine the important physiological consequences of decompression scenarios in aircraft and the pathophysiological situations in which the equilibrium of gas tensions along the pulmonary capillary are particularly critical.

Appendix

The current model is a modification of our previous one21 based on the seminal model by Grodins et al.11 Grodins’s model used ordinary differential equations to describe the material balance of O₂ and CO₂ in alveolar gas, brain tissue fluid, peripheral tissue fluid, and blood compartments. For example, he described the movement of O₂ between alveolar gas and blood in terms of the volume flows of O₂ in and out of the lungs and the flow of O₂ in the venous blood going to the lungs and the arterialized blood leaving them. We wrote similar equations, but because of the changes in masses and volumes that occur during a rapid decompression to altitude, it was convenient to write all of these equations in terms of masses of O₂ and CO₂. Hence, for example, our conventional (equilibrium) alveolar material-balance equation for O₂ is,

$$ \dot{f}_{\text{A}} {\text{O}}_{2} = {\frac{{\dot{V}_{\text{A}} \times \left( {f_{\text{I}} {\text{O}}_{2} - f_{\text{A}} {\text{O}}_{2} } \right) \times \left( {{\frac{n}{V}}} \right) + Q_{\text{L}} \times \left( {nC_{\text{v}} {\text{O}}_{2} - nC_{\text{A}} {\text{O}}_{2} } \right)}}{{V_{\text{A}} \times \left( {{\frac{n}{V}}} \right)}}}, $$

(A1)

where $\dot{f}$ is time rate-of-change of fraction, V _A and $\dot{V}_{\text{A}}$ are alveolar volume and ventilation, respectively, I and v stand for inspired and venous, respectively, Q _L is unshunted (capillary) lung blood flow, and nC is blood content (mass concentration) of gases (mmol gas L⁻¹). The factor, n/V, refers to the conversion of volume to mass which changes with barometric pressure according to the perfect-gas law. The alveolar partial pressure of O₂ is

$$ P_{\text{A}} {\text{O}}_{2} = P_{\text{B}} \times f_{\text{A}} {\text{O}}_{2} , $$

(A2)

where P _B is barometric pressure. Other partial pressures are computed similarly.

Figure A1 shows the factors involved in this alveolar material balance. As seen, diffusion of O₂ across the capillary membrane produces an increase in partial pressure in the capillary from venous (v) to arterial (ec) ends, an uptake of O₂ into RBCs and subsequent chemical binding of O₂ to hemoglobin. If these processes are sufficiently rapid, equilibrium is attained and P _ecO₂ is equal to P _AO₂. This assumption leads to Eq. (A1).

Figure A1

Illustration of the factors involved in O₂ transfer from alveoli to red blood cells (RBCs). The result is an increase in O₂ content and partial pressure down the length of the pulmonary capillary

The mass balance equations for the other compartments were described similarly to Eq. (A1) except no volume–mass conversion was needed. In addition, it was necessary to account for the changes in alveolar gas masses due to altitude changes. Rapid decompression produces a rather immediate expulsion of air from the lungs in proportion to the decrease in barometric pressure18 subsequently affecting the mass of gas breathed in and out. Hence the masses of O₂, CO₂, and the total mass of gas in the alveoli were each altered as the atmospheric pressure changed.

If equilibrium is not attained, Eq. (A1) is not solved. Instead, because P _ecO₂ is now dependent upon capillary membrane diffusion capacity (D _m), the diffusion capacity for O₂ movement into RBCs, and incorporation with hemoglobin $\left( {\theta_{{{\text{O}}_{2} }} \times V_{\text{c}} } \right)$ and the capillary blood flow (Q _c), a new formulation is required. Hence, as described by Wagner and West,20 the combined diffusion capacity for the lung (D _L) is described by

$$ D_{\text{L}} = {\frac{{D_{\text{m}} \times \theta_{{{\text{O}}_{2} }} \times V_{\text{c}} }}{{D_{\text{m}} + \theta_{{{\text{O}}_{2} }} \times V_{\text{c}} }}}, $$

(A3)

where an empirical fit to $\theta_{{{\text{O}}_{2} }}$ data19 was determined to be

$$ \theta_{{{\text{O}}_{ 2} }} = {\frac{2.73}{{1 + \left( {{\frac{\text{SAT}}{94}}} \right)^{31.9} }}}, $$

(A4)

and SAT is the % O₂ saturation of hemoglobin.

The rate at which O₂ diffuses into capillary blood $\left( {\dot{C}_{\text{c}} {\text{O}}_{2} } \right)$ is

$$ \dot{C}_{\text{c}} {\text{O}}_{2} = D_{\text{L}} \times \left( {P_{\text{A}} {\text{O}}_{2} - P_{\text{c}} {\text{O}}_{2} } \right), $$

(A5)

where the capillary O₂ partial pressure (P _cO₂) changes from the mixed venous value (P _vO₂) to P _ecO₂ as blood travels down the length of the pulmonary capillary.

The content of O₂ in end-capillary blood was determined from

$$ C_{\text{ec}} {\text{O}}_{2} = C_{\text{v}} {\text{O}}_{2} \left( {t_{\text{c}} } \right) + \int\limits_{0}^{{t_{\text{c}} }} {\dot{C}_{\text{c}} {\text{O}}_{2} \left( {t^{\prime}} \right)dt^{\prime}} , $$

(A6)

where

$$ t_{\text{c}} = {\frac{{V_{\text{c}} }}{{Q_{\text{c}} }}}. $$

(A7)

In these latter equations, t _c is the time for blood to move from the venous to arterial ends of the capillary and t′ is the integration time variable; C _vO₂ is determined from P _vO₂ and the hemoglobin–O₂ saturation curve as we previously described.21 In the disequilibrium simulation, Eq. (A6) is solved repetitively every t _c s. The value at the end of each cycle is used to compute P _ecO₂ from the hemoglobin–O₂ saturation curve. Then, this value is saved until the end of the next t _c-integration cycle and the saved value used to compute P _aO₂. Figure A2 shows the simulation procedure.

Figure A2

Simulation procedure. The lung plant equations are solved at each time (t) step using values of end capillary (ec) O₂ content and partial pressure determined from the capillary integration procedure. The latter are solved at the same time step as the lung plant equations, but time (t′) goes from 0 at the beginning of an integration cycle to t _c, the end of the cycle. The final values are held until they are updated at the end of the next cycle. The held values are used in the integration of the lung-plant equations

The simulation of these equations was performed using the VisSim program as previously described.21 The integration routine selected from the ones built into VisSim was a fourth order Runge–Kutta algorithm. The time step was 0.001 s. For example, for a V _c of 75 mL and a Q _c of 100 mL s⁻¹, t _c would be 0.75 s. Hence, the final value of P _cO₂, saved as P _ecO₂, is held and passed to the lung-plant equations until its value is updated at the end of the next capillary-O₂ integration procedure. If t _c remained constant, P _ecO₂ would be updated every 0.75 s or 750 integration steps.