Simple accurate mathematical models of blood HbO₂ and HbCO₂ dissociation curves at varied physiological conditions: evaluation and comparison with other models

Abstract

Purpose

Equations for blood oxyhemoglobin (HbO₂) and carbaminohemoglobin (HbCO₂) dissociation curves that incorporate nonlinear biochemical interactions of oxygen and carbon dioxide with hemoglobin (Hb), covering a wide range of physiological conditions, are crucial for a number of practical applications. These include the development of physiologically-based computational models of alveolar-blood and blood-tissue O₂–CO₂ transport, exchange, and metabolism, and the analysis of clinical and in vitro data.

Methods and results

To this end, we have revisited, simplified, and extended our previous models of blood HbO₂ and HbCO₂ dissociation curves (Dash and Bassingthwaighte, Ann Biomed Eng 38:1683–1701, 2010), validated wherever possible by available experimental data, so that the models now accurately fit the low HbO₂ saturation ($S_{{{\text{HbO}}_{ 2} }}$) range over a wide range of values of $P_{{{\text{CO}}_{ 2} }}$, pH, 2,3-DPG, and temperature. Our new equations incorporate a novel $P_{{{\text{O}}_{ 2} }}$-dependent variable cooperativity hypothesis for the binding of O₂ to Hb, and a new equation for P ₅₀ of O₂ that provides accurate shifts in the HbO₂ and HbCO₂ dissociation curves over a wide range of physiological conditions. The accuracy and efficiency of these equations in computing $P_{{{\text{O}}_{ 2} }}$ and $P_{{{\text{CO}}_{ 2} }}$ from the $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ levels using simple iterative numerical schemes that give rapid convergence is a significant advantage over alternative $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ models.

Conclusion

The new $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ models have significant computational modeling implications as they provide high accuracy under non-physiological conditions, such as ischemia and reperfusion, extremes in gas concentrations, high altitudes, and extreme temperatures.

Appendix: Efficient iterative schemes for numerical computations of \(P_{{{\text{O}}_{ 2} }}\) from \(S_{{{\text{HbO}}_{ 2} }}\) and \(P_{{{\text{CO}}_{ 2} }}\) from \(S_{{{\text{HbCO}}_{ 2} }}\)

Two efficient numerical methods are presented below for the inversion of $P_{{{\text{O}}_{ 2} }}$ from $S_{{{\text{HbO}}_{ 2} }}$ and $P_{{{\text{CO}}_{ 2} }}$ from $S_{{{\text{HbCO}}_{ 2} }}$, based on fixed-point and quasi-Newton–Raphson iteration methods (Scheme-1 and Scheme-2, respectively) (Heath 2002; Pozrikidis 2008).

The iterative schemes for the computation of $P_{{{\text{O}}_{ 2} }}$ from $S_{{{\text{HbO}}_{ 2} }}$ are given by:

Scheme-1:

$$P_{{{\text{O}}_{ 2} }}^{\text{New}} = P_{50} \left( {\frac{{S_{{{\text{HbO}}_{ 2} }}^{\text{Input}} }}{{1 - S_{{{\text{HbO}}_{ 2} }}^{\text{Input}} }}} \right)^{{\frac{1}{{nH(P_{{{\text{O}}_{ 2} }}^{\text{Old}} )}}}}$$

(A-1a)

Scheme-2:

$$\begin{aligned} P_{{{\text{O}}_{ 2} }}^{\text{New}} &= P_{{{\text{O}}_{ 2} }}^{\text{Old}} - \left( {\frac{{S_{{{\text{HbO}}_{ 2} }} (P_{{{\text{O}}_{ 2} }}^{\text{Old}} ) - S_{{{\text{HbO}}_{ 2} }}^{\text{Input}} }}{{S^{\prime}_{{{\text{HbO}}_{ 2} }} (P_{{{\text{O}}_{ 2} }}^{\text{Old}} )}}} \right)\\&= P_{{{\text{O}}_{ 2} }}^{\text{Old}} - \left( {\frac{{0.02P_{{{\text{O}}_{ 2} }}^{\text{Old}} \left( {S_{{{\text{HbO}}_{ 2} }} (P_{{{\text{O}}_{ 2} }}^{\text{Old}} ) - S_{{{\text{HbO}}_{ 2} }}^{\text{Input}} } \right)}}{{S_{{{\text{HbO}}_{ 2} }} (P_{{{\text{O}}_{ 2} }}^{\text{Old}} + 0.01P_{{{\text{O}}_{ 2} }}^{\text{Old}} ) - S_{{{\text{HbO}}_{ 2} }} (P_{{{\text{O}}_{ 2} }}^{\text{Old}} - 0.01P_{{{\text{O}}_{ 2} }}^{\text{Old}} )}}} \right) \end{aligned}$$

(A-1b)

where $S_{{{\text{HbO}}_{ 2} }}^{\text{Input}}$ is the input $S_{{{\text{HbO}}_{ 2} }}$ (given), $S_{{{\text{HbO}}_{ 2} }} (P_{{{\text{O}}_{ 2} }}^{\text{Old}} )$ is the value of $S_{{{\text{HbO}}_{ 2} }}$ evaluated at $P_{{{\text{O}}_{ 2} }}^{\text{Old}}$, and $S^{\prime}_{{{\text{HbO}}_{ 2} }} (P_{{{\text{O}}_{ 2} }}^{\text{Old}} )$ is the derivative of $S_{{{\text{HbO}}_{ 2} }}$ w.r.t. $P_{{{\text{O}}_{ 2} }}$ evaluated at $P_{{{\text{O}}_{ 2} }}^{\text{Old}}$. Either Eq. 1a or Eq. 6 can be used as the expression for $S_{{{\text{HbO}}_{ 2} }}$. In the second version of Eq. A-1b, it is only necessary to perform function evaluation, because $S^{\prime}_{{{\text{HbO}}_{ 2} }} (P_{{{\text{O}}_{ 2} }}^{\text{Old}} )$ has been estimated using a central-difference formula for first-order derivatives (Pozrikidis 2008), which eliminates the need to differentiate the expression for $S_{{{\text{HbO}}_{ 2} }}$, which may be complicated. If the Hill coefficient nH is held constant, Eq. A-1a itself provides the analytical inversion for the computation of $P_{{{\text{O}}_{ 2} }}$ from $S_{{{\text{HbO}}_{ 2} }}$. For $P_{{{\text{O}}_{ 2} }}$-dependent nH (Eq. 11), the iteration scheme of Eq. A-1a converges within 3 to 5 iterations with 10⁻³ accuracy, using any starting value for $P_{{{\text{O}}_{ 2} }}$. For the same accuracy, the iteration scheme of Eq. A-1b converges within 5–8 iterations using P ₅₀ from Eq. 10 as the starting value for $P_{{{\text{O}}_{ 2} }}$.

The analogous iterative schemes for the computation of $P_{{{\text{CO}}_{ 2} }}$ from $S_{{{\text{HbCO}}_{ 2} }}$ are given by:

Scheme-1:

$$P_{\text{CO2}}^{\text{New}} = \left( {\frac{{S_{\text{HbCO2}}^{\text{Input}} }}{{1 - S_{\text{HbCO2}}^{\text{Input}} }}} \right)\left( {\frac{1}{{\alpha_{\text{CO2}} K_{\text{HbCO2}} (P_{\text{CO2}}^{\text{Old}} )}}} \right)$$

(A-2a)

Scheme-2:

$$\begin{aligned} P_{\text{CO2}}^{\text{New}} &= P_{\text{CO2}}^{\text{Old}} - \left( {\frac{{S_{\text{HbCO2}} (P_{\text{CO2}}^{\text{Old}} ) - S_{\text{HbCO2}}^{\text{Input}} }}{{S^{\prime}_{\text{HbCO2}} (P_{\text{CO2}}^{\text{Old}} )}}} \right)\\&= P_{\text{CO2}}^{\text{Old}} - \left( {\frac{{0.02P_{\text{CO2}}^{\text{Old}} \left( {S_{\text{HbCO2}} (P_{\text{CO2}}^{\text{Old}} ) - S_{\text{HbCO2}}^{\text{Input}} } \right)}}{{S_{\text{HbCO2}} (P_{\text{CO2}}^{\text{Old}} + 0.01P_{\text{CO2}}^{\text{Old}} ) - S_{\text{HbCO2}} (P_{\text{CO2}}^{\text{Old}} - 0.01P_{\text{CO2}}^{\text{Old}} )}}} \right) \end{aligned}$$

(A-2b)

Similar iterative methods can be used to compute $P_{{{\text{O}}_{ 2} }}$ from total [O₂] and $P_{{{\text{CO}}_{ 2} }}$ from total [CO₂]. Note that the total [O₂] and the total [CO₂] are defined in Table 1.