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.

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Abbreviations

\(\alpha_{{{\text{O}}_{ 2} }}\) :

Solubility of oxygen in water

\(\alpha_{{{\text{CO}}_{ 2} }}\) :

Solubility of carbon dioxide in water

[O2]:

Concentration of free oxygen

[CO2]:

Concentration of free carbon dioxide

[H+]:

Concentration of hydrogen ions (protons)

[DPG]:

Concentration of 2,3-diphosphoglycerate (2,3-DPG)

T :

Temperature

pH:

−log10([H+])

\(P_{{{\text{O}}_{ 2} }}\) :

Partial pressure of oxygen

\(P_{{{\text{CO}}_{ 2} }}\) :

Partial pressure of carbon dioxide

P 50 :

Partial pressure of oxygen for 50 % HbO2 saturation

Hb:

Hemoglobin

HbO2 :

Oxyhemoglobin

HbCO2 :

Carbaminohemoglobin

\(K_{{{\text{HbO}}_{ 2} }}\) :

Apparent equilibrium constant for the binding of oxygen to hemoglobin

\(K_{{{\text{HbCO}}_{ 2} }}\) :

Apparent equilibrium constant for the binding of carbon dioxide to hemoglobin

\(S_{{{\text{HbO}}_{ 2} }}\) :

Saturation of hemoglobin with oxygen

\(S_{{{\text{HbCO}}_{ 2} }}\) :

Saturation of hemoglobin with carbon dioxide

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Acknowledgments

We thank the reviewers for helpful and insightful comments that have enhanced the overall quality of the manuscript. This work was supported by the National Institute of Health Grants P50-GM094503 and P01-GM066730. The extension of P 50 model to extreme/wider physiological conditions (e.g. pH > 8.5) was motivated by RKD’s email correspondence with Stefan Kleiser (University Hospital Zurich), a user of the 2010 Dash and Bassingthwaighte \(S_{{{\text{HbO}}_{ 2} }}\) and \(S_{{{\text{HbCO}}_{ 2} }}\) models.

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Correspondence to Ranjan K. Dash or James B. Bassingthwaighte.

Additional information

Communicated by Carsten Lundby.

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

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.

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Dash, R.K., Korman, B. & Bassingthwaighte, J.B. Simple accurate mathematical models of blood HbO2 and HbCO2 dissociation curves at varied physiological conditions: evaluation and comparison with other models. Eur J Appl Physiol 116, 97–113 (2016). https://doi.org/10.1007/s00421-015-3228-3

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Keywords

  • O2 and CO2 binding to hemoglobin
  • O2 and CO2 saturation of hemoglobin
  • Oxyhemoglobin and carbaminohemoglobin dissociation curves
  • Nonlinear O2–CO2 interactions
  • Bohr and Haldane effects
  • Mathematical modeling