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.

Method 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_HbO2) range over a wide range of values of P_O2, P_CO2, pH, 2,3-DPG, and temperature. Our new equations incorporate a novel P_O2-dependent variable cooperativity hypothesis for the binding of O₂ to Hb, and a new equation for P₅₀ 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_O2 and P_CO2 from the S_HbO2 and S_HbCO2 levels using simple iterative numerical schemes that give rapid convergence is a significant advantage over alternative S_HbO2 and S_HbCO2 models.

Conclusion

The new S_HbO2 and S_HbCO2 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.

Introduction

The delivery of oxygen to tissues for metabolism and energy production in parenchymal cells is regulated by a complex system of physicochemical processes in the microcirculation (Dash & Bassingthwaighte, 2006; Bassingthwaighte et al., 2012). In systemic capillaries, the release of oxygen from hemoglobin (Hb) inside red blood cells (RBCs) depends in part on the simultaneous release of carbon dioxide into the flowing blood. In the lungs, the loss of CO₂ from RBCs increases O₂ uptake, which in turn enhances CO₂ dissociation from Hb. This interacting process also depends on the buffering of CO₂ by bicarbonate ions (HCO₃⁻), acid-base regulation, and the Hb-mediated nonlinear biochemical O₂-CO₂ interactions inside RBCs (Geers & Gros, 2000; Rees & Andreassen, 2005; Dash & Bassingthwaighte, 2006; Bassingthwaighte et al., 2012). In this regard, a decrease in pH or an increase in CO₂ partial pressure (P_CO2) in systemic capillaries decreases the affinity of O₂ for Hb and HbO₂ saturation (S_HbO2), and increases the delivery of O₂ to peripheral tissues (the Bohr effect). On the other hand, an increase in O₂ partial pressure (P_O2) in pulmonary capillaries results in the release of hydrogen ions (H⁺) from Hb, which in turn decreases the affinity of Hb for CO₂, reducing HbCO₂ saturation (S_HbCO2) and aiding elimination of CO₂ by the lungs (the Haldane effect). Thus, both the Bohr and Haldane effects are important in defining the Hb-mediated nonlinear biochemical O₂-CO₂ interactions inside RBCs (Siggaard-Andersen & Garby, 1973; Tyuma, 1984; Matejak et al., 2015). Raising blood temperature (T) also lowers the affinity of O₂ for Hb, and increases the delivery of O₂ to tissues. Consequently, the integrated computational modeling of O₂ transport, exchange and metabolism in tissue-organ systems must account for the coupled transport and exchange of CO₂, HCO₃⁻, H⁺, and heat in the microcirculation (Dash & Bassingthwaighte, 2006). This is particularly important in highly metabolic tissue-organ systems such as heart and skeletal muscle during exercise (von Restorff et al., 1977), where arterio-venous (AV) temperature increases of 1 °C or so, P_CO2 increases of nearly 10 mmHg, and pH decreases of about 0.1 pH unit, all combine to push the HbO₂ dissociation curve to the right. Under these conditions, P_O2 may decrease by as much as 80 mmHg and S_HbO2 may decrease to less than 10% along capillaries which may be shorter than one millimetre in length. To properly account for the mass balance throughout the capillary-tissue exchange region, it is necessary to compute the changes associated with the nonlinear biochemical O₂-CO₂ interactions in blood as influenced by P_O2, P_CO2, pH, and T, thus the need for accurate and efficient forward and invertible S_HbO2 and S_HbCO2 equations.

In order to quantify the Bohr and Haldane effects as well as the synergistic effects of 2,3-DPG and T, Dash and Bassingthwaighte (2004; 2010) have developed new mathematical models of O₂ and CO₂ saturation of Hb (S_HbO2 and S_HbCO2) based on equilibrium binding of O₂ and CO₂ to Hb inside RBCs. They are in the form of an invertible Hill-type equation with apparent binding constants K_HbO2 and K_HbCO2 which depend on the levels of P_O2, P_CO2, pH, 2,3-DPG, and T in blood. The invertibility of these new equations enables analytical calculations of P_O2 from S_HbO2 and P_CO2 from S_HbCO2 and vice-versa. This is especially important for the integrated computational modeling of simultaneous transport and exchange of O₂ and CO₂ in alveolar-blood and blood-tissue exchange systems (Dash & Bassingthwaighte, 2006). The HbO₂ dissociation curves computed from the S_HbO2 model are in very good agreement with previously published experimental and theoretical curves in the literature over a wide range of physiological conditions (Kelman, 1966b; Severinghaus, 1979; Winslow et al., 1983; Siggaard-Andersen et al., 1984; Buerk & Bridges, 1986). However, as with any theoretical model, there are some limitations in the 2010 Dash and Bassingthwaighte S_HbO2 and S_HbCO2 models. In particular, the S_HbO2 model is only accurate for values of S_HbO2 lying between 0.3 and 0.98 (with a Hill coefficient nH of 2.7), even if accounting well in that range for the effects of pH, P_CO2, 2,3-DPG and T. Furthermore, the S_HbO2 and S_HbCO2 models require complicated calculations of the indices n₁, n₂, n₃, and n₄ involved in the expression for the apparent equilibrium constant in a single-step O₂-Hb binding reaction (Dash & Bassingthwaighte, 2010). These limitations are addressed here by simplifying the S_HbO2 and S_HbCO2 models further and extending the accuracy of the S_HbO2 model to the whole S_HbO2 range for the varied physiological conditions. The resulting S_HbO2 and S_HbCO2 models are also tested using diverse experimental data available in the literature on the HbO₂ and HbCO₂ dissociation curves for a wide range of physiological conditions (Joels & Pugh, 1958; Naeraa et al., 1963; Bauer & Schroder, 1972; Hlastala et al., 1977; Matthew et al., 1977; Reeves, 1980). This study does not include the buffering of CO₂ in blood as that is covered in our previous work on blood-tissue gas exchange (Dash & Bassingthwaighte, 2006) and by Wolf (2013) for whole-body acid-base and electrolyte balance. Interested readers are referred to our previous article (Dash & Bassingthwaighte, 2010) for a historical perspective and details of the mathematical models of S_HbO2 not covered in this paper.

Methods

Simple Mathematical Expressions for S_HbO2 and S_HbCO2

Based on the detailed nonlinear biochemical interactions of O₂ and CO₂ with Hb inside RBCs, Dash and Bassingthwaighte (2010) derived the following mechanistic mathematical expressions for the fractional saturation of Hb with O₂ and CO₂ (S_HbO2 and S_HbCO2, respectively):

$$S_{\text{HbO}2}=\frac{K_{\text{HbO}2}[{\mathrm{O}}_2]}{1+K_{\text{HbO}2}[{\mathrm{O}}_2]};S_{\text{HbCO}2}=\frac{K_{\text{HbCO}2}[{\text{CO}}_2]}{1+K_{\text{HbCO}2}[{\text{CO}}_2]}$$ (1a,b)

Here the concentrations are in moles per liter or M with respect to the water space of RBCs. The concentration of free O₂ or free CO₂ in the water space of RBCs may also be expressed in terms of the partial pressure of O₂ or CO₂ so that [O₂] = α_O2P_O2 and [CO₂] = α_CO2P_CO2, where α_O2 and α_CO2 are the solubilities of O₂ and CO₂ in water, computed from their measured values in plasma (Austin et al., 1963; Hedley-Whyte & Laver, 1964) and corrected for the effects of temperature (Kelman, 1966a; 1967; Dash & Bassingthwaighte, 2010):

$${\alpha }_{\mathrm{O}2}=\left[1.37-1.37\times {10}^{-2}(T-37)+5.8\times {10}^{-4}{(T-37)}^2\right]\left[{10}^{-6}/W_{\text{pl}}\right]\mathrm{M}/\text{mmHg}$$ (2a)

$${\alpha }_{\text{CO}2}=\left[3.07-5.7\times {10}^{-2}(T-37)+2\times {10}^{-3}{(T-37)}^2\right]\left[{10}^{-5}/W_{\text{pl}}\right]\mathrm{M}/\text{mmHg}$$ (2b)

Here W_pl = 0.94 is the fractional water space of plasma and T is in degrees centigrade (°C). Thus =α_O21.46×10⁻⁶ M/mmHg and α_CO2 = 3.27×10⁻⁵ M/mmHg at 37 °C (see Table 1). The expressions for [O₂] and [CO₂] in terms of P_O2 and P_CO2 and vice-versa are freely interchanged throughout this paper. The apparent equilibrium constants of Hb with O₂ and CO₂ (K_HbO2 and K_HbCO2 with units of M⁻¹), based on single-step bindings of O₂ and CO₂ to Hb, are reproduced here from Dash and Bassingthwaighte (2010) in a simplified form, introducing the new variables Φ₁ − Φ₄:

$$K_{\text{HbO}2}=\frac{K_4^{\prime }(K_3^{\prime }[{\text{CO}}_2]{\mathrm{\Phi }}_2+{\mathrm{\Phi }}_4)}{K_2^{\prime }[{\text{CO}}_2]{\mathrm{\Phi }}_1+{\mathrm{\Phi }}_3};K_{{\text{HbCO}}_2}=\frac{K_2^{\prime }{\mathrm{\Phi }}_1+K_3^{\prime }{K}_4^{\prime }[{\mathrm{O}}_2]{\mathrm{\Phi }}_2}{{\mathrm{\Phi }}_3+K_4^{\prime }[{\mathrm{O}}_2]{\mathrm{\Phi }}_4}$$ (3a,b)

where the molar concentrations of the gases O₂ and CO₂ are used; the terms Φ₁ − Φ₄ involving the interactions of H⁺ with Hb-bound O₂ and CO₂ are given by:

$${\mathrm{\Phi }}_1=1+\frac{K_2^{″}}{[{\mathrm{H}}^{+}]};{\mathrm{\Phi }}_2=1+\frac{K_3^{″}}{[{\mathrm{H}}^{+}]};{\mathrm{\Phi }}_3=1+\frac{[{\mathrm{H}}^{+}]}{K_5^{″}};{\mathrm{\Phi }}_4=1+\frac{[{\mathrm{H}}^{+}]}{K_6^{″}}$$ (4a–d)

Table 1.

Model parameter values used in the simulations, most of which are as presented in the Dash and Bassingthwaighte (2010) paper; model parameters that are re-estimated based on fittings of the model to the experimental data in Figs. 1 and 3 are shown with footnotes. Unless otherwise noted, the kinetic parameter values are at T = 37 °C.

Parameter	Definition	Value	Unit
$K_1^{″}$	Ionization constant of H2CO3	5.5×10−4	M
$K_1^{\prime }$	Equilibrium constant for the CO2 hydration reaction: ${\text{CO}}_2+{\mathrm{H}}_2\mathrm{O}\leftrightarrow {\mathrm{H}}_2{\text{CO}}_3(K_1^{\prime }=K_1/K_1^{″})$	1.4×10−3	Unitless
K1	Equilibrium constant for the overall CO2 hydration reaction: ${\text{CO}}_2+{\mathrm{H}}_2\mathrm{O}\leftrightarrow {{\text{HCO}}_3}^{-}+{\mathrm{H}}^{+}(K_1=K_1^{\prime }·K_1^{″})$; pK1 = −log10(K1) = 6.1 − 0.0434(pHpl−7.4) + 0.0014(T−37)(pHpl−7.4)	7.94×10−7 (pHpl = 7.4; T = 37°C)	M
$K_2^{″}$	Ionization constant of HbNHCOOH	1×10−6	M
$K_2^{\prime }$	Equilibrium constant for the CO2-Hb binding reaction: ${\text{CO}}_2+{\text{HbNH}}_2\leftrightarrow \text{HbNHCOOH}(K_2^{\prime }=K_2/K_2^{″})$	21.5# 23.65*	M−1
K2	Equilibrium constant for the overall CO2-Hb binding reaction: ${\text{CO}}_2+{\text{HbNH}}_2\leftrightarrow {\text{HbNHCOO}}^{-}+{\mathrm{H}}^{+}(K_2=K_2^{\prime }·K_2^{″})$	21.5×10−6# 23.65×10−6*	Unitless
$K_3^{″}$	Ionization constant of HbO2NHCOOH	1×10−6	M
$K_3^{\prime }$	Equilibrium constant for the CO2-HbO2 binding reaction: ${\text{CO}}_2+{\text{HbO}}_2{\text{NH}}_2\leftrightarrow {\text{HbO}}_2\text{NHCOOH}(K_3^{\prime }=K_3/K_3^{″})$	11.3# 14.7*	M−1
K3	Equilibrium constant for the overall CO2-HbO2 binding reaction: ${\text{CO}}_2+{\text{HbO}}_2{\text{NH}}_2\leftrightarrow {\text{HbO}}_2{\text{NHCOO}}^{-}+{\mathrm{H}}^{+}(K_3=K_3^{\prime }·K_3^{″})$	11.3×10−6# 14.7×10−6*	Unitless
$K_{4,\mathrm{S}}^{\prime }$	Equilibrium constant for the overall O2-Hb binding reaction: O2 + HbNH2 ↔ HbO2NH2 at standard physiological conditions (otherwise it is a function of PO2, PCO2, pHrbc, [DPG]rbc and T)	2.03×105	M−1
$K_5^{″}$	Ionization constant of HbNH3+	2.4×10−8# 2.64×10−8*	M
$K_6^{″}$	Ionization constant of HbO2NH3+	1.2×10−8# 1.56×10−8*	M
nH	Hill coefficient with variable cooperativity hypothesis for O2 binding to Hb: nH = α − β×10−PO2/γ (Eq. 11)	Variable	Unitless
α	Parameter governing PO2-dependent variable Hill coefficient nH with variable cooperativity hypothesis for O2 binding to Hb	2.82$	Unitless
β	Parameter governing PO2-dependent variable Hill coefficient nH with variable cooperativity hypothesis for O2 binding to Hb	1.20$	Unitless
γ	Parameter governing PO2-dependent variable Hill coefficient nH with variable cooperativity hypothesis for O2 binding to Hb	29.25$	mmHg
P50,S	Level of PO2 at which Hb is 50% saturated by O2 at standard physiological levels of PCO2, pHrbc, [DPG]rbc and T	26.8$	mmHg
PO2,S	Standard partial pressure of O2 in blood	100	mmHg
PCO2,S	Standard partial pressure of CO2 in blood	40	mmHg
pHpl,S	Standard pH in plasma	7.4	Unitless
pHrbc,S	Standard pH in RBCs, related to the standard plasma pH by: pHrbc,S = 0.795pHpl,S +1.357	7.24	Unitless
[DPG]rbc,S	Standard 2,3-DPG concentration in RBCs	4.65×10−3	M
TS	Standard temperature of blood	37	°C
pHrbc	pH in RBCs, related to plasma pH by: pHrbc = 0.795pHpl + 1.357)	Variable	Unitless
Rrbc	Gibbs-Donnan ratio for electrochemical equilibrium of protons or bicarbonate ions across the RBC membrane (Rrbc = [H+]pl/[H+]rbc = [HCO3−]rbc/[HCO3−]pl; Rrbc = 10−(pHpl−pHrbc) = 10−(0.205·pHpl−1.357))	Variable (0.692 at pHpl = 7.4)	Unitless
αO2,S	Solubility of O2 in water at standard temperature (37 °C)	1.46×10−6	M/mmHg
αCO2,S	Solubility of CO2 in water at standard temperature (37 °C)	3.27×10−5	M/mmHg
HctS	Standard hematocrit (volume fraction of RBCs in blood)	0.45	Unitless
[Hb]bl,S	Standard hemoglobin concentration in blood	2.33×10−3	M
[Hb]rbc,S	Standard hemoglobin concentration in RBCs ([Hb]rbc=[Hb]bl/Hct)	5.18×10−3	M
Wpl	Fractional water space of plasma	0.94	Unitless
Wrbc	Fractional water space of RBCs	0.65	Unitless
Wbl	Fractional water space of blood: Wbl = (1−Hct)·Wpl + Hct·Wrbc	0.81	Unitless
[O2]tot	Total O2 concentration of whole blood: [O2]tot = Wbl·αO2·PO2 + 4·Hct·[Hb]rbc·SHbO2	Variable	M
[CO2]tot	Total CO2 concentration of whole blood: [CO2]tot = Wbl·αCO2·PCO2 + ((1−Hct)Wpl+Hct·Wrbc·Rrbc)(K1·αCO2·PCO2/[H+]pl) + 4·Hct·[Hb]rbc·SHbCO2 (parameter K1 defined above)	Variable	M

^#Estimated based on model fittings to the data of Bauer and Schröder (1972) at T = 37 °C (see Fig. 3E)

^*Estimated based on model fittings to the data of Matthew et al. (1977) at T = 30 °C (see Fig. 3F)

^$Estimated based on model fittings to the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) in normal human blood at standard physiological conditions

Here [H⁺] = 10^−pH in RBCs. Given the pH of plasma, the pH of RBCs can be obtained using the simple relationship established by Siggaard-Andersen and colleagues (Siggaard-Andersen, 1971; Siggaard-Andersen & Salling, 1971): pH_rbc = 0.795pH_pl + 1.357, giving the Gibbs-Donnan ratio for electrochemical equilibrium of H⁺ and HCO₃⁻ across the RBC membrane as a function of pH_pl: R_rbc = 10^{−(pH_pl−pH_rbc)} = 10^{−(0.205pH_pl−1.357)} (see Table 1). In our previous S_HbO2 and S_HbCO2 models (Dash & Bassingthwaighte, 2010), R_rbc was considered as a constant: R_rbc = 0.69, a value corresponding to pH_pl = 7.4 and pH_rbc = 7.24 (see Table 1).

With these relationships (Eqs. 1–4), the total O₂ and CO₂ concentrations in whole blood can be calculated, as described in the Appendix of the Dash and Bassingthwaighte (2010) paper (see Table 1), and hence are not covered here in detail.

Dependency of $K_4^{\prime }$ on the Physiological Variables of Interest

Except for $K_4^{\prime }$, the binding association/dissociation constants are all specified in Table 1. $K_4^{\prime }$ represents the single-step binding association constant of O₂ with each heme chain of Hb (HmNH₂) according to the following hypothetical reaction scheme (Dash & Bassingthwaighte, 2010):

$${\mathrm{O}}_2+{\text{HmNH}}_2\stackrel{K_4^{\prime }}{\rightleftharpoons }{\mathrm{O}}_2{\text{HmNH}}_2$$ (5)

The fundamental S-shape of the HbO₂ dissociation curve results from the form of Eq. 1a and the values taken by the product K_HbO2 [O₂], which depends on $K_4^{\prime }$. For any given range of [O₂], the shifts in the HbO₂ dissociation curve with respect to varying physiological conditions arise from the dependence of K_HbO2 on [CO₂], [H⁺] (via Φ₁−Φ₄), and $K_4^{\prime }$. Similarly, the shape of the HbCO₂ dissociation curve results from the form of Eq. 1b and the values taken by the product K_HbCO2 [CO₂], which also depends on $K_4^{\prime }$. For any given range of [CO₂], the shifts in the HbCO₂ dissociation curve with respect to varying physiological conditions arise from the dependence of K_HbCO2 on [O₂], [H⁺] (via Φ₁−Φ₄), and $K_4^{\prime }$. From the above description, it is clear that determining $K_4^{\prime }$ is a critical step in the application of Eqs. 1a and 1b. In our previous S_HbO2 and S_HbCO2 models (Dash & Bassingthwaighte, 2010), $K_4^{\prime }$ is expressed as a complicated function of [O₂], [CO₂], [H⁺], [DPG] and T involving the exponents n₀, n₁, n₂, n₃ and n₄, which, except for n₀ (a constant equal to 1.7), are themselves complicated functions of these physiological variables and P₅₀ of O₂. In the section below, we show how a very simple expression for $K_4^{\prime }$ can be obtained without involving the exponents n₀−n₄, significantly simplifying the S_HbO2 and S_HbCO2 models.

Derivation of Simple Mathematical Expression for $K_4^{\prime }$

Hill’s exponent model for S_HbO2 is based on an n^th-order one-step binding of Hb with O₂:

$$S_{\text{HbO}2}=\frac{{(P_{\mathrm{O}2}/P_{50})}^n}{1+{(P_{\mathrm{O}2}/P_{50})}^n}$$ (6)

Here n = nH, the Hill coefficient, is approximately 2.7 for normal human blood; P₅₀ is the value of P_O2 at which Hb is 50 percent saturated by O₂. In Eq. 6, the shifts of the HbO₂ dissociation curve produced by varying physiological conditions arise because P₅₀ is a function of P_CO2, pH, [DPG] and T, as described below. If Eq. 1a and Eq. 6 are to give identical HbO₂ dissociation curves, we must have K_HbO2 α_O2P_O2= (P_O2/P₅₀)^nH, which when simplified gives:

$$K_4^{\prime }=\frac{{\left({\alpha }_{\mathrm{O}2}{P}_{\mathrm{O}2}\right)}^{nH-1}\left(K_2^{\prime }{\alpha }_{\text{CO}2}{P}_{\text{CO}2}{\mathrm{\Phi }}_1+{\mathrm{\Phi }}_3\right)}{{\left({\alpha }_{\mathrm{O}2}{P}_{50}\right)}^{nH}\left(K_3^{\prime }{\alpha }_{\text{CO}2}{P}_{\text{CO}2}{\mathrm{\Phi }}_2+{\mathrm{\Phi }}_4\right)}$$ (7)

While it is simple to compute S_HbO2 from Eq. 6, this is not true for S_HbCO2. However, Eq. 7 provides an expression for $K_4^{\prime }$ in terms of P_O2, P_CO2, pH and P₅₀, which along with Eq. 1b and Eq. 3b describes a simplified model for the HbCO₂ dissociation curve. Note also that Eq. 7 can be written in the following alternative form:

$$K_4^{\prime }=\left(\frac{1}{{\alpha }_{\mathrm{O}2}{P}_{\mathrm{O}2}}\right)\left(\frac{S_{{\text{HbO}}_2}}{1-S_{{\text{HbO}}_2}}\right)\left(\frac{K_2^{\prime }{\alpha }_{\text{CO}2}{P}_{\text{CO}2}{\mathrm{\Phi }}_1+{\mathrm{\Phi }}_3}{K_3^{\prime }{\alpha }_{\text{CO}2}{P}_{\text{CO}2}{\mathrm{\Phi }}_2+{\mathrm{\Phi }}_4}\right)$$ (8)

so that as S_HbO2 approaches 1, $K_4^{\prime }$ becomes infinitely large and K_HbCO2 becomes equal to $K_3^{\prime }{\mathrm{\Phi }}_2/{\mathrm{\Phi }}_4$ in Eq. 3b. If a “Division by Zero” error is to be avoided when calculating K_HbCO2 in computer programs based on these equations, it is necessary to identify situations in which S_HbO2 might equal 1 (at very high P_O2), bypass Eq. 7, and immediately set $K_3^{\prime }{\mathrm{\Phi }}_2/{\mathrm{\Phi }}_4$.

Simple Mathematical Expression for P₅₀ of O₂

P₅₀ is a function of P_CO2, pH, [DPG] and T. Dash and Bassingthwaighte (2010) obtained a polynomial expression for P₅₀ by varying one of these variables at a time, while keeping the other three fixed at their standard physiological values. The resulting polynomial expressions for P_{50,Δ pH}, P_{50, ΔCO2}, P_{50, ΔDPG}, and P_{50, ΔT} were fitted to the reported P₅₀ data from the studies of Buerk and Bridges (1986) and Winslow et al. (1983). These polynomial expressions from Dash and Bassingthwaighte (2010) are further refined here based on additional experimental data from Joels and Pugh (1958), Naeraa et al. (1963), Hlastala et al. (1977), and Reeves (Reeves, 1980) that provide appropriate shifts in S_HbO2 with varying pH, P_CO2 and T:

$$P_{50,\mathrm{\Delta }\text{pH}}=P_{50,\mathrm{S}}-25.535(\text{pH}-{\text{pH}}_{\mathrm{S}})+10.646{(\text{pH}-{\text{pH}}_{\mathrm{S}})}^2-1.764{(\text{pH}-{\text{pH}}_{\mathrm{S}})}^3$$ (9a)

$$P_{50,\mathrm{\Delta }\text{CO}2}=P_{50,\mathrm{S}}+1.273\times {10}^{-1}(P_{\text{CO}2}-P_{\text{CO}2,\mathrm{S}})+1.083\times {10}^{-4}{(P_{\text{CO}2}-P_{\text{CO}2,\mathrm{S}})}^2$$ (9b)

$$P_{50,{\mathrm{\Delta }}_{\text{DPG}}}=P_{50,\mathrm{S}}+795.63([\text{DPG}]-{[\text{DPG}]}_{\mathrm{S}})-19660.89{([\text{DPG}]-{[\text{DPG}]}_{\mathrm{S}})}^2$$ (9c)

$$P_{50,\mathrm{\Delta }\mathrm{T}}=P_{50,\mathrm{S}}+1.435(T-T_{\mathrm{S}})+4.163\times {10}^{-2}{(T-T_{\mathrm{S}})}^2+6.86\times {10}^{-4}{(T-T_{\mathrm{S}})}^3$$ (9d)

Here the standard physiological values are denoted by the subscript “S” and are listed in Table 1. The accuracy of the expression for P_{50, ΔpH} at extreme pH levels has been further improved in Eq. 9a by incorporating the cubic term. Previous researchers (Kelman, 1966b; Severinghaus, 1979; Siggaard-Andersen et al., 1984; Buerk & Bridges, 1986) have shown that for the calculation of P₅₀ to be valid when multiple physiological variables are allowed to vary simultaneously, the contribution of each physiological variable to the resulting P₅₀ is by multiplication of normalized individual P₅₀’s. Consequently, the expression for P₅₀ that best describes the simultaneous varying physiological conditions is given by:

$$P_{50}=P_{50,\mathrm{S}}(P_{50,\mathrm{\Delta }\mathrm{H}}/P_{50,\mathrm{S}})(P_{50,\mathrm{\Delta }\text{CO}2}/P_{50,\mathrm{S}})(P_{50,\mathrm{\Delta }\text{DPG}}/P_{50,\mathrm{S}})(P_{50,\mathrm{\Delta }\mathrm{T}}/P_{50,\mathrm{S}})$$ (10)

Incorporation of the Variable Cooperativity Hypothesis

In our previous S_HbO2 and S_HbCO2 models (Dash & Bassingthwaighte, 2010), the Hill coefficient nH was fixed at 2.7 (or n₀ was fixed at 1.7), making the S_HbO2 accurate only between 30% and 98%. Next, we extend the accuracy of the modified S_HbO2 model for the whole saturation range by incorporating an empirical P_O2-dependent variable cooperativity hypothesis for O₂ binding to Hb. Specifically, this is achieved by expressing the Hill coefficient nH at low P_O2 as a simple exponential function of P_O2:

$$nH=\alpha -\beta \times {10}^{-(P_{O2}/\gamma )}$$ (11)

where α, β and γ are parameters that govern an apparent cooperativity of O₂ for Hb. Eq. 11 suggests that at low P_O2, nH is close to α−β, but increases exponentially towards α with the rate γ as P_O2 increases. These three parameters along with P_50,S are estimated here based on fittings of the S_HbO2 model to the available experimental data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) and Winslow et al. (1977) on normal human blood HbO₂ dissociation curves at standard physiological conditions. With nH accurately defined by Eq. 11, suitable shifts in the HbO₂ and HbCO₂ dissociation curves are achieved through our new expressions for (Eqs. 9 and 10) and $K_4^{\prime }$ (Eq. 7).

Efficient Numerical Inversion of the S_HbO2 and S_HbCO2 Expressions for Practical Usage

Eq. 1a for HbO₂ saturation (S_HbO2) is not convenient for analytical inversion, because the apparent equilibrium constant K_HbO2 for the binding of O₂ to Hb depends on $K_4^{\prime }$, which in turn depends on P_O2 (see Eq. 7). However, since Eq. 1a is equivalent to Eq. 6, which depends on P₅₀, which in turn is independent of P_O2 and is only a function of pH, P_CO2, [DPG] and T, Eq. 6 can be analytically inverted to compute P_O2 from S_HbO2, subject to the condition that the Hill coefficient nH is a constant. When nH is allowed to vary with P_O2, Eq. 6 can no longer be inverted analytically. However, numerical inversion is possible using efficient iterative schemes such as those presented in the Appendix to this paper. These converge within 8 or less iterations when appropriate starting values are used for P_O2. The same approach holds for the inversion of Eq. 1b in the computation of P_CO2 from S_HbCO2 (see Appendix), which may not be important clinically, but is of relevance in the integrated computational modeling of simultaneous transport and exchange of respiratory gases in physiological systems. Two efficient numerical methods are presented in the Appendix for the inversion of each gas, based on fixed-point and quasi-Newton-Raphson iteration methods. Similar iterative methods can be used to compute P_O2 and P_CO2 from the total [O₂] and the total [CO₂], respectively (these last two parameters are defined in Table 1).

Results

The effect of varying the Hill coefficient nH as a function of P_O2 on the modified S_HbO2 model simulations at standard physiological conditions is demonstrated through Fig. 1, in which Figs. 1(A,C,E) are based on the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979), while Figs. 1(B,D,F) are based on the data of Winslow et al. (1977). The first diagram in each set shows how the old and new S_HbO2 models (constant and variable nH) fit the data in the P_O2 range of 0–150 mmHg. The inset in each of these two figures shows the fit in the lower P_O2 range of 0–30 mmHg. It is immediately apparent that the previous S_HbO2 model, which does not incorporate the variable cooperativity hypothesis, produces a poor fit in the lower 40% saturation range. In Figs. 1(C,D), the residual error in the computed S_HbO2 relative to the experimental S_HbO2 is shown for each model for the range of values taken by S_HbO2. When the saturation is less than 40%, the residual error is much improved for the new S_HbO2 model (<0.05% for variable nH compared to ~3–% for constant nH), and the average residual error over the whole saturation range is virtually zero. Figs. 1(E,F) show how this improvement has been achieved by varying the Hill coefficient nH in the low P_O2 range through the application of Eq. 11 (solid line), compared with the constant value used for nH in the old S_HbO2 model (dashed line). We note here that the nH rate parameter γ is related to the P_50,S value, which differs slightly for the two different data sets (26.8 mmHg vs. 29.1 mmHg). To achieve the best fit, the parameters α and β also differ slightly between the two different sets of data. This improved accuracy in S_HbO2 at low P_O2 may not be relevant clinically, but is important for the computational modeling and mechanistic understanding of in-vitro and in vivo blood gas data under non-physiological conditions, such as ischemia and reperfusion, extremes in gas concentrations, high altitudes, and extreme temperatures.

Figure 1. Illustration of the accuracy of the modified S_HbO2 model under standard physiological conditions with P_O2-dependent variable cooperativity hypothesis for O₂-Hb binding.

(A,B) Comparison of the model-simulated HbO₂ dissociation curves with constant and variable cooperativity hypotheses for O₂-Hb binding (constant and variable Hill coefficient nH) to available experimental data in the literature, on normal human blood at standard physiological conditions. The simulations in panel A are compared to the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979), and those in panel B are compared to the data of Winslow et al. (1977). The simulations in the main plots are compared to the data over the whole S_HbO2 range, while those in the inset plots are compared to the data for S_HbO2 ≤ 0.5, effectively demonstrating the improved accuracy of the modified S_HbO2 model with variable nH in simulating the data in the lower S_HbO2 range. (C,D) The percentage deviations (100 × ΔS_HbO2) of the model-simulated S_HbO2 values from the experimental S_HbO2 values for the constant and variable nH models plotted as functions of S_HbO2 for the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) and Winslow et al. (1977). The incorporation of the variable cooperativity hypothesis for O₂-Hb binding (variable nH) has improved the accuracy of the S_HbO2 model. (E,F) The P_O2-dependent variation of nH corresponding to the constant and variable cooperativity hypotheses for O₂-Hb binding that are obtained based on fittings of the modified S_HbO2 model to the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) and Winslow et al. (1977). The insets in plots (E,F) show the expressions for the variable nH and the governing parameter values for the two data sets.

The P₅₀ values computed from Eqs. 9 and 10 under varying physiological conditions are plotted in Figs. 2(A–D) and are compared to those computed from alternative models from the literature (Kelman, 1967; Buerk & Bridges, 1986; Dash & Bassingthwaighte, 2010). The alternative models do not include corrections based on the experimental data of Joels and Pugh (1958), Naeraa et al. (1963), Hlastala et al. (1977) and Reeves (1980); so for the purpose of this comparison, these corrections have been omitted from Eqs. 9(a–d) by applying appropriate scaling factors of 0.833, 0.588 and 1.02 for P_{50, ΔpH}, P_{50, ΔCO2} and P_{50, ΔT}, respectively. These simulations show that our scaled P₅₀ values agree well with those computed from the model of Buerk and Bridges (1986), that are based on the studies of Winslow et al. (1983). However, they do differ from the P₅₀ values computed from the model of Kelman (1966b), where it fails to fit the data at low pH and at either very low or very high P_CO2. Our scaled P₅₀ values are also improved from our old P₅₀ values for pH > 8.5 with the incorporation of the cubic term in Eq. 9a. In addition, the inclusion of corrections based on the diverse experimental data sets of Joels and Pugh (1958), Naeraa et al. (1963), Hlastala et al. (1977), and Reeves (1980) in Eqs. 9(a–d) provides further improvement in the P₅₀ values over a wide range of variation in the relevant physiological variables (see description below for Fig. 3). A new model recently published by Mateják et al. (2015), while this article was under preparation, also fits some of these experimental data well under altered physiological conditions. Their approach, a modification of Adair’s four-step algorithm (Adair, 1925), considers the problems in depth, and, like ours, relates the O₂-Hb and CO₂-Hb binding effects to the acid-base chemistry of blood based on pH and P_CO2, as proposed by Siggaard-Andersen and others over the years (Rossi-Bernardi & Roughton, 1967; Forster et al., 1968; Siggaard-Andersen, 1971; Siggaard-Andersen & Salling, 1971; Bauer & Schroder, 1972; Siggaard-Andersen et al., 1972a; Siggaard-Andersen et al., 1972b; Siggaard-Andersen & Garby, 1973; Siggaard-Andersen et al., 1984; Siggaard-Andersen & Siggaard-Andersen, 1990). Specifically, they effectively express the four Adair coefficients in terms of the physiological variables of interest (i.e. pH, P_CO2, and T they do not consider the effects of 2,3-DPG), providing appropriate shifts in the HbO₂ dissociation curve with altered physiological conditions.

Figure 2. Comparison of P₅₀ values under various physiological conditions based on simulations of different P₅₀ models.

(A–D) Comparison of the modified P₅₀ model (Eqs. 9 and 10) with the previous P₅₀ models from the literature (Kelman, 1966b; Buerk & Bridges, 1986; Dash & Bassingthwaighte, 2010), in which the model-simulated P₅₀ values are plotted as functions of one variable with the other variables fixed at their standard physiological values; i.e. P₅₀ as a function of pH_rbc (A), P_CO2 (B), [DPG]_rbc (C), and T (D). The P₅₀ values have been computed from Eqs. 9 and 10 under varying physiological conditions with appropriate multiplicative factors (0.833, 0.588 and 1.02 for P_50,ΔpH, P_50,ΔCO2 and P_50,ΔT, respectively) to allow a comparison with the previous models, which have not been adjusted to account for the data of Joels and Pugh (1958), Naeraa et al. (1963), Hlastala et al. (1977) and Reeves (1980).

Figure 3. Comparison of our improved S_HbO2 and S_HbCO2 model simulations to the diverse experimental data available in the literature under non-standard physiological conditions.

(A) HbO₂ dissociation curves obtained from our improved S_HbO2 model (i.e. modified S_HbO2 model with variable cooperativity hypothesis for O₂-Hb binding (variable nH)) compared with the data of Joels and Pugh (1958) obtained at various pH_pl and P_CO2 levels with T = 37 °C. The inset shows the expression for the variable nH and the governing parameter values that produce the best fit of the model to these data sets. In addition, these model fittings characterize the pH and P_CO2 dependencies of P₅₀ in Eqs. 9a and 9b. (B) S_HbO2 levels obtained from our improved S_HbO2 model compared to the data of Naeraa et al. (1963) obtained as a function of pH_rbc for different P_O2 and P_CO2 levels at 37 °C. Other details are as for panel A. (C,D) HbO₂ dissociation curves obtained from our improved S_HbO2 model compared with the data of Hlastala et al. (1977) and Reeves (1980) obtained for various values of T with pH_pl and P_CO2 fixed at 7.4 and 40 mmHg, respectively. The insets show the expression for the variable nH and the governing parameter values that enable best fits of the model to these data sets. In addition, these model fittings characterize the temperature dependency of P₅₀ in Eq. 9d. (E,F) S_HbCO2 levels obtained from our improved S_HbCO2 model compared with the data of Bauer and Schröder (1972) and Matthew et al. (1977) obtained as a function of pH_rbc in the oxygenated (high P_O2) and deoxygenated (zero P_O2) blood with P_CO2 fixed at 40 mmHg (T = 37 °C) in the former experiments and total [CO₂] fixed at 55 mM (T = 30 °C) in the latter experiments. These model fittings provide the estimates of the equilibrium constants associated with the binding of CO₂ to oxygenated and deoxygenated Hb as well as the ionization constants of oxygenated and deoxygenated Hb.

The accuracy and robustness of our simplified and extended models of S_HbO2 and S_HbCO2 under varying physiological conditions (P₅₀ defined by Eqs. 9 and 10, and variable nH defined by Eq. 11) has also been tested against diverse experimental data available from the literature (Joels & Pugh, 1958; Naeraa et al., 1963; Bauer & Schroder, 1972; Hlastala et al., 1977; Matthew et al., 1977; Reeves, 1980), and is illustrated in Fig. 3. Our refined S_HbO2 model (i.e. the simplified S_HbO2 model with variable nH) is able to reproduce the S_HbO2 data of Joels and Pugh (1958) and Naeraa et al. (1963) obtained for a range of values of pH and P_CO2 at 37 °C (Figs. 3A and 3B) as well as the S_HbO2 data of Hlastala et al. (1977) and Reeves (1980) obtained over a range of values of T at fixed P_CO2 and pH (Figs. 3C and 3D). These data allow a more accurate determination of the coefficients for use in the polynomial expressions for P_{50, ΔpH}, P_{50, ΔCO2}, P_{50, ΔDPG}, and P_50,ΔT, and hence characterize the dependence of P₅₀ on pH, P_CO2, [DPG] and T, and the corresponding shifts in the HbO₂ dissociation curves. Similarly, our refined model of S_HbCO2 is able to reproduce the S_HbCO2 data of Bauer and Schröder (1972) and Matthew et al. (1977) obtained as a function of pH_rbc in oxygenated and deoxygenated blood with either constant P_CO2 of 40 mmHg at 37°C (Fig. 3E) or constant total [CO₂] of 55 mM at 30 °C (Fig. 3F). These model fittings provide accurate estimates of the equilibrium constants associated with the binding of CO₂ to oxygenated and deoxygenated Hb as well as the ionization constants of oxygenated and deoxygenated Hb at 30 °C and 37 °C (see Table 1). These estimates differ slightly from those used in our previous models of S_HbO2 and S_HbCO2 (Dash & Bassingthwaighte, 2010), but are in close agreement with those reported by Bauer and Schröder (1972) and Rossi-Bernardi and Roughton (1967). It can be noted here that the estimates of the parameters associated with the P_O2-dependent variable nH for these different data sets are all similar, further signifying the importance of the variable cooperativity hypothesis in accurately simulating HbO₂ dissociation curves in the whole saturation range under varying physiological conditions.

Simulations of the apparent HbO₂ binding constant $K_4^{\prime }$, HbO₂ dissociation curves, and HbCO₂ dissociation curves over a wide range of physiological conditions (P_O2, P_CO2, pH, [DPG], and T) are based on our refined models of $K_4^{\prime }$, S_HbO2 and S_HbCO2 with the P_O2-dependent variable cooperativity hypothesis for O₂-Hb binding, and are shown in Fig. 4. The models of S_HbO2 and S_HbCO2 simulate the Bohr and Haldane effects, and the synergistic effects of 2,3-DPG and T on the HbO₂ and HbCO₂ dissociation curves. These simulations are consistent with the previous simulations by Dash and Bassingthwaighte (2010), except for their greater accuracy. The shifts in the HbO₂ dissociation curves seen in Figs. 4(E–H) are correlated with the variations in the P₅₀ values plotted in Figs. 2(A–D) (solid lines), as defined by Eqs. 9(a–d) and validated by diverse experimental data in Figs. 3(A–D). Similarly, the shifts in the HbCO₂ dissociation curves seen in Figs. 4(I–L) are correlated with the extent of CO₂ binding to Hb under stipulated physiological conditions, as validated by diverse experimental data in Figs. 3(E–F). These simulations clearly show the different effect of each variable on O₂ and CO₂ binding to Hb; pH is shown to significantly influence the O₂ and CO₂ binding, compared with the smaller effects of P_CO2, 2,3-DPG, and T.

Figure 4. Simulations of the apparent HbO₂ binding constant $K_4^{\prime }$ (A–D), HbO₂ dissociation curves (E–H), and HbCO₂ dissociation curves (I–L) under varying physiological conditions based on our modified models of $K_4^{\prime }$, S_HbO2 and S_HbCO2 with the P_O2-dependent variable cooperativity hypothesis for O₂-Hb binding.

In plots (A–D) and (E–H), the simulations of $K_4^{\prime }$ and S_HbO2 are shown as functions of P_O2, respectively, while in plots (I–L), the simulations of S_HbCO2 are shown as a function of P_CO2. The values of the variables used in each simulation are shown as insets in each panel. Three of the variables are fixed at their standard physiological values, while the fourth one is allowed to increase from a low value to a high value by predetermined increments, shown by the sequence [low: increment: high]. Specifically, in plots (A,E,I), pH_rbc varies, while either P_CO2 or P_O2, [DPG]_rbc and T are fixed; in plots (B,F,J), either P_CO2 or P_O2 varies, while pH_rbc, [DPG]_rbc and T are fixed; in plots (C,G,K), [DPG]_rbc varies, while pH_rbc, either P_CO2 or P_O2, and T are fixed; in plots (D,H,L), T varies, while pH_rbc, either P_O2 or P_CO2, and [DPG]_rbc are fixed. In each plot, the long arrow shows the direction of the shift produced by increasing the value of the fourth variable. The shifts in the HbO₂ dissociation curves seen in plots (E–H) are correlated with the variations in the P₅₀ values plotted in Figs. 2(A–D) (solid lines), as defined by Eqs. 9(a–d) and validated by diverse experimental data in Figs. 3(A–D). Similarly, the shifts in the HbCO₂ dissociation curves seen in plots (I–L) are correlated with the extent of CO₂ binding with Hb under stipulated physiological conditions, as validated by diverse experimental data in Figs. 3(E–F).

The accurate and efficient numerical inversion of our refined S_HbO2 model under standard physiological conditions is demonstrated in Fig. 5, based on the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) and Winslow et al. (1977). The inset plots in Figs. 5A and 5B clearly show the improved agreement between the inverted P_O2 values and the experimental P_O2 values at lower S_HbO2 levels (<40%) when the Hill coefficient nH is allowed to vary as a function P_O2 according to Eq. 11, rather than being held constant. A similar conclusion may be drawn from Figs. 5C and 5D. The results for the inverse problem are consistent with the results for the forward problem shown in Fig. 1. The comprehensive simulations of the numerical inversions under varying physiological conditions are shown in Fig. 6. These include the simulations of P_O2 as a function of S_HbO2 and the simulations of P_CO2 as a function of S_HbCO2 over a wide range of values of the relevant physiological variables. In each scenario, three variables are fixed at their standard physiological levels, while the fourth one is allowed to increase by predetermined increments over a large range, as in the simulations for the forward problem in Fig. 4. It is apparent that our iterative numerical inversion schemes are able to effectively simulate P_O2 and P_CO2 levels from the S_HbO2 and S_HbCO2 levels over a wide range of physiological conditions.

Figure 5. Illustration of the accurate numerical inversion of the modified S_HbO2 model under standard physiological conditions with the P_O2-dependent variable cooperativity hypothesis for O₂-Hb binding.

(A,B) Comparison of the numerically-inverted P_O2 values from the S_HbO2 values with constant and variable cooperativity hypotheses for O₂-Hb binding (constant and variable Hill coefficient nH) to available experimental data in the literature on normal human blood at standard physiological conditions. The numerical inversions in panel A are compared to the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979), and those in panel B are compared to the data of Winslow et al. (1977). The numerical inversions in the main plots are compared to the data over the whole S_HbO2 range, while those in the inset plots are compared to the data for S_HbO2 ≤ 0.5, effectively demonstrating the accurate numerical inversion of the modified S_HbO2 model with variable nH in simulating the data in the lower S_HbO2 range. (C,D) The percentage deviations (100 × Δlog₁₀(P_O2)) of the numerically-inverted log₁₀(P_O2) values from the experimental log₁₀(P_O2) values for the constant and variable nH models plotted as functions of log₁₀(P_O2) for the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) and Winslow et al. (1977). The P_O2 dependencies of nH for these two data sets are as shown in Figs. 1(E,F). The incorporation of the variable cooperativity hypothesis for O₂-Hb binding (variable nH) has improved the accuracy of the numerically-inverted P_O2 values from the S_HbO2 values.

Figure 6. Illustration of the effective numerical inversion of our improved S_HbO2 and S_HbCO2 equations under varying physiological conditions.

In plots (A–D), the simulations of P_O2 as a function of S_HbO2 are shown, while in plots (E–H), the simulations of P_CO2 as a function of S_HbCO2 are shown. In each scenario, three variables are fixed at their standard physiological levels, while the fourth one is allowed to increase by predetermined increments over a large range, as described in detail in the legend to Fig. 4. The inversion computations were carried out using a variant of the Newton-Raphson method (i.e. quasi-Newton-Raphson method) with appropriate initial guesses that guaranteed convergence. These plots are the mirror-images of the corresponding plots in Fig. 4 (but with different scales), illustrating the accuracy of the numerical inversion schemes.

Discussion

We have shown here how shifts in the HbO₂ dissociation curve due to variations in P_CO2, pH, 2,3-DPG, and T (where the variations occur in one variable at a time or several variables simultaneously) may be handled easily in the Hill equation by using an equation of state for P₅₀ which incorporates the contribution from each of these four variables. While other equations exist for P₅₀, to the best of our knowledge, none incorporates contributions from all of the four variables under discussion or is as extensively validated using independent experimental data. Moreover, we believe that Eq. 10 is the most accurate equation of state for P₅₀ currently available. Although standard versions of Hill’s two-parameter equation for S_HbO2 (using only P₅₀ and nH) have been widely used in the past, it is recognized that when nH is set equal to 2.7, as is common practice, the equation is inaccurate for S_HbO2 below 30 per cent and above 98 per cent (Dash & Bassingthwaighte, 2010), thus stimulating the development of improved descriptions by Adair (1925), Kelman (1966b), Severinghaus (1979), Siggaard-Andersen et al. (1984), Buerk and Bridges (1986), and others. In this paper, we have addressed this important deficiency in the standard Hill equation by expressing the Hill coefficient nH as a simple exponential function of P_O2 (Eq. 11), with P₅₀ given by Eq. 10. Figures 1, 3, and 5 show that this change improves the agreement between the model and available experimental data at both standard physiological conditions and at other values of pH, P_CO2, 2,3-DPG, and temperature that characterize pathophysiological states.

So the question that arises is this: Using these modifications, how does Hill’s equation compare with other commonly used mathematical models of the HbO₂ dissociation curve? In Figs. 7A and 7C, we show the fit of our new refined Hill-based equation for S_HbO2 to the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) under standard physiological conditions and compare that to the fit obtained using the S_HbO2 models of Adair (1925), Kelman (1967), Buerk and Bridges (1986), Severinghaus (1979), and Siggaard-Andersen et al. (1984; 1990) for both the forward (computation of S_HbO2 from P_O2) and inverse (computation of P_O2 from S_HbO2) situations. The associated residual errors are shown in Figs. 7B and 7D. Although all the models shown in Fig. 7A seem to provide a good fit to the data (except for Kelman (1966b) for some P_O2 values and Buerk and Bridges (1986) for high P_O2), the residual error shown in Fig. 7B is least for our present model and the models of Severinghaus (1979) and Siggaard-Andersen et al. (1984; 1990). Fig. 7C shows that all the models also produce a good fit to the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) on inversion, but the error plotted in Fig. 7D indicates that our model is the most accurate for P_O2 greater than 10^0.5 (i.e. P_O2 > 3) mmHg. Note that the distinction between the various models for the inverse problem is more apparent because of the use of a log-log scale. Apart from the two exceptions mentioned above, when compared with the data most of these models provide accuracy greater than 99.5% over the whole saturation range.

Figure 7. Comparison of our improved S_HbO2 model with alternative S_HbO2 models from the literature in simulating available experimental data at standard physiological conditions (both forward and inverse problems).

In each case, the model for comparison has first been parametrized to produce a “best fit” based on the experimental data. (A) HbO₂ dissociation curves obtained from our improved S_HbO2 model compared with those obtained from other commonly used S_HbO2 models (Adair, 1925; Kelman, 1966b; Severinghaus, 1979; Siggaard-Andersen et al., 1984; Buerk & Bridges, 1986), in simulating the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) on normal human blood at standard physiological conditions. (B) The percentage deviations (100 × ΔS_HbO2) of the model-simulated S_HbO2 values from the experimental S_HbO2 values based on our improved S_HbO2 model and the alternative models, plotted as functions of S_HbO2 for the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979). (C) Comparisons of the numerically-inverted P_O2 values from the S_HbO2 values obtained from our improved S_HbO2 model and the alternative models, in simulating the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979) on normal human blood at standard physiological conditions. (D) The percentage deviations (100 × Δlog₁₀(P_O2)) of the numerically-inverted log₁₀(P_O2) values from the experimental log₁₀(P_O2) values based on our improved S_HbO2 model and the alternative S_HbO2 models, plotted as functions of log₁₀(P_O2) for the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979). The comparison may be repeated using the data of Winslow et al. (1977). This involves the re-parametrization of each model to obtain the best fit to the data. The result has not been shown here as it does not contribute anything further to the conclusions already drawn.

We also note here that the other S_HbO2 models shown in Fig. 7 were parametrized based on the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979), and hence a comparison of the models with respect to the data of Winslow et al. (1977) cannot be made, unless they are re-parameterized. This is not the purpose of Fig. 7. Rather, the purpose of Fig. 7 is to illustrate how our new refined S_HbO2 model is “at least” comparable in terms of accuracy to many other S_HbO2 models used widely in the literature. Fig. 7 shows that this is indeed the case, yet our model is simpler than many others and efficiently invertible.

The recent model of Mateják et al. (2015), based on their version of the Adair equation (1925), is reported to fit well, the data of Severinghaus and colleagues (Roughton et al., 1972; Roughton & Severinghaus, 1973; Severinghaus, 1979), and is also shown to fit the temperature and pH-dependent S_HbO2 data of Reeves (1980) and Naeraa (1963), and carbaminohemoglobin data of Bauer and Schröder (1972) and Matthew (1977). Our present S_HbO2 and S_HbCO2 models fit these diverse data sets as accurately, and also fit the P_CO2, pH, and temperature-dependent data of Joels and Pugh (1958) and Hlastala et al. (1977), which were not used by Mateják et al. (18). Thus, the present S_HbO2 and S_HbCO2 models are well-validated based on a larger set of experimental data. More importantly, not only is our new refined S_HbO2 model more accurate over the whole saturation range (important for the integrated computational modeling of alveolar-blood and blood-tissue O₂-CO₂ transport, exchange, and metabolism), it is also much simpler than our previous S_HbO2 model, as there is no need to perform the complex computations to derive the indices n₁, n₂, n₃, and n₄.

The HbCO₂ dissociation curve is obtained in our treatment from Eq. 1b. The expression for K_HbCO2 is derived from Eq. 3b and Eq. 4. When S_HbO2 equals 1, K_HbCO2 equals $K_3^{\prime }{\mathrm{\Phi }}_2/{\mathrm{\Phi }}_4$. Otherwise $K_4^{\prime }$ is required and is given by Eq. 7. Fortuitously, all the information regarding the variables P_CO2, pH, 2,3-DPG and T is contained in Eq. 10, the equation of state for P₅₀. Thus, in our treatment, both S_HbO2 and S_HbCO2 rely on this equation for information about P_CO2, pH, 2,3-DPG and T. Other authors incorporate this information in different ways. For example, Severinghaus (1979) provides one equation for the temperature coefficient of P_O2 and another for the change in lnP_O2 per unit change in pH. If instead of equating Eq. 6 with Eq. 1a, we equate the Severinghaus equation for S_HbO2 with Eq. 1a, we obtain the following expression for $K_4^{\prime }$:

$$K_4^{\prime }=\frac{\left(150+23400P_{O2}^2\right)\left(K_2^{\prime }{\alpha }_{\text{CO}2}{P}_{\text{CO}2}{\mathrm{\Phi }}_1+{\mathrm{\Phi }}_3\right)}{{\alpha }_{O2}\left(K_3^{\prime }{\alpha }_{\text{CO}2}{P}_{\text{CO}2}{\mathrm{\Phi }}_2+{\mathrm{\Phi }}_4\right)}$$ (12)

The value for P_O2 used in Eq. 12 is that obtained after correction for temperature and pH. A similar approach may be adopted in the case of the Adair equation (Adair, 1925) to transfer information regarding pH, 2,3-DPG and T to the CO₂-Hb reaction system. Thus, Eq. 1b can be adapted for use with a variety of models of the HbO₂ dissociation curve.

Inversion of S_HbO2 and S_HbCO2 equations is an essential feature in modeling of alveolar-blood and blood-tissue O₂-CO₂ exchange at a sophisticated level. Algebraic inversion is possible with the Severinghaus S_HbO2 equation (1979), but not in our models, once the Hill coefficient nH is allowed to vary. Iterative approaches are generally used for this. The Appendix provides fixed-point and quasi-Newton-Raphson iterative methods that would also be applicable to the models of Mateják et al. (2015) and others that are not analytically invertible.

Conclusions

We have simplified and extended our previously developed mathematical models of blood HbO₂ and HbCO₂ dissociation curves (Dash & Bassingthwaighte, 2010) to make them accurate over the whole saturation range for a wide range of values of P_O2, P_CO2, pH, 2,3-DPG, and temperature. The extended S_HbO2 model features a P_O2-dependent variable Hill coefficient nH for the binding of O₂ to Hb, and incorporates a modified P₅₀ model that provides accurate shifts in the HbO₂ dissociation curve over a wide range of physiological conditions, validated by diverse experimental data sets available in the literature. The information contained in this modified equation for P₅₀ is transferred to the HbCO₂ dissociation curve via $K_4^{\prime }$, the apparent equilibrium constant in a single-step binding reaction of O₂ and Hb. The coupling of the Hb dissociation curves for O₂ and CO₂ may also be accomplished using models of S_HbO2 other than the Hill equation. Finally, the extended S_HbO2 and S_HbCO2 models are conveniently invertible for efficient computation of P_O2 from S_HbO2 and P_CO2 from S_HbCO2, using simple iterative numerical schemes with appropriate starting values that guarantee convergence. The calculations involved in our new S_HbO2 and S_HbCO2 models may be performed on handheld devices. More importantly, they can conveniently be used in the integrated computational modeling of alveolar-blood and blood-tissue O₂-CO₂ transport, exchange, and metabolism for analysis and mechanistic understanding of in-vitro and in-vivo blood gas data under physiological and non-physiological conditions, such as ischemia and reperfusion, extremes in gas concentrations, high altitudes, and extreme temperatures.

Frequently Used Abbreviations

α_O2

Solubility of oxygen

α_CO2

Solubility of carbon dioxide

[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_O2

Partial pressure of oxygen

P_CO2

Partial pressure of carbon dioxide

P₅₀

Partial pressure of oxygen for 50% HbO₂ saturation

Hb

Hemoglobin

HbO₂

Oxyhemoglobin

HbCO₂

Carbaminohemoglobin

K_HbO2

Apparent equilibrium constant of hemoglobin and oxygen binding

K_HbCO2

Apparent equilibrium constant of hemoglobin and carbon dioxide binding

S_HbO2

Saturation of hemoglobin with oxygen

S_HbCO2

Saturation of hemoglobin with carbon dioxide

Appendix: Efficient Iterative Schemes for Numerical Computations of P_O2 from S_HbO2 and P_CO2 from S_HbCO2

Two efficient numerical methods are presented below for the inversion of P_O2 from S_HbO2 and P_CO2 from S_HbCO2, 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_O2 from S_HbO2 are given by:

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

$$\begin{array}{l}\text{Scheme}-2:P_{\mathrm{O}2}^{\text{New}}=P_{\mathrm{O}2}^{\text{Old}}-\left(\frac{S_{\text{HbO}2}(P_{\mathrm{O}2}^{\text{Old}})-S_{\text{HbO}2}^{\text{Input}}}{S_{\text{HbO}2}^{\prime }(P_{\mathrm{O}2}^{\text{Old}})}\right)\\ =P_{\mathrm{O}2}^{\text{Old}}-\left(\frac{0.02P_{\mathrm{O}2}^{\text{Old}}\left(S_{\text{HbO}2}(P_{\mathrm{O}2}^{\text{Old}})-S_{\text{HbO}2}^{\text{Input}}\right)}{S_{\text{HbO}2}(P_{\mathrm{O}2}^{\text{Old}}+0.01P_{\mathrm{O}2}^{\text{Old}})-S_{\text{HbO}2}(P_{\mathrm{O}2}^{\text{Old}}-0.01P_{\mathrm{O}2}^{\text{Old}})}\right)\end{array}$$ (A-1b)

where $S_{\text{HbO}2}^{\text{Input}}$ is the input S_HbO2 (given), $S_{\text{HbO}2}(P_{\mathrm{O}2}^{\text{Old}})$ is the value of S_HbO2 evaluated at $P_{\mathrm{O}2}^{\text{Old}}$, and $S_{\text{HbO}2}^{\prime }(P_{\mathrm{O}2}^{\text{Old}})$ is the derivative of S_HbO2 w.r.t. P_O2 evaluated at $P_{\mathrm{O}2}^{\text{Old}}$. Either Eq. 1a or Eq. 6 can be used as the expression for S_HbO2. In the second version of Eq. A-1b, it is only necessary to perform function evaluation, because $S_{\text{HbO}2}^{\prime }(P_{\mathrm{O}2}^{\text{Old}})$ has been estimated using a central-difference formula for first-order derivatives (Pozrikidis, 2008), which eliminates the need to differentiate the expression for S_HbO2, 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_O2 from S_HbO2. For P_O2-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_O2. For the same accuracy, the iteration scheme of Eq. A-1b converges within 5 to 8 iterations using P₅₀ from Eq. 10 as the starting value for P_O2.

The analogous iterative schemes for the computation of P_CO2 from S_HbCO2 are given by:

$$\text{Scheme}-1:P_{\text{CO}2}^{\text{New}}=\left(\frac{S_{\text{HbCO}2}^{\text{Input}}}{1-S_{\text{HbCO}2}^{\text{Input}}}\right)\left(\frac{1}{{\alpha }_{\text{CO}2}{K}_{\text{HbCO}2}(P_{\text{CO}2}^{\text{Old}})}\right)$$ (A-2a)

$$\begin{array}{l}\text{Scheme}-2:P_{\text{CO}2}^{\text{New}}=P_{\text{CO}2}^{\text{Old}}-\left(\frac{S_{\text{HbCO}2}(P_{\text{CO}2}^{\text{Old}})-S_{\text{HbCO}2}^{\text{Input}}}{S_{\text{HbCO}2}^{\prime }(P_{\text{CO}2}^{\text{Old}})}\right)\\ =P_{\text{CO}2}^{\text{Old}}-\left(\frac{0.02P_{\text{CO}2}^{\text{Old}}\left(S_{\text{HbCO}2}(P_{\text{CO}2}^{\text{Old}})-S_{\text{HbCO}2}^{\text{Input}}\right)}{S_{\text{HbCO}2}(P_{\text{CO}2}^{\text{Old}}+0.01P_{\text{CO}2}^{\text{Old}})-S_{\text{HbCO}2}(P_{\text{CO}2}^{\text{Old}}-0.01P_{\text{CO}2}^{\text{Old}})}\right)\end{array}$$ (A-2b)

Similar iterative methods can be used to compute P_O2 from total [O₂] and P_CO2 from total [CO₂]. Note that the total [O₂] and the total [CO₂] are defined in Table 1.