Erratum to: Blood HbO₂ and HbCO₂ Dissociation Curves at Varied O₂, CO₂, pH, 2,3-DPG and Temperature Levels

Abstract

New mathematical model equations for O₂ and CO₂ saturations of hemoglobin ($S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$) are developed here from the equilibrium binding of O₂ and CO₂ with hemoglobin inside RBCs. They are in the form of an invertible Hill-type equation with the apparent Hill coefficients $K_{{{\text{HbO}}_{ 2} }}$ and $K_{{{\text{HbCO}}_{ 2} }}$ in the expressions for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ dependent on the levels of O₂ and CO₂ partial pressures ($P_{{{\text{O}}_{ 2} }}$ and $P_{{{\text{CO}}_{ 2} }}$), pH, 2,3-DPG concentration, and temperature in blood. The invertibility of these new equations allows $P_{{{\text{O}}_{ 2} }}$ and $P_{{{\text{CO}}_{ 2} }}$ to be computed efficiently from $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ and vice versa. The oxyhemoglobin (HbO₂) and carbamino-hemoglobin (HbCO₂) dissociation curves computed from these equations are in good agreement with the published experimental and theoretical curves in the literature. The model solutions describe that, at standard physiological conditions, the hemoglobin is about 97.2% saturated by O₂ and the amino group of hemoglobin is about 13.1% saturated by CO₂. The O₂ and CO₂ content in whole blood are also calculated here from the gas solubilities, hematocrits, and the new formulas for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$. Because of the mathematical simplicity and invertibility, these new formulas can be conveniently used in the modeling of simultaneous transport and exchange of O₂ and CO₂ in the alveoli–blood and blood–tissue exchange systems.

Introduction

As blood passes through capillaries, the affinity of hemoglobin (Hb) for O₂ and CO₂ changes along the length of the capillary. Each O₂ and CO₂ reduce the affinity of Hb for the other. Changes in pH and temperature have synergistic effects. In metabolizing tissue, the blood warms, becomes more acidic, and carries more CO₂ as it progresses along the capillary; the rising temperature, the diminishing pH, and the rising $P_{{{\text{CO}}_{ 2} }}$ all reduce the affinity of Hb for O₂ and foster O₂ release from Hb into the tissue. The loss of O₂ from Hb into the tissue fosters the uptake of CO₂ by Hb, though this effect is small compared to the buffering by bicarbonate. In the lungs, the reduction in temperature, the loss of CO₂, and the concordant rising of pH all foster increasing the affinity of Hb for O₂. Thus the local influences in lung versus tissue capillaries are ideally suited to maximize the delivery of O₂ from alveolar air to tissues and the removal of CO₂ from tissue to alveolar air. The other solute having a significant influence on the binding of O₂ to Hb is 2,3-diphosphoglycerate (2,3-DPG); raising [2,3-DPG] levels, as occurs with altitude and in diabetes, reduces the O₂ binding to Hb, shifting the oxyhemoglobin (HbO₂) dissociation curve to higher P₅₀s, just like higher CO₂, lower pH and higher temperature do.

Early Development of Oxyhemoglobin Dissociation Descriptions

Numerous mathematical models have been proposed in the literature to describe the standard and nonstandard HbO₂ “equilibrium” dissociation curves since the pioneering work of Hill13 and Adair.1 These are reviewed extensively by Roughton,29 Antonini and Brunori,2 Baumann et al.,5 and Popel.27 We shall henceforth, for brevity, call the HbO₂ “equilibrium” dissociation curves the HbO₂ dissociation curves (ODC). Hill13 originally postulated an nth-order one-step kinetic hypothesis for O₂ binding to Hb to derive the simplest model for standard ODC involving only two parameters. Hill’s equation for the ODC describes the oxygen saturation $S_{{{\text{O}}_{ 2} }}$ as a function of oxygen partial pressure $P_{{{\text{O}}_{ 2} }}$ relative to the half-saturation level P ₅₀:

$$ S_{{{\text{O}}_{2} }} = {\frac{{K_{{{\text{O}}_{2} }} P_{{{\text{O}}_{2} }}^{n} }}{{1 + K_{{{\text{O}}_{2} }} P_{{{\text{O}}_{2} }}^{n} }}} = {\frac{{\left( {{{P_{{{\text{O}}_{2} }} } \mathord{\left/ {\vphantom {{P_{{{\text{O}}_{2} }} } {P_{50} }}} \right. \kern-\nulldelimiterspace} {P_{50} }}} \right)^{n} }}{{1 + \left( {{{P_{{{\text{O}}_{2} }} } \mathord{\left/ {\vphantom {{P_{{{\text{O}}_{2} }} } {P_{50} }}} \right. \kern-\nulldelimiterspace} {P_{50} }}} \right)^{n} }}} $$

(1)

where $K_{{{\text{O}}_{ 2} }}$ is the Hill coefficient and n is the Hill exponent. They are related by $K_{{{\text{O}}_{ 2} }}$ = (P ₅₀)⁻ⁿ where P ₅₀ is the level of $P_{{{\text{O}}_{ 2} }}$ at which Hb is 50% saturated by O₂. The value n = 2.7 was found to fit well to the data for normal human blood in the saturation range of 20–98%29 for which the value of P ₅₀ is about 26.8 mmHg. This gives $K_{{{\text{O}}_{ 2} }}$ = 1.3933 × 10⁻⁴ mmHg⁻ⁿ. Hill’s equation is analytically invertible.

Subsequently, Adair1 postulated a more realistic four-step kinetic hypothesis (known as the intermediate compound hypothesis) and derived a more accurate formula for standard ODC involving four distinct parameters. Because of its better accuracy, Adair’s equation has been particularly useful in the analysis of experimental data at very low and very high $P_{{{\text{O}}_{ 2} }}$’s.28,30,40 Winslow et al.39 developed an algorithm for computing the nonstandard ODCs by analyzing fresh human whole blood data over a range of O₂ and CO₂ partial pressures, pH, and [DPG]/[Hb] concentration ratio using the Adair’s equation. O’Riordan et al.26 compared nine different models, including Hill’s equation and Adair’s equation, by fitting them to the data for normal human whole blood. Hill’s equation was found to give good characterization of the data over the saturation range of 20–98%, confirming the earlier finding of Roughton,29 which is the range of major physiological interest. However, Adair’s equation was found to be accurate at saturations approaching 100% and was good also down to a little less than 10% saturation (see also Baumann et al.,5 Roughton,28 Roughton et al.,29 Roughton and Severinghaus30).

The Need for a Model with Practical Accuracy and Convenience

The aim of this study is to provide an expression describing the relationship between hemoglobin saturation and $P_{{{\text{O}}_{ 2} }}$ over a wide range of not only $P_{{{\text{O}}_{ 2} }}$ but $P_{{{\text{CO}}_{ 2} }}$, pH, 2,3-DPG, and temperature, a total of five variables. It is furthermore important that this expression be invertible so that one can convert from observations on a blood sample to the relevant chemical driving forces and to the total contents of oxygen and carbon dioxide in the blood. Our efforts in searching for extensive data sets covering large ranges of these five variables met with failure: Winslow et al.’s40 data were by far the most extensive, but the original data tables have not been preserved, though of course they are summarized by the P ₅₀’s he reported. Consequently we have been reduced to fitting these “summaries” defined through other models rather than performing optimizations to parameterize our new, more broadly defined model against original experimental observations. This compromise is acceptable because of the many observations and analyses (referenced above and in the following paragraphs) on which it is based.

Development of Further Oxyhemoglobin Models

Margaria et al.25 and Margaria24 modified Adair’s equation by expressing the four Adair constants in terms of two distinct parameters, one representing the O₂ affinity for first three heme sites and the other representing an increased affinity for fourth oxygenation. Subsequently, Kelman18 proposed an empirical formula (also see Kelman19) for converting O₂ tension into its saturation; it is a little more complicated than Adair’s, using seven distinct parameters. For nonstandard physiological conditions, a virtual O₂ tension was computed as a function of pH, $P_{{{\text{CO}}_{ 2} }}$, and temperature from the experimental data and curve-fitting results of Severinghaus.32 Kelman’s formula gives negative values of $S_{{{\text{O}}_{2} }}$ for $P_{{{\text{O}}_{ 2} }}$ < 10 mmHg, the physically unrealistic values indicating failure of the algorithm in the region 0 mmHg < $P_{{{\text{O}}_{ 2} }}$ < 10 mmHg. Later, Kelman20 proposed a quadratic formula for $S_{{{\text{O}}_{ 2} }}$ for this range. It is worth pointing out here that the models proposed by Adair,1 Margaria,24 and Kelman18 are not analytically invertible. Therefore, to obtain $P_{{{\text{O}}_{ 2} }}$ from $S_{{{\text{O}}_{ 2} }}$, an iterative numerical method had to be employed.

Severinghaus33 developed a simple and accurate empirical formula for standard ODC by modifying Hill’s equation. For nonstandard physiological conditions, appropriate $P_{{{\text{O}}_{ 2} }}$ factors for pH, base excess and temperature were used, assuming as usual that these variations do not alter the shape of the curve. The significance of this model is that it fits the normal human blood data to within ±0.0055 ${\text{S}}_{{{\text{O}}_{ 2} }}$ in the range 0 < $S_{{{\text{O}}_{ 2} }}$ < 1. Later, Ellis9 and Severinghaus34 established that Severinghaus’s33 model is analytically invertible. The motivation behind Severinghaus’s33 new model is that Roughton was never satisfied with the Adair’s equation as he could not use the normal human blood data to generate the needed unique set of Adair’s constants and get a good O₂ saturation curve (e.g., see Roughton et al.28,30 and Roughton and Severinghaus30). Adair’s equation does not accommodate the Hb affinity change for O₂ which occurs when the second O₂ is bound to Hb and the shape of the Hb molecule changes. Severinghaus’s33 new cubic formula accounts for this affinity change and fits the data far better. Later, Siggaard-Andersen et al.37 developed a different empirical formula for standard and nonstandard ODCs which fits very well to the model and data of Severinghaus.33 However, Siggaard-Andersen et al.’s37 equation is not analytically invertible and requires an iterative numerical method for inversion (e.g., they used a Newton–Raphson method).

Easton8 proposed a new paradigm involving two parameters for characterizing the standard ODC. His mathematical description was based on the assumption that the formation of HbO₂, and hence O₂ saturation of Hb, is exponentially related to the O₂ partial pressure. Buerk6 modified Easton’s formula and fitted it to normal human and dog blood data30,33,40 using a linear regression algorithm; the modified Easton’s model was found to fit well to the data in the saturation range of 0 to 95%, providing the same accuracy as that of Adair’s model. Buerk and Bridges7 further modified Easton’s formula and developed an algorithm for computing the nonstandard ODCs with varying pH, CO₂ partial pressure, [DPG]/[Hb] concentration ratio, and temperature. The dissociation curves computed through this revised formula were found to agree well with those computed from the algorithms of Kelman18,19 and Winslow et al.39 The model proposed by Easton8 and subsequently modified by Buerk6 and Buerk and Bridges7 is analytically invertible, so one can efficiently calculate $S_{{{\text{O}}_{ 2} }}$ from $P_{{{\text{O}}_{ 2} }}$ and vice versa. However, since these models are good only up to 95% $S_{{{\text{O}}_{ 2} }}$, we have developed the present model which is good above this level.

Carboxyhemoglobin

There are few mathematical models available in the literature for computing the CO₂ saturation of hemoglobin and CO₂ content in whole blood. Kelman21 described an algorithm for computing the whole blood CO₂ content from the levels of pH, CO₂ tension, O₂ saturation and temperature in blood. Forster et al.10 and Forster11 have studied the rate of reaction of CO₂ with Hb to form HbCO₂ (carbamino-hemoglobin) at various physiological conditions. Hill et al.14–16 and Salathe et al.31 have computed the concentration of HbCO₂ during O₂ and CO₂ exchange through mathematical modeling by accounting for the physical and biochemical processes including the acid–base balance. Later, Singh et al.,38 extending their earlier work,36 developed mathematical formulas for nonstandard HbO₂ and HbCO₂ dissociation curves from the equilibrium binding of O₂ and CO₂ with Hb inside RBCs; these were similar to Hill’s equation but included the effects of $P_{{{\text{O}}_{ 2} }}$, $P_{{{\text{CO}}_{ 2} }}$ and pH. However, the effects of 2,3-DPG and temperature were not established. More recently, Huang and Hellums17 developed a computational model for convective-diffusive gas (O₂ and CO₂) transport in the microcirculation and in oxygenators by accounting for the acid–base regulation and the Bohr and Haldane effects (increasing $P_{{{\text{CO}}_{ 2} }}$ or pH reduces O₂ affinity of Hb and increasing $P_{{{\text{O}}_{ 2} }}$ reduces CO₂ affinity of Hb).

The Consequence of This Study

This study fulfills an important requirement in the physiological studies of simultaneous O₂ and CO₂ transport and exchange by including the influences of Hb-mediated nonlinear O₂–CO₂ interactions14–17,31 and the related changes in pH that occur with passage through capillaries in tissues and in the lung. Accounting for both Bohr and Haldane effects is crucial in the modeling of simultaneous transport and exchange of O₂ and CO₂ in the circulatory system. For example, an important clinical application using the results of the present work is in using ¹⁵O-oxygen positron emission tomography (PET) and in the analysis of signals from BOLD (blood oxygen level dependent) MRI (magnetic resonance imaging). Our governing equations account for O₂ saturation of Hb as well as CO₂ saturation of Hb, which are coupled or linked to each other through the kinetics of O₂ and CO₂ binding to Hb. With this motivation, by considering a detailed mathematical analysis of the equilibrium binding of O₂ and CO₂ with Hb inside RBCs, including the nonlinear O₂–CO₂ interactions and the effects of pH, 2,3-DPG and temperature, we end up with relatively simple model equations for nonstandard HbO₂ and HbCO₂ dissociation curves, as well as the O₂ and CO₂ contents in whole blood.

The equations for O₂ and CO₂ saturations of Hb ($S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$) are of the form of a Hill-type equation which is invertible. The apparent Hill coefficients $K_{{{\text{HbO}}_{ 2} }}$ and $K_{{{\text{HbCO}}_{ 2} }}$ in the expression for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ are explicitly dependent on the levels of O₂ and CO₂ partial pressures, pH, 2,3-DPG concentration, and temperature in blood. The results show that, at normal physiological conditions, Hb is about 97.2% saturated by O₂ and the amino group of Hb is about 13.1% saturated by CO₂. The invertibility of our model equations for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ allows their convenient usage in computationally complex models of simultaneous transport and exchange of O₂ and CO₂ in the pulmonary and systemic circulations. The mathematical modeling language (MML) code for our model, which is implemented in our Java Simulation (JSim) interface, is available for download and public use at http://physiome.org/Models/GasTransport/.

Mathematical Formulation

Governing Biochemical Reactions

The dynamics of hemoglobin-facilitated transport of O₂ and CO₂ in blood and their nonlinear interactions are governed by the following biochemical reactions inside RBCs.2,14,29,31,38 Hemoglobin, Hb, consists of four heme-amino chains, two α and two β chains; each contains a heme group, Hm, which binds to an O₂ molecule and has a terminal amino group, –NH₂, which can bind to a CO₂ molecule to form an ionizable carbamino terminus, –NHCOOH. We consider the α and β chains to be identical in their binding with CO₂. The four heme sites for O₂ binding show cooperativity so that an HbO₂ saturation curve has a Hill exponent of about 2.7. Therefore, we consider Hb as of 4 Hm (i.e., Hb = Hm₄). The governing reactions are:

• CO₂ hydration reaction—${\text{HCO}}_{3} ^{ - },$ buffering of CO₂:

• $$ {\text{CO}}_{2} + {\text{H}}_{2} {\text{O}}\underset{{kb_{1}^{\prime} }}{\overset{{kf_{1}^{\prime} }}{\leftrightarrows }}{\text{H}}_{2} {\text{CO}}_{3} \overset {K_{1}^{{\prime\prime}} } \leftrightarrows {\text{HCO}}_{3} ^{ - } + {\text{H}}^{ + } $$

  (2a)

• CO₂ binding to HmNH₂ chains—HmNHCOO⁻ buffering of CO₂:

• $$ {\text{CO}}_{2} + {\text{HmNH}}_{2} \underset{{kb_{2}^{\prime} }}{\overset{{kf_{2}^{\prime} }}{\leftrightarrows}}{\text{HmNHCOOH}}\overset {K_{2}^{\prime \prime} } \leftrightarrows {\text{HmNHCOO}}^{ - } + {\text{H}}^{ + } . $$

  (2b)

• CO₂ binding to O₂HmNH₂ chains—O₂HmNHCOO⁻ buffering of CO₂:

• $$ {\text{CO}}_{2} + {\text{O}}_{2} {\text{HmNH}}_{2} \underset{{kb_{3}^{\prime} }}{\overset{{kf_{3}^{\prime} }}{\leftrightarrows}}{\text{O}}_{2} {\text{HmNHCOOH}}\overset {K_{3}^{\prime \prime} } \leftrightarrows {\text{O}}_{2} {\text{HmNHCOO}}^{ - } + {\text{H}}^{ + } . $$

  (2c)

• O₂ binding to HmNH₂ chains—one-step kinetics using the $P_{{{\text{O}}_{ 2} }}$-dependent values of the rates of association and dissociation to accounts for the cooperativity.

• $$ {\text{O}}_{2} + {\text{HmNH}}_{2} \underset{{kb_{4}^{\prime} }}{\overset{{kf_{4}^{\prime} }}{\leftrightarrows}}{\text{O}}_{2} {\text{HmNH}}_{2} . $$

  (2d)

• Ionization of HmNH₂ chains—pH buffering:

• $$ {\text{HmNH}}_{3}^{ + } \overset {K_{5}^{\prime \prime} } \leftrightarrows {\text{HmNH}}_{2} + {\text{H}}^{ + } . $$

  (2e)

• Ionization of O₂HmNH₂ chains—pH buffering:

• $$ {\text{O}}_{2} {\text{HmNH}}_{3}^{ + } \overset {K_{6}^{\prime \prime} } \leftrightarrows {\text{O}}_{2} {\text{HmNH}}_{2} + {\text{H}}^{ + } . $$

  (2f)

HmNH₂ and O₂HmNH₂ refer to the reduced and oxygenated heme sites attached to the amino chain, ${\text{HmNH}}_{3}^{ + }\;{\text{and}}\;{\text{O}}_{2} {\text{HmNH}}_{3}^{+}$ denote their ionized forms; HmNHCOOH and O₂HmNHCOOH refer to the reduced and oxygenated carbamino chain, HmNHCOO⁻ and O₂HmNHCOO⁻ denote their ionized forms; $kf^{\prime}_{i}$ and $kb^{\prime}_{i}$ are the rate constants of forward and backward direction reactions; $K_{1}^{\prime\prime},\;K_{2}^{\prime\prime},\;K_{3}^{\prime\prime},\;K_{5}^{\prime\prime}\;\text{and}\;K_{6}^{\prime\prime}$ are the ionization constants of H₂CO₃, HmNHCOOH, O₂HmNHCOOH, ${\text{HmNH}}_{3} ^{ + }\;{\text{and}}\;{\text{O}}_{2} {\text{HmNH}}_{3} ^{ + }$. The units of $kf_{1}^{\prime},\;kf_{2}^{\prime},\;kf_{3}^{\prime},\;\text{and}\;kf_{4}^{\prime}$ are M⁻¹ s⁻¹; $kb_{1}^{\prime},\;kb_{2}^{\prime},\;kb_{3}^{\prime},\;\text{and}\;kb_{4}^{\prime}$ are in s⁻¹; $K_{1}^{\prime\prime},\;K_{2}^{\prime\prime},\;K_{3}^{\prime\prime},\;K_{5}^{\prime\prime}\;\text{and}\;K_{6}^{\prime\prime}$ are in M.

In plasma, the CO₂ hydration reaction (2a) is slow, but it is fast within RBCs because there is carbonic anhydrase in the cytosol and on the membrane. The uptake of O₂ and CO₂ by Hb inside RBCs is governed by reactions (2b), (2c) and (2d). Reactions (2e) and (2f) act as a buffer system and control the concentration of H⁺ in RBCs. The interaction between O₂ and CO₂ is mediated by the proton H⁺ within RBCs via reactions (2a) to (2f).

The binding of CO₂ with the four amino groups of a hemoglobin molecule is noncooperative in nature. So the kinetics of CO₂ uptake by Hb can be represented by reactions (2b) and (2c) with the rate and equilibrium constants fixed. However, the binding of O₂ with Hb takes place in four intermediate steps and is cooperative in nature due to the interactions between the binding heme sites.2 To account for this through our one-step kinetic approach in reaction (2d), we use the equilibrium “constant” $K_{4}^{\prime}(=kf_{4}^{\prime}/kb_{4}^{\prime})$ a function of O₂ partial pressure $P_{{{\text{O}}_{2} }}$; $K_{4}^{\prime}$ also depends on the levels of pH, $P_{{{\text{CO}}_{2} }}$, 2,3-DPG concentration, and temperature inside the RBCs.4

Equilibrium Relations

The reactions (2a)–(2f) are not instantaneous and can be described through a set of ordinary differential equations. However, these reactions approach equilibrium within about 20 ms, so the equilibrium descriptions through algebraic equations are very close to the truth. Therefore, from here on, we use the ratios of the on-to-off rate constants, that is, the equilibrium constants, instead of the rate constants. At equilibrium, we obtain the following set of algebraic relations from the set of reactions (2a)–(2f), where we denote [·] as the concentration of a species in the water space of RBCs:

$$ K_{1}^{\prime} \left[ {{\text{CO}}_{2} } \right] = \left[ {{\text{H}}_{2} {\text{CO}}_{3} } \right], $$

(3a)

$$ K_{1}^{\prime \prime} \left[ {{\text{H}}_{2} {\text{CO}}_{3} } \right] = \left[ {{\text{HCO}}_{3}^{ - } } \right]\left[ {{\text{H}}^{ + } } \right], $$

(3b)

$$ K_{2}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left[ {{\text{HmNH}}_{2} } \right] = \left[ {\text{HmNHCOOH}} \right], $$

(3c)

$$ K_{2}^{\prime \prime} \left[ {\text{HmNHCOOH}} \right] = \left[ {{\text{HmNHCOO}}^{ - } } \right]\left[ {{\text{H}}^{ + } } \right], $$

(3d)

$$ K_{3}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left[ {{\text{O}}_{2} {\text{HmNH}}_{2} } \right] = \left[ {{\text{O}}_{2} {\text{HmNHCOOH}}} \right], $$

(3e)

$$ K_{3}^{\prime \prime} \left[ {{\text{O}}_{2} {\text{HmNHCOOH}}} \right] = \left[ {{\text{O}}_{2} {\text{HmNHCOO}}^{ - } } \right]\left[ {{\text{H}}^{ + } } \right], $$

(3f)

$$ K_{4}^{\prime} \left[ {{\text{O}}_{2} } \right]\left[ {{\text{HmNH}}_{2} } \right] = \left[ {{\text{O}}_{2} {\text{HmNH}}_{2} } \right], $$

(3g)

$$ K_{5}^{\prime \prime} \left[ {{\text{HmNH}}_{3}^{ + } } \right] = \left[ {{\text{HmNH}}_{2} } \right]\left[ {{\text{H}}^{ + } } \right], $$

(3h)

$$ K_{6}^{\prime \prime} \left[ {{\text{O}}_{2} {\text{HmNH}}_{3}^{ + } } \right] = \left[ {{\text{O}}_{2} {\text{HmNH}}_{2} } \right]\left[ {{\text{H}}^{ + } } \right], $$

(3i)

where the equilibrium constants $K_{1}^{\prime},\;K_{2}^{\prime},\;K_{3}^{\prime},\;\text{and}\;K_{4}^{\prime}$ are defined by

$$ K_{1}^{\prime} = {\frac{{kf_{1}^{\prime} }}{{kb_{1}^{\prime} }}}\left[ {{\text{H}}_{2} {\text{O}}} \right],\quad K_{2}^{\prime} = {\frac{{kf_{2}^{\prime} }}{{kb_{2}^{\prime} }}},\quad K_{3}^{\prime} = {\frac{{kf_{3}^{\prime} }}{{kb_{3}^{\prime} }}},\quad K_{4}^{\prime} = {\frac{{kf_{4}^{\prime} }}{{kb_{4}^{\prime} }}}. $$

(4)

Pure water (H₂O) has a molecular weight of 18 g/mole so that its concentration in plasma, which is 94% water, is about 55.56 × 0.94 = 52.23 M which is very high as compared to the total solute concentration (280 mM) in plasma. This leads to $K_{1}^{\prime}$ being practically a constant. The units of $K_{2}^{\prime},\;K_{3}^{\prime}\;\text{and}\;K_{4}^{\prime}$ are M⁻¹; $K_{1}^{\prime}$ is unitless; $K_{4}^{\prime}$ is dependent on [O₂], [CO₂], [H⁺], [2,3-DPG] and T.

The concentrations of total hemoglobin, total O₂-bound hemoglobin, and total CO₂-bound hemoglobin in RBCs are given by

$$ \begin{aligned} \left[ {\text{Hb}} \right] = & 4\left( {\left[ {{\text{HmNH}}_{2} } \right] + \left[ {{\text{HmNH}}_{3}^{ + } } \right] + \left[ {\text{HmNHCOOH}} \right] + \left[ {{\text{HmNHCOO}}^{ - } } \right]} \right. \\ & \quad + \left. {\left[ {{\text{O}}_{2} {\text{HmNH}}_{2} } \right] + \left[ {{\text{O}}_{2} {\text{HmNH}}_{3}^{ + } } \right] + \left[ {{\text{O}}_{2} {\text{HmNHCOOH}}} \right] +\,\left[ {{\text{O}}_{2} {\text{HmNHCOO}}^{ - } } \right]} \right), \\ \end{aligned} $$

(5a)

$$ \begin{aligned} \left[ {{\text{HbO}}_{2} } \right] = & 4\left( {\left[ {{\text{O}}_{2} {\text{HmNH}}_{2} } \right] + \left[ {{\text{O}}_{2} {\text{HmNH}}_{3}^{ + } } \right] + \left[ {{\text{O}}_{2} {\text{HmNHCOOH}}} \right]} \right. \\ & \quad \left. { + \left[ {{\text{O}}_{2} {\text{HmNHCOO}}^{ - } } \right]} \right), \\ \end{aligned} $$

(5b)

$$ \begin{aligned} \left[ {{\text{HbCO}}_{2} } \right] = & 4\left( {\left[ {\text{HmNHCOOH}} \right] + \left[ {{\text{HmNHCOO}}^{ - } } \right] + \left[ {{\text{O}}_{2} {\text{HmNHCOOH}}} \right]} \right. \\ & \quad \left. { + \left[ {{\text{O}}_{2} {\text{HmNHCOO}}^{ - } } \right]} \right). \\ \end{aligned} $$

(5c)

The values of –NHCOOH dissociation constants $K_{2}^{\prime\prime}\;{\text{and}}\;K_{3}^{\prime\prime}$ are usually higher than 10⁻⁶ M10,11 while the concentration of H⁺ in RBCs is about 5.75 × 10⁻⁸ M (i.e., pH in RBCs is about 7.24 when pH in plasma is about 7.4). So the concentrations of HmNHCOOH and O₂HmNHCOOH are usually two orders of magnitude smaller than those of HmNHCOO⁻ and O₂HmNHCOO⁻.14 Nevertheless, extending the work of Singh et al.,38 we include the contributions of HmNHCOOH and O₂HmNHCOOH for conceptual completeness and improved accuracy.

Using Eqs. (3c)–(3i) in Eqs. (5a), (5b) and (5c), we obtain the following explicit expressions for the concentrations of total hemoglobin, total O₂-bound hemoglobin, and total CO₂-bound hemoglobin in RBCs:

$$ \begin{aligned} \left[ {\text{Hb}} \right] = & 4\left[ {{\text{HmNH}}_{2} } \right]\left[ {\left( {K_{2}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left\{ {1 + {\frac{{K_{2}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\} + \left\{ {1 + {\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{K_{5}^{\prime \prime} }}}} \right\}} \right)} \right. \\ & \quad \left. { +\,K_{4}^{\prime} \left[ {{\text{O}}_{2} } \right]\left( {K_{3}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left\{ {1 + {\frac{{K_{3}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\} + \left\{ {1 + {\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{K_{6}^{\prime \prime} }}}} \right\}} \right)} \right], \\ \end{aligned} $$

(6a)

$$ \left[ {{\text{HbO}}_{2} } \right] = 4\left[ {{\text{HmNH}}_{2} } \right]\left[ {K_{4}^{\prime} \left[ {{\text{O}}_{2} } \right]\left( {K_{3}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left\{ {1 + {\frac{{K_{3}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\} + \left\{ {1 + {\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{K_{6}^{\prime \prime} }}}} \right\}} \right)} \right], $$

(6b)

$$ \left[ {{\text{HbCO}}_{2} } \right] = 4\left[ {{\text{HmNH}}_{2} } \right]\left[ {\left( {K_{2}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left\{ {1 + {\frac{{K_{2}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\}} \right) + K_{4}^{\prime} \left[ {{\text{O}}_{2} } \right]\left( {K_{3}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left\{ {1 + {\frac{{K_{3}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\}} \right)} \right]. $$

(6c)

Expressions for ${\text{S}}_{{{\text{HbO}}_{ 2} }}$ and ${\text{S}}_{{{\text{HbCO}}_{ 2} }}$

The fractional O₂ and CO₂ saturations of Hb ($S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$) is obtained from Eqs. (6a), (6b) and (6c) as the ratios [HbO₂]/[Hb] and [HbCO₂]/[Hb]. Putting these in the form of the Hill13 equation gives the advantage of their being analytically invertible, allowing the O₂ and CO₂ concentrations ([O₂] and [CO₂]), or their partial pressures ($P_{{{\text{O}}_{ 2} }}$ and $P_{{{\text{CO}}_{ 2} }}$), to be calculated efficiently from their fractional saturations ($S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$) and vice versa (see Appendix B). These expressions for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ are formulated as for single-site binding and a Hill exponent of unity and therefore using concentration-dependent Hill coefficients:

$$ S_{{{\text{HbO}}_{2} }} = {\frac{{\left[ {{\text{HbO}}_{2} } \right]}}{{\left[ {\text{Hb}} \right]}}} = {\frac{{K_{{{\text{HbO}}_{2} }} [{\text{O}}_{2} ]}}{{1 + K_{{{\text{HbO}}_{2} }} [{\text{O}}_{2} ]}}}, $$

(7a)

$$ S_{{{\text{HbCO}}_{2} }} = {\frac{{\left[ {{\text{HbCO}}_{2} } \right]}}{{\left[ {\text{Hb}} \right]}}} = {\frac{{K_{{{\text{HbCO}}_{2} }} [{\text{CO}}_{2} ]}}{{1 + K_{{{\text{HbCO}}_{2} }} [{\text{CO}}_{2} ]}}}, $$

(7b)

where the apparent Hill coefficients $K_{{{\text{HbO}}_{2} }}$ and $K_{{{\text{HbCO}}_{2} }}$ (with units M⁻¹) account for the influences of $P_{{{\text{O}}_{ 2} }}$, $P_{{{\text{CO}}_{ 2} }}$, pH, [2,3-DPG] and T as well as the nonlinear O₂–CO₂ interactions on the binding of O₂ and CO₂ with Hb inside RBCs. These concentration-dependent Hill coefficients are quite different from the constant-valued Hill coefficient of Eq. (1). The expressions for $K_{{{\text{HbO}}_{ 2} }}$ and $K_{{{\text{HbCO}}_{ 2} }}$ are given by

$$ K_{{{\text{HbO}}_{2} }} = {\frac{{K_{4}^{\prime} \left( {K_{3}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left\{ {1 + {\frac{{K_{3}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\} + \left\{ {1 + {\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{K_{6}^{\prime \prime} }}}} \right\}} \right)}}{{\left( {K_{2}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left\{ {1 + {\frac{{K_{2}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\} + \left\{ {1 + {\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{K_{5}^{\prime \prime} }}}} \right\}} \right)}}}, $$

(8a)

$$ K_{{{\text{HbCO}}_{2} }} = {\frac{{\left( {K_{2}^{\prime} \left\{ {1 + {\frac{{K_{2}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\}K_{3}^{\prime} K_{4}^{\prime} + \left\{ {1 + {\frac{{K_{3}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\}\left[ {{\text{O}}_{2} } \right]} \right)}}{{\left( {\left\{ {1 + {\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{K_{5}^{\prime \prime} }}}} \right\} + K_{4}^{\prime} \left\{ {1 + {\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{K_{6}^{\prime \prime} }}}} \right\}\left[ {{\text{O}}_{2} } \right]} \right)}}}. $$

(8b)

$K_{{{\text{HbO}}_{ 2} }}$ and $K_{{{\text{HbCO}}_{ 2} }}$ depend on [2,3-DPG] and T through their dependency on $K_{4}^{\prime}$ (see below). The P₅₀ for 50% HbO₂ saturation is not a constant, but also a function of $P_{{{\text{CO}}_{ 2} }}$, pH, [2,3-DPG] and T. Note that the form of Eqs. (7a) and (7b) is first order and that the Hill exponent inferred by analogy to Eq. (1) is unity. This means that the fundamental S-shape of the HbO₂ dissociation curve is built into the dependency of $K_{{{\text{HbO}}_{ 2} }}$ on [O₂]. The linkage between $K_{{{\text{HbO}}_{ 2} }}$ and $K_{{{\text{HbCO}}_{ 2} }}$ can be deduced from the fact that the equilibrium constants $K_{2}^{\prime},\;K_{2}^{\prime\prime}\;\text{and}\;K_{5}^{\prime\prime}$ for deoxygenated Hb are higher than $K_{3}^{\prime},\;K_{3}^{\prime\prime}\;\text{and}\;K_{6}^{\prime\prime}$ for oxygenated Hb.

Expression for $K_{4}^{\prime}$

The forward rate constant $Kf_{4}^{\prime}$ for the association of O₂ with HmNH₂ and the backward rate constant $Kb_{4}^{\prime}$ for the dissociation of O₂ from O₂HmNH₂ depend on the levels of $P_{{{\text{O}}_{ 2} }}$, $P_{{{\text{CO}}_{ 2} }}$, pH, [2,3-DPG] and T in RBCs.4 We characterize the dependency of the equilibrium constant $K_{4}^{\prime}\,=\,kf_{4}^{\prime}/kb_{4}^{\prime}$ on [O₂], [CO₂], [H⁺], [2,3-DPG] and T through the following power-law proportionality equation:

$$ \begin{aligned} K_{4}^{\prime} = & K_{4}^{\prime \prime} \left\{ {{\frac{{\left[ {{\text{O}}_{2} } \right]}}{{\left[ {{\text{O}}_{2} } \right]_{\text{S}} }}}} \right\}^{{n_{0} }} \left\{ {{\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{\left[ {{\text{H}}^{ + } } \right]{\text{S}}}}}} \right\}^{{ - n_{1} }} \left\{ {{\frac{{\left[ {{\text{CO}}_{2} } \right]}}{{\left[ {{\text{CO}}_{2} } \right]_{\text{S}} }}}} \right\}^{{ - n_{2} }} \left\{ {{\frac{{\left[ {\text{DPG}} \right]}}{{\left[ {\text{DPG}} \right]_{\text{S}} }}}} \right\}^{{ - n_{3} }} \left\{ {{\frac{T}{{T_{\text{S}} }}}} \right\}^{ - n4} \\ & = K_{4}^{\prime \prime} \left\{ {{\frac{{\left[ {{\text{O}}_{2} } \right]}}{{146\;\mu {\text{M}}}}}} \right\}^{{n_{0} }} \left\{ {{\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{57.5\;{\text{nM}}}}}} \right\}^{{ - n_{1} }} \left\{ {{\frac{{\left[ {{\text{CO}}_{2} } \right]}}{{1.31\;{\text{mM}}}}}} \right\}^{{ - n_{2} }} \left\{ {{\frac{{\left[ {\text{DPG}} \right]}}{{4.65\;{\text{mM}}}}}} \right\}^{{ - n_{3} }} \left\{ {{\frac{T}{{310^{ \circ } \;{\text{K}}}}}} \right\}^{ - n4} , \\ \end{aligned} $$

(9)

where the proportionality equilibrium constant $K_{4}^{\prime\prime}$ and the empirical exponents n _i , i = 0, 1, …, 4, are to be determined. The subscript “S” refers to the values under standard physiological conditions in the arterial system: [O₂]_S = 146 μM (or $P_{{{\text{O}}_{2,{\rm S}}}}$ = 100 mmHg), [CO₂]_S = 1.31 mM (or $P_{{{\text{CO}}_{2,{\rm S}} }}$ = 40 mmHg), [H⁺]_S = 57.5 nM (or pH_S = 7.24), [2,3-DPG]_S = 4.65 mM, and T _S = 310 K = 37 °C in RBCs (see Table 1). At these conditions $K_{4}^{\prime}\,=\,K_{4}^{\prime\prime};{\text{so}}\;K_{4}^{\prime\prime}$ is also in the units of M⁻¹. The standard hematocrit, Hct, is about 0.45 and the Hb concentration in blood, [Hb]_bl, is about 0.15 g/mL or 2.33 mM taking the molecular weight of Hb to be 64458.19 Thus, [Hb]_rbc is about 5.18 mM. Buerk and Bridges7 have used the molar concentration ratio [2,3-DPG]/[Hb] as 0.9 which corresponds to [2,3-DPG] = 4.65 mM.

TABLE 1 The representative values of the parameters used in the model

Now we see that Eqs. (8a) and (8b) along with Eq. (9) completely determine the apparent Hill coefficients $K_{{{\text{HbO}}_{ 2} }}$ and $K_{{{\text{HbCO}}_{ 2} }}$. Singh et al.38 did not account for the effects of [CO₂], [2,3-DPG] and T on the equilibrium constant $K_{4}^{\prime}$ in their modeling. So their case corresponds to Eq. (9) with n ₂ = n ₃ = n ₄ = 0 and nonzero n ₀ and n ₁.

Expressions for $\alpha_{{{\text{O}}_{ 2} }}$ and $\alpha_{{{\text{CO}}_{ 2} }}$

The equations for fractional saturations, $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$, and apparent Hill coefficients, $K_{{{\text{HbO}}_{ 2} }}$ and $K_{{{\text{HbCO}}_{ 2} }}$, can be expressed in terms of O₂ and CO₂ partial pressures, $P_{{{\text{O}}_{2} }}$ and $P_{{{\text{CO}}_{2} }}$, by expressing the molar concentrations [O₂] and [CO₂] in water space of RBCs in terms of $P_{{{\text{O}}_{2} }}$ and $P_{{{\text{CO}}_{2} }}$ using Henry’s law: [O₂] = $\alpha_{{{\text{O}}_{ 2} }}$ $P_{{{\text{O}}_{2} }}$ and [CO₂] = $\alpha_{{{\text{CO}}_{ 2} }}$ $P_{{{\text{CO}}_{2} }}$, where $\alpha_{{{\text{O}}_{ 2} }}$ and $\alpha_{{{\text{CO}}_{ 2} }}$ are the solubilities of O₂ and CO₂ in water. At body temperature (T = 37 °C), $\alpha_{{{\text{O}}_{ 2} }}$ = 1.46 × 10⁻⁶ M mmHg⁻¹ and $\alpha_{{{\text{CO}}_{ 2} }}$ = 3.27 × 10⁻⁵ M mmHg⁻¹. The variation of $\alpha_{{{\text{O}}_{ 2} }}$ and $\alpha_{{{\text{CO}}_{ 2} }}$ with temperature (T) can be expressed through the following quadratic curve-fit equations19,21 based on the experimental data of Hedley-Whyte and Laver12 on $\alpha_{{{\text{O}}_{ 2} }}$ and Austin et al.3 on $\alpha_{{{\text{CO}}_{ 2} }}$, correcting for the plasma fractional water content, W _pl:

$$ \alpha_{{{\text{O}}_{2} }} = \left[ {1.37 - 0.0137\left( {T - 37} \right) + 0.00058\left( {T - 37} \right)^{2} } \right] \times \left[ {10^{ - 6} /W_{\text{pl}} } \right]\quad {\text{M}}/{\text{mmHg}}, $$

(10a)

$$ \alpha_{{{\text{CO}}_{2} }} = \left[ {3.07 - 0.057\left( {T - 37} \right) + 0.002\left( {T - 37} \right)^{2} } \right] \times \left[ {10^{ - 5} /W_{\text{pl}} } \right]\quad {\text{M}}/{\text{mmHg}}, $$

(10b)

where W _pl = 0.94. The first term for $\alpha_{{{\text{O}}_{ 2} }}$ is (1.37/0.94) × 10⁻⁶ or 1.46 × 10⁻⁶ M mmHg⁻¹ and for $\alpha_{{{\text{CO}}_{ 2} }}$ is (3.07/0.94) × 10⁻⁵ or 3.27 × 10⁻⁵ M mmHg⁻¹, as mentioned above. The solubility of O₂ and CO₂ decreases as temperature increases.

The HbO₂ and HbCO₂ dissociation curves described by Eqs. (7a)–(7b) and (8a)–(8b) require knowing the equilibrium constants $K_{2}^{\prime},\;K_{3}^{\prime},\;K_{2}^{\prime\prime}\;{\text{to}}\;K_{6}^{\prime\prime}$ and the empirical exponents n ₀, n ₁, n ₂, n ₃ and n ₄. From the saturations, $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$, the total O₂ and CO₂ contents in whole blood can be calculated as described in Appendix A.

Results: Parameter Estimation

The O₂ and CO₂ saturations of Hb ($S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$) and their whole blood contents ([O₂]_bl and [CO₂]_bl) depend on the physiological state variables $P_{{{\text{O}}_{ 2} }}$, $P_{{{\text{CO}}_{ 2} }}$, pH, Hct, [2,3-DPG]_rbc and T, empirical exponents n ₀, n ₁, n ₂, n ₃ and n ₄, and equilibrium constants $K_{1}^{\prime}\;{\text{to}}\;K_{3}^{\prime},\;K_{1}^{\prime\prime}\;{\text{to}}\;K_{6}^{\prime\prime}$ and R _rbc. The literature does not provide consistent values for the equilibrium constants. Singh et al.38 estimated some of them in their theoretical studies of HbO₂ and HbCO₂ dissociation curves, but their estimates do not match with the experimental values obtained by Roughton,29 Forster et al.10 and Forster11 which are the values generally accepted.2,14–16,31 Singh et al.38 assumed that the Hb molecule has only one amino chain, –NH₂, and is capable of binding to only one CO₂ molecule, but Hb has four –NH₂ chains and can bind to four CO₂ molecules. Thus, Singh et al.’s estimates are not acceptable for the kinetic reactions (2a)–(2f). In the present study, we choose or calculate the equilibrium constant $K_{1},\;K_{1}^{\prime},\;K_{1}^{\prime\prime},\;K_{2},\;K_{2}^{\prime},\;K_{2}^{\prime\prime},\;K_{3},\;K_{3}^{\prime},\;K_{3}^{\prime\prime},\;K_{5}^{\prime\prime},\;K_{6}^{\prime\prime},$ and R _rbc appropriately from the literature (see Table 1 for references) and then estimate the proportionality equilibrium constant $K_{4}^ {\prime \prime}$ and the empirical exponents n ₀, n ₁, n ₂, n ₃ and n ₄ so as to obtain the appropriate forms and shifts in the HbO₂ dissociation curves with respect to the levels of pH_rbc, $P_{{{\text{CO}}_{ 2} }}$, [DPG]_rbc and T in accord with experimental observations and their summaries captured in the models of Kelman18 and Buerk and Bridges.7 The generality and comprehensiveness of the calculations might seem disadvantaged by their apparent complexity, but in fact the code for the equations is quite simple algebra and is viewable (and downloadable) within three clicks of http://physiome.org/Models/GasTransport.

Calculation of Equilibrium Constants

We choose the values $K_{1}\,=\,K_{1}^{\prime}K_{1}^{\prime\prime}\,=\,7.43\,\times\,10^{-7}\,{\text{M}},\;K_{1}^{\prime\prime}\,=\,5.5\,\times\,10^{-4}\,{\text{M}};K_{2}\,=\,K_{2}^{\prime}K_{2}^{\prime\prime}\,=\,2.95\,\times\,10^{-5},\;K_{2}^{\prime\prime}\,=\,1\,\times\,10^{-6}\,{\text{M}},\;K_{3}\,=\,K_{3}^{\prime}K_{3}^{\prime\prime}\;=\;2.51\,\times\,10^{-5},\;K_{3}^{\prime\prime}\,=\,1\,\times\,10^{-6}\,{\text{M}},\;K_{5}^{\prime\prime}=\,2.63\,\times\,10^{-8}\,{\text{M}},\;{\text{and}}\;K_{6}^{\prime\prime}\,=\,1.91\,\times\,10^{-8}\,{\text{M}}$ (see Table 1 for references); K ₁ is the equilibrium constant of the overall CO₂ hydration reaction (2a); K ₂ and K ₃ are the equilibrium constants of the overall reaction of CO₂ with the reduced and oxygenated Hb, reactions (2b) and (2c). These give $K_{1}^{\prime}$ = 1.35 × 10⁻³, $K_{2}^{\prime}$ = 29.5 M⁻¹ and $K_{3}^{\prime}$ = 25.1 M⁻¹, as shown in Table 1. Thus, the equilibrium constants $K_{2},\;K_{2}^{\prime}$ and $K_{5}^{\prime\prime}$ for reduced Hb are higher than $K_{3},\;K_{3}^{\prime}$ and $K_{6}^{\prime \prime}$ for oxygenated Hb. This indicates that the reduced Hb has a greater ability to bind to CO₂ than does the oxygenated Hb (e.g., see Figs. 24 and 26 of Roughton,29 and Fig. 10.11 of Antonini and Brunori.2 The estimation of the proportionality equilibrium constant $K_{4}^{\prime\prime}$ and the empirical exponents n ₀, n ₁, n ₂, n ₃ and n ₄ are described below.

Calculation of Oxygen P ₅₀

The P ₅₀ for O₂ (the level of $P_{{{\text{O}}_{ 2} }}$ at which Hb is 50% saturated by O₂) is conventionally used as a measure of O₂ affinity of Hb and shift in the HbO₂ dissociation curve (ODC). We first calculate the P ₅₀ values for nonstandard ODCs as a function of pH_rbc, $P_{{{\text{CO}}_{ 2} }}$, [DPG]_rbc and T from the model of Buerk and Bridges7 which agree well with those obtained experimentally by Winslow et al.39 We then employ the method of regression analysis to determine the curve-fit polynomials for these computed P ₅₀ values recursively with one of the variables varying and the other three fixed at their standard physiological values. We found that the P ₅₀(pH_rbc), P ₅₀($P_{{{\text{CO}}_{ 2} }}$) and P ₅₀([DPG]_rbc) are best-fitted by quadratic polynomials, whereas P ₅₀(T) is best-fitted by a cubic polynomial. This is demonstrated through Fig. 1. The combined best-fit polynomial for P ₅₀(pH_rbc,$P_{{{\text{CO}}_{ 2} }}$,[DPG]_rbc,T) is given by

$$ \begin{aligned} P_{50} & = 26.8 - 21.279\left( {{\text{pH}}_{\text{rbc}} - 7.24} \right) + 8.872\left( {{\text{pH}}_{\text{rbc}} - 7.24} \right)^{2} \\ & \quad + 0.0482\left( {P_{{{\text{CO}}_{2} }} - 40} \right) + 3.64{\text{E}} - 5\left( {P_{{{\text{CO}}_{2} }} - 40} \right)^{2} \\ & \quad + 795.63\left( {\left[ {\text{DPG}} \right]_{\text{rbc}} -\,0.00465} \right) - 19660.89\left( {\left[ {\text{DPG}} \right]_{\text{rbc}} -\,0.00465} \right)^{2} \\ & \quad + 1.4945\left( {T - 37} \right)\,+\,0.04335\left( {T - 37} \right)^{2} + 0.0007\left( {T - 37} \right)^{3} . \\ \end{aligned} $$

(11)

FIGURE 1

The plot of oxygen P ₅₀ (pH_rbc, $P_{{{\text{CO}}_{2} }}$, [DPG]_rbc, T) with one of the variables varying and the other three fixed at their standard physiological values. The P ₅₀’s computed from Buerk and Bridges’7 model are shown by solid points, from our best-fit polynomial (11) are shown by solid lines, and from Kelman’s empirical equation (12) are shown by dashed lines

The empirical equation for P ₅₀(pH_rbc, $P_{{{\text{CO}}_{ 2} }}$, T) from Kelman’s18 model is given by

$$ P_{50} = 26.8 \times 10^{ \wedge } \left[ {0.4\left( {7.24 - {\text{pH}}_{\text{rbc}} } \right) + 0.06\log \left( {P_{{{\text{CO}}_{2} }} /40} \right) + 0.024\left( {T - 37} \right)} \right]. $$

(12)

The P ₅₀(pH_rbc) and P ₅₀($P_{{{\text{CO}}_{ 2} }}$) computed from Eq. (11) are more accurate than those computed from Eq. (12). They agree closely near the standard physiological conditions, but deviate when the conditions are far different from the standard conditions, as seen in Figs. 1a and 1b. The P ₅₀(T) formulas agree well over the whole range of T, as seen in Fig. 1d. Kelman did not account for the effect of [DPG]_rbc.

Figure 1 shows the variation of P ₅₀ with pH_rbc for $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg, [DPG]_rbc = 4.65 mM and T = 37 °C in panel A; P ₅₀ with $P_{{{\text{CO}}_{ 2} }}$ for pH_rbc = 7.24, [DPG]_rbc = 4.65 mM and T = 37 °C in panel B; P ₅₀ with [DPG]_rbc for pH_rbc = 7.24, $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg and T = 37 °C in panel C; and P ₅₀ with T for pH_rbc = 7.24, $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg and [DPG]_rbc = 4.65 mM in panel D. The P ₅₀ values from the model of Buerk and Bridges7 are shown by solid points and the corresponding best-fit polynomials are shown by solid lines. The P ₅₀ values from the model of Kelman18 are shown by dashed lines. This figure illustrates that the O₂ affinity of Hb decreases (i.e., the P ₅₀ level increases) as pH decreases (or [H⁺] increases) and $P_{{{\text{CO}}_{ 2} }}$, [DPG] or T increases. Also, the O₂ affinity is significantly affected by the pH and temperature T. The values of P ₅₀ computed from Eq. (11) are used here to estimate the values of $K_{4}^{\prime \prime}$, n ₀, n ₁, n ₂, n ₃ and n ₄.

Equations for Parameter Estimation

Now eliminating $K_{{{\text{HbO}}_{ 2} }}$ from Eqs. (7a) and (8a), then substituting the expression for $K_{4}^{\prime \prime}$ from Eq. (9), and finally evaluating the resulting equation at 50% HbO₂ saturation ($S_{HbO_{2}}$ = 0.5), we obtain the following equation relating $K_{4}^{\prime \prime}$, n ₀, n ₁, n ₂, n ₃ and n ₄ to P ₅₀:

$$ {\frac{{K_{4}^{\prime \prime} K_{\text{fact}} }}{{K_{\text{ratio}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} }}} = {\frac{1}{{\left( {\alpha_{{{\text{O}}_{2} }} P_{50} } \right)^{{1 + n_{0} }} }}}, $$

(13)

where K _ratio and K _fact below characterize the nonlinear O₂–CO₂ interactions:

$$ K_{\text{ratio}} = {\frac{{\left( {K_{2}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left\{ {1 + {\frac{{K_{2}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\} + \left\{ {1 + {\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{K_{5}^{\prime \prime} }}}} \right\}} \right)}}{{\left( {K_{3}^{\prime} \left[ {{\text{CO}}_{2} } \right]\left\{ {1 + {\frac{{K_{3}^{\prime \prime} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\} + \left\{ {1 + {\frac{{\left[ {{\text{H}}^{ + } } \right]}}{{K_{6}^{\prime \prime} }}}} \right\}} \right)}}} $$

(14a)

$$ K_{\text{fact}} = \left\{ {{\frac{{\left[ {{\text{H}}^{ + } } \right]_{\text{S}} }}{{\left[ {{\text{H}}^{ + } } \right]}}}} \right\}^{{n_{1} }} \left\{ {{\frac{{\left[ {{\text{CO}}_{2} } \right]_{\text{S}} }}{{\left[ {{\text{CO}}_{2} } \right]}}}} \right\}^{{n_{2} }} \left\{ {{\frac{{\left[ {\text{DPG}} \right]_{\text{S}} }}{{\left[ {\text{DPG}} \right]}}}} \right\}^{{n_{3} }} \left\{ {{\frac{{T_{\text{S}} }}{T}}} \right\}^{{n_{4} }} , $$

(14b)

P₅₀ = P₅₀ ([H⁺], [CO₂], [DPG], T) on the right hand side of Eq. (13) is based on the model of Buerk and Bridges7 which is plotted in Fig. 1 and is given by the polynomial (11). We use Eqs. (13), (14a) and (14b) to estimate the values of $K_{4}^{\prime \prime}$, n₀, n₁, n₂, n₃ and n₄.

Estimation of $K_{4}^{\prime \prime}$ and n ₀

Now the proportionality equilibrium constant $K_{4}^{\prime \prime}$ and empirical exponent n ₀ can be estimated simultaneously by fixing the physiological state variables at their standard values: [H⁺] = [H⁺]_S (or pH = pH_S), [CO₂] = [CO₂]_S (or $P_{{{\text{CO}}_{ 2} }}$ = $P_{{{\text{O}}_{ 2} }}$), [DPG] = [DPG]_S, and T = T _S. Then Eq. (13) becomes independent of the exponents n ₁, n ₂, n ₃ and n ₄ (because all the ratios in K _fact are 1) and so depends only on the exponent n ₀. On comparing the resulting equation with the relation between the Hill exponent and Hill coefficient (given below Eq. 1), we obtain the following estimations for $K_{4}^{\prime \prime}$ and n ₀:

$$ K_{4}^{\prime \prime} = {\frac{{K_{{{\text{ratio}},{\text{S}}}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} }}{{\left( {\alpha_{{{\text{O}}_{{2,\;{\text{S}}}} }} P_{{50,\;{\text{S}}}} } \right)^{{1 + n_{0} }} }}}\quad {\text{and}}\quad n_{0} = 1.7. $$

(15)

Under standard physiological conditions in the arterial system, we have [O₂] = [O₂]_S = 146 μM (or $P_{{{\text{O}}_{ 2} }}$ = $P_{{{\text{O}}_{ 2, {\text{S}}}}}$ = 100 mmHg), [CO₂] = [CO₂]_S = 1.31 mM (or $P_{{{\text{CO}}_{ 2} }}$ = $P_{{{\text{CO}}_{ 2, {\text{S}}}}}$ = 40 mmHg), [H⁺] = [H⁺]_S = 57.5 nM (or pH = pH_S = 7.24), [DPG] = [DPG]_S = 4.65 mM, and T = T _S = 37 °C in RBCs; pH = 7.24 in RBCs corresponds to pH = 7.4 in plasma, because the Gibbs–Donnan ratio R _rbc for the electrochemical equilibrium condition across the RBC membrane is about 0.6914–16,31,38; the value [DPG] = 4.65 mM corresponds to the ratio [DPG]/[Hb] = 0.9 which is used as the standard ratio by Winslow et al.39 and Buerk and Bridges.7 Under these conditions, P ₅₀ = P _50,S = 26.8 mmHg for human blood,6,7,39,40 which when substituted into Eq. (15) gives the estimation $K_{4}^{\prime \prime}$ = 202123 M⁻¹ that is independent of the choices of n ₁, n ₂, n ₃ and n ₄. It can be noted here that one can use $P_{{{\text{O}}_{ 2, {\text{S}}}}}$ = P _50,S = 26.8 mmHg as the reference $P_{{{\text{O}}_{ 2} }}$ instead of $P_{{{\text{O}}_{ 2, {\text{S}}}}}$ = 100 mmHg. However, this will give a different estimate of $K_{4}^{\prime \prime}$, while the estimate n ₀ = 1.7 will remain unchanged.

The current model can be made to fit the Adair model (or Severinghaus’s33 model) by adjusting parameters so that Eqs. (7a), (8a) and (9) fit Adair’s equation (or Severinghaus’s33 equation). In this case, the exponent n ₀ will be obtained as a function of $P_{{{\text{O}}_{ 2} }}$. However, we avoid this because the $S_{{{\text{HbO}}_{ 2} }}$ to $P_{{{\text{O}}_{ 2} }}$ relationship will then no longer be analytically invertible. See Appendix A for the calculation of blood O₂ content from saturation, $S_{{{\text{HbO}}_{ 2} }}$, and $P_{{{\text{O}}_{ 2} }}$, and see Appendix B for calculating $P_{{{\text{O}}_{ 2} }}$ from $S_{{{\text{HbO}}_{ 2} }}$ and the P ₅₀, taking into account $P_{{{\text{CO}}_{ 2} }}$, pH, [2,3-DPG], Hct, and T.

Estimation of n ₁, n ₂, n ₃ and n ₄

The empirical exponents n ₁, n ₂, n ₃ and n ₄ are estimated as independent of each other. These are estimated recursively by varying one of the physiological state variables and fixing the others at their standard values. For example, the exponent n ₁ is estimated by varying the level of pH and fixing the levels of $P_{{{\text{CO}}_{ 2} }}$, [DPG] and T at their standard values $P_{{{\text{CO}}_{ 2,{\text{S}}}}}$, [DPG]_S and T _S. In this case, Eq. (13) becomes independent of the exponents n ₂, n ₃ and n ₄ with the pH or [H⁺] as the only varying variable. From this resulting equation, the functional expression for n ₁ as a function of pH or [H⁺] can be obtained. In a similar fashion, the functional expressions for n ₂ as a function of $P_{{{\text{CO}}_{ 2} }}$ or [CO₂], n ₃ as a function of [DPG], and n ₄ as a function of T can also be obtained. These are given by the following exact expressions:

$$ n_{1} \left( {{\text{pH}}_{\text{rbc}} } \right) = {\frac{{\log \left[ {{{K_{{{\text{ratio}},1}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} } \mathord{\left/ {\vphantom {{K_{{{\text{ratio}},1}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} } {K_{4}^{\prime \prime} \left( {\alpha_{{{\text{O}}_{{2,{\text{S}}}} }} P_{50,1} } \right)^{{1 + n_{0} }} }}} \right. \kern-\nulldelimiterspace} {K_{4}^{\prime \prime} \left( {\alpha_{{{\text{O}}_{{2,{\text{S}}}} }} P_{50,1} } \right)^{{1 + n_{0} }} }}} \right]}}{{{\text{pH}}_{\text{rbc}} - {\text{pH}}_{\text{rbc,S}} }}}, $$

(16a)

$$ n_{2} \left( {P_{{{\text{CO}}_{2} }} } \right) = {\frac{{\log \left[ {{{K_{{{\text{ratio}},2}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} } \mathord{\left/ {\vphantom {{K_{{{\text{ratio}},2}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} } {K_{4}^{\prime \prime} \left( {\alpha_{{{\text{O}}_{{2,{\text{S}}}} }} P_{50,2} } \right)^{{1 + n_{0} }} }}} \right. \kern-\nulldelimiterspace} {K_{4}^{\prime \prime} \left( {\alpha_{{{\text{O}}_{{2,{\text{S}}}} }} P_{50,2} } \right)^{{1 + n_{0} }} }}} \right]}}{{{ \log }\left( {{{P_{{{\text{CO}}_{{2,{\text{S}}}} }} } \mathord{\left/ {\vphantom {{P_{{{\text{CO}}_{{2,{\text{S}}}} }} } {P_{{{\text{CO}}_{2} }} }}} \right. \kern-\nulldelimiterspace} {P_{{{\text{CO}}_{2} }} }}} \right)}}}, $$

(16b)

$$ n_{3} \left( {\left[ {\text{DPG}} \right]_{\text{rbc}} } \right) = {\frac{{\log \left[ {{{K_{{{\text{ratio}},3}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} } \mathord{\left/ {\vphantom {{K_{{{\text{ratio}},3}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} } {K_{4}^{\prime \prime} \left( {\alpha_{{{\text{O}}_{{2,{\text{S}}}} }} P_{50,3} } \right)^{{1 + n_{0} }} }}} \right. \kern-\nulldelimiterspace} {K_{4}^{\prime \prime} \left( {\alpha_{{{\text{O}}_{{2,{\text{S}}}} }} P_{50,3} } \right)^{{1 + n_{0} }} }}} \right]}}{{\log \left( {{{\left[ {\text{DPG}} \right]_{\text{rbc,S}} } \mathord{\left/ {\vphantom {{\left[ {\text{DPG}} \right]_{\text{rbc,S}} } {\left[ {\text{DPG}} \right]_{\text{rbc}} }}} \right. \kern-\nulldelimiterspace} {\left[ {\text{DPG}} \right]_{\text{rbc}} }}} \right)}}}, $$

(16c)

$$ n_{4} \left( T \right) = {\frac{{\log \left[ {{{K_{{{\text{ratio}},4}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} } \mathord{\left/ {\vphantom {{K_{{{\text{ratio}},4}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} } {K_{4}^{\prime \prime} \left( {\alpha_{{{\text{O}}_{{2,{\text{S}}}} }} P_{50,4} } \right)^{{1 + n_{0} }} }}} \right. \kern-\nulldelimiterspace} {K_{4}^{\prime \prime} \left( {\alpha_{{{\text{O}}_{{2,{\text{S}}}} }} P_{50,4} } \right)^{{1 + n_{0} }} }}} \right]}}{{\log \left( {T_{\text{S}} /T} \right)}}}, $$

(16d)

where the P ₅₀’s are given by the polynomial expression (11) and the second subscript “1”, “2”, “3” or “4” indicates that a particular one of the variables pH_rbc, $P_{{{\text{CO}}_{ 2} }}$, [DPG]_rbc or T is varying.

Figure 2 shows the variation of n ₁ with pH_rbc for $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg, [DPG]_rbc = 4.65 mM and T = 37 °C in panel A; n ₂ with $P_{{{\text{CO}}_{ 2} }}$ for pH_rbc = 7.24, [DPG]_rbc = 4.65 mM and T = 37 °C in panel B; n ₃ with [DPG]_rbc for pH_rbc = 7.24, $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg and T = 37 °C in panel C; and n ₄ with T for pH_rbc = 7.24, $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg and [DPG]_rbc = 4.65 mM in panel D. It is depicted that the exponent n ₄ is considerably larger compared to the other exponents n ₁, n ₂ and n ₃, whereas the exponent n ₂ is very small. This is because the effect of temperature T on the O₂ affinity is very significant and that of $P_{{{\text{CO}}_{ 2} }}$ is relatively small. These calculated functional forms of the exponents n ₁, n ₂, n ₃ and n ₄ are used here to compute $S_{{{\text{HbO}}_{ 2} }}$, $S_{{{\text{HbCO}}_{ 2} }}$, [O₂]_bl and [CO₂]_bl. However, for simplicity, the standard values n _1,S = 1.06, n _2,S = 0.12, n _3,S = 0.37, and n _4,S = 4.65 (see Table 1; calculated using the standard physiological conditions) can also be used. In this case, the shifts in the HbO₂ saturation curves will not be very accurate. The results are summarized through Figs. 3–6, presented below.

FIGURE 2

The exponents for Eq. (9); the plots of n ₁(pH_{rb c}), n ₂($P_{{{\text{CO}}_{2} }}$), n ₃([DPG]_rbc) and n ₄(T) with the other three variables fixed at their standard physiological values. These are computed from Eqs. (16a)–(16d) using the estimated value of $K_{4}^{\prime \prime}$ and n ₀ and the best-fit polynomial (11) for oxygen P ₅₀

FIGURE 3

The comparison of the HbO₂ dissociation curves computed from our model, Kelman18 model, and Buerk and Bridges7 model for different values of pH_rbc with $P_{{{\text{CO}}_{2} }}$, [DPG]_rbc and T fixed at their standard physiological values

Dissociation Curves and Blood Gas Contents

Oxyhemoglobin (HbO₂) Dissociation Curves

Figure 3 shows the comparison of the HbO₂ dissociation curves computed from current model, Kelman18 model and Buerk and Bridges7 model for pH_rbc = 6.92, 7.24 and 7.56 with $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg, [DPG]_rbc = 4.65 mM and T = 37 °C. It is seen that these curves are in fairly good agreement with each other over the entire saturation range, although our curves are consistently slightly below the curves of Kelman and Buerk and Bridges when $S_{{{\text{HbO}}_{ 2} }}$ < 30%. In this range, our curves are not very accurate, since our model is based on the Hill’s equation, which is considered to be accurate only in the saturation range of 20 to 98%,29 and one might therefore prefer their models at very low $P_{{{\text{O}}_{ 2} }}$ even though inversion is more difficult. However, for $S_{{{\text{HbO}}_{ 2} }}$ > 30%, our curves agree closely with the curves of Buerk and Bridges which were fit to actual human and dog blood HbO₂ saturation data of Roughton et al.,28 Roughton and Severinghaus,30 Winslow et al.,40 and Sveringhaus.33 Also Kelman’s model does not give accurate shifts in HbO₂ dissociation curves when physiological variables deviate far from their standard values.

Figure 4 shows the variation of $S_{{{\text{HbO}}_{ 2} }}$ with $P_{{{\text{O}}_{ 2} }}$ for different values of pH_rbc with $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg, [DPG]_rbc = 4.65 mM and T = 37 °C in panel A; different values of $P_{{{\text{CO}}_{ 2} }}$ with pH_rbc = 7.24, [DPG]_rbc = 4.65 mM and T = 37 °C in panel B; different values of [DPG]_rbc with pH_rbc = 7.24, $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg and T = 37 °C in panel C; and different values of T with pH_rbc = 7.24, $P_{{{\text{CO}}_{ 2} }}$ = 40 mmHg and [DPG]_rbc = 4.65 mM in panel D. The HbO₂ dissociation curve shifts to the right (i.e., the P ₅₀ for O₂ increases, or O₂ affinity of Hb decreases) as pH decreases (or [H⁺] increases) and $P_{{{\text{CO}}_{ 2} }}$, [DPG] or T increases. This diminution of Hb affinity for O₂ as pH decreases or $P_{{{\text{CO}}_{ 2} }}$ increases is known as the Bohr effect, so our model provides a quantitative description of the Bohr effect. The shifts in the HbO₂ dissociation curve with changes in $P_{{{\text{CO}}_{ 2} }}$ or [DPG] are small as compared to those occurring with changes in pH and T; the O₂ affinity of Hb is most sensitive to temperature T. Undoubtedly the fit of these models to complete data sets (such as the unfortunately unavailable set of Winslow et al.40 would result in improved parameter estimates, but these curves should be accurate roughly to 2 or 3% for $P_{{{\text{CO}}_{ 2} }}$’s above 20 mmHg.

FIGURE 4

The quantitative behavior of the HbO₂ dissociation curves at various physiological conditions (i.e., with varying levels of pH_rbc, $P_{{{\text{CO}}_{2} }}$, [DPG]_rbc and T) as computed from Eqs. (7a), (8a), (9), (15), and (16a)–(16d)

Carbamino-Hemoglobin (HbCO₂) Dissociation Curves

Figure 5 shows the variation of $S_{{{\text{HbCO}}_{ 2} }}$ with $P_{{{\text{CO}}_{ 2} }}$ for different values of pH_rbc with $P_{{{\text{O}}_{ 2} }}$ = 100 mmHg, [DPG]_rbc = 4.65 mM and T = 37 °C in panel A; different values of $P_{{{\text{O}}_{ 2} }}$ with pH_rbc = 7.24, [DPG]_rbc = 4.65 mM and T = 37 °C in panel B; different values of [DPG]_rbc with pH_rbc = 7.24, $P_{{{\text{O}}_{ 2} }}$ = 100 mmHg and T = 37 °C in panel C; and different values of T with pH_rbc = 7.24, $P_{{{\text{O}}_{ 2} }}$ = 100 mmHg and [DPG]_rbc = 4.65 mM in panel D. It is seen that the behavior of the HbCO₂ dissociations curves (hyperbolic) is quite different than the HbO₂ dissociation curves (sigmoid). Also, at fixed values of $P_{{{\text{O}}_{ 2} }}$, $P_{{{\text{CO}}_{ 2} }}$, pH, [DPG] and T, the Hb saturation of CO₂ ($S_{{{\text{HbCO}}_{2} }}$) is considerably lower than the Hb saturation of O₂ ($S_{{{\text{HbO}}_{2} }}$). This is because the CO₂ binding to Hb is noncooperative in nature,2 whereas the O₂ binding to Hb is cooperative.

Figure 5a depicts that the CO₂ saturation of Hb ($S_{{{\text{HbCO}}_{2} }}$) is greatly affected by pH, and is dependent only on the shift in affinity for single site binding. But raising [H⁺], reducing pH, greatly reduces the CO₂ affinity for Hb, reducing the carbamino formation: so the P ₅₀ for CO₂ shifts rightward. Likewise, Figs. 5b and 5d show that raising $P_{{{\text{O}}_{ 2} }}$ and T shift the P ₅₀ for CO₂ rightward, indicating a reduction in carbamino formation and CO₂ affinity for Hb. This shift in P ₅₀ for CO₂ is higher at lower values of $P_{{{\text{O}}_{ 2} }}$ and T and negligible at higher values of $P_{{{\text{O}}_{ 2} }}$ and T. The alteration of the CO₂ affinity of Hb with respect to $P_{{{\text{O}}_{ 2} }}$ is known as the Haldane effect, so our model provides a quantitative description of the Haldane effect.

FIGURE 5

The quantitative behavior of the HbCO₂ dissociation curves at various physiological conditions (i.e., with varying levels of pH_rbc, $P_{{{\text{O}}_{2} }}$ [DPG]_rbc and T) as computed from Eqs. (7b), (8b), (9), (15), and (16a)–(16d)

The curve of $S_{{{\text{HbCO}}_{ 2} }}$ asymptotes at high $P_{{{\text{O}}_{ 2} }}$ to a limiting curve (Fig. 5b) because HbO₂ saturation is almost complete by $P_{{{\text{O}}_{ 2} }}$ = 100 mmHg beyond which O₂ has no direct effect on CO₂ binding. For temperature (Fig. 5d), the apparent asymptotic behavior is due to a combination of factors: a reduction in the concentration of dissolved CO₂ due a decrease in solubility of CO₂ and an increase in the apparent Hill coefficient $K_{{{\text{HbCO}}_{ 2} }}$ due to a decrease in HbO₂ saturation. These opposing trends in [CO₂] and $K_{{{\text{HbCO}}_{ 2} }}$ with raising T diminish the shift in the HbCO₂ dissociation curves. Figure 5c depicts that the effect of [DPG]_rbc on the CO₂ affinity of Hb can be neglected, as $K_{{{\text{HbCO}}_{ 2} }}$ and $K_{{{\text{HbCO}}_{ 2} }}$ are almost independent of [DPG]_rbc.

CO₂ Content in Whole Blood, [CO₂]_bl

Figure 6 shows the variation of [CO₂]_bl (ml of CO₂ per 100 mL of blood) with $P_{{{\text{CO}}_{ 2} }}$, for different values of $P_{{{\text{O}}_{ 2} }}$ and [DPG]_rbc with pH_rbc = 7.24, T = 37 °C, and Hct = 0.45 in panel A; for different values of pH_rbc with T = 37 °C, Hct = 0.45, and all $P_{{{\text{O}}_{ 2} }}$ and [DPG]_rbc in panel B; for different values of T with pH_rbc = 7.24, Hct = 0.45, and all $P_{{{\text{O}}_{ 2} }}$ and [DPG]_rbc in panel C; and for different values of Hct with pH_rbc = 7.24, T = 37 °C, and all $P_{{{\text{O}}_{ 2} }}$ and [DPG]_rbc in panel D. Generally [CO₂]_bl varies linearly with $P_{{{\text{CO}}_{ 2} }}$, and varies negligibly with $P_{{{\text{O}}_{ 2} }}$ and [DPG]_rbc. This is because most of the CO₂ is carried as bicarbonate, and very little in the form of carbamino. Figure 6b shows that the [CO₂]_bl decreases as pH decreases and $[{\text{HCO}}_{3} ^{ - }]$ decreases, by Eq. (A.3). An increase in the temperature T (Fig. 6c) reduces the CO₂ solubility and leads to a decrease in [CO₂] and then [HCO₃ ⁻]. Figure 6d shows that an increase in Hct leads to a decrease of plasma space, an increase in average pH in blood (since pH_rbc = 7.24 and pH_pl = 7.4), and hence a decrease in [HCO₃ ⁻]. These lead to a decrease in total CO₂ content in whole blood.

FIGURE 6

Total CO₂ content of whole blood ([CO₂]_bl, mL CO₂ per 100 mL blood) as a function of $P_{{{\text{CO}}_{2} }}$ at various physiological conditions (i.e., with varying levels of pH_rbc, $P_{{{\text{O}}_{2} }}$, [DPG]_rbc, T and Hct) as computed through Eqs. (A.2) and (A.3)

Discussion and Conclusions

The transport and exchange of O₂ and CO₂ in the circulatory system is highly influenced by the competitive binding of O₂ and CO₂ with hemoglobin (i.e., the Hb-mediated nonlinear O₂–CO₂ interactions) and by the levels of pH (acidity), 2,3-DPG concentration, and temperature in RBCs. Thus, in the modeling of simultaneous transport and exchange of O₂ and CO₂, one must consider suitable model equations for O₂ and CO₂ saturations of Hb ($S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$) which are coupled or linked to each other through the kinetics of O₂ and CO₂ binding to Hb. Also, for computational efficiency, in simulating the blood-tissue gas exchange processes in changing physiological states, the $S_{{{\text{HbO}}_{ 2} }}$ to $P_{{{\text{O}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ to $P_{{{\text{CO}}_{ 2} }}$ relationships should be analytically invertible. Again, since the $S_{{{\text{HbO}}_{ 2} }}$ is measurable spectrophotometrically, estimating $P_{{{\text{O}}_{ 2} }}$ from $S_{{{\text{HbO}}_{ 2} }}$ is of practical, even clinical, consequence. However, as $S_{{{\text{HbCO}}_{ 2} }}$ can not currently be estimated spectrophotometrically, there is no practical utility in calculating $P_{{{\text{CO}}_{ 2} }}$ from $S_{{{\text{HbCO}}_{ 2} }}$.

The models for standard HbO₂ dissociation curve, valid for only fixed/standard values of pH, $P_{{{\text{CO}}_{ 2} }}$, [DPG]/[Hb] and T,1,6,8,13 are not very efficient to use in the simulation of dynamic blood–tissue gas exchange processes. To make them applicable, one must multiply all $P_{{{\text{O}}_{ 2} }}$ values of the standard curve by a factor which is composite for the variations in pH, $P_{{{\text{CO}}_{ 2} }}$, [DPG] and T (e.g., as in Kelman18 and Severinghaus33). The numerical algorithm of Winslow et al.39 for computing the nonstandard HbO₂ dissociation curves is very complicated, because it needs numerical evaluation of all the four Adair constants as functions of pH, $P_{{{\text{CO}}_{ 2} }}$ and [DPG]/[Hb] by fitting the Adair’s equation with the experimental data on HbO₂ saturation in human whole blood. Their quadratic fit regression analysis for the four Adair constants needs numerical evaluation of a total of 72 coefficients. The effect of temperature (T) were not established, but would need an estimation of 24 additional coefficients to extend their algorithm. Like Adair’s equation, the empirical equations of Kelman18 and Siggaard-Andersen et al.37 for nonstandard HbO₂ dissociation curves require numerical inversion for computing $P_{{{\text{O}}_{ 2} }}$ from $S_{{{\text{HbO}}_{ 2} }}$, which is computationally expensive. The model of Buerk and Bridges,7 being analytically invertible, is more efficient. However, it does not describe the upper 5% of the dissociation curve accurately. Besides, all these models of HbO₂ dissociation curves need a suitable coupled model of HbCO₂ dissociation curves for integrating into a computational model of simultaneous or dynamic O₂ and CO₂ transport and exchange in the microcirculation.

The model of Singh et al.38 for nonstandard HbO₂ and HbCO₂ dissociation curves is based on the equilibrium binding of O₂ and CO₂ with Hb, but it does not account for the effects of 2,3-DPG concentration and temperature in blood. Also Singh et al.’s estimates of the equilibrium constants for the kinetic reactions do not agree well with the experimental values reported earlier.2,10,11,14–16,29,31 Their formulation for kinetic reactions of O₂ and CO₂ with Hb was based on the assumption that the Hb molecule has only one heme-amino chain, Hm-NH₂, and is capable of binding to only one CO₂ molecule, and therefore requires correction, since Hb actually has four Hm-NH₂ chains and can bind to four CO₂ molecules.

In an attempt to address these issues, a more appropriate and relatively simple mathematical model equations (Eqs. 7a and 7b) for O₂ and CO₂ saturations of Hb ($S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$) are developed in this paper. These are derived by considering the equilibrium binding of O₂ and CO₂ with Hb inside RBCs,2,14,29,31,38 including the Hb-mediated nonlinear O₂–CO₂ interactions and the effects of pH, 2,3-DPG, and temperature. Unlike in the previous models mentioned above, Hb molecule is considered to have four heme-amino chains, Hm-NH₂, each capable of binding to one O₂ molecule and one CO₂ molecule. The binding of O₂ is considered to be cooperative and that of CO₂ as noncooperative. The new model equations for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ are of the form of Hill’s equation, which has the extra advantage of being analytically invertible, allows the O₂ and CO₂ partial pressures ($P_{{{\text{O}}_{ 2} }}$ and $P_{{{\text{CO}}_{ 2} }}$) to be computed directly from their saturations ($S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$) and vice versa (see Appendix B). However, this new model is highly accurate only above 30% ($S_{{{\text{HbO}}_{ 2} }}$ or a $P_{{{\text{O}}_{ 2} }}$ of 20 mmHg, and leads to 1 to 2% underestimation of the saturation from the $P_{{{\text{O}}_{ 2} }}$ in computing intracapillary profiles of O₂ and CO₂ from models of simultaneous transport and exchange of O₂ and CO₂. Improvement is needed here, though the low range is not very commonly encountered in normal circumstances; the Severinghaus33 model would be an improvement within the range of conditions covered by that model.

The apparent Hill coefficients $K_{{{\text{HbO}}_{ 2} }}$ and $K_{{{\text{HbCO}}_{ 2} }}$ in the expressions for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ (Eqs. 8a and 8b) are explicitly dependent on the levels of $P_{{{\text{O}}_{ 2} }}$, $P_{{{\text{CO}}_{ 2} }}$, pH, [2,3-DPG] and T in RBCs and are dependent on the equilibrium constants of the chemical reactions involved in the binding O₂ and CO₂ with Hb. So these establish the linkage between $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ and the nonlinear O₂–CO₂ interactions. The cooperativity of O₂ binding and the effects of nonstandard physiological conditions are established in this model by considering the equilibrium constant for the reaction of O₂ with an Hm-NH₂ chain as a suitable function of $P_{{{\text{O}}_{ 2} }}$, $P_{{{\text{CO}}_{ 2} }}$, pH, [2,3-DPG] and T involving six adjustable parameters (see Eq. 9), including one proportionality equilibrium constant $K_{4}^{\prime\prime}$ and five empirical exponents n ₀, n ₁, n ₂, n ₃ and n ₄. These were estimated using the P ₅₀ (pH_rbc $P_{{{\text{CO}}_{ 2} }}$ [DPG]_rbc T) values from the model of Buerk and Bridges7 for nonstandard HbO₂ dissociation curves which agree closely with those obtained theoretically by Kelman18 and experimentally by Winslow et al.39 for normal human whole blood. These estimates are also influenced by the equilibrium constants for the other kinetic reactions in the uptake of CO₂ and ionization of Hm-NH₂ chains. These equilibrium constants are chosen or calculated appropriately to be consistent with those referred to largely in the literature.2,14–16,31

Our new model could be fitted to other sets of experimental data for characterizing the O₂ and CO₂ saturation of Hb. The report of Winslow et al.39 was based on a large dataset, and in their report, the data were summarized by the P ₅₀ values; the original data are no longer available, unfortunately, else they would have served as an excellent test of our descriptive equations. From an optimization using such large datasets, we would obtain more precise estimates of equilibrium constants and P ₅₀ (pH_rbc, $P_{{{\text{CO}}_{ 2} }}$, [DPG]_rbc, T) values, and presumably improved estimates of our model parameters, particularly the exponents n ₁, n ₂, n ₃ and n ₄.

The new model equations for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ are used here to calculate the O₂ and CO₂ contents in whole blood (see Appendix A). The HbO₂ and HbCO₂ dissociation curves and the O₂ and CO₂ contents in whole blood computed through these new equations are in good agreement with the published experimental and theoretical results in the literature. The result shows that at normal physiological conditions in arterial blood, the $P_{{{\text{O}}_{ 2} }}$ is about 100 mmHg and so Hb is about 97.2% saturated by O₂ while the amino group of Hb is about 13.1% saturated by CO₂. The invertibility (see Appendix B) of our new equations for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ allows their convenient use in computationally complex models of simultaneous or dynamic transport and exchange of O₂ and CO₂ in the alveoli–blood and blood–tissue exchange systems. This model has been implemented in our Java Simulation (JSim) interface, the mathematical modeling language (MML) code of which is available for download and public use at our physiome website, http://physiome.org/Models/GasTransport.

Our new model is unique in the sense that it is derived from the equilibrium binding of O₂ and CO₂ with Hb and it accounts for all the factors (e.g., $P_{{{\text{O}}_{ 2} }}$, $P_{{{\text{CO}}_{ 2} }}$, pH, [DPG] and T) that affect the O₂ and CO₂ binding to Hb. It also establishes the linkage between $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ as well as the Hb-mediated nonlinear O₂–CO₂ interactions (or Bohr and Haldane effects), the effects of which are important in simulating the complex simultaneous or dynamic transport and exchange of O₂ and CO₂ in the microcirculation. The kinetic and equilibrium constants, compiled from the literature or estimated in this paper, can be used in the dynamically modeling of the complex gas exchange process in vivo. These can also used to compute the concentrations of all the reaction products at equilibrium from Eqs. (3a) to (3i).

In summary, our new multidimensional model equations for $S_{{{\text{HbO}}_{ 2} }}$ and $S_{{{\text{HbCO}}_{ 2} }}$ are relatively simple, lending themselves to simple equation-solving by spreadsheet or by a simple program in a handheld computer. Severinghaus32 provided a slide rule for this purpose, and later in his 1979 paper,33 he summarized all the earlier calculations through appropriate model equations, which gives the earlier references, and which stimulated Ellis9 to provide the inverted calculation ($P_{{{\text{O}}_{ 2} }}$ from $S_{{{\text{O}}_{ 2} }}$). The Appendix B provides a recipe for the inversion of our new $S_{{{\text{HbO}}_{ 2} }}$ to $P_{{{\text{O}}_{ 2} }}$ relationship that allows calculation of whole blood O₂ content from the HbO₂ saturation (e.g., provided spectrophotometrically) and other chemically or physically determined measures of pH, $P_{{{\text{O}}_{ 2} }}$ $P_{{{\text{CO}}_{ 2} }}$ or bicarbonate, [2,3-DPG], and temperature (T) in blood. Of particular value in analyzing the ¹⁵O-oxygen time course data from PET (positron emission tomography) using convection–diffusion distributed models (e.g., Li et al.23 is the analytical invertibility of the Hill-type expression for $S_{\text{HbO}_{2}}$, as given by Eqs. (B.1) to (B.3) in the Appendix B.

We measured the computation times for numerical inversion of Kelman’s18 equation. It took 20–25 times more computer time (because of the numbers of iterations required) than the analytical inversion of Hill’s equation for computing $P_{{{\text{O}}_{ 2} }}$ from $S_{{{\text{HbO}}_{ 2} }}$. This is more significant in the convection–diffusion distributed modeling of O₂ and CO₂ transport in blood–tissue exchange systems which involves large number of spatial and temporal grid points. Again, in the estimation of parameters, using such axially distributed models, accounting for the gradients in [O₂], [CO₂], pH and T along the capillary length, will require another level of iteration during the optimization process: this means that computational efficiency is doubly important.

Appendices

Appendix A

Calculation of O₂ Content in Whole Blood

To calculate the O₂ content of whole blood, we need to sum the two forms of O₂ in blood: O₂ dissolved in plasma water and RBC water and O₂ as oxyhemoglobin in RBCs. Thus, the molar O₂ concentration of whole blood, [O₂]_bl, can be calculated as

$$ \left[ {{\text{O}}_{2} } \right]_{\text{bl}} = W_{\text{bl}} \alpha_{{{\text{O}}_{2} }} P_{{{\text{O}}_{2} }} + 4\left[ {\text{Hb}} \right]_{\text{bl}} S_{{{\text{HbO}}_{2} }} , $$

(A.1)

where W _bl is the fractional water space of blood and is related to that of plasma and RBCs (W _pl and W _rbc) by W _bl = (1 − Hct) W _pl + Hct W _rbc; [Hb]_bl is the molar concentration of hemoglobin in whole blood and is related to that in RBCs, [Hb]_rbc, by [Hb]_bl = Hct [Hb]_rbc; Hct is the blood hematocrit. It is assumed that the O₂ partial pressures in plasma and RBCs are equal. The factor 4 in Eq. (A.1) indicates that four molecules of O₂ bind to one molecule of Hb. The whole blood O₂ content in mL of O₂ per mL of blood is 22.256 times [O₂]_bl.

At standard physiological conditions, Hct is about 0.45 and [Hb]_bl is about 0.15 g/mL or 2.33 mM taking the molecular weight of Hb to be 64,458.19 Correspondingly, [Hb]_rbc is about 5.18 mM that is assumed to be fixed and independent of Hct. With W _pl = 0.94, W _rbc = 0.65 and Hct = 0.45, we have W _bl = 0.81 mL water per mL blood. With these data and at a $P_{{{\text{O}}_{ 2} }}$of 100 mmHg, the O₂ saturation of Hb, $S_{{{\text{HbO}}_{ 2} }}$, is about 97.2%, the O₂ content of whole blood is about 0.26 mL O₂ per 100 mL blood in free form plus about 20.18 mL O₂ per 100 mL blood bound to Hb, making a total O₂ content in whole blood of 20.44 mL O₂ per 100 mL blood.

Calculation of CO₂ Content in Whole Blood

To calculate the CO₂ content of whole blood, we need to sum the four forms of CO₂ in blood: CO₂ in dissolved form, as carbonic acid (H₂CO₃), as bicarbonate ions $({\text{HCO}}_{3} ^{ - }),$ and as carbamino-hemoglobin (HbCO₂). The dissociation constant $K_{1}^{\prime\prime}$ for H₂CO₃ is about 5.5 × 10⁻⁴ M14–16 while the concentration of H⁺ in plasma is about 3.98 × 10⁻⁸ M and in RBCs is about 5.75 × 10⁻⁸ M. So the concentration of H₂CO₃ is usually four orders of magnitude smaller than that of $({\text{HCO}}_{3} ^{ - }).$ Therefore, its contribution to the CO₂ content can be neglected.

Again four moles of CO₂ bind with one mole of Hb in RBCs. So with the assumption that the CO₂ partial pressures in plasma and RBCs are equal, the molar CO₂ concentration of whole blood, [CO₂]_bl, can be calculated as

$$ \left[ {{\text{CO}}_{2} } \right]_{\text{bl}}\,= W_{\text{bl}} \alpha_{{{\text{CO}}_{2} }} P_{{{\text{CO}}_{2} }} + \left[ {{\text{HCO}}_{3}^{ - } } \right]_{\text{bl}}\,+\,4\left[ {\text{Hb}} \right]_{\text{bl}} S_{{{\text{HbCO}}_{2} }} , $$

(A.2)

where the total bicarbonate in blood, [HCO₃ ⁻]_bl, is obtained by using the Henderson-Hasselbalch equation and Gibbs–Donnan electrochemical equilibrium condition as:

$$ \begin{aligned} \left[ {{\text{HCO}}_{3}^{ - } } \right]_{\text{bl}} & = \left( {1 - Hct} \right)W_{\text{pl}} \left[ {{\text{HCO}}_{3}^{ - } } \right]_{\text{pl}} + HctW_{\text{rbc}} \left[ {{\text{HCO}}_{3}^{ - } } \right]_{\text{rbc}} \\ & = \left\{ {\left( {1 - Hct} \right)W_{\text{pl}} + Hct\;W_{\text{rbc}} R_{\text{rbc}} } \right\}\left\{ {{\frac{{K_{1} \alpha_{{{\text{CO}}_{2} }} P_{{{\text{CO}}_{2} }} }}{{\left[ {{\text{H}}^{ + } } \right]_{\text{pl}} }}}} \right\}. \\ \end{aligned} $$

(A.3)

Here $K_{1}=K_{1}^{\prime} K_{1}^{\prime \prime}$ is the equilibrium constant for overall CO₂ hydration reaction (2a) and R _rbc = [H⁺]_pl/H⁺]_rbc = $[{\text{HCO}}_{3} ^{ - } ]_{{{\text{rbc}}}} /[{\text{HCO}}_{3} ^{ - } ]_{{{\text{pl}}}}$ is the Gibbs–Donnan ratio for electrochemical equilibrium condition across the RBC membrane. The value of R _rbc is about 0.69.14–16,31,38 Multiplying [CO₂]_bl by 22.256 converts the units of molar to mL of CO₂ per mL of blood.

From the data used in Hill et al.,14–16 [H₂O] $kf_{1}^{\prime}$ ≈ 0.12 s⁻¹ and $kb_{1}^{\prime}$ ≈ 89 s⁻¹, so $K_{1}^{\prime} = [{\rm H}_{2}{\rm O}]kf_{1}^{\prime}/kb_{1}^{\prime}$ ≈ 1.35 × 10⁻³. Again, $K_{1}^{\prime \prime}$ ≈ 5.5 × 10⁻⁴ M, so $K_{1} = K_{1}^{\prime} K_{1}^{\prime \prime}$ ≈ 7.43 × 10⁻⁷ M, as shown in Table 1. Thus, pK₁ = −log(K ₁) ≈ 6.13, which is close to the standard value pK₁ = 6.1 reported in text books and literature. However, pK₁ depends on the levels of pH and temperature T in plasma. This dependency was expressed by the following curve-fit equation by Kelman21 based on the experimental data of Austin et al.3:

$$ {\text{pK}}_{1} = 6.09 - 0.0434\left( {{\text{pH}}_{\text{pl}} - 7.4} \right) + 0.0014\left( {T - 37} \right)\left( {{\text{pH}}_{\text{pl}} - 7.4} \right). $$

(A.4)

At pH_pl = 7.4 and T = 37, Eq. (A.4) gives pK₁ = 6.09 which is close to the value pK₁ = 6.13 calculated above and agrees well with the data of Severinghaus et al.35

At standard physiological conditions, the CO₂ content of whole blood in free form is about 2.35 mL CO₂ per 100 mL blood (i.e., 1.06 mM), in bicarbonate form is about 42.69 mL CO₂ per 100 mL blood (i.e., 19.18 mM), and in carbamino form is about 2.72 mL CO₂ per 100 mL blood (i.e., 1.22 mM), making a total CO₂ content in whole blood of 47.76 mL CO₂ per 100 mL blood (i.e., 21.45 mM). These estimates agree with the data reported in the literature.21,22,29 The CO₂ saturation of Hb, $S_{{{\text{HbCO}}_{ 2} }}$, is computed to be about 13.1% at a $P_{{{\text{CO}}_{2} }}$of 40 mmHg.

Appendix B

Algorithm for Inverting the Relationships Between $S_{{{\text{HbO}}_{ 2} }}$ and $P_{{{\text{O}}_{ 2} }}$

Equation (7a) for the fractional saturation $S_{{{\text{HbO}}_{ 2} }}$ is not convenient for analytical inversion because the apparent Hill coefficient $K_{{{\text{HbO}}_{ 2} }}$ depends on [O₂] (see Eq. 8a). However, with the help of Eq. (9), the expression for $S_{{{\text{HbO}}_{ 2} }}$ can be rewritten in the following form, which is amenable for analytical inversion:

$$ S_{{{\text{HbO}}_{2} }} = {\frac{{K^{\prime}_{{{\text{HbO}}_{2} }} \left[ {{\text{O}}_{2} } \right]^{{1 + n_{0} }} }}{{1 + K^{\prime}_{{{\text{HbO}}_{2} }} \left[ {{\text{O}}_{2} } \right]^{{1 + n_{0} }} }}}, $$

(B.1)

where $K_{{{\text{HbO}}_{ 2} }}^{\prime}$ (with units ${\rm M}^{-(1+n_{0})}$) is given by Eqs. (13) and (14):

$$ K^{\prime}_{{{\text{HbO}}_{2} }} = {\frac{{K_{4}^{\prime \prime} K_{\text{fact}} }}{{K_{\text{ratio}} \left[ {{\text{O}}_{2} } \right]_{\text{S}}^{{n_{0} }} }}} = {\frac{1}{{\left( {\alpha_{{{\text{O}}_{2} }} P_{50} } \right)^{{1 + n_{0} }} }}}. $$

(B.2)

The P ₅₀ is defined as before by Eq. (11) and the kinetic terms K _ratio and K _fact are defined by Eqs. (14a) and (14b). The Hill exponent in the expression for $S_{{{\text{HbO}}_{ 2} }}$ is now 1 + n ₀ and apparent Hill coefficient $K_{{{\text{HbO}}_{ 2} }}^{\prime}$ is now independent of [O₂]. This makes $S_{{{\text{HbO}}_{ 2} }}$ analytically invertible, when $P_{{{\text{CO}}_{2} }}$, pH, [DPG] and T are known. The inverted equation for [O₂] is given by

$$ \left[ {{\text{O}}_{2} } \right] = \left[ {{\frac{{S_{{{\text{HbO}}_{2} }} }}{{K^{\prime}_{{{\text{HbO}}_{2} }} \left( {1 - S_{{{\text{HbO}}_{2} }} } \right)}}}} \right]^{{{\frac{1}{{1 + n_{0} }}}}} = \alpha_{{{\text{O}}_{2} }} P_{50} \left[ {{\frac{{S_{{{\text{HbO}}_{2} }} }}{{1 - S_{{{\text{HbO}}_{2} }} }}}} \right]^{{{\frac{1}{{1 + n_{0} }}}}} . $$

(B.3)

It is clear from Eq. (B.2) that $K_{{{\text{HbO}}_{ 2} }}^{\prime}$ can be determined completely using only the P ₅₀ = P ₅₀ ([H⁺], [CO₂], [DPG], T) data which is given by Eq. (11). This avoids the calculations of K _ratio and K _fact which involves the complex computations of the empirical indices n ₁, n ₂, n ₃, and n ₄ using Eqs. (16a) to (16d). This also simplifies the computation of $S_{{{\text{HbO}}_{ 2} }}$ from [O₂] and vice versa significantly. However, in the computation of $S_{{{\text{HbCO}}_{ 2} }}$ from [CO₂] and vice versa (not shown in this appendix), the calculations of K _fact and K _ratio are essential as the P ₅₀ data for 50% HbCO₂ saturation is not available in the literature.

Here Eq. (B.1) further suggests that, in nonstandard physiological conditions, $S_{{{\text{HbO}}_{ 2} }}$ can be computed from the original Hill’s equation (Eq. 1) by scaling the [O₂] axis by a factor 1/$K_{{{\text{HbO}}_{ 2} }}^{\prime}$ (see Eq. B.2) which is composite for the variations in pH, $P_{{{\text{CO}}_{ 2} }}$, [2,3-DPG], and temperature (T), as in Kelman18 and Severinghaus.33 The same conclusion is valid for the computation of $S_{{{\text{HbCO}}_{ 2} }}$ from [CO₂].

The inversion procedure for determining $P_{{{\text{O}}_{ 2} }}$ and then total O₂ content in whole blood from the observations on $S_{{{\text{HbO}}_{ 2} }}$, $P_{{{\text{CO}}_{ 2} }}$, pH_pl, [DPG]_bl, Hct, and T in blood is as follows:

• A. Calculate [H⁺]_pl = $10^{-{\rm pH}_{\rm pl}}$, [H⁺]_rbc = [H⁺]_pl/R _rbc, pH_rbc = −log[H⁺]_rbc, and [DPG]_rbc = [DPG]_bl/Hct, where R _rbc = 0.69.

• B. Calculate P ₅₀ = P ₅₀ ($P_{{{\text{CO}}_{ 2} }}$, pH_rbc, [DPG]_rbc, T) from Eq. (11).

• C. Calculate [O₂] from Eq. (B.3), and then $P_{{{\text{O}}_{ 2} }}$ = [O₂]/$\alpha_{{{\text{O}}_{ 2} }}$, where $\alpha_{{{\text{O}}_{ 2} }}$ is given by Eq. (10a), and n ₀ is 1.7. This is the inversion step.

• D. Calculate [Hb]_bl = Hct × [Hb]_rbc = Hct × 5.18 mM, assuming that the RBC has standard hemoglobin concentration of 5.18 mM and that there is no methemoglobin or other abnormal type of hemoglobin.

• E. Calculate W _bl = W _pl (1 − Hcts) + W _rbc Hct = 0.94 (1 − Hct) + 0.65 Hct, as the fractional water content of blood.

• F. Calculate [O₂]_bl = W _bl [O₂]_bl + 4 [Hb]_bl $S_{{{\text{HbO}}_{ 2} }}$, as the total O₂ content in whole blood in M. To convert from [O₂]_bl M to mL gaseous O₂ per mL blood, use mL O₂ gas/100 mL blood = 22,256 × 100/1000 × [O₂]_bl = 2225 × [O₂]_bl.

This procedure can be set up in a spreadsheet, programmable hand calculator, or in any computer program such as the JSim MML code (available from the website http://www.physiome.org/Models/GasTransport/ for Linux, Unix, Macintosh and Windows). JSim, the general modeling system, and extensive manuals for it, can be downloaded from http://www.physiome.org/jsim/ for free.