## Abstract

This paper presents an analytical expression for the diffusing capacity (Θ_{t}) of the red blood cell (RBC) for any reactive gas in terms of size and shape of the RBC, thickness of the unstirred plasma layer surrounding the RBC, diffusivities and solubilities of the gas in RBC and boundary layer, hematocrit, and the slope of the dissociation curve. The expression for Θ_{t} has been derived by spatial averaging of the fundamental convection-diffusion-reaction equation for O_{2} in the RBC and has been generalized to all cell shapes and for other reactive gases such as CO, NO, and CO_{2}. The effects of size and shape of the RBC, thickness of the unstirred plasma layer, hemoglobin concentration, and hematocrit on Θ_{t} have been analyzed, and the analytically obtained expression for Θ_{t} has been validated by comparison with different sets of existing experimental data for O_{2} and CO_{2}. Our results indicate that the discoidal shape of the human RBC with average dimensions of 1.6-μm thickness and 8-μm diameter is close to optimal design for O_{2} uptake and that the true reaction velocity in the RBC is suppressed significantly by the mass transfer resistance in the surrounding unstirred layer. In vitro measurements using rapid-mixing technique, which measures Θ_{t} in the presence of artificially created large boundary layers, substantially underpredicts the in vivo diffusing capacity of the RBC in the diffusion-controlled regime. Depending on the conditions in the RBC, uptake of less reactive gases (such as CO) undergoes transition from reaction-limited to diffusion-limited regime. For a constant set of morphological parameters, the theoretical expression for Θ_{t} predicts that Θ_{t,NO} > Θ_{t,}_{CO2} > Θ_{t,}_{O2} > Θ_{t,CO}.

- red blood cells
- diffusing capacity
- hemoglobin
- oxygen
- carbon monoxide
- nitric oxide
- carbon dioxide

the diffusing capacity of the red blood cell (RBC), Θ, is the effective (or mass transfer-disguised) reaction rate between a reactive gas and hemoglobin in the RBC and depends on three physical and chemical processes, namely, internal mass transfer in the RBC due to finite rates of diffusion of the gas and hemoglobin inside the RBC, external mass transfer (due to diffusional gradients of the gas) in the stagnant plasma layer surrounding the RBC, and the actual rate of reaction between the dissolved gas and hemoglobin within the RBC.

The term Θ first appeared in the footnote of a 1954 paper of Forster et al. (14). In 1957, Roughton and Forster in their classic paper (46) proposed that resistances offered by the capillary membrane (1/D_{M}) and reaction rate in the RBC (1/ΘVc) are in series and therefore could be summed up to give the total resistance to gas transfer (1/Dl) between the alveolar gas and the RBC, and this relationship is given by (1) where Dl is the overall diffusing capacity of the lung, D_{M} is the diffusing capacity of the membrane separating the alveolar air from the blood, Vc is the total volume in milliliters of the blood in the lung capillaries exposed to alveolar air, and Θ is the diffusing capacity of the RBC. In an attempt to account for the diffusional resistance in the plasma, Crapo et al. (8) defined D_{M} in *Eq. 1* as (2) with (3) (4) where D_{t} is the tissue component of diffusing capacity; D_{p} is the plasma component of diffusing capacity; *K*_{t} and *K*_{p} are the permeation coefficients of the gas in the tissue and the plasma, respectively; *S*_{a} and *S*_{c} are the surface areas of air-tissue interface and tissue-blood interface, respectively; and τ_{ht} and τ_{hp} are the (harmonic) mean thicknesses of the tissue and the plasma, respectively.

In a historical review of the overall diffusing capacity (Dl) and its components (D_{M} and ΘVc), Hughes and Bates (31) summarized the experimental efforts for evaluating Θ (for O_{2} and CO) in the last 50 years. As mentioned in this paper, Holland (28), Forster (13), Reeves and Park (44), and Borland and Cox (4) used direct and indirect ways of measuring diffusing capacity of the RBC.

Staub et al. (49) measured the rate of reaction between O_{2} and hemoglobin in the RBC using a continuous-flow rapid-reaction apparatus and suggested an empirical formula for diffusing capacity of O_{2} (Θ_{O2}) based on the reaction rate, which is given by (5) where *k*_{c} is the rate constant for the reaction with units of Torr per second, S is the fractional O_{2} saturation of the RBC, and α/760 is the solubility of O_{2} in milliliters per milliliter of blood per Torr. Because *k*_{c} and S are functions of the partial pressure of O_{2} in the RBC (〈P_{rbc}〉), Θ given by *Eq. 5* is also a function of 〈P_{rbc}〉. Using the data of Staub et al. (49) on *k*_{c}, the O_{2} dissociation curve, and *Eq. 5*, Flummerfelt and Crandall (11) obtained the following empirical relation to quantify Θ as a function of 〈P_{rbc}〉: where (6) The implications of the data of Staub et al. (49) shall be discussed in detail in a later section, when we compare our model predictions with experimental results.

The fundamental mechanism of uptake of a reactive gas in the RBC involves coupled convection-diffusion-reaction (CDR) of the gas and hemoglobin in the RBC and diffusion of the gas in the stagnant plasma layer. In this paper, we derive an analytical expression of the diffusing capacity (Θ) for reactive gases in RBCs by spatially averaging the equations over the volume of the RBC. For the sake of simplicity, we first derive Θ for O_{2}, and (because the fundamental reaction and transport mechanisms are same for all reactive gases), we generalize it (using appropriate modifications) for other reactive gases such as CO, NO, and CO_{2}. The expression for Θ is generalized to RBCs of different shapes, and the effects of size and shape of the RBC, thickness of the unstirred layer, and hematocrit on Θ are also analyzed.

The model equations for this transport process of O_{2} have been summarized in a review article by Groebe and Thews (22), in which they pointed out the effects of facilitated diffusion of O_{2} due to coupling between the balances of O_{2} and hemoglobin through the reaction term. “Facilitated diffusion of oxygen in the presence of hemoglobin” (37) under unsteady-state conditions and nonequilibrium state had earlier been studied in greater detail by Kreuzer and Hoofd (36, 37). However, numerical calculations (39, 48, 51) revealed that the reaction between O_{2} and hemoglobin attains chemical equilibrium, and the use of the O_{2} dissociation curve to relate the concentration of hemoglobin and bound O_{2} in the RBC is quantitatively accurate. In the prior literature, there are several successful modeling attempts made at quantifying the O_{2} uptake in the RBC. It must be mentioned that most of these papers have solved the coupled CDR equations in the RBC numerically, using finite-difference or finite-element schemes (6, 21, 38, 39, 48, 53, 56). An earlier attempt at solving the problem analytically (34) proved to be ineffective, because the problem was oversimplified by completely ignoring the coupling between diffusion of O_{2} and hemoglobin in the RBC. It must be noted that an analytical solution per se of the full problem is not possible because of its nonlinear nature.

### Glossary

*b*(=*H*/2)- Average half thickness of the RBC
*D*- Average diameter of the RBC
*D*_{rbc,j}- Diffusion coefficient of gas
*j*in the RBC^{1} *D*_{bl,j}- Diffusion coefficient of gas
*j*in the unstirred plasma layer^{1} *D*_{Hb}- Diffusion coefficient of hemoglobin in RBC
- Dl
- Overall diffusing capacity of the lung
- D
_{M} - Diffusing capacity of the membrane separating the alveolar air from the blood
- h
- Hematocrit
- [Hb]
_{T} - Total intraerythrocytic hemoglobin (free + bound) concentration
- M
_{j} - Relative affinity of hemoglobin for reactive gas
*j*compared with O_{2} - P
_{rbc,j} - Partial pressure of physically dissolved reactive gas
*j*in the RBC^{1} - P
_{pl,j} - Partial pressure of physically dissolved reactive gas
*j*in the plasma^{1} - S
_{j} - Fractional saturation of the reactive gas
*j*in hemoglobin^{1} - (
*S*/V)_{rbc} - Surface area-to-volume ratio of the RBC
- Sh
_{i} - Dimensionless internal mass transfer coefficient of the RBC
- Vc
- Pulmonary blood volume
- α
_{E,j} - Solubility of gas
*j*in the RBC^{1} - α
_{bl,j} - Solubility of gas
*j*in the unstirred plasma layer^{1} - β
_{j} - Slope of the dissociation curve of the reactive gas
*j*^{1} - δ
- Thickness of the unstirred plasma layer surrounding the RBC
- η
- Mass transfer coefficient in the unstirred plasma layer
- Θ
_{t,j} - Diffusing capacity of the RBC for the reactive gas
*j*per unit blood volume^{1}

## MATHEMATICAL MODEL

Figure 1 shows the schematic of O_{2} uptake in a pulmonary capillary of radius *a* and length *L*. Transport of O_{2} from the alveolus (where the partial pressure of O_{2} is denoted by Pa_{O2}) to the RBC occurs across the membranes (alveolar epithelium and capillary endothelium) and the plasma. As shown in Fig. 1, the human RBC is a biconcave disc, which alters shape and orientation as it convects along blood vessels of different diameters. Average dimensions of this discoid are 8 μm in diameter and 1.6 μm in thickness, and Fig. 2 shows the discoid (or finite cylinder) model of an RBC of diameter *D* and thickness (or height) *H*.

### Governing Equations

The diffusion-reaction equations in Lagrangian coordinates for a single RBC of any arbitrary geometry and volume Ω, and external surface area ∂Ω are given by (7) (8) where P_{rbc} is the partial pressure of physically dissolved O_{2} in the RBC, S is the (fractional) hemoglobin O_{2} saturation, ∇^{2} is the three-dimensional Laplacian in the local coordinate in the RBC, *D*_{rbc} and *D*_{Hb} are the diffusion coefficients of O_{2} and hemoglobin inside the RBC, R(P_{rbc},S) is the net rate of conversion of physically dissolved O_{2} to hemoglobin bound O_{2} by chemical reaction [see appendix a for details on kinetics of reaction between O_{2} and hemoglobin and an expression for R(P_{rbc},S)], [Hb]_{T} is the total intraerythrocytic hemoglobin (free + bound), and α_{E} is the solubility of O_{2} in the RBC. *Equations 7* and *8* are subject to the initial condition given by (9) (10) and the boundary conditions given by (12) and (14) where Pv̄_{O2} is the mixed venous partial pressure of O_{2} and 〈P_{pl}〉 is the spatially averaged partial pressure of dissolved O_{2} in the plasma. It has been shown (34, 39) that a thin unstirred plasma layer is formed around the surface of the RBC and retards O_{2} transfer from the plasma, and η in *Eq. 12* is the mass transfer coefficient in the unstirred layer that quantifies the transfer resistance between the RBC and the plasma. An expression for η in terms of measurable variables could be derived by solving the coupled CDR equation of the RBC with the unstirred (boundary) layer surrounding it (please see appendix b for derivation) and is obtained as (15) where δ is the thickness of the boundary layer, α_{bl} is the solubility of dissolved O_{2} in the stagnant layer, and *D*_{bl} is the diffusion coefficient of O_{2} in the layer.

Although it is possible to solve *Eqs. 7*–*13* numerically (36) using the reaction kinetics shown in appendix a, we use the dissociation relation between O_{2} and hemoglobin (appendix a shows the dissociation relation used in this paper), which is expressed as (16) and the slope of the dissociation curve is given by (17) This approach has the dual advantage of eliminating one of the variables (P_{rbc} or S) from the model equations (*Eqs. 7*–*13*), while including the special case of infinitely fast reaction (which cannot be obtained by solving *Eqs. 7*–*13* numerically).

Eliminating R(P_{rbc},S) from *Eqs. 7* and *8*, we obtain (18) Calculating (∂^{2}S/∂P^{2}) by using the O_{2} dissociation, we find that ∂^{2}S/∂Pis very small in the normal range of P_{rbc} values. For example, ∂^{2}S/∂P= 7.6 × 10^{−4} at P_{rbc} = 40 Torr and decreases monotonically to −2.5 × 10^{−5} at P_{rbc} = 100 Torr. Thus *D*_{Hb} ∂^{2}S/∂P∼ 10^{−11} cm^{2}·s^{−1}·Torr^{−2} in human lung, as a result of which the second term in the right-hand side of *Eq. 18* ≪ than the first. *Equation 18* could therefore be simplified to (19) with initial and boundary conditions being given by *Eqs. 9*, *11*, and *12*, where De is given by (20) We are interested in deriving an analytical expression of the O_{2} diffusing capacity of the RBC in terms of the average partial pressure of O_{2} in the RBC, 〈P_{rbc}〉 (rather than the detailed profile of P_{rbc}), and would also like to obtain a generic solution that is applicable to RBCs of different shapes. To this end, instead of solving *Eq. 19* numerically, we average it spatially over the volume of the RBC. We skip the details of the spatial averaging procedure in the body of the paper and have presented them in appendix c.

The spatially averaged form of *Eq. 19* in Eulerian coordinates is given by (21) with the initial condition where *v*_{rbc} is the velocity of the RBC, *x* is the axial coordinate along the length of the capillary, (*S*/V)_{rbc} is the surface area-to-volume ratio of the RBC, and Sh_{i} is the internal Sherwood number (or dimensionless internal mass transfer coefficient) of the RBC, which has been tabulated for different RBC shapes and geometries in Table 1. An analytical expression for Sh_{i} for a discoid-shaped RBC as a function of its ratio of height (*H*) to diameter (*D*) has been given in *Eq. 124* in appendix c. Figure 3 shows the variation of Sh_{i} with aspect ratio *H*/*D* for a finite cylinder or discoid (shown in Fig. 2). Normal RBCs with average dimensions of 8 μm in diameter and 1.6 μm in thickness or height have *H*/*D* = 0.2 and therefore (from *Eq. 124*) have Sh_{i} = 2.

### Analytical Expression for O_{2}-Diffusing Capacity of RBC

The total O_{2} carried by the RBC (per unit time per unit volume) is the summation of the dissolved free O_{2} in the RBC and the O_{2} carried in bound form (i.e., HbO_{2}). Thus the convective derivative may be expressed as (22) where β = dS/d〈P_{rbc}〉. Equating *Eqs. 21* and *22*, the equation for total O_{2} transport from plasma to a single RBC is obtained as (23) where Θ is the diffusing capacity of a single RBC, which is given by (24) As mentioned earlier, the diffusing capacity Θ includes the effect of diffusional gradients within the RBC [through the term (1/Sh_{i}D_{rbc})(V/*S*)_{rbc} in the denominator of *Eq. 24*], diffusional resistance in the stagnant plasma layer (through the term δ/D_{bl} in the denominator of *Eq. 24*), and the effect of facilitated transport of O_{2} due to reactive coupling with hemoglobin {through the term (D_{Hb}/D_{rbc})·[Hb]_{T}·(β/α_{E})}.

The flux of O_{2} from the plasma to a RBC per unit volume of blood, N_{rbc}, is given by (25) where (26) and h is the hematocrit. Thus we obtain the diffusing capacity of the RBC per unit volume of blood (Θ_{t}) as (27) *Equation 27* represents an analytical expression for the diffusing capacity of the RBC of any shape (quantified by the dimensionless internal mass transfer coefficient of the RBC, Sh_{i}), volume V, surface area *S*, surrounded by an unstirred plasma layer of thickness δ, where α_{E} and *D*_{rbc} are the solubility and diffusivity of O_{2} in the RBC, respectively; α_{bl} and *D*_{bl} are the O_{2} solubility and diffusivity in the unstirred layer, respectively; *D*_{Hb} is the diffusion coefficient of hemoglobin in RBC; [Hb]_{T} is the total hemoglobin concentration in the RBC; and β is the slope of the O_{2} dissociation curve.

In *Eq. 27*, δ/*D*_{bl} quantifies the external mass transfer resistance of the RBC, whereas (V/*S*)_{rbc}/(Sh_{i}*D*_{rbc}) is a measure of the internal mass transfer resistance in the RBC.

Although derived using the specific example of O_{2}, the expression for diffusing capacity of the RBC (*Eq. 27*) is generic and could be used for all reactive gases in the transport-limited regime (i.e., when reaction equilibrium is rapidly achieved). The effects of simultaneous binding of two (or more) reactive gases to the same ligand or the coupling between the transport of two gases could be captured by appropriately modifying the slope of the dissociation curve of the reactive gas, β. For example, the effect of Pco_{2} or pH on the diffusing capacity of O_{2} (Θ_{t,}_{O2}) could be easily incorporated by evaluating β in *Eq. 17* by using a dissociation relation in which S = S(P_{rbc},pH), such as the one proposed by Gomez (19). [Please see appendix a for the dissociation relation proposed by Gomez. However, in the calculations that follow, we ignore this effect and use the dissociation relation proposed by Severinghaus (47), which is also given in appendix a]. Similarly, the effect of simultaneous binding of two gases (such as CO and O_{2} or NO and O_{2}) to hemoglobin on Θ_{t} could be captured by using the appropriate expression of β in *Eq. 27*. This has been illustrated in a later section (*Other Reactive Gases*).

## RESULTS AND DISCUSSION

### Effects of Morphological Parameters on Oxygen Diffusing Capacity

In this section, we explore the effect of different parameters such as size and shape of RBCs, thickness of the stagnant plasma layer, and hematocrit on the O_{2} uptake of the RBC, using the expression for Θ_{t} given in *Eq. 27*.

Figure 4 illustrates the effect of facilitated transport on Θ_{t}. Calculations for Fig. 4 were performed using a discoid model, for a typical RBC of volume = 94.1 μm^{3} and surface area = 134.1 μm^{2}, with (V/*S*)_{rbc} = 0.7017 μm and Sh_{i} = 2. As could be seen from the figure, in the presence of facilitated diffusion, the enhancement of O_{2} uptake by the RBC could be as large as 6.3 times (attained at a 〈P_{rbc}〉 = 20 Torr). Groebe and Thews (22) report a maximum enhancement factor of 7 obtained at 〈P_{rbc}〉 = 20 Torr. As shown in Fig. 4, in the absence of facilitated diffusion ([Hb]_{T} = 0), Θ_{t} is independent of the O_{2} partial pressure in the RBC and is given by (28) Figure 5 illustrates the effect of the shape of RBCs on its O_{2} diffusing capacity by comparing the values of Θ_{t} for discoidal and spherical shapes of RBCs of the same volume and the flat plate model with thickness equivalent to the average thickness of a standard RBC. For the discoid model, calculations were performed for a typical RBC of volume = 94.1 μm^{3} and surface area = 134.1 μm^{2}, with (V/*S*)_{rbc} = 0.7017 μm and Sh_{i} = 2. For the spherical model, the volume of the RBC was taken as 94.1 μm^{3} and Sh_{i} = 5/3. For the flat plate model, the thickness of the RBC, 2*b*, was taken as 1.6 μm and Sh_{i} = 3. For all cases, the thickness of the stagnant layer, δ, was taken (39) as 0.75 μm. Comparison of results shows that the discoidal shape of the RBC is close to optimal design as far as O_{2} uptake is concerned, whereas the spherical shape is the least efficient one. The reason for this observation could be attributed to the fact that, for a given volume of an RBC, the discoid shape provides the maximum surface area per unit volume, whereas the spherical shape provides the minimum. Therefore, nonmammalian RBCs that are ellipsoidal in shape have lower O_{2} diffusing capacity than mammalian ones (which are biconcave discs). It could also be noted from Fig. 5 that the difference between the discoidal and flat plate models is negligible, thus validating previous attempts (6, 38, 39) of modeling the normal RBC as a thin flat plate or sheet.

Figure 6 shows the effect of the thickness of the stagnant layer (δ) on Θ_{t} for a discoidal model of a RBC of (V/*S*)_{rbc} = 0.7017 μm. As could be guessed intuitively, a thicker stagnant layer would result in enhanced external mass transfer resistance, thus decreasing the value of Θ_{t}. As could be observed from Fig. 6, the thickness of the stagnant layer affects Θ_{t} more at lower O_{2} partial pressures than at higher ones, because the effect of facilitated transport decreases steadily as O_{2} tension in the RBC increases. In the limit of δ → 0 (i.e., when there is no stagnant plasma layer surrounding the RBC), the external mass transfer resistance to O_{2} uptake vanishes and Θ_{t} (from *Eq. 27*) is given by (29) For a discoidal RBC with a (V/*S*)_{rbc} = 0.7017 μm and δ = 0.75 μm, the ratio of internal to external resistance is approximately equal to 2:3. Therefore, as shown in Fig. 6, decreasing the stagnant layer thickness from 0.75 μm to 0 increases Θ_{t} by a factor of 2.5.

Figure 7 explores the effect of the thickness on the RBC on Θ_{t} by using the flat plate model for δ = 0.75 μm. Although a 25% increase in red cell thickness (over its normal value of 1.6 μm) reduces its O_{2} diffusing capacity by (a maximum of) 25%, a 25% reduction in thickness increases Θ_{t} by 44%. Therefore, for a given red cell volume and hematocrit, thinner cells have significantly higher O_{2} diffusing capacity than normal ones.

Figure 8 illustrates the effect of anemia on the O_{2} uptake capacity of the RBC. Because Θ_{t} (in *Eq. 27*) is proportional to the hematocrit h, its value decreases linearly with decreasing hematocrit, as shown in Fig. 8. The nature of the curves in Fig. 8 suggests that O_{2} uptake by the RBC is further reduced if anemia (or low hematocrit) is accompanied by low 〈P_{rbc}〉, caused by ventilation-perfusion heterogeneities or under pathophysiological conditions.

### Experimental Verification of Model Predictions

In this section, we compare our model predictions of Θ_{t} with two sets of experimental results existing in the literature.

One of the earliest experimental measurements of Θ_{t} was done by Staub et al. (49). Staub and coworkers measured the O_{2} uptake using the classical Hartridge-Roughton continuous-flow rapid-reactions apparatus. The internal diameter of the observation tube used by them is 10 mm, whereas the inner diameter of a typical pulmonary capillary is around 10 μm. As a result, thick boundary layers are formed around the RBC in the observation tube, leading to large external mass transfer resistances that reduce the O_{2} diffusing capacity of the RBC considerably. Figure 9 compares the values of Θ_{t} obtained using *Eq. 27* for the discoidal model with the experimental data of Staub et al. (In Fig. 9, open circles represent Staub et al.'s data and the dotted line represents model predictions for δ = 2 μm). As could be noted from the figure, the theoretical curve for Θ_{t} for a boundary layer thickness of δ = 2 μm corresponds closely to the experimental data of Staub et al., suggesting the presence of large plasma boundary layers around the RBC in their experimental measurements.

In the 70 years that have followed the development of the Hartridge-Roughton technique, the rapid-mixing technique has undergone technical improvements. Despite these advances, uncertainty exists about the measurements made with this technique primarily because of the thickness of the unstirred plasma boundary layer surrounding the RBC. Several authors (6, 29, 32, 52, 57) have reported values of unstirred layer thickness that vary between 1 and 15 μm, and Weibel (55) pointed out that the mean (harmonic) thickness of the unstirred layer in pulmonary capillaries is 0.5 μm. Thus the values of Θ_{t} measured by rapid-mixing technique are disguised by mass-transfer limitations in the boundary layer and are therefore significantly lower than those that might be present in the pulmonary capillaries. Realizing this, Heidelberger and Reeves (25) used a planar monolayer of whole blood sandwiched between two Gore-Tex membranes to study O_{2} uptake while varying the partial pressure of O_{2} from 0 to 104 Torr. Figure 9 compares values of Θ_{t} based on the experimental measurements of Heidelberger and Reeves (26) with the model predictions for an unstirred layer thickness of 0.5 μm, as suggested by Weibel. In Fig. 9, squares represent Heidelberger and Reeves’ data and the solid line represents the model predictions for δ = 0.5 μm. As could be noted from the figure, both qualitative and quantitative agreement is obtained between theoretical and experimental values of Θ_{t}.

### Other Reactive Gases

Although derived using the specific example of O_{2}, the expression for diffusing capacity of the RBC (*Eq. 27*) is generic and applicable to other reactive gases with appropriate modification. In this section, we obtain Θ_{t} for other reactive gases of practical interest, such as CO, CO_{2}, and NO.

#### Carbon monoxide.

The overall kinetics for reaction between a reactive gas *X* (e.g., O_{2}, NO, CO) and hemoglobin in the RBC is given by (30) and (31) where *k*′_{X} and *k*_{X} are the forward (or association) and reverse (or dissociation) rates, respectively, and the equilibrium constant *K*_{X} = *k*′_{X}/*k*_{X}. Table 2 reproduces the values of *k*′_{X}, *k*_{X}, and *K*_{X} for O_{2}, CO, and NO binding to high-affinity form (R state) of human deoxyHBA, from Olson et al. (42). It could be noted from Table 2 whereas the association rate constants for O_{2} and NO are of the same order of magnitude, that of CO is only 10% that of O_{2} (*k*′_{CO}/*k*′_{O2} = 0.09). On the other hand, the dissociation rate constants of both CO and NO are very small compared with O_{2} (*k*_{CO} = 4 × 10^{−4} *k*_{O2}, *k*_{NO} = 1.5 × 10^{−6} *k*_{O2}), as a result of which both CO and NO have much higher affinity for hemoglobin than O_{2} (M_{CO} = 234, M_{NO} = 6.25 × 10^{5}). This implies that although the overall reaction for CO and NO with hemoglobin can practically be considered as irreversible, the rate of binding of CO to hemoglobin is slow because *k*′_{CO} is small. As pointed out by Olson et al. (42), the rate-limiting step for CO binding is internal bond formation with heme iron. Also, as observed by Johnson et al. (33), “because of the marked differences in reaction velocities between CO and NO with hemoglobin, measurements of Dl_{CO} and Dl_{NO} provide uniquely different and complementary information about the pulmonary capillary bed.”

It was first recognized by Gibson and Roughton (17) and Roughton and Forster (46) that, at low CO tension, the uptake of CO by the RBC is purely limited by the rate of binding of CO to oxyhemoglobin. This was later also verified by the experiments of Reeves and Park (44) in which they measured spectrophotometrically the simultaneous rates of uptake of CO and O_{2} in intact RBCs contained in whole blood thin films (created by spreading <1 μl of whole blood between two Gore-Tex membranes) that minimize extracellular diffusion barriers due to unstirred layers, which were estimated to be between 0.1 and 0.5 μm. They applied a simultaneous step change of Po_{2} from 40 → 100 and Pco from 0 → 2.1 Torr and observed that the combination of deoxyhemoglobin with O_{2} is largely completed within the first 40 ms before significant CO uptake occurs. On the basis of their measurements of unsteady-state uptake of CO in the RBC, Reeves and Park observed that “in vivo, in normoxia and hyperoxia, red cell CO uptake rate is wholly reaction rate limited and that pulmonary capillary red cell CO diffusion equilibrium is rapidly achieved.”

However, it was observed by Reeves and Park (44) that as “Po_{2} falls from 100 toward zero there is a marked increase in diffusing capacity of CO (Θ_{CO}) from 1 to 2.2 ml·ml^{−1}·min^{−1}·Torr^{−1},” due to “CO binding to unliganded heme sites rather than the CO for oxyhemoglobin replacement reaction that occurs at higher O_{2} pressures.” Thus, under hypoxic conditions, both CO and O_{2} bind to reduced hemoglobin through competitive-parallel reactions. In addition, when Pco (i.e., CO concentration) is high, the rate of CO binding to deoxyhemoglobin, which is given proportional to Pco, also increases, shifting the CO uptake process from a reaction- (or kinetic-) limited regime to a diffusion-limited regime.

Thus the uptake process of CO by hemoglobin occurs in two separate regimes, namely reaction (or kinetically) limited and diffusion limited, depending on the values of Pco and Po_{2}. Because the physics of the uptake process in these two regimes are completely different, we derive expressions for Θ_{CO} in the two regimes separately.

##### Reaction-limited regime.

In the reaction-controlled regime, the binding of CO to hemoglobin is essentially a slow irreversible replacement reaction between oxyhemoglobin and CO, the kinetics of which has been modeled by Gibson and Roughton (17) as where and the overall reaction rate of CO binding to oxyhemoglobin (R_{COHb}) was obtained by Gibson and Roughton (17) as (32) where *k*_{4} was measured by Gibson (15) in human blood at a pH of 7.1 and a temperature of 19°C as 26 s^{−1} with activation energy of 18.4 kcal, and *k*′_{4}/*l*′_{4} as 3, the ratio being independent of temperature. As calculated by Reeves and Park (44) using the data of Gibson, *k*_{4} for human hemoglobin at 37°C is 157 s^{−1}.

As in the case of O_{2} binding to hemoglobin, the diffusion-reaction equations in Lagrangian coordinates for binding of CO to oxyhemoglobin in a single RBC is given by (33) (34) where P_{rbc,CO} is the partial pressure of free O_{2} in the RBC, [COHb] is the concentration of carboxyhemoglobin (COHb) in the RBC, *D*_{rbc,CO} and *D*_{Hb} are the diffusion coefficients of CO and COHb inside the RBC, R_{COHb} is the net rate of binding of free CO to oxyhemoglobin, and α_{E,CO} is the solubility of CO in the RBC. *Equations 33*–*34* are subject to the initial conditions (for the case of a single-breath measurement of CO diffusing capacity) given by (35) and the boundary conditions given by (36) (37) where η = (α_{bl,CO}*D*_{bl,CO})/δ (please see appendix b for derivation).

Applying the spatial averaging procedure outlined in appendix c, we average *Eqs. 33*–*37* over the volume of a single RBC to obtain the spatially averaged equations in Eulerian coordinates as (38) (39) where the symbols P_{rbc,CO}, [COHb], and R_{COHb} now represent volume-averaged quantities.

Experimental observations of Reeves and Park (44) have shown that the uptake of CO in the RBC in low-Pco conditions is purely reaction limited, with diffusion equilibrium being rapidly achieved. Here, we verify this observation using a time-scale analysis. For the case of a single-breath test of CO diffusing capacity, Pco = 2.1 Torr, and the characteristic reaction time, the characteristic internal diffusion time (t_{D}_{i}_{,CO}), and the characteristic external diffusion time (t_{D}_{e}_{,CO}) (for a unstirred layer thickness of 0.5 μm), Therefore, the ratio of total diffusion time to reaction time, ℜ, is given by Because ℜ ≪ 1 (or, in other words, total diffusion time ≪ reaction time) for the conditions under which a single-breath diffusing capacity test of CO is performed, it is evident that diffusion equilibrium is attained very rapidly. Thus, under such low CO concentrations (in normoxia and hyperoxia), the uptake of CO is not limited by intra- or extracellular diffusion in the RBC but by the rate of reactive binding between CO and oxyhemoglobin. Therefore, under conditions of diffusion equilibrium and such low values of Pco, internal as well as external diffusional gradients of CO in the RBC are negligible, and it is reasonable to assume (40) where Pco is the partial pressure of inspired CO in the single-breath CO diffusing capacity test. Using the above simplifying assumption (*Eq. 40*), we ignore *Eq. 38* and evaluate the diffusing capacity of CO using *Eq. 39* alone at constant values of Pco and Po_{2}. Because (41) *Eq. 39* could be written as (42) with the initial condition being given by (43) Integrating *Eqs. 42*–*43* at constant values of Pco and Po_{2}, we obtain (44) where (45) Using *Eqs. 42*, *44*, and *45*, the time-dependent uptake rate of CO is given by (46) By definition, Θ_{CO} for a single RBC is given by (47) i.e., (48) and the diffusing capacity of CO per unit volume of blood, Θ_{t,CO}, is obtained by simply multiplying Θ_{CO} by the hematocrit h and is given by (49) Figure 10 shows the variation of Θ_{t,CO} with time by plotting *Eq. 49* for a 10-s-long single-breath test of CO diffusing capacity, where Pco = 2.1 Torr. Plots of Θ_{t,CO} for different values Po_{2} are shown in Fig. 10. As could be seen from the figure, Θ_{t,CO} is maximum at *t* = 0 and decreases exponentially with increasing time. Figure 10 also shows that Θ_{t,CO}|_{t=0} decreases significantly as Po_{2} increases, where Θ_{t,CO}|_{t=0} is obtained from *Eq. 49* as (50) The time-averaged value of Θ_{t,CO} obtained over a (10-s) single-breath measurement of CO diffusing capacity is given by where *t*_{test} is the duration of the CO diffusing capacity measurement.

Based on experimental measurements, several empirical expressions for evaluation of Θ_{t,CO} are available in the literature (13, 28, 44, 46), which have been tabulated in Table 3. Figure 11 compares the analytical expression of Θ̄_{t,CO} obtained above (*Eq. 51*) as well the initial value of diffusing capacity of CO (Θ_{t,CO}|_{t=0}), given by *Eq. 50*, with the four expressions given in Table 3. In evaluation of Θ_{t,CO}|_{t=0} using *Eq. 50* and Θ̄_{t,CO}|_{t=0} using *Eq. 51*, the following values have been used corresponding to a single-breath test of CO diffusing capacity: Pco = 2.1 Torr, *t*_{test} = 10 s, *k*′_{4}/*l*′_{4} = 3, as obtained by Gibson and Roughton (17), and *k*_{4} = 157 s^{−1} at 37°C, as calculated by Reeves and Park (44) using Gibson and Roughton's data. As could be seen from Fig. 11, the analytical expressions of Θ̄_{t,CO} and Θ_{t,CO}|_{t=0}, given by *Eqs. 51* and *50*, respectively, agree very well, both qualitatively and quantitatively, with the experimental measurements available in the literature. Needless to mention that in the kinetically controlled regime, Θ_{t,CO} decreases as Po_{2} increases, a feature that, although not obvious from *Eq. 51*, is evident from Fig. 11.

It is interesting to observe that, in the range of Po_{2} < 300 Torr, the experimental results of Roughton and Forster (46), Forster (13), and Holland (28) (*curves B*, *C*, and *D*, respectively, in Fig. 11) are bounded by the *curves F* and *E* of Θ_{t,CO}|_{t=0} and Θ̄_{t,CO}, given by *Eqs. 50* and *51*, respectively. Although *curves B*–*D* in Fig. 11 are based on measurements of initial rates of reaction between CO and oxyhemoglobin, the measured values of Θ_{t,CO} are lower than the analytically obtained one, Θ_{t,CO}|_{t=0} (*curve F*). This could be attributed to the presence of unstirred layers of finite thickness in the experimental measurements, whereas the analytical expression (*Eq. 50*) has been obtained for the ideal case in which no unstirred layer is present. It may also be noted from Fig. 10 that Θ_{t,CO} decreases exponentially with increasing time, and, therefore, the initial value of Θ_{t,CO} (Θ_{t,CO}|_{t=0}) is invariably higher than the time-averaged value of Θ_{t,CO} (Θ̄_{t,CO}). For Po_{2} > 300 Torr, this decrease is slow, as a result of which Θ_{t,CO} ≈ Θ̄_{t,CO}.

##### Diffusion-limited regime.

As discussed above, under hypoxic conditions, both CO and O_{2} bind to deoxyhemoglobin simultaneously, and both the reactions are parallel, competitive, and diffusion limited. To obtain an expression for Θ_{t,CO} in this regime, we use the Haldane relations (23) to express the relation between COHb and Pco and the dependence on O_{2}, under equilibrium conditions, which are given by (52) (53) where So_{2} and Sco are the fractional saturation of O_{2} and CO, respectively; Po_{2} and Pco are the average partial pressures of O_{2} and CO, respectively; M_{CO} is the relative affinity of hemoglobin for CO compared with O_{2} (given in Table 2); and the function H is the same function as the usual dissociation curve for O_{2} expressed as So_{2} = H(P_{rbc}) as in *Eq. 88*. Using *Eqs. 52*–*53*, we obtain (54) In the diffusion-limited regime, the reactions of CO and O_{2} with deoxyhemoglobin attain equilibrium, and, therefore, the treatment of this case is similar to that of O_{2}. The Θ_{t,CO} in this regime is given (as in the case of Θ_{t,}_{O2}) by (56) where β_{CO}, the slope of the dissociation curve, given by *Eq. 55*, captures the effect that the binding of O_{2} with deoxyhemoglobin exerts on CO binding.

Table 4 compares the values of Θ_{t,CO} (in ml·ml^{−1}·min^{−1}·Torr^{−1}) obtained by using *Eq. 56* for an unstirred layer thickness of 0.5 μm with the experimental measurements of Θ_{t,CO} by Reeves and Park (44) in the diffusion-limited regime (i.e., hypoxic with high Pco). Agreement is found to be good, and deviations between experimental results and model predictions are within the bounds of experimental errors. It must be stated that, in this mass-transfer-limited regime, Θ_{t,CO} (like Θ_{t,}_{O2}) is sensitive to the unstirred layer thickness.

It must be mentioned that a quantitative analysis of simultaneous binding of O_{2} and CO to deoxyhemoglobin was performed by Nicolson and Roughton (41) in 1950. The major difference between the cited work and the present one is that whereas the former is numerical calculation of increase in carboxyhemoglobin saturation in the RBC under reaction nonequilibrium conditions (i.e., in the reaction-limited regime), *Eq. 56* presented above provides an analytical expression for diffusing capacity of the RBC under reaction equilibrium conditions (i.e., in the diffusion-limited regime), when the Haldane relations hold good (*Eqs. 52*–*53*). It may be argued that CO and O_{2} bind to deoxyhemoglobin simultaneously only when CO and O_{2} tensions are comparable, in which case the uptake occurs in the diffusion-limited regime (as discussed above). Moreover, although Nicolson and Roughton's work was restricted to a slab geometry of the RBC, the present approach accommodates different red cell shapes and is applicable to other reacting gases such as NO and CO_{2}, as we shall illustrate in the following sections.

#### Nitric oxide.

Table 2 presents the values of *k*′_{NO}, *k*_{NO}, and M_{NO} for NO binding to high-affinity form (R state) of human deoxyHBA, from Olson et al. (42). M_{NO}, the relative affinity of hemoglobin for NO compared with O_{2} is approximately equal to 6.25 × 10^{5}, whereas the rate of forward reaction *k*′_{NO} is of the same order of magnitude as that O_{2}. Thus it could be concluded that (like O_{2}) the binding of NO to deoxyhemoglobin is fast and not reaction limited but diffusion limited, whereas, because of its high affinity toward hemoglobin (i.e., because M_{NO} ≪ 1), the binding of NO to hemoglobin (unlike O_{2}) is assumed to be irreversible. Thus, for the case of simultaneous binding of O_{2} and NO to deoxyhemoglobin, both the reactions are parallel and diffusion limited. We use the Haldane relations (23) to express the relation between HbNO and NO partial pressure (Pno) and the dependence on O_{2}, under equilibrium conditions, which are given by (57) (58) where *Eq. 58* is a modified form of *Eq. 53* that accounts for the high reactivity and affinity of NO with hemoglobin. Simplifying *Eqs. 57*–*58*, we obtain the fraction saturation of NO as (59) and β_{NO}, the slope of the NO dissociation curve as (60) Perillo et al. (43) used chemiluminescent techniques to measure pulmonary diffusing capacity of NO. They used an initial inspiration of 5–10 ppm of NO and the average value of Pno (steady-state partial pressure of NO in the alveoli) measured in seven normal human subjects was 1.8 × 10^{−6} Torr. At such typically used low values of Pno, β_{NO} could be simplified to (61) (62) and because the uptake of NO due to reactive binding far exceeds that in the dissolved form, Θ_{t,NO} could also be simplified to (63) (64) for a discoidal RBC (i.e., Sh_{i} = 2) with δ = 0.75 μm and (V/*S*)_{rbc} = 0.7017 μm, or, in other words, (65) indicating that, for highly reactive gases like NO, the diffusional resistance offered by the RBC, i.e., (Θ_{t,NO})^{−1}, is negligible, and the total resistance offered by the capillary, Dl_{NO}^{−1}, is purely due to the resistance provided by the alveolar capillary membrane.

In the limit of vanishingly small stagnant layer thickness (i.e., δ → 0), *Eq. 63* is given by (66) and for Po_{2} of 100 Torr, Θ_{t,NO} for a discoidal RBC (i.e., Sh_{i} = 2), with (V/*S*)_{rbc} = 0.7017 μm, is given by

It is also interesting to analyze the effect of O_{2} partial pressure on Θ_{t,NO}, which as given by *Eq. 63* is expected to decrease linearly as Po_{2} increases. As shown in *Eq. 65*, at Po_{2} of 100 Torr (i.e., normal human breathing in room air, inspired O_{2} fraction = 21%), Θ = 0.61 × 10^{−6} min/Torr. When pure O_{2} is used, i.e., inspired O_{2} fraction is increased to 100%, Θincreases to 4.4 × 10^{−6} min/Torr, which is practically ≃ 0. Therefore, it might be concluded that, for most cases of practical interest, Θ→ 0 and remains almost unaffected by the O_{2} partial pressure.

#### Carbon dioxide.

It must be mentioned that though the expression for Θ_{t}, (i.e., *Eq. 27*) has been derived for gases that bind with hemoglobin, it could easily be used with slight modifications to obtain the diffusing capacity of gases that react with other red cell constituents (instead of hemoglobin). In this section, we illustrate how to evaluate Θ_{t} by using a modified form of *Eq. 27* for such gases by using the example of CO_{2}.

CO_{2} is present in the blood in three different forms, namely, physically dissolved in plasma and RBC, bicarbonate ions (HCO_{3}^{−}) in plasma and RBC, and carbamino compounds (R-NHCOO^{−}) in RBC only. Apart from its transport in dissolved form, CO_{2} in the RBC is transported primarily as bicarbonate ions (i.e., [R-NHCOO^{−}] ≪ [HCO_{3}^{−}]), which are formed as a result of hydration of CO_{2} catalyzed by the enzyme carbonic anhydrase (CA). The hydration reaction (68) is almost instantaneous owing to the presence of CA activity, which is very high within the RBC (acceleration factor A_{CA} ∼ 6,000–10,000).

Because the rate of the above reaction primarily determines the rate of CO_{2} transport in blood, we evaluate β_{CO2} (and therefore Θ_{t,CO2}) on the basis of this reaction.

When a slightly modified form of *Eq. 27* is used, the diffusing capacity of the RBC for CO_{2} (Θ_{t,}_{CO2}) is given by (69) where α_{E,CO2} is the physical solubility of CO_{2} in the RBC, and (70) where [HCO_{3}^{−}]_{RBC} is the content of CO_{2} in bicarbonate form in the RBC. Assuming [R-NHCOO^{−}] ≪ [HCO_{3}^{−}], the total CO_{2} present in the blood as HCO_{3}^{−} is given by (71) The total amount of bicarbonate ions in the blood ([HCO_{3}^{−}]_{total} given by *Eq. 71*) is partitioned between the two phases, namely the RBC and the plasma, and the partition coefficient, which is obtained by using the principle of equality of electrochemical potentials in each phase (Gibbs-Donnan equilibrium), is given by (10) (72) where [HCO_{3}^{−}]_{RBC} and [HCO_{3}^{−}]_{plasma} are the contents of bicarbonate ions in the RBC and the plasma, respectively; pH_{p} is the pH in the plasma; and S is the fractional O_{2} saturation in the RBC. Using *Eqs. 71* and *72*, we obtain (73) Using *Eqs. 70* and *73*, we obtain β_{CO2} (at any fixed O_{2} saturation) as (74) and using *Eqs. 69* and *74*, we obtain Θ_{t,}_{CO2} as (75) We use the data on [CO_{2}]_{total} vs. Pco_{2} given by Comroe et al. (7) at pH_{p} = 7.4, and S = 0.7 and S = 0.975, respectively, to calculate ∂[CO_{2}]_{total}/∂Pco_{2}. Using these data, Θ_{t,}_{CO2} for a discoid-shaped RBC with (V/*S*)_{rbc} = 0.7017 μm and boundary layer thickness δ = 0.75 μm, is obtained as (76) (77) where Θ_{t,}_{CO2} is in ml·ml^{−1}·min^{−1}·Torr^{−1} and Pco_{2} is in Torr.

Figure 12 presents the plots of Θ_{t,}_{CO2} vs. Pco_{2} for 70% and 97.5% O_{2} saturation, using *Eqs. 76* and *77*, respectively. It could be noticed that, unlike in the case of O_{2}, Θ_{t,}_{CO2} decreases monotonically with increasing CO_{2} tension, and the effect of O_{2} saturation per se on CO_{2} diffusing capacity is negligible. In vivo, however, the binding of O_{2} to reduced hemoglobin in pulmonary capillaries releases Bohr protons (H^{+}), and this tends to shift *Eq. 68* to the left and alter the estimate of Θ_{t,}_{CO2}. As could be noted from the figure, the magnitude of Θ_{t,}_{CO2} is up to 20 times larger than that of O_{2} or CO, a range of values that is accepted in the existing literature. Experimental measurements of CO_{2} diffusing capacity in isolated dog lungs by Enns and Hill (9) show that the CO_{2} diffusing capacity is 5–15 times greater than that of CO or O_{2}.

### Limiting Cases

On the basis of the above analysis, we summarize the limiting cases in the uptake of any reactive gas by the RBC by isolating the dominant physical phenomena.

#### Facilitated transport.

As shown above, for highly reactive gases like NO, the uptake process is primarily transport limited, with the rate of facilitated transport being at least one order of magnitude higher than that of physical diffusion. In this limit, Θ_{t} is given by (78) *Equation 78* gives the effective reaction velocity in the RBC in the absence of any external mass-transfer limitation (or disguise).

For O_{2}, the above limit could be attained during exercise, i.e., at lower values of 〈P_{rbc}〉 (as is evident from Fig. 4), when facilitated transport dominates the uptake process. As is shown in Fig. 4, as 〈P_{rbc}〉 increases facilitated transport decreases considerably, and for 〈P_{rbc}〉 > 80 Torr, β (and therefore facilitated diffusion) decreases linearly with increasing 〈P_{rbc}〉.

#### Pure diffusion.

In the limit of high O_{2} tension, i.e., 〈P_{rbc}〉 > 200 Torr, facilitated transport is negligible and the uptake process is purely through diffusion of physically dissolved O_{2}. In this limit, Θ_{t} is given by (79)

#### External mass transfer.

If there is no stagnant plasma layer surrounding the RBC, external mass transfer resistance is absent. In this limit of δ → 0, Θ_{t} is given by (80) As shown in the case of O_{2} and NO, for a discoidal RBC of δ = 0.75 μm and (V/*S*)_{rbc} = 0.7017 μm, the ratio of internal to external mass transfer resistance is ∼2:3. Therefore, decreasing the stagnant layer thickness from 0.75 μm to 0 increases Θ_{t} by a factor of 2.5.

On the other hand, if the surrounding plasma is completely unstirred resulting in a large boundary layer around the RBC (as in the case of measurements done with a stopped-flow apparatus), then external mass transfer resistance ≪ internal mass transfer resistance and the uptake process is limited by the external mass transfer resistance due to the boundary layer. In this case, Θ_{t} is inversely proportional to the boundary layer thickness δ and independent of the internal mass transfer coefficient, Sh_{i}, and is given by (81)

#### Internal mass transfer.

If internal mass transfer resistance ≫ external mass transfer resistance, or in other words (82) Θ_{t} attains the same limit as given by *Eq. 80*. On the other hand, if internal mass transfer resistance ≪ external mass transfer resistance (i.e., the inequality sign in *Eq. 82* is reversed), Θ_{t} attains the limit given by *Eq. 81*.

### Mass Transfer Disguise of Reaction Velocity in the RBC

As defined earlier, the diffusing capacity of the RBC (Θ_{t}) is the mass transfer-disguised effective rate of reaction between a reactive gas and hemoglobin in the RBC. The rate of uptake of most reactive gases is determined not by the intrinsic reaction rate of the gas with one (or more) of the RBC constituents but by the mass transfer-disguised effective reaction rate, Θ_{t}. In other words, most gases of practical interest such as O_{2}, CO_{2}, NO, etc., react with RBC constituents fast enough so that the overall uptake process is mass transfer (diffusion and perfusion) limited. In terms of time scales, this implies that the time scales for internal and external mass transfer in the RBC are greater than that of reaction.

As illustrated in this paper, Θ_{t} is influenced significantly by both internal and external mass transfer resistances as well as by its affinity with hemoglobin and its reactivity (which determines the slope of the dissociation curve, β). The in vivo thickness of the unstirred plasma layer surrounding the RBC (δ), which determines the external mass transfer resistance of the RBC, has been estimated to vary between 0.5 and 1 μm. The classical Hartridge-Roughton continuous-flow apparatus measures Θ_{t} in vitro in the presence of artificially created large (1–15 μm) boundary layers. As is evident from the present analysis, for gases that are in the transport-limited regime (such as O_{2}, CO_{2}, NO), the artificially created boundary layers could underpredict Θ_{t} by severalfold, whereas for less reactive gases that are in the reaction-controlled regime (such as CO), unstirred layers have negligible effect on Θ_{t}. In other words, the more reactive the gas, more sensitive it is to external mass transfer limitations (i.e., to unstirred layer thickness) and, therefore, the larger is the experimental error. Heidelberger and Reeves (25) and Reeves and Park (44) circumvented this problem by measuring Θ_{t,}_{O2} and Θ_{t,CO}, respectively, of a planar monolayer of whole blood sandwiched between Gore-Tex membranes. As has been shown in Fig. 9, the theoretical expression for Θ_{t} (*Eq. 27*) could be used to determine the unstirred layer thickness in the in vitro experiments, which is otherwise difficult to measure.

### Limitations of the Model

The present model has the following limitations.

We have ignored the effects of pulsatility of capillary blood flow on the boundary layer thickness (δ) surrounding the RBC, which, in turn, affects the diffusing capacity. The effects of pulsatile nature of capillary blood flow and volume on gas exchange have been discussed in detail by Bidani et al. (3).

We have assumed the Pco to be constant while deriving the analytical expression for diffusing capacity of CO in the reaction-controlled regime.

Synergetic effects of O_{2} and CO_{2} binding concurrently to hemoglobin have been neglected.

The binding reaction between the gas and hemoglobin (H_{2}O in case of CO_{2}) has been assumed to be fast enough so that the time scale required to attain chemical equilibrium is much smaller than the diffusion time scale in the RBC, or, in other words, the uptake occurs in the diffusion/transport-limited regime. This assumption is valid for most reactive gases of practical interest (except CO and CO_{2} under certain conditions, as shown in Table 4). The hydration of CO_{2} in the RBC is catalyzed by CA activity and is therefore assumed to be instantaneous. Despite the general validity of this assumption, it has been shown by Bidani (2) that the CA activity is vulnerable to inhibition. In such cases, the slope of CO_{2} dissociation curve would depend on the acceleration factor (A_{CA}) of the catalyst, and β_{CO2} (in *Eq. 74*) would be a function of A_{CA}, along with pH_{p} and S.

CO_{2} transport, in vivo, is somewhat more complex because of mass transport (HCO_{3}^{−} and Cl^{−} exchange across the red cell membrane), different buffer capacities for plasma and RBC, and availability of the CA activity to plasma (2).

Our analysis on NO does not include recent controversies (20, 35, 40) of alternate sites or forms of NO binding to hemoglobin.

## CONCLUSIONS

In this paper, we have derived an analytical expression for the diffusing capacity of the RBC for reactive gases. We start with the fundamental CDR equations that describe the transport and reaction rate processes of the reactive gas and hemoglobin in the RBC. We apply a spatial averaging procedure to the CDR equations to average them over the volume of the RBC and obtain simplified low-dimensional models that describe the net uptake of the gas in the RBC in terms of diffusing capacity (Θ_{t}). As an outcome of the averaging process, an analytical expression of Θ_{t} (*Eq. 27*) is obtained in terms of size and shape of RBC, thickness of the stagnant plasma layer surrounding it, diffusivities and solubilities of the gas in the RBC and unstirred plasma layer, diffusivity and concentration of hemoglobin in the RBC, and hematocrit as well the O_{2} dissociation curve.

Using *Eq. 27*, we explore the effects of the above-mentioned parameters on Θ_{t} for O_{2} and therefore on the rate of O_{2} uptake in the RBC. Calculations of Θ_{t,}_{O2} for different shapes of RBCs show that the discoidal shape of the RBC is close to optimal design as far as O_{2} uptake is concerned, whereas the spherical shape is the least efficient one. *Equation 27* is useful to study the effects of deformation of the RBC on its O_{2} uptake capacity, especially under pathophysiological conditions, like sickle cell anemia or diminished caliber of the microcirculation. We quantify the decrease in Θ_{t,}_{O2} caused by increase in the thickness of the surrounding stagnant plasma layer and find that the thickness of the stagnant layer affects Θ_{t,}_{O2} more at lower O_{2} partial pressures than at higher ones. As far as the thickness of the RBC is concerned, we find that, for a given RBC volume and hematocrit, thinner cells have significantly higher O_{2} diffusing capacity than normal RBCs. Using *Eq. 27*, we also analyze the effects of anemia on Θ_{t,}_{O2}. The diffusing capacity of the RBC is found to decrease linearly with decreasing hematocrit and is reduced further if anemia or low hematocrit is accompanied by low 〈P_{rbc}〉, caused by ventilation-perfusion heterogeneities or under pathophysiological conditions.

Comparisons of model predictions (for different values of unstirred layer thickness, δ) have been made with the experimental data of Staub et al. (49) and Heidelberger and Reeves (25, 26). Although Staub et al.'s data agree qualitatively with our theoretical predictions for an unstirred layer thickness (δ) of 2 μm, Heidelberger and Reeves’ data give good qualitative as well as quantitative agreement with model predictions for δ = 0.5 μm. It has been shown that low values of Θ_{t} measured in vitro by using continuous-flow apparatus for rapid reactions are likely due to artificially created large unstirred layers surrounding the RBC. The uptake of most reactive gases such as O_{2}, CO_{2}, and NO (and also CO under certain conditions) occurs in the mass transfer limited regime, in which Θ_{t} is particularly sensitive to the unstirred layer thickness.

The analytical expression of Θ_{t} (given by *Eq. 27*) is valid for all reactive gases for which the uptake occurs in the mass-transfer-limited regime and has been applied to obtain values of Θ_{t} for CO (under hypoxic conditions), CO_{2}, and NO. The Θ_{t} values for CO agree very well with experimental data available in the literature, and Θ_{t} values for CO_{2} are found to be larger than that of O_{2} by one order of magnitude, whereas Θ_{t} for NO are ∼10^{6}, validating traditional views that the resistance provided by the RBC in case of NO binding is negligible and (Θ_{t}|_{NO})^{−1} ≅ 0.

Analysis of diffusing capacity of CO led us to the conclusion that at low values of Pco (as in measurements of a single-breath test of CO diffusing capacity) under normoxic or hyperoxic condition, the uptake process is reaction controlled, whereas in the other limit of high Pco and hypoxic condition it is diffusion controlled. In fact, it can be shown that such transition from a reaction-limited regime to a diffusion- (or transport-) limited regime also occurs for other reactive gases like O_{2} and CO_{2}, depending on the conditions present in the RBC. On the basis of the above analysis, we identify the controlling regimes of the different reactive gases (O_{2}, CO, CO_{2}, NO) in the RBC and tabulate them in Table 5. It is hoped that better understanding the transition of the uptake process of different reactive gases at the level of RBCs from one controlling regime to another would improve our ability to quantify the overall pulmonary gas uptake.

## APPENDIX A

The overall rate kinetics for reaction between O_{2} and hemoglobin in the RBC is given by (83) and R(P_{rbc},S) in *Eq. 7* could be written as (84) (85) where S = [HbO_{2}]/([Hb]+[HbO_{2}]) = [HbO_{2}]/[Hb]_{T}. Experimentally, *k*′ has been measured (16) as (86) (87) where N is the O_{2} capacity of 1 ml RBCs, and We use the dissociation relation proposed by Severinghaus (47), given by (88) where S is the fractional saturation of O_{2} in the RBC, which is related to the concentration of O_{2} in RBC (C_{rbc}) by (89) and P_{rbc} (in Torr in *Eqs. 88*–*89*) is the partial pressure of dissolved O_{2} in the RBC.

To include the effect of plasma pH on Θ_{t,}_{O2}, the O_{2} saturation relation proposed by Gomez (19) could be used instead of *Eq. 88*, which is given by (90) where (91) (92) S is the O_{2} saturation (fractional) of hemoglobin, P_{rbc} is O_{2} partial pressure in the RBC (in Torr), T is the temperature (in °C) and pH_{p} is the plasma pH.

Parameter values used in this paper have been taken from Groebe and Thews (21):

Total hemoglobin concentration in the RBC [Hb]_{T} = 2.03 × 10^{−2} (mol/l)

Solubility of O_{2} in RBC α_{E} = 1.56 × 10^{−6} (mol·l^{−1}·Torr^{−1})

Solubility of O_{2} in stagnant layer α_{bl} = 1.3 × 10^{−6} (mol·l^{−1}·Torr^{−1})

Diffusion coefficient of hemoglobin in the RBC *D*_{Hb} = 1.44 × 10^{−7} (cm^{2}/s)

Diffusion coefficient of O_{2} in RBC *D*_{rbc} = 9.5 × 10^{−6} (cm^{2}/s)

Diffusion coefficient of O_{2} in stagnant layer *D*_{bl} = 1.3 × 10^{−5} (cm^{2}/s)

Diffusion coefficient of NO in RBC *D*_{NO} = 3.3 × 10^{−5} (cm^{2}/s)

α_{O2}/α_{CO} = 1.33

## APPENDIX B

### Derivation of Mass Transfer Coefficient in the Unstirred Plasma Layer

Here, we illustrate the derivation of the external mass transfer coefficient in the plasma boundary layer surrounding the RBC. Figure 13 shows the schematic of a RBC surrounded by a stagnant plasma layer. The steady-state mass balance equation for the above system is given by (93) where P_{rbc} is the partial pressure of O_{2} in RBC phase, P_{bl} is the partial pressure of O_{2} in the boundary layer, *k* is the pseudo-first-order rate constant, and *D*_{rbc} and *D*_{bl} are the diffusion coefficients of O_{2} in the RBC and boundary layers, respectively. Here *b* is the half thickness of the RBC, δ is the thickness of the unstirred layer, and α is the solubility of O_{2} in the RBC boundary layer. The solution of *Eq. 93* is given by (94) (95) where (96) (97) and the flux is given by (98) Therefore, the mass transfer coefficient is given by (99) i.e., (100) The mass transfer coefficient remains practically the same, even in the presence of nonlinear kinetics.

## APPENDIX C

### Spatial Averaging of Convection-Diffusion-Reaction Equation in the RBC

The simplified diffusion-reaction equation (in Lagrangian coordinates) for dissolved O_{2} and HbO_{2} concentrations in the RBC is given by (101) with initial and boundary conditions being given by *Eqs. 9*, *11*, and *12*, where De is given by (102) We first illustrate the averaging process for an infinite slab geometry and then generalize it for other shapes of RBCs. *Equations 9*, *11*, *12*, and *19* for the case of a thin disc of thickness 2*b* are written in terms of dimensionless coordinates and numbers as (103) with (104) (105) (106) where , and Pa_{O2} is the alveolar partial pressure of O_{2}. Here, *Bi* is the ratio of external mass transfer resistance in the stagnant layer to the internal mass transfer resistance in the RBC and is given by (108) *Bi* depends on the thickness of the unstirred layer (δ) and that of the RBC (*b*) and the diffusivity of O_{2} in it (D_{bl}). If *Bi* → 0 (either as η → 0 or as *b* → 0), i.e., if the diffusion time scale in the RBC (*t*_{D-RBC}) ≪ the time scale for mass transfer across the unstirred layer (*t*_{trans-bl}), there exist no gradients inside the RBC, local diffusional equilibrium exists and p_{rbc} = 〈p_{rbc}〉 everywhere inside the RBC. Mathematically speaking, for *Bi* = 0, *Eqs. 103*–*106* have a zero eigenvalue with a constant eigenfunction. For *Bi* > 0 (i.e., *Bi* is small but finite), diffusional gradients exist inside the RBC, and we express the dimensionless partial pressure of O_{2} in the RBC (which now depends both on the transverse coordinates and time) as (109) where 〈p_{rbc}〉 is the spatially averaged partial pressure of O_{2} in the RBC and p′_{rbc} is a fluctuation about this average such that it satisfies the solvability criterion (110) The fluctuation p′_{rbc}(ξ,*t*) is solved by substituting *Eq. 109* in *Eq. 103*, which gives (111) with boundary conditions (112) (113) *Equation 103* on being averaged (or integrated) over the thickness (2*b*) of the RBC gives (114) where and p_{rbc,s} = p_{rbc}(ξ = 1,*t*), which could be obtained by evaluating the local fluctuation p′_{rbc} by solving *Eqs. 110*–*113* and using the relation (115) This is achieved by subtracting *Eq. 114* from *Eq. 111* and using *Eq. 115*, which gives (116) *Equation 116* is solved along with *Eqs. 110*, *112*, and *113* by using perturbation expansion of the fluctuation as (117) *Equation 114* is called the “global equation,” *Eq. 116* is called the “local equation,” and the averaging technique illustrated above by using the simple example of diffusion and reaction in a flat plate is called “spatial averaging by the Liapunov-Schmidt method.” This averaging technique can be applied to any diffusion-convection-reaction or diffusion-reaction system in which diffusion is dominant at the small scale and at diffusion-equilibrium the system has a zero eigenvalue with a constant eigenfunction. In such cases, the above method can be used to eliminate spatial degrees of freedom near zero eigenvalues by averaging over the small scale and yet retain all the important physics of the small scale in the averaged equations. Further details of spatial averaging by Liapunov-Schmidt method could be obtained from related publications (1, 5).

The spatially averaged equations (following the solution of *Eq. 116*) for the above example are given by (118) (119) where 〈p_{rbc}〉 and 〈p_{pl}〉 are the spatially averaged partial pressures of O_{2} in the RBC and the plasma, respectively, and p_{rbc,s} is the O_{2} partial pressure in the RBC membrane. The averaged equations (*Eqs. 118* and *119*) could be written as a single equation by eliminating the partial pressure of O_{2} at the red cell membrane, P_{rbc,s}, and the single averaged equation is given in dimensional form by (120) where 〈P_{rbc}〉 and 〈P_{pl}〉 are the (dimensional) average partial pressures of O_{2} in the RBC and the plasma, respectively, and other symbols retain their usual meanings. As is obvious, the averaged equations (*Eqs. 118* and *119*) retain information about the size and shape of the RBC, and the thickness of the unstirred layer in terms of *Bi* and Sh_{i}, as well as internal and external diffusional gradients of the RBC in terms of differences between the different O_{2} partial pressures, namely, 〈P_{rbc}〉, 〈P_{rbc,s}〉, and 〈P_{pl}〉.

Although the above formulation has been derived in Lagrangian (or moving) frame of reference, it could easily be transformed into Eulerian (or fixed) coordinates by replacing the total derivative d/d*t* in *Eq. 120* by the substantial derivative D/D*t* (=d/d*t* + *v*d/d*x*), and the averaged equation for the RBC is given by (121) (122) with the initial condition where *v*_{rbc} is the velocity of the RBC and *x* is the axial coordinate along the length of the capillary. The advantage of an Eulerian description is that the slip between the RBC and the plasma could easily be accounted for by allowing the velocity of RBC (*v*_{rbc}) to be different from that of the plasma (*v*_{pl}).

It must be mentioned that the spatial averaging procedure illustrated above is independent of the shape and/or geometry of the RBC, and the averaged equations (*Eqs. 118* and *119*) remain the same irrespective of the shape of the RBC. The shape of the RBC is accounted for in the averaged models through the (dimensionless) internal mass transfer coefficient, Sh_{i} (in *Eq. 119*), which is obtained for different shapes of the RBC by inverting the Laplacian ∇^{2}P_{rbc} in *Eq. 7* for different geometries, or, in other words, by solving the following local equation instead of *Eq. 116*: (123) Table 1 shows the shape factors (Sh_{i}) for different shapes of RBCs, obtained by solving *Eq. 123* for different geometries. For a finite cylinder of height *H* and diameter *D* (as shown in Fig. 2), the internal mass transfer coefficient is given as a function of *H*/*D* by (from unpublished notes of V. Balakotaiah) (124) where (125) (126) and *I*_{0} and *I*_{1} are modified Bessel functions. Figure 3 uses *Eq. 124* to plot the variation of Sh_{i} with *H*/D. As shown in the figure, in the limit of *H*/D → 0, from *Eq. 124*, Sh_{i} → 3, which is the internal mass transfer coefficient for a flat plate, as shown in Table 1. In the other limit of *H*/D → ∞, Sh_{i} → 2, which is the internal mass transfer coefficient for an infinite cylinder. Normal RBCs with average dimensions of 8 μm in diameter and 1.6 μm in thickness or height have *H*/D = 0.2 and therefore (from Fig. 3) have Sh_{i} = 2.004.

The averaged model (*Eq. 122*) for an RBC of any shape is given by (127) with the initial condition where (*S*/V)_{rbc} is the surface area-to-volume ratio of a single RBC, Sh_{i} is the dimensionless internal mass transfer coefficient of the RBC (listed in Table 1 for different geometries), and (128) (129) (130) (131) In dimensional form, *Eq. 127* is given by (132)

## GRANTS

The work of V. Balakotaiah and S. Chakraborty was supported by the University of Houston through a Moores Professorship. The work of A. Bidani and S. Chakraborty was supported by National Heart, Lung, and Blood Institute Grant HL-51421.

,

## Acknowledgments

We thank the reviewers for helpful comments.

## Footnotes

↵1 The absence of the subscript

*j*implies*j*= O_{2}The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

- Copyright © 2004 the American Physiological Society