HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

This Article Abstract Full Text (PDF) Free All Versions of this Article: 97/6/2284    most recent 00469.2004v1 Submit a response Alert me when this article is cited Alert me when eLetters are posted Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Web of Science (8) Citing Articles via Google Scholar Google Scholar Articles by Chakraborty, S. Articles by Bidani, A. Search for Related Content PubMed PubMed Citation Articles by Chakraborty, S. Articles by Bidani, A.

Diffusing capacity reexamined: relative roles of diffusion and chemical reaction in red cell uptake of O₂, CO, CO₂, and NO

¹Pulmonary, Critical Care and Sleep Medicine, Department of Internal Medicine, The University of Texas Medical School, Houston 77030, and ²Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4004

Submitted 4 May 2004 ; accepted in final form 16 August 2004

	ABSTRACT

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

 
This paper presents an analytical expression for the diffusing capacity (_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 _t has been derived by spatial averaging of the fundamental convection-diffusion-reaction equation for O₂ in the RBC and has been generalized to all cell shapes and for other reactive gases such as CO, NO, and CO₂. The effects of size and shape of the RBC, thickness of the unstirred plasma layer, hemoglobin concentration, and hematocrit on _t have been analyzed, and the analytically obtained expression for _t has been validated by comparison with different sets of existing experimental data for O₂ and CO₂. 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 O₂ 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 _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 _t predicts that _t,NO > _t,CO2 > _t,O2 > _t,CO.

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

The term 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/Vc) 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 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 _ht and _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 Vc), Hughes and Bates (31) summarized the experimental efforts for evaluating (for O₂ 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 O₂ and hemoglobin in the RBC using a continuous-flow rapid-reaction apparatus and suggested an empirical formula for diffusing capacity of O₂ (_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 O₂ saturation of the RBC, and /760 is the solubility of O₂ in milliliters per milliliter of blood per Torr. Because k_c and S are functions of the partial pressure of O₂ in the RBC (P_rbc), given by Eq. 5 is also a function of P_rbc. Using the data of Staub et al. (49) on k_c, the O₂ dissociation curve, and Eq. 5, Flummerfelt and Crandall (11) obtained the following empirical relation to quantify 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 () for reactive gases in RBCs by spatially averaging the equations over the volume of the RBC. For the sake of simplicity, we first derive for O₂, 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 CO₂. The expression for 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 are also analyzed.

The model equations for this transport process of O₂ have been summarized in a review article by Groebe and Thews (22), in which they pointed out the effects of facilitated diffusion of O₂ due to coupling between the balances of O₂ 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 O₂ and hemoglobin attains chemical equilibrium, and the use of the O₂ dissociation curve to relate the concentration of hemoglobin and bound O₂ in the RBC is quantitatively accurate. In the prior literature, there are several successful modeling attempts made at quantifying the O₂ 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 O₂ 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 RBC¹

D_bl,j

Diffusion coefficient of gas j in the unstirred plasma layer¹

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 O₂

P_rbc,j

Partial pressure of physically dissolved reactive gas j in the RBC¹

P_pl,j

Partial pressure of physically dissolved reactive gas j in the plasma¹

S_j

Fractional saturation of the reactive gas j in hemoglobin¹

(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

_E,j

Solubility of gas j in the RBC¹

_bl,j

Solubility of gas j in the unstirred plasma layer¹

_j

Slope of the dissociation curve of the reactive gas j¹

Thickness of the unstirred plasma layer surrounding the RBC

Mass transfer coefficient in the unstirred plasma layer

_t,j

Diffusing capacity of the RBC for the reactive gas j per unit blood volume¹

	MATHEMATICAL MODEL

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

 
Figure 1 shows the schematic of O₂ uptake in a pulmonary capillary of radius a and length L. Transport of O₂ from the alveolus (where the partial pressure of O₂ 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.

View larger version (72K): [in this window] [in a new window]   Fig. 1. Schematic showing transport of O2 from alveolus to red blood cell (RBC) across alveolar membrane and plasma. PAO2, alveolar partial pressure of O2; a and L, radius and length, respectively, of pulmonary capillary.

View larger version (10K): [in this window] [in a new window]   Fig. 2. Discoid model of the red blood cell. b, Average half thickness; H, height; D, diameter.

The diffusion-reaction equations in Lagrangian coordinates for a single RBC of any arbitrary geometry and volume , and external surface area are given by

(7)

(8)

where P_rbc is the partial pressure of physically dissolved O₂ in the RBC, S is the (fractional) hemoglobin O₂ saturation, ² is the three-dimensional Laplacian in the local coordinate in the RBC, D_rbc and D_Hb are the diffusion coefficients of O₂ and hemoglobin inside the RBC, R(P_rbc,S) is the net rate of conversion of physically dissolved O₂ to hemoglobin bound O₂ by chemical reaction [see APPENDIX A for details on kinetics of reaction between O₂ and hemoglobin and an expression for R(P_rbc,S)], [Hb]_T is the total intraerythrocytic hemoglobin (free + bound), and _E is the solubility of O₂ in the RBC. Equations 7 and 8 are subject to the initial condition given by

(9)

(10)

and the boundary conditions given by

(11)

(12)

and

(13)

(14)

where P_O2 is the mixed venous partial pressure of O₂ and P_pl is the spatially averaged partial pressure of dissolved O₂ 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 O₂ transfer from the plasma, and 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 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 is the thickness of the boundary layer, _bl is the solubility of dissolved O₂ in the stagnant layer, and D_bl is the diffusion coefficient of O₂ 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 O₂ 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 (²S/P²) by using the O₂ dissociation, we find that ²S/Pis very small in the normal range of P_rbc values. For example, ²S/P= 7.6 x 10^–4 at P_rbc = 40 Torr and decreases monotonically to –2.5 x 10^–5 at P_rbc = 100 Torr. Thus D_Hb ²S/P 10^–11 cm²·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 O₂ diffusing capacity of the RBC in terms of the average partial pressure of O₂ 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.

View larger version (12K): [in this window] [in a new window]   Fig. 3. Variation of (dimensionless) internal mass transfer coefficient, Shi, of a finite cylinder of height H and diameter D, with the ratio H/D.

The total O₂ carried by the RBC (per unit time per unit volume) is the summation of the dissolved free O₂ in the RBC and the O₂ carried in bound form (i.e., HbO₂). Thus the convective derivative may be expressed as

(22)

where = dS/dP_rbc. Equating Eqs. 21 and 22, the equation for total O₂ transport from plasma to a single RBC is obtained as

(23)

where is the diffusing capacity of a single RBC, which is given by

(24)

As mentioned earlier, the diffusing capacity includes the effect of diffusional gradients within the RBC [through the term (1/Sh_iD_rbc)(V/S)_rbc in the denominator of Eq. 24], diffusional resistance in the stagnant plasma layer (through the term /D_bl in the denominator of Eq. 24), and the effect of facilitated transport of O₂ due to reactive coupling with hemoglobin {through the term (D_Hb/D_rbc)·[Hb]_T·(/_E)}.

The flux of O₂ 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 (_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 , where _E and D_rbc are the solubility and diffusivity of O₂ in the RBC, respectively; _bl and D_bl are the O₂ 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 is the slope of the O₂ dissociation curve.

In Eq. 27, /D_bl quantifies the external mass transfer resistance of the RBC, whereas (V/S)_rbc/(Sh_iD_rbc) is a measure of the internal mass transfer resistance in the RBC.

Although derived using the specific example of O₂, 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, . For example, the effect of PCO₂ or pH on the diffusing capacity of O₂ (_t,O2) could be easily incorporated by evaluating 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 O₂ or NO and O₂) to hemoglobin on _t could be captured by using the appropriate expression of in Eq. 27. This has been illustrated in a later section (Other Reactive Gases).

	RESULTS AND DISCUSSION

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 O₂ uptake of the RBC, using the expression for _t given in Eq. 27.

Figure 4 illustrates the effect of facilitated transport on _t. Calculations for Fig. 4 were performed using a discoid model, for a typical RBC of volume = 94.1 µm³ and surface area = 134.1 µm², 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 O₂ 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), _t is independent of the O₂ partial pressure in the RBC and is given by

(28)

Figure 5 illustrates the effect of the shape of RBCs on its O₂ diffusing capacity by comparing the values of _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 µm³ and surface area = 134.1 µm², 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 µm³ and Sh_i = 5/3. For the flat plate model, the thickness of the RBC, 2b, was taken as 1.6 µm and Sh_i = 3. For all cases, the thickness of the stagnant layer, , 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 O₂ 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 O₂ 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.

View larger version (29K): [in this window] [in a new window]   Fig. 4. Effect of total hemoglobin (monomer) concentration ([Hb]T) and facilitated diffusion of O2 on diffusing capacity (t).

View larger version (22K): [in this window] [in a new window]   Fig. 5. Effect of the shape of the RBC on its O2 diffusing capacity.

(29)

For a discoidal RBC with a (V/S)_rbc = 0.7017 µm and = 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 _t by a factor of 2.5.

View larger version (25K): [in this window] [in a new window]   Fig. 6. Effect of the thickness () of the stagnant plasma layer on the O2 diffusing capacity of the red blood cell.

View larger version (24K): [in this window] [in a new window]   Fig. 7. Effect of the thickness of the red blood cell on its O2 diffusing capacity.

View larger version (25K): [in this window] [in a new window]   Fig. 8. Effect of hematocrit (h) on the O2 diffusing capacity of the red blood cell.

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

One of the earliest experimental measurements of _t was done by Staub et al. (49). Staub and coworkers measured the O₂ 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 O₂ diffusing capacity of the RBC considerably. Figure 9 compares the values of _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 = 2 µm). As could be noted from the figure, the theoretical curve for _t for a boundary layer thickness of = 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.

View larger version (24K): [in this window] [in a new window]   Fig. 9. Comparison of model predictions of red blood cell O2 diffusing capacity (t,O2) with experimental data of Staub et al. (49) () and Heidelberger and Reeves (25) ().

Other Reactive Gases

Although derived using the specific example of O₂, 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 _t for other reactive gases of practical interest, such as CO, CO₂, and NO.

Carbon monoxide.   The overall kinetics for reaction between a reactive gas X (e.g., O₂, 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 O₂, 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 O₂ and NO are of the same order of magnitude, that of CO is only 10% that of O₂ (k'_CO/k'_O2 = 0.09). On the other hand, the dissociation rate constants of both CO and NO are very small compared with O₂ (k_CO = 4 x 10^–4 k_O2, k_NO = 1.5 x 10^–6 k_O2), as a result of which both CO and NO have much higher affinity for hemoglobin than O₂ (M_CO = 234, M_NO = 6.25 x 10⁵). 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."

However, it was observed by Reeves and Park (44) that as "PO₂ falls from 100 toward zero there is a marked increase in diffusing capacity of CO (_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 O₂ pressures." Thus, under hypoxic conditions, both CO and O₂ 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 PO₂. Because the physics of the uptake process in these two regimes are completely different, we derive expressions for _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 k₄ 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, k₄ for human hemoglobin at 37°C is 157 s^–1.

As in the case of O₂ 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 O₂ 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 _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 = (_bl,COD_bl,CO)/ (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_DI,CO),

and the characteristic external diffusion time (t_DE,CO) (for a unstirred layer thickness of 0.5 µm),

Therefore, the ratio of total diffusion time to reaction time, , is given by

Because << 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 PO₂. 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 PO₂, we obtain

(44)

where

(45)

Using Eqs. 42, 44, and 45, the time-dependent uptake rate of CO is given by

(46)

By definition, _CO for a single RBC is given by

(47)

i.e.,

(48)

and the diffusing capacity of CO per unit volume of blood, _t,CO, is obtained by simply multiplying _CO by the hematocrit h and is given by

(49)

Figure 10 shows the variation of _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 _t,CO for different values PO₂ are shown in Fig. 10. As could be seen from the figure, _t,CO is maximum at t = 0 and decreases exponentially with increasing time. Figure 10 also shows that _t,CO|_t=0 decreases significantly as PO₂ increases, where _t,CO|_t=0 is obtained from Eq. 49 as

(50)

The time-averaged value of _t,CO obtained over a (10-s) single-breath measurement of CO diffusing capacity is given by

(51)

where t_test is the duration of the CO diffusing capacity measurement.

View larger version (19K): [in this window] [in a new window]   Fig. 10. Variation of diffusing capacity of CO (t,CO) (given by Eq. 49) with time during a 10-s-long single-breath test of CO diffusing capacity, for different values of PO2.

View larger version (29K): [in this window] [in a new window]   Fig. 11. Variation of t,CO with PO2 at partial pressure of CO (PCO) = 2.1 Torr. (Curves A–D refer to empirical expressions of t,CO available in the literature obtained by using experimental data, which have been listed in Table 3. Curves F and E represent values from the present study based on initial rate and average rate over 10 s, respectively.)

Diffusion-limited regime.   As discussed above, under hypoxic conditions, both CO and O₂ bind to deoxyhemoglobin simultaneously, and both the reactions are parallel, competitive, and diffusion limited. To obtain an expression for _t,CO in this regime, we use the Haldane relations (23) to express the relation between COHb and PCO and the dependence on O₂, under equilibrium conditions, which are given by

(52)

(53)

where SO₂ and SCO are the fractional saturation of O₂ and CO, respectively; PO₂ and PCO are the average partial pressures of O₂ and CO, respectively; M_CO is the relative affinity of hemoglobin for CO compared with O₂ (given in Table 2); and the function H is the same function as the usual dissociation curve for O₂ expressed as SO₂ = H(P_rbc) as in Eq. 88. Using Eqs. 52–53, we obtain

(54)

(55)

In the diffusion-limited regime, the reactions of CO and O₂ with deoxyhemoglobin attain equilibrium, and, therefore, the treatment of this case is similar to that of O₂. The _t,CO in this regime is given (as in the case of _t,O2) by

(56)

where _CO, the slope of the dissociation curve, given by Eq. 55, captures the effect that the binding of O₂ with deoxyhemoglobin exerts on CO binding.

Table 4 compares the values of _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 _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, _t,CO (like _t,O2) is sensitive to the unstirred layer thickness.

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 O₂ is approximately equal to 6.25 x 10⁵, whereas the rate of forward reaction k'_NO is of the same order of magnitude as that O₂. Thus it could be concluded that (like O₂) 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 O₂) is assumed to be irreversible. Thus, for the case of simultaneous binding of O₂ 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 O₂, 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 _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 x 10^–6 Torr. At such typically used low values of PNO, _NO could be simplified to

(61)

(62)

and because the uptake of NO due to reactive binding far exceeds that in the dissolved form, _t,NO could also be simplified to

(63)

(64)

for a discoidal RBC (i.e., Sh_i = 2) with = 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., (_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., 0), Eq. 63 is given by

(66)

and for PO₂ of 100 Torr, _t,NO for a discoidal RBC (i.e., Sh_i = 2), with (V/S)_rbc = 0.7017 µm, is given by

(67)

It is also interesting to analyze the effect of O₂ partial pressure on _t,NO, which as given by Eq. 63 is expected to decrease linearly as PO₂ increases. As shown in Eq. 65, at PO₂ of 100 Torr (i.e., normal human breathing in room air, inspired O₂ fraction = 21%), = 0.61 x 10^–6 min/Torr. When pure O₂ is used, i.e., inspired O₂ fraction is increased to 100%, increases to 4.4 x 10^–6 min/Torr, which is practically 0. Therefore, it might be concluded that, for most cases of practical interest, 0 and remains almost unaffected by the O₂ partial pressure.

Carbon dioxide.   It must be mentioned that though the expression for _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 _t by using a modified form of Eq. 27 for such gases by using the example of CO₂.

CO₂ is present in the blood in three different forms, namely, physically dissolved in plasma and RBC, bicarbonate ions (HCO₃^–) in plasma and RBC, and carbamino compounds (R-NHCOO^–) in RBC only. Apart from its transport in dissolved form, CO₂ in the RBC is transported primarily as bicarbonate ions (i.e., [R-NHCOO^–] << [HCO₃^–]), which are formed as a result of hydration of CO₂ 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 CO₂ transport in blood, we evaluate _CO2 (and therefore _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 CO₂ (_t,CO2) is given by

(69)

where _E,CO2 is the physical solubility of CO₂ in the RBC, and

(70)

where [HCO₃^–]_RBC is the content of CO₂ in bicarbonate form in the RBC. Assuming [R-NHCOO^–] << [HCO₃^–], the total CO₂ present in the blood as HCO₃^– is given by

(71)

The total amount of bicarbonate ions in the blood ([HCO₃^–]_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 [HCO₃^–]_RBC and [HCO₃^–]_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 O₂ saturation in the RBC. Using Eqs. 71 and 72, we obtain

(73)

Using Eqs. 70 and 73, we obtain _CO2 (at any fixed O₂ saturation) as

(74)

and using Eqs. 69 and 74, we obtain _t,CO2 as

(75)

We use the data on [CO₂]_total vs. PCO₂ given by Comroe et al. (7) at pH_p = 7.4, and S = 0.7 and S = 0.975, respectively, to calculate [CO₂]_total/PCO₂. Using these data, _t,CO2 for a discoid-shaped RBC with (V/S)_rbc = 0.7017 µm and boundary layer thickness = 0.75 µm, is obtained as

(76)

(77)

where _t,CO2 is in ml·ml^–1·min^–1·Torr^–1 and PCO₂ is in Torr.

Figure 12 presents the plots of _t,CO2 vs. PCO₂ for 70% and 97.5% O₂ saturation, using Eqs. 76 and 77, respectively. It could be noticed that, unlike in the case of O₂, _t,CO2 decreases monotonically with increasing CO₂ tension, and the effect of O₂ saturation per se on CO₂ diffusing capacity is negligible. In vivo, however, the binding of O₂ 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 _t,CO2. As could be noted from the figure, the magnitude of _t,CO2 is up to 20 times larger than that of O₂ or CO, a range of values that is accepted in the existing literature. Experimental measurements of CO₂ diffusing capacity in isolated dog lungs by Enns and Hill (9) show that the CO₂ diffusing capacity is 5–15 times greater than that of CO or O₂.

View larger version (16K): [in this window] [in a new window]   Fig. 12. Variation of CO2 diffusing capacity of the RBC (t,CO2) with partial pressure of CO2.

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, _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 O₂, 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, (and therefore facilitated diffusion) decreases linearly with increasing P_rbc.

Pure diffusion.   In the limit of high O₂ tension, i.e., P_rbc > 200 Torr, facilitated transport is negligible and the uptake process is purely through diffusion of physically dissolved O₂. In this limit, _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 0, _t is given by

(80)

As shown in the case of O₂ and NO, for a discoidal RBC of = 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 _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, _t is inversely proportional to the boundary layer thickness 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)

_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), _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 (_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, _t. In other words, most gases of practical interest such as O₂, CO₂, 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, _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, ). The in vivo thickness of the unstirred plasma layer surrounding the RBC (), 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 _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 O₂, CO₂, NO), the artificially created boundary layers could underpredict _t by severalfold, whereas for less reactive gases that are in the reaction-controlled regime (such as CO), unstirred layers have negligible effect on _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 _t,O2 and _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 _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 () 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 O₂ and CO₂ binding concurrently to hemoglobin have been neglected.

The binding reaction between the gas and hemoglobin (H₂O in case of CO₂) 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 CO₂ under certain conditions, as shown in Table 4). The hydration of CO₂ 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 CO₂ dissociation curve would depend on the acceleration factor (A_CA) of the catalyst, and _CO2 (in Eq. 74) would be a function of A_CA, along with pH_p and S.

CO₂ transport, in vivo, is somewhat more complex because of mass transport (HCO₃^– 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

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 (_t). As an outcome of the averaging process, an analytical expression of _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 O₂ dissociation curve.

Using Eq. 27, we explore the effects of the above-mentioned parameters on _t for O₂ and therefore on the rate of O₂ uptake in the RBC. Calculations of _t,O2 for different shapes of RBCs show that the discoidal shape of the RBC is close to optimal design as far as O₂ 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 O₂ uptake capacity, especially under pathophysiological conditions, like sickle cell anemia or diminished caliber of the microcirculation. We quantify the decrease in _t,O2 caused by increase in the thickness of the surrounding stagnant plasma layer and find that the thickness of the stagnant layer affects _t,O2 more at lower O₂ 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 O₂ diffusing capacity than normal RBCs. Using Eq. 27, we also analyze the effects of anemia on _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, ) 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 () of 2 µm, Heidelberger and Reeves’ data give good qualitative as well as quantitative agreement with model predictions for = 0.5 µm. It has been shown that low values of _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 O₂, CO₂, and NO (and also CO under certain conditions) occurs in the mass transfer limited regime, in which _t is particularly sensitive to the unstirred layer thickness.

The analytical expression of _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 _t for CO (under hypoxic conditions), CO₂, and NO. The _t values for CO agree very well with experimental data available in the literature, and _t values for CO₂ are found to be larger than that of O₂ by one order of magnitude, whereas _t for NO are 10⁶, validating traditional views that the resistance provided by the RBC in case of NO binding is negligible and (_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 O₂ and CO₂, 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 (O₂, CO, CO₂, 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

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

(83)

and R(P_rbc,S) in Eq. 7 could be written as

(84)

(85)

where S = [HbO₂]/([Hb]+[HbO₂]) = [HbO₂]/[Hb]_T. Experimentally, k' has been measured (16) as

(86)

(87)

where N is the O₂ 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 O₂ in the RBC, which is related to the concentration of O₂ in RBC (C_rbc) by

(89)

and P_rbc (in Torr in Eqs. 88–89) is the partial pressure of dissolved O₂ in the RBC.

To include the effect of plasma pH on _t,O2, the O₂ saturation relation proposed by Gomez (19) could be used instead of Eq. 88, which is given by

(90)

where

(91)

(92)

S is the O₂ saturation (fractional) of hemoglobin, P_rbc is O₂ 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 x 10^–2 (mol/l)

Solubility of O₂ in RBC _E = 1.56 x 10^–6 (mol·l^–1·Torr^–1)

Solubility of O₂ in stagnant layer _bl = 1.3 x 10^–6 (mol·l^–1·Torr^–1)

Diffusion coefficient of hemoglobin in the RBC D_Hb = 1.44 x 10^–7 (cm²/s)

Diffusion coefficient of O₂ in RBC D_rbc = 9.5 x 10^–6 (cm²/s)

Diffusion coefficient of O₂ in stagnant layer D_bl = 1.3 x 10^–5 (cm²/s)

Diffusion coefficient of NO in RBC D_NO = 3.3 x 10^–5 (cm²/s)

_O2/_CO = 1.33

	APPENDIX B

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 O₂ in RBC phase, P_bl is the partial pressure of O₂ in the boundary layer, k is the pseudo-first-order rate constant, and D_rbc and D_bl are the diffusion coefficients of O₂ in the RBC and boundary layers, respectively. Here b is the half thickness of the RBC, is the thickness of the unstirred layer, and is the solubility of O₂ 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.

View larger version (13K): [in this window] [in a new window]   Fig. 13. Stagnant plasma layer around the RBC.

	APPENDIX C

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

The simplified diffusion-reaction equation (in Lagrangian coordinates) for dissolved O₂ and HbO₂ 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 2b are written in terms of dimensionless coordinates and numbers as

(103)

with

(104)

(105)

(106)

where

(107)

, and PA_O2 is the alveolar partial pressure of O₂. 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 () and that of the RBC (b) and the diffusivity of O₂ in it (D_bl). If Bi 0 (either as 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 O₂ 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 O₂ in the RBC and p'_rbc is a fluctuation about this average such that it satisfies the solvability criterion

(110)

The fluctuation p'_rbc(,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 (2b) of the RBC gives

(114)

where and p_rbc,s = p_rbc( = 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 O₂ in the RBC and the plasma, respectively, and p_rbc,s is the O₂ 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 O₂ 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 O₂ 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 O₂ 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/dt in Eq. 120 by the substantial derivative D/Dt (=d/dt + vd/dx), 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 ²P_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 I₀ and I₁ 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

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

 
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

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

 
We thank the reviewers for helpful comments.

	FOOTNOTES

 

Address for reprint requests and other correspondence: V. Balakotaiah, Dept. of Chemical Engineering, University of Houston, Houston, TX 77204 (E-mail: bala{at}uh.edu) or A. Bidani, Dept. of Internal Medicine, Univ. of Texas Medical School, Houston, TX 77030 (E-mail: akhil.bidani{at}uth.tmc.edu)

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.

¹ The absence of the subscript j implies j = O₂

	REFERENCES

TOP ABSTRACT MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C GRANTS ACKNOWLEDGMENTS REFERENCES

 

1. Balakotaiah V and Chakraborty S. Averaging theory and low-dimensional models for chemical reactors and reacting flows. Chem Eng Sci 58: 4769–4786, 2003.
2. Bidani A. Analysis of abnormalities of capillary CO₂ exchange in vivo. J Appl Physiol 70: 1686–1699, 1991.
3. Bidani A, Flummerfelt RW, and Crandall ED. Analysis of the effects of pulsatile capillary blood flow and volume on gas exchange. Respir Physiol 35: 27–42, 1978.
4. Borland CDR and Cox Y. Effect of varying alveolar oxygen partial pressure on diffusing capacity for nitric oxide and carbon monoxide, membrane diffusing capacity and lung capillary volume. Clin Sci (Lond) 81: 759–765, 1991.
5. Chakraborty S and Balakotaiah V. Low dimensional models for describing mixing effects in laminar flow tubular reactors. Chem Eng Sci 57: 2545–2564, 2002.
6. Coin JT and Olson JS. The rate of oxygen uptake by human red blood cells. J Biol Chem 254: 1178–1190, 1979.
7. Comroe JH Jr, Forster RE II, Dubois AB, Briscoe WA, and Carlsen E. The Lung: Clinical Physiology and Pulmonary Function Tests (2nd ed.). Chicago, IL: Year Book Medical, 1962.
8. Crapo JD, Crapo RO, Jensen RL, Mercer RR, and Weibel ER. Evaluation of lung diffusing capacity by physiological and morphometric techniques. J Appl Physiol 64: 2083–2091, 1988.
9. Enns E and Hill EP. CO₂ diffusing capacity in isolated dog lung lobes and the role of carbonic anhydrase. J Appl Physiol 54: 483–490, 1983.
10. Fitzsimons EJ and Sendroy J. Distribution of electrolytes in human blood. J Biol Chem 236: 1595–1601, 1961.
11. Flummerfelt RW and Crandall ED. An analysis of external respiration in man. Math Biosci 3: 205–230, 1968.
12. Forster RE. Diffusion of gases. In: Handbook of Physiology. Respiration. Bethesda, MD: Am. Physiol. Soc., 1964, sect. 3, vol. I, chapt. 33, p. 839–872, 1964.
13. Forster RE. Diffusion of gases across the alveolar membrane. In: Handbook of Physiology. The Respiratory System. Gas Exchange. Bethesda, MD: Am. Physiol. Soc., 1987, sect. 3, vol. IV, chapt. 5, p. 71–88.
14. Forster RE, Fowler WS, and Bates DV. Considerations on the uptake of carbon monoxide by the lungs. J Clin Invest 33: 1128–1134, 1954.
15. Gibson QH. The kinetics of reactions between hemoglobin and gases. Prog Biophys Biophys Chem 9: 1–53, 1959.
16. Gibson QH, Kreuzer F, Meda E, and Roughton FJW. The kinetics of human haemoglobin in solution and in the red cell at 37°C. J Physiol 129: 65–89, 1955.
17. Gibson QH and Roughton FJW. The kinetics of dissociation of the first oxygen molecule from fully saturated oxyhemoglobin in sheep blood solutions. Proc R Soc Lond B Biol Sci 143: 310–344, 1955.
18. Goldstick TK, Ciuryla VT, and Zuckerman L. Diffusion of oxygen in plasma and blood. Adv Exp Med Biol 75: 183–198, 1976.
19. Gomez DM. Consideration of oxygen hemoglobin equilibrium in the physiological state. Am J Physiol 200: 135–142, 1961.
20. Gow AJ and Stamler JS. Reactions between nitric oxide and hemoglobin under physiological conditions. Nature 391: 169–173, 1998.
21. Groebe K and Thews G. Effects of cell spacing and red cell movement upon oxygen release under conditions of maximally working skeletal muscle. Adv Exp Med Biol 248: 175–184, 1989.
22. Groebe K and Thews G. Basic mechanisms of diffusive and diffusion-related oxygen transport in biological systems: a review. In: Oxygen Transport to Tissue, edited by Erdmann W and Bruley JF. New York: Plenum, 1992, vol. 317, p. 21–33.
23. Haldane JBS. The dissociation of oxyhaemoglobin in human blood during partial CO poisoning. J Physiol 45: 22–24, 1912–1913.
24. Hartridge H and Roughton FJW. The velocity with which carbon monoxide displaces oxygen from combination with hemoglobin. Proc R Soc Lond B Biol Sci 94: 336–367, 1923.
25. Heidelberger E and Reeves RB. O₂ transfer kinetics in a whole blood unicellular thin layer. J Appl Physiol 68: 1854–1864, 1990.
26. Heidelberger E and Reeves RB. Factors affecting whole blood O₂ transfer kinetics: implications for (O₂). J Appl Physiol 68: 1865–1874, 1990.
27. Hill EP, Hill JR, Power GG, and Longo LD. Carbon monoxide exchanges between the human fetus and mother: a mathematical model. Am J Physiol Heart Circ Physiol 232: H311–H323, 1977.
28. Holland RAB. Rate at which CO replaces O₂ from O₂Hb in red cells of different species. Respir Physiol 7: 43–63, 1969.
29. Holland RAB, Shibata H, Scheid P, and Piiper J. Kinetics of O₂ uptake and release by red cells in stopped flow apparatus: effects of unstirred layers. Respir Physiol 59: 71–91, 1985.
30. Hsia CCW, McBrayer DG, and Ramanathan M. Critique of conceptual basis of diffusing capacity estimates: a finite element analysis. J Appl Physiol 79: 1039–1047, 1995.
31. Hughes JMB and Bates DV. Historical review: the carbon monoxide diffusing capacity (DL_CO) and its membrane (Dm) and red cell (·Vc) components. Respir Physiol Neurobiol 138: 115–142, 2003.
32. Huxley V and Kutchai H. The effect of red cell membrane and a diffusion boundary layer on the rate of oxygen uptake by human erythrocyte. J Physiol 316: 75–83, 1981.
33. Johnson RL Jr, Heigenhauser GJF, Hsia CCW, Jones NL, and Wagner PD. Determinants of gas exchange and acid-base balance during exercise. Handbook of Physiology. Exercise: Regulation and Integration of Multiple Systems. Bethesda, MD: Am. Physiol. Soc., 1996, sect. 12, chapt. 12, p. 515–584.
34. Kagawa T and Mochizuki M. Numerical solution of partial differential equation describing oxygenation rate of red blood cell. Jpn J Physiol 32: 197–218, 1982.
35. Kim-Shapiro DB. Hemoglobin-nitric oxide cooperativity: is NO the third respiratory ligand? Free Radic Biol Med 36: 402–412, 2004.
36. Kreuzer KH. Facilitated diffusion of oxygen and its possible significance: a review. Respir Physiol 9: 1–30, 1970.
37. Kreuzer KH and Hoofd LJC. Facilitated diffusion of oxygen in the presence of hemoglobin. Respir Physiol 8: 280–302, 1970.
38. Kutchai H. Role of the red cell membrane in oxygen uptake. Respir Physiol 23: 121–132, 1975.
39. Mochizuki M. A theoretical study on the velocity factor of oxygenation of the red cell. Jpn J Physiol 32: 197–218, 1966.
40. Ng ESM and Kubes P. The physiology of S-nitrosothiols: carrier molecules for nitric oxide. Can J Physiol Pharmacol 81: 759–764, 2003.
41. Nicolson P and Roughton FJW. A theoretical study of the influence of diffusion and chemical reaction velocity on the rate of exchange of carbon monoxide and oxygen between the red blood corpuscle and the surrounding fluid. Proc R Soc Lond B Biol Sci 138: 241–264, 1950.
42. Olson JS, Foley EW, Maillet DH, and Paster EV. Measurement of rate constants for reactions of O₂, CO, and NO with hemoglobin. In: Methods in Molecular Medicine. Hemoglobin Disorders: Molecular Methods and Protocols, edited by Nagel RL. Totowa, NJ: Humana, 2003. vol. 82, p. 65–81.
43. Perillo IB, Hyde RW, Olszowka AJ, Pietropaoli AT, Perkins PT, Forster RE II, Utell MJ, and Frampton MW. Chemiluminescent measurements of nitric oxide pulmonary diffusing capacity and alveolar production in humans. J Appl Physiol 91: 1931–1940, 2001.
44. Reeves RB and Park HK. CO uptake kinetics of red cells and CO diffusing capacity. Respir Physiol 88: 1–21, 1992.
45. Roughton FJW. Transport of oxygen and carbon dioxide. In: Handbook of Physiology. Respiration. Bethesda, MD: Am. Physiol. Soc., 1964, sect. 3, vol. I, chapt. 31, p. 767–825.
46. Roughton FJW and Forster RE. Relative importance of diffusion and chemical reaction in determining rate of exchange of gases in human lung. J Appl Physiol 11: 290–302, 1957.
47. Severinghaus JW. Simple, accurate equations for human blood oxygen dissociation computations. J Appl Physiol 46: 599–602, 1979.
48. Singh MP, Khetarpal K, and Sharan M. A theoretical model for studying the rate of oxygenation of blood in pulmonary capillaries. J Math Biol 9: 305–330, 1980.
49. Staub NC, Bishop JM, and Forster RE. Importance of diffusion and chemical reaction rates in O₂ uptake in the lungs. J Appl Physiol 17: 21–27, 1962.
50. Thews G. Die Sauerstoffdiffusion in den Lungencapillaren. In: Physiologie und Pathophysiologie des Gasaustausches in der Lunge, edited by Bartels H and Witzleb E. Berlin: Springer-Verlag, 1961, p. 1–19.
51. Thews G. Oxygen diffusion in the brain: a contribution to the oxygen supply of the organs. Arch Ges Physiol 271: 197–205, 1960.
52. Vandergriff KD and Olson JS. A quantitative description in three dimensions of oxygen uptake by human red cells. Biophys J 45: 825–835, 1984.
53. Wang CH and Popel AS. Effect of red blood cell shape on oxygen transport in capillaries. Math Biosci 116: 89–110, 1993.
54. Weibel ER. Morphological basis of alveolar capillary exchange. Physiol Res 53: 419–495, 1973.
55. Weibel ER. The Pathway for Oxygen: Structure and Function in the Mammalian Respiratory System. Cambridge, MA: Harvard Univ. Press, 1984.
56. Whiteley JP, Gavaghan DJ, and Hahn CEW. Some factors affecting oxygen uptake by the red blood cells in the pulmonary capillaries. Math Biosci 169: 153–172, 2001.
57. Yamaguchi K, Nguyen-Phu D, Scheid P, and Piiper J. Kinetics of O₂ uptake and release by human erythrocytes studied by a stopped-flow technique. J Appl Physiol 58: 1215–1224, 1985.

This article has been cited by other articles:

				K.S Burrowes, A.J Swan, N.J Warren, and M.H Tawhai Towards a virtual lung: multi-scale, multi-physics modelling of the pulmonary system Phil Trans R Soc A, September 28, 2008; 366(1879): 3247 - 3263. [Abstract] [Full Text] [PDF]

				S. N. Glenet, C. De Bisschop, F. Vargas, and H. J. P. Guenard Deciphering the nitric oxide to carbon monoxide lung transfer ratio: physiological implications J. Physiol., July 15, 2007; 582(2): 767 - 775. [Abstract] [Full Text] [PDF]

				P. Pacher, J. S. Beckman, and L. Liaudet Nitric Oxide and Peroxynitrite in Health and Disease Physiol Rev, January 1, 2007; 87(1): 315 - 424. [Abstract] [Full Text] [PDF]

				C. Borland, H. Dunningham, F. Bottrill, and A. Vuylsteke Can a membrane oxygenator be a model for lung NO and CO transfer? J Appl Physiol, May 1, 2006; 100(5): 1527 - 1538. [Abstract] [Full Text] [PDF]

This Article Abstract Full Text (PDF) Free All Versions of this Article: 97/6/2284    most recent 00469.2004v1 Submit a response Alert me when this article is cited Alert me when eLetters are posted Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Web of Science (8) Citing Articles via Google Scholar Google Scholar Articles by Chakraborty, S. Articles by Bidani, A. Search for Related Content PubMed PubMed Citation Articles by Chakraborty, S. Articles by Bidani, A.