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 A corrigendum has been published All Versions of this Article: 95/6/2333    most recent 00560.2003v1 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 ISI 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 ISI Web of Science (11) Citing Articles via Google Scholar Google Scholar Articles by Wooten, E. W. Search for Related Content PubMed PubMed Citation Articles by Wooten, E. W.

Calculation of physiological acid-base parameters in multicompartment systems with application to human blood

Department of Radiology, Magnetic Resonance Imaging Division, Baylor University Medical Center, Dallas, Texas 75246; and Radiology Associates, PA, Little Rock, Arkansas 72205

Submitted 27 May 2003 ; accepted in final form 7 August 2003

	ABSTRACT

TOP ABSTRACT THEORY METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
A general formalism for calculating parameters describing physiological acid-base balance in single compartments is extended to multicompartment systems and demonstrated for the multicompartment example of human whole blood. Expressions for total titratable base, strong ion difference, change in total titratable base, change in strong ion difference, and change in Van Slyke standard bicarbonate are derived, giving calculated values in agreement with experimental data. The equations for multicompartment systems are found to have the same mathematical interrelationships as those for single compartments, and the relationship of the present formalism to the traditional form of the Van Slyke equation is also demonstrated. The multicompartment model brings the strong ion difference theory to the same quantitative level as the base excess method.

base excess; strong ion difference; Van Slyke equation; Stewart method; blood titration

SID theory has been applied to plasma (6, 9, 10, 15, 16, 55), and it has been shown previously (57) that plasma BE and plasma SID can be derived from a common formalism and that BE and the change in SID (SID) are numerically the same for plasma, provided that the concentrations of plasma noncarbonate buffers remain constant (43, 57). When this condition is not met, plasma C_B and plasma SID differ by an added constant. If, however, the reference state is chosen to coincide with the new (abnormal) noncarbonate buffer concentrations, the equivalence of C_B and SID is restored (57).

A complete quantitative description of the acid-base status of an organism requires that both intra- and extracellular effects in multiple compartments be taken into consideration (47). To this end, Siggaard-Andersen (46–48) defined BE for plasma, erythrocyte fluid, whole blood, and extracellular fluid. In contrast, the published applications of Stewart's SID theory thus far (e.g., see Refs. 1, 7, 20, 26, 39, 42) have been confined to plasma because no corresponding extant theory for multicompartment systems, such as whole blood or extracellular fluid, exists.

In the following, the formalism used to derive expressions for C_B and SID in the single-compartment case of plasma (57) is extended to systems with multiple compartments. General equations for C_B, SID, C_B, and SID are obtained for multicompartment systems and applied to the specific example of human whole blood, after the relevant model for the single compartment of human erythrocyte fluid is derived. These equations give results approximating experimentally determined values. The formulas for multicompartment systems are shown to have the same mathematical interrelationships as those demonstrated previously for single compartments (57), and the relationship between the form of the expressions derived here and the form of the Van Slyke equation traditionally used to calculate BE is also demonstrated. SID theory for multicompartment systems is thus shown to be precise to the same level of approximation as the traditional equations used for BE (33, 47–49), thereby bringing SID theory to the same quantitative level as the BE method.

	THEORY

TOP ABSTRACT THEORY METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Single-compartment systems. Previously, it was shown that the master equation approach developed by Guenther (21) could be adapted to obtain expressions for the two acid-base parameters, C_B and SID. In a general notational form, the acid-base concentration parameters P() in a single aqueous compartment can be written as

(1)

where refers, for example, to plasma ( = P) or erythrocyte fluid ( = E); P() represents either C_B() or SID() in compartment . Similarly, is the average value per molecule of some physical property of noncarbonate buffer species n (21, 57). In the case of base excess, is the average number of proton binding sites per molecule, _n(). In SID theory, is the opposite of the average charge per molecule of species n, . The calculation of in terms of conditional molar equilibrium constants and H⁺ concentration ([H⁺]) has been described in Ref. 57. C is the equilibrium concentration of proton acceptor sites of carbonate species (also equaling the opposite of the charge concentration of carbonate species), C_n() is the analytical concentration of noncarbonate buffer species n, and D is the difference between aqueous free proton and free hydroxyl ion concentrations, all in compartment (57).

Following the convention used before (57), and first advocated by Constable (6), terms that are small under physiological conditions are neglected to give

(2)

where is the concentration of bicarbonate in compartment . Because of the linearity of these expressions over the physiological pH range from 6.8 to 7.8 (47, 57), an approximate straight line form of P() can be derived as (57)

(3)

Here represents either _max(n), the maximum number of proton acceptor sites on species n for BE, or –_min(n), the opposite of the minimum possible charge on species n for SID. b_n Is a constant for species n and at a given pH can be computed from Eqs. 2 and 3 to be

(4)

It is also worth noting

(5)

as this relationship links the BE and SID theories (57). An additional relevant relationship obtained from the results of Ref. 57 is

(6)

which provides the specific relationship between C_B and SID as obtained previously (57)

(7)

where _max(n) is the maximum possible charge of species n.

Physiological pH is determined by the simultaneous solution of any of the expressions for P() and the Henderson-Hasselbalch equation in a given compartment .

(8)

where, for human plasma, pK' = 6.103 and S is the equilibrium constant between aqueous dissolved CO₂ and CO₂ in the gas phase, and equals 0.0306 at 37°C when [H⁺] is in moles per liter, is in millimoles per liter, and PCO₂ is in Torr (4).

Multicompartment systems. For a multicompartment system M with single subcompartments separated by semipermeable membranes, the acid-base parameter P(M) for the system can be expressed as a linear combination of the parameter values P() in the various single subcompartments (28, 37, 47), assuming that sufficient time has elapsed after a perturbation to reach a new steady state (47)

(9)

noting that for the volume fractions ()

(10)

It follows that for constant volume fractions

(11)

where denotes a change. The derivation of explicit expressions for P() has been described in detail previously (57). The corresponding expression to Eq. 7 is then obtained as

(12)

and from this relation follows, similar to the result for a single compartment (57)

(13)

One of the issues encountered in multicompartment systems is that variables such as pH() are often only experimentally available for a single compartment. For example, in routine clinical work, the plasma values pH(P) and in equilibrium with the erythrocyte phase and interstitial fluid are the variables typically measured, with the other pH() and seldom being obtained directly. It is therefore necessary to calculate the concentrations of species in one compartment from those in another. Fortunately, for many species of interest, concentrations can be calculated across the relevant membrane via the Nernst equation (12, 28, 40, 47)

(14)

where () is the membrane potential between compartment and the plasma compartment in volts, R is the gas constant (8.3144 J/mol-K), T is absolute temperature, c is the charge on the species under consideration, F is Faraday's constant (96, 485 Coulomb/mol), and r_m() is the Donnan ratio for a given species m between compartment and plasma.

Application to whole blood. As a specific example of a multicompartment system, human whole blood (M = B) is considered. Whole blood is made up of contributions from the two single compartments of plasma ( = P) and erythrocyte fluid ( = E). This definition, together with Eq. 10, yields from Eq. 9

(15)

(E) can be equated with the hematocrit (47, 48). The Nernst equation (Eq. 14) can be used to calculate the erythrocyte pH and [] from the corresponding plasma concentrations to calculate P(E). It follows from Eq. 11 that

(16)

under the assumption of constant hematocrit.

As an alternative to using the Nernst equation to calculate erythrocyte pH and [], other approaches specific to the determination of intraerythrocyte concentration have also been employed (47, 48). For example, in the case of the carbonate system equilibrating across the human erythrocyte membrane, the following expression has been suggested for the Donnan ratio r_c(E) for bicarbonate as a function of the plasma pH (48)

(17)

The empirically determined relationship

(18)

has also been used for the pH distribution (18, 47).

Thus, by using a knowledge of the noncarbonate buffer constituents of the plasma and erythrocyte compartments, the concentrations of those constituents, their effective molar equilibrium constants, along with pH(P), , and their distribution functions across the erythrocyte plasma membrane, C_B, SID, C_B, and SID for whole blood can be calculated as outlined above.

Relationship to the Van Slyke equation. BE was initially read from a nomogram, but with the advent of automated blood-gas analyzers it is now traditionally calculated from the Van Slyke equation developed by Siggaard-Andersen (47, 48). A small point of confusion exists regarding the precise definition of the Van Slyke equation, as Siggaard-Andersen originally referred to both the equation for the pH, [] equilibration line (the equation for C_B plotted as [] vs. pH at constant C_B), as well as to the equation relating pH and BE, as the Van Slyke equation (48). Because the latter is usually called the Van Slyke equation (33, 49), this is the convention that will be used here, and the former will be referred to as the CO₂ equilibration curve (47, 48).

It is instructive at this point to consider the relationship of the present model to the traditional form of the Van Slyke equation for whole blood, a recent version of which is given by Siggaard-Andersen and Fogh-Andersen (49) as

(19)

where BE(B) is the BE of whole blood, C_Hb(B) is the hemoglobin concentration of whole blood, and and pH(P) are the changes in the plasma bicarbonate concentration and in the plasma pH, respectively. C_Hb° is a constant, which depends on the erythrocyte fluid hemoglobin concentration C_Hb(E) and the bicarbonate Donnan ratio r_c(E), both of which are assumed to have constant normal values (48), according to

(20)

Because Siggaard-Andersen and coworkers (48, 49) defined hemoglobin concentrations and buffer values in terms of the oxyhemoglobin monomer, a value for C_Hb° of 43 mM is obtained on substituting the appropriate human normal values, including a value for r_c(E) of 0.51. In some earlier references, a value of 0.57 was used, giving a C_Hb° of 48 mM (47). '(B), an effective buffer value for whole blood (48), is the slope of the CO₂ equilibration curve for whole blood in a vs. pH(P) coordinate system at constant noncarbonate buffer concentration and constant total titratable base (47, 48). '(B) is calculated via (49)

(21)

where '_Hb is the apparent molar buffer value of the oxyhemoglobin monomer in whole blood and is assigned a value of 2.3 in humans (49). (P), with a default value of 7.7 mM in humans (49), is the buffer value of plasma computed from (49)

(22)

The _i are the true molar buffer values of the noncarbonate buffers in plasma (including albumin, inorganic phosphate, and globulins) with concentrations C_i(P).

Equation 19 can also be derived from the present formalism. As shown previously (57), the assumption of constant noncarbonate buffer concentrations from Eq. 3 for a given compartment gives

(23)

where

(24)

analogous to Eq. 22. is the molar buffer value of species n, assumed to be constant over the physiological pH range, and () is the buffer value of the noncarbonate buffer contribution in compartment (57). Substituting Eq. 23 into Eq. 11 gives

(25)

under the assumption of constant hematocrit. This equation can be recast in terms of plasma concentrations and pH(P) in equilibrium with the other compartments (true plasma) to give

(26)

where r_c() is the bicarbonate Donnan ratio between compartment and plasma and is assumed to have a constant value. By factoring out the term in r_c(), Eq. 26 can then be written in the form

(27)

with

(28)

'(M), the effective buffer value of system M, is the slope of the CO₂ equilibration curve for system M in a vs. pH(P) coordinate system at constant noncarbonate buffer concentration and constant C_B (47, 48). Equation 27 represents a general multicompartment form of the Van Slyke equation.

For the specific example of whole blood, Eq. 27 can be rewritten by using Eqs. 10 and 17 as

(29)

Likewise, substituting Eqs. 10, 17, and 18 into Eq. 28 and rearranging gives

(30)

The hematocrit, (E), can be expressed as the ratio of the whole blood and erythrocyte concentrations (48)

(31)

allowing Eq. 29, with the use of Eq. 20, to be written as

(32)

which is of the same form as Eq. 19 and traditionally used to calculate BE. '(B) can also be written, after substituting Eq. 31 into Eq. 30, in a similar manner to Eq. 21

(33)

with effective buffer values for plasma

(34)

and erythrocyte fluid

(35)

As discussed before, the term '(M)pH(P) of Eq. 27 can be thought of as a correction term to , which corrects for respiratory effects (44, 57). The corrected in this case is called the Van Slyke standard bicarbonate (VSSB). The change in VSSB (VSSB) is the difference in plasma bicarbonate concentration remaining after the PCO₂ has been altered to bring the pH(P) back to 7.40 at constant C_B or SID (46–49, 57). Because the normal and abnormal CO₂ equilibration curves are parallel under constant noncarbonate buffer concentrations, VSSB will be equal to C_B and SID (57).

As discussed by Siggaard-Andersen (46, 47), Eq. 27 can therefore be written as

(36)

where VSSB(P) is the VSSB of plasma in equilibrium with the other compartments of system M, and VSSB(M) is the VSSB of M. The VSSB(P) for system M will be numerically different from the VSSB(P) for separated plasma, since in the former case of system M this concentration represents the VSSB of plasma in equilibrium with the other single compartments (true plasma) and therefore will have a steeper CO₂ equilibration curve than will separated plasma (11).

Use of the above information regarding multicompartment systems together with Eq. 13 implies that, in general, as found in the single compartment case (57)

(37)

In the case of constant noncarbonate buffer concentrations [C_n() = 0], however

(38)

but at the same time, it is also straightforward to deduce that if the reference state is chosen to include the new (abnormal) concentrations [C_n'()], then in general

(39)

where the primes refer to the new reference state (57).

	METHODS

TOP ABSTRACT THEORY METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Calculations using the mathematical models derived in THEORY were carried out for human whole blood in a manner similar to that described previously for human plasma (57). Computation of acid-base parameters was performed by using Microsoft Excel 2002 running on a Compaq Presario 8000Z computer equipped with an AMD Athlon XP 1.733-GHz processor. pH was stepped in 0.01 increments to calculate the variables _n(), C_B(), C_B(), C_B(M), C_B(M), SID(), SID(), SID(M), SID(M), and as well as to generate graphs of these same variables vs. pH. was calculated by taking the tangent to the _n() vs. pH curves (57). () was calculated from Eq. 24, and (M) was calculated by taking the tangent to the P(M) curve when carbonate species concentrations are set to 0.0 mM. All calculations were carried out at 37°C, except where otherwise noted. In a departure from previous work (57), no correction for ionic strength was performed, given the small changes produced in the final calculated values (57). In addition, following the approach of Siggaard-Andersen (47), no correction for osmotic effects was explicitly included.

All protein amino acids, the heme moieties of hemoglobin, and the carboxyl group of 2,3-diphospho-D-glycerate (DPG) were assumed to behave as independent monoprotic acids (15, 16, 29, 30, 57). Inorganic phosphate was treated as a triprotic acid, and the two phosphates of DPG were assumed to behave as independent diprotic acids (15, 16, 29, 57).

For human plasma, it was assumed that albumin and inorganic phosphate were sufficient to account for all of the noncarbonate buffer activity (15, 16, 55, 57), and simulations of acid-base balance used the human albumin and inorganic phosphate dissociation constants given by Figge et al. (15). Bicarbonate concentrations were derived from the Henderson-Hasselbalch equation (Eq. 8) together with plasma pH and PCO₂ (4). The effective equilibrium constants for human albumin, phosphate, and carbonate at 37°C are given in Table 1.

For human erythrocyte fluid, a slightly different approach was required because no complete theoretical or experimental equilibrium constant data under erythrocyte physiological conditions could be found. The calculation was carried out for oxygenated blood, and it was assumed, in accordance with the work of Siggaard-Andersen (47) and Reeves (38), that the noncarbonate buffer species of human oxygenated erythrocyte fluid are primarily oxyhemoglobin and free DPG. For the equilibrium constants of the and chains in the human oxyhemoglobin tetramer, effective pK_a values calculated from a modified Tanford-Kirkwood theory were available at 25°C and an ionic strength of 0.10 M from the work of Matthew et al. (30). A similar publication, again by Matthew et al. (29), provided the effective pK_a of free DPG at 25°C and ionic strength 0.10 M. At the same time, more recent experimental pK_a data for the solvent of the solvent-accessible histidine imidazole groups of human carbon monoxyhemoglobin were available from a study by Fang et al. (13) at 29°C in 0.1 M HEPES buffer plus 0.1 M chloride. These experimental values were used for the solvent-accessible histidine imidazole groups instead of the theoretical values determined from the modified Tanford-Kirkwood theory (30). -Histamine (-His)⁹⁷ and -His¹¹⁶, considered nontitratable by Matthew et al. (30), were found to be titratable experimentally by Fang at al. (13).

The effective oxyhemoglobin proton dissociation constants K_l were then corrected to 37°C by employing the van't Hoff relation (3, 9)

(40)

where H° is the standard enthalpy of ionization for the ionizable groups of human hemoglobin, all of which were assumed to have the same H°, in accordance with the findings of Stadie and Martin (51). The H° was calculated by changing the enthalpy in 1 cal/mol increments to minimize the function

(41)

is the theoretically calculated average charges of the hemoglobin tetramer, and is the experimentally determined oxyhemoglobin values for k = 31 experimental pH data points at 37°C obtained from the data of Raftos et al. (36). The numerical values for the data points, originally presented in graphical form in Fig. 4 of Ref. 36, were provided courtesy of Dr. J. Raftos. These values were then corrected for the small contribution from a PCO₂ of 0.25 Torr under the experimental conditions of that study (36) by calculating the bicarbonate contribution from PCO₂ and pH by using Eq. 8, then adding this to the experimental value to give . The minimum S² of 1.3 thus obtained provided a H° of 10.6 kcal/mol (44.5 kJ/mol).

Based on work by Reeves (38), the order of magnitude of the standard enthalpy of ionization for DPG was assumed to be 1 kcal/mol and thus not expected to cause a significant temperature dependence; therefore, these values were left uncorrected for temperature. The uncorrected human oxyhemoglobin and DPG effective pK_a values are also listed in Table 1.

Values for pH(E) were obtained from Eq. 14 and the plasma pH(P), under the assumption of a membrane potential of –13.0 mV (47, 48). was calculated from Eq. 17, together with and pH(P).

Whole blood acid-base parameters were calculated by using the above models for human plasma and erythrocyte fluid, which were then incorporated into Eqs. 15 and 16, assuming a constant hematocrit of 44% (4). For calculation of using Eqs. 19 and 32, a value of 43 mM was used for C_Hb°.

Numerical values for the comparison experimental data for human oxyhemoglobin at 25°C were obtained from Table I of Ref. 41 and assumed to be free of carbonate. The comparison numerical data for human oxyhemoglobin at 37°C were obtained from Raftos et al. (36) as discussed above. The numerical values for the experimentally derived titration curves of whole blood at PCO₂ = 28.7 Torr, 40.0 Torr, and 66.0 Torr were read directly from Fig. 10 of Ref. 47.

The designations "normal plasma" and "normal erythrocyte fluid" used in the DISCUSSION, tables, and figures refer to the normal human values pH(P) = 7.40, = 24.25 mM, PCO₂(P) = 40.0 Torr, pH(E) = 7.19, = 12.49 mM, and the concentrations of noncarbonate buffer species (4, 22, 47, 48, 57) given in Table 2. "Normal whole blood" likewise refers to the corresponding human multicompartment model with a constant hematocrit of 44%.

Finally, although Eq. 2 provides exact expressions for C_B(M) and SID(M), the linear forms obtained from Eq. 3 are more easily manipulated. These were substituted into Eq. 15, which after rearrangement of the bicarbonate term in a manner similar to that used to for Eq. 29 gave

(42)

where the index i extends over the plasma and the index j over the erythrocyte noncarbonate buffers. To obtain explicit numerical forms, the values in Table 2 were then substituted into Eq. 42. Equation 31 was employed to obtain the hemoglobin term as a function of C_Hb(B). The approximation that the bicarbonate Donnan ratio is a constant at r_c(E) = 0.51 over the physiological pH range was also used, as well as that

(43)

also found to be a good approximation over the physiological range, giving explicit linear forms for C_B(B) and SID(B) as functions of pH(P).

(44)

(45)

	RESULTS

TOP ABSTRACT THEORY METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Theoretical titration curves for human oxyhemoglobin with corresponding experimental data from Rollema et al. at 25°C (41) and from Raftos et al. at 37°C (36) are shown in Fig. 1. Theoretical titration curves for human whole blood at PCO₂= 28.7, 40, and 66 Torr are shown in Fig. 2 together with experimental data obtained by Siggaard-Andersen (45, 47). In keeping with the practice of graphing the equations for C_B and SID in a vs. pH(P) coordinate system as advocated by Davenport (11), Fig. 3 shows such plots illustrated for human whole blood and plasma. To compare the traditional Van Slyke equation (Eq. 19) with that derived from pH and effective conditional molar equilibrium constants (Eq. 32), a graph of human whole blood vs. pH(P) is provided in Fig. 4, together with the corresponding curve obtained from the more complete Eqs. 2 and 15 at constant P(B).

View larger version (13K): [in this window] [in a new window]   Fig. 1. Average charge per molecule (Hb) vs. erythrocyte pH [pH(E)] under carbonate-free conditions for the human oxyhemoglobin tetramer showing theoretical curves (lines) and experimental data from Ref. 41 at 25°C () and from Ref. 36 at 37°C ().

View larger version (15K): [in this window] [in a new window]   Fig. 2. Change in total titratable base (CB) vs. plasma pH [pH(P)] titration curves for human whole blood at 37°C and variable PCO2. Theoretical curves and experimentally derived values from Ref. 47 at PCO2 = 28.7 (), 40.0 (), and 66.0 () Torr are shown.

View larger version (14K): [in this window] [in a new window]   Fig. 3. Plasma concentration () vs. pH(P) graphs for human whole blood (a) and plasma (b), indicated by solid lines. The normal physiological state is indicated by the arrow at pH(P) = 7.40, where the solid lines intersect. A dotted line is shown for an acid-base disturbance of whole blood with a metabolic component (a') in a patient with pH(P) = 7.16 and . The arrow at pH(P) = 7.16 also indicates where this line crosses that for plasma. All lines have normal noncarbonate buffer concentrations. To obtain P = CB = change in strong ion difference (SID) = change in Van Slyke standard bicarbonate for a given body fluid (VSSB), any of the lines is traced back to pH(P) = 7.40 to get the corresponding . Then 24.25 mM is subtracted from this result to obtain VSSB(P) for that body fluid. This value is then scaled by the appropriate distribution function as indicated in Eq. 36 to obtain the body fluid parameter, giving P(P) = 0.0 mM and P(B) = –5.3 mM.

View larger version (13K): [in this window] [in a new window]   Fig. 4. vs. pH(P) graphs of human whole blood with normal noncarbonate buffer concentrations for the traditional Van Slyke equation () from Ref. 49 (Eq. 19), the Van Slyke equation from Eq. 32 (dotted line), and the more exact Eq. 15 (solid line).

Various calculated parameter values are compared with the results of experimental data and other models in Table 3. Calculated values are under normal conditions as defined in METHODS.

The derivation of the explicit linear forms for P(M)as described in METHODS gave the following results from Eq. 42

The extent to which these expressions approximate the results from Eqs. 2 and 15 is demonstrated in Fig. 5 for human whole blood and plasma. To calculate the variables for plasma, (E) and C_Hb(B) are both set to zero. Also note that, in these explicit linear forms, the hemoglobin concentration is expressed as the concentration of tetramer.

View larger version (18K): [in this window] [in a new window]   Fig. 5. CB and SID vs. pH(P) for human whole blood and plasma at 37°C, PCO2 = 40.0 Torr, and normal noncarbonate buffer concentrations by using the more complete Eq. 15 (symbols) and the approximate forms of Eqs. 44 and 45. Dotted lines, plasma CB + SID; solid lines, whole blood CB + SID.

	DISCUSSION

TOP ABSTRACT THEORY METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
The accuracy of the single and multicompartment models can be assessed by how well they predict experimentally accessible parameters. In general, agreement is good. A summary of theoretical values obtained for various different parameters is presented in Table 3, where these values are also compared with previously published data. A few comments are in order regarding specific values, as well as some of the features of Figs. 1, 2, 3, 4, 5.

Results for the plasma single compartment using the present formalism have been given previously (57), and the extent to which this model and others like it reproduce experimental data has been discussed by several authors (15, 55, 57). It has been shown that the model produces excellent agreement with experimental data for human plasma, even when the plasma globulin component is neglected; for example, the SID values of 39 mM from Ref. 57 and the SID of 40.4 mM from Ref. 9 agree well with the value determined by Singer and Hastings (50) of 41.7 mM. This is despite the apparent disagreement between the buffer value (P) = 5.7 mM obtained from the present model and the (P) = 7.7 mM published by Siggaard-Andersen (47, 49).

Although probably the weakest portion of the model currently, the theory for the erythrocyte compartment nonetheless produces values commensurate with previously published data, although returning values were slightly higher than those published for the charge on the human oxyhemoglobin tetramer (e.g., –2.10 vs. –2.54) and slightly lower for the tetramer molar buffer value (e.g., 10.2 vs. 10.8). Similarly, the theoretical charge on DPG of –4.6 is lower than the –4.1 given in Table 3. The molar buffer value (), however, is similar to the 0.69 published by Reeves (38) at 25°C but lower than the 1.0 obtained from Raftos et al. at 37°C (36). The H° determined here of 10.6 kcal/mol is similar to the 10 kcal/mol obtained by Stadie and Martin (51). A H° of 9 kcal/mol was found by Antonini et al. (2) for the alkaline groups and –1.5 kcal/mol for the acid groups of human hemoglobin. The validity of the approximation of Stadie and Martin that all ionizable groups of hemoglobin have the same standard enthalpy of ionization probably lies in the fact that, over the physiological pH range, groups with pK_a outside the physiological pH range largely have a fixed charge, and therefore the overall molecular charge is relatively insensitive to small temperature-induced changes in the effective pK_a of these groups. Although the agreement of the model with the data from Rollema et al. (41) at 25°C is seen to be moderately good over the physiological range, the agreement at low pH is not as impressive. Still, it is striking how well the theory predicts the experimental titration curves for human oxyhemoglobin shown in Fig. 1, given the number of assumptions inherent in the calculation. Dedicated calculations of the effective pK_a of hemoglobin and DPG under erythrocyte physiological temperature and ionic strength conditions would most likely give better agreement, and the need for more accurate effective pK_a may stimulate further research in this area.

Despite the mild discrepancies in the erythrocyte portion of the model, the experimental titration curves (45, 47) for human whole blood shown in Fig. 2 are faithfully reproduced by the theory for three different PCO₂ values. The normal values calculated for SID in human plasma, erythrocyte fluid, and whole blood also agree reasonably well with those obtained by direct calculation (Table 3). Additionally, the titration curves for C_B(B) and SID(B) are shown to be well approximated, to within 1 mM over the physiological range, by the linear forms of Eqs. 44 and 45. This is also the case for C_B(P) and SID(P), as previously demonstrated (57). These results place confidence in the ability of the present model to predict the acid-base behavior of the multicompartment physiological system human whole blood. It bears mentioning, however, that acid-base parameters may be different in different species (6, 9), requiring a separate analysis for each.

In addition to reproducing the experimental human whole blood titration curves, the model also reproduces the traditional Van Slyke equation. Furthermore, the relationship of the present formalism to that of Siggaard-Andersen (47–49) has been demonstrated, providing Eq. 27 as a general multicompartment form of the Van Slyke equation. Figure 4 compares the values for in human whole blood calculated with the traditional Van Slyke equation (Eq. 19) by using the parameters of Siggaard-Andersen and Fogh-Andersen (49) with that calculated by using the Van Slyke expression determined with the present model (Eq. 32). The vs. pH(P) curve calculated for whole blood with the complete expression (Eqs. 2 and 15) is also shown. Exact agreement between the approximate values of Eqs. 19 and 32 is present, and these deviate significantly from the values calculated with the more complete expression only at low pH(P). This deviation is a consequence, as discussed before by Siggaard-Andersen (47), of the pH dependence of parameters such as the Donnan ratio for bicarbonate and the buffer values of the noncarbonate species, which are assumed to have constant values in the approximate formulas (47). It is also a reflection of the use of Eq. 18 for the pH distribution in the derivation of the traditional form, as opposed to Eq. 14 in the more complete expression.

It also bears mentioning that the value of 7.7 mM used for (P) in Eq. 21, although similar to the effective value of 7.3 mM for '(P) derived via Eq. 34, should not actually be the same as the true buffer value for plasma in Eq. 22. Similarly, the (E) of 63 mM given in Ref. 48 should not give a '(E) of 2.3 according to Eq. 35. It is important to point out, however, that the expression quoted by Siggaard-Andersen in the form of Eq. 21 actually appears to be an empirical relationship, and it is thus possible that the appearance of the factor 7.7 mM is coincidental.

Figure 3 demonstrates the important point that, in a multicompartment system, a complete quantitative treatment of the metabolic component of an acid-base disturbance requires that one account for all of the nonvolatile excess acid or base, considering the relative contributions from each compartment. For example, suppose that a patient presents with pH(P) = 7.16, , PCO₂ = 73.4 Torr, and normal noncarbonate buffer concentrations. In this case, Fig. 3 demonstrates that the calculated P (equal to BE, C_B, SID, and VSSB by Eq. 38) gives P(P) = 0.0 mM for separated plasma and P(B) = –5.3 mM for whole blood. One would therefore predict only a respiratory disturbance in the former case, whereas there is a metabolic component in the latter (11, 47, 57). Consideration of only the plasma compartment may therefore produce an erroneous assessment of acid-base status, although it should be noted that the in vitro equations for both plasma and whole blood differ from the in vivo equilibration curves (44, 47), with the in vivo curve intermediate in slope between plasma and whole blood (44, 47). For BE with constant noncarbonate buffer concentrations, the corresponding in vivo condition has been approximated by an extracellular fluid model in which whole blood is diluted 2:1 in its own plasma (47, 49).

In his development of the BE theory, Siggaard-Andersen sought an expression for C_B directly (46, 47). By conceptualizing BE as the change in the Van Slyke standard bicarbonate, he derived an expression for VSSB, which has come to be known as the Van Slyke equation (48). Via this approach, the crucial piece of data that needed to be measured or calculated was the slope of the CO₂ equilibration curve (46, 47). After an expression for the change in Van Slyke standard bicarbonate in terms of the slope of the CO₂ equilibration curve was derived, an empirical linear approximation for this curve was then inserted at the appropriate point in the derivation (46, 47) to give C_B, which was defined as BE. With the use of a different approach, Eq. 16 is obtained via a full theoretical treatment by first seeking an equation for C_B, then calculating C_B. This provides an expression for BE (Eq. 32) nearly equal to the traditional Eq. 19. This formula, by Eq. 38, is also equal to SID and VSSB in the setting of constant noncarbonate buffer concentrations.

In contrast to Siggaard-Andersen's approach to calculating BE, Stewart calculated SID as a variable in its own right. Based on this, Stewart considered the effect of noncarbonate buffer concentrations on SID, leading others to interpret hyper- or hypoproteinemic states as forms of acid-base disorders by examining SID, the corresponding variable to C_B, as a deviation from its normal value (23, 31, 42). Siggaard-Andersen and co-workers (49) have rejected this notion. Although differences in noncarbonate buffer concentrations do enter into the calculation of BE via Eq. 21, this has the effect of producing Eq. 39 and altering the slope, but not the normal physiological position, of the CO₂ equilibration curve (32, 49, 57). Because this approach produces small changes in the final calculated BE, the default values are commonly used independent of the actual noncarbonate buffer concentrations (11, 49).

Although the Stewart theory is often said to have an advantage over BE in that it claims to examine the only independent variables of acid-base physiology (24, 25, 52, 53), this contention has been challenged (5, 49, 56). However, the clear theoretical disadvantage that the Stewart approach has had relative to the BE method is that its previous formulations did not treat the complete acid-base disorder quantitatively, because only the plasma compartment was considered. The present treatment of multicompartment systems brings the SID theory to the same quantitative level, and within the same degree of precision, as the traditional BE theory. It is also worth noting that the more complete expressions simplify under certain limiting conditions to the Henderson-Hasselbalch equation, as shown by Constable (6), and to the Van Slyke equation, as shown in the present work.

The model presented here provides a unifying formalism for the BE and SID approaches, and provides flexibility in the quantitative description of acid-base disorders, by allowing calculation of the absolute quantities C_B and SID for single compartments or systems with multiple compartments, including the examples of human plasma, erythrocytes, and whole blood demonstrated. Both exact expressions and linear approximations can be used for calculation. With the use of these equations, acid-base status can be assessed from the vantage point of both C_B and SID, considering changes in noncarbonate buffer concentration either explicitly (Eq. 37) or implicitly (Eq. 39). Given an extant theory for calculating absolute values of acid-base parameters in the whole organism, factors involved in acid-base disturbances can be more fully explored.

In conclusion, the general formalism previously developed for calculating parameters describing physiological acid-base balance in single compartments has been extended to multicompartment systems. Together with the previously obtained model for human plasma, a model for the human erythrocyte single compartment was used to obtain expressions for the multicompartment case of human whole blood. Calculations using these equations produced values that approximate a wide range of experimental data, providing confidence in the model. The equations for multicompartment systems were found to have the same mathematical interrelationships as those for single compartments. The relationship of the present formalism to the traditional form of the Van Slyke equation was also demonstrated, and a general multicompartment form of the Van Slyke equation was given. With the multicompartment model, the SID theory is brought to the same quantitative level as the BE method.

	ACKNOWLEDGMENTS

TOP ABSTRACT THEORY METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
I thank Dr. Howard Corey for rekindling my interest in physiological acid-base balance and Dr. Peter Hildenbrand for helpful comments. I thank Dr. Julia Raftos for graciously providing the numerical values of experimental data, originally presented in graphical form as Fig. 4 of Raftos et al. (36), for use in Fig. 1.

	FOOTNOTES

 

Address for reprint requests and other correspondence: E. W. Wooten, Radiology Associates, PA, 500 S. Univ. Ave., Little Rock, AR 72205.

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.

	REFERENCES

TOP ABSTRACT THEORY METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 

1. Alfaro V, Torras R, Ibanez J, and Palacios L. A physical-chemical analysis of the acid-base response to chronic obstructive pulmonary disease. Can J Physiol Pharmacol 74: 1229–1235, 1996.[Web of Science][Medline]
2. Antonini E, Wyman J, Brunori M, Fronticelli C, Bucci E, and Rossi-Fanelli A. Studies on the relations between molecular and functional properties of hemoglobin. V. The influence of the Bohr effect in human and in horse hemoglobin. J Biol Chem 240: 1096–1103, 1965.[Free Full Text]
3. Atkins P and de Paula J. Physical Chemistry (7th ed.). New York: Freeman, 2002.
4. Burus CA and Ashwood ER (Editors). Tietz Textbook of Clinical Chemistry (2nd ed.). Philadelphia, PA: Saunders, 1994.
5. Cameron JN. Acid-base homeostasis: past and present perspectives. Phys Zool 62: 845–849, 1989.
6. Constable PD. A simplified strong ion model for acid-base equilibria: application to horse plasma. J Appl Physiol 83: 297–311, 1997.[Abstract/Free Full Text]
7. Constable PD. Clinical assessment of acid-base status: strong ion difference theory. Vet Clin N Am 15: 447–471, 1999.
8. Constable PD. Clinical assessment of acid-base status: comparison of the Henderson-Hasselbalch and strong ion approaches. Vet Clin Path 29: 115–128, 2000.
9. Constable PD. Total weak acid concentration and effective dissociation constant of nonvolatile buffers in human plasma. J Appl Physiol 91: 1364–1371, 2001.[Abstract/Free Full Text]
10. Constable PD. Calculation of variables describing plasma nonvolatile weak acids for use in the strong ion approach to acid-base balance in cattle. AJVR 63: 482–490, 2001.
11. Davenport HW. The ABC of Acid-Base Chemistry (6th ed.). Chicago, IL: Univ. of Chicago Press, 1974.
12. DeLand EC. The Classical Structure of Blood Biochemistry—A Mathematical Model. Rand Memo RM-4962-PR, Santa Monica, CA, 1966.
13. Fang TY, Zou M, Simplaceanu V, Ho NT, and Ho C. Assessment of roles of surface histidyl residues in the molecular basis of the Bohr effect and of 143 histidine in the binding of 2,3-bisphosphoglycerate in human normal adult hemoglobin. Biochemistry 38: 13423–13432, 1999.[Medline]
14. Fencl V and Lieth DE. Stewart's quantitative acid-base chemistry: applications in biology and medicine. Respir Physiol 91: 1–16, 1993.[Web of Science][Medline]
15. Figge J, Mydosh T, and Fencl V. Serum proteins and acid-base equilibria: a follow-up. J Lab Clin Med 120: 713–719, 1992.[Web of Science][Medline]
16. Figge J, Rossing TH, and Fencl V. The role of serum proteins in acid-base equilibria. J Lab Clin Med 117: 453–467, 1991.[Web of Science][Medline]
17. Fogh-Andersen N, Bjerrum PJ, and Siggaard-Andersen O. Ionic binding, net charge, and Donnan effect of human serum albumin as a function of pH. Clin Chem 39: 48–52, 1993.[Abstract]
18. Funder J and Wieth JO. Chloride and hydrogen ion distribution between human red cells and plasma. Acta Physiol Scand 68: 234–245, 1966.
19. Gary-Bobo CM and Solomon AK. Properties of hemoglobin solutions in red cells. J Gen Physiol 52: 825–853, 1968.[Abstract/Free Full Text]
20. Gilfix B, Bique M, and Magder S. A physical chemical approach to the analysis of acid-base balance in the clinical setting. J Crit Care 8: 187–197, 1993.[Web of Science][Medline]
21. Guenther WB. Unified Equilibrium Calculations. New York: Wiley, 1991.
22. Hlady SB and Rink TJ. pH equilibrium across the red cell membrane. In: Membrane Transport in Red Blood Cells, edited by Ellory JC and Lew VI. London: Academic, 1977, p. 115–135.
23. Jabor A and Kazda A. Modelling of acid-base equilibria. Acta Anaesthesiol Scand 39, Suppl 107: 119–122, 1995.
24. Kellum JA. Metabolic acidosis in the critically ill: lessons from physical chemistry. Kidney Int 53, Suppl 66: S81–S86, 1998.
25. Kellum JA. Determinants of blood pH in health and disease. Crit Care 4: 6–14, 2000.[Web of Science][Medline]
26. Kellum JA, Bellomo R, Kramer DJ, and Pinsky MR. Etiology of metabolic acidosis during saline resuscitation in endotoxemia. Shock 9: 364–368, 1998.[Web of Science][Medline]
27. Kowalchuk JM and Scheuermann BW. Acid-base regulation: a comparison of quantitative methods. Can J Physiol Pharmacol 72: 818–826, 1994.[Web of Science][Medline]
28. Lloyd BB and Michel CC. A theoretical treatment of the carbon dioxide dissociation curve of true plasma in vitro. Respir Physiol 1: 107–120, 1966.[Web of Science][Medline]
29. Matthew JB, Friend SH, and Gurd FRN. Electrostatic effects in hemoglobin: electrostatic energy associated with allosteric transition and effector binding. Biochemistry 20: 571–580, 1981.[Medline]
30. Matthew JB, Hanania GIH, and Gurd FRN. Electrostatic effects in hemoglobin: hydrogen ion equilibria in human deoxy- and oxyhemoglobin A. Biochemistry 18: 1919–1928, 1979.[Medline]
31. McAuliffe JJ, Lind LJ, Lieth DE, and Fencl V. Hypoproteinemic alkalosis. Am J Med 81: 86–90, 1986.[Web of Science][Medline]
32. Moon JB. Abnormal base excess curves. Pediat Res 1: 333–340, 1967.[Web of Science][Medline]
33. Morgan TJ, Clark C, and Endre ZH. Accuracy of base excess—an in vitro evaluation of the Van Slyke equation. Crit Care Med 28: 2932–2936, 2000.[Web of Science][Medline]
34. Narins RG (Editor). Maxwell and Kleeman's Clinical Disorders of Fluid and Electrolyte Metabolism (5th ed.). New York: McGraw-Hill, 1994.
35. Narins RG and Emmett M. Simple and mixed acid-base disorders: a practical approach. Medicine (Baltimore) 59: 161–187, 1980.[Medline]
36. Raftos JE, Bulliman BT, and Kuchel PW. Evaluation of an electrochemical model of erythrocyte pH buffering using ³¹P nuclear magnetic resonance data. J Gen Physiol 95: 1183–1204, 1990.[Abstract/Free Full Text]
37. Rees SE, Andreassen S, Hovorka R, Summers R, and Carson ER. Acid-base chemistry of the blood—a general model. Comp Meth Prog Biomed 51: 107–119, 1996.[Web of Science][Medline]
38. Reeves RB. Temperature-induced changes in blood acid-base status: Donnan R_Cl and red cell volume. J Appl Physiol 40: 762–767, 1976.[Abstract/Free Full Text]
39. Rehm M, Orth V, Scheingraber S, Kreimeier U, Brechtelsbauer H, and Finsterer U. Acid-base changes caused by 5% albumin vs. 6% hydroxyethyl starch solution in patients undergoing acute normovolemic hemodilution. Anesthesiology 93: 1174–1183, 2000.[Web of Science][Medline]
40. Rohwer JM, Kuchel PW, and Maher AD. Thermokinetic modeling. Membrane potential as a dependent variable in ion transport processes. Mol Biol Rep 29: 217–225, 2002.[Web of Science][Medline]
41. Rollema HS, de Bruin SH, Janssen LHM, and van Os GAJ. The effect of potassium chloride on the Bohr effect in hemoglobin. J Biol Chem 250: 1333–1339, 1975.[Abstract/Free Full Text]
42. Rossing TH, Maffeo N, and Fencl V. Acid-base effects of altering plasma protein concentration in human blood in vitro. J Appl Physiol 61: 2260–2265, 1986.[Abstract/Free Full Text]
43. Schlichtig R. [Base excess] vs. [strong ion difference]. Which is more helpful? Adv Exp Med Biol 411: 91–95, 1997.[Web of Science][Medline]
44. Severinghaus JW. Siggaard-Andersen and the "Great Trans-Atlantic Acid-Base Debate." Scand J Lab Invest 53, Suppl 214: 99–104, 1993.
45. Siggaard-Andersen O. The pH, log PCO₂ blood acid-base nomogram revised. Scand J Clin Lab Invest 14: 598–604, 1962.[Web of Science][Medline]
46. Siggaard-Andersen O. Titratable acid or base of body fluids. Ann NY Acad Sci 133: 41–58, 1966.[Web of Science][Medline]
47. Siggaard-Andersen O. The Acid-base Status of the Blood (4th ed.). Baltimore, MD: Williams and Wilkins, 1974.
48. Siggaard-Andersen O. The Van Slyke equation. Scand J Clin Lab Invest Suppl 146: 15–20, 1977.
49. Siggaard-Andersen O and Fogh-Andersen N. Base excess of buffer base (strong ion difference) as a measure of a non-respiratory acid-base disturbance. Acta Anaesthesiol Scand 39, Suppl 107: 123–128, 1995.
50. Singer RB and Hastings AB. An improved clinical method for the estimation of disturbances in the acid-base balance of human blood. Medicine (Baltimore) 27: 223–242, 1948.[Medline]
51. Stadie WC and Martin KA. The thermodynamic relations of the oxygen- and base-combining properties of blood. J Biol Chem 60: 191–235, 1924.[Free Full Text]
52. Stewart PA. Independent and dependent variables of acid-base control. Respir Physiol 33: 9–26, 1978.[Web of Science][Medline]
53. Stewart PA. Modern quantitative acid-base chemistry. Can J Physiol Pharmacol 61: 1441–1461, 1983.
54. Watanabe K, Miyamoto M, and Imai Y. An estimation of buffer values of human whole blood by titration experiment under the open condition for carbon dioxide gas. Jap J Physiol 51: 671–677, 2001.[Web of Science][Medline]
55. Watson PD. Modeling the effects of proteins on pH in plasma. J Appl Physiol 86: 1421–1427, 1999.[Abstract/Free Full Text]
56. Wooten EW. Strong ion difference theory: more lessons from physical chemistry. Kidney Int 54: 1769–1770, 1998.[Web of Science][Medline]
57. Wooten EW. Analytic calculation of physiological acid-base parameters in plasma. J Appl Physiol 86: 326–334, 1999.[Abstract/Free Full Text]

This article has been cited by other articles:

				I. Kurtz, J. Kraut, V. Ornekian, and M. K. Nguyen Acid-base analysis: a critique of the Stewart and bicarbonate-centered approaches Am J Physiol Renal Physiol, May 1, 2008; 294(5): F1009 - F1031. [Abstract] [Full Text] [PDF]

				E. M. Omron, J. A. Kellum, and J. Gomez Effects of Hypercapnea on BP in Rats Chest, November 1, 2007; 132(5): 1717 - 1717. [Full Text] [PDF]

This Article Abstract Full Text (PDF) Free A corrigendum has been published All Versions of this Article: 95/6/2333    most recent 00560.2003v1 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 ISI 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 ISI Web of Science (11) Citing Articles via Google Scholar Google Scholar Articles by Wooten, E. W. Search for Related Content PubMed PubMed Citation Articles by Wooten, E. W.