# My opinion

Previous quantitative approaches that model acid-base physiology and used to predict the equilibrium pH rely on the mathematical convenience of electroneutrality/charge balance considerations (1-4). This fact has caused confusion in the literature, and has led to the assumption that charge balance/electroneutrality is a causal factor in modulating proton buffering (Stewart formulation). In our recent study, we reported the derivation of a new mathematical model to predict the equilibrium pH based on the partitioning of H+ buffering in a multiple-buffered aqueous solution (5). The goal of our study was to determine whether it is possible to derive a mathematical model that is both predictive and mechanistic. Specifically, the goal of our paper was not to derive a predictive formula per se, but to derive a predictive formula based on the underlying physical chemistry involved (partitioning of H+ buffering) without utilizing the mathematical convenience of electroneutrality/charge balance considerations as had previous authors. Our reasoning was based on the consideration that if a derivation based only on partitioning of H+ buffering was indeed possible, this would demonstrate convincingly that electroneutrality/charge balance considerations are not only mathematically not required, but are de facto not fundamental in determining the pH from a chemical standpoint.

We were motivated to pursue this approach because we had previously shown that although charge balance is a convenient mathematical tool that can be utilized to calculate and predict the equilibrium pH, charge balance (electroneutrality considerations) is not a fundamental physicochemical parameter that is mechanistically involved in predicting or determining the equilibrium pH value of a solution (5,6). Indeed, if strong ion difference (SID, a term used in the Stewart formulation which is based on electroneutrality and charge balance considerations) were to have a mechanistic role in determining the equilibrium pH, it must do so by imparting a fixed macroscopic charge to the solution which will in turn cause the [H+] to attain a given value in order to maintain macroscopic electroneutrality. However, we demonstrated that for a given change in SID due to the addition of HCl to a NaCl-containing solution, electroneutrality is maintained (i.e. [Na+] + [H+] - [Cl-] - [OH-] = 0) at all pre-equilibrium and equilibrium pH values, and that the equilibrium pH is only determined by the dissociation constant of water, K’w (5).

The significance and novelty of our model and certain technical aspects of our pH measurements have recently been questioned (7-9). These authors referred to previous formulas derived by other investigators to predict and analyze the equilibrium pH of an aqueous solution. These include the predictive formula published by Rang and Herman et al, Charlot equation, Guenther’s n-bar equation, de Levie equation, Morel’s tableau method, and the quantitative approaches discussed in the classic texts by Bjerrum, Ricci, Stumm, Ramette and Butler (4,10,11-20). However, we stress that none of these authors has achieved the goal of basing their derivations solely on the underlying mechanistic chemistry involved, i.e. proton partitioning among various buffers.

In response to an inquiry with regard to how the initial reactant concentrations were calculated in our study (5,8), they were calculated as follows: At each titration step, the reactant concentrations of each sample containing the mixture of buffers were first calculated based on the measured pH of the sample prior to the addition of HCl as follows:

Since [H+]sample [A-]sample = K’a [HA]sample and [ATOT]sample = [A-]sample + [HA]sample :

[A-]sample = ([ATOT]sample x K’a)/([H+]sample + K’a) and [HA]sample = [ATOT]sample – [A-]sample

Based on the water association/dissociation equilibrium reaction:

[OH-]sample = K’w/ [H+]sample

After the addition of HCl, the initial reactant concentrations as displayed in Table 1 were calculated by accounting for the amount of H+ and OH- added and the dilutional effect of the added volume (5):

[ATOT]sample = (0.01 x 0.02)/Total Vol where Vol = volume

[A-]i = ([A-]sample x Volsample)/ Total Vol and [HA]i = ([HA]sample x Volsample)/ Total Vol

[H+]i = ([H+]sample x Volsample + [H+]HCl x VolHCl) / Total Volume

where [H+]HCl = H+ concentration of HCl solution; VolHCl = volume of HCl added; and HCl is assumed to be completely dissociated.

[OH-]i = ([OH-]sample x Vol sample + [OH-]HCl x Vol HCl) / Total Volume

where [OH-]HCl = OH- concentration of the HCl solution = K’w/[H+]HCl

All the initial reactant concentrations displayed in Table 1 were determined based on the above calculations, and the calculated values were rounded to the fourth decimal place (5). There is no dilution error in Experiment 4 as suggested (8). In Experiment 4, the total volume of the solution for the initial data set was 20.04 ml because more HCl was initially needed to titrate the pH of the solution to the calibration range of our pH measurements. Thereafter, the total volume changed in increment of 0.02 ml. Moreover, the suggestion that buffer B is diluted by another method than buffer A is an impossibility (8). The dilution is performed exactly as described in the Methods section of our article. It is important to note that both buffers A and B were mixed in the same solution and not in different solutions. Therefore, the same volume of HCl was added to the same solution containing the mixture of buffers A and B. In reviewing the data in Experiment 2, there are typographical errors in the reported [HB]. The correct [HB] in Experiment 2 should be: 1.0503E-03, 1.5365E-03, 2.0763E-03, 2.6531E-03, 3.3220E-03, 4.0649E-03, 4.9081E-03, 5.7530E-03, 6.6020E-03, 7.4804E-03, and 8.3978E-03 respectively. However, the reported predicted pH values are correct as originally stated; this can be easily verified by entering the values of [HB] listed here into our mathematical model and solving for pH.

It was also suggested that two buffers in the first experiment in our article were exposed to different ionic strengths (8). First, in our study (5), the apparent equilibrium constant K’ of each buffer was calculated based on the thermodynamic equilibrium constant K and the ionic strength of the solution: pK’ = pK – 0.51√I (Eq. 23). The ionic strength of the solution was calculated based on Eq. 24: I = ½ ∑ cZ2. Therefore, the same exact value for ionic strength was entered into Eq. 23 to calculate the apparent equilibrium constant K’ of the two buffers. Second, one needs to consider the temperature dependence of the pK of any buffer pair. The pK of PIPES (Amresco, Solon, OH) at 25°C is 6.80 and not 6.76 (21). Using this pK value of 6.80, the ionic strength is ((6.80 – 6.7559457)/0.51)2 = 0.00746, which is the same ionic strength calculated for Buffer B. In reply to the footnote regarding buffer B (8), the pK of HEPES (Amresco, Solon, OH) at 25°C is 7.55 and not 7.48 (21). Indeed, in our study, the buffers and pH electrode were incubated at 25°C in a temperature regulated water bath to ensure that the pK’s of the buffers used in our experiments were the same as those reported by our supplier.

The validity of our data was questioned suggesting that the equilibrium pH as calculated by the Henderson-Hasselbalch equation is higher than the measured equilibrium pH (8). However, we note that this author actually calculated the equilibrium pH by entering the initial (pre-equilibrium) reactant concentrations, [A-]i and [HA]i, into the Henderson-Hasselbalch equation. Since the Henderson-Hasselbalch equation and any equation modeling acid-base equilibrium reactions only holds true for reactant concentrations at equilibrium, one has to enter the equilibrium reactant concentrations into the Henderson-Hasselbalch equation to calculate the equilibrium pH. In our study, the equilibrium reactant concentrations were expressed in terms of the initial reactant concentrations, i.e. [A-]e = [A-]i – y and [HA]e = [HA]i + y. When one enters the equilibrium reactant concentrations, [A-]i – y and [HA]i + y, into the Henderson-Hasselbalch equation to calculate the equilibrium pH, the measured and calculated pH values agree.

Our analysis was also criticized from the viewpoint that the goal of acid-base equilibrium calculations in clinical medicine ought to be aimed at quantifying and characterizing the metabolic component of an acid-base disorder (e.g. base excess) rather than defining the equilibrium pH (9,22). We disagree. If the goal is to quantify and characterize the metabolic component of an acid-base disorder, then we feel that quantification and characterization of the partitioning of excess H+ among the various buffers as provided by our model can provide important additional insight. However, in addition to the metabolic component, the respiratory component is also important clinically. Moreover, in clinical medicine, defining the equilibrium pH in certain circumstances is more essential than defining the base excess in guiding the treatment of mixed acid-base disorders. For example, in a patient with mixed metabolic acidosis and chronic respiratory alkalosis, the goal of therapy is to normalize the systemic pH rather than to correct the base excess. Indeed, therapy aimed at correction of the base excess will result in worsening systemic alkalemia in this clinical setting. Other examples of this kind are purposefully omitted for the sake of brevity.

Complexity is apparently also an issue in that it was suggested that the classic analytical methods utilized to predict the equilibrium pH are less complex and cumbersome than our current mathematical model (9). Although our mathematical model may be more complex than certain classic analytical methods, in our view complexity is not a sufficient criterion for choosing between mathematical models. Furthermore, proponents of the Stewart strong ion approach have long argued that although the Stewart strong ion approach is more complex and cumbersome than the Henderson-Hasselbalch approach, the Stewart strong ion formulation is superior to the Henderson-Hasselbalch approach since SID is purported to play a mechanistic role in acid-base physiology. This contrasts with the same author’s previous analysis: “However, like the BE approach and like any other method derived from considerations involving the calculation of interval change via the assessment of initial and final equilibrium states, the Stewart method does not produce mechanistic information. These are basically bookkeeping methods. To believe otherwise risks falling prey to the computo, ergo est (I calculate it, therefore it is) fallacy” (23). We view the latter statement as a justification for the need for our study and model. Therefore, although our mathematical model may be more complex than other formulas, it is the first to be based solely on the underlying mechanistic physical chemistry involved.

We also disagree that the Stewart strong ion formulation is predictive in physiological fluids (9). There is no mathematical model that is predictive in vivo (including our new model) since the equilibrium partial pressure of CO2 (PCO2) in physiological fluids cannot be predicted as a result of the modulation of alveolar ventilation in various acid-base disorders. In this regard, neither the Stewart strong ion model, Henderson-Hasselbalch equation nor any other mathematical model is predictive in bicarbonate-buffered physiological fluids in vivo. Specifically, both the Stewart strong ion formulation and Henderson-Hasselbalch equation consist of an equilibrium term, PCO2, and are therefore not predictive in physiological fluids where the equilibrium PCO2 may differ from its initial value in acid-base disorders.

Finally, we must disagree with the contention regarding the superiority of the Stewart strong ion model over the Henderson-Hasselbalch equation (7). Both the Henderson-Hasselbalch and Stewart strong ion approaches are based on equilibrium reactant concentrations. In a multiple buffered solution, the isohydric principle (a well accepted principle in acid-base chemistry) underscores the fact that any buffer pair (assuming the pK’ is accurately known) can be utilized to calculate the equilibrium pH value. This fact alone necessitates that the Henderson-Hasselbalch equation and Stewart strong ion model are theoretically identical quantitatively in terms of their accuracy in calculating the equilibrium pH. Indeed, recent analysis has demonstrated that the Henderson-Hasselbalch equation and Stewart strong ion model are identical quantitatively in terms of their accuracy in calculating the equilibrium pH in a multiple buffered solution (6). If the Stewart strong ion model is quantitatively superior to the Henderson-Hasselbalch equation as suggested (7), then in our view it is incumbent on those holding this view to provide a valid mathematical explanation as to why the Stewart strong ion model can be mathematically simplified to the Henderson-Hasselbalch equation as shown previously (6). In addition, those who maintain that the Stewart approach and by inference the SID calculation is mechanistically superior in interpreting acid-base phenomenology need to provide a valid physicochemical explanation as to why, for a given change in SID, electroneutrality is maintained (i.e. [Na+] + [H+] - [Cl-] - [OH-] = 0) regardless of the actual value of [H+] as demonstrated in the example given in Table 3 of our study (5). This example highlights the lack of a causal connection between changes in SID and [H+].

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21. http://www.igena.com.pl/pdf/buffers.pdf

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