## Abstract

Currently, three strong ion models exist for the determination of plasma pH. Mathematically, they vary in their treatment of weak acids, and this study was designed to determine whether any significant differences exist in the simulated performance of these models. The models were subjected to a “metabolic” stress either in the form of variable strong ion difference and fixed weak acid effect, or vice versa, and compared over the range 25 ≤ Pco_{2} ≤ 135 Torr. The predictive equations for each model were iteratively solved for pH at each Pco_{2} step, and the results were plotted as a series of log(Pco_{2})-pH titration curves. The results were analyzed for linearity by using ordinary least squares regression and for collinearity by using correlation. In every case, the results revealed a linear relationship between log(Pco_{2}) and pH over the range 6.8 ≤ pH ≤ 7.8, and no significant difference between the curve predictions under metabolic stress. The curves were statistically collinear. Ultimately, their clinical utility will be determined both by acceptance of the strong ion framework for describing acid-base physiology and by the ease of measurement of the independent model parameters.

- acid-base balance
- strong ion difference
- mathematical model
- pH

on the basis of earlier work by Singer and Hastings in 1948 (15), Stewart published his landmark theories of acid-base chemistry in 1981 (18), and over the last 20 years or so his physicochemical system has steadily gained acceptance (7, 11, 21). Despite its strengths, his approach has been criticized for its singular treatment of the total plasma concentration of nonvolatile weak acids represented by A_{tot}, the net maximum negative charge concentration (19). The main contributors to A_{tot} are known to be albumin and phosphate, and in Stewart's algorithm these are modeled by using the single dissociation constant *K*_{A}. This criticism is reinforced because the values assigned to A_{tot} and *K*_{A} are poorly documented (10, 20). Subsequent weak acid models have sought to address these shortcomings (5, 6, 20). The most successful of these are the models by Figge et al. and Watson. In the Figge model, both the albumin and phosphoric acid systems are represented completely, whereas in the Watson model, the author uses simplified algorithms for both. When these modifications are incorporated into the original Stewart framework, we have three models for the physicochemical representation of the human acid-base system. They are the original Stewart model, the Figge model (5, 6), and the Watson model (20). The analysis that follows will examine the theoretical performance of these models under conditions of simulated biochemical stress. Intramodel linearity and intermodel correlation will be used to assess model performance.

### Glossary

#### Terminology and Definitions

- Strong cation
- A cation that is fully dissociated at physiological pH (e.g., Na
^{+}) - Strong anion
- An anion that is fully dissociated at physiological pH (e.g., Cl
^{−}) - SID
- Strong ion difference = ∑ (Strong cations) − ∑ (Strong anions)
*K′*_{W}- Ion product for water
*K*_{c1}- Combined equilibrium and solubility constants linking Pco
_{2}and [HCO_{3}^{−}] *K*_{c2}- Dissociation constant for the transformation of HCO
_{3}^{−}into CO *K*_{A}- Dissociation constant for A
_{tot} *K*_{p1},*K*_{p2},*K*_{p3}- Respective dissociation constants in the phosphoric acid system
*K*_{Asp}- Dissociation constant for the 98 aspartic and glutamic acid groups on albumin
*K*_{Tyr}- Dissociation constant for the 18 tyrosine groups on albumin
*K*_{Lys}- Dissociation constant for the 77 low-pK lysine and arginine groups on albumin
- (
*K*_{Alb})_{i} - The
*i*th dissociation constant in the albumin system representing 16 histidine groups, 1 cysteine group, 1 terminal amine group, and 1 terminal carboxyl group, making*n*= 19 in total

Brackets denote concentration.

## SUMMARY OF THE MODELS

All of the models use the effects of SID, Pco_{2}, and total [weak acid] as independent variables to predict the dependent variables [H^{+}] and [HCO_{3}^{−}] and thence describe the acid-base status of human plasma. Furthermore, each model is described by a series of equilibrium equations that, when combined with a mathematical statement of electroneutrality, form the final algorithm.

Several equations for mass conservation are shared between the models.

These are: with different equations proposed to account for the influence of weak acids.

The general statement of electroneutrality is which gives rise to the general form of each algorithm where WA is the individual model for the effect of weak acids in each case.

### Stewart Model

This is the simplest of the three models.

The statement of system electroneutrality is The equations for weak acid dissociation and conservation are resulting in the weak acid term [A^{−}] = [A_{tot}]*K*_{A}/(*K*_{A} + [H^{+}]), which, when substituted into the general model, gives the Stewart model.

### Watson Model

This model divides the effect of the weak acids into two main groups: the relatively minor effect of the phosphoric acid system (P_{i}^{−}) and the major effect of albumin (Alb^{−x}).

The statement of system electroneutrality is The dissociation equations for the phosphoric acid system are with the unifying mass conservation equation where Z_{Phos} is the charge per mole of total phosphate and P_{i,Tot} is the total serum phosphate concentration. Because the values of *K*_{p1} and *K*_{p3} lie well outside the physiological range (p*K*_{p1} = 2.12, p*K*_{p3} = 12.66; Ref. 6), a simplified model for the phosphoric acid system can be constructed using only *K*_{p2}, that is: Z_{Phos} = 2 − [H^{+}]/(*K*_{p2} + [H^{+}]).

A slightly different approach is taken when modeling the total effect due to albumin. Here, three parameters are used, all determined from published data (20). They are the net negative fixed charge (A= −21 eq/mol), the number of histidine residues actively taking part in proton buffering (A_{H,Tot} = 16/mol) and the dissociation constant for the imidazole side chain (*K*_{H} = 1.77 × 10^{−7} eq/l).

Therefore, [A] = [albumin] × A/66,500 and [A_{H,Tot}] = [albumin] × A_{H,Tot}/66,500, with [albumin] representing the concentration of serum albumin in grams per liter.

These result in the following model for albumin: The weak acid term is therefore which is substituted, as before, to complete the Watson model.

### Figge Model

This is the most complex model.

The statement of system electroneutrality is shared with the Watson model: The differences lie in the complete treatment of the phosphoric acid and albumin systems.

The respective charges per mole (Z) are calculated from the following formulas: and [P_{i}] is calculated as before and [Alb^{−x}] = Z_{Alb} × [albumin]/66,500 where [albumin] is measured in grams per liter.

Once again, substitution into the general equation results in the Figge model.

## MATHEMATICAL AND STATISTICAL METHODOLOGY

The analysis is based on the derivation of a simulated pH for each Pco_{2} in an appropriate range for normal, acidemic, and alkalemic plasma and then using this pH to construct log-linear titration curves that enable direct statistical comparison between the models. Because each of the models expresses the dependent variable [H^{+}] implicitly and also because each model cannot be directly solved for [H^{+}], an iterative process is required. For the sake of mathematical simplicity, each model was rearranged by using SID as the “dependent” variable and differentiated to find dSID/d[H^{+}], thus permitting the derivation of [H^{+}] by Raphson iteration. Once derived, [H^{+}] is converted to pH by the conventional means of finding its negative logarithm to base ten.

To quantify the individual and combined effects of nonrespiratory parameters on pH prediction, three main groups were examined. These are *group 1*, fixed [albumin]/A_{tot} with variable SID; *group 2*, fixed SID with variable [albumin]/A_{tot}; and *group 3*, variable SID and [albumin]/A_{tot}.

For simulations within each group, Pco_{2} ranged from 25 to 135 Torr in 1-Torr increments and [P] was set at 1.16 mmol/l in the Figge and Watson models. Where applicable, A_{tot} was calculated by using the published value of 0.378 mmol/g albumin (17).

### Group 1

[Albumin] was set at 42.0 g/l in the Figge and Watson models. A_{tot} was calculated as ∼15.9 mmol/l for the Stewart model. Each simulation was performed at four SID values to simulate very acidemic (23.3 meq/l), acidemic (30.3 meq/l), normal (38.3 meq/l), and alkalemic plasma (46.3 meq/l). It was felt that this range would encompass the vast majority of clinical scenarios. Approximate equivalents are tabulated in Table 1.

### Group 2

SID was set at 38.3 meq/l in each subgroup. Each simulation was performed at three [albumin]/A_{tot} values to simulate acidemic (60.0 g/l or 22.7 mmol/l), normal (42.0 g/l or 15.9 mmol/l), and alkalemic plasma (25.0 g/l or 9.5 mmol/l). Once again, approximate equivalents are tabulated in Table 2.

### Group 3

There were four subgroups investigated, two acidemic and two alkalemic, for each model. The acidemic groups were either normal SID, low [albumin]/A_{tot} or low SID, normal [albumin]/A_{tot} adjusted to give a pH = 7.31 (Pco_{2} = 40 Torr). Conversely, the alkalemic groups were either normal SID, high [albumin]/A_{tot} or high SID, normal [albumin]/A_{tot} adjusted to give a pH = 7.47 (Pco_{2} = 40 Torr). The approximate equivalents are tabulated in Table 3.

A Monte Carlo simulation was written and compiled in Pascal to perform 2,000 simulations on each Pco_{2} point for each perturbation of each model. The resulting output (model pH and standard deviation) was collected and imported into a propriety statistical software package for analysis. Measurement means and variances from clinical or company literature were used as input parameters for the iterative algorithms. Simulation normal variates were generated by the Box-Muller method.

The resulting pHs obtained in each group were plotted against Pco_{2} as a semilog plot, to give the expected linear, negative slope, log-CO_{2} titration curves (1, 14, 22).

Statistical analysis was by regression of the derived Watson or Figge pH on the derived Stewart pH at a 95% level of confidence with comparison to a line of identity. Simple pairwise correlation was also performed. Results were accepted as significant if the resulting *P* value was <0.05.

The software packages used were TurboPascal for Windows (v1.5), OpenOffice for the spreadsheet analysis, and STATA for the statistical analysis.

## RESULTS

The models were analyzed in two ways.

First, to gauge general model fit and linearity, the simulation was run with all the parameter standard deviations set to zero. This resulted in sets of standard curves for each model at each nonrespiratory perturbation. The curve slopes were tabulated and plotted.

Second, the simulation was run with the standard deviations set at agreed laboratory values. Once again sets of curves were generated and analyzed statistically for significance.

### Model Fit With Zero Standard Deviation

#### Group 1: fixed [albumin]/A_{tot}, variable SID.

Each model was analyzed by linear regression against log(Pco_{2}). There was no significant difference in regression slope within the four subgroups. As seen in Table 4, though, there is a significant difference in slope between each of the groups with the slope tending toward zero as the SID falls. This is in keeping with the observations of many other authors and may be, to a certain extent, adaptive.

The log(Pco_{2}) regression results are summarized graphically in Fig. 1. It is obvious that the regression lines for each model within each subgroup virtually overlie one another. There isvery minor divergence of the curves at high Pco_{2} in very acidemic plasma (∼0.01 of a pH unit at pH 6.63 and Pco_{2} 135 Torr), although this is clinically inconsequential.

Each model is represented at each SID value ranging from alkalemic (*top curves*) to very acidemic (*bottom curves*). The titration curves converge toward a theoretical virtual point at high Pco_{2} and low pH. There are no significant alinearities nor differences in mapping; the subgroup curves virtually overlie one another and graphically appear as single thick lines.

#### Group 2: fixed SID, variable [albumin]/A_{tot}.

Once again, each model was analyzed by linear regression against log(Pco_{2}). There was no significant difference in regression slopes within the three subgroups. As seen in Table 5, though, there is a significant difference in slope between each of the groups with the slope tending toward zero as the [albumin]/A_{tot} rises.

The log(Pco_{2}) regression results are summarized graphically in Fig. 2. It is obvious that the regression lines for each model within each subgroup virtually overlie one another.

#### Group 3: variable SID and [albumin]/A_{tot}.

Once again, each model was analyzed using linear regression against log(Pco_{2}). As before, there was no significant difference in regression slopes within the two subgroups (see Table 6).

The log(Pco_{2}) regression results are summarized graphically in Fig. 3. It is obvious that the regression lines for each model within each subgroup virtually overlie one another.

### Statistical Analysis

Before formal analysis, mean and standard deviations for the independent parameters had to be selected to permit accurate simulation. Values were sought from the published literature and from manufacturers' specifications (Table 7). Where compound variables were involved (e.g., SID), pooled variance was calculated from individual standard deviations and checked with a separate long-run simulation.

The main simulator was primed with these specifications and set to run for 2,000 cycles. Results were collected and analyzed using STATA with the predicted Watson and Figge pH values compared directly with the predicted Stewart pH (as the “gold” standard) by linear regression and correlation. In every case the results were highly correlated (*r* > 0.99) with a regression slope equal to 1.00. That is, there was no significant difference between either model compared with the Stewart model; they are, for all intents and purposes, identical. A representative subset of results is illustrated in Fig. 4.

## DISCUSSION

The three models examined describe the acid-base behavior of human plasma by using a robust mechanistic, physicochemical system. Each model uses the independent effects of carbon dioxide, the plasma electrolytes (calculated as SID), and the plasma weak acids. Through a series of simultaneous equations based on the law of mass action, the law of mass conservation, and a separate statement of electroneutrality, the dependent variables pH and bicarbonate can be derived. The major difference between the models is in their treatment of the plasma weak acids.

In its original form, the Stewart algorithm models the collective effect of weak acids using the variable A_{tot}. The clinical utility of the model is severely hampered largely because A_{tot} is a difficult entity to quantify (17). Also, because A_{tot} is represented by the single dissociation constant *K*_{A}, it is an entity with a nonlinear dissociation curve against pH. This causes theoretical problems because we know that the weak acid dissociation curves are linear in the biological pH range (6.8 ≤ pH ≤ 7.8) (1, 14). Nevertheless, this theoretical concern does not seem to translate into reality when this particular model is constructed. Furthermore, it is known that, physiologically, the weak acids exert their effect on pH through complex interactions involving the polyprotic macromolecule albumin and the phosphoric acid system. As mentioned above, these interactions result in linear titration curves over biologically relevant pH ranges. For this reason alone, models that incorporate the effects of albumin and phosphates will give a more accurate reflection of hydrogen ion regulation.

In 1992, Figge et al. (6) developed a model for albumin based on microenvironmental pK values determined by nuclear magnetic resonance spectroscopy. They used it to compare calculated pH (pH_{c}) against measured pH (pH_{m}) in both human albumin and human serum protein solutions. Their results, using linear regression of pH_{c} on pH_{m}, revealed no significant difference from the line of identity, and their model remains, to this day, the most complete model of human albumin available for these purposes. They also showed that, among the serum proteins, albumin completely accounts for the weak acid effect at physiological pH. Almost as a footnote in their discussion, they suggested that “it might be possible to use a derivative of the mathematic (sic) model to help classify and interpret acid-base disorders in clinical settings.”

Seven years later in 1999, Watson developed a similarly accurate, although simplified, single-association-constant model derived from the known properties of human albumin (20). In it, the effect of albumin was divided into a net fixed negative charge resulting from the charge balance between various amino acid groups at physiological pH and the net variable negative charge resulting from the imidazole groups on histidine (pK ≈ 7.0). Using the albumin solution data published by Figge (5) to calculate pH, and comparing that pH to the pH_{c} of Figge, they showed a statistically significant difference between the two (regression slope = 1.032, *P* < 0.05). The main disparity was in the extreme alkaline end of the range where some of the Watson points were far enough above the line of identity to exert significant leverage on the slope of the regression line. However, it was felt by the authors that the deviation of pH (≈+0.04) was clinically and practically insignificant.

In each of the Figge and Watson papers, the human albumin and the phosphoric acid system algorithms were directly substituted for A_{tot} in the general Stewart framework. This resulted in the construction of final models that take Pco_{2}, the various commonly measured electrolyte concentrations used to calculate SID, [PO], and [albumin] as their input parameters. Given that they reflect in vivo acid-base reality, both of these models are capable of being used in routine clinical practice.

This present paper has examined the effects of simulated fixed alterations in nonrespiratory parameters (SID and albumin/A_{tot}) during a stepped Pco_{2} rise on the resulting CO_{2} titration curves for the Figge and Watson models compared with the original Stewart model. In every case, the models retained linearity over the log(CO_{2})/pH ranges while showing no statistically significant deviation from the line of identity. These three models are, for all practical purposes, identical. From a clinical perspective, and compared with the Stewart model, the models of Figge and Watson are far easier to implement as they use commonly measured parameters to determine pH.

Finally, both Figge and Watson compared their weak acid models with artificial in vitro solutions initially containing only albumin and then in samples containing albumin and globulins. They demonstrated very close correlation between pH_{c} and pH_{m}. However, neither author compared the predicted results in an in vivo whole blood situation. It is possible, using the data published by Brackett et al. (2), to construct these comparative in vivo titration curves using samples with no known “metabolic” disturbance (Fig. 5). When the slopes of these titration lines are examined, we see that the present models predict a lower pH with increasing Pco_{2} above 30 Torr. This is probably due to the incomplete modeling of the weak acids. For example, hemoglobin is known to buffer at least six protons in a saturation-dependent manner on the β-chain COOH-terminal histidines, on the α-chain NH_{4}-terminal valines, and possibly on other general α-chain histidines (9). Therefore, inclusion of this major buffer in future whole blood rather than plasma models may account for the slope difference.

Whether either the Figge or Watson model gains clinical utility is dependent on many factors. The present interpretation of acid-base parameters is based on the use of the Henderson-Hasselbalch equation, which involves analysis of codependent perturbations in Pco_{2} and either standard bicarbonate (North American school) or standard base excess (Copenhagen school) to gain some insight into the human biochemical milieu. Despite the transatlantic divisions in opinion (13), the great strength of the Henderson-Hasselbalch acid-base model is, that in clinical practice at least, it works (3, 4). Until strong ion theory and its associated models exert a stronger influence over the current Henderson-Hasselbalch approach, it may be relegated forever to the “interesting but not clinically useful” category of physiological theory (8, 12, 16).

In summary, from the results, we may conclude that there is no significant difference between the models in pH prediction across a Pco_{2} range from 25 to 135 Torr with simultaneous “nonrespiratory stress.” In terms of clinical utility, both the Watson and Figge models permit the input of standard measured values (Pco_{2}, electrolyte values to calculate SID, serum [albumin], and plasma [phosphate]) and, as such, are more practical than the original Stewart model. With its simple but accurate modeling of albumin, the Watson model lends itself to easier electronic programming.

## APPENDIX

Equilibrium constants (6, 17, 19):

Ion product for water | K′_{W} | 4.40E-14 | (eq/l)^{2} |

Carbonic acid system | K_{cl} | 2.46E-11 | (eq/l)^{2}/Torr |

K_{c2} | 6.00E-11 | (eq/l) | |

A_{TOT} | K_{A} | 3.00E-07 | (eq/l) |

Phosphoric acid system | K_{p1} | 1.22E-02 | (mol/l) |

K_{p2} | 2.19E-07 | (mol/l) | |

K_{p3} | 1.66E-12 | (mol/l) | |

Human serum albumin | |||

Aspartic + glutamic acids | K_{Asp} | 1.00E-04 | (eq/l) |

Tyrosine | K_{Tyr} | 2.51E-10 | (eq/l) |

Lysine + arginine | K_{Lys} | 3.98E-10 | (eq/l) |

Histidine 1 | (K_{Alb})_{1} | 7.58E-08 | (eq/l) |

Histidine 2 | (K_{Alb})_{2} | 6.03E-08 | (eq/l) |

Histidine 3 | (K_{Alb})_{3} | 7.94E-08 | (eq/l) |

Histidine 4 | (K_{Alb})_{4} | 3.23E-08 | (eq/l) |

Histidine 5 | (K_{Alb})_{5} | 9.77E-08 | (eq/l) |

Histidine 6 | (K_{Alb})_{6} | 4.90E-08 | (eq/l) |

Histidine 7 | (K_{Alb})_{7} | 1.78E-07 | (eq/l) |

Histidine 8 | (K_{Alb})_{8} | 4.37E-07 | (eq/l) |

Histidine 9 | (K_{Alb})_{9} | 1.41E-05 | (eq/l) |

Histidine 10 | (K_{Alb})_{10} | 1.74E-06 | (eq/l) |

Histidine 11 | (K_{Alb})_{11} | 6.76E-07 | (eq/l) |

Histidine 12 | (K_{Alb})_{12} | 1.86E-07 | (eq/l) |

Histidine 13 | (K_{Alb})_{13} | 1.51E-06 | (eq/l) |

Histidine 14 | (K_{Alb})_{14} | 5.01E-08 | (eq/l) |

Histidine 15 | (K_{Alb})_{15} | 6.31E-06 | (eq/l) |

Histidine 16 | (K_{Alb})_{16} | 5.01E-08 | (eq/l) |

Cysteine | (K_{Alb})_{17} | 3.16E-09 | (eq/l) |

Amino terminus | (K_{Alb})_{18} | 1.00E-08 | (eq/l) |

Carboxy terminus | (K_{Alb})_{19} | 7.94E-04 | (eq/l) |

## Footnotes

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- Copyright © 2005 the American Physiological Society