Clinical Application of Stewart's Theory & Strong Ion Difference
ACVIM 2008
Robert J. Callan, DVM, MS, PhD, DACVIM
Fort Collins, CO, USA


The physiology of acid-base balance is a complex subject that requires a solid understanding of fundamental acid-base chemistry, respiratory physiology, and the ability to integrate multiple interrelated factors. Successful teaching of this subject is challenging and is often reflected by the difficulty students and graduate veterinarians have in applying the concepts to clinical case evaluation and management. Part of the difficulty may arise from the quantitative nature of the subject and the difficulty in integrating the numerous variables that influence acid-base physiology. In 1981, P.A. Stewart proposed the strong ion difference model for acid-base chemistry. There are several excellent reviews describing and comparing classical base excess and strong ion models of acid-base physiology.1-3 While the strong ion model is mathematically and quantitatively complex, beneath the computational complexity lies an elegant model that lends itself to effective evaluation of clinical acid-base physiology and medical intervention. Understanding both classical and strong ion difference models will allow the clinician to better utilize the advantages of each method and conceptualize acid-base disturbances in medicine.


Historically, evaluation of acid-base disturbances was limited by the methods of measurement and what analytes could be measured in blood and other body fluids. In the early 1900's, Henderson described physiologic acid-base phenomenon relative to the dissociation of carbonic acid:

H2O + CO2 ↔ H2CO3 ↔ H+ + HCO3-

He was able to describe the chemical equilibrium mathematically and developed the Henderson Equation:


Hasselbalch then introduced the relationship of the partial pressure of CO2 (Pco2) to the solubility of CO2 (Sco2) in aqueous solutions and adapted the pH (-log[H+]) notation to this equation resulting in the Henderson-Hasselbalch Equation:


From this equation, determining the value of any two of the variables (pH, HCO3-, and Pco2) allows calculation of the third. This equation forms the fundamental basis for classical acid-base interpretation. Respiratory acid-base disturbances can be described by Pco2 and metabolic disturbances are related to [HCO3-]. However, it must be stressed that [HCO3-] is not independent of Pco2 as indicated by the chemical equilibrium equation (1).1 While the Henderson-Hasselbalch equation explains the relationship of Pco2 with pH and [HCO3-], it gives no further information on the other factors governing these variables.

Base Excess

Since [HCO3-] is partially dependent on Pco2, determination of [HCO3-] is not sufficient in describing the metabolic component of acid-base disturbances. In a patient with no respiratory abnormality, [HCO3-] is an accurate reflection of the metabolic disturbance. Base excess (BE) was developed as a means of assessing the magnitude of variation in [HCO3-] that was not due to changes in Pco2 and thus better represents the metabolic component of acid-base disturbances, particularly if pulmonary function is abnormal. BE is defined as the number of milliequivalents of acid or base that are needed to titrate 1 liter of blood to pH 7.40 at 37°C while the Pco2 is held constant at 40mmHg. This accounts for the effect of Pco2 on [HCO3-] and standardizes the metabolic changes in the [HCO3-] relative to a specific respiratory physiological state.

BE is influenced by the buffering effect of hemoglobin in whole blood. In vivo, the ion equilibration pool includes both whole blood and interstitial fluid compartments, reducing the effective hemoglobin concentration. Thus, BE should be corrected for the hemoglobin concentration of blood. The effect of hemoglobin also explains why BE can be falsely lowered (metabolic acidosis) in cases of acute respiratory acidosis where the acidosis caused by rapid hypercapnea is buffered greater in whole blood than it is in vivo. Standard Base Excess (SBE) accounts for this phenomena and calculates the base excess to a standard hemoglobin concentration of 5g/dl.

If an acid-base disturbance is purely respiratory in nature, then SBE = 0. Positive values for SBE indicate metabolic alkalosis and negative values indicate metabolic acidosis. Current blood gas analyzers measure blood pH and Pco2. Based on those values, [HCO3-] is calculated and BE or SBE is determined from a Siggaard-Anderson nomogram or derivations of the Van Slyke formula.

Anion Gap

Anion gap (AG) is commonly used to evaluate whether a metabolic acidosis is associated with an increase in unmeasured anions. AG is used to evaluate the acid-base contribution of unmeasured cations (UC) and unmeasured anions (UA) in serum. Unmeasured cations in this context are any positive charged ion other than Na+ and K+ and may include Ca2+ and Mg2+. Unmeasured anions are any negatively charged ion other than Cl- and is predominantly composed of charged proteins but also may include lactate, ketoacids (β-OH butyrate, acetoacetate), sulfate, phosphate, oxalate, urates, and other physiological or exogenously administered anions. The standard equation for the anion gap is:

AG = [Na+] + [K+] - [Cl-] - [HCO3-]

Normal AG varies slightly by species and in cattle ranges from 8.9-15 mEq/L.4 An increase in AG reflects an increase in unmeasured anions. AG is influenced by albumin or total protein concentration as well as the other anions listed above. Thus, anion gap must always be interpreted relative to serum protein concentration. Because of the confounding effect of changes in total protein or albumin, AG is not considered a reliable indicator of unmeasured anions in patients with abnormal protein concentrations. Anion gap can be adjusted for albumin or total protein in cattle using the equations:4,5

Albumin Adjusted AG = AG + 3.4 x (3.1 - [albumin g/dl])

Total Protein Adjusted AG = AG + 1.9 x (5.4 - [total protein g/dl])

Strong Ion Model

The strong ion model of acid-base balance was described in 1983 by Stewart.6 There are several published reviews describing the theory and mathematical computation of Stewart's quantitative acid base model.1-3,7 Quantitative acid-base chemistry is based on the physical laws of:

1.  Electroneutrality: The sum of all cationic charges must equal the sum of all anionic charges.

2.  Conservation of Mass: [A-] + [HA] remains constant [ATot].

3.  Laws of Mass Action: Ka[HA] = [H+][A-] for the dissociation equilibrium HA ↔ H+ + A-.

The model proposes that pH of plasma is determined by three independent factors and their associated equilibrium constants:

1.  Pco2

2.  Net Strong Ion Charge (Strong Ion Difference, SID)

3.  Concentration of nonvolatile weak buffers (ATot)

The strong ion model recognizes that pH and HCO3- are dependent on the Pco2 (Henderson-Hasselbalch equation), the strong ion difference (SID), and the ionization of weak acids. Strong ions are defined as all organic and inorganic salts that are fully dissociated at physiological pH. The SID is defined as the difference between all strong cations and strong anions. Nonvolatile weak buffers include proteins and inorganic phosphates as well as other organic and inorganic salts with dissociation constants near physiological pH.

Figure 1.
Figure 1.

Basic Gamblegram representing the primary components of the strong ion model of acid base balance.

While the model is mathematically complex, it provides a conceptual framework for understanding mixed acid base interactions than can not be accomplished with the Henderson-Hasselbalch and Base Excess methods. This conceptual framework is effectively demonstrated by use of a graphical representation of the components of quantitative acid-base equilibrium called a Gamblegram (Figure 1). Gamblegrams depict the relationship of positive and negative charged ions that regulate acid-base balance in a solution. When used with the strong ion model, the independent variables of strong cations are compared with strong anions and ionized weak buffers (A-) along with the dependent variable, HCO3-. The primary criterion of a Gamblegram is that the sum of the cation charges must equal the sum of the anion charges (electroneutrality). Using this graphical representation, changes in HCO3- secondary to changes in SID or ATot are inferred and the influence on metabolic acid-base disturbances is demonstrated. An excellent software program that can be used to examine acid-base disturbances in a quantitative manner is AcidBasics II (available at

Conventional evaluation of acid-base balance using [HCO3-] and BE allows acid-base disturbances to be divided into four mechanisms, 1) Respiratory Acidosis (upwards arrow Pco2), 2) Respiratory Alkalosis (downwards arrowPco2), 3) Metabolic Acidosis (downwards arrowBE), and 4) Metabolic Alkalosis (upwards arrowBE). The strong ion model allows further differentiation of metabolic acid-base disturbances into those caused by changes in SID and those caused by changes in ATot. More importantly, the strong ion model allows for the characterization of mixed metabolic acid-base disturbances that could otherwise be overlooked by evaluation of AG and BE alone. Changes in SID can be characterized further through evaluation of Fencl-Stewart equations or strong ion gap (SIG) as described below.

Simplified Strong Ion Model

The simplified strong ion model described by Constable2, 8 reduces Stewart's strong ion model equation to a logarithmic equation relating pH to the same three independent variables PCO2, SID, and ATot (same as Stewart) but only three constants (Ka, K1', and S) instead of five.


This simplified model is particularly useful from a research standpoint in determining the estimated ATot, Ka, and pKa values for blood from different species (Table 1).4,8-13 One of the findings of these studies is that there are considerable species differences in these parameters. Thus, extrapolation of ATot and Ka between species should be taken with caution.

Table 1. Measured SID, estimated ATot, Ka, pKa and albumin and total protein conversion factors for normal animals in different species.













17.2 ± 3.5

0.80 X 10-7









24.3 ± 4.6

0.67 X 10-7






-0.0093 (alb)



17.4 ± 8.6

0.17 X 10-7








15.0 ± 0.32

15.0 ± 2.8

2.22 X 10-7








40.5 ± 1.2


0.87 X 10-7








41.1 ± 2.7

19.2 ± 6.1

0.84 X 10-7







Determination of ATot and Ka allows for the calculation of species specific estimates of [A-] that could be substituted in Fencl-Stewart equations and estimation of the strong ion gap described below. The formulas governing the relationship between [ATot] and [A-] are:

HA ↔ H+ + A- such that: 1) [ATot] = [HA] + [A-]2) Ka = [H+][A-]/[HA]3) [A-] = [ATot]/(1 + 10pKa-pH)

Fencl-Stewart Equations

Fencl and others have described and applied quantitative methods to evaluate the impact of individual factors of the strong ion model to acid-base balance.1,7,14-19 This approach is based on the concept that net changes in the [HCO3-] from normal are a reflection of the impact of Pco2, SID, and ATot. BE (or SBE) is corrected for the influence of Pco2 on [HCO3-] and thus is a reflection of the metabolic components SID and ATot. SID is further described as the influence of dilutional acidosis (characterized by downwards arrow[Na+]) and concentration alkalosis (characterized by upwards arrow[Na+]) and changes in the [Cl-] relative to a standard [Na+]. This method allows the clinician to assess the contribution of free water ([Na+]), chloride, and weak buffer disturbances on acid-base balance and also estimate the contribution of net unmeasured ions ([unmeasured cations]--[unmeasured anions]). The common Fencl-Stewart equations used in human and veterinary medicine are:

Base excess cause


Common formula

Generalized formula

Free water effect


0.3 ([Na+obs] - 140)

z ([Na+obs] - [Na+norm])

Chloride changes


[Cl-norm] - [Cl-cor]

[Cl-norm] - [Cl-cor]

Corrected chloride


[Cl-obs] x 140/[Na+obs]

[Cl-obs] x [Na+norm]/[Na+obs]

Changes in ATot


3.4 x (4.5 - [Alb g/dl])

[A-norm] - [A-obs]

Net unmeasured ions (NUI)


BEnet - (BEfw + BEcl + BEATot)

BEnet - (BEfw + BEcl + BEATot)

The Fencl-Stewart formulas used in veterinary medicine are generally adapted from human medicine and can be applied with some modifications based on differences in normal electrolyte balance between species. Species specific values for z, the change in SID caused by a change in [Na+], have been described.7 It may also be appropriate to substitute species specific normal mean electrolyte, albumin, or total protein values into the appropriate equations. Lastly, more accurate assessments may be attained by using species specific ATot calculations (see box below), however this has not been critically examined.

Species Specific Determination of z and BEATot

1.  z = SIDnorm/[Na+norm] or ([Na+norm] - [Cl-norm])/[Na+norm]

2.  BEATot = [A-norm] - [A-obs] = (ATot-alb)([Albnorm] - [Albobs])/(1+10(pKa-pH))
= (ATot-tp)([TPnorm] - [TPobs])/(1+10pKa-pH))

Note that [Alb] and [TP] are in g/L for the ATot-alb and ATot-tp constants listed in the table above.

In the Fencl-Stewart equations, lactate, ketoacids, and increases in inorganic phosphate are included as part of the BENUI and may result in a lower (more negative) value. Calcium and magnesium also contribute to the BENUI and may result in a higher (more positive) value. The Fencl-Stewart equations can be expanded further to incorporate the affects of other strong anions (such as lactate), strong cations (Ca2+, Mg2+) or abnormal elevations of inorganic phosphate if measured values are available. At this time there is no evidence that inclusion of these additional ions provides additional clinical utility.

Strong Ion Gap

The contribution of unmeasured cations and unmeasured anions to acid-base disturbances can be estimated in many different ways. Anion gap provides one estimate and is most useful if the contribution of weak buffers (ATot, total protein, albumin, inorganic phosphate) are within normal limits. The BENUI as described above is estimated by subtracting BEfw, BEcl, and BEATot from the calculated SBE which is determined from measurements of Pco2 and pH. Strong Ion Gap (SIG) is defined as:2,20,21

SIG = [strong cations] - [strong anions]

The SIG can be estimated as the difference between [A-] and AG. Conceptually, SIG is similar to Net Unmeasured Ions (NUI = [unmeasured cations] - [unmeasured anions]) which would include strong cations and strong anions as well as weak buffers that are not measured in [A-]. For the purposes of clinical evaluation, SIG and NUI can be considered to be similar, with a negative values representing an excess of unmeasured anions. In normal patients, the SIG or NUI should be near 0. NUI values < -5 mEq/L are associated with increased morbidity and mortality in human pediatric ICU patients.14 Figure 2 shows the relationship of the measured SID (SIDmeas = [Na+] + [K+] - [Cl-]), AG, NUI, and SIG. SIG can be estimated using formulas and constants derived from the simplified strong ion model.

SIG = [A-] - AG = (ATot-alb)[Albobs]/(1+10(pKa-pH)) + [Cl-] + [HCO3-] - [Na+] - [K+]

Note that in the formula above, SIG includes the contribution of lactate and all other unmeasured cations and anions. As mentioned for BENUI, these additional cations and anions could be included in the calculated estimate of SIG but it is not know if this provides additional clinical utility.

Figure 2.
Figure 2.

Gamblegram representing specific components of the strong ion model of acid base balance and their relationship to measured SID, AG, and SIG.

Simplified Quantitative Methods

One of the detractions of the Fencl-Stewart and SIG equations are their complexity. Approaches utilizing simpler estimates have been described in human medicine and show some clinical utility.22, 23 Both of these methods combine the sodium and chloride (strong ion) effect and utilize constants that are convenient for simple calculations in order to estimate the NUI or SIG. The simplified method described by Story et al. shows good agreement with no bias when applied to human intensive care unit patients.23 Simplified methods may provide utility when initially screening blood electrolyte and acid-base analyses but need further investigation.


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16. Fencl V, et al. Annu Rev Med 1989;4017.

17. Leith DE, in Proceedings of the 9th ACVIM Forum. (New Orleans, LA, 1991), pp. 611.

18. Figge J, et al. J Lab Clin Med 1992;120(5):713.

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21. Kellum JA, et al. J Crit Care 1995;10(2):51.

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Speaker Information
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Robert Callan, DVM, MS, PhD, DACVIM
Colorado State University
Ft. Collins, CO