The Stewart Approach to Acid-Base Analysis

Owen T. Fink, DVM; Melinda S. Camus, DVM; Bruce E. LeRoy, DVM, PhD

Department of Medicine, College of Veterinary Medicine, University of Pennsylvania (Fink) and the Department of Pathology (Camus, LeRoy), College of Veterinary Medicine, University of Georgia, Athens, GA 30602-7388

Introduction

In the late 1970’s, a Canadian physiologist, Peter Stewart, introduced an alternate approach to understanding acid-base physiology. Prior to Stewart, the conventional view of acid base centered on the role of the bicarbonate buffer system, expressed mathematically via the Henderson-Hasselbalch equation. Stewart formulated a different approach to acid base that took into account many more components of physiologic fluids. Because it factored in many variables, this system was more technically complex than the conventional approach and required computers to completely solve the equations. The Stewart system had, however, the advantage of giving precise quantitative acid-base information based on a patient’s physiologic variables. A critical point of Stewart’s new approach was that it shifted emphasis away from the bicarbonate system and showed that a crucial determinant of acid-base equilibrium is the so-called strong ion difference. Since Stewart first published his approach, there has been much debate as to which approach provides the best quantitative information and the most accurate conceptual view of acid-base mechanisms.

Summary of the conventional approach to acid-base

The conventional (aka traditional, or bicarbonate centered) approach to acid-base analysis focuses on the CO₂-HCO₃- buffering system and its primacy in body fluids. This approach was developed over time through the work of such investigators as Sorenson, Bronsted, Lowry, Henderson and Hasselbalch. The principal chemical reaction is

CO₂ + H₂0<--> H₂CO₃<--> H + + HCO₃ -

Which leads to the equilibrium condition

[H +] = (K 1’ x [CO₂])/[HCO₃ -]

Or in its more familiar form

pH = pK + log [HCO₃ -]/(S x PCO₂)

This, the “Henderson-Hasselbalch Equation”, expresses the pH as a function of the concentration of bicarbonate, the partial pressure of CO₂, the dissociation constant (K) and the solubility of CO₂ (S). This mathematical framework led to the conceptualization that changes in [H +] are due to changes in [HCO₃ -] or PCO₂. By the 1960’s, this concept was firmly rooted and had led to the well-known respiratory/metabolic classification scheme for acid-base disorders. Primary changes in CO₂ were termed “respiratory” while primary changes in [HCO₃ -] were called “metabolic.” While this scheme is helpful for characterizing a disorder and can help direct therapy, the Henderson-Hasselbalch approach, as it is commonly used in practice, provides little quantitative data on a patient’s acid-base status. This fact has led to the proliferation of many empirical rules-of-thumb used to direct therapy. While these rules usually give appropriate results, they can hardly be said to constitute a rigorous system. Additionally, a flaw in using the rules-of thumb is that they make assumptions about levels of protein, phosphate, chloride and other ions, and hence are prone to error in situations where these values are abnormal.

Summary of the Stewart approach to acid-base

The Stewart approach to acid-base analysis (aka the “physiochemical approach”) is a mathematically rigorous system that describes the acid-base behavior of aqueous solutions. The goal of the Stewart approach is to establish the quantitative relationships that determine [H +] in any aqueous solution. Stewart's system rests upon 1) establishing a system of equations describing the behavior of aqueous solutions (for example, an aqueous solution is always electrically neutral) and then 2) using mathematics to find solutions to the set of equations. In principle, it is a logical, even simple approach. However, in all but the most basic cases, the mathematics involved in actually solving the equations is very difficult, and would have been virtually impossible prior to computers. Luckily, the equations are readily solved using computers and it is no coincidence that the Stewart approach emerged in the computing age. Critically no extensive mathematical knowledge is required to be able to understand, appreciate and use the Stewart approach.

A few helpful definitions

Acidic: A solution is acidic if [H +] > [OH -]. A substance is an acid if, when added to a solution, it leads to an increase in [H +], all other variables being equal.

Basic: A solution is basic if [H +] < [OH -].A substance is a base if, when added to a solution, it leads to an decrease in [H +], all other variables being equal.

Strong electrolyte: A substance that completely dissociates when dissolved in water, so that no molecules of the parent substance are present. NaCl is an example of a strong electrolyte, and Na + and Cl - are referred to as strong ions. By definition then, a salt water solution contains no molecules of NaCl, only free Na + and free Cl -. As such, the name “sodium chloride solution” is a bit of a misnomer. A strong ion will not participate in chemical reactions in aqueous solution. Na + in water stays as free floating Na + and will not react with other species.

Weak electrolyte: A substance that only partially dissociates when dissolved in water, so that molecules of the parent substance as well as the products of dissociation all exist together in solution. A weak acid is a weak electrolyte that dissociates to form H +. The dissociation of a weak acid satisfies the equation HA <-->H + + A - and, assuming an equilibrium state, has an equilibrium constant, K a, such that [H +] x [A -] = K a x [HA]. The value of K a, and thus the degree of dissociation of a weak electrolyte, changes depending on the conditions in the solution.

Independent variable: A variable that is deliberately manipulated and can be changed in a known way. Its value does not depend on other variables, and does not change in response to changes in other variables. It is the answer to the question “What do I change in my experiment?”

Dependent variable: A variable whose value changes in response to changes in the independent variables. It is the answer to the question “What do I observe?”

As an example, consider the job of building a six foot long, three foot high brick wall. The thickness of the wall and the number of masons are independent variables (for example, the number of masons is predetermined and does not depend on how thick the wall is to be). The time it takes to build the wall, however, is a dependent variable. It depends on both the required wall thickness and on the number of masons working.

Fundamental principles

Electroneutrality: An aqueous solution contains the same number of negative and positive charges.A consequence of this principle is that it is not possible to add a single species of ion to a solution all by itself. Some other species of opposite charge must always be added at the same time, and its amount and identity must be incorporated into calculations of the final result of such additions.

Conservation of mass: The amount of a substance in an aqueous solution remains constant unless 1) the substance is added or removed from the outside or 2) the substance is created or destroyed in a chemical reaction in the solution.

The case of pure water

The mathematical analysis in this section may seem simplistic to the point of condescension. However the same methodology used here will be applied to more complex situations further on. It is the methodology rather than the math that is of crucial importance.

Pure water contains H₂O, H + and OH - and undergoes the reaction:

H 20 <--> H + + OH -

At equilibrium, the concentrations of these three species are related by the equation

[H +] x [OH -] = K w [H₂O]

In this equation, K w is the dissociation constant. H₂O does not dissociate significantly, hence K w is extremely small. In biological solutions, the concentration of H + is at least four orders of magnitude smaller than the concentration of other relevant ions. So little H + and OH - are present in water that, were this any other molecule, we would ignore the dissociation products altogether and approximate the solution as 100% H₂O. The reason we do not ignore them is because, though its concentration is small, the biological impact of H + is, to quote Stewart, “out of all proportion to its magnitude.” The tiny degree of dissociation does, however, allow us to make the approximation that [H₂O] in water is constant. We therefore define a new value, K’ w, such that:

[H +] x [OH -] = K w [H₂O] = K’ w

Electroneutrality says that the solution must be electrically neutral. Because H₂O is has no charge, the principle of electroneutrality means that [H +] = [OH -]. (Note that this means that pure water, is by definition acid-base neutral). We now have two equations and two unknowns.

[H +] x [OH -] = K’ w

[H +] = [OH -]

The second equation allows us to make the substitution [OH -] = [H +]into the first equation, yielding

[H +] 2 = K’ w or [H +] = K’ w ½

Since our goal is to determine [H +], this equation completes the goal for pure water. As we can see, in pure water, K’ w alone determines [H +]. K’ w is a constant, but it does change significantly with temperature. The following table shows values of K’ w and [H +] in various solutions.

Since we know that pure water is always acid-base neutral, it is clear that pH = 7.0 does not mean neutral. Only at 25 ˚C does neutral water have a pH of 7.0. At body temperature (37 ˚C), neutral water has a pH of 6.7, revealing that blood plasma (pH = 7.4) is a significantly basic solution (indeed in plasma, [OH -] = 28 x [H +]).

Strong electrolytes in water

Consider an aqueous solution made by adding amounts of NaOH and HCl to water. Since Na + and Cl - are strong ions, by definition they completely dissociate in water. The only species in the solution are H 20, H +, OH -, Na + and Cl -. There is no NaCl and the commonly written equation Na + + Cl - <-->NaCl is false. As such the Na + and Cl - participate in no chemical reactions. From above, the equation governing the dissociation of H 20 into H + and OH - still holds true.

[H +] x [OH -] = K’ w

Since Na + and Cl - participate in no chemical reactions, their only effect on the acid-base balance of the solution comes from their charges. These must be taken into account for electroneutrality.

[Na +] + [H +] = [Cl -] + [OH -]

The quantities [Na +] and [Cl -] are known. For example if I add one mol of HCl to pure water and the final volume of the solution is one liter, then the resultant concentration of Cl - is 1 mol/L. Is the resultant concentration of H + also 1 mol/L? No, nor is it the sum of the amount of H + in the pure water prior to the addition of HCl plus the amount of H + added. This is a clear example of why strong electrolytes are independent variables and [H +] is a dependent variable.

Thus, we again have two equations and two unknowns ([H +] and [OH -]).

Water dissociation [H +] x [OH -] = K’ w

Electroneutrality [Na +] + [H +] = [Cl -] + [OH -]

The second equation allows us to make the substitution [OH -] = [H +] + [Na +] – [Cl -] into the first equation, yielding

[H+] 2 + ([Na +] – [Cl -]) x [H +]- K’ w = 0

This is a second order polynomial (“quadratic”) equation of the form ax 2 + bx + c = 0 where a = 1, b = ([Na +] – [Cl -]) and c = -K’ w. It is readily solved for the variable ([H +]) via the quadratic formula and the simplified solution is

x = ((b/2a) 2 – c/a) ½ - (b/2a) or

[H +] = ((([Na +] – [Cl -])/2) 2 + K’ w) ½ - (([Na +] – [Cl -])/2)

Our goal of determining [H +] in aqueous solutions containing strong electrolytes is complete. We can refine this expression, however, by noting that [Na +] and [Cl -] appear in the equation only as([Na +] – [Cl -]). If additional strong electrolytes (such as K +, SO 4 -, etc.) had been added, we would find that, by extension, these would only appear in the solution for [H +] in the form: the sum of the concentrations of all strong cations minus sum of the concentrations of all strong anions. Thus we define a new term, the strong ion difference ([SID]) that equals exactly this. The expression for [H +] simplifies to

[H +] = ((SID/2) 2 + K’ w) ½ - (SID/2)

What does this simplification mean? It means that SID is the only independent variable in determining the [H +] of a solution containing only strong electrolytes. How much Na + or K + or Cl - in particular is present does not matter—it is only the net difference in cations and anions that affects [H +]. This result follows from the fact that, as noted above, strong ions dissolved in water do not participate in chemical reactions and are important only because of their charge. The SID can thus be viewed as the net positive electrical charge in the solution due to the presence of strong ions.

There is an important conceptual implication to the SID. Consider this example. Conventionally, when you add HCl to water to make it more acidic, you imagine that you are adding H +, and that this is resulting in increased acidity. The equation that we have derived above turns this idea on its head. It says that instead of thinking about adding H +, you should focus on the fact that you are adding a strong anion (Cl -) without adding a strong cation. Thus you are decreasing the SID, which causes an increased [H +], the value of which can be precisely determined by the equation above. Performing this mathematical exercise reveals that when adding HCl to a solution, the resultant change in[H +] bears no simple or obvious relationship to the amount of H + added. Also salient is the fact that [OH -] in the solution changes an amount comparable to the change in [H +], even though no OH - was added. These results underscore that [H +] and [OH -] are dependent variables and show the importance of the SID.

Weak electrolytes in water

When weak electrolytes are added to the solution, the situation becomes more complex, both conceptually and mathematically. Consider the prototype weak acid, HA. This will dissociate in the following reaction

HA <--> H ++ A -

At equilibrium the reaction is governed by the relationship

[H +] x [A -] = K a [HA]

The principle of the conservation of mass says that the total amount of conjugate acid (either in the protonated or unprotonated form) does not change. Thus we define “total weak acid” (Atot) such that

[Atot] = [A -] + [HA]

We now have a set of four equations:

Water dissociation [H +] x [OH -] = K’ w

Weak acid dissociation [H +] x [A -] = K a [HA]

Weak acid conservation [Atot] = [A -] + [HA]

Electroneutrality [H +] + [SID] = [OH -] + [A -]

These equations contain four unknown dependent variables ([H +], [OH -], [HA] and [A -]) and two externally controlled independent variables ([SID] and [Atot]). Just as we did previously, we can solve these equations for the unknowns via substitutions. The result is four third-order polynomial (cubic) equations—one equation for each of the independent variables. Prior to computers, these equations required extensive time and effort to solve. Computers, however, solve them easily. All the operator must do is input the two independent variables, [SID] and [Atot]. The computer then returns values for all of the unknowns (including for our primary goal, [H +]). Additionally, in biological systems, the variable [Atot] is tightly regulated. Thus it is appropriate to examine changes in the dependent variables due only to changes in the remaining independent variable, [SID]. In simple terms, though the acid-base relationship is complex in the presence of weak acids, changes in [H +] are still principally a function of changes in [SID].

More complex aqueous solutions

The Stewart derivation continues in this vein, analyzing progressively more complex solutions that begin to approximate biological fluids. These are completed using the same methodology as the more simple cases above. The next step is to analyze an aqueous solution composed of:

Strong ions, carbon dioxide and no other weak acids (this approximates interstitial fluid)

In this case there are:

Four equations

Four dependent variables: [H +], [OH -], [HCO₃ -] and [CO 3 -2]

Two independent variables: [SID] and PCO₂

These again combine to form third-order polynomials. The solution to the equations reveals that, in this aqueous solution, [H +] is a function of two independent variables, [SID] and the partial pressure of CO₂ (PCO₂). Additionally, though the exact form of the solution equation for [H +] is complex, at physiologic conditions it can be very closely approximated by the more simple equation [H +] = K c x PCO₂ / [SID]. Additionally, solving the equations for a different dependent variable, [HCO3-], reveals the interesting conclusion that, at physiologic conditions, [HCO₃ -] is closely approximated by [SID] and is independent of PCO₂.

The final aqueous solution to examine is one that contains

Strong ions, carbon dioxide and other weak acids (this approximates intracellular fluid and blood plasma)

In this case there are

Six equations

Six dependent variables: [H +], [OH -], [HCO₃ -], [CO 3 -2], [HA] and [A -])

Three independent variables: [SID], PCO₂ and [Atot]

Though complex, this set of equations can be solved, and the result shows that [H +] is a function of the three independent variables, [SID], PCO₂ and [Atot]. With so many variables, there are many ways to graphically represent these relationships. One clinically applicable way is to set [SID] and [Atot] based on known blood values in a given patient. [H +] (or pH) can then be plotted vs. PCO₂. A program that performs this function is available online at

http://www.anaesthetist.com/icu/elec/ionz/Findex.htm#index.htm

(click on the link “Java Applet” and then scroll down to the last white calculator box on the page)

Key points of the Stewart approach

1) The Stewart approach works by setting up a system of equations based on properties of aqueous solutions and then solving this system to find [H +]. In a solution such as plasma that contains water, strong electrolytes and weak electrolytes this set of equations leads to polynomial equations that are mathematically complex to solve but are readily solved by using a computer.

2) [H +], [OH -], [HCO₃^-2], [CO₃^-2], [HA] and [A -] are dependent variables. As such it is conceptually inaccurate to view changes in [H +] as being caused by extrinsic addition or subtraction of these variables.

3) The relevant independent variables are [SID], PCO₂ and [Atot]. It is changes in these variables that bring about changes in [H +] and the other dependent variables.

4) The rigorous quantitative analysis of aqueous solutions frequently reveals conclusions that defy qualitative or intuitive expectations. This is because an aqueous solution is a complex system, and changing one variable can often have unexpected effects on many other aspects of the system. As Stewart writes: “The properties of mixtures are seldom simply the sum of properties of their components. In fact, the properties of complex systems…are often quite contrary to qualitative predictions.”

Frequently asked questions regarding the contrasting explanations between conventional and the Stewart approaches

Question: What happens when you add hydrochloric acid (HCl) to water?

Conventional approach: You add H +, so this causes an increased [H +], leading to acidosis.

Stewart approach: You are adding a strong anion (Cl -) without adding a strong cation. Therefore the SID decreases. This is a net negative change in charge due to SID. To maintain electroneutrality the solution must liberate H +, leading to acidosis. In the Stewart view, you don’t attempt to directly follow the H + that you add. At first this may seem convoluted (“I added HCl and [H +] increased. Why make it more complicated?”). However the apparent simplicity of the traditional view breaks down when one asks by how much did [H +] change? When, after adding HCl you measure [H +], it is immediately apparent that [H +] does not change in an easily predictable way (as does [Cl -]. This is because there are many factors that will influence what exactly happens to the added H +. In mathematical terms, [H +] is a dependent variable and depends on many factors. SID, on the other hand, is an independent variable.

Question: Why does 0.9% NaCl solution cause acidosis when added to plasma?

Conventional approach: Since this fluid has a greater [Cl -] than plasma, the kidneys increase the preferential retention of Cl - compared to HCO₃ -. This results in a loss of HCO₃ - and a net acidosis.

Stewart approach: Plasma normally has a positive SID. Since normal saline solution has an SID of 0 ([Na +] = [Cl -] = 154 mEq/L with no other strong ions present), adding this to plasma results in a net decrease in the SID. From an electroneutrality perspective, this is a decrease in the net positive charge due to SID. To maintain electroneutrality, the solution must liberate H +, increasing the [H +] and resulting in acidosis. Note that this effect occurred in particular because of the greater chloride concentration in the normal saline solution (154 mEq/L) compared to the plasma (110 mEq/L), whereas the sodium concentrations are quite close (154 mEq/L in the normal saline solution compared to 140 mEq/L in the plasma). Thus adding the normal saline solution to the plasma will also increase the chloride concentration, and hence the scenario is termed “hyperchloremic acidosis.”

Question: Adding water to plasma causes so-called “dilutional acidosis.” Why does this occur?

Conventional approach: Adding free water dilutes out the HCO₃ -, leading to acidosis. (Though this explanation does not explain what equal dilution of H+ does not offset the effect).

Stewart approach: Consider adding 1L of free water to 1L of plasma. The concentrations of Na + and Cl - will decreased from 140 mEq/L to 70 mEq/L and from 110 mEq/L to 55 mEq/L respectively, and the SID will decrease from 30 mEq/L to 15 mEq/L. This is effectively 15 mEq/L fewer positive charges from the SID contribution. To make up for this loss in positive charge and maintain electroneutrality, H + must be liberated, and acidosis results. This view has interesting implications for the treatment of dilutional acidosis. While traditional treatment consists of administration of 0.9% NaCl solution or hypertonic NaCl solution, analysis using the Stewart approach shows that while these therapies would help to correct both the hyponatremia and hypochloremia, they would actually worsen the acidosis by decreasing still further the SID. What is indicated instead is the administration of strong cations without strong anions, most commonly given as NaOH solution.

Q: How is stomach acid produced?

Conventional approach: Parietal cells secrete HCl into the stomach fluid, increasing its acidity.

Stewart approach: Parietal cells transport a strong anion (Cl -) from the plasma into the stomach fluid without transporting a strong cation. This decreases the SID in the stomach fluid, which causes it to be more acidic. To maintain electroneutrality, either a positive charge must move with the Cl - or a negative charge must move opposite it. This means that either H + moves with the Cl -, or that OH - or HCO₃ - move opposite Cl - from the stomach acid into the plasma. Which of these occurs is unimportant, and in fact can often not be experimentally determined (exchanging OH - for Cl - is indistinguishable from secreting both Cl - and H +). The only important fact is that parietal cells decrease the stomach fluid SID by secreting a strong anion and excluding concomitant movement of a strong cation.

Conclusions

In the time since Stewart first published his approach, there has been much debate as to its validity or necessity. Without doubt the Stewart approach is compelling in its clarity of derivation, but clarity does not necessarily equal verity. Several key points of the Stewart approach have been called into question. As an example, the classification of certain variables as independent or dependent is, in fact an empirical rather than a mathematical choice. The equation y = f(x), for example, can just as easily be written x = f -1 (y). Stewart is thus not wrong in treating certain variables as independent or dependent, but his selections as such cannot be said to be absolutely required.

Importantly, it has been shown that the conventional and Stewart approaches are essentially different derivations of the same mathematical endpoint. If the conventional and Stewart approaches are not different in their mathematics nor in their predictive capability, how are they different? The answer is that they place conceptual emphasis on different areas. The conventional approach focuses directly on the kinetics of H + and HCO₃ -, while the Stewart approach focuses much more on the [SID]. In certain examples, the Stewart conception is very compelling. Dilutional acidosis, for example, can be easily understood in terms of changes in the [SID]. In other cases, however, the conceptual mechanism envisioned in the conventional approach seems much more plausible than an [SID]-based approach. For example, in the case of gastric acid secretion, the discovery of Na +/H + ATP-ase transporters in the parietal cell throws doubt on Stewart’s claim that it is the transport of Cl - and not H + directly that leads to acidification of gastric fluid.

If the two approaches give the same answer, why learn the Stewart approach? For the clinician, learning the Stewart approach will broaden one’s view of the factors that influence acid base balance. When examining a blood gas analysis, for example, the clinician steeped in the conventional approach would likely not think to assess the Na + - Cl - difference (which approximates [SID]) as relevant in acid-base balance. Knowing that [SID] is a major factor affecting acid-base will aid the clinician in more accurate assessment and more appropriate therapy choices.

Finally, one may ask the fundamental question: if two approaches are mathematically equivalent, is it necessary or even appropriate to seek a mechanistic conceptualization? Put another way, science is a predictive, not an explicative pursuit. After all, Newton did not believe that he discovered what or why gravity is, nor would he have claimed that gravity must obey an equation. Instead he simply found that the equation F = GMm/r 2 accurately predicted the behavior of falling objects. To paraphrase Einstein, the scientist’s job is to seek laws that predict the movement of the universe’s clock hands. Speculating about the inner workings of the clock, however, is the business of philosophy not physics.

References

Constable. Clinical Assessment of Acid-Base Status: Comparison of the Henderson-Hasselbalch and Strong Ion Approaches. Veterinary Clinical Pathology (2000; 4: 115-128)

This article is a review of basic principles of both conventional and Stewart approaches. It also addresses the “modified Stewart” model, which is not discussed in this text.

Kurtz, Kraut, Vahram, and Nguyen. Acid-Base Analysis: A Critique of the Stewart and Bicarbonate-Centered Approaches. American Journal of Physiology - Renal Physiology (January 9, 2008).

This recent article is an excellent comparison of the conventional and Stewart approaches. It also summarizes the critiques that have been made of Stewart approach.

Lloyd and Freebairn. Using Quantitative Acid-Base Analysis in the ICU. Critical Care and Resuscitation (2006; 8:19-30)

This article contrasts the bicarbonate based approach using rule-of-thumb with the Stewart quantitative approach. It goes through several examples of analyzing a patient’s acid-base status by using a computer program that runs a quantitative analysis.

Stewart PA.How to Understand Acid-Base. A Quantitative Acid-Base Primer for Biology and Medicine.1981. Edward Arnold

This is Stewart’s original text. It presents a logical, lucid explanation of both the motivation and derivation of his approach to acid-base.

http://www.acidbase.org/index.php?show=sb

This website has the entirety of Stewart’s original text.

http://www.anaesthetist.com

This website has summaries of the conventional and Stewart approaches. The Stewart section includes some excellent interactive tools that plot graphs of the Stewart equations. For example, entering [SID] and [Atot] produces a graph of pH verses PCO₂.

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