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The basics of Stewart´s acid-base approach
Usually, Stewart’s acid-base approach considers blood plasma as an isolated physicochemcial system
[1]. The following plasma constituents have acidic or basic properties and are included in the model: the pCO 2/
HCO3– buffer system, the strong (i.e. fully dissociated) acids and bases (represented by the strong ion
difference), the weak acids (albumin and phosphate) and water as the solvent. By applying standard methods of
physical chemistry for pH calculations, i.e. combining equations for chemical equilibria (dissociation of water,
the bicarbonate system and weak acids), mass and charge balance, Stewart developed an equation relating the
concentration of H+ ([H+]) to three variables. These variables are partial pressure of carbon dioxid (pCO2), the
strong ion difference (SID, i.e. the difference in concentration between the strong anions and cations;
representing the strong acids and bases) and the total concentration of weak acids (A TOT). For the derivation of
Watson’s modification of Stewart’s formula see appendix:
[SID]  [H + ] 
K c  K 2(H2CO 3 )  pCO 2
K c  pCO 2
K
K  [A tot ]
 2
 w+  A
0
+
+ 2
[H ]
[H ]
[H ] K A  [H + ]
Measuring these three variables enables to calculate the pH and have therefore been termed, “independent
variables”. On the other hand, the dependent vaiables are [HCO3–], [CO32–], [OH–], [H+] (or pH), [A–] (anion
concentration of the dissociated weak acids) and [HA] (concentration of the undissociated weak acids). In this
model, only changes in the independent variables can affect hydrogen ion concentration, i.e. pH. It was asserted
that this equation not only enables quantification of acid-base status, but also describes the mechanisms on
which acid-base regulation and homeostasis are based [2-8].
Evolution of Stewart´s mechanistic acid-base approach
To understand how the misconception of Stewart´s approach evolved, one has to consider the history
and development of this approach. More specifically, one needs to understand:
a) The mathematical relationship of independent and dependent acid-base variables in Stewart´s
original work: In his original work by applying standard methods of physical chemistry, Stewart created a
mathematical formula considering all acid-base relevant components of plasma, which includes HCO3–, CO32-,
OH–, H+, SID, A- and HA. He found that this formula can be rearranged so that H+, and therefore all variables,
can be calculated by the three variables SID, pCO2 and ATOT, which were termed, “independent variables”. All
other variables calculated from the independent variables were termed,”dependent variables”.
b) The transformation of this mathematical relationship into a chemical mechanism: This mathematical
correlation of independent and dependent variables led to the concept, in which the acid-base state is solely
determined by the independent variables. Therefore, the mechanistic considerations of acid-base derangements
were reduced to changes in these three parameters.
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c) The physiological explanations for acid-base derangements: Stewart and others regard the three
independent variables as chemical / physiological entities, which are themselves regulated by the body to
control acid-base balance and enable an easy understanding of every acid-base phenomenon. Kellum summed
this up in one sentence: “… in order to understand how the body regulates pH, we need only ask how it
regulates these three independent variables (SID, pCO2 and ATOT)”. Kellum also pointed out that the
independent variables not only describe H+ but are causally related [5].
d) The central role of SID and water dissociation: This implies that a decrease in [SID] (e.g. addition of
a chloride rich solution as in saline induced dilutional acidosis) causes an increase in [H +], i.e. causes dilutional
acidosis. The mechanistic explanation was based on the maintenance of electroneutrality as described in the
introduction. Analogous explanations were derived for many other acid-base disorders.
e) False definition of acids and bases: Finally, the correlation that an increase in chloride causes an
increase in H+ led to the misconception that chloride is considered as an acid [9].
Misconceptions by Stewart’s mechanistic approach
In the field of dilutional acidosis, the explanation by Stewart’s approach of how dilution of plasma
causes metabolic acidosis is questionable for two reasons. First, H20 dissociation produces equal amounts of
cations (H+) and anions (OH-) and is therefore electrically, neutral; hence, it cannot compensate for increasing
negative charges (anions). Second, it is dubious that a difference in charge (in this case [SID]) would
mechanistically influence H20 dissociation. These misconceptions of Stewart’s mechanistic approach are based
on the confusion of mathematical correlation and causal connection as shown in the following.
If we symbolize the strong ion difference in Stewart’s equation (see appendix, equation 12) as [SID], we
suggest that SID itself is a component that can be measured and takes part in acid-base equilibria. We must not
forget that [SID] is merely a difference in charge. Furthermore, it is not just the difference between cations and
anions, but the difference in charge of strong ions arbitrarily defined depending on the model (Na+, Cl- and K+,
or including also Ca2+, Mg2+ and lactate-). It can neither be measured directly by analytical methods nor be
sensed or directly regulated within living organisms. [SID] is neither a physical nor a physiological entity.
The origin of [SID] is mathematical. Standard acid-base calculations in chemistry as used in Stewart’s
model are based on mass and charge balance. The maintenance of electroneutrality is a mathematical condition
which is required to solve the equation; however, physicochemically it is meaningless. Every dissociation
process by itself is electrically neutral. Every neutral molecule can only dissociate into the same amounts of
positively and negatively charged ions. Therefore, the following statement by Kellum has two fundamental
misconceptions: “When the SID is decreased, such as by increasing Cl- or decreasing Na+, powerful effects are
imposed on the dependent variables, including H+ and OH-. The net result is that water dissociation increases
and H+ ion concentration increases” [10]. First, an isolated change in the concentration of Cl- or Na+ (SID) is
possible in mathematics but is physicochemically implausible because it is impossible to perform an isolated
addition of e.g. chloride ions into a solution. There has to be a counter ion (e.g. H+) which must not be
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neglected. Second, if an isolated change in SID would be possible, the “powerful effects” of charge imbalance
could not be balanced by increased water dissociation because it cannot generate isolated positive charges (H +)
due to the equal amount of negative charge (OH-) produced by water dissociation. Therefore, charge cannot
have any impact on dissociation equilibria because every dissociation is electrically neutral per se. The
correlation between [SID] and [H+], as graphically shown by Kellum [10], is quantitative but cannot be used as
evidence for its causative role.
Furthermore, the alternative definition of anions as acids and cations as bases [8] was based on earlier
acid-base definitions used in medicine by Naunyn and Van Slyke [11] until the 1950s. However, a detailed look
at acid-base history reveals that this definition is based on a terminological mistake and was only used in
medicine and never occurred in chemistry. In the nineteenth century, it was known that the alkali-metals such
as sodium or potassium form alkaline solutions, whereas chlorine forms acidic solutions. By electrical
decomposition of acidic and basic solutions, the elements could be recovered again. Thus, alkali-metals were
called “base-forming” and chlorine “acid-forming”. The description of the process of electrolysis in terms of
cations and anions by Faraday finally led to a terminology where cations were called “base-forming” and
anions “acid-forming”. As Relman showed in an editorial in 1954 [12] at the turn of the twenty century in
clinical chemistry textbooks, this terminology was transformed to referring cations as bases and anions as acids.
Unfortunately, this abbreviation led to an acid-base definition, which stands in contrast to the definition in
chemistry. This semantic schism and terminological inaccuracy resulted in controversies and confusion for
many years and it seems to reappear with the use of Stewart’s approach. Long before Stewart’s approach
appeared, Christensen published in 1957 a hypothetic dialogue between one proponent of a “ion-centred” and
one of a “hydrogen-centred” view of acid-base balance, which already covered many of the aspects of the
today’s controversy on Stewart’s approach [13].
The arguments raised by Corey [8] supporting the statement that ions affect water dissociation are
misleading (Grotthuss mechanism, chosmotropes, chaotropes, etc.). These arguments focus on the concept of
chemical activity and this concept is traditionally ignored in medicine. These “equilibrium modifying factors”
are not at all accounted for in Stewart’s approach or in other acid-base approaches in medicine. At best, these
factors may influence the position of chemical equilibria but as “equilibrium modifying factors” they cannot
explain any mechanistic cause of chemical reactions.
Cause and effect in acid-base reactions
Beyond the description of the formal chemical mechanism involved in acid-base derangements, one
must ask for the reasons why such chemical acid-base reactions occur at all. In the context of dilutional
acidosis, the statement that decreases in [SID] and increases in chloride concentration are the actual causes of
dilutional acidosis [5] is not reasonable. The causes (reasons) of a chemical reaction are the driving forces. In
general, the direction of all spontaneous reactions is towards decreases in free energy (Gibbs energy) [14]. The
Gibbs energy comprises the competing effects of the enthalpic and entropic driving forces involved in
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thermodynamic processes, such as acid-base reactions. The enthalpic driving force is the release of heat
(exothermic reaction) and the entropic driving force is the gain in degrees of freedom at a molecular level. In a
simple reaction, such as the strong exothermic neutralization reaction (H+ and OH- react to form water), it is
easy to identify the underlying driving force (formation of water), whereas in more complex reactions such as
dilution reactions in buffered solutions, the detailed driving forces will not be established easily. However, the
idea of reaching a minimum in Gibbs energy is entirely general, which means that if the change in Gibbs energy
of a reaction is zero (ΔrG = 0), the system is in equilibrium. The thermodynamic equilibrium constant (K) gives
the equilibrium composition of the reaction mixture. Therefore, the net driving force in complex acid-base
reactions is to fulfil equilibrium concentrations given by the dissociation constants K (this state equals a
minimum in Gibbs energy). Looking at these chemical equilibrium equations (see electronic supplement), there
are no strong ions included. This makes their causative role implausible.
Dependent and independent variables
In Stewart’s formula, the variables used to solve the equation are called the “independent variables”
whereas all the others are called “dependent variables”. Stewart concluded from this mathematical dependence
that “H+ movements into or out of a solution do not provide quantitative explanations for changes in [H+] . . .
do not determine the value of [H+], and by themselves cannot explain changes in it” [2] and Kellum completed
“Thus, these three independent variables, are related causally to the H+concentration, rather than merely being
correlated. The distinction between independent and dependent, and between causation and correlation, is as
important to acid-base physiology as any other area of science” [15]. However, the question as to which
variable is dependent or independent is not mathematical, but is rather empirical in nature. The ultimate goal in
science in determining which variable is dependent or independent is to uncover a mechanism for a particular
observation.
In the closed CO2/HCO3- buffer system a change in the mass/charge balance (as in dilutional acidosis)
alters both the CO2 and HCO3- concentration. In vivo, the pCO2 can be regulated in the pulmonary capillary
bed by respiration. CO2 can diffuse freely between most compartments within the body and therefore cannot be
regulated within individual compartments separately. Transmembrane transporters of HCO3- or H+ can, by the
HCO3-/CO2 reaction, per se lead to a CO2 consumption or generation without changing the whole body CO2
mass balance. When [HCO3-] changes, [SID] must also change (e.g. via [Na+]), since HCO3- can only be added
or removed with a counter ion, except in a negligible quantity by electrogenic ion transport across membranes.
Thus, considering an isolated (i.e. independent) change is neither useful for [SID] nor for [HCO3-], since in
reality this is impossible. The total body HCO3- balance can be regulated by the kidney’s net acid excretion (as
titrable acid, NH4+ and HCO3-). Therefore, the selective definition of “independent variables” and “dependent
variables” depends on the system and the uncovered mechanism for a particular phenomenon.
Doberer et al. – Critique of Stewart’s approach – Electronic supplementary material
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Appendix
Equations for acid-base calculation in human blood:
Water dissociation:
K w  [ H  ]  [OH - ]
(E1)
HCO3- formation:
[H  ]  [HCO 3 ]
Kc 
pCO 2
(E2)
CO32- formation:
K 2(H2CO 3 )
[H  ]  [CO 32 ]

[HCO 3- ]
(E3)
Weak acid dissociation and conservation:
[H  ]  [A - ]
;
KA 
[HA]
[A tot ]  [HA ]  [A - ]
(E4; E5)
charge balance:
[SID]  [ Na + ]  [Cl- ]
(E6)
[SID]  [H + ]  [HCO 3- ]  2  [CO 32- ]  [OH - ]  [A - ]  0
(E7)
Using equations E1 to E5 equation E7 can be expressed as:
[SID]  [H + ] 
K c  K 2(H2CO 3 )  pCO 2
K c  pCO 2
K
K  [A tot ]
 2
 w+  A
0
+
+ 2
[H ]
[H ]
[H ] K A  [H + ]
(E8)
Rearrangement gives the quartic equation:
[H + ] 4  [H + ]3  [SID]  K A   [H + ] 2  [SID]  K A  pCO 2  K c  [A tot ]  K A  K w  


 [H + ]  pCO 2  K c  K A  2  pCO 2  K c  K 2(H2CO 3 )  K w  K A  2  pCO 2  K 2(H2CO 3 )  K c  K A  0
(E9)
Equilibrium constants (values for plasma of 37 °C as used by Watson [16] and Reeves [17]):
Kw
autoionization constant of water applying to plasma
2.39·10-14 mol2/L2
Kc
combined equilibrium and solubility constant for CO2
2.45·10-11 mol2/(L2·mmHg)
K2(H2CO3) second dissociation constant of carbonic acid
5.76·10-11 mol/L
KA
1.77·10-07 mol/L
dissociation constant of weak acids
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Equations for acid-base calculation in a SO2/HSO3- system:
Water dissociation (see E1):
HSO3- formation:
K c(SO 2 ) 
[H  ]  [HSO 3 ]
pSO 2
(E10)
[H  ]  [SO 32 ]
[HSO 3- ]
(E11)
SO32- formation:
K 2(H2SO 3 ) 
charge balance:
[ Na  ]  [H + ]  [HSO 3- ]  2  [SO 32- ]  [OH - ]  0
(E12)
Using equations E1, E10 and E11 equation E12 can be expressed as:
[ Na  ]  [H + ] 
K c(SO 2 )  K 2(H2SO 3 )  pSO 2
K c  pSO 2
K

2

 w+  0
+
+ 2
[H ]
[H ]
[H ]
(E13)
Rearrangement gives the cubic equation:



[H + ]3  [H + ]2  [ Na  ]  [H + ]  pSO 2  K c(SO 2 )  K w  2  pSO 2  K 2(H2SO3 )  K c(SO 2 )  0
(E14)
Equilibrium constants for 25 °C [18]:
Kw
autoionization constant of water
1.00·10-14 mol2/L2
Kc(SO2)
combined equilibrium and solubility constant for SO2
3.09·10-05 mol2/(L2·mmHg)
K2(H2SO3) second dissociation constant of sulfurous acid
6.31·10-08 mol/L
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Equations for acid-base calculation in a NH3/NH4+ system:
Water dissociation (see E1):
NH4+ formation:
[ NH 4 ]  [OH - ]
pNH 3
(E15)
[ NH 4 ]  [H + ]  [Cl - ]  [OH- ]  0
(E16)
K c(NH3 ) 
charge balance:
Using equations E1 and E15 equation E16 can be expressed as:
K c(NH3 )  pNH3  [H + ]
Kw
 [H + ]  [Cl - ] 
Kw
0
[H + ]
(E17)
Rearrangement gives the quadratic equation:

 pNH3  K c(NH3 )

[H + ]2  
 1  [H + ]  [Cl - ] - K w  0
Kw


(E18)
Equilibrium constants for 25 °C [18]:
Kw
autoionization constant of water
1.00·10-14 mol2/L2
Kc(NH3)
combined equilibrium and solubility constant for NH3
1.39·10-06 mol2/(L2·mmHg)
Doberer et al. – Critique of Stewart’s approach – Electronic supplementary material
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