Abstract
Analytic expressions for plasma total titratable base, base excess (ΔC_{B}), strongion difference, change in strongion difference (ΔSID), change in Van Slyke standard bicarbonate (ΔVSSB), anion gap, and change in anion gap are derived as a function of pH, total buffer ion concentration, and conditional molar equilibrium constants. The behavior of these various parameters under respiratory and metabolic acidbase disturbances for constant and variable buffer ion concentrations is considered. For constant noncarbonate buffer concentrations, ΔSID = ΔC_{B} = ΔVSSB, whereas these equalities no longer hold under changes in noncarbonate buffer concentration. The equivalence is restored if the reference state is changed to include the new buffer concentrations.
 acidosis
 alkalosis
 base excess
 strongion difference
 anion gap
 delta anion gap
characterization of acidbase balancein the body is of central importance in medicine, since it may provide valuable information about the status of a patient or provide clues about the underlying pathophysiology of a patient’s disease process (16). Mathematical models of physiological acidbase balance help the clinician conceptualize the processes involved, in order to better diagnose and treat the patient.
Several models, algorithms, and methods have been proposed to evaluate clinical acidbase status. These have classically been divided into those that are based primarily on measured bicarbonate concentration ([
HCO3−
]) (16, 17) vs. those that use base excess (BE) (24) to evaluate nonrespiratory acidbase disorders. In the former approach, the anion gap (AG) and the change in anion gap (ΔAG) are also calculated to gain further insight into the origins of a metabolic acidbase disturbance (5, 31).
More recently, Stewart and others (68, 13, 29) have popularized the use of the strongion difference (SID) method to describe acidbase. This is the same idea originally set forth by Singer and Hastings under the name “buffer base” (27). The SID is the sum of positiveion concentrations minus the sum of negativeion concentrations for those ions that do not participate in proton transfer reactions. The Stewart approach (29) is a very general physicochemical method that uses charge and mass balance to deduce an expression for proton concentration. Similarly, the BE method is another very general physicochemical approach, but one that uses proton balance to calculate changes in proton concentration by using the Van Slyke equation (25).
In the following sections, a general formalism for calculating total titratable acid and base is given, and it is shown that a linear approximation to the expression for total titratable base (C_{B}) yields the Van Slyke equation. Next, a similar equation is developed for SID, and it is shown that a linear approximation to the complete equation for SID has the same form as the Van Slyke equation. Mathematical relationships between the various parameters commonly used to assess acidbase status are then derived, yielding insight into the interrelationships between the different methods for assessing physiological acidbase balance.
GENERAL THEORY
Glossary
Anion gap
AG′
Anion gap corrected for respiratory effects
ΔAG
Change in anion gap (delta gap)
α_{j(i)}
Fraction of species i with j protons bound
BE
Base excess
β
Buffer value
b
_{i}
Constant that together with C_{i} determines C
C
Constant in linear approximation to complete equations
C
Concentration of proton acceptor sites of the carbonate buffers
C_{B}
Total titratable base
ΔC_{B}
Change in C_{B}, including noncarbonate buffer base
ΔC′_{B}
Change in C_{B} referenced to new buffer ion concentration C′_{i}
C_{H}
Total titratable acid
C_{i}
Analytical concentration of buffer ion species i
C′_{i}
New (abnormal) analytical concentration of buffer ion
D
[H^{+}] − [OH^{−}]
̅e̅
̅
_{i}
Average number of proton acceptor sites per molecule of speciesi
K
_{l(i)}
Conditional molar equilibrium constant for the lth dissociation step of species i
̅n̅
̅
_{i}
Average number of protons per molecule of species i
̅n̅
̅
_{max(i)}
Maximum number of protons per molecule for species i
S
Equilibrium constant between dissolved CO_{2} and CO_{2} in the gas phase
SID
Strongion difference
SID_{m}
Strongion difference for measured ions
SID_{u}
Strongion difference for unmeasured ions
VSSB
Van Slyke standard bicarbonate
VSSB′
Van Slyke standard bicarbonate referenced to C′_{i}
̅z̅
̅
_{i}
Average charge per molecule of species i
̅z̅
̅
_{max(i)}
Maximum charge on species i
Minimum charge on species i
z
_{p}
Charge on unmeasured positive ion p
z
_{q}
Charge on unmeasured negative ion q
Acidbase equilibrium in aqueous solution has been exhaustively treated in the classic works of Ricci (18), Butler (2), and Kolthoff and Elving (14). Most recently, Guenther (10) has extended and simplified their work to give a master equation for solving complex acidbase problemsCH−D=∑i Cin¯i
Equation 1where C_{H} is total titratable acid and Dis Ricci’s difference functionD=[H+]−[OH−]
Equation 2Unless otherwise specified, parameters are expressed in terms of concentrations (indicated by brackets) rather than activities.
̅n̅
̅
_{i} is the average number of protons per molecule of species i. C_{i} is the analytical concentration of speciesi, the sum of all concentrations of subspecies, regardless of protonation stateCi=∑j(i)=0n¯max(i) [Hj(i)Ai]
Equation 3where
n¯
_{max(i)} is the maximum value of
n¯
for speciesi and j(i) is the proton ligand number for species i.
Thus Eq. 1
states that if one knows the total concentrations of all of the various species and their ligand numbers, the C_{H}can be calculated. The right side of Eq. 1
represents the bound titratable acid, whereas D is the net free titratable acid.
The
n¯
_{i} can be calculated from the fraction of species α_{j(i)} asn¯i=∑j(i)=0n¯max(i) j(i)αj(i)
Equation 4The α_{j(i)} are calculated viaαj (i)=[Hj (i)Ai]∑j (i)=0n¯max(i) [Hj (i)Ai]
Equation 5
The α_{j(i)} can in turn be recast as functions of [H^{+}] and equilibrium constants K(see for example Ref. 10)αj(i)=[H+] j(i) ∏l(i)=0n¯max(i)−j(i) Kl(i)∑j (i)=0n¯max(i) [H+] j(i) ∏l (i)=0n¯max(i)−j(i) Kl (i)
Equation 6where the K
_{l(i)}’s are conditional molar equilibrium constants (10) andK
_{l(i)} is the lth step for the dissociation of H^{+} from speciesi. The symbol Π is a standard mathematical notation denoting that a product ofK
_{l(i)} is taken over the prescribed limits. K
_{0} is defined to be unity. Also note that∑j (i)=0n¯max(i) αj(i)=1
Equation 7
From Eq. 1, therefore, changes in C_{H} can be separated conceptually into changes in the C_{i} or changes in pH. A related concept is the total titratable base C_{B}, which, by analogy with Eq. 1, isCB=∑i Cie¯i−D
Equation 8where
̅e̅
̅
_{i} is the average number of proton acceptor sites per molecule of species i and is calculated bye¯i=∑j (i)=0n¯max(i) [n¯max(i)−j(i)αj (i)]
Equation 9
Equations 1, 4, 8, and
9
yield the obvious relationsn¯i+e¯i=n¯max(i)
Equation 10andCH+CB=∑i Cin¯max(i)
Equation 11
PHYSIOLOGICAL ACIDBASE THEORY
To agree with the standard definitions employed in the usual treatment of physiological acidbase, Eq. 8
is used to calculate acidbase balance. Because of the crucial role of CO_{2}equilibrium in physiology, it is useful to separate the carbonate and noncarbonate contributions to C_{B}
CB=C+∑i Ci ei−D
Equation 12where the C_{i} and
̅e̅
̅
_{i} now refer to the noncarbonate buffers. This definition is retained throughout the rest of the calculations. C represents the carbonate contributionC=2[CO32−]+[HCO3−]+[PrNHCOO−]
Equation 13where PrNHCOO^{−} is the carbamate derivative formed from interaction of CO_{2} with protein (24). Under physiological conditions, the [
CO32−
] and [PrNHCOO^{−}] terms are small and can be neglected. Similarly, the D term of Eq. 12
is small and can be neglected. Under these approximations, Eq. 12
becomesCB=[HCO3−]+∑i Cie¯i
Equation 14Combining Eqs. 4, 9, and
14
givesCB=[HCO3−]−∑i Cin¯i+∑i Cin¯max(i)
Equation 15Differentiation of Eq. 15
with respect to pH yields∂CB∂pH=∂[HCO3−]∂pH−∑i Ci∂n¯i∂pH
Equation 16It turns out that because of the large number of buffer groups on plasma proteins ∂
̅n̅
̅
_{i}/∂pH is approximately linear over the physiological pH range, so thatEq. 16, using Eq. 15, can be recast in integrated form asCB=[HCO3−]−∑i Ci ∂n¯i∂pH pH+∑i Cin¯max(i)+C
Equation 17The constant C depends on the concentrations and identities of the various species i and is given byC=−∑i Ci bi
Equation 18where b
_{i} is a constant that depends on the difference between the pH at which the slope is determined and the negative base ten logarithm of the dissociation constant (pK ) of species i. Equation 17
is referred to as the Van Slyke equation (25).
Physiological pH is determined under the simultaneous solution of the Van Slyke equation and the HendersonHasselbalch equation (1a)pH=pK′+log[HCO3−]S⋅PCO2
Equation 19where pK′ = 6.103 and S is the equilibrium constant between dissolved CO_{2} and CO_{2} in the gas phase and equals 0.0306 at 37°C, when H^{+} is in moles per liter,
HCO3−
is in millimoles per liter, and Pco
_{2} is in Torr.
SID THEORY
In 1983, Stewart published an article (29) in which he developed a quantitative model of acidbase, employing a parameter first used by Singer and Hastings (27). Stewart called this parameter the “strongion difference,” whereas Singer and Hastings referred to this same parameter as “buffer base.” The advantage of the buffer base parameter is that it is independent of changes in Pco
_{2}.
Strong ions are defined as those that do not participate in acidbase reactions; that is, they are spectator ions. Ions that do involve net proton exchange are called buffer ions. The SID is, therefore, given bySID=∑k zk[Skzk]
Equation 20where z
_{k} is the charge on strong ion S_{k}. In terms of the physiological ions which are typically present, SID usually isSID=[Na+]+[K+]+2[Mg2+]+2[Ca2+]
−[Cl−]−[lactate−]−2[SO42−]
Equation 21By virtue of the principle of electroneutrality, SID can also be calculated, in analogy to Eq. 12, bySID=C−∑i Ciz¯i−D
Equation 22where
z¯
_{i} is the average charge per molecule for species i and is given byz¯i=∑j (i)=0n¯max(i) [z¯min(i)+j(i)αj (i)]
Equation 23
z¯
_{min(i)}is the minimum possible charge for species i. Equation22
is a more general form of the equations of Stewart (29) and Figge et al. (7). After neglecting the terms with small values under physiological conditions as before, Eq. 22
gives, in analogy with Eq. 14
SID=[HCO3−]−∑i Ci z¯i
Equation 24In analogy with Eq. 15
SID=[HCO3−]−∑iCin¯i−∑i Ci z¯min(i)
Equation 25Several additional relations are worth pointing out, includingn¯i+z¯min(i)=z¯i
Equation 26Hence∂n¯i∂pH=∂z¯i∂pH
Equation 27and by similar arguments to those leading to Eq. 17
SID=[HCO3−]−∑i Ci∂n¯i∂pH pH−∑i Ci z¯min(i)+C
Equation 28Note that Eq. 28
has the same general form asEq. 17. Equation 28
is also an analytical expression for the simplified equation of Figge et al. (7). Combining Eqs. 15, 25, and
26
gives the additional relationshipCB=SID+∑i Ci z¯max(i)
Equation 29where
z¯
_{max(i)} is the maximum possible charge for species i.
METHODS
Theoretical simulations of acidbase balance in plasma, using the mathematical models above, were performed. In accordance with the results of Figge et al. (7), it was assumed that albumin and phosphate were sufficient to account for all of the noncarbonate buffer activity of plasma.
C_{B} vs. pH, SID vs. pH, and [
HCO3−
] vs. pH curves for different values of Pco
_{2} for both the SID and C_{B} cases were calculated. Microsoft Excel 97 running on a Compaq Deskpro computer equipped with a Pentium II processor was used for the calculations. pH was stepped in 0.01unit increments to calculate the dependent variable. An ionic strength of 0.17 for plasma was assumed (1a), and activity coefficients were calculated from the Davies equation (4) at 37°C. This gave activity coefficients off
_{1} = 0.75, f
_{2} = 0.31, andf
_{3} = 0.072 for ions with charges of ±1, ±2, and ±3, respectively. Concentrations of buffer ions are expressed in millimoles per liter.
Acid dissociation constants for the ionizable groups on albumin were those determined by Figge et al. (7, 8), and it was further assumed that the constants in their study were determined via concentrations for the protein but activity for H^{+}. Carbonate and phosphate equilibrium constants were obtained from Refs. 1a and 21, respectively. The relevant equilibrium constants are listed in Table1. For albumin,
n¯
_{max(i)} = 212 and
z¯
_{min(i)} = −118; for phosphate,
n¯
_{max(i)} = 3 and
z¯
_{min(i)} = −3.
Table 1.
pK and equilibrium constant values for carbonate and noncarbonate buffers of human plasma at 37°C
The designation “normal plasma” within Figs. 15 and text denotes plasma with C_{i} for albumin of 0.66 mM (4.4 g/dl) and C_{i} for phosphate of 1.16 mM (3.6 mg/dl). The normal values for pH, Pco
_{2}, and [
HCO3−
] are assumed to be 7.40, 40.0 Torr, and 24.25 mM, respectively, corresponding to a C_{B} = 101 mM and a SID = 39 mM. The constant C was calculated fromEq. 18
and the above values to give −110 mM for normal plasma, with a corresponding b
_{i} of 160 for albumin and 3.3 for phosphate.
ACIDBASE PARAMETERS AT CONSTANT NONCARBONATE BUFFER CONCENTRATION
Equation 12
represents an exact expression for C_{B}, subject to several caveats. First, since protein conformation is pH dependent, the K
_{l(i)} will also be pH dependent, especially outside the physiological pH range. Second, as noted before, the K
_{l(i)}values used here are really pseudoequilibrium constants, since they were not all determined directly; some were determined as parameters fitted to a model and others were assigned “average values” (7,8). Finally, some authors have previously pointed out that, in general, the notions of both C_{B} and SID are somewhat illdefined, since it depends on pH whether a given species behaves as a strong ion or a buffer ion (23, 26). In practice, however, there is very little ambiguity over the physiological pH range; therefore, these parameters may actually be regarded as well defined under physiological conditions. For the purposes of calculation, what is required is that all of the species that are involved in net proton transfer over the pH range of interest must be included in the right side of Eq. 8
or 22. With these approximations in mind, the behavior of the functions under various physiological perturbations can be considered.
Over the physiological pH range between pH = 6.8 and 7.8, Eq.14
is often plotted as [
HCO3−
] vs. pH at constant C_{B}. Equation 19
is also plotted on the same graph for various Pco
_{2} values, as shown in Fig. 1. Equation 14
is seen to be nearly a straight line over this same pH range and can therefore be approximated by Eq. 17.The theoretical error introduced through the use of the linear approximations rather than the complete equations was found to be <1.0 mM over the pH range from 6.0 to 8.0.
As mentioned in physiological acidbase theory,physiological pH is determined by the simultaneous solution of the Van Slyke equation (Eq. 17
) and the HendersonHasselbalch equation (Eq. 19
); that is, at their crossing points, as shown in the case of human plasma plotted in Fig. 1. This type of plot has been called a Davenport diagram (3).
The Van Slyke equation shows how [
HCO3−
] changes with Pco
_{2} at a constant concentration of noncarbonate buffer, so that C_{i} remains constant while
̅n̅
̅
_{i}increases or decreases. This change in [
HCO3−
], for example moving from pointA to B in Fig. 1, is due to the Pco
_{2}/[
HCO3−
] equilibrium and represents a pure respiratory derangement. Metabolic disturbance at constant Pco
_{2} is illustrated by the move from B to C in Fig. 1, in which the entire Van Slyke curve is shifted with a concomitant change in C_{B}.
These changes are described under constant noncarbonate buffer (ΔC_{i} = 0) using Eq. 17
byΔCB=Δ[HCO3−]+β⋅ΔpH
Equation 30where β represents the buffer valueβ=−∑i Ci∂n¯i∂pH
Equation 31β can be calculated by a termbyterm differentiation ofEq. 4
or by fitting the data to a straight line. The data of Figge et al. (7) used here give a molar buffer value (−∂
̅n̅
̅
_{i}/∂ pH) of 8.3 for albumin. The original data of Tanford et al. (30) give a molar buffer value 8.0. The molar buffer value of phosphate is calculated to be 0.29, compared with 0.309 given by SiggaardAndersen and FoghAndersen (26). The corresponding theoretical and experimental (26) values of β for albumin are 5.5 and 5.3 mM, respectively, whereas the theoretical and experimental (26) values of β for phosphate are 0.34 and 0.36 mM, respectively. If theK
_{l(i)} values are defined in terms of activity for protons and concentration for the remaining species, the calculated values are identical to the experimental values.
Pure respiratory disturbances, those with ΔC_{B} = 0, give a Δ[
HCO3−
] ofΔ[HCO3−]=−β⋅ΔpH
Equation 32whereas metabolic disturbances are calculated via Eq.30, which can be conceptualized in two ways. The first is as a calculation of the ΔC_{B}, which is the BE as compared with the normal value. The second is to focus on the right side of Eq.30
and to think of β ⋅ ΔpH as a correction term to the measured Δ[
HCO3−
], which corrects for changes in Pco
_{2} (respiratory effects). The correction term corrects the [
HCO3−
] to what it would be at pH = 7.40 (point D of Fig. 1). The [
HCO3−
] when corrected to pH = 7.40 is called the “Van Slyke standard bicarbonate,” and the corresponding change in [
HCO3−
] is the change in Van Slyke standard bicarbonate (VSSB) or ΔVSSB (23,25). The ΔVSSB is illustrated by the difference between pointsA and D in Fig. 1. Since the two lines in Fig. 1 are parallelΔCB=ΔVSSB=BE
Equation 33The equation for SID (Eq. 24
) when plotted in a [
HCO3−
] vs. pH coordinate system at constant SID gives a plot identical to Fig. 1, recognizing that the slopes and yintercepts will be the same but that SID and C_{B} differ by a constant (Eq. 29
). The straightline plots in Fig. 1 also show how [
HCO3−
] varies with Pco
_{2} at constant SID. The same arguments apply to SID that apply to the C_{B} case above, so thatΔSID=Δ[HCO3−]+β⋅ΔpH
Equation 34and, thereforeΔSID=ΔCB=ΔVSSB=BE
Equation 35as stated in SiggaardAndersen’s treatise (24) and also as concluded by Schlichtig (19). Figure 2shows how SID and C_{B} vary as a function of pH for normal plasma and a metabolic acidosis. As will be shown in acidbase parameters at variable noncarbonate buffer concentrations, the equalities of Eq. 35
break down if the noncarbonate buffer concentration does not remain constant.
SiggaardAndersen and others have previously argued (22, 26) that BE, as a parameter by itself, is the most relevant arbiter of whether there is a net metabolic acidbase disturbance and to what degree; however, examination of the change in the concentrations of other ions can also give additional diagnostic information. The parameter most often used for extracting that information is the anion gap AG (5).
The AG is usually defined asAG=[Na+]−[Cl−]−[HCO3−]
Equation 36and it is generally interpreted as the “unmeasured anions,” although in actuality the AG isAG=−∑p zp[Sp zp]−∑q zq[Sq zq]
Equation 37where the sums extend over the unmeasured positive ions p and the unmeasured negative ions q. The AG, therefore, represents the opposite of the net unmeasured charge concentration. Because the net unmeasured negative charge is typically much greater than the unmeasured positive charge, AG is positive. Whether the AG is low, normal, or high helps the clinician by dividing the differential diagnosis into those associated with low, normal, or high gaps, as has been described in Refs. 5, 16, and 17.
The “delta gap” (ΔAG) is used to uncover mixed metabolic disturbances (17, 31). It is given byΔAG=AG−12
Equation 38where AG is calculated from measured values via Eq.36, and the constant 12 mM represents the normal value for the gap. If the ΔAG does not equal the Δ[
HCO3−
] calculated fromΔ[HCO3−]=[HCO3−]−24
Equation 39where [
HCO3−
] represents the measured value and the constant 24 mM represents the normal bicarbonate value, then a mixed metabolic disorder is present. This mixed nonrespiratory disorder will be made up of two disorders with different AGs, for example, a metabolic alkalosis from vomiting and a metabolic acidosis from diabetic ketoacidosis.
Equations 17
and
28
show how to incorporate a similar analysis into the BE and Stewart approach. SID can be divided into measured and unmeasured components, as suggested by Jones (12) and by Figge et al. (7).SID=SIDm+SIDu
=[HCO3−]−∑i Ci ∂n¯i∂pH pH−∑i Ci z¯min(i)+C
Equation 40where SID_{m} is the measured SID and SID_{u} is the unmeasured SID. Hence, using Eq.30
ΔAG′=−ΔSIDu=ΔSIDm−BE
=ΔSIDm−Δ[HCO3−]−β⋅ΔpH
Equation 41This is the delta gap for a metabolic disturbance, remembering that the BE at constant C_{i} represents the change in [
HCO3−
] corrected for respiratory effects. The AG′ is the difference between the measured SID and the VSSB. Therefore, if ΔAG′ = BE = 0, then there is no net disturbance. If ΔAG′ = 0 and BE ≠ 0, then there is a “measured ioncompensated” disturbance, for example, a hyperchloremic metabolic acidosis. If ΔAG′ = −BE, then there is a differential diagnosis consistent with the presence of “unmeasured ions,” for example, a wide gap metabolic acidosis. Finally, if ΔAG′ is between 0 and −BE, then there is a mixed delta gap problem with both measured ioncompensated and unmeasured ion disturbances present.
These concepts are illustrated graphically in Fig.3, showing the AG′ in the BE formalism and how SID changes with metabolic disturbances, through a change in SID_{m} or SID_{u}, or potentially both.
ACIDBASE PARAMETERS AT VARIABLE NONCARBONATE BUFFER CONCENTRATION
When the concentrations of buffer ions C_{i} change, the relationships between the acidbase parameters discussed above also change. Using Eq. 15, the change in C_{B} due only to the change in C
_{i} can be calculated by assuming that [
HCO3−
], Pco
_{2}, and pH are held constant at their normal valuesΔCB=−∑i n¯iΔCi+∑i n¯max(i)ΔCi
Equation 42Here,
̅n̅
̅
_{i} is understood to be evaluated at pH = 7.40. Alternatively, using the linear approximation of Eq. 17
ΔCB=−∑i ∂n¯i∂pH ΔCi (7.40)+∑i n¯max(i) ΔCi+ΔC
Equation 43The change in titratable base due only to ΔC_{i} is illustrated graphically in Fig.4.
The total change in C_{B} when the above variables are not held constant isΔCB=Δ[HCO3−]−∑i Δ(Cin¯i)+∑i n¯max(i) ΔCi
Equation 44or in the linear approximationΔCB=Δ[HCO3−]
−∑i ∂n¯i∂pH Δ (Ci pH)+∑i n¯max(i) ΔCi+ΔC
Equation 45Again, this is shown graphically in Fig. 4.
The relevant analogous equations for SID can be derived in the same way to giveΔSID=−∑i n¯iΔCi−∑i z¯min(i) ΔCi
Equation 46for the change due to ΔC_{i} alone and for the total changeΔSID=Δ[HCO3−]−∑i Δ(Cin¯i)−∑i z¯min(i)ΔCi
Equation 47In their linear approximationsΔSID=−∑i ∂n¯i∂pH ΔCi (7.40)
−∑i z¯min(i) ΔCi+ΔC
Equation 48andΔSID=Δ[HCO3−]
−∑i ∂n¯i∂pH Δ(Ci pH)−∑i z¯min(i) ΔCi+ΔC
Equation 49The ΔSID due only to a change in ΔC_{i}is also illustrated in Fig. 4.
Under variations in noncarbonate buffer concentration, the ΔVSSB is calculated in the linear approximation viaΔVSSB′=ΔCB′=Δ[HCO3−]−∑i Ci′ ∂n¯i∂pH ΔpH
Equation 50as shown graphically in Fig. 5. Here C′_{i} represents the new (different from normal) C_{i}. ΔC′_{B} is interpreted as the BE referenced to the new buffer ion concentration C′_{i} (dotted line in Fig. 5). ΔC′_{B} will, of course, be equal to the correspondingly defined ΔSID′.
Fig. 5.
Davenport diagram ([
] vs. pH) for normal plasma (solid straight line containing pointA), metabolic acidosis for ΔC_{i} = 0 mM and ΔVSSB = −10 mM (solid straight line containing pointB), plasma with normal values except albumin concentration = 0.33 mM (dashed line containing point A), and the same corresponding metabolic acidosis with albumin concentration = 0.33 mM (dotted line containing point C). ΔC′_{B}, change in C_{B} referenced to new buffer ion concentration C′_{i}.
It should be clear from Eqs. 44, 47, and
50
that the equalities of Eq. 35
no longer hold. In fact, Eq. 29
shows that in general for ΔC_{i} ≠ 0ΔCB=ΔSID+∑i z¯max(i)ΔCi
Equation 51and, therefore, in generalΔSID≠ΔCB≠ΔVSSB
Equation 52whereas, in generalΔSID′=ΔCB′=ΔVSSB′
Equation 53
DISCUSSION
Disagreement about the best parameter to describe acidbase balance in the body has dominated this area of physiology for more than three decades (1, 20, 22, 26). For the most part, the disagreement has been between advocates of the Boston method employing measured [
HCO3−
] and the Copenhagen method employing BE. This controversy has, in fact, been given a name “The Great TransAtlantic AcidBase Debate” (1). Recently, on another front, Stewart and others (11, 28, 29) have argued that SID is the best parameter for expressing acidbase derangements and have even suggested that the very concept of pH be abandoned.
The limitations of the Boston method have been eloquently discussed by Severinghaus (22). The basic problem with this method is that the effects of buffer ions are not taken into account directly, and thus changes in [
HCO3−
] are treated as independent of Pco
_{2}. In reality, though, [
HCO3−
] changes acutely as a function of Pco
_{2}, which is purely due to the chemical equilibrium and independent of any compensation by the kidneys. As a result, changes in bicarbonate due to nonrespiratory causes may be underestimated or overestimated. This same approach also examines changes in ion concentration via AG and ΔAG to obtain additional information; however, these variables are also subject to the same problem, since [
HCO3−
] is also involved in their calculation.
The BE approach is an attempt to correct for respiratory changes in [
HCO3−
]. A point of confusion is that the definition of BE is nebulous. BE implies, and is stated in the literature to be, the change in total titratable base, ΔC_{B} (24, 25). The term BE is also used synonymously with ΔVSSB. As demonstrated in acidbase parameters at variable noncarbonate buffer concentration, however, the equivalence is justified only if C_{i} is constant or if the standard reference state is changed to that with the new buffer ion concentration C′_{i}. That section alluded to a situation in which pH, [
HCO3−
], and Pco
_{2} conceivably could all be normal while the noncarbonate buffer concentration may not be normal. As demonstrated in Fig. 4, because of the large number of buffer groups on a protein, small absolute changes in protein concentration will have large accompanying changes in C_{B} or SID. This may be true even within the normal albumin reference range. Also, as seen in Fig. 4, the change in SID for such a case will be smaller than the change in C_{B}. SiggaardAndersen and FoghAndersen have suggested (26) that changes in protein concentration should not be considered acidbase disorders, and have considered the Stewart approach problematic in that regard. The calculations in acidbase parameters at variable noncarbonate buffer concentrationsdemonstrate the same potential problem with BE when defined as ΔC_{B}.
By comparison, because the Van Slyke equation is fairly flat over the physiological pH range, ΔVSSB is relatively constant with changes in buffer ion concentration. Figure 5 shows that the error introduced by using C_{i} to calculate ΔVSSB rather than C′_{i} to calculate ΔVSSB′ will usually be less than ±1 mM. In addition, although many find the characterization of metabolic disturbances in terms of AG and ΔAG behavior useful, advocates of the BE approach tend to denigrate the use of these two parameters (26), and so extracting information about ion disturbances other than H^{+} is not dealt with in the BE approach. As demonstrated in acidbase parameters at constant noncarbonate buffer concentrations, however, the concept of the AG and ΔAG can be recast within the framework of the BE and Stewart formalisms to give the same useful information.
Stewart and his followers (28, 29) advocate the description of acidbase status in terms of SID, C_{i}, and Pco
_{2}, based on the claim that these are the only independent variables of acidbase physiology. It bears pointing out, however, that SID is a surrogate variable for the difference in analytic concentrations between strong base and strong acid, the variables found in the standard treatments of acidbase equilibrium from the 1950s. The very same calculations of acidbase balance can be made, yielding the very same results, without explicit inclusion of spectator ions in the calculation (2, 10, 14, 18).
The Stewart approach has mainly found utility via the work of Figge et al. (7, 8), who calculated SID from pH, C_{i}, and Pco
_{2}, basically from Eq. 24. The only difference is that Figge et al. include the small terms D and [
CO32−
] in their calculations and they also calculate the carbonate terms from pH and Pco
_{2} (7). Next, they calculate SID_{u}from the difference between SID_{m} and SID, including [Na^{+}], [K^{+}], [Mg^{2+}], [Ca^{2+}], [Cl^{−}], and [lactate^{−}] in their calculation of SID_{m}. The value for SID_{u} is then interpreted clinically much like the AG. This model has been tested both in vitro (68) and in vivo (9, 15) and found to give reliable predictions.
The central relationship linking the BE concept with the Stewart approach is Eq. 27, which states that the rate of change in charge with respect to pH is equal to the rate of change in protonation with respect to pH. From that principle comes the fact that C_{B} and SID are the same within an added constant and that BE and ΔSID are also the same within an added constant.
The foregoing discussion should clarify the debate about whether SID or BE is a better measure of acidbase disturbance in plasma. They are really two different kinds of parameter; one represents a change in concentration and the other does not. SID, as a standalone parameter, does not seem that useful and is certainly no more useful than C_{B} or C_{H} from a theoretical standpoint. Similarly, BE is no more or less useful than ΔSID, but BE and ΔSID seem to be more relevant parameters, since they do represent the deviations from normal, which is what one is usually concerned about practically.
Conclusions
A general formalism for calculating parameters describing physiological acidbase balance in plasma has been presented. Analytic expressions have been derived for the parameters commonly used to address nonrespiratory physiological acidbase disturbances. It was shown that the Van Slyke equation arises naturally from this formalism and that an equation with the same form can be derived for the SID. It was also shown that the anion gap and delta anion gap concepts can be expressed within the framework of the base excess and Stewart approaches. The mathematical relationships between the various parameters under various physiological perturbations were discussed, and theoretical simulations for human plasma were performed. Based on these results, it appears that there is no real theoretical advantage of using SID over C_{B}, or ΔSID over ΔC_{B}. Calculation of ΔSID and ΔC_{B} incorporates changes in all buffer bases, including the potentially large change accompanying changes in protein concentration, whereas calculation of ΔC′_{B} and ΔSID′ does not include changes in the noncarbonate buffers directly. In addition, the assumption that the ΔVSSB is equivalent to ΔC_{B} only holds if the noncarbonate buffer concentration remains constant or if the VSSB′ is calculated from the new noncarbonate buffer concentrations. The results presented here should help clarify some of the confusion about the meaning of the various parameters and their suitability for describing clinical acidbase disorders.
Acknowledgments
I thank Drs. John Johnson, Vito Rocco, and Alfred George for helpful discussions. I also thank Dr. A. George for the use of his computer facility.
Footnotes

Financial support from the Mallinckrodt Institute of Radiology is gratefully acknowledged.

Present address and address for reprint requests: E. W. Wooten, Dept. of Radiology, BarnesJewish Hospital, 510 South Kingshighway, St. Louis, MO 63110.

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.
 Copyright © 1999 the American Physiological Society
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