Abstract
Constable, Peter D. A simplified strong ion model for acidbase equilibria: application to horse plasma. J. Appl. Physiol. 83(1): 297–311, 1997.—The HendersonHasselbalch equation and Stewart’s strong ion model are currently used to describe mammalian acidbase equilibria. Anomalies exist when the HendersonHasselbalch equation is applied to plasma, whereas the strong ion model does not provide a practical method for determining the total plasma concentration of nonvolatile weak acids ([A_{tot}]) and the effective dissociation constant for plasma weak acids (K
_{a}). A simplified strong ion model, which was developed from the assumption that plasma ions act as strong ions, volatile buffer ions (
HCO3−
), or nonvolatile buffer ions, indicates that plasma pH is determined by five independent variables:
PCO2
, strong ion difference, concentration of individual nonvolatile plasma buffers (albumin, globulin, and phosphate), ionic strength, and temperature. The simplified strong ion model conveys on a fundamental level the mechanism for change in acidbase status, explains many of the anomalies when the HendersonHasselbalch equation is applied to plasma, is conceptually and algebraically simpler than Stewart’s strong ion model, and provides a practical in vitro method for determining [A_{tot}] andK
_{a} of plasma. Application of the simplified strong ion model to CO_{2}tonometered horse plasma produced values for [A_{tot}] (15.0 ± 3.1 meq/l) and K
_{a}(2.22 ± 0.32 × 10^{−7} eq/l) that were significantly different from the values commonly assumed for human plasma ([A_{tot}] = 20.0 meq/l, K
_{a} = 3.0 × 10^{−7} eq/l). Moreover, application of the experimentally determined values for [A_{tot}] andK
_{a} to published data for the horse (known
PCO2
, strong ion difference, and plasma protein concentration) predicted plasma pH more accurately than the values for [A_{tot}] andK
_{a} commonly assumed for human plasma. Speciesspecific values for [A_{tot}] andK
_{a} should be experimentally determined when the simplified strong ion model (or strong ion model) is used to describe acidbase equilibria.
 acidbase balance
 acidosis
 alkalosis
 alphastat
 strong ion difference
two methods are currently used clinically to describe the physicochemical determinants of plasma pH in mammals: the HendersonHasselbalch equation (20) and Stewart’s strong ion model (4446). The purpose of this study is to briefly discuss the strengths and weaknesses of the HendersonHasselbalch equation and Stewart’s strong ion model and to develop a simplified strong ion model that is conceptually and algebraically simpler than Stewart’s strong ion model. The simplified strong ion model also explains many of the anomalies observed when the HendersonHasselbalch equation is applied to plasma.
HENDERSONHASSELBALCH EQUATION
The traditional approach used to clinically describe mammalian acidbase equilibria focuses on how
PCO2
,
HCO3−
concentration ([
HCO3−
]), the negative logarithm of the equilibrium constant (p
K1′
), and the solubility of CO_{2} in plasma (
SCO2
) interact to determine the plasma pH (35). This relationship is most commonly expressed as the HendersonHasselbalch equation (20, 23, 31)pH=pK1′+log[HCO3−]SCO2PCO2
Equation 1 where p
K1′
is a collective equilibrium constant for the reactionCO2(aq)+H2O⇌H2CO3⇌H++HCO3−
Equation 2The HendersonHasselbalch equation has proven to be invaluable in aiding our understanding of mammalian acidbase physiology and is routinely and widely used in the clinical management of acidbase abnormalities in humans and animals (2, 7, 25, 40). However, it was evident as early as 1922 that factors other than
PCO2
, [
HCO3−
], p
K1′
, and
SCO2
influence plasma pH (52).
SCO2
varies with ionic strength, temperature, and protein concentration, and accurate values are available for mammalian plasma (3). Determination of accurate p
K1′
values for plasma has been more problematic, inasmuch as the experimental value for p
K1′
in plasma (called the apparent dissociation constant) differs marginally from the value obtained in aqueous, nonplasma solutions (1, 8, 19, 21, 27, 29,31, 34, 38, 39). Moreover, like all equilibrium constants based on molalities, the value for p
K1′
is dependent on the ionic strength (21) and temperature (8). A number of studies have demonstrated that the apparent value for p
K1′
in plasma is also influenced by pH (1, 27, 29, 34, 38), protein concentration (27,29), and Na^{+} concentration (22), leading to routine adjustment of the p
K1′
for plasma by nomograms (40, 41), tables (3, 27, 34), and polynomial equations (19,22). The mechanistic basis for these adjustments is unknown.
Numerous experiments have demonstrated that the in vitro log
PCO2
pH equilibration curve for plasma is well approximated by a straight line over the normal physiological range (2, 40, 52) (Fig. 1). The HendersonHasselbalch equation partially explains this finding, inasmuch as rearrangement of Eq. 1
provideslogPCO2=−pH+log[HCO3−]K1′SCO2
Equation 3 indicating that the log
PCO2
pH relationship is linear with an intercept value of log([
HCO3−
]/
K1′
SCO2
). Experimental studies have also found that the linear relationship between log
PCO2
and pH is displaced by changes in the protein concentration (2) or the addition of Na^{+} or Cl^{−} (2, 39) (Fig. 1), suggesting that the intercept value has changed. Other studies have found that the in vitro relationship between log
PCO2
and pH becomes nonlinear in markedly acidic plasma (40), suggesting that the intercept value is pH dependent and is nonlinear during in vivo CO_{2} equilibration studies (6, 10) (Fig. 1). The HendersonHasselbalch equation provides no explanation for these phenomena.
Fig. 1.
Line plots of linear in vitro (•, ○, ▴, ▵) and curvilinear in vivo (dots) log
pH realtionship for human plasma. •, Plasma with a protein concentration of 7 g/dl (normal [A_{tot}] and [SID^{+}]); ○, plasma with a protein concentration of 13 g/dl (increased [A_{tot}]) (data from Ref. 2); ▴, plasma with a decrease in [SID^{+}] of 25 meq/l; ▵, plasma with an increase in [SID^{+}] of 50 meq/l (data from Ref. 2). Dots, curvilinear in vivo log
pH relationship (data from Refs.6 and 10). [A_{tot}], total plasma concentration of nonvolatile weak acid; [SID^{+}], strong ion difference.
Because the HendersonHasselbalch equation does not satisfactorily explain why the apparent value of pK
_{1} in plasma depends on pH, protein concentration, and Na^{+} concentration and why a nonlinear relationship exists between log
PCO2
and pH in vitro over a wide range of pH and in vivo during CO_{2}equilibration studies, the approach can only be accurately applied to mammalian plasma at approximately normal pH, protein concentration, and Na^{+} concentration. Moreover, the empiric nature of the adjustments to the value of
K1′
in plasma indicates that the HendersonHasselbalch equation is more descriptive than mechanistic.
STRONG ION MODEL
Dissatisfaction with the HendersonHasselbalch approach prompted Singer and Hastings (41) to propose in 1948 that plasma pH was determined by two independent factors,
PCO2
and net strong ion charge, equivalent to the strong ion difference ([SID^{+}]) (41). Stewart (4446) later proposed that a third variable, the total plasma concentration of nonvolatile weak acids ([A_{tot}]), also exerted an independent effect on plasma pH. By combining equations for conservation of charge, conservation of mass, and dissociation equilibrium reactions, Stewart developed a polynomial equation relating the plasma H^{+} concentration [H^{+}] to three independent variables (
PCO2
, [SID^{+}], and [A_{tot}]) and five “constants” (K
_{a},
Kw′
,
K1′
,K
_{3}, and
SCO2
) (45,46)[H+]4+([SID+]+Ka)[H+]3+(Ka([SID+]−[Atot])
−Kw′−K1′SCO2PCO2)[H+]2
−[Ka(Kw′+K1′SCO2PCO2)−K3K1′SCO2PCO2][H+]
−KaK3K1′SCO2PCO2=0
Equation 4
whereK
_{a} is the effective equilibrium dissociation constant for plasma weak acids,
Kw′
is the ion product of water,
K1′
is the apparent equilibrium constant for the HendersonHasselbalch equation,
SCO2
is the solubility of CO_{2} in plasma, andK
_{3} is the apparent equilibrium dissociation constant for
HCO3−
.
Although the strong ion model offers a unique insight into the pathophysiology of acidbase derangements in mammals and is mechanistic (11, 25), Stewart’s approach has not been widely accepted, because it does not provide a practical method for determining [A_{tot}] andK
_{a} (7). The most commonly used value for [A_{tot}] is 20 meq/l {calculated from the net protein charge, where [A_{tot}] (in meq/l) = 2.4 × [total protein] = 8.3 g/dl, where [total protein] is total protein concentration} (25, 50); however, a recent study suggested that the correct value for [A_{tot}] in human plasma is ∼14 meq/l {calculated as [A_{tot}] (meq/l) = 1.7 × [total protein] = 7 g/dl + 1.8 × phosphate concentration = 1 mmol/l} (13). A number of different values forK
_{a} (2 × 10^{−7}, 3 × 10^{−7}, 4 × 10^{−7}, and 4 × 10^{−8} eq/l) have been suggested (4446), with 3 × 10^{−7} eq/l being the most commonly used value (11, 15, 25, 26, 30, 53). It is unclear which values for [A_{tot}] andK
_{a} should be used when the strong ion model is applied to nonhuman plasma, inasmuch as it is likely that species differences in values for [A_{tot}] andK
_{a} exist (15). From an experimental viewpoint, the strong ion model is considered by some authors to offer no significant improvement over the conventional HendersonHasselbalch equation (7, 22). However, from a clinical viewpoint, the strong ion model is invaluable, in that it offers a novel insight into the pathophysiology of mixed acidbase disorders (11, 15, 25). In particular, the effects of hypoproteinemia and hyperproteinemia on acidbase status (35) can be satisfactorily explained only by the strong ion model.
In summary, deficiencies exist in present methods to describe mammalian acidbase equilibria. Accordingly, Stewart’s strong ion model was conceptually and algebraically reduced in the hope that a simpler model would 1) explain the apparent dependence of plasma p
K1′
on pH, protein concentration, and Na^{+} concentration;2) explain why the log
PCO2
pH relationship for plasma is displaced by changes in plasma protein, Na^{+}, and Cl^{−} concentration and is nonlinear in vivo and in markedly acidic plasma;3) provide a practical method for experimentally determining values for [A_{tot}] (in meq/l) andK
_{a} (in eq/l) in plasma; and 4) provide an acidbase model that unites the HendersonHasselbalch equation and strong ion model.
MATERIALS AND METHODS
Model development.
The simplified model reduces the chemical reactions in plasma to that of simple ions in solution. This assumption can be made because the major plasma cations (Na^{+}, K^{+}, Ca^{2+}, and Mg^{2+}) and anions (Cl^{−},
HCO3−
, protein, lactate^{−}, and sulfate^{2−}) bind each other in a salttype manner (9, 49, 52). Plasma ions that enter into oxidationreduction reactions, complex ion interactions, and precipitation reactions are not categorized as simple ions (9, 49). Plasma ions such as Cu^{2+}, Fe^{2+}, Fe^{3+}, Zn^{2+}, Co^{2+}, and Mn^{2+}, which are not simple ions (49), are assumed to be quantitatively unimportant in determining plasma pH, primarily because their plasma concentrations are low.
Simple ions in plasma can be differentiated into two types: nonbuffer ions (strong ions or strong electrolytes) and buffer ions (Table1). Strong ions are considered to be fully dissociated at physiological pH (4) and therefore exert no buffering effect. Strong ions do, however, exert an electrical effect, because the sum of completely dissociated cations does not equal the sum of completely dissociated anions (45). Stewart (4446) termed this difference the strong ion difference (SID), which is always positive in plasma. Because strong ions do not participate in chemical reactions in plasma at physiological pH, for practical purposes the strong ions can be regarded as a collective unit of charge, the SID^{+}. The concentration of this charge in plasma is expressed as [SID^{+}] (in meq/l).
Table 1.
Categorization of simple ions in equine plasma and approximate values for their normal concentration
In contrast to strong ions, buffer ions are derived from plasma weak acids and bases that are not fully dissociated at physiological pH. The BronstedLowry theory defines an acid as any substance that can donate protons. The dissociation reaction for a weak acidconjugate base pair, HA and A^{−}, isHA⇌H++A−
Equation 5and at equilibrium,K
_{a} can be calculated from the law of mass action (23)Ka=aH+[A−]/[HA]
Equation 6where
aH+
represents H^{+} activity and [HA] and [A^{−}] represent the plasma concentrations of weak acid and conjugate base, respectively. The value forK
_{a} will depend on temperature and ionic strength, inasmuch as it is being defined in terms of the activity of H^{+} and [A^{−}] and [HA] (molarity). For a weak acid to act as an effective buffer, its pK
_{a}must lie within the range of pH ±1.5 (9, 22, 32). On this basis, substances in plasma that act as weak acids at physiological pH have a pK
_{a} between 5.9 and 8.9 (Table 2). Ions derived from weak acids with a pK
_{a}outside this range are classified as nonbuffer ions (strong ions; Tables 1 and 2).
Table 2.
Approximate pK_{a} values for acids that produce nonbuffer ions (strong ions) or buffer ions in plasma at physiological pH
Conceptually, the buffer ions can be subdivided into volatile buffer ion (bicarbonate) and nonvolatile buffer ions (nonbicarbonate). Bicarbonate is considered separately, because this buffer system is an open system in arterial plasma (25); rapid changes in
PCO2
and, hence, arterial plasma bicarbonate concentration can be readily induced through alterations in respiratory activity (25). In contrast, the nonbicarbonate buffer system is a closed system containing a relatively fixed quantity of buffer. Another important physiological distinction between these two buffer systems is that an open buffer system such as bicarbonate can be effective beyond the limits of pH = pK
_{a} ± 1.5. Finally, it should be appreciated that bicarbonate is a homogeneous buffer ion, whereas the nonvolatile buffer ion (A^{−}) represents a diverse and heterogeneous group of plasma buffers consisting primarily of dissociable imidazole and αamino groups on plasma proteins with a smaller contribution from phosphatecontaining weak acids and citrate (Tables 1 and 2). It should be emphasized that the heterogeneous group of nonvolatile buffer ions is being treated as if it were a single buffer with a classical sigmoidal titration curve. This modeling assumption is validated later and is consistent with the alphastat theory for acidbase regulation, which proposes that nonvolatile plasma buffers can be modeled as a single imidazole group (32). The derivation of Stewart’s strong ion model requires the same modeling assumption.
On the basis of the information stated above, plasma contains three types of charged entities: SID^{+},
HCO3−
, and A^{−}. The requirement for electroneutrality dictates that at all times [SID^{+}] equals the sum of [
HCO3−
] and nonvolatile [A^{−}], such that[SID+]−[HCO3−]−[A−]=0
Equation 7
Equation7
obviously assumes that all ionized entities in plasma can be classified as a strong ion (SID^{+}), a volatile buffer ion (
HCO3−
), or a nonvolatile buffer ion (A^{−}). This assumption forms the basis for the simplified strong ion model. The electroneutrality equation is similar to that developed by Singer and Hastings in 1948 (41) but differs from that developed by Stewart (4446), who preferred the following[SID+]−[HCO3−]−[A−]−[CO32−]−[OH−]+[H+]=0
Equation 8where [
CO32−
] and [OH^{−}] are
CO32−
and OH^{−} concentration, respectively. In plasma, [SID^{+}], [
HCO3−
], and [A^{−}] are present in milliequivalents per liter, whereas [
CO32−
] exists in microequivalents per liter and [OH^{−}] and [H^{+}] exist in nanoequivalents per liter. Because of the large differences in the magnitudes of the factors in Stewart’s electroneutrality equation,Eq. 8
does not appear to offer any significant improvement over Eq. 7
. The simplified strong ion model therefore assumes that the ionic charges carried by [
CO32−
], [OH^{−}], and [H^{+}] are quantitatively unimportant. This assumption is validated later.
Another assumption in the simplified strong ion model (and Stewart’s strong ion model) is that HA and A^{−} do not take part in plasma reactions that result in the net destruction or creation of HA or A^{−}. This is because when HA dissociates, it ceases to be HA (therefore reducing the plasma [HA]) and becomes A^{−} (therefore increasing the plasma [A^{−}]). The sum of [HA] and [A^{−}] (called A_{tot}) therefore remains constant through conservation of mass (45). This is expressed as a mass balance statement[Atot]=[HA]+[A−]
Equation 9The units of [HA] and [A^{−}] are millimoles per liter and not milliequivalents per liter as used by Stewart (4446), because mass, not charge, is conserved. In plasma under physiological conditions, HA consists of four dissociable groups: imidazole, αamino, phosphate, and citric acid (Table 2). Human plasma contains ≥9.51 mmol/l of dissociable imidazole groups and ≥2.38 mmol/l of dissociable αamino groups, because1) there are 16 dissociable imidazole groups and 4 dissociable αamino groups per albumin molecule (Table 3) (47),2) there is 0.59 mmol of albumin per liter of plasma on the basis of a plasma albumin concentration of 4.1 g/dl and a molecular weight for albumin of 69,000 (47), and3) the number of dissociable imidazole and αamino groups in plasma is greater than or equal to that for albumin. Human plasma also contains 1.29 mmol/l of dissociable phosphate groups, on the basis of a plasma phosphate concentration of 4 mg/dl, and <0.6 mmol/l of dissociable citric acid. [A_{tot}] for human plasma is therefore ≥13.8 mmol/l, inasmuch as [A_{tot}]_{plasma}= [A_{tot}]_{imidazole}+ [A_{tot}]_{αamino}+ [A_{tot}]_{phosphate}+ [A_{tot}]_{citric acid}. To facilitate further calculations, it is desirable to express [A_{tot}] in terms of milliequivalents per liter instead of millimoles per liter. This can be accomplished by using the equilibrium constant for acid dissociation and attributing a valence to [HA] and [A^{−}] for the four dissociable groups (see appendix
). The derivation suggests that the value of [A_{tot}] for human plasma, when expressed in terms of milliequivalents per liter, varies with plasma pH and is ∼18.0 meq/l at physiological pH (appendix
). A more accurate estimate for [A_{tot}], in terms of milliequivalents per liter, cannot be calculated by this method, inasmuch as detailed information for protein composition is not available for plasma proteins other than albumin.
Table 3.
Approximate intrinsic pK_{a} values and number of dissociable groups on plasma albumin
We now have enough information to express pH in terms of the plasma constituents. Substituting [HA] in Eq.9
into Eq. 6
producesKa=aH+[A−][Atot]−[A−]
Equation 10rearrangement producesaH+=Ka [Atot][A−]−1
Equation 11Substituting for [A^{−}] fromEq. 7
and taking the reciprocal of both sides produces1aH+=1Ka 1[Atot][SID+]−[HCO3−]−1
Equation 12Taking the logarithm of both sides provides−logaH+=−logKa−log[Atot][SID+]−[HCO3−]−1
Equation 13and because pH = −log
aH+
and pK
_{a} = −log K
_{a}
pH=pKa−log[Atot][SID+]−[HCO3−]−1
Equation 14
Equation 14
provides a simple expression relating plasma pH to four variables: pK
_{a}, [A_{tot}], [SID^{+}], and [
HCO3−
]. Unfortunately, not all variables in Eq. 14
are independent, inasmuch as [
HCO3−
] is dependent on another variable,
PCO2
(45). Because it is valuable to express pH in terms of independent variables (4446), Eq. 14
was algebraically manipulated (see appendix
) to provide an equation relating pH to Stewart’s three independent variables (
PCO2
, [SID^{+}], and [A_{tot}]), the solution beingpH=log2[SID+]K1′SCO2PCO2+Ka[Atot]−Ka[SID+]+(K1′SCO2PCO2+Ka[SID+]+Ka[Atot])2−4K a2[SID+][Atot]
Equation 15
Equation15
indicates that plasma pH is determined by three independent variables (
PCO2
, [SID^{+}], and [A_{tot}]) and three “variable constants” (K
_{a},
K1′
, and
SCO2
). The latter three factors are considered variable constants, becauseK
_{a} and
K1′
, like all apparent equilibrium constants, are affected by temperature and ionic strength and
SCO2
is affected by temperature, ionic strength, and protein concentration (3).
Under the condition
PCO2
= 0, at which time [
HCO3−
] = 0 and [SID^{+}] = [A^{−}] by virtue of Eq. 7, Eq. 15
reduces to the law of mass action for a weak acid (Eq.6
). Under the condition [A_{tot}] = 0, at which time [A^{−}] = 0 and [SID^{+}] = [
HCO3−
] by virtue ofEq. 7, Eq. 15
reduces to the HendersonHasselbalch equation (Eq.1
). The latter may be more readily appreciated ifEq. 14
is rearranged in terms of [
HCO3−
] and substituted into Eq. 1
pH=pK1′+log[SID+]−Ka[Atot]/(Ka+10−pH)SCO2PCO2
Equation 16In contrast to the individual conditions
PCO2
= 0 or [A_{tot}] = 0,Eqs. 15
and
16
indicate that the mathematical condition [SID^{+}] = 0 cannot exist, inasmuch as in this case there is no solution for plasma pH, because the logarithm of a number ≤0 does not exist. However, because of the law of electroneutrality and the fact that volatile and nonvolatile plasma buffers are negatively charged, as [SID^{+}] approaches zero, [
HCO3−
] or [A^{−}] must also approach zero but be closer to zero by virtue of Eq.7
. This means that the simplified strong ion model reduces to the equilibrium reaction for plasma weak acids or the HendersonHasselbalch equation before the condition [SID^{+}] = 0 exists. In summary, the simplified strong ion model reduces to appropriately simpler models under the conditions
PCO2
= 0 or [A_{tot}] = 0, whereas Stewart’s strong ion model (Eq.4
) is not appreciably simplified under these conditions.
As stated previously, the electroneutrality equation used to derive the new acidbase model differs from that used by Stewart. Because his electroneutrality equation contains six unknowns, Stewart’s approach requires six simultaneous equations to solve for [H^{+}], specificallyEqs. 1, 6, 8, 9
, and two additional equilibrium equations (45, 46)Kw′=[H+][OH−]
Equation 17
K3[HCO3−]=[H+][CO32−]
Equation 18where
Kw′
is the ion product of water andK
_{3} represents the equilibrium dissociation constant for bicarbonate. Stewart (46) stated that “nothing less than the whole set of six equations is sufficient” to explain pH behavior. A solution forEq. 4
can also be generated through algebraic manipulation and simplification, as detailed inappendix
. The solution demonstrates that Stewart’s fourthorder polynomial equation (Eq.4
), which is derived from six equations and eight factors, can be algebraically simplified to Eq.14
, which was derived from four equations and six factors. In other words, Eq. 15
, derived from the simplified strong ion model, produces values for plasma pH identical to those produced by Eq.4
, derived from Stewart’s strong ion model, but from fewer variables.
A comparison between the plasma pH predicted by Stewart’s polynomial equation (Eq. 4
) and the simplified strong ion model (Eq. 15
) for solutions of widely varying
PCO2
, [SID^{+}], and [A_{tot}] confirms that the algebraic reduction detailed in appendix
is valid, inasmuch as the equations produce identical results, allowing for rounding error (Table4). The finding also confirms the assumption made earlier that the simpler electroneutrality equation is valid. The conclusion that leads directly from this observation is that the dissociation equilibrium between
HCO3−
and
CO32−
and the dissociation equilibrium of water do not play a quantitatively important role in the physicochemical determination of plasma pH.
Table 4.
Comparison of plasma pH predicted by Stewart’s strong ion model (Eq.4) and that predicted by the simplified strong ion model (Eq. 15) for equine plasma from known Pco
_{2}, [SID^{+}], and [A_{tot}]
Clinical application of the new model.
Some limitations exist in the practical clinical application of the simplified strong ion model, primarily because of difficulties in obtaining accurate values for [SID^{+}], [A_{tot}], andK
_{a}. Similar difficulties exist with Stewart’s strong ion model (30). The factors [SID^{+}], [A_{tot}], andK
_{a} cannot be easily measured in plasma, and their values must therefore be estimated, assumed, or derived from the plasma constituents.
Determination of [SID^{+}] requires identification and measurement of all strong ions in plasma (Tables 1and 2). This can be an arduous and difficult task, since unidentified strong ions may be present (11, 28). Despite these shortcomings, a clinically practical estimate of [SID^{+}] can be obtained by determining the plasma concentration of at least four strong ions (Na^{+}, K^{+}, Cl^{−}, and lactate^{−}), whereby [SID^{+}] = [Na^{+}] + [K^{+}] − [Cl^{−}] − [lactate^{−}], where [Na^{+}], [K^{+}], [Cl^{−}], and [lactate^{−}] are Na^{+}, K^{+}, Cl^{−}, and lactate^{−} concentrations (14,15, 43, 53). Other investigators have employed different equations to estimate [SID^{+}], e.g., [SID^{+}] = [Na^{+}] + [K^{+}] − [Cl^{−}] (12), [SID^{+}] = [Na^{+}] + [K^{+}] + [Ca^{2+}] − [Cl^{−}] − [lactate^{−}] (26,30), [SID^{+}] = [Na^{+}] + [K^{+}] + [Ca^{2+}] + [Mg^{2+}] − [Cl^{−}] (11, 12), [SID^{+}] = [Na^{+}] + [K^{+}] + [Ca^{2+}] + [Mg^{2+}] − [Cl^{−}] − 1.5 (13), and [SID^{+}] = [Na^{+}] + [K^{+}] + [Mg^{2+}] − [Cl^{−}] − [citrate^{−}] (35), where [Ca^{2+}], [Mg^{2+}], and [citrate^{−}] are concentrations of Ca^{2+}, Mg^{2+}, and citrate^{−}. All these different mathematical approaches provide an estimate of [SID^{+}] instead of the exact value, because 1) each method assumes that the sum of the unmeasured strong cations approximates the sum of the unmeasured strong anions,2) unmeasured strong ions may become quantitatively important in specific pathological states (28),3) each method does not directly incorporate the effect of sulfate, which is a strong anion with an approximate plasma concentration of 1 meq/l (30), and4) each individual measurement is subject to error, thereby leading to a larger cumulative error in [SID^{+}].
An estimate for [A_{tot}] in milliequivalents per liter can be obtained for normal human plasma by multiplying total protein concentration (in g/dl) by 2.4 (25, 50) or the albumin concentration (in g/dl) by 4.0 (35), inasmuch as [A_{tot}] essentially represents the ionic equivalent of plasma proteins and phosphate. This method may be inaccurate in human plasma (13) or when applied to nonhuman plasma, inasmuch as the protein charge, albumintoglobulin ratio, and inorganic phosphate concentration vary among species (12,13, 15, 30, 49, 50) (appendix
). Instead of estimating general values for [A_{tot}] andK
_{a}, speciesspecific values can be experimentally determined by nonlinear regression using Eq. 15
of the simplified strong ion model and known values for pH,
PCO2
, and [SID^{+}] obtained from plasma equilibrated with different
PCO2
. This requires measurement of pH and
PCO2
, estimation of [SID^{+}] from [Na^{+}], [K^{+}], [Cl^{−}], and [lactate^{−}], and measurement of the predominant volatile buffers in plasma (total protein, albumin, globulin, and phosphate) to express [A_{tot}] in a meaningful manner. Nonlinear regression can also be applied to Stewart’s strong ion equation (Eq. 4
) to solve for [A_{tot}] andK
_{a}; however, this approach may fail to provide a solution or produce unrealistic values for [A_{tot}] andK
_{a}, suggesting an overspecified model or the presence of multicollinearity (16). An alternative method for determiningK
_{a} based on computer modeling of ionizable groups has been used to predictK
_{a} for human albumin (13); however, this procedure is laborious, requires detailed knowledge of the structure and composition of albumin for each species, and appears to produce an estimate forK
_{a} (0.5 × 10^{−7} eq/l) (13) of human plasma that differs from that of imidazole (1.9 × 10^{−7} eq/l) (32).
Data acquisition for experimental determination of [A_{tot}] + K_{a} in equine plasma.
Venous blood was collected anaerobically from six healthy adult horses (3 females, 3 males) into tubes containing heparin sodium and centrifuged, and the plasma was harvested. Plasma samples were equilibrated at 37°C for 20 min with a water vaporsaturated gas containing CO_{2} (range 6–70 Torr) by a tonometer (model IL237, Instrumentation Laboratory, Lexington, MA). Various mixtures of two CO_{2} gases (2% CO_{2}17% O_{2}81% N_{2} and 10% CO_{2}7% O_{2}83% N_{2}) were used to provide a wide range of
PCO2
. The pH and
PCO2
of the tonometered plasma samples were determined at 37°C by a pH/blood gas analyzer (model 238, Ciba Corning, Halstead, UK). Plasma concentrations of Na^{+}, K^{+}, Cl^{−}, albumin, total protein, and phosphate were determined by automated methods (model 704 Automatic Analyzer, Hitachi, Tokyo, Japan). Plasma [lactate^{−}] was determined by spectrophotometric methods (Sigma Chemical, St. Louis, MO). The [SID^{+}] was calculated as [SID^{+}] = [Na^{+}] + [K^{+}] − [Cl^{−}] − [lactate^{−}], with all values in milliequivalents per liter.
Nonlinear regression was applied using the simplified strong ion model (Eq. 15
) and known values for pH,
PCO2
, [SID^{+}], S_{CO2}, and K′_{1} to solve for [A_{tot}] andK
_{a}. The value used for
SCO2
in plasma at 37°C was 0.0307 Torr^{−1} (3). The value for p
K1′
at 37°C and an ionic strength of 0.16 (mammalian extracellular fluid) was obtained from the sum of pK
_{s} (6.038; Table II, interpolated, Ref. 19) and the negative logarithm of the activity coefficient of H^{+} (0.091) (31), producing a value of 6.129. Nonlinear regression was also performed using Stewart’s electroneutrality equation (Eq.8
) in the following form:
PCO2
= {[H^{+}]/(
K1′
S_{CO2}+K
_{3}
K′_{1}
SCO2
/[H^{+}])} × ([SID^{+}] − {K
_{a}[A_{tot}]/(K
_{a}+ [H^{+}])} + [H^{+}] −K
_{w}/[H^{+}]). The values used forK
_{3} andK
_{w} were 6 × 10^{−11} eq/l and 4.4 × 10^{−14}eq^{2}/l^{2}, respectively. Marquardt’s expansion algorithm (PROC NLIN, SAS Institute) was used for the nonlinear regression procedure (16, 37) on the basis of initial values for [A_{tot}] of 5–30 meq/l in 5 meq/l increments and forK
_{a} of 1–9 × 10^{−7} eq/l in 1 × 10^{−7} eq/l increments.
Nonlinear regression (using the simplified strong ion modelEq. 15
) was also applied to published values for pH,
PCO2
, and [SID^{+}] obtained from CO_{2} equilibration of equine plasma albumin, globulin, and serum protein solutions (50, 51). Inasmuch as equilibration in these studies was accomplished at 38°C, temperatureadjusted values for
SCO2
(0.0301 Torr^{−1}) (3) and p
K1′
[6.1201; obtained from the sum of pK
_{s} (6.0300; Table II, interpolated, Ref. 19) and the negative logarithm of the activity coefficient of H^{+}(0.0901) (31)] were used. Calculated values for [A_{tot}] andK
_{a} were expressed as estimated means ± SE of the estimate.
Model validation.
The simplified strong ion model was validated by applying the mean values for [A_{tot}] andK
_{a} obtained by the method described above to published blood gas and serum biochemical data derived from horses and ponies given endotoxin or strong electrolyte solutions such as sodium bicarbonate, sodium chloride, sodium lactate, and dilute hydrochloric acid (17, 18, 36). The total protein concentration was estimated from the albumin concentration, assuming that total protein concentration (g/dl) = 2.09 × albumin concentration (g/dl), in the studies (17, 18) where the total protein concentration was not reported. The plasma lactate concentration was calculated from the whole blood lactate concentration, assuming a hematocrit of 42% and using the following adjustment (24): [lactate]_{plasma} = [lactate]_{blood}/(1 − 0.56 × hematocrit), where [lactate]_{plasma} and [lactate]_{blood} are lactate concentrations in plasma and blood. The plasma lactate concentration was assumed to have a constant value (1.2 meq/l) in the experimental study (36) where it was not measured. The [SID^{+}] was calculated as follows: [SID^{+}] = [Na^{+}] + [K^{+}] − [Cl^{−}] − [lactate^{−}], with all values in milliequivalents per liter. The values for p
K1′
and
SCO2
were 6.129 and 0.0307 Torr^{−1}, respectively. Equation 15
was then used to predict the plasma pH from the reported values for
PCO2
, [SID^{+}], and total protein concentration obtained during the experimental studies. The calculated pH values were then compared with the measured pH values by linear regression analysis for each data set, and the means ± SD for the slope and intercept were determined.
RESULTS
Experimental determination of [A_{tot}] and K_{a}.
Nonlinear regression using the simplified strong ion model (Eq. 15
) produced a value for horse plasma [A_{tot}] of 15.0 ± 3.1 (SD) meq/l and for horse plasma and a value for K
_{a} of 2.22 ± 0.32 × 10^{−7} eq/l (Table5). The 95% confidence interval for [A_{tot}] andK
_{a} for horse plasma did not contain the values commonly used for human plasma ([A_{tot}] = 20 meq/l,K
_{a} = 3.0 × 10^{−7} eq/l).
Table 5.
[A_{tot}] and K_{a} determined from nonlinear regression of equine plasma equilibrated with different Pco
_{2}
When calculated solely from the total protein concentration, [A_{tot}] (in meq/l) = (2.24 ± 0.42) × [total protein] (in g/dl). Because this value assigns the [A_{tot}] contribution of inorganic phosphate to total protein, this formula should be used only in horse plasma with normal phosphate concentration. The direct contributions of total protein and phosphate to [A_{tot}] were 12.3 and 2.7 meq/l, respectively, on the basis of a mean total protein concentration of 6.7 g/dl, a mean phosphate concentration of 4.6 mg/dl, and assignment of valences to phosphate as described inappendix
. A more complete formula for estimating [A_{tot}] in horse plasma with a normal albumintoglobulin ratio (0.90 ± 0.12) but an abnormal phosphate concentration is therefore as follows: [A_{tot}] (in meq/l) = (1.84 ± 0.42) × [total protein] (in g/dl) + 0.59 [phosphate] (in mg/dl), where [phosphate] is phosphate concentration. The error estimate for [A_{tot}] was attributed entirely to total protein, because the error in estimating [A_{tot}] from phosphate was comparatively much smaller.
Nonlinear regression using Stewart’s strong ion model (Eq. 8
) produced values for [A_{tot}] andK
_{a} in three plasma samples similar to those obtained with the simplified strong ion model, an unrealistic value forK
_{a} (6.4 × 10^{−9} eq/l) in one plasma sample, and did not produce a mathematical solution in the two remaining plasma samples (Table 5).
Nonlinear regression using the simplified strong ion model (Eq. 15
) produced a mean estimate for [A_{tot}] of purified horse serum protein (2.05 × [total protein], in g/dl) that was within the 95% confidence interval for the value calculated above for a nonphosphatecontaining solution: (1.84 ± 0.46) × [total protein] (in g/dl) (Table6). The estimated values for [A_{tot}] {(1.4 ± 0.6) × [globulin] (in g/dl)}, where [globulin] is globulin concentration, andK
_{a} [(3.4 ± 1.9) × 10^{−7}eq/l] of horse globulin were not significantly different from the values obtained for horse plasma (Table 6). The calculated estimate for [A_{tot}] (1.84 × [total protein], in g/dl) of a solution containing purified horse albumin ([A_{tot}] = 2.25 × [albumin], in g/dl), where [albumin] is albumin concentration, and horse globulin ([A_{tot}] = 1.4 × [globulin], in g/dl) with a normal albumintoglobulin ratio was within the 95% confidence interval for the value obtained for a nonphosphatecontaining plasma protein solution.
Table 6.
Calculation of [A_{tot}] and K_{a} for equine albumin, globulin, and serum protein using published values for Pco
_{2}, [SID^{+}], pH, ionic strength, temperature, pK ′_{1}, and S_{CO2}
Model validation.
Data from the published studies covered a physiological range of
PCO2
(36–54 Torr), [SID^{+}] (22.2–52.9 meq/l), and total protein concentration (4.6–7.3 g/dl). By use of the values experimentally determined by the simplifed strong ion model for horse plasma ([A_{tot}] = 2.24 × [total protein], in g/dl;K
_{a} = 2.22 × 10^{−7} eq/l), an excellent correlation between calculated pH and measured pH was observed for all experimental studies (Table7, Fig. 2). The values (means ± SD) for the slope (1.11 ± 0.12) and intercept (−0.84 ± 0.88) did not differ significantly from the line of identity. The mean difference between the estimated and actual pH was −0.004 (range −0.054 to +0.049).
Table 7.
Summary of linear regression analysis of the relationship between calculated and measured plasma pH
Fig. 2.
Scatterplot of relationship between calculated plasma pH (calculated from reported values for
, [SID^{+}], and total protein concentration) and measured plasma pH for horses (data from Refs. 17, 18, and 36). •, Calculated pH values using values obtained for [A_{tot}] and effective dissociation constant for plasma weak acids (K
_{a}) of horse plasma obtained by simplified strong ion model; ○, calculated pH values using Stewart’s commonly assumed values for [A_{tot}] andK
_{a}; ▵, calculated pH values using commonly assumed [A_{tot}] andK
_{a} values for human plasma. Solid line, line of identity; dashed lines, mean linear regression lines for pH calculated using values assumed by Stewart or for human plasma. TP, total protein.
By use of the commonly accepted human plasma values for [A_{tot}] (2.4 × [total protein], in g/dl) andK
_{a} (3.0 × 10^{−7} eq/l), the values (means ± SD) for the slope (1.18 ± 0.10) and intercept (−1.36 ± 0.73) differed significantly (P < 0.05) from the line of identity (Table 7, Fig. 2). The mean difference between the estimated and actual pH was −0.027 (range −0.084 to 0.000).
By use of a fixed value for [A_{tot}] (20.0 meq/l) and the most commonly used value forK
_{a} (3.0 × 10^{−7} eq/l), the values (means ± SD) for the slope (1.43 ± 0.33) and intercept (−3.32 ± 2.44) also differed significantly from the line of identity (Table 7, Fig. 2). The mean difference between the estimated and actual pH was −0.135 (range −0.242 to −0.054).
DISCUSSION
The simplified strong ion model provides a quantitative mechanistic acidbase model that explains many of the anomalies of the HendersonHasselbalch equation. The model explains why the apparent value for p
K1′
in plasma is dependent on pH, protein concentration, and Na^{+} concentration and provides a mechanistic explanation for the temperature dependence of plasma pH. The simplified strong ion model provides a practical method for experimentally determining [A_{tot}] andK
_{a} that produces values for horse plasma that are significantly different from those most commonly used for human plasma. Finally, the model simplifies to the HendersonHasselbalch equation when applied to aqueous nonprotein solutions, thereby providing an acidbase model that unites the HendersonHasselbalch equation and strong ion model.
Explanation for anomalies in the HendersonHasselbalch equation.
Rearrangement of Eq. 16
provides the following expressionlogPCO2
Equation 19
=−pH+log[SID+]−Ka[Atot]/(Ka+10−pH)K1′SCO2
which should be compared with Eq. 3
from the HendersonHasselbalch equation. The simplified strong ion model predicts that, over a physiological range of plasma pH at constant temperature and ionic strength, the in vitro log
PCO2
pH equilibration curve will be linear, because [SID^{+}], S_{CO2}, and pK′_{1}are constant andK
_{a}[A_{tot}]/(K
_{a}+ 10^{−pH}) is approximately constant. In alkalotic plasma solutions,K
_{a}[A_{tot}]/(K
_{a}+ 10^{−pH}) approaches [A_{tot}], inasmuch as (K
_{a} + 10^{−pH}) ≃K
_{a}, becauseK
_{a} (2.22 × 10^{−7}) > 10^{−pH}. In markedly acidic plasma solutions,K
_{a}[A_{tot}]/(K
_{a}+ 10^{−pH}) and, therefore, the intercept value of the log
PCO2
pH relationship becomes pH dependent, because 10^{−pH} > K_{a}. The simplified strong ion model therefore explains the nonlinearity of the log
PCO2
pH relationship in markedly acidic plasma.
As discussed previously, the HendersonHasselbalch equation fails to explain why the log
PCO2
pH relationship is displaced by changes in protein, Na^{+}, and Cl^{−} concentration (2, 39) (Fig. 1). The simplified strong ion model, through Eq.19
, predicts that, in solutions with increased protein concentration (increased [A_{tot}]),K
_{a}[A_{tot}]/(K
_{a}+ 10^{−pH}) is increased, thereby displacing the log
PCO2
pH curve to the left (Fig. 1). Addition of Na^{+} increases the [SID^{+}], thereby shifting the curve to the right (Fig. 1), whereas addition of Cl^{−} decreases the [SID^{+}], thereby shifting the curve to the left (Fig. 1).
The simplified strong ion model also explains why the apparent p
K1′
in plasma is dependent on pH, protein concentration, and Na^{+} concentration. Values for the apparent p
K1′
in plasma are usually obtained by titrating a plasma sample with a known amount of hydrochloric acid, thereby changing [SID^{+}]. This technique assumes that plasma protein dissociation (the ratio of [A^{−}] to [HA]) remains constant, regardless of the
PCO2
value (39) and that the known Δ[SID^{+}] is equal to and opposite from Δ[
HCO3−
]. The simplified strong ion model suggests that the [A^{−}]to[HA] ratio does not remain constant during in vitro CO_{2} equilibration, inasmuch as [A_{tot}] and [SID^{+}] will remain constant with changes in
PCO2
, but pH, [A^{−}], and [HA] will vary, inasmuch as they are dependent variables. The apparent dependence of plasma p
K1′
on pH, when p
K1′
is determined by acid titration, therefore, results from the dependence of [A^{−}] on
PCO2
and [SID^{+}]. A similar explanation can be offered for the effect of plasma protein and Na^{+} concentration on the apparent p
K1′
, inasmuch as protein concentration is the predominant determinant of [A_{tot}] and Na^{+} concentration determines [SID^{+}]. Changes in [A_{tot}] or [SID^{+}] alter [A^{−}] and, therefore, the ratio of [A^{−}] to [HA]. The explanations above suggest that p
K1′
, when used in Stewart’s strong ion model or the simplified strong ion model, should be corrected only for temperature and ionic strength.
The simplified strong ion model predicts that p
K1′
, determined by acid titration, will not be influenced by pH in aqueous nonplasma solutions, inasmuch as Eqs. 15
and
16
simplify to the HendersonHasselbalch equation in nonplasma solutions (where [A_{tot}] = 0). Studies demonstrating a pH dependence of apparent
pK1′
in serum, plasma, and cerebrospinal fluid (1, 27, 34, 38), which contain various concentrations of nonvolatile buffers ([A_{tot}] > 0) but no pH dependence in aqueous nonplasma solutions ([A_{tot}] = 0) (1, 19,27), support this prediction. Finally, the curvilinear nature of the in vivo log
PCO2
pH relationship (Fig.1) results from changes in [SID^{+}] reflected by alterations in plasma Na^{+}, K^{+}, and Cl^{−} concentrations (6, 10), inasmuch as acidbase status is regulated to ensure a constant protein charge state (7).
Temperature dependence of plasma pH.
Equation 14
indicates that plasma pH varies directly with plasma pK
_{a}. It therefore follows that ΔpH/ΔT will approximate ΔpK
_{a}/ΔT, where T is temperature. This is consistent with the alphastat hypothesis, because the ΔpH/ΔT of mammalian plasma (−0.015 to −0.020 unit/°C) is similar to the ΔpK
_{a}/ΔT of imidazole (−0.016 unit/°C) (32). The simplified strong ion model therefore provides a direct explanation for the temperature dependence of plasma pH, in that plasma pH varies with temperature, primarily because the value forK
_{a} varies with temperature, and plasma pH is dependent onK
_{a}. Temperatureinduced changes in
K1′
and
SCO2
play a much smaller role in the temperatureinduced changes in pH. The effect of temperature should not be neglected in studies utilizing Stewart’s strong ion model or the simplified strong ion model, inasmuch as an increase in temperature of 4°C (a common occurrence during strenuous exercise) will decrease plasma pH by ∼0.06 unit, primarily through temperatureinduced changes inK
_{a}.
The 95% confidence interval for the effectiveK
_{a} of horse plasma at 37°C (2.22 ± 0.32 × 10^{−7} eq/l) includes the value predicted for imidazole at 37°C (1.90 × 10^{−7} eq/l) on the basis of a pK
_{a} of 6.95 for imidazole at 25°C, a heat of enthalpy of 7,700 cal/mol (32), and correction of this value for temperature by the van’t Hoff equation. The close agreement between theK
_{a} values for horse plasma and imidazole is consistent with Reeve’s hypothesis that plasma nonvolatile buffers can be modeled as a single imidazole group over the physiological range of pH (32).
Experimental determination of [A_{tot}] and K_{a}.
The simplified strong ion model provides a practical in vitro method for experimentally determining [A_{tot}] andK
_{a}. Of interest is the finding that the experimentally determined values for [A_{tot}] (15.0 ± 3.1 meq/l) and K
_{a}(2.22 ± 0.32 × 10^{−7} eq/l) of horse plasma were significantly different from the values most commonly used for human plasma ([A_{tot}] = 20 meq/l, K
_{a} = 3.0 × 10^{−7} eq/l) (45,46). Figure 2 demonstrates that the experimentally determined values for [A_{tot}] andK
_{a} more accurately predict pH for horse plasma than values derived from human plasma. This emphasizes the point that speciesspecific values for [A_{tot}] andK
_{a} should be experimentally determined when Stewart’s strong ion model or the simplified strong ion model is used to describe acidbase equilibria.
The nonlinear regression technique used in this study to estimate [A_{tot}] andK
_{a} was complicated by the presence of multicollinearity. The correlation between regression parameter estimates for [A_{tot}] andK
_{a} exceeded 0.95 for all analyses, indicating severe multicollinearity (16). The presence of large standard errors for the parameter estimates, despite excellent goodness of fit values (R
^{2} ≥ 0.998) (Table 6), and occasional unreasonable parameter estimates or inability to provide a parameter estimate (Table 5) are also suggestive of multicollinearity. Structural multicollinearity is inherent in the simplified strong ion and Stewart’s strong ion approaches because of the mathematical relationship between [A_{tot}] andK
_{a} demonstrated in Eq. 10
. Additional structural multicollinearity exists in Stewart’s strong ion approach because of the mathematical relationship between [OH^{−}] and [H^{+}] (Eq. 17
), between [
CO32−
] and [
HCO3−
] (Eq. 18
), and between [
CO32−
] and [H^{+}] (Eq. 18
). Recommended methods for analyzing data containing multicollinearities include using ridge regression techniques (such as Marquardt’s approach used in this study), reformulating the mathematical equation (the equations used demonstrated the least multicollinearity), and eliminating parameters from the regression model (16, 37). On the basis of the derivation inappendix
and the results in Table 4, it appeared that two parameters ([OH^{−}] and [
CO32−
]) could be removed from Stewart’s strong ion model. When this was done (equivalent to reducing the strong ion model to the simplified strong ion model), realistic estimates for [A_{tot}] andK
_{a} were obtained for six of six tonometered horse plasma samples compared with three of six samples when Stewart’s approach was used (Table 5). In other words, experimental determination of [A_{tot}] andK
_{a} is facilitated by use of the simplified strong ion model.
Independent determinants of plasma pH.
Equation 15
indicates that six factors (
PCO2
, [SID^{+}], [A_{tot}],K
_{a},
K1′
, and
SCO2
) physicochemically determine plasma pH. Not all these factors exert an independent effect on plasma pH, inasmuch as the apparent dissociation constants K
_{a} and
K1′
are dependent on temperature and ionic strength,
SCO2
is dependent on temperature, ionic strength, and protein concentration, and [A_{tot}] andK
_{a} are dependent on the relative contributions of individual nonvolatile plasma buffers (such as albumin, globulin, and phosphate). The independent factors that determine plasma pH are therefore
PCO2
, [SID^{+}], concentration of individual nonvolatile plasma buffers (albumin, globulin, and phosphate), ionic strength, and temperature. A change in any one of these variables will produce a direct and predictable change in plasma pH.
Limitations of the simplified strong ion model.
The major limitations of the simplified model are identical to those of Stewart’s strong ion model in that1) an accurate value for [SID^{+}] can be difficult to obtain, 2) values for [A_{tot}] andK
_{a} are pH dependent when expressed in terms of milliequivalents per liter,3) values for [A_{tot}] andK
_{a} depend on the relative concentrations of the four nonvolatile plasma buffers (imidazole, αamino,
H2PO4−
, and citric acid), and 4) the heterogeneous group of nonvolatile plasma buffers with an approximately linear titration curve is being modeled as a single buffer with a classic sigmoidal titration curve. Despite these limitations, the simplified strong ion model can be used clinically, in that it predicts plasma pH within 0.05 unit (with a mean prediction within 0.001 unit) from measured values for
PCO2
, [SID^{+}], and total protein concentration (Fig. 2, Table 7).
PCO2
can be measured accurately to within 1 Torr, resulting in an error of 0.01 unit in the predicted pH. [SID^{+}] can be measured within 3 meq/l when calculated from the Na^{+}, K^{+}, Cl^{−}, and lactate^{−} concentrations, the error resulting from cumulative measurement errors and the presence of unmeasured strong ions. This produces an error of 0.05 unit in the predicted pH. [A_{tot}] can be measured within 10%, the error resulting from changes in the albumintoglobulin ratio or a marked increase in the phosphate concentration. This produces an error in pH of 0.02 unit. The simplified strong ion model should therefore produce a maximum error in the predicted pH of ∼0.08 unit, a value that exceeds the observed maximum error (0.06 unit) when the model was applied to published data (Table 7, Fig. 2). Other studies have shown that Stewart’s strong ion model predicts plasma pH within a similar error margin (13, 30).
The pH dependence of [A_{tot}] andK
_{a} is theoretically of some concern but is practically inconsequential. As demonstrated in appendix
, [A_{tot}] is pH dependent when expressed in terms of milliequivalents per liter. The effect of this pH dependence on [A_{tot}] (in meq/l) is very small, however, inasmuch as the predominant determinant of [A_{tot}] (in meq/l) is the net charge produced by fully dissociated groups on plasma proteins (appendix
, Table 3). A decrease in plasma pH from 7.40 to 6.80 alters the calculated value of [A_{tot}] from 18.0 to 17.5 meq/l, a change of 2.2%. An increase in plasma pH from 7.40 to 7.70 alters the calculated value of [A_{tot}] from 18.0 to 18.5 meq/l, a change of 2.8%. As calculated above, these changes in [A_{tot}] will result in an error in predicted pH of <0.01. For practical purposes, [A_{tot}] can therefore be considered constant over the physiological range of pH (6.8–7.7). An explanation as to why plasmaK
_{a} also varies with pH is the functional categorization used in this study to differentiate strong ions from buffer ions, namely, whether the individual pK
_{a}falls within the range of pH ±1.5. For example, nuclear magnetic resonance examination of human serum albumin indicates that the pK
_{a} of individual imidazole groups ranges from 5.2 to 7.9, with an overall mean of ∼6.9 (5). An increase in pH from 7.4 to 7.7 causes imidazole residues with a pK
_{a} between 5.9 and 6.2 to effectively lose their ability to function as a nonvolatile buffer, potentially altering the apparent plasmaK
_{a}. The resultant effect on predicted pH is small, however, inasmuch as an increase inK
_{a} from 2 × 10^{−7} to 3 × 10^{−7} eq/l changes plasma pH by <0.01. The dependence of [A_{tot}] andK
_{a} on plasma pH does not invalidate the simplified strong ion model and Stewart’s strong ion model; instead it limits the pH range to which both models can be accurately applied. Validation of the simplified strong ion model (Table 7, Fig. 2) indicates that the experimentally determined values for [A_{tot}] andK
_{a} are accurate in the horse for pH 7.20–7.60. It is unknown whether these values remain accurate outside this pH range.
Concern has been raised over the effect of changes in the relative concentrations of albumin, globulin, and phosphate on [A_{tot}] andK
_{a} (11), the effect of citrate being ignored because of its relatively low plasma concentration. Removal of phosphate from plasma will decrease [A_{tot}] by ∼20% but not change K
_{a}, inasmuch as phosphate normally contributes 2.7 meq/l to [A_{tot}] (appendix
) and has aK
_{a} (2 × 10^{−7}) similar to that of imidazole and plasma. A fivefold increase in plasma phosphate concentration will also not change plasmaK
_{a} but will increase [A_{tot}] by 10.8 meq/l. This will result in a large increase in [A_{tot}] and, therefore, a decrease in plasma pH. The effect of the globulin concentration on [A_{tot}] andK
_{a} requires consideration, inasmuch as the estimated values for [A_{tot}] andK
_{a} of horse globulin may differ from those of plasma, although a significant difference was not observed in this study. This suggests that an altered albumintoglobulin ratio could alter the effective values for [A_{tot}] andK
_{a} in the horse. This is not surprising, in that the amino acid composition of globulin (particularly the composition of dissociable imidazole and αamino groups) probably differs from that of albumin. Because of the concordance between estimates for [A_{tot}] obtained for normal horse plasma and solutions of purified horse serum protein, the following equation is suggested to estimate [A_{tot}] for horse plasma with abnormal concentrations of albumin, globulin, or phosphate[Atot](meq/l)=2.25[albumin](g/dl)
Equation 20
+1.40[globulin](g/dl)+0.59[phosphate](mg/dl)
The titration curve of plasma protein over the physiological range of pH (6.6–7.8) is approximately linear (40, 4751). This result has been attributed to the titration of dissociable imidazole and αamino groups that possess different intrinsic pK
_{a} values (7). The simplified strong ion model (and Stewart’s strong ion model) reduces the heterogeneous group of dissociable plasma buffers to a single imidazole group with a clearly identifiable pK
_{a}. This modeling assumption is consistent with the alphastat theory for acidbase regulation (32); however, the model appears to be inconsistent with experimental observation, in that over a wide range of pH this modeling assumption would produce a sigmoidal, rather than a linear, relationship between net protein charge and pH (7) (Fig.3). However, close examination of the titration curves for albumin, globulin, and serum protein modeled as homogeneous buffers reveals that the net protein chargepH relationship can be well approximated by a straight line over the pH range used in titration studies (Fig. 3). Moreover, the validation study indicates that the simplification is accurate for horse plasma over a physiological pH range of 7.2–7.6. It remains to be determined whether this simplification remains valid over a wider range of pH.
Fig. 3.
Titration curves for horse albumin, globulin, and total protein modeled as a single imidazole group with a specific pK
_{a} (values determined by nonlinear regression using simplified strong ion model from values in Refs. 50 and 51). Superimposed on each modeled titration curve are individual data points (○) for net protein chargepH relationship determined in Refs. 50 and 51. Vertical lines represent range of pH used for titration. A^{−}, conjugate base; HA, weak acid.
Generalizability of simplified strong ion model.
The approach used to develop the simplified strong ion model can be applied to any biological fluid consisting of strong ions, volatile buffer ions, and nonvolatile buffer ions, provided that the effects of complex ion interactions, oxidationreduction reactions, and precipitation reactions in the fluid are quantitatively unimportant. If these criteria are not met, the simplified strong ion model should not be applied, because one of the model’s assumptions (all quantitatively important chemical reactions are those of simple ions in solution) has been violated. The simplified strong ion model can therefore be adapted to describe acidbase equilibria in peritoneal, pleural, pericardial, interstitial, synovial, and cerebrospinal fluid, as well as in erythrocytes. The model should not be applied to urine, because precipitation reactions and complex ion interactions occur in this medium. Similar difficulties may occur when the model is applied to the intracellular environment.
Footnotes

Address for reprint requests: P. D. Constable, Dept. of Veterinary Clinical Medicine, College of Veterinary Medicine, University of Illinois at UrbanaChampaign, 1008 West Hazelwood Dr., Urbana, IL 61801.
 Copyright © 1997 the American Physiological Society
Appendix
An estimate for the value of [A_{tot}] in milliequivalents per liter can be obtained by determining the molar concentration and attributing a valence to [HA] and [A^{−}] for the four nonvolatile plasma buffers and then summing the resultant values for [HA] and [A^{−}] when expressed in milliequivalents per liter. Accurate data are available for human and bovine albumin (47, 48), and the following estimate for [A_{tot}] is calculated for a solution resembling human plasma that has no globulin.
For imidazole at pH 7.40, the ratio of [HA] to [A^{−}] equals 0.2, inasmuch as K
_{a} = 2 × 10^{−7} eq/l. For human albumin the value for [HA] in milliequivalents per liter can be calculated as[HA]imidazole=(net valence of albumin,in eq/M)×(proportion of dissociable groups in albumin that are imidazole)
×(proportion of imidazole groups in [HA]state at pH 7.4)×(mol of albumin in solution,in mmol/l)
Equation A1
Human albumin contains 16 dissociable imidazole groups and 4 dissociable αamino groups (47), with the fully dissociated groups producing a net valence of −26 eq (Table 3). Because the valence for HA_{imidazole} = +1, the following equations can be derived for a solution containing 4.1 g/dl albumin (mol wt of human albumin = 69,000)[HA]imidazole=(+1−26 eq/M)
×(16/20)×(2.67/16)×0.59 mmol/l=−1.97 meq/l
Equation A2
[A−]imidazole=(0−26 eq/M)
×(16/20)×(13.33/16)×0.59 mmol/l=−10.22 meq/l
Equation A3
For αamino groups at pH 7.40, the ratio of [HA] to [A^{−}] equals 2.0, inasmuch as K
_{a} = 2 × 10^{−8} eq/l. Because the valence for [HA]_{αamino} = 0, the following equations can be derived for a solution containing 4.1 g/dl albumin[HA]αamino=(0−26 eq/M)
×(4/20)×(2.67/4)×0.59 mmol/l=−2.05 meq/l
Equation A4
[A−]αamino=(−1−26 eq/M)
×(4/20)×(1.33/4)×0.59 mmol/l=−1.06 meq/l
Equation A5
For phosphate at pH 7.40, the ratio of [
H2PO4−
] to [
HPO42−
] (where [
H2PO4−
] and [
HPO42−
] are concentrations of
H2PO4−
and
HPO42−
) equals 0.2, inasmuch asK
_{a} = 2 × 10^{−7} eq/l. For human plasma with a phosphate concentration of 4 mg/dl (1.29 mmol/l), the following equations can be calculated[HA]H2PO4−=(−1 eq/M)
×(0.167/1)×1.29 mmol/l=−0.22 meq/l
Equation A6
[A−]HPO4−=(−2 eq/M)
×(0.833/1)×1.29 mmol/l=−2.15 meq/l
Equation A7
For citric acid at pH 7.40, the ratio of [R ⋅ COOH] to [R ⋅ COO^{−}] equals 1.2, inasmuch asK
_{a} = 7.9 × 10^{−7} eq/l. For human plasma with a citrate concentration of 0.6 mmol/l, the following equations can be calculated[HA]R⋅COO−=(0 eq/M)×(0.547/1)×0.6 mmol/l=0 meq/l
Equation A8
[A−]R⋅COO−=(−1 eq/M)×(0.453/1)
×0.6 mmol/l=−0.28 meq/l
Equation A9
The value for [A_{tot}] in milliequivalents per liter for an electrolyte solution containing human albumin can now be estimated[Atot]=Σ[HA]i+Σ[A−]i
≃[(−1.97)+(−2.05)+(−0.22)+(0)]
+[(−10.22)+(−1.06)+(−2.15)+(−0.28)]≃−18.0 meq/l
Equation A10
where [HA]_{i}and [A^{−}]_{i}are the ith value for [HA] and [A^{−}].
The value derived for [A_{tot}] in milliequivalents per liter is approximate, inasmuch as the number of dissociated groups in albumin is approximate (47, 48). A similar approach for a solution resembling bovine plasma (Table 3) (48), which has no globulin (assuming [albumin] = 3.2 g/dl; mol wt of albumin = 65,000; [phosphate] = 1.29 mmol/l) produces an estimate for [A_{tot}] of 12.8 meq/l, which differs from that obtained for human albumin solution. This suggests that the value for [A_{tot}] will vary among species.
Appendix
Equation 12
can be rearranged to provide([SID+]−[HCO3−])(aH++Ka)=Ka[Atot]
Equation B1Which can be expanded to[SID+]aH++Ka[SID+]
−Ka[Atot]−aH+[HCO3−]−Ka[HCO3−]=0
Equation B2
Substituting for [
HCO3−
] from the overall equilibrium reaction for the HendersonHasselbalch equation (Eq. 1
), such that [
HCO3−
] =
K1′
SCO2
PCO2
/
aH+
provides[SID+]aH++Ka[SID+]−Ka[Atot]−aH+K1′SCO2PCO2/aH+
−KaK1′SCO2PCO2/aH+=0
Equation B3
Multiplying both sides of Eq.EB3
by [H^{+}] (i.e.,a
_{H+}) provides[SID+](aH+)2+(Ka[SID+]−Ka[Atot]−K1′SCO2PCO2)(aH+)
−KaK1′SCO2PCO2=0
Equation B4
The general solution of the quadratic equationax
^{2} +bx +c = 0 isx = (−b ±
b2−4ac
)/2a. The solution for Eq. EB4
is thereforeaH+=−(Ka[SID+]−Ka[Atot]−K 1SCO2PCO2)±(Ka[SID+]−Ka[Atot]−K1′SCO2PCO2)2 +4[SID+]KaK1′SCO2PCO22[SID+]
Equation B5Expansion and rearrangement providesaH+=K1′SCO2PCO2+Ka[Atot]−Ka[SID+]±(Ka[SID+])2−2Ka2[SID+][Atot]+(Ka[Atot])2 +(K1′SCO2PCO2)2+2[SID+]KaK1′SCO2PCO2 +2[Atot]KaK1′SCO2PCO22[SID+]
Equation B6which is equivalent toaH+=K1′SCO2PCO2+Ka[Atot]−Ka[SID+]±(Ka[SID+])2+2Ka2[SID+][Atot]+(Ka[Atot])2+(K1′SCO2PCO2)2+2Ka[SID+]K1′SCO2PCO2+2Ka[Atot]K1′SCO2PCO2−4Ka2[SID+][Atot]2[SID+]
Equation B7which can be further simplified toaH+=K1′SCO2PCO2+Ka[Atot]−Ka[SID+]±(K1′SCO2PCO2+Ka[SID+]+Ka[Atot])2 −4Ka2[SID+][Atot]2[SID+]
Equation B8Taking the logarithm of the reciprocal of both sides of Eq.EB8
produces only one real solutionpH=log2[SID+]K1′SCO2PCO2+Ka[Atot]−Ka[SID+]+(K1′SCO2PCO2+Ka[SID+]+Ka[Atot])2−4K a2[SID+][Atot]
Equation B9
Appendix
Stewart developed the following equation relating [H^{+}] to 3 independent variables (
PCO2
, [SID^{+}], [A_{tot}]) and 4 “constants” (K
_{a},
Kw′
,K
_{3}, andK
_{c}), whereK
_{c} =
K1′
×
SCO2
[H+]4+([SID+]+Ka)[H+]3+(Ka[SID+]−Ka[Atot]
−Kw′−KcPCO2)[H+]2−[Ka(Kw′+KcPCO2)
−K3KcPCO2][H+]−KaK3KcPCO2=0
Equation C1
Simplification of Eq. EC1
for mammalian plasma at 37°C requires recognition that the equation can be rewritten as[H+]4+w[H+]3+x[H+]2+y[H+]+z=0
Equation C2and that using appropriate units, the approximate values of [SID^{+}] and [A_{tot}] are 4 × 10^{−2} and 2 × 10^{−2} eq/l, respectively,
PCO2
is ∼4 × 10^{1} Torr, [H^{+}] is 40 × 10^{−9} eq/l,K
_{a} is on the order of 2 × 10^{−7}eq/l, K
_{c} andK
_{3} are ∼2.5 × 10^{−11}eq ⋅ l^{−1} ⋅ Torr^{−1}and 6.0 × 10^{−11} eq/l, respectively, and
Kw′
is 4.4 × 10^{−14}eq^{2}/l^{2}. On this basisw=[SID+]+Ka=[SID+],since 2×10−7≪4×10−2
x=Ka[SID+]−Ka[Atot]−Kw′−KcPCO2
=Ka[SID+]−Ka[Atot]−KcPCO2
since 4.4×10−14≪(2×10−7×4×10−2
−2×10−7×2×10−2−2.5×10−11×4×101)
y=−[Ka(Kw′+KcPCO2)−K3KcPCO2]
=−KaKcPCO2−K3KcPCO2,
since 4.4×10−14≪2.5×10−11×4×101
=−KcPCO2(Ka+K3)
=−KaKcPCO2,since 6.0×10−11≪2×10−7
z=−KaK3KcPCO2
Substituting the above into Eq. EC2
provides[H+]4+[SID+][H+]3+(Ka[SID+]−Ka[Atot]
−KcPCO2)[H+]2−KaKcPCO2[H+]−KaK3KcPCO2=0
Equation C3
Substituting for
PCO2
fromEq. 1
provides[H+]4+([SID+]−[HCO3−])[H+]3+Ka([SID+]−[Atot]
−[HCO3−])[H+]2−KaK3[HCO3−][H+]=0
Equation C4
which simplifies to[H+]3+([SID+]−[HCO3−])[H+]2+Ka([SID+]−[Atot]
−[HCO3−])[H+]−KaK3[HCO3−]=0
Equation C5
Simplification of Eq. EC5
for physiological plasma requires recognition that Eq.EC5
can be rewritten as[H+]3+a[H+]2+b[H+]+c=0
Equation C6The approximate values of each term in Eq.EC6
can now be calculated, given that [
HCO3−
] = 2.5 × 10^{−2} eq/l, [
CO32−
] = 3.8 × 10^{−5} eq/l, and [OH^{−}] = 1.1 × 10^{−6} eq/l (calculated from above)[H+]3=[40×10−9]3=6.4×10−23
a[H+]2=1.5×10−2×[40×10−9]2=2.4×10−17
b[H+]=2×10−7([SID+]−[Atot]−[HCO3−])(40×10−9)
=(8×10−15)([SID+]−[Atot]−[HCO3−])
=(8×10−15)([CO32−]+[OH−]−[H+])
=(8×10−15)(3.8×10−5+1.1×10−6−4×10−8)
=3.1×10−19
c=2×10−7×6×10−11×2.5×10−2=3.0×10−19
The [H^{+}]^{3}term can therefore be ignored, and Eq.EC6
can be simplified to([SID+]−[HCO3−])[H+]2−Ka([Atot]
+[HCO3−]−[SID+])[H+]−KaK3[HCO3−]=0
Equation C7
The general solution of the quadratic equationax
^{2} +bx +c = 0 isx = (−b ±
b2−4ac
)/2a. The solution for Eq. EC7
is therefore[H+]=Ka([Atot]+[HCO3−]−[SID+])±Ka2([Atot]+[HCO3−]−[SID+])2 +4KaK3[HCO3−]([SID+]−[HCO3−])2([SID+]−[HCO3−])
Equation C8which is equivalent to[H+]
=Ka([Atot]+[HCO3−]−[SID+])±Ka [Atot]2+2[Atot][HCO3−]−2[Atot][SID+]+[SID+]2+(1−4K3/Ka)[HCO3−]2−2(1−2K3/Ka)[HCO3−][SID+]2([SID+]−[HCO3−])
Equation C9
This can be simplified to[H+]
=Ka([Atot]+[HCO3−]−[SID+])±Ka [Atot]2+2[Atot][HCO3−]−2[Atot][SID+]+[SID+]2+[HCO3−]2−2[HCO3−][SID+]2([SID+]−[HCO3−])
Equation C10
on the basis that(1−4K3/Ka)[HCO3−]2=(1−4×6×10−11/2×10−7)[HCO3−]2
=(1−1.3×10−3)[HCO3−]2
≃[HCO3−]2
and that(1−2K3/Ka)[HCO3−][SID+]=(1−2×6×10−11/2×10−7)
×[HCO3−][SID+]
=(1−6×10−4)[HCO3−][SID+]
≃[HCO3−][SID+]
EquationC10 can be expressed as[H+]=Ka([Atot]+[HCO3−]−[SID+])±Ka([Atot]+[HCO3−]−[SID+])22([SID+]−[HCO3−])
Equation C11The only real solution of Eq. EC11
is[H+]=2Ka([Atot]−[SID+]+[HCO3−])2([SID+]−[HCO3−])
Equation C12which simplifies to[H+]=Ka [Atot][SID+]−[HCO3−]−1
Equation C13Taking the logarithm of the reciprocal of both sides of Eq.EC13
producespH=pKa−log[Atot][SID+]−[HCO3−]−1
Equation C14which is similar to Eq. 14
developed from the simplified strong ion model, assuming that Stewart’s parameter [H^{+}] approximates the H^{+} activity.