J Appl Physiol 95: 2333–2344, 2003.
First published August 15, 2003; 10.1152/japplphysiol.00560.2003.
Calculation of physiological acid-base parameters in
multicompartment systems with application to human blood
E. Wrenn Wooten
Department of Radiology, Magnetic Resonance Imaging Division, Baylor University
Medical Center, Dallas, Texas 75246; and Radiology Associates, PA, Little Rock, Arkansas 72205
Submitted 27 May 2003; accepted in final form 7 August 2003
base excess; strong ion difference; Van Slyke equation; Stewart method; blood titration
THERE HAS BEEN RENEWED INTEREST in the theoretical
description of physiological acid-base chemistry and in
the parameters used to describe acid-base balance (7,
8, 14, 23, 24, 27, 37). Although the Henderson-Hasselbalch equation (34, 35) is often used alone to evaluate
acid-base disturbances, the base excess (BE) approach
has traditionally been viewed as the most complete
method to quantify the metabolic component of an
acid-base disorder (49). The BE was originally obtained
from a nomogram (45, 47), but with automation of body
fluid analysis, a computational approach involving the
use of proton balance to calculate BE as the change in
total titratable base (⌬CB) of a given body fluid via the
Van Slyke equation was developed (47, 48). More recently, the strong ion difference (SID) method, proposed by Stewart in 1978 (52), has also been increasingly used as an alternate means to characterize acidbase status. SID is the difference between the sum of
positive-charge concentrations and the sum of negative-charge concentrations for those ions that do not
participate in proton-transfer reactions. This parameter is the same as that proposed by Singer and Hastings in 1948 under the name “buffer base” (50). SID
Address for reprint requests and other correspondence: E. W.
Wooten, Radiology Associates, PA, 500 S. Univ. Ave., Little Rock, AR
72205.
http://www.jap.org
theory uses charge and mass balance to deduce an
expression for proton concentration (52, 53).
SID theory has been applied to plasma (6, 9, 10, 15,
16, 55), and it has been shown previously (57) that
plasma BE and plasma SID can be derived from a
common formalism and that BE and the change in SID
(⌬SID) are numerically the same for plasma, provided
that the concentrations of plasma noncarbonate buffers remain constant (43, 57). When this condition is not
met, plasma ⌬CB and plasma ⌬SID differ by an added
constant. If, however, the reference state is chosen to
coincide with the new (abnormal) noncarbonate buffer
concentrations, the equivalence of ⌬CB and ⌬SID is
restored (57).
A complete quantitative description of the acid-base
status of an organism requires that both intra- and
extracellular effects in multiple compartments be
taken into consideration (47). To this end, SiggaardAndersen (46–48) defined BE for plasma, erythrocyte
fluid, whole blood, and extracellular fluid. In contrast,
the published applications of Stewart’s SID theory
thus far (e.g., see Refs. 1, 7, 20, 26, 39, 42) have been
confined to plasma because no corresponding extant
theory for multicompartment systems, such as whole
blood or extracellular fluid, exists.
In the following, the formalism used to derive expressions for CB and SID in the single-compartment
case of plasma (57) is extended to systems with multiple compartments. General equations for CB, SID, ⌬CB,
and ⌬SID are obtained for multicompartment systems
and applied to the specific example of human whole
blood, after the relevant model for the single compartment of human erythrocyte fluid is derived. These
equations give results approximating experimentally
determined values. The formulas for multicompartment systems are shown to have the same mathematical interrelationships as those demonstrated previously for single compartments (57), and the relationship between the form of the expressions derived here
and the form of the Van Slyke equation traditionally
used to calculate BE is also demonstrated. SID theory
for multicompartment systems is thus shown to be
precise to the same level of approximation as the traditional equations used for BE (33, 47–49), thereby
The costs of publication of this article were defrayed in part by the
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8750-7587/03 $5.00 Copyright © 2003 the American Physiological Society
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Wooten, E. Wrenn. Calculation of physiological acid-base
parameters in multicompartment systems with application
to human blood. J Appl Physiol 95: 2333–2344, 2003. First
published August 15, 2003; 10.1152/japplphysiol.00560.
2003.—A general formalism for calculating parameters describing physiological acid-base balance in single compartments is extended to multicompartment systems and demonstrated for the multicompartment example of human
whole blood. Expressions for total titratable base, strong ion
difference, change in total titratable base, change in strong
ion difference, and change in Van Slyke standard bicarbonate
are derived, giving calculated values in agreement with experimental data. The equations for multicompartment systems are found to have the same mathematical interrelationships as those for single compartments, and the relationship
of the present formalism to the traditional form of the Van
Slyke equation is also demonstrated. The multicompartment
model brings the strong ion difference theory to the same
quantitative level as the base excess method.
2334
PHYSIOLOGICAL ACID-BASE PARAMETERS
bringing SID theory to the same quantitative level as
the BE method.
THEORY
Single-compartment systems. Previously, it was
shown that the master equation approach developed by
Guenther (21) could be adapted to obtain expressions
for the two acid-base parameters, CB and SID. In a
general notational form, the acid-base concentration
parameters P(␹) in a single aqueous compartment ␹
can be written as
P共␹兲 ⫽ C ␹ ⫹
Cn共␹兲␰៮ n共␹兲 ⫺ D␹
(1)
兺
n
兺
n
where [HCO3⫺]␹ is the concentration of bicarbonate in
compartment ␹. Because of the linearity of these expressions over the physiological pH range from 6.8 to
7.8 (47, 57), an approximate straight line form of P(␹)
can be derived as (57)
⳵␰៮ n
Cn共␹兲
pH共␹兲
P共␹兲 ⫽ 关HCO 3⫺兴␹ ⫹
⳵pH
n
冉兺
⫹
冊
兺 C 共␹兲␰៮
n
m共n兲
n
⫺
兺 C 共␹兲b
n
n
(3)
n
Here ␰៮ m(n)(␹) represents either e៮ max(n), the maximum
number of proton acceptor sites on species n for BE, or
⫺z៮ min(n), the opposite of the minimum possible charge
on species n for SID. bn Is a constant for species n and
at a given pH can be computed from Eqs. 2 and 3 to be
⳵␰៮ n
bn ⫽
(4)
pH共␹兲 ⫺ ␰៮ n共␹兲 ⫹ ␰៮ m共n兲
⳵pH
It is also worth noting
⳵e៮ n
⳵z៮ n
⫽⫺
⳵pH
⳵pH
(5)
J Appl Physiol • VOL
z៮ max共n兲 ⫽ e៮ max共n兲 ⫹ z៮ min共n兲
(6)
which provides the specific relationship between CB
and SID as obtained previously (57)
C B共␹兲 ⫽ SID共␹兲 ⫹
兺 C 共␹兲z៮
n
(7)
max共n兲
n
where z៮ max(n) is the maximum possible charge of species n.
Physiological pH is determined by the simultaneous
solution of any of the expressions for P(␹) and the
Henderson-Hasselbalch equation in a given compartment ␹.
pH共␹兲 ⫽ pK⬘ ⫹ log
关HCO3⫺兴␹
S 䡠 PCO2共␹兲
(8)
where, for human plasma, pK⬘ ⫽ 6.103 and S is the
equilibrium constant between aqueous dissolved CO2
and CO2 in the gas phase, and equals 0.0306 at 37°C
when [H⫹] is in moles per liter, [HCO3⫺]P is in millimoles per liter, and PCO2 is in Torr (4).
Multicompartment systems. For a multicompartment
system M with single subcompartments separated by
semipermeable membranes, the acid-base parameter
P(M) for the system can be expressed as a linear
combination of the parameter values P(␹) in the various single subcompartments ␹ (28, 37, 47), assuming
that sufficient time has elapsed after a perturbation to
reach a new steady state (47)
P共M兲 ⫽
兺 ␾共␹兲P共␹兲
(9)
␹
noting that for the volume fractions ␾(␹)
兺 ␾共␹兲 ⫽ 1
(10)
␹
It follows that for constant volume fractions
⌬P共M兲 ⫽
兺 ␾共␹兲⌬P共␹兲
(11)
␹
where ⌬ denotes a change. The derivation of explicit
expressions for ⌬P(␹) has been described in detail previously (57). The corresponding expression to Eq. 7 is
then obtained as
C B共M兲 ⫽ SID共M兲 ⫹
兺 兺 ␾共␹兲C 共␹兲z៮
n
␹
max共n兲
(12)
n
and from this relation follows, similar to the result for
a single compartment (57)
⌬CB共M兲 ⫽ ⌬SID共M兲 ⫹
兺 兺 ␾共␹兲z៮
max共n兲
␹
⌬Cn共␹兲 (13)
n
One of the issues encountered in multicompartment
systems is that variables such as pH(␹) are often only
experimentally available for a single compartment. For
example, in routine clinical work, the plasma values
pH(P) and [HCO3⫺]P in equilibrium with the erythro-
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where ␹ refers, for example, to plasma (␹ ⫽ P) or
erythrocyte fluid (␹ ⫽ E); P(␹) represents either CB(␹)
or SID(␹) in compartment ␹. Similarly, ␰៮ n is the average value per molecule of some physical property of
noncarbonate buffer species n (21, 57). In the case of
base excess, ␰៮ n is the average number of proton binding
sites per molecule, e៮ n(␹). In SID theory, ␰៮ n(␹) is the
opposite of the average charge per molecule of species
n, ⫺z៮ n(␹). The calculation of ␰៮ n(␹) in terms of conditional molar equilibrium constants and H⫹ concentration ([H⫹]) has been described in Ref. 57. C␹ is the
equilibrium concentration of proton acceptor sites of
carbonate species (also equaling the opposite of the
charge concentration of carbonate species), Cn(␹) is the
analytical concentration of noncarbonate buffer species
n, and D␹ is the difference between aqueous free proton
and free hydroxyl ion concentrations, all in compartment ␹ (57).
Following the convention used before (57), and first
advocated by Constable (6), terms that are small under
physiological conditions are neglected to give
P共␹兲 ⫽ 关HCO 3⫺兴␹ ⫹
Cn共␹兲␰៮ n共␹兲
(2)
as this relationship links the BE and SID theories (57).
An additional relevant relationship obtained from the
results of Ref. 57 is
2335
PHYSIOLOGICAL ACID-BASE PARAMETERS
cyte phase and interstitial fluid are the variables typically measured, with the other pH(␹) and [HCO3⫺]␹
seldom being obtained directly. It is therefore necessary to calculate the concentrations of species in one
compartment from those in another. Fortunately, for
many species of interest, concentrations can be calculated across the relevant membrane via the Nernst
equation (12, 28, 40, 47)
⌬␺共␹兲 ⫽ ⫺
关H⫹兴␹
RT
RT
ln rm共␹兲 ⫽ ⫺
ln ⫹
cF
cF
关H 兴P
⫽⫺
RT
关HCO3⫺兴␹
(14)
ln
cF
关HCO3⫺兴P
P共B兲 ⫽ 关1 ⫺ ␾共E兲兴P共P兲 ⫹ ␾共E兲P共E兲
(15)
⌽(E) can be equated with the hematocrit (47, 48). The
Nernst equation (Eq. 14) can be used to calculate the
erythrocyte pH and [HCO3⫺] from the corresponding
plasma concentrations to calculate P(E). It follows from
Eq. 11 that
⌬P共B兲 ⫽ 关1 ⫺ ␾共E兲兴⌬P共P兲 ⫹ ␾共E兲⌬P共E兲
(16)
under the assumption of constant hematocrit.
As an alternative to using the Nernst equation to
calculate erythrocyte pH and [HCO3⫺], other approaches specific to the determination of intraerythrocyte concentration have also been employed (47, 48).
For example, in the case of the carbonate system equilibrating across the human erythrocyte membrane, the
following expression has been suggested for the Donnan ratio rc(E) for bicarbonate as a function of the
plasma pH (48)
r c 共E兲 ⫽
关HCO3⫺兴E
⫽ 0.835 䡠 10关1.492⫺0.23pH共P兲兴
关HCO3⫺兴P
(17)
The empirically determined relationship
⌬pH共E兲
⫽ 0.77
⌬pH共P兲
(18)
has also been used for the pH distribution (18, 47).
Thus, by using a knowledge of the noncarbonate
buffer constituents of the plasma and erythrocyte compartments, the concentrations of those constituents,
their effective molar equilibrium constants, along with
pH(P), [HCO3⫺]P, and their distribution functions
J Appl Physiol • VOL
冋
BE共B兲 ⫽ 1 ⫺
册
CHb共B兲
共⌬关HCO3⫺兴P ⫹ ␤⬘共B兲⌬pH共P兲兲
CHb°
(19)
where BE(B) is the BE of whole blood, CHb(B) is the
hemoglobin concentration of whole blood, and
⌬[HCO3⫺]P and ⌬pH(P) are the changes in the plasma
bicarbonate concentration and in the plasma pH, respectively. CHb° is a constant, which depends on the
erythrocyte fluid hemoglobin concentration CHb(E) and
the bicarbonate Donnan ratio rc(E), both of which are
assumed to have constant normal values (48), according to
C Hb°⫽
CHb共E兲
关1 ⫺ rc共E兲兴
(20)
Because Siggaard-Andersen and coworkers (48, 49)
defined hemoglobin concentrations and buffer values
in terms of the oxyhemoglobin monomer, a value for
CHb° of 43 mM is obtained on substituting the appropriate human normal values, including a value for
rc(E) of 0.51. In some earlier references, a value of 0.57
was used, giving a CHb° of 48 mM (47). ␤⬘(B), an
effective buffer value for whole blood (48), is the slope
of the CO2 equilibration curve for whole blood in a
[HCO3⫺]P vs. pH(P) coordinate system at constant noncarbonate buffer concentration and constant total titratable base (47, 48). ␤⬘(B) is calculated via (49)
␤⬘共B兲 ⫽ CHb共B兲 䡠 ␤⬘Hb ⫹ ␤共P兲
(21)
where ␤⬘Hb is the apparent molar buffer value of the
oxyhemoglobin monomer in whole blood and is assigned a value of 2.3 in humans (49). ␤(P), with a
default value of 7.7 mM in humans (49), is the buffer
value of plasma computed from (49)
95 • DECEMBER 2003 •
␤共P兲 ⫽
兺 C 共P兲␤
i
i
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i
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where ⌬␺(␹) is the membrane potential between compartment ␹ and the plasma compartment in volts, R is
the gas constant (8.3144 J/mol-K), T is absolute temperature, c is the charge on the species under consideration, F is Faraday’s constant (96, 485 Coulomb/mol),
and rm(␹) is the Donnan ratio for a given species m
between compartment ␹ and plasma.
Application to whole blood. As a specific example of a
multicompartment system, human whole blood (M ⫽
B) is considered. Whole blood is made up of contributions from the two single compartments of plasma (␹ ⫽
P) and erythrocyte fluid (␹ ⫽ E). This definition, together with Eq. 10, yields from Eq. 9
across the erythrocyte plasma membrane, CB, SID,
⌬CB, and ⌬SID for whole blood can be calculated as
outlined above.
Relationship to the Van Slyke equation. BE was initially read from a nomogram, but with the advent of
automated blood-gas analyzers it is now traditionally
calculated from the Van Slyke equation developed by
Siggaard-Andersen (47, 48). A small point of confusion
exists regarding the precise definition of the Van Slyke
equation, as Siggaard-Andersen originally referred to
both the equation for the pH, [HCO3⫺] equilibration line
(the equation for CB plotted as [HCO3⫺] vs. pH at
constant CB), as well as to the equation relating pH
and BE, as the Van Slyke equation (48). Because the
latter is usually called the Van Slyke equation (33, 49),
this is the convention that will be used here, and the
former will be referred to as the CO2 equilibration
curve (47, 48).
It is instructive at this point to consider the relationship of the present model to the traditional form of the
Van Slyke equation for whole blood, a recent version of
which is given by Siggaard-Andersen and FoghAndersen (49) as
2336
PHYSIOLOGICAL ACID-BASE PARAMETERS
The ␤i are the true molar buffer values of the noncarbonate buffers in plasma (including albumin, inorganic
phosphate, and globulins) with concentrations Ci(P).
Equation 19 can also be derived from the present
formalism. As shown previously (57), the assumption
of constant noncarbonate buffer concentrations from
Eq. 3 for a given compartment gives
⌬P共␹兲 ⫽ ⌬关HCO 3⫺兴␹ ⫹ ␤共␹兲⌬pH共␹兲
兵␤共P兲 ⫹ ␾共E兲关0.77␤共E兲 ⫺ ␤共P兲兴其
兵1 ⫺ 关1 ⫺ rc共E兲兴␾共E兲其
(30)
The hematocrit, ⌽(E), can be expressed as the ratio
of the whole blood and erythrocyte concentrations (48)
␾共E兲 ⫽
(23)
CHb共B兲
CHb共E兲
(31)
allowing Eq. 29, with the use of Eq. 20, to be written as
where
␤共␹兲 ⫽
⳵␰៮ n
C n 共␹兲
⳵pH
兺
n
(24)
analogous to Eq. 22. ␦␰៮ n/␦pH is the molar buffer value
of species n, assumed to be constant over the physiological pH range, and ␤(␹) is the buffer value of the
noncarbonate buffer contribution in compartment ␹
(57). Substituting Eq. 23 into Eq. 11 gives
兺 ␾共␹兲兵⌬关HCO
␹
⫺
3 ␹
兴 ⫹ ␤共␹兲⌬pH共␹兲其 (25)
under the assumption of constant hematocrit. This
equation can be recast in terms of plasma concentrations [HCO3⫺]P and pH(P) in equilibrium with the other
compartments (true plasma) to give
冋兺
⌬P共M兲 ⫽ ⌬关HCO 3⫺兴P
␹
册
␾共␹兲rc共␹兲
冋兺
⫹ ⌬pH共P兲
␹
␾共␹兲␤共␹兲
册
⳵pH共␹兲
⳵pH共P兲
(26)
␹
册
␾共␹兲rc共␹兲 兵⌬关HCO3⫺兴P ⫹ ␤⬘共M兲⌬pH共P兲其
(27)
with
⳵pH共␹兲
兺 ␾共␹兲␤共␹兲 ⳵pH共P兲
␤⬘共M兲 ⫽
兺 ␾共␹兲r 共␹兲
␹
(28)
c
␹
␤⬘(M), the effective buffer value of system M, is the
slope of the CO2 equilibration curve for system M in a
[HCO3⫺]P vs. pH(P) coordinate system at constant noncarbonate buffer concentration and constant CB (47,
48). Equation 27 represents a general multicompartment form of the Van Slyke equation.
For the specific example of whole blood, Eq. 27 can be
rewritten by using Eqs. 10 and 17 as
⌬P共B兲 ⫽ 兵1 ⫺ 关1 ⫺ rc共E兲兴␾共E兲其 兵⌬关HCO3⫺兴P
⫹ ␤⬘共B兲⌬pH共P兲其 (29)
Likewise, substituting Eqs. 10, 17, and 18 into Eq. 28
and rearranging gives
J Appl Physiol • VOL
册
CHb共B兲
兵⌬关HCO3⫺兴P ⫹ ␤⬘共B兲⌬pH共P兲其
CHb°
(32)
which is of the same form as Eq. 19 and traditionally
used to calculate BE. ␤⬘(B) can also be written, after
substituting Eq. 31 into Eq. 30, in a similar manner to
Eq. 21
␤⬘共B兲 ⫽ CHb共B兲 䡠 ␤⬘共E兲 ⫹ ␤⬘共P兲
(33)
with effective buffer values for plasma
␤⬘共P兲 ⫽ ␤共P兲
CHb共E兲
CHb共E兲 ⫺ 关1 ⫺ rc共E兲兴 䡠 CHb共B兲
(34)
and erythrocyte fluid
␤⬘共E兲 ⫽
where rc(␹) is the bicarbonate Donnan ratio between
compartment ␹ and plasma and is assumed to have a
constant value. By factoring out the term in rc(␹), Eq.
26 can then be written in the form
冋兺
冋
⌬P共B兲 ⫽ 1 ⫺
0.77␤共E兲 ⫺ ␤共P兲
CHb共E兲 ⫺ 关1 ⫺ rc共E兲兴 䡠 CHb共B兲
(35)
As discussed before, the term ␤⬘(M)⌬pH(P) of Eq. 27
can be thought of as a correction term to ⌬[HCO3⫺]P,
which corrects for respiratory effects (44, 57). The
corrected [HCO3⫺]P in this case is called the Van Slyke
standard bicarbonate (VSSB). The change in VSSB
(⌬VSSB) is the difference in plasma bicarbonate concentration remaining after the PCO2 has been altered to
bring the pH(P) back to 7.40 at constant CB or SID
(46–49, 57). Because the normal and abnormal CO2
equilibration curves are parallel under constant noncarbonate buffer concentrations, ⌬VSSB will be equal
to ⌬CB and ⌬SID (57).
As discussed by Siggaard-Andersen (46, 47), Eq. 27
can therefore be written as
⌬P共M兲 ⫽
冋兺
␹
册
␾共␹兲rc共␹兲 关⌬VSSB共P兲兴 ⫽ ⌬VSSB共M兲
(36)
where VSSB(P) is the VSSB of plasma in equilibrium
with the other compartments of system M, and
VSSB(M) is the VSSB of M. The VSSB(P) for system M
will be numerically different from the VSSB(P) for
separated plasma, since in the former case of system M
this concentration represents the VSSB of plasma in
equilibrium with the other single compartments (true
plasma) and therefore will have a steeper CO2 equilibration curve than will separated plasma (11).
Use of the above information regarding multicompartment systems together with Eq. 13 implies that, in
general, as found in the single compartment case (57)
95 • DECEMBER 2003 •
⌬C B共M兲 ⫽ ⌬SID共M兲 ⫽ ⌬VSSB共M兲
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⌬P共M兲 ⫽
⌬P共M兲 ⫽
␤⬘共B兲 ⫽
2337
PHYSIOLOGICAL ACID-BASE PARAMETERS
In the case of constant noncarbonate buffer concentrations [⌬Cn(␹) ⫽ 0], however
⌬C B共M兲 ⫽ ⌬SID共M兲 ⫽ ⌬VSSB共M兲
(38)
but at the same time, it is also straightforward to
deduce that if the reference state is chosen to include
the new (abnormal) concentrations [Cn⬘(␹)], then in
general
⌬C B⬘共M兲 ⫽ ⌬SID⬘共M兲 ⫽ ⌬VSSB⬘共M兲
(39)
imidazole groups of human carbon monoxyhemoglobin were
available from a study by Fang et al. (13) at 29°C in 0.1 M
HEPES buffer plus 0.1 M chloride. These experimental values
were used for the solvent-accessible histidine imidazole groups
instead of the theoretical values determined from the modified
Tanford-Kirkwood theory (30). ␤-Histamine (␤-His)97 and
␤-His116, considered nontitratable by Matthew et al. (30), were
found to be titratable experimentally by Fang at al. (13).
The effective oxyhemoglobin proton dissociation constants
Kl were then corrected to 37°C by employing the van’t Hoff
relation (3, 9)
where the primes refer to the new reference state (57).
ln
METHODS
J Appl Physiol • VOL
冊
(40)
where ⌬H° is the standard enthalpy of ionization for the
ionizable groups of human hemoglobin, all of which were
assumed to have the same ⌬H°, in accordance with the
findings of Stadie and Martin (51). The ⌬H° was calculated
by changing the enthalpy in 1 cal/mol increments to minimize the function
S2 ⫽
兺 关z៮
c
Hb
e
共E兲k ⫺ z៮ Hb
共E兲k兴2
(41)
k
c
z៮ Hb
(E)k is the theoretically calculated average charges of
e
the hemoglobin tetramer, and z៮ Hb
(E)k is the experimentally
determined oxyhemoglobin values for k ⫽ 31 experimental
pH data points at 37°C obtained from the data of Raftos et al.
(36). The numerical values for the data points, originally
presented in graphical form in Fig. 4 of Ref. 36, were provided
courtesy of Dr. J. Raftos. These values were then corrected
for the small contribution from a PCO2 of 0.25 Torr under the
experimental conditions of that study (36) by calculating the
bicarbonate contribution from PCO2 and pH by using Eq. 8,
e
then adding this to the experimental value to give z៮ Hb
(E)k.
2
The minimum S of 1.3 thus obtained provided a ⌬H° of 10.6
kcal/mol (44.5 kJ/mol).
Based on work by Reeves (38), the order of magnitude of
the standard enthalpy of ionization for DPG was assumed to
be ⬃1 kcal/mol and thus not expected to cause a significant
temperature dependence; therefore, these values were left
uncorrected for temperature. The uncorrected human oxyhemoglobin and DPG effective pKa values are also listed in
Table 1.
Values for pH(E) were obtained from Eq. 14 and the
plasma pH(P), under the assumption of a membrane potential of ⫺13.0 mV (47, 48). [HCO3⫺]E was calculated from Eq.
17, together with [HCO3⫺]P and pH(P).
Whole blood acid-base parameters were calculated by using the above models for human plasma and erythrocyte
fluid, which were then incorporated into Eqs. 15 and 16,
assuming a constant hematocrit of 44% (4). For calculation of
[HCO3⫺]P using Eqs. 19 and 32, a value of 43 mM was used
for CHb°.
Numerical values for the comparison experimental data
for human oxyhemoglobin at 25°C were obtained from Table
I of Ref. 41 and assumed to be free of carbonate. The comparison numerical data for human oxyhemoglobin at 37°C
were obtained from Raftos et al. (36) as discussed above. The
numerical values for the experimentally derived titration
curves of whole blood at PCO2 ⫽ 28.7 Torr, 40.0 Torr, and 66.0
Torr were read directly from Fig. 10 of Ref. 47.
The designations “normal plasma” and “normal erythrocyte fluid” used in the DISCUSSION, tables, and figures refer to
the normal human values pH(P) ⫽ 7.40, [HCO3⫺]P ⫽ 24.25
mM, PCO2(P) ⫽ 40.0 Torr, pH(E) ⫽ 7.19, [HCO3⫺]E ⫽ 12.49
mM, and the concentrations of noncarbonate buffer species
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Calculations using the mathematical models derived in
were carried out for human whole blood in a manner
similar to that described previously for human plasma (57).
Computation of acid-base parameters was performed by using Microsoft Excel 2002 running on a Compaq Presario
8000Z computer equipped with an AMD Athlon XP 1.733GHz processor. pH was stepped in 0.01 increments to calculate the variables z៮ n(␹), CB(␹), ⌬CB(␹), CB(M), ⌬CB(M),
SID(␹), ⌬SID(␹), SID(M), ⌬SID(M), and [HCO3⫺]P as well as
to generate graphs of these same variables vs. pH. ⳵␰៮ n/⳵pH
was calculated by taking the tangent to the z៮ n(␹) vs. pH
curves (57). ␤(␹) was calculated from Eq. 24, and ␤(M) was
calculated by taking the tangent to the P(M) curve when
carbonate species concentrations are set to 0.0 mM. All calculations were carried out at 37°C, except where otherwise
noted. In a departure from previous work (57), no correction
for ionic strength was performed, given the small changes
produced in the final calculated values (57). In addition,
following the approach of Siggaard-Andersen (47), no correction for osmotic effects was explicitly included.
All protein amino acids, the heme moieties of hemoglobin,
and the carboxyl group of 2,3-diphospho-D-glycerate (DPG)
were assumed to behave as independent monoprotic acids
(15, 16, 29, 30, 57). Inorganic phosphate was treated as a
triprotic acid, and the two phosphates of DPG were assumed
to behave as independent diprotic acids (15, 16, 29, 57).
For human plasma, it was assumed that albumin and
inorganic phosphate were sufficient to account for all of the
noncarbonate buffer activity (15, 16, 55, 57), and simulations
of acid-base balance used the human albumin and inorganic
phosphate dissociation constants given by Figge et al. (15).
Bicarbonate concentrations were derived from the Henderson-Hasselbalch equation (Eq. 8) together with plasma pH
and PCO2 (4). The effective equilibrium constants for human
albumin, phosphate, and carbonate at 37°C are given in
Table 1.
For human erythrocyte fluid, a slightly different approach
was required because no complete theoretical or experimental equilibrium constant data under erythrocyte physiological conditions could be found. The calculation was carried out
for oxygenated blood, and it was assumed, in accordance with
the work of Siggaard-Andersen (47) and Reeves (38), that the
noncarbonate buffer species of human oxygenated erythrocyte fluid are primarily oxyhemoglobin and free DPG. For the
equilibrium constants of the ␣ and ␤ chains in the human
oxyhemoglobin tetramer, effective pKa values calculated
from a modified Tanford-Kirkwood theory were available at
25°C and an ionic strength of 0.10 M from the work of
Matthew et al. (30). A similar publication, again by Matthew et
al. (29), provided the effective pKa of free DPG at 25°C and ionic
strength 0.10 M. At the same time, more recent experimental
pKa data for the solvent of the solvent-accessible histidine
THEORY
冉
K2
⌬H° 1
1
⫽⫺
⫺
K1
R T2 T1
2338
PHYSIOLOGICAL ACID-BASE PARAMETERS
Table 1. pK and equilibrium constant values for carbonate and noncarbonate buffers
of human oxygenated whole blood
Species
pK
8.50
4.00
9.60
9.40
7.12
7.22
7.10
7.49
7.01
7.31
6.75
6.36
4.85
5.76
6.17
6.73
5.82
7.3
5.2
7.3
8.00
3.10
2.91
3.36
3.48
2.99
3.86
2.53
4.21
2.69
3.92
4.00
4.02
3.95
3.63
2.98
4.29
4.15
2.34
2.79
4.24
4.51
3.08
3.22
2.39
4.34
4.49
3.35
2.73
2.19
3.55
3.98
3.46
3.81
4.04
10.24
10.55
Hemoglobin (cont.)
␤ Tyr35
␤ Tyr130
␤ Tyr145
␣ Lys7
␣ Lys11
␣ Lys16
␣ Lys40
␣ Lys56
␣ Lys60
␣ Lys61
␣ Lys90
␣ Lys99
␣ Lys127
␣ Lys139
␤ Lys8
␤ Lys17
␤ Lys59
␤ Lys61
␤ Lys65
␤ Lys66
␤ Lys82
␤ Lys95
␤ Lys120
␤ Lys132
␤ Lys144
␣ Arg31
␣ Arg92
␣ Arg141
␤ Arg30
␤ Arg40
␤ Arg104
␣ His20
␣ His45
␣ His50
␣ His72
␣ His89
␣ His112
␤ His2
␤ His77
␤ His97
␤ His116
␤ His117
␤ His143
␤ His146
␣ Val1
␤ Val1
2,3-Diphosphoglycerate
Carboxyl
Phosphate#1
Phosphate#2
Inorganic phosphate
H3PO4
H2PO4⫺
HPO42⫺
Carbonate
PCO2
pK
11.00
10.31
10.71
11.70
10.75
11.21
10.75
10.84
10.74
10.82
10.60
10.99
13.03
11.95
10.57
11.52
10.48
10.65
10.65
11.02
9.40
10.75
10.58
11.86
10.54
13.55
12.21
13.97
13.72
12.69
12.52
7.06
6.09
6.90
7.27
6.30
7.48
6.41
7.73
7.66
6.26
6.42
5.73
6.47
7.30
6.80
pKC ⫽ 4.58
pK1 ⫽ 3.19
pK2 ⫽ 6.69
pK1 ⫽ 3.16
pK2 ⫽ 6.39
pK1 ⫽ 1.91
pK2 ⫽ 6.66
pK3 ⫽ 11.78
S ⫽ 0.0306
pK⬘ ⫽ 6.103
Solvent-accessible histidine imidazole values for oxyhemoglobin are at 29°C (13). The remaining oxyhemoglobin (30) and 2-3-diphospho(DPG) (29) values are at 25°C, and the albumin (15), inorganic phosphate (15), and carbonate (4) values are at 37°C. Numbers
in parentheses under albumin indicate number of amino acid residues present in the protein.
D-glycerate
J Appl Physiol • VOL
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Albumin
Cysteine
Aspartic acid and glutamic acid (98)
Tyrosine (18)
Arginine and lysine (77)
Histidine#1
Histidine#1
Histidine#3
Histidine#4
Histidine#5
Histidine#6
Histidine#7
Histidine#8
Histidine#9
Histidine#10
Histidine#11
Histidine#12
Histidine#13
Histidine#14
Histidine#15
Histidine#16
NH2-terminal
COOH-terminal
Hemoglobin
␣ Asp6
␣ Asp47
␣ Asp64
␣ Asp74
␣ Asp75
␣ Asp85
␣ Asp94
␣ Asp126
␤ Asp21
␤ Asp47
␤ Asp52
␤ Asp73
␤ Asp79
␤ Asp94
␤ Asp99
␣ Glu23
␣ Glu27
␣ Glu30
␣ Glu116
␤ Glu6
␤ Glu7
␤ Glu22
␤ Glu26
␤ Glu43
␤ Glu90
␤ Glu101
␤ Glu121
␣ Arg141
␤ His146
␣ heme1
␣ heme2
␤ heme1
␤ heme2
␤ Cys93
␣ Tyr24
Species
PHYSIOLOGICAL ACID-BASE PARAMETERS
2339
Table 2. Single-compartment concentrations and
physical constants for the noncarbonate buffers of
normal human oxygenated whole blood
Ch(␹), mM e៮ max(n) z៮ min(n) z៮ max(n)
Species
Albumin
Hemoglobin
DPG
Inorganic phosphate
0.66
5.3
6.0
1.16
212
162
5
3
⫺118
⫺76
⫺5
⫺3
94
86
0
0
⳵␰៮ n
bn
⳵pH
159
8.0
147.5 10.2
5.4 0.70
3.4 0.30
See Single-compartment systems for definitions.
P共B兲 ⫽ 兵1 ⫺ 关1 ⫺ rc共E兲兴␾共E兲其关HCO3⫺兴P ⫹ 关1 ⫺ ␾共E兲兴
⳵␰៮ i
Ci共P兲
pH共P兲 ⫹
Ci共P兲␰៮ m共i兲 ⫺
Ci共P兲bi ⫹ ␾共E兲
⳵pH
i
i
i
⳵␰៮ j
Cj共E兲
pH共E兲 ⫹
Cj共E兲␰៮ m共j兲 ⫺
Cj共E兲bj
(42)
⳵pH
j
j
j
再冋兺
再冋兺
册
册
兺
兺
兺
兺
冎
冎
where the index i extends over the plasma and the index j
over the erythrocyte noncarbonate buffers. To obtain explicit
numerical forms, the values in Table 2 were then substituted
into Eq. 42. Equation 31 was employed to obtain the hemoglobin term as a function of CHb(B). The approximation that
the bicarbonate Donnan ratio is a constant at rc(E) ⬇ 0.51
over the physiological pH range was also used, as well as that
pH共E兲 ⬇ pH共P兲 ⫺ 0.21
(43)
also found to be a good approximation over the physiological
range, giving explicit linear forms for CB(B) and SID(B) as
functions of pH(P).
Fig. 1. Average charge per molecule (Z៮ Hb) vs. erythrocyte pH [pH(E)]
under carbonate-free conditions for the human oxyhemoglobin tetramer showing theoretical curves (lines) and experimental data from
Ref. 41 at 25°C (䊐) and from Ref. 36 at 37°C (■).
plasma. To compare the traditional Van Slyke equation (Eq. 19) with that derived from pH and effective
conditional molar equilibrium constants (Eq. 32), a
graph of human whole blood [HCO3⫺]P vs. pH(P) is
provided in Fig. 4, together with the corresponding
curve obtained from the more complete Eqs. 2 and 15
at constant P(B).
Various calculated parameter values are compared
with the results of experimental data and other models
in Table 3. Calculated values are under normal conditions as defined in METHODS.
The derivation of the explicit linear forms for P(M) as
described in METHODS gave the following results from
Eq. 42
C B共B兲 ⫽ 关1 ⫺ 0.49␾共E兲兴关HCO3⫺兴P ⫹ 关1 ⫺ ␾共E兲兴兵CAlb共P兲关8.0 pH共P兲 ⫹ 53兴 ⫹ CPhos共P兲关0.30 pH共P兲 ⫺ 0.4兴其
(44)
⫹ CHb共B兲兵10.2关pH共P兲 ⫺ 0.21兴 ⫹ 14.5其 ⫹ ␾共E兲CDPG共E兲兵0.70关pH共P兲 ⫺ 0.21兴 ⫺ 0.4其
SID共B兲 ⫽ 关1 ⫺ 0.49␾共E兲兴关HCO3⫺兴P ⫹ 关1 ⫺ ␾共E兲兴兵CAlb共P兲关8.0 pH共P兲 ⫺ 41兴 ⫹ CPhos共P兲关0.30 pH共P兲 ⫺ 0.4兴其
(45)
⫹ CHb共B兲兵10.2关pH共P兲 ⫺ 0.21兴 ⫺ 71.5其 ⫹ ␾共E兲CDPG共E兲兵0.70关pH共P兲 ⫺ 0.21兴 ⫺ 0.4其
RESULTS
Theoretical titration curves for human oxyhemoglobin with corresponding experimental data from
Rollema et al. at 25°C (41) and from Raftos et al. at
37°C (36) are shown in Fig. 1. Theoretical titration
curves for human whole blood at PCO2⫽ 28.7, 40, and
66 Torr are shown in Fig. 2 together with experimental data obtained by Siggaard-Andersen (45, 47). In
keeping with the practice of graphing the equations
for CB and SID in a [HCO3⫺]P vs. pH(P) coordinate
system as advocated by Davenport (11), Fig. 3 shows
such plots illustrated for human whole blood and
J Appl Physiol • VOL
The extent to which these expressions approximate the
results from Eqs. 2 and 15 is demonstrated in Fig. 5 for
human whole blood and plasma. To calculate the variables for plasma, ␾(E) and CHb(B) are both set to zero.
Also note that, in these explicit linear forms, the hemoglobin concentration is expressed as the concentration of tetramer.
DISCUSSION
The accuracy of the single and multicompartment
models can be assessed by how well they predict experimentally accessible parameters. In general, agree-
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(4, 22, 47, 48, 57) given in Table 2. “Normal whole blood”
likewise refers to the corresponding human multicompartment model with a constant hematocrit of 44%.
Finally, although Eq. 2 provides exact expressions for
CB(M) and SID(M), the linear forms obtained from Eq. 3 are
more easily manipulated. These were substituted into Eq. 15,
which after rearrangement of the bicarbonate term in a
manner similar to that used to for Eq. 29 gave
2340
PHYSIOLOGICAL ACID-BASE PARAMETERS
ment is good. A summary of theoretical values obtained
for various different parameters is presented in Table
3, where these values are also compared with previously published data. A few comments are in order
regarding specific values, as well as some of the features of Figs. 1–5.
Results for the plasma single compartment using the
present formalism have been given previously (57), and
the extent to which this model and others like it reproduce experimental data has been discussed by several
authors (15, 55, 57). It has been shown that the model
produces excellent agreement with experimental data
for human plasma, even when the plasma globulin
component is neglected; for example, the SID values of
39 mM from Ref. 57 and the SID of 40.4 mM from Ref.
9 agree well with the value determined by Singer and
Hastings (50) of 41.7 mM. This is despite the apparent
disagreement between the buffer value ␤(P) ⫽ 5.7 mM
obtained from the present model and the ␤(P) ⫽ 7.7
mM published by Siggaard-Andersen (47, 49).
Although probably the weakest portion of the model
currently, the theory for the erythrocyte compartment
nonetheless produces values commensurate with previously published data, although returning values
were slightly higher than those published for the
charge on the human oxyhemoglobin tetramer (e.g.,
⫺2.10 vs. ⫺2.54) and slightly lower for the tetramer
molar buffer value (e.g., 10.2 vs. 10.8). Similarly, the
theoretical charge on DPG of ⫺4.6 is lower than the
⫺4.1 given in Table 3. The molar buffer value (⳵␰DPG/
⳵pH), however, is similar to the 0.69 published by
Reeves (38) at 25°C but lower than the 1.0 obtained
from Raftos et al. at 37°C (36). The ⌬H° determined
here of 10.6 kcal/mol is similar to the 10 kcal/mol
obtained by Stadie and Martin (51). A ⌬H° of 9 kcal/
J Appl Physiol • VOL
Fig. 3. Plasma HCO3⫺ concentration ([HCO3⫺]P) vs. pH(P) graphs for
human whole blood (a) and plasma (b), indicated by solid lines. The
normal physiological state is indicated by the arrow at pH(P) ⫽ 7.40,
where the solid lines intersect. A dotted line is shown for an acidbase disturbance of whole blood with a metabolic component (a⬘) in a
patient with pH(P) ⫽ 7.16 and [HCO3⫺]P ⫽ 25.60 mM. The arrow at
pH(P) ⫽ 7.16 also indicates where this line crosses that for plasma.
All lines have normal noncarbonate buffer concentrations. To obtain
⌬P ⫽ ⌬CB ⫽ change in strong ion difference (⌬SID) ⫽ change in Van
Slyke standard bicarbonate for a given body fluid (⌬VSSB), any of
the lines is traced back to pH(P) ⫽ 7.40 to get the corresponding
[HCO3⫺]P. Then 24.25 mM is subtracted from this result to obtain
⌬VSSB(P) for that body fluid. This value is then scaled by the
appropriate distribution function as indicated in Eq. 36 to obtain the
body fluid parameter, giving ⌬P(P) ⫽ 0.0 mM and ⌬P(B) ⫽ ⫺5.3 mM.
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Fig. 2. Change in total titratable base (⌬CB) vs. plasma pH [pH(P)]
titration curves for human whole blood at 37°C and variable PCO2.
Theoretical curves and experimentally derived values from Ref. 47 at
PCO2 ⫽ 28.7 (}), 40.0 (E), and 66.0 (F) Torr are shown.
mol was found by Antonini et al. (2) for the alkaline
groups and ⫺1.5 kcal/mol for the acid groups of human
hemoglobin. The validity of the approximation of
Stadie and Martin that all ionizable groups of hemoglobin have the same standard enthalpy of ionization
probably lies in the fact that, over the physiological pH
range, groups with pKa outside the physiological pH
range largely have a fixed charge, and therefore the
overall molecular charge is relatively insensitive to
small temperature-induced changes in the effective
pKa of these groups. Although the agreement of the
model with the data from Rollema et al. (41) at 25°C is
seen to be moderately good over the physiological
range, the agreement at low pH is not as impressive.
Still, it is striking how well the theory predicts the
experimental titration curves for human oxyhemoglobin shown in Fig. 1, given the number of assumptions
inherent in the calculation. Dedicated calculations of
the effective pKa of hemoglobin and DPG under erythrocyte physiological temperature and ionic strength
conditions would most likely give better agreement,
and the need for more accurate effective pKa may
stimulate further research in this area.
Despite the mild discrepancies in the erythrocyte
portion of the model, the experimental titration curves
(45, 47) for human whole blood shown in Fig. 2 are
2341
PHYSIOLOGICAL ACID-BASE PARAMETERS
Fig. 5. CB and SID vs. pH(P) for human whole blood and plasma at
37°C, PCO2 ⫽ 40.0 Torr, and normal noncarbonate buffer concentrations by using the more complete Eq. 15 (symbols) and the approximate forms of Eqs. 44 and 45. Dotted lines, plasma CB ⫹ SID; solid
lines, whole blood CB ⫹ SID.
faithfully reproduced by the theory for three different
PCO2 values. The normal values calculated for SID in
human plasma, erythrocyte fluid, and whole blood also
agree reasonably well with those obtained by direct
calculation (Table 3). Additionally, the titration curves
for CB(B) and SID(B) are shown to be well approximated, to within 1 mM over the physiological range, by
the linear forms of Eqs. 44 and 45. This is also the case
for CB(P) and SID(P), as previously demonstrated (57).
These results place confidence in the ability of the
present model to predict the acid-base behavior of the
multicompartment physiological system human whole
blood. It bears mentioning, however, that acid-base
parameters may be different in different species (6, 9),
requiring a separate analysis for each.
In addition to reproducing the experimental human
whole blood titration curves, the model also reproduces
the traditional Van Slyke equation. Furthermore, the
relationship of the present formalism to that of Siggaard-Andersen (47–49) has been demonstrated, providing Eq. 27 as a general multicompartment form of
the Van Slyke equation. Figure 4 compares the values
for [HCO3⫺]P in human whole blood calculated with the
traditional Van Slyke equation (Eq. 19) by using the
parameters of Siggaard-Andersen and Fogh-Andersen
(49) with that calculated by using the Van Slyke expression determined with the present model (Eq. 32).
The [HCO3⫺]P vs. pH(P) curve calculated for whole
blood with the complete expression (Eqs. 2 and 15) is
also shown. Exact agreement between the approximate
values of Eqs. 19 and 32 is present, and these deviate
significantly from the values calculated with the more
complete expression only at low pH(P). This deviation
is a consequence, as discussed before by SiggaardAndersen (47), of the pH dependence of parameters
such as the Donnan ratio for bicarbonate and the
buffer values of the noncarbonate species, which are
assumed to have constant values in the approximate
formulas (47). It is also a reflection of the use of Eq. 18
for the pH distribution in the derivation of the tradi-
J Appl Physiol • VOL
Table 3. Theoretically derived normal parameter
values compared with previously published data
Parameter
Present Study
CB(P)
SID(P)
CB(E)
SID(E)
CB(B)
SID(B)
␤(P)
␤(E)
␤(B)
␤⬘(P)
␤⬘(E)
␤⬘(B)
⳵␰៮ Alb
⳵pH
⳵␰៮ Phos
101
39
507
51
279
44
5.7
58
29
7.3b
9.5 (2.4c)
29
Previous Values
101 (57)
39 (57), 41.7 (50), 40.4(9)
—
55.7 (50), 54.6 (36), 52 (19)
—
47.9 (50)
7.7 (49)
63 (47)
26 (47)a, 28 (54)
7.7b (49)
2.3d (47)
29 (47)
8.0
8.2 (15), 8.0 (9), 7.6 (17)
⳵pH
⳵␰៮ Hb
0.30
0.31 (49)
⳵pH
⳵␰៮ DPG
⳵pH
z៮ Alb (P)
z៮ Phos (P)
z៮ Hb (E)
z៮ DPG (E)
10.2e
0.70
⫺18.3
⫺1.84
⫺2.10e
⫺4.62
10.8 (36)e, 11.2 (51)e, 11.3 (2)e
1.0 (36), 0.688 (37)
⫺19.2 (36), ⫺18.1 (55), ⫺18.3 (15), ⫺17 (17)
⫺1.8 (36), ⫺1.8 (55)
⫺2.54 (36)e, ⫺3 (22)e
⫺4.1 (47), ⫺3.72 (36), ⫺4 (22)
Values for concentration parameters are in mM. Numbers in
parentheses are Refs. See text for definitions. aCalculated from Eq. 3,
Section 2.2.5 of Ref. 47. bCorresponding values in Eqs. 21 and 33. See
text for discussion. cDefined in terms of the hemoglobin monomer to
agree with definition in Ref. 47. dEquivalent to the ␤⬘Hb term in Eq.
21. eDefined in terms of the hemoglobin tetramer.
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Fig. 4. [HCO3⫺]P vs. pH(P) graphs of human whole blood with normal
noncarbonate buffer concentrations for the traditional Van Slyke
equation (}) from Ref. 49 (Eq. 19), the Van Slyke equation from Eq.
32 (dotted line), and the more exact Eq. 15 (solid line).
2342
PHYSIOLOGICAL ACID-BASE PARAMETERS
J Appl Physiol • VOL
In contrast to Siggaard-Andersen’s approach to calculating BE, Stewart calculated SID as a variable in its
own right. Based on this, Stewart considered the effect
of noncarbonate buffer concentrations on SID, leading
others to interpret hyper- or hypoproteinemic states as
forms of acid-base disorders by examining ⌬SID, the
corresponding variable to ⌬CB, as a deviation from its
normal value (23, 31, 42). Siggaard-Andersen and coworkers (49) have rejected this notion. Although differences in noncarbonate buffer concentrations do enter
into the calculation of BE via Eq. 21, this has the effect
of producing Eq. 39 and altering the slope, but not the
normal physiological position, of the CO2 equilibration
curve (32, 49, 57). Because this approach produces
small changes in the final calculated BE, the default
values are commonly used independent of the actual
noncarbonate buffer concentrations (11, 49).
Although the Stewart theory is often said to have an
advantage over BE in that it claims to examine the
only independent variables of acid-base physiology (24,
25, 52, 53), this contention has been challenged (5, 49,
56). However, the clear theoretical disadvantage that
the Stewart approach has had relative to the BE
method is that its previous formulations did not treat
the complete acid-base disorder quantitatively, because only the plasma compartment was considered.
The present treatment of multicompartment systems
brings the SID theory to the same quantitative level,
and within the same degree of precision, as the traditional BE theory. It is also worth noting that the more
complete expressions simplify under certain limiting
conditions to the Henderson-Hasselbalch equation, as
shown by Constable (6), and to the Van Slyke equation,
as shown in the present work.
The model presented here provides a unifying formalism for the BE and SID approaches, and provides
flexibility in the quantitative description of acid-base
disorders, by allowing calculation of the absolute quantities CB and SID for single compartments or systems
with multiple compartments, including the examples
of human plasma, erythrocytes, and whole blood demonstrated. Both exact expressions and linear approximations can be used for calculation. With the use of
these equations, acid-base status can be assessed from
the vantage point of both ⌬CB and ⌬SID, considering
changes in noncarbonate buffer concentration either
explicitly (Eq. 37) or implicitly (Eq. 39). Given an
extant theory for calculating absolute values of acidbase parameters in the whole organism, factors involved in acid-base disturbances can be more fully
explored.
In conclusion, the general formalism previously developed for calculating parameters describing physiological acid-base balance in single compartments has
been extended to multicompartment systems. Together
with the previously obtained model for human plasma,
a model for the human erythrocyte single compartment
was used to obtain expressions for the multicompartment case of human whole blood. Calculations using
these equations produced values that approximate a
wide range of experimental data, providing confidence
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tional form, as opposed to Eq. 14 in the more complete
expression.
It also bears mentioning that the value of 7.7 mM
used for ␤(P) in Eq. 21, although similar to the effective
value of 7.3 mM for ␤⬘(P) derived via Eq. 34, should not
actually be the same as the true buffer value for
plasma in Eq. 22. Similarly, the ␤(E) of 63 mM given in
Ref. 48 should not give a ␤⬘(E) of 2.3 according to Eq.
35. It is important to point out, however, that the
expression quoted by Siggaard-Andersen in the form of
Eq. 21 actually appears to be an empirical relationship,
and it is thus possible that the appearance of the factor
7.7 mM is coincidental.
Figure 3 demonstrates the important point that, in a
multicompartment system, a complete quantitative
treatment of the metabolic component of an acid-base
disturbance requires that one account for all of the
nonvolatile excess acid or base, considering the relative
contributions from each compartment. For example,
suppose that a patient presents with pH(P) ⫽ 7.16,
[HCO3⫺]P ⫽ 25.60 mM, PCO2 ⫽ 73.4 Torr, and normal
noncarbonate buffer concentrations. In this case, Fig. 3
demonstrates that the calculated ⌬P (equal to BE,
⌬CB, ⌬SID, and ⌬VSSB by Eq. 38) gives ⌬P(P) ⫽ 0.0
mM for separated plasma and ⌬P(B) ⫽ ⫺5.3 mM for
whole blood. One would therefore predict only a respiratory disturbance in the former case, whereas there is
a metabolic component in the latter (11, 47, 57). Consideration of only the plasma compartment may therefore produce an erroneous assessment of acid-base status, although it should be noted that the in vitro
equations for both plasma and whole blood differ from
the in vivo equilibration curves (44, 47), with the in
vivo curve intermediate in slope between plasma and
whole blood (44, 47). For BE with constant noncarbonate buffer concentrations, the corresponding in vivo
condition has been approximated by an extracellular
fluid model in which whole blood is diluted 2:1 in its
own plasma (47, 49).
In his development of the BE theory, SiggaardAndersen sought an expression for ⌬CB directly (46,
47). By conceptualizing BE as the change in the Van
Slyke standard bicarbonate, he derived an expression
for ⌬VSSB, which has come to be known as the Van
Slyke equation (48). Via this approach, the crucial
piece of data that needed to be measured or calculated
was the slope of the CO2 equilibration curve (46, 47).
After an expression for the change in Van Slyke standard bicarbonate in terms of the slope of the CO2
equilibration curve was derived, an empirical linear
approximation for this curve was then inserted at the
appropriate point in the derivation (46, 47) to give ⌬CB,
which was defined as BE. With the use of a different
approach, Eq. 16 is obtained via a full theoretical
treatment by first seeking an equation for CB, then
calculating ⌬CB. This provides an expression for BE
(Eq. 32) nearly equal to the traditional Eq. 19. This
formula, by Eq. 38, is also equal to ⌬SID and ⌬VSSB in
the setting of constant noncarbonate buffer concentrations.
PHYSIOLOGICAL ACID-BASE PARAMETERS
in the model. The equations for multicompartment
systems were found to have the same mathematical
interrelationships as those for single compartments.
The relationship of the present formalism to the traditional form of the Van Slyke equation was also demonstrated, and a general multicompartment form of the
Van Slyke equation was given. With the multicompartment model, the SID theory is brought to the same
quantitative level as the BE method.
I thank Dr. Howard Corey for rekindling my interest in physiological acid-base balance and Dr. Peter Hildenbrand for helpful
comments. I thank Dr. Julia Raftos for graciously providing the
numerical values of experimental data, originally presented in
graphical form as Fig. 4 of Raftos et al. (36), for use in Fig. 1.
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