Rev.int.med.cienc.act.fís.deporte
vol. 11  número 41  marzo 2011  ISSN:
15770354
Peinado,
A.B., Díaz, V.; Benito, P.J.; Álvarez, M. y Calderón, F.J.
(2011). Simplificación de la ecuación de Stewart para valorar el estado
ácidobase. Revista
Internacional de Medicina y Ciencias de la Actividad Física y el Deporte vol.
10 (41) pp. 115126. Http://cdeporte.rediris.es/revista/revista41/artecuacion197.htm
ORIGINAL
ECUACIÓN SIMPLIFICADA DE STEWART
PARA VALORAR EL ESTADO ÁCIDOBASE EN EJERCICIO
A SIMPLIFICATION OF THE STEWART EQUATION TO DETERMINE
ACIDBASE STATUS IN EXERCISE
Peinado, A.B.^{1}; Díaz, V.^{2}; Benito,
P.J.^{3}; Álvarez, M.^{4} y Calderón, F.J.^{5}
^{1}Doctora en Ciencias de la Actividad
Física y del Deporte. Profesora
Ayudante Doctor, Facultad de Ciencias de la Actividad
Física y del Deporte – INEF. Universidad Politécnica de Madrid, España (anabelen.peinado@upm.es).
^{2}Doctor en Ciencias de la Actividad Física y del Deporte. Investigador postdoctoral, Facultad de Ciencias de la Actividad Física y del Deporte – INEF. Universidad Politécnica de Madrid, España (victor.diaz@upm.es). Instituto de Fisiología Veterinaria y Zurich Center for Integrative Human Physiology (ZIHP), Universidad de Zurich, Suiza.
^{3}Doctor en Ciencias de la
Actividad Física y del Deporte. Profesor Titular Interino, Facultad de Ciencias
de la Actividad Física y del Deporte – INEF. Universidad Politécnica de Madrid,
España (pedroj.benito@upm.es).
^{4}Licenciada en Ciencias de la Actividad Física y del Deporte. Instituto de Fisiología Veterinaria y Zurich Center for Integrative Human Physiology (ZIHP), Universidad de Zurich, Suiza. (mariaalvarezsanchez@gmail.com).
^{5}Doctor en Medicina y
Cirugía. Profesor Titular de Universidad, Facultad de Ciencias de la Actividad
Física y del Deporte – INEF. Universidad Politécnica de Madrid, España (franciscojavier.calderon@upm.es).
Código UNESCO: 2411
Fisiología Humana
Clasificación del Consejo de Europa: 13.
Fisiología del deporte
Recibido 9 de
septiembre de 2009
Aceptado 3 de
noviembre de 2009
RESUMEN
El objetivo del estudio fue
simplificar la ecuación de Stewart y verificar la validez de la ecuación
propuesta. Veinticuatro varones realizaron un test a carga constante de 30
minutos en tapiz rodante. Fueron tomadas muestras de sangre capilar en reposo y
en los minutos 10, 20 y 30 del test. Los parámetros ácidobase fueron
analizados con un analizador de gases en sangre y el lactato por método
enzimático. La [H^{+}] fue calculada usando la ecuación de Stewart y la
ecuación propuesta. La diferencia de medias entre la ecuación propuesta y la de
Stewart fue de 0,004 nmol.L^{1} para la [H^{+}]. Sin embargo,
la diferencia de medias entre las ecuaciones y los valores medidos fue mayor de
8 nmol.L^{1} para la [H^{+}] (p<0,001). La ecuación
propuesta puede ser usada para estimar la [H^{+}] en lugar de la
ecuación de Stewart, aunque los valores estimados son significativamente
diferentes a los valores medidos.
PALABRAS CLAVE: ion hidrógeno, diferencia de iones fuertes,
equilibrio ácidobase, lactato, ejercicio a carga constante.
ABSTRACT
The aim of the present study was to simplify the Stewart equation and to
test the validity of the proposed form. Twentyfour men performed a constant
load exercise test for 30 min on a treadmill. Capillary blood samples were
taken at rest, and again 10, 20 and 30 min into the test. Acidbase variables
were measured using a bloodgas analyser and lactate levels were measured enzymatically. The [H^{+}] was calculated using the
Stewart equation: A[H^{+}]^{4}+B[H^{+}]^{3}+C[H^{+}]^{2}+D[H^{+}]+E=0,
and using a proposed, simplified version of this equation: A[H^{+}]^{2}+B[H^{+}]+C=0.
The difference in the mean [H^{+}] results obtained with the two
equations was 0.004 nmol·L^{1}. However, the difference between the
means of the equationderived results and the measured values was highly
significant at >8 nmol·L^{1}
(p<0.001). The proposed equation can be used to estimate [H^{+}]
instead of the full Stewart equation, although the values obtained are
significantly different to those actually measured.
KEY WORDS: acidbase equilibrium, constant load, hydrogen ion, lactate, strong
ions.
1. INTRODUCTION
The traditional approach of the acidbase balance is most commonly
expressed as the HendersonHasselbach equation (1). Moreover, it has been
explained in quantitative terms by the relationship between the pH of the
plasma and the intensity of exercise: as the intensity of exercise increases so
too does the plasma acid concentration, and therefore also the hydrogen ion
concentration ([H^{+}]). At the same time there is a reduction in the
bicarbonate concentration ([HCO_{3}^{}]) (2). However, this explanation, although
simple to understand, has considerable limitations. Firstly, it does not take
into account the variables affecting the acidbase status may have different
values in the three compartments affected (the intracellular, erythrocyte and
plasma compartments), and secondly, it does not take into account that the
relationships between these three compartments  variations in one might lead
to changes in the other two.
The independent variables that determine the acidbase balance of
biological solutions are the partial pressure of carbon dioxide (PCO_{2}),
the difference in the concentration of strong ions ([SID]) (completely
dissociated organic and inorganic ions), and the concentration of partially
dissociated weak acids ([A_{TOT}]) (38). The main weak acids involved
are proteins (especially albumin and globulin) and phosphates (9). The influence
of these variables on [H^{+}] can be determined using the Stewart or
Fencl equations (4), as well as with the simplified strong ion model derived by
Constable (10). These quantitative approaches are used to explain the acidbase
behavior of simple and complex solutions and offers a novel insight into the
pathophysiology of mixed acidbase disorders (1, 10, 11), being very important
from a clinical viewpoint. That proposed by Stewart is a fourth degree
polynomic equation: A[H^{+}]^{4} + B[H^{+}]^{3} + C[H^{+}]^{2} + D[H^{+}]
+ E = 0 (3, 12). Solving this equation,
however, has the disadvantage that it requires the use of mathematical programs
that are difficult to use and it is possible to eliminate the coefficientes D
and E due to their scant contribution to the final result. In addition, the
equation should include only those terms that are important in the
phenomenology of the procedure (10). The
aim of the present work was to simplify this equation and to analyse its
validity by comparing the [H^{+}] values obtained with the traditional
and simplified forms, and then by comparing the values predicted by both with
measurements of capillary blood [H^{+}] made during a constant load
exercise test.
2. MATERIALS AND
METHODS
2.1. SUBJECTS
The study subjects were 24 healthy men (age 26.7±4.9 years, height
176.1±6.3 m, body weight 72.8±6.7 kg), all students of
Sports and Physical Activity Sciences, and all of whom were familiar with
treadmill testing. The subjects were
explained the nature of the study and informed consent was obtained from each
participant, in accordance with the guidelines of the World Medical Association
regarding human investigation as outlined in the Helsinki declaration.
2.2. PROTOCOL
The subjects performed two constant load tests on a treadmill
(H/P/Cosmos Pulsar 3P 4.0®; H/P/Cosmos Sports & Medical,
NussdorfTraunstein, Germany). In the first test, the treadmill was set at a
fixed slope of 1% and was accelerated by 0.2 km·h^{1} every 12 s until
the subject became exhausted. This test
was performed in order to determine the two ventilatory thresholds. The second
test involved a stable treadmill rate at the load corresponding to the mid
point between these two ventilatory thresholds (i.e., a constant load exercise
test) (1315). The volume and
composition of the expired air was determined using a Jaeger Oxycon Pro^{®}
apparatus (Erich Jaeger, Hoechberg, Germany).
2.3. BLOOD SAMPLES
AND ANALYSIS
Capillary blood samples were taken from the fingertip during the
constant load exercise test at 0, 10, 20 and 30 min; all samples were collected
using 100 µl heparinised (heparin electrolyte balanced) capillary tubes. Part of each sample (75 µl of whole blood)
was used to determine the values for the variables affecting the acidbase status
(pH, PCO_{2}, HCO_{3}^{}), and to determine the
concentrations of electrolytes (Na^{+}, K^{+}, Ca^{2+ }and
Cl^{}). This was performed using an ABL 77® blood
gas analyser (Radiometer, Copenhagen, Denmark). The remainder of each samples
(25 µl of whole blood) was used to determine the
lactate concentration ([Lac^{}]) by an enzymatic method using the YSI
1500® kit (Yellow Springs Instruments Co., Yellow Springs, USA). Both
analytical systems were calibrated before each test. The lactate analyser was
calibrated using knowed solutions with a [Lac^{}] of 5 mmol·L^{1}
and 15 mmol·L^{1}. The blood gas analyser was calibrated automatically
following instructions of the manufacturer.
2.4. CALCULATIONS
The Stewart equation (3,
12) was used to determine [H^{+}]:
Equation 1
A[H^{+}]^{4} + B[H^{+}]^{3} + C[H^{+}]^{2} + D[H^{+}]
+ E = 0
where A = 1, B = K_{A}
+ [SID], C = (K_{A} [SID]  [A_{TOT}])  (K_{c} · PCO_{2}
+ K_{w}^{`}), D = [K_{A} (K_{c} · PCO_{2}
+ K_{w}^{`}) + (K_{3} · K_{C} · PCO_{2})],
and E = K_{A} · K_{3} · _{ }K_{C }· PCO_{2}. The [SID] was
calculated using the values for the electrolytes obtained with the blood gas
analyser and using the following formula: [SID] = ([Na^{+}]+ [K^{+}] + [Ca^{2+}]) – ([Cl^{}]
+ [Lac^{}]) (4). For [A_{TOT}],
the mean value of 18.2 mequiv·L^{1} reported by other authors (4, 5)
was used.
K_{A} (3.0 x 10^{7} ([equiv·L^{1}]), K_{C
}(2.46 x 10^{11} [equiv·L^{1}]^{2}/Torr), K_{3}
(6.0 x 10^{11} [equiv·L^{1}]), and K´w (4.4 x 10^{14}
[equiv·L^{1}]^{2}) are the dissociation constants of the weak
acids, of carbonic acid, of bicarbonate, and of water respectively. The above values for these constants were
those used by other authors
(36, 12, 16). Matlab v.7.1.0.246 software (MathWorks, Inc.
Natick, USA) was used to solve the Stewart equation.
Given the scant contribution of coefficients D and E to the final
result, these were eliminated from the equation to give. Then, [H^{+}] was calculated using a
proposed, simplified version of the Stewart equation. Firstly:
Equation 2
A[H^{+}]^{4} +
B[H^{+}]^{3} + C[H^{+}]^{2} = 0
which was then
simplified to:
A[H^{+}]^{2} + B[H^{+}] + C = 0
2.5. STATISTICAL
ANALYSIS
One way ANOVA was used to compare the pH and [H^{+}] values
measured with the blood gas analyser and those estimated by the Stewart and
simplified Stewart equations. When
significant differences were detected, a posthoc
Scheffé test was performed. To test the validity of the simplified
equation, linear regression analysis was performed, Pearson correlation
coefficients were calculated, and, following the method of Bland and
Altman (17), graphs were produced to
show the differences between the means
of the measured pH and [H^{+}] values and those calculated using the
two forms of the Stewart equation. The determination coefficient (r^{2})
was used to estimate the proportion of the variance explained by the proposed
equation. All statistical calculations
were undertaken using SPSS v.12.0 software (SPSS Worldwide Headquarters,
Chicago, IL) for Windows. Significance was set at α < 0.05.
The mean values
measured for pH and [H^{+}] were significantly different to those
calculated by the traditional and simplified Stewart equations. However, no
significant differences were seen between both equations.
Table 1 shows the
means ± SD for the [SID], PCO_{2} and [Lac^{}] measures
obtained by the blood gas and the lactate analyser (see above).
Table 1. Descriptive data. Data are shown as means ± SD for the
[SID], PCO_{2} and [Lac^{}]. 


[SID] (mequiv.L^{1}) 
[Lac^{}] (mmol.L^{1}) 
PCO_{2} (mmHg)_{} 
At rest 
39.1 ± 12.15 
1.52 ± 0.53 
36.21 ± 3.50 
Min 10 
32.64 ± 3.58 
4.87 ± 2.09 
35.79 ± 3.54 
Min 20 
33.5 ± 3.93 
4.85 ± 1.97 
31.88 ± 2.71 
Min 30 
34.42 ± 4.20 
4.31 ± 2.08 
30.92 ± 3.68 
Table 2 shows the
means ± SD for the pH and [H^{+}] determined by measurement, by the
traditional Stewart equation, and by the proposed, simplified Stewart equation
(Equations 1 and simplified equation 2 respectively).
Table 2. Differences between the measured pH and [H^{+}] values, those
estimated using the Stewart equation and the proposed, simplified Stewart
equation. Data are shown as means ± SD. pH_{m}, [H^{+}]_{m}: Measured pH and [H^{+}] values. pH_{stw}, [H^{+}]_{stw}: Estimated
values using the Stewart equation. pH_{p},
[H^{+}]_{p}: Estimated values using the proposed, simplified
Stewart equation. * Significantly different to measured pH (p<0.05). ^{†
}Significantly different to measured [H^{+}] (p<0.05). 


pH_{m} 
[H^{+}]_{m} (nmol.L^{1}) 
pH_{stw} 
[H^{+}]_{stw} (nmol.L^{1}) 
pH_{p} 
[H^{+}]_{p} (nmol.L^{1})_{} 
At rest 
7.41±0.02 
39.02±1.50 
7.30±0.02^{*} 
50.60±2.40^{†} 
7.30±0.02^{*} 
50.60±2.40^{†} 
Min 10 
7.32±0.06 
48.92±7.44 
7.26±0.04^{*} 
55.34±5.15^{†} 
7.26±0.04^{*} 
55.34±5.15^{†} 
Min 20 
7.34±0.06 
45.77±6.59 
7.27±0.05^{*} 
54.05±5.56^{†} 
7.27±0.05^{*} 
54.06±5.56^{†} 
Min 30 
7.36±0.05 
44.11±4.83 
7.28±0.05^{*} 
52.72±5.78^{†} 
7.28±0.05^{*} 
52.72±5.78^{†} 
Figure 1 shows the
dispersion diagrams. A high correlation
was obtained between the [H^{+}] values calculated by the traditional
and simplified Stewart equations (r = 0.999; p<0.001); however the
correlation coefficients between the measured [H^{+}] values and those
determined by the two forms of the Stewart equation were both <0.50 (r =
0.491; p<0.001, and r = 0.492; p<0.001), respectively).
The mean measured pH
and [H^{+}] values were significantly different to those calculated by
either Stewart equations (Table 2). However, no significant differences were
seen between the pH and [H^{+}] values calculated with either of the
Stewart equations at any particular time point (0, 10, 20 or 30 min) (Table
2). The average difference between the
mean [H^{+}] values determined by the traditional and proposed Stewart
equations was 0.004 ± 0.013 nmol.L^{1}, while that between the mean measured [H^{+}]
values and the traditional Stewart equationdetermined results was 8.723 ± 6.032 nmol.L^{1}. Similar differences were obtained in comparisons
between the mean measured and the proposed Stewart equationdetermined results
(8.727 ± 6.031 nmol.L^{1}). Figure 1 shows the Bland and Altman graphs
for [H^{+}], in which these comparisons can be seen.
Fig. 1. The graphs at the left represent the linear regression
analysis (continuous line) and the line of complete similarity (dotted line)
for [H^{+}]. A) Stewart ecuation vs. proposed, simplified
Stewart equation (nmol·L^{1}). B) Measured values vs. proposed,
simplified Stewart equation (nmol·L^{1}). C) Measured values vs.
traditional Stewart equation (nmol·L^{1}).
We have found a
little contribution of coefficients D and E of the Stewart´s equation, which
allow the possibility to work with a second degree equation obtaining similar
results. The results of the present work differ to those obtained by other
authors (4, 6, 1820). Significant differences
were seen between the measured and Stewart equationestimated (either form) [H^{+}].
However, no significant differences were seen between the [H^{+}]^{ }determined
by the traditional and proposed Stewart equations.^{ } The Bland and Altman procedure (17) confirmed
the validity of the proposed equation (Fig. 1A), which can therefore be used in
place of the mathematically more complex traditional Stewart equation.
The differences
between the mean [H^{+}] values measured in the capillary blood and
those determined using either equation were >8 nmol·L^{1}; this
contrasts with the results obtained by Kowalchuk and Scheuermann (1994, 1995)
who reported the difference in the mean measured and traditional Stewart
equationderived [H^{+}] values to be <3 nmol·L^{1} (2.1±
7.2 nmol.L^{1}). Heenan y Wolfe (2000) reported similarly small
differences in their study of pregnant women.
However, the differences detected by these other authors were also
significant. In addition, (4) found a
strong correlation between the mean measured and estimated [H^{+}]
values (r =0.81), while in the present work the correlation coefficient was r
<0.50. Other authors report significant differences between mean measured [H^{+}]
values and those calculated using the traditional Stewart equation (19, 20),
along with a correlation coefficient of
r = 0.99 (18). However, none of these authors used Bland and Altman
graphs to validate the Stewart equation. In the present work, these graphs
showed a poor agreement between the measured [H^{+}] values and those
determined by the traditional (Fig. 1C) and proposed, simplified (Fig. 1B)
equations. Taking in accuount the poor agreement and low correlation in our
study (r = 0.49) is possible to argue that both equations are inappropriate to
determine the [H^{+}], but the Stewart´s approach offers a deeper
knowledge of the acidbase status (22). The reasons to
explain why Stewart´s equation could fail when determinig the pH are difficult
to explain and could be related with the impossibilty to measure all the strong
ions or determine A_{TOT}. Also, the temperature and ionic
strength of the plasma are influencing the values of the equilibrium constants
(see also below).
When the plasma [H^{+}]
is >55 nmol·L^{1}, the differences between the measured and
calculated values have been reported to increase (4), but in the present study
the difference remained the same throughout.
In the study of Kowalchuk et al. (1988) calculated and measured [H^{+}]
were closely comparable, apart from arterial plasma at rest, where there was a
marked variability between the subjects.
The reasons for the
differences between the results of this study and those of other authors could
be due to several factors. Although
several works have shown no differences in acidbase measurements depending on
the type of blood (23, 24), we used capillary blood while Weinstein et al.
(1991) used venous blood, Kowalchuk and
Scheuermann (1994) used arterialised venous blood, and Fedde and Pieschl (1995)
used arterial blood. Secondly, [A_{TOT}] was not measured in the
present work; rather, the mean value reported by (4, 5) was used; [A_{TOT}]
(or [P_{TOT}]) appears to influence the results much less than [SID] or
PCO_{2} (18). In addition, errors in the measurement of [A_{TOT}]
do not appear to influence the calculated [H^{+}] values when the [SID]
is close to 40 mequiv·L^{1} (4, 16). Finally, errors in the
measurement of the independent variables with most influence on [H^{+}]
 PCO_{2} and [SID] – may explain some of these differences. Both
variables determine the coefficients of the traditional and proposed Stewart
equations. The values obtained for these variables were similar to those obtained
in all other studies. Although the
coefficients D and E have been eliminated in the propoesed equation, we
consider that the physiological impact of deleting them is low because PCO_{2}
is the only independient variable included in them.
The methodology used
to measure the pH, PCO_{2}, and the strong ion (Na^{+}, K^{+},
Ca^{+2} and Cl^{}) and lactate concentrations was similar to
that employed by (4). Kowalchuk and Scheuermann (1995) reported that
differences between the measured and calculated [H^{+}] values in their
study might be due to the fact not all the strong ions are measured, leading to
an inaccurate [SID]. Finally, errors in the values of the equilibrium constants
used to determine the coefficients of the Stewart equation may explain some of
these differences. In the present study,
values recommended in the literature were used, which correspond to a blood
plasma temperature of 37ºC (3, 5, 9, 12). These
constants, however, are dependent on the temperature and ionic strength of the
plasma. Nonetheless, the use of incorrect values is only associated with small
errors since the Stewart equation is largely insensitive to inaccuracies in the
majority of the dissociation constants (4, 5, 16). In the present work, the
same values were assumed for the equilibrium constants independent of the
difference between the central and peripheral temperatures during exercise. In
addition, although the constant load exercise was assumed
to have been performed at a constant body temperature, it is likely that it was
not the same at the beginning and the end of the exercise. However, the effect of increasing temperature
on the factors required to determine the Stewart equation constants is
negligible from a biological point of view (25). Further, temperature variation
cannot explain the differences between the mean measured and Stewart equationderived
(either form) [H^{+}] values.
None of the studies
mentioned above discuss the mathematical procedure used to solve the polynomial
Stewart equation, yet this is important when trying to determine the possible
causes of differences in the measured and estimated [H^{+}] values.
5. CONCLUSIONS
6. PHYSIOLOGICAL RELEVANCE
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Rev.int.med.cienc.act.fís.deporte vol. 11  número 41  marzo 2011  ISSN: 15770354