Clinical Science and Molecular Medicine (1974) 46, 113-123.
THE RELATIONSHIP BETWEEN ARTERIAL Pco, A N D
HYDROGEN I O N CONCENTRATION IN CHRONIC
METABOLIC ACIDOSIS A N D ALKALOSIS
J. M. BONE, J. COWIE, A N N E T . L A M B I E
AND
J. S . ROBSON
Medical Renal Unit, Department of Medicine of the University of Edinburgh,
Royal InJirmary, Edinburgh
(Received 10 September 1973)
SUMMARY
1. An analysis was made of the relationship which exists between arterial [H'],
Pco, and [HC03-] in twenty-five patients with stable metabolic acidosis and
alkalosis and in three normal subjects.
2. Contrary to previous reports, the relationship between Pco, and [H'] was nonlinear and could best be described in terms of a rectangular hyperbola (Pco, =
962/([H'] - 12).
3. The relationship between Pco, and [HC03-] was curvilinear and best described by the quadratic function 23.8(Pc0,)~ - 12 Pco,[HCO,-] -962 [HC03-] = 0.
4. The small acute changes in [H'],.[HCO,-] and Pco, produced by infusion of
the weak organic acid 5,5-dimethyl2,4-oxazolidinedione(DMO) could be predicted
from the curvilinear regression.
Key words : Pco,, [H'], metabolic acidosis, metabolic alkalosis, ventilation.
In 1917 J. P. Peters first recognized that the ventilatory response to metabolic acidosis could be
monitored by following the changes in the alveolar carbon dioxide partial pressure. In addition
he showed that in stable states of acidosis there was a close relationship between alveolar
Pco, and the CO, combining power of the blood.
More recently technical refinements have facilitated measurements of pH and PCO,in
arterial blood and further attempts have been made to define the normal ventilatory response
in patients with metabolic acidosis and alkalosis in terms of the relationship which exists, in
arterial blood, between Pco, and pH (or [H']) or between Pco, and plasma bicarbonate
concentration ([HC03-]).
On the basis of such clinical studies linear relationships have been described between PCO,
and [H'] (Albert, Dell & Winters, 1967; Flenley, 1971a, b) and between Pco, and [HCO,-]
(Elkington, 1966; Albert et al., 1967; Pierce, Fedson, Brigham, Mitra, Sack & Mondal, 1970;
Van Ypersele de Strihou & Frans, 1970) and these have been generally accepted.
Correspondence:Dr Anne T. Lambie, Medical Renal Unit, Royal Infirmary, Edinburgh EH3 9YW.
113
114
J. M. Bone et al.
The relationship of the three variables is given by equation (1).
[H'] = K'.
S. Pco,
1Hc03-1
K is the equilibrium constant for the first dissociation of carbonic acid and when [H'] is
expressed in nmol/l, [HC03-] in mmol/l and Pco, in mmHg, K has a value of 794 in plasma
at 38°C. The solubility factor S for carbon dioxide is 0.0301 mmol l-'mmHg-l. In view of the
form of this expression it is improbable that the relationship between any two of the three
variables will take the form of a simple linear regression when all three are changing.
It follows from equation (1) that:
When [HCO,-] remains constant this can be re-written:
PCO, = C[H+]
where C is a constant incorporating K , S and [HCO,-1. In these circumstances the relationship between Pco, and [H'] is linear, the slope of the line being proportional to [HC03-].
In metabolic acidosis and alkalosis in which [HCO,-] is not constant, the slope of the line
varies and a curvilinear relationship between Pcoz and [H'] would be predicted. The equations
for the linear regressions derived by Albert et al. (1967) and by Flenley (1971b) are given in
Table 1. In the first of these studies the duration of the acidosis varied considerably and in some
cases was as short as 24 h, so that doubt exists as to whether all the patients were in a steady
state of acid-base balance. Moreover, half of the observations were made upon blood from
children under 2 years of age, in whom it is likely that the relationship between [H'] and Pco,
differs from that found in adults (Albert & Winters, 1966). More important is the fact that the
range of values for [H'] encountered in this series was limited to between 48 and 78 nmol/l,
since this may have led to erroneous conclusions regarding the form of the relationship. The
range of values analysed by Flenley (1971b) was much greater; in the data which he assembled
[H'] varied from 30 to 80 nmol/l. Included in the analysis, however, are data from subjects
studied by Msller (1959), Weller, Swan & Merrill (1953), Bergofsky, Lehr & Fishman (1962)
and Chiesa, Stretton, Massond & Howell (1969), who were almost certainly not in a steady state,
and this may have influenced the form of the calculated regression.
Inspection of equations (1) and (2) suggests that the relationship between Pco, and [HC03-]
in patients with disturbances of acid-base balance is also unlikely to be described by a simple
linear regression. In most of the studies listed in Table I, in which linear regressions were
derived, the range of values for [HCO,-] was limited and no cases of metabolic alkalosis were
included. The widest range is found in the work of Van Ypersele de Strihou & Frans (1971),
but these studies contain no examples of severe acidosis or alkalosis. In addition, the patients
with cholera reported by Pierce et al, (1970), and some of the subjects of Albert et al. (1967),
were almost certainly not in a steady state of acid-base balance. The older literature contains
other, slightly less satisfactory studies (Lennon & Lemann, 1966; Msller, 1959), in which a
linear relationship was reported to exist between PCO, and [HC03-] in metabolic disorders of
acid-base balance. Lennon & Lemann (1966) used venous blood and some of Msller's patients
appear to have had pulmonary disorders. In the present study we have analysed the values for
Children and adults with acute and chronic
metabolic acidosis (diarrhoea, renal disease)
Values taken from literature. Adults and children
with metabolic acidosis (diarrhoea, ketoacidosis, renal disease). Normal subjects taking
NaHC03, NH4CI or ethacrynic acid
Adults with chronic metabolic acidosis (renal
disease)
Children and adults with acute and chronic
metabolic acidosis (diarrhoea, renal disease)
Adults with acute metabolic acidosis (cholera)
Normal subjects. Adults on chronic haemodialysis
with metabolic acidosis and with metabolic
alkalosis induced by high bath [acetate]
Albert et al. (1967)
Elkington (1966)
Albert et al. (1967)
Van Ypersele
de Strihou &
Frans (1970)
Pierce et al. (1970)
Flenley (1971a, b)
Subjects
Author
P C O=
~ 8*8+1*51[HCOs-]+40
P C O=
~ 52.95-0.50 [H+] 2*76(''
(r = 0.83)
[H+] = 86*8+1.17 P C O ~
Equation
3-9
1 SE.
30-80
48-78
Range of Range of
[HCOs-l
[H+l
(mmol/l) (nmol/l)
P C O=
~ 8.36+1*54[HCOj-]+ 1.1
4-16
(r = 0.97)
20
P C O=
~ 12*38+1*11[HC03-]~1*45 8-13
(r = 098)
(12 acidotic,
8 control)
337
P C O=
~ 15.4+092[HC0~-]+2.6
11-38
60
27
143
60
No.
of
observations
TABLE
1. Published relationships between [H+]and Pcoz
20
E'
t
2
%
9a
C'
;;;
2
R
$
3
%
a
w
s'
H
116
J. M. Bone et al.
arterial pH, Pco, and [HCO,-] obtained in twenty-five patients with stable disturbances of
metabolic origin, comprising a wide range of acidosis and alkalosis. In the great majority of
patients the arterial pH and Pco, were known to have been steady for at least 24 h. In the few
patients in whom treatment could not be delayed there was good clinical evidence that the
disturbance had developed gradually over the previous few days. Data from three normal
healthy subjects have been included. In thirteen of the patients the effect of a small increase in
blood [H'] produced by infusion of the weak organic acid 5,5-dimethyl2,4-oxazolidinedione
(DMO) was studied (obtained from L. Light and Co.).
PATIENTS A N D METHODS
Twenty patients had metabolic acidosis due to chronic renal failure and five had metabolic
alkalosis arising from loss of acid or excessive intake of base. In no case was the metabolic
alkalosis due to primary potassium depletion. No patient had airways obstruction, pulmonary
oedema or parenchymal lung disease. In only one patient was the arterial pH less than 7.10.
Arterial blood was drawn directly into a heparinized syringe. In the three healthy subjects
and two of the patients with metabolic acidosis, blood was obtained by direct puncture of the
brachial artery with local anaesthesia; in all other cases blood was taken from an indwelling
arterial cannula inserted before the beginning of the study. Blood pH was measured immediately with a Radiometer G 298A glass capillary electrode with a KC1 liquid junction, at a
temperature of 38"C, and a Radiometer pH meter, standardized with BDH standard buffers
(7.380+0-005 pH units at 38°C; 6.840+0.005 pH units at 38°C). A factor of 0.01 was added to
the observed blood pH to correct for alteration of the junction potential at the KCl junction by
precipitated red cells (Andersen, 1961). The reproducibility of duplicate readings was within
0.005 pH unit. The Pco, was determined simultaneously by the Siggaard Andersen microequilibration technique with Radiometer BMS, equipment (Siggaard Andersen, Engel, Jorgensen & Astrup, 1960). The plasma bicarbonate concentration was calculated by using the Henderson-Hasselbalch equation and the measured values for pH and Pco, with values for pK'
determined in this laboratory (Cowie, Lambie & Robson, 1962).
Thirteen patients received an intravenous infusion of DMO. Approximately 0.8 mmol of
DMO/kg body weight was dissolved in sterile 5% dextrose solution and infused over a period
of 20-30 min. This provides about 0.8 mmol of H'/kg, i.e. a total of approximately 50 mmol of
H'. Arterial blood was taken before the DMO was given and 4 h after the beginning of the
infusion. All patients who received DMO gave their consent for the procedure after it had been
explained to them and for arterial cannulation. The procedure has been approved by the
Advisory Ethical Committee of the hospital.
Regressions were calculated by using the method of least squares.
RESULTS
The data from the twenty-five patients with stable metabolic acidosis and alkalosis and the
three healthy subjects are shown in Table 2 together with the data obtained after infusion of
DMO. Fig. 1 shows the relationship between arterial blood PCO, and [H'] in stable conditions of acid-base balance. As the arterial [H'] increases there is a reduction in Pco, resulting from an increase in ventilation; a fall in arterial [H'] is associated with a decrease in
PCO, in metabolic acidosis and alkalosis
117
TABLE
2. Arterial blood acid-base measurements
Stable conditions
Subject
no.
Metabolic alkalosis
1
2
3
4
5
Healthy subjects
1
2
3
Metabolic acidosis
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
pH
After DMO
[H+]
P C O ~ [HCOj-]
(nmol/l) (mmHg) (mmol/l)
7.56
7.53
7.49
7.49
7.45
29
30
32
32
34
47
52
58
44
46
42
43
43
33
32
7,42
7.41
7.41
38
39
39
43
38
39
27
24
25
7.40
7.37
7.34
7.34
7.34
7.32
7.30
7-30
7.29
7.28
7.27
7.25
7.22
7.20
7.19
7-14
7.14
7.13
7.12
7.09
40
43
46
46
46
48
50
50
51
52
54
56
60
63
65
73
73
74
76
81
33
24
29
28
27
23
34
17
21
23
22
23
21
26
26
15
19
10
13
19
20
13
15
15
14
12
16
8
10
10
10
10
8
10
9
5
6
3
4
5
pH
[H+]
PcO~
(nmol/l) (mmHg)
7.53
7.53
7,53
7.45
30
30
30
34
7.24
58
55
65
72
69
79
87
87
74
-
-
7.26
7-19
7.14
7.16
7.10
7.06
7.06
7-13
-
40
46
59
[HC03-]
(mmol/l)
34
38
48
-
-
41
29
17
7
-
-
18
17
21
21
9.5
15
9.5
12.5
-
8
6
7
7
2.5
4
2.5
4
-
ventilation and an increase in PCO,. In Fig. l(a) a single linear regression, Pco, = 64.5-0.69
[H'], has been calculated to fit the data. Clearly this is inadequate since there is a preponderance of points above the line at either end of the range, and below the line in the middle.
The sum of squares of deviations about the regression is 1226.
A much better description of the data is given by a single curvilinear regression in the form
of a rectangular hyperbola, shown in Fig. l(b), which is continuous over the whole range of
values. The equation of the hyperbola, derived by the method of least squares, is
Pco -
962 +5.4 (SE)
[H'I- 12-
118
J. M . Bone et al.
PH
PH
7.4
7.3
7.2
7.5
7.4
7.3
7.2
I
I
I
I
I
I
I
I
..
60-
50-
-..4 0
7.5
,
I
7.05
I
Pco2=962/(CH*1-12)5s' 759
-
m
\ ..
r
E
k
7.1
.i
3020 -
IO(a)
3b
$0
40
Go
$0
7b
do
80
C H(nrnol/i)
~
;o
40
io
$0
70
8'0
9'0
FIG.1. Arterial blood [H+]and Pco2in stable metabolic acidosis and alkalosis.Data are expressed
(a) in the form of a single linear regression,Pco2 = 64-5-0.69 [H+],sum of squares (s.s.) = 1226,
and (b) in the form of a hyperbola, Pco2 = 962/([H+]- 12), S.S. = 759. The continuous lines (-)
indicate 1 S.E.
and there is a reduction in the sum of squares to 759. An almost identical hyperbolic curve can
be derived by using the method of Hey & Hey (1960), in which the functional relationship
between [H'] and Pco, is defined by the equation
926
pcOz = [H+]-13
(sum of squares 353).
The changes that occur after infusion of approximately 50 mmol of the weak acid DMO are
shown in Fig. 2. The small changes in [H"] and Pcoz might have been predicted from the
curvilinear regression. In patients with metabolic alkalosis the change in [H'] is small but
Pcoz falls by approximately 5 mmHg; in patients with metabolic acidosis the increase in [H+]
is greater and the fall in Pco, less. The curvilinear regression derived from all data, including
those from patients who had received DMO 4 h previously, was not appreciably different from
that shown in Fig. l(b) and Fig. 2.
Once the equation of the rectangular hyperbola has been determined it is possible to obtain
an expression relating Pco, to [HC03-] using the Henderson equation (equation 1) and the
equation of the hyperbola.
Thus
Combining the constants K' = 794 and
CI =
0.03
23.8 PCO,
IH+] = [HC03-]
Pco2 in metabolic acidosis and alkalosis
& ) A
I
I
I
I
40
50
60
70
119
I
I
a0
90
[Ht](nmol/l)
FIG.2. Arterial blood [H+] and Pcoz in stable metabolic acidosis and alkalosis ( 0 ) and 4 h after
infusion of approximately 50 mmol of DMO (0).
Substituting for [H'] in the equation for the hyperbola
962
Pco - [(23*8Pco~/[HCO~-])
- 121
which, on rearranging, gives
23.8 ( P c o ~ ) ~12 P C O[HC03-]
~
-962 [HC03-]
=
0.
The curve represented by this quadratic function is shown in Fig. 3, together with the data
from the individual patients and healthy subjects. The linear regression derived by Van Ypersele de Strihou & Frans (1970), for patients with mild acidosis and alkalosis, is also shown.
Over the range covered by their investigations this regression agrees closely with the curve
calculated from our data.
In Fig. 4 data derived from a number of published studies dealing with metabolic disturbances of acid-base balance are shown in relationship to the hyperbolic curve derived from
analysis of our results (Cowie et al., 1962; Bradley & Semple, 1962; Stone, 1962; Bergofsky
et al., 1962; Pauli, Riedwil, Reubi & Wegmuller, 1963; Lambie, Anderton, Cowie, Simpson,
Tothill & Robson, 1965; Posner, Swanson & Plum, 1965; Falchuk, Lamb & Tenney, 1966;
Goldring, Cannon, Heinemann & Fishman, 1968; Fend, Vale & Broch, 1969; Chiesa et al.,
1969; Zimmett, Taft, Ennis & Sheath, 1970). The subjects of these studies were patients with
untreated metabolic acidosis, due to a variety of causes, or healthy volunteers taking sodium
bicarbonate or diuretics. The criteria used for selection of subjects were similar to our own but,
120
J. M. Bone et al.
[HCOS-I(rnmol/U
FIG.3. Arterial blood [HCO,-] andPcoz in stablemetabolicacidosis and alkalosis ( 0 )and4 h after
data from Van
infusion of approximately 50 mmol of DMO (0). - - - - -, Present data; -,
Ypersele de Strihou & Frans (1970). The parallel continuous curves indicate 1 SE.
in some cases, slightly less rigid. For this reason, in the studies on healthy volunteers reported
by Bergofsky et al. (1962) and Chiesa et al. (1969) only the values obtained during control
periods are shown in Fig. 4 or included in the analysis. Clearly there is no discrepancy between
our own results and those of others. For the patients with extremely severe acidosis reported
by Posner et al. (1965) and Zimmett et al. (1970) the extrapolated hyperbolic regression appears
to describe the relationship between PCO, and [H'] satisfactorily. The hyperbolic curve
computed from the data collected from the literature conforms to the equation PCO, =
1169/([HfJ- 8) 6.2 (SE) and does not differ appreciably from the curve that describes our
own data and which is shown Fig. 4. There is also good agreement between a relatively small
portion of our curve and the linear regression reported by Albert et al. (1967).
DISCUSSION
Analysis of a wide range of values for arterial [H'], Pco, and [HCO,-1, measured in patients
with stable chronic metabolic acidosis or alkalosis, suggests that the relationship between [H+]
and PCO, is best described by a rectangular hyperbola. The best description of the relationship
between [HCO,-] and Pco, is provided by a quadratic function. These findings are not
unexpected since non-linear relationships between [H'] and Pco, and [HCO,-] and Pcoz
might be predicted from the Henderson equation, but, in the past, linear regressions have been
derived to describe the relationships between these pairs of variables and have been accepted.
A linear regression often appears to be the most obvious way of defining the relationship
PCO, in metabolic acidosis and alkalosis
121
PH
7.5
7,4
7.3
7.2
I
I
I
60L
50 -
-I"
-E
7.1
I
7.05
7.0
I
I
*.
\w
'0.
40 -
30-
N
20 4,
$-I33
10-
.-I60
v
20
x)
40
50
60
80
70
90
100
120
I30
[Ht] (nrnol/l)
FIG.4. Relationship between arterial [H+]and Pco2 in metabolic acidosis and alkalosis. 0 , Values
derived from literature (Cowie et al., 1962; Bradley & Semple, 1962; Stone, 1962; Bergofsky et al.,
1962; Pauli et al., 1963; Lambie et al., 1965; Posner et al., 1965; Falchuk et al., 1966; Goldring
et al., 1968; Fencl et al., 1969; Chiesa et al., 1969; Zimmett et al., 1970). The hyperbola derived
The linear regression
from the data of the present paper is shown by the continuous line (-).
of Albert et al. (1967) is shown by the broken line (-).
Mean value for [H+]and Pco2 in
patients with 'stable' metabolic acidosis, described by Pierce et al. (1970).
- - - +,
between two variables and the existence of a curvilinear relationship is easily overlooked.
This is particularly likely to happen when the ranges of values examined is limited, as it was
in most reported studies. A curvilinear relationship may also be obscured when accumulated
data from different laboratories are examined, because of variations in analytical methods and
because of the variety of clinical and experimental material. In view of this it is of interest that
analysis of published data obtained from selected studies upon subjects who were in a reasonably stable state of metabolic acidosis or alkalosis shows that the relationship between [H']
and Pcoz can be described by a hyperbolic function similar to that derived from our own series
of cases.
The linear regressions reported by Albert et al. (1967) and by Van Ypersele de Strihou &
Frans (1970) correspond closely to the relevant sections of the curves described in this paper,
differing from them only at the ends of the range of values for [H']. They can therefore be
used in clinical practice, in all but the most severe disturbances, to determine whether a particular value of Pcoz is appropriate to the degree of acidosis or alkalosis. However, preoccupation with linear relationships has obscured the fact that arterial PCO, is inversely proportional
to [H']. In subjects in a steady state of acid-base balance Pcoz is inversely proportional to
122
J. M. Bone et al.
alveolar ventilation; although a discussion of the factors that control ventilation is outside
the scope of this paper, the existence of a reciprocal relationship between PCO, and [H'] in
patients with stable chronic metabolic acidosis and alkalosis supports the idea that the concentration of H' in arterial blood, or in some compartment of the body fluid in which
proportionate changes in [H'] occur, is an important factor in regulating ventilation.
ACKNOWLEDGMENT
The authors are extremely grateful for the help of Dr D. A. Williams, Department of Statistics,
University of Edinburgh.
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