Titrations of CO
2
in water, plasma and blood
General Principles
The blood contains many substances that act as buffers, both acidic (e.g. phosphates and bicarbonate) and basic (e.g. His residues of plasma proteins and hemoglobin) in nature. The ocncentration of the hydrogen ion in solution is very low, in the oprder of tens of nM, and thus one can safely assume that acid-base equilibria are exchanges of hydrogen ions between different buffers, while the direct contribution of hydrogen (or hydronium) ions can be neglected. It is possible to simplify the description of the system to only two buffers: that composed by CO
2
and bicarbonate (with its acidic dissociation constant Kc), and the sum of all other buffers (non -carbonic buffers), to which an "average acidic dissociation constant Kb is assigned. The chemical reactions of the system can thus be described as follows:
The above expression apply to
in vivo
as well as
in vitro
conditions. In view of the large number of unknown variables, the above equation can be solved only by introducing further parameters and considerations, which differ depending on the physiological condition considered or the type of experiment carried out
in vitro
.
The titration of the bicarbonate/CO
2
buffer has been carried out under different experimental conditions and using different methods; its relevance to medicine is evident in view of the role of CO
2
in respiration and acid-base balance. However, titrates at the same time the carbonic and non-carbonic buffers and the interpretation of the experiment demands a theory that describes both. In this exercise we shall consider three different types of experiment, which give very different results, and we shall demonstrate that they are fully compatible with each other, provided that they are interpreted in a unique and coherent theoretical framework:
1) The titration of the two buffers by varying the partial pressure of CO
2
(PCO
2
); in this case the sum of de-protonated components of the two buffers is constant, because of the reaction reported above (which shows that one can convert CO
2
to bicarbonate only at the expense of converting B to BH
+
and
vice-versa
: Sdp = [HCO
3
-
] + [B].
2) The titration of the two buffers with HCl or NaOH at constant PCO
2
.
3) The titration of the two buffers with HCl or NaOH in the absence of a gas phase, in which case the total CO
2
= [CO
2
] + [HCO
3
-
] is constant.
To simplify the calculations we shal consider a volume of 1 litre of solution, or plasma "separated" (from the erythrocytes), or blood. In this way molarity and number of moles of solute will have the same value. We shall express the concentrations in mM and the CO
2
partial pressure in mmHg. We shall consider starting conditions similar to the physiological one: pH=7.4; bicarbonate=24 mM; PCO
2
=40 mmHg. The pKa of CO
2
a T=37
o
C is 6,1 and the solubility constant of the gas is 0,03 mM/mmHg. In the case of
plasma
we shall include, in addition to CO
2
, a second "average" buffer, to represent the contribution of albumin, phosphate, etc. to which we assign pKa=6,8 and [B]
tot
=[BH
+
]+[B]=24 mM. Finally, the case of
blood
is similar to that of plasma but demands a higher value for the concentration of buffer B in order to account for the contribution of hemoglobin: [B]
tot
(in mEq/L) = Hb (in g/dL) x 4.4 + 15. MOreover, in the case of blood we need to include a parameter that corrects for the bicarbonate concentration given that this parameter differs in the plasma and in the erythrocytes: λ = [HCO
3
-
]
blood
/ [HCO
3
-
]
plasma
= 0,85. In the case of blood, pH is measured in the plasma, but titration affects the bicarbonate of blood: thus we need to be careful to distinguish those calculations which require either.
1)
Titration with PCO
2
(Sdp= cost.)
One liter of a 24 mM solution of sodium bicarbonate ([HCO
3
-
]
in.
) is equilibrated with CO
2
at different partial pressures (at T=37
o
C). The equation that describes this experiment is:
[H
3
O
+
] = Kc x 0,03 x PCO
2
/ [HCO
3
-
]
in.
.
To a first approximation we may assume that in this experiment [HCO
3
-
] is constant and equal to [HCO
3
-
]
in.
.
If the same experiment is carried out on
plasma
, we need to take into account also the other buffer(s). In this case the exchange of hydrogen ions between the two buffers obeys this relationship:
CO
2
+ H
2
O + B <=> HCO
3
-
+ BH
+
.
As a consequence the
sum of deprotonated components of the two buffers is constant: Sdp = [HCO
3
-
] + [B]
. The values of PCO
2
and [CO
2
]
tot
are the independent variables of the system.
The equation that describes this experiment is:
[H
3
O
] = Kc x 0,03 x PCO
2
/ [HCO
3
-
] = Kb x [BH] / [B]
We can rewrite the above equation taking advantage of the relationship [HCO
3
-
] = Sdp - [B]:
[H
3
O
] = Kc x 0,03 x PCO
2
/ (Sdp - [B]) = Kb x ([B]
tot
- [B]) / [B].
We obtain the following second degree equation:
Kb [B]
2
- [B] (Kb Sdp + Kb [B]
tot
+ Kc 0,03 PCO
2
) + Kb [B]
tot
Sdp = 0
from which we easily calculate [B] and, with this value, all other parameters of the system.
In the case of
whole blood
we obtain equations similar to those obtained for plasma but we need to take into account that: (i) [B]
tot
has a higher value than for plasma because hemoglobin (Hb) has a very high buffer capacity. The formula to obtain [B]
tot
is as follows: [B]
tot
(mEq/L) = [Hb] (g/dL) x 4.4 + 15. At a value of [Hb]=14 g/dL we obtain [B]
tot
=77 mEq/L. (ii) The total concentration of bicarbonate in the blood is lower than in the plasma, because approx. 40% of the blood volume is accounted for by the erythrocytes whose intracellular concentration of bicarbonate is lower than that of plasma. The ratio [HCO
3
-
]
blood
/ [HCO
3
-
]
plasma
is λ = 0.85. In our equations we need at times [HCO
3
-
]
blood
(when we calculate the consumption/production of the anion or the Sdp), and at other times [HCO
3
-
]
plasma
(when we measure the pH of plasma). Thus we replace [HCO
3
-
]=(Sdp - [B]) with [HCO
3
-
]=(Sdp - [B]) / λ.
We obtain the following equation:
Kb [B]
2
- [B] (Kb Sdp + Kb [B]
tot
+ Kc 0,03 PCO
2
λ) + Kb [B]
tot
Sdp = 0
The results of these experiments are as follows:
CO
2
water
CO
2
plasma
CO
2
blood
PCO
2
(mmHg)
total CO
2
(mM)
pH
total CO
2
(mM)
pH
total CO
2
(mM)
pH
10
24.3
8
21.1
7.95
12.6
7.79
20
24.6
7.7
22.7
7.67
16.3
7.59
30
24.9
7.53
24
7.51
19.2
7.48
40
25.2
7.4
25.2
7.4
21.6
7.4
50
25.5
7.3
26.2
7.32
23.7
7.34
60
25.8
7.22
27.2
7.25
25.5
7.29
70
26.1
7.16
28.2
7.2
27.2
7.25
80
26.4
7.1
29
7.15
28.8
7.22
90
26.7
7.05
29.8
7.11
30.2
7.18
The
buffer capacity
is defined for titrations in which the total concentration of the buffer is constant (Cs+Ca = constant), and the titratant is
not
the acidic component of the buffer. Nevertheless it is possible to define the somewhat equivalent ratio Δ[CO
2
]/ΔpH that at pH=7,4 turns out to be: 3 mEq/L in water, 3.75 mEq/L for plasma e 5 mEq/L for blood, when measured on the PCO
2
and 3 mEq/L in water, 12.5 mEq/L for plasma e 35 mEq/L for blood, when measured on the total CO
2
(latter values are quite similar to those obtained when titrating with HCl in the absence of the gaseuous phase).
Comparison with experimental data:
. A classical experiment of titration of blood with CO
2
has been reported by
Christensen, Douglas e Haldane
in 1914 (their experimental points are reported as circles in the figure below); the line calculated using the model described above is in blue; the red line represents the solubility of CO
2
in water. The titrations of bicarbonate and plasma are not reported because there are no experimental data with which they can be compared.
2)
Titration with HCl at constant PCO
2
One liter of a 24 mM solution of sodium bicarbonate, or plasma, or whole blood are equilibrated with CO
2
at a constant pressure of 40 mmHg (at T=37
o
C) and titrated with HCl. The gas phase has variable volume, so that the CO
2
absorbed or released does not change the PCO
2
. The equations that describe this type of esperiment are as follows:
for the bicarbonate solution: [H
3
O
+
] = Kc x 0,03 x PCO
2
/ ([HCO
3
-
]-[HCl])
for plasma: [H
3
O
+
] = Kc x 0,03 x PCO
2
/ ([HCO
3
-
] -X) = Kb x (BH + [HCl] - X) / ([B] - [HCl] + X)
for blood: [H
3
O
+
] = Kc x 0,03 x PCO
2
/ (([HCO
3
-
] - X) / λ) = Kb x (BH + [HCl] - X) / ([B] - [HCl] + X)
We remark that [HCO
3
-
] in blood represents the total bicarbonate, not the plasma fraction.
The results one obtains are as follows:
CO
2
in water
CO
2
in plasma
CO
2
in blood
HCl (mM)
total CO
2
(mM)
pH
total CO
2
(mM)
pH
total CO
2
(mM)
pH
0
25.2
7.4
25.2
7.4
21.5
7.4
2
23.2
7.36
23.5
7.37
20.3
7.38
4
21.2
7.32
21.8
7.34
19.1
7.35
6
19.2
7.28
20.1
7.3
18
7.32
8
17.2
7.22
18.5
7.26
16.9
7.29
10
15.2
7.17
17
7.22
15.9
7.26
12
13.2
7.1
15.5
7.18
14.9
7.23
14
11.2
7.02
14
7.13
14
7.2
16
9.2
6.92
12.6
7.08
13.1
7.17
The buffer capacity at pH=7,4 is: Δ[HCl]/ΔpH = 50 mEq/L for bicarbonate in water, Δ[HCl]/ΔpH = 66.67 mEq/L for plasma e Δ[HCl]/ΔpH = 100 mEq/L for blood.
Comparison with experimental data
: the values of buffer capacity obtained in this titration can be compared with that reported by
Watanabe et al. Japn. J. Physiol. 2001: 51; 671-677
: Δ[HCl]/ΔpH = 90 mEq/L for blood at pH=7.4 and PCO
2
= 40 mmHg. With respect to the values obtained in the absence of the gaseous phase (see below) these values are strongly overestimated because the total concentration of the carbonic buffer is not constant, and the consumption of bicarbonate is not associated to a parallel increase of CO
2
.
A comparison between the original data by Watanabe et al. and the simulation presented here is reported in the figure below.
The gray area represents the distribution of the (very numerous) original samples obtained from different subjects; the black line is the simulation. It should be noted that: (i) the titration was obtained using NaOH instead of HCl; (ii) the authors used an acidic anticoagulant, and because of this fact the starting value of pH is 7.1 instead of 7.4; (iii) at pH>7.8 most probably other aminoacid residues or solution components contribute some buffer capacity to the sample, not taken into account in the model that uses a single value of Pka to simulate all blood buffers.
3)
Titration with HCl at constant CO
2 tot
(i.e. in the absence of the gaseous phase)
One liter of a 24 mM solution of sodium bicarbonate, or plasma, or blood are equilibrated with CO
2
at P=40 mmHg and T=37
o
C; then a sample of the liquid is transferred to a vessel in the absence of the gaseous phase and is titrated with HCl.
The equations that describe this type of experiment are as follows:
for the bicarbonate solution: [H
3
O
+
] = Kc x ([CO
2
]i + [HCl]) / ([HCO
3
-
]i - [HCl])
for plasma: [H
3
O
+
] = Kc x ([CO
2
]i + X) / ([HCO
3
-
]i - X) = Kb x ([BH]i + [HCl] - X) / ([B]i - [HCl] + X)
for blood: [H
3
O
+
] = (Kc x 0,03 x PCO
2
+ X) / (([HCO
3
-
] - X) / λ) = Kb x (BH + [HCl] - X) / ([B] - [HCl] + X)
Here again, in the case of blood [HCO
3
-
] represents the total bicarbonate, not the plasma fraction; the suffixes "i" refer to the initial values of the parameters, before the beginning of the titration.
We obtain the following results:
HCO
3
-
in water
HCO
3
-
in plasma (separated from red cells)
HCO
3
-
in blood plasma
HCl (mM)
HCO
3
-
(mM)
pH
HCO
3
-
(mM)
pH
HCO
3
-
(mM)
pH
0
24
7.4
24
7.4
24
7.4
1
23
7.12
23.8
7.32
23.9
7.37
2
22
6.94
23.5
7.24
23.8
7.34
3
21
6.8
23.3
7.18
23.6
7.31
4
20
6.69
23
7.12
23.5
7.28
5
19
6.59
22.7
7.06
23.4
7.26
6
18
6.5
22.4
7
23.3
7.23
7
17
6.42
22.1
6.95
23.3
7.2
8
16
6.34
21.8
6.9
23.2
7.18
This is the only experiment in which the buffer capacity can be measured according to the definiton of
Van Slyke in J. Biol. Chem. 1922
. Under these conditions the buffer capacity at pH=7,4 is: Δ[HCl]/ΔpH = 3.57 mEq/L for the bicarbonate solution, Δ[HCl]/ΔpH = 12.5 mEq/L for plasma e Δ[HCl]/ΔpH = 33.33 mEq/L for blood.
Comparison with experimental data
: the buffer capacity values obtained in this simulation can be compared with those obtained by
Ellison et al., Clin. Chem. 1958: 4; 452-461
under the same experimental conditions: Δ[HCl]/ΔpH = 15-16 mEq/L for plasma e 31-38 mEq/L for blood.
PHYSIOLOGICAL CONSIDERATIONS: THE HALDANE EFFECT
The blood titrations described in this analysis were always carried out on samples of oxygenated hemoglobin (HbO
2
) because the gas phase with which the sample was equilibrated contained O
2
at the same partial pressure as air.
In vivo
, changes o PCO
2
are coupled with changes of PO
and of the oxygen saturation of hemoglobin. As a consequence of these, and of the Bohr effect, the pK of some buffer residues changes; oxygenation is associated to the release of Bohr protons, that are buffered (also) by bicarbonate. This effect was discovered by the british physiologist John Scott Haldane and is namend after him:
(H
+
)Hb + O
2
+ HCO
3
-
+ B
plasma
<=> HbO
2
+ CO
2
+ H
2
O + BH
+
plasma
The Haldane effect increases the offload of CO
2
in the lungs and the upload of CO
2
from the tissues, as well as the artero-venous difference of CO
2
by a factor of approx. 30%. To all practical purposes the Haldane effect is analogous to a titration of plasma with HCl. The maximum Bohr coefficient, at constant pH=7.4 corresponds to the release of 0,5 mol of hydrogn ion released per mol of bound oxygen (the so-called Bohr coefficient).
Simulating the Haldane effect under physiological conditions is complex because it requires to introduce a third buffer in the equations (see the reaction above); moreover, under physiological conditions hemoglobin does not release all the bound oxygen, thus it exists in two states: oxygenated and (partly) deoxygenated with different pKa. It is however quite easy and equally instructive to simulate the effect of the full oxygenation/deoxygenation of Hb in an artificial closed system, in the absence of a gaseous phase, composed of only two buffers Hb (at 14 g/dL in order to simulate blood) and CO
2
/bicarbonate (at 25 mM). In this artificial system we shall assume that the red cells have been hemolyzed so that Hb is released in solution and the factor λ is not required.
The equation that describes this system is:
[H
3
O
+
] = Kc ([CO
2
]
tot
- [HCO
3
-
) / [HCO
3
-
] = Kb ([B]
tot
- Sdp + [HCO
3
-
]) / (Sdp - [HCO
3
-
])
Since the gas phase is not present in this system Sdp, [B]
tot
, and [CO
2
]
tot
are all constant, while Kb varies with the oxygenated-deoxygenated state of Hb.
The second degree equation to be solved is:
[HCO
3
-
]
2
(Kc - Kb) - [HCO
3
-
] (Kc [CO
2
]
tot
+ KC Sdp+ Kb [B]
tot
- Kb Sdp) + Kc [CO
2
]
tot
Sdp = 0
from the equation above we calculate [HCO
3
-
] and, from its value, all other variables of the system. The results of the simulation are as follows:
oxygenated state of Hb
stato deoxygenated state of Hb
pK di Hb
6,8
7,05
pH
7.4
7.63
CO
2
1.21 mM
0.73 mM
PCO
2
40 mmHg
24.3 mmHg
HCO
3
-
23.99 mM
24.47 mM
CO
2
totale
25.2 mM
25.2 mM
Δ CO
2
0.48 mEq/L
protonated buffer residues of Hb
12.27 mEq/L
12.75 mEq/L
deprotonated buffer residues of Hb
49.33 mEq/L
48.85 mEq/L
total buffer residues of Hb
61.6 mEq/L
61.6 mEq/L
Sdp
73.32 mEq/L
Questa simulazione
in vitro
dimostra che l'ossigenazione dell'emoglobina causa diminuzione del pH e aumento della PCO
2
, ma non considera il fatto che nel capillare venoso viene rilasciata CO
2
: infatti la desossigenazione dell'emoglobina
in vivo
si associa ad aumento della PCO
2
e della CO
2
totale (che in questa simulazione era mantenuta costante). Si noti che in questa simulazione si e' ipotizzata una soluzione di emoglobina in acqua alla concentrazione di 14 g/dL insieme a 25,2 mmoli/L di anidride carbonica totale; mancano quindi il contributo al potere tampone totale dato dalle proteine plasmatiche e dai fosfati (gli unici tamponi sono Hb e CO
2
) e manca il fattore λ che descrive il rapporto tra il bicarbonato plasmatico e quello totale.
CLINICAL LABORATORY: THE STANDARD PARAMETERS
Since the first blood gas analyses it became obvious that alterations of blood pH may depend on respiratory and non-respiratory causes. PCO
2
was recognized as an index of lung function, but indexes of non-respiratory functions or disease were more difficult to identify. In 1916 Hasselbalch, working in Copenhagen, proposed the firat of the "standard" parameters:
standard pH
. In order to measure the standard parameters, after measuring the arterial blood parameters PCO
2
, HCO
3
-
, and pH, the physician equilibrated the blood sample under
standard conditions
, defined as T=37
o
C, PCO
2
=40 mmHg, PO
2
=140 mmHg and N
2
and water vapour to 1 atm and measured again the same parameters. In practice Hasselbalch considered this procedure as a way to eliminate the lung contribution to pH homeostasis. Whatever abnormality remained could be attributed to non-respiratory causes (either pathological or compensatory).
The first standard parameter, introduced in the clinical use by Hasselbalch himself was
standard pH
.
Standard bicarbonate
, introduced by Astrup and Siggaard-Andersen followed, and finally
base excess
by the same authors.
Standard pH and standard bicarbonate are to be interpreted as titrationf of blood with PCO
2
at constant Sdp (case 1 in the above descrption). Base excess is a more complex parameter and is obtained as follows: first the patient's blood is equilibrated under standard conditions, and standard pH is recorded; next the sample is titrated to pH=7.4, at constant pCO
2
with NaOH or HCl (case 2 in the above description). If pH
standard
> 7.4 the sample is titrated with HCl and the amount of acid required (in mEq/L) is called the
base excess
of the sample. If pH
standard
< 7.4 the sample is titrated with NaOH and the amount of base required (in mEq/L) is called the
base deficit
of the sample (or the sample is said to have a
negative base excess
).
To summarize: pH
standard
> 7.4 = positive base excess; pH
standard
< 7.4 = negative base excess.
The table below reports examples of the use of standard parameters (
caution: I did not yet check the accuracy of the calculations! Take it as a qualitative example
):
Example values of standard parameters in health and disease
diagnosis
values prior to any manipulation
standard values
pH
pCO
2
[HCO
3
-
]
pH
pCO
2
[HCO
3
-
]
BE
healthy
7.4
40 mmHg
24 mM
7.4
40 mmHg
24 mM
zero
acute resp. acidosis
7.2
70 mmHg
26 mM
7.4
40 mmHg
24 mM
zero
acute resp. alkalosis
7.5
28 mmHg
21 mM
7.4
40 mmHg
24 mM
zero
chronic resp. acidosis
7.35
70 mmHg
38 mM
7.55
40 mmHg
34 mM
positive (compensatory)
chronic resp. alkalosis
7.48
18 mmHg
13 mM
7.27
40 mmHg
18 mM
negative (compensatory)
metabolic acidosis
7.2
20 mmHg
8 mM
7.06
40 mmHg
11 mM
negative (causative)
metabolic alkalosis
7.47
50 mmHg
35 mM
7.52
40 mmHg
32 mM
positive (causative)