Derivation of Theoretical Equations of the CO2 Dissociation Curve and the Carbamate Fraction in the Haldane Effect

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1 Japanese Journal of Physiology, 33, ,1983 Derivation of Theoretical Equations of the CO2 Dissociation Curve and the Carbamate Fraction in the Haldane Effect Masaji MocHIzuKI, Hirotsugu TAKIWAKI, Tomoko KAGAWA, and Hiroshi TAZAWA Department of Physiology, Yamagata University School of Medicine, Yamagata, Yamagata, Japan Abstract The simultaneous Henderson-Hasselbalch equations in plasma and red cell were solved in order to obtain the C02 dissociation curve of oxygenated blood. In order to solve the above two equations the following equation was added, in which the relationship between the intracellular (4pH) cand the extracellular ph change (dphp) was defined as follows: phc=(1+a)dphp, where 1 +a is a factor to be determined from experimental data on Donnan's ratio for H+. From the solution, the ratio of bicarbonate shift to the C02 quantity released out of or combined with hemoglobin was calculated. The solution was validated by comparing the above ratio between the theoretical and experimental data. The C02 contents calculated at 12 Torr in whole blood, red cell, and plasma compartments show good agreement with the respective analyzed values. When the buffer values of hemoglobin and plasma buffer protein were 70.0 and 7.5 mmol/ (liter plasma ph), respectively, a= dphp, and the Donnan's ratio for HC03- was assumed to be 0.7 at ph=7.33, the theoretical C02 dissociation curve fitted well with the experimental curve. The C02 dissociation curve of deoxygenated blood was expressed by adding the measured Haldane effect to the C02 content of oxygenated blood. This additive characteristic in turn made it possible to estimate carbamate contribution in the Haldane effect. Key Words: C02 dissociation curve, buffer value, Henderson-Hasselbalch equation, Haldane effect, carbamate contribution. In a previous paper (MocuIzuKI et al., 1982) it was clarified that the C02 dissociation curve could be expressed by an exponential function of PC02, whose exponent and multiplying factor had a consistent relation to each other. Therefore, a certain relation such as the Henderson-Hasselbalch equation was Received for publication November 11,

2 580 M. MOCHIZUKI, H. TAKIWAKI, T. KAGAWA, and H. TAZAWA conjectured to exist not only in plasma, but also in the red cell. VAN SLYKE et al. (1923) reported that bicarbonate quantities released out of or combined with hemoglobin and plasma protein with changing Pco2 were given by linear functions of ph, whose gradients, ~c and 8p, were referred to as the buffer values. Furthermore, the difference in ph between the red cell and plasma was given by logarithm of the Donnan's ratio for H+, log r(h+). Therefore, it was thought possible to obtain an approximate relationship between the amount of bicarbonate shift and hematocrit, if the intracellular bicarbonate content could be estimated from the bound C02 and the carbamate concentration measured by previous authors. Taking the above experimental evidences into account, we attempted to obtain the theoretical equations of the C02 dissociation curve of whole blood. The relation between the bicarbonate shift and the hematocrit has already been measured experimentally by us as reported in the preceding paper (TAKIWAKI et al., 1983). It was clarified there that the intracellular ph depended on both the bicarbonate concentration and the amount of alkali (or H+) bound with hemoglobin. In an equilibrium state at a constant Pco2, the amount of alkali is linearly related to the ph, which is increased with increasing bicarbonate concentration. When the Pco2 is decreased, the amount of alkali bound with hemoglobin increases together with the ph. Consequently, the bicarbonate ions are dehydrated by H+ ions released from hemoglobin by alkali ions, and thus, the increased rate of the ph is slowed, attaining an equilibrium state. Hence, an active interest has been taken in deriving theoretically the equations which elucidate the above relation quantitatively. Actually, the intra- and extracellular C02 dissociation curves were separately obtained in oxygenated blood from four simultaneous equations; two Henderson-Hasselbalch equations in the red cell and plasma, an equation relating to the intra- and extracellular ph changes, and the one to evaluate the bicarbonate change from the buffer value and ph change (VAN SLYKE et al., 1923). Then, the ratio of the bicarbonate shift to the change in bound C02 content (A) was calculated from the dissociation curve equations by varying the hematocrit. The above A ratio was fairly compatible with the preceding experimental data (TAKIWAKI et al., 1983). In addition, the C02 dissociation curve was agreeable with the data of MocHIzuKI et al. (1982) and TAZAWA et al. (1983). The relation between the intra- and extracellular ph's used in the above derivation was obtained from the experimental data on the H+-Donnan ratio measured by previous authors (GARBY and MELDON, 1977; VEDA and TYUMA, 1982). The intracellular ph was calculated from the Henderson-Hasselbalch equations using the intracellular bicarbonate concentration, which was evaluated by subtracting carbamate content from the bound C02. The carbamate content of oxygenated blood was assumed to be 0.5 mmol/liter, blood independently of Pco2 according to BAUER and SCHRODER (1972). The intracellular bicarbonate change was calculated from the ph change by using the lc value ranging from 60 to 70 mmol/(liter RBC ph), where the j3p was kept constant at 7.5 mmol/(liter plasma rn ph) (TAKIWAKI Japanese Journal of Physiology

3 THEORETICAL EQUATIONS FOR BOUND CO2 IN BLOOD 581 et al., 1983). The comparison between the theoretical and measured dissociation curves was made by varying the pc value. The j3c in the above range was 5 to 15 mmol/(liter RBC. ph) higher than the value reported by SIGGAARD-ANDERSEN (1974), but was compatible with the data of VAN SLYKE et al. (1923), suggesting pertinency of the theoretical equations. The C02 content in the deoxygenated blood is higher than in the oxygenated blood by the Haldane effect. In general, when the Pc02 changes, the ph rise (or fall) occurs with the fall (or rise) of bicarbonate concentration. However, when the deoxygenation proceeds at a constant Pc02, the ph rises in parallel with the bicarbonate concentration. Thus the C02 dissociation curve of the deoxygenated blood was expressed by adding the measured Haldane effect to that of the oxygenated blood. Putting the measured plasma and red cell components of the Haldane effect into the equations of the Haldane effect, the carbamate and bicarbonate contributions in the Haldane effect were estimated. The obtained carbamate fraction was about 10% smaller than the value of ROSSI-BERNARDI and ROUGHTON (1967), but about 10% higher than that of BAUER and SCHRODER (1972). In the present paper, the derivation of the equations for the C02 dissociation curve and its compatibility with the experimental data will be described. Furthermore, the derivation of the equations for the carbamate content in the Haldane effect will be described together with the experimental support to the calculated value. SYMBOLS AND NOTATIONS Cb : C02 content in whole blood (mmol/liter blood, or vol %) Cc : C02 content in the red cell (vol %) Cp : C02 content in the plasma (vol %) (C)c:intracellular bound C02 (mmol/liter blood) [HC03-]c(x): intracellular bicarbonate concentration at PCO2=x Torr (mmol/liter H2O) [HC03-]p(x): extracellular bicarbonate concentration at PCO2=x Torr (mmol/liter H2O) (HC03-)c(x): intracellular bicarbonate content at Pco2=x Torr (mmol/ liter blood) (HC03-)p(x): extracellular bicarbonate content at Pco2=x Torr (mmol/ liter blood) HE: Haldane effect (mmol/liter blood) HEp : plasma component of Haldane effect (mmol/liter blood) Ht(x) : hematocrit at Pco 2 =x Torr 4Ht=Ht(P0)-Ht(x), or JHt= Ht(ox)- Ht(deox) Po: initial Pc02 phc(x) : intracellular ph at P02 =x Torr Vol. 33, No. 4, 1983

4 582 M. MOCHIZUKI, H. TAKIWAKI, T. KAGAWA, and H. TAZAWA php(x): extracellular ph at Pco2=x Torr 4pHc=pHc(P0)-pHc(x), or dphc=phc(ox)-phc(deox) dphp=php(p0)-php(x), or dphp=php(ox)-php(deox) d pk'=pk'(p0)-pk'(x) R : carbamate fraction in the Haldane effect (mmol/liter blood) Wc(x): intracellular water concentration at Pco2=x Torr (kg/liter RBC) Wp(x) : extracellular water concentration at PC o 2 =x Torr (kg/liter plasma) J Wc= Wc(P0)- Wc(x) d Wp= Wp(P0)- Wp(x) k= dpk'/dphp x : any Pc o 2 (Torr) ac: C02 solubility in the red cell (mmol/(liter RBC Torr)) ap : C02 solubility in the plasma (mmol/(liter plasma Torr)) 9c: buffer value of hemoglobin (mmol/(liter RBC ph)) pp : buffer value of plasma protein (mmol/(liter plasma ph)) A : ratio of bicarbonate shift to the change in intracellular bound C02 A': A value in the Haldane effect a=(4 phc-dphp)/dphp The parameters to be put in and derived from the dissociation curve equations and Haldane effect equations are summarized in Table 1, A and B, respectively. The equations in Table 1 were used for calculating the parameters to be derived. Table 1. A. Parameters to be given and derived from the dissociation curve equations. Japanese Journal of Physiology

5 THEORETICAL EQUATIONS FOR BOUND CO2 IN BLOOD 583 DERIVATION OF THEORETICAL EQUATIONS FOR CO2 DISSOCIATION CURVE The theoretical equations of the C02 dissociation curve was derived so as to be able to obtain the C02 content at varying Pco2 by putting the bound C02 contents in plasma and red cells measured at a certain Pco2. The change in C02 content was calculated by referring to the buffer value and ph change. Thus, the theoretical equations were composed of the two Henderson-Hasselbalch equations in the red cell and plasma, an equation relating the change in intracellular ph to that in extracellular ph, and the one to evaluate the bicarbonate content change from the ph change using the buffer values of hemoglobin and plasma protein. Denoting the initial Pco2 by Po Torr, the Henderson-Hasselbalch equations at Po in both the red cell and plasma are given by, and phc(p0)=pk'c(p0)+log [HC03-]c(P0)-log ac-log P0, (1) php(p0)=pk'p(p0)---log [HC03-]p(P0)-log ap-log P0, (2) where ac and ap are C02 solubilities, and [HC03-]c and [HC03-]p are bicarbonate concentrations in mmol/liter water in the red cells and plasma, respectively. Similarly, when Pco2=x (Torr), the same equations are given by, and phc(x)=pk'c(x)h-log [HC03-]c(x)-log ac-log x, (3) php(x)=pk'p(x)-log [HC03-]p(x)-log ap-log x, (4) respectively. Subtracting Eq. (3) from Eq. (1), the difference in log Pco2 between x and Po is derived as follows: to g P x = 4 H- d K'c~ to [HC03~]c(x) 5 o p c p g HC03- c Po ~ () where dphc=ph0(p0)-phc(x), and dpk'c=pk'c(p0)-pk'c(x). Similarly, subtracting Eq. (4) from Eq. (2), the following equation is obtained: to g Px = d H-d K' -~- to [HC03~lp(x) 6 o p P p p g [ HC03 ]p( Po ) ' where dphp is the difference in ph, and dpk'p is that in pk'p between the two Pco2's of F0 and x. In order to express the bicarbonate concentrations in water by using the bicarbonate contents per unit volume of blood and the buffer values the following treatment was made. Using the buffer values, j3c (mmol/(liter RBC ph)) and j3p (mmol/(liter plasma ph)), the changes in bound C02 buffered by hemoglobin and plasma protein will be given by, j3c dphc and /3p dphp, respectively. Let the bicarbonate content in the red cell compartment per 1 liter of blood at Po be (HC03-)c (P0), the water concentration in 1 liter RBC at Po and x be Wc(P0) and Wc(x), and the difference between Wc(P0) and Wc(x), d Wc, and the hematocrit at Vol. 33, No. 4, 1983

6 584 M. MOCHIZUKI, H. TAKIWAKI, T. KAGAWA, and H. TAZAWA Po and x be Ht (Po) and Ht(x), respectively, and the difference in hematocrit, Ht (P0)-Ht(x)=4Ht. Substituting the intracellular HCO3- content, (HC03-)c for the bicarbonate HC03- concentration, [HC03!]c of Eq. (5), the ratio of the intracellular bicarbonate at Pco2=x to that at Po is given by [HCO3 [HCO3 ic(x) ( 1 +Ht(x) (1-A -)j3c 4pHc \ Wc(P0). Ht(P0) ' ( 7 jc(po) = (HC03)c(Po We(x) Ht(x) ) where A is the ratio of the shifted bicarbonate quantity to the j3c 4pHc. Let the HC03- content in the plasma of 1 liter blood at Po be (HC03-)p(P0), the water concentrations in 1 liter plasma at Po and x be Wp (Po) and Wp(x), and the difference between them, 4 Wp. Then, the similar ratio of the extracellular bicarbonate concentration is obtained as follows : [HC03-]p(x) = 1+(1-Ht(x))fip' dphp-~- Ht(x)Ac' 4pHc [HCO3-]P(P0) (HC03-)p(P0) Wp(P0), l -Ht(Po) (8) Wp(x) 1-Ht(x) From the definitions of 4 We and 4 Wp, the intra- and extracellular water concentrations at Po, Wc(P0) and Wp(P0) are written as: Wc(P0)= Wc(x)+4 Wc, and Wp(P0)= Wp(x)-4Wp. (9) Furthermore, since 4Ht= Wc(P0)Ht(P0)- Wc(x)Ht(x) = Wp(x)(1-Ht(x))- Wp(Po)(1--Ht(P0)), the following relations are derived between 4 Wc, 4 Wp, and 4Ht, 4 W - (1- Wc(Po))4Ht Ht( and 4 W J(1- Wp(Po))4Ht. 10 x) p 1- Ht(x) ( ) On the other hand, Wc(P0)Ht(P0)/ Wc(x)Ht(x) and Wp(PQ)(1- Ht(P0))/ Wp(x)(1 - Ht(x)) in Eqs. (7) and (8) are given by 1+ 4 Ht/ Wc(x)Ht(x) and Ht/ Wp(x) (1-Ht(x)), respectively. Therefore, putting the equations from (7) to (10) into Eqs. (5) and (6), the difference in log Pco2 between Po and x are given by the following equations: and log x =4pHc-4pK'c+log 1+ 4Ht Po x log P a Wc(x) Ht(x) =4pH P -4pK'p+l0g1-- 4Ht Wp(x)(1-Ht(x)) + Ht(x)(1-A) 8c 4pHc (HCO3~)c(Po) (1- Ht(x))~p 4 php + Ht(x)A 4pHc (HC03-)p(P0) Since logarithm of the Donnan ratio of H~, log r(h+) ). (11) (12) is equal to the difference Japanese Journal of Physiology

7 THEORETICAL EQUATIONS FOR BOUND CO2 IN BLOOD 585 between the intra- and extracellular ph, ph-ph, and dphp is given by the relation between dphc log r(h+)(pho)-log r(h+)=41og r(h+)=dphc-dphp, (13) where pho is ph P at P02 =P0, dph is the difference between pho and ph P at any Pco2, and dphc, the intracellular ph difference corresponding to dphp. In general, using the measured relation between r(h) and ph, the zllog r(h) is given experimentally by Thus, from Eqs. (13) and (14) the dphc is given by dlog r(h+)=q dphp. (14) dphc=(1+o)4php. (15) In Fig. 1 are shown the measured log r(h) and q-values illustrated against ph. The closed circles with bars and the open circles with cross bars are the averages and S. D. of log r(h) values obtained from the data of previous authors (GARBY and MELDON, 1977) and USDA and TYUMA (1982), respectively. The latter log r(h) values were obtained in blood with three Pco2 of 10, 40, and 80 Torr. Since the latter measuring conditions were closer to those used to obtain the C02 dissoci- Fig. 1. Logarithm of the Donnan ratio for H~ ion and the 6 value used in the theoretical equations. The closed circles with bars and the open circles with cross bars are the averages and S.D. of log r(h) values obtained from the data of previous authors (GARBY and MELDON, 1977) and UEDA and TYUMA (1982), respectively. The latter log r(hp) values were obtained in blood with three different Pco2's of 10, 40, and 80 Torr. The a value was calculated from the latter log r(h) value. Vol. 33, No. 4, 1983

8 586 M. MOCHIZUKI, H. TAKIWAKI, T. KAGAWA, and H. TAZAWA ation curve, the a-value of Eqs. (13) and (14) was evaluated from the latter log r(h) values by putting ph,=7.4, as shown by the chain line in Fig. 1. The a-value thereof is given by a= dphp. (16) According to SEVERINGHAUS et al. (1956), dpk'p is given by d pk'p=-k 4pHP= dphp. (17) As for the d pk'c no previous data were available. Thus in the present study the similar relation to Eq. (17) was assumed to hold for the d pk'c. Putting the relations of Eqs. (15), (16), and (17) into Eqs. (11) and (12), 4pHc is eliminated and the following simultaneous equations are derived for obtaining the relations between dphp, A, Ht, and x: and } )Ht +Ht(x)(1 +a)(1-a)i3c JPHP P tog x o =(1 +a+k) dphp+l0g1- W e x Ht x (HC03-)c Po ' x --=(1 -{--k d H ~- l0 1- dht log Po ) p P g W ('x 1-Ht x (18) +(1-Ht(x))~9p dphp+ht(x)(1+a) (HC0 3-):p(P0) A ~3c dphp 19 In the above equations the contributions of Ht(x), Wc(x), and Wp(x) to ph P and A values are fairly small, and the error caused by using the average of the previous author's data is permissible. Thus, when the Ht, 4Ht, and the bicarbonate contents in the red cell and plasma at Po are known besides BSc and 3p, the A and dphp values at x are calculated from Eqs. (18) and (19) without measuring the We and Wp in individual subjects. Using the A and dphp values thus obtained, (C02)c(x) and (HC03-)p(x) are evaluated as follows: and (C02)c(x)=(C02)c(Po)+Ht(x)(1-F-a)(1-A)Rc dphp, (20) (HC03-)p(x)=(HCO3-)p(P0)+{(1-Ht(x))j3p +Ht(x)(1+a) A j3c} dphp. (21) Thus, the CO2 content in the whole blood is obtained simply by adding Eqs. (20) and (21) together with physically dissolved CO2 which is given by multiplying [Ht(x)ac+(1-Ht(x))ap] by x. When the bound CO2 contents in the plasma and red cell at Pco2=Po are known, the intra- and extracellular bound CO2 at any Pco2 are derived from Eqs. (20) and (21) by using the hematocrit at the same Pco2 and the a and buffer values measured by previous authors. Japanese Journal of Physiology

9 THEORETICAL EQUATIONS FOR BOUND CO2 IN BLOOD 587 EQUATIONS FOR CARBAMATE CONTRIBUTION IN THE HALDANE EFFECT The change in bound CO2 due to the Haldane effect basically depends on the 02 saturation(so2). It is unlikely that the above CO2 change occurs as the result of the change of the buffer value which is caused primarily by the S02 change. This is because.(1) in order to fit the calculated dissociation curve of the deoxygenated blood to the experimental value, the jlc value should be smaller than that used in the oxygenated blood, and (2) the phc increases as the deoxygenation proceeds, so that a negative buffer value becomes necessary to explain the bicarbonate increase during the deoxygenation. Eventually, the CO2 content of the deoxygenated blood was given by adding the Haldane effect to that of the oxygenated blood as follows : Cb(deox, x)=cb(ox, x)+he, (22) where Cb is the CO2 content in the whole blood, and HE, the change in bound CO2 due to the Haldane effect per liter blood. One of the advantages of separating the Haldane effect from the usual hydration and dehydration reaction of CO2 seems to consist in making possible the analysis of the Haldane effect itself. Even though the Haldane effect is not primarily determined by the ph change, the ph's in the red cell and plasma receive the influence of the Haldane effect. Thus, the similar equations to Eqs. (18) and (19) are derived in respect to the Haldane effect as follows : and -dphc= -phc(ox)+phc(deox), -l0 g 1+ 4Ht (1-2')(HE-R) 23 We Ht HC03- -c ox) ' ( ) -4pHP=-pHP(ox)+pHP(deox), 10 1 _ AHt 2'(HE-R) - = g Wp(1- Ht) + (HCO3 )p( ox) ' (24) where Wc, Wp, and Ht are the respective values in the deoxygenated blood at Pco2 -x, 4Ht is the difference in Ht, or Ht(ox)-Ht(deox), and R, the carbamate content in 1 liter blood and in order to distinguish the 2 value in the Haldane effect from that in the PC02 change, 2' was used in the Haldane effect. Since the change in ph due to the Haldane effect is as small as about 0.04 (MocHizuKi et a!.,1982), the 4pK' of Eq. (16) is neglected in Eqs. (23) and (24). When the plasma component of the Haldane effect, HEp, is known as given by HEp=2'(HE-R), (25) the 2' and R values are evaluated from Eqs. (23), (24), and (25) together with Eqs. (15) and (16). Vol. 33, No. 4, 1983

10 588 M. MOCHIZUKI, H. TAKIWAKI, T. KAGAWA, and H. TAZAWA EVALUATIONS OF A IN Pc02 CHANGE, THE BUFFER VALUE AND DISSOCIATION CURVE First, the A value calculated from Eqs. (18) and (19) was compared with the experimental value of TAKIWAKI et al. (1983) for obtaining the reliable j9c value. All the parameter values necessary for the calculations were assumed as follows: the buffer value of separated plasma was 7.5 mmol/(liter plasma ph) (TAKIWAKI et al., 1983). The average values of the initial PC02, the initial bound C02 in the red cell and plasma were 57.1 Torr, mmol/liter RBC, and mmol/ liter plasma, respectively (TAKIWAKI et a!.,1983). Since neither the 02 saturation nor the carbamate content were measured in the initial blood samples, the intracellular bicarbonate concentration was calculated by referring to the Donnan ratio for HC03-, r(hc03-), which was assumed to be 0.7 (FiTZSIMONS and SENDROY, 1961; TAZAWA et al., 1983). The carbamate content of oxygenated blood was assumed to be 0.5 mmol/liter blood referring to the calculated data of TAZAWA et al. (1983), and it was kept constant throughout a PC02 range of 12 to 60 Torr according to FERGUSON and ROUGHTON (1934) and STADIE and O'BRIEN (1937). Consequently, the average intra- and extracellular bicarbonate contents at 57.1 Torr PC02 were calculated to be mmol/liter RBC and to be mmol/ liter plasma. The intracellular bound C02 at 12 Torr was evaluated by adding 0.5 mmol/liter blood to the calculated HC03- content in the red cell. As for the reliability of using such carbamate contents, a detailed discussion will be made in the following paper (TAZAWA et al., 1983). For the water concentrations in the red cell and plasma, the experimental data obtained by TAZAWA et al. (1983) were applied. That is, and Wc(x)=(0.714 Ht(57. 1)+ JHt)/Ht(x), Fig. 2. Relation between the hematocrit and the average difference in hematocrit between Po and x, (4Ht) measured by TAKIWAKI et al. (1983). Po and x were 57.1 and 12.0 Torr, respectively. Japanese Journal of Physiology

11 THEORETICAL EQUATIONS FOR BOUND C02 IN BLOOD 589 Wp=(0.939 (1-Ht(57.1)-4Ht)/(1-Ht(x)), whereas for JHt=Ht(57.1)-Ht(x), the measured values illustrated in Fig. 2 were used. By processing Eqs. (18) and (19) by using the above parameters together with the a value of Eq. (16) and Ht=0.455, we obtained the relationships between the,3c value, the A and the blood C02 content at 12 Torr Pco2, Cb(12), as illustrated in Fig. 3. The A value increases along with the /3c value, whereas the Cb(12) decreases. In Takiwaki's paper the measured A value at Ht=0.455 was 0.565, on an average, and its S. D. was 0.023, moreover, the measured Cb(12)was 12.5 mmol/liter blood, on an average, and its S. D. was 0.66 mmol/liter blood. Therefore, from the relation of Fig. 3, the 0c value was approximated to be about 70 mmol/(liter RBC ph). When the above j3c value and the a value of Eq. (16) were used, the A value agreed fairly well with the experimental values at the hematocrit of about 0.45 and 0.74, but deviated downwards at the hematocrit of about 0.23, as shown in Fig. 4. The plotted points are the experimental values obtained by TAKIWAKI et al. (1983), and the solid line was calculated from Eqs. (18) and (19). The buffer value of 70 mmol/(liter RBC ph) is certainly higher than the value measured by SIGGAARD-ANDERSEN (1974) at Pco2=0 Torr, but is fairly compatible with the value obtained by VAN SLYKE et al. (1923). Next, we calculated approximately the C02 content of oxygenated blood when the Pco2 was changed from the venous levels to 12 Torr at three different hematocrit Fig. 3. The calculated A and Cb(12) values plotted against the buffer value of hemoglobin where Ht= The A value is the ratio of bicarbonate shift to the released C02 quantity out of hemoglobin. Vol. 33, No. 4, 1983

12 590 M. MOCHIZUKI, H. TAKIWAKI, T. KAGAWA, and H. TAZAWA Fig. 4. Relation between the hematocrit and the 2 ratio. The plotted points were obtained from the data of TAKIWAKI et al., and the broken line was calculated from Eqs. (18) and (19), by using 4Ht of Fig. 2. j3c=70 mmol/(liter RBC ph) and the 6 value of Eq. (16). Fig. 5. Relation between the measured and calculated CO2 contents in blood at 12 Torr. The calculation was made by using the measured intra- and extracellular bicarbonate contents at the venous PCO2 level, and j3c=70 mmol/(liter RBC ph). Japanese Journal of Physiology

13 THEORETICAL EQUATIONS FOR BOUND CO2 IN BLOOD 591 values, using Eqs. (18) and (19). The acid-base status of the initial bloods was cited from the experimental data of the 8 individual subjects measured by TAKIWAKI et al. (1983). The parameter values such as 4Ht, a, 5p, and j3c were assumed to be those as used in Figs. 3 and 4. The carbamate fraction of the Haldane effect was estimated to be 1.2 mmol/liter blood (TAZAWA et al., 1983), therefore, that of the venous blood whose 02 saturation was approximated to be about 50 % was presumed to be 1.1 mmol/liter blood, of which 0.5 mmol/liter blood was thought to remain in the red cell, even when the cell was oxygenated. Figure 5 shows the correlation between the calculated and the actually measured C02 contents at P02= 12 Torr and at three different Ht (TAKIwAKI et al., 1983). The chain line is the regression line which is given by Cb(calc)=0.933 Cb(meas)+0.568, where the correlation coefficient is Likewise, calculations of (COZ)c and (HC03)p -were made, which were also compared with the measured ones (TAKIWAKI et al., 1983) (Fig. 6). The calculated bound C02 in the cell was about 0.5 mmol/ liter blood smaller than the measured value, and the regression line was given by (C02)c(calc)=1.078 (C02)c(meas) (mmol/liter blood). The calculated extracellularhc03-content The regression line was given by agreed well with the meausred content. Fig. 6. Calculated intracellular bound CO2 (closed circle) and extracellular bicarbonate (open circle) plotted against the measured values. The same parameter values were used as those in the calculation of Fig. 5. Vol. 33, No. 4, 1983

14 592 M. MOCHIZUKI, H. TAKIWAKI, T. KAGAWA, and H. TAZAWA (HC03-)p(calc)=0.974 (HC03-)p(meas) (mmol/liter blood). The correlation coefficients were and 0.996, respectively. In the previous paper (MocHlzuKl et al., 1982) the CO2 dissociation curve of the normal blood was represented by the following experimental equation: Cb=1.15 B^ Pcc,'2 (vol %), (26) where B is a constant specific to the sampled blood. In Eq. (26), a curvilinear relationship exists between the B value and CO2 content at PC02=40Torr, Cb(40), as shown by the lower curve in Fig. 7. On the other hand, in the simultaneous Eqs. (18) and (19), the curvature of the dissociation curve is mainly determined by the buffer value of hemoglobin, Rc. Thus, we calculated the relation between the j9c and Cb(40) values. On calculation, we assumed that 1) the a value be expressed by Eq. (16), 2) the ratio, (C02)c(40)/(HCO3-)p(40), be kept constant at 0.45 in a Cb(40) range of 43 to 48 vol % (MocHlzuKl et al., 1982), and 3) the ph dependency of the hematocrit be given according to TAZAWA et al. (1983) by Ht(x)=Ht(40) dphp. (27) The result is shown by the upper curve of Fig. 7. The ~9c was distributed in a range of 60 to 70 mmol/(liter RBC ph). The typical calculated CO2 dissociation curves in the whole blood, the red cell and plasma are illustrated in Fig. 8. The parameter values used in the calcu- Fig. 7. Relation between the B value of Eq. (26), the j3c satisfying the dissociation curve o f Eq. (26) and the CO2 content at PC02 of 40 Torr. Japanese Journal of Physiology

15 THEORETICAL EQUATIONS FOR BOUND CO2 IN BLOOD 593 Fig. 8. Typical CO2 dissociation curve of whole blood (Cb), and its plasma (Cp) and red cell (Cc) components. lation are as follows : Cb(40)=44.0 vol %, 13c=65.7 mmol/(liter RBC ph), (HC03-)c=5.23 mmol/liter blood,,3p=7.5 mmol/(liter RBC ph), R=0.5 mmol/liter blood, a= dphp, (HC03-)p=12.85 mmol/liter blood. The above carbamate fraction, R, was referred to the experimental data of BAUER and SCHRODER (1972) and TAZAWA et al. (1983). Assuming Cb(40) to be 44 vol %, (HC03-)c and (HC03-)p and i9c were determined as above by use of (C02)c/ (C02)p=0.45 and the relation of Fig. 7. The CO2 dissociation curve of this standard blood was given by the following equation, which was approximately similar to Eq. (26) : Cb=1.07 B-2.54s Pco2B (vol %). CARBAMATE CONTRIBUTION AND THE A' VALUE IN THE HALDANE EFFECT For analyzing the Haldane effect, the following two CO2 dissociation curve equations were used for the oxygenated and deoxygenated bloods: and Ox.: Cb=3.14 x Pco2.49 (mmol/liter blood), Deox.: Cb=4.26 x P (mmol/liter blood). The upper curve of Fig. 9 is the difference between the above two equations, that is, the Haldane effect. It gradually increases with Pco2. Vol. 33, No. 4, 1983

16 594 M. MOCHIZUKI, H. TAKIWAKI, T. KAGAWA, and H. TAZAWA The plasma component of the Haldane effect, HEp, has been evaluated also from the previous data (MocHizuiu et al., :1982). Between log Pco2 and ph there were the following relations in the oxygenated and deoxygenated bloods, respectively : and Ox.: log Pco2=2.037-D(pH-7.085), Deox.: log Pco2=2.144-D(pH-7.045). Furthermore, between [HC03-]p and ph there were the following relations : and Ox.: log [HC03-]p=1.531-F(pH-7.011), Deox.: log [HC03-]p=1.532-F(pH-7.116). The average ph and [HCO3-]pat Pco2=40Torr in the normal blood measured by TAZAWA et al. (1983) were and 7.427, and and mmol/liter plasma, in the oxygenated and deoxygenated bloods, respectively. From the above data it follows that in the oxygenated blood, D=1.48 and F=0.434, and in the deoxygenated blood, D=1.42 and F= Putting these values in the above equations [HC03-]p can be evaluated along the Pco2. Moreover, subtracting [HC03-]p of the oxygenated blood from that of the deoxygenated blood, the plasma component of the Haldane effect, HEp was obtained. The calculated HEp is illustrated by the second upper curve of Fig. 9, which increases gradually with Pco2 Fig. 9. Relation between the calculated carbamate fraction and the P02. HE and HEp are the Haldane effect and its plasma component used in the calculation. Japanese Journal of Physiology

17 THEORETICAL EQUATIONS FOR BOUND CO2 IN BLOOD 595 Fig. 10. Relation between the A' and R/HE ratios and the Pco2 obtained from the data of Fig. 9. (HC03-)c(ox) and (HC03-)p(ox) needed in calculations of Eqs. (23) and (24) were evaluated along the Pco2 from the plasma and red cell components of the dissociation curve shown in Fig. 8. The intracellular bicarbonate content was evaluated by subtracting the carbamate content from the bound C02, where the carbamate fraction in the oxygenated blood was assumed to be 40 % of that of the Haldane effect. Since the latter carbamate was obtained after assuming the former carbamate, an iterative method was adopted to increase the accuracy. From the relations between log Pco2 and ph, in both the oxygenated and deoxygenated bloods, dphp=ph(ox)-ph(deox), was calculated. As for the remaining parameters the following values were used: JHt=-0.004, We=0.715 kg/liter RBC and Wp=0.939 kg/liter plasma, and Q= The calculated carbamate fraction is shown by the lower curve of Fig. 9. The A value was constant at about 0.7, as shown in Fig. 10, in addition, that was fairly high compared with that observed when the Pco2 was changed. The carbamate showed a slow peak in a Pco2 range of 30 to 50 Torr, as reported by FERGUSON and ROUGHTON (1934) and STADIE and O'BRIEN (1937). The ratio of the carbamate to the total Haldane effect was about 0.45 and a little lower than that reported by ROSSI-BERNARDI and ROUGHTON (1967) and a little higher than that reported by BAUER and SCHRODER (1972). DISCUSSION In the preceding experiment (TAKIWAKI et al., 1983) the A value was measured by reducing the Pco2 of venous blood from its initial level to 12 Torr. Oxygenation procedure in prior to the Pco2 reduction was omitted so as to prevent the sample Vol. 33, No.4,1983

18 596 M. MOCHIZUKI, H. TAKIWAKI, T. KAGAWA, and H. TAZAWA from hemolysis by frequent tonometry. Consequently, the C02 release was brought about by the increase in 02 saturation as well as the Pco2 reduction. The amount of C02 released out of the red cell was, therefore, partly derived from the carbamate fraction in the Haldane effect. Since the carbamate is mainly bound with hemoglobin and released upon the oxygenation, the A value is thought to decrease, when the oxygenation occurs together with the outward C02 diffusion. The A' value in the Haldane effect (Fig. 10) is greater than the 2 in the Pco2 reduction (Fig. 4). This means that when the C02 release out of the red cell is increased by the oxygenation, HC03- ions in the plasma enter into the red cell at the high ~' ratio. Namely, in the Haldane effect, the C02 release out of carbamate fraction and the inward HC03- shift with the high A' ratio occur simultaneously, and exert opposite effects to the total 2 ratio. Therefore, the 2 value measured by decreasing Pco2 will not receive a great influence of the oxygenation of the red cell. When the 02 saturation of the venous blood is roughly estimated to be 50%, the amount of carbamate fraction becomes about 0.6 mmol/liter blood or so (Fig. 9), and it is ca. 40% of the total Haldane effect (Fig. 10). Taking the samll amount of carbamate and the high 2' ratio into account, the influence of the carbamate release to 2 ratio may be disregarded in the experimental data of TAKIWAKI et al. (1983). That is, their data may be accurate enough to validate the 2 and fic values used in the present paper. The hematocrit change, 4Ht, observed by changing Pco2 is given from Eq. (27) as a function of dph by 4Ht= pHP. (28) By using Eq. (28) together witn Wc=0.715, Wp=0.939, and Ht=0.455, the following numerical relation is derived : Consequently, if it is assumed that, dht + dht We Ht Wp(1- Ht) pHP. dphc=4php, or, a=0, (29) The following equation can easily be derived from Eqs. (18) and (19): (1-2) 48c _ Rp+2 j9c Ht/(1-Ht) (HC0 3r)c/Ht (HC03- c 1-Ht) ( ) Thus, when the numerical values of (HC03-)c/Ht, (HC03-)p/(1-Ht), Ht, and j3c are put in the above equation, the 2 value for a==0 is easily obtained. In the preceding paper (TAKIwAKI et al., 1983), the,2 value at Ht=0.45 was distributed in a range of 0.53 to Similar 2 values are calculated by using the pc value in a range of 52 to 60 mmol/(liter RBC ph), when the same parameter values as those in Figs. 3 to 5 are used together with a=0. SIGGAARD-ANDERSEN (1974) reported that the j3c value of oxygenated hemoglobin, which was measured in red cell Japanese Journal of Physiology

19 THEORETICAL EQUATIONS FOR BOUND CO2 IN BLOOD 597 suspension with no C02, decreased from 68 to 37 mmol/(liter RBC ph), as the ph increased from 7 to 8. The /3c for a=0, thus, is fairly close to the Siggaard-Andersen's data. In addition, the C02 contents calculated from Eqs. (18) and (30) along the PC02 were also fairly compatible with the measured values. When a=o, on the other hand, the simultaneous equations for the Haldane effect are simplified, yielding (1-A)(HE-R) - A(HE-R) -- 4Ht _ 4Ht 31 (HC03~)c(ox) (HC03-)p(ox) We Ht Wp(1 -- Ht)' ) The carbamate fraction at a=0 can be calculated from Eqs. (25) and (31) by using the same parameter values as those of Fig. 9, and it was found to be only 0.1 mmol/ liter blood smaller than the value at Q= dphp, as shown in Fig. 9. Hitherto, except for the Donnan ratio for bound C02, no significant differences have been observed between a= dphp and a=o. The similar C02 dissociation curves to the typical curves in Fig. 7 were calculated, by using a=0,,3c=54 mmol/(liter RBC ph), but by leaving other parameter values unchanged. The Donnan ratio for the bound C02 was then calculated from the C02 contents in the red cell and plasma of the above dissociation curve by referring to the respective water concentrations. An example of the calculated ratio is shown in Fig. 11. When a= ' dphp, the Donnan ratio decreases as the ph is increased. However, when a=o, the slope of it becomes significantly flatter as shown by the broken line. The chain line in the figure was cited from the data of FITzsIMoNs and SENDROY (1961), which was fairly compatible with the ratio obtained by a= ' 4pHP. Fig. 11. Donnan ratio for the bound CO2 calculated when 6 =0 (closed circle) and a = ' dphp (open circle) illustrated against the ph in plasma. The chain line was cited from the experimental data of Fitzsimons and Sendroy. Vol. 33, No. 4, 1983

20 598 M. MOCHIZUKI, H., TAKIWAKI, T. KAGAWA, and H. TAZAWA The 9c value used in the calculation of Fig. 8 was 65.7 mmol/(liter RBC ph), which is certainly higher than the Siggaard-Andersen's value. But it is compatible with the value measured by VAN SLYKE et al. (1923) in the presence of CO2. When the water shift is disregarded, the buffer value of the true plasma is expressed from Eq. (21) by, 1-Ht d(hc03-)p d HP - ~p.~( 1 +Q)A 8c 1-Ht ' (32) p Thus, the validity of the j3c and A values can be appreciated to some extent by comparing the calculated and measured plasma buffer values. When Qc=68 mmol/ (liter RBC ph), j9p=7.5 mmol/(liter plasma ph), Ht=0.45, and a=-0.21, the A ratio is evaluated from Fig. 3 to be about Putting these values into Eq. (32), the plasma buffer value is calculated to be 31.6 mmol/(liter plasma. ph), which is fairly compatible with the experimental data thus far obtained by many authors. In the calculation of CO2 contents in Figs. 5 and 6, the assumed carbamate in the Haldane effect of the venous blood was constantly 0.6 mmol/liter blood, and that in the oxygenated blood, 0.5 mmol/liter blood. Despite such a rough approximation, the CO2 contents at 12 Torr Pco2 calculated from the venous level at three different hematocrits agreed fairly well with the measured ones (Fig. 5). This agreement seems to support the validity of the simultaneous equations of (16), (18), and (19). As for the carbamate contribution to the Haldane effect, it has hitherto been almost impossible to evaluate it, even when the plasma and red cell components of the Haldane effect are known. In the present study, the additive characteristic given by Eq. (22) demonstrated the possibility of evaluating the carbamate content. The Haldane effect and its plasma component were calculated using the data of Pco2, ph, and Cco2 in oxygenated and deoxygenated bloods which have been measured by MOcHIzuKI et al. (1982). As shown in Figs. 9 and 10, the carbamate content and its contribution to the Haldane effect were fairly identical with the recent data obtained by TAZAWA et al. (1983). FERGUSON and ROUGHTON (1934), STADIE and O'BRIEN (1937), FERGUSON (1936), and ROUGHTON (1964) reported that the carbamate contribution was about 70 % of the Haldane effect. However, later ROSSI-BERNARDI and ROUGHTON (1967) reported the lesser fraction than the previous values, or 55 % of the Haldane effect. The carbamate fraction obtained in the present analysis was in a range of 40 to 50 %, and was still lower than the data of Rossi Bernardi and Roughton. BAUER and SCHRODER (1972) assumed that the carbamate fraction measured in hemoglobin solution would be higher than that meausred in the red cell becuase of low DPG concentration in the former solution. The intracellular carbamate estimated by them was about 33 % of the total Haldane effect. They estimated the intracellular bicarbonate content from the extracellular content, by using the Donnan ratio for C1 ion in both the oxygenated and deoxygenated bloods, and then calculated the carbamate fraction. Japanese Journal of Physiology

21 THEORETICAL EQUATIONS FOR BOUND CO2 IN BLOOD 599 According to Fn'zslMoNs and SENDROY (1961) the Donnan ratio forhc03- behaves differently from that of C1`, when the blood is deoxygenated. Therefore, the assumption; r(hc03-)=r(cl`), seems ambiguous, the details of which will be discussed in the following paper. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. REFERENCES BAUER, C. and SCHRODER, E. (1972) Carbamino compounds of hemoglobin in human adult and fetal blood. J. Physiol. (Loud.), 227: FERGUSON, J. K. W. (1936) Carbamino compounds of CO2 with human hemoglobin and their role in the transport of CO2. J. Physiol. (Loud.), 88: FERGUSON, J. K. W, and ROUGHTON, F. J. W. (1934) The chemical relationships and physiological importance of carbamino compounds of CO2 with hemoglobin. J. Physiol. (Lond.), 83: FITZSIMONS, E. J. and SENDROY, J., Jr. (1961) Distribution of electrolytes in human blood. J. Biol. Chem., 236: GARBY, L. and MELDON, J. (1977) The Respiratory Function of Blood, Plenum Medical Book Co., New York and London, pp MOCHIZUKI, M., TAZAWA, H., and TAMURA, M. (1982) Mathematical formulation of CO2 dissociation curve and buffer line of human blood at rest. Jpn. J. Physiol., 32: ROSSI-BERNARDI, L, and ROUGHTON, F. J. W. (1967) The specific influence of carbon dioxide and carbamate compounds on the buffer power and Bohr effects in human hemoglobin solutions. J. Physiol. (Lond.),189: RoUGHTON, F. J. W. (1964) Transport of oxygen and carbon dioxide. In: Handbook of Physiology. Respiration 1, Am. Physiol. Soc., Washington, D. C., Ch. 31, pp SEVERINGHAUS, J. W., STUPFEL, M., and BRADLEY, A. (1956) Variations of serum carbonic acid pk' with ph and temperature. J. App!. Physiol., 9: SIGGAARD-ANDERSEN, 0. (1974) The Acid-Base Status of the Blood, Munksgaard, Copenhagen, pp STADIE, W. C. and O'BRIEN, H. (1937) The carbamate equilibrium. II. The equilibrium of oxyhemoglobin and reduced hemoglobin. J. Biol. Chem., 117: TAKIWAKI, H., MOCHIZUKI, M., and NIIZEKI, K. (1983) Relationship between hematocrit and CO2 contents in whole blood and true plasma. Jpn. J. Physiol., 33: TAZAWA, H., MOCHIZUKI, M., TAMURA, M., and KAGAWA, T. (1983) Quantitaitve analyses of the CO2 dissociation curve of oxygenated blood and the Haldane effect in human blood. Jpn. J. Physiol., 33: VEDA, Y. and TYUMA, I. (1982) Measurements of intra- and extracellular ph, and Donnan effect in whole blood. J. Physiol. Soc. Jpn., 44: 496. VAN SLYKE, D. D., WU, H., and MCLEAN, F. C. (1923) Studies of gas and electrolyte equilibria in the blood. V. Factors controlling the electrolyte and water distribution in the blood. J. Biol. Chem., 106: Vol. 33, No. 4, 1983