The Figge-Fencl Quantitative Physicochemical Model
of Human Acid-Base Physiology (Version 3.0)

by James J. Figge, MD, MBA, FACP

Copyright 2003 - 2013 James J. Figge.
Published 8 October, 2012; updated on 28 April, 2013; updated on 27 October, 2013;
on www.Figge-Fencl.org and www.acid-base.org.

Preface.

This model is dedicated to Dr. Vladimir Fencl (1923 - 2002) and his friend and colleague, Dr. Peter A. Stewart (1921 - 1993).

Abstract.

The Figge-Fencl quantitative physicochemical model of human acid-base physiology in blood plasma (version 3.0) is presented below. Following Stewart, the model incorporates three fundamental physicochemical principles as they apply to a single body fluid compartment (such as arterial blood plasma) under steady-state conditions: the law of conservation of mass is always obeyed; electrical neutrality is always maintained; and all statements of chemical equilibria are simultaneously satisfied.

Dissociation equilibria for the carbon dioxide - bicarbonate - carbonate system are explicitly included. The expression employed for the carbon dioxide - bicarbonate equilibrium is mathematically equivalent to the Henderson-Hasselbalch equation. Hence, the Henderson-Hasselbalch equation is always satisfied in the Figge-Fencl model (version 3.0). This is a necessary but not sufficient condition to describe the acid-base status of a given body fluid compartment.

Dissociation equilibria for major species of weak acids and their conjugate bases are incorporated into the Figge-Fencl model (version 3.0). These include the phosphoric acid - phosphate system, and dissociable amino acid side chains of albumin. Plasma globulins play only a minor role in acid-base balance and are not explicitly included in the current version of the Figge-Fencl model.

Human serum albumin is treated as a polyprotic macromolecule with multiple equilibrium dissociation constants corresponding to different classes of amino acid side chains. As in the Figge-Mydosh-Fencl model, pK(a) values for 13 of 16 albumin histidine residues in the Figge-Fencl model (version 3.0) are based on NMR spectroscopy data (temperature-corrected from 25 to 37 degrees Celsius). The Figge-Fencl model (version 3.0) features three major advances over the Figge-Mydosh-Fencl model. First, the Figge-Fencl model (version 3.0) takes into account the neutral-to-base (N-B) structural transition of human serum albumin over the pH range of 6 to 9. Second, the Figge-Fencl model (version 3.0) explicitly accounts for the contribution of all 59 lysine residues in albumin. The model features a small number of lysine residues with unusually low pK(a) values, the existence of which is suggested by spectroscopic and x-ray crystallographic data. Third, the Figge-Fencl model (version 3.0) accounts for the anomalously low average pK(a) value of glutamic and aspartic acid residues in albumin. This feature allows the model to provide useful functionality down to a pH of approximately 4.0.

The Figge-Fencl model (version 3.0) gives results equivalent to those of the Figge-Mydosh-Fencl model within the pH range of biologic interest (6.9 to 7.9). The Figge-Fencl model (version 3.0) has been optimized against the data of Figge, Rossing and Fencl for albumin-containing solutions using a least-squares algorithm. The Figge-Fencl model (version 3.0) demonstrates an improved least-squares fit to these data compared with the Figge-Mydosh-Fencl model. In addition, the albumin titration curve that is generated by the Figge-Fencl model (version 3.0) closely tracks with the experimental data points of Fogh-Andersen, Bjerrum, and Siggaard-Andersen within the pH range of 4 to 9, providing an additional level of validation. Hence, the model allows for robust analysis of acid-base phenomena over the pH window of 4 to 9.

At pH 7.40, the Figge-Fencl model (version 3.0) predicts that the charge contributed by 4.4 g / dL of albumin is approximately -12.3 mEq / L. The Figge-Fencl model (version 3.0) predicts that the titration curve of human serum albumin (charge displayed by albumin versus pH) is approximately linear over the pH range of biologic interest (6.9 to 7.9). Within this pH interval, the Figge-Fencl model (version 3.0) predicts that the approximate charge contributed by albumin in human plasma at a given pH is:

[ Albx- ] = -10 x [ Albumin ] x ( 0.123 x pH - 0.631 );

where [ Albx- ] is in mEq / L and [ Albumin ] is in g / dL. This exactly replicates the result obtained for the Figge-Mydosh-Fencl model over this pH range.

The molar buffer capacity of albumin is calculated by taking the additive inverse of the first derivative of [ Albx- ] with respect to pH:

- d [ Albx- ] / d pH

At pH = 7.40, the molar buffer capacity of albumin is 8.2 Eq / mol / pH. Thus, for [ Albumin ] of 4.4 g / dL, the buffer capacity of albumin is: 5.4 mEq / L / pH. The buffer capacity of albumin is used in the formulation of the classic van Slyke equation for plasma. The approximate linearity of the albumin titration curve over the pH range of biologic interest is critical for the successful derivation of the van Slyke equation for plasma.

The Figge-Fencl quantitative physicochemical model (version 3.0) can be used in conjunction with a computer application to illustrate the effects of independent variables on acid-base status in plasma-like solutions containing albumin. The application program will solve the following function for pH:

pH = fpH { SID, PCO2, [ Pitot ], [ Albumin ] }



Introductory Notes.

[1] Peter A. Stewart introduced a quantitative physicochemical model of acid-base balance in blood plasma [ reference 1 ]. Stewart incorporated the principle of electrical neutrality and accounted for the electrical charges contributed by all ionized species. Dissociation equilibria for the carbon dioxide - bicarbonate - carbonate system were explicitly included. The expression employed by Stewart for the carbon dioxide - bicarbonate equilibrium is mathematically equivalent to the Henderson-Hasselbalch equation. All nonvolatile weak acids (H2PO4-, and plasma proteins) were characterized by a single equilibrium dissociation constant in Stewart's model. The text of Stewart's classic book entitled "How to Understand Acid-Base: A Quantitative Acid-Base Primer for Biology and Medicine" is available online at http://www.acidbase.org/.

[2] Figge, Rossing and Fencl [ reference 2 ] produced electrolyte solutions resembling human serum that contained albumin as the sole protein moiety. Data collected from these solutions were used in a least-squares algorithm to develop a more robust quantitative physicochemical model. This model treated albumin as a polyprotic macromolecule with multiple equilibrium dissociation constants corresponding to different classes of amino acid side chains (i.e., Arg + Lys, Asp + Glu, Cys, His, Tyr, amino terminus, carboxyl terminus). This model was successful in calculating the pH of albumin-containing electrolyte solutions as well as the pH of filtrands of serum.

[3] Figge, Mydosh and Fencl [ reference 3 ] further refined the quantitative physicochemical model by incorporating pK(a) values for albumin histidine residues as determined by NMR spectroscopy [ reference 4 and reference 5 ]. The pK(a) values were temperature-corrected to 37 degrees Celsius in the model [ reference 3 ]. Based on a model compound, a correction factor of -0.27 was used to adjust the histidine pK(a) values from 25 degrees Celsius (298 K) to 37 degrees Celsius (310 K):

pK(a)310 = pK(a)298 - 0.27

This model accounted for the effects of the microenvironments within the macromolecule of albumin on the pK(a) values of individual histidine residues. The same temperature-corrected values are employed in the Figge-Fencl quantitative physicochemical model of acid-base physiology (version 3.0), presented below. The imidazole groups of histidine residues are of particular importance because they can titrate within the pH range of biologic interest. Although the Figge-Mydosh-Fencl model was successful in many aspects, it did not account for the presence of all 59 lysine residues in human serum albumin.

[4] Human serum albumin undergoes several structural transitions as a function of pH. The N-B (neutral-to-base) transition occurs between pH 6 and 9, which is important because this includes the physiologic pH range. Human serum albumin is organized into three structurally homologous domains, denoted 1, 2 and 3 [ reference 6 ]. A variety of spectroscopic methods including far-UV circular dichroism (CD), near-UV CD, and fluorescence emission (from tryptophan and tyrosine residues) have been employed to study the structural transitions of human albumin and the three recombinant domains of albumin [ reference 7 ]. The role of histidine residues in the N-B transition has been studied by NMR spectroscopy [ reference 4 and reference 5 ]. The fluorescence spectrum of bovine serum albumin has also been studied in the pH range of 3 to 10 [ reference 8 ]. The N-B transition has been described in terms of a two-state model [ reference 9 ]. The N state exists at lower pH values, and the B state at high pH. This interpretation is supported by the far-UV CD data which demonstrate a slight reduction in alpha-helical content of albumin as the pH ranges from 7.4 to 9.0 [ reference 7 ]. Furthermore, near-UV CD data demonstrate that albumin domains 1 and 2 undergo a tertiary structural isomerization in the pH range of the N-B transition [ reference 7 ]. Domain 3 is not involved in the N-B structural transition. Based on model calculations, there are most likely five conformation-linked histidine residues that undergo a downward pK shift as albumin transitions from the N to the B conformation [ reference 5 ]. The five histidine residues that participate in the N-B transition have been assigned to domain 1 of human serum albumin based on an analysis of NMR data [ reference 5 ]. Calcium ions can exert a strong influence on the N-B transition. Calcium ions induce a downward shift in the pK(a) of several histidine residues at constant pH and a concomitant downward shift in the midpoint pH of the N-B transition [ reference 4, reference 5, and reference 9 ]. Consequently, the addition of calcium results in the release of protons and a shift from the N to the B conformation.

[5] The Figge-Fencl quantitative physicochemical model of acid-base physiology (version 3.0), presented below, incorporates an empiric function that models the N-B transition by downshifting the pK(a) values of five histidine residues located within domain 1 of human serum albumin. The magnitude of the pK downshift is 0.4 units, as estimated in reference 4. In the presence of physiologic calcium ion concentrations (2.5 mM), the midpoint of the pH range for the N-B transition is approximately 6.9 [ reference 9 ].

[6] Buried lysine residues with unusually low pK(a) values (e.g., pK(a) of 5.7) have been documented in the literature [ reference 10 ]. The available evidence suggests that there are potentially six lysine residues in human serum albumin that have an unusually low pK(a). This interpretation is supported by data from tryptophan and tyrosine fluorescence emission studies [ reference 7 ]. Human serum albumin contains a single tryptophan residue at position 214, which is located within domain 2. Tryptophan fluorescence can be excited by a wavelength of 295 nm, which does not excite tyrosine residues. When tryptophan-specific fluorescence is studied in the context of intact human serum albumin, there is a decrease in fluorescence intensity as the pH is raised from 7.4 to 9.0 [ reference 7 ]. When tyrosine fluorescence is studied from recombinant domain 3, there is a decrease in fluorescence intensity as the pH is raised from 6 to 9 [ reference 7 ]. The decrease of the tyrosine fluorescence signal intensity could be partially explained by deprotonation of the phenolic hydroxyl group of some tyrosine side chains. Furthermore, deprotonated epsilon-amino groups of lysine side chains are known to quench the fluorescence signal from both tryptophan and tyrosine residues. Therefore, deprotonated lysine residues in close proximity to tryptophan and tyrosine are candidate quenching groups. Based on the known x-ray crystal structure of human albumin [ reference 11, reference 12, Protein Data Bank entries 1UOR and 1AO6 ], the epsilon-amino groups of lysine-525, 414, 432, and 534 are within five Angstroms of tyrosine-401, 411, 452, and 497, respectively. These tyrosine residues are all located within domain 3 of human serum albumin. Structural rearrangements associated with the N-B transition (which do not affect domain 3) are not likely to be a factor in the decrease of the fluorescence signal from these particular tyrosine residues. This suggests that the adjacent lysine epsilon-amino groups could be at least partially responsible for quenching the tyrosine fluorescence signal as they deprotonate within the pH range of 6 to 9. Likewise, lysine-199 and lysine-195 are positioned 3.7 and 7.4 Angstroms, respectively, from tryptophan-214. Thus, the decrease in the fluorescence signal from tryptophan-214 might be due to changes in secondary or tertiary structure of domain 2, and/or quenching from one or both adjacent lysine epsilon-amino groups as they deprotonate within the slightly alkaline pH range. These data suggest that there are potentially six lysine epsilon-amino groups in human albumin that exhibit an anomalously low pK(a) value. Similar fluorescence data were presented regarding bovine serum albumin [ reference 8 ].

[7] The exact number of low-titrating lysine residues in human serum albumin is not known. Trial pK(a) values for the low-titrating lysine residues in the Figge-Fencl model (version 3.0) are estimated from the albumin titration curve of Halle and Lindman [ reference 13 ] (corrected from 22 to 37 degrees C). Based on this analysis, there are nine low-titrating lysine residues distributed in two clusters. One small cluster features pK(a) values near 6. The second, larger cluster, features pK(a) values near 8. The number and exact distribution of low-titrating lysine residues are optimized in the Figge-Fencl model (version 3.0) using a least-squares algorithm against the data of Figge, Rossing and Fencl for albumin-containing solutions at 37 degrees C. The computer program incorporates 11 sets of trial pK(a) values for low-titrating lysine residues. The set number is designated in the computer program by the parameter, i, which assumes values of 0 to 100 (inclusive) in increments of 10.
Set 1 (i = 0): {LYS1 = 5.8; LYS2 = 6.15; LYS3 = 7.500; LYS4 = 7.675; LYS5 = 7.850; LYS6 = 8.025};
Set 2 (i = 10): {LYS1 = 5.8; LYS2 = 6.15; LYS3 = 7.510; LYS4 = 7.685; LYS5 = 7.860; LYS6 = 8.035};
Set 3 (i = 20): {LYS1 = 5.8; LYS2 = 6.15; LYS3 = 7.520; LYS4 = 7.695; LYS5 = 7.870; LYS6 = 8.045}; . . .;
Set 11 (i = 100): {LYS1 = 5.8; LYS2 = 6.15; LYS3 = 7.600; LYS4 = 7.775; LYS5 = 7.950; LYS6 = 8.125}.
The number of lysine residues with a pK(a) of LYS1 is designated as N1; the number with a pK(a) of LYS2 is designated as N2; etc. Each parameter, N1 through N6, is allowed to assume a value of 0, 1, or 2. All possible combinations of values are tested for inclusion in the model. Up to nine low-titrating lysine residues are allowed in the model, so that (N1 + N2 + N3 + N4 + N5 + N6) < 10. This constraint results in 701 possible combinations for N1 through N6. The pK(a) for normally-titrating lysine residues in the model, LYS7, is assigned trial values of 10.3; 10.4; 10.5; . . .; 10.8 by the computer program, in accordance with standard textbook values [ references 14 and 15 ]. The value of LYS7 is calculated in the program from the parameter, L7, which assumes values of 103 to 108 (inclusive). This model accounts for the contribution of all 59 lysine residues in human serum albumin. Hence, the number of normally-titrating lysine residues, N7, is given by: N7 = 59 - N1 - N2 - N3 - N4 - N5 - N6. Arginine residues are considered separately from lysine. Each of the 24 arginine resides in albumin is assigned a pK(a) of 12.5, in accordance with standard textbook values [ references 14 and 15 ].

[8] In the Figge-Fencl quantitative model (version 3.0), presented below, each of the 18 tyrosine residues in albumin is assigned a pK(a) value of 11.7, in accordance with a spectrophotometrically determined value [ reference 13 ]. The formal possibility of anomalously low pK(a) values for a subset of tyrosine residues is not addressed in this model.

[9] In the Figge-Fencl quantitative model (version 3.0), presented below, the 36 aspartic acid and 62 glutamic acid residues in albumin are each assigned a pK(a) of 3.9. This is in recognition of the anomalously low average pK(a) value of glutamic and aspartic acid residues in albumin. Cysteine is assigned a pK(a) of 8.5, the amino terminus is assigned a pK(a) of 8.0, and the carboxyl terminus is assigned a pK(a) of 3.1. These are consistent with standard textbook values [ references 14 and 15 ] and were previously employed in Figge, Mydosh and Fencl [ reference 3 ].

[10] An incompletely characterized histidine residue has a temperature-corrected pK(a) < 5.2 and is assigned a value of HIS14 = 5.10 in the Figge-Fencl model (version 3.0). A second incompletely characterized histidine residue has a temperature-corrected pK(a) in the range of 6.7 to 7.7 (inclusive). The computer program assigns trial pK(a) values for HIS15 of 6.7; 6.8; 6.9; . . .; 7.7. The value of HIS15 is calculated from the parameter, N0, which assumes values of 67 to 77 (inclusive) in the computer program. The pK(a) value of one histidine residue that could not be determined by NMR is designated as HIS16 and is arbitrarily assigned a value of 6.2 in the model, in keeping with standard textbook values [ references 14 and 15 ].

[11] A detailed model for the contribution of plasma globulins is difficult to develop due to the marked heterogeneity of plasma globulin species. Based on liquid phase preparative isoelectric focusing of native human immunoglobulin molecules, the distribution of isoelectric points for IgG is 4.35 to 9.95, with a dominant peak between pH 7 and 9.95, centered at pH 8.2 [ reference 16 ]. Thus a significant fraction of IgG molecules will carry a positive charge within the physiologic pH range. Hence, at physiologic pH values, it is expected that the positive charges contributed by IgG molecules will at least partially offset the negative charges carried by alpha- and beta-globulin fractions as well as IgA, the majority of IgM and the remainder of IgG molecules.

[12] K1, K2 and K3 are the apparent equilibrium dissociation constants for phosphoric acid for plasma [ Reference 17 ]. pK1 = 1.915; pK2 = 6.66; pK3 = 11.78.

[13] The constant Kc1, governing the carbon dioxide - bicarbonate equilibrium, is derived directly from parameters in the Henderson-Hasselbalch equation. The solubility of CO2 in plasma is: 0.230 mmol / L / kPa x 0.13332236842105 kPa / Torr = 0.0307 mmol / L / Torr.
Kc1 = (10-6.1) (0.0307) / (1000) = 2.44 x 10-11. Hence, calculations using Kc1 in the Figge-Fencl model (version 3.0) yield results identical to those calculated with the Henderson-Hasselbalch equation.

[14] The constant Kc2, the second dissociation constant for carbonic acid, is calculated from the formula (equation 9) given by Harned and Scholes [ reference 18 ]. At 37 degrees Celsius (310 K), the formula yields: log Kc2 = (-2902.39 / 310) + 6.4980 - (0.02379 x 310) = -10.239. Hence, at zero ionic strength, pKc2 = 10.239. The correction factor for an ionic strength of 0.15 M is approximately 0.022. Hence pKc2 = 10.261, and kc2 = 5.5E-11. Due to the fact that carbonate ion has a charge of -2, Kc2 when expressed in Eq / L is 1.1E-10.

[15] The computer program is written in BASIC, using double-precision floating point arithmetic. This program optimizes the values of i, L7, and N0 through N7 using a least-squares algorithm against the data of Figge, Rossing and Fencl [ reference 2 ]. The algorithm finds the subset of all solutions that satisfy the following criteria: (a) the absolute value of the mean deviation, { Σ(calculated pH - measured pH) } / 65, is 0.0035 or less; and, (b) the slope of the regression line (calculated pH versus measured pH) is 1.0000 +/- 0.0068. From that subset, the single solution that minimizes the sum of squares of deviations is selected. The program analyzes a total of 508,926 combinations of values for i, L7, N0, and N1 through N6 (11 x 6 x 11 x 701 = 508,926). The program runs in about 6 hours on an Intel(R) Pentium(R) 2.13 GHz processor. It performs approximately 4 x 10^11 (four hundred billion) double-precision floating point calculations. This large number arises from the fact that an iterative procedure is required to solve the model equation for each combination of variables, and this is repeated 65 times to accommodate each data point.

[16] A model summary and a simple formula for calculating the predicted charge displayed by albumin over the pH range of 6.9 to 7.9 are also presented. The molar buffer capacity of albumin is calculated by taking the additive inverse of the first derivative of [ Albx- ] with respect to pH: - d [ Albx- ] / d pH. The first derivative is evaluated at pH = 7.40. Following this, a formal mathematical representation of the optimized model is presented. An application program is also available for performing calculations and simulations with the optimized Figge-Fencl quantitative physicochemical model (version 3.0).

References:

1. Stewart, PA. Modern quantitative acid-base chemistry. Canadian Journal of Physiology and Pharmacology 1983; 61:1444-1461. [ Abstract on PubMed ].

2. Figge J, Rossing TH, Fencl V. The role of serum proteins in acid-base equilibria. J Lab Clin Med. 1991; 117: 453-467. [ Abstract on PubMed ].

3. Figge J, Mydosh T, Fencl V. Serum proteins and acid-base equilibria: a follow-up. J Lab Clin Med. 1992; 120: 713-719. [ Abstract on PubMed ].

4. Labro JFA, Janssen LHM. A proton nuclear magnetic resonance study of human serum albumin in the neutral pH region. Biochim Biophys Acta. 1986; 873: 267-278. [ Abstract on PubMed ].

5. Bos OJM, Labro JFA, Fischer MJE, Wilting J, Janssen LHM. The molecular mechanism of the neutral-to-base transition of human serum albumin. Acid/base titration and proton nuclear magnetic resonance studies on a large peptic and a large tryptic fragment of albumin. J Biol Chem. 1989; 264: 953-959. [ PDF download of the full text is available online at http://www.jbc.org/cgi/content/abstract/264/2/953 ].

6. Dockal M, Carter DC, Ruker F. The three recombinant domains of human serum albumin. Structural characterization and ligand binding properties. J Biol Chem. 1999; 274: 29303-29310. [ Full text is available online at http://www.jbc.org/cgi/content/full/274/41/29303 ].

7. Dockal M, Carter DC, Ruker F. Conformational transitions of the three recombinant domains of human serum albumin depending on pH. J Biol Chem. 2000; 275: 3042-3050. [ Full text is available online at http://www.jbc.org/cgi/content/full/275/5/3042 ].

8. Halfman CJ, Nishida T. Influence of pH and electrolyte on the fluorescence of bovine serum albumin. Biochim Biophys Acta. 1971; 243: 284-293. [ Citation on PubMed ].

9. Janssen LHM, Van Wilgenburg MT, Wilting J. Human serum albumin as an allosteric two-state protein. Evidence from effects of calcium and warfarin on proton binding behaviour. Biochim Biophys Acta. 1981; 669: 244-250. [ Abstract on PubMed ].

10. Fitch CA, Karp DA, Lee KK, Stites WE, Lattman EE, Garcia-Moreno E B. Experimental pK(a) values of buried residues: analysis with continuum methods and role of water penetration. Biophys J. 2002; 82: 3289-3304. [ Full text is available online at http://www.biophysj.org/cgi/content/full/82/6/3289 ].

11. He XM, Carter DC. Atomic structure and chemistry of human serum albumin. Nature. 1992; 358: 209-215. [ Abstract on PubMed ].

12. Sugio S, Kashima A, Mochizuki S, Noda M, Kobayashi K. Crystal Structure of Human Serum Albumin at 2.5 Angstrom Resolution. Protein Eng. 1999; 12: 439-446. [ Full text is available online at http://peds.oupjournals.org/cgi/content/full/12/6/439 ].

13. Halle B, Lindman B. Chloride ion binding to human plasma albumin from chlorine-35 quadrupole relaxation. Biochemistry. 1978; 17: 3774-3781. [ Abstract on PubMed ].

14. Stryer L. Biochemistry. San Francisco: W.H. Freeman. First Edition. 1975. Pages 44 and 86.

15. Berg JM, Tymoczko JL, Stryer L. Biochemistry. New York: W.H. Freeman. Sixth Edition. 2007. Pages 33 and Appendix B.

16. Prin C, Bene MC, Gobert B, Montagne P, Faure GC. Isoelectric restriction of human immunoglobulin isotypes. Biochemica et Biophysica Acta. 1995; 1243: 287-290. [ Abstract on PubMed ].

17. Sendroy J, Hastings B. Studies of the solubility of calcium salts. II. The solubility of tertiary calcium phosphate in salt solutions and biological fluids. J Biol Chem. 1927; 71: 783-796. [ Full text is available online at http://www.jbc.org/cgi/reprint/71/3/783 ].

18. Harned HS, Scholes SR. The ionization constant of HCO3- from 0 to 50 Degrees. J Am Chem Soc. 1941; 63: 1706-1709.

19. Siggaard-Andersen O, Fogh-Andersen N. Base excess or buffer base (strong ion difference) as a measure of non-respiratory acid-base disturbance. Acta Anesthesiol Scand. 1995; 39 (Suppl 106): 123-128.

[ See also O. Siggaard-Andersen's web site at http://www.siggaard-andersen.dk/; the full text article is available under the Bibliography. ].

20. Fencl V, Jabor A, Kazda A, Figge J. Diagnosis of metabolic acid-base disturbances in critically ill patients. Am J Respir Crit Care Med. 2000; 162: 2246-2251. [ Full text is available online at http://www.atsjournals.org/doi/full/10.1164/ajrccm.162.6.9904099 ].

[ See also the online supplement, which is accessible at http://www.atsjournals.org/doi/suppl/10.1164/ajrccm.162.6.9904099 ].

Computer Program.

Sub Model()

Rem: Figge-Fencl Quantitative Physicochemical Model
Rem: of Human Acid-Base Physiology (Version 3.0).
Rem:
Rem: Program by James J. Figge, MD, MBA, FACP.
Rem: Copyright 2003 - 2013 James J. Figge. Update published 28 April, 2013;
Rem: Update of computer program published 27 October, 2013.

Close #1
Dim pHm(65), SID(65), PCO2(65), Pi(65), Alb(65)

rownum = 1
colnum = 1
rownum = ActiveCell.Row
colnum = ActiveCell.Column

Worksheets("Sheet1").Activate

sum1 = 0
sum2 = 0
sum3 = 0
sum4 = 0
sum5 = 0

For rownum = 1 To 65

pHm(rownum) = ActiveSheet.Cells(rownum, 2)
SID(rownum) = ActiveSheet.Cells(rownum, 3)
PCO2(rownum) = ActiveSheet.Cells(rownum, 4)
Pi(rownum) = ActiveSheet.Cells(rownum, 5)
Alb(rownum) = ActiveSheet.Cells(rownum, 6)

sum1 = sum1 + pHm(rownum)
sum2 = sum2 + SID(rownum)
sum3 = sum3 + PCO2(rownum)
sum4 = sum4 + Pi(rownum)
sum5 = sum5 + Alb(rownum)

Next rownum

Rem: Kc1 is derived from the parameters in the Henderson-Hasselbalch
Rem: equation. pK = 6.1; a = 0.230 mM / kPa; 1 Torr = 0.13332236842105 kPa
Rem: The value of Kc1 is 2.44E-11 (Eq / L)^2 / Torr.

Rem: Kc2 is calculated from Harned and Scholes (1941) for 37 degrees C and ionic
Rem: strength 0.15 M. The value of Kc2 is 5.5E-11 mol / L x 2 = 1.1E-10 Eq / L.

Rem: K1, K2, and K3 for the phosphoric acid - phosphate system are from Sendroy and
Rem: Hastings (1927).

Const kw = 0.000000000000044

Const Kc1 = 0.0000000000244
Const Kc2 = 0.00000000011

Const K1 = 0.0122
Const K2 = 0.000000219
Const K3 = 0.00000000000166

Const LYS1 = 5.8
Const LYS2 = 6.15

Const L3 = 7500
Const L4 = 7675
Const L5 = 7850
Const L6 = 8025

Rem: HIS 14 has a pK of less than 5.2; the pK value is set at 5.1
Rem: HIS 15 has a pK in the range of 6.7 to 7.7 (inclusive)
Rem: HIS 16 has an unknown pK; the pK value is arbitrarily set at 6.2

Const HIS14 = 5.1
Const HIS16 = 6.2

minss = 9999999

For i = 0 To 100 Step 10

LYS3 = (L3 + i) / 1000
LYS4 = (L4 + i) / 1000
LYS5 = (L5 + i) / 1000
LYS6 = (L6 + i) / 1000

For L7 = 103 To 108
LYS7 = L7 / 10

For N0 = 67 To 77

HIS15 = N0 / 10

For N1 = 0 To 2
For N2 = 0 To 2
For N3 = 0 To 2
For N4 = 0 To 2
For N5 = 0 To 2
For N6 = 0 To 2

If N1 + N2 + N3 + N4 + N5 + N6 > 9 Then GoTo getnext

N7 = 59 - N1 - N2 - N3 - N4 - N5 - N6

ss = 0
s = 0
abvs = 0
sx = 0
sxx = 0
sy = 0
syy = 0
sxy = 0

For j = 1 To 65

High = 14
Low = 1

calculatepH:
pH = (High + Low) / 2
Rem: H is hydrogen ion activity (also used as an approximation of [H+])
H = 10 ^ -pH

HCO3 = Kc1 * PCO2(j) / H
CO3 = Kc2 * HCO3 / H

FNX = K1 * H * H + 2 * K1 * K2 * H + 3 * K1 * K2 * K3
FNY = H * H * H + K1 * H * H + K1 * K2 * H + K1 * K2 * K3
FNZ = FNX / FNY
P = Pi(j) * FNZ

Netcharge = SID(j) + 1000 * (H - kw / H - HCO3 - CO3) - P

Rem: NB accounts for histidine pK shift due to the NB transition
NB = 0.4 * (1 - (1 / (1 + (10 ^ (pH - 6.9)))))

Rem: Calculate charge on albumin
Rem: alb2 accumulates results

Rem: cysteine residue
alb2 = -1 / (1 + 10 ^ (-(pH - 8.5)))

Rem: glutamic acid and aspartic acid residues
alb2 = alb2 - 98 / (1 + 10 ^ (-(pH - 3.9)))

Rem: tyrosine residues
alb2 = alb2 - 18 / (1 + 10 ^ (-(pH - 11.7)))

Rem: arginine residues
alb2 = alb2 + 24 / (1 + 10 ^ (pH - 12.5))

Rem: lysine residues
alb2 = alb2 + N1 / (1 + 10 ^ (pH - LYS1))
alb2 = alb2 + N2 / (1 + 10 ^ (pH - LYS2))
alb2 = alb2 + N3 / (1 + 10 ^ (pH - LYS3))
alb2 = alb2 + N4 / (1 + 10 ^ (pH - LYS4))
alb2 = alb2 + N5 / (1 + 10 ^ (pH - LYS5))
alb2 = alb2 + N6 / (1 + 10 ^ (pH - LYS6))

alb2 = alb2 + N7 / (1 + 10 ^ (pH - LYS7))

Rem: 16 different histidine residues
Rem: correction factor to convert HIS pK(a) from 25 deg C to 37 deg C is approx -0.27
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 7.12 + NB))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 7.22 + NB))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 7.1 + NB))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 7.49 + NB))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 7.01 + NB))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 7.31))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 6.75))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 6.36))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 4.85))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 5.76))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 6.17))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 6.73))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 5.82))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - HIS14))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - HIS15))
alb2 = alb2 + 1 / (1 + 10 ^ (pH - HIS16))

Rem: amino terminus
alb2 = alb2 + 1 / (1 + 10 ^ (pH - 8))

Rem: carboxyl terminus
alb2 = alb2 - 1 / (1 + 10 ^ (-(pH - 3.1)))

alb2 = alb2 * 1000 * 10 * Alb(j) / 66500

Netcharge = Netcharge + alb2

If Abs(Netcharge) < 0.0000001 Then GoTo complete
If Netcharge < 0 Then High = pH
If Netcharge > 0 Then Low = pH
GoTo calculatepH

complete:

ss = ss + (pHm(j) - pH) * (pHm(j) - pH)
s = s + (pHm(j) - pH)
abvs = abvs + Abs(pHm(j) - pH)
sx = sx + pHm(j)
sxx = sxx + pHm(j) * pHm(j)
sy = sy + pH
syy = syy + pH * pH
sxy = sxy + pHm(j) * pH

Next j

If ss > minss Then GoTo getnext

n = 65

If Abs(10000 * s / n) > 35 Then GoTo getnext

Slope = (n * sxy - sx * sy) / (n * sxx - sx * sx)

If Abs(10000 * Slope - 10000) > 68 Then GoTo getnext

minss = ss

Open "model-results" For Output As #1
Print #1, "Checksum1 =", sum1
Print #1, "Checksum2 =", sum2
Print #1, "Checksum3 =", sum3
Print #1, "Checksum4 =", sum4
Print #1, "Checksum5 =", sum5
Print #1, " "

Print #1, "abvs / n =", abvs / 65, "s= ", s, "ss= ", ss
Print #1, " "
Print #1, "HIS14 =", HIS14, "HIS15 = ", HIS15, "HIS16 = ", HIS16
Print #1, " "

Print #1, "LYS1 = ", LYS1, "; N1 = ", N1
Print #1, "LYS2 = ", LYS2, "; N2 = ", N2
Print #1, "LYS3 = ", LYS3, "; N3 = ", N3
Print #1, "LYS4 = ", LYS4, "; N4 = ", N4
Print #1, "LYS5 = ", LYS5, "; N5 = ", N5
Print #1, "LYS6 = ", LYS6, "; N6 = ", N6
Print #1, "LYS7 = ", LYS7, "; N7 = ", N7

n = 65
Slope = (n * sxy - sx * sy) / (n * sxx - sx * sx)
incpt = (sy * sxx - sx * sxy) / (n * sxx - sx * sx)

vincpt = sy / n - Slope * sx / n

r = (n * sxy - sx * sy) / Sqr(n * sxx - sx * sx) / Sqr(n * syy - sy * sy)

Var = (syy - incpt * sy - Slope * sxy) / (n - 2)

varslope = n * Var / (n * sxx - sx * sx)

stndevslope = Sqr(varslope)

Rem: t(n-2, alpha/2) for n=65 is 2.3870, where alpha=0.02
t = 2.387

Lconfint = Slope - t * stndevslope
Uconfint = Slope + t * stndevslope

Print #1, " "
Print #1, "slope = ", slope
Print #1, "intercept = ", incpt
Print #1, "intercept = ", vincpt, "(verify)"
Print #1, "r = ", r
Print #1, "r^2 = ", r * r
Print #1, "Variance = ", Var
Print #1, "Variance of slope = ", varslope
Print #1, "Stnd Deviation of slope = ", stndevslope
Print #1, "98% confidence interval for the slope = ", Lconfint, " to ", Uconfint

Close #1

getnext:
Next N6
Next N5
Next N4
Next N3
Next N2
Next N1
Next N0
Next L7
Next i

End Sub

REM: Data from Figge J, Rossing TH, Fencl V. J Lab Clin Med.
REM: 1991; 117:453-467 (Table A).
REM: Data must be entered into a spreadsheet for use in the program.

DATA 01, 7.388, 49.8, 39.3, 1.1, 7.2
DATA 02, 7.383, 45.4, 40.0, 1.0, 7.0
DATA 03, 7.521, 45.4, 26.1, 1.0, 7.0
DATA 04, 7.389, 45.4, 38.1, 1.0, 7.0
DATA 05, 7.217, 45.4, 62.9, 1.0, 7.0
DATA 06, 7.315, 32.2, 23.4, 1.2, 6.6
DATA 07, 7.194, 32.2, 35.0, 1.2, 6.6
DATA 08, 6.979, 32.2, 68.6, 1.2, 6.6
DATA 09, 7.819, 71.3, 28.9, 1.1, 6.8
DATA 10, 7.716, 71.3, 37.7, 1.1, 6.8
DATA 11, 7.504, 71.3, 65.4, 1.1, 6.8
DATA 12, 7.850, 70.2, 26.4, 1.0, 6.8
DATA 13, 7.719, 70.2, 37.9, 1.0, 6.8
DATA 14, 7.513, 70.2, 65.3, 1.0, 6.8
DATA 15, 7.447, 45.9, 30.8, 1.0, 7.1
DATA 16, 7.375, 45.9, 38.4, 1.0, 7.1
DATA 17, 7.094, 45.9, 85.0, 1.0, 7.1
DATA 18, 7.935, 70.2, 22.5, 1.0, 6.8
DATA 19, 7.716, 70.2, 40.2, 1.0, 6.8
DATA 20, 7.423, 70.2, 83.5, 1.0, 6.8
DATA 21, 7.746, 45.8, 22.1, 1.0, 3.5
DATA 22, 7.518, 45.8, 39.9, 1.0, 3.5
DATA 23, 7.218, 45.8, 85.9, 1.0, 3.5
DATA 24, 7.446, 24.2, 21.9, 0.9, 3.4
DATA 25, 7.226, 24.2, 39.8, 0.9, 3.4
DATA 26, 7.018, 24.2, 69.7, 0.9, 3.4
DATA 27, 7.676, 63.7, 40.2, 1.0, 3.6
DATA 28, 7.369, 63.7, 86.7, 1.0, 3.6
DATA 29, 7.711, 75.5, 38.6, 1.0, 6.7
DATA 30, 7.702, 76.4, 38.2, 1.0, 7.3
DATA 31, 7.630, 65.9, 37.9, 1.0, 7.3
DATA 32, 7.572, 60.2, 35.1, 0.7, 7.6
DATA 33, 7.718, 58.9, 41.0, 1.0, 1.7
DATA 34, 7.510, 58.9, 67.9, 1.0, 1.7
DATA 35, 7.399, 58.9, 88.0, 1.0, 1.7
DATA 36, 7.684, 70.4, 38.2, 1.0, 7.0
DATA 37, 7.477, 70.4, 65.4, 1.0, 7.0
DATA 38, 7.390, 70.4, 85.7, 1.0, 7.0
DATA 39, 7.551, 53.5, 38.7, 1.0, 6.2
DATA 40, 7.348, 53.5, 68.4, 1.0, 6.2
DATA 41, 7.240, 53.5, 86.3, 1.0, 6.2
DATA 42, 7.598, 51.2, 41.2, 0.9, 1.9
DATA 43, 7.378, 51.2, 69.8, 0.9, 1.9
DATA 44, 7.307, 51.2, 87.7, 0.9, 1.9
DATA 45, 7.320, 32.5, 22.5, 1.0, 8.0
DATA 46, 7.144, 32.5, 38.9, 1.0, 8.0
DATA 47, 7.006, 32.5, 59.2, 1.0, 8.0
DATA 48, 7.416, 28.5, 22.7, 1.0, 2.9
DATA 49, 7.213, 28.5, 40.3, 1.0, 2.9
DATA 50, 7.068, 28.5, 58.6, 1.0, 2.9
DATA 51, 7.460, 22.8, 23.1, 1.0, 1.6
DATA 52, 7.246, 22.8, 40.2, 1.0, 1.6
DATA 53, 7.083, 22.8, 60.6, 1.0, 1.6
DATA 54, 7.125, 23.7, 22.6, 1.0, 5.7
DATA 55, 6.968, 23.7, 40.0, 1.0, 5.7
DATA 56, 6.849, 23.7, 58.0, 1.0, 5.7
DATA 57, 7.254, 21.4, 23.0, 1.0, 3.5
DATA 58, 7.051, 21.4, 40.7, 1.0, 3.5
DATA 59, 6.924, 21.4, 58.3, 1.0, 3.5
DATA 60, 7.654, 67.5, 39.7, 1.0, 7.2
DATA 61, 7.508, 67.5, 56.9, 1.0, 7.2
DATA 62, 7.347, 67.5, 87.0, 1.0, 7.2
DATA 63, 7.706, 62.5, 40.1, 1.0, 3.8
DATA 64, 7.561, 62.5, 57.5, 1.0, 3.8
DATA 65, 7.386, 62.5, 91.2, 1.0, 3.8

END

Optimized Model Parameters.

Contents of output file: 'model-results':

Checksum1 = 481.218
Checksum2 = 3194.9
Checksum3 = 3210.8
Checksum4 = 65.1
Checksum5 = 342.8

abvs / n = 2.51751200069458E-02
s= -0.221307340304365
ss= 7.00002270873043E-02

HIS14 = 5.1
HIS15 = 6.7
HIS16 = 6.2

LYS1 = 5.800; N1 = 2
LYS2 = 6.150; N2 = 2
LYS3 = 7.510; N3 = 2
LYS4 = 7.685; N4 = 2
LYS5 = 7.860; N5 = 1
LYS6 = 8.035; N6 = 0
LYS7 = 10.30; N7 = 50

slope = 1.00678874215608
intercept = -4.68547320397751E-02
intercept = -4.68547320394102E-02 (verify)
r = 0.991565222939416
r^2 = 0.983201591342893
Variance = 1.09623723925786E-03
Variance of slope = 2.74891760302995E-04
Stnd Deviation of slope = 1.65798600809233E-02
98% confidence interval for the slope = 0.967212616142919 to 1.04636486816925


The sum of squares of the differences between pH (measured) and pH (calculated) is 0.07000.

The optimized parameters are as follows:

The pK(a)ís of three histidine residues not determined by NMR spectroscopy:
HIS14 = 5.10
HIS15 = 6.70
HIS16 = 6.20

The low-titrating lysine residues were assigned the following pK(a) values:
LYS1: pK(a) = 5.800; N1 = 2
LYS2: pK(a) = 6.150; N2 = 2
LYS3: pK(a) = 7.510; N3 = 2
LYS4: pK(a) = 7.685; N4 = 2
LYS5: pK(a) = 7.860; N5 = 1

The normally-titrating lysine residues were assigned the following pK(a) value:
LYS7: pK(a) = 10.30

Figge-Fencl Model (Version 3.0) Summary.

The model includes the following features for human serum albumin:

1 Cys residue; pK(a) = 8.5

98 Glu and Asp residues; pK(a) = 3.9

18 Tyr residues; pK(a) = 11.7

24 Arg residues; pK(a) = 12.5

59 Lys residues; 2 with pK(a) = 5.800; 2 with pK(a) = 6.150; 2 with pK(a) = 7.510; 2 with pK(a) = 7.685; 1 with pK(a) = 7.860; and 50 with pK(a) = 10.30

16 His residues; with pK(a)'s of 7.12; 7.22; 7.10; 7.49; 7.01; 7.31; 6.75; 6.36; 4.85; 5.76; 6.17; 6.73; 5.82; 5.10; 6.70; and 6.20 (note that the pK(a)'s of the first five His residues will each downshift by 0.4 pH units due to the structural rearrangement associated with the N-B transition).

amino terminus; pK(a) = 8.0

carboxyl terminus; pK(a) = 3.1


Charge on Albumin at pH 7.40 as Calculated per the Model.

At pH 7.40, calculations using the model predict that albumin will carry a net charge of -18.538 Eq / mol. Therefore, in human plasma at pH 7.40 with [ Albumin ] = 4.4 g / dL, the charge attributed to albumin per the model is -12.3 mEq / L, calculated as follows:

[ Albx- ] = ( -18.538 Eq / mol ) x ( 10 dL / L ) x ( 1000 mEq / Eq ) x ( 4.4 g / dL) / ( 66500 g / mol ) = -12.3 mEq / L;

where 66500 g / mol is the molecular weight of albumin.


Charge on Albumin over the pH Range of 6.9 to 7.9: Linear Regression.

The Figge-Fencl model (version 3.0) predicts that the titration curve of human serum albumin (charge displayed by albumin, expressed as mEq per g of albumin, versus pH) is approximately linear over the pH range of biologic interest (6.9 to 7.9) [ see the albumin titration curve ]. Therefore, a standard least-squares linear regression algorithm can be used to predict the approximate charge contributed by albumin in human plasma over this pH interval:

[ Albx- ] = -10 x [ Albumin ] x ( 0.123 x pH - 0.631 );

where [ Albx- ] is in mEq / L and [ Albumin ] is in g / dL. Therefore, at pH 7.40, the charge contributed by 4.40 g / dL of albumin is approximately -12.3 mEq / L.


Molar Buffer Capacity of Albumin as Calculated per the Model.

The molar buffer capacity of albumin is calculated by taking the additive inverse of the first derivative of [ Albx- ] with respect to pH:

- d [ Albx- ] / d pH

At pH = 7.40, d [ Albx- ] / d pH = -8.2; hence, the molar buffer capacity of albumin is 8.2 Eq / mol / pH. This value is similar to the value of 8.0 Eq / mol / pH calculated by Siggaard-Andersen and Fogh-Andersen in the literature [ reference 19 ]. Thus, for [ Albumin ] of 4.4 g / dL, the buffer value of albumin at that concentration and at pH 7.40 is:

( 4.4 g / dL ) x ( 8.2 Eq / mol / pH) x ( mol / 66500 g ) x ( 1000 mEq / Eq ) x ( 10 dL / L) = 5.4 mEq / L / pH

The average value of the molar buffer capacity of albumin over the pH range of 6.9 to 7.9 can be calculated using the slope of the least-squares regression line over that interval:

( 0.123 mEq / g / pH ) x ( 66500 g / mol ) x ( Eq / 1000 mEq ) = 8.2 Eq / mol / pH

Hence, the average molar buffer capacity over this pH interval is equal to the specific value at the midpoint (pH = 7.40).

The approximate linearity of the albumin titration curve over the pH range of biologic interest (6.9 - 7.9) is critical for the successful derivation of the classic van Slyke equation for plasma.

As explained in the online supplement to reference 20 [accessible at http://www.atsjournals.org/doi/suppl/10.1164/ajrccm.162.6.9904099 ] and in reference 19, the molar buffer capacity of albumin is used in the formulation of the van Slyke equation for human plasma. More information about the van Slyke equation can be found on O. Siggaard-Andersen's web site at http://www.siggaard-andersen.dk/.

Formal Mathematical Representation of the Figge-Fencl Quantitative Physicochemical Model of Human Acid-Base Physiology in Blood Plasma (Version 3.0).
Copyright 2003 - 2013 James J. Figge.
Published 8 October, 2012; update published 28 April, 2013; update published 27 October, 2013.



SID + 1000 x ( (aH+) - Kw / (aH+) - Kc1 x PCO2 / (aH+) - Kc1 x Kc2 x PCO2 / (aH+)2 )

- [ Pitot ] x Z

+ { -1 / ( 1 + 10 ^ - ( pH - 8.5 ) )

- 98 / ( 1 + 10 ^ - ( pH - 3.9 ) )

- 18 / ( 1 + 10 ^ - ( pH - 11.7 ) )

+ 24 / ( 1 + 10 ^ + ( pH - 12.5 ) )

+ 2 / ( 1 + 10 ^ + ( pH - 5.80 ) )

+ 2 / ( 1 + 10 ^ + ( pH - 6.15 ) )

+ 2 / ( 1 + 10 ^ + ( pH - 7.51 ) )

+ 2 / ( 1 + 10 ^ + ( pH - 7.685 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.86 ) )

+ 50 / ( 1 + 10 ^ + ( pH - 10.3 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.12 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.22 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.10 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.49 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.01 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.31 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.75 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.36 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 4.85 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 5.76 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.17 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.73 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 5.82 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 5.10 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.70 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.20 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 8.0 ) )

- 1 / ( 1 + 10 ^ - ( pH - 3.1 ) ) } x 1000 x 10 x [ Albumin ] / 66500 = 0.



Where:

(aH+) = 10-pH ; (aH+) is the hydrogen ion activity, also used as an approximation of hydrogen ion concentration, [H+];

Z = ( K1 x (aH+)2 + 2 x K1 x K2 x (aH+) + 3 x K1 x K2 x K3 ) / ( (aH+)3 + K1 x (aH+)2 + K1 x K2 x (aH+) + K1 x K2 x K3 );

NB = 0.4 x ( 1 - 1 / ( 1 + 10 ^ ( pH - 6.9 ) ) );

Strong Ion Difference, SID, is given in mEq / L;

PCO2 is given in Torr;

Total concentration of inorganic phosphorus-containing species, [ Pitot ], is given in mmol / L;

[ Albumin ] is given in g / dL;

Kw = 4.4E-14 ( Eq / L )2;

Kc1 = 2.44E-11 ( Eq / L )2 / Torr;

Kc2 = 1.1E-10 ( Eq / L );

K1 = 1.22E-2 ( mol / L );

K2 = 2.19E-7 ( mol / L );

K3 = 1.66E-12 ( mol / L );

66500 g /mol is the molecular weight of albumin.


The above expression defines a function, fpH, which can be used to calculate the pH of plasma for any valid set of values for SID, PCO2, [ Pitot ], and [ Albumin ]:


pH = fpH { SID, PCO2, [ Pitot ], [ Albumin ] }


The function is too complex to be solved by hand and must be solved via an iterative approach on a computer. An application program is available for this purpose.

Link to Statistical Validation of Model 3.0

Link to Albumin Titration Curve (Model 3.0)

Link to Online Model 3.0 Application

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