The Figge-Fencl Quantitative Physicochemical Model
of Human Acid-Base Physiology

by James J. Figge, MD, MBA, FACP

Copyright 2003 - 2009 James J. Figge.
Updated 22 December, 2007; 28 December, 2008.
Updated version published 15 January, 2009 on www.Acid-Base.org and www.Figge-Fencl.org.

The Figge-Fencl Quantitative Physicochemical Model of Human Acid-Base Physiology is also described in the following publication:

Figge J. Role of Non-Volatile Weak Acids (Albumin, Phosphate and Citrate). In Stewart's Textbook of Acid-Base. Kellum JA and Elbers PWG, editors. Amsterdam: AcidBase.org. 2009. Chapter 11, pages 217 - 232. [ The Formal Mathematical Representation of the Figge-Fencl Model is published in the Appendix, pages 230 - 232 ].

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 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. 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. These include the phosphoric acid - phosphate system, the citric acid - citrate 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 are based on NMR spectroscopy data (temperature-corrected from 25 to 37 degrees Celsius using the van't Hoff equation). The Figge-Fencl model also takes into account the neutral-to-base (N-B) structural transition of human serum albumin over the pH range of 6 to 9, and the model 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. Based on the albumin titration curve of Halle and Lindman (temperature-corrected from 22 to 37 degrees C) it is estimated that there are up to nine low-titrating lysine residues with pK(a) values distributed within the range of 5.8 to 8.4 (inclusive) in two clusters; one cluster is near 6.0 and the second cluster is near 8.0. Final pK(a) values in the Figge-Fencl model are established using an iterative least-squares fit to the data of Figge, Rossing and Fencl for albumin-containing solutions. All permitted combinations of trial pK(a) values for 59 lysine and 2 histidine residues are systematically analyzed in successive model iterations. The optimal set of parameters is the single set that minimizes the sum of the squares of the deviations between measured and calculated pH.

At pH 7.40, the Figge-Fencl model predicts that the charge contributed by 4.4 g / dL of albumin is approximately -11.8 mEq / L. Within the pH range of biologic interest (6.9 to 7.9) the Figge-Fencl model predicts that the approximate negative charge contributed by albumin in human plasma at a given pH is:

[ ZAlbumin ] = -10 x [ Albumin ] x (0.1204 x pH - 0.625);

where [ ZAlbumin ] is in mEq / L and [ Albumin ] is in g / dL. Over the stated pH range, the molar buffer capacity of albumin is 8.0 Eq / mol / pH unit. Thus, for [ Albumin ] of 4.4 g / dL, the buffer capacity of albumin is: 5.3 mEq / L / pH unit. The buffer capacity of albumin is used in the formulation of the classic van Slyke equation for plasma.

The Figge-Fencl quantitative physicochemical model 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 ], [ Citratetot ] }.



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-, plasma proteins, and divalent citrate ion) 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 ]. This model accounted for the effects of the microenvironments within the macromolecule of albumin on the pK(a) values of individual histidine residues. Temperature-corrected pK(a) values for histidine are also employed in the Figge-Fencl quantitative physicochemical model of acid-base physiology. Temperature correction from 25 to 37 degrees Celsius (298 to 310 degrees K) is accomplished using the van't Hoff equation, which yields a correction factor of -0.2, as follows. The standard molar enthalpy for the dissociation of a proton from an imidazole group of histidine will vary according to the protein microenvironment. A representative value is 6,940 cal / mol. R is the gas constant, 1.987 cal / deg K / mol. The natural logarithm of 10, ln(10), is 2.302585. The van't Hoff equation to adjust the pK(a) of a histidine residue from 298 to 310 degrees K is as follows:
pK(a)310 = pK(a)298 + (6940 / ln(10) / R)(1 / 310 - 1 / 298) = pK(a)298 - 0.2.
The imidazole groups of histidine residues are of particular importance because they can titrate within the pH range of biologic interest.

[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, 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. Therefore, the Figge-Fencl quantitative physicochemical model of acid-base physiology, as presented below, uses a least-squares procedure to estimate the total number. Based on the albumin titration curve of Halle and Lindman [ reference 13 ] (corrected from 22 to 37 degrees C) it is estimated that there are up to nine low-titrating lysine residues with pK(a) values distributed within the range of 5.8 to 8.4 (inclusive), with one cluster near 6.0, and a second cluster near 8.0. Hence, the model allows up to nine low-titrating lysine residues to populate trial pK(a) values distributed as follows: N1 residues are assigned a pK(a) of LYS1 = 5.8; N2 residues a pK(a) of LYS2 = 6.0; N3 residues a pK(a) of LYS3 = 7.6; N4 residues a pK(a) of LYS4 = 7.8; N5 residues a pK(a) of LYS5 = 8.0; N6 residues a pK(a) of LYS6 = 8.2; and N7 residues a pk(a) of LYS7 = 8.4. The parameters N1, N2, N4, N5 and N6 are each allowed to vary from 0 to 2, inclusive, and N3 and N7 from 0 to 1, inclusive, with the provision that N1 + N2 + N3 + N4 + N5 + N6 + N7 cannot exceed 9. The optimized values of the parameters N1, N2, N3, N4, N5, N6 and N7 are determined by a least-squares algorithm using a computer program as described below. There are a total of 59 lysine residues in human albumin. Consequently, the number of normally-titrating lysine residues in the model is (59 - N1 - N2 - N3 - N4 - N5 - N6 - N7). The pK(a) of lysine residues in polypeptides is typically near 10.5. Consequently, the parameter LYS8, representing the pK(a) of a normally-titrating lysine residue, is allowed to assume a value of 10.3, 10.4 or 10.5. These fall within the range of published textbook values for the pK(a) of lysine [ references 14 and 15 ]. This model accounts for the contribution of all 59 lysine residues in human serum albumin. 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 a standard textbook value [ references 14 and 15 ].

[8] In the Figge-Fencl quantitative model 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 presented below, the 36 aspartic acid and 62 glutamic acid residues in albumin are each assigned a pK(a) of 4.0, 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] The pK(a) values of two histidine residues that could not be determined by NMR are designated as HIS15 and HIS16. HIS15 is constrained by the NMR data and is allowed to vary from 6.8 to 7.8, inclusive. HIS16 is allowed to vary from 5.5 to 6.5, inclusive, encompassing a range of standard textbook values for histidine after temperature correction to 37 degrees C [ references 14 and 15 ]. The values of HIS15 and HIS16 are determined as part of the least-squares algorithm in the computer program below. A third incompletely characterized histidine residue is assigned a temperature-corrected pK(a) value of 5.2, consistent with available NMR data.

[11] The computer program for the albumin model is written in Visual BASIC. The parameters, HIS15, HIS16, N1 through N7, and LYS8 are allowed to vary through all possible combinations of values (within the stated admissible limits). Consequently, there are a total of 340,494 combinations of values for HIS15, HIS16, N1 to N7, and LYS8 that are tested in the model ([3 x 3 x 2 x 3 x 3 x 3 x 2 - 34] x 11 x 11 x 3 = 340,494). A least-squares algorithm is employed using the data of Figge, Rossing and Fencl for albumin-containing solutions [ reference 2 ]. Since there are 65 data points in this database, the model calculations are run a total of 22,132,110 times in the computer program. The optimal set of parameters is the single set that minimizes the sum of the squares of the deviations between measured and calculated pH. The program takes under ten hours to run in compiled form on a 2.80 GHz processor. The single set of optimized parameters is displayed below, following the computer program listing.

[12] 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 17 ]. 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.

[13] 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 from this analysis. Following this, a formal mathematical representation of the optimized model is presented. An application program is also available for performing calculations with the optimized Figge-Fencl quantitative physicochemical model.

[14] Concentrations of substances are noted by brackets, [ X ], and the appropriate units will be provided. Except as noted below, [ X ] will be a non-negative value. However, in the case of substances carrying a variable charge per mole, the charge will be calculated and the algebraic sign associated with the charge will already be incorporated in the numerical value of the concentration. To avoid ambiguity, the symbol [ Zx ] will be employed when the algebraic sign associated with the charge is already incorporated in the numerical value of the concentration. This applies to the following calculated value:
[ ZAlbumin ].

[15] K1, K2 and K3 are the apparent equilibrium dissociation constants for phosphoric acid for plasma [ Reference 19 ]. pK1 = 1.915; pK2 = 6.66; pK3 = 11.78.

[16] C1, C2 and C3 are the apparent equilibrium dissociation constants for citric acid. Values of these constants at 37 degrees Celsius were taken from reference 20: pK1 = 3.11; pK2 = 4.75; pK3 = 6.43. The constants were then adjusted to an ionic strength of 0.16 M for use in the Figge-Fencl model. The adjusted values are: pK1 = 2.98; pK2 = 4.37; pK3 = 5.79. The adjustments were made using the equation pK' = pK + nA(I1/2)/{1 + 1.6(I1/2)}; where n = (2 x charge of the acid - 1); I = 0.16; and A is the constant of the Debye-Hueckel equation. At 37 degrees Celsius, A = 1.825 x 106{0.993/( (74.27)3 (310)3 )}1/2 = 0.521.

[17] 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 / mmHg = 0.0307 mmol / L / mmHg. Kc1 = (10-6.1) (0.0307) / (1000) = 2.44 x 10-11. Hence, calculations using Kc1 in the Figge-Fencl model yield results identical to those calculated with the Henderson-Hasselbalch equation.

[18] The constant Kc2, the second dissociation constant for carbonic acid, is calculated from the formula (equation 9) given by Harned and Scholes [ reference 21 ]. 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.

References:

1. Stewart, PA. Modern quantitative acid-base chemistry. Can J Physiol Pharmacol. 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. Fifth Edition. 2002. Pages 50 and 74.

16. 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. ].

17. 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 ].

18. 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 ].

19. 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 ].

20. http://analytical.biochem.purdue.edu/221/wwwboard/handouts/supplemental/buffer.pdf

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

Computer Program.

Sub Model()

Rem: Figge-Fencl Quantitative Physicochemical Model
Rem: of Human Acid-Base Physiology.
Rem:
Rem: Program by James J. Figge, MD, MBA, FACP. Updated 22 December, 2007;
Rem: Updated 28 December, 2008.
Rem: Copyright 2003 - 2009 James J. Figge.

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

Const kw = 0.000000000000044

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

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 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

Const LYS3 = 7.6
Const LYS4 = 7.8
Const LYS5 = 8
Const LYS6 = 8.2
Const LYS7 = 8.4

Const HIS14 = 5.2

minss = 9999999

For g1 = 680 To 780 Step 10
HIS15 = g1 / 100

For g2 = 550 To 650 Step 10
HIS16 = g2 / 100

For q = 1030 To 1050 Step 10
LYS8 = q / 100

For n1 = 0 To 2
For n2 = 0 To 2

For n3 = 0 To 1
For n4 = 0 To 2
For n5 = 0 To 2
For n6 = 0 To 2
For n7 = 0 To 1

LowTitrate = n1 + n2 + n3 + n4 + n5 + n6 + n7

If LowTitrate > 9 Then GoTo GetNext

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+ concentration)
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 - 4))))

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)))

alb2 = alb2 + (59 - LowTitrate) / (1 + (10 ^ (pH - LYS8)))

Rem: 16 different histidine residues
Rem: Correction factor to convert HIS pKa from 25 deg C to 37 deg C is approx -0.2
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 7.19 + NB)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 7.29 + NB)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 7.17 + NB)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 7.56 + NB)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 7.08 + NB)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 7.38)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 6.82)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 6.43)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 4.92)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 5.83)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 6.24)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 6.8)))
alb2 = alb2 + 1 / (1 + (10 ^ (pH - 5.89)))
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
minss = ss

Open "C:\Documents and Settings\James\My Documents\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

Print #1, "LYS8 = ", LYS8

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 n7
Next n6
Next n5
Next n4
Next n3
Next n2
Next n1

Next q

Next g2
Next g1

End Sub

Data from Figge J, Rossing TH, Fencl V. J Lab Clin Med.
1991; 117:453-467 (Table A).
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 = 2.70805008875636E-02
s= -0.721372312510383
ss = 7.67876879398109E-02

HIS14 = 5.2
HIS15 = 6.8
HIS16 = 5.5

LYS1 = 5.8; n1 = 2
LYS2 = 6.0; n2 = 2
LYS3 = 7.6; n3 = 1
LYS4 = 7.8; n4 = 2
LYS5 = 8.0; n5 = 2
LYS6 = 8.2; n6 = 0
LYS7 = 8.4; n7 = 0
LYS8 = 10.3

slope = 0.998653587250776
intercept = 2.10660055814287E-02
intercept = 2.10660055825596E-02 (verify)
r = 0.991464333427831
r^2 = 0.983001524459492
Variance = 1.09166100868144E-03
Variance of slope = 2.7374422760327E-04
Stnd Deviation of slope = 1.65452176656359E-02
98% confidence interval for the slope = 0.959160152682903 to 1.03814702181865



The sum of squares of the differences between pH (measured) and pH (calculated) is 0.0768.

The optimized parameters are as follows:

The pK(a)’s of two histidine residues not determined by NMR spectroscopy:
HIS15 = 6.80
HIS16 = 5.50

The low-titrating lysine residues were assigned the following pK(a) values:
LYS1: pK(a) = 5.80; N1 = 2
LYS2: pK(a) = 6.00; N2 = 2
LYS3: pK(a) = 7.60; N3 = 1
LYS4: pK(a) = 7.80; N4 = 2
LYS5: pK(a) = 8.00; N5 = 2
LYS6: pK(a) = 8.20; N6 = 0
LYS7: pK(a) = 8.40; N7 = 0

The normally-titrating lysine residues were assigned the following pK(a) value:
LYS8: pK(a) = 10.30

Model 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) = 4.0

18 Tyr residues; pK(a) = 11.7

24 Arg residues; pK(a) = 12.5

59 Lys residues; 2 with pK(a) = 5.80; 2 with pK(a) = 6.00; 1 with pK(a) = 7.60; 2 with pK(a) = 7.80; 2 with pK(a) = 8.00; and 50 with pK(a) = 10.30

16 His residues; with pK(a)'s of 7.19; 7.29; 7.17; 7.56; 7.08; 7.38; 6.82; 6.43; 4.92; 5.83; 6.24; 6.80; 5.89; 5.20; 6.80; and 5.50 (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 negative charge of -17.8446 Eq / mol. Therefore, in human plasma at pH 7.40 with [Albumin] = 4.4 g / dL, the charge attributed to albumin per direct calculations from the model is -11.8 mEq / L, calculated as follows:

[ ZAlbumin ] = (-17.8446 Eq / mol) x (10 dL / L) x (1000 mEq / Eq) x (4.4 g / dL) / (66500 g / mol) = -11.8 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 predicts that the titration curve of human serum albumin (negative charge displayed by albumin, expressed as mEq per g of albumin, versus pH) is approximately linear over the pH range of 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 range:

[ ZAlbumin ] = -10 x [ Albumin ] x (0.1204 x pH - 0.625);

where [ ZAlbumin ] 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 -11.7 mEq / L.


Molar Buffer Capacity of Albumin as Calculated per the Model.

The molar buffer capacity of albumin within the pH range of 6.9 to 7.9 can be calculated from the slope of the linear regression fit given above:

(0.1204 mEq / g / pH unit) x (66500 g / mol) x (Eq / 1000 mEq) = 8.0 Eq / mol / pH unit.

This value is identical to that given by Siggaard-Andersen and Fogh-Andersen [16].

For [ Albumin ] of 4.4 g / dL, the buffer value of albumin at that concentration is:

(0.1204 mEq / g / pH unit) x (4.4 g / dL) x (10 dL / L) = 5.3 mEq / L / pH unit.

As explained in the online supplement to reference 18 [accessible at http://www.atsjournals.org/doi/suppl/10.1164/ajrccm.162.6.9904099 ] and in reference 16, the buffer capacity of albumin is used in the formulation of the classic 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.
Copyright 2003 - 2009 James J. Figge.
Updated 22 December, 2007; Updated 28 December, 2008. Updated version published 15 January, 2009.


SID + 1000 x ( (aH+) - Kw / (aH+) - Kc1 x Pco2 / (aH+) - Kc1 x Kc2 x Pco2 / (aH+)2 )

- [ Pitot ] x Zp - [ Citratetot ] x Zc

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

- 98 / ( 1 + 10 ^ - ( pH - 4.0 ) )

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

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

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

+ 2 / ( 1 + 10 ^ + ( pH - 6.00 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.60 ) )

+ 2 / ( 1 + 10 ^ + ( pH - 7.80 ) )

+ 2 / ( 1 + 10 ^ + ( pH - 8.00 ) )

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

+ 1 / ( 1 + 10 ^ + ( pH - 7.19 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.29 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.17 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.56 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.08 + NB ) )

+ 1 / ( 1 + 10 ^ + ( pH - 7.38 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.82 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.43 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 4.92 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 5.83 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.24 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.80 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 5.89 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 5.20 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 6.80 ) )

+ 1 / ( 1 + 10 ^ + ( pH - 5.50 ) )

+ 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+];

Zp = ( 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 );

Zc = ( C1 x (aH+)2 + 2 x C1 x C2 x (aH+) + 3 x C1 x C2 x C3 ) / ( (aH+)3 + C1 x (aH+)2 + C1 x C2 x (aH+) + C1 x C2 x C3 );

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

[ Citratetot ] is given in mmol / L;

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

Pco2 is given in mmHg;

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 / mmHg;

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

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

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

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

C1 = 1.05E-3 ( mol / L );

C2 = 4.27E-5 ( mol / L );

C3 = 1.62E-6 ( 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-like solutions containing albumin for any valid set of values for SID, Pco2, [ Pitot ], [ Albumin ], and [ Citratetot ]:


pH = fpH { SID, Pco2, [ Pitot ], [ Albumin ], [ Citratetot ] }


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

Link to Validation against Independent Data

Link to Albumin Titration Curve

Link to Online Model Application

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