European Journal of Applied Physiology

, Volume 116, Issue 1, pp 97–113 | Cite as

Simple accurate mathematical models of blood HbO2 and HbCO2 dissociation curves at varied physiological conditions: evaluation and comparison with other models

  • Ranjan K. DashEmail author
  • Ben Korman
  • James B. BassingthwaighteEmail author
Original Article



Equations for blood oxyhemoglobin (HbO2) and carbaminohemoglobin (HbCO2) dissociation curves that incorporate nonlinear biochemical interactions of oxygen and carbon dioxide with hemoglobin (Hb), covering a wide range of physiological conditions, are crucial for a number of practical applications. These include the development of physiologically-based computational models of alveolar-blood and blood-tissue O2–CO2 transport, exchange, and metabolism, and the analysis of clinical and in vitro data.

Methods and results

To this end, we have revisited, simplified, and extended our previous models of blood HbO2 and HbCO2 dissociation curves (Dash and Bassingthwaighte, Ann Biomed Eng 38:1683–1701, 2010), validated wherever possible by available experimental data, so that the models now accurately fit the low HbO2 saturation (\(S_{{{\text{HbO}}_{ 2} }}\)) range over a wide range of values of \(P_{{{\text{CO}}_{ 2} }}\), pH, 2,3-DPG, and temperature. Our new equations incorporate a novel \(P_{{{\text{O}}_{ 2} }}\)-dependent variable cooperativity hypothesis for the binding of O2 to Hb, and a new equation for P 50 of O2 that provides accurate shifts in the HbO2 and HbCO2 dissociation curves over a wide range of physiological conditions. The accuracy and efficiency of these equations in computing \(P_{{{\text{O}}_{ 2} }}\) and \(P_{{{\text{CO}}_{ 2} }}\) from the \(S_{{{\text{HbO}}_{ 2} }}\) and \(S_{{{\text{HbCO}}_{ 2} }}\) levels using simple iterative numerical schemes that give rapid convergence is a significant advantage over alternative \(S_{{{\text{HbO}}_{ 2} }}\) and \(S_{{{\text{HbCO}}_{ 2} }}\) models.


The new \(S_{{{\text{HbO}}_{ 2} }}\) and \(S_{{{\text{HbCO}}_{ 2} }}\) models have significant computational modeling implications as they provide high accuracy under non-physiological conditions, such as ischemia and reperfusion, extremes in gas concentrations, high altitudes, and extreme temperatures.


O2 and CO2 binding to hemoglobin O2 and CO2 saturation of hemoglobin Oxyhemoglobin and carbaminohemoglobin dissociation curves Nonlinear O2–CO2 interactions Bohr and Haldane effects Mathematical modeling 


\(\alpha_{{{\text{O}}_{ 2} }}\)

Solubility of oxygen in water

\(\alpha_{{{\text{CO}}_{ 2} }}\)

Solubility of carbon dioxide in water


Concentration of free oxygen


Concentration of free carbon dioxide


Concentration of hydrogen ions (protons)


Concentration of 2,3-diphosphoglycerate (2,3-DPG)





\(P_{{{\text{O}}_{ 2} }}\)

Partial pressure of oxygen

\(P_{{{\text{CO}}_{ 2} }}\)

Partial pressure of carbon dioxide


Partial pressure of oxygen for 50 % HbO2 saturation







\(K_{{{\text{HbO}}_{ 2} }}\)

Apparent equilibrium constant for the binding of oxygen to hemoglobin

\(K_{{{\text{HbCO}}_{ 2} }}\)

Apparent equilibrium constant for the binding of carbon dioxide to hemoglobin

\(S_{{{\text{HbO}}_{ 2} }}\)

Saturation of hemoglobin with oxygen

\(S_{{{\text{HbCO}}_{ 2} }}\)

Saturation of hemoglobin with carbon dioxide



We thank the reviewers for helpful and insightful comments that have enhanced the overall quality of the manuscript. This work was supported by the National Institute of Health Grants P50-GM094503 and P01-GM066730. The extension of P 50 model to extreme/wider physiological conditions (e.g. pH > 8.5) was motivated by RKD’s email correspondence with Stefan Kleiser (University Hospital Zurich), a user of the 2010 Dash and Bassingthwaighte \(S_{{{\text{HbO}}_{ 2} }}\) and \(S_{{{\text{HbCO}}_{ 2} }}\) models.


  1. Adair GS (1925) The hemoglobin system VI. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529–545Google Scholar
  2. Austin WH, Lacombe E, Rand PW, Chatterjee M (1963) Solubility of carbon dioxide in serum from 15 to 38 C. J Appl Physiol 18:301–304PubMedGoogle Scholar
  3. Bassingthwaighte JB, Beard DA, Carlson BE, Dash RK, Vinnakota K (2012) Modeling to link regional myocardial work, metabolism and blood flows. Ann Biomed Eng 40:2379–2398PubMedPubMedCentralCrossRefGoogle Scholar
  4. Bauer C, Schroder E (1972) Carbamino compounds of haemoglobin in human adult and foetal blood. J Physiol 227:457–471PubMedPubMedCentralCrossRefGoogle Scholar
  5. Buerk DG, Bridges EW (1986) A simplified algorithm for computing the variation in oxyhemoglobin saturation with pH, PCO2, T and DPG. Chem Eng Commun 47:113–124CrossRefGoogle Scholar
  6. Dash RK, Bassingthwaighte JB (2004) Blood HbO2 and HbCO2 dissociation curves at varied O2, CO2, pH, 2,3-DPG and temperature levels. Ann Biomed Eng 32:1676–1693PubMedCrossRefGoogle Scholar
  7. Dash RK, Bassingthwaighte JB (2006) Simultaneous blood-tissue exchange of oxygen, carbon dioxide, bicarbonate, and hydrogen ion. Ann Biomed Eng 34:1129–1148PubMedPubMedCentralCrossRefGoogle Scholar
  8. Dash RK, Bassingthwaighte JB (2010) Erratum to: blood HbO2 and HbCO2 dissociation curves at varied O2, CO2, pH, 2,3-DPG and temperature levels. Ann Biomed Eng 38:1683–1701PubMedPubMedCentralCrossRefGoogle Scholar
  9. Forster RE, Constantine HP, Craw MR, Rotman HH, Klocke RA (1968) Reaction of CO2 with human hemoglobin solution. J Biol Chem 243:3317–3326PubMedGoogle Scholar
  10. Geers C, Gros G (2000) Carbon dioxide transport and carbonic anhydrase in blood and muscle. Physiol Rev 80:681–715PubMedGoogle Scholar
  11. Heath MT (2002) Scientific computing: an introductory survey. The McGraw-Hill Companies Inc, BostonGoogle Scholar
  12. Hedley-Whyte J, Laver MB (1964) O2 Solubility in Blood and Temperature Correction Factors for P O2. J Appl Physiol 19:901–906PubMedGoogle Scholar
  13. Hlastala MP, Woodson RD, Wranne B (1977) Influence of temperature on hemoglobin-ligand interaction in whole blood. J Appl Physiol Respir Environ Exerc Physiol 43:545–550PubMedGoogle Scholar
  14. Joels N, Pugh LG (1958) The carbon monoxide dissociation curve of human blood. The Journal of Physiology 142:63–77PubMedPubMedCentralCrossRefGoogle Scholar
  15. Kelman GR (1966a) Calculation of certain indices of cardio-pulmonary function, using a digital computer. Respir Physiol 1:335–343PubMedCrossRefGoogle Scholar
  16. Kelman GR (1966b) Digital computer subroutine for the conversion of oxygen tension into saturation. J Appl Physiol 21:1375–1376PubMedGoogle Scholar
  17. Kelman GR (1967) Digital computer procedure for the conversion of PCO2 into blood CO2 content. Respir Physiol 3:111–115PubMedCrossRefGoogle Scholar
  18. Mateják M, Kulhanek T, Matousek S (2015) Adair-based hemoglobin equilibrium with oxygen, carbon dioxide and hydrogen ion activity. Scand J Clin Lab Invest 75:113–120PubMedCrossRefGoogle Scholar
  19. Matthew JB, Morrow JS, Wittebort RJ, Gurd FR (1977) Quantitative determination of carbamino adducts of alpha and beta chains in human adult hemoglobin in presence and absence of carbon monoxide and 2,3-diphosphoglycerate. J Biol Chem 252:2234–2244PubMedGoogle Scholar
  20. Naeraa N, Petersen ES, Boye E (1963) The influence of simultaneous, independent changes in pH and carbon dioxide tension on the in vitro oxygen tension-saturation relationship of human blood. Scand J Clin Lab Invest 15:141–151PubMedCrossRefGoogle Scholar
  21. Pozrikidis C (2008) Numerical computations in science and engineering. Oxford University Press, New YorkGoogle Scholar
  22. Rees SE, Andreassen S (2005) Mathematical models of oxygen and carbon dioxide storage and transport: the acid-base chemistry of blood. Crit Rev Biomed Eng 33:209–264PubMedCrossRefGoogle Scholar
  23. Reeves RB (1980) The effect of temperature on the oxygen equilibrium curve of human blood. Respir Physiol 42:317–328PubMedCrossRefGoogle Scholar
  24. Rossi-Bernardi L, Roughton FJ (1967) The specific influence of carbon dioxide and carbamate compounds on the buffer power and Bohr effects in human haemoglobin solutions. J Physiol 189:1–29PubMedPubMedCentralCrossRefGoogle Scholar
  25. Roughton FJW, Deland EC, Kernohan JC, Severinghaus JW (1972) Some recent studies of the oxyhemoglobin dissociation curve of human blood under physiological conditions and the fitting of the Adair equation to the standard curve. In: Rørth M, Astrup P (eds) Proceedings of the oxygen affinity of hemoglobin and red cell acid base status, proceedings of the alfred benzon symposium IV held at the premises of the royal danish academy of sciences and letters, pp 73–81Google Scholar
  26. Roughton FJ, Severinghaus JW (1973) Accurate determination of O2 dissociation curve of human blood above 98.7 percent saturation with data on O2 solubility in unmodified human blood from 0 degrees to 37 degrees C. J Appl Physiol 35:861–869PubMedGoogle Scholar
  27. Severinghaus JW (1979) Simple, accurate equations for human blood O2 dissociation computations. J Appl Physiol Respir Environ Exerc Physiol 46:599–602PubMedGoogle Scholar
  28. Siggaard-Andersen O (1971) Oxygen-linked hydrogen ion binding of human hemoglobin. Effects of carbon dioxide and 2,3-diphosphoglycerate. I. Studies on erythrolysate. Scand J Clin Lab Invest 27:351–360PubMedCrossRefGoogle Scholar
  29. Siggaard-Andersen O, Garby L (1973) The Bohr effect and the Haldane effect. Scand J Clin Lab Invest 31:1–8PubMedCrossRefGoogle Scholar
  30. Siggaard-Andersen O, Salling N (1971) Oxygen-linked hydrogen ion binding of human hemoglobin. Effects of carbon dioxide and 2,3-diphosphoglycerate. II. Studies on whole blood. Scand J Clin Lab Invest 27:361–366PubMedCrossRefGoogle Scholar
  31. Siggaard-Andersen O, Siggaard-Andersen M (1990) The oxygen status algorithm: a computer program for calculating and displaying pH and blood gas data. Scand J Clin Lab Invest Suppl 203:29–45PubMedCrossRefGoogle Scholar
  32. Siggaard-Andersen O, Rorth M, Norgaard-Pedersen B, Andersen OS, Johansen E (1972a) Oxygen-linked hydrogen ion binding of human hemoglobin. Effects of carbon dioxide and 2,3-diphosphoglycerate. IV. Thermodynamical relationship between the variables. Scand J Clin Lab Invest 29:303–320PubMedCrossRefGoogle Scholar
  33. Siggaard-Andersen O, Salling N, Norgaard-Pedersen B, Rorth M (1972b) Oxygen-linked hydrogen ion binding of human hemoglobin. Effects of carbon dioxide and 2,3-diphosphoglycerate. Scand J Clin Lab Invest 29:185–193PubMedCrossRefGoogle Scholar
  34. Siggaard-Andersen O, Wimberley PD, Gothgen I, Siggaard-Andersen M (1984) A mathematical model of the hemoglobin-oxygen dissociation curve of human blood and of the oxygen partial pressure as a function of temperature. Clin Chem 30:1646–1651PubMedGoogle Scholar
  35. Tyuma I (1984) The Bohr effect and the Haldane effect in human hemoglobin. Jpn J Physiol 34:205–216PubMedCrossRefGoogle Scholar
  36. von Restorff W, Holtz J, Bassenge E (1977) Exercise induced augmentation of myocardial oxygen extraction in spite of normal coronary dilatory capacity in dogs. Pflugers Arch 372:181–185CrossRefGoogle Scholar
  37. Winslow RM, Swenberg ML, Berger RL, Shrager RI, Luzzana M, Samaja M, Rossi-Bernardi L (1977) Oxygen equilibrium curve of normal human blood and its evaluation by Adair’s equation. J Biol Chem 252:2331–2337PubMedGoogle Scholar
  38. Winslow RM, Samaja M, Winslow NJ, Rossi-Bernardi L, Shrager RI (1983) Simulation of continuous blood O2 equilibrium curve over physiological pH, DPG, and PCO2 range. J Appl Physiol Respir Environ Exerc Physiol 54:524–529PubMedGoogle Scholar
  39. Wolf MB (2013) Whole body acid-base and fluid-electrolyte balance: a mathematical model. Am J Physiol Renal Physiol 305:F1118–F1131PubMedCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Physiology, Biotechnology and Bioengineering CenterMedical College of WisconsinMilwaukeeUSA
  2. 2.Department of Anaesthesia and Pain MedicineRoyal Perth HospitalPerthAustralia
  3. 3.Department of BioengineeringUniversity of WashingtonSeattleUSA

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