Mathematical Models of Oxygen and Carbon Dioxide Storage and Transport: The Acid-Base Chemistry of Blood

Abstract

This article describes a mathematical model of the acid-base chemistry of blood. The model is formulated from first principles by considering the "components" of blood and the reaction equations in the plasma and erythrocyte fractions. Equations are formulated to describe the total concentration of blood components, the physicochemical properties, and the equilibrium position of reactions. The model includes 28 equations and 12 parameters. All equations can be solved from six variables included in the model. The model uses simple mathematics, without introducing intermediate concepts or linear coefficients necessary for algebraic solution. Model equations are solved simultaneously using numerical methods. Model parameters are estimated and the model verified for plasma, fully oxygenated blood, and deoxygenated blood. Published data are used to estimate model parameters and normal conditions and to verify model simulations. The model reproduces experimental results, including addition or removal of CO2, or strong acid to plasma; CO2, strong acid or haemoglobin to blood; and the effects of deoxygenating blood. The model can also be used as the basis for models of whole body CO2 transport as illustrated in the accompanying article. As such, it is possible to simulate the effects on blood of physiological changes in ventilation or metabolism.

Converting venous acid-base and oxygen status to arterial in patients with lung disease

“…Figure 1 illustrates the method for calculating arterial acid-base and oxygenation status from measurements in the venous blood and pulse oximetry, described in detail previously [7]. The method calculates arterial values using mathematical models [9] to simulate the transport of venous blood back through the tissues until simulated arterial oxygenation matches that measured by pulse oximetry. In this simulation, oxygen is added to and carbon dioxide removed from venous blood using a constant value of the respiratory quotient (RQ), selected in the present study to be 0.82 [7], and no strong acid is removed from the blood on its simulated passage from the veins to the arteries, i.e.…”

The Intelligent Ventilator (INVENT) project: The role of mathematical models in translating physiological knowledge into clinical practice

“…This evaluation was reported in Rees and Andreassen [62] figure 8. This latter data set is the characteristic distribution of HCO 3 between blood and interstitial fluid [47], which was the basis for the criticism of BE leading to the Great-Transatlantic debate.…”

“…The approach taken here to modeling the acid-base of blood has been to formulate models with The parameterization of this model is described in detail in Rees and Andreassen [62]. and BE in the combined interstitial fluid and tissue compartment as the weighted sum of the individual concentrations.…”

“…The purpose of the model of acid-base chemistry described in [62] was to both have the functionality of the models of Siggaard-Andersen, i.e. including red blood cells and BohrHaldane effects; with the formulation of Stewart, i.e.…”

“…Whilst his equations do not therefore have the coverage of Siggaard-Andersen's in describing blood, it can be argued that they are more transparent and therefore accessible to the non-mathematically inclined. This type of formulation also has the benefit that appropriate, additive, state variables can be selected, a requirement which has been important for the development of INVENT From an acid-base perspective the major contribution of this work has been to illustrate the similarity between the approaches of Siggaard-Andersen and Stewart [52] and to develop a model with the coverage of Sigaard-Andersen and the structure of Stewart [62,63]. This model has been validated against literature data illustrating that it could simulate the addition and removal of CO 2 strong acid/base, and haemoglobin; and the effects of oxygenation or deoxygenation, including reproduction of values of the Haldane and BE coefficients.…”