Oxyhemoglobin dissociation curve as expo-exponential paradigm of asymmetric sigmoid function

Erratum to: Blood HbO2 and HbCO2 dissociation curves at varied O2, CO2, pH, 2,3-DPG and temperature levels

New mathematical model equations for O(2) and CO(2) saturations of hemoglobin (S(HbO)(2) and S(HbCO)(2) are developed here from the equilibrium binding of O(2) and CO(2) with hemoglobin inside RBCs. They are in the form of an invertible Hill-type equation with the apparent Hill coefficients KHbO(2) and KHbCO(2) in the expressions for SHbO(2) and SHbCO(2) dependent on the levels of O(2) and CO(2) partial pressures (P(O)(2) and P(CO)(2)), pH, 2,3-DPG concentration, and temperature in blood. The invertibility of these new equations allows PO(2) and PCO(2) to be computed efficiently from S(HbO)(2) and S(HbCO)(2) and vice versa. The oxyhemoglobin (HbO(2)) and carbamino-hemoglobin (HbCO(2)) dissociation curves computed from these equations are in good agreement with the published experimental and theoretical curves in the literature. The model solutions describe that, at standard physiological conditions, the hemoglobin is about 97.2% saturated by O(2) and the amino group of hemoglobin is about 13.1% saturated by CO(2). The O(2) and CO(2) content in whole blood are also calculated here from the gas solubilities, hematocrits, and the new formulas for S(HbO)(2) and S(HbCO)(2). Because of the mathematical simplicity and invertibility, these new formulas can be conveniently used in the modeling of simultaneous transport and exchange of O(2) and CO(2) in the alveoli-blood and blood-tissue exchange systems.

Blood HbO2 and HbCO2 Dissociation Curves at Varied O2, CO2, pH, 2,3-DPG and Temperature Levels

New mathematical model equations for O2 and CO2 saturations of hemoglobin (S(HbO2) and S(HbCO2)) are developed here from the equilibrium binding of O2 and CO2 with hemoglobin inside RBCs. They are in the form of an invertible Hill-type equation with the apparent Hill coefficients K(HbO2) and K(HbCO2) in the expressions for S(HbO2) and S(HbCO2) dependent on the levels of O2 and CO2 partial pressures (P(O2) and P(CO2), pH, 2,3-DPG concentration, and temperature in blood. The invertibility of these new equations allows P(O2) and P(CO2) to be computed efficiently from S(HbO2) and S(Hbco2) and vice-versa. The oxyhemoglobin (HbO2) and carbamino-hemoglobin (HbCO2) dissociation curves computed from these equations are in good agreement with the published experimental and theoretical curves in the literature. The model solutions describe that, at standard physiological conditions, the hemoglobin is about 97.2% saturated by O2 and the amino group of hemoglobin is about 13.1% saturated by CO2. The O2 and CO2 content in whole blood are also calculated here from the gas solubilities, hematocrits, and the new formulas for S(HbO2) and S(HbCO2). Because of the mathematical simplicity and invertibility, these new formulas can be conveniently used in the modeling of simultaneous transport and exchange of O2 and CO2 in the alveoli-blood and blood-tissue exchange systems.

Gompertz pharmacokinetic model for drug disposition

PURPOSE

Disposition of drugs among compartments of the body usually occurs at changing rates that are commonly modeled as sums of exponential terms with different rate constants. This paper describes an alternative. Gompertz kinetics, in which the rates can change systematically.

METHODS

Differential equations were developed and solved that fit typical examples taken from the literature. The three or four constants required for a visually satisfactory fit to data could readily be found by successive adjustment "by hand," but strategies and results are presented for computer fitting of the data.

RESULTS

In four examples, the amount remaining in the blood decreases as an exponentially declining fraction of the amount present at any moment, but the antecedent processes responsible for that amount differ as follows: (a) In simple i.v. disposition (e.g., lidocaine) concentration falls as a decelerated exponential decay. (b) Delayed i.v. disposition (e.g., hexobarbital) requires, as well, a decelerated exponential growth function. (c) In simple disposition after oral administration, the concentration in the blood initially increases at a decelerating rate. (d) In biphasic oral disposition (e.g., Li+ carbonate), the initial Gompertz growth is followed by decelerated exponential decay.

CONCLUSIONS

Gompertz kinetics provides an accurate and parsimonious mathematical model describing drug disposition.

Using hippocampal slices to study how aging alters ion regulation in brain tissue

Aging alters ion regulation in brain tissue. This article describes methods useful for studying such age-related changes in the rat hippocampal slice preparation. Topics considered include (a) selection of appropriate age groups of rats for aging studies, (b) a description of methods for preparing and maintaining hippocampal slices, (c) measurement of intracellular pH with the H+-sensitive dye carboxy-SNARF-1, and (d) measurement of extracellular pH and K+ with cation-selective microelectrodes.

Is the pulmonary pressure-volume curve symmetrical with respect to the inflection point?

The following is the abstract of the article discussed in the subsequent letter:

Venegas, José G., R. Scott Harris, and Brett A. Simon. A comprehensive equation for the pulmonary pressure-volume curve. J. Appl. Physiol. 84(1): 389–395, 1998.—Quantification of pulmonary pressure-volume (P-V) curves is often limited to calculation of specific compliance at a given pressure or the recoil pressure (P) at a given volume (V). These parameters can be substantially different depending on the arbitrary pressure or volume used in the comparison and may lead to erroneous conclusions. We evaluated a sigmoidal equation of the form, V = a + b[1 +  e −(P− c)/ d ]−1, for its ability to characterize lung and respiratory system P-V curves obtained under a variety of conditions including normal and hypocapnic pneumoconstricted dog lungs ( n = 9), oleic acid-induced acute respiratory distress syndrome ( n = 2), and mechanically ventilated patients with acute respiratory distress syndrome ( n = 10). In this equation, a corresponds to the V of a lower asymptote, b to the V difference between upper and lower asymptotes, c to the P at the true inflection point of the curve, and d to a width parameter proportional to the P range within which most of the V change occurs. The equation fitted equally well inflation and deflation limbs of P-V curves with a mean goodness-of-fit coefficient ( R 2) of 0.997 ± 0.02 (SD). When the data from all analyzed P-V curves were normalized by the best-fit parameters and plotted as (V −  a)/ b vs. (P −  c)/ d, they collapsed into a single and tight relationship ( R 2 = 0.997). These results demonstrate that this sigmoidal equation can fit with excellent precision inflation and deflation P-V curves of normal lungs and of lungs with alveolar derecruitment and/or a region of gas trapping while yielding robust and physiologically useful parameters.

Gompertz growth in number dead confirms medflies and nematodes show excess oldster survival

A recent report (Easton, 1995) showed that, at least for Mediterranean fruit flies, a Gompertz growth equation based on the increase in number of individuals that die is a better predictor of survival data than is the classical Gompertz survivorship model based on the decrease in number that survive (analysis of medfly data of Carey et al., 1992). In the growth model, the rate of increase of the number dead (i.e., the death rate) decreases exponentially with age. The poor fit of the classical model predicts "excess survival" of older members, but, when the scale of the better-fitting growth model is increased 2400x, such excess is now also evident as a small but distinctly separate cohort of the medfly subjects. The smaller population appears to be about 0.01% of the larger, and the death rate decreases about one-fourth as fast. Survival of the nematode C. elegans (Brooks et al., 1994) is also better predicted by the growth model, which also shows excess survival of the worms at great age.

Dynamic model of oxygen transport for transcutaneous PO2 analysis

A dynamic model of oxygen transport through the outer skin layers and a polarographic sensor was developed for the analysis of transcutaneous oxygen tension (tcPO2). It provides a basis for quantifying the factors that determine the relationship between tcPO2 and arterial oxygen tension (PaO2). Model simulations show the importance of stratum papillare metabolic oxygen consumption; the oxygen permeability of the skin relative to that of the sensor membrane and electrolyte; and temperature and the oxyhemoglobin dissociation curve. These simulations were consistent with experimental data obtained by using microcathode transcutaneous oxygen sensors, which were placed on the skin of 10 healthy adults. Furthermore, the model indicates that accurate evaluation of arterial oxygen tension by using transcutaneous measurements requires continuous estimation of skin perfusion. On the basis of tcPO2 measurements made during arterial occlusion, simulations indicate that quantitative evaluation of the metabolic oxygen consumption of the viable skin tissues is possible only when the oxygen permeabilities of the skin and sensor are known.

A mathematical model for the computation of the oxygen dissociation curve in human blood

The mathematical relations developed by various researchers for the oxygen dissociation curve are reviewed. Using well-known mechanisms of chemical kinetics of various species in the blood, we have developed a mathematical formula to compute the oxygen dissociation curve in the blood showing its dependence on the pH and PCO2. The functional form, proposed here, is much simpler in comparison to those available in the literature for use in the mathematical modelling of O2 transport in the pulmonary and systemic circulations. In the process, the well-known Hill's equation has been generalized showing an explicit dependence on PCO2 and pH. It is shown that the oxygen dissociation curve computed from our comparatively simpler equation, fits in fairly well with the documented data and shows realistic shift with PCO2 and pH.

Microelectrode studies of facilitated O2 transport across hemoglobin and myoglobin layers

Experimentally measured PO2 profiles across layers of hemoglobin and myoglobin solutions were compared with profiles predicted from facilitated transport theory assuming chemical equilibrium. Measurements across myoglobin layers were in excellent agreement with theory, but measurements across hemoglobin layers departed from theory at low PO2. This departure was greatest for salt-free hemoglobin solution, which may be caused by an electrical potential formed by a pH gradient in the layer as oxyhemoglobin is deoxygenated.

Mathematical model of cardiac mechanogram rhythmicity (based on mollusc heart)

1. The time course of force development by the heart is modelled by Gompertz kinetics from the product of two terms: a cumulative increase in relative number of activated "contractile units", and an exponential decrease in contractile force. 2. For each beat, an "initial condition" is specified by an "intrinsic tension" parameter, and a specific rate of change of tension; cardioactive agents change these specifications. 3. Depending on parameter values, heartbeats are predicted that are constant, or in which the frequency, amplitude and baseline tension are appropriate to inhibited or augmented cardiac activity.

Prediction of X-ray induced mitotic delay and recovery of G2 cells

A mathematical model is presented that predicts the delay of mitosis caused by X-irradiation of an asynchronous, exponentially growing cell culture (data of Schneiderman & Schneiderman, 1984). In the model, based on Gompertz kinetics, the driving function to generate the curves is a simple exponential decay expression. For the delayed mitotic progress curves, this function characterizes the distribution of the time required for cells to enter mitosis.

Interpretation of oxygen disappearance curves measured in blood perfused tissues

We have developed a two compartment (tissue and blood) lumped parameter model to interpret oxygen disappearance curves (O2 DCs) measured in vivo with PO2 microelectrodes in tissues which are perfused with blood. To include the properties of the oxyhemoglobin equilibrium curve (HEC), we used an algorithm we have recently developed for both standard and nonstandard conditions. The new blood and tissue model is more useful than a previous analysis using the Hill equation for blood and constant oxygen consumption (VO2). The model can be adapted for constant (zero-order) consumption, Michelis-Menten kinetics, or for double cytochrome systems. Examples for the former include brain, and for the latter, carotid body. The models are discussed in relationship to experimental microelectrode measurements in gerbil brain and in cat carotid body after blood flow occlusion.

Modelling whole blood oxygen equilibrium: comparison of nine different models fitted to normal human data

The ability of nine different models, prominent in the literature, to meaningfully characterize the oxygen-hemoglobin equilibrium curve (OHEC) of normal individuals was examined. Previously reported data (N = 33), obtained using the DCA-1 (Radiometer, Copenhagen), and new data (N = 8), obtained using the Hemox-Analyzer (TCS, Southampton, PA), from blood samples of normal, non-smoking volunteers were used and these devices were found to give statistically similar results. The OHECs were digitized and fitted to the models using least-squares techniques developed in this laboratory. The "goodness-of-fit" was determined by the root-mean-squared (RMS) error, the number of parameters, and the parameter redundancy, i.e., correlation between the parameters. The best RMS error did not necessarily indicate the best model. Most literature models consist of ratios of similar-order polynomials. These showed considerable parameter redundancy which made the curve fitting difficult. The best fits gave RMS errors as low as 0.2% saturation. The Hill model gave a good characterization over the saturation range 20%-98% with RMS errors of about 0.6% saturation. On the other hand, good characterizations over the entire range were given by several other models. The relative advantages and disadvantages of each model have been compared as well as the difficulties in fitting several of the models. No single model is best under all circumstances. The best model depends upon the particular circumstances for which it is to be utilized.

An evaluation of Easton's paradigm for the oxyhemoglobin equilibrium curve

A new paradigm for the oxyhemoglobin equilibrium curve proposed by Easton (1979) has been fit to human and dog blood saturation data by a simple linear regression algorithm. The equation derived from Easton's paradigm is characterized by only two parameters, and can fit saturation data between 0 and 95% with a root mean square error less than 0.5%. The upper 5% of the curve is not adequately described. Easton's equation is more accurate than the empirical Hill (1910) equation and approaches the accuracy of the more complicated Adair (1925) equation in this range.