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Mathematische simulation des respirationssystems

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Mathematical Simulation of the Respiratory System. The respiratory system is described as a feedback control system. The controller consists of the peripheral chemoreceptors and the central chemosensitive structures, the respiratory centre in the medulla oblongata and the thorax-lung pump which they drive. The controlled system is comprised of three compartments (lung, brain and the remaining tissue) connected by the blood circulation. The controlled values are arterial pH and arterial O2 partial pressure and cerebral extracellular pH. Earlier models have been improved by: (1) the dead space description, (2) the thermodynamic formulation of the CO2 dissociation equation and the simple but accurate O2 dissociation equation of the blood, (3) the alteration of the CO2 dissociation equation for the brain and the remaining tissue to accommodate recent results, (4) the application of the one-receptor-theory of central chemosensitivity, (5) the pH dependence of brain circulation, (6) the bicarbonate exchange between blood and extracellular fluid of the brain and (7) the introduction of variable circulation times. Respiratory and metabolic disturbances of the respiratory system are analyzed.

The mathematical formulation of the respiratory system is a differential difference equation system.

In the steady state the experimental results are reproduced fairly well. A slight discrepancy is found in the simulation of metabolic acidosis. Apparently we have assumed the sensitivity of the peripheral chemoreceptors to be too large so that the respiratory response is not correctly predicted.

In the numerical solution there is an overshoot in the on-transient and a damped oscillation in the off-transient of the alveolar CO2 partial pressure during respiratory acidosis. We have varied the parameters to make deviations small. The best agreement seems to result, if the central treshold is near the normal extracellular pH of the brain. A further deviation from experimental findings is that the cerebral CO2 and H+ concentration, the blood circulation of the brain, the alveolar O2 partial tension and the ventilation show a slight oscillation in the off-transient. Except for these discrepancies the experimental results, especially the stability of the extracellular pH of the brain, are reproduced fairly well. During hypoxia there are deviations form the experimental results if the central residual activity is constant and the central threshold deviates from the normal extracellular pH of the brain. But if the central residual activity is pH dependent and if the central threshold is equal to the normal extracellular pH of the brain, then the time course of VE and the other variables agree fairly well with experimental results. There is also a good correspondence between the theoretical and experimental data during hyperoxia. During metabolic acidosis the time constant of the bicarbonate exchange between blood and extracellular fluid of the brain is important. If a time constant of one minute is assumed, then the predicted and the experimental results correspond sufficiently well.

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  1. Institut für Physiologie I, Ruhr-Universität Bochum, Postfach 2148, D-4630, Bochum, Bundesrepublik Deutschland

    T. Middendorf &  H. H. Loeschcke

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  1. T. Middendorf
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Mit Unterstützung durch die Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 114 (Bionach).

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Middendorf, T., Loeschcke, H.H. Mathematische simulation des respirationssystems. J. Math. Biology 3, 149–177 (1976). https://doi.org/10.1007/BF00276203

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