[Mathematical simulation of the respiratory system (author's transl)]

Mathematical models of periodic breathing and their usefulness in understanding cardiovascular and respiratory disorders

Periodic breathing is an unusual form of breathing with oscillations in minute ventilations and with repetitive apnoeas or near apnoeas. Reported initially in patients with heart failure or stroke, it was later recognized to occur especially during sleep. The recurrent hypoxia and surges of sympathetic activity that often occur during the apnoeas have serious health consequences. Mathematical models have helped greatly in the understanding of the causes of recurrent apnoeas. It is unlikely that every instance of periodic breathing has the same cause, but many result from instability in the feedback control involved in the chemical regulation of breathing caused by increased controller and plant gains and delays in information transfer. Even when it is not the main cause of the periodic breathing, unstable control modifies the ventilatory pattern and sometimes intensifies the recurrent apnoeas. The characteristics of disturbances to breathing and their interaction with the control system can be critical in determining ventilation responses and the occurrence of periodic breathing. Large abrupt changes in ventilation produced, for example, in the transition from waking to sleep and vice versa, or in the transition from breathing to apnoea, are potent factors causing periodic breathing. Mathematical models show that periodic breathing is a 'systems disorder' produced by the interplay of multiple factors. Multiple factors contribute to the occurrence of periodic breathing in congestive heart failure and cerebrovascular disease, increasing treatment options.

Central chemosensitivity and the reaction theory

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Transient CO2 elimination and storage as functions of the ventilatory response to CO2

The effect on CO2 storage and elimination of variations in the slope and intercept of the ventilatory response to CO2 curve was examined. Theoretical and experimental results show that although CO2 elimination rate following a transient ventilatory disturbance is decreased at low ventilatory response slopes, this decrease can be compensated by elevated PCO2 intercepts, or thresholds. Conversely, high CO2 elimination rate following a ventilatory disturbance due to a high ventilatory response slope can be off-set by a depressed PCO2 threshold. The results suggest that elevated thresholds which often accompany depressed ventilatory response slopes may be part of a compensatory mechanism for minimizing transient hypercapnia and acidosis.

Cooperation of peripheral and central chemosensitive mechanisms in the control of the extracellular pH in brain in non-respiratory acidosis

The mathematical model of the respiratory control system in man of Middendorf and Loeschcke (1976 a, b) opens the possibility to stimulate the constellation of parameters in non-respiratory acidosis. Several investigators agree that the pH in CSF or in the extracellular fluid of the brain stays remarkably constant in this situation and it can be shown that this is a result of a precise control rather than the consequence of a sluggishly reacting system. Application of the model assuming constant extracellular brain pH allowed to calculate the relative sensitivities to pH changes of the central and the peripheral sensory mechanisms generating respiratory drive. Assuming air breathing and a normal critical arterial O2-pressure and otherwise normal parameters of respiration, circulation and blood composition (except diminished buffer base) the central chemosensitivity to a pH change turned out to be 25 times the peripheral. This factor is critically dependent on the ratio of the bicarbonate change in extracellular brain fluid to that in arterial blood. The coinciding data of Fencl (1971) and of Kronenberg and Cain (1968) were used for the calculation.