Convexity, Jensen's inequality and benefits of noisy mechanical ventilation

Abstract

Mechanical ventilators breathe for you when you cannot or when your lungs are too sick to do their job. Most ventilators monotonously deliver the same-sized breaths, like clockwork; however, healthy people do not breathe this way. This has led to the development of a biologically variable ventilator--one that incorporates noise. There are indications that such a noisy ventilator may be beneficial for patients with very sick lungs. In this paper we use a probabilistic argument, based on Jensen's inequality, to identify the circumstances in which the addition of noise may be beneficial and, equally important, the circumstances in which it may not be beneficial. Using the local convexity of the relationship between airway pressure and tidal volume in the lung, we show that the addition of noise at low volume or low pressure results in higher mean volume (at the same mean pressure) or lower mean pressure (at the same mean volume). The consequence is enhanced gas exchange or less stress on the lungs, both clinically desirable. The argument has implications for other life support devices, such as cardiopulmonary bypass pumps. This paper illustrates the benefits of research that takes place at the interface between mathematics and medicine.

Figure 1

Comparison of noisy and monotonous ventilation strategies under a four-parameter logistic model. a=0 ml (volume at lower asymptote), b=1200 ml (volume range), c=30 cm H₂O (pressure at inflection point) and d=7 cm H₂O (index of linear compliance). The pressure for the monotonous strategy is 18 cm H₂O (open blue circle). The pressures for the noisy strategy are uniformly distributed between 10 and 26 cm H₂O, with a mean of 18 cm H₂O (red circle). Probability density functions are on the margins (red curves). The mean volume for the noisy strategy is 205.73 ml (red circle), greater than 183.13 ml, the constant (or mean) volume for the monotonous strategy (open blue circle).

Figure 2

A normalized snapshot of a normal breathing pattern (300 breaths). Normal breathing patterns exhibit autocorrelation and have fractal characteristics. This figure is based on tidal volumes that were acquired from a healthy, spontaneously breathing individual over a number of breaths. The observed volumes were centred at zero (by subtracting the mean) and scaled to keep the resulting values within the range −1 to +1. Such a signal can drive a ventilator using engineered software and hardware, with an appropriately scaled version of the signal being added as noise to a monotonous tidal volume.