John F. Brewster

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

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 Jensens 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.

Fractal ventilation enhances respiratory sinus arrhythmia

BackgroundProgramming a mechanical ventilator with a biologically variable or fractal breathing pattern (an example of 1/f noise) improves gas exchange and respiratory mechanics. Here we show that fractal ventilation increases respiratory sinus arrhythmia (RSA) – a mechanism known to improve ventilation/perfusion matching.MethodsPigs were anaesthetised with propofol/ketamine, paralysed with doxacurium, and ventilated in either control mode (CV) or in fractal mode (FV) at baseline and then following infusion of oleic acid to result in lung injury.ResultsMean RSA and mean positive RSA were nearly double with FV, both at baseline and following oleic acid. At baseline, mean RSA = 18.6 msec with CV and 36.8 msec with FV (n = 10; p = 0.043); post oleic acid, mean RSA = 11.1 msec with CV and 21.8 msec with FV (n = 9, p = 0.028); at baseline, mean positive RSA = 20.8 msec with CV and 38.1 msec with FV (p = 0.047); post oleic acid, mean positive RSA = 13.2 msec with CV and 24.4 msec with FV (p = 0.026). Heart rate variability was also greater with FV. At baseline the coefficient of variation for heart rate was 2.2% during CV and 4.0% during FV. Following oleic acid the variation was 2.1 vs. 5.6% respectively.ConclusionThese findings suggest FV enhances physiological entrainment between respiratory, brain stem and cardiac nonlinear oscillators, further supporting the concept that RSA itself reflects cardiorespiratory interaction. In addition, these results provide another mechanism whereby FV may be superior to conventional CV.

The Design of Blocked Fractional Factorial Split-Plot Experiments

In industrial screening experiments, if some factors are hard to vary and others are easy to vary, randomization restrictions may lead to the use of fractional factorial split-plot (FFSP) designs. Blocked FFSP (BFFSP) designs arise when all runs cannot be performed under homogeneous conditions. This article introduces three approaches for blocking FFSP designs. A catalog of designs is presented in which BFFSP designs are ranked according to the minimum aberration criterion. In addition, additional optimality criteria are used to assist in the ranking procedure. The case study that motivated this research is also discussed in detail.

Blocked Fractional Factorial Split-Plot Experiments for Robust Parameter Design

Fractional factorial experiments are commonly used for robust parameter design and, for ease of use, such experiments are often run as split-plot designs. If the control factors are at the subplot level and the noise factors are at the whole-plot level, this also results in gains in efficiency. If all runs of the fractional factorial split-plot design cannot be run under homogeneous conditions, such designs are frequently blocked. In this paper, we explore the choice of blocked fractional factorial split-plot designs for use in robust parameter design. A ranking scheme for such designs is developed and, using a search algorithm, a catalog of 32-run optimal designs is provided. Two situations are considered, one in which the control factors are at the subplot level and one in which the control factors are at the whole-plot level. An example from the aerospace sector is used to illustrate the concepts.

Optimal Foldover Plans for Two-Level Fractional Factorial Split-Plot Designs

We consider the construction of foldovers of two-level fractional factorial split-plot designs. As a follow-up strategy, the foldover technique is useful when the objective is to systematically de-alias effects of interest after the initial fractional factorial split-plot design has been run. Due to the sequential nature in which the runs of the initial and foldover designs are conducted, the resulting combined design (the design consisting of both the initial design and its foldover) is a blocked fractional factorial split-plot design. Using the minimum aberration criterion for blocked fractional factorial split-plot designs, in conjunction with other optimality criteria, we provide a catalog of optimal foldover plans for initial minimum aberration fractional factorial split-plot designs consisting of 16 and 32 runs.

Variable ventilation and Jensen's inequality: citation corrections

We would like to commend Huhle and co-authors [1] for their extensive review of variable ventilation featuring citations from our pioneering work. We would suggest a couple of corrections to the cited articles, however. The authors reference important work by Venegas and colleagues [2], including Figure 4 in their manuscript. This figure explains the convex and concave portions of the sigmoidal pressure-volume curve indicating where variable ventilation has beneficial and detrimental effects, respectively, as suggested by Jensen’s theorem. No mention of Jensen’s ‘theorem’ or ‘inequality’ is discussed in the paper by Venegas et al. This explanation is advanced in a paper by us and not cited (‘Convexity, Jensen’s inequality and benefits of noisy mechanical ventilation’ [3]). The data supplement in our paper gives a comprehensive mathematical explanation for conditions where non-linear systems can and cannot benefit from the addition of noise. Mechanical ventilation is but one example where noisy life support systems may provide benefit to patients.

The Role of Academic Statisticians in Quality Improvement Strategies of Small and Medium Sized Companies

This paper will describe the relationships and interactions that have developed between academic statisticians in the Department of Statistics at the University of Manitoba and small to medium sized companies. The size of the local community and the limited resources available to many of these small companies mean that few knowledgeable commercial consultants are readily available. As a result, these companies have turned to the statisticians at the University of Manitoba for assistance. The paper will provide some details on the type and range of activities that the Department has undertaken. The benefits of these activities to the local industrial community as well as the benefits to the academic statisticians and their students are described.