Sensitivity and uncertainty analyses for a sars model with time-varying inputs and outputs.

This paper presents a statistical study of a deterministic model for the transmission dynamics and control of severe acute respiratory syndrome (SARS). The effect of the model parameters on the dynamics of the disease is analyzed using sensitivity and uncertainty analyses. The response (or output) of interest is the control reproduction number, which is an epidemiological threshold governing the persistence or elimination of SARS in a given population. The compartmental model includes parameters associated with control measures such as quarantine and isolation of asymptomatic and symptomatic individuals. One feature of our analysis is the incorporation of time-dependent functions into the model to reflect the progressive refinement of these SARS control measures over time. Consequently, the model contains continuous time-varying inputs and outputs. In this setting, sensitivity and uncertainty analytical techniques are used in order to analyze the impact of the uncertainty in the parameter estimates on the results obtained and to determine which parameters have the largest impact on driving the disease dynamics.

Mathematical modelling to centre low tidal volumes following acute lung injury: a study with biologically variable ventilation.

BACKGROUND: With biologically variable ventilation [BVV--using a computer-controller to add breath-to-breath variability to respiratory frequency (f) and tidal volume (VT)] gas exchange and respiratory mechanics were compared using the ARDSNet low VT algorithm (Control) versus an approach using mathematical modelling to individually optimise VT at the point of maximal compliance change on the convex portion of the inspiratory pressure-volume (P-V) curve (Experimental). METHODS: Pigs (n = 22) received pentothal/midazolam anaesthesia, oleic acid lung injury, then inspiratory P-V curve fitting to the four-parameter logistic Venegas equation F(P) = a + b[1 + e-(P-c)/d]-1 where: a = volume at lower asymptote, b = the vital capacity or the total change in volume between the lower and upper asymptotes, c = pressure at the inflection point and d = index related to linear compliance. Both groups received BVV with gas exchange and respiratory mechanics measured hourly for 5 hrs. Postmortem bronchoalveolar fluid was analysed for interleukin-8 (IL-8). RESULTS: All P-V curves fit the Venegas equation (R2 > 0.995). Control VT averaged 7.4 +/- 0.4 mL/kg as compared to Experimental 9.5 +/- 1.6 mL/kg (range 6.6 - 10.8 mL/kg; p < 0.05). Variable VTs were within the convex portion of the P-V curve. In such circumstances, Jensen's inequality states "if F(P) is a convex function defined on an interval (r, s), and if P is a random variable taking values in (r, s), then the average or expected value (E) of F(P); E(F(P)) > F(E(P))." In both groups the inequality applied, since F(P) defines volume in the Venegas equation and (P) pressure and the range of VTs varied within the convex interval for individual P-V curves. Over 5 hrs, there were no significant differences between groups in minute ventilation, airway pressure, blood gases, haemodynamics, respiratory compliance or IL-8 concentrations. CONCLUSION: No difference between groups is a consequence of BVV occurring on the convex interval for individualised Venegas P-V curves in all experiments irrespective of group. Jensen's inequality provides theoretical proof of why a variable ventilatory approach is advantageous under these circumstances. When using BVV, with VT centred by Venegas P-V curve analysis at the point of maximal compliance change, some leeway in low VT settings beyond ARDSNet protocols may be possible in acute lung injury. This study also shows that in this model, the standard ARDSNet algorithm assures ventilation occurs on the convex portion of the P-V curve.

Fractal ventilation enhances respiratory sinus arrhythmia.

BACKGROUND: Programming 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. METHODS: Pigs 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. RESULTS: Mean 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. CONCLUSION: These 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.

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 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.