Limit cycle stability analysis and adaptive control of a multi-compartment model for a pressure-limited respirator and lung mechanics system

Abstract

Acute respiratory failure due to infection, trauma or major surgery is one of the most common problems encountered in intensive care units, and mechanical ventilation is the mainstay of supportive therapy for such patients. In this article, we develop a general mathematical model for the dynamic behaviour of a multi-compartment respiratory system in response to an arbitrary applied inspiratory pressure. Specifically, we use compartmental dynamical system theory and Poincaré maps to model and analyse the dynamics of a pressure-limited respirator and lung mechanics system, and show that the periodic orbit generated by this system is globally asymptotically stable. Furthermore, we show that the individual compartmental volumes, and hence the total lung volume, converge to steady-state end-inspiratory and end-expiratory values. Finally, we develop a model reference direct adaptive controller framework for the multi-compartmental model of a pressure-limited respirator and lung mechanics system where the plant and reference model involve switching and time-varying dynamics. We then apply the proposed adaptive feedback controller framework to stabilise a given limit cycle corresponding to a clinically plausible respiratory pattern.

Keywords:

• mechanical ventilation
• multi-compartment model
• bronchial tree
• limit cycle analysis
• periodic orbits
• stability of periodic orbits
• Poincaré maps
• adaptive control

Acknowledgements

This research was supported in part by the US Army Medical Research and Material Command under Grant 08108002 and NSF under Grant ECS-0601311.

Notes

Note

1. Note that since (3) is a time-varying dynamical system it is typical to denote its solution as ŝ(t, t ₀, x ₀) to indicate the dependence on both the initial time t ₀ and the initial state x ₀. In this article, we assume that t ₀ = 0 and define s(t, x ₀) ≜ ŝ(t, 0, x ₀).