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Mathematical Optimisation of Diver Ascent Profiles at a Constant Risk of Decompression Illness

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Mathematical Optimisation of Diver Ascent Profiles at a Constant Risk of Decompression Illness MATHEMATICAL OPTIMISATION OF DIVER ASCENT PROFILES AT A CONSTANT RISK OF DECOMPRESSION ILLNESS A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy m Computational and Applied Mathematics by B. J. Horn University of Canterbury 2003 11 rro my parents for encouraging a rove of [earning &V ~3~ 'blt ,Slf , HIS)3 iii Acknowledgements I wish to thank my supervisors Professor Graeme Wake (University of Canterbury) and Gavin Anthony (QinetiQ Alverstoke) for their encouragement and always being available for discussion, information and advice. I would also like to thank Dr. Chris Price for discussing the finer points of optimisation with me and Dr. Britta Basse for being a sounding board for ideas and sharing her MATLAB knowledge. I would also like to gratefully acknowledge the research grant and information provided by QinetiQ Alverstoke that made the research possible. This thesis would not have been completed without the never ending support and proof reading ability of my husband Clive who could always pick out the areas that needed clarification. As well as my kids who provided some light relief from the challenges of research. Finally I would like to acknowledge the ladies of the UCSA Creche, who welcomed my family into the creche community and provided a supportive, safe and fun environment for my daughters that allowed me to finish this research. - 1 JUN 2004 v Abstract Divers use decompression schedules that provide a stepped ascent to the surface from their maximum depth to help prevent the occurrence of decompression illness. The risk of decompression illness resulting from these schedules varies across different dives and the models used to generate them. The diver is unaware of this variance in risk. This thesis describes an investigation into the feasibility of producing optimised iso­ probabilistic decompression schedules that minimise the time it takes for the diver to reach the surface from maximum depth. In particular, 1.3 bar constant partial pressure of oxygen in helium dives are considered. The US Linear Exponential Multi-gas (LEM) model is used to describe the risk of decompression illness for a given dive. The Sequential Quadratic Programming (SQP) method is used to minimise the ascent time given non-linear risk constraints and a maximum dive time constraint. Two approaches to describing the ascent profile have been investigated. The first scheme finds the stop times at each possible stop depth to produce optimised schedules. The total time for decompression is a function of the sum of the stop times. The second scheme defines the ascent profile as a three parameter hyperbolic tangent equation. The SQP method finds the three parameters that produce optimised decompression schedules once the curve is converted to a schedule of decompression stops. The schedules produced by the SQP method, using a curve to describe the ascent profile, show that it is feasible to produce optimised iso-probabilistic tables that are operationally practical given an acceptable physiological risk model. Comparison with the QinetiQ 90 tables with a nominal 2% operational risk of decompression illness show that the method could provide reductions in the ascent time subject to manned testing. vi vii Table of contents Acknowledgements ................................................................................................................. iii Abstract ................................................................................................................................... v 1 Introduction 1.1 Research aim ...................................................................................................................... 1 1.2 Diving terminology ............................................................................................................ 2 1.3 Decompression illness ........................................................................................................ 2 1.4 Reducing the risk of decompression illness ....................................................................... 4 1.5 Why are new decompression methods required for constant partial pressure of oxygen in helium apparatus? .............................................................................................. 6 1.6 Scope of thesis ................................................................................................................... 7 1.7 Notation ............................................................................................................................. 8 1.8 Caution ............................................................................................................................... 8 2 Decompression methods 2.1 Notation ............................................................................................................................. 9 2.2 Introduction ...................................................................................................................... 10 2.3 Overview of decompression methods .............................................................................. 10 2.4 Representing the dive ....................................................................................................... 12 2.5 Physiological models ....................................................................................................... 15 2.6 A perfusion, diffusion or bubble limited process? ........................................................... 25 2.7 The contribution of oxygen to decompression illness risk .............................................. 27 2.8 Decompression illness risk models .................................................................................. 28 2.9 Safe Ascent Criteria ......................................................................................................... 32 2.10 Conclusions ...................................................................................................................... 35 3 Data 3.1 Introduction ...................................................................................................................... 37 3.2 The nature of dive trial data ............................................................................................. 37 3.3 Dive trial database ............................................................................................................ 38 3.4 Comparison of model predictions with trial data ............................................................. 48 3.5 Conc1usions ...................................................................................................................... 50 4 Review of the US Probabilistic Linear Exponential Multi-gas Model 4.1 Notation ........................................................................................................................... 51 4.2 Introduction ...................................................................................................................... 52 4.3 LEM model overview ...................................................................................................... 52 4.4 Physiological model ......................................................................................................... 53 4.5 Physiological assumptions ............................................................................................... 58 4.6 Risk model ....................................................................................................................... 60 4.7 LEM parameter set ........................................................................................................... 62 4.8 Sensitivity of LEM model to the format of partial pressure of oxygen input.. ................. 63 4.9 PROB3 analysis of oxygen in helium dives ..................................................................... 68 4.10 Conclusions ...................................................................................................................... 74 viii 5 Are current decompression methods iso-probabilistic? 5.1 Introduction ...................................................................................................................... 77 5.2 Comparison of risk associated with published decompression tables .............................. 77 5.3 Decompression advantages of higher partial pressures of oxygen in helium ................... 82 5.4 Iso-probability decompression tables ............................................................................... 85 5.5 What is an acceptable risk of decompression illness? ...................................................... 85 5.6 Conclusions ...................................................................................................................... 86 6 Application of sequential quadratic programming to decompression optimisation 6.1 Notation ............................................................................................................................ 87 6.2 Introduction ...................................................................................................................... 88 6.3 The general optimisation problem applied to decompression optimisation ...................... 88 6.4 Introduction to Sequential Quadratic Programming ......................................................... 90 6.5 Convergence and stopping criteria ................................................................................... 94 6.6 Evaluation
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