Development and Analysis of a Simple Grey-Box Model of Central Sleep Apnea Ali Kazerani

126 Pages·2013·10.52 MB·English

Development and Analysis of a Simple Grey-Box
Model of Central Sleep Apnea
by
Ali Kazerani
B.A.Sc., University of Waterloo (2010)
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2013
© Massachusetts Institute of Technology 2013. All rights reserved.
A u th or ........... . .... ......................................
Department of Electrical Engineering and Computer Science
May 22, 2013
C ertified by ....... ..............................
I
John N. Tsitsiklis
Professor
Thesis Supervisor
Certified by....L......... .......... ....................
George C. Verghese
Professor
Thesis Supervisor
Accepted by.
/6/0 Leslie A. Kolodziejski
Chairman, Departmental Committee on Graduate Students

Development and Analysis of a Simple Grey-Box Model of Central
Sleep Apnea
by
Ali Kazerani
Submitted to the Department of Electrical Engineering and Computer Science
on May 23, 2013, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
In this thesis, we develop and analyze a simple grey-box model that describes the pathophysiology
of central sleep apnea (CSA). We construct our model following a thorough survey of published
approaches. Special attention is given to PNEUMA, a complex, comprehensive model of human
respiratory and cardiovascular physiology that brings together many existing physiological models.
We perform a sensitivity analysis, concluding that signals of interest in PNEUMA are insensitive
to changes in all but approximately twenty parameters. This justifies our goal of developing a
small, simple model that captures approximately the same behaviour among signals of interest.
The simplicity of our model not only makes it accessible to analytical and intuitive exploration, but
also opens up the possibility that its parameters could be reliably estimated from a patient's data
records. This could be of great value in developing patient-specific or state-specific treatments for
CSA.
Our model describes the dynamics of the alveolar gas exchange, blood gas transport, and cerebral
gas exchange processes, which together determine the cerebral and arterial partial pressures of
carbon dioxide, given ventilation as input. Our model of the ventilatory controller senses both
the cerebral and arterial carbon dioxide partial pressures and issues a ventilatory drive command
from which the level of ventilation is determined, closing the loop. We develop a linearized small-
signal model of our system and determine conditions for its stability. We conclude by comparing
the stability predictions suggested by our linear analysis to the stability properties of our original
nonlinear model, with promising results.
Thesis Supervisor: John N. Tsitsiklis
Title: Professor
Thesis Supervisor: George C. Verghese
Title: Professor

Acknowledgments
Mom (Parichehr) and Dad (Mehrdad), no words will ever be enough. Thank you, for absolutely
everything. To my brother, Iman: thank you for, amongst so many other things, never failing to
make me smile. I love you all so dearly.
To my research advisors, Professors John Tsitsiklis and George Verghese: I am remarkably
fortunate to work with you. Thank you for demonstrating, without fail, every professional and
personal virtue imaginable, along with something resembling omniscience. And of course, thank
you in particular for all the guidance you offered as I wrote (and you read) this thesis.
Thanks also to Dr. Thomas Heldt, who taught me so much both in class and during research
meetings.
I would like to thank the Periodic Breathing Foundation for a generous gift to MIT, which made
this work possible. I offer my sincere thanks also to Robert Daly, for the inspiration he provided
and for sharing his experience and expertise with us in discussions, meetings, and tours.
I am incredibly grateful to all those who guided and helped me while, not so very long ago, I was
searching for a research home. My thanks in particular to Professors Leslie Kolodziejski and Alan
Oppenheim, for looking after me through the tumult. I would also like to thank Janet Fischer for
all her help and apparently inexhaustible patience. And I will forever owe a great debt to Professor
Terry Orlando, who spent a great many hours over many months helping me find and make my way
in this place. My journey would have been very, very different if I had not met him.
PNEUMA, a physiological model implementation that I explore in this thesis, is supported and
distributed by the University of Southern California Biomedical Simulations Resource.

Contents
1 Introduction 13
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Contributions and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 PNEUMA 21
2.1 Introduction to PNEUMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Simulating Stable and Unstable Breathing in PNEUMA . . . . . . . . . . . . . . . . 23
2.3 Issues with PNEUMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Preparing PNEUMA for Numerical Experiments . . . . . . . . . . . . . . . . . . . . 29
2.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 Our Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.2 Some Additional Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.3 R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 A New, Simple Model 43
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 Pulmonary Gas Exchange Plant, PA . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.2 Lung-to-Carotid Transport Plant, Pa. . . . . . . . . . . . . . . . . . . . . ... 45
3.1.3 Cerebral Gas Exchange Plant, Pb . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.4 Chemoreflex Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.5 Ventilation Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7
8 CONTENTS
3.2 Model Properties ...................................... 48
3.2.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.2 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Model Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Pulmonary Gas Exchange Model . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.2 Cerebral Gas Exchange Model . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.3 Lung-to-Carotid Transport Model . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.4 Chemoreflex Control of Ventilation . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.4.1 The System Operating Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.5.1 Incorporating Pad6 Approximants . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5.2 Using Linearized Plant Models . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Stability Analysis 101
4.1 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.1 The Linearized Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.2 Stability of the Linear Model System . . . . . . . . . . . . . . . . . . . . . . . 104
4.2 Influence of Central and Peripheral Gains on Stability . . . . . . . . . . . . . . . . . 111
4.3 Stability of our Nonlinear Model System . . . . . . . . . . . . . . . . . . . . . . . . . 116
5 Conclusion 121
5.1 Future Work . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 122
References 125
List of Figures
1.1 Lung volume waveforms for normal breathing and CSA. . . . . . . . . . . . . . . . . 15
2.1 PNEUMA's top-level block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 PNEUMA-simulated waveforms representing normal, steady conditions in sleep. . . . 24
2.3 PNEUMA-simulated waveforms representing CSA. . . . . . . . . . . . . . . . . . . . 25
2.4 The continuous tidal volume and arterial carbon dioxide tension waveforms. . . . . . 31
2.5 J and i. ............ ... ... . ...................... 33
2.6 Sensitivity versus parameter rank, for v' . ........................ 36
2.7 Sensitivity versus parameter rank, for &. . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 Sensitivity versus parameter rank, for T. . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.9 Baseline and perturbed -i3w aveforms for the six highest-ranked parameters. ..... 39
2.10 Baseline and perturbed - waveforms for the six highest-ranked parameters. . . . . . 40
3.1 Block diagram of our new, simple model. . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Autoregulation of cerebral blood flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Step response of Lange model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Step responses of D1M and DOM models best fitting the D2M step response . . . . . 67
3.5 Responses of D2M, D1M, and DOM models to sinusoidal input. . . . . . . . . . . . . 69
3.6 Responses of Pad6-approximated transport models to sinusoidal input. . . . . . . . . 71
3.7 Phase delay versus period, for adjusted and unadjusted Pade-approximated transport
m od els. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.8 Filtered Pa,C02 and Sa,0 2 waveforms from PNEUMA simulation of CSA. . . . . . . . 79
9
10 LIST OF FIGURES
3.9 [jPa,CO2 - Pa,C02,TH]+, [SaO 2,TH - Sa,02]+, and [Pa,CO2 - Pa,CO2,TH]+ [Sa,0 2,TH - Sa,0 2]±
wa veform s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.10 GPNEU.M A,a jPa,C0 2 - Pa,C02,TH]+ [Sa,02,TH - Sa,021+ and Ga jPa,C02 - Pa,C02,TH]+
wa veform s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.11 Ventilation, tidal volume, and respiratory rate versus chemical drive, in PNEUMA. . 82
3.12 The subsystem made up of the gas exchange and transport model systems. . . . . . . 91
3.13 Steady-state CSA simulation results from our model (black) and PNEUMA (red). . 94
3.14 Simulation results from our model when Pad6-approximated transport models are used. 95
3.15 Simulation results from our model when linearized gas exchange models are used. . . 96
4.1 Our large-signal linearized model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Our small-signal linearized model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 The XA= 0 hyperbola in the first quadrant of the (G', G') plane. . . . . . . . . . . . 115
4.4 Stable and unstable regions in the (G', G') plane. . . . . . . . . . . . . . . . . . . . . 116
4.5 Stability boundary, along with numerically-determined stability of points in the
(G' ,G') plane, using m = n = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.6 Stability boundary using m= n = 1, along with numerically-determined stability of
points in the (G', G') plane, using m = n = 2 and m = n = 3. . . . . . . . . . . . . . 118
4.7 Stability of the nonlinear and linearized models. . . . . . . . . . . . . . . . . . . . . . 119