Homeostasis Revisited in the Genesis of Stress Reactivity
by
Pedram Ataee
B.Sc., Electrical Engineering, University of Tehran, Iran, 2005
M.Sc., Electrical Engineering, University of Tehran, Iran, 2007
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Doctor of Philosophy
in
THE FACULTY OF GRADUATE AND POSTDOCTORAL
STUDIES
(Electrical and Computer Engineering)
The University Of British Columbia
(Vancouver)
March 2014
c Pedram Ataee, 2014
Abstract
Autonomic-cardiac regulation operates through interactions between the autonomic
nervous system (ANS) and the cardiovascular system (CVS). In order to maintain
homeostasis in the CVS, the ANS adjusts it effectors, such as the stiffness of blood
vessels and the pace of heartbeats, against physical and psychological stressors, so
that it can maintain adequate blood flow. This allows oxygen and nutrients to be
delivered to organs and enables the performance of other essential functions.
Autonomic-cardiac regulation can be described by a mathematical model and
it can be analyzed under different scenarios such as a stressful condition or an increased arterial stiffness. This may help researchers to obtain new understandings
of the autonomic-cardiac regulation. This thesis is built upon a physiology-based
mathematical model of autonomic-cardiac regulation describing the regulation of
heart rate (HR) and blood pressure (BP), using a set of nonlinear, coupled differential equations with delay.
Non-invasive and subject-specific monitoring of autonomic-cardiac regulation
has the potential to improve current treatments of autonomic-cardiac disorders.
A parameter estimation method has been used to specify time-varying subjectspecific model parameters associated with autonomic-cardiac regulation. The proposed method will help to improve monitoring of autonomic-cardiac variables,
such as sympathetic and parasympathetic nerve activities affecting the heart and
sympathetic nerve activity affecting the arterial tree.
The complex dynamic interactions between nonlinearities and delays in the
autonomic-cardiac regulation may result in the onset of instabilities in BP and HR
regulation. In this thesis, we propose a model-based approach to stability analysis
and introduce a quantitative stability indicator of the autonomic-cardiac regulaii
tion. We can prevent irregularities in cardiovascular rhythms (e.g., HR and BP) by
knowing their causes and developing an intelligent method to control them.
An artificial bionic baroreflex can be an effective treatment for baroreflex failure in, for example, individuals with severe orthostatic hypotension. We propose a
method to design an artificial bionic baroreflex by mimicking the baroreflex mechanism in the body. This could then be potentially used to adjust existing neurostimulator devices that regulate BP.
iii
Preface
The work presented in this thesis has been partially published in different journals
or conference proceedings. The list of these publications is provided below. I have
been the main author for all publications and have had the main role in generating
the ideas, developing the methodologies, processing the data, and analyzing the
results.
The work presented in Chapter 3 has been partially published in the Proceedings of Computer in Cardiology Conference in 2010 [1], and has been accepted for
publication in the IEEE Transactions on Biomedical Engineering [2].
Parts of the work presented in Chapter 4 have been published in the Proceedings of the 33rd Annual International Conference of the IEEE EMBS in 2011 [3].
Chapter 5 is based on the work published in the Proceedings of the 35th Annual
International Conference of the IEEE EMBS in 2013 [4].
Chapter 6 is based on the work published in the Proceedings of the 34th Annual
International Conference of the IEEE EMBS in 2012 [5].
The conclusions provided in Chapter 7 are based on the papers published in
IEEE Transactions on Biomedical Engineering [2], Proceedings of the Annual International Conference of the IEEE EMBS [3–5], Computer in Cardiology Conference [1], and American Control Conference [6].
iv
The list of publications resulted in this thesis is as follows:
Journal Articles
• P. Ataee, J.O. Hahn, Dumont, G.A., and W.T. Boyce. Non-Invasive SubjectSpecific Monitoring of Autonomic-Cardiac Regulation. IEEE Transactions
on Biomedical Engineering, accepted, 2013.
Refereed Conference Papers
• P. Ataee, J.O. Hahn, C. Brouse, G.A. Dumont, and W.T. Boyce. Identifica-
tion of cardiovascular baroreflex for probing homeostatic stability. Computing in Cardiology, (37):141-144, 2010.
• P. Ataee, J.O. Hahn, G.A. Dumont, and W.T. Boyce. A Systemic Approach
to Local Stability Analysis of Cardiovascular Baroreflex. 33rd Annual International Conference of the IEEE EMBS, pages 700-703, 2011.
• P. Ataee, L. Belingard, G.A. Dumont, H.A. Noubari, and W.T. Boyce. Autonomic-Cardiorespiratory Regulation: A Physiology-Based Mathematical
Model. 34th Annual International Conference of the IEEE EMBS, pages
3805-3808, 2012.
• P. Ataee, G.A. Dumont, H.A. Noubari, W.T. Boyce, J.M. Ansermino. A
Novel Approach to the Design of an Artificial Bionic Baroreflex. 35th Annual International Conference of the IEEE EMBS, pages 3813-3816, 2013.
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
xix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Our Approach . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2
Challenges . . . . . . . . . . . . . . . . . . . . . . . . .
2
Scope of Application . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1
Autonomic-Cardiac Reactivity Assessment . . . . . . . .
5
1.2.2
Clinical Decision Support Systems . . . . . . . . . . . . .
5
1.2.3
Artificial Bionic Baroreflex . . . . . . . . . . . . . . . . .
7
Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . .
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
9
1.2
1.3
1.4
vi
2
Literature Review on Autonomic-Cardiorespiratory Regulation . .
10
2.1
Physiological Background . . . . . . . . . . . . . . . . . . . . .
12
2.1.1
Cardiorespiratory System . . . . . . . . . . . . . . . . . .
12
2.1.2
Autonomic Nervous System . . . . . . . . . . . . . . . .
12
2.1.3
Baroreceptor Reflex . . . . . . . . . . . . . . . . . . . .
15
2.1.4
Chemoreceptor Reflex . . . . . . . . . . . . . . . . . . .
17
2.1.5
Lung-Stretch Receptor Reflex . . . . . . . . . . . . . . .
18
Autonomic-Cardiac Monitoring . . . . . . . . . . . . . . . . . .
18
2.2.1
Standard Heart Rate Variability Measures . . . . . . . . .
19
2.2.2
Respiratory Sinus Arrythmia . . . . . . . . . . . . . . . .
20
2.2.3
Pre-Ejection Period . . . . . . . . . . . . . . . . . . . . .
20
Standard Clinical Tests . . . . . . . . . . . . . . . . . . . . . . .
21
2.3.1
Lower Body Negative Pressure . . . . . . . . . . . . . . .
21
2.3.2
Orthostatic Hypotension . . . . . . . . . . . . . . . . . .
21
2.3.3
Mental Stress . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4
Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . .
22
2.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Subject-Specific Monitoring of Autonomic-Cardiac Regulation . . .
26
3.1
Methods and Algorithm . . . . . . . . . . . . . . . . . . . . . . .
28
3.1.1
Experimental Dataset . . . . . . . . . . . . . . . . . . . .
29
3.1.2
Mathematical Model . . . . . . . . . . . . . . . . . . . .
29
3.1.3
Sensitivity Analysis
. . . . . . . . . . . . . . . . . . . .
32
3.1.4
System Identification . . . . . . . . . . . . . . . . . . . .
35
3.1.5
Validation . . . . . . . . . . . . . . . . . . . . . . . . . .
36
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . .
38
3.2.1
Sensitivity Analysis
. . . . . . . . . . . . . . . . . . . .
38
3.2.2
System Identification . . . . . . . . . . . . . . . . . . . .
40
3.2.3
Limitations of the Proposed Approach . . . . . . . . . . .
48
3.2.4
Autonomic-Cardiac Regulation Monitoring . . . . . . . .
50
Conclusions and Future Work . . . . . . . . . . . . . . . . . . . .
51
Model-Based Stability Analysis of Autonomic-Cardiac Regulation .
54
2.2
2.3
3
3.2
3.3
4
vii
4.1
Methods and Algorithm . . . . . . . . . . . . . . . . . . . . . . .
56
4.1.1
Physiology-Based Model: Delayed Differential Equations
56
4.1.2
Delay-Free Realization . . . . . . . . . . . . . . . . . . .
58
4.1.3
Identification of Equilibrium States . . . . . . . . . . . .
59
4.1.4
Stability Analysis . . . . . . . . . . . . . . . . . . . . . .
62
4.1.5
Simulation Data . . . . . . . . . . . . . . . . . . . . . .
64
4.1.6
Validation of the Proposed Approach . . . . . . . . . . .
65
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . .
69
4.2.1
Identification of Equilibrium States . . . . . . . . . . . .
69
4.2.2
Proposed Stability Metrics . . . . . . . . . . . . . . . . .
69
4.2.3
Multi-dimensional Stability Analysis . . . . . . . . . . .
71
4.2.4
Limitations . . . . . . . . . . . . . . . . . . . . . . . . .
73
Conclusion and Future Work . . . . . . . . . . . . . . . . . . . .
73
A Novel Approach to the Design of an Artificial Bionic Baroreflex .
75
5.1
Methods and Algorithm . . . . . . . . . . . . . . . . . . . . . . .
76
5.1.1
5.1.2
Experimental Data . . . . . . . . . . . . . . . . . . . . .
Mathematical Model . . . . . . . . . . . . . . . . . . . .
77
77
5.1.3
System Identification . . . . . . . . . . . . . . . . . . . .
79
5.1.4
Artificial Bionic Baroreflex . . . . . . . . . . . . . . . . .
80
5.1.5
Robustness Analysis . . . . . . . . . . . . . . . . . . . .
83
5.2
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . .
83
5.3
Conclusions and Future Work . . . . . . . . . . . . . . . . . . . .
86
4.2
4.3
5
6
Mathematical Modeling of Autonomic-Cardiorespiratory Regulation
88
6.1
Methods and Algorithm . . . . . . . . . . . . . . . . . . . . . . .
89
6.1.1
Experimental Dataset . . . . . . . . . . . . . . . . . . . .
89
6.1.2
Physiological Background . . . . . . . . . . . . . . . . .
90
6.1.3
Autonomic-Cardiac Regulation . . . . . . . . . . . . . .
91
6.1.4
Autonomic-Cardiorespiratory Regulation . . . . . . . . .
93
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . .
96
6.2.1
Model Validation . . . . . . . . . . . . . . . . . . . . . .
96
6.2.2
Mechanical vs. Neuromechanical Couplings . . . . . . .
98
6.2
viii
6.2.3
6.3
7
Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 100
Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 100
Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . 101
7.1
Summary: Work Accomplished
. . . . . . . . . . . . . . . . . . 101
7.2
Future-Work: The Road Ahead . . . . . . . . . . . . . . . . . . . 103
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
ix
List of Tables
Table 3.1
Parameters in the mathematical model of autonomic-cardiac regulation [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Table 3.2
Sensitivity-based parameter classification. . . . . . . . . . . .
38
Table 3.3
Statistical properties of the identified baroreflex-modulated Sympathetic Nerve Activity (SNA) and Parasympathetic Nerve Activity (PSNA ): mean±std . . . . . . . . . . . . . . . . . . . . .
Table 4.1
52
Parameters in the mathematical model of autonomic-cardiac regulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Table 5.1
Model parameters of autonomic-cardiac regulation. . . . . . .
78
Table 5.2
Individualized nominal values of high-sensitivity parameters in
three subjects versus corresponding population nominal values.
86
Table 6.1
Model parameters of autonomic-cardiac regulation. . . . . . .
93
Table 6.2
Respiratory system impacts on VL , Heart Rate (HR), VR, and ∆V . 96
Table 6.3
A numerical measure of perturbation caused by mechanical coupling effects J2 and neuromechanical coupling effects J1 . . . .
x
98
List of Figures
Figure 1.1
A schematic model of a hemodynamic stability monitoring
system using a subject-specific mathematical model . . . . . .
6
Figure 1.2
A schematic diagram of the artificial bionic baroreflex. . . . .
7
Figure 2.1
An extensive block diagram model of autonomic-cardiorespiratory
regulation. Parameters shown in red are outputs of the Autonomic Nervous System (ANS) as well as inputs for parts of
autonomic-cardiorespiratory regulation (i.e., the closed-loop
autonomic-cardiac regulation is opened at this level). . . . . .
11
Figure 2.2
A schematic diagram of the cardiorespiratory system [8]. . . .
13
Figure 2.3
Various factors affect autonomic regulation of the heart, including but not limited to respiration, thermoregulation, humoral regulation, Blood Pressure (BP), and Cardiac Output
Figure 2.4
(CO ) [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic diagram of the baroreflex mechanism. The Nu-
14
cleus Tractus Solitarius (NTS ) excites the parasympathetic motor neurons and inhibits the sympathetic motor neurons [10].
16
Figure 2.5
Two aspects of baroreflex characteristic . . . . . . . . . . . .
17
Figure 2.6
A schematic example of electrocardiography (ECG) signal and
Figure 3.1
impedance cardiography (dZ/dt) signal [11]. . . . . . . . . . .
20
Schematic diagram of the autonomic-cardiac regulation model.
30
xi
Figure 3.2
An extensive block-diagram model of autonomic-cardiorespiratory
regulation [see Chapter 2] with emphasis on the parts studied
in this chapter. The shaded parts are not described in the mathematical model Equation 3.1-Equation 3.2. . . . . . . . . . .
Figure 3.3
Measured versus the model-estimated signals (Case No.: 289);
blue is measured signals and black is model-estimated signals.
Figure 3.4
Distribution
of the index IµEval
j
37
for the estimated high-sensitivity
parameters in a set of 500 idealized simulations. . . . . . . . .
Figure 3.5
31
39
The overall sensitivity (mean and standard deviation) of autonomiccardiac model parameters over 100 sensitivity analysis runs
with nominal values selected from +/-20% the associated nominal values introduced in Table 3.1. . . . . . . . . . . . . . . .
40
Figure 3.6
Experimental results from MIMIC dataset (Case No.: 476). . .
43
Figure 3.7
Experimental results from MIMIC dataset (Case No.: 486). . .
44
Figure 3.8
Experimental results from MIMIC dataset (Case No.: 289). . .
45
Figure 3.9
Experimental results from MIMIC dataset (Case No.: 477). . .
46
Figure 3.10 System identification results on the orthostatic hypotension dataset
to monitor SNA and PSNA during a tilt test. The two top panels
show the measured vs. model-estimated HR and BP signals,
and the three bottom panels show the identification results:
α Ts , βH Ts , and VH Tp . . . . . . . . . . . . . . . . . . . . . . .
Figure 3.11 Measured versus model-estimated HR and BP signals with and
without the use of measured CO signal (Case No.: 289); blue
are measured signals, black are model-estimated signals with
measured CO and red are model-estimated signals without measured CO. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 4.1
47
49
Comparison of equilibrium states estimated using the proposed
analytical approach Equation 4.25 against numerical optimization (left panel) and nonlinear simulation (right panel). . . . .
xii
66
Figure 4.2
Two metrics for stability margin Sm and S p over changes of a
model parameter from 50% to 200% of its nominal value for a
healthy physiological condition with and without stress. Sm is
the blue solid line; S p is the green dashed line. A normal condition (i.e., VH , βH , and α were fixed at their nominal values).
Figure 4.3
67
Two metrics for stability margin, Sm and S p , over changes of a
model parameter from 50% to 200% of its nominal value for
a healthy physiological condition with and without stress. Sm
is the blue solid line; S p is the green dashed line. A stressful
condition (i.e., a 50% lower VH and 100% higher βH and α
compared to their nominal values). . . . . . . . . . . . . . . .
Figure 4.4
68
The proposed stability metric, Sm , over 2-D parameter spaces
from 50% to 150% of their nominal values for a normal physiological condition. The quantitative stability margin metric,
Sm , at each point of the 2-D parameter space is mapped into a
pixel-intensity level. A higher pixel-intensity level is related to
lower stability margin, and vice versa.
. . . . . . . . . . . .
72
Schematic model of autonomic-cardiac regulation with emphasis on the baroreflex . . . . . . . . . . . . . . . . . . . . .
77
Figure 5.2
Schematic model of the proposed artificial bionic baroreflex .
79
Figure 5.3
BP
Figure 5.1
measurement (BP setpoint) vs. the results of the artificial
bionic baroreflex (simulated
number 477.
Figure 5.4
BP
80
measurement (BP setpoint) vs. the results of the artificial
number 486.
BP
for individual with subject
. . . . . . . . . . . . . . . . . . . . . . . . . .
bionic baroreflex (simulated
Figure 5.5
BP )
BP )
for individual with subject
. . . . . . . . . . . . . . . . . . . . . . . . . .
81
measurement (BP setpoint) vs. the results of the artificial
bionic baroreflex (simulated
number 476.
BP )
for individual with subject
. . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
81
Figure 5.6
The results of robustness analysis for an individual with subject number 477. The solid line shows an average value of 100
simulated signals obtained by the proposed control strategy,
whereas the shaded area indicates the corresponding standard
deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
Figure 5.7
The calculated control signal, P0 , in three subjects . . . . . . .
85
Figure 6.1
Physiological measurement during Lower Body Negative Pressure (LBNP ) experiment in an individual; mean BP, Stroke Volume (SV ), and
HR
were calculated according to the
BP
wave-
form and Electrocardiogram (ECG ) recordings. . . . . . . . .
Figure 6.2
Schematic diagram of interactions between cardiovascular, respiratory and nervous systems. . . . . . . . . . . . . . . . . .
Figure 6.3
90
92
An extensive block-diagram model of autonomic-cardiorespiratory
regulation [see Chapter 2] with emphasis on parts described in
Equation 6.7-Equation 6.8. The shaded parts are not described
in the mathematical model. . . . . . . . . . . . . . . . . . . .
Figure 6.4
94
Power Spectral Density (PSD ) difference of Heart Rate Variability (HRV) among simulated (two methods) vs. measured
HR signals at the different stages of the LBNP experiment; the
shaded area shows the respiratory frequency band. . . . . . .
97
Figure 6.5
Neuromechanical coupling effects of respiration on HR and BP.
99
Figure 6.6
Mechanical coupling effects of respiration on HR and BP. . . .
99
xiv
Glossary
ABP
Arterial Blood Pressure
AP
Action Potentials
ANS
Autonomic Nervous System
BP
Blood Pressure
CO
Cardiac Output
CRS
Cardiorespiratory System
CSF
Cerebrospinal Fluid
CVS
Cardiovascular System
CNS
Central Nervous System
ECG
Electrocardiogram
HF
High Frequency
HR
Heart Rate
HRV
Heart Rate Variability
HUT
Head-Up Tilt
ILV
Instantaneous Lung Volume
LBNP
Lower Body Negative Pressure
xv
LF
Low Frequency
NTS
Nucleus Tractus Solitarius
PEP
Pre-Ejection Period
PNS
Parasympathetic Nervous System
PSNA
Parasympathetic Nerve Activity
PSD
Power Spectral Density
RR
Respiration Rate
RS
Respiratory System
RSA
Respiratory Sinus Arrhythmia
SA
Sinoatrial
SCI
Spinal Cord Injury
SNA
Sympathetic Nerve Activity
SNS
Sympathetic Nervous System
SV
Stroke Volume
TPR
Total Peripheral Resistance
TV
Tidal Volume
xvi
Acknowledgments
I would like to express my deepest appreciation to my supervisors, Professors Guy
A. Dumont, Hossein A. Noubari, and W. Tom Boyce, for their inspiration, encouragement, patience and unconditional support. They have provided me not only
wonderful support, but also enough freedom to explore my interests and find my
way to the end. I would also like to express my gratitude to Dr. J. Mark Ansermino,
who provided me wonderful opportunities to be involved with clinical experiences
and who shared his precious medical knowledge with us. This thesis could not
have been completed without the help of all of these people. I have learned how
to elaborate and conduct research in a complex field, how to collaborate with other
researchers as a team and how to conduct research as an individual, how to target a real-life problem to remove a burden from society, and how to be patient
throughout the course of my PhD program.
I have been fortunate to collaborate with Dr. Jin-Oh Hahn during, and after, his
stay at the University of British Columbia. He has certainly been an excellent mentor for me; his advice has always been relevant and effective. His professionalism,
modesty, and hard work have been helped me to collaborate with him productively
through different stages of my PhD program.
Many thanks also to my friends and colleagues at the laboratory of Electrical and Computer Engineering in Medicine, in alphabetical order: Chris Brouse,
Matthias Görges, Walter Karlen, Sara Khosravi, Mande Leung, Joanne Lim, Behnam
Molavi, Prasaad Shrawane, Kouhyar Tavakkolian, Klaske van Heusden, Aryannah
Umedaly, Ali Shahidi-Zandi, Ping Yang and all my other fellow students and colleagues. They have provided a pleasant academic and social environment in the
laboratory and a wonderful teamwork culture that helped me to resolve my profesxvii
sional and personal issues during this program. I also want to thank my dear friends
Amin Aziznia, Pouyan Abouzar, Sona Kazemi, and Kaveh Shafiee who supported
me in my unkind moments, and laughed with me in my wonderful moments during
my stay in Vancouver.
Lastly, I wish to express my genuine gratitude to my wonderful parents, my
constant source of energy, for their never-ending love, support, and guidance throughout my entire life. I also thank my two lovely sisters, Maryam and Sara, for their
support during every stage of my life.
xviii
Dedication
To my family, an insufficient token of my appreciation of their unwavering love
and faithfulness
xix
Chapter 1
Introduction
Cardiovascular disease is the leading cause of mortality and morbidity worldwide;
more than one million individuals in the United States suffer a heart attack each
year [12]. According to the World Health Organization (WHO), hypertension is
estimated to cause 7.5 million deaths worldwide, which is about 12.8% of the
total of all deaths [13]. Although hypertension is a major risk factor for coronary heart disease and hemorrhagic stroke [13] and is extremely common, it is
still poorly understood [14]. For example, it has been recently shown that many
patients need newly conceived Blood Pressure (BP)-stabilizing drugs as well as
BP -lowering
drugs [14].
1.1 Problem Statement
This project aimed to investigate a potential autocatalytic loop, also referred to
as positive feedback, within the autonomic-cardiac regulation, which leads to an
unstable (i.e., fluctuating)
BP .
To investigate such physiological conditions, we
selected a subject-specific model-based approach to analyze the stability of the
autonomic-cardiac regulation.
1.1.1 Our Approach
We investigated a large number of mathematical models describing autonomiccardiac regulation to find a physiology-based mathematical model. A physiologybased mathematical model with two coupled differential equations [7] to describe
1
the autonomic-cardiac regulation has been selected. We revised this model regarding the baroreflex mechanism to increase its physiological consistency. To individualize the mathematical model, we then developed a parameter identification
technique to estimate time-varying and subject-specific model parameters including sympathetic and parasympathetic activation, by using routine clinical measurements including Heart Rate (HR ) and BP.
Further, we developed a systematic framework to analyze the system stability. To investigate the potential system-level causes of instability (e.g., a potential
positive feedback) in the autonomic-cardiac regulation, we introduced an index
showing the stability margin of the autonomic-cardiac regulation. We then used
the subject-specific mathematical model for the autonomic-cardiac regulation to
design a closed-loop artificial bionic baroreflex.
Finally, we recognized the significance of respiratory effects in autonomiccardiac regulation. Therefore, to describe the respiratory effects on autonomiccardiac regulation, the mathematical model was improved to include respiratoryrelated terms.
1.1.2 Challenges
In this section, some challenges that we have confronted in this research are explained and categorized into three parts: mathematical modeling, system identification, and stability analysis.
Mathematical Modeling- The major purposes of developing a mathematical model
for a dynamic physiological system are to improve our understanding of the system, to reveal new insights into physiological mechanisms within the system, and
to predict the behavior of the system in different clinical conditions [15]. A dynamic physiological system can be described by different types of mathematical
models including statistical models or differential equations. In this context, a
white-box1 physiology-based mathematical model (e.g., differential equations) has
more advantages than a black-box2 model (e.g., statistical models) [16–18]. De1A
white-box model is a mathematical model developed based on a priori information about the
system.
2 A black-box model is a mathematical model solely developed based on its input, output and
transfer function without any knowledge of its internal dynamics.
2
veloping a mathematical model with minimal complexity to simplify mathematical
analysis, as well as developing the model sufficiently detailed to reproduce as much
clinically relevant data as possible are the main challenges of the modeling process
[19]. In fact, a mathematical model with a complex structure and a large number
of parameters may produce significantly more accurate simulation results that are
consistent with experimental observations; however, such a model generates many
complexities in the mathematical analysis and may cuase parameter identification
become impossible.
Developing an accurate mathematical model of the autonomic-cardiac regulation and then a simulator environment could reduce the need for invasive clinical experiments on the system as well as some clinical expenses. For instance, a
physiology-based pharmacokinetic model may be used to predict the absorption,
distribution, and excretion of synthetic or natural chemical substances (e.g., a brand
new drug) in the human body. In fact, the pharmacokinetic model gives clinicians
the ability to investigate how a new drug impacts system outcomes, e.g., dosagerelated effects of a drug on other physiological variables can be investigated without any of drug being injected into the body. Recently, many mathematical models
for the autonomic-cardiorespiratory regulation have been introduced. In this thesis, we intend to develop a physiology-based mathematical model of autonomiccardiorespiratory regulation.
System Identification- The prediction-error framework is the dominant approach in system identification theory and its applications with a focus on multivariable and closed-loop systems [20]. Once we have a mathematical model of
autonomic-cardiac regulation based on the physical laws describing the various
components and interconnection structure, system identification is used to estimate
unknown parameters in the model using the measured signals. Considering that all
of the model parameters were not identifiable, we had to examine whether the predicted outputs were sensitive to each parameter to obtain a group of identifiable
parameters [20].
In general, two types of sensitivity analysis approaches have been introduced
in the literature, local and global sensitivity analyses [21]. The local sensitivity of
a system output due to a model parameter is computed by the first-order partial
3
derivatives of the system output with respect to that model parameter. Similarly,
the global sensitivity used to quantify the overall effects of the parameters on the
system output is computed by perturbing parameters within large ranges.
Since the introduced mathematical model of autonomic-cardiac regulation contains a set of coupled nonlinear and delayed differential equations, we used a finite
difference approximation method to compute the global sensitivity and to separate
the model parameters into two groups: high-sensitivity and low-sensitivity parameters.
Stability Analysis- An improper dynamic change may cause an oscillatory
system, such as the respiratory system, to stop oscillating or to oscillate irregularly.
On the other hand, dynamic changes in a non-oscillatory system, such as blood
pressure regulation, may cause undesirable oscillation [22]. In fact, a large number
of physiological disorders are characterized by improper changes in the dynamics
of corresponding physiological systems, which result in unstable or irregular system behavior [22]. In the autonomic-cardiorespiratory regulation, model parameter changes (e.g., an increase in the time delay of sensory afferent pathways) may
cause an onset of oscillations (limit cycle or even instability) in
BP
and
HR
which
is not relevant to its normal regulatory task [16, 23]. An instability or irregularity
in a physiological system is called homeostatic imbalance (i.e., a disturbance in
homeostasis3 ). The homeostatic imbalance may occur as a result of the complex
dynamic interactions among nonlinearities and delays in a physiological system.
It is crucial to maintain a certain degree of stability margin in the autonomiccardiorespiratory regulation as a major in-vivo physiological control mechanism
for individuals with, for example, treatment-resistant hypertension since they are
susceptible to cardiovascular instability [26]. The system-level cause of instability
and the stability margin of the autonomic-cardiorespiratory regulation can be investigated using model-based stability analysis. To perform model-based stability
analysis, we must first develop an accurate physiology-based mathematical model
of the system. To actively monitor and then control the system’s stability and to
provide the patients with appropriate preventive interventions, it is important to
3 Homeostasis is the capability of living systems to maintain a physiological parameter fixed at a
setpoint by means of dynamic regulatory mechanisms in the face of external or internal challenges
[24, 25].
4
identify the root causes of instability, and to predict the system’s transition to instability. In this thesis, we propose a model-based approach for stability analysis of
autonomic-cardiorespiratory regulation to determine impacts of the parameter configurations that cause complex undesirable behavior and to determine the stability
margin of the physiological system.
1.2 Scope of Application
1.2.1 Autonomic-Cardiac Reactivity Assessment
Assessment of autonomic-cardiac reactivity can be used in many fields. Autonomiccardiac reactivity is the deviation of an autonomic-cardiac parameter, enforced by
an individual’s Autonomic Nervous System (ANS), from its normal value in response to a stimulus (e.g., environmental stress) [27]. In this context, reactivity
is defined as an individual’s physiological response to an environmental challenge
(e.g., stressful condition) compared with his resting state [28].
Autonomic-cardiac reactivity can be used as a criterion to assess the severity
of injury in individuals with Spinal Cord Injury (SCI ) as well as an indicator of life
satisfaction in individuals with high thoracic and cervical
SCI
[29, 30]. Moreover,
it can be used to improve the current classification systems for the Paralympics to
ensure a fair competition among Paralympians [31].
Exaggerated stress-related autonomic-cardiac reactivity puts children at risk
of, for example, cognitive impairments and poor emotion regulation. The process
that creates these effects can be investigated by finding an autocatalytic loop (i.e.,
positive feedback) within autonomic-cardiac regulation that is stimulated under
stressful conditions [32].
1.2.2 Clinical Decision Support Systems
Clinical decision support systems are computer-based intelligent systems that can
potentially provide subject-specific recommendations for a clinician to increase
patient safety and improve health outcome [33]. Subject-specific mathematical
models may eventually play a significant role in providing subject-specific recommendations in clinical decision support systems. Subject-specific mathematical
5
Figure 1.1: A schematic model of a hemodynamic stability monitoring system using a subject-specific mathematical model
models can be used to predict an individual’s physiological response to, for example, a specific medication dosage or a surgery procedure [34].
Patients with pre-existing conditions including cardiovascular diseases and SCI
undergoing major surgical procedures with anesthesia are at the risk of hemodynamic instability [35]. It is important for anesthesiologists to be able to predict the
risk of hemodynamic instability in their patients. Therefore, the stability margin
of a patient’s autonomic-cardiac regulation could be continuously monitored and
predicted during surgery [35]. Hemodynamic instability is mostly associated with
an unstable (i.e., fluctuating)
BP ;
however, fluctuations in HR, central venous pres-
sure, and Cardiac Output (CO) may also be referred to as hemodynamic instabilities
[35]. BP instability is determined by transient fluctuations in BP, which are usually
caused by a specific stimulus such as surgery, drug injection, emotional stress, or
postural change [14]. A quantitative metric of BP instability could be used to prevent hemodynamic instability during surgery. Increased BP instability is also a risk
factor for vascular dementia, which can be prevented by prescribing medicines that
6
Figure 1.2: A schematic diagram of the artificial bionic baroreflex.
reduce variability in BP [14].
We investigated
BP
instability by using a model-based analysis of autonomic-
cardiac regulation. Pre-existing conditions in the autonomic-cardiac regulation as
well as physiological changes during surgery can be described using a subjectspecific mathematical model with different parameter configurations. Figure 1.1
depicts a schematic model of the hemodynamic stability monitoring system that
was studied in this research.
1.2.3 Artificial Bionic Baroreflex
The disruption of the autonomic regulation (e.g., baroreflex failure) critically affects the quality of life for individuals with neurological disorders (e.g., Shy-Drager
syndrome) or traumatic
SCI s
which results in severe orthostatic hypotension [36].
The baroreflex characteristic is also altered in individuals with chronic hypertension, preventing proper
BP
constantly exposed to high
regulation [10, 37]. The fact that the baroreceptors are
BP
may impair the baroreflex mechanism, resulting in
a significant loss of baroreceptor sensitivity. Therefore, the impaired baroreflex
can not attenuate the effects of rapid perturbation in arterial pressure during, for
example, a posture change from lying to standing, possibly resulting in loss of consciousness [36]. A novel therapeutic approach including artificial bionic baroreflex
must be investigated for the treatment of severe baroreflex failure.
An artificial bionic baroreflex could be used for treatment of individuals with
baroreflex failure by using an external mechanism that activates the sympathetic
efferent nerves. An artificial bionic baroreflex is a functional replacement of the
7
baroreflex that consists of arterial pressure sensors as well as an automatic nerve
stimulator, which generates a pulse train signal to stimulate sympathetic nerves
[38]. Sunagawa [39, 40] proposed several anatomical sites, such as the carotid
sinus and the spinal cord, to manipulate the Sympathetic Nervous System (SNS )
. Yamasaki [38] also investigated a bionic baroreflex by stimulating sympathetic
nerves through an epidural catheter located at the level of the lower thoracic spinal
cord [38]. Accordingly,
BP
could be normally regulated in different physiological
conditions, providing a higher quality of life for individuals with baroreflex failure.
1.3 Thesis Contributions
The major contributions of this thesis are as follows:
• Develops a solid framework to analyze system stability and investigates the
presence of positive feedback in the autonomic-cardiac regulation using a
physiology-based subject-specific mathematical model of autonomic-cardiac
regulation.
• Develops a novel non-invasive model-based method to estimate and then
monitor autonomic-cardiac regulation based on a computationally efficient
system identification method by using routine clinical measurements:
BP ,
HR ,
and CO.
• Presents a systematic approach to investigate the system-level cause of instability in the autonomic-cardiac regulation as a major in-vivo physiological
control mechanism based on a stability index that determines the stability
margin for a parameter configuration.
• Introduces a novel model-based approach to the design of an artificial bionic
baroreflex that can be used to restore normal arterial pressure regulation in
individuals with baroeflex failure by mimicking the in-vivo baroreflex mechanism.
• Introduces a novel physiology-based mathematical model of autonomic-cardiorespiratory regulation described by a set of three nonlinear, coupled dif-
8
ferential equations, each of which describes regulations of
HR , BP ,
and In-
stantaneous Lung Volume (ILV ).
• Develops a software package that simulates macro level interactions in the
autonomic-cardiac regulation to investigate potential instability conditions
in BP and HR, and a software package that assesses the autonomic reactivity
by monitoring sympathetic (cardiac and arterial) and parasympathetic activation.
1.4 Thesis Outline
Chapter 2 presents the background material on the autonomic-cardiorespiratory
regulation as well as reviews of the previously published mathematical models.
Chapter 3 describes the details of an identification technique conducted on the
physiology-based mathematical model of autonomic-cardiac regulation. A parametric sensitivity analysis used to classify the model parameters into high-sensitivity,
low-sensitivity, and invariant groups is also described in this chapter. The feasibility and potential of the proposed subject-specific monitoring technique are demonstrated and discussed using two datasets: the MIMIC dataset and an orthostatic
hypotension dataset.
Chapter 4 presents a systematic approach to the stability analysis of the autonomiccardiac regulation. A Lyapunov-based systematic approach to analyze the system
stability in the neighborhood of the equilibrium state is developed in this chapter.
Further, a quantitative metric of stability margin capable of comparing different
parameter configurations regarding their stability has been introduced. Chapter 5
introduces a novel model-based approach to the design of a closed-loop artificial
bionic baroreflex. In this chapter, an individual’s in-vivo autonomic-cardiac regulation is described by a subject-specific mathematical model. In Chapter 6, a
physiologically-based mathematical model of the autonomic-cardiorespiratory regulation is introduced. Further, the significance of respiratory dynamics in autonomiccardiac regulation is studied.
9
Chapter 2
Literature Review on
Autonomic-Cardiorespiratory
Regulation
Autonomic-cardiorespiratory regulation operates through interactions between the
ANS and the Cardiorespiratory System (CRS ). The ANS maintains homeostasis in
the cardiorespiratory system against physical stressor, such as exercise and orthostatic hypotension, and psychological stressor, such as fear and anxiety [41–43].
A recently developed theory proposes that a physiological system only restores its
stability (allostasis) against a stressor rather than regulating a physiological parameter to a fixed setpoint (homeostasis) [24]. The
ANS
responses to physical and
psychological stressors are dictated by the individual’s autonomic reactivity characteristics [44, 45]. In fact, the
ANS
responds to different conditions by adjusting
cardiorespiratory parameters, including Respiration Rate (RR),
Peripheral Resistance
(TPR )1 ,
BP , HR ,
and Total
to deliver adequate oxygenated blood-flow to organs
in different conditions [45]. Figure 2.1 shows the complex interconnected structure
of short-term autonomic-cardiorespiratory regulation.
The present chapter will provide a brief overview of autonomic-cardiorespiratory regulation including the
CRS,
the
ANS ,
1 Total
and the autonomic regulation mecha-
peripheral resistance (i.e., systemic vascular resistance) refers to the resistance to blood
flow from the systemic vasculature, excluding the pulmonary vasculature [10].
10
11
Figure 2.1: An extensive block diagram model of autonomic-cardiorespiratory regulation. Parameters shown in red
are outputs of the ANS as well as inputs for parts of autonomic-cardiorespiratory regulation (i.e., the closed-loop
autonomic-cardiac regulation is opened at this level).
nisms of the Cardiovascular System (CVS ) and the Respiratory System (RS), followed by a review of the related work in the field of autonomic-cardiorespiratory
modeling. Further, we briefly introduce several common clinical tests to investigate autonomic-cardiorespiratory regulation with different stressors including orthostatic hypotension, Lower Body Negative Pressure (LBNP ), and mental stress
tests. We used readily available orthostatic hypotension [46] and MIMIC datasets
[47] to assess the proposed parameter identification method described in Chapter 3
and used an
LBNP
test to assess the proposed mathematical model of autonomic-
cardiorespiratory regulation described in Chapter 6.
2.1 Physiological Background
2.1.1 Cardiorespiratory System
The cardiorespiratory system consists of the
CVS
and the respiratory system. The
cardiorespiratory system transports nutrients, oxygen, carbon dioxide, hormones,
and blood cells in the body. The major functions of the cardiorespiratory system are
to provide vital needs for metabolic activities, to protect the body from infection,
to maintain homeostasis in thermoregulation, and to maintain fluid balance within
the body cells.
The
CVS
consists of the heart, blood, and two networks of blood vessels: pul-
monary circulation and systemic circulation. The pulmonary circulation carries
deoxygenated blood from the heart to the lungs and returns oxygenated blood to
the heart. The systemic circulation carries oxygenated blood from the heart to the
body tissues and returns oxygen-depleted blood back to the heart (Figure 2.2). The
respiratory system consists of the lungs, airways, and respiratory muscles (e.g., the
diaphragm). During respiration, carbon dioxide accumulated in the blood is exchanged with oxygen inhaled from the external environment through the diffusion
mechanisms within the lungs [48].
2.1.2 Autonomic Nervous System
The nervous system is divided into the somatic nervous system, which controls
organs under voluntary actions, and the
ANS ,
12
which mostly regulates involuntary
Figure 2.2: A schematic diagram of the cardiorespiratory system [8].
organ functions. The
ANS
maintains physiological parameters of the cardiorespi-
ratory system within their functional ranges. The ANS is divided into two separate
branches, the Parasympathetic Nervous System (PNS ) and the Sympathetic Nervous System (SNS ), based on anatomical and functional differences. The PNS is
dominant in “rest and digest” states, and the
SNS
is aroused in “fight or flight”
states [49, 50].
In most cases, these systems are reciprocally activated (i.e., when one system
is activated, the other is usually depressed) with antagonistic impacts [51]. However, there are conditions during which SNS and PNS may be activated (e.g., sexual
arousal) or inhibited (e.g., anesthesia) together [50, 52].
The medulla oblongata (often referred to as the medulla) is the brain’s primary site for regulation of sympathetic and parasympathetic (vagal) outflows. The
medulla is located in the lowest part of the brain and the lowest portion of the brain-
13
Figure 2.3: Various factors affect autonomic regulation of the heart, including
but not limited to respiration, thermoregulation, humoral regulation, BP,
and CO [9].
stem above the spinal cord [10]. Within the medulla, a visceral sensory nucleus,
known as the Nucleus Tractus Solitarius (NTS ), receives sensory information from
different systemic and central receptors (e.g., baroreceptors and chemoreceptors)
as well as higher brain centers (e.g., the hypothalamus). The hypothalamus plays
a particularly important role in determining cardiovascular responses to emotion
and stress. Efferent fibers of sympathetic and vagal nerves innervate the heart and
blood vessels, where they modulate the activity of these target organs. The heart
is innervated by both sympathetic and vagal divisions, which exert a regulatory
influence on
HR
by influencing the activity of the heart’s primary pacemaker, the
Sinoatrial (SA )-node [51] (Figure 2.3). An increase in
HR
could arise from either
increased cardiac sympathetic activity or decreased cardiac vagal inhibition.
The
ANS
adjusts the cardiorespiratory parameters through several involuntar-
ily mechanisms, mostly negative-feedback control mechanisms (also referred to
as reflexes), based on continuously integrated measurement of vital physiological
14
variables captured by specialized biological receptors [51, 53, 54]. For example,
perturbations in BP (e.g., orthostatic hypotension) are measured by the baroreceptors, and the baroreceptor reflex is primarily responsible for short-term
BP
regula-
tion. Further, perturbations in blood oxygen or carbon dioxide concentration (e.g.,
exercise-induced hypercapnia) are measured by the chemoreceptors, and then the
chemoreceptor reflex regulates oxygen and carbon dioxide concentration in the
blood.
The measured physiological variables are transmitted to the ANS, and the ANS
acts against perturbations by sending control commands through sympathetic and
parasympathetic pathways to a set of effectors including the heart and blood vessels
[16, 17, 23]. For instance, a rise in sympathetic activation tone elevates Cardiac
Output (CO)2 by increasing cardiac contractility (contraction force of the heart) and
the pace of the heartbeat, and elevates
TPR
by decreasing the diameter of blood
vessels (i.e., vasoconstriction). Conversely, a rise in parasympathetic activation
tone decreases
CO
by decreasing
HR
[55].
2.1.3 Baroreceptor Reflex
The baroreceptor reflex (baroreflex) is a short-term homeostatic mechanism in
the autonomic cardiorespiratory regulation. It includes specialized sensory neurons (also known as baroreceptors), efferent and afferent neural pathways, and the
brainstem [41]. The baroreceptors, mostly located in the carotid sinus and the aortic arch, are stretch-sensitive mechanoreceptors that are excited by the stretch of
blood vessels. They are sensitive to both absolute stretch (mean BP) and the rate of
stretch variation (pulsatile
and pulsatile
BP
BP );
however, the response characteristics to mean
BP
are different (Figure 2.5b). In this work, we modelled absolute
stretch baroreceptors that respond to mean BP.
A series of Action Potentials (AP) are fired and conveyed to the
NTS
(refer to
Section 2.1.2) in response to deformations in the arterial wall according to a nonlinear response curve [56–58]. For example, a
BP
rise causes the walls of vessels
with baroreceptors to expand and the baroreceptors to increase firing rate of APs
[10]. The greater the stretch, the more rapidly baroreceptors fire APs . The NTS
2 CO
is the amount of blood pumped through the circulatory system in a minute by the heart.
15
Figure 2.4: Schematic diagram of the baroreflex mechanism. The NTS excites the parasympathetic motor neurons and inhibits the sympathetic
motor neurons [10].
uses the frequency of received
APs
as a measure of
BP
[16]. The baroreceptor
firing-rate pattern also adapts to alterations in physiological conditions, causing
both short-term and long-term changes in
BP .
For example, baroreflex responses
will be adjusted under a stressful condition to maintain BP in a proper range (shortterm) [10], while the baroreflex loses its sensitivity to high BP in an individual with
chronic hypertension (long-term) [10]. Further, the baroreflex is less sensitive to
a fall in
BP
than to a rise in
BP ,
also known as the hysteresis3 effect, as shown in
Figure 2.5a [60].
In a healthy individual, the baroreflex maintains
using a set of sensors and effectors that adjusts
ity under a closed-loop negative
feedback4
baroreflex responds to a decrease in
BP
BP
within a narrow range by
HR , TPR ,
and cardiac contractil-
mechanism (see Figure 2.4) [41]. The
(mean, pulsatile, or both) by increasing
sympathetic outflow and decreasing vagal outflow, while it acts differently on sym3 Hysteresis is defined as “dependency of the steady-state response curve of a deterministic system
on the direction of the parameter change (increase or decrease)” [59].
4 Feedback is the property of a control system to use its output as (a part of) its input [61].
16
(a) The hysteresis effect in baroreflex fir(b) Two different response curves of baroreing rate to increasing and decreasing BP in an flex firing rate in response to pulsatile and mean
anesthetized dog [65].
BP [56].
Figure 2.5: Two aspects of baroreflex characteristic
pathetic and vagal outflow in an elevated BP [16, 18]. The baroreflex characteristic
driven by the ANS is time-varying and subject-varying, i.e., it changes in different
physiological conditions, and it differs among individuals [41, 56]. Homeostatic
imbalance in the CVS can be caused by impaired baroreflex response to an external
or internal stressor, which results in an oscillation and an instability in
HR
and
BP
regulation [18, 62]. For example, Mayer waves are low-frequency oscillations in
HR
with an approximate frequency of 0.1 Hz. These waves are caused by the sym-
pathetic (delayed) feedback control of the BP through the baroreflex [7, 63, 64]. It
has been shown that if sympathetic activity becomes chemically blocked, Mayer
waves are significantly reduced [41]. Therefore, investigation of the baroreflex
characteristics within the autonomic-cardiorespiratory regulation can be of significant importance in the context of homeostatic imbalance.
2.1.4 Chemoreceptor Reflex
The chemoreceptor reflex (chemoreflex) is a cardiorespiratory reflex that has evolved
to maintain systemic blood gas (O2 and CO2 ) levels within a functional range
[66, 67]. The chemoreflex can be divided into two reflexes with two different sets
of receptors: the peripheral chemoreceptors and the central chemoreceptors [68].
The peripheral chemoreceptors are located in the carotid bodies at the bifurcation
of the carotid arteries, and the central chemoreceptors are located in the medulla
[69].
The peripheral chemoreceptors respond primarily to a fall in partial pressure
of oxygen in arterial blood PaO2 (hypoxia) [69, 70]. At normal levels of PaO2 ,
17
some neural activity arises from the peripheral chemoreceptors, while, in arterial
hyperoxia (i.e., abnormally high PaO2 ), this activity is slightly reduced in a healthy
individual. However, in arterial hypoxemia (i.e., abnormally low PaO2 ), the intensity of neural activity varies in a nonlinear manner according to the severity of
the condition, causing an increase in the depth and rate of breathing. Hypoxia
also causes sympathetically mediated vasoconstriction in most arterioles (except
for coronary and brain arterioles) to maintain BP and circulation [71].
The central chemoreceptors respond primarily to a rise in partial pressure of
carbon dioxide in the arterial blood PaCO2 (hypercapnia) [69, 70]. In other words,
the response of the peripheral chemoreflex to arterial PaCO2 is less important than
that of the central chemoreflex [68]. The central chemoreceptors are exposed to
Cerebrospinal Fluid (CSF ) and are not in direct contact with the arterial blood [72].
Nevertheless, alterations in arterial PaCO2 are rapidly transmitted to the
CSF .
An
increase in the concentration of CO2 in the CSF causes hyperventilation [73].
2.1.5 Lung-Stretch Receptor Reflex
The lung-stretch receptor reflex (often referred to as the Hering-Breuer reflex) is a
cardiorespiratory reflex that triggers lung inflation and deflation by using mechanoreceptors located on the lung to provide information on the degree of lung expansion
or contraction. For example, inspiration causes the lung-stretch receptors and their
afferent nerves to activate and then project to the
NTS .
The activation of receptor
afferent nerves causes vagal cardiac outflow to inhibit and sympathetic outflow to
excite. This reflex may play a major role in ventilation by regulating breathing rate
and depth in newborns. However, more recent work indicates that this reflex is
largely inactive in adults unless the tidal volume exceeds one liter, as in exercise
[48].
2.2 Autonomic-Cardiac Monitoring
The ANS responds differently during exposure to physical and psychological challenges including stressful, emotional, and threatening conditions. The response
deviation of a physiological variable (e.g.,
HR )
from a control value that results
from an individual’s ANS response to a stimulus is called autonomic, or ANS, reac18
tivity and is associated with physical and psychological health [27, 32, 45, 74–76].
For example, researchers in behavioral pediatrics have shown that increased ANS or
autonomic reactivity (i.e., exaggerated physiological responses to stress) puts children at risk for a variety of physical and mental disorders, including poor emotion
regulation and cognitive impairments [32]. Further, cardiac vagal tone has been
proposed as a physiological marker of stress vulnerability (i.e., an individual’s differences in response thresholds to the identical challenging condition) [77]. In general, any change in
HR ,
also referred to as Heart Rate Variability (HRV), has been
used as an indicator of autonomic reactivity [43, 78]. However, the ability to monitor and interpret HRV is dependent on measuring technology, the HRV quantifying
method and the knowledge of underlying mechanisms [79]. Autonomic reactivity indices are classified into data-driven and model-based groups. In this thesis,
we mostly investigated autonomic reactivity using a model-based technique. Autonomic reactivity can be assessed by using some data-driven measures, as well
[74].
2.2.1 Standard Heart Rate Variability Measures
Standard methods for measuring
HRV
can be divided into time-domain and fre-
quency-domain methods. Time-domain measures are calculated directly from the
R-R interval signal such as the standard deviation and the standard deviation of the
successive differences of R-R intervals describing the overall variation and shortterm variation, respectively [80, 81]. The frequency-domain measures are calculated using the power spectral density of the R-R intervals. The well-accepted
measures are the powers of Low Frequency (LF ) (0.04-0.15 Hz) and High Frequency (HF) (0.15-0.4 Hz) bands in absolute and relative values, the normalized
powers of
LF
and
HF
modulated by both
PNS
bands, and the
SNS
and
PNS
LF
to
HF
power ratio [80, 81].
activities, while
HF
LF
power is
power is modulated only by
activities [82, 83]. Therefore, it is commonly assumed that the LF to HF power
ratio provides a measure of "sympathovagal balance" [82, 83].
19
Figure 2.6: A schematic example of electrocardiography (ECG) signal and
impedance cardiography (dZ/dt) signal [11].
2.2.2 Respiratory Sinus Arrythmia
Respiratory Sinus Arrhythmia (RSA ) is a periodic oscillation in HR, which is caused
by respiration [84]. This periodic oscillation can be triggered during inhalation
and exhalation, and results in an
RSA
HR
increase and decrease, respectively [84–86].
has been used as an index of cardiac vagal control as well as an index of
respiratory-circulatory interactions [84]. Several time-based and frequency-based
indices for assessment of the
methods,
RSA
RSA
have been introduced [87–89]. In the spectral
is mostly calculated using the power spectrum of R-R interval data
and an individual’s respiratory bandwidth [50, 85]. For example, Quas et al. [32]
quantified
RSA
as the natural logarithm of the variance of the R-R interval within
the respiration bandwidth.
2.2.3 Pre-Ejection Period
Pre-Ejection Period (PEP ) is the duration of isovolumetric ventricular contraction
in the left ventricle [74, 88].
PEP
is quantified as the time interval between the
onset of ventricular depolarization (indicated by the ECG Q-wave) and the onset
of left ventricular ejection (indicated by the B-point of the impedance cardiography signal) (Figure 2.6).
PEP
is an indirect, non-invasive measure of sympathetic
influence on cardiac rhythm, as a lower PEP score indicates higher cardiac sympathetic activity [32, 74, 76, 90]. However,
PEP
has been shown to be more reliable
to investigate within-subjects differences rather than between-subject differences
[90, 91].
20
2.3 Standard Clinical Tests
To study autonomic-cardiac monitoring techniques, several clinical experiments including LBNP, orthostatic hypotension, and mental stress were introduced in the literature, each of which specifically targets an aspect of autonomic-cardiorespiratory
regulation. That is,
LBNP ,
orthostatic hypotension, and mental stress tests mainly
affect Stroke Volume (SV ), Arterial Blood Pressure (ABP), and parasympathetic
nerves activation. These clinical experiments are briefly explained as follows.
2.3.1 Lower Body Negative Pressure
In an
LBNP
test, the lower body of a subject (for just above the pelvis) is placed
supine in a sealed chamber [92, 93]. After a resting control period, negative pressure is imposed mostly in 10 mmHg increments for a specific interval. The gradual
pressure decrease in the LBNP chamber continues until either completion of the test
or the onset of presyncope symptoms including light-headedness, nausea, sweating, dizziness, or blurred vision [93]. Further, a sudden decrease in systolic BP
(>25 mmHg) or HR (>15 bpm) is the symptoms of presyncope. In the literature,
the
LBNP
test is widely used to investigate post-spaceflight orthostatic intolerance
as well as severe hemorrhage in humans [94].
2.3.2 Orthostatic Hypotension
Orthostatic hypotension, also referred to as Head-Up Tilt (HUT), is a sustained
reduction of either systolic
BP
(> 20 mmHg) or diastolic
BP
BP
(>10 mmHg) within
three minutes of standing or HUT [95]. The magnitude of orthostatic
BP
reduction
is dependent on the baseline BP. Orthostatic hypotension is a clinical condition that
can severely affect quality of life in individuals with
SCI .
After a postural change
from supine to standing position, the venous return to the heart falls because of
gravitationally mediated redistribution of blood volume in the circulation system.
The venous return fall results in a decrease in SV and CO. In response, sympathetic
outflow to the heart and blood vessels increases and cardiac vagal outflow decreases
[95]. These autonomic-cardiac mechanisms increase vascular tone, HR and cardiac
contractility, and stabilize
BP
[95].
21
2.3.3 Mental Stress
In a mental stress test, challenge tasks are designed to elicit
ANS
responses in a
group of individuals (especially children) to different types of stressors: social,
cognitive, sensory, and emotional. For example, the social challenge task can be a
structured interview about a child’s family and friends.
The cognitive challenge task can be a digit-span recitation task in which a child
is asked to recall sequences of numbers. The sensory challenge task can be a tasteidentification task in which two drops of concentrated lemon juice are placed on
a child’s tongue, and the child is asked to recognize the taste. The emotionalchallenge task can be consisted of watching an emotion-evoking movie to elicit
fear in a child. The details of such a mental stress test are thoroughly explained in
[88].
2.4 Mathematical Modeling
To describe autonomic-cardiorespiratory regulation, a variety of mathematical models using either black-box or white-box (physiology-based) approaches have been
proposed [23, 55, 96]. The nonlinearity of the baroreflex and medulla responses,
the various time delays, and the number of different feedback loops create a large
number of challenges [7]. Further, the subject-varying and time-varying properties
of model parameters in each mathematical model have been neglected many times
to reduce challenges. By introducing a physiology-based mathematical model of
the autonomic-cardiorespiratory regulation that includes a set of ordinary differential equations, we will be able to simulate the physiology deliberately, and investigate system-level causes of a physiological observation.
Vooren et al. [55] proposed a model for short-term
BP
control without breath-
ing modulation which was tuned for supine posture. The model represented the
systemic circulation and consisted of three sections: a hemodynamic section simulated by a Windkessel model and Starling heart, a baroreceptor section simulated
by a linear function within the range between a threshold of 90 mmHg and a saturation level of 150 mmHg, and an autonomic control section simulated based on
the first-order system dynamic.
Saul et al. [97] proposed a mathematical model describing the closed loop
22
cardiorespiratory regulation to test complex links among
RR , HR ,
and
ABP .
This
model consists of the SA node, HR baroreflex, and mechanical effects of respiration
on
ABP
but ignores two significant aspects of hemodynamic regulation: the effect
of modulation of
TPR
via the baroreflex and the influence of cardiopulmonary re-
ceptors. Further, it describes the relation between all physiological variables by
using the frequency analysis technique.
In 2003, Ursino and Magosso [66] proposed a mathematical model of shortterm cardiovascular regulation to investigate the reliability of using
HRV
to study
the action of the autonomic regulatory mechanisms (vagal and sympathetic). The
proposed mathematical model included the pulsating heart, the systemic and pulmonary circulation, the mechanical effect of respiration on venous return, two
groups of receptors (arterial baroreceptors and lung-stretch receptors), the sympathetic and vagal efferent branches, and a very low-frequency vasomotor noise.
Fowler and McGuinness [7] proposed a nonpulsatile lumped-parameter5 model,
which consists of two coupled differential equations with nonlinear and delayed
dynamic interactions, each of which describes the dynamics of
lation. The pulmonary system and the small delay of the
PNS
HR
and
BP
regu-
were neglected in
this work. The sensitivity of Mayer waves to sympathetic delay and gain, to sympathetic control of peripheral resistance, and to sympathetic control of HR were
explored. This model is an extension of the mathematical model introduced by
Ottesen [16] with an added intrinsically controlled
HR ,
and baroreflex control of
peripheral resistance.
Ringwood and Malpas [96] developed a nonlinear model based on a linear
feedback model comprising delay and lag terms for the vasculature, and a linear
proportional derivative controller and an amplitude-limiting sigmoidal nonlinearity, which could belong to either the neural controller or the vasculature itself. They
showed that variations in the nonlinearity characteristics may account for growth
or decay in the BP oscillations as well as situations where the oscillations can thoroughly disappear. Further, they studied a BP oscillation between 0.1 Hz and 0.4 Hz
potentially caused by a resonant feedback in the baroreflex loop.
5 The lumped parameter model simplifies the description of the behavior of spatially distributed
physical systems into a topology consisting of discrete entities that approximate the behavior of the
distributed system under certain assumptions.
23
Cavalcanti [23] used bifurcation theory in nonlinear systems to explain the high
sensitivity of the
HR
oscillatory pattern to model parameter changes, specifically
parameter changes in the arterial baroreflex model. In this work, the basic mechanisms that generate
HRV
pump with a constant
such as the systemic circulation, a non-pulsatile cardiac
SV 6
and nonlinear negative feedback simulating an arterial
baroreflex closed-loop control of the HR were studied. The proposed model of the
short-term autonomic control (i.e., the arterial baroreflex) consists of two distinct
delayed branches of
mean
ABP
SNS
and
PNS
(2.8s and 0.8s). Dynamic linking between the
and mean aortic flow is described based on the classic three-element
Windkessel model, and aortic flow is expressed as SV times HR.
Seidel and Herzel [41, 98] proposed a hybrid model to capture both beat-tobeat and continuous dynamics of the autonomic-cardiorespiratory regulation and
to investigate its dynamic properties. The mathematical model consists of delay
differential equations that can describe physiological rhythms on timescales from
fractions of a second to a few minutes. Since chemoreceptors, temperature regulation, dynamics of renal hormones, and the circadian cycle were not included in
the model, the long-term variations in the physiological variables cannot be studied. They showed that an increase of the delays in conveying SNS signals leads via
Hopf bifurcation to the HR oscillations (called Mayer waves).
Ottesen [16] extended a model of uncontrolled CVS by adding an explicit modeling of the baroreflex-feedback mechanism to investigate the chronotropic effect
(HR regulation) and the inotropic (ventricle contractility regulation). Besides, the
system stability with special attention to the effect of the value of the time delay was studied. A well-established physiological theory was used in this work.
The introduced model of the baroreflex-feedback mechanism inserted some nonlinearity to the model as well as a time delay. Moreover, the
CVS
was simulated
using an expanded Windkessel model. In this work, Ottesen showed that the timedelay enforced some instability to the system, while the exact location of instability
windows were sensitive to the values of other parameters in the model.
In 2006, Olufsen et al. [99] developed a mathematical model describing
HR
dynamics as a function of BP during postural change from sitting to standing. This
6 SV
is the amount of blood pumped by the left ventricle of the heart in one contraction.
24
model ignored the
BP
feedback impacts on
HR
changes and regulation of
TPR ,
vascular tone, and cardiac contractility. The introduced mathematical model is divided into four sub-models connected in series. The first sub-model is an afferent
trigger model, which uses the finger
BP
as an input to predict the firing rates of
baroreflex afferent fibers. The second sub-model, representing the Central Nervous System (CNS), uses baroreceptor afferent nerve activity as an input to predict sympathetic and parasympathetic firing in response to the rate of change of
the mean BP.The third sub-model uses sympathetic and parasympathetic responses
as an input to predict concentrations of the neurotransmitters norepinephrine and
acetylcholine. The fourth sub-model, the effector model, uses concentrations of
neurotransmitters as an input to predict
HR .
2.5 Conclusion
In this chapter, we briefly reviewed the physiological background of autonomic-cardiorespiratory regulation by focusing on the baroreflex mechanism. We then described some standard data-driven measures of autonomic-cardiac reactivity, including RSA and PEP , as well as several standard clinical tests to study autonomiccardiac reactivity. At the end, we provided an extensively review of mathematical models to describe autonomic-cardiorespiratory regulation that have been proposed in the literature. It has been observed that a physiology-based and closedform mathematical model of autonomic-cardiac regulation has not been studied
properly in the past. Considering that a physiology-based and closed-form mathematical model of autonomic-cardiac regulation is developed in this work. This
mathematical model is used to investigate the system stabilty using an analytical,
rather than numerical, stability analysis algorithm with both accuracy and computational efficiency. This model is also used to develop an artificial bionic baroreflex
for treatment of baroreflex failure.
25
Chapter 3
Subject-Specific Monitoring of
Autonomic-Cardiac Regulation
Autonomic-cardiac regulation operates through interactions between the
the
CVS .
The
ANS
dictates homeostasis in the
CVS
ANS
and
in order to maintain adequate
blood flow to deliver oxygen and nutrients to organs by adjusting its effectors
against internal (e.g., orthostatic hypotension) and external (e.g., hemorrhage) perturbations [41–43]. Specifically, the
ANS
adjusts
BP , CO ,
and
HR
using different
mechanisms, e.g., adjusting Sympathetic Nerve Activity (SNA ) and Parasympathetic Nerve Activity (PSNA ) on sinoatrial node, cardiac contractility, and peripheral resistance [17, 23, 51]. In particular,
homeostasis in the
CVS
CVS
SNA
and PSNA are controlled to maintain
against physical and/or emotional stressors acting on the
[16, 56]. In this regard, autonomic-cardiac regulation is closely linked to car-
diovascular disorders. Indeed, it has been suggested that the capacity of autonomiccardiac regulation (i.e., a measure of sympathovagal balance [90]) is an important
predictor of an individuals’s health outcome [90, 100].
Subject-specific monitoring of autonomic-cardiac regulation has the potential
to improve current treatments of autonomic-cardiac disorders such as chronic drugresistant hypertension and
SCI .
For example, the capacity of autonomic-cardiac
regulation can be monitored and used to reduce excessive intake of anti-hypertensive drugs in individuals with chronic drug-resistant hypertension. The capacity
of autonomic-cardiac regulation is also a criterion used to assess the severity of
26
injury in individuals with
SCI
as well as an indicator of life satisfaction in indi-
viduals with high thoracic and cervical
SCI
[29, 30]. Moreover, the capacity of
autonomic-cardiac regulation can be used to improve the current classification systems for the Paralympics to ensure a fair competition among Paralympians [31].
In addition, autonomic-cardiac regulation monitoring can be used to categorize the
severity of the spinal cord injury, in regard to
HR
and
BP
regulation, with better
accuracy. It can also be beneficial to assign proper special care to individuals with
chronic hypertension as well as SCI .
Several data-driven indices of autonomic-cardiac regulation, including the RSA
and
PEP ,
have been introduced using the autonomic blockade research methodol-
ogy [101].
RSA
and PEP are used as indices of PSNA [45, 87] and SNA [32, 45] on
the cardiac cycle, respectively. However, there are critical limitations to the use of
RSA
on
and PEP measures [86, 102]. For example, RSA is not a good measure of PSNA
HR
during mechanical ventilation or severe physical activity.
RSA
has limited
capability to differentiate the inter-individual differences of the PNS [86].
PEP
can
only indicate the SNA on the heart, but it cannot be used to assess the sympathetic
outflows to blood vessels. More importantly, these two measures are both calculated based on the heart rhythm; however, previous investigations (e.g., [67]) show
that the SNA and PSNA may be better estimated by looking into multiple effector
mechanisms.
Subject-specific and model-based estimation techniques to identify and monitor autonomic-cardiac variables including blood flow and blood pressure have
drawn attention [55, 103–108]. Nevertheless, only a small number of studies have
addressed identification of the subject-specific, time-varying model parameters in
autonomic-cardiac regulation by using model-based estimation techniques (e.g.,
[99]).
The objective of this study is to develop and validate a model-based approach
to subject-specific monitoring of autonomic-cardiac regulation. Note that the proposed method does not rely on the dataset containing invasive CO measurement, as
discussed in this chapter. Moreover, as explained in Section 3.1.5, the use of
CO
measurement in the proposed method is not necessary; however, it will increase
the parameter estimation accuracy. The proposed approach allows us to monitor temporal changes in autonomic-cardiac regulation by continuously identifying
27
time-varying changes in the autonomic-cardiac model parameters, including
and
PSNA
on the heart (modulating
HR )
and
SNA
SNA
on the arterial tree (modulating
peripheral resistance). The validity of the proposed approach was tested by using
a number of experimental data from the MIMIC (Multiparameter Intelligent Monitoring in Intensive Care) database and the orthostatic hypotension tests performed
in the Center for Hypotension at New York Medical College.
3.1 Methods and Algorithm
We used a physiologically-based mathematical model of autonomic-cardiac regulation described by a set of coupled nonlinear and delayed differential equations.
The mathematical model consists of 12 subject-specific parameters (VH , βH , α , P0 ,
α0 , ∆V , γ , Ca , τ , δH , H0 , R0a ; see Table 3.1) and two outputs: HR and BP. This
model was chosen for its relative simplicity, which is crucial for successful system identification with routinely available clinical measurements. However, it is
emphasized that the general idea constituting the proposed approach is applicable
to more complicated physiologically-based autonomic-cardiac regulation models
upon the availability of additional measurements. Figure 3.2 illustrates the aspects
of autonomic-cardiorespiratory regulation introduced in Chapter 2 that were studied in this chapter.
Given the task of identifying complex autonomic-cardiac regulation using the
limited information included in routine clinical measurements, it was determined
that only high-sensitivity parameters (whose changes significantly affect the system outputs, i.e., HR and BP) be individualized with the aid of system identification.
To achieve this aim, parametric sensitivity analysis was used to classify the model
parameters into two groups, high-sensitivity and low-sensitivity groups, according
to the physiology underlying autonomic-cardiac regulation and the significance of
each model parameter in terms of its impacts on the system outputs.
Then, a system identification problem formulated as a nonlinear optimization
was solved to estimate high-sensitivity model parameters associated with autonomiccardiac regulation; whereas, low-sensitivity parameters were fixed at their nominal
values. The high-sensitivity parameters can be estimated properly by using HR and
BP measurements, which can be non-invasively obtained in real clinical practice.
28
In addition to
HR
and
BP , CO
obtained using a direct measurement or an indirect
analysis can be helpful in order to increase the fidelity of the estimated parameters.
3.1.1 Experimental Dataset
We used experimental data from the MIMIC database and the orthostatic hypotension tests, each of which is described in detail below.
The MIMIC dataset is described in detail in [47] and is freely available on the
PhysioNet website [109]. It contains multiple physiologic signals of 121 subjects
recorded from monitors in the intensive care units (ICUs) at the Beth Israel Hospital, Boston, MA [110]. In this dataset, the measured data of each subject usually
contains ECG signals recorded by surface ECG leads and sampled at 500 Hz and
ABP
signal recorded by invasive radial artery catheterization and sampled at 125
Hz [110]. The data also contain several signals including systolic, diastolic, and
mean
ABP
as well as beat-to-beat
Hz [110]. The
CO
HR ,
which are computed and sampled at 0.9765
signal, recorded using thermodilution technique, is also avail-
able for some subjects. In this work, four 1-hour data segments containing
HR ,
mean ABP, and CO signals corresponding to four different subjects were extracted
and subsequently used for analysis.
The orthostatic hypotension dataset used in this study were collected from the
Center for Hypotension at New York Medical College and are described in [46].
A single lead ECG, beat-to-beat continuous
BP ,
respiratory plethysmography and
capnography recordings as well as a Modelflow estimate of beat-to-beat
CO
(using
a proprietary arterial pulse contour method) are included for each subjects undergoing head-up tilt table testing (upto 70◦ ). We used experimental data from two
subjects to establish initial proof-of-concept of the proposed approach to monitoring autonomic-cardiac regulation.
3.1.2 Mathematical Model
We adopted a model of autonomic-cardiac regulation proposed by Ottesen [16] and
Fowler and McGuinness [7]. This model is schematically shown in Figure 3.1, and
is described by two differential equations Equation 3.1-Equation 3.2. The model
consists of two coupled nonlinear and delayed differential equations, each of which
29
Figure 3.1: Schematic diagram of the autonomic-cardiac regulation model.
describes the dynamics of HR and BP regulation:
βH Ts
−VH Tp + δH H0 − H(t)
1 + γ Tp
P(t)
H(t)∆V
Ṗ(t) = − 0
+
,
Ra (1 + α Ts )Ca
Ca
Ḣ(t) =
where H is
HR
and P is
BP .
(3.1)
(3.2)
Ts = g1 (P(t − τ )) is the sympathetic control with the
strength βH on the heart and α on the peripheral resistance, and Tp = 1−g1 (P(t)) is
the parasympathetic control with the strength VH on the heart, where g1 (x) =
1
.
1+x4
Note that Ts (sympathetic control function) and Tp (parasympathetic control function) are dependent of BP, and the time delay associated with sympathetic pathway
is denoted by τ . Since the PNS is relatively fast-acting in comparison with the SNS,
the time delay associated with the PNS is neglected. This model is devised based on
the physiologic mechanisms underlying autonomic-cardiac regulation and therefore is equipped with parameters having physiologic implications that dictate essential short-term regulation mechanisms of HR and BP such as baroreflex control
of HR and peripheral resistance. The definitions and nominal values of the parameters in Equation 3.1-Equation 3.2 are adopted from Fowler and McGuinness [7]
and summarized in Table 3.1.
To emulate the in-vivo sympathetic and parasympathetic control functions, Ts
and Tp with any explicit algebraic definition must always satisfy the following
properties [16]:
• 0 < Ts < 1; Ts ≃ 1 for P(t − τ ) small, and Ts ≃ 0 for P(t − τ ) large
30
Figure 3.2: An extensive block-diagram model of autonomiccardiorespiratory regulation [see Chapter 2] with emphasis on the
parts studied in this chapter. The shaded parts are not described in the
mathematical model Equation 3.1-Equation 3.2.
• 0 < Tp < 1; Tp ≃ 0 for P(t) small, and Tp ≃ 1 for P(t) large
•
∂ Tp
∂ Ts
> 0 and
< 0; i.e., Ts and Tp are monotonic functions
∂P
∂P
Though the model g1 (x) =
1
1+x4
may reproduce baroreflex activity with accept-
able accuracy, it has limited capability to be adapted at the individual level since
it does not involve any tunable parameters. In this regard, we replaced it by the
well-known sigmoid function [16, 18, 111–113], which includes parameters to represent inter-individual differences in baroreflex activity. That is, g1 (P) =
1
1+P4
in
(Equation 3.1)-(Equation 3.2) is replaced with g2 (P) = 1 − σ (P). σ (P) is defined
31
Table 3.1: Parameters in the mathematical model of autonomic-cardiac regulation [7].
Parameter
Ca
R0a
∆V
H0
τ
VH
βH
α
γ
δH
Definition
arterial compliance
minimum arterial resistance
stroke volume
intrinsic HR
sympathetic delay
parasympathetic control of HR
sympathetic control of HR
sympathetic effect on Ra
parasympathetic damping of βH
relaxation time
Nominal Value
1.55 mlmmHg−1
0.6 mmHgsml −1
50 ml
100 min−1
3s
1.17 s−2
0.84 s−2
1.3
0.2
1.7 s−1
as follows:
σ (P) =
1
1 + e−α0 (P−P0)
50 ≤ P ≤ 200.
(3.3)
The sigmoid function σ (P) is characterized using two variables, setpoint P0 and
1
sensitivity α0 . In contrast to the Hill function ( 1+P
4 ) which lacks physiological
implications, the sigmoid function can be easily mapped to human physiology. For
example, α0 shows the amount of baroreflex compensatory response against
BP
perturbation at P0 [67]. Since the maximum slope of the sigmoid function occurs
at P0 , the baroreflex compensatory response assumes its maximum at P0 . Further,
P0 shows the central tendency of the mean ABP [67].
3.1.3 Sensitivity Analysis
Since the number of unknown parameters is significantly greater than the number of independent equations in the autonomic-cardiac regulation model (i.e., an
undetermined system), complete estimation of subject-specific model parameters
without some constraints is not likely to be feasible [103, 114, 115]. To make
the parameter identification feasible, we can fix some parameter values that are
less relevant to model predictions by applying some a priori physiological knowledge or using sensitivity analysis. We can also use inequality constraints on high32
sensitivity parameters imposed by a feasible physiological range to effectively limit
the range of values for the solutions on unknown parameters [114, 116]. To systematically select a subset of the model parameters amenable to identification from the
observation of system outputs, we performed a sensitivity analysis on the model
parameters and investigated the impacts of each model parameter on the system
outputs.
It is noted that H0 and R0a were excluded from this analysis, since they are not
expected to vary much within the time scale of interest (from hours to days) within
an individual. Indeed, H0 denotes the intrinsic or “denervated”
expected to change much in an individual. In addition,
TPR
R0a
HR ,
which is not
denotes the minimum
when the sympathetic excitation to the arterial tree (i.e., vasoconstriction) is
minimal, which should be characterized mostly by the geometry (and properties to
some extent) of the arterial vessels that is not supposed to vary significantly within
the time scale of interest. In this work, therefore, H0 and R0a were classified into
the category of invariant parameters. Thus, the nominal values listed in Table 3.1
were assigned to H0 and R0a during the system identification procedure.
We then classified the remaining model parameters into high-sensitivity and
low-sensitivity groups based on the results of the sensitivity analysis, thereby identifying a subset of parameters with significant impact on the system outputs that
must be individualized by the system identification procedure. Essentially, even
a small change in these high-sensitivity parameters yields a large change in outputs, whereas the outputs are not significantly impacted by (even large) changes in
low-sensitivity parameters.
Traditionally, parametric sensitivity in dynamic systems is analyzed in the frequency domain [117]. However, the frequency-domain technique is not applicable
to the autonomic-cardiac regulation model Equation 3.1-Equation 3.2, mainly due
to the nonlinearities in the model. To resolve this challenge, this work seeks to
carry out sensitivity analysis in the time domain. To this aim, the sensitivity functions for HR and BP are defined as follows:
H(t, µ j ) − H(t, µ j,0 ) H
× µj
S (t, µ j ) = H(t, µ j )
−
µ j µ j,0
33
(3.4)
(3.5)
P(t, µ j ) − P(t, µ j,0 ) × µj
S (t, µ j ) = P(t, µ j )
µ j − µ j,0
P
(3.6)
In Equation 3.4-Equation 3.6, SX (t, µ j ); X = H, P is the instantaneous sensitivity
of X at time t due to perturbation of parameter µ j . In other words, SH (t, µ j ) and
SP (t, µ j ) represent percent changes in
HR
and
BP
at time t due to a certain per-
centage perturbation of parameter µ j from its nominal value. The total sensitivity
function Equation 3.7 is obtained by combining the sensitivity functions of both
system outputs, i.e., HR (H) and BP (P):
S(t, µ j ) =
SH (t, µ j ) + SP (t, µ j )
,
2
(3.7)
where we decided to use equal weights of 0.5 to both SH and SP since we aim to
analyze autonomic-cardiac regulation as a whole rather than
BP
or
HR
regulation
separately. Due to the nonlinearity in the autonomic-cardiac regulation model,
the S(t, µ j ) can assume different values depending on the amount and direction of
perturbations given to the independent variable µ j . To develop a robust sensitivity
metric against variations in magnitude of perturbations in the parameter µ j , we
elaborated on the sensitivity function Equation 3.7 by considering the variation in
µ j up to +/-50% in 1% increments, which yields the sensitivity metric as a function
of time Equation 3.8:
v
u
u
u
S j (t) = t
3
2 µ j,0
∑
S2 (t, µ j )
(3.8)
µ j = 21 µ j,0
Finally, a scalar metric S j in Equation 3.9 is calculated by aggregating the sensitivity values S j (t) in time to obtain the overall sensitivity of a particular parameter
on the system output:
v
u t f inal
u
S j = t ∑ S2j (t)
(3.9)
tinitial
The parameters are classified into high-sensitivity and low-sensitivity groups
based on the values of S j .
34
3.1.4 System Identification
To estimate subject-specific high-sensitivity parameters in Equation 3.1-Equation 3.2,
a system identification method was developed based on an optimization problem
minimizing the normalized L1 -error between measured versus model-estimated HR
and BP signals. The error function (i.e., the objective function) was specified as follows:
EP + EH
J=
;
2
Xs (t, M) − Xm (t) ,
EX = ∑ Xm (t)
n
(3.10)
t=0
where Xm (t) and Xs (t, M) (X = H, P) are measured and model-estimated output signals, respectively, and M is the set of high-sensitivity parameters in the autonomiccardiac regulation model, i.e., M = {VH , βH , α , P0 , ∆V }.
For each 30s-long data segment, system identification was performed by opti-
mizing the high-sensitivity parameters in Equation 3.1-Equation 3.2 so that the error function Equation 3.10 is minimized, while low-sensitivity parameters as well
as H0 and R0a were fixed at their corresponding nominal values (Table 3.1). The
optimization problem was solved using the fmincon routine with an active-set algorithm in the MATLAB Optimization Toolbox [118], which finds the constrained
minimum of the multivariable nonlinear scalar function J in Equation 3.10 by using the Quasi-Newton approximation that determines the search direction using an
approximation of the Hessian matrix during each optimization iteration[118]. The
set of optimized high-sensitivity parameters minimizing the error function were
used as their optimal estimates associated with the corresponding data segment.
To perform system identification for the first data segment, the high-sensitivity
parameters (VH , βH , α , P0 , and ∆V ) were initialized by assigning random values from a uniform distribution in the neighborhood (+/- 10%) of their respective
nominal values. In all of the remaining data segments, the high-sensitivity parameters were initialized by the corresponding estimates in the previous data segment.
We assumed that the high-sensitivity model parameters vary slowly within each
30s-long segment, so that they can be approximated as constants within each data
segment. In fact, we assumed that the hemodynamic state of the subject is stable in
each 30s interval, and thus the model parameters to be identified can be regarded as
35
constants. If a data segment involves abrupt changes in physiologic and/or mental
states, the model parameters may represent the ”average" state corresponding to
the data segment.
The model-estimated
HR
and
BP
signals were calculated by solving the model
equations Equation 3.1-Equation 3.2 on the interval [0 s,30 s] using the estimates of
high-sensitivity model parameters in each 30s-long segment during the optimization process. In each segment, the model-estimated HR and BP in the previous segment were assigned as initial conditions for HR and BP, whereas the first measured
HR
and
and
BP
BP
samples were used as initial conditions to solve model-estimated
HR
in the first segment. In order to prevent the divergence of high-sensitivity
model parameters out of the physiologically relevant range during the course of
the optimization procedure, the range of each model parameter was constrained as
follows:
µ j,Nom
< µ j < 2µ j,Nom ;
2
µ j ∈ M,
(3.11)
where µNom is the nominal value of µ taken from Table 3.1.
3.1.5 Validation
To establish the performance of the proposed approach, it is necessary to evaluate
the accuracy and repeatability of the system identification procedure. To evaluate
the accuracy of the parameter estimates, the proposed system identification method
was first applied to the idealized data. The idealized data are a set of 30s-long simulated HR and BP signals under the idealized setting with nominal model parameter
values in the absence of any structural uncertainty (i.e., model mismatch). We performed a total of 500 system identification trials, which resulted in 500 sets of
high-sensitivity parameter estimates. For each system identification trial, the highsensitivity parameters were randomly initialized in the neighborhood (+/- 50%) of
their corresponding nominal values while the low-sensitivity parameters were fixed
at their nominal values. The accuracy of the system identification method was assessed by examining the distributions of the index Equation 3.12, which is the ratio
36
Figure 3.3: Measured versus the model-estimated signals (Case No.: 289);
blue is measured signals and black is model-estimated signals.
of each individual parameter estimate and its actual counterpart.
IµEval
=
j
µ Est
j
µ Nom
j
,
1 ≤ j ≤ 12
(3.12)
To evaluate the repeatability of the system identification method, it was applied
50 times to each 30s-long segment of the experimental data corresponding to four
subjects obtained from the MIMIC dataset (case numbers 476, 486, 289 and 477).
For each 30s-long data segment, 50 system identification trials were carried out
with 50 distinct, randomized initial conditions for the high-sensitivity parameters.
The mean and standard deviation of the estimated high-sensitivity parameters associated with each 30s-long data segment were studied to assure the repeatability
of the proposed system identification method.
In comparison with HR and BP that can be measured easily in clinical practice,
CO
measurement is usually accessible only from critically ill patients. To assess
37
Table 3.2: Sensitivity-based parameter classification.
High-sensitivity
P0
∆V
VH
βH
α
the benefit of using
CO
Low-sensitivity
γ
Ca
τ
α0
δH
Invariant
H0
R0a
measurements in the system identification procedure (i.e.,
to quantify how much the availability of
CO
data can improve its performance),
we applied the proposed system identification procedure to both idealized and experimental data in the absence/presence of
available, the
CO
by
HR ,
SV
CO
data. In case the
CO
data were
(∆V ) was not estimated but was calculated directly by dividing
thereby eliminating uncertainty associated with ∆V . Otherwise, it was
fixed at its nominal value in the absence of
CO
data. Our expectation was that the
use of CO measurement may lead to better model-estimated HR and BP signals, and
accordingly, the estimates of the autonomic-cardiac model parameters with better
accuracy.
3.2 Results and Discussion
3.2.1 Sensitivity Analysis
The overall sensitivity metric S j of the autonomic-cardiac model parameters are
illustrated in Figure 3.5. The model parameters were classified into high-sensitivity
(P0 , ∆V , VH , βH , and α ) and low-sensitivity parameters (γ , Ca , τ , α0 , and δH ) based
on their overall sensitivity metric Equation 3.9 (see Figure 3.5). It is noted that all
the parameters pertaining to
ANS
(VH , βH , α and P0 ) were assigned to the high-
sensitivity group as anticipated. These high-sensitivity parameters were used to fit
the model-estimated
HR
and
BP
to their measured counterparts during the system
identification procedure.
38
for the estimated high-sensitivity
Figure 3.4: Distribution of the index IµEval
j
parameters in a set of 500 idealized simulations.
Since the sensitivity analysis was performed in the neighborhood of the nominal values shown in Table 3.1, the classification results obtained based on S j may
be affected by the nominal values. To investigate the effect of perturbations in
the nominal values on S j and to assess the consistency of the parameter classification strategy employed in this chapter, we repeated the sensitivity analysis for 100
sets of distinct nominal values. Each set contained nominal values randomly selected from +/-20% intervals in the neighborhood of the associated nominal values
listed in Table 3.1. Figure 3.5 shows the mean and standard deviation of 100 calculated S j values associated with each model parameter. It can be concluded that the
classification shown in Table 3.2 would not be affected significantly by reasonable
perturbations in the nominal model parameter values.
39
Figure 3.5: The overall sensitivity (mean and standard deviation) of
autonomic-cardiac model parameters over 100 sensitivity analysis runs
with nominal values selected from +/-20% the associated nominal values introduced in Table 3.1.
3.2.2 System Identification
1) Idealized Data: Overall, the proposed system identification procedure performed
well. Figure 3.4 shows the distribution of the index Equation 3.12 for the estimated
high-sensitivity parameters (VH , βH , α and P0 ) obtained from the 500 system identification trials on the idealized data. The distributions corresponding to VH , βH ,
α and P0 are mostly centered around the unity (see Figure 3.4), suggesting that
high-sensitivity parameters are accurately estimated based on our proposed system
identification procedure even when the low-sensitivity parameters were fixed at
their nominal values. Since the system identification could accurately estimate the
autonomic-cardiac model parameters in the idealized dataset, its potential to accurately estimate the model parameters in the experimental dataset can be regarded
40
as promising.
2) MIMIC Dataset: The top panels of Figure 3.6-Figure 3.9 depict the measured signals (BP,
HR ,
and
CO )
for 4 individuals taken from the MIMIC dataset,
and the bottom panels depict the baroreflex-modulated SNA and PSNA (VH Tp , βH Ts ,
and α Ts ) derived from the estimated model parameters for the corresponding four
individuals, where the solid line shows an average value of 50 estimated baroreflexmodulated
SNA
and
PSNA ,
whereas the shaded area indicates the corresponding
standard deviation. The relatively small standard deviation (which suggests the
uncertainty related to system identification) supports the repeatability of the proposed system identification procedure.
We further scrutinized the experimental results to examine whether the identified autonomic-cardiac model parameters are reasonable based on a priori knowledge on the behavior of autonomic-cardiac regulation as follows.
Consider Figure 3.6. First, gradual decrease in
BP
during t=100 s-500 s is
the consequence of gradual decrease in α Ts (baroreflex-modulated
SNA
on arterial
tree), while a sudden drop (≃ −5%) of CO at t=360s is compensated by the increase
in α Ts at t=360 s to maintain stable
HR
and
BP .
Moreover, an abrupt increase in
(≃ +25%) at around t=1000 s ( 3.6a) is caused by an abrupt increase in βH Ts
(baroreflex-modulated SNA on the heart) at t=1000 s. Note that HR and BP are
HR
dependent variables according to the model (Equation 3.1)-(Equation 3.2) but
CO
is an independent variable.
Now consider Figure 3.7. At t=1740 s, a sudden drop (≃-30%) in the measured
CO
is observed. However, there is no specific change in
HR
and
BP .
Therefore, it
is reasonable not to observe any changes in βH Ts and VH Tp (baroreflex-modulated
PSNA
on the heart), but to observe a sudden increase in α Ts to maintain stability
of the system by increasing TPR. There are abrupt increases in
HR
BP
(≃+30%) and
(≃+5%) at t=2700 s, which are caused by the increase in α Ts and decrease in
VH Tp . Overall, these results suggest, to a large extent, the physiologic relevance
and consistency of the proposed system identification procedure. Similar interpretation could be made for the remaining datasets to which the proposed approach
was applied.
Finally, it is important to note that
SNA
and
PSNA
may or may not act antago-
nistically. Indeed, βH Ts and VH Tp at t=1000 s in Figure 3.6b act antagonistically,
41
whereas the opposite pattern can be observed in Figure 3.7b. For example, a large
reduction in VH Tp occurs at t = 2700s, which is accompanied by a slight reduction
in βH Ts . In fact, simultaneous activation of
SNA
and
PSNA
is not uncommon ac-
cording to the recently proposed notion of co-activation [90]. Indeed, a series of
studies performed by Cacioppo and Berntson shows that psychological processes
and higher neuro-behavioral substrates can result in independent activation or even
co-activation of SNA and PSNA [12, 90].
3) Orthostatic Hypotension Dataset: During the postural changes from supine
to upright positions (which can be caused either by tilting the bed or standing),
blood volume is redistributed in the lower extremities due to gravity [95]. As a
consequence, blood volume returning to the heart in each cardiac cycle (venous
return) falls resulting in diminished
CO , SNA
stabilize
SV
on the heart increases while
ABP
by modulating
HR
and
CO .
PSNA
In response to this decrease in
on the heart increases decreases to
and SV . The activation of SNA on the arterial tree
is different for standing and tilting situations, because the body’s skeletal muscle
contraction helps to maintain venous return in an appropriate range during standing
whereas muscle contraction has no role during tilting.
Figure 3.10 shows the results of applying the proposed system identification
procedure to the orthostatic hypotension test data of two subjects. In regards to
the first subject, the two upper panels of Figure 3.10a show that the subject is
in the supine position starts being tilted at t=400 s, and is then brought back to
the supine position at t=1060 s after the tilting maneuver ends. The three lower
panels in Figure 3.10a show that βH Ts increases and VH Tp decreases to increase
HR to maintain homeostasis in BP regulation once the tilting maneuver commences.
After the tilting maneuver ends, HR decreases, while BP increases with some delay
(Figure 3.10a). Note that this observation can be caused only by an increase in
VH Tp and a decrease in βH Ts to decrease
HR
and an increase in α Ts to increase
mean ABP through TPR. The three lower panels in Figure 3.10a clearly show that
the proposed approach could estimate successfully these anticipated changes in
SNA
and PSNA acting on both the heart and the arterial tree. The results shown for
the second subject in Figure 3.10b exhibit trends consistent with those of the first
subject and they both support the potential of the proposed approach in monitoring
the autonomic-cardiac regulation.
42
(a) Measured vs. model-estimated signals:
BP, HR,
and CO
(b) Identification results: α Ts , βH Ts , and VH Tp
Figure 3.6: Experimental results from MIMIC dataset (Case No.: 476).
43
(a) Measured vs. model-estimated signals:
BP, HR,
and CO
(b) Identification results: α Ts , βH Ts , and VH Tp
Figure 3.7: Experimental results from MIMIC dataset (Case No.: 486).
44
(a) Measured vs. model-estimated signals:
BP, HR,
and CO
(b) Identification results: α Ts , βH Ts , and VH Tp
Figure 3.8: Experimental results from MIMIC dataset (Case No.: 289).
45
(a) Measured vs. model-estimated signals:
BP, HR,
and CO
(b) Identification results: α Ts , βH Ts , and VH Tp
Figure 3.9: Experimental results from MIMIC dataset (Case No.: 477).
46
(a) Subject I
(b) Subject II
Figure 3.10: System identification results on the orthostatic hypotension
dataset to monitor SNA and PSNA during a tilt test. The two top panels show the measured vs. model-estimated HR and BP signals, and
the three bottom panels show the identification results: α Ts , βH Ts , and
VH Tp .
47
3.2.3 Limitations of the Proposed Approach
Despite its promising preliminary results, this study has a number of limitations as
discussed below.
First, comparing the system identification results with and without the use of
CO
measurements, the results with CO measurements incorporated into the system
identification procedure were superior to the results without
CO
measurements.
This means that the proposed approach may benefit from the availability of
CO
measurements. Figure 3.11 shows a typical result comparing model-estimated
HR
and BP signals with and without CO measurements (Case No.: 289). The thermodilution technique (which is accepted as the gold standard
CO
nique [119]) was used in this individual to measure
Figure 3.11 shows that
the accuracy of the model-estimated
the measured
CO
HR
and
BP
CO .
measurement tech-
signals can be improved by using
signal in the system identification procedure. Accordingly, it is
expected that the accuracy of the estimated baroreflex-modulated
SNA
and
PSNA
(i.e., α Ts , βH Ts , and VH Tp ) will be enhanced as well. Considering that currently
available techniques for direct measurement of
CO
with acceptable accuracy, in-
cluding the thermodilution technique, are highly invasive [120], the use of CO data
for the purpose of system identification may not be practical. Non-invasive techniques including echo-cardiography [121], electrical velocimetry [122] and the use
of pulse contour methods to estimate CO from arterial BP waveform using the morphological feature (e.g., [120, 123, 124]), can be considered potential alternatives.
Second, the possible interdependence between model parameters was not explored in this paper. For example, physiology dictates that
related to arterial compliance (Ca ) and
by the afterload if
TPR
TPR
(in particular,
SV
R0a ),
(∆V ) is essentially
e.g.,
SV
is affected
increases. The incorporation of a priori knowledge on
the interdependence between model parameters may have improved the outcomes
of system identification. Further, it is important to emphasize that the structural
coupling among α , Ca and R0a , and between ∆V and Ca in Equation 3.2 could be
avoided using the classification of model parameters used in this study (Table 3.2).
Indeed, α could be uniquely identified since R0a and Ca were fixed at their nominal
values. Likewise, ∆V could also be uniquely identified since Ca was fixed at its
nominal value.
48
Figure 3.11: Measured versus model-estimated HR and BP signals with and
without the use of measured CO signal (Case No.: 289); blue are measured signals, black are model-estimated signals with measured CO
and red are model-estimated signals without measured CO.
Third, due to the nonlinear dynamic nature of autonomic-cardiac regulation,
the sensitivity of the model parameters may depend on their respective nominal
values. This, in turn, suggests that the classification of the autonomic-cardiac parameters as presented in Table 3.2 may also depend on the values of the model
parameters (or equivalently, the underlying physiologic state). The nominal model
parameter values used in this study were well-suited for an average adult in a stable
resting state. However, the model parameters may need to be re-classified when applying the proposed system identification approach to subject groups under highly
non-nominal physiologic and/or mental conditions.
Fourth, to make maximal use of the limited information contained in the HR
and BP data, we chose to identify a subset of parameters characterizing the autonomiccardiac regulation model that largely impacts the HR and BP signals. This ap-
49
proach is, in fact, not uncommon when identifying systems involving many parameters (e.g., [125]). It was claimed that fixing the invariant parameters may be
well-justified physiologically (see Section 3.1.3 for details). However, the lowsensitivity parameters may vary in time, and physiologic justification of fixing
these parameters at constant values is not trivial. Regardless, the effect of fixing
these low-sensitivity parameters on the estimates of high-sensitivity SNA and PSNA
parameters is expected to be non-significant, since the low-sensitivity parameters
cannot alter the HR and BP signals (which are used in the system identification procedure to estimate the
SNA
and
PSNA
parameters) much due to their small impact
on these signals.
Lastly, although this study provides an initial evidence and proof-of-concept
for the physiologic relevance of the estimates of autonomic-cardiac regulation parameters (as demonstrated by the physiologically anticipated changes in the estimated SNA and PSNA parameters in response to head-up tilt tests; see Figure 3.10),
the clinical strength of the proposed method for diagnostic/therapeutic procedures
is yet to be investigated deeply . In this regard, this study must be regarded as
a preliminary feasibility study to estimate autonomic-cardiac regulation parameters based on easily accessible clinical measurements; additional in-depth work is
necessary before the clinical value of the proposed method can actually be claimed.
3.2.4 Autonomic-Cardiac Regulation Monitoring
The model parameters of autonomic-cardiac regulation (e.g., VH , βH , α , and P0 )
and, therefore, baroreflex-modulated
SNA
and
PSNA
(α Ts , βH Ts , and VH Tp ) are
subject-specific, time-varying (short-term and long-term), and health-dependent.
The model parameters are subject-specific because physiologic differences of the
ANS
and
CVS
among different individuals result in different statistical properties
(e.g., mean and variance) in the baroreflex-modulated
SNA
and
PSNA
(Table 3.3).
The model parameters of autonomic-cardiac regulation are also time-varying in
both short-term and long-term periods. For example, the model parameters are
continuously adjusted in response to physical and emotional stressors in order to
maintain the stability of vital physiologic variables such as HR and BP (short-term).
Additionally, as humans age, neural reflexes become slower, thereby resulting in
50
larger delays in the sympathetic pathways (τ ) associated with older adults (longterm). We showed that autonomic-cardiac regulation is health-dependent since individuals with different types of stress reactivity can be differentiated by the characteristic of autonomic-cardiac regulation, especially the baroreflex characteristic
(Ts and Tp ) [6]. Therefore, a subject-specific monitoring method for autonomiccardiac regulation can be used to identify the underlying regulation mechanism
and to diagnose the
ANS -
and
CVS -related
deficiencies [99, 126]. Further, we can
deduce valuable physiologic information from autonomic-cardiac regulation monitoring based on the variation of estimated parameters in different subjects under the
same physiologic condition, and similarly within a subject under different physiologic conditions.
The non-invasive direct measurement method for
SNA
and
PSNA
is not avail-
able. Therefore, non-invasive indirect measurement methods such as electrocardiogram-based indices (e.g.,
RSA
and
PEP ),
galvanic skin response method, and
model-based measurement methods have been used to estimate
For example, the galvanic skin response method measures
SNA
SNA
and
PSNA .
using the electri-
cal conductance of the skin which changes according to the skin’s moisture level,
which is itself altered by changes in the SNA. It has been proposed that the RSA
shows activity of the PNS and the PEP shows activity of the SNS [32]. However,
the efficacy of RSA and PEP are limited in comparison with model-based measurement methods for SNA and PSNA since they are calculated solely on the basis of the
electrocardiogram signal, while model-based measurement methods are developed
based on the complex physiological structure of autonomic-cardiac regulation.
In this work, we therefore proposed a model-based monitoring method for
autonomic-cardiac regulation using a mathematical model that takes into account
important physiologic parameters in the regulation mechanism. The proposed
method can potentially be used to improve non-invasive autonomic-cardiac regulation monitoring.
3.3 Conclusions and Future Work
In this chapter, we presented initial evidence and proof-of-concept for a novel
subject-specific model-based monitoring method for autonomic-cardiac regulation
51
Table 3.3: Statistical properties of the identified baroreflex-modulated
and PSNA : mean±std
Case No.
476
486
289
477
βH .Ts
0.40 ± 0.08
0.62 ± 0.10
0.45 ± 0.07
0.64 ± 0.07
VH .Tp
0.27 ± 0.06
0.28 ± 0.13
0.25 ± 0.03
0.25 ± 0.09
SNA
α .Ts
1.21 ± 0.17
1.04 ± 0.40
0.75 ± 0.06
0.39 ± 0.12
that uses a computationally efficient system identification method with routine clinical measurements: HR and BP. We used CO measurement in this chapter as provided in the MIMIC dataset; however, the proposed method has not been developed
with the assumption of incorporating
CO
measurement (refer to Section 3.2.3).
Note that some non-invasive CO estimation techniques including electrical velocimetry and echocardiography have also been introduced recently. The proposed
method is effective in estimating the time-varying and subject-specific characteristics of autonomic-cardiac regulation, since it accommodates the complex nature
of the regulation mechanism through a mathematical model rather than calculating arguable features directly extracted from signal measurements. Our proposed
model-based monitoring method has the potential to eliminate the limitations of
competing methods currently available (e.g.,
RSA
and
PEP )
such as lack of inter-
individual separability and lack of fidelity during mechanical ventilation or severe
physical exercise.
In the future, more intensive experimental validation of the proposed method
must be performed to further assess its efficacy in monitoring autonomic-cardiac
regulation. The method should also be improved by incorporating the impact of
the respiratory system on the regulation mechanisms for HR and BP, which will
consequently lead to an enhancement in the fidelity of the system identification
results (i.e., in terms of α Ts , βH Ts , and VH Tp ). Note that the system identification technique must be revised to be applicable to the mathematical model with
respiratory effects. In addition, the proposed method must be compared with the
conventional markers of
SNA
and
PSNA ,
including the time-domain measures of
the HRV. Further, the use of multi-dimensional autonomic-cardiac spaces (such as
VH Tp -βH Ts , VH Tp -α Ts , and βH Ts -α Ts ) may be investigated to assess and differen52
tiate the capacity of autonomic-cardiac regulation in different individuals. Finally,
an extensive clinical study must be conducted to determine the clinical usefulness
of the proposed method for diagnostic and therapeutic procedures.
53
Chapter 4
Model-Based Stability Analysis of
Autonomic-Cardiac Regulation
Autonomic-cardiac regulation operates through interactions between the
the
CVS .
ANS
and
The ANS mostly regulates involuntary organ function and maintains
homeostasis in the
CVS
against physical (e.g., exercise and orthostatic hypoten-
sion) and psychological (e.g., fear and anxiety) stressors [41–43]. The ANS adjusts
cardiorespiratory parameters, including BP, HR, vascular resistance and respiratory
rate (RR), to deliver adequate oxygenated blood flow to organs in different conditions [45]. In general, autonomic-cardiac regulation is regarded as stable if BP and
HR
converge to equilibrium states after a certain amount of transient time when
it is exposed to a stressor, whereas it is considered unstable if
HR
and
BP
exhibit
non-decaying or slowly-decaying oscillations or diverge from their normal values.
It is well known that undesirable changes in the dynamics of the autonomic-cardiac regulation (e.g., an excessive increase in the time delay of sensory afferent
pathways) can result in an onset of instabilities in
BP
and
HR
[16, 23]. Cavalcanti
et. al. [18, 23] showed that perturbations in autonomic-cardiac parameters affect
the stability of the CVS. Ottesen [16] showed that switching between stability and
instability of the
CVS
can occur depending on the value of the time delay associ-
ated with the baroreflex feedback mechanism. He also demonstrated that complex
dynamic interactions between nonlinearities and delays in autonomic-cardiac regulation may cause instability [16]. Abbiw-Jackson [127] reported that an increase
54
in the gain of the baroreflex feedback loop controlling venous volume may cause
the onset of oscillation in
BP.
Deboer et. al. [128] also showed that the time
delay in the baroreceptor feedback loop may be the cause of the Mayer waves
(low-frequency oscillations in the mean arterial
BP ).
As a result, stability analysis
of autonomic-cardiac regulation may be beneficial in improving current diagnostic
and treatment methods for ANS-CVS disorders.
Model-based stability analysis is useful to examine the system-level causes of
instability and the stability margin in autonomic-cardiac regulation [23, 128]. For
example, several model-based analyses of the baroreflex mechanism have revealed
that mechanisms underlying the baroreflex are responsible for the Mayer waves
[16–18]. In another model-based analysis, it was shown that autonomic-cardiac
regulation may remain stable or be driven to instability in response to changes in
the baroreflex parameters, even if the baroreflex delays remain constant, i.e., the
stability of the autonomic-cardiac regulation is sensitive to both time delays and
other parameters associated with baroreflex [16]. A model-based study was also
used to show that baroreflex modulation does not promptly return to a steady state
in hypertensive elderly individuals during postural change from sitting to standing
[99]. Model-based analysis of autonomic-cardiac regulation was also used in investigating the reliability of the heart period variability index to study the autonomic
regulatory mechanisms [66]. However, to the best of our knowledge, existing results using the model-based approach for stability analysis of the autonomic-cardiac regulation have limited capability for quantifying stability margins, although
some previous studies have qualitatively examined the impacts of parameter configurations on the stability margin of
ANS -CVS
[3, 16]. Moreover, it is crucial to
maintain a certain degree of stability margin in autonomic-cardiac regulation for
individuals with, for example, treatment-resistant hypertension.
In a recent study, we developed an optimization-based system identification
approach to characterize autonomic-cardiac regulation mechanisms based on a
physiology-based
ANS -CVS
model [1], which we used to conduct a preliminary
feasibility study on the model-based stability analysis of autonomic-cardiac regulation [3]. In this work, we present a model-based approach to investigate the stability of autonomic-cardiac regulation. A unique strength of the proposed approach
is its capability to determine the stability margin of autonomic-cardiac regulation
55
quantitatively and computationally efficiently once the model parameter configuration is given. Specifically, the proposed approach quantifies the stability margin of
autonomic-cardiac regulation via two key contributions: 1) an analytical method
to determine the equilibrium states of the autonomic-cardiac regulation and 2) a
systematic approach to analyze the system stability in the vicinity of the equilibrium state. First, we validated our approach by comparing our analysis results to
well-established physiological concepts; we then used this approach to explore potential model parameter configurations that can incur instability in autonomic-cardiac regulation. We also demonstrated that the proposed approach can determine
the equilibrium state and quantify its stability with a high level of accuracy. This
approach is very powerful in identifying the system-level cause of instability in the
autonomic-cardiac regulation by virtue of its capability to determine the stability
margin associated with model parameter configurations.
4.1 Methods and Algorithm
In this section, the physiology-based mathematical model of autonomic-cardiac
regulation that is used for the proposed stability analysis is described. The delayfree realization of the mathematical model is then devised to be used in further
analysis. The proposed stability analysis consists of three steps. In the first step, the
equilibrium state of autonomic-cardiac regulation, in terms of BP and HR, is identified as a closed-form steady-state solution of the mathematical model Equation 4.10
presented (see Section 4.1.3). In the second step, the model of autonomic-cardiac
regulation is linearized around the equilibrium state to obtain the Jacobian matrix
of the system that can be used to assess its stability in the neighborhood of the equilibrium state. In the last step, the stability margin of autonomic-cardiac regulation
is quantified using the eigenvalues of its Jacobian matrix (see Section 4.1.4). We
validated our proposed stability analysis using a simulation dataset.
4.1.1 Physiology-Based Model: Delayed Differential Equations
A wide variety of mathematical models for autonomic-cardiac regulation with different levels of complexity have been proposed in the literature. Examples include
a three-element Windkessel model of
CVS
56
with baroreflex represented by a series
connection of delayed first-order linear dynamics and a sigmoid nonlinear function
[23], a simple nonlinear feedback control system containing an amplitude-limiting
nonlinearity added to a linear feedback model comprising delay and lag terms for
the vasculature and a linear proportional-derivative controller for the
ANS
[96],
a set of two coupled nonlinear and delayed differential equations describing
and
BP
HR
regulation mechanisms [7], and a relatively complex model consisting of
a Windkessel model and Starling heart for the hemodynamic section, a saturated
linear function for baroreceptor section, and a set of first-order systems for autonomic control section [55]. Considering that a high-fidelity, physiology-based,
and closed-form mathematical model is required to develop an analytical, rather
than numerical, stability analysis algorithm with both accuracy and computational
efficiency, we adopted a model described by Fowler [7]. The autonomic-cardiac
regulation model used in this chapter is described by:
βH Ts
−VH Tp + δH H0 − H(t)
1 + γ Tp
P(t)
H(t)∆V
Ṗ(t) = − 0
+
,
Ra (1 + α Ts )Ca
Ca
Ḣ(t) =
where H is
HR ,
and P is mean arterial
BP .
(4.1)
(4.2)
The definitions and nominal values
of the parameters in Equation 4.1-Equation 4.2 are summarized in Table 4.1. In
this model, the sympathetic and parasympathetic modulating functions generated
by the baroreflex control mechanism are denoted by Ts and Tp , respectively. The
time delay associated with the sympathetic pathway is denoted by τ , whereas the
parasympathetic delay was assumed to be negligible [37]. In this study, we neglected the inhibitory impact of the parasympathetic system on the sympathetic
system by setting γ ≃ 0 in Equation 4.1-Equation 4.2, as it is well known that its
effect on overall autonomic-cardiac regulation is generally small [7, 16]. The mathematical model Equation 4.1-Equation 4.2 is then rewritten as follows:
Ḣ(t) = βH Ts −VH Tp + δH H0 − H(t)
P(t)
H(t)∆V
Ṗ(t) = − 0
+
.
Ra (1 + α Ts )Ca
Ca
(4.3)
(4.4)
The sympathetic and parasympathetic modulating functions Ts and Tp can be
57
Table 4.1: Parameters in the mathematical model of autonomic-cardiac regulation.
Parameter
Ca
R0a
∆V
H0
τ
VH
βH
α
γ
δH
Definition
arterial compliance
minimum arterial resistance
stroke volume
intrinsic HR
sympathetic delay
vagal tone
sympathetic control of HR
sympathetic effect on Ra
vagal damping of βH
relaxation time
Nominal Value
1.55 mlmmHg−1
0.6 mmHgsml −1
50 ml
100 min−1
3s
1.17 s−2
0.84 s−2
1.3
0.2
1.7 s−1
modeled as sigmoid functions with amplitude-limiting characteristic [96]. A sigmoid function, σ (P), is characterized by a setpoint and a sensitivity coefficient
[113, 129] as follows:
σ (P) =
where P0 and α0 are the
1
1 + e−α0 (P−P0)
50 ≤ P ≤ 200.
(4.5)
setpoint and the sensitivity of the baroreflex mech
anism, respectively. Substituting Ts = 1 − σ P(t − τ ) and Tp = σ P(t) into
BP
Equation 4.3-Equation 4.4 yields:
h
i
Ḣ(t) = βH 1 − σ P(t − τ ) −VH σ P(t) + δH H0 − H(t)
Ṗ(t) = −
R0a
H(t)∆V
P(t)
h
.
i + C
a
1 + α 1 − σ P(t − τ ) Ca
(4.6)
(4.7)
4.1.2 Delay-Free Realization
To alleviate analytical and computational challenges that can potentially arise in the
course of stability analysis of autonomic-cardiac regulation, the transport delays associated with the sympathetic and parasympathetic responses were replaced by approximations. Note that this essentially simplifies the original infinite-dimensional
58
model Equation 4.6-Equation 4.7 to a finite-dimensional model. For this purpose,
we employed the first-order Pad é approximation to eliminate the delayed stated
variable P(t − τ ) as follows. First, a new state variable X is defined as follows:
Pτ (t) = P(t − τ )
⇒
L
Pτ (s) = P(s)e−τ s ≃ P(s)
⇒
Pτ (s) ≃ P(s)(−1 +
1 − τ2 s
1 + τ2 s
2
)
1 + τ2 s
2
Pτ (s) ≃ −P(s) + P(s)
1 + τ2 s
2P(s)
Pτ (s) + P(s) ≃
|
{z
} 1 + τ2 s
⇒
⇒
X(s)
X (s) = Pτ (s) + P(s)
⇒
L −1
⇒
P(t − τ ) = X (t) − P(t)
(4.8)
which serves as the output equation relating P(t − τ ) to P(t) and X . The state
equation dictating the dynamics of X is obtained as follows:
X (s) ≃
2P(s)
1 + τ2 s
⇒
L −1
⇒
⇒
τ
X (s) + X (s) ≃ 2P(s)
2
τ
X (t) + Ẋ(t) ≃ 2P(t)
2
2
Ẋ(t) ≃ 2P(t) − X (t)
τ
(4.9)
Using Equation 4.8 and Equation 4.9, the delay-free realization of the autonomic-cardiac regulation model Equation 4.6-Equation 4.7 can be obtained as Equation 4.10,
shown below.
4.1.3 Identification of Equilibrium States
At an equilibrium state of the system described by Equation 4.10, time derivatives
of the the state variables are zero, i.e., Ẇ(t) = 03×1 , and the state variables reach
their respective steady-state values P(t) = P(t − τ ) = Pf , H(t) = H f , and X (t) = X f .
59

f1 H(t), P(t), X (t)

 


 




Ẇ(t) =  Ṗ(t)  ≃  f2 H(t), P(t), X (t) 


 

Ẋ(t)
f3 H(t), P(t), X (t)

 βH 1 − σ X (t) − P(t) −VH σ P(t) + δH H0 − H(t)
P(t)
H(t)∆V




− + C

(4.10)
= 
a
R0a 1 + α 1 − σ (X (t) − P(t)) Ca



2
2P(t) − X (t)
τ

Ḣ(t)


Therefore, Equation 4.10 at the equilibrium state can be rewritten into:
h
i
0 = βH 1 − σ (X f − Pf ) −VH σ (Pf ) + δH H0 − H f
(4.11)
0 = −
(4.12)
Pf
H ∆V
h
i + f
Ca
R0a 1 + α 1 − σ (X f − Pf ) Ca
2
2Pf − X f
τ
0 =
(4.13)
Note that, according to Equation 4.13, X f = 2Pf and Equation 4.11-Equation 4.12
reduce to a set of two algebraic equations as shown below:
1 βH 1 − σ (Pf ) −VH σ (Pf ) + δH H0
δH
= H f ∆V R0a 1 + α 1 − σ (Pf )
=
Hf
Pf
(4.14)
(4.15)
which further simplifies to:
Hf
Pf
1 βH + δH H0 − σ (Pf ) βH +VH
δH
= H f ∆V R0a 1 + α − α σ (Pf ) .
=
(4.16)
(4.17)
Deriving closed-form solutions of the equilibrium state (H f and Pf ) from these
nonlinear equations is not trivial. Employing a numerical optimization method
60
using MATLAB Optimization Toolbox [118] or solving the set of nonlinear differential equations using a delay differential equation solver in MATLAB [130]
may be considered suitable options. However, there are three potential drawbacks:
large convergence time especially for a slowly varying system, relatively expensive
computational load and potential convergence to local minima.
To avoid these drawbacks, we propose a method to derive a closed-form, analytical solution for the equilibrium states using a linearized form of Equation 4.16Equation 4.17. To linearize the nonlinear term σ (Pf ) in Equation 4.16-Equation 4.17,
σ (Pf ) in Equation 4.5 can be approximated into the following piecewise linear
function in which σ (Pf ) is replaced by a set of three linear functions representing
its behavior in low, normal and high BP regions:


 k1 P + c1 ; Pmin ≤ P ≤ P1
σ (P) ≃ σlin (P) =
k2 P + c2 ; P1 ≤ P ≤ P2


k3 P + c3 ; P2 ≤ P ≤ Pmax
(4.18)
where Pmin and Pmax were assigned as 50 and 200 in this work. Further, P1 and P2
are estimated based on a constrained optimization that minimizes an error between
the sigmoid function σ (P) and its linear approximation σlin (P). The constraints
are: 1) two lines k1 P + c1 and k3 P + c3 pass through [Pmax , 1] and [Pmin , 0], respectively, and 2) the slope of the line k2 P + c2 is equal to the slope of σ (P) at the
inflection point
∂ 2 σ (P)
∂ P2
= 0, which yields k2 =
∂ σ (p)
∂P
=
α
4
.
According to Equation 4.18, the steady-state equations Equation 4.16-Equation 4.17
can be rewritten into a set of linear equations for each region with corresponding
slope ki and y-intercept ci , i ∈ {1, 2, 3}, as follows:
Hf
Pf
1 βH + δH H0 − (ki Pf + ci )(VH + βH )
δH
= H f ∆V R0a 1 + α − α (ki Pf + ci )
=
(4.19)
(4.20)
which can ultimately be reduced to:
Hf
Pf
= A6 A1 − A2 Pf
= H f A5 A3 − A4 Pf .
61
(4.21)
(4.22)
where A1 = βH + δH H0 − ciVH − ci βH , A2 = ki (VH + βH ), A3 = 1 + α − α ci , A4 =
α ki , A5 = ∆V R0a , and A6 = δ1H .
Equation 4.21-Equation 4.22 can be reformulated into the following quadratic
equation solely based on Pf :
Pf = A5 A3 − A4 Pf A6 A1 − A2 Pf
|
{z
}
(4.23)
Hf
or,
aPf2 + bPf + c = 0
(4.24)
where a = −A4 A2 A5 A6 , b = A4 A1 A5 A6 + A3 A2 A5 A6 + 1, and c = −A1 A3 A5 A6 . The
closed-form solution for Pf is then obtained as follows:
Pf1,2
√
−b ± b2 − 4ac
=
2a
(4.25)
Once Pf is determined, H f can be easily calculated as a function of Pf using
Equation 4.21:
H f1,2 = A6 A1 − A2 Pf1,2
(4.26)
It is noted that, since Pf and H f are not known a priori, their candidate values
must be determined for the three regions specified in Equation 4.18. The three
pairs of Pf and H f thus obtained must then be validated against the corresponding
regions. For instance, Pf and H f determined from σ (P) = k2 P + c2 is regarded as
valid if P1 < Pf < P2 .
4.1.4 Stability Analysis
To the best of our knowledge, there is no well-accepted method for global stability analysis of nonlinear dynamic systems with delays, which include the autonomic-cardiac regulation model used in this study. However, according to the
Hartman-Grobman Theorem [131], stability properties of a nonlinear system in
the vicinity of an isolated equilibrium state can be determined by investigating
the properties of its linearization in the neighborhood of the equilibrium. Note
62
that the equilibrium states obtained by our analysis are isolated in the sense that
they are uniquely determined for the autonomic-cardiac regulation model once
its parameter configuration is provided. In order to exploit linear systems theory to solve our problem, we developed the delay-free realization Equation 4.10 of
the delayed nonlinear autonomic-cardiac regulation model. The stability of autonomic-cardiac regulation can be assessed by calculating the Jacobian matrix (JJ ) or
the state matrix of the nonlinear system Equation 4.10 at an estimated equilibrium
state W f = [H f , Pf , X f ]T as follows:

∂ ( f1 , f2 , f3 ) J (W f ) =
∂ (H, P, X ) W f

−δH





J (W f ) =  ∆V
 C
 a


0
−βH
∂ f1
∂H

 ∂ f2
=
∂H

∂ f
3
∂H
∂ σ (X − P)
∂ σ (P)
−VH
∂P
∂P
∂ f1
∂P
∂ f2
∂P
∂ f3
∂P

∂ f1
∂X 

∂ f2 

∂X 

∂ f3 
∂ X W=W f
(4.27)

∂ σ (X − P)

∂X


(X
−
P)
∂
σ

0
−PRaCa α

∂X
(4.28)
2 

0

RaCa 1 + α [1 − σ (X − P)] 

−2
τ
W=W f
−βH
∂ σ (X − P)
−R0aCa [1 + α (1 − σ (X − P))] − PR0aCa α
∂P
2
0
RaCa 1 + α [1 − σ (X − P)]
4
τ
Taking partial derivatives of the delay-free realization Equation 4.10 at a given
equilibrium state W f = [H f , Pf , X f ]T yields the Jacobian matrix Equation 4.28 where
e−α0 (Pf −P0 )
∂ σ (P)
.
|W=W f = α0
∂P
[1 + e−α0 (Pf −P0 ) ]2
(4.29)
According to the Hartman-Grobman theorem [131], the original nonlinear system (i.e., the autonomic-cardiac regulation) is stable in the neighborhood of an
equilibrium state if all the eigenvalues λi (i = 1, 2, 3) of the state matrix (JJ ) have
negative real parts, whereas it is unstable if any of its eigenvalues has a positive real
part. To quantitatively investigate conditions suggested by the Hartman-Grobman
theorem and also to calculate the stability margin of the system, we propose the
63
following stability margin metric, Sm :
Sm = max real(λi )
i=1,2,3
(4.30)
where real(·) denotes the real part of its argument, and λi is the i-th eigenvalue. Sm
represents the stability margin of the original system at the point of linearization,
and Sm < 0 is required for a stable system. In fact, Sm is a quantitative index representing the stability margin of the autonomic-cardiac regulation whose absolute
value measures the distance between the dominant system pole and the imaginary
axis. The system generally forfeits its stability margin, i.e., approaches to instability, as Sm becomes closer to zero.
4.1.5 Simulation Data
In this study, we generated two simulation-based datasets to validate the proposed
approach for estimating the equilibrium states and analyzing the stability of autonomic-cardiac regulation. First, to validate our approach to estimate the
BP
HR
and
equilibrium states, we generated 100 parameter configurations for the auto-
nomic-cardiac regulation model Equation 4.1-Equation 4.2 in which each model
parameter was determined randomly from a uniform distribution within 80% and
120% of the corresponding nominal value (see Table 4.1). Second, to validate our
approach to analyze the stability of autonomic-cardiac regulation, we considered
two distinct mental conditions: normal and stressed. The normal condition was
simulated by assigning normal parameter values listed in Table 4.1 to the model
parameters, while the stressed condition was simulated with appropriate changes
in the sympathetic and parasympathetic reflex parameters. Specifically, VH was
decreased by 50%, whereas βH and α were increased by 100% [50]. For each
of the mental conditions, we generated 12 sets of 100 parameter configurations.
In each set only a single parameter in the autonomic-cardiac regulation model
Equation 4.1-Equation 4.2 was altered from 50% to 200% of its nominal value,
while other parameters were fixed at their respective (i.e., normal or stressed) nominal values.
64
4.1.6 Validation of the Proposed Approach
Using the datasets described above, the validity of the proposed approach was
examined as follows. First, for each of the 100 parameter configurations generated to validate the proposed approach for estimating the equilibrium state of
autonomic-cardiac regulation, the equilibrium state determined by the proposed
approach was compared with those obtained numerically via numerical optimization and nonlinear simulation. For this purpose, we first computed
HR
and
BP
equilibrium states using Equation 4.25-Equation 4.26 according to the proposed
approach. Then, to obtain
HR
and
BP
equilibrium states via numerical optimiza-
tion, we solved Equation 4.14-Equation 4.15 for H f and Pf using the MATLAB
Optimization Toolbox [118].
Second, the nonlinear dynamic autonomic-cardiac regulation model Equation 4.6Equation 4.7 was simulated with MATLAB’s delay-differential equation solver
(dde23), from which HR and BP equilibrium states were determined as the steadystate values of simulated
HR
and
BP
time series obtained directly from the orig-
inal nonlinear autonomic-cardiac regulation model. The fidelity of the equilibrium states obtained from the proposed approach was assessed by its consistency
with those obtained from numerical optimization and nonlinear simulation via the
Bland-Altman analysis.
Finally, in order to validate the proposed approach for analyzing the stability
margin of autonomic-cardiac regulation, the proposed analytical stability metric,
Sm , was compared to an empirical metric, S p , obtained directly from a nonlinear
simulation Equation 4.31, which was defined based on the absolute amount of fluctuation of BP P(t) around its steady state:
30
Sp = ∑
t=1
P(t) − P(t)
P(t)
,
(4.31)
where P(t) was calculated by solving the original nonlinear system model Equation 4.6Equation 4.7 using the dde23 routine in MATLAB [118]. For the dataset generated
to validate the proposed approach for analyzing the stability of autonomic-cardiac regulation, the proposed metric, Sm , was calculated using Equation 4.28 and
Equation 4.30. The empirical metric, S p , was calculated using Equation 4.31.
65
(a) BP
(b) HR
Figure 4.1: Comparison of equilibrium states estimated using the proposed
analytical approach Equation 4.25 against numerical optimization (left
panel) and nonlinear simulation (right panel).
In addition to comparing Sm with S p , the validity of the proposed stability metric was further assessed using a priori knowledge of the relationship between autonomic-cardiac regulation model parameters and its stability. In particular, we tested
whether or not the proposed stability margin metric deteriorated as parasympathetic
tone (VH ) decreased and/or sympathetic tone (βH ) increased, as reported in the literature [50]. We also tested if the stability margin metric degrades as sympathetic
delay is increased [18].
66
Figure 4.2: Two metrics for stability margin Sm and S p over changes of a
model parameter from 50% to 200% of its nominal value for a healthy
physiological condition with and without stress. Sm is the blue solid
line; S p is the green dashed line. A normal condition (i.e., VH , βH , and
α were fixed at their nominal values).
67
Figure 4.3: Two metrics for stability margin, Sm and S p , over changes of a
model parameter from 50% to 200% of its nominal value for a healthy
physiological condition with and without stress. Sm is the blue solid
line; S p is the green dashed line. A stressful condition (i.e., a 50% lower
VH and 100% higher βH and α compared to their nominal values).
68
4.2 Results and Discussion
4.2.1 Identification of Equilibrium States
The Bland-Altman analysis clearly indicates that the equilibrium states determined
by the proposed analytical approach are highly consistent with those obtained from
numerical optimization and nonlinear simulation. Specifically, with respect to the
nonlinear simulation results, bias and 95% confidence interval associated with
BP
equilibrium states were only 1.0mmHg and 0.4mmHg, respectively, and bias
and 95% confidence interval associated with
HR
were both less than 0.5bpm (see
Figure 4.1). Therefore, we can conclude that the proposed analytical approach
could estimate equilibrium states very accurately with relatively low computational burden, when compared with those estimated by numerical optimization and
nonlinear simulation. Note that the low computational burden of the proposed
method will be observed during high-dimensional stability analysis of the timevarying mathematical model. Further, the proposed approach does not suffer from
issues associated with local minima since it is not an optimization-based method.
Finally, the proposed approach computes the equilibrium states associated with
each parameter configuration independently of the dynamic characteristics of autonomic-cardiac regulation, which often cause problems when nonlinear simulation
is used to obtain equilibrium states of slowly varying systems.
4.2.2 Proposed Stability Metrics
Figure 4.2-Figure 4.3 show the behaviour of Sm and S p calculated for the dataset
that we generated to validate the proposed approach for analyzing the stability of
autonomic-cardiac regulation. Using this dataset with a wide range of variation
in each parameter (i.e., 50% to 200%), we can investigate the pure effect of a
single model parameter on the stability margin of autonomic-cardiac regulation
in different physiologic conditions. Sm and S p values in response to variations
in a single parameter in the autonomic-cardiac regulation model are depicted in
Figure 4.2-Figure 4.3 for normal and stressful conditions. Overall, the tendency in
behaviors of the proposed stability margin metric, Sm , and the empirical metric, S p ,
were qualitatively consistent in most cases. It is noted that although inconsistency
69
in pattern between Sm and S p was observed for Ca in the normal condition, it was
regarded as noncritical since the sensitivity of the metrics to Ca was relatively small
compared with those to other parameters. The discrepancy in pattern between Sm
and S p associated with γ is due to the fact that γ is set to zero in the model used to
develop the proposed approach to stability analysis, whereas its value is not zero in
the model used for simulating autonomic-cardiac regulation. Therefore, the effect
of a single model parameter on the stability of autonomic-cardiac regulation can
be examined by analyzing the stability margin metric, Sm , over a desired parameter
space.
Figure 4.2-Figure 4.3 also suggest that the proposed stability metric, Sm , exhibits behavior consistent with well-known physiologic knowledge on the relationship between the stability of autonomic-cardiac regulation and sympathetic/parasympathetic tones and delays. In particular, the stability margin of autonomic-cardiac regulation is expected to decrease as cardiac vagal tone (VH ) decreases or
cardiac sympathetic tone (βH ) increases [90]. Figure 4.2-Figure 4.3 show that the
magnitude of Sm decreases with decreasing VH and increasing βH in both nominal (Figure 4.2) and stressful (Figure 4.3) conditions, as anticipated. It is known
that the stability margin of autonomic-cardiac regulation decreases as sympathetic
delay (τ ) increases [18]. Indeed, the magnitude of Sm is shown to decrease with
increasing sympathetic delay; the system may become unstable for a large enough
delay. In essence, along the consistency with S p , these observations support the
validity of the proposed approach to analyze the stability of autonomic-cardiac
regulation.
The results suggest that the effect of autonomic-cardiac parameters on stability
margin is not always monotonous, i.e., an increase (or a decrease) in a model parameter does not always cause a strict decrease or increase in the stability margin.
In Figure 4.2-Figure 4.3, the stability margin is shown to be related monotonously
to most parameters in the autonomic-cardiac regulation model, including Ca , τ , α0 ,
VH , βH , α , and δh , but it is shown to be enhanced or deteriorated depending on the
value of R0a , P0 , and H0 . These parameters can be critical in determining the stability margin of autonomic-cardiac regulation, since they complicate the analysis of
autonomic-cardiac stability. Further, we also observe in Figure 4.3 that a large peripheral resistance may help the stabilizing effort of autonomic-cardiac regulation
70
under stressful conditions.
4.2.3 Multi-dimensional Stability Analysis
Comparing Figure 4.2 and Figure 4.3 suggests that the reliance of stability margin on each individual model parameter pertaining to autonomic-cardiac regulation
is distinct for different physiologic conditions, e.g., normal condition or stressful
condition. For instance, the stability of autonomic-cardiac regulation is largely affected by R0a in Figure 4.2, but the effect of R0a is relatively small in Figure 4.3.
Further, a decrease in the nominal value of δh may cause instability during a stressful condition, whereas the same change in δh causes a decrease in the stability
margin only during a normal condition. The patterns in the reliance of the stability margin on H0 and γ are largely different between Figure 4.2 and Figure 4.3.
Since γ represents the inhibitory strength of the
PNS
on the cardiac sympathetic
tone βH , the significance of γ on the stability margin will be increased during a
stressful condition with an increased βH . These observations indicate that physiologic conditions (as represented by a particular parameter configuration in the autonomic-cardiac regulation model) must be accounted for when studying the impact
of autonomic-cardiac parameters on the stability of autonomic-cardiac regulation.
Considering that autonomic-cardiac regulation is a multi-parameter nonlinear
system with delay, one-dimensional stability analysis (i.e., stability analysis over
changes in a single model parameter) may not provide a comprehensive perspective on the stability of autonomic-cardiac regulation. For instance, the pattern of
reliance of Sm on H0 is dependent on the entire parameter configuration as indicated in Figure 4.2-Figure 4.3. Because of this, it is preferable to analyze the
stability properties of autonomic-cardiac regulation and its stability margin against
simultaneous changes in multiple parameters, i.e., the stability properties of autonomic-cardiac regulation should be examined in a multi-dimensional parameter space. An important strength of the proposed approach is that it can examine
the effect of changes in multiple parameters on the stability of autonomic-cardiac
regulation. Figure 4.4 graphically illustrates the reliance of the proposed stability
metric, Sm , Equation 4.30 on simultaneous changes of two parameters (baroreflex
set point P0 and another parameter). Accordingly, the stability properties of au-
71
Figure 4.4: The proposed stability metric, Sm , over 2-D parameter spaces
from 50% to 150% of their nominal values for a normal physiological condition. The quantitative stability margin metric, Sm , at each point
of the 2-D parameter space is mapped into a pixel-intensity level. A
higher pixel-intensity level is related to lower stability margin, and vice
versa.
tonomic-cardiac regulation against changes in two distinct model parameters can
be easily predicted. In essence, Figure 4.4 properly demonstrates the complexities
associated with multi-dimensional stability analysis of autonomic-cardiac regulation, i.e., interaction among autonomic-cardiac model parameters in determining
its stability. Figure 4.4, for example, depicts that a specific amount of change in
P0 can have different influences on the stability of autonomic-cardiac regulation in
the presence of simultaneous changes in other parameters. Indeed, large P0 results
in smaller stability margin in response to increasing H0 , whereas large P0 yields
a larger stability margin in response to increasing Ca . Overall, Figure 4.4 clearly
demonstrates the importance of analyzing the stability of autonomic-cardiac regulation in multi-dimensional parameter space; mush in-depth work on this issue in
follow-up studies is warranted.
Autonomic-cardiac regulation can also be studied using a hybrid dynamical
systems framework that describes a physical system with a combination of continuous and discrete parts to represent time- and event-based behaviors [132]. Autonomiccardiac regulation is a complex, nonlinear physiological system that can be approximated with several relatively simple, linear mathematical models in different op-
72
erating points. Different physiologic conditions (e.g., normal condition or stressful
condition) described with a specific set of model parameters generates some complexities in stability analysis. Using the framework of hybrid dynamical systems
reduces complexities to analyze the system stability, and it may increase the accuracy of mathematical modeling to capture physiological behaviors [133]. The linearized mathematical model introduced in this chapter will be beneficial to analyze
the stability of autonomic-cardiac regulation using a hybrid system framework.
4.2.4 Limitations
This study has a number of limitations, as discussed below.
First, the mathematical model may not capture every significant mechanism in
HR
and BP regulation. For example, baroreflex control of SV , respiratory coupling
on
CVS
(refer to Chapter 6), and chemoreflex mechanism are not described in the
current mathematical scheme, and therefore, they were not included in the stability
analysis results.
Second, we assumed that all of the parameters in autonomic-cardiac regulation are independent of one another in the simulated data; this may not be true in
reality. Thus, it is possible that a small portion of the simulated data we used to
validate the proposed approach may not be good reproductions of reality. In-depth
understanding of interactions and dependence among the parameters is required to
resolve this issue.
Third, to study hemodynamic instability in an individual, we must first develop
a subject-specific mathematical model (refer to Chapter 3) and then perform stability analysis. For example, we used empirical minimum/maximum
BP
values to
linearize the sigmoidal baroreflex characteristic Equation 4.18. In the future, the
proposed method must be improved by specifying the model parameters for each
subject.
4.3 Conclusion and Future Work
In this chapter, we proposed a model-based analytical approach to stability analysis of autonomic cardiac regulation. Based on a physiology-based model of au73
tonomic-cardiac regulation, we developed an analytical and computationally efficient method to estimate the equilibrium states of the system, and we developed a
systematic approach to stability analysis of autonomic-cardiac regulation that can
provide a quantitative metric of stability margin. The efficacy of the proposed approach was examined using a series of simulation experiments. Future work will
include developing 1) an approach to analyze global stability of autonomic-cardiac
regulation, 2) computationally efficient strategies to identify parameter configurations associated with autonomic-cardiac instability in multi-dimensional parameter
space, and 3) novel intervention and therapeutic strategies to maintain the stability
of autonomic-cardiac regulation.
74
Chapter 5
A Novel Approach to the Design
of an Artificial Bionic Baroreflex
The ANS maintains homeostasis in the CVS through many negative feedback mechanisms including the baroreflex (the major short-term blood pressure control mechanism) to deliver adequate oxygenated blood flow to organs in response to physical
(e.g., exercise and orthostatic hypotension) and psychological (e.g., fear and anxiety) stressors [41–43].
In the CVS, instantaneous arterial
BP
is sensed by baroreceptors located on the
major arteries. Accordingly, a series of commands is produced by the baroreflex
and transmitted to the heart, arteries, and other organs to maintain homeostasis
in the
CVS.
An artificial bionic baroreflex consists of pressure sensors to measure
arterial BP and a neurostimulator that generates an electrical pulse train to stimulate
sympathetic and parasympathetic nerves regulated by a computerized device [36,
134].
The gravitational effect on circulation during postural changes provokes a baroreflex response to prevent hypotension and hypoperfusion of the brain [39]. Therefore, baroreflex failure in individuals with severe orthostatic hypotension (e.g., individuals with traumatic SCI s) may result in loss of consciousness during a sittingto a standing-position change resulting in a severely impaired quality of life [36,
135]. Moreover, the prevalence of drug-resistant hypertension (i.e.,
BP
remains
above 140/90 mmHg in spite of the concurrent use of three anti-hypertensive med75
ications [58, 136]) has increased in recent years [136, 137]. An artificial bionic
baroreflex is aimed to be an effective treatment for baroreflex failure in individuals
with drug-resistant hypertension and severe orthostatic hypotension.
In [39], the open-loop transfer function of the baroreflex was identified using white noise perturbation after anatomically isolating the carotid sinuses by assuming that the baroreflex works linearly in some physiological pressure range.
Kawada and Sugimachi [135] presented encouraging results to regarding the prevention of orthostatic hypotension in anesthetized cats by using epidural spinal
cord stimulation and frequency analysis. The nerve stimulation devices can be
implanted or percutaneously inserted into the skin’s surface.
We proposed a method to design an artificial bionic baroreflex by mimicking
the in-vivo baroreflex mechanism. This method can be used to adjust existing neurostimulator devices to regulate
BP
within an individual’s
CVS
(Figure 5.1). The
proposed method consists of two parts: a sigmoidal characteristic that mimics the
modulating baroreflex functions on the
SNA
and
PSNA
and an adaptation mecha-
nism that adjusts the sigmoidal characteristic to different physiological conditions
(e.g., exercise and sleep) as well as pathological conditions (e.g., hypertension and
cardiovascular disorders). The adaptation mechanism resetting the baroreflex characteristic is devised according to the physiological adjustment mechanism of the
in-vivo baroreflex. Further, we analyzed the robustness of the proposed controller
scheme in regard to the model uncertainty showing the inter-individual differences
in autonomic-cardiac regulation.
5.1 Methods and Algorithm
In this section, we first briefly explain the experimental data obtained from the
MIMIC dataset, which is used in this study. We then present a physiology-based
mathematical model of autonomic-cardiac regulation described by two coupled
nonlinear and delayed differential equations. Subsequently, the system identification technique introduced in [1] and used to develop subject-specific models for
three subjects is briefly explained. In this study, the subject-specific mathematical
model has been used instead of the individual’s in-vivo autonomic-cardiac regulation (Figure 5.2). Finally, the proposed method to design an artificial bionic
76
Figure 5.1: Schematic model of autonomic-cardiac regulation with emphasis
on the baroreflex
baroreflex is described and is followed by a robustness analysis of the proposed
control strategy.
5.1.1 Experimental Data
We examined the proposed method using experimental data of autonomic-cardiac
regulation in three subjects taken from the MIMIC dataset [47]. A 1-hour sample of
HR
and
BP
signals in each individual is extracted and divided into 30s-long
data segments to be used in the system identification section. The MIMIC dataset
contains physiologic signals including
HR , BP ,
and
CO
in different lengths contin-
uously recorded at approximately 1 Hz from intensive care unit (ICU) monitors.
The MIMIC dataset is freely available on the PhysioNet website [109].
5.1.2 Mathematical Model
We introduced a physiology-based mathematical model of the autonomic-cardiac
regulation in Chapter 3 by using two coupled differential equations Equation 5.1Equation 5.2 having nonlinear and delayed dynamic interactions, each of which
77
Table 5.1: Model parameters of autonomic-cardiac regulation.
Parameter
Ca
R0a
∆V
H0
τ
VH
βH
α
γ
δH
Definition
arterial compliance
minimum arterial resistance
stroke volume
intrinsic HR
sympathetic delay
vagal tone
sympathetic control of HR
sympathetic effect on Ra
vagal damping of βH
relaxation time
Nominal Value
1.55 mlmmHg−1
0.6 mmHgsml −1
50 ml
100 min−1
3s
1.17 s−2
0.84 s−2
1.3
0.2
1.7 s−1
describes the dynamics of HR and BP regulation as follows:
Ḣ(t) = βH Ts −VH Tp + δH H0 − H(t)
H(t)∆V
P(t)
+
Ṗ(t) = − 0
Ra (1 + α Ts )Ca
Ca
where H is
HR ,
and P is mean arterial
BP .
(5.1)
(5.2)
The definitions and nominal values of
the parameters in Equation 5.1-Equation 5.2 are summarized in Table 5.1. In this
model, the modulating baroreflex functions on the
SNA
and
PSNA
are denoted by
Ts and Tp , respectively.
The modulating baroreflex functions on the SNA and PSNA (i.e., Ts and Tp ) can
be modeled using a sigmoid function σ (P) with an amplitude-limiting characteristic [96]. σ (P) is defined as follows:
σ (P) =
1
1 + e−α0 (P−P0)
50 ≤ P ≤ 200.
(5.3)
The sigmoid function σ (P) is characterized using two variables, setpoint P0 and
sensitivity α0 [113, 129]. To simulate the in-vivo sympathetic and parasympathetic
modulating functions, we substitute Ts = 1 − σ P(t − τ ) and Tp = σ P(t) into
Equation 5.1-Equation 5.2.
Parametric sensitivity analysis is conducted on the model to classify the model
parameters into high-sensitivity and low-sensitivity groups based on their relative
78
Figure 5.2: Schematic model of the proposed artificial bionic baroreflex
impacts on the system outputs. H0 and R0a were initially classified into the category
of invariant parameters since they are essentially constant within an individual in
a short-time interval. The remaining model parameters were classified into highsensitivity (VH , βH , α , ∆V , and P0 ) and low-sensitivity (α0 , γ , Ca , τ , δH ) groups,
according to the results of the sensitivity analysis, to select a subset of parameters
with significant impact on the system outputs (i.e., high-sensitivity group). The
detailed description of the mathematical model and parametric sensitivity analysis
is explained in Chapter 3.
5.1.3 System Identification
To estimate subject-specific high-sensitivity parameters in Equation 5.1-Equation 5.2,
a system identification method was developed based on an optimization problem
minimizing the normalized L1 -error between measured versus model-estimated HR
and
BP
signals. The system identification was performed by optimizing the high-
sensitivity parameters such that the error function Equation 5.4 became minimized
in each 30s-long data segment, whereas low-sensitivity and invariant parameters
were fixed at their corresponding population nominal values (Table 5.1). The error
79
Figure 5.3: BP measurement (BP setpoint) vs. the results of the artificial
bionic baroreflex (simulated BP) for individual with subject number 477.
function (i.e., the objective function) was specified as follows:
EP + EH
;
J=
2
Xs (t, M) − Xm (t) ,
EX = ∑ Xm (t)
n
(5.4)
t=0
where Xm (t) and Xs (t, M) (X = H, P) are measured and model-estimated output signals, respectively, and M is the set of high-sensitivity parameters in the autonomiccardiac regulation model, i.e., M = {VH , βH , α , P0 , ∆V }. The optimization problem
was solved using the fmincon routine with an active-set algorithm in the MATLAB
Optimization Toolbox [118], which finds the constrained minimum of a multivariable nonlinear scalar function J using Quasi-Newton approximation. The set of
optimized high-sensitivity parameters minimizing the error function was used as
estimates of high-sensitivity parameters for the corresponding data segment. The
system identification method has been thoroughly described in Chapter 3.
5.1.4 Artificial Bionic Baroreflex
The artificial bionic baroreflex is a negative-feedback control system containing a
set of sensors that measures
BP ,
a computerized device that determines the con-
trol action and a set of electrodes that stimulates the sympathetic and parasympa-
80
Figure 5.4: BP measurement (BP setpoint) vs. the results of the artificial
bionic baroreflex (simulated BP) for individual with subject number 486.
Figure 5.5: BP measurement (BP setpoint) vs. the results of the artificial
bionic baroreflex (simulated BP) for individual with subject number 476.
81
thetic efferent nerves. This system continuously measures
BP
and computes the
frequency of a pulse train required to stimulate sympathetic and parasympathetic
efferent nerves. The measured
time-varying
BP
BP
(BPm ) must be compared continuously to the
setpoints (BPsp ), the
BP
level providing the need of body organs
for oxygenated-blood, to be used in the control scheme. If the BPm differs from the
BPsp , the baroreflex characteristic is adjusted so that the
BP
to gradually reaches
the BPsp . Note that determining the BPsp signal is a challenge, and the proposed
method has been developed based on a given BPsp signal. In theory, the timevarying BP setpoints must be estimated based on the major vital needs of the body,
e.g., the oxygenated blood-flow supply to the brain during physical (e.g., exercise
and orthostatic hypotension) and psychological (e.g., fear and anxiety) stressors.
In this study, individuals were replaced by subject-specific mathematical models describing autonomic-cardiac regulation of each subject, and simulated
BP
(BPsim ) was used instead of BPm continuously compared against BPsp . The baroreflex effects on the autonomic-cardiac regulation are achieved by modulating
and
PSNA
(VH , βH , and α ) using Ts and T p . For example, when the
BP
SNA
must de-
crease to reach the setpoint, T p must reset such that its magnitude at the same
BP
level becomes higher. This causes HR to decrease and then BP to decrease. As T p
is a sigmoidal curve, P0 must reset to a lower value in order to obtain a higher HR
decelerating parasympathetic effect and vice versa. Therefore, P0 , then sigmoidal
characteristic, must be updated by the adjustment rule as follows:
P0 (t + ∆) = P0 (t) + k · (BPm − BPsp)
(5.5)
where k is a positive coefficient representing the pace or the strength of the adjusting mechanism in response to an error in the
BP
regulation, and ∆ is an interval
in which the baroreflex characteristic needs to be updated. The adjustment rule
Equation 5.5 is initialized by P0 = 100. Moreover, to be consistent with baroreflex
physiology, P0 is constrained between 50 mmHg and 200 mmHg. A very large
k may cause overshoot in the control system, while a very small k may cause a
large settling time, preventing proper adjustment of the baroreflex characteristic to
track the BPsp . This coefficient is empirically tuned to k = 0.08 by considering
both overshoot and settling time of the control system. The time inetrval, ∆, in
82
Equation 5.5 is 30s in this study; however, it can be set to a larger or smaller value
depending on the pace of variation in BPsp .
5.1.5 Robustness Analysis
To validate the robustness of the proposed control strategy in regard to model uncertainty as well as inter-individual differences, we generated 100 sets of model
parameters showing 100 different mathematical models of autonomic-cardiac regulation. In each set of model parameters, the high-sensitivity parameters were assigned to random values from a uniform distribution in the neighborhood (+/-50%)
of their respective individualized nominal values (Table 5.2), while the remaining
parameters were fixed at their population nominal values (Table 5.1). The
BP
set-
points were also assigned to the BPm of the individual whose nominal parameter
values were selected to generate the 100 sets of random values.
5.2 Results and Discussion
Since we aimed to use a subject-specific mathematical model for each individual, the mathematical model Equation 5.1-Equation 5.2 was specified by estimating individualized nominal values for high-sensitivity parameters in each subject,
whereas the remaining parameters were assigned by their population nominal values (Table 5.1). As the MIMIC dataset contains CO measurement, we calculated
an individualized nominal value of ∆V for each subject instead of either estimating ∆V by the proposed identification technique or using the population nominal
value. Accordingly, we obtained a time series of VH , βH , and α with a 1-hour
length using the system identification technique; the average value of these parameters over a 1-hour length was calculated to be assigned as individualized nominal
values (Table 5.2). Note that the individualized high-sensitivity parameters must
be updated at every 1-hour (or any other length initially assumed) interval of data
in future studies. We obtained three sets of model parameters representing three
subjects in the 1-hour interval to evaluate the proposed method for designing an
artificial bionic baroreflex.
To evaluate the proposed method, we compared the simulated
BP
obtained
based on the control strategy Equation 5.5 versus the BP setpoints Figure 5.3-Figure 5.5.
83
(a) BP measurement (BP setpoint) vs. the results of the artificial bionic baroreflex
(simulated BP)
(b) The calculated control signal P0
Figure 5.6: The results of robustness analysis for an individual with subject
number 477. The solid line shows an average value of 100 simulated
signals obtained by the proposed control strategy, whereas the shaded
area indicates the corresponding standard deviation.
In each panel of Figure 5.3-Figure 5.5, the top figure shows the
BP
measurements
used as BP setpoints versus simulated BP obtained by the proposed artificial bionic
baroreflex and the bottom figure shows the tracking error over the 1-hour interval.
In each subject shown in Figure 5.3-Figure 5.5, the tracking error is considerably
higher during t=0 s - 100 s because of the P0 initialization. After t>100 s, P0 converges to a proper value to control the
BP
regulation system. Since we assumed
that the control strategy would not respond very rapidly, several abrupt changes
in the
BP
setpoints during t=1000 s - 1300 s in Figure 5.3 and t=1700 s - 1800 s
in Figure 5.5 which may be originated due to the measurement noise caused the
84
Figure 5.7: The calculated control signal, P0 , in three subjects
tracking error to become large. Figure 5.7 shows the calculated P0 for 3 subjects.
The large tracking error between t=2700 s and t=3200 s in Figure 5.4 and the saturated P0 at 200 s during the same interval indicates that the control strategy was
not able to track the setpoints successfully in that interval.
Figure 5.6 shows the exemplary results of robustness analysis for an individual
(Subject I). Figure 5.6a indicates that the proposed control strategy meets the setpoint tracking specification (low tracking error) regardless of variability in model
parameters as well as parameter identification error described by a large uncertainty
in the model parameters (VH , βH , and α ). Indeed, we showed that the proposed
control strategy is robust against model uncertainty by tolerating large variations
in P0 (Figure 5.6b).
To evaluate the proposed approach in a clinical setting, we must perform a
clinical study in which the regulation mechanism of the baroreflex on sympathetic
and vagal nerves are replaced with an external controller. The closed-loop in-vivo
85
Table 5.2: Individualized nominal values of high-sensitivity parameters in
three subjects versus corresponding population nominal values.
VH
βH
α
∆V
I (477)
0.65
1.5
0.7
46
Subject
II (486) III (476)
1.37
2.13
0.87
0.68
1.36
1.55
40
36
Population
1.17
0.84
1.3
50
baroreflex must be opened at the level of efferent or afferent nerves according to the
level/type of injury in the baroreflex mechanism, and an electrical stimulator must
be subcutaneously implanted to be overridden the corresponding nerves. The electrical stimulator is a rate responsive pulse generator, and the stimulation frequency
and magnitude must be adjusted based on calculated P0 (Figure 5.7).
As mentioned above, the electrode placement site will be different according to
the level/type of injury in the baroreflex mechanism. For example, afferent baroreceptor nerves may need to be overridden in individuals with chronic drug-resistant
hypertension, while efferent nerves may need to override in individuals with
SCI .
Sensory information, such as BP and HR measurement, is needed in the closed-loop
control scheme. For example, BP can be sensed by in-vivo mechanism (e.g., afferent baroreceptor nerves) or artificial receptors (e.g., implanted microstrain gauges)
according to the specific physiological condition of the individual. Since there is
no in-vivo HR measurement mechanism, the artificial HR sensors may improve the
efficiency of the proposed approach. In order to determine the optimal site of electrode placement to stimulate sympathetic and vagal nerves, we must investigate
ease of access, significance of effect, and possible side effects [38].
5.3 Conclusions and Future Work
This chapter presented the feasibility and potential for a computationally efficient
closed-loop control scheme to design an artificial bionic baroreflex that can be used
in the treatment of baroreflex failure. The recently introduced open-loop BP control
schemes will be improved by the proposed closed-loop technique to accommodate
the time-varying needs of
BP
level in different daily life conditions. In the future,
86
the proposed approach should be validated extensively in clinical settings.
87
Chapter 6
Mathematical Modeling of
Autonomic-Cardiorespiratory
Regulation
Autonomic-cardiorespiratory regulation operates through interactions between the
ANS , the cardiovascular system, and the respiratory system. The ANS maintains
homeostasis in the cardiorespiratory system in order to deliver adequate oxygenated
blood flow to organs against physical (e.g., exercise and orthostatic hypotension)
and psychological (e.g., fear and anxiety) stressors [41–43]. The
ANS
consists of
two branches, the PNS , which is dominant in “rest and digest" states, and the SNS,
which is aroused in “fight or flight" states. The
ANS
regulates
RR , ILV , BP , CO ,
and HR using different mechanisms, e.g., adjusting SNA and PSNA on the sinoatrial
node, cardiac contractility, and peripheral resistance [17, 23, 51].
A variety of mathematical models to describe autonomic-cardiac regulation
using black-box and white-box (physiology-based) approaches have been proposed
previously [23, 55, 96]. However, a physiology-based mathematical model for the
respiratory system impacts has been investigated to a certain extent [43]. Further,
the respiratory system impacts on
and
BP
HR
(i.e., respiratory sinus arrhythmia or
RSA )
(i.e., venous return variation) have either been neglected [55] or simply
modeled by a non-physiology function (e.g., a sine function) [64].
In this chapter, we introduce a physiology-based mathematical model of auto88
nomic-cardiorespiratory regulation described by a set of three coupled nonlinear
and delayed differential equations, each of which describes the regulation of
BP , and RR. A unique
HR ,
strength of the proposed model is its physiology-based mod-
eling approach to describe most of the internal mechanisms within in-vivo systems.
Recently, we proposed a relatively improved model of autonomic-cardiac regulation [1] based on the work of Fowler et. al. [7]. However, the respiratory system
dynamics and effects such as venous return variation during respiration phases,
lung stretch-receptor reflex and respiratory generator center were not studied in
[1].
6.1 Methods and Algorithm
In this section, we first describe the experimental dataset collected in this study.
Then, we present the physiological background associated with major causes of
HR
and
BP
fluctuations. A mathematical model of autonomic-cardiac regulation
(without respiratory system effects) described by a set of two coupled nonlinear
and delayed differential equations is also introduced. We then present an improvement in the mathematical model that describes neuromechanical and mechanical
coupling of cardiovascular and respiratory systems, i.e, lung stretch-receptor reflex
and venous return variations. We also introduce a differential equation to model
RR regulation that mainly originates from the medullary respiratory center in the
brainstem, which is influenced by voluntary actions and chemoreflex.
6.1.1 Experimental Dataset
We collected autonomic-cardiorespiratory signals including
ECG , BP
waveform,
and Tidal Volume (TV ) from 18 healthy subjects without any cardiovascular disorder history during an
LBNP
experiment followed by a respiration maneuver using
the Pneumocard and Portapress devices. The experiment was approved by Simon
Fraser University board of ethics (Appl. # 2012s0078; Dated November 26,2012)
and consent form was signed by the participants. Figure 6.1 depicts an example of
recorded signals during different stages of the LBNP test.
89
Figure 6.1: Physiological measurement during LBNP experiment in an individual; mean BP, SV , and HR were calculated according to the BP waveform and ECG recordings.
6.1.2 Physiological Background
HR
fluctuations around the mean
the
SNS
and
PNS
HR
(also referred to as
HRV )
are generated by
in the cardiorespiratory control system. In healthy individuals,
the HRV spectrum shows two predominant peaks: one at low frequency around
0.1 Hz (Mayer waves) associated with arterial pressure biofeedback. The other
one, at higher frequency around 0.25 Hz (corresponding to respiration frequency),
is called
RSA . RSA
is mainly generated through two mechanisms: neural-based
modulation of cardiac vagal activation by the medullary respiratory center and
neuromechanical-based modulation of cardiac vagal activation by the lung stretchreceptor reflex [84].
RSA
has been observed at the approximate respiratory fre-
quency even in the absence of respiration due to the activation of the medullary
respiratory center [85]. The lung stretch-receptor reflex inhibits and excites cardiac vagal activation tone during inspiration (lung inflation) and expiration (lung
90
deflation) respectively, causing a decrease and an increase in heart periods during
respiratory cycles [84, 85]. The synchrony of heart period fluctuations (i.e., HRV)
and respiration cycles caused by the lung stretch-receptor reflex has the potential to
increase the efficacy of the pulmonary gas exchange between capillary blood flow
and alveolar gas volume by matching perfusion to ventilation within each respiratory cycle [84, 138].
Blood pressure variability (BPV) is caused mainly by HRV as well as the direct
mechanical effects of respiration (either spontaneous or mechanical) on
The HRV influences
BP
BP
[139].
through the heart period baroreflex mechanism. Further,
the direct mechanical effect of respiration causes variation of venous return in each
respiratory cycle. During spontaneous inspiration, the chest wall expands and the
diaphragm descends resulting in lower intrapleural pressure1 and therefore expansion of the lungs and cardiac chambers [48]. This expansion causes an increase
in cardiac pre-load2 and
SV
due to the Frank-Starling mechanism as well as a de-
crease in right atrial pressure that is necessary for obtaining the required pressure
gradient for the venous return [48, 67]. Consequently, venous return increases during spontaneous inspiration and decreases during spontaneous expiration. During
mechanical ventilation, the chest wall and diaphragm are not displaced; however,
the lungs are inflated due to an external air force that causes different consequences
such as an increase in intrapleural pressure during mechanical inspiration. Similarly, venous return decreases during mechanical inspiration and increases during
mechanical expiration (Table 6.2).
6.1.3 Autonomic-Cardiac Regulation
We introduced a physiology-based mathematical model of autonomic-cardiac regulation in [1] using two coupled differential equations Equation 6.1-Equation 6.2
having nonlinear and delayed dynamic interactions, each of which describe the
1 The pressure within the thoracic space between the organs (lungs, heart, vena cava) and the chest
wall is called intrapleural pressure.
2 Cardiac pre-load is the end-diastolic volume (EDV) of the ventricle at the beginning of systole.
91
Figure 6.2: Schematic diagram of interactions between cardiovascular, respiratory and nervous systems.
92
Table 6.1: Model parameters of autonomic-cardiac regulation.
Parameter
Ca
R0a
∆V
H0
τ
VH
βH
α
γ
δH
Definition
arterial compliance
minimum arterial resistance
stroke volume
intrinsic HR
sympathetic delay
vagal tone
sympathetic control of HR
sympathetic effect on Ra
vagal damping of βH
relaxation time
Nominal Value
1.55 mlmmHg−1
0.6 mmHgsml −1
50 ml
100 min−1
3s
1.17 s−2
0.84 s−2
1.3
0.2
1.7 s−1
dynamics of HR and BP regulation:
Ḣ(t) =
βH Ts
1+γ Tp
−VH Tp + δH H0 − H(t)
P(t)
= − R0 (1+
+ H(t)∆V
Ca .
α Ts )Ca
Ṗ(t)
a
where H is HR and P is mean arterial
BP .
(6.1)
(6.2)
Ts = 1 − σ P(t − τ ) and Tp = σ P(t)
are sympathetic modulating function and parasympathetic modulating function respectively, generated by the baroreflex control mechanism. Note that Ts and Tp
are both purely
BP
dependent while the
SNS
and
PNS
are also modulated by other
physiological variables (e.g., O2 and CO2 concentration in blood) or psychophysiological states (e.g., fear and anger). The time delay associated with the sympathetic
pathway is denoted by τ . σ (P) is defined as follows:
σ (P) =
where P0 and α0 are the
respectively.
1
1 + e−α0 (P−P0)
BP
50 ≤ P ≤ 200.
(6.3)
setpoint and the sensitivity of baroreflex mechanism,
6.1.4 Autonomic-Cardiorespiratory Regulation
In this study, we improve our previous mathematical model of autonomic-cardiac
regulation Equation 6.1-Equation 6.2 by modeling two major interactions of car93
Figure 6.3: An extensive block-diagram model of autonomiccardiorespiratory regulation [see Chapter 2] with emphasis on
parts described in Equation 6.7-Equation 6.8. The shaded parts are not
described in the mathematical model.
diovascular and respiratory systems, i.e., mechanical and neuromechanical. Further, we introduce a differential equation representing the dynamic of respiration
rhythm originated in the medullary respiratory center.
The mechanical coupling of the cardiovascular and respiratory systems causes
an increase in venous return and, consequently, an increase in
SV ,
during sponta-
neous inspiration and mechanical expiration. On the other hand, venous return, and
therefore
SV ,
decrease during spontaneous expiration and mechanical inspiration
(Table 6.2). We modeled this pure mechanical effect by adding (during mechanical respiration) or subtracting (during spontaneous respiration) k2V̇L with positive
94
coefficient k2 to the SV (∆V ) as follows:
Ṗ(t) = −
P(t)
R0a (1 + α Ts )Ca
+
H(t)(∆V ± k2V̇L )
.
Ca
(6.4)
The neuromechanical coupling of the respiratory and cardiovascular systems
is generated by the lung stretch-receptor reflex. This reflex causes an increase in
HR
in
during inspiration, while ILV consistently increases (i.e., V̇L >0), and a decrease
HR
during expiration, while ILV consistently decreases (i.e., V̇L <0) (Table 6.2).
The lung stretch-receptor reflex inhibits and excites cardiac vagal activation tone
during inspiration and expiration, respectively [84]. We model this mechanism
by subtracting a respiration-related term consisting of a rate of change in ILV, V̇L ,
multiplied by a positive coefficient k1 , to cardiac vagal activation tone VH in the HR
equation as follows:
Ḣ(t) =
βH Ts
− (VH − k1V̇L )Tp + δH H0 − H(t)
1 + γ Tp
During inspiration while V̇L >0, inhibition effects of the
therefore
HR
(6.5)
on
HR
reduce, and
increases. Similarly, during expiration, while V̇L <0,
HR
decreases in
PNS
response to a rise in inhibition effects of the PNS .
The medullary respiratory center of each individual generates a relatively constant rhythm, R0 , which is modulated in different conditions such as low O2 or high
CO2 concentration in blood, sleep, and emotions (e.g., fear and anxiety). Specifically, the respiration rhythm, R0 , is modulated by chemoreflex stimulation caused
by changes in chemoreceptors responses throughout the body. The chemoreflex
operates mainly in response to the CO2 levels rather than O2 levels [48]. We model
the dynamics of respiration rate, denoted by R, generated in the respiratory center
as follows:
h
i
Ṙ(t) = k3 1 + σco2 R0 − R(t) + u(t)
(6.6)
where σco2 is a sigmoid function representing chemoreflex modulating function on
R0 and u(.) is a voluntary component of RR regulation. Note that the only voluntary
term in autonomic-cardiorespiratory regulation is the individual’s ability to change
95
Table 6.2: Respiratory system impacts on VL , HR, VR, and ∆V .
Spontaneous
Inhale Exhale
↑
↓
↑
↓
↑
↓
↑
↓
VL (Instantaneous Lung Volume)
HR (Heart Rate)
VR (Venous Return)
∆V (Stroke Volume)
Mechanical
Inhale Exhale
↑
↓
↑
↓
↓
↑
↓
↑
RR. Finally, we propose the mathematical model of autonomic-cardiorespiratory
regulation illustrated in Figure 6.3 as follows:
Ḣ(t) =
βH Ts
− (VH − k1V̇L )Tp + δH H0 − H(t)
1 + γ Tp
Ṗ(t) = −
P(t)
R0a (1 + α Ts )Ca
+
(6.7)
H(t)(∆V ± k2V̇L )
Ca
(6.8)
i
h
Ṙ(t) = k3 1 + σco2 R0 − R(t) + u(t)
(6.9)
where nominal values of the non-respiratory related parameters are shown in Table 6.1,
and nominal values of k1 and k2 are 0.073 l −1 s−1 and 3.12 ms, respectively. Nominal values of k1 and k2 are assigned such that the respiration-related terms k1V̇L and
k2V̇L generate 10% perturbation on the amplitude of VH and ∆V , respectively.
6.2 Results and Discussion
6.2.1 Model Validation
In this work, we proposed a physiology-based mathematical model of autonomic-cardiorespiratory regulation built upon a mathematical model of autonomic-cardiac
regulation described in Chapter 3. We included respiratory terms in the mathematical model according to the corresponding location and dynamic of the respiratory effect on autonomic-cardiac regulation. We potentially can use the proposed
mathematical model to investigate the effects of respiration on
96
HR
and
BP
regula-
Figure 6.4: PSD difference of HRV among simulated (two methods) vs. measured HR signals at the different stages of the LBNP experiment; the
shaded area shows the respiratory frequency band.
tion and fluctuation. To validate the proposed mathematical model, we must show
that adding respiratory terms in the mathematical model improves the accuracy of
model-estimated signals.
Since respiration plays a major role in HRV and Power Spectral Density (PSD )
analysis commonly used in HRV studies, the model-estimated HR obtained from the
autonomic-cardiorespiratory regulation model and HR obtained from the autonomiccardiac regulation model were tested against measured
HRV .
HR
using
PSD
Figure 6.4 shows that ∆PSD of model-estimated and measured
analysis of
HR
is close
to zero within the respiratory frequency band (mostly located at 0.2±0.05 Hz) in
the autonomic-cardiorespiratory regulation model. We can conclude that a more
accurate mathematical model was obtained by including the respiratory dynamic
in the autonomic-cardiac regulation model.
97
Table 6.3: A numerical measure of perturbation caused by mechanical coupling effects J2 and neuromechanical coupling effects J1 .
k1 /k1,Nom
50%
75%
100%
125%
150%
k2 /k2,Nom
50%
75%
100%
125%
150%
J1
0.18
0.27
0.36
0.44
0.53
J2
0.76
1.14
1.52
1.89
2.28
6.2.2 Mechanical vs. Neuromechanical Couplings
To investigate the neuromechanical coupling effects of respiration k1V̇L on HR and
BP ,
we assigned k2 =0 and all the non-respiratory related parameters to their nominal values, whereas k1 was changed from 50% to 150% (25% increment) of its
nominal value. Further, to compute the perturbation of
HR
and
BP
merely caused
by neuromechanical coupling effects k1V̇L , the average sum of absolute normalized
errors of HR and BP J1 was used as follows:
30 X
EP,1 + EH,1
k1 6=0 (t) − Xk1 =0 (t) J1 =
; EX,1 = ∑ 2
Xk1 =0 (t)
t=0
whereas Xk1 6=0 (t) and Xk1 =0 (t) are
HR
or
BP
(6.10)
(X = H, P) with and without neu-
romechanical coupling effects, respectively. To solve the mathematical model
Equation 6.7-Equation 6.8 for each given set of parameteres, we first generated
an arbitrary ILV signal for a 30 s-length segment with constant rate of change in
lung volume V̇L =2 ls−1 during inhaling and exhaling phases (Figure 6.5). Then,
we numerically solved the mathematical model Equation 6.7-Equation 6.8 using a
DDE (delay differential equation) solver in MATLAB to obtain perturbed
BP
HR
and
and
BP ,
for five different values of k1 (Figure 6.5).
Similarly, J2 is the average sum of absolute normalized errors of
HR
while k2 was changed from 50% to 150% (25% increment) of its nominal value.
This study shows that
HR
and
BP
perturbation caused by mechanical coupling ef-
fects J2 is higher than perturbation caused by neuromechanical coupling effects J1
(Table 6.3).
98
Figure 6.5: Neuromechanical coupling effects of respiration on HR and BP.
Figure 6.6: Mechanical coupling effects of respiration on HR and BP.
99
6.2.3 Limitations
Despite the novelty of this study to describe the autonomic-cardiorespiratory system using a physiology-based mathematical model, it has one major limitation, as
discussed below. The nominal values of two parameters, k1 and k2 , presented in
this work must be reconsidered. In fact, we must perform an experiment to calculate numerically two parameters, k1 and k2 , which represents open-loop gains
from
ILV
to VH and from
ILV
to
SV ,
respectively. Accordingly, the significance of
mechanical vs. neuromechanical couplings should be revisited.
6.3 Conclusions and Future Work
In this chapter, we resolved the lack of accuracy in the autonomic-cardiac regulation model proposed in Chapter 3. The mathematical model was revised in regard
to the respiratory system effects by taking the major respiratory impacts, including
lung stretch-receptor reflex and venous return variation on HR and BP, into consideration. Future work will include extending the mathematical model to increase the
model’s accuracy and improving the proposed identification technique described in
Chapter 3 to use capabilities of the proposed autonomic-cardiorespiratory model.
We will aim to eliminate the effects of respiration on PSNA k1V̇L . This will help to
extract a pure parasympathetic activation caused by different mental states and environmental stimulus. Further, the effects of spontaneous (during consciousness)
and mechanical (during anaesthesia) respiration on
HR
and
BP
regulation can be
investigated individually. Similarly, results of the identification technique will be
improved for anaesthetized and awake individuals.
100
Chapter 7
Conclusion and Future Work
7.1 Summary: Work Accomplished
In this thesis, we studied autonomic-cardiac regulation with and without respiratory coupling within a series of investigations using different techniques including
mathematical modeling, system identification, stability analysis, and control design. We summarize and conclude the major points and achievements in this chapter.
Mathematical Modeling- In Chapter 3, we adopted a model of autonomiccardiac regulation consisting of two coupled nonlinear and delayed differential
equations, each of which describes the dynamics of
HR
and
BP
regulation. We
improved the existing model to mathematically describe respiratory-based mechanisms in Chapter 6. We revised the model in regard to the mechanical and neuromechanical couplings of the respiration system and autonomic-cardiac regulation
including the lung stretch receptor reflex and the venous return variation, which had
not been investigated properly in the past. The revised model can physiologically
present a source of oscillatory patterns observed in
tem, which is commonly known as
RSA .
HR
due to the respiration sys-
The proposed mathematical model must
be improved further to describe other significant mechanisms in HR and BP regulation. For example, baroreflex control of SV and the renin-angiotensin system have
not been described mathematically in this manuscript.
101
System Identification- A parameter identification technique for estimating and
then monitoring
SNA
and
PSNA
using routine clinical measurements,
HR
and
BP ,
was introduced in Chapter 3. We presented a proof-of-concept for the proposed
identification technique using two clinical datasets: the MIMIC dataset collected
at Beth Israel Hospital, Boston, MA [110] and the orthostatic hypotension dataset
collected at New York Medical College [46]. We examined the repeatability of the
identification outcome using the MIMIC dataset and the physiological consistency
using the orthostatic hypotension dataset. Despite the promising preliminary results, the proposed method has several limitations. For example, the identification
technique may benefit from the availability of
CO
measurement that is not com-
monly measured in the clinical setup; however, some non-invasive
CO
estimation
techniques including electrical velocimetry and echo-cardiography have recently
been introduced (refer to Section 3.2.3). Further, the possible interdependence between model parameters was not deeply explored.
Stability Analysis- A systematic approach to stability analysis of autonomiccardiac regulation was proposed in Chapter 4. The proposed method was derived
according to the mathematical model of autonomic cardiac regulation with two
coupled nonlinear and delayed differential equations. We introduced a stability
index to compare numerically different parameter configurations and to monitor
the stability margin of
CVS
during any clinical condition enforced by a parameter
configuration. We can investigate the stability margin of a large number of clinical conditions using the proposed index to recognize a possible disorder in the
autonomic-cardiac regulation that could not be recognized easily without a modelbased stability analysis. For example, the stability margin of the autonomic-cardiac
regulation can be investigated in individuals with high arterial stiffness and low intrinsic
HR
in stressful conditions by using the proposed stability index. Further,
this study may be used to determine dosage or type of BP stabilizing drugs, a new
concept, that has been proposed recently [14]. An extensive clinical study demonstrating the potential significance of the proposed stability analysis should be performed.
102
Artificial Bionic Baroreflex- In Chapter 5, we developed a method for designing an artificial bionic baroreflex capable of restoring normal arterial pressure
regulation by mimicking the in-vivo baroreflex mechanism. The individual’s invivo autonomic-cardiac regulation was described by a subject-specific mathematical model. To individualize the mathematical model, we used the proposed system
identification technique in Chapter 3. A unique strength of the proposed method
is its capability to determine the modulating baroreflex functions on the sympathetic and parasympathetic nerves. The proposed method potentially can be used
to design an advanced pacemaker, a medical device that regulates the heartbeat
sequence according to the individual’s current physical and psychophysical condition. Current pacemakers provide only constant-rate stimulation for the heart. An
extensive clinical study of the proposed bionic baroreflex is needed to evaluate the
significance of this work.
7.2 Future-Work: The Road Ahead
This section suggests a number of possibilities for future work, categorized according to the chapters of this thesis.
Mathematical Modeling- The proposed mathematical model that includes the
respiration system could be applied to the frequency analysis of HRV. In the power
spectrum analysis of
HRV , LF
power, modulated by both
SNA
and
PSNA ,
and
HF
power, modulated by PSNA , have been used together as measures to monitor what
has been called the sympathovagal balance (e.g.,
model-based method to analyze the
of the
HRV
HRV
LF /HF
ratio) on the heart. A
spectrum may reveal some unseen parts
dynamic, which could be useful for investigaing other possible causes
of variation in these frequency bands
System Identification- Considering that we used a relatively fast parameter identification technique, we can potentially use the proposed sympathetic and
parasympathetic monitoring technique in the treatment of individuals with
ANS
disorders using biofeedback techniques. A large number of studies have attempted
to use biofeedback to alter HRV . Further, we can investigate extensively the consis-
103
tencey of the proposed system identification results on sympathetic and parasympathetic activation and other markers of SNS and PNS including HRV -based markers
[82, 93].
Stability Analysis- In this thesis, we showed that the presence of a negative
feedback (i.e., baroreflex mechanism) in autonomic-cardiac regulation may not always result in stability due to, for example, the system nonlinearity. It has been
shown in [140] that negative feedback can cause expanding oscillations in certain
circumstances. Further, cyclic interaction among elements of a system with two
negative feedbacks may cause an instability in the system. Note that two or any
even number of negative interactions generate a positive circuit in the system [141].
In autonomic-cardiorespiratory regulation described by ordinary differential equations, circuits can be defined in terms of the elements of the Jacobian matrix [142].
Considering that a compact DDE mathematical model for autonomic-cardiac regulation described in Chapter 6 and the corresponding Jacobian matrix was presented
in Chapter 4, a possible future study is an investigation of occurence of positive
feedback (circuit) in autonomic-cardiac regulation.
Artificial Bionic Baroreflex- The method proposed in Chapter 5 can be used
to design of an artificial bionic baroreflex that would regulate
with resistant hypertension or
SCI
BP
in individuals
[143]. With this method, the sympathetic and
parasympathetic nerves must be stimulated such that the measured (or, simulated)
BP
tracks
BP
setpoints.
HR
changes due to the sympathetic and parasympathetic
activation were not studied. A control strategy in which
BP
and
HR
are regulated
simultaneously and a control strategy that considers constraints other than
HR
setpoints can be developed in the future.
104
BP
and
Bibliography
[1] P. Ataee, J. Hahn, C. Brouse, G. Dumont, and W. Boyce, “Identification of
cardiovascular baroreflex for probing homeostatic stability,” in Proceedings
of the Computing in Cardiology, 2010, pp. 141–144. → pages iv, 55, 76,
89, 91
[2] P. Ataee, J. Hahn, G. Dumont, and W. Boyce, “Non-invasive
subject-specific monitoring of autonomic-cardiac regulation,” IEEE
Transactions on Biomedical Engineering (submitted), 2013. → pages iv
[3] P. Ataee, J. Hahn, G. Dumont, and W. Boyce, “A systematic approach to
local stability analysis of cardiovascular baroreflex,” in Proceedings of the
33rd Annual International Conference of the IEEE EMBS, 2011, pp.
700–703. → pages iv, 55
[4] P. Ataee, G. A. Dumont, H. A. Noubari, W. T. Boyce, and J. M. Ansermino,
“A novel approach to the design of an artificial bionic baroreflex,” in
Proceedings of the 35th Annual International Conference of the IEEE
EMBS, 2013, pp. 3813–3816. → pages iv
[5] P. Ataee, L. Belingard, G. Dumont, H. Noubari, and W. Boyce,
“Autonomic-cardiorespiratory regulation: a physiology-based
mathematical model,” in Proceedings of the 34th Annual International
Conference of the IEEE EMBS, 2012, pp. 3805–3808. → pages iv
[6] P. Ataee, G. Dumont, and W. Boyce, “Baroreflex modeling in the genesis of
stress reactivity using sigmoidal characteristic,” in Proceedings of the
American Control Conference, 2010, pp. 3566–3571. → pages iv, 51
[7] A. Fowler and M. McGuinness, “A delay recruitment model of the
cardiovascular control system,” Journal of Mathematical Biology, vol. 51,
pp. 508–526, 2005. → pages x, 1, 17, 22, 23, 29, 30, 32, 57, 89
105
[8] “The venous system within the cardiovascular system.” [Online]. Available:
http://www.urgomedical.com/Pathophysiologies/Compression/ → pages
xi, 13
[9] “Heart rate variability analysis scientific background.” [Online]. Available:
http://www.biocomtech.com/hrvscientific → pages xi, 14
[10] R. Klabunde, Cardiovascular Physiology Concepts.
& Wilkins, 2004. → pages xi, 7, 10, 14, 15, 16
Lippincott Williams
[11] H. Burgess, “Estimating cardiac autonomic activity during sleep:
impedance cardiography, spectral analysis, and Poincar plots,” Clinical
Neurophysiology, vol. 115, no. 1, pp. 19–28, 2004. → pages xi, 20
[12] G. Berntson, G. Norman, L. Hawkley, and J. Cacioppo, “Spirituality and
Autonomic Cardiac Control,” Annals of Behavioral Medicine, vol. 35, pp.
198–208, 2008. → pages 1, 42
[13] “Raised blood pressure.” [Online]. Available:
http://www.who.int/gho/ncd/risk_factors/blood_pressure_prevalence_text/en/
→ pages 1
[14] P. Rothwel, “Limitations of the usual blood-pressure hypothesis and
importance of variability, instability, and episodic hypertension,” The
Lancet, vol. 375, no. 9718, pp. 938 – 948, 2010. → pages 1, 6, 7, 102
[15] H. Kitano, “Computational systems biology,” Nature, vol. 420, no. 6912,
pp. 206–10, 2002. → pages 2
[16] J. Ottesen, “Modelling of the baroreflex-feedback mechanism with
time-delay,” Journal of Mathematical Biology, vol. 36, pp. 41–63, 1997. →
pages 2, 4, 15, 16, 17, 23, 24, 26, 29, 30, 31, 54, 55, 57
[17] E. Magosso, V. Biavati, and M. Ursino, “Role of the baroreflex in
cardiovascular instability: a modeling study,” Cardiovascular Engineering,
vol. 1, pp. 101–115, 2001. → pages 15, 26, 88
[18] S. Cavalcanti and E. Belardinelli, “Modeling of cardiovascular variability
using a differential delay equation,” IEEE Transactions on Biomedical
Engineering, vol. 43, pp. 982–9, 1996. → pages 2, 17, 31, 54, 55, 66, 70
[19] D. Beard, J. Bassingthwaighte, and A. Greene, “Computational modeling
of physiological systems,” Physiological Genomics, vol. 23, no. 1, pp. 1–3;
discussion 4, 2005. → pages 3
106
[20] M. Gevers, “A personal view of the development of system identification:
A 30-year journey through an exciting field,” IEEE Control Systems
Magazine, vol. 26, no. 6, pp. 93–105, 2006. → pages 3
[21] Z. Zi, “Sensitivity analysis approaches applied to systems biology models,”
IET Systems Biology, vol. 5, no. 6, pp. 336–6, 2011. → pages 3
[22] L. Glass and M. C. Mackey, “Pathological conditions resulting from
instabilities in physiological control systems,” Annals of the New York
Academy of Sciences, vol. 316, no. 1, pp. 214–235, 1979. → pages 4
[23] S. Cavalcanti, “Arterial baroreflex influence on heart rate variability: a
mathematical model-based analysis,” Medical & Biological Engineering &
Computing, vol. 38, pp. 189–197, 2000. → pages 4, 15, 22, 24, 26, 54, 55,
57, 88
[24] D. Goldstein and B. McEwen, “Allostasis, homeostats, and the nature of
stress.” Stress, vol. 5, no. 1, pp. 55–8, 2002. → pages 4, 10
[25] J. Coote, “Homeostasis and stress,” Clinical Autonomic Research, vol. 15,
pp. 247–248, 2005. → pages 4
[26] I. J. M. Scheffers, A. A. Kroon, J. Schmidli, J. Jordan, T. J. J. M., M. J.
Mohaupt, F. C. Luft, H. Haller, J. Menne, S. Engeli, J. Ceral, S. Eckert,
A. Erglis, K. Narkiewicz, T. Philipp, and P. W. Leeuw, “Novel Baroreflex
Activation Therapy in Resistant Hypertension: Results of a European
Multi-Center Feasibility Study,” Journal of the American College of
Cardiology, vol. 56, no. 15, pp. 1254–1258, 2010. → pages 4
[27] S. Pearson, A. Alkon, M. Treadwell, B. Wolff, K. Quirolo, and W. Boyce,
“Autonomic reactivity and clinical severity in children with sickle cell
disease,” Clinical Autonomic Research, vol. 15, no. 6, pp. 400–7, 2005. →
pages 5, 19
[28] A. Alkon, W. Boyce, N. Davis, and B. Eskenazi, “Developmental changes
in autonomic nervous system resting and reactivity measures in Latino
children from 6 to 60 months of age,” Journal of Developmental and
Behavioral Pediatrics, vol. 32, no. 9, pp. 668–77, 2011. → pages 5
[29] A. Laird, A. Finch, P. Waite, and P. Carrive, “Peripheral changes above and
below injury level lead to prolonged vascular responses following high
spinal cord injury,” American Journal of Physiology - Heart and
Circulatory Physiology, vol. 294, no. 2, pp. H785–H792, 2008. → pages 5,
27
107
[30] R. Teasell, J. Arnold, A. Krassioukov, and G. Delaney, “Cardiovascular
consequences of loss of supraspinal control of the sympathetic nervous
system after spinal cord injury,” Archives of Physical Medicine and
Rehabilitation, vol. 81, no. 4, pp. 506–16, 2000. → pages 5, 27
[31] P. Mills and A. Krassioukov, “Autonomic function as a missing piece of the
classification of Paralympic athletes with spinal cord injury,” Spinal Cord,
vol. 49, no. 7, pp. 768–76, 2011. → pages 5, 27
[32] J. Quas, A. Bauer, and W. Boyce, “Physiological reactivity, social support,
and memory in early childhood,” Child Development, vol. 75, no. 3, pp.
797–814, 2004. → pages 5, 19, 20, 27, 51
[33] A. Garg, N. Adhikari, M. H., M. Rosas-Arellano, P. Devereaux, J. Beyene,
J. Sam, and R. Haynes, “Effects of computerized clinical decision support
systems on practitioner performance and patient outcomes: A systematic
review,” JAMA, vol. 293, no. 10, pp. 1223–1238, 2005. → pages 5
[34] K. Kawamoto, C. Houlihan, E. Balas, and D. Lobach, “Improving clinical
practice using clinical decision support systems: a systematic review of
trials to identify features critical to success,” British Medical Journal, vol.
330, no. 7494, p. 765, Apr. 2005. → pages 6
[35] J. Bailey, “Hemodynamic instability,” 2003. [Online]. Available:
clinicalview.gehealthcare.com/download.php?obj_id=195&browser →
pages 6
[36] T. Sato, T. Kawada, T. Shishido, M. Sugimachi, J. Alexander, and
K. Sunagawa, “Novel Therapeutic Strategy Against Central Baroreflex
Failure : A Bionic Baroreflex System,” Circulation, vol. 100, no. 3, pp.
299–304, 1999. → pages 7, 75
[37] M. La Rovere, G. Pinna, and G. Raczak, “Baroreflex sensitivity:
measurement and clinical implications,” Annals of Noninvasive
Electrocardiology, vol. 13, no. 2, pp. 191–207, 2008. → pages 7, 57
[38] F. Yamasaki, T. Ushida, T. Yokoyama, M. Ando, K. Yamashita, and T. Sato,
“Artificial baroreflex: clinical application of a bionic baroreflex system,”
Circulation, vol. 113, no. 5, pp. 634–9, Feb. 2006. → pages 8, 86
[39] K. Sunagawa and M. Sugimachi, “Development of artificial bionic
baroreflex system,” in Proceedings of the 32nd Annual International
Conference of the IEEE EMBS, vol. 2010, 2010, pp. 3446–8. → pages 8,
75, 76
108
[40] Y. Yanagiya, T. Sato, T. Kawada, M. Inagaki, T. Tatewaki, C. Zheng,
A. Kamiya, H. Takaki, M. Sugimachi, and K. Sunagawa, “Bionic epidural
stimulation restores arterial pressure regulation during orthostasis,” Journal
of Applied Physiology, vol. 97, no. 3, pp. 984–90, Sep. 2004. → pages 8
[41] A. Dudkowska and D. Makowiec, “Seidel-Herzel model of human
baroreflex in cardiorespiratory system with stochastic delays,” Journal of
Mathematical Biology, vol. 57, pp. 111–137, 2008. → pages 10, 15, 16, 17,
24, 26, 54, 75, 88
[42] R. Dampney, J. Horiuchi, and L. McDowall, “Hypothalamic mechanisms
coordinating cardiorespiratory function during exercise and defensive
behaviour,” Autonomic Neuroscience: Basic & Clinical, vol. 142, no. 1-2,
pp. 3–10, 2008. → pages
[43] J. Elghozi and C. Julien, “Sympathetic control of short-term heart rate
variability and its pharmacological modulation,” Fundamental & Clinical
Pharmacology, vol. 21, no. 4, pp. 337–47, 2007. → pages 10, 19, 26, 54,
75, 88
[44] S. Porges, “Love: an emergent property of the mammalian autonomic
nervous system,” Psychoneuroendocrinology, vol. 23, no. 8, pp. 837–61,
1998. → pages 10
[45] W. Boyce, J. Quas, A. Alkon, N. Smider, M. Essex, and D. Kupfer,
“Autonomic reactivity and psychopathology in middle childhood,” The
British Journal of Psychiatry, vol. 179, no. 2, pp. 144–50, 2001. → pages
10, 19, 27, 54
[46] A. Ocon, M. Medow, I. Taneja, and J. Stewart, “Respiration drives phase
synchronization between blood pressure and RR interval following loss of
cardiovagal baroreflex during vasovagal syncope,” Heart and Circulatory
Physiology: American Journal of Physiology, vol. 300, no. 2, pp. H527–40,
2011. → pages 12, 29, 102
[47] G. Moody and R. Mark, “A database to support development and
evaluation of intelligent intensive care monitoring,” in Proceedings of the
Computers in Cardiology, 1996, pp. 657–660. → pages 12, 29, 77
[48] J. West, Respiratory physiology: the essentials. Philadelphia: Wolters
Kluwer Health/Lippincott Williams & Wilkins, 2008. → pages 12, 18, 91,
95
109
[49] M. Henson, B. Ogunnaike, J. Schwaber, and F. Doyle, “The Baroreceptor
Reflex: A Biological Control System with Applications in Chemical
Process Control,” Industrial & Engineering Chemistry Research, vol. 33,
pp. 2453–2466, 1994. → pages 13
[50] S. Porges, “Cardiac vagal tone: a physiological index of stress,”
Neuroscience and biobehavioral reviews, vol. 19, pp. 225–33, 1995. →
pages 13, 20, 64, 66
[51] B. Appelhans and L. Luecken, “Heart rate variability as an index of
regulated emotional responding,” Review of General Psychology, vol. 10,
pp. 229–240, 2006. → pages 13, 14, 15, 26, 88
[52] S. Porges, “The Polyvagal Theory: phylogenetic contributions to social
behavior,” Physiology & Behavior, vol. 79, no. 3, pp. 503–13, 2003. →
pages 13
[53] T. Thrasher, “Baroreceptors and the long-term control of blood pressure,”
Experimental Physiology, vol. 89, no. 4, p. 331, 2004. → pages 15
[54] R. Dampney, M. Coleman, M. Fontes, Y. Hirooka, J. Horiuchi, Y. Li,
J. Polson, P. Potts, and T. Tagawa, “Central mechanisms underlying
short-and long-term regulation of the cardiovascular system,” Clinical and
Experimental Pharmacology and Physiology, vol. 29, no. 4, pp. 261–268,
2002. → pages 15
[55] H. Van de Vooren, M. Gademan, C. Swenne, B. TenVoorde, M. Schalij, and
E. Van der Wall, “Baroreflex sensitivity, blood pressure buffering, and
resonance: what are the links? Computer simulation of healthy subjects
and heart failure patients,” Journal of Applied Physiology, vol. 102, pp.
1348–1356, 2006. → pages 15, 22, 27, 57, 88
[56] D. Mohrman, Cardiovascular Physiology.
2006. → pages 15, 17, 26
New York: McGraw-Hill,
[57] P. Hammer and J. Saul, “Resonance in a mathematical model of baroreflex
control: arterial blood pressure waves accompanying postural stress,”
Regulatory, Integrative and Comparative Physiology: American Journal of
Physiology, vol. 288, no. 6, pp. R1637–48, 2005. → pages
[58] J. Filippone and J. Bisognano, “Baroreflex stimulation in the treatment of
hypertension,” Current Opinion in Nephrology and Hypertension, vol. 16,
no. 5, pp. 403–8, 2007. → pages 15, 76
110
[59] A. Mitrophanov and E. Groisman, “Positive feedback in cellular control
systems,” BioEssays : news and reviews in molecular, cellular and
developmental biology, vol. 30, no. 6, pp. 542–55, 2008. → pages 16
[60] C. Willie, P. Ainslie, C. Taylor, H. Jones, P. Sin, and Y. Tzeng,
“Neuromechanical features of the cardiac baroreflex after exercise,”
Hypertension, vol. 57, no. 5, pp. 927–33, 2011. → pages 16
[61] J. Wagner and G. Stolovitzky, “Stability and Time-Delay Modeling of
Negative Feedback Loops,” in Proceedings of the IEEE, vol. 96, 2008, pp.
1398–1410. → pages 16
[62] T. Ketch, “Four Faces of Baroreflex Failure: Hypertensive Crisis, Volatile
Hypertension, Orthostatic Tachycardia, and Malignant Vagotonia,”
Circulation, vol. 105, no. 21, pp. 2518–2523, 2002. → pages 17
[63] H. Huikuri, J. Perkiömäki, R. Maestri, and G. Pinna, “Clinical impact of
evaluation of cardiovascular control by novel methods of heart rate
dynamics,” Philosophical Transactions of the Royal Society A:
Mathematical, Physical and Engineering Sciences, vol. 367, no. 1892, p.
1223, 2009. → pages 17
[64] P. McSharry, M. McGuinness, and A. Fowler, “Confronting a
cardiovascular system model with heart rate and blood pressure data,” in
Proceedings of the Computers in Cardiology, vol. 32. IEEE, 2005, pp.
587–590. → pages 17, 88
[65] J. Ottesen and M. Olufsen, “Functionality of the baroreceptor nerves in
heart rate regulation,” Computer Methods and Programs in Biomedicine,
vol. 101, no. 2, pp. 208–19, Feb. 2011. [Online]. Available:
http://www.ncbi.nlm.nih.gov/pubmed/21167621 → pages 17
[66] M. Ursino and E. Magosso, “Role of short-term cardiovascular regulation
in heart period variability: a modeling study,” Am J Physiol Heart Circ
Physiol, vol. 284, no. 4, pp. H1479–1493, 2003. → pages 17, 23, 55
[67] G. Berntson and J. Cacioppo, Integrative physiology: Homeostasis,
allostasis and the orchestration of systemic physiology. Cambridge Univ
Pr, 2007, pp. 433–452. → pages 17, 27, 32, 91
[68] J. Halliwill, B. Morgan, and N. Charkoudian, “Peripheral chemoreflex and
baroreflex interactions in cardiovascular regulation in humans,” The
Journal of Physiology, vol. 552, no. Pt 1, pp. 295–302, 2003. → pages 17,
18
111
[69] J. Duffin, “The role of the central chemoreceptors: A modeling
perspective,” Respiratory Physiology and Neurobiology, vol. 173, no. 3, pp.
230 – 243, 2010. → pages 17, 18
[70] T. Kara, K. Narkiewicz, and V. Somers, “Chemoreflexes-physiology and
clinical implications,” Acta physiologica Scandinavica, vol. 177, no. 3, pp.
377–84, 2003. → pages 17, 18
[71] G. Berntson, K. Quigley, and D. Lozano, Cardiovascular
psychophysiology. Cambridge Univ Pr, 2007, vol. 32, no. 5, pp. 182–210.
→ pages 18
[72] M. Levitzky, Ed., Respiratory Physiology.
→ pages 18
McGraw-Hill Medical, 2007.
[73] U. Lucangelo, P. Pelosi, W. Zin, and A. Aliverti, Eds., Respiratory System
and Artificial Ventilation. Milano: Springer Milan, 2008. → pages 18
[74] A. Alkon, S. Lippert, N. Vujan, M. Rodriquez, W. Boyce, and B. Eskenazi,
“The ontogeny of autonomic measures in 6- and 12-month-old infants,”
Developmental Psychobiology, vol. 48, pp. 197Ű–208, 2006. → pages 19,
20
[75] A. Sgoifo, F. Braglia, T. Costoli, E. Musso, P. Meerlo, G. Ceresini, and
A. Troisi, “Cardiac autonomic reactivity and salivary cortisol in men and
women exposed to social stressors: relationship with individual ethological
profile,” Neuroscience and Biobehavioral Reviews, vol. 27, no. 1-2, pp.
179–88, 2003. → pages
[76] T. Beauchaine, “Vagal tone, development, and Gray’s motivational theory:
Toward an integrated model of autonomic nervous system functioning in
psychopathology,” Development and Psychopathology, vol. 13, pp.
183–214, 2001. → pages 19, 20
[77] S. Porges, “Vagal tone: a physiologic marker of stress vulnerability,”
Pediatrics, vol. 90, p. 498, 1992. → pages 19
[78] Z. Chen, P. Purdon, E. Pierce, G. Harrell, E. Brown, and R. Barbieri,
“Assessment of Baroreflex Control of Heart Rate During General
Anesthesia Using a Point Process Method,” in Proceedings of the IEEE
International Conference on Acoustics, Speech and Signal Processing.
Ieee, 2009, pp. 333–336. → pages 19
112
[79] S. Porges, “The polyvagal perspective,” Biological Psychology, vol. 74,
no. 2, pp. 116–43, 2007. → pages 19
[80] U. Rajendra Acharya, K. Paul Joseph, N. Kannathal, C. Lim, and J. Suri,
“Heart rate variability: a review,” Medical and Biological Engineering and
Computing, vol. 44, no. 12, pp. 1031–1051, 2006. → pages 19
[81] J. Niskanen, M. Tarvainen, P. Ranta-Aho, and P. Karjalainen, “Software for
advanced HRV analysis,” Computer Methods and Programs in
Biomedicine, vol. 76, no. 1, pp. 73–81, 2004. → pages 19
[82] P. Stein and Y. Pu, “Heart rate variability, sleep and sleep disorders,” Sleep
Medicine Reviews, vol. 16, no. 1, pp. 47–66, 2011. → pages 19, 104
[83] A. Malliani, “Heart rate variability: from bench to bedside,” European
journal of internal medicine, vol. 16, no. 1, pp. 12–20, Feb. 2005. → pages
19
[84] F. Yasuma and J. Hayano, “Respiratory sinus arrhythmia: why does the
heartbeat synchronize with respiratory rhythm?” Chest, vol. 125, pp.
683–690, 2004. → pages 20, 90, 91, 95
[85] G. Berntson, J. Cacioppo, and K. Quigley, “Respiratory sinus arrhythmia:
autonomic origins, physiological mechanisms, and psychophysiological
implications,” Psychophysiology, vol. 30, no. 2, pp. 183–96, 1993. →
pages 20, 90, 91
[86] P. Grossman and E. Taylor, “Toward understanding respiratory sinus
arrhythmia: relations to cardiac vagal tone, evolution and biobehavioral
functions,” Biological Psychology, vol. 74, no. 2, pp. 263–85, 2007. →
pages 20, 27
[87] M. Mantaras, S. Carrasco, M. Gaitan, and R. Gonzalez, “Comparison of
indexes of respiratory sinus arrhythmia,” in Proceedings of the 25th Annual
International Conference of the IEEE EMBS, Cancun, Mexico, 2003, pp.
232–235. → pages 20, 27
[88] N. Bush, A. Alkon, J. Obradović, J. Stamperdahl, and W. Boyce,
“Differentiating challenge reactivity from psychomotor activity in studies
of children’s psychophysiology: considerations for theory and
measurement,” Journal of Experimental Child Psychology, vol. 110, no. 1,
pp. 62–79, 2011. → pages 20, 22
113
[89] T. Dinh, H. Perrault, P. Calabrese, A. Eberhard, and G. Benchetrit, “New
statistical method for detection and quantification of respiratory sinus
arrhythmia,” IEEE Transactions on Biomedical Engineering, vol. 46, no. 9,
pp. 1161–5, 1999. → pages 20
[90] G. Berntson, G. Norman, L. Hawkley, and J. Cacioppo, “Cardiac
autonomic balance versus cardiac regulatory capacity,” Psychophysiology,
vol. 45, pp. 643–52, 2008. → pages 20, 26, 42, 70
[91] J. Houtveen, P. Groot, and E. Geus, “Effects of variation in posture and
respiration on RSA and pre-ejection period,” Psychophysiology, vol. 42,
no. 6, pp. 713–9, 2005. → pages 20
[92] N. Selvaraj, K. Shelley, D. Silverman, N. Stachenfeld, and K. Chon,
“Autonomic control mechanism of maximal lower body negative pressure
application,” in Proceedings of the 34th Annual International Conference
of the IEEE EMBS, 2012, pp. 3120–3. → pages 21
[93] K. Lee, D. Buchanan, A. Flatau, and W. Franke, “Reproducibility of the
heart rate variability responses to graded lower body negative pressure,”
European Journal of Applied Physiology, vol. 92, no. 1, pp. 106–113,
2004. → pages 21, 104
[94] W. Cooke, K. Ryan, and V. Convertino, “Lower body negative pressure as a
model to study progression to acute hemorrhagic shock in humans,”
Journal of Applied Physiology, vol. 96, no. 4, pp. 1249–61, 2004. → pages
21
[95] R. Freeman, W. Wieling, F. Axelrod, D. Benditt, E. Benarroch,
I. Biaggioni, W. Cheshire, T. Chelimsky, P. Cortelli, C. Gibbons,
D. Goldstein, R. Hainsworth, M. Hilz, G. Jacob, H. Kaufmann, J. Jordan,
L. Lipsitz, B. Levine, P. Low, C. Mathias, S. Raj, D. Robertson,
P. Sandroni, I. Schatz, R. Schondorff, J. Stewart, and J. van Dijk,
“Consensus statement on the definition of orthostatic hypotension, neurally
mediated syncope and the postural tachycardia syndrome,” Clinical
Autonomic Research, vol. 21, no. 2, pp. 69–72, 2011. → pages 21, 42
[96] J. Ringwood and S. Malpas, “Slow oscillations in blood pressure via a
nonlinear feedback model,” American Journal of Physiology - Regulatory,
Integrative and Comparative Physiology, vol. 280, pp. R1105–15, 2001. →
pages 22, 23, 57, 58, 78, 88
114
[97] J. Saul, R. Berger, P. Albrecht, S. Stein, M. Chen, and R. Cohen, “Transfer
function analysis of the circulation: unique insights into cardiovascular
regulation,” American Journal of Physiology, vol. 261, pp. H1231–45,
1991. → pages 22
[98] H. Seidel and H. Herzel, “Bifurcations in a nonlinear model of the
baroreceptor-cardiac reflex,” Physica D: Nonlinear Phenomena, vol. 115,
pp. 145–160, 1998. → pages 24
[99] M. Olufsen, H. Tran, J. Ottesen, and Others, “Modeling baroreflex
regulation of heart rate during orthostatic stress,” Regulatory, Integrative
and Comparative Physiology: American Journal of Physiology, vol. 291,
no. 5, pp. R1355–68, 2006. → pages 24, 27, 51, 55
[100] F. Lombardi, G. Sandrone, S. Pernpruner, R. Sala, M. Garimoldi, S. Cerutti,
G. Baselli, M. Pagani, and A. Malliani, “Heart rate variability as an index
of sympathovagal interaction after acute myocardial infarction,” American
Journal of Cardiology, vol. 60, no. 16, pp. 1239 – 1245, 1987. → pages 26
[101] G. Berntson, J. Cacioppo, P. Binkley, B. Uchino, K. Quigley, and
A. Fieldstone, “Autonomic cardiac control. III. Psychological stress and
cardiac response in autonomic space as revealed by pharmacological
blockades,” Psychophysiology, vol. 31, no. 6, pp. 599–608, 1994. → pages
27
[102] J. Cacioppo, G. Berntson, P. Binkley, K. Quigley, B. Uchino, and
A. Fieldstone, “Autonomic cardiac control. ii. noninvasive indices and
basal response as revealed by autonomic blockades,” Psychophysiology,
vol. 31, no. 6, pp. 586–598, 1994. → pages 27
[103] M. Olufsen and J. Ottesen, “A practical approach to parameter estimation
applied to model predicting heart rate regulation,” Journal of Mathematical
Biology, 2012. → pages 27, 32
[104] M. Olufsen, A. Alston, H. Tran, J. Ottesen, and V. Novak, “Modeling heart
rate regulation-part I: sit-to-stand versus head-up tilt,” Cardiovascular
Engineering, vol. 8, no. 2, pp. 73–87, 2008. → pages
[105] J. Ottesen, “On the Track of Syncope Induced by Orthostatic Stress Feedback Mechanisms Regulating the Cardiovascular System,” in
Proceedings of the 7th IFAC Symposium on Modelling and Control in
Biomedical Systems, 2009, pp. 187–191. → pages
115
[106] S. Cavalcanti, S. Cavani, A. Ciandrini, and G. Avanzolini, “Mathematical
modeling of arterial pressure response to hemodialysis-induced
hypovolemia,” Computers in Biology and Medicine, vol. 36, pp. 128–44,
2006. → pages
[107] C. Taylor and C. Figueroa, “Patient-specific modeling of cardiovascular
mechanics,” Annual Review of Biomedical Engineering, vol. 11, pp.
109–34, 2009. → pages
[108] M. Aoi, B. Matzuka, and M. Olufsen, “Toward online, noninvasive,
nonlinear assessment of cerebral autoregulation,” in Proceedings of the
33rd Annual International Conference of the IEEE EMBS, Boston, MA,
2011, pp. 2410–2413. → pages 27
[109] A. Goldberger, L. Amaral, L. Glass, J. Hausdorff, P. Ivanov, R. Mark,
J. Mietus, G. Moody, C. Peng, and H. Stanley, “Physiobank, physiotoolkit,
and physionet: components of a new research resource for complex
physiologic signals,” Circulation, vol. 101, no. 23, pp. e215–e220, 2000.
→ pages 29, 77
[110] G. Moody, R. Mark, and A. Goldberger, “PhysioNet: a Web-based resource
for the study of physiologic signals,” IEEE Engineering in Medicine and
Biology Magazine, vol. 20, no. 3, pp. 70–5, 2001. → pages 29, 102
[111] F. Melchior, R. Srinivasan, and J. Charles, “Mathematical modeling of
human cardiovascular system for simulation of orthostatic response,”
American Journal of Physiology, vol. 262, no. 6, pp. H1920–33, 1992. →
pages 31
[112] D. Eckberg, “Arterial baroreflexes and cardiovascular modeling,”
Cardiovascular Engineering, vol. 8, no. 1, pp. 5–13, 2008. → pages
[113] L. McDowall and R. Dampney, “Calculation of threshold and saturation
points of sigmoidal baroreflex function curves,” Heart and Circulatory
Physiology: American Journal of Physiology, vol. 291, no. 4, pp. H2003–7,
2006. → pages 31, 58, 78
[114] C. Cobelli and J. DiStefano, “Parameter and structural identifiability
concepts and ambiguities: a critical review and analysis,” American
Journal of Physiology, vol. 239, no. 1, pp. R7–24, 1980. → pages 32, 33
[115] A. Cintrón-Arias, H. T. Banks, A. Capaldi, and A. L. Lloyd, “A sensitivity
matrix based methodology for inverse problem formulation,” Journal of
Inverse and Ill-posed Problems, vol. 17, no. 6, pp. 1–20, 2009. → pages 32
116
[116] O. Chis, J. Banga, and E. Balsa-Canto, “Structural identifiability of systems
biology models: a critical comparison of methods,” PloS One, vol. 6,
no. 11, p. e27755, 2011. → pages 33
[117] M. Eslami, Theory of sensitivity in dynamic systems: an introduction.
Berlin: Springer Berlin Heidelberg, 1994. → pages 33
[118] MATLAB Optimization Toolbox User’s Guide, MathWorks, 2010. → pages
35, 61, 65, 80
[119] W. Ganz, R. Donoso, H. Marcus, J. Forrester, and H. Swan, “A new
technique for measurement of cardiac output by thermodilution in man,”
American Journal of Cardiology, vol. 27, no. 4, pp. 392–6, 1971. → pages
48
[120] N. Fazeli and J. Hahn, “Estimation of cardiac output and peripheral
resistance using square-wave-approximated aortic flow signal,” Frontiers in
Physiology, vol. 3, no. July, pp. 1–10, 2012. → pages 48
[121] H. Ihlen, J. Amlie, J. Dale, K. Forfang, S. Nitter-Hauge, J. Otterstad,
S. Simonsen, and E. Myhre, “Determination of cardiac output by Doppler
echocardiography,” British Heart Journal, vol. 51, no. 1, pp. 54–60, 1984.
→ pages 48
[122] N. Zoremba, J. Bickenbach, B. Krauss, R. Rossaint, R. Kuhlen, and
G. Schälte, “Comparison of electrical velocimetry and thermodilution
techniques for the measurement of cardiac output,” Acta Anaesthesiologica
Scandinavica, vol. 51, no. 10, pp. 1314–9, 2007. → pages 48
[123] R. Mukkamala, A. Reisner, H. Hojman, R. Mark, and R. Cohen,
“Continuous cardiac output monitoring by peripheral blood pressure
waveform analysis,” IEEE Transactions on Biomedical Engineering,
vol. 53, no. 3, pp. 459–67, 2006. → pages 48
[124] T. Parlikar, T. Heldt, G. Ranade, and G. Verghese, “Model-based estimation
of cardiac output and total peripheral resistance,” in Proceedings of the
Computers in Cardiology, 2007, pp. 379–382. → pages 48
[125] J. Chase, C. Starfinger, C. Hann, J. Revie, D. Stevenson, G. Shaw, and
T. Desaive, “Model-based prediction of the patient-specific response to
adrenaline,” The Open Medical Informatics Journal, vol. 4, pp. 149–63,
2010. → pages 50
117
[126] M. Olufsen, J. Ottesen, H. Tran, L. Ellwein, L. Lipsitz, and V. Novak,
“Blood pressure and blood flow variation during postural change from
sitting to standing: model development and validation,” Journal of Applied
Physiology, vol. 99, no. 4, pp. 1523–37, 2005. → pages 51
[127] R. Abbiw-Jackson and W. Langford, “Gain-induced oscillations in blood
pressure,” Journal of Mathematical Biology, vol. 37, no. 3, pp. 203–34,
1998. → pages 54
[128] R. DeBoer, J. Karemaker, and J. Strackee, “Hemodynamic fluctuations and
baroreflex sensitivity in humans: a beat-to-beat model,” American Journal
of Physiology, vol. 253, pp. H680–9, 1987. → pages 55
[129] V. Cooper, S. Pearson, C. Bowker, M. Elliott, and R. Hainsworth,
“Interaction of chemoreceptor and baroreceptor reflexes by hypoxia and
hypercapnia - a mechanism for promoting hypertension in obstructive sleep
apnoea,” The Journal of Physiology, vol. 568, no. 2, pp. 677–87, 2005. →
pages 58, 78
[130] L. Shampine and S. Thompson, “Solving ddes in matlab,” Applied
Numerical Mathematics, vol. 37, no. 4, pp. 441 – 458, 2001. → pages 61
[131] L. Perko, Differential Equations and Dynamical Systems.
→ pages 62, 63
Springer, 2001.
[132] J. Kretschmer, T. Haunsberger, E. Drost, E. Koch, and K. Möller,
“Simulating physiological interactions in a hybrid system of mathematical
models,” Journal of Clinical Monitoring and Computing, pp. 1–11, 2013.
→ pages 72
[133] P. Antsaklis, “A brief introduction to the theory and applications of hybrid
systems,” Proceedings of the IEEE, vol. 88, no. 7, pp. 879–887, 2000. →
pages 73
[134] K. Hosokawa, T. Ide, T. Tobushi, K. Sakamoto, K. Onitsuka, T. Sakamoto,
T. Fujino, K. Saku, and K. Sunagawa, “Bionic baroreceptor corrects
postural hypotension in rats with impaired baroreceptor,” Circulation, vol.
126, no. 10, pp. 1278–85, 2012. → pages 75
[135] T. Kawada and M. Sugimachi, “Artificial neural interfaces for bionic
cardiovascular treatments,” Journal of Artificial Organs, vol. 12, no. 1, pp.
17–22, 2009. → pages 75, 76
118
[136] D. Calhoun, D. Jones, S. Textor, D. Goff, T. Murphy, R. Toto, A. White,
W. Cushman, W. White, D. Sica, K. Ferdinand, T. Giles, B. Falkner, and
R. Carey, “Resistant hypertension: diagnosis, evaluation, and treatment. A
scientific statement from the American Heart Association Professional
Education Committee of the Council for High Blood Pressure Research,”
Hypertension, vol. 51, no. 6, pp. 1403–19, 2008. → pages 76
[137] S. Navaneethan, T. Lohmeier, and J. Bisognano, “Baroreflex stimulation: A
novel treatment option for resistant hypertension,” Journal of the American
Society of Hypertension, vol. 3, no. 1, pp. 69–74, 2009. → pages 76
[138] J. Hayano, F. Yasuma, A. Okada, S. Mukai, and T. Fujinami, “Respiratory
sinus arrhythmia: a phenomenon improving pulmonary gas exchange and
circulatory efficiency,” Circulation, vol. 94, no. 4, pp. 842–847, 1996. →
pages 91
[139] M. Khoo, “Modeling of autonomic control in sleep-disordered breathing,”
Cardiovascular Engineering, vol. 8, no. 1, pp. 30–41, 2008. → pages 91
[140] O. Cinquin and J. Demongeot, “Roles of positive and negative feedback in
biological systems,” Comptes Rendus Biologies, vol. 325, pp. 1085–95,
2002. → pages 104
[141] R. Thomas and M. Kaufman, “Multistationarity, the basis of cell
differentiation and memory. I. Structural conditions of multistationarity and
other nontrivial behavior,” Chaos, vol. 11, no. 1, pp. 170–179, 2001. →
pages 104
[142] R. Thomas and M. Kaufman, “Multistationarity, the basis of cell
differentiation and memory. II. Logical analysis of regulatory networks in
terms of feedback circuits,” Chaos (Woodbury, N.Y.), vol. 11, pp. 180–195,
2001. → pages 104
[143] A. Phillips, A. Krassioukov, P. Ainslie, and D. Warburton, “Baroreflex
Function after Spinal Cord Injury,” Journal of Neurotrauma, vol. 15, pp.
1–15, 2012. → pages 104
119