Homeostasis Revisited in the Genesis of Stress ReactivitybyPedram AtaeeB.Sc., Electrical Engineering, University of Tehran, Iran, 2005M.Sc., Electrical Engineering, University of Tehran, Iran, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Electrical and Computer Engineering)The University Of British Columbia(Vancouver)March 2014c? Pedram Ataee, 2014AbstractAutonomic-cardiac regulation operates through interactions between the autonomicnervous system (ANS) and the cardiovascular system (CVS). In order to maintainhomeostasis in the CVS, the ANS adjusts it effectors, such as the stiffness of bloodvessels and the pace of heartbeats, against physical and psychological stressors, sothat it can maintain adequate blood flow. This allows oxygen and nutrients to bedelivered to organs and enables the performance of other essential functions.Autonomic-cardiac regulation can be described by a mathematical model andit can be analyzed under different scenarios such as a stressful condition or an in-creased arterial stiffness. This may help researchers to obtain new understandingsof the autonomic-cardiac regulation. This thesis is built upon a physiology-basedmathematical model of autonomic-cardiac regulation describing the regulation ofheart rate (HR) and blood pressure (BP), using a set of nonlinear, coupled differ-ential equations with delay.Non-invasive and subject-specific monitoring of autonomic-cardiac regulationhas the potential to improve current treatments of autonomic-cardiac disorders.A parameter estimation method has been used to specify time-varying subject-specific model parameters associated with autonomic-cardiac regulation. The pro-posed method will help to improve monitoring of autonomic-cardiac variables,such as sympathetic and parasympathetic nerve activities affecting the heart andsympathetic nerve activity affecting the arterial tree.The complex dynamic interactions between nonlinearities and delays in theautonomic-cardiac regulation may result in the onset of instabilities in BP and HRregulation. In this thesis, we propose a model-based approach to stability analysisand introduce a quantitative stability indicator of the autonomic-cardiac regula-iition. We can prevent irregularities in cardiovascular rhythms (e.g., HR and BP) byknowing their causes and developing an intelligent method to control them.An artificial bionic baroreflex can be an effective treatment for baroreflex fail-ure in, for example, individuals with severe orthostatic hypotension. We propose amethod to design an artificial bionic baroreflex by mimicking the baroreflex mech-anism in the body. This could then be potentially used to adjust existing neurostim-ulator devices that regulate BP.iiiPrefaceThe work presented in this thesis has been partially published in different journalsor conference proceedings. The list of these publications is provided below. I havebeen the main author for all publications and have had the main role in generatingthe ideas, developing the methodologies, processing the data, and analyzing theresults.The work presented in Chapter 3 has been partially published in the Proceed-ings of Computer in Cardiology Conference in 2010 [1], and has been accepted forpublication in the IEEE Transactions on Biomedical Engineering [2].Parts of the work presented in Chapter 4 have been published in the Proceed-ings 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 AnnualInternational Conference of the IEEE EMBS in 2013 [4].Chapter 6 is based on the work published in the Proceedings of the 34th AnnualInternational Conference of the IEEE EMBS in 2012 [5].The conclusions provided in Chapter 7 are based on the papers published inIEEE Transactions on Biomedical Engineering [2], Proceedings of the Annual In-ternational Conference of the IEEE EMBS [3?5], Computer in Cardiology Confer-ence [1], and American Control Conference [6].ivThe 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 Subject-Specific Monitoring of Autonomic-Cardiac Regulation. IEEE Transactionson 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. Comput-ing in Cardiology, (37):141-144, 2010.? P. Ataee, J.O. Hahn, G.A. Dumont, and W.T. Boyce. A Systemic Approachto Local Stability Analysis of Cardiovascular Baroreflex. 33rd Annual Inter-national Conference of the IEEE EMBS, pages 700-703, 2011.? P. Ataee, L. Belingard, G.A. Dumont, H.A. Noubari, and W.T. Boyce. Au-tonomic-Cardiorespiratory Regulation: A Physiology-Based MathematicalModel. 34th Annual International Conference of the IEEE EMBS, pages3805-3808, 2012.? P. Ataee, G.A. Dumont, H.A. Noubari, W.T. Boyce, J.M. Ansermino. ANovel Approach to the Design of an Artificial Bionic Baroreflex. 35th An-nual International Conference of the IEEE EMBS, pages 3813-3816, 2013.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Our Approach . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Scope of Application . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Autonomic-Cardiac Reactivity Assessment . . . . . . . . 51.2.2 Clinical Decision Support Systems . . . . . . . . . . . . . 51.2.3 Artificial Bionic Baroreflex . . . . . . . . . . . . . . . . . 71.3 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9vi2 Literature Review on Autonomic-Cardiorespiratory Regulation . . 102.1 Physiological Background . . . . . . . . . . . . . . . . . . . . . 122.1.1 Cardiorespiratory System . . . . . . . . . . . . . . . . . . 122.1.2 Autonomic Nervous System . . . . . . . . . . . . . . . . 122.1.3 Baroreceptor Reflex . . . . . . . . . . . . . . . . . . . . 152.1.4 Chemoreceptor Reflex . . . . . . . . . . . . . . . . . . . 172.1.5 Lung-Stretch Receptor Reflex . . . . . . . . . . . . . . . 182.2 Autonomic-Cardiac Monitoring . . . . . . . . . . . . . . . . . . 182.2.1 Standard Heart Rate Variability Measures . . . . . . . . . 192.2.2 Respiratory Sinus Arrythmia . . . . . . . . . . . . . . . . 202.2.3 Pre-Ejection Period . . . . . . . . . . . . . . . . . . . . . 202.3 Standard Clinical Tests . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Lower Body Negative Pressure . . . . . . . . . . . . . . . 212.3.2 Orthostatic Hypotension . . . . . . . . . . . . . . . . . . 212.3.3 Mental Stress . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . 222.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Subject-Specific Monitoring of Autonomic-Cardiac Regulation . . . 263.1 Methods and Algorithm . . . . . . . . . . . . . . . . . . . . . . . 283.1.1 Experimental Dataset . . . . . . . . . . . . . . . . . . . . 293.1.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . 293.1.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . 323.1.4 System Identification . . . . . . . . . . . . . . . . . . . . 353.1.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . 383.2.2 System Identification . . . . . . . . . . . . . . . . . . . . 403.2.3 Limitations of the Proposed Approach . . . . . . . . . . . 483.2.4 Autonomic-Cardiac Regulation Monitoring . . . . . . . . 503.3 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 514 Model-Based Stability Analysis of Autonomic-Cardiac Regulation . 54vii4.1 Methods and Algorithm . . . . . . . . . . . . . . . . . . . . . . . 564.1.1 Physiology-Based Model: Delayed Differential Equations 564.1.2 Delay-Free Realization . . . . . . . . . . . . . . . . . . . 584.1.3 Identification of Equilibrium States . . . . . . . . . . . . 594.1.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . 624.1.5 Simulation Data . . . . . . . . . . . . . . . . . . . . . . 644.1.6 Validation of the Proposed Approach . . . . . . . . . . . 654.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 694.2.1 Identification of Equilibrium States . . . . . . . . . . . . 694.2.2 Proposed Stability Metrics . . . . . . . . . . . . . . . . . 694.2.3 Multi-dimensional Stability Analysis . . . . . . . . . . . 714.2.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 734.3 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . 735 A Novel Approach to the Design of an Artificial Bionic Baroreflex . 755.1 Methods and Algorithm . . . . . . . . . . . . . . . . . . . . . . . 765.1.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . 775.1.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . 775.1.3 System Identification . . . . . . . . . . . . . . . . . . . . 795.1.4 Artificial Bionic Baroreflex . . . . . . . . . . . . . . . . . 805.1.5 Robustness Analysis . . . . . . . . . . . . . . . . . . . . 835.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 835.3 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 866 Mathematical Modeling of Autonomic-Cardiorespiratory Regulation 886.1 Methods and Algorithm . . . . . . . . . . . . . . . . . . . . . . . 896.1.1 Experimental Dataset . . . . . . . . . . . . . . . . . . . . 896.1.2 Physiological Background . . . . . . . . . . . . . . . . . 906.1.3 Autonomic-Cardiac Regulation . . . . . . . . . . . . . . 916.1.4 Autonomic-Cardiorespiratory Regulation . . . . . . . . . 936.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 966.2.1 Model Validation . . . . . . . . . . . . . . . . . . . . . . 966.2.2 Mechanical vs. Neuromechanical Couplings . . . . . . . 98viii6.2.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 1006.3 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 1007 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . 1017.1 Summary: Work Accomplished . . . . . . . . . . . . . . . . . . 1017.2 Future-Work: The Road Ahead . . . . . . . . . . . . . . . . . . . 103Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105ixList of TablesTable 3.1 Parameters in the mathematical model of autonomic-cardiac reg-ulation [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Table 3.2 Sensitivity-based parameter classification. . . . . . . . . . . . 38Table 3.3 Statistical properties of the identified baroreflex-modulated Sym-pathetic Nerve Activity (SNA) and Parasympathetic Nerve Ac-tivity (PSNA): mean?std . . . . . . . . . . . . . . . . . . . . . 52Table 4.1 Parameters in the mathematical model of autonomic-cardiac reg-ulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Table 5.1 Model parameters of autonomic-cardiac regulation. . . . . . . 78Table 5.2 Individualized nominal values of high-sensitivity parameters inthree subjects versus corresponding population nominal values. 86Table 6.1 Model parameters of autonomic-cardiac regulation. . . . . . . 93Table 6.2 Respiratory system impacts on VL, Heart Rate (HR), VR, and ?V . 96Table 6.3 A numerical measure of perturbation caused by mechanical cou-pling effects J2 and neuromechanical coupling effects J1. . . . 98xList of FiguresFigure 1.1 A schematic model of a hemodynamic stability monitoringsystem using a subject-specific mathematical model . . . . . . 6Figure 1.2 A schematic diagram of the artificial bionic baroreflex. . . . . 7Figure 2.1 An extensive block diagram model of autonomic-cardiorespiratoryregulation. Parameters shown in red are outputs of the Auto-nomic Nervous System (ANS) as well as inputs for parts ofautonomic-cardiorespiratory regulation (i.e., the closed-loopautonomic-cardiac regulation is opened at this level). . . . . . 11Figure 2.2 A schematic diagram of the cardiorespiratory system [8]. . . . 13Figure 2.3 Various factors affect autonomic regulation of the heart, in-cluding but not limited to respiration, thermoregulation, hu-moral regulation, Blood Pressure (BP), and Cardiac Output(CO) [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.4 Schematic diagram of the baroreflex mechanism. The Nu-cleus Tractus Solitarius (NTS) excites the parasympathetic mo-tor neurons and inhibits the sympathetic motor neurons [10]. 16Figure 2.5 Two aspects of baroreflex characteristic . . . . . . . . . . . . 17Figure 2.6 A schematic example of electrocardiography (ECG) signal andimpedance cardiography (dZ/dt) signal [11]. . . . . . . . . . . 20Figure 3.1 Schematic diagram of the autonomic-cardiac regulation model. 30xiFigure 3.2 An extensive block-diagram model of autonomic-cardiorespiratoryregulation [see Chapter 2] with emphasis on the parts studiedin this chapter. The shaded parts are not described in the math-ematical model Equation 3.1-Equation 3.2. . . . . . . . . . . 31Figure 3.3 Measured versus the model-estimated signals (Case No.: 289);blue is measured signals and black is model-estimated signals. 37Figure 3.4 Distribution of the index IEval? j for the estimated high-sensitivityparameters in a set of 500 idealized simulations. . . . . . . . . 39Figure 3.5 The overall sensitivity (mean and standard deviation) of autonomic-cardiac model parameters over 100 sensitivity analysis runswith nominal values selected from +/-20% the associated nom-inal values introduced in Table 3.1. . . . . . . . . . . . . . . . 40Figure 3.6 Experimental results from MIMIC dataset (Case No.: 476). . . 43Figure 3.7 Experimental results from MIMIC dataset (Case No.: 486). . . 44Figure 3.8 Experimental results from MIMIC dataset (Case No.: 289). . . 45Figure 3.9 Experimental results from MIMIC dataset (Case No.: 477). . . 46Figure 3.10 System identification results on the orthostatic hypotension datasetto monitor SNA and PSNA during a tilt test. The two top panelsshow the measured vs. model-estimated HR and BP signals,and the three bottom panels show the identification results:?Ts, ?HTs, and VHTp. . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.11 Measured versus model-estimated HR and BP signals with andwithout the use of measured CO signal (Case No.: 289); blueare measured signals, black are model-estimated signals withmeasured CO and red are model-estimated signals without mea-sured CO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 4.1 Comparison of equilibrium states estimated using the proposedanalytical approach Equation 4.25 against numerical optimiza-tion (left panel) and nonlinear simulation (right panel). . . . . 66xiiFigure 4.2 Two metrics for stability margin Sm and Sp over changes of amodel parameter from 50% to 200% of its nominal value for ahealthy physiological condition with and without stress. Sm isthe blue solid line; Sp is the green dashed line. A normal con-dition (i.e., VH , ?H , and ? were fixed at their nominal values). 67Figure 4.3 Two metrics for stability margin, Sm and Sp, over changes of amodel parameter from 50% to 200% of its nominal value fora healthy physiological condition with and without stress. Smis the blue solid line; Sp is the green dashed line. A stressfulcondition (i.e., a 50% lower VH and 100% higher ?H and ?compared to their nominal values). . . . . . . . . . . . . . . . 68Figure 4.4 The proposed stability metric, Sm, over 2-D parameter spacesfrom 50% to 150% of their nominal values for a normal phys-iological condition. The quantitative stability margin metric,Sm, at each point of the 2-D parameter space is mapped into apixel-intensity level. A higher pixel-intensity level is related tolower stability margin, and vice versa. . . . . . . . . . . . . 72Figure 5.1 Schematic model of autonomic-cardiac regulation with em-phasis on the baroreflex . . . . . . . . . . . . . . . . . . . . . 77Figure 5.2 Schematic model of the proposed artificial bionic baroreflex . 79Figure 5.3 BP measurement (BP setpoint) vs. the results of the artificialbionic baroreflex (simulated BP) for individual with subjectnumber 477. . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 5.4 BP measurement (BP setpoint) vs. the results of the artificialbionic baroreflex (simulated BP) for individual with subjectnumber 486. . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 5.5 BP measurement (BP setpoint) vs. the results of the artificialbionic baroreflex (simulated BP) for individual with subjectnumber 476. . . . . . . . . . . . . . . . . . . . . . . . . . . 81xiiiFigure 5.6 The results of robustness analysis for an individual with sub-ject number 477. The solid line shows an average value of 100simulated signals obtained by the proposed control strategy,whereas the shaded area indicates the corresponding standarddeviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 5.7 The calculated control signal, P0, in three subjects . . . . . . . 85Figure 6.1 Physiological measurement during Lower Body Negative Pres-sure (LBNP) experiment in an individual; mean BP, Stroke Vol-ume (SV), and HR were calculated according to the BP wave-form and Electrocardiogram (ECG) recordings. . . . . . . . . 90Figure 6.2 Schematic diagram of interactions between cardiovascular, res-piratory and nervous systems. . . . . . . . . . . . . . . . . . 92Figure 6.3 An extensive block-diagram model of autonomic-cardiorespiratoryregulation [see Chapter 2] with emphasis on parts described inEquation 6.7-Equation 6.8. The shaded parts are not describedin the mathematical model. . . . . . . . . . . . . . . . . . . . 94Figure 6.4 Power Spectral Density (PSD) difference of Heart Rate Vari-ability (HRV) among simulated (two methods) vs. measuredHR signals at the different stages of the LBNP experiment; theshaded area shows the respiratory frequency band. . . . . . . 97Figure 6.5 Neuromechanical coupling effects of respiration on HR and BP. 99Figure 6.6 Mechanical coupling effects of respiration on HR and BP. . . . 99xivGlossaryABP Arterial Blood PressureAP Action PotentialsANS Autonomic Nervous SystemBP Blood PressureCO Cardiac OutputCRS Cardiorespiratory SystemCSF Cerebrospinal FluidCVS Cardiovascular SystemCNS Central Nervous SystemECG ElectrocardiogramHF High FrequencyHR Heart RateHRV Heart Rate VariabilityHUT Head-Up TiltILV Instantaneous Lung VolumeLBNP Lower Body Negative PressurexvLF Low FrequencyNTS Nucleus Tractus SolitariusPEP Pre-Ejection PeriodPNS Parasympathetic Nervous SystemPSNA Parasympathetic Nerve ActivityPSD Power Spectral DensityRR Respiration RateRS Respiratory SystemRSA Respiratory Sinus ArrhythmiaSA SinoatrialSCI Spinal Cord InjurySNA Sympathetic Nerve ActivitySNS Sympathetic Nervous SystemSV Stroke VolumeTPR Total Peripheral ResistanceTV Tidal VolumexviAcknowledgmentsI would like to express my deepest appreciation to my supervisors, Professors GuyA. Dumont, Hossein A. Noubari, and W. Tom Boyce, for their inspiration, encour-agement, patience and unconditional support. They have provided me not onlywonderful support, but also enough freedom to explore my interests and find myway 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 experiencesand who shared his precious medical knowledge with us. This thesis could nothave been completed without the help of all of these people. I have learned howto elaborate and conduct research in a complex field, how to collaborate with otherresearchers as a team and how to conduct research as an individual, how to tar-get a real-life problem to remove a burden from society, and how to be patientthroughout the course of my PhD program.I have been fortunate to collaborate with Dr. Jin-Oh Hahn during, and after, hisstay at the University of British Columbia. He has certainly been an excellent men-tor for me; his advice has always been relevant and effective. His professionalism,modesty, and hard work have been helped me to collaborate with him productivelythrough different stages of my PhD program.Many thanks also to my friends and colleagues at the laboratory of Electri-cal and Computer Engineering in Medicine, in alphabetical order: Chris Brouse,Matthias G?rges, Walter Karlen, Sara Khosravi, Mande Leung, Joanne Lim, BehnamMolavi, Prasaad Shrawane, Kouhyar Tavakkolian, Klaske van Heusden, AryannahUmedaly, Ali Shahidi-Zandi, Ping Yang and all my other fellow students and col-leagues. They have provided a pleasant academic and social environment in thelaboratory and a wonderful teamwork culture that helped me to resolve my profes-xviisional and personal issues during this program. I also want to thank my dear friendsAmin Aziznia, Pouyan Abouzar, Sona Kazemi, and Kaveh Shafiee who supportedme in my unkind moments, and laughed with me in my wonderful moments duringmy stay in Vancouver.Lastly, I wish to express my genuine gratitude to my wonderful parents, myconstant source of energy, for their never-ending love, support, and guidance through-out my entire life. I also thank my two lovely sisters, Maryam and Sara, for theirsupport during every stage of my life.xviiiDedicationTo my family, an insufficient token of my appreciation of their unwavering loveand faithfulnessxixChapter 1IntroductionCardiovascular disease is the leading cause of mortality and morbidity worldwide;more than one million individuals in the United States suffer a heart attack eachyear [12]. According to the World Health Organization (WHO), hypertension isestimated to cause 7.5 million deaths worldwide, which is about 12.8% of thetotal of all deaths [13]. Although hypertension is a major risk factor for coro-nary heart disease and hemorrhagic stroke [13] and is extremely common, it isstill poorly understood [14]. For example, it has been recently shown that manypatients need newly conceived Blood Pressure (BP)-stabilizing drugs as well asBP-lowering drugs [14].1.1 Problem StatementThis project aimed to investigate a potential autocatalytic loop, also referred toas positive feedback, within the autonomic-cardiac regulation, which leads to anunstable (i.e., fluctuating) BP. To investigate such physiological conditions, weselected a subject-specific model-based approach to analyze the stability of theautonomic-cardiac regulation.1.1.1 Our ApproachWe investigated a large number of mathematical models describing autonomic-cardiac regulation to find a physiology-based mathematical model. A physiology-based mathematical model with two coupled differential equations [7] to describe1the autonomic-cardiac regulation has been selected. We revised this model regard-ing the baroreflex mechanism to increase its physiological consistency. To indi-vidualize the mathematical model, we then developed a parameter identificationtechnique to estimate time-varying and subject-specific model parameters includ-ing sympathetic and parasympathetic activation, by using routine clinical measure-ments including Heart Rate (HR) and BP.Further, we developed a systematic framework to analyze the system stabil-ity. To investigate the potential system-level causes of instability (e.g., a potentialpositive feedback) in the autonomic-cardiac regulation, we introduced an indexshowing the stability margin of the autonomic-cardiac regulation. We then usedthe subject-specific mathematical model for the autonomic-cardiac regulation todesign a closed-loop artificial bionic baroreflex.Finally, we recognized the significance of respiratory effects in autonomic-cardiac regulation. Therefore, to describe the respiratory effects on autonomic-cardiac regulation, the mathematical model was improved to include respiratory-related terms.1.1.2 ChallengesIn this section, some challenges that we have confronted in this research are ex-plained and categorized into three parts: mathematical modeling, system identifi-cation, and stability analysis.Mathematical Modeling- The major purposes of developing a mathematical modelfor a dynamic physiological system are to improve our understanding of the sys-tem, to reveal new insights into physiological mechanisms within the system, andto predict the behavior of the system in different clinical conditions [15]. A dy-namic physiological system can be described by different types of mathematicalmodels including statistical models or differential equations. In this context, awhite-box1 physiology-based mathematical model (e.g., differential equations) hasmore advantages than a black-box2 model (e.g., statistical models) [16?18]. De-1A white-box model is a mathematical model developed based on a priori information about thesystem.2A black-box model is a mathematical model solely developed based on its input, output andtransfer function without any knowledge of its internal dynamics.2veloping a mathematical model with minimal complexity to simplify mathematicalanalysis, as well as developing the model sufficiently detailed to reproduce as muchclinically 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 numberof parameters may produce significantly more accurate simulation results that areconsistent with experimental observations; however, such a model generates manycomplexities in the mathematical analysis and may cuase parameter identificationbecome impossible.Developing an accurate mathematical model of the autonomic-cardiac regula-tion and then a simulator environment could reduce the need for invasive clini-cal experiments on the system as well as some clinical expenses. For instance, aphysiology-based pharmacokinetic model may be used to predict the absorption,distribution, and excretion of synthetic or natural chemical substances (e.g., a brandnew drug) in the human body. In fact, the pharmacokinetic model gives cliniciansthe ability to investigate how a new drug impacts system outcomes, e.g., dosage-related effects of a drug on other physiological variables can be investigated with-out any of drug being injected into the body. Recently, many mathematical modelsfor the autonomic-cardiorespiratory regulation have been introduced. In this the-sis, we intend to develop a physiology-based mathematical model of autonomic-cardiorespiratory regulation.System Identification- The prediction-error framework is the dominant ap-proach in system identification theory and its applications with a focus on multi-variable and closed-loop systems [20]. Once we have a mathematical model ofautonomic-cardiac regulation based on the physical laws describing the variouscomponents and interconnection structure, system identification is used to estimateunknown parameters in the model using the measured signals. Considering that allof the model parameters were not identifiable, we had to examine whether the pre-dicted outputs were sensitive to each parameter to obtain a group of identifiableparameters [20].In general, two types of sensitivity analysis approaches have been introducedin the literature, local and global sensitivity analyses [21]. The local sensitivity ofa system output due to a model parameter is computed by the first-order partial3derivatives of the system output with respect to that model parameter. Similarly,the global sensitivity used to quantify the overall effects of the parameters on thesystem output is computed by perturbing parameters within large ranges.Since the introduced mathematical model of autonomic-cardiac regulation con-tains a set of coupled nonlinear and delayed differential equations, we used a finitedifference approximation method to compute the global sensitivity and to separatethe model parameters into two groups: high-sensitivity and low-sensitivity param-eters.Stability Analysis- An improper dynamic change may cause an oscillatorysystem, 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 bloodpressure regulation, may cause undesirable oscillation [22]. In fact, a large numberof physiological disorders are characterized by improper changes in the dynamicsof corresponding physiological systems, which result in unstable or irregular sys-tem behavior [22]. In the autonomic-cardiorespiratory regulation, model parame-ter changes (e.g., an increase in the time delay of sensory afferent pathways) maycause an onset of oscillations (limit cycle or even instability) in BP and HR whichis not relevant to its normal regulatory task [16, 23]. An instability or irregularityin a physiological system is called homeostatic imbalance (i.e., a disturbance inhomeostasis3). The homeostatic imbalance may occur as a result of the complexdynamic interactions among nonlinearities and delays in a physiological system.It is crucial to maintain a certain degree of stability margin in the autonomic-cardiorespiratory regulation as a major in-vivo physiological control mechanismfor individuals with, for example, treatment-resistant hypertension since they aresusceptible to cardiovascular instability [26]. The system-level cause of instabilityand the stability margin of the autonomic-cardiorespiratory regulation can be in-vestigated using model-based stability analysis. To perform model-based stabilityanalysis, we must first develop an accurate physiology-based mathematical modelof the system. To actively monitor and then control the system?s stability and toprovide the patients with appropriate preventive interventions, it is important to3Homeostasis is the capability of living systems to maintain a physiological parameter fixed at asetpoint by means of dynamic regulatory mechanisms in the face of external or internal challenges[24, 25].4identify the root causes of instability, and to predict the system?s transition to in-stability. In this thesis, we propose a model-based approach for stability analysis ofautonomic-cardiorespiratory regulation to determine impacts of the parameter con-figurations that cause complex undesirable behavior and to determine the stabilitymargin of the physiological system.1.2 Scope of Application1.2.1 Autonomic-Cardiac Reactivity AssessmentAssessment of autonomic-cardiac reactivity can be used in many fields. Autonomic-cardiac reactivity is the deviation of an autonomic-cardiac parameter, enforced byan individual?s Autonomic Nervous System (ANS), from its normal value in re-sponse to a stimulus (e.g., environmental stress) [27]. In this context, reactivityis 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 severityof injury in individuals with Spinal Cord Injury (SCI) as well as an indicator of lifesatisfaction in individuals with high thoracic and cervical SCI [29, 30]. Moreover,it can be used to improve the current classification systems for the Paralympics toensure a fair competition among Paralympians [31].Exaggerated stress-related autonomic-cardiac reactivity puts children at riskof, for example, cognitive impairments and poor emotion regulation. The processthat creates these effects can be investigated by finding an autocatalytic loop (i.e.,positive feedback) within autonomic-cardiac regulation that is stimulated understressful conditions [32].1.2.2 Clinical Decision Support SystemsClinical decision support systems are computer-based intelligent systems that canpotentially provide subject-specific recommendations for a clinician to increasepatient safety and improve health outcome [33]. Subject-specific mathematicalmodels may eventually play a significant role in providing subject-specific rec-ommendations in clinical decision support systems. Subject-specific mathematical5Figure 1.1: A schematic model of a hemodynamic stability monitoring sys-tem using a subject-specific mathematical modelmodels can be used to predict an individual?s physiological response to, for exam-ple, a specific medication dosage or a surgery procedure [34].Patients with pre-existing conditions including cardiovascular diseases and SCIundergoing major surgical procedures with anesthesia are at the risk of hemody-namic instability [35]. It is important for anesthesiologists to be able to predict therisk of hemodynamic instability in their patients. Therefore, the stability marginof a patient?s autonomic-cardiac regulation could be continuously monitored andpredicted during surgery [35]. Hemodynamic instability is mostly associated withan 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 usuallycaused by a specific stimulus such as surgery, drug injection, emotional stress, orpostural change [14]. A quantitative metric of BP instability could be used to pre-vent hemodynamic instability during surgery. Increased BP instability is also a riskfactor for vascular dementia, which can be prevented by prescribing medicines that6Figure 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 aswell as physiological changes during surgery can be described using a subject-specific mathematical model with different parameter configurations. Figure 1.1depicts a schematic model of the hemodynamic stability monitoring system thatwas studied in this research.1.2.3 Artificial Bionic BaroreflexThe disruption of the autonomic regulation (e.g., baroreflex failure) critically af-fects the quality of life for individuals with neurological disorders (e.g., Shy-Dragersyndrome) or traumatic SCIs which results in severe orthostatic hypotension [36].The baroreflex characteristic is also altered in individuals with chronic hyperten-sion, preventing proper BP regulation [10, 37]. The fact that the baroreceptors areconstantly exposed to high BP may impair the baroreflex mechanism, resulting ina significant loss of baroreceptor sensitivity. Therefore, the impaired baroreflexcan not attenuate the effects of rapid perturbation in arterial pressure during, forexample, a posture change from lying to standing, possibly resulting in loss of con-sciousness [36]. A novel therapeutic approach including artificial bionic baroreflexmust be investigated for the treatment of severe baroreflex failure.An artificial bionic baroreflex could be used for treatment of individuals withbaroreflex failure by using an external mechanism that activates the sympatheticefferent nerves. An artificial bionic baroreflex is a functional replacement of the7baroreflex that consists of arterial pressure sensors as well as an automatic nervestimulator, which generates a pulse train signal to stimulate sympathetic nerves[38]. Sunagawa [39, 40] proposed several anatomical sites, such as the carotidsinus and the spinal cord, to manipulate the Sympathetic Nervous System (SNS). Yamasaki [38] also investigated a bionic baroreflex by stimulating sympatheticnerves through an epidural catheter located at the level of the lower thoracic spinalcord [38]. Accordingly, BP could be normally regulated in different physiologicalconditions, providing a higher quality of life for individuals with baroreflex failure.1.3 Thesis ContributionsThe major contributions of this thesis are as follows:? Develops a solid framework to analyze system stability and investigates thepresence of positive feedback in the autonomic-cardiac regulation using aphysiology-based subject-specific mathematical model of autonomic-cardiacregulation.? Develops a novel non-invasive model-based method to estimate and thenmonitor autonomic-cardiac regulation based on a computationally efficientsystem identification method by using routine clinical measurements: HR,BP, and CO.? Presents a systematic approach to investigate the system-level cause of insta-bility in the autonomic-cardiac regulation as a major in-vivo physiologicalcontrol mechanism based on a stability index that determines the stabilitymargin for a parameter configuration.? Introduces a novel model-based approach to the design of an artificial bionicbaroreflex that can be used to restore normal arterial pressure regulation inindividuals with baroeflex failure by mimicking the in-vivo baroreflex mech-anism.? Introduces a novel physiology-based mathematical model of autonomic-car-diorespiratory regulation described by a set of three nonlinear, coupled dif-8ferential 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 theautonomic-cardiac regulation to investigate potential instability conditionsin BP and HR, and a software package that assesses the autonomic reactivityby monitoring sympathetic (cardiac and arterial) and parasympathetic acti-vation.1.4 Thesis OutlineChapter 2 presents the background material on the autonomic-cardiorespiratoryregulation as well as reviews of the previously published mathematical models.Chapter 3 describes the details of an identification technique conducted on thephysiology-based mathematical model of autonomic-cardiac regulation. A para-metric sensitivity analysis used to classify the model parameters into high-sensitivity,low-sensitivity, and invariant groups is also described in this chapter. The feasibil-ity and potential of the proposed subject-specific monitoring technique are demon-strated and discussed using two datasets: the MIMIC dataset and an orthostatichypotension dataset.Chapter 4 presents a systematic approach to the stability analysis of the autonomic-cardiac regulation. A Lyapunov-based systematic approach to analyze the systemstability in the neighborhood of the equilibrium state is developed in this chapter.Further, a quantitative metric of stability margin capable of comparing differentparameter configurations regarding their stability has been introduced. Chapter 5introduces a novel model-based approach to the design of a closed-loop artificialbionic baroreflex. In this chapter, an individual?s in-vivo autonomic-cardiac reg-ulation is described by a subject-specific mathematical model. In Chapter 6, aphysiologically-based mathematical model of the autonomic-cardiorespiratory reg-ulation is introduced. Further, the significance of respiratory dynamics in autonomic-cardiac regulation is studied.9Chapter 2Literature Review onAutonomic-CardiorespiratoryRegulationAutonomic-cardiorespiratory regulation operates through interactions between theANS and the Cardiorespiratory System (CRS). The ANS maintains homeostasis inthe cardiorespiratory system against physical stressor, such as exercise and ortho-static hypotension, and psychological stressor, such as fear and anxiety [41?43].A recently developed theory proposes that a physiological system only restores itsstability (allostasis) against a stressor rather than regulating a physiological param-eter to a fixed setpoint (homeostasis) [24]. The ANS responses to physical andpsychological stressors are dictated by the individual?s autonomic reactivity char-acteristics [44, 45]. In fact, the ANS responds to different conditions by adjustingcardiorespiratory parameters, including Respiration Rate (RR), BP, HR, and TotalPeripheral Resistance (TPR)1, to deliver adequate oxygenated blood-flow to organsin different conditions [45]. Figure 2.1 shows the complex interconnected structureof short-term autonomic-cardiorespiratory regulation.The present chapter will provide a brief overview of autonomic-cardiorespira-tory regulation including the CRS, the ANS, and the autonomic regulation mecha-1Total peripheral resistance (i.e., systemic vascular resistance) refers to the resistance to bloodflow from the systemic vasculature, excluding the pulmonary vasculature [10].10Figure 2.1: An extensive block diagram model of autonomic-cardiorespiratory regulation. Parameters shown in redare outputs of the ANS as well as inputs for parts of autonomic-cardiorespiratory regulation (i.e., the closed-loopautonomic-cardiac regulation is opened at this level).11nisms of the Cardiovascular System (CVS) and the Respiratory System (RS), fol-lowed by a review of the related work in the field of autonomic-cardiorespiratorymodeling. Further, we briefly introduce several common clinical tests to investi-gate autonomic-cardiorespiratory regulation with different stressors including or-thostatic hypotension, Lower Body Negative Pressure (LBNP), and mental stresstests. We used readily available orthostatic hypotension [46] and MIMIC datasets[47] to assess the proposed parameter identification method described in Chapter 3and used an LBNP test to assess the proposed mathematical model of autonomic-cardiorespiratory regulation described in Chapter 6.2.1 Physiological Background2.1.1 Cardiorespiratory SystemThe cardiorespiratory system consists of the CVS and the respiratory system. Thecardiorespiratory system transports nutrients, oxygen, carbon dioxide, hormones,and blood cells in the body. The major functions of the cardiorespiratory system areto provide vital needs for metabolic activities, to protect the body from infection,to maintain homeostasis in thermoregulation, and to maintain fluid balance withinthe body cells.The CVS consists of the heart, blood, and two networks of blood vessels: pul-monary circulation and systemic circulation. The pulmonary circulation carriesdeoxygenated blood from the heart to the lungs and returns oxygenated blood tothe heart. The systemic circulation carries oxygenated blood from the heart to thebody tissues and returns oxygen-depleted blood back to the heart (Figure 2.2). Therespiratory system consists of the lungs, airways, and respiratory muscles (e.g., thediaphragm). During respiration, carbon dioxide accumulated in the blood is ex-changed with oxygen inhaled from the external environment through the diffusionmechanisms within the lungs [48].2.1.2 Autonomic Nervous SystemThe nervous system is divided into the somatic nervous system, which controlsorgans under voluntary actions, and the ANS, which mostly regulates involuntary12Figure 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 separatebranches, the Parasympathetic Nervous System (PNS) and the Sympathetic Ner-vous System (SNS), based on anatomical and functional differences. The PNS isdominant 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 systemis activated, the other is usually depressed) with antagonistic impacts [51]. How-ever, there are conditions during which SNS and PNS may be activated (e.g., sexualarousal) or inhibited (e.g., anesthesia) together [50, 52].The medulla oblongata (often referred to as the medulla) is the brain?s pri-mary site for regulation of sympathetic and parasympathetic (vagal) outflows. Themedulla is located in the lowest part of the brain and the lowest portion of the brain-13Figure 2.3: Various factors affect autonomic regulation of the heart, includingbut 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 fromdifferent systemic and central receptors (e.g., baroreceptors and chemoreceptors)as well as higher brain centers (e.g., the hypothalamus). The hypothalamus playsa particularly important role in determining cardiovascular responses to emotionand stress. Efferent fibers of sympathetic and vagal nerves innervate the heart andblood vessels, where they modulate the activity of these target organs. The heartis innervated by both sympathetic and vagal divisions, which exert a regulatoryinfluence on HR by influencing the activity of the heart?s primary pacemaker, theSinoatrial (SA)-node [51] (Figure 2.3). An increase in HR could arise from eitherincreased 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 toas reflexes), based on continuously integrated measurement of vital physiological14variables captured by specialized biological receptors [51, 53, 54]. For example,perturbations in BP (e.g., orthostatic hypotension) are measured by the barorecep-tors, 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 thechemoreceptor reflex regulates oxygen and carbon dioxide concentration in theblood.The measured physiological variables are transmitted to the ANS, and the ANSacts against perturbations by sending control commands through sympathetic andparasympathetic pathways to a set of effectors including the heart and blood vessels[16, 17, 23]. For instance, a rise in sympathetic activation tone elevates CardiacOutput (CO)2 by increasing cardiac contractility (contraction force of the heart) andthe pace of the heartbeat, and elevates TPR by decreasing the diameter of bloodvessels (i.e., vasoconstriction). Conversely, a rise in parasympathetic activationtone decreases CO by decreasing HR [55].2.1.3 Baroreceptor ReflexThe baroreceptor reflex (baroreflex) is a short-term homeostatic mechanism inthe autonomic cardiorespiratory regulation. It includes specialized sensory neu-rons (also known as baroreceptors), efferent and afferent neural pathways, and thebrainstem [41]. The baroreceptors, mostly located in the carotid sinus and the aor-tic arch, are stretch-sensitive mechanoreceptors that are excited by the stretch ofblood vessels. They are sensitive to both absolute stretch (mean BP) and the rate ofstretch variation (pulsatile BP); however, the response characteristics to mean BPand pulsatile BP are different (Figure 2.5b). In this work, we modelled absolutestretch baroreceptors that respond to mean BP.A series of Action Potentials (AP) are fired and conveyed to the NTS (refer toSection 2.1.2) in response to deformations in the arterial wall according to a non-linear response curve [56?58]. For example, a BP rise causes the walls of vesselswith baroreceptors to expand and the baroreceptors to increase firing rate of APs[10]. The greater the stretch, the more rapidly baroreceptors fire APs . The NTS2CO is the amount of blood pumped through the circulatory system in a minute by the heart.15Figure 2.4: Schematic diagram of the baroreflex mechanism. The NTS ex-cites the parasympathetic motor neurons and inhibits the sympatheticmotor neurons [10].uses the frequency of received APs as a measure of BP [16]. The baroreceptorfiring-rate pattern also adapts to alterations in physiological conditions, causingboth short-term and long-term changes in BP. For example, baroreflex responseswill be adjusted under a stressful condition to maintain BP in a proper range (short-term) [10], while the baroreflex loses its sensitivity to high BP in an individual withchronic hypertension (long-term) [10]. Further, the baroreflex is less sensitive toa fall in BP than to a rise in BP, also known as the hysteresis3 effect, as shown inFigure 2.5a [60].In a healthy individual, the baroreflex maintains BP within a narrow range byusing a set of sensors and effectors that adjusts HR, TPR, and cardiac contractil-ity under a closed-loop negative feedback4 mechanism (see Figure 2.4) [41]. Thebaroreflex responds to a decrease in BP (mean, pulsatile, or both) by increasingsympathetic outflow and decreasing vagal outflow, while it acts differently on sym-3Hysteresis is defined as ?dependency of the steady-state response curve of a deterministic systemon the direction of the parameter change (increase or decrease)? [59].4Feedback 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-ing rate to increasing and decreasing BP in ananesthetized dog [65].(b) Two different response curves of barore-flex firing rate in response to pulsatile and meanBP [56].Figure 2.5: Two aspects of baroreflex characteristicpathetic and vagal outflow in an elevated BP [16, 18]. The baroreflex characteristicdriven by the ANS is time-varying and subject-varying, i.e., it changes in differentphysiological conditions, and it differs among individuals [41, 56]. Homeostaticimbalance in the CVS can be caused by impaired baroreflex response to an externalor internal stressor, which results in an oscillation and an instability in HR and BPregulation [18, 62]. For example, Mayer waves are low-frequency oscillations inHR 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]. Ithas been shown that if sympathetic activity becomes chemically blocked, Mayerwaves are significantly reduced [41]. Therefore, investigation of the baroreflexcharacteristics within the autonomic-cardiorespiratory regulation can be of signifi-cant importance in the context of homeostatic imbalance.2.1.4 Chemoreceptor ReflexThe chemoreceptor reflex (chemoreflex) is a cardiorespiratory reflex that has evolvedto 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 setsof receptors: the peripheral chemoreceptors and the central chemoreceptors [68].The peripheral chemoreceptors are located in the carotid bodies at the bifurcationof the carotid arteries, and the central chemoreceptors are located in the medulla[69].The peripheral chemoreceptors respond primarily to a fall in partial pressureof oxygen in arterial blood PaO2 (hypoxia) [69, 70]. At normal levels of PaO2,17some neural activity arises from the peripheral chemoreceptors, while, in arterialhyperoxia (i.e., abnormally high PaO2), this activity is slightly reduced in a healthyindividual. However, in arterial hypoxemia (i.e., abnormally low PaO2), the inten-sity of neural activity varies in a nonlinear manner according to the severity ofthe condition, causing an increase in the depth and rate of breathing. Hypoxiaalso causes sympathetically mediated vasoconstriction in most arterioles (exceptfor coronary and brain arterioles) to maintain BP and circulation [71].The central chemoreceptors respond primarily to a rise in partial pressure ofcarbon dioxide in the arterial blood PaCO2 (hypercapnia) [69, 70]. In other words,the response of the peripheral chemoreflex to arterial PaCO2 is less important thanthat of the central chemoreflex [68]. The central chemoreceptors are exposed toCerebrospinal Fluid (CSF) and are not in direct contact with the arterial blood [72].Nevertheless, alterations in arterial PaCO2 are rapidly transmitted to the CSF. Anincrease in the concentration of CO2 in the CSF causes hyperventilation [73].2.1.5 Lung-Stretch Receptor ReflexThe lung-stretch receptor reflex (often referred to as the Hering-Breuer reflex) is acardiorespiratory reflex that triggers lung inflation and deflation by using mechanore-ceptors located on the lung to provide information on the degree of lung expansionor contraction. For example, inspiration causes the lung-stretch receptors and theirafferent nerves to activate and then project to the NTS. The activation of receptorafferent nerves causes vagal cardiac outflow to inhibit and sympathetic outflow toexcite. This reflex may play a major role in ventilation by regulating breathing rateand depth in newborns. However, more recent work indicates that this reflex islargely inactive in adults unless the tidal volume exceeds one liter, as in exercise[48].2.2 Autonomic-Cardiac MonitoringThe ANS responds differently during exposure to physical and psychological chal-lenges including stressful, emotional, and threatening conditions. The responsedeviation of a physiological variable (e.g., HR) from a control value that resultsfrom an individual?s ANS response to a stimulus is called autonomic, or ANS, reac-18tivity and is associated with physical and psychological health [27, 32, 45, 74?76].For example, researchers in behavioral pediatrics have shown that increased ANS orautonomic reactivity (i.e., exaggerated physiological responses to stress) puts chil-dren at risk for a variety of physical and mental disorders, including poor emotionregulation and cognitive impairments [32]. Further, cardiac vagal tone has beenproposed as a physiological marker of stress vulnerability (i.e., an individual?s dif-ferences in response thresholds to the identical challenging condition) [77]. In gen-eral, any change in HR, also referred to as Heart Rate Variability (HRV), has beenused as an indicator of autonomic reactivity [43, 78]. However, the ability to mon-itor and interpret HRV is dependent on measuring technology, the HRV quantifyingmethod and the knowledge of underlying mechanisms [79]. Autonomic reactiv-ity indices are classified into data-driven and model-based groups. In this thesis,we mostly investigated autonomic reactivity using a model-based technique. Au-tonomic reactivity can be assessed by using some data-driven measures, as well[74].2.2.1 Standard Heart Rate Variability MeasuresStandard methods for measuring HRV can be divided into time-domain and fre-quency-domain methods. Time-domain measures are calculated directly from theR-R interval signal such as the standard deviation and the standard deviation of thesuccessive differences of R-R intervals describing the overall variation and short-term variation, respectively [80, 81]. The frequency-domain measures are calcu-lated using the power spectral density of the R-R intervals. The well-acceptedmeasures are the powers of Low Frequency (LF) (0.04-0.15 Hz) and High Fre-quency (HF) (0.15-0.4 Hz) bands in absolute and relative values, the normalizedpowers of LF and HF bands, and the LF to HF power ratio [80, 81]. LF power ismodulated by both SNS and PNS activities, while HF power is modulated only byPNS activities [82, 83]. Therefore, it is commonly assumed that the LF to HF powerratio provides a measure of "sympathovagal balance" [82, 83].19Figure 2.6: A schematic example of electrocardiography (ECG) signal andimpedance cardiography (dZ/dt) signal [11].2.2.2 Respiratory Sinus ArrythmiaRespiratory Sinus Arrhythmia (RSA) is a periodic oscillation in HR, which is causedby respiration [84]. This periodic oscillation can be triggered during inhalationand exhalation, and results in an HR increase and decrease, respectively [84?86].RSA has been used as an index of cardiac vagal control as well as an index ofrespiratory-circulatory interactions [84]. Several time-based and frequency-basedindices for assessment of the RSA have been introduced [87?89]. In the spectralmethods, RSA is mostly calculated using the power spectrum of R-R interval dataand 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 withinthe respiration bandwidth.2.2.3 Pre-Ejection PeriodPre-Ejection Period (PEP) is the duration of isovolumetric ventricular contractionin the left ventricle [74, 88]. PEP is quantified as the time interval between theonset of ventricular depolarization (indicated by the ECG Q-wave) and the onsetof left ventricular ejection (indicated by the B-point of the impedance cardiogra-phy signal) (Figure 2.6). PEP is an indirect, non-invasive measure of sympatheticinfluence on cardiac rhythm, as a lower PEP score indicates higher cardiac sympa-thetic activity [32, 74, 76, 90]. However, PEP has been shown to be more reliableto investigate within-subjects differences rather than between-subject differences[90, 91].202.3 Standard Clinical TestsTo study autonomic-cardiac monitoring techniques, several clinical experiments in-cluding LBNP, orthostatic hypotension, and mental stress were introduced in the lit-erature, each of which specifically targets an aspect of autonomic-cardiorespiratoryregulation. That is, LBNP, orthostatic hypotension, and mental stress tests mainlyaffect Stroke Volume (SV), Arterial Blood Pressure (ABP), and parasympatheticnerves activation. These clinical experiments are briefly explained as follows.2.3.1 Lower Body Negative PressureIn an LBNP test, the lower body of a subject (for just above the pelvis) is placedsupine in a sealed chamber [92, 93]. After a resting control period, negative pres-sure is imposed mostly in 10 mmHg increments for a specific interval. The gradualpressure decrease in the LBNP chamber continues until either completion of the testor the onset of presyncope symptoms including light-headedness, nausea, sweat-ing, 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 intoleranceas well as severe hemorrhage in humans [94].2.3.2 Orthostatic HypotensionOrthostatic hypotension, also referred to as Head-Up Tilt (HUT), is a sustained BPreduction of either systolic BP (> 20 mmHg) or diastolic BP (>10 mmHg) withinthree minutes of standing or HUT [95]. The magnitude of orthostatic BP reductionis dependent on the baseline BP. Orthostatic hypotension is a clinical condition thatcan severely affect quality of life in individuals with SCI. After a postural changefrom supine to standing position, the venous return to the heart falls because ofgravitationally mediated redistribution of blood volume in the circulation system.The venous return fall results in a decrease in SV and CO. In response, sympatheticoutflow to the heart and blood vessels increases and cardiac vagal outflow decreases[95]. These autonomic-cardiac mechanisms increase vascular tone, HR and cardiaccontractility, and stabilize BP [95].212.3.3 Mental StressIn a mental stress test, challenge tasks are designed to elicit ANS responses in agroup of individuals (especially children) to different types of stressors: social,cognitive, sensory, and emotional. For example, the social challenge task can be astructured interview about a child?s family and friends.The cognitive challenge task can be a digit-span recitation task in which a childis asked to recall sequences of numbers. The sensory challenge task can be a taste-identification task in which two drops of concentrated lemon juice are placed ona child?s tongue, and the child is asked to recognize the taste. The emotional-challenge task can be consisted of watching an emotion-evoking movie to elicitfear in a child. The details of such a mental stress test are thoroughly explained in[88].2.4 Mathematical ModelingTo describe autonomic-cardiorespiratory regulation, a variety of mathematical mod-els using either black-box or white-box (physiology-based) approaches have beenproposed [23, 55, 96]. The nonlinearity of the baroreflex and medulla responses,the various time delays, and the number of different feedback loops create a largenumber of challenges [7]. Further, the subject-varying and time-varying propertiesof model parameters in each mathematical model have been neglected many timesto reduce challenges. By introducing a physiology-based mathematical model ofthe autonomic-cardiorespiratory regulation that includes a set of ordinary differen-tial equations, we will be able to simulate the physiology deliberately, and investi-gate 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 thesystemic circulation and consisted of three sections: a hemodynamic section sim-ulated by a Windkessel model and Starling heart, a baroreceptor section simulatedby a linear function within the range between a threshold of 90 mmHg and a sat-uration level of 150 mmHg, and an autonomic control section simulated based onthe first-order system dynamic.Saul et al. [97] proposed a mathematical model describing the closed loop22cardiorespiratory regulation to test complex links among RR, HR, and ABP. Thismodel consists of the SA node, HR baroreflex, and mechanical effects of respirationon ABP but ignores two significant aspects of hemodynamic regulation: the effectof modulation of TPR via the baroreflex and the influence of cardiopulmonary re-ceptors. Further, it describes the relation between all physiological variables byusing the frequency analysis technique.In 2003, Ursino and Magosso [66] proposed a mathematical model of short-term cardiovascular regulation to investigate the reliability of using HRV to studythe action of the autonomic regulatory mechanisms (vagal and sympathetic). Theproposed mathematical model included the pulsating heart, the systemic and pul-monary circulation, the mechanical effect of respiration on venous return, twogroups of receptors (arterial baroreceptors and lung-stretch receptors), the sym-pathetic and vagal efferent branches, and a very low-frequency vasomotor noise.Fowler andMcGuinness [7] proposed a nonpulsatile lumped-parameter5 model,which consists of two coupled differential equations with nonlinear and delayeddynamic interactions, each of which describes the dynamics of HR and BP regu-lation. The pulmonary system and the small delay of the PNS were neglected inthis work. The sensitivity of Mayer waves to sympathetic delay and gain, to sym-pathetic control of peripheral resistance, and to sympathetic control of HR wereexplored. This model is an extension of the mathematical model introduced byOttesen [16] with an added intrinsically controlled HR, and baroreflex control ofperipheral resistance.Ringwood and Malpas [96] developed a nonlinear model based on a linearfeedback model comprising delay and lag terms for the vasculature, and a linearproportional derivative controller and an amplitude-limiting sigmoidal nonlinear-ity, which could belong to either the neural controller or the vasculature itself. Theyshowed that variations in the nonlinearity characteristics may account for growthor decay in the BP oscillations as well as situations where the oscillations can thor-oughly disappear. Further, they studied a BP oscillation between 0.1 Hz and 0.4 Hzpotentially caused by a resonant feedback in the baroreflex loop.5The lumped parameter model simplifies the description of the behavior of spatially distributedphysical systems into a topology consisting of discrete entities that approximate the behavior of thedistributed system under certain assumptions.23Cavalcanti [23] used bifurcation theory in nonlinear systems to explain the highsensitivity of the HR oscillatory pattern to model parameter changes, specificallyparameter changes in the arterial baroreflex model. In this work, the basic mech-anisms that generate HRV such as the systemic circulation, a non-pulsatile cardiacpump with a constant SV 6 and nonlinear negative feedback simulating an arterialbaroreflex closed-loop control of the HR were studied. The proposed model of theshort-term autonomic control (i.e., the arterial baroreflex) consists of two distinctdelayed branches of SNS and PNS (2.8s and 0.8s). Dynamic linking between themean ABP and mean aortic flow is described based on the classic three-elementWindkessel model, and aortic flow is expressed as SV times HR.Seidel and Herzel [41, 98] proposed a hybrid model to capture both beat-to-beat and continuous dynamics of the autonomic-cardiorespiratory regulation andto investigate its dynamic properties. The mathematical model consists of delaydifferential equations that can describe physiological rhythms on timescales fromfractions of a second to a few minutes. Since chemoreceptors, temperature regu-lation, dynamics of renal hormones, and the circadian cycle were not included inthe model, the long-term variations in the physiological variables cannot be stud-ied. They showed that an increase of the delays in conveying SNS signals leads viaHopf bifurcation to the HR oscillations (called Mayer waves).Ottesen [16] extended a model of uncontrolled CVS by adding an explicit mod-eling of the baroreflex-feedback mechanism to investigate the chronotropic effect(HR regulation) and the inotropic (ventricle contractility regulation). Besides, thesystem stability with special attention to the effect of the value of the time de-lay was studied. A well-established physiological theory was used in this work.The introduced model of the baroreflex-feedback mechanism inserted some non-linearity to the model as well as a time delay. Moreover, the CVS was simulatedusing an expanded Windkessel model. In this work, Ottesen showed that the time-delay enforced some instability to the system, while the exact location of instabilitywindows were sensitive to the values of other parameters in the model.In 2006, Olufsen et al. [99] developed a mathematical model describing HRdynamics as a function of BP during postural change from sitting to standing. This6SV is the amount of blood pumped by the left ventricle of the heart in one contraction.24model ignored the BP feedback impacts on HR changes and regulation of TPR,vascular tone, and cardiac contractility. The introduced mathematical model is di-vided into four sub-models connected in series. The first sub-model is an afferenttrigger model, which uses the finger BP as an input to predict the firing rates ofbaroreflex afferent fibers. The second sub-model, representing the Central Ner-vous System (CNS), uses baroreceptor afferent nerve activity as an input to pre-dict sympathetic and parasympathetic firing in response to the rate of change ofthe mean BP.The third sub-model uses sympathetic and parasympathetic responsesas an input to predict concentrations of the neurotransmitters norepinephrine andacetylcholine. The fourth sub-model, the effector model, uses concentrations ofneurotransmitters as an input to predict HR.2.5 ConclusionIn this chapter, we briefly reviewed the physiological background of autonomic-car-diorespiratory regulation by focusing on the baroreflex mechanism. We then de-scribed some standard data-driven measures of autonomic-cardiac reactivity, in-cluding RSA and PEP, as well as several standard clinical tests to study autonomic-cardiac reactivity. At the end, we provided an extensively review of mathemati-cal models to describe autonomic-cardiorespiratory regulation that have been pro-posed in the literature. It has been observed that a physiology-based and closed-form mathematical model of autonomic-cardiac regulation has not been studiedproperly in the past. Considering that a physiology-based and closed-form math-ematical model of autonomic-cardiac regulation is developed in this work. Thismathematical model is used to investigate the system stabilty using an analytical,rather than numerical, stability analysis algorithm with both accuracy and compu-tational efficiency. This model is also used to develop an artificial bionic baroreflexfor treatment of baroreflex failure.25Chapter 3Subject-Specific Monitoring ofAutonomic-Cardiac RegulationAutonomic-cardiac regulation operates through interactions between the ANS andthe CVS. The ANS dictates homeostasis in the CVS in order to maintain adequateblood flow to deliver oxygen and nutrients to organs by adjusting its effectorsagainst internal (e.g., orthostatic hypotension) and external (e.g., hemorrhage) per-turbations [41?43]. Specifically, the ANS adjusts BP, CO, and HR using differentmechanisms, e.g., adjusting Sympathetic Nerve Activity (SNA) and Parasympa-thetic Nerve Activity (PSNA) on sinoatrial node, cardiac contractility, and periph-eral resistance [17, 23, 51]. In particular, SNA and PSNA are controlled to maintainhomeostasis in the CVS against physical and/or emotional stressors acting on theCVS [16, 56]. In this regard, autonomic-cardiac regulation is closely linked to car-diovascular disorders. Indeed, it has been suggested that the capacity of autonomic-cardiac regulation (i.e., a measure of sympathovagal balance [90]) is an importantpredictor of an individuals?s health outcome [90, 100].Subject-specific monitoring of autonomic-cardiac regulation has the potentialto improve current treatments of autonomic-cardiac disorders such as chronic drug-resistant hypertension and SCI. For example, the capacity of autonomic-cardiacregulation can be monitored and used to reduce excessive intake of anti-hyperten-sive drugs in individuals with chronic drug-resistant hypertension. The capacityof autonomic-cardiac regulation is also a criterion used to assess the severity of26injury 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 ofautonomic-cardiac regulation can be used to improve the current classification sys-tems for the Paralympics to ensure a fair competition among Paralympians [31].In addition, autonomic-cardiac regulation monitoring can be used to categorize theseverity of the spinal cord injury, in regard to HR and BP regulation, with betteraccuracy. It can also be beneficial to assign proper special care to individuals withchronic hypertension as well as SCI.Several data-driven indices of autonomic-cardiac regulation, including the RSAand 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] onthe cardiac cycle, respectively. However, there are critical limitations to the use ofRSA and PEP measures [86, 102]. For example, RSA is not a good measure of PSNAon HR during mechanical ventilation or severe physical activity. RSA has limitedcapability to differentiate the inter-individual differences of the PNS [86]. PEP canonly indicate the SNA on the heart, but it cannot be used to assess the sympatheticoutflows to blood vessels. More importantly, these two measures are both calcu-lated based on the heart rhythm; however, previous investigations (e.g., [67]) showthat the SNA and PSNA may be better estimated by looking into multiple effectormechanisms.Subject-specific and model-based estimation techniques to identify and mon-itor autonomic-cardiac variables including blood flow and blood pressure havedrawn attention [55, 103?108]. Nevertheless, only a small number of studies haveaddressed identification of the subject-specific, time-varying model parameters inautonomic-cardiac regulation by using model-based estimation techniques (e.g.,[99]).The objective of this study is to develop and validate a model-based approachto subject-specific monitoring of autonomic-cardiac regulation. Note that the pro-posed method does not rely on the dataset containing invasive CO measurement, asdiscussed in this chapter. Moreover, as explained in Section 3.1.5, the use of COmeasurement in the proposed method is not necessary; however, it will increasethe parameter estimation accuracy. The proposed approach allows us to moni-tor temporal changes in autonomic-cardiac regulation by continuously identifying27time-varying changes in the autonomic-cardiac model parameters, including SNAand PSNA on the heart (modulating HR) and SNA on the arterial tree (modulatingperipheral resistance). The validity of the proposed approach was tested by usinga number of experimental data from the MIMIC (Multiparameter Intelligent Mon-itoring in Intensive Care) database and the orthostatic hypotension tests performedin the Center for Hypotension at New York Medical College.3.1 Methods and AlgorithmWe used a physiologically-based mathematical model of autonomic-cardiac regu-lation described by a set of coupled nonlinear and delayed differential equations.The mathematical model consists of 12 subject-specific parameters (VH , ?H , ? , P0,?0, ?V , ? , Ca, ? , ?H , H0, R0a; see Table 3.1) and two outputs: HR and BP. Thismodel was chosen for its relative simplicity, which is crucial for successful sys-tem identification with routinely available clinical measurements. However, it isemphasized that the general idea constituting the proposed approach is applicableto more complicated physiologically-based autonomic-cardiac regulation modelsupon the availability of additional measurements. Figure 3.2 illustrates the aspectsof autonomic-cardiorespiratory regulation introduced in Chapter 2 that were stud-ied in this chapter.Given the task of identifying complex autonomic-cardiac regulation using thelimited information included in routine clinical measurements, it was determinedthat only high-sensitivity parameters (whose changes significantly affect the sys-tem 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 modelparameters into two groups, high-sensitivity and low-sensitivity groups, accordingto the physiology underlying autonomic-cardiac regulation and the significance ofeach model parameter in terms of its impacts on the system outputs.Then, a system identification problem formulated as a nonlinear optimizationwas solved to estimate high-sensitivity model parameters associated with autonomic-cardiac regulation; whereas, low-sensitivity parameters were fixed at their nominalvalues. The high-sensitivity parameters can be estimated properly by using HR andBP measurements, which can be non-invasively obtained in real clinical practice.28In addition to HR and BP, CO obtained using a direct measurement or an indirectanalysis can be helpful in order to increase the fidelity of the estimated parameters.3.1.1 Experimental DatasetWe used experimental data from the MIMIC database and the orthostatic hypoten-sion tests, each of which is described in detail below.The MIMIC dataset is described in detail in [47] and is freely available on thePhysioNet website [109]. It contains multiple physiologic signals of 121 subjectsrecorded from monitors in the intensive care units (ICUs) at the Beth Israel Hospi-tal, Boston, MA [110]. In this dataset, the measured data of each subject usuallycontains ECG signals recorded by surface ECG leads and sampled at 500 Hz andABP signal recorded by invasive radial artery catheterization and sampled at 125Hz [110]. The data also contain several signals including systolic, diastolic, andmean ABP as well as beat-to-beat HR, which are computed and sampled at 0.9765Hz [110]. The CO 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 extractedand subsequently used for analysis.The orthostatic hypotension dataset used in this study were collected from theCenter for Hypotension at New York Medical College and are described in [46].A single lead ECG, beat-to-beat continuous BP, respiratory plethysmography andcapnography recordings as well as a Modelflow estimate of beat-to-beat CO (usinga proprietary arterial pulse contour method) are included for each subjects under-going head-up tilt table testing (upto 70?). We used experimental data from twosubjects to establish initial proof-of-concept of the proposed approach to monitor-ing autonomic-cardiac regulation.3.1.2 Mathematical ModelWe adopted a model of autonomic-cardiac regulation proposed by Ottesen [16] andFowler and McGuinness [7]. This model is schematically shown in Figure 3.1, andis described by two differential equations Equation 3.1-Equation 3.2. The modelconsists of two coupled nonlinear and delayed differential equations, each of which29Figure 3.1: Schematic diagram of the autonomic-cardiac regulation model.describes the dynamics of HR and BP regulation:H?(t) = ?HTs1+ ? Tp?VHTp+?H(H0?H(t))(3.1)P?(t) = ? P(t)R0a(1+?Ts)Ca+ H(t)?VCa, (3.2)where H is HR and P is BP. Ts = g1(P(t? ?)) is the sympathetic control with thestrength ?H on the heart and ? on the peripheral resistance, and Tp = 1?g1(P(t)) isthe parasympathetic control with the strength VH on the heart, where g1(x) = 11+x4 .Note that Ts (sympathetic control function) and Tp (parasympathetic control func-tion) are dependent of BP, and the time delay associated with sympathetic pathwayis denoted by ? . 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 onthe physiologic mechanisms underlying autonomic-cardiac regulation and there-fore is equipped with parameters having physiologic implications that dictate es-sential short-term regulation mechanisms of HR and BP such as baroreflex controlof HR and peripheral resistance. The definitions and nominal values of the param-eters 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, Tsand Tp with any explicit algebraic definition must always satisfy the followingproperties [16]:? 0< Ts < 1; Ts ? 1 for P(t? ?) small, and Ts ? 0 for P(t? ?) large30Figure 3.2: An extensive block-diagram model of autonomic-cardiorespiratory regulation [see Chapter 2] with emphasis on theparts studied in this chapter. The shaded parts are not described in themathematical model Equation 3.1-Equation 3.2.? 0< Tp < 1; Tp ? 0 for P(t) small, and Tp ? 1 for P(t) large? ?Tp?P > 0 and?Ts?P < 0; i.e., Ts and Tp are monotonic functionsThough the model g1(x) = 11+x4 may reproduce baroreflex activity with accept-able accuracy, it has limited capability to be adapted at the individual level sinceit does not involve any tunable parameters. In this regard, we replaced it by thewell-known sigmoid function [16, 18, 111?113], which includes parameters to rep-resent inter-individual differences in baroreflex activity. That is, g1(P) = 11+P4 in(Equation 3.1)-(Equation 3.2) is replaced with g2(P) = 1??(P). ?(P) is defined31Table 3.1: Parameters in the mathematical model of autonomic-cardiac regu-lation [7].Parameter Definition Nominal ValueCa arterial compliance 1.55 mlmmHg?1R0a minimum arterial resistance 0.6 mmHgsml?1?V stroke volume 50 mlH0 intrinsic HR 100 min?1? sympathetic delay 3 sVH parasympathetic control of HR 1.17 s?2?H sympathetic control of HR 0.84 s?2? sympathetic effect on Ra 1.3? parasympathetic damping of ?H 0.2?H relaxation time 1.7 s?1as follows:?(P) = 11+ e??0(P?P0) 50? P? 200. (3.3)The sigmoid function ?(P) is characterized using two variables, setpoint P0 andsensitivity ?0. In contrast to the Hill function ( 11+P4 ) which lacks physiologicalimplications, the sigmoid function can be easily mapped to human physiology. Forexample, ?0 shows the amount of baroreflex compensatory response against BPperturbation at P0 [67]. Since the maximum slope of the sigmoid function occursat 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 AnalysisSince the number of unknown parameters is significantly greater than the num-ber of independent equations in the autonomic-cardiac regulation model (i.e., anundetermined system), complete estimation of subject-specific model parameterswithout some constraints is not likely to be feasible [103, 114, 115]. To makethe parameter identification feasible, we can fix some parameter values that areless relevant to model predictions by applying some a priori physiological knowl-edge or using sensitivity analysis. We can also use inequality constraints on high-32sensitivity parameters imposed by a feasible physiological range to effectively limitthe range of values for the solutions on unknown parameters [114, 116]. To system-atically select a subset of the model parameters amenable to identification from theobservation of system outputs, we performed a sensitivity analysis on the modelparameters and investigated the impacts of each model parameter on the systemoutputs.It is noted that H0 and R0a were excluded from this analysis, since they are notexpected to vary much within the time scale of interest (from hours to days) withinan individual. Indeed, H0 denotes the intrinsic or ?denervated? HR, which is notexpected to change much in an individual. In addition, R0a denotes the minimumTPR when the sympathetic excitation to the arterial tree (i.e., vasoconstriction) isminimal, which should be characterized mostly by the geometry (and properties tosome extent) of the arterial vessels that is not supposed to vary significantly withinthe time scale of interest. In this work, therefore, H0 and R0a were classified intothe category of invariant parameters. Thus, the nominal values listed in Table 3.1were assigned to H0 and R0a during the system identification procedure.We then classified the remaining model parameters into high-sensitivity andlow-sensitivity groups based on the results of the sensitivity analysis, thereby iden-tifying a subset of parameters with significant impact on the system outputs thatmust be individualized by the system identification procedure. Essentially, evena small change in these high-sensitivity parameters yields a large change in out-puts, whereas the outputs are not significantly impacted by (even large) changes inlow-sensitivity parameters.Traditionally, parametric sensitivity in dynamic systems is analyzed in the fre-quency domain [117]. However, the frequency-domain technique is not applicableto the autonomic-cardiac regulation model Equation 3.1-Equation 3.2, mainly dueto the nonlinearities in the model. To resolve this challenge, this work seeks tocarry out sensitivity analysis in the time domain. To this aim, the sensitivity func-tions for HR and BP are defined as follows:SH(t,? j) =????H(t,? j)?H(t,? j,0)? j ?? j,0????? ? jH(t,? j)(3.4)(3.5)33SP(t,? j) =????P(t,? j)?P(t,? j,0)? j ?? j,0????? ? jP(t,? j)(3.6)In Equation 3.4-Equation 3.6, SX(t,? j); X = H,P is the instantaneous sensitivityof X at time t due to perturbation of parameter ? j. In other words, SH(t,? j) andSP(t,? j) represent percent changes in HR and BP at time t due to a certain per-centage perturbation of parameter ? j from its nominal value. The total sensitivityfunction Equation 3.7 is obtained by combining the sensitivity functions of bothsystem outputs, i.e., HR (H) and BP (P):S(t,? j) =SH(t,? j)+SP(t,? j)2, (3.7)where we decided to use equal weights of 0.5 to both SH and SP since we aim toanalyze autonomic-cardiac regulation as a whole rather than BP or HR regulationseparately. Due to the nonlinearity in the autonomic-cardiac regulation model,the S(t,? j) can assume different values depending on the amount and direction ofperturbations given to the independent variable ? j. To develop a robust sensitivitymetric against variations in magnitude of perturbations in the parameter ? j, weelaborated on the sensitivity function Equation 3.7 by considering the variation in? j up to +/-50% in 1% increments, which yields the sensitivity metric as a functionof time Equation 3.8:S j(t) =?????32 ? j,0?? j= 12 ? j,0S2 (t,? j) (3.8)Finally, a scalar metric S j in Equation 3.9 is calculated by aggregating the sen-sitivity values S j(t) in time to obtain the overall sensitivity of a particular parameteron the system output:S j =????t f inal?tinitialS2j(t) (3.9)The parameters are classified into high-sensitivity and low-sensitivity groupsbased on the values of S j.343.1.4 System IdentificationTo estimate subject-specific high-sensitivity parameters in Equation 3.1-Equation 3.2,a system identification method was developed based on an optimization problemminimizing the normalized L1-error between measured versus model-estimated HRand BP signals. The error function (i.e., the objective function) was specified as fol-lows:J = EP+EH2 ; EX =n?t=0????Xs(t,M)?Xm(t)Xm(t)????, (3.10)where Xm(t) and Xs(t,M) (X =H,P) are measured and model-estimated output sig-nals, respectively, and M is the set of high-sensitivity parameters in the autonomic-cardiac regulation model, i.e., M = {VH ,?H ,? ,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 er-ror function Equation 3.10 is minimized, while low-sensitivity parameters as wellas H0 and R0a were fixed at their corresponding nominal values (Table 3.1). Theoptimization problem was solved using the fmincon routine with an active-set al-gorithm in the MATLAB Optimization Toolbox [118], which finds the constrainedminimum of the multivariable nonlinear scalar function J in Equation 3.10 by us-ing the Quasi-Newton approximation that determines the search direction using anapproximation of the Hessian matrix during each optimization iteration[118]. Theset of optimized high-sensitivity parameters minimizing the error function wereused as their optimal estimates associated with the corresponding data segment.To perform system identification for the first data segment, the high-sensitivityparameters (VH , ?H , ? , P0, and ?V ) were initialized by assigning random val-ues from a uniform distribution in the neighborhood (+/- 10%) of their respectivenominal values. In all of the remaining data segments, the high-sensitivity param-eters were initialized by the corresponding estimates in the previous data segment.We assumed that the high-sensitivity model parameters vary slowly within each30s-long segment, so that they can be approximated as constants within each datasegment. In fact, we assumed that the hemodynamic state of the subject is stable ineach 30s interval, and thus the model parameters to be identified can be regarded as35constants. If a data segment involves abrupt changes in physiologic and/or mentalstates, the model parameters may represent the ?average" state corresponding tothe data segment.The model-estimated HR and BP signals were calculated by solving the modelequations Equation 3.1-Equation 3.2 on the interval [0 s,30 s] using the estimates ofhigh-sensitivity model parameters in each 30s-long segment during the optimiza-tion process. In each segment, the model-estimated HR and BP in the previous seg-ment were assigned as initial conditions for HR and BP, whereas the first measuredHR and BP samples were used as initial conditions to solve model-estimated HRand BP in the first segment. In order to prevent the divergence of high-sensitivitymodel parameters out of the physiologically relevant range during the course ofthe optimization procedure, the range of each model parameter was constrained asfollows:? j,Nom2< ? j < 2? j,Nom; ? j ?M, (3.11)where ?Nom is the nominal value of ? taken from Table 3.1.3.1.5 ValidationTo establish the performance of the proposed approach, it is necessary to evaluatethe accuracy and repeatability of the system identification procedure. To evaluatethe accuracy of the parameter estimates, the proposed system identification methodwas first applied to the idealized data. The idealized data are a set of 30s-long sim-ulated HR and BP signals under the idealized setting with nominal model parametervalues in the absence of any structural uncertainty (i.e., model mismatch). We per-formed a total of 500 system identification trials, which resulted in 500 sets ofhigh-sensitivity parameter estimates. For each system identification trial, the high-sensitivity parameters were randomly initialized in the neighborhood (+/- 50%) oftheir corresponding nominal values while the low-sensitivity parameters were fixedat their nominal values. The accuracy of the system identification method was as-sessed by examining the distributions of the index Equation 3.12, which is the ratio36Figure 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.IEval? j =?Estj?Nomj, 1? j ? 12 (3.12)To evaluate the repeatability of the system identification method, it was applied50 times to each 30s-long segment of the experimental data corresponding to foursubjects obtained from the MIMIC dataset (case numbers 476, 486, 289 and 477).For each 30s-long data segment, 50 system identification trials were carried outwith 50 distinct, randomized initial conditions for the high-sensitivity parameters.The mean and standard deviation of the estimated high-sensitivity parameters as-sociated with each 30s-long data segment were studied to assure the repeatabilityof 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 assess37Table 3.2: Sensitivity-based parameter classification.High-sensitivity Low-sensitivity InvariantP0 ? H0?V Ca R0aVH ??H ?0? ?Hthe benefit of using CO 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 ex-perimental data in the absence/presence of CO data. In case the CO data wereavailable, the SV (?V ) was not estimated but was calculated directly by dividingCO by HR, thereby eliminating uncertainty associated with ?V . Otherwise, it wasfixed at its nominal value in the absence of CO data. Our expectation was that theuse of CO measurement may lead to better model-estimated HR and BP signals, andaccordingly, the estimates of the autonomic-cardiac model parameters with betteraccuracy.3.2 Results and Discussion3.2.1 Sensitivity AnalysisThe overall sensitivity metric S j of the autonomic-cardiac model parameters areillustrated in Figure 3.5. The model parameters were classified into high-sensitivity(P0, ?V ,VH , ?H , and ?) and low-sensitivity parameters (? ,Ca, ? , ?0, and ?H) basedon their overall sensitivity metric Equation 3.9 (see Figure 3.5). It is noted that allthe parameters pertaining to ANS (VH , ?H , ? and P0) were assigned to the high-sensitivity group as anticipated. These high-sensitivity parameters were used to fitthe model-estimated HR and BP to their measured counterparts during the systemidentification procedure.38Figure 3.4: Distribution of the index IEval? j for the estimated high-sensitivityparameters in a set of 500 idealized simulations.Since the sensitivity analysis was performed in the neighborhood of the nomi-nal values shown in Table 3.1, the classification results obtained based on S j maybe affected by the nominal values. To investigate the effect of perturbations inthe nominal values on S j and to assess the consistency of the parameter classifica-tion strategy employed in this chapter, we repeated the sensitivity analysis for 100sets of distinct nominal values. Each set contained nominal values randomly se-lected from +/-20% intervals in the neighborhood of the associated nominal valueslisted in Table 3.1. Figure 3.5 shows the mean and standard deviation of 100 calcu-lated S j values associated with each model parameter. It can be concluded that theclassification shown in Table 3.2 would not be affected significantly by reasonableperturbations in the nominal model parameter values.39Figure 3.5: The overall sensitivity (mean and standard deviation) ofautonomic-cardiac model parameters over 100 sensitivity analysis runswith nominal values selected from +/-20% the associated nominal val-ues introduced in Table 3.1.3.2.2 System Identification1) Idealized Data: Overall, the proposed system identification procedure performedwell. Figure 3.4 shows the distribution of the index Equation 3.12 for the estimatedhigh-sensitivity parameters (VH , ?H , ? and P0) obtained from the 500 system iden-tification trials on the idealized data. The distributions corresponding to VH , ?H ,? and P0 are mostly centered around the unity (see Figure 3.4), suggesting thathigh-sensitivity parameters are accurately estimated based on our proposed systemidentification procedure even when the low-sensitivity parameters were fixed attheir nominal values. Since the system identification could accurately estimate theautonomic-cardiac model parameters in the idealized dataset, its potential to accu-rately estimate the model parameters in the experimental dataset can be regarded40as promising.2) MIMIC Dataset: The top panels of Figure 3.6-Figure 3.9 depict the mea-sured signals (BP, HR, and CO) for 4 individuals taken from the MIMIC dataset,and the bottom panels depict the baroreflex-modulated SNA and PSNA (VHTp, ?HTs,and ?Ts) derived from the estimated model parameters for the corresponding fourindividuals, where the solid line shows an average value of 50 estimated baroreflex-modulated SNA and PSNA, whereas the shaded area indicates the correspondingstandard deviation. The relatively small standard deviation (which suggests theuncertainty related to system identification) supports the repeatability of the pro-posed system identification procedure.We further scrutinized the experimental results to examine whether the identi-fied autonomic-cardiac model parameters are reasonable based on a priori knowl-edge on the behavior of autonomic-cardiac regulation as follows.Consider Figure 3.6. First, gradual decrease in BP during t=100 s-500 s isthe consequence of gradual decrease in ?Ts (baroreflex-modulated SNA on arterialtree), while a sudden drop (??5%) of CO at t=360s is compensated by the increasein ?Ts at t=360 s to maintain stable HR and BP. Moreover, an abrupt increase inHR (? +25%) at around t=1000 s ( 3.6a) is caused by an abrupt increase in ?HTs(baroreflex-modulated SNA on the heart) at t=1000 s. Note that HR and BP aredependent variables according to the model (Equation 3.1)-(Equation 3.2) but COis an independent variable.Now consider Figure 3.7. At t=1740 s, a sudden drop (?-30%) in the measuredCO is observed. However, there is no specific change in HR and BP. Therefore, itis reasonable not to observe any changes in ?HTs and VHTp (baroreflex-modulatedPSNA on the heart), but to observe a sudden increase in ?Ts to maintain stabilityof the system by increasing TPR. There are abrupt increases in BP (?+30%) andHR (?+5%) at t=2700 s, which are caused by the increase in ?Ts and decrease inVHTp. Overall, these results suggest, to a large extent, the physiologic relevanceand consistency of the proposed system identification procedure. Similar interpre-tation could be made for the remaining datasets to which the proposed approachwas applied.Finally, it is important to note that SNA and PSNA may or may not act antago-nistically. Indeed, ?HTs and VHTp at t=1000 s in Figure 3.6b act antagonistically,41whereas the opposite pattern can be observed in Figure 3.7b. For example, a largereduction in VHTp occurs at t = 2700s, which is accompanied by a slight reductionin ?HTs. 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 ofstudies performed by Cacioppo and Berntson shows that psychological processesand higher neuro-behavioral substrates can result in independent activation or evenco-activation of SNA and PSNA [12, 90].3) Orthostatic Hypotension Dataset: During the postural changes from supineto 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 aconsequence, blood volume returning to the heart in each cardiac cycle (venousreturn) falls resulting in diminished SV and CO. In response to this decrease inCO, SNA on the heart increases while PSNA on the heart increases decreases tostabilize ABP by modulating HR and SV. The activation of SNA on the arterial treeis different for standing and tilting situations, because the body?s skeletal musclecontraction helps to maintain venous return in an appropriate range during standingwhereas muscle contraction has no role during tilting.Figure 3.10 shows the results of applying the proposed system identificationprocedure to the orthostatic hypotension test data of two subjects. In regards tothe first subject, the two upper panels of Figure 3.10a show that the subject isin the supine position starts being tilted at t=400 s, and is then brought back tothe supine position at t=1060 s after the tilting maneuver ends. The three lowerpanels in Figure 3.10a show that ?HTs increases and VHTp decreases to increaseHR 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 inVHTp and a decrease in ?HTs to decrease HR and an increase in ?Ts to increasemean ABP through TPR. The three lower panels in Figure 3.10a clearly show thatthe proposed approach could estimate successfully these anticipated changes inSNA and PSNA acting on both the heart and the arterial tree. The results shown forthe second subject in Figure 3.10b exhibit trends consistent with those of the firstsubject and they both support the potential of the proposed approach in monitoringthe autonomic-cardiac regulation.42(a) Measured vs. model-estimated signals: BP, HR, and CO(b) Identification results: ?Ts, ?HTs, and VHTpFigure 3.6: Experimental results from MIMIC dataset (Case No.: 476).43(a) Measured vs. model-estimated signals: BP, HR, and CO(b) Identification results: ?Ts, ?HTs, and VHTpFigure 3.7: Experimental results from MIMIC dataset (Case No.: 486).44(a) Measured vs. model-estimated signals: BP, HR, and CO(b) Identification results: ?Ts, ?HTs, and VHTpFigure 3.8: Experimental results from MIMIC dataset (Case No.: 289).45(a) Measured vs. model-estimated signals: BP, HR, and CO(b) Identification results: ?Ts, ?HTs, and VHTpFigure 3.9: Experimental results from MIMIC dataset (Case No.: 477).46(a) Subject I(b) Subject IIFigure 3.10: System identification results on the orthostatic hypotensiondataset to monitor SNA and PSNA during a tilt test. The two top pan-els show the measured vs. model-estimated HR and BP signals, andthe three bottom panels show the identification results: ?Ts, ?HTs, andVHTp.473.2.3 Limitations of the Proposed ApproachDespite its promising preliminary results, this study has a number of limitations asdiscussed below.First, comparing the system identification results with and without the use ofCO measurements, the results with CO measurements incorporated into the systemidentification procedure were superior to the results without CO measurements.This means that the proposed approach may benefit from the availability of COmeasurements. Figure 3.11 shows a typical result comparing model-estimated HRand BP signals with and without CO measurements (Case No.: 289). The thermod-ilution technique (which is accepted as the gold standard CO measurement tech-nique [119]) was used in this individual to measure CO. Figure 3.11 shows thatthe accuracy of the model-estimated HR and BP signals can be improved by usingthe measured CO signal in the system identification procedure. Accordingly, it isexpected that the accuracy of the estimated baroreflex-modulated SNA and PSNA(i.e., ?Ts, ?HTs, and VHTp) will be enhanced as well. Considering that currentlyavailable techniques for direct measurement of CO with acceptable accuracy, in-cluding the thermodilution technique, are highly invasive [120], the use of CO datafor the purpose of system identification may not be practical. Non-invasive tech-niques including echo-cardiography [121], electrical velocimetry [122] and the useof pulse contour methods to estimate CO from arterial BP waveform using the mor-phological feature (e.g., [120, 123, 124]), can be considered potential alternatives.Second, the possible interdependence between model parameters was not ex-plored in this paper. For example, physiology dictates that SV (?V ) is essentiallyrelated to arterial compliance (Ca) and TPR (in particular, R0a), e.g., SV is affectedby the afterload if TPR increases. The incorporation of a priori knowledge onthe interdependence between model parameters may have improved the outcomesof system identification. Further, it is important to emphasize that the structuralcoupling among ? , Ca and R0a, and between ?V and Ca in Equation 3.2 could beavoided using the classification of model parameters used in this study (Table 3.2).Indeed, ? could be uniquely identified since R0a and Ca were fixed at their nominalvalues. Likewise, ?V could also be uniquely identified since Ca was fixed at itsnominal value.48Figure 3.11: Measured versus model-estimated HR and BP signals with andwithout the use of measured CO signal (Case No.: 289); blue are mea-sured signals, black are model-estimated signals with measured COand 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 nominalvalues. This, in turn, suggests that the classification of the autonomic-cardiac pa-rameters as presented in Table 3.2 may also depend on the values of the modelparameters (or equivalently, the underlying physiologic state). The nominal modelparameter values used in this study were well-suited for an average adult in a stableresting state. However, the model parameters may need to be re-classified when ap-plying the proposed system identification approach to subject groups under highlynon-nominal physiologic and/or mental conditions.Fourth, to make maximal use of the limited information contained in the HRand BP data, we chose to identify a subset of parameters characterizing the autonomic-cardiac regulation model that largely impacts the HR and BP signals. This ap-49proach is, in fact, not uncommon when identifying systems involving many pa-rameters (e.g., [125]). It was claimed that fixing the invariant parameters may bewell-justified physiologically (see Section 3.1.3 for details). However, the low-sensitivity parameters may vary in time, and physiologic justification of fixingthese parameters at constant values is not trivial. Regardless, the effect of fixingthese low-sensitivity parameters on the estimates of high-sensitivity SNA and PSNAparameters is expected to be non-significant, since the low-sensitivity parameterscannot alter the HR and BP signals (which are used in the system identification pro-cedure to estimate the SNA and PSNA parameters) much due to their small impacton these signals.Lastly, although this study provides an initial evidence and proof-of-conceptfor the physiologic relevance of the estimates of autonomic-cardiac regulation pa-rameters (as demonstrated by the physiologically anticipated changes in the esti-mated 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 proceduresis yet to be investigated deeply . In this regard, this study must be regarded asa preliminary feasibility study to estimate autonomic-cardiac regulation parame-ters based on easily accessible clinical measurements; additional in-depth work isnecessary before the clinical value of the proposed method can actually be claimed.3.2.4 Autonomic-Cardiac Regulation MonitoringThe model parameters of autonomic-cardiac regulation (e.g., VH , ?H , ? , and P0)and, therefore, baroreflex-modulated SNA and PSNA (?Ts, ?HTs, and VHTp) aresubject-specific, time-varying (short-term and long-term), and health-dependent.The model parameters are subject-specific because physiologic differences of theANS 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 inboth short-term and long-term periods. For example, the model parameters arecontinuously adjusted in response to physical and emotional stressors in order tomaintain the stability of vital physiologic variables such as HR and BP (short-term).Additionally, as humans age, neural reflexes become slower, thereby resulting in50larger delays in the sympathetic pathways (?) associated with older adults (long-term). We showed that autonomic-cardiac regulation is health-dependent since in-dividuals with different types of stress reactivity can be differentiated by the char-acteristic of autonomic-cardiac regulation, especially the baroreflex characteristic(Ts and Tp) [6]. Therefore, a subject-specific monitoring method for autonomic-cardiac regulation can be used to identify the underlying regulation mechanismand to diagnose the ANS- and CVS-related deficiencies [99, 126]. Further, we candeduce valuable physiologic information from autonomic-cardiac regulation moni-toring based on the variation of estimated parameters in different subjects under thesame physiologic condition, and similarly within a subject under different physio-logic conditions.The non-invasive direct measurement method for SNA and PSNA is not avail-able. Therefore, non-invasive indirect measurement methods such as electrocar-diogram-based indices (e.g., RSA and PEP), galvanic skin response method, andmodel-based measurement methods have been used to estimate SNA and PSNA.For example, the galvanic skin response method measures SNA 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 RSAshows 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 measure-ment methods for SNA and PSNA since they are calculated solely on the basis of theelectrocardiogram signal, while model-based measurement methods are developedbased on the complex physiological structure of autonomic-cardiac regulation.In this work, we therefore proposed a model-based monitoring method forautonomic-cardiac regulation using a mathematical model that takes into accountimportant physiologic parameters in the regulation mechanism. The proposedmethod can potentially be used to improve non-invasive autonomic-cardiac reg-ulation monitoring.3.3 Conclusions and Future WorkIn this chapter, we presented initial evidence and proof-of-concept for a novelsubject-specific model-based monitoring method for autonomic-cardiac regulation51Table 3.3: Statistical properties of the identified baroreflex-modulated SNAand PSNA: mean?stdCase No. VH .Tp ?H .Ts ?.Ts476 0.27? 0.06 0.40? 0.08 1.21? 0.17486 0.28? 0.13 0.62? 0.10 1.04? 0.40289 0.25? 0.03 0.45? 0.07 0.75? 0.06477 0.25? 0.09 0.64? 0.07 0.39? 0.12that uses a computationally efficient system identification method with routine clin-ical measurements: HR and BP. We used CO measurement in this chapter as pro-vided in the MIMIC dataset; however, the proposed method has not been developedwith the assumption of incorporating CO measurement (refer to Section 3.2.3).Note that some non-invasive CO estimation techniques including electrical ve-locimetry and echocardiography have also been introduced recently. The proposedmethod is effective in estimating the time-varying and subject-specific character-istics of autonomic-cardiac regulation, since it accommodates the complex natureof the regulation mechanism through a mathematical model rather than calculat-ing arguable features directly extracted from signal measurements. Our proposedmodel-based monitoring method has the potential to eliminate the limitations ofcompeting methods currently available (e.g., RSA and PEP) such as lack of inter-individual separability and lack of fidelity during mechanical ventilation or severephysical exercise.In the future, more intensive experimental validation of the proposed methodmust be performed to further assess its efficacy in monitoring autonomic-cardiacregulation. The method should also be improved by incorporating the impact ofthe respiratory system on the regulation mechanisms for HR and BP, which willconsequently lead to an enhancement in the fidelity of the system identificationresults (i.e., in terms of ?Ts, ?HTs, and VHTp). Note that the system identifica-tion technique must be revised to be applicable to the mathematical model withrespiratory effects. In addition, the proposed method must be compared with theconventional markers of SNA and PSNA, including the time-domain measures ofthe HRV. Further, the use of multi-dimensional autonomic-cardiac spaces (such asVHTp-?HTs, VHTp-?Ts, and ?HTs-?Ts) may be investigated to assess and differen-52tiate the capacity of autonomic-cardiac regulation in different individuals. Finally,an extensive clinical study must be conducted to determine the clinical usefulnessof the proposed method for diagnostic and therapeutic procedures.53Chapter 4Model-Based Stability Analysis ofAutonomic-Cardiac RegulationAutonomic-cardiac regulation operates through interactions between the ANS andthe CVS. The ANS mostly regulates involuntary organ function and maintainshomeostasis in the CVS against physical (e.g., exercise and orthostatic hypoten-sion) and psychological (e.g., fear and anxiety) stressors [41?43]. The ANS adjustscardiorespiratory parameters, including BP, HR, vascular resistance and respiratoryrate (RR), to deliver adequate oxygenated blood flow to organs in different condi-tions [45]. In general, autonomic-cardiac regulation is regarded as stable if BP andHR converge to equilibrium states after a certain amount of transient time whenit is exposed to a stressor, whereas it is considered unstable if HR and BP exhibitnon-decaying or slowly-decaying oscillations or diverge from their normal values.It is well known that undesirable changes in the dynamics of the autonomic-car-diac regulation (e.g., an excessive increase in the time delay of sensory afferentpathways) can result in an onset of instabilities in BP and HR [16, 23]. Cavalcantiet. al. [18, 23] showed that perturbations in autonomic-cardiac parameters affectthe stability of the CVS. Ottesen [16] showed that switching between stability andinstability of the CVS can occur depending on the value of the time delay associ-ated with the baroreflex feedback mechanism. He also demonstrated that complexdynamic interactions between nonlinearities and delays in autonomic-cardiac reg-ulation may cause instability [16]. Abbiw-Jackson [127] reported that an increase54in the gain of the baroreflex feedback loop controlling venous volume may causethe onset of oscillation in BP. Deboer et. al. [128] also showed that the timedelay 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 analysisof autonomic-cardiac regulation may be beneficial in improving current diagnosticand treatment methods for ANS-CVS disorders.Model-based stability analysis is useful to examine the system-level causes ofinstability and the stability margin in autonomic-cardiac regulation [23, 128]. Forexample, several model-based analyses of the baroreflex mechanism have revealedthat mechanisms underlying the baroreflex are responsible for the Mayer waves[16?18]. In another model-based analysis, it was shown that autonomic-cardiacregulation may remain stable or be driven to instability in response to changes inthe baroreflex parameters, even if the baroreflex delays remain constant, i.e., thestability of the autonomic-cardiac regulation is sensitive to both time delays andother parameters associated with baroreflex [16]. A model-based study was alsoused to show that baroreflex modulation does not promptly return to a steady statein hypertensive elderly individuals during postural change from sitting to standing[99]. Model-based analysis of autonomic-cardiac regulation was also used in inves-tigating the reliability of the heart period variability index to study the autonomicregulatory mechanisms [66]. However, to the best of our knowledge, existing re-sults using the model-based approach for stability analysis of the autonomic-car-diac regulation have limited capability for quantifying stability margins, althoughsome previous studies have qualitatively examined the impacts of parameter con-figurations on the stability margin of ANS-CVS [3, 16]. Moreover, it is crucial tomaintain a certain degree of stability margin in autonomic-cardiac regulation forindividuals with, for example, treatment-resistant hypertension.In a recent study, we developed an optimization-based system identificationapproach to characterize autonomic-cardiac regulation mechanisms based on aphysiology-based ANS-CVS model [1], which we used to conduct a preliminaryfeasibility study on the model-based stability analysis of autonomic-cardiac regu-lation [3]. In this work, we present a model-based approach to investigate the sta-bility of autonomic-cardiac regulation. A unique strength of the proposed approachis its capability to determine the stability margin of autonomic-cardiac regulation55quantitatively and computationally efficiently once the model parameter configura-tion is given. Specifically, the proposed approach quantifies the stability margin ofautonomic-cardiac regulation via two key contributions: 1) an analytical methodto determine the equilibrium states of the autonomic-cardiac regulation and 2) asystematic approach to analyze the system stability in the vicinity of the equilib-rium state. First, we validated our approach by comparing our analysis results towell-established physiological concepts; we then used this approach to explore po-tential model parameter configurations that can incur instability in autonomic-car-diac regulation. We also demonstrated that the proposed approach can determinethe equilibrium state and quantify its stability with a high level of accuracy. Thisapproach is very powerful in identifying the system-level cause of instability in theautonomic-cardiac regulation by virtue of its capability to determine the stabilitymargin associated with model parameter configurations.4.1 Methods and AlgorithmIn this section, the physiology-based mathematical model of autonomic-cardiacregulation that is used for the proposed stability analysis is described. The delay-free realization of the mathematical model is then devised to be used in furtheranalysis. The proposed stability analysis consists of three steps. In the first step, theequilibrium state of autonomic-cardiac regulation, in terms of BP and HR, is identi-fied as a closed-form steady-state solution of the mathematical model Equation 4.10presented (see Section 4.1.3). In the second step, the model of autonomic-cardiacregulation is linearized around the equilibrium state to obtain the Jacobian matrixof the system that can be used to assess its stability in the neighborhood of the equi-librium state. In the last step, the stability margin of autonomic-cardiac regulationis quantified using the eigenvalues of its Jacobian matrix (see Section 4.1.4). Wevalidated our proposed stability analysis using a simulation dataset.4.1.1 Physiology-Based Model: Delayed Differential EquationsA wide variety of mathematical models for autonomic-cardiac regulation with dif-ferent levels of complexity have been proposed in the literature. Examples includea three-element Windkessel model of CVS with baroreflex represented by a series56connection of delayed first-order linear dynamics and a sigmoid nonlinear function[23], a simple nonlinear feedback control system containing an amplitude-limitingnonlinearity added to a linear feedback model comprising delay and lag terms forthe vasculature and a linear proportional-derivative controller for the ANS [96],a set of two coupled nonlinear and delayed differential equations describing HRand BP regulation mechanisms [7], and a relatively complex model consisting ofa Windkessel model and Starling heart for the hemodynamic section, a saturatedlinear function for baroreceptor section, and a set of first-order systems for au-tonomic control section [55]. Considering that a high-fidelity, physiology-based,and closed-form mathematical model is required to develop an analytical, ratherthan numerical, stability analysis algorithm with both accuracy and computationalefficiency, we adopted a model described by Fowler [7]. The autonomic-cardiacregulation model used in this chapter is described by:H?(t) = ?HTs1+ ? Tp?VHTp+?H(H0?H(t))(4.1)P?(t) = ? P(t)R0a(1+?Ts)Ca+ H(t)?VCa, (4.2)where H is HR, and P is mean arterial BP. The definitions and nominal valuesof the parameters in Equation 4.1-Equation 4.2 are summarized in Table 4.1. Inthis model, the sympathetic and parasympathetic modulating functions generatedby the baroreflex control mechanism are denoted by Ts and Tp, respectively. Thetime delay associated with the sympathetic pathway is denoted by ? , whereas theparasympathetic delay was assumed to be negligible [37]. In this study, we ne-glected the inhibitory impact of the parasympathetic system on the sympatheticsystem by setting ? ? 0 in Equation 4.1-Equation 4.2, as it is well known that itseffect on overall autonomic-cardiac regulation is generally small [7, 16]. The math-ematical model Equation 4.1-Equation 4.2 is then rewritten as follows:H?(t) = ?HTs?VHTp+?H[H0?H(t)](4.3)P?(t) = ? P(t)R0a(1+?Ts)Ca+ H(t)?VCa. (4.4)The sympathetic and parasympathetic modulating functions Ts and Tp can be57Table 4.1: Parameters in the mathematical model of autonomic-cardiac regu-lation.Parameter Definition Nominal ValueCa arterial compliance 1.55 mlmmHg?1R0a minimum arterial resistance 0.6 mmHgsml?1?V stroke volume 50 mlH0 intrinsic HR 100 min?1? sympathetic delay 3 sVH vagal tone 1.17 s?2?H sympathetic control of HR 0.84 s?2? sympathetic effect on Ra 1.3? vagal damping of ?H 0.2?H relaxation time 1.7 s?1modeled as sigmoid functions with amplitude-limiting characteristic [96]. A sig-moid function, ?(P), is characterized by a setpoint and a sensitivity coefficient[113, 129] as follows:?(P) = 11+ e??0(P?P0) 50? P? 200. (4.5)where P0 and ?0 are the BP setpoint and the sensitivity of the baroreflex mech-anism, respectively. Substituting Ts = 1? ?(P(t ? ?))and Tp = ?(P(t))intoEquation 4.3-Equation 4.4 yields:H?(t) = ?H[1??(P(t? ?))]?VH?(P(t))+?H[H0?H(t)](4.6)P?(t) = ? P(t)R0a(1+?[1??(P(t? ?))])Ca+ H(t)?VCa. (4.7)4.1.2 Delay-Free RealizationTo alleviate analytical and computational challenges that can potentially arise in thecourse of stability analysis of autonomic-cardiac regulation, the transport delays as-sociated with the sympathetic and parasympathetic responses were replaced by ap-proximations. Note that this essentially simplifies the original infinite-dimensional58model Equation 4.6-Equation 4.7 to a finite-dimensional model. For this purpose,we employed the first-order Pade? approximation to eliminate the delayed statedvariable P(t? ?) as follows. First, a new state variable X is defined as follows:P?(t) = P(t? ?) L? P?(s) = P(s)e??s ? P(s)1? ?2 s1+ ?2 s? P?(s)? P(s)(?1+21+ ?2 s)? P?(s)??P(s)+P(s)21+ ?2 s? P?(s)+P(s)? ?? ?X(s)? 2P(s)1+ ?2s? X(s) = P?(s)+P(s)L ?1? P(t? ?) = X(t)?P(t) (4.8)which serves as the output equation relating P(t ? ?) to P(t) and X . The stateequation dictating the dynamics of X is obtained as follows:X(s)? 2P(s)1+ ?2s? X(s)+ ?2X(s)? 2P(s)L ?1? X(t)+ ?2 X?(t)? 2P(t)? X?(t)? 2?[2P(t)?X(t)](4.9)Using Equation 4.8 and Equation 4.9, the delay-free realization of the auto-nomic-cardiac regulation model Equation 4.6-Equation 4.7 can be obtained as Equation 4.10,shown below.4.1.3 Identification of Equilibrium StatesAt an equilibrium state of the system described by Equation 4.10, time derivativesof the the state variables are zero, i.e., W?(t) = 03?1, and the state variables reachtheir respective steady-state values P(t)=P(t??)=Pf ,H(t)=H f , and X(t)=X f .59W?(t) =??????H?(t)P?(t)X?(t)?????????????f1(H(t),P(t),X(t))f2(H(t),P(t),X(t))f3(H(t),P(t),X(t))??????=???????H[1??(X(t)?P(t))]?VH ?(P(t))+?H(H0?H(t))? P(t)R0a(1+?[1??(X(t)?P(t))])Ca+ H(t)?VCa2?[2P(t)?X(t)]??????(4.10)Therefore, Equation 4.10 at the equilibrium state can be rewritten into:0 = ?H[1??(X f ?Pf )]?VH ?(Pf )+?H(H0?H f)(4.11)0 = ? PfR0a(1+?[1??(X f ?Pf )])Ca+ H f ?VCa(4.12)0 = 2?[2Pf ?X f](4.13)Note that, according to Equation 4.13, X f = 2Pf and Equation 4.11-Equation 4.12reduce to a set of two algebraic equations as shown below:H f =1?H(?H[1??(Pf )]?VH ?(Pf )+?HH0)(4.14)Pf = H f ?V R0a(1+?[1??(Pf )])(4.15)which further simplifies to:H f =1?H(?H +?HH0??(Pf )[?H +VH])(4.16)Pf = H f ?V R0a(1+? ?? ?(Pf )). (4.17)Deriving closed-form solutions of the equilibrium state (H f and Pf ) from thesenonlinear equations is not trivial. Employing a numerical optimization method60using MATLAB Optimization Toolbox [118] or solving the set of nonlinear dif-ferential 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 expensivecomputational load and potential convergence to local minima.To avoid these drawbacks, we propose a method to derive a closed-form, an-alytical solution for the equilibrium states using a linearized form of Equation 4.16-Equation 4.17. To linearize the nonlinear term ?(Pf ) in Equation 4.16-Equation 4.17,?(Pf ) in Equation 4.5 can be approximated into the following piecewise linearfunction in which ?(Pf ) is replaced by a set of three linear functions representingits behavior in low, normal and high BP regions:?(P)? ?lin(P) =?????k1P+ c1; Pmin ? P? P1k2P+ c2; P1 ? P? P2k3P+ c3; P2 ? P? Pmax(4.18)where Pmin and Pmax were assigned as 50 and 200 in this work. Further, P1 and P2are estimated based on a constrained optimization that minimizes an error betweenthe sigmoid function ?(P) and its linear approximation ?lin(P). The constraintsare: 1) two lines k1P+ c1 and k3P+ c3 pass through [Pmax,1] and [Pmin,0], respec-tively, and 2) the slope of the line k2P+ c2 is equal to the slope of ?(P) at theinflection point ?2?(P)?P2 = 0, which yields k2 =??(p)?P = ?4 .According to Equation 4.18, the steady-state equations Equation 4.16-Equation 4.17can be rewritten into a set of linear equations for each region with correspondingslope ki and y-intercept ci, i ? {1,2,3}, as follows:H f =1?H(?H +?HH0? (kiPf + ci)(VH +?H))(4.19)Pf = H f ?V R0a(1+? ??(kiPf + ci))(4.20)which can ultimately be reduced to:H f = A6(A1?A2Pf)(4.21)Pf = H fA5(A3?A4Pf). (4.22)61where A1 = ?H +?HH0? ciVH ? ci?H , A2 = ki(VH +?H), A3 = 1+? ??ci, A4 =?ki, A5 = ?V R0a, and A6 = 1?H .Equation 4.21-Equation 4.22 can be reformulated into the following quadraticequation solely based on Pf :Pf = A5(A3?A4Pf)A6(A1?A2Pf)? ?? ?H f(4.23)or,aP2f +bPf + c= 0 (4.24)where a=?A4A2A5A6, b= A4A1A5A6+A3A2A5A6+1, and c=?A1A3A5A6. Theclosed-form solution for Pf is then obtained as follows:Pf1,2 =?b??b2?4ac2a (4.25)Once Pf is determined, H f can be easily calculated as a function of Pf usingEquation 4.21:H f1,2 = A6(A1?A2Pf1,2)(4.26)It is noted that, since Pf and H f are not known a priori, their candidate valuesmust be determined for the three regions specified in Equation 4.18. The threepairs of Pf and H f thus obtained must then be validated against the correspondingregions. For instance, Pf and H f determined from ?(P) = k2P+ c2 is regarded asvalid if P1 < Pf < P2.4.1.4 Stability AnalysisTo the best of our knowledge, there is no well-accepted method for global sta-bility analysis of nonlinear dynamic systems with delays, which include the au-tonomic-cardiac regulation model used in this study. However, according to theHartman-Grobman Theorem [131], stability properties of a nonlinear system inthe vicinity of an isolated equilibrium state can be determined by investigatingthe properties of its linearization in the neighborhood of the equilibrium. Note62that the equilibrium states obtained by our analysis are isolated in the sense thatthey are uniquely determined for the autonomic-cardiac regulation model onceits parameter configuration is provided. In order to exploit linear systems the-ory to solve our problem, we developed the delay-free realization Equation 4.10 ofthe delayed nonlinear autonomic-cardiac regulation model. The stability of auto-nomic-cardiac regulation can be assessed by calculating the Jacobian matrix (J) orthe state matrix of the nonlinear system Equation 4.10 at an estimated equilibriumstate W f = [H f ,Pf ,X f ]T as follows:J(W f ) =? ( f1, f2, f3)? (H,P,X)????W f=????????? f1?H? f1?P? f1?X? f2?H? f2?P? f2?X? f3?H? f3?P? f3?X????????W=W f(4.27)J(W f ) =?????????????H ??H??(X ?P)?P ?VH??(P)?P ??H??(X ?P)?X?VCa?R0aCa[1+?(1??(X ?P))]?PR0aCa???(X?P)?P[R0aCa(1+?[1??(X ?P)])]2?PR0aCa???(X ?P)?X[R0aCa(1+?[1??(X ?P)])]20 4??2????????????W=W f(4.28)Taking partial derivatives of the delay-free realization Equation 4.10 at a givenequilibrium stateW f = [H f ,Pf ,X f ]T yields the Jacobian matrix Equation 4.28 where??(P)?P |W=W f = ?0e??0(Pf?P0)[1+ e??0(Pf?P0)]2. (4.29)According to the Hartman-Grobman theorem [131], the original nonlinear sys-tem (i.e., the autonomic-cardiac regulation) is stable in the neighborhood of anequilibrium state if all the eigenvalues ?i (i = 1,2,3) of the state matrix (J) havenegative real parts, whereas it is unstable if any of its eigenvalues has a positive realpart. To quantitatively investigate conditions suggested by the Hartman-Grobmantheorem and also to calculate the stability margin of the system, we propose the63following stability margin metric, Sm:Sm = maxi=1,2,3[real(?i)](4.30)where real(?) denotes the real part of its argument, and ?i is the i-th eigenvalue. Smrepresents 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 rep-resenting the stability margin of the autonomic-cardiac regulation whose absolutevalue measures the distance between the dominant system pole and the imaginaryaxis. The system generally forfeits its stability margin, i.e., approaches to instabil-ity, as Sm becomes closer to zero.4.1.5 Simulation DataIn this study, we generated two simulation-based datasets to validate the proposedapproach for estimating the equilibrium states and analyzing the stability of auto-nomic-cardiac regulation. First, to validate our approach to estimate the HR andBP equilibrium states, we generated 100 parameter configurations for the auto-nomic-cardiac regulation model Equation 4.1-Equation 4.2 in which each modelparameter was determined randomly from a uniform distribution within 80% and120% of the corresponding nominal value (see Table 4.1). Second, to validate ourapproach to analyze the stability of autonomic-cardiac regulation, we consideredtwo distinct mental conditions: normal and stressed. The normal condition wassimulated by assigning normal parameter values listed in Table 4.1 to the modelparameters, while the stressed condition was simulated with appropriate changesin the sympathetic and parasympathetic reflex parameters. Specifically, VH wasdecreased by 50%, whereas ?H and ? were increased by 100% [50]. For eachof the mental conditions, we generated 12 sets of 100 parameter configurations.In each set only a single parameter in the autonomic-cardiac regulation modelEquation 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) nom-inal values.644.1.6 Validation of the Proposed ApproachUsing the datasets described above, the validity of the proposed approach wasexamined as follows. First, for each of the 100 parameter configurations gen-erated to validate the proposed approach for estimating the equilibrium state ofautonomic-cardiac regulation, the equilibrium state determined by the proposedapproach was compared with those obtained numerically via numerical optimiza-tion and nonlinear simulation. For this purpose, we first computed HR and BPequilibrium states using Equation 4.25-Equation 4.26 according to the proposedapproach. 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 MATLABOptimization Toolbox [118].Second, the nonlinear dynamic autonomic-cardiac regulation model Equation 4.6-Equation 4.7 was simulated with MATLAB?s delay-differential equation solver(dde23), from which HR and BP equilibrium states were determined as the steady-state values of simulated HR and BP time series obtained directly from the orig-inal nonlinear autonomic-cardiac regulation model. The fidelity of the equilib-rium states obtained from the proposed approach was assessed by its consistencywith those obtained from numerical optimization and nonlinear simulation via theBland-Altman analysis.Finally, in order to validate the proposed approach for analyzing the stabilitymargin of autonomic-cardiac regulation, the proposed analytical stability metric,Sm, was compared to an empirical metric, Sp, obtained directly from a nonlinearsimulation Equation 4.31, which was defined based on the absolute amount of fluc-tuation of BP P(t) around its steady state:Sp =30?t=1???P(t)?P(t)???P(t) , (4.31)where P(t)was calculated by solving the original nonlinear system model Equation 4.6-Equation 4.7 using the dde23 routine in MATLAB [118]. For the dataset generatedto validate the proposed approach for analyzing the stability of autonomic-car-diac regulation, the proposed metric, Sm, was calculated using Equation 4.28 andEquation 4.30. The empirical metric, Sp, was calculated using Equation 4.31.65(a) BP(b) HRFigure 4.1: Comparison of equilibrium states estimated using the proposedanalytical approach Equation 4.25 against numerical optimization (leftpanel) and nonlinear simulation (right panel).In addition to comparing Sm with Sp, the validity of the proposed stability met-ric was further assessed using a priori knowledge of the relationship between auto-nomic-cardiac regulation model parameters and its stability. In particular, we testedwhether or not the proposed stability margin metric deteriorated as parasympathetictone (VH ) decreased and/or sympathetic tone (?H) increased, as reported in the lit-erature [50]. We also tested if the stability margin metric degrades as sympatheticdelay is increased [18].66Figure 4.2: Two metrics for stability margin Sm and Sp over changes of amodel parameter from 50% to 200% of its nominal value for a healthyphysiological condition with and without stress. Sm is the blue solidline; Sp is the green dashed line. A normal condition (i.e., VH , ?H , and? were fixed at their nominal values).67Figure 4.3: Two metrics for stability margin, Sm and Sp, over changes of amodel parameter from 50% to 200% of its nominal value for a healthyphysiological condition with and without stress. Sm is the blue solidline; Sp is the green dashed line. A stressful condition (i.e., a 50% lowerVH and 100% higher ?H and ? compared to their nominal values).684.2 Results and Discussion4.2.1 Identification of Equilibrium StatesThe Bland-Altman analysis clearly indicates that the equilibrium states determinedby the proposed analytical approach are highly consistent with those obtained fromnumerical optimization and nonlinear simulation. Specifically, with respect to thenonlinear simulation results, bias and 95% confidence interval associated withBP equilibrium states were only 1.0mmHg and 0.4mmHg, respectively, and biasand 95% confidence interval associated with HR were both less than 0.5bpm (seeFigure 4.1). Therefore, we can conclude that the proposed analytical approachcould estimate equilibrium states very accurately with relatively low computa-tional burden, when compared with those estimated by numerical optimization andnonlinear simulation. Note that the low computational burden of the proposedmethod will be observed during high-dimensional stability analysis of the time-varying mathematical model. Further, the proposed approach does not suffer fromissues associated with local minima since it is not an optimization-based method.Finally, the proposed approach computes the equilibrium states associated witheach parameter configuration independently of the dynamic characteristics of auto-nomic-cardiac regulation, which often cause problems when nonlinear simulationis used to obtain equilibrium states of slowly varying systems.4.2.2 Proposed Stability MetricsFigure 4.2-Figure 4.3 show the behaviour of Sm and Sp calculated for the datasetthat we generated to validate the proposed approach for analyzing the stability ofautonomic-cardiac regulation. Using this dataset with a wide range of variationin each parameter (i.e., 50% to 200%), we can investigate the pure effect of asingle model parameter on the stability margin of autonomic-cardiac regulationin different physiologic conditions. Sm and Sp values in response to variationsin a single parameter in the autonomic-cardiac regulation model are depicted inFigure 4.2-Figure 4.3 for normal and stressful conditions. Overall, the tendency inbehaviors of the proposed stability margin metric, Sm, and the empirical metric, Sp,were qualitatively consistent in most cases. It is noted that although inconsistency69in pattern between Sm and Sp was observed for Ca in the normal condition, it wasregarded as noncritical since the sensitivity of the metrics toCa was relatively smallcompared with those to other parameters. The discrepancy in pattern between Smand Sp associated with ? is due to the fact that ? is set to zero in the model used todevelop the proposed approach to stability analysis, whereas its value is not zero inthe model used for simulating autonomic-cardiac regulation. Therefore, the effectof a single model parameter on the stability of autonomic-cardiac regulation canbe examined by analyzing the stability margin metric, Sm, over a desired parameterspace.Figure 4.2-Figure 4.3 also suggest that the proposed stability metric, Sm, ex-hibits behavior consistent with well-known physiologic knowledge on the rela-tionship between the stability of autonomic-cardiac regulation and sympathetic/-parasympathetic tones and delays. In particular, the stability margin of autonomic-car-diac regulation is expected to decrease as cardiac vagal tone (VH ) decreases orcardiac sympathetic tone (?H) increases [90]. Figure 4.2-Figure 4.3 show that themagnitude of Sm decreases with decreasing VH and increasing ?H in both nomi-nal (Figure 4.2) and stressful (Figure 4.3) conditions, as anticipated. It is knownthat the stability margin of autonomic-cardiac regulation decreases as sympatheticdelay (?) increases [18]. Indeed, the magnitude of Sm is shown to decrease withincreasing sympathetic delay; the system may become unstable for a large enoughdelay. In essence, along the consistency with Sp, these observations support thevalidity of the proposed approach to analyze the stability of autonomic-cardiacregulation.The results suggest that the effect of autonomic-cardiac parameters on stabilitymargin is not always monotonous, i.e., an increase (or a decrease) in a model pa-rameter 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 monotonouslyto most parameters in the autonomic-cardiac regulation model, includingCa, ? , ?0,VH , ?H , ? , and ?h, but it is shown to be enhanced or deteriorated depending on thevalue of R0a, P0, and H0. These parameters can be critical in determining the stabil-ity margin of autonomic-cardiac regulation, since they complicate the analysis ofautonomic-cardiac stability. Further, we also observe in Figure 4.3 that a large pe-ripheral resistance may help the stabilizing effort of autonomic-cardiac regulation70under stressful conditions.4.2.3 Multi-dimensional Stability AnalysisComparing Figure 4.2 and Figure 4.3 suggests that the reliance of stability mar-gin on each individual model parameter pertaining to autonomic-cardiac regulationis distinct for different physiologic conditions, e.g., normal condition or stressfulcondition. For instance, the stability of autonomic-cardiac regulation is largely af-fected 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 ?h may cause instability during a stress-ful condition, whereas the same change in ?h causes a decrease in the stabilitymargin only during a normal condition. The patterns in the reliance of the stabil-ity margin on H0 and ? are largely different between Figure 4.2 and Figure 4.3.Since ? represents the inhibitory strength of the PNS on the cardiac sympathetictone ?H , the significance of ? on the stability margin will be increased during astressful condition with an increased ?H . These observations indicate that physio-logic conditions (as represented by a particular parameter configuration in the auto-nomic-cardiac regulation model) must be accounted for when studying the impactof autonomic-cardiac parameters on the stability of autonomic-cardiac regulation.Considering that autonomic-cardiac regulation is a multi-parameter nonlinearsystem with delay, one-dimensional stability analysis (i.e., stability analysis overchanges in a single model parameter) may not provide a comprehensive perspec-tive on the stability of autonomic-cardiac regulation. For instance, the pattern ofreliance of Sm on H0 is dependent on the entire parameter configuration as in-dicated in Figure 4.2-Figure 4.3. Because of this, it is preferable to analyze thestability properties of autonomic-cardiac regulation and its stability margin againstsimultaneous changes in multiple parameters, i.e., the stability properties of au-tonomic-cardiac regulation should be examined in a multi-dimensional parame-ter space. An important strength of the proposed approach is that it can examinethe effect of changes in multiple parameters on the stability of autonomic-cardiacregulation. Figure 4.4 graphically illustrates the reliance of the proposed stabilitymetric, Sm, Equation 4.30 on simultaneous changes of two parameters (baroreflexset point P0 and another parameter). Accordingly, the stability properties of au-71Figure 4.4: The proposed stability metric, Sm, over 2-D parameter spacesfrom 50% to 150% of their nominal values for a normal physiologi-cal condition. The quantitative stability margin metric, Sm, at each pointof the 2-D parameter space is mapped into a pixel-intensity level. Ahigher pixel-intensity level is related to lower stability margin, and viceversa.tonomic-cardiac regulation against changes in two distinct model parameters canbe easily predicted. In essence, Figure 4.4 properly demonstrates the complexitiesassociated with multi-dimensional stability analysis of autonomic-cardiac regula-tion, i.e., interaction among autonomic-cardiac model parameters in determiningits stability. Figure 4.4, for example, depicts that a specific amount of change inP0 can have different influences on the stability of autonomic-cardiac regulation inthe presence of simultaneous changes in other parameters. Indeed, large P0 resultsin smaller stability margin in response to increasing H0, whereas large P0 yieldsa larger stability margin in response to increasing Ca. Overall, Figure 4.4 clearlydemonstrates the importance of analyzing the stability of autonomic-cardiac regu-lation in multi-dimensional parameter space; mush in-depth work on this issue infollow-up studies is warranted.Autonomic-cardiac regulation can also be studied using a hybrid dynamicalsystems framework that describes a physical system with a combination of continu-ous and discrete parts to represent time- and event-based behaviors [132]. Autonomic-cardiac regulation is a complex, nonlinear physiological system that can be approx-imated with several relatively simple, linear mathematical models in different op-72erating points. Different physiologic conditions (e.g., normal condition or stressfulcondition) described with a specific set of model parameters generates some com-plexities in stability analysis. Using the framework of hybrid dynamical systemsreduces complexities to analyze the system stability, and it may increase the accu-racy of mathematical modeling to capture physiological behaviors [133]. The lin-earized mathematical model introduced in this chapter will be beneficial to analyzethe stability of autonomic-cardiac regulation using a hybrid system framework.4.2.4 LimitationsThis study has a number of limitations, as discussed below.First, the mathematical model may not capture every significant mechanism inHR and BP regulation. For example, baroreflex control of SV, respiratory couplingon CVS (refer to Chapter 6), and chemoreflex mechanism are not described in thecurrent mathematical scheme, and therefore, they were not included in the stabilityanalysis results.Second, we assumed that all of the parameters in autonomic-cardiac regula-tion are independent of one another in the simulated data; this may not be true inreality. Thus, it is possible that a small portion of the simulated data we used tovalidate the proposed approach may not be good reproductions of reality. In-depthunderstanding of interactions and dependence among the parameters is required toresolve this issue.Third, to study hemodynamic instability in an individual, we must first developa subject-specific mathematical model (refer to Chapter 3) and then perform sta-bility analysis. For example, we used empirical minimum/maximum BP values tolinearize the sigmoidal baroreflex characteristic Equation 4.18. In the future, theproposed method must be improved by specifying the model parameters for eachsubject.4.3 Conclusion and Future WorkIn this chapter, we proposed a model-based analytical approach to stability anal-ysis of autonomic cardiac regulation. Based on a physiology-based model of au-73tonomic-cardiac regulation, we developed an analytical and computationally effi-cient method to estimate the equilibrium states of the system, and we developed asystematic approach to stability analysis of autonomic-cardiac regulation that canprovide a quantitative metric of stability margin. The efficacy of the proposed ap-proach was examined using a series of simulation experiments. Future work willinclude developing 1) an approach to analyze global stability of autonomic-cardiacregulation, 2) computationally efficient strategies to identify parameter configura-tions associated with autonomic-cardiac instability in multi-dimensional parameterspace, and 3) novel intervention and therapeutic strategies to maintain the stabilityof autonomic-cardiac regulation.74Chapter 5A Novel Approach to the Designof an Artificial Bionic BaroreflexThe ANS maintains homeostasis in the CVS through many negative feedback mech-anisms including the baroreflex (the major short-term blood pressure control mech-anism) to deliver adequate oxygenated blood flow to organs in response to physical(e.g., exercise and orthostatic hypotension) and psychological (e.g., fear and anxi-ety) stressors [41?43].In the CVS, instantaneous arterial BP is sensed by baroreceptors located on themajor arteries. Accordingly, a series of commands is produced by the baroreflexand transmitted to the heart, arteries, and other organs to maintain homeostasisin the CVS. An artificial bionic baroreflex consists of pressure sensors to measurearterial BP and a neurostimulator that generates an electrical pulse train to stimulatesympathetic and parasympathetic nerves regulated by a computerized device [36,134].The gravitational effect on circulation during postural changes provokes a barore-flex response to prevent hypotension and hypoperfusion of the brain [39]. There-fore, baroreflex failure in individuals with severe orthostatic hypotension (e.g., in-dividuals with traumatic SCIs) may result in loss of consciousness during a sitting-to a standing-position change resulting in a severely impaired quality of life [36,135]. Moreover, the prevalence of drug-resistant hypertension (i.e., BP remainsabove 140/90 mmHg in spite of the concurrent use of three anti-hypertensive med-75ications [58, 136]) has increased in recent years [136, 137]. An artificial bionicbaroreflex is aimed to be an effective treatment for baroreflex failure in individualswith drug-resistant hypertension and severe orthostatic hypotension.In [39], the open-loop transfer function of the baroreflex was identified us-ing white noise perturbation after anatomically isolating the carotid sinuses by as-suming that the baroreflex works linearly in some physiological pressure range.Kawada and Sugimachi [135] presented encouraging results to regarding the pre-vention of orthostatic hypotension in anesthetized cats by using epidural spinalcord stimulation and frequency analysis. The nerve stimulation devices can beimplanted or percutaneously inserted into the skin?s surface.We proposed a method to design an artificial bionic baroreflex by mimickingthe in-vivo baroreflex mechanism. This method can be used to adjust existing neu-rostimulator devices to regulate BP within an individual?s CVS (Figure 5.1). Theproposed method consists of two parts: a sigmoidal characteristic that mimics themodulating 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 andcardiovascular disorders). The adaptation mechanism resetting the baroreflex char-acteristic is devised according to the physiological adjustment mechanism of thein-vivo baroreflex. Further, we analyzed the robustness of the proposed controllerscheme in regard to the model uncertainty showing the inter-individual differencesin autonomic-cardiac regulation.5.1 Methods and AlgorithmIn this section, we first briefly explain the experimental data obtained from theMIMIC dataset, which is used in this study. We then present a physiology-basedmathematical model of autonomic-cardiac regulation described by two couplednonlinear and delayed differential equations. Subsequently, the system identifica-tion technique introduced in [1] and used to develop subject-specific models forthree subjects is briefly explained. In this study, the subject-specific mathematicalmodel has been used instead of the individual?s in-vivo autonomic-cardiac reg-ulation (Figure 5.2). Finally, the proposed method to design an artificial bionic76Figure 5.1: Schematic model of autonomic-cardiac regulation with emphasison the baroreflexbaroreflex is described and is followed by a robustness analysis of the proposedcontrol strategy.5.1.1 Experimental DataWe examined the proposed method using experimental data of autonomic-cardiacregulation in three subjects taken from the MIMIC dataset [47]. A 1-hour sam-ple of HR and BP signals in each individual is extracted and divided into 30s-longdata segments to be used in the system identification section. The MIMIC datasetcontains 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 ModelWe introduced a physiology-based mathematical model of the autonomic-cardiacregulation in Chapter 3 by using two coupled differential equations Equation 5.1-Equation 5.2 having nonlinear and delayed dynamic interactions, each of which77Table 5.1: Model parameters of autonomic-cardiac regulation.Parameter Definition Nominal ValueCa arterial compliance 1.55 mlmmHg?1R0a minimum arterial resistance 0.6 mmHgsml?1?V stroke volume 50 mlH0 intrinsic HR 100 min?1? sympathetic delay 3 sVH vagal tone 1.17 s?2?H sympathetic control of HR 0.84 s?2? sympathetic effect on Ra 1.3? vagal damping of ?H 0.2?H relaxation time 1.7 s?1describes the dynamics of HR and BP regulation as follows:H?(t) = ?HTs?VHTp+?H[H0?H(t)](5.1)P?(t) = ? P(t)R0a(1+?Ts)Ca+ H(t)?VCa(5.2)where H is HR, and P is mean arterial BP. The definitions and nominal values ofthe parameters in Equation 5.1-Equation 5.2 are summarized in Table 5.1. In thismodel, the modulating baroreflex functions on the SNA and PSNA are denoted byTs and Tp, respectively.The modulating baroreflex functions on the SNA and PSNA (i.e., Ts and Tp) canbe modeled using a sigmoid function ?(P) with an amplitude-limiting characteris-tic [96]. ?(P) is defined as follows:?(P) = 11+ e??0(P?P0) 50? P? 200. (5.3)The sigmoid function ?(P) is characterized using two variables, setpoint P0 andsensitivity ?0 [113, 129]. To simulate the in-vivo sympathetic and parasympatheticmodulating functions, we substitute Ts = 1??(P(t? ?))and Tp = ?(P(t))intoEquation 5.1-Equation 5.2.Parametric sensitivity analysis is conducted on the model to classify the modelparameters into high-sensitivity and low-sensitivity groups based on their relative78Figure 5.2: Schematic model of the proposed artificial bionic barorefleximpacts on the system outputs. H0 and R0a were initially classified into the categoryof invariant parameters since they are essentially constant within an individual ina short-time interval. The remaining model parameters were classified into high-sensitivity (VH , ?H , ? , ?V , and P0) and low-sensitivity (?0, ? , Ca, ? , ?H) groups,according to the results of the sensitivity analysis, to select a subset of parameterswith significant impact on the system outputs (i.e., high-sensitivity group). Thedetailed description of the mathematical model and parametric sensitivity analysisis explained in Chapter 3.5.1.3 System IdentificationTo estimate subject-specific high-sensitivity parameters in Equation 5.1-Equation 5.2,a system identification method was developed based on an optimization problemminimizing the normalized L1-error between measured versus model-estimated HRand BP signals. The system identification was performed by optimizing the high-sensitivity parameters such that the error function Equation 5.4 became minimizedin each 30s-long data segment, whereas low-sensitivity and invariant parameterswere fixed at their corresponding population nominal values (Table 5.1). The error79Figure 5.3: BP measurement (BP setpoint) vs. the results of the artificialbionic baroreflex (simulated BP) for individual with subject number 477.function (i.e., the objective function) was specified as follows:J = EP+EH2; EX =n?t=0????Xs(t,M)?Xm(t)Xm(t)????, (5.4)where Xm(t) and Xs(t,M) (X =H,P) are measured and model-estimated output sig-nals, respectively, and M is the set of high-sensitivity parameters in the autonomic-cardiac regulation model, i.e., M = {VH ,?H ,? ,P0,?V}. The optimization problemwas solved using the fmincon routine with an active-set algorithm in the MATLABOptimization Toolbox [118], which finds the constrained minimum of a multivari-able nonlinear scalar function J using Quasi-Newton approximation. The set ofoptimized high-sensitivity parameters minimizing the error function was used asestimates of high-sensitivity parameters for the corresponding data segment. Thesystem identification method has been thoroughly described in Chapter 3.5.1.4 Artificial Bionic BaroreflexThe artificial bionic baroreflex is a negative-feedback control system containing aset 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-80Figure 5.4: BP measurement (BP setpoint) vs. the results of the artificialbionic baroreflex (simulated BP) for individual with subject number 486.Figure 5.5: BP measurement (BP setpoint) vs. the results of the artificialbionic baroreflex (simulated BP) for individual with subject number 476.81thetic efferent nerves. This system continuously measures BP and computes thefrequency of a pulse train required to stimulate sympathetic and parasympatheticefferent nerves. The measured BP (BPm) must be compared continuously to thetime-varying BP setpoints (BPsp), the BP level providing the need of body organsfor oxygenated-blood, to be used in the control scheme. If the BPm differs from theBPsp, the baroreflex characteristic is adjusted so that the BP to gradually reachesthe BPsp. Note that determining the BPsp signal is a challenge, and the proposedmethod has been developed based on a given BPsp signal. In theory, the time-varying 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., exerciseand orthostatic hypotension) and psychological (e.g., fear and anxiety) stressors.In this study, individuals were replaced by subject-specific mathematical mod-els describing autonomic-cardiac regulation of each subject, and simulated BP(BPsim) was used instead of BPm continuously compared against BPsp. The barore-flex effects on the autonomic-cardiac regulation are achieved by modulating SNAand PSNA (VH , ?H , and ?) using Ts and Tp. For example, when the BP must de-crease to reach the setpoint, Tp must reset such that its magnitude at the same BPlevel becomes higher. This causes HR to decrease and then BP to decrease. As Tpis a sigmoidal curve, P0 must reset to a lower value in order to obtain a higher HRdecelerating parasympathetic effect and vice versa. Therefore, P0, then sigmoidalcharacteristic, 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 adjust-ing mechanism in response to an error in the BP regulation, and ? is an intervalin which the baroreflex characteristic needs to be updated. The adjustment ruleEquation 5.5 is initialized by P0 = 100. Moreover, to be consistent with baroreflexphysiology, P0 is constrained between 50 mmHg and 200 mmHg. A very largek may cause overshoot in the control system, while a very small k may cause alarge settling time, preventing proper adjustment of the baroreflex characteristic totrack the BPsp. This coefficient is empirically tuned to k = 0.08 by consideringboth overshoot and settling time of the control system. The time inetrval, ?, in82Equation 5.5 is 30s in this study; however, it can be set to a larger or smaller valuedepending on the pace of variation in BPsp.5.1.5 Robustness AnalysisTo validate the robustness of the proposed control strategy in regard to model un-certainty as well as inter-individual differences, we generated 100 sets of modelparameters showing 100 different mathematical models of autonomic-cardiac reg-ulation. In each set of model parameters, the high-sensitivity parameters were as-signed to random values from a uniform distribution in the neighborhood (+/-50%)of their respective individualized nominal values (Table 5.2), while the remainingparameters 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 parametervalues were selected to generate the 100 sets of random values.5.2 Results and DiscussionSince we aimed to use a subject-specific mathematical model for each individ-ual, the mathematical model Equation 5.1-Equation 5.2 was specified by estimat-ing individualized nominal values for high-sensitivity parameters in each subject,whereas the remaining parameters were assigned by their population nominal val-ues (Table 5.1). As the MIMIC dataset contains CO measurement, we calculatedan individualized nominal value of ?V for each subject instead of either estimat-ing ?V by the proposed identification technique or using the population nominalvalue. Accordingly, we obtained a time series of VH , ?H , and ? with a 1-hourlength using the system identification technique; the average value of these param-eters over a 1-hour length was calculated to be assigned as individualized nominalvalues (Table 5.2). Note that the individualized high-sensitivity parameters mustbe updated at every 1-hour (or any other length initially assumed) interval of datain future studies. We obtained three sets of model parameters representing threesubjects in the 1-hour interval to evaluate the proposed method for designing anartificial bionic baroreflex.To evaluate the proposed method, we compared the simulated BP obtainedbased 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 P0Figure 5.6: The results of robustness analysis for an individual with subjectnumber 477. The solid line shows an average value of 100 simulatedsignals obtained by the proposed control strategy, whereas the shadedarea indicates the corresponding standard deviation.In each panel of Figure 5.3-Figure 5.5, the top figure shows the BP measurementsused as BP setpoints versus simulated BP obtained by the proposed artificial bionicbaroreflex 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 considerablyhigher during t=0 s - 100 s because of the P0 initialization. After t>100 s, P0 con-verges to a proper value to control the BP regulation system. Since we assumedthat the control strategy would not respond very rapidly, several abrupt changesin the BP setpoints during t=1000 s - 1300 s in Figure 5.3 and t=1700 s - 1800 sin Figure 5.5 which may be originated due to the measurement noise caused the84Figure 5.7: The calculated control signal, P0, in three subjectstracking 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 sat-urated P0 at 200 s during the same interval indicates that the control strategy wasnot 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 set-point tracking specification (low tracking error) regardless of variability in modelparameters as well as parameter identification error described by a large uncertaintyin the model parameters (VH , ?H , and ?). Indeed, we showed that the proposedcontrol strategy is robust against model uncertainty by tolerating large variationsin P0 (Figure 5.6b).To evaluate the proposed approach in a clinical setting, we must perform aclinical study in which the regulation mechanism of the baroreflex on sympatheticand vagal nerves are replaced with an external controller. The closed-loop in-vivo85Table 5.2: Individualized nominal values of high-sensitivity parameters inthree subjects versus corresponding population nominal values.Subject PopulationI (477) II (486) III (476)VH 0.65 1.37 2.13 1.17?H 1.5 0.87 0.68 0.84? 0.7 1.36 1.55 1.3?V 46 40 36 50baroreflex must be opened at the level of efferent or afferent nerves according to thelevel/type of injury in the baroreflex mechanism, and an electrical stimulator mustbe subcutaneously implanted to be overridden the corresponding nerves. The elec-trical stimulator is a rate responsive pulse generator, and the stimulation frequencyand magnitude must be adjusted based on calculated P0 (Figure 5.7).As mentioned above, the electrode placement site will be different according tothe level/type of injury in the baroreflex mechanism. For example, afferent barore-ceptor nerves may need to be overridden in individuals with chronic drug-resistanthypertension, while efferent nerves may need to override in individuals with SCI.Sensory information, such as BP and HR measurement, is needed in the closed-loopcontrol scheme. For example, BP can be sensed by in-vivo mechanism (e.g., affer-ent baroreceptor nerves) or artificial receptors (e.g., implanted microstrain gauges)according to the specific physiological condition of the individual. Since there isno in-vivo HR measurement mechanism, the artificial HR sensors may improve theefficiency of the proposed approach. In order to determine the optimal site of elec-trode placement to stimulate sympathetic and vagal nerves, we must investigateease of access, significance of effect, and possible side effects [38].5.3 Conclusions and Future WorkThis chapter presented the feasibility and potential for a computationally efficientclosed-loop control scheme to design an artificial bionic baroreflex that can be usedin the treatment of baroreflex failure. The recently introduced open-loop BP controlschemes will be improved by the proposed closed-loop technique to accommodatethe time-varying needs of BP level in different daily life conditions. In the future,86the proposed approach should be validated extensively in clinical settings.87Chapter 6Mathematical Modeling ofAutonomic-CardiorespiratoryRegulationAutonomic-cardiorespiratory regulation operates through interactions between theANS, the cardiovascular system, and the respiratory system. The ANS maintainshomeostasis in the cardiorespiratory system in order to deliver adequate oxygenatedblood flow to organs against physical (e.g., exercise and orthostatic hypotension)and psychological (e.g., fear and anxiety) stressors [41?43]. The ANS consists oftwo 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 sinoatrialnode, cardiac contractility, and peripheral resistance [17, 23, 51].A variety of mathematical models to describe autonomic-cardiac regulationusing black-box and white-box (physiology-based) approaches have been proposedpreviously [23, 55, 96]. However, a physiology-based mathematical model for therespiratory system impacts has been investigated to a certain extent [43]. Further,the respiratory system impacts on HR (i.e., respiratory sinus arrhythmia or RSA)and BP (i.e., venous return variation) have either been neglected [55] or simplymodeled by a non-physiology function (e.g., a sine function) [64].In this chapter, we introduce a physiology-based mathematical model of auto-88nomic-cardiorespiratory regulation described by a set of three coupled nonlinearand delayed differential equations, each of which describes the regulation of HR,BP, and RR. A unique 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 regula-tion [1] based on the work of Fowler et. al. [7]. However, the respiratory systemdynamics 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 AlgorithmIn this section, we first describe the experimental dataset collected in this study.Then, we present the physiological background associated with major causes ofHR and BP fluctuations. A mathematical model of autonomic-cardiac regulation(without respiratory system effects) described by a set of two coupled nonlinearand delayed differential equations is also introduced. We then present an improve-ment in the mathematical model that describes neuromechanical and mechanicalcoupling of cardiovascular and respiratory systems, i.e, lung stretch-receptor reflexand venous return variations. We also introduce a differential equation to modelRR regulation that mainly originates from the medullary respiratory center in thebrainstem, which is influenced by voluntary actions and chemoreflex.6.1.1 Experimental DatasetWe collected autonomic-cardiorespiratory signals including ECG, BP waveform,and Tidal Volume (TV) from 18 healthy subjects without any cardiovascular disor-der history during an LBNP experiment followed by a respiration maneuver usingthe Pneumocard and Portapress devices. The experiment was approved by SimonFraser University board of ethics (Appl. # 2012s0078; Dated November 26,2012)and consent form was signed by the participants. Figure 6.1 depicts an example ofrecorded signals during different stages of the LBNP test.89Figure 6.1: Physiological measurement during LBNP experiment in an indi-vidual; mean BP, SV, and HR were calculated according to the BP wave-form and ECG recordings.6.1.2 Physiological BackgroundHR fluctuations around the mean HR (also referred to as HRV) are generated bythe SNS and PNS in the cardiorespiratory control system. In healthy individuals,the HRV spectrum shows two predominant peaks: one at low frequency around0.1 Hz (Mayer waves) associated with arterial pressure biofeedback. The otherone, at higher frequency around 0.25 Hz (corresponding to respiration frequency),is called RSA. RSA is mainly generated through two mechanisms: neural-basedmodulation of cardiac vagal activation by the medullary respiratory center andneuromechanical-based modulation of cardiac vagal activation by the lung stretch-receptor reflex [84]. RSA has been observed at the approximate respiratory fre-quency even in the absence of respiration due to the activation of the medullaryrespiratory center [85]. The lung stretch-receptor reflex inhibits and excites car-diac vagal activation tone during inspiration (lung inflation) and expiration (lung90deflation) respectively, causing a decrease and an increase in heart periods duringrespiratory 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 toincrease the efficacy of the pulmonary gas exchange between capillary blood flowand alveolar gas volume by matching perfusion to ventilation within each respira-tory cycle [84, 138].Blood pressure variability (BPV) is caused mainly by HRV as well as the directmechanical effects of respiration (either spontaneous or mechanical) on BP [139].The HRV influences BP through the heart period baroreflex mechanism. Further,the direct mechanical effect of respiration causes variation of venous return in eachrespiratory cycle. During spontaneous inspiration, the chest wall expands and thediaphragm descends resulting in lower intrapleural pressure1 and therefore expan-sion of the lungs and cardiac chambers [48]. This expansion causes an increasein 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 pressuregradient for the venous return [48, 67]. Consequently, venous return increases dur-ing spontaneous inspiration and decreases during spontaneous expiration. Duringmechanical ventilation, the chest wall and diaphragm are not displaced; however,the lungs are inflated due to an external air force that causes different consequencessuch as an increase in intrapleural pressure during mechanical inspiration. Simi-larly, venous return decreases during mechanical inspiration and increases duringmechanical expiration (Table 6.2).6.1.3 Autonomic-Cardiac RegulationWe introduced a physiology-based mathematical model of autonomic-cardiac reg-ulation in [1] using two coupled differential equations Equation 6.1-Equation 6.2having nonlinear and delayed dynamic interactions, each of which describe the1The pressure within the thoracic space between the organs (lungs, heart, vena cava) and the chestwall is called intrapleural pressure.2Cardiac pre-load is the end-diastolic volume (EDV) of the ventricle at the beginning of systole.91Figure 6.2: Schematic diagram of interactions between cardiovascular, respi-ratory and nervous systems.92Table 6.1: Model parameters of autonomic-cardiac regulation.Parameter Definition Nominal ValueCa arterial compliance 1.55 mlmmHg?1R0a minimum arterial resistance 0.6 mmHgsml?1?V stroke volume 50 mlH0 intrinsic HR 100 min?1? sympathetic delay 3 sVH vagal tone 1.17 s?2?H sympathetic control of HR 0.84 s?2? sympathetic effect on Ra 1.3? vagal damping of ?H 0.2?H relaxation time 1.7 s?1dynamics of HR and BP regulation:H?(t) = ?HTs1+? Tp ?VHTp+?H(H0?H(t))(6.1)P?(t) =? P(t)R0a(1+?Ts)Ca +H(t)?VCa. (6.2)where H is HR and P is mean arterial BP. Ts = 1??(P(t? ?))and Tp = ?(P(t))are sympathetic modulating function and parasympathetic modulating function re-spectively, generated by the baroreflex control mechanism. Note that Ts and Tpare both purely BP dependent while the SNS and PNS are also modulated by otherphysiological variables (e.g., O2 and CO2 concentration in blood) or psychophysio-logical states (e.g., fear and anger). The time delay associated with the sympatheticpathway is denoted by ? . ?(P) is defined as follows:?(P) = 11+ e??0(P?P0) 50? P? 200. (6.3)where P0 and ?0 are the BP setpoint and the sensitivity of baroreflex mechanism,respectively.6.1.4 Autonomic-Cardiorespiratory RegulationIn this study, we improve our previous mathematical model of autonomic-cardiacregulation Equation 6.1-Equation 6.2 by modeling two major interactions of car-93Figure 6.3: An extensive block-diagram model of autonomic-cardiorespiratory regulation [see Chapter 2] with emphasis onparts described in Equation 6.7-Equation 6.8. The shaded parts are notdescribed in the mathematical model.diovascular and respiratory systems, i.e., mechanical and neuromechanical. Fur-ther, we introduce a differential equation representing the dynamic of respirationrhythm originated in the medullary respiratory center.The mechanical coupling of the cardiovascular and respiratory systems causesan increase in venous return and, consequently, an increase in SV, during sponta-neous inspiration and mechanical expiration. On the other hand, venous return, andtherefore SV, decrease during spontaneous expiration and mechanical inspiration(Table 6.2). We modeled this pure mechanical effect by adding (during mechani-cal respiration) or subtracting (during spontaneous respiration) k2V?L with positive94coefficient k2 to the SV (?V ) as follows:P?(t) =? P(t)R0a(1+?Ts)Ca+ H(t)(?V ? k2V?L)Ca. (6.4)The neuromechanical coupling of the respiratory and cardiovascular systemsis generated by the lung stretch-receptor reflex. This reflex causes an increase inHR during inspiration, while ILV consistently increases (i.e., V?L>0), and a decreasein HR during expiration, while ILV consistently decreases (i.e., V?L<0) (Table 6.2).The lung stretch-receptor reflex inhibits and excites cardiac vagal activation toneduring inspiration and expiration, respectively [84]. We model this mechanismby subtracting a respiration-related term consisting of a rate of change in ILV, V?L,multiplied by a positive coefficient k1, to cardiac vagal activation toneVH in the HRequation as follows:H?(t) = ?HTs1+ ? Tp? (VH ? k1V?L)Tp+?H(H0?H(t))(6.5)During inspiration while V?L>0, inhibition effects of the PNS on HR reduce, andtherefore HR increases. Similarly, during expiration, while V?L<0, HR decreases inresponse to a rise in inhibition effects of the PNS.The medullary respiratory center of each individual generates a relatively con-stant rhythm, R0, which is modulated in different conditions such as low O2 or highCO2 concentration in blood, sleep, and emotions (e.g., fear and anxiety). Specifi-cally, the respiration rhythm, R0, is modulated by chemoreflex stimulation causedby changes in chemoreceptors responses throughout the body. The chemoreflexoperates mainly in response to the CO2 levels rather than O2 levels [48]. We modelthe dynamics of respiration rate, denoted by R, generated in the respiratory centeras follows:R?(t) = k3[(1+?co2)R0?R(t)]+u(t) (6.6)where ?co2 is a sigmoid function representing chemoreflex modulating function onR0 and u(.) is a voluntary component of RR regulation. Note that the only voluntaryterm in autonomic-cardiorespiratory regulation is the individual?s ability to change95Table 6.2: Respiratory system impacts on VL, HR, VR, and ?V .Spontaneous MechanicalInhale Exhale Inhale ExhaleVL (Instantaneous Lung Volume) ? ? ? ?HR (Heart Rate) ? ? ? ?VR (Venous Return) ? ? ? ??V (Stroke Volume) ? ? ? ?RR. Finally, we propose the mathematical model of autonomic-cardiorespiratoryregulation illustrated in Figure 6.3 as follows:H?(t) = ?HTs1+ ? Tp? (VH ? k1V?L)Tp+?H(H0?H(t))(6.7)P?(t) =? P(t)R0a(1+?Ts)Ca+ H(t)(?V ? k2V?L)Ca(6.8)R?(t) = k3[(1+?co2)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?1s?1 and 3.12 ms, respectively. Nomi-nal values of k1 and k2 are assigned such that the respiration-related terms k1V?L andk2V?L generate 10% perturbation on the amplitude of VH and ?V , respectively.6.2 Results and Discussion6.2.1 Model ValidationIn this work, we proposed a physiology-based mathematical model of autonomic-car-diorespiratory regulation built upon a mathematical model of autonomic-cardiacregulation described in Chapter 3. We included respiratory terms in the mathe-matical model according to the corresponding location and dynamic of the respira-tory effect on autonomic-cardiac regulation. We potentially can use the proposedmathematical model to investigate the effects of respiration on HR and BP regula-96Figure 6.4: PSD difference of HRV among simulated (two methods) vs. mea-sured HR signals at the different stages of the LBNP experiment; theshaded area shows the respiratory frequency band.tion and fluctuation. To validate the proposed mathematical model, we must showthat adding respiratory terms in the mathematical model improves the accuracy ofmodel-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 theautonomic-cardiorespiratory regulation model and HR obtained from the autonomic-cardiac regulation model were tested against measured HR using PSD analysis ofHRV. Figure 6.4 shows that ?PSD of model-estimated and measured HR is closeto zero within the respiratory frequency band (mostly located at 0.2?0.05 Hz) inthe autonomic-cardiorespiratory regulation model. We can conclude that a moreaccurate mathematical model was obtained by including the respiratory dynamicin the autonomic-cardiac regulation model.97Table 6.3: A numerical measure of perturbation caused by mechanical cou-pling effects J2 and neuromechanical coupling effects J1.k1/k1,Nom J1 k2/k2,Nom J250% 0.18 50% 0.7675% 0.27 75% 1.14100% 0.36 100% 1.52125% 0.44 125% 1.89150% 0.53 150% 2.286.2.2 Mechanical vs. Neuromechanical CouplingsTo investigate the neuromechanical coupling effects of respiration k1V?L on HR andBP, we assigned k2=0 and all the non-respiratory related parameters to their nom-inal values, whereas k1 was changed from 50% to 150% (25% increment) of itsnominal value. Further, to compute the perturbation of HR and BP merely causedby neuromechanical coupling effects k1V?L, the average sum of absolute normalizederrors of HR and BP J1 was used as follows:J1 =EP,1+EH,12 ; EX ,1 =30?t=0????Xk1 6=0(t)?Xk1=0(t)Xk1=0(t)????(6.10)whereas Xk1 6=0(t) and Xk1=0(t) are HR or BP (X = H,P) with and without neu-romechanical coupling effects, respectively. To solve the mathematical modelEquation 6.7-Equation 6.8 for each given set of parameteres, we first generatedan arbitrary ILV signal for a 30 s-length segment with constant rate of change inlung volume V?L=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 aDDE (delay differential equation) solver in MATLAB to obtain perturbed HR andBP for five different values of k1 (Figure 6.5).Similarly, J2 is the average sum of absolute normalized errors of HR and BP,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).98Figure 6.5: Neuromechanical coupling effects of respiration on HR and BP.Figure 6.6: Mechanical coupling effects of respiration on HR and BP.996.2.3 LimitationsDespite the novelty of this study to describe the autonomic-cardiorespiratory sys-tem using a physiology-based mathematical model, it has one major limitation, asdiscussed below. The nominal values of two parameters, k1 and k2, presented inthis work must be reconsidered. In fact, we must perform an experiment to cal-culate numerically two parameters, k1 and k2, which represents open-loop gainsfrom ILV to VH and from ILV to SV, respectively. Accordingly, the significance ofmechanical vs. neuromechanical couplings should be revisited.6.3 Conclusions and Future WorkIn this chapter, we resolved the lack of accuracy in the autonomic-cardiac regula-tion model proposed in Chapter 3. The mathematical model was revised in regardto the respiratory system effects by taking the major respiratory impacts, includinglung stretch-receptor reflex and venous return variation on HR and BP, into consid-eration. Future work will include extending the mathematical model to increase themodel?s accuracy and improving the proposed identification technique described inChapter 3 to use capabilities of the proposed autonomic-cardiorespiratory model.We will aim to eliminate the effects of respiration on PSNA k1V?L. This will help toextract a pure parasympathetic activation caused by different mental states and en-vironmental stimulus. Further, the effects of spontaneous (during consciousness)and mechanical (during anaesthesia) respiration on HR and BP regulation can beinvestigated individually. Similarly, results of the identification technique will beimproved for anaesthetized and awake individuals.100Chapter 7Conclusion and Future Work7.1 Summary: Work AccomplishedIn this thesis, we studied autonomic-cardiac regulation with and without respira-tory coupling within a series of investigations using different techniques includingmathematical modeling, system identification, stability analysis, and control de-sign. We summarize and conclude the major points and achievements in this chap-ter.Mathematical Modeling- In Chapter 3, we adopted a model of autonomic-cardiac regulation consisting of two coupled nonlinear and delayed differentialequations, each of which describes the dynamics of HR and BP regulation. Weimproved the existing model to mathematically describe respiratory-based mech-anisms in Chapter 6. We revised the model in regard to the mechanical and neu-romechanical couplings of the respiration system and autonomic-cardiac regulationincluding the lung stretch receptor reflex and the venous return variation, which hadnot been investigated properly in the past. The revised model can physiologicallypresent a source of oscillatory patterns observed in HR due to the respiration sys-tem, which is commonly known as RSA. The proposed mathematical model mustbe improved further to describe other significant mechanisms in HR and BP regula-tion. For example, baroreflex control of SV and the renin-angiotensin system havenot been described mathematically in this manuscript.101System Identification-A parameter identification technique for estimating andthen monitoring SNA and PSNA using routine clinical measurements, HR and BP,was introduced in Chapter 3. We presented a proof-of-concept for the proposedidentification technique using two clinical datasets: the MIMIC dataset collectedat Beth Israel Hospital, Boston, MA [110] and the orthostatic hypotension datasetcollected at New York Medical College [46]. We examined the repeatability of theidentification outcome using the MIMIC dataset and the physiological consistencyusing the orthostatic hypotension dataset. Despite the promising preliminary re-sults, the proposed method has several limitations. For example, the identificationtechnique may benefit from the availability of CO measurement that is not com-monly measured in the clinical setup; however, some non-invasive CO estimationtechniques including electrical velocimetry and echo-cardiography have recentlybeen introduced (refer to Section 3.2.3). Further, the possible interdependence be-tween model parameters was not deeply explored.Stability Analysis- A systematic approach to stability analysis of autonomic-cardiac regulation was proposed in Chapter 4. The proposed method was derivedaccording to the mathematical model of autonomic cardiac regulation with twocoupled nonlinear and delayed differential equations. We introduced a stabilityindex to compare numerically different parameter configurations and to monitorthe stability margin of CVS during any clinical condition enforced by a parameterconfiguration. We can investigate the stability margin of a large number of clin-ical conditions using the proposed index to recognize a possible disorder in theautonomic-cardiac regulation that could not be recognized easily without a model-based stability analysis. For example, the stability margin of the autonomic-cardiacregulation can be investigated in individuals with high arterial stiffness and low in-trinsic 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 newconcept, that has been proposed recently [14]. An extensive clinical study demon-strating the potential significance of the proposed stability analysis should be per-formed.102Artificial Bionic Baroreflex- In Chapter 5, we developed a method for de-signing an artificial bionic baroreflex capable of restoring normal arterial pressureregulation by mimicking the in-vivo baroreflex mechanism. The individual?s in-vivo autonomic-cardiac regulation was described by a subject-specific mathemati-cal model. To individualize the mathematical model, we used the proposed systemidentification technique in Chapter 3. A unique strength of the proposed methodis its capability to determine the modulating baroreflex functions on the sympa-thetic and parasympathetic nerves. The proposed method potentially can be usedto design an advanced pacemaker, a medical device that regulates the heartbeatsequence according to the individual?s current physical and psychophysical condi-tion. Current pacemakers provide only constant-rate stimulation for the heart. Anextensive clinical study of the proposed bionic baroreflex is needed to evaluate thesignificance of this work.7.2 Future-Work: The Road AheadThis section suggests a number of possibilities for future work, categorized accord-ing to the chapters of this thesis.Mathematical Modeling- The proposed mathematical model that includes therespiration system could be applied to the frequency analysis of HRV. In the powerspectrum analysis of HRV, LF power, modulated by both SNA and PSNA, and HFpower, modulated by PSNA, have been used together as measures to monitor whathas been called the sympathovagal balance (e.g., LF/HF ratio) on the heart. Amodel-based method to analyze the HRV spectrum may reveal some unseen partsof the HRV dynamic, which could be useful for investigaing other possible causesof variation in these frequency bandsSystem Identification- Considering that we used a relatively fast parame-ter identification technique, we can potentially use the proposed sympathetic andparasympathetic monitoring technique in the treatment of individuals with ANSdisorders using biofeedback techniques. A large number of studies have attemptedto use biofeedback to alter HRV. Further, we can investigate extensively the consis-103tencey of the proposed system identification results on sympathetic and parasym-pathetic 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 negativefeedback (i.e., baroreflex mechanism) in autonomic-cardiac regulation may not al-ways result in stability due to, for example, the system nonlinearity. It has beenshown in [140] that negative feedback can cause expanding oscillations in certaincircumstances. Further, cyclic interaction among elements of a system with twonegative feedbacks may cause an instability in the system. Note that two or anyeven number of negative interactions generate a positive circuit in the system [141].In autonomic-cardiorespiratory regulation described by ordinary differential equa-tions, circuits can be defined in terms of the elements of the Jacobian matrix [142].Considering that a compact DDE mathematical model for autonomic-cardiac regu-lation described in Chapter 6 and the corresponding Jacobian matrix was presentedin Chapter 4, a possible future study is an investigation of occurence of positivefeedback (circuit) in autonomic-cardiac regulation.Artificial Bionic Baroreflex- The method proposed in Chapter 5 can be usedto design of an artificial bionic baroreflex that would regulate BP in individualswith resistant hypertension or SCI [143]. 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Homeostasis revisited in the genesis of stress reactivity Ataee, Pedram 2014
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Title | Homeostasis revisited in the genesis of stress reactivity |
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Ataee, Pedram |
Publisher | University of British Columbia |
Date Issued | 2014 |
Description | 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 subject-specific 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 regulation. 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. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2014-03-13 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial 2.5 Canada |
DOI | 10.14288/1.0165898 |
URI | http://hdl.handle.net/2429/46220 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2014-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc/2.5/ca/ |
AggregatedSourceRepository | DSpace |
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