Trajectory analysis in Health care by Hollar

David W Hollar Trajectory Analysis in Health Care Trajectory Analysis in Health Care David W Hollar Trajectory Analysis in Health Care David W Hollar Health Administration Pfeiffer University Morrisville, North Carolina, USA ISBN 978-3-319-59625-9 ISBN 978-3-319-59626-6 DOI 10.1007/978-3-319-59626-6 (eBook) Library of Congress Control Number: 2017944366 © Springer International Publishing AG 2018 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland For our daughter, Brooke Hollar, future clinician, courageous follower of Christ, and leader by example Preface The objective of this book is to introduce health researchers, epidemiologists, health policy makers, and clinicians to Trajectory Analysis, a term that I use to refer to the nonlinear analysis of processes This science was developed by a number of scientists, most notably the French mathematician Henri Poincare and the Russian mathematician Aleksandr Lyapunov during the late 1800s During the twentieth century, the science gained traction in the physical sciences with the development of quantum mechanics and mathematical descriptions of fluid mechanics, further applied during the latter part of the century to Chaos Theory and the analysis of defects in nanoparticles and crystals Trajectory Analysis relies heavily on the measurement of continuous change, which is measured using differential equations, although phase transitions across critical transition points are generally non-integrable I have kept the mathematics to a minimum, but several central equations and rough derivations are provided to demonstrate importance and applicability to health research Health research and epidemiology currently enjoys many powerful statistical tools, but few address process and change, to which I constantly refer to Rene Thom’s work on Catastrophe Theory, where he emphasized the topology of patterns, processes, and change and criticized statistical analyses that involved “clouds of points.” Health research also suffers from the lack of consistent, longitudinal data on the physiology and behaviors of people as well as the endless variables that impact health Most importantly, I stress a systems perspective on change, process, and analysis, and particularly using these tools to effect positive health change Chapter provides an overview of the physical principles and universality of nonlinear dynamics in health and our environment Chapter represents a needs assessment of health research, particularly the need for comprehensive, multiple variable, and longitudinal analyses to demonstrate long-term health care conditions, contributing variables, and ultimate outcomes Decision-making models are emphasized, along with an introduction to Thom’s Catastrophe Theory and Ilya Prigogine’s work on non-reversibility in changing systems Chapter addresses the major problem in vii viii Preface health research, recidivism, and introduces the nonlinear approaches to countering this recurring issue for health interventions Chapter provides a very general overview of epidemiological methods I refer the reader to the several cited excellent references and textbooks for more comprehensive discussions of these methods Nevertheless, Chap provides an introduction to two valuable statistical methods, structural equation models and hierarchical linear models, and it discusses the importance of Sewall Wright’s pioneering work on path coefficients and the role of direct and indirect variable effects/associations in multiple variable statistical regression analyses leading up to these modeling approaches Chapter provides a more detailed discussion of the problems involved in processes and trajectories, focusing on Prigogine’s research contributions Chapter illustrates the importance of energy potentials in all living processes and in the necessary transitions to better health Chapter diverges somewhat but illustrates a different viewpoint on probability: negative probability and the role of both actions and non-actions among variables in a process Along these lines, I present chaos theory, the systems perspective, ecological examples, and Poincare’s return maps and “sensitive dependence on initial conditions” in Chaps and 10 Continuing from Chap 10, I introduce topological aspects of Trajectory Analysis by considering health behaviors as processes that operate on physical as well as conceptual surfaces as they evolve over time Chapter 12 derives the important Jacobian matrix and its characteristic roots, the Lyapunov exponents, that directly measure trajectory changes Connected with these equations, Chap 13 describes phase transitions that are necessary for physiological and health behavior change, using the physical principle of the Rankine-Hugoniot Jump conditions Chapter 14 provides applied examples of nonlinear dynamics for cardiology and neuroscience interventions, although many of the latter primarily involve animal models Chapter 15 represents another divergence but illustrates again the universality and intricacies of nonlinear causes and effects across large time scales that ultimately contributed to our current existence and health scenarios Chapter 16 illustrates straightforward computations and simulations of nonlinear trajectory changes using Wilensky’s freeware NetLogo, an Agent-Based Modeling platform Finally, Chap 17 pulls everything together with perspectives and eight central principles of Trajectory Analysis that apply directly to health care I use many non-health references from diverse scientific disciplines to support the described methods and theory, so be prepared for somewhat of a wild ride Nevertheless, I trust that these sources will be informative and illuminating, and that you will freely explore these fascinating works There are many other excellent sources that unfortunately had to be omitted for purposes of brevity and clarity This is a rich subject area with so many possibilities for advancing health research Morrisville, NC, USA January 27, 2017 David W Hollar Acknowledgments I thank my wife, Paige, and daughter, Brooke, for their tremendous support during the writing of this book For Brooke, it was a true team effort in extraordinary ways! I also thank Virginia Dean and my good colleagues Dr Barnett Parker, Dr Nur Onvural, and Dr Jennifer Rowland I also thank my many colleagues with Pfeiffer University and the Billy Graham Evangelistic Association The journey in writing this work was seeded by a March 1989 National Science Foundation Chautauqua short course on Cellular Automata taught by Professor Max Dresden at SUNYStony Brook It was further motivated during my doctoral coursework under the supervision of Professors Bert Goldman, John Hattie, Mary Olson, Sam Miller, and Jim Lancaster at the University of North Carolina—Greensboro and through later interdisciplinary research in Disability and Children’s Genetics at Wright State University and the University of Tennessee I thank Janet Kim, Acquisitions Editor at Springer, for encouraging this project I thank Paramasivam Vijay Shanker for manuscript styling and proofing In all good things, we give thanks to God ix Contents Introduction: The Universality of Physical Principles in the Analysis of Health and Disease References Longitudinal and Nonlinear Dynamics “Trajectory” Analysis in Health Care: Opportunities and Necessity 2.1 Background 2.2 Necessary and Sufficient Conditions 2.3 Decision-Making in Longitudinal Research 2.4 Nonlinear Dynamics References 10 12 14 18 The Problem of Recidivism in Healthcare Intervention Studies 3.1 Periodic Behavior 3.2 Stages of Change Models 3.3 Education, Race, Socioeconomics 3.4 Biopsychosocial Models 3.5 Health Literacy Issues in Recidivism 3.6 Examples 3.7 Behaviors Locked in Periodic Patterns 3.8 Tinbergen’s Four Questions and Ethology 3.9 The Issue of Creating Bifurcations References 21 22 24 26 27 29 29 30 31 32 34 Epidemiological Methods 4.1 Types of Studies 4.2 Non-experimental Studies 4.3 Demographic Considerations 37 38 42 44 xi xii Contents 4.4 Methods of Analysis 4.5 Summary References 45 46 46 The Method of Path Coefficients 5.1 Background 5.2 Path Coefficients 5.3 Structural Equation Models 5.4 Hierarchical Linear Models 5.5 Examples 5.6 Agent-Based Models 5.7 Nonlinearity 5.8 Causal Inference and Complexity 5.9 Validity and Reliability (Accuracy and Precision) 5.10 Summary References 49 49 51 54 59 62 63 64 66 67 69 70 Stability and Reversibility/Irreversibility of Health Conditions 6.1 Irreversible Change and the Arrow of Time 6.2 Levels of Functioning 6.3 Measuring Disturbances to Functioning 6.4 Human Development 6.5 Summary References 73 74 76 77 79 83 84 Energy Levels and Potentials 7.1 Energy Is Central to Life Processes, Health, and Change 7.2 Quantum Metabolism and Health 7.3 Systems Topology and Ecology 7.4 Catastrophes 7.5 Energetic Jumps and Interventions 7.6 Stability and Instability in Health 7.7 Summary References 87 87 88 90 91 93 96 98 98 On Negative Probabilities and Path Integrals 8.1 Healthcare Analysis and Medical Errors 8.2 Population Health Distributions 8.3 Superposition of Wave Phases (States) and Negative Probability 8.4 Applications of Negative Probabilities 8.5 Cancer 8.6 Balancing Health and Probabilities 8.7 The Wave Function 104 106 108 109 109 101 101 102 248 17 Review of Basic Principles manifolds/surfaces Most importantly, Trajectory Analysis will be essential for the future of health care, when researchers, clinicians, and policy makers will have to merge health assessments and interventions/treatments with individualized genomics, epigenomics, metabolomics, and the diverse microbiome inhabiting the body 17.1 Principles The major principles that comprise Trajectory Analysis include the following: Change is continuous and often cyclic/periodic in nature, with angular frequencies measuring the phase of a health process or condition; Multiple correlating variables with direct and indirect effects contribute to the trajectories of health outcomes, either driving (facilitating) or dissipating (inhibiting) the health condition or behavior; Variable effects can be positive, negative, or neutral, and they can include both actions and non-actions (negative probabilities); Body systems are resilient, but with specific circumstances (e.g., low immune function, high stress), health can change and conditions emerge due to sensitive dependence on these initial (or new) conditions; Longitudinal health trajectories continuously change, but their return paths usually return to normal within a brief but finite time, or they may diverge, with the Jacobian matrix and its characteristic root Lyapunov exponents measuring the change; Health varies, but a genuine transition from a positive health state to a negative health state, or vice-versa, that is maintained requires a Type Phase Shift as measured by a Phase Response Curve (PRC) All health processes operate via energy potentials, such that the two optimum strategies to achieve a Type Health Phase Shift are: (1) a Rankine–Hugoniot Jump with the input of energy or resources (and maintenance thereafter), and (2) superimposition of correlating driving variables with the health condition to shift and maintain the desired condition; and Poor health is associated with molecular and system instabilities/chaos, as measured by a Lyapunov exponent λ > and especially approaching λ ~ 3, or a decreasing coherence length ξ Hayflick (2007) and Davies, Demetrius, and Tuszynski (2012) argued molecular instability as central characteristics of aging and cancer Pecora and Carroll (1990) demonstrated that chaotic systems can be stabilized by synchronizing their aberrant oscillations with driving signals, forcing the chaotic system to superimpose and correlate with the introduced signal Similarly, Ott, Grebogi, and Yorke (1990) argued that small disturbances to a chaotic system can nudge the system into varied periodic cycles From an applied perspective, Buzsa´ki and Wang (2012) provided evidence that gamma oscillations (35–45 Hz) might be useful in regulating various 17.2 Methods 249 neurological conditions, including sleep/wake cycles Applications of cardiac arrhythmia phase resetting via cardiac catheter electrical stimulation of the heart were described in Chap 14 The above eight principles are universal for living and nonliving processes Thus, they represent an experimental approach to support current and epidemiological methods for improving health research, policy, and clinical practice They provide us with approaches to conceptualizing physiological as well as psychological health processes 17.2 Methods The central approaches to measuring changes in health trajectories involve the collection of continuous, longitudinal data with many time points so that trends can be plotted, mapped, and analyzed The contributions of multiple driving and dissipative variables can be assessed via multiple regression statistical analysis and higher order structural equation models as well as hierarchical linear models For trajectory analysis, we move further with the analysis of cycles, return maps, and deviations from trajectories We utilized differential equations, which we have simplified to focus on Jacobian matrices and the calculation of Lyapunov exponents and system phases Cvitanovic et al (2004, pp 132–134) defined the Lyapunov exponent λ as the positive diagonal elements of the Jacobian matrix and that can be calculated as (see Chap 12, Eq 12.2): ẳ 1=tịlnjxt1 ịj=jxt0 ịjị 17:1ị which matches closely the Glass and Mackey (1988, p 54) derivation of λ (Eq 11.10) For system phase, Glass and Mackey (1988, p 105–106) defined the initial system state or phase as θ0 ¼ 0, shifted phases following a disturbance are θn ¼ tn/t0, and the change of phase (see Chap 11, Eq 11.8): ẳ tn t0 ị=t0 17:2ị Using Eq (17.2), we can map the change of phase Δθ (ordinate) versus each phase point over time t (abscissa) to construct a Phase Response Curve (PRC) A Type phase shift will be cyclic, whereas an effective Type phase shift will show separation of curves Finally, the deviations of trajectories follow hyperbolic mappings of pressure curves at critical transition points, such that λ ¼ corresponds to a circle, whereas increasing positive λ topologically corresponds to the increasing eccentricity ε of a stretching circle As λ positively increases, the circle becomes elliptical, then parabolic It becomes hyperbolic near λ ~ and the phase response curve pattern of chaos Simultaneously, λ is inversely proportional to the coherence length ξ, as 250 17 Review of Basic Principles two resonating systems in stability have greater correlations or coherence when λ is near zero Therefore, we have our final relationship for understanding trajectory change: λ $ ε $ ξÀ1 ð17:3Þ This relationship also is proportional to Kolmogorov entropy for a system and to the fractal dimension of repeatability at various levels within a system Currently, applied applications of trajectory analysis in health care are scarce due to the lack of extensive continuous, longitudinal data on individuals that is representative, consistent, and offering informed consent of study participants Epidemiological trend and time series analyses exist and are extremely useful to health policy experts and clinicians However, the periodicities of these trends and the contributions of many independent variables have not been widely addressed in these models Simulation and Agent-Based Models have a rich history for theory development and modeling system behaviors (Dresden & Wong, 1975; Grimm & Railsback, 2005) These approaches represent the mainstay of current trajectory analysis, although we are seeing their direct applications in cardiology and neuroscience (Chap 14) 17.3 The Future Public health and medicine face many challenges with growing, diverse human populations that are living longer Figure 17.1 shows Gompertz mortality curves with increasing age for American males (Fig 17.1a) and females (Fig 17.1b) The mortality rates are provided on a logarithmic scale to linearize the relationship (Gompertz, 1825; Riklefs & Finch, 1995) Examining each curve in Fig 17.1, the increase in the probability of dying stays relatively flat with only slight increases from age 20 to 40, after which there is a doubling of the probability roughly every years for both males and females Gompertz’ (1825) work is the basis for the actuarial tables that life insurance companies use to compute premium rates, besides individual risk indicators The curves are remarkably consistent across race, culture, and nationalities, although the curve is more compressed toward younger ages for disadvantaged countries and regions (Riklefs & Finch, 1995) What is most striking about the mortality curves is the age range 14–24 This range represents the largest increase in the probability of dying for the entire lifespan, and this bump exists for all populations studied (Riklefs & Finch, 1995) The probability of dying for males increases fourfold, and the probability of dying for females doubles during this life period The probable causes include variations in cognitive development maturation, raging hormones, reproductive competition, and especially engagement in risk behaviors and experimentation, including the impact of these behaviors on others While most people survive this period, some people not The instabilities of this age range are clearly defined along with the 17.3 The Future 251 A Males LOG DATA - MALE PROBABILITY OF DYING 00 -1.00 -2.00 -3.00 -4.00 20.0 40.0 60.0 80.0 100.0 120.0 100.0 120.0 AGE INTERVAL BY MIDYEAR B Females LOG DATA - MALE PROBABILITY OF DYING 00 -1.00 -2.00 -3.00 -4.00 20.0 40.0 60.0 80.0 AGE INTERVAL BY MIDYEAR Fig 17.1 Gompertz logarithm mortality rate curves by age in years for American males (a) and females (b) Data from the CDC National Center for Health Statistics 2002 Death index plotted by the author using SPSS version 18.0 252 17 Review of Basic Principles probable causes Therefore, trajectory analysis with longitudinal interventions (education, support mechanisms) certainly applies for this sensitive period of human development as well as for the entire lifespan With the 7-year doubling period past age 40, we can examine the variables that promote increased risk to physical decline, disease, and disability so that more individuals experience healthy aging instead of accumulating conditions (Hayflick, 2007) 17.4 Context At the beginning of the twentieth century, all human populations were susceptible to infectious diseases, poverty, and associated suffering Figure 17.2 shows an eroding 111-old creek stone that was used as a grave marker for one of the author’s ancestors, who died at the age of years Her mother chiseled a barely perceptible eulogy into the flat stone, “Our Little Darling at rest.” Such a scenario was not uncommon even in the Western World during the early twentieth century Fortunately, the advent of antibiotics and drug discovery, public health programs, immunizations, water sanitation, medical technologies, and standardization of best clinical practices revolutionized health care in the West and globally However, health disparities remain widespread due to poverty, even in Western nations, and due to environmental catastrophes, wide socioeconomic disparities, war, and other human crises worldwide The remarkable process of life represents a far-from-equilibrium, nonholonomic maintained system (Courbage, 1983; Nicolis & Prigogine, 1981) that has been continuous, without interruption for approximately 3.8 billion years on the fragile Fig 17.2 A forgotten child’s grave in 2017 after 111 years Photograph by the author 17.4 Context 253 biospheric film covering the earth’s solid and liquid surfaces, the only known planet where this process has occurred around approximately 1023 stars in the universe (van Dokkum & Conroy, 2010) This continuous process has survived numerous biogeochemical crises and accompanying massive extinctions of both terrestrial and extraterrestrial origins (Vernadsky, 1997; Brenchley & Harper, 1998; Hoffman, Kaufman, Halverson, & Schrag, 1998; Erwin, 2008), today producing tremendous variation both within and between species (Darwin, 1859; Wilson, 1975) Despite the mass dominance of prokaryotes (i.e., bacteria) throughout the history of life, the past 600 million years of life have been characterized by the emergence of multicellular, endosymbiotic living network development having high complexity in the eukaryotes: protists, fungi, plants, and animals (Maynard Smith & Szathmary, 1999; Seielstad, 1989; Wilson, 1975) All life and all physical events in the universe are predicated on energetic potentials and driving forces to maintain these potentials The earth is not a closed system, nor is the solar system despite the staggeringly vast interstellar distances and near complete vacuum conditions between even nearby stars For our solar system, the sun (Fig 17.3) overwhelmingly is the driving energetic force, releasing 2.4  1039 MeV s-1 of energy and 1.8  1038 neutrinos sÀ1 isotropically (Rolfs & Rodney, 1988, p 491) About 64  109 neutrinos cmÀ2 pass through the earth’s surface each second, although it is the dissipated energy that drives all life on earth Furthermore, the sun, its planets, a trillion comets, and other orbiting objects orbit over 250  106 years at 2.5  1017 km distance the central black hole singularity of the Milky Way galaxy at a relative velocity of 16.5 km sÀ1 and oscillating above and below the galactic plane at a periodicity of 66  106 years (Bash, 1986; Frisch, Fig 17.3 The Sun, 4:28:10 GMT on February 2016 Photograph captured and enlarged on February 2016 using the open source software JHelioviewer® (www.jhelioviewer.org), European Space Agency and National Aeronautics and Space Administration 254 17 Review of Basic Principles 1993), the latter no doubt affected by some unknown external driving force from the galaxy’s past Some of the tiny but relatively substantial percentage of energy that reaches the earth’s surface is absorbed by the thylakoid membranes of photosynthetic bacteria and algal/plant chloroplasts, with the natural semiconductor chlorophylls, carotenoids, and xanthophyll’s triggering a series of coupled oxidation/reduction reactions that ultimately reduce nicotinamide-based cofactors for further coupling to water, carbon dioxide/oxygen, glucose, and other metabolic cycling reactions in cells Without thylakoids to capture solar energy, animals, protozoans, and fungi steal glucose and metabolites from plants and from each other to further couple metabolic activities and the reverse cellular respiration activities (oxidation/reduction potentials again) that drive energy release in eukaryotic mitochondria or bacterial cell membranes Therefore, living systems cycle energy along potential gradients in order to survive (Eigen & Schuster, 1979), thereby driving each other, with the predominantly ultimate driver being the sun (Fig 17.3) We have hypercycles of periodic events and order impacting each other throughout the universe, either directly or indirectly, via chemical and physical processes of photosynthesis, cellular respiration, and solar nucleosynthesis This connectedness and the ramifications for subtle changes in each of us over the course of our lives are staggering We detect minute radio signals from distant stars, galaxies, and quasars, even potential evidence of extraterrestrial intelligence (Seielstad, 1989) using radiotelescopes such as the instrument shown in Fig 17.4 that astronomer Frank Fig 17.4 Two radiotelescopes at the National Radio Astronomy Observatory, Green Bank, West Virginia Photograph by author, May 1988 17.6 Summary 255 Drake used for the first SETI experiment during the early 1960s These large telescopes collect tiny photons of electromagnetic radiation that have traveled for millions and even billions of years How much more readily available are the health measurement tools that are at our disposal! Trajectory analysis illustrates the wide effects of many variables both direct and indirect With increasing complexity, we are seeing that even complicated classical systems exhibit quantum behaviors (Vattay, Kauffman, & Niiranen, 2014) that are consistent with Nicolis and Prigogine’s (1981) arrow of time and the irreversibility of correlating (coherent) systems that lead to change at all levels Except for our spiritual aspects that are beyond measurement, we physically are part of the universe and are subject to all of its effects Health conditions are not limited to arbitrary classifications that currently are intensively studied Instead, we need to focus on multiscale events and processes that create change in health, many conditions that can be mediated or moderated based upon our mapping of these dynamic events 17.5 Perspective Lightman and Gingerich (1991, p 255) defined a scientific anomaly as “an observed fact that is difficult to explain in terms of the existing conceptual framework,” and they suggest the use of retrorecognition as a psychological tool to address unexplained facts/givens within new frameworks and to provide improved theories of knowledge As we perform health research, taking alternate perspectives and employing tools from other scientific disciplines (i.e., benchmarking) enables us to see problems and potential solutions in new lights Kahneman (2003) challenged researchers and clinicians to engage in increased Systems Reasoning as we critically attack problems and fallacies in decisionmaking and problem-solving Trajectory analysis requires new conceptual frameworks to study complex systems of variables that interact and affect health or any type of action in the universe 17.6 Summary We described eight basic principles of Trajectory Analysis that are applicable to health care The science of nonlinear and trajectory dynamics is growing rapidly, although it remains mostly theoretical Cardiology and neuroscience have introduced these concepts to improve human health Our next steps are to expand this science into health care to support current statistical and epidemiological methods 256 17 Review of Basic Principles References Bash, F (1986) Present, past and future velocity of nearby stars: The path of the sun in 108 years In R Smoluchowski, J N Bahcall, & M S Matthews (Eds.), The galaxy and the solar system Tucson: University of Arizona Press Brenchley, P J., & Harper, D A T (1998) Palaeoecology: Ecosystems, environments and evolution London: Chapman & Hall Buzsa´ki, G., & Wang, X.-J (2012) Mechanisms of gamma oscillations Annual Review of Neuroscience, 35, 203–225 Courbage, M (1983) Intrinsic irreversibility of Kolmogorov dynamical systems Physica, 122A, 459–482 Cvitanovic, P., Artuso, R., Dahlqvist, P., Mainieri, R., Tanner, G., Vattay, G., et al (2004) Chaos: Classical and quantum, version 14.4.1 (April 21, 2013) Accessed 01 Feb 2015 at ChaosBook org Darwin, C R (1859) On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life London: John Murray Davies, P., Demetrius, L A., & Tuszynski, J A (2012) Implications of quantum metabolism and natural selection for the origin of cancer cells and tumor progression AIP Advances, 2, 011101 http://dx.doi.org/10.1063/1.3697850 Dresden, M., & Wong, D (1975) Life games and statistical models Proceedings of the National Academy of Sciences USA, 72(3), 956–960 Eigen, M., & Schuster, P (1979) The hypercycle: A principle of natural self organization Berlin: Springer Verlag Erwin, D H (2008) Extinction: How life on earth nearly ended 250 million years ago Princeton, NJ: Princeton University Press Frisch, P C (1993) G-star astropauses: A test for interstellar pressure The Astrophysical Journal, 407, 198–206 Glass, L., & Mackey, M C (1988) From clocks to chaos: The rhythms of life Princeton, NJ: Princeton University Press Gompertz, B (1825) On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies Philosophical Transactions of the Royal Society of London, 115, 513–583 Grimm, V., & Railsback, S F (2005) Individual-based modeling and ecology Princeton, NJ: Princeton University Press Hayflick, L (2007) Entropy explains aging, genetic determinism explains longevity, and undefined terminology explains misunderstanding both PLoS Genetics, 3(12), 2351–2354 Hoffman, P F., Kaufman, A J., Halverson, G P., & Schrag, D P (1998) A neoproterozoic snowball earth Science, 281, 1342–1346 Kahneman, D (2003) Maps of bounded rationality: Psychology for behavioral economics The American Economic Review, 93(5), 1449–1475 Lightman, A., & Gingerich, O (1991) When anomalies begin? Science, 255, 690–695 Maynard Smith, J., & Szathmary, E (1999) The origins of life: From the birth of life to the origin of language New York: Oxford University Press Nicolis, G., & Prigogine, I (1981) Symmetry breaking and pattern selection in far-from-equilibrium systems Proceedings of the National Academy of Sciences USA, 78(2), 659–663 Ott, E., Grebogi, C., & Yorke, J A (1990) Controlling chaos Physical Review Letters, 64(11), 1196–1199 Pecora, L M., & Carroll, T L (1990) Synchronization in chaotic systems Physical Review Letters, 64(8), 821–824 Riklefs, R E., & Finch, C E (1995) Aging: A natural history New York: Scientific American Library, WH Freeman & Co Rolfs, C E., & Rodney, W S (1988) Cauldrons in the cosmos: Nuclear astrophysics Chicago: University of Chicago Press References 257 Seielstad, G A (1989) At the heart of the web: The inevitable genesis of intelligent life Boston: Harcourt Brace Jovanovich Thom, R (1972) Structural stability and morphogenesis: An outline of a general theory of models New York: W.A Benjamin/Westview Thom, R (1977) Structural stability, catastrophe theory, and applied mathematics: The John von Neumann lecture, 1976 SIAM Review, 19(2), 189–201 van Dokkum, P G., & Conroy, C (2010) A substantial population of low-mass stars in luminous elliptical galaxies Nature, 468, 940–942 Vattay, G., Kauffman, S., & Niiranen, S (2014) Quantum biology on the edge of quantus chaos PloS One, 9(3), e89017 doi:10.1371/journal.pone.0089017 Vernadsky, V I (1997) The biosphere (trans: Langmuir, D.) M McMenamin (Ed.), New York: Nevraumont/Copernicus Wilson, E O (1975) Sociobiology: The new synthesis Cambridge, MA: Harvard/Belknap Wilson, E O (1998) Consilience: The unity of knowledge New York: Alfred A Knopf Index A Actions, 2, 24, 29, 101, 102, 112, 114, 127, 136, 138, 147, 150, 151, 157, 207, 244, 247, 248, 255 Addiction, 62, 113, 245 Adenosine monophosphate (AMP), 224 Adenosine triphosphate (ATP), 32, 88, 89, 91, 113, 149, 201 Adrenal, 125, 180, 221, 222 Agent based models (ABMs), 63, 64, 232, 245 Aggression, 80, 81, 154, 223, 225 Aging, 3, 8, 57, 66, 74, 75, 88, 95, 96, 109, 112–114, 122, 126, 127, 140, 154, 156, 181, 183, 190, 201, 205, 211, 213, 215, 222, 223, 248, 252 Allostatic load, 21, 98, 180, 222 Alveolar cells, 191 Angular frequency (ω), 111, 135, 139, 152, 248 Aperiodic, 16, 33, 108, 118, 136, 138, 141, 151, 154, 222 Arrow of time, 7, 8, 14, 74–76, 83, 255 Arteries, 97, 132, 205 Atrioventricular node, 2, 33 Attractor, 15, 118–121, 157, 172, 200 Axons, 2, 197, 202, 203 B Bacteria, 3, 73, 97, 113, 150, 214–216, 219, 224, 239, 243, 253, 254 Barriers, 28, 31, 32, 76, 77, 122, 127, 133, 160, 185, 189, 235 Behaviors, 1–3, 5, 10, 15, 16, 21–33, 39, 42, 44, 50, 51, 53, 54, 59, 61–63, 68, 69, 75, 78–83, 88, 90, 91, 93, 95, 96, 98, 101, © Springer International Publishing AG 2018 D.W Hollar, Trajectory Analysis in Health Care, DOI 10.1007/978-3-319-59626-6 106, 117, 121, 122, 126–128, 131–137, 139–141, 143, 144, 147, 160, 164–166, 168–170, 172, 176, 180, 183, 188, 191, 192, 198, 201, 204, 211, 217–224, 232, 237, 240, 243–245, 247, 248, 250, 255 Belousov-Zhabotinsky reactions, 199 Bioenergetics, 1, 88, 89, 95, 97, 98, 105, 109, 113, 128, 205, 214 Biopsychosocial, 21, 24, 27–29, 31, 51, 77, 159, 168, 180 Bone remodeling, 92, 110, 182 Bonhoeffer, D., 101, 114 C Cancer, 3, 13, 30, 31, 44, 51, 88, 92, 96, 97, 105, 108, 109, 112–114, 123, 126, 127, 150, 190, 201, 206, 214, 219, 221, 243, 248 Cardiac catheters, 198, 203, 249 Cardiology, 90, 159, 160, 169, 192, 197–207, 231, 250, 255 Cartesian, 151, 167 Case-control studies, 38, 42, 43 Catastrophe theory, 78, 89, 90, 92, 95, 183, 231 Causal, 4, 30, 44, 49, 51, 54, 57, 66, 154 Cause-and-effect, 9–11, 49, 57, 66, 79, 132 Cellular respiration, 4, 91, 113, 254 Chaos, 5, 15, 16, 139, 142, 151, 166, 172, 198, 201, 231, 248, 249 Chaos theory, 117–119, 121–123, 125–128 Characteristic root, 248 Christian, J.J., 125, 138, 180, 221, 223, 238 Circadian rhythms, 2, 95, 110, 113, 136, 204 Coefficient of variation (R2), 50 259 260 Coherence length (ξ), 77, 78, 81, 83, 96, 111, 118, 122, 124, 128, 129, 136, 142, 159, 190, 192, 204, 247–249 Community intervention trials, 38, 40 Competition, 17, 82, 90, 92, 97, 121, 171, 182, 212, 214, 216, 219, 222, 233, 235, 250 Complexities, 3, 9, 22, 63, 66, 74, 93, 97, 125, 127, 128, 143, 151, 168, 169, 180, 182, 201, 203, 212, 243, 255 Consilience, 1, 4, 83, 211, 247 Counterbalanced designs, 38, 40, 41 County Health Rankings, 43, 171, 173, 175, 176 Critical points, 77, 78, 83, 94, 95, 122, 141, 180, 182, 184–191, 200 Critical threshold, 179 Cross-sectional studies, 38, 42, 43, 138 Curie, P., 1, 183 Cyclic, 2, 22, 29, 75, 113, 128, 142, 152, 153, 156, 164, 168, 172, 199, 202, 204, 235, 248, 249 Cytokines, 216, 218–222 D Dawes, R M., 12, 13, 97, 123 Deoxyribonucleic acid (DNA), 66, 75, 97, 212 Dependence, 118 Dirac, P.A.M., 105, 184, 186–191 Disadvantaged, 250 Discontinuities, 61, 92, 140, 156, 158, 180, 185, 200 Dissipative, 16, 77, 110, 141, 151–154, 157, 235, 249 Disturbances, 2, 8, 18, 22, 32, 33, 59, 77–79, 95, 98, 117, 118, 121, 127, 131, 142, 154–157, 159, 168, 187, 191, 192, 197–199, 203–205, 207, 213, 215, 231, 248, 249 E Eccentricity (ε), 2, 18, 94, 119, 131, 151, 159, 166, 249 Ecologic studies, 38, 42, 43 Ecology, 37, 89–91, 95, 97, 123, 124, 150, 170, 171, 211, 243, 247 Eigen, M., 2, 61, 91, 107, 113, 124, 151, 182, 183, 187, 211–213, 216, 254 Einstein, A., 7, 88, 190 Electrocardiogram (ECG/EKG), 22, 33, 110, 121, 132, 198, 201, 204 Index Electroencephalogram (EEG), 110, 136, 168, 169, 204 Electron, 8, 89, 91, 93–96 Energy, 1, 8, 16–18, 32, 73–75, 77, 78, 81, 105–107, 109, 110, 117, 147, 149, 151, 152, 159, 181, 184, 185, 187–192, 200, 201, 205, 233, 248, 253, 254 Energy levels, 87, 88, 90–98, 121, 141 Enthalpy, 184 Entropy, 1, 8, 74, 77, 78, 81, 83, 91, 94–97, 105, 109, 113, 117, 118, 121, 129, 154, 159, 250 Enzymes, 74, 107, 181, 192, 204 Epidemiology, 1, 3, 4, 10, 11, 18, 24, 37–46, 79, 88, 90, 97, 98, 105, 106, 122, 123, 128, 138, 140–143, 154, 156, 159, 168, 189, 231, 247, 249, 250, 255 Epigenetics, 2, 8, 9, 44, 69, 74–76, 79, 80, 83, 96, 97, 102, 109, 113, 132, 139, 154, 156, 160, 200, 212, 222, 245 Equilibrium, 8, 32, 73, 75, 77, 81, 89, 93–96, 98, 110, 123, 185 Eukaryote, 91, 109, 113, 140, 147, 214, 215, 253, 254 Exercises, 3, 21, 24, 25, 73, 88, 96, 103, 132, 139, 153, 158, 159, 168, 182, 202 Experimental studies, 24, 38–41 F Facilitators, 28, 29, 31, 66, 76, 128 Fallacies, 10–12, 75, 102, 255 Far-from-equilibrium, 73, 75, 83, 94–96, 98, 190, 252 Feynman, R., 74, 78, 98, 105–107, 112–114, 132, 140, 189 Field trials, 38, 40 Fluid mechanics, 243 G General adaptation syndrome, 21, 125 Generalized Linear Model (GLM), 60, 69 Genetics, 9, 37, 44, 45, 51, 57, 68, 75, 76, 79, 80, 83, 88, 93, 97, 98, 102, 103, 106, 107, 109, 113, 123, 132, 139, 149, 150, 153, 154, 156, 160, 170, 181, 182, 192, 212, 214–217, 223, 225, 245 Genomic, 9, 44, 66, 88, 95, 98, 170, 206, 244, 248 Geospatial, 170, 171 Gompertz, B., 126, 250, 251 Index 261 H Harmonics, 22, 104, 110, 133–139, 190, 242 Hayflick, L., 8, 14, 88, 95, 96, 112, 201, 211, 213, 215, 221, 248, 252 Health, 1, 7, 21, 37, 50, 74, 87, 101, 133, 147, 197, 211, 247 Healthy People 2010, 3, 138 Healthy People 2020, 173, 244 Healthy People 2030, 244 Heart rate variation (HRV), 201, 202 Hierarchical linear models (HLM), 45, 59–61, 63, 65, 66, 69, 232, 249 Homoscedasticity, 50 Human leukocyte antigen (HLA), 126, 170, 217, 218, 222, 224, 225 Hydrodynamics, 205 Hypercycles, 2, 95, 108, 182, 212–214, 216, 219, 220, 247, 254 Hypothalamic-pituitary-adrenal axis (HPA), 221–224 Hypothalamic-pituitary-thyroid hormone axis (HPT), 221 L LDL cholesterol, 60, 123 Left ventricular ejection fraction (LVEF), 206 Long terminal repeats (LTR), 217 Longitudinal, 1, 3, 7–14, 16–18, 22, 28–31, 38, 46, 50, 57, 59, 63, 65, 82, 93, 98, 123, 127, 133, 136–138, 156, 159, 160, 191, 198, 202, 231, 232, 244, 245, 247–250, 252 Lorenz attractor, 118–121, 157 Low birth weight, 171–176 Lyapunov exponent (λ), 1, 78, 83, 95, 96, 111, 117–119, 122, 124, 127, 129, 131, 136, 142, 156, 157, 159, 163–177, 182, 183, 187, 188, 190, 192, 198, 201, 202, 204, 231, 237, 240–243, 247–249 Lyapunov, A., 1, 77, 78, 81, 83, 95, 96, 117–120, 122, 124, 127, 129, 131, 136, 142, 156, 157, 159, 163–176, 182, 183, 187, 188, 190, 192, 198, 201, 202, 204, 231, 237, 240–243, 247–249 Lymphocytes, 97, 114, 126, 217–221 I Immune system, 92, 108, 122, 157, 216, 219–221 Immunity, 126, 217–221 Individual-based model (IBM), 63 See also Agent Based Model (ABM) Insect vectors, 240 Instabilities, 88, 95–98, 109, 112, 114, 119, 125, 127, 141, 142, 150, 156, 171, 198, 201, 202, 213, 238, 248, 250 Institute of Medicine (IOM), 101, 102 Institute of Medicine report, 102 Instrumental Activities of Daily Living (IADL), 27 Interferon (IFN ), 217 Interleukin (IL), 221 International Classification of Functioning, Disability and Health (ICF), 31, 76, 77 Irreversibility, 83, 185, 191, 255 M Macroscales, 89, 122, 123, 140, 141, 214 Macroscopic, 7, 74, 78, 88, 92, 96, 160 Major histocompatibility complex (MHC), 44, 170, 215–220, 223–226 Malaria, 240 Manifolds, 3, 16, 18, 23, 90, 91, 138, 148–152, 157, 159, 160, 163, 164, 167, 169, 181, 198, 232, 247 Medical errors, 10, 101, 102 Mendelian Inheritance in Man, Microarray, 203 Microscale, 101, 105, 123, 140 Microscopic, 88, 95, 97, 160 Mitochondria, 1, 66, 88, 90, 91, 95, 97, 105, 109, 113, 140, 147–149, 190, 201–203, 205, 214, 215, 221, 254 Morbidity, 2, 21, 30, 38, 42, 44, 69 Mortality, 2, 21, 30, 38, 42–44, 69, 138, 174, 250 Multiple-drug resistant Staphylococcus aureus (MRSA), 97 Myocytes, 147, 154, 197, 199–203 J Jacobian matrix, 23, 46, 159, 163–166, 172, 192, 247–249 Jump conditions, 179–185, 187–192 K Kahneman, D., 10–12, 81–83, 102, 105, 255 Kinetic energy, 87, 88, 105, 106 N National Center for Biotechnology Information (NCBI), National Center for Health Statistics (NCHS), 3, 138, 171 262 National Health and Nutrition Examination Survey (NHANES), 43 Necrosis factor, 217, 218 Negative probabilities, 1, 101–114, 117, 248 NetLogo, 64, 65, 232–234, 237, 242, 245 Neural networks, 64, 113 Neurology, 197, 203, 204 Neurons, 142, 203, 204 Neuroscience, 132, 141, 159, 160, 169, 198, 199, 231, 250 Nicotinamide adenine dinucleotide (NAD), 66 Nonaction, 114 Non-experimental studies, 38, 42–44 Nonlinear dynamics, 7–14, 16–18, 46, 117, 121, 148, 164, 169, 171, 191, 198–201, 203, 206, 231, 243, 247 Nonlinearity, 57, 61, 64, 65, 205 Nutrition, 2, 3, 21, 69, 88, 96, 97, 138, 139, 150, 154, 182, 219, 223 O Old field succession, 125, 215 Olfactory receptor (OR), 224 Overpopulation, 125, 138, 235, 238 P Path coefficients, 49, 51–54, 56, 58, 62, 69, 75, 78, 83, 89, 98, 101, 112, 113, 133, 148, 234, 247 Path integrals, 112, 113 Pembrey, M., 2, 69 Periodicity, 22, 90, 118, 119, 121, 123, 126, 128, 131–133, 139–143, 152, 156, 159, 166, 171, 172, 198, 253 Perturbations, 8, 16, 17, 127, 131, 132, 168, 192 Phase reset, 33, 73, 157, 198–202, 213, 249 Phase resetting, Phase response curves (PRCs), 113, 142, 156, 172, 190, 198–200, 238, 243, 247–249 Phase space, 14, 17, 18, 121, 122, 141, 143, 150, 155–160, 165, 169, 172, 189 Phase transition curve (PTC), 156 Phase transitions, 1, 32, 166, 181–185, 187–192 Photosynthesis, 113, 254 Physiology/physiological, 2, 3, 9, 14, 39, 73, 77, 81, 83, 87, 88, 91–93, 95, 96, 102, 103, 112–114, 122, 123, 125–127, 129, 132, 135, 140–144, 150, 151, 153, 154, 157, 160, 164, 166, 170, 176, 180–183, 188, 191, 192, 198–201, 203–205, 207, Index 216, 219, 221, 223, 226, 232, 233, 244, 245, 247, 249 Planetary, 1, 2, 4, 89–91, 95, 109, 112, 128, 129, 131, 133, 135, 140, 141, 147, 205, 211, 212, 214–217, 219, 226, 233, 253, 254 Poincare return maps, 22, 23, 95, 118, 131–144, 156, 157, 159, 163, 172 Poincare, H., 14, 117, 118, 123, 129, 243 Polar coordinate, 151 Populations, 3, 8, 30, 38, 40, 42–44, 62–64, 69, 76, 82, 98, 102, 103, 106, 124–126, 137, 138, 154, 170, 171, 182, 187, 216, 217, 222, 223, 233–235, 237–245, 250, 252 Post-traumatic stress disorder (PTSD), 133, 150 Posttest, 4, 38, 39, 41, 42, 140 Potential energies, 87, 92–96, 121 PQRST waves, 2, 33, 132 Pretest, 4, 38, 39, 41, 42, 140 Prigogine, I., 7, 8, 14, 16, 17, 51, 73, 74, 77, 78, 83, 96, 185, 188, 190, 252, 255 Probability, 8, 12, 13, 32, 39, 50, 65, 74, 101, 104–109, 112–114, 117, 124, 126, 132, 156, 159, 168, 181, 250 Process, 1, 7, 22, 51, 87, 88, 101, 102, 105–109, 112–114, 117, 131, 147, 163, 180, 211, 233, 247 Prokaryote, 212, 214, 253 Proton-motive force, 1, 91, 109, 147, 201 PTC See Phase transition curve (PTC) Public health, 1, 3, 4, 7, 9, 24, 31, 37, 44, 50, 54, 57, 75, 82, 88, 90, 97, 98, 123, 148, 154, 159, 171, 174, 188, 250, 252, BNF–245 Python, 119, 164, 167 Q Quantum mechanics, 243 Quantum metabolism, 88, 89, 143 Quasi-experimental, 38, 40, 41 R Randomized clinical trials (RCTs), 39 Rankine-Hugoniot (R-H), 189, 190, 248 Reactive oxygen species (ROS), 66, 201, 215, 221, 222 Receiver operator characteristic (ROC) curve, 13, 97 Recidivism, 21, 22, 24–33, 61, 93 Regression analysis, 12, 45, 49, 50, 64, 131 Reliability, 58, 59, 67–69 Repeller, 15 Index Resonances, 2, 5, 16, 22, 78, 93, 95, 96, 128, 141, 202 Return maps, 1, 14, 18, 131, 156, 157, 159, 163, 172, 249 Reversibility, 83, 185 Ribonucleic acid (RNA), 9, 203, 217 Ruelle, D., 15, 18, 77, 83, 94, 95, 117–119, 157, 159, 163 S Schuster, P., 2, 61, 91, 151, 182, 187, 211–213, 216, 254 Selye, H., 21, 125, 180, 220 Sensitive dependence, 16, 17, 22, 117–119, 121–123, 125–128, 143, 156, 166, 198, 226, 232, 234, 243, 248 Sheep-wolf predation, 233–235 Sinoatrial node, 73, 121, 199, 201, 202 Sirtuin, 66 Socioeconomic, 8, 31, 44, 45, 60, 62, 63, 170, 171, 175, 180, 252 Solomon Four-Group design, 38, 39 Solzhenitsyn, A., 80 Spring, 110, 126, 133, 134, 141, 151, 235 Stabilities, 2, 3, 16–18, 73–83, 91, 92, 96, 98, 109, 119, 122, 125, 127, 128, 133, 138, 139, 141, 142, 150, 153, 156, 163, 164, 166, 168, 170, 172–176, 183, 192, 198, 201, 202, 204, 205, 212, 247, 250 Stages of change models, 24–26 Stress, 2, 4, 21, 25, 49, 54, 61, 73, 112, 125, 138, 144, 153, 179, 200–202, 205, 206, 214, 215, 219–225, 248 Structural equation model (SEM), 54–59, 61–63, 65, 68, 69, 83, 233, 249 Superposition, 104–107, 113, 114, 124, 135–139, 141 Surfaces, 3, 14, 23, 78, 89, 90, 95–97, 117, 124, 133, 138, 147–160, 164, 168, 169, 191, 199, 205, 212, 218, 224, 231, 235, 248 Surfactant, 191, 192 System perspectives, 98, 122, 123, 143, 159, 170, 215, 226, 247 Systems/Type reasoning, 12 T Thermodynamics, 8, 17, 74, 89, 91, 92, 97, 107, 154, 183, 243 Thom, R., 1, 2, 16, 17, 52, 57, 78, 89, 91, 92, 96, 121, 131, 148, 185, 215, 231 263 Thresholds, 9, 30, 79, 92, 97, 109, 121, 124, 138, 154, 156, 179, 180, 213 Tinbergen, N., 9, 31, 32, 37, 108 Toroid (-al), 119, 148–150 Trajectories, 38, 39, 45, 46, 53, 54, 59, 62–66, 74, 89, 90, 93–98, 101, 105, 109, 111, 113, 114, 117, 131, 132, 147, 163–166, 168, 169, 200, 203, 205, 206, 213, 231, 232, 237, 240, 243–245, 247–250, 252, 255 Trajectory analysis, 5, 7–14, 16–18, 22, 29–31, 33, 38, 45, 53, 62, 74, 78, 122, 136, 148, 160, 164, 166, 168, 190, 197, 203, 205, 226, 231, 243–245, 247–250, 252, 255 Transition points, 89, 92, 166, 182–185, 188, 190–192, 249 Trans-membrane potential, 199 Transtheoretical model, 24, 25 Tuberculosis, 239 Tumor necrosis factor (TNF), 169, 218, 221 Type phase reset, 157, 192, 198, 200, 248 Type phase reset, 157, 199 U U.S Centers for Disease Control and Prevention, 171 U.S Department of Health and Human Services, V Validity, 11, 38, 39, 41–43, 46, 53, 54, 57, 66–69, 79, 93, 137, 160, 202 Violence, 3, 80, 134, 150, 154, 224 Viruses, 7, 73, 87, 90, 97, 124, 158, 182, 212, 216, 217, 219, 222, 240, 245 Vomeronasal Organ (VNO), 224, 225 W Wall Shear Stress (WSS), 206 Wave functions, 78, 83, 109–112, 141, 154, 157, 176 Wilson, E.O., 4, 5, 68, 83, 124, 125, 211, 247, 253 World Health Organization (WHO), 51, 76, 80, 154 Wright, S., 49, 52, 112, 113, 133 Z Zika, 240 .. .Trajectory Analysis in Health Care David W Hollar Trajectory Analysis in Health Care David W Hollar Health Administration Pfeiffer University Morrisville, North Carolina, USA ISBN... reality but © Springer International Publishing AG 2018 D.W Hollar, Trajectory Analysis in Health Care, DOI 10.1007/978-3-319-59626-6_2 Longitudinal and Nonlinear Dynamics Trajectory Analysis also... 2008) Trajectory analysis in health care involves the mapping of sequences of events and multiple variables contributing to health outcomes for individuals As such, the term trajectory is used in