Nonlinear Systems in Medicine The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Higgins, John P. 2002. Nonlinear systems in medicine. The Yale Journal of Biology and Medicine 75(5-6): 247-260. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:4791064 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA YALE JOURNAL OF BIOLOGY AND MEDICINE 75 (2002), pp. 247-260. Copyright © 2003. All nghts reserved. MEDICAL REVIEW Nonlinear Systems in Medicine John P. Higginsa Cardiology Section, Noninvasive Cardiac Laboratories, VA Boston Healthcare System, Boston, Massachusetts Many achievements in medicine have comefrom applying linear theory to problems. Most current mnethods ofdata analysis use linear models, which are based on proportionality betweeni two vari- ables and/or relationships described by linear differential equations. However, nonlinear behavior commonly occurs within human systems due to their complex dynamic nature; this cannot be described adequately by linear models. Nonlinear thinking has grown among physiologists andphysicians over the past century, and non- linear system theories are beginning to be applied to assist in interpreting, explaining, andpredict- ing biologicalphenomena. Chaos theory describes elements manifesting behavior that is extremely sensitive to initial conditions, does not repeat itself andyet is deterministic. Complexity' theory goes one step beyond chaos and is attempting to explain complex behavior that emerges within dynamic nonlinear systems. Nonlinear modeling still has not been able to explain all ofthe complexitypresent in human systems, and further models still need to be refined and developed. However, nonlinear modeling is helping to explain some system behaviors that linear systems cannot and thuis will auigment our understanding ofthe nature ofcomplex dynamic systems within the human bocdy in health and in disease states. INTRODUCTION simple differential equations to the input. Yet systems within nature, including the human A system is a collection of interacting body, frequently lack mechanical periodicity elements. Behavior ofthe system is distinct or linear dynamics and thus are referred to from the behavior of its parts or elements as nonlinear systems [2]. Within nonlinear (Figure 1). These elements interact with each systems, output is usually not proportional other directly and indirectly to modulate the to input, and output for the same input value system fmnction. may not be constalnt over time [3]. Further- The reductionist or mechanistic view of more, in contrast to linear systems, breaking nature involves reducing systems into their a nonlinear system down into its elements component parts (elements) in an attempt to (parts) and analyzing those parts under con- understand them [1]. This is the basis oflinear trolled conditions does not accurately reflect system analysis, where output is proportional the complex behavior present, nor capture to or can be determined through applying the dynamic relationships operating between aTo whom all correspondence should be addressed: John P. Higgins M.D., M. Phil., Co- Director of Noninvasive Cardiac Laboratories, Instructor in Medicine - Harvard Medical School, Cardiology Section, 4C108, VA Boston Healthcare System, 1400 VFW Parkway, Boston, MA 02132; Tel.: 617-323-7700 ext. 6830; Fax: 617-363-5550; E-mail: [email protected]. 247 248 Higgins: Nonlinear systems in medicine S Y S T E M INPUT OUTPUT Figure 1. A system is composed of elements or parts (circle, square, rectangle, triangle, diamond) that manifest their own behavior, and can interact with each other (arrows). In addition, feedback loops may be present between elements (triangle output feeds back to diamond and square). The interaction and modulation of these elements under different conditions and different times result in a dynamic system, which can respond to the input at a particular time under particular conditions with a specific output. various elements [4, 5]. By analogy, know- "attractors." This leads to "Emergence," ing the structure ofwater (H20) gives you no which describes the order that arises from clue as to why water goes down a plughole what on initial inspection appears to be dis- in a vortex [6]. Likewise, with all the order, firther, such emergence can arise from knowledge ofthe individual musicians and local and simple rules within the system [6]. their instruments in isolation, one could The immune system, for example, con- never predict the high degree of interde- sists of various elements (macrophages, pendence and harmony of an orchestra monocytes, neutrophils) that interact with playing Beethoven's Fifth Symphony [7]. each other by means of messengers or sig- System behavior may be simple or nals (immunoglobulins, cytokines, inter- complex, static or dynamic. Simple systems leukins). This system is forever in a state of follow basic rules; thus with knowledge of flux, with complex offensive and defenses the elements that make up the system and maneuvers mounted against a foray of the rules that govern them, one can accu- invaders. Even when exposed to an identical rately predict the system behavior under stimulus, this system can respond differently various conditions. and various behaviors emerge depending on In contrast, a complex nonlinear sys- multiple external and internal influences. tem has been defined as "a system or whole Behavior ofthe elements (parts) ofthe consisting ofan extremely large and variable system may be periodic, chaotic, or random. number ofcomponent parts, where the indi- In addition, one element can exhibit one or vidual components display marked variabil- more types of behavior depending on the ity over time, and are characterized by a state of the other elements of the system high degree of connectivity or interdepen- and the overall system status at the time dence between variables" [8]. Rather than examined. exhibiting random behavior, most complex Not only are nonlinear systems impor- nonlinear systems will tend towards and tant to the collection and interpretation of manifest certain states more often, called data, but such nonlinear connectivity and Higgins: Nonlinear systems in medicine 249 variability within a system may be a requi- In the Twentieth Century, nonlinear site for health. Breakdown ofthese normal theory came into the spotlight by accident. nonlinear rhythms may produce "patho- In 1961, Edward Lorenz, a mathematician- logical rhythms," which may underlie dis- meteorologist working at the Massachusetts ease states [9]. Improved identification and Institute of Technology, observed what he recognition of such rhythms may help in believed was order masquerading as ran- diagnosing illness at an earlier stage [7]. In domness [16]. He used a simple mathemat- addition, timely interventions may augment ical model of weather patterns and a com- healthy rhythms and suppress pathological puter capable ofperforming multiple itera- rhythms and so maintain health [10-13]. tions (repetitions). After accidentally inputting an incorrect decimal point in a number, he noted that small variations in MATERIALS AND METHODS initial conditions (temperature or atmospheric In preparation for this paper, a review pressure) would cascade through various ofthe English-language scientific literature iterations into remarkably different output was performed primarily by searching the (weather conditions) [12, 17]. MEDLINE and EMBASE databases for the This and other observations by Lorenz time period 1966 through 2002. Keywords were the earliest reference to chaos theory used in the search included "complex," [18]. Robert May wrote about chaos in "dynamic," "systems," "nonlinear," "linear," regard to deterministic nonlinearbehavior in "chaos theory," and "complexity theory." 1974, but he credits James Yorke with using In addition, the bibliographies ofarticles found the term "chaos" to describe behavior man- were also searched for relevant articles. ifesting similar features to what Lorenz Also, standard texts on nonlinear systems had observed a decade earlier [19, 20]. and chaos theory were reviewed, as were After penetrating much ofthe physical nonlinear and comiiplex systems web sites. sciences, only recently has nonlinear sys- tems theory been seriously investigated and applied within the biological sciences. HISTORY OF NONLINEAR SYSTEMS Specifically, nonlinear systems are those In attempt to understand system struc- where output is not directly proportional to ture and function, different approaches have input, or cannot be described by linear dif- been used. The reductionist or mechanistic ferential equations; such output generally view of nature, which has been referred to cannot be modeled easily if at all [3, 21, as the "Newtonian Paradigm," "Cartesian 22]. A brief description of system element reductionism," or simply "reductionism," behavior will be followed by a discussion of involves reducing systems into their com- chaos theory and complexity theory, with ponent parts in an attempt to understand examples from medicine, and concluding them. This approach is fundamental to linear remarks. modeling [1]. One of the first to allude to nonlinear systems was James Clerk Maxwell, who late BEHAVIOR OF ELEMENTS OF A in the