The Relevance of Chaos Theory to Medical Analysis

The term “chaos” typically engenders a sense of complete randomness or disorder, evoking turbulent images such as a traffic accident or a noisy birthday party. But there is another definition for this term that is not yet as well-known: a mathematical one. The premise of the chaotic system is such that it approaches “asymptotic conditions” but never reaches them, with this scientific definition being best summarized by a quotation from Chaos Theory Tamed, where Garnett P. Williams interprets it as “sustained and disorderly-looking long-term evolution that satisfies certain special mathematical criteria and that occurs in a deterministic non-linear system.”2 As will be shown, this definition only appears to be complicated, and it it is important to note that its general definition reflects the fact that chaos is truly an interdisciplinary topic. Physics, chemistry, social sciences, and biology all have the potential to be dramatically transformed in the future through chaos theory.5 We have known for a long time that nothing is ever linear in nature.4 Yet, there appears to be such structure and order present, from the never-ending spirals on a head of broccoli to the rhythmic scales of a snake. This is why the use of models of chaos theory has an extremely high prospect for success: its applicability to the non-linear systems that exist today makes it an area of modeling that we can rely on to deepen our understanding of a multitude of fields.

To discuss chaos theory proper, there are four terms intrinsically linked with chaos that must be defined in order to be incorporated into new scientific fields: “system,” “non-linear,” “deterministic,” and “dynamical.”2 A system is simply an assemblage of interacting parts, such as the weather or the stock market. A linear system is one in which a small set of inputs can be used to predict any possible outputs; in contrast, a non-linear system is much more difficult to predict, as they respond more violently and unpredictably to initial stimuli than linear ones. In a deterministic system, the smallest differences in initial conditions will make major changes in its future outcomes, as is the case in the well-known “butterfly effect”: a flap of a butterfly’s wings in China has the potential to cause a hurricane in North America.4 And finally, a dynamical system can change and evolve over time.

Today’s medical field appears to be completely dependent on scalar models. Doctors limit themselves by placing their patients into vague categories.1 While undoubtedly useful to sort patients into treatable sets, groupings result in expanded usage of averages, means, and approximations, which are not always helpful in treating specific cases.1 Even prescriptions are determined through generalized ranges and sometimes patients are treated using qualities such as ethnicity and age, but not necessarily through what is appropriate specifically for each individual.1 Such inefficiency exists in the world of medicine because the human body is a chaotic system.4

Practically everything about an individual’s body is chaotic. One of the clearest examples of this is the beating of the heart.1 Within a single cardiac cycle the heart experiences seemingly chaotic or random electrical activity; interestingly enough, scientists found that a group of subjects with congestive heart failure actually experienced periods of time where there was no chaos in their heartbeat.3 Judging from these results, it appears that a lack of chaos in the body is a cause for concern. An absence of chaos can also be a sign of aging, as our bodies and the systems within them lose their chaotic characteristics as we grow older. Our hearts beat more regularly, our thermoregulation becomes less varied, and our brains experience a lack of erratic electrical activity.1 The presence of chaos in the human body allows it to rebound from a variety of potential problems; instead of striving to remain at stable conditions, a youthful body can use this sort of inner chaos to regulate to “near-normal behavior.”4 Examining the elements of chaos within the body becomes the tipping point in our understanding of individualized medicine.

The most significant contribution that chaos theory could make in the field of medicine would have to do with the creation of chaotic models that would be able to predict progression of aging or diseases within the body. The true beauty of such systems lies in how simple mathematical equations can lead to complete chaos. Scientists have already started down this path in several fields, such as with the creation of non-linear equations that could be used to predict the progression of heart failure.3 Applying chaotic models to other systems of the body such as the immune system could help us in understanding how it ages.

When using chaotic equations, one could use either iterative or continuous models.5 One of the better-known examples of an iterative, or repeating, chaotic model also happens to be extremely simple to understand: the logistic equation. Logistic equations are typically used as one-dimensional systems for modeling changes in biological populations. The iterative nature of the logistic equations allows the model to be observed over specific time intervals, and could be used to predict changes in populations within the body, like the amount of white blood cells in the blood stream or the concentration of bacteria within a certain organ.2

While the logistic equation would be perfect for iterative models in the body, differential equations would suit continuous chaotic models such as the Lorenz attractor.5 Lorenz found that his system always followed the same general pattern, but never actually repeated.  Unlike the logistic model, the differential model has chaos solely in its attractor, the asymptotic set of values around which all variables in the model avoid. As a result, such chaotic attractors end up having an infinite precision, as they transcend into “fractals,” infinitely repeating dimensions that fall between integers. This accuracy makes differential equations potential candidates for modeling parts of the body such as the nervous system. By measuring the dimensionality of the space occupied by data points, one could monitor the parts of a biological model that change deterministically – not randomly.8 Developing such systems to comprehend the relation between chaos and longevity would equip us with the knowledge of how individuals, and not just general groups, age.

Another application for chaos theory besides models for predicting aging is in the actual dosage of prescription medicines.1 Rather than treating symptoms with medication at a fixed time interval and a fixed dosage, a better understanding of the chaos in our bodies would help us adapt treatment to make it more efficient, whether that be by manipulating the time intervals according to chaotic systems, or changing the dosage itself.1,6 An example of this could be delivering small pulses of proteins at pre-determined times to correct protein production in aging subjects.9

Ultimately, chaotic models must be aggressively researched to identify those that are applicable to the various types of chaos that exist in the human body. The next major step in advancing other sciences is to take a step back into their mathematical roots. This is where the so-called “digital generation” comes into play. Linking the computational powers of computers and the advanced mathematics it allows us to pursue with the vastly researched and developed sciences that exist today will allow us to turn to a new era of complete interdependence between math and science, improving the current knowledge we have today and potentially bettering our current medical system.

References

1. Kumar A, Hegde BM. Chaos theory: impact on and applications in medicine. Nitte University Journal of Heath Science 2012; 2(4)
2. Williams GP Chaos Theory Tamed. Washington, DC: Joseph Henry Press; 1997.
3. Wagner CD, Persson PB. Chaos in the cardiovascular system: an update. Oxford Journals Cardiovascular Research 1998; 40(2):257 – 264.
4. Dalgleish A, The relevance of non-linear mathematics (chaos theory) to the treatment of cancer, the role of the immune response and the potential for vaccines. Oxford Journals QJM 1999; 92(6):347 – 359.
5. Strogatz SH Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. New York, NY: Collins Publishers; 1994.
6. Atkinson C. Linear Systems of Ordinary Differential Equations. [Homepage on the Internet]. 2008 [cited 2012 Nov 4]. Available from: University of Navarra, Web site: http://www.tecnun.es/asignaturas/metmat/Texto/En_web/Sistemas_lineales/Linear_systems_of_ordinary_differential_equations.pdf
7. Montgomery R, A new solution to the three-body problem. Notices of the AMS 2001; 48(5):471 – 481.
8. Skinner JE, Molnar M, Vybiral T, Mitra M. Application of chaos theory to biology and medicine. Integrative Physiological and Behavioral Science 1992; 27(1):39 – 53.
9. Skinner JE, Low-dimensional chaos in biological systems. Nature Biotechnology 1994; :596 – 600.
10. Lavoix H. Images to illustrate complexity and strategic foresight and warning. [homepage on the Internet]. 2012 [cited 2013 Jan 2]. Available from: http://www.redanalysis.org/2012/03/20/images-to-illustrate-complexity-science-strategic-foresight-and-warning/

Vivek Sriram is a student at The Harker School. Follow The Triple Helix Online on Twitter and join us on Facebook.

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