Abstract: In this paper' the importance of nonlinearityin medicine is analysed. The methods and notions ofnonlinear dynamics permit to describe both thestructure and the behaviour of complex biologicalobjects. This gives more tools to clinicists in order tounderstand the reasons of pathological behaviour andpredict possible changes.

INTRODUCTION

During the last couple of decades, nonlinear dynamicshas been developing into a real interdisciplinary science.

Based on mathematical exactness, the mathematical

models in physics, chemistry, biology, etc. have revealed

many novel characteristics of the world around us.

Medicine is not the exception in this long list. However, a

practitioner can always ask "What is the use ofmathematics when we have to cure a living person?"

Indeed, the modelling of such complicated biologicalphenomena like living organisms, is confronted by many

obstacles. For example, the whole is composed of manyparts, all interacting with each other, the parts are

influenced by processes of the different nature, there are

different scales, and the control by consciousness is still a

black box. And this curious "nonpropert5/" - nonlinearityis everywhere. From a certain viervpoint of nonlinear

systems, biological objects could be defined as those

including different processes rvith different titne and lengtltscales, all spiced with nonlinearity.

Nonlinearity actually means that the superposition is

not possible. In many physical sciences the linear theory

has been serving mankind truly for a long time and onlynowadays we have discovered the importance of being

nonlinear [U. It means actually that many linear models

have had a certain resemblance to the reality My special

thesis here is that this not the case in biology and

medicine. If a process/object is really nonlinear [2] then alinear model or a linear description simply fails. The

model or description should be nonlinear from the very

beginning. The best examples here are nerve pulse

propagation [3] and heart arrhythmias [4].In this talk, an attempt is made to characterise

nonlinear systems in medicine from a viewpoint of a

theoretician. The attention is focused on the modelling ofvarious phenomena from the dynamical as well as from the

descriptional (fractality) viewpoints. The model of the

heart serves as an example. Some specific problems are

collected to demonstrate the richness of the contemporarymethods of nonlinear dynamics. However, the models

might only open our eyes to certain phenomena, the realityis still more complicated and the complexity theory makes

its first steps [5].

10.06

MODELS

There are many nonlinear mathematical models used to

describe the behaviour of biological objects f6, 7,81. A listof models which by no means is exhaustive includes, forexample:

* theory of nerve pulse propagation [3];:t theory of heart [4,9];r, fractal physiology [8];,* brain modelling [6];'r flow in the blood-vessel systems [l0];* modelling of light reception [6];* modelling of epidemics 16,71],* genetic control models [6];etc., etc.

Roughly speaking, the models could be divided into twogroups:

'* explaining the structure of complex biologicalobjects;

* describing the behaviour of complex biologicalobjects.

The first group of models uses widely the concept offractality [7, I l], the second - the concepts of nonlinear

dynamics [2, l3].

NONLINEARITY

As explained above, nonlinearity means a loss ofsuperposition in the mathematical sense. In nonlinear

dynamics, leading to chaotic regimes, there are many

special properties, like I l]:

* sensitivity to initial conditions;* the l/f Fourier spectrum of a signal;* special types of bifurcations leading sometimes to

chaotic regimes;* several quantitative characteristics (fractal

dirnensions, Lyapunov exponents, etc.) that could be

used for distinguishing between various structures

and/or processes.

Starting from the practical side, many of these

properties could also be interpreted in clinical terms

[4,8, 9]. Several fruitful notions like chaos, fractality,attractors, etc., could be used as novel tools in diagnostics.

This is quite natural because the complex biologicalprocesses are often on the edge of chaos [5]. However, thisextremely interesting situation could be easily

characterised by certain quantities (Lyapunov exponents,

Nonlinear Svstems in Medicine

Itiri EngelbrechtInstitute of Cybernetics, Estonian Academy of Sciences

Akadeemia 21, EE0026 Tallinn, Estonia

Medical & Biological Engineering & Computing Vol. 34, Supplement 1, Part 1, 1996The 1Oth Nordic-Baltic Öonference on Biomedical Enlineering, June 9-13, 1996, Tampere, Finland 1 5

for example) that allow to distinguish between thequasiperiodic and chaotic processes.

It seems that actually the biological objects are socomplex that nonlinearity is their natural property. Or, inother wordso one cannot derive any suitable mathematicalmodel, say in medicine, just using linear approximations.To prove that we could follow the ideas related to nervepulse propagation and heart dynamics. The story of thenerve pulse propagation is explicitly illuminated byHodgkin [1a]. The ion currents responsible for propagationjust are nonlinear by their character. Or, in other words,the effects of potassium on membrane conductance, thepotassium leakage during the passage of a nerve irnpulse,the sodium entry at the same time - all are described bynonlinear mathematical models. Characteristic to thistheory is that contrary to the present accent onnonlinearity, Hodgkin and Huxley have just constructed amodel which "...is clearly satisfactory" [14, p. 300]. Thecelebrated Hodgkin-Huxley model [3, 14] and itssimplified variant in the form of FitzHugh-Nagumo [15]are good examples of this "satisfactory" theory. The heartmodels are certainly very much related to the nenre pulsepropagation. Starting from the early van der Pol model

[6] until the DiFrancesco-Noble model (see [9]) - all themodels are genuinely nonlinear, exposing the richness oftheir solutions. The processes related to the K-currents, thehyper-polarisation-activated pacemaker current, networksof atrial cells, etc. cannot be described only by using simpleadditive procedures.

Although nonlinear dynamics started from famousthree-body systems and other physical problems, therichness of its methods can be nowadays widely used inmedicine. It applies equally for theoretical models andtheir analysis [8] as well as for the analysis of time-seriesu7t.

The small group in the Institute of Cybernetics of theEstonian Academy of Sciences has been studying thenonlinear processes of nerye pulse propagation.A simplified nerve pulse equation [18, 19] has been usedfor modelling heart arrhythmias [20, 2l] and the interest tomodelling of blood-vessel systems [11] has been growing.

CONCLUSIONS

Nonlinear dynamics marks a special chapter of sciencethat has been only recently opened and understood. As faras the world around us is nonlinear, there is nothingspecific in this notion. One could predict with a certainboldness that maybe instead of calling something"nonlinear", we should use just the notion "nonnal", while"linear" would be replaced by "sirnplified" [2]. Holever,nowadays nonlinear systerns in medicine mark the areawhere new ideas from dynamics are fertilised with clinicalknowledge. Much is to be done in the nearest future.

REFERENCES

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t3l A.L. Hodgkin, The Conduction of the NerrousImpulse. Liverpool: University Press, 1964.

t4l L. Glass, P. Hunter, and A. McCulloch (eds./, Theoryof Heart. New York et al.: Springer, 1991.

t5l R. Lewin, Complexip. London: J.M. Dent Ltd., 1993.

t6l A.S. Perelson, B. Goldstein, M. Dembo, and J.A.Jacquez, Nonlinearity in Biolog,t and Medicine.Amsterdam: Elsevier, 1988.

I7l A.V. Holden (ed)., Chaos. Manchester: UniversityPress, 1987.

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t9l D. Noble, "The development of mathematical modelsof the heart". Chaos, Solitons & Fractals, vol. 5, pp.32t-333,1995.

[0] D.A. McDonald, Blood Flow in Arteries. London:Edward Arnold Publ., 1974.

U U J. Kalda, "Fractal model of blood vessel qystem",Fractals, No. I (2), pp. l9I-197,1993.

[12] J.M.T. Thompson and H.B. Stewart, NonlinearDynamics and Chaos. Chicheser et al.: Wiley, 1986.

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U4l A. Hodgkin, Chance & Design. Cambridge:University Press, 1994.

[5] J. Nagumo, S. Arimoto, and S. Yoshizawa, "Anactive pulse transmission line simulating neryeaxon". Proc. IRE,vol.50, pp. 2061-2070,1962.

U6l B. van der Pol and J. van der Mark, "The heartbeatconsidered as a relaxation oscillation and an electricalmodel of the heart". Phil. Mag., Suppl. 6, pp. 763-775, Ig2g.

[17] A. Babloyantz and P. Mauer, "Correlations in heartrate variability: assessment of nature and extent ofarrhythmias", in: EUROSPACE'9S, Bologna:Monduzzi, 1995, pp. 161-171.

[8] J. Engelbrecht, "On theory of pulse transmission in anerve fibre". Proc. Roy. Soc. London, A375, pp. 195-209, lggl.

[19] J. Engelbrecht. An Introduction to AsymmetricSolitary ll/aves. Harlow: Longman, 1983.

[20] J. Engelbrecht and O. Kongas, "Mathematicalmodelling of the heartbeat". Proc. Estonian Acad.Sci. Phys. Math., vol. 42,pp. 124-127,1993.

l2ll J. Engelbrecht and O. Kongas, "Driven oscillations inmodelling of heart dynamics", Appl. Analysis, vol.57, pp. ll9-I44,1995.

Medical& Biological Engineering & Computing Vol.34, Supplement 1, part 1, 1996The 1oth Nordic-Baltic Conference on Biomedical Enlineering, June g-13, 1996, Tampere, Finland

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