Fractal Physiology

Fractal Physiology William Deering and Bruce J. West Department of Physics University of North Texas

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n all areas of medical research there is a common physiological theme. Complexity is the salient feature shared by all such systems; afeature that is attracting more and more attention in physical systems as well. Until recently, scientists have assumed that understanding such systems in different contexts, or even understanding various physiological systems in the same organism, would require completely different models. One of the most exciting prospects f o r t h e new scaling addressed herein is that it may well provide a unifying theme to many investigations, which up until now have been considered unrelated. The swirling spiral of the cochlea and the finely branched structure of the bronchial tree suggest the complex interrelations among biological development, form, and f u n c t i o n . R e lationships that depend on scale can have profound imysiology. Consider, for exthe standard problems in increases as the cube of a

typical length, but surface area increases only as the square. Accordingly, if one species is twice the size of another, it is likely to be eight times heavier, but have only four times the surface area. Thus, the larger plants and animals must compensate for their bulk; respiration depends on surface area for the exchange of gases, as does cooling by evaporation from the skin, and nutrition by absorption through membranes. One way to add surface to a given volume is to make the exterior more irregular, as with branches and leaves; another way is to hollow out the interior. The human lung, with about 300 million alveoli, approaches the more favorable ratio of surface to volume enjoyed by our evolutional ancestors, the single-celled microbes. The classical scaling concepts in biology, while of great importance, are not capable of accounting for the irregular surfaces and structures seen in hearts, lungs, intestines and brains. Experiments suggest that biological processes are not continuous, homogeneous, and regular, but rather are discontinuous, inhomogeneous, and irregular. Thus, a new way of analyzing such processes is required. This new perspective is that of fractal geometry and fractal statistics. Herein, we briefly review how these approaches have enabled scientists to peer behind the veil obscuring our understanding of a number of physiological processes.

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athematical Fractals A basic theme in science is that of the invariance or symmetry of laws

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with respect to various types of transformations that may be performed in ordinary space and time, or even in certain abstract mathematical spaces. I n dynamics, there are the familiar symmetries in space and time which lead to conservation laws. For example, translational invariance implies conservation of linear momentum, whereas invariance with respect to rotation implies conservation of angular momentum. Using the known symmetries of a given system provides a framework to which the details of structure and motion must conform. In some cases, it is not known initially just what symmetry a system possesses bkcause of lack of fundamental information about interactions a m o n g t h e elements of the system. However, a time-honored strategy for constructing the equations of motion for a complex system is t o a s s u m e a symmetry property, and then determine the forces required to maintain this symmetry over time. This approach is the variational principle, and it has been applied to such quantities as the system’s entropy, mass, and energy. This strategy for determining the dynamic properties of a system has been very useful in bioengineering, where concepts such as efficiency have been introduced. A biosystem can be assumed to operate so as to maximize its efficiency. The equations that produce this effect are sought through a variational principle. In other areas, where it might at first seem that symmetry is of little use, it tums out that there is often an unnoticed symmetry framework present that determines system behavior. As an example, the motions of the individual molecules in an equilibrium gas in the lungs are random, and symmetry seems to be the last thing one would think about to describe the gas exchange process. Considered another way, however, equivalent randomness occurs everywhere in the gas, so there is a statistical symmetry with respect to spatial translation. The entire gas can be generated by taking a small element of volume, one that contains many particles

with an equilibrium distribution of velocities characterized by a constant temperature, and copying it in all directions throughout the given volume. In this way, a disorganized system may be self-similar under scale changes and can be thought of as being formed by repetitious translations of a generating element. The idea of forming a large set of points from a smaller generating set is used below to provide one definition of dimension for sets of points. In the 1960s, Benoit Mandelbrot began discussing a new geometry of nature [I], one that embraces the irregular shapes of objects such as coastlines, lightning bolts, cloud surfaces, and molecular trajectories. It was soon realized that these geometrical ideas could be applied in other areas, including nontraditional ones [2], from outside the physical sciences. A common feature of these objects. which Mandelbrot called ,fi-actuls, is that their boundaries are so irregular that it is not easy to understand how to apply s i m p l e metrical ideas and operations to them. Such fundamental concepts as dimension and length measurement must be generalized. Therefore, let us consider some of the metric peculiarities of a few unusual mathematical objects, which we subsequently use to describe some biomedical systems. The physical phenomena mentioned above seem to be without symmetry over many orders of magnitude. However, that in itself is found to be the symmetry of self-similarity, just as in the case of the gas in equilibrium discussed above. That is, these phenomena are self-similar under scaling transformations. Mathematically, s c a l i n g transformations are multiplicative, as compared to the translational transformations mentioned above, which are additive. A system that is invariant under time translation means that its properties at time t are unchanged under the replacement t + t + z, with z being any time interval. The multiplicative linear scaling transformationof the form x’ = a x , changes the original distances through multiplication by the constant, a . This

New scaling methods may provide a unifying theme to various investigations heretofore considered unrelated

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transformation expands or contracts the observed scales in the same way at each point. A more general scaling transformation is x’= axP, with a and B both being constants, so that different values of x, e.g., locations in space, are weighted differently. Large x is affected more than small x for I3 > I , and vice versa for I3 < 1. This latter type of scaling was used by D’Arcy Thompson [3] in scaling anatomical structures. It appears quite often in the form of allometric growth laws in botany, as well as in biology. This particular kind of scaling has been successfully used in biology for over a century. However, the significance of the existence of such allometric growth laws have not always been properly appreciated. This scaling often refers to a process that has “infinite” levels of substructure that repeat in an ever decreasing cascade to even smaller scales, over which one averages to obtain the allometric growth law. Processes such as these are often described by functions that are continuous but not differentiable. In the late 18OOs,most mathematicians felt that a continuous function must have a derivative “almost everywhere,” which means the derivative is singular only on a set of points whose total length or measure vanishes. However, some mathematicians wondered if functions existed that were continuous but did not have a derivative at any point (continuous everywhere but differentiable nowhere). It is interesting to note that mathematicians at the time were reluctant to consider such unusual functions as being worthy of serious research attention. Similarly, there is some resistance today to the shifting emphasis that has occurred in research, and is just beginning in the classroom, with regard to placing general nonlinear analysis in the curriculum. The motivation for considering such pathological functions was initiated within mathematics and not in the physical or biological sciences [4]. After all, what possible use can there be for a function that is so jagged that it has no tangents at all? In 1872, Karl Weierstrass (18151897) gave a lecture to the Berlin Academy in which he presented functions that had the aforementioned continuity and differentiation properties [ 5 ] .Thus, these functions had the symmetry of self-similarity. Twenty-six years later, Ludwig Boltzmann, who elucidated the microscopic basis of entropy, said that physicists could have invented such functions in order to treat such things as collisions among molecules in gases and fluids. Boltzmann had a great deal of experience thinking about such things as discontinuous changes of particle velocities that occur in kinetic theory. He had spent many years in trying to develop a micro-

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1. Construction of the Cantor set by removing the middle third from each line segment between successive generations, z. scopic theory of gases. He was successful, only to have his colleagues reject his theory. Although it led to acceptable results (and provided a suitable microscopic definition of entropy), it really shouldn’t have, because it was based on time-reversible equations; that is, entropy distinguishes the past from the future. It was assumed in the kinetic theory of gases that molecules were unchanged as a result of interactions with other molecules, and collisions were instantaneous events as would occur if the molecules were impenetrable and perfectly elastic. As a result, it seemed quite natural that the trajectories of molecules would sometimes undergo discontinuous changes. Robert Brown, in 1827, had observed the random motion of a speck of pollen immersed in a water droplet. Discontinuous changes of motion were indicated, but the mechanism causing these changes was not understood. Albert Einstein published a paper in 1905 that explained the source of Brownian motion as due to random collisions of the pollen mote with the lighter particles of the medium. Jean Baptiste Perrin, of the University of Paris, later experimentally verified Einstein’s result and received the Nobel Prize for his work in 1926. Perrin, giving a physicist’s view of mathematics in 1906, carefully said that curves without tangents are more common than those special, but interesting ones, like the circle, that have tangents. Thus, there are valid physical reasons for looking for these types of functions, but the scientific reasons appeared only long after the mathematical discovery by Weierstrass. Soon after the Weierstrass function appeared, Georg Cantor ( I 845- 19 18) constructed a set of points that provided another surprising result for mathematicians [6]. Cantor provided the example of a proper subset of the unit interval { 0,l } , which contained an uncountable number of points (as did the entire unit interval), and which also had zero measure (the entire interval had a length of unity).

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the deleted set of points has a measure of unity, just as does the original interval {0,1 1. The elements of the Cantor set are effectively self-similar; note that any stage in the process can be obtained from taking only the left half of the next stage and multiplying by three. That is, the entire set can be obtained by appropriately rescaling any portion of the set. If the mathematicians were surprised at the existence of such a set, imagine the shock experienced by scientists in the last two decades as they discovered phenomena described by these abstract mathematical objects. Another example of mathematical pathology is the snowflake curve [SI first constructed by Helge Von Koch (1906). This closed plane curve has an infinite length but encloses a finite area. Starting with an equilateral triangle (the generator) in Fig. 2. the second stage is generated by replacing the middle third of each line in the generator by the generator. Continuing this process results in a curve that, in the limit, is infinitely long and continuous but without tangents anywhere. The curve is self-similar and has a topological dimension of unity, since it is topologically equivalent to a circle. The circumference of the figure at the nth generation is (4/3)”, which diverges as n approaches infinity. Many prominent mathematicians sought to avoid these functions, while recognizing theirexistence. PoincarC (1854-1912), who contributed to all areas of mathematics, and laid the basis of much of what we today call chaos theory and nonlinear

The initial stage (z = 0) of the Cantor set (Fig. 1) is a line segment of unit length. The next stage (z = 1 ) is obtained by discarding the middle third of the line, leaving the two intervals (0, 1/31 and (2/3, 1 I.The z = 2 stage is obtained by removing the middle third of each of the two intervals, leaving the four intervals (0, 1/91, {2/9, 1/31. {2/3, 7/91> {8/9. 1 I as shown for the z = 2 stage. Repeating this process (z -+ m) eventually produces the Cantor set, called “Cantor dust” [7]. consisting of an uncountably infinite number of points that are separated from one another. That ia. in every neighborhood of one of the Cantor points, no matter how small, there are points that do not belong to the Cantor set. So, the set is not continuous, even though it has the same number of points as the entire continuous interval { 0.1 1. In fact, the Cantor set is so full of holes that its measure is zero and

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dynamics, referred to these functions as a “gallery of monsters.” Hermite ( 1 8221901) was of similar mind and tried to convince Lebesgue ( 1 875- 1941),inventor of the modem theory of integration, not to publish in this area. We can immediately find application of these ideas in physiology. In Fig. 3 is a mathematicians version of the mammalian lung, i.e., a branching structure that attempts to put end points in the neighborhood of every point in space. In this way, the one dimensional line “fills” the two dimensional area in an optimal way in order to perform the gas exchange function for which this ‘‘lung’’ is designed. A more realistic version of the fractal lung is given in Fig. 4, where the “dimension” of the structure corresponds to that measured from casts of mammalian lungs.

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Fractal geometry and statistics offer a viable way to analyze discontinuous, inhomogeneous

tegervalue forthe dimension of a 5 elf - 5 im 11ar obJXt For the Cantor \et, the zth generdtion has N = 2‘ line segments, each having length q = 3- I , so that using Eq. 2 we obtain the similarity dimension d = In 2/ln 3 = 0.6309. Thus, this dimension classifies the set as being between a line (d = 1) and a point (d = 0): it is “fractal dust.” The total length, L(z), of the set after z generations is 213 of the length L(z - 1 ) of the previous generation. so that

processes

imensionalities of Sets of Points he similarity dimension of a set of points may be defined for self-similar sets. Such sets can be covered by translating a generating element throughout the set, which is the basic idea in making a measurement of the length of a continuous object with a ruler. Consider a straight line segment. Dividing the segment into N self-similar pieces by applying a ruler of length q , the length of the interval is then L(q) = Nq. If L = 1, then the ruler must have length q = l/N to exactly cover the line. Similarly, an area L2 can Ipe covered by N elements, each of area so that L’ = Nq2, and q = I/”’* for L = 1 . In three dimensions, a unit cube is covered by q = l/N1’3elementary cubes. Generalizing, a d-dimensional object is covered by N selfsimilar objects, with

This equation has the solution L(z) = (2/3)‘L(O), and as z -+ the length of the Cantor set exponentially goes to zero. 00,

Another way to see this uses the dimensionality of the set by writing the length of the z-th generation in terms of q = 1/3’, instead of z. Then:

and, with N(q) = $ from Eq. 1:

which vanishes as q 4 0 because ( 1 - d) > 0. This expression for L shows the explicit dependence of the length of the curve on the size of the measuring ruler (q) that is used, which is a property of self-similar objects. In this case, the length vanishes as the ruler length vanishes, but as the next example shows, this is not the general situation. For the snowflake curve, starting with a line of unit length, the generator is the set of 4 line segments, each of length 1/3. Thus, N = 4‘ and the ruler length after z generations is q = 1/3‘, giving the dimension d = In 4/ln 3 = 1.2618. Now, using Eq. 5 yields lim L(q) = as r\ 4 0, because d > 1 . 00,

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atural Fractals A fractal property can be spatial, as in the fixed geometry of the mathematical examples above; it can be temporal, as in a series of data taken from a system over an interval of time; and it can be exact or statistical. Forexample, in Fig. 5. we depict the time series for the beatto-beat interval of normal sinus rhythm of a mammalian heart. Regions of the time series are magnified (Fig. 5a) to emphasize the suggested “self-similarity” of the fluctuations of the data. This property

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The equation can be used to define the dimension, d, of a set in terms of the number N of elementary covering elements (of length, area, volume, etc.), which are constructed from basic intervals of length q. Solving for d gives:

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3. Schematic model of the mammalian lung [7].

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4. Computer simulation of a fractal lung, in which the boundary conditions influence morphogenesis. The boundary was derived from a chest radiograph. The model data are in good agreement with actual structural data [9]. is more clearly revealed in the inverse power law spectrum for this time series (Fig. 5b). As is usual in the application of mathematical models to nature, natural fractals are more restricted than mathematical ones. The ideal elements (infinite lines, smooth planes, etc.) of Euclidean geometry are never realized in nature, and neither are the ideal elements of fractal geometry, although the latter is closer to nature than the former. The definitions of the dimension of the mathematical fractals given previously require that the size of the elementary covering element vanish. These definitions of similarity dimensions are examples of a general dimension defined by Hausdorff, which is now encompassed by the term fractal dimension [lo], applicable even to sets that may not be self-similar over all ranges of space or time. Independence of scale for properties of systems containing mass is limited to finite spatial and/or temporal intervals. For example, molecular diameters and typical periods of molecular motions may set lower limits to self-similarity, although those may be extreme restrictions in aparticular situation. However, it is still possible and useful to apply the general idea to a natural system and define its fractal dimension. Knowing that the fractal dimension of an object or process is closer to unity than it is to two, for example, indicates a system or process is closer to being a smooth curve, with respect to some set of variables, than it is to being a smooth plane. The dimension of a naturally occurring fractal is a quantitative measure of a qualitative property of a structure that is self-similar over some region of space or interval of time. There may be more than one region of self-similarity, depending

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on the parameters of the system. A finger of frost on a window pane is a crystalline solid that has a fractal dimension less than 2, but greater than 1. This dimension is a consequence of the process being constrained to unfold within a two-dimensional space; if the process had unfolded in free space, the result would be falling snow, with a fractal dimension greater than 2, but less than 3. To investigate the fractal structure experimentally, it is necessary to be able to relate the results of observations to fractal measures, such as dimension. One useful dimension for this purpose is the cluster ,fractal dinletision, D, which is a measure of how completely a cluster of objects fills the region where it resides, as for example a collection of small spherical balls in a spherical volume. The parameter D may be defined for a physical object with a lower length limit, q. If a system is composed of spherical physical objects of radius q, then the number of objects placed side by side that are required to cover a distance, R, is R/2q. If, for example, the smallest sphere in the cluster of objects has radius Ro, the number of particles in the cluster is:

in analogy to Eq I . The equation above defines the clirster diniension, D. If we assign the same mass to all the spherical objects in the cluster, o can be considered to be the mass density and N the total mass. The parameter D may then also be called the mass dimension. The nonlinear dynamical processes that can produce fractals are currently the subject of much research, but they are not understood in many cases. However, there have been recent, successful computer

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simulations of the growth of dendritic structures [ 11, 121, in which the structures are “grown” by allowing particles to start a large distance away from a target or seed cluster of particles, and diffuse (randomwalk) toward the seed cluster. Such processes are referred to as diffusionlimited aggregation (DLA), and the mathematical models rely on the assumed diffusive character of the random motion. In steady state, the diffusion equation for the particle number density, p, becomes Laplace’s equation V2p = 0. This equation is solved numerically on a grid, with the solutions forced to satisfy certain boundary conditions. (Solutions of Laplace’s equation can have neither maxima nor minima in a region in which the equation holds. A field that satisfies the equation in a region has no sources, and therefore no “lumpiness” in the region. It is interesting that such a condition of smoothness is used as part of the mathematical model that generates irregular forms). The aggregation of particles at the seed causes changes in the boundaries and results in complicated applied mathematics problems. An effective radial density for such clusters can be defined by writing Eq. 6 for an arbitrary r < R and dividing by the B Euclidean volume a r , where the positive integer, E, is the ordinary dimension number of the underlying Euclidean space. The resulting density:

shows that, because D < E, the fractal density decreases with distance from the origin. The cluster dimension refers to an average density or mass distribution, but it does not carry information about the shape of the cluster. Observations of DLA-type patterns have been reported for many phenomena including dielectric breakdown, metallic colloid clustering, growth of metal leaves on a two-dimensional surface, and viscous fingering in porous media. The scattering of light, x-rays, and neutrons has been used to determine the fractal dimensions of clusters by employing the usual method of generalizing scattering expressions in Euclidean spaces to fractional dimensional spaces. Therefore, we expect the scattered intensity, I(k) to obey a powerlaw equation I(k) - k-b, where k is the spatial frequency (reciprocal of the wavelength) that lies in the range l/R << k << l/q, where R is the scale of the entire cluster, and q is the scale of the individual particle. This scaling means the wavelength of the radiation is very large compared to the particle diameters, and very small compared to the entire cluster. For silica particles, it is found that D = 2. I , while light scattering from IgG-type

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immunoglobulins [14], which form clusters when heated, gives D = 2.56.

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ower laws, Noise and Fractal Time Signals

The power law relating intensity of scattered radiation to spatial frequency mentioned above, reminds us of the numerous power law relations in science that have the self-similarity property. For example, the inverse-square force, which is fundamental in gravitation and in electricity and magnetism, has no intrinsic scale: it has the same form at all scales under a linear scaling transformation. For any type of field (sound, electromagnetic, etc.), noise is the part of a signal due to random influences; for example, the distortion of a radio program due to static, or the erratic displacements due to random collisions of a Brownian particle. In the description of a dynamic process, the various time scales (frequencies) contributing to the process make up a characteristic spectrum. This spectrum shows how the energy of the process is distributed over different kinds of motions. For example, a simple harmonic oscillator has a spectrum that consists of a single frequency. The spectrum associates an amplitude with that frequency, which is the square of the maximum displacement (proportional to the energy) of the oscillator. If our system is made more complicated by adding more interacting particles and extemal influences, the spectrum of the motions broadens to include nonzero amplitudes at different frequencies. Quite often, the only information available about a complex dynamical system is a measured time series; e.g., the concentration of a particular chemical species in an ongoing chemical reaction, or the displacement of the plucked string of a musical instrument over time, the velocity of the wind at the airport weather station, and so on. This time series can be used to construct a power spectrum that can, in some cases, be used to determine the properties of the system generating the time series. This type of reasoning is often what a physician does when interpreting an electrocardiogram. An uncorrelated random time series, in contrast to the harmonic oscillator, has a power spectrum that has a constant energy level at all frequencies. This is white noise. It is called white because all frequencies contribute equally as they do in white light; it is called noise because the frequency components are random, one with respect to another. A fractal random time series is quite different. There is no characteristic scale for a fractal process, and its frequencies make up what is known as an inverse power-law spectrum, given by the term IIP, where f is a frequency, and a is a

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acteristic time-scale for a normally beating heart. A similar result in terms of space-frequency (inverse wavelength) is found for the static spatial structure of the mammalian lung [ 161, which indicates a lack of characteristic length. The lack of regular motion of the heart upsets the conventional view of normalcy of the motion of that system. That view, one of homeostasis [ 171, holds that normal physiological systems strive to maintain constancy in their intemal function, and therefore attempt to reduce fluctuations in their variables. If perturbed, such systems would

positive number. The fractal dimension for a random fractal process with such a spectrum is d = 2/(a-l), for 1 < a 1 3 . The inverse power-law spectrum is often a manifestation of the fractal or scale-invariant nature of the underlying process. Measuring heart rate by listening or by feeling, would lead one to describe the normal heart rate as periodic or regular. This description implies a power spectrum that consists of sharp peaks at isolated frequencies, rather than the broad spectrum indicated by the data of Fig. 5 . The real data suggest that there is no char-

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5. Normal sinus rhythm (data from a healthy, active, 61-year-old woman) is not regular, but rather shows apparently erratic fluctuations across multiple time scales (a). Power spectrum of this typical heart rate time series shows broad band of frequencies with inverse power-law scaling, consistent with a fractal process (b) [15]. Note that normal subjects at rest will show a superimposed frequency peak at about 0.2 Hz, corresponding to the respiratory rate; however, a broadband l/f-like spectrum as shown here is observed even at bed-rest in normal subjects.

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References I. Mandelbrot BB: The Fractal Geonirtry of Ntrtrrrr, W. H. Freeman. New York, 1983. 2. Goldberger AL, Rigney DR, West BJ: Chaos and fractals in human physiology. Scientific. Aniericun. 262, No. 2:42-49. 1990.

physics from the University of Rochester in 1969. His current research interests lie in nonequilibrium statistical mechanics, descriptions of complex systems, quantum manifestations of chaos, and the applications of nonlinear dynamics systems theory to biomedical phenomena.

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6. Plotted data of Weibel and Gomez [19], for dogs, rats and hamsters, on a log-log graph, indicates the average bronchial diameters follow inverse power-law, not exponential, scaling. In addition, there appears to be a harmonic (periodic) variation of the data ooints about the Dower-law regression line 1211. 0

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William Deer-ing is Associate Professor and Vice Chairman of the Department of Physics at the University of N o r t h T e x a s . He r e c e i v e d a B . A . in physics from T e x a s Christian University in 1956, a master’s in 1960, and a Ph.D. in physics in 1964 from New Mexico State University. I n addition to directing graduate programs in the department, he conducts research in semiconductor physics, nonlinear processes in classical and quantum systems, and statistical physics.Address for correspondence: PO Box 5368, Denton, TX 76203

can be derived that closely fit all essential features of the data 1201. Once again, fractal geometry shows its superiority in t h e d e s c r i p t i o n s o f s i g n i f i c a n t problems that are unsubmissive t o Euclidean geometry. In the last two decades, we have seen the emergence of a new descriptive scheme that can be used successfully to deal with many of the problems of the geometry of the natural world. The widespread applicability of fractals makes it a tool for all science and other fields as well. It is clear that the further application of fractals and their underlying nonlinear dynamics, together with high speed computation, will continue to bring together biomedical, physical and mathematical sciences for work on problems that were formerly thought to be outside some of the artificially set ranges in each field [ 2 2 ] .

try to retum as soon as possible to the unperturbed state. The broad spectrum of the time series of Fig. 5 clearly does not support the homeostatic model, which requires sharp isolated peaks representing stable rates. The question of why such lack of regularity is a common feature in biological systems has been addressed [18], and shows that statistical fractals provide a mechanism for the structural basis of complex structures, which are more stable than those generated by classical scaling. Classical scaling may be illustrated in terms of the mammalian lung. Assuming that the diameter of an airway scales as d(z) = qd(z - 1) between successive generations [ 191, where q is a constant, then the diameter at the z-th generation in terms of that at z = 0, is the exponential relation d(z) = d(O)exp(-rz), where r = In(l/q) introduces a length scale. The first ten generations of the human lung approximately satisfy the exponential relation, but thereafter there is a marked deviation from this form. When carefully analyzed [20], it is found that the data can be approximated by an inverse power law in z, with a superimposed harmonic variation (Fig. 6) [21]. The power law dependence implies there is no length scale, which eliminates the exponential relation above. To account for the observed properties, it was assumed that at each generation of the bronchial tree, there is present a multiplicity of scales described by the distribution function, p(T). If a scaling relation is expressed in terms of the average of r, an expression for the average diameter of the airways

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3. Thompson DW: O i i ( ; r o ~ ~ t / i u i ~ d F o ~ ~ n i , S e c o n d edition. Cambridge Univ Press, Cambridge, 1963. 4. Stewart I: The PI-ohlenisqf Mathematics. Oxford University Press, 1985. 5. Korner TW: Forrrier Analysis. Cambridge University Press, Cambridge, 1988. 6. Schroeder M: Finc.tal.\, Chuos. P m w LOMT. W. H. Freeman. New York, 1991. 7. Mandelbrot BB: Frac,tuls:Forni, Chanc.~. atid Din7en.sion. W. H. Freeman, San Francisco, 1977. X. Kasner E, Newman JR: Matheniutic,.s and the lniuginution Sirnon andShii~ter. Neb(, York, 1965. 9. Nelson TR, Manchester DK: Morphological modelling using fractal geometries. IEEE Tf-urz.5 Medlniug. 7:439, 1988. 10. Feder J: Fi-trc.tals. Plenum, New York, 1988. 1 1 . Mandelbrot BB, Evertsz C: The potential distribution around growing fractal clusters. Naf i i w 348. 8 Nov 1990. 12. Meakin P: Formation of fractal clusters and networks by irreverhible diffusion-limited aggregation.P/i~sRe~~Lett51:1I19-l 122, 1983. 1.3. Schafer DW, Martin JE, Wiltzius P, Canne11 DS: Fractal geometry of fractal aggregates. P l i w Re,, Lett. 52: 237 1-2374. 14. Feder J, Jossang T, Rossenqvist E: Scaling behavior and cluster fractal dimension by light vxttering from aggregating proteins. Phys Rei, Let/. 53. 1403-1406. 1984. IS. Goldberger AL, West BJ: Fractals in physiology and medicine. Yale J Biol Med, 60:42 1-435. 1987. 16. West BJ: Fractal forms in physiology. Ifiternational .I M o d P h j s B 4( IO): 1629-1669, 1990. 17. Dyson F: Irrfiirrte if7 all Dii.edons. Harper & Row, New York, 1988. 18. West BJ: Physiology in fractal dimensions: error tolerance. Annuls of Biomiedicul Engineer/n,y. 18: 135- 149. 1990. 19. Weibel ER, Gomez DM: Architecture of the human lung. . k i c w e 137577. 1962. 20. West BJ, Bhargava V, Goldberger AL: Beyond the principle of similitude: renormalization in the bronchial tree. J Appl Physiol60: 188197. 1986. 21. West BJ, Goldberger AL: Physiology in fractal dimensions.Aniei.ic.unSc.ient,.st, 75(4):354-364, 1987. 22. West R J : Frac~roilPh~srolo~qy and Chaos in Mrvlicinc~,World Scientific, New Jersey, 1990.

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