Fractals and Physiology: Nature s Biology and Geometry

Transcription

1 Fractals and Physiology: Nature s Biology and Geometry Victoria Fairchild University Honors in Applied Mathematics Spring 2014 Advisor: Dr. Stephen Casey Department of Mathematics and Statistics The American University 4400 Massachusetts Ave., N.W. Washington, D.C Abstract Fractal geometry is present at nearly every turn of nature and can be used to describe and analyze natural phenomena that fail to be explained sufficiently by traditional Euclidean geometry. Within this paper the foundational principles of fractals will be developed by means of illustrating examples and then applied to physiology. It will be shown that fractal analysis can be useful in the description and interpretation of various physiological structures and processes. Specifically, this paper will describe the fractal organization of the lungs and the detection of heart rate variability using fractal analysis. The program FracLac created by NIH which utilizes the box-counting method will be used in order to determine fractal dimension of data that cannot be developed in a mathematical manner easily. This paper is not intended to be a comprehensive account of the research being done in the field, but rather an introduction to the utility of fractals in physiology. 1

2 FRACTALS AND PHYSIOLOGY 2 1 Introduction to Nonlinearity Nonlinear dynamics is a branch of the sciences which is used to study complex systems and understand many issues encountered when considering physiological form and function. The concept of a fractal arises naturally in these systems, and can provide valuable information regarding these types of systems. We first introduce nonlinear systems. Perhaps the easiest way to understand the concept of nonlinearity is through understanding what it is not. By comprehending the properties of linear systems, we can better understand a nonlinear system and the properties that are not shared with its linear complement. The first property of linearity is that of proportionality. According to this property, the response of an action to each separate factor is proportional to its value. Take pushing a heavy object, for instance. The distance the object moves is proportional to how hard one pushes the object, and therefore the property of proportionality is fulfilled. Another property of linearity is that of independence. The property of independence asserts that the overall response of a system to a particular action is equal to the sum of the results of the impacts of each separate factor. Considering the previous example, we can see that with more individuals pushing the heavy object, the greater the speed of the object [12]. According to systems theory, which is used in the analysis of complex linear systems, a system is considered to be linear if the output of an operation is directly proportional to the input and the relationship between the applied force and the response of a physical system can be expressed as: R = αf + β where R is the response, F is the applied force, and α and β are constants. Rewriting this equation to include N independent applied forces represented by the vector F = (F 1, F 2, F 3,..., F n ), the response of the system is linear if there exists a vector of independent constant components

3 FRACTALS AND PHYSIOLOGY 3 α = (α1, α 2, α 3,..., α n ) such that: R = α F = N α j F j j=1 It is important to note that the concept of linearity can be expressed in both an algebraic and geometric manner. In a system with two variables a geometric expression of linearity implies that if a graph is constructed with each axis denoting the values of one variable the relation will appear as a straight line. This relationship appears as higher order flat surface when more than two variables are present in a system. Although geometric and algebraic methods of expressing linearity are equivalent in essence, the two approaches have different implications. Geometric methods of expressing linearity employ a static graph of a function whereas the algebraic notion of linearity has to do with the response of a system to a force, which in turn suggests that the system is dynamic [12]. Unlike linear systems, even the simplest nonlinear systems violate the principles of proportionality and independence. One example of a nonlinear system seen in population biology is the logistic equation y = ax(1 x) [4]. The quadratic term of this equation gives rise to its nonlinearity, and describes a parabola. When iterating the seemingly simple logistic equation it can be seen that complex dynamics exist, and that this single equation is capable of generating steady states, oscillations, or erratic behaviors [4]. This example shows how for nonlinear systems, proportionality does not hold. Another property of nonlinear systems that is unlike linear systems is that nonlinear systems composed of multiple components cannot be understood through analyzing each individual component. In other words, the property of independence previously explained does not hold. This characteristic can be seen when looking at certain biological situations such as the cross-talk of neurons in the brain [4]. The coupled effects of each component (in this example, each neuron) cause an output that cannot be explained with a

4 FRACTALS AND PHYSIOLOGY 4 linear model. Rather, they may show types of behaviors characteristic of nonlinear systems. Some of these types of behavior include periodic waves, abrupt changes, and potentially chaos [4]. 2 Introduction to Fractals The concept of fractals originated in the 17th century with studies conducted by Karl Weierstrass, Georg Cantor, and Felix Hausdroff. The term fractal was first coined by Benoit Mandelbrot comes from the Latin word fractus meaning uneven [10]. In the simplest of terms, fractals are shapes that can be broken down into smaller shapes where these smaller shapes resemble the original shape. Fractals are present in a multitude of areas including biology, medicine, soil mechanics, and technical analysis [7]. Throughout this paper the mathematical foundations of fractals will be developed as will various uses and applications of fractals in biology and physiology. Fractal forms possess a few key related features. First, a fractal is a set with a detailed fine structure. Magnification of a fractal reveals increasing amounts of detail which differs from the differentiable curves seen in calculus [2]. Further, fractals have scale invariance meaning that the smaller scale structure of fractals resembles the structure of the larger scale. In other words, the small and large scale structure of fractals are said to be self-similar. In terms of construction, these sets are produced recursively and cannot be described easily in terms of Euclidean geometry [1]. Because of the repeated application of self-similar scales, the length of a fractal line is dependent upon the scale of measurement [6]. As will be displayed during this paper, irregular but complex structures of various physiological structures display the geometric features of fractals. Thus, fractal mathematics can be used to describe shapes in nature that are failed by traditional geometric description.

5 FRACTALS AND PHYSIOLOGY Dimensionality Fractal is a term used in order to describe an object that has a fractional or fractal dimension [6]. Fractional dimension refers to an object or structure that does not have an integer dimension as seen in classical geometry (i.e. a line with dimension 1, a rectangle with dimension 2, a cube with dimension 3, etc.). Visually, a fractal curve appears wrinkly and greater detail becomes apparent within the curve upon closer inspection and magnification [6]. Because of this property, the length of a fractal line will increase as the size of the ruler or measuring stick decreases. The fractal dimension serves as a quantitative measure of self-similarity and scaling, and tells us how many new pieces of the curve are revealed at more detailed resolutions [7]. Various versions of the fractal dimension exist however the Kolmogorov-Mandelbrot Dimension D KM, one of the least technical measures of dimension, will be developed herein. Casey and Reingold explain the process of determining fractal dimension stepwise as follows. Considering any closed and bounded set in n-dimensional Euclidean space R n where r is any positive number and N (r) is the minimal number of closed line segments, balls, or spheres of radius r needed to cover the set. In order to determine the fractal dimension of the set we must calculate a number D such that as r 0, N (r) r D 1. This relationship is referred to as the scaling relationship. Thus, D KM of a set X will be: D KM = lim log N(r) log 1/r This value represents the number necessary to preserve the scaling relationship previously described. The continuous variable r can be replaced by the discrete variable r n = ρ n, 0 < ρ < 1, n = 1, 2,... Hence, if: = lim log N(r n) log (1/r n )

6 FRACTALS AND PHYSIOLOGY 6 then = D KM [2]. The process of determining D KM manually will be followed through at a later point. 2.2 Power Law Relationship Scale-dependence of a fractal can be modeled by a power law relationship. To visualize this, a plot of the log of the ruler length vs. the log of the measured length can be generated. If we let r represent the scale and L(r) represent the length measured the following can be produced: L(r) = c r β In this equation c and β are constants for any particular fractal. By taking the logarithm of both sides we obtain: log L(r) = β log r + log c Where β is the gradient or slope of the straight line representing the scaling relationship [7]. This type of power law scaling relationship is characteristic of a fractal object or process. A visual representation of this scaling can be seen below. Figure 1: For a fractal line there is no characteristic length since the measured length will differ as the size of the measuring stick varies.

7 FRACTALS AND PHYSIOLOGY Self-Similarity Self-similarity implies that for a fractal a particular property at one magnification is the same as that particular property at an alternative magnification. Mathematically this can be represented by the following: L(λ r) = c(λ r) β = c λ β r β In this equation L(λ r) is the length measured at a scale of λ r where λ is the stretching factor. If we substitute L(r) = c r β and set a constant of proportionality k = λ β we are left with L(λ r) = k L(r) Such indicates that if a property is self similar then measurements of that property at different scales are proportional [7]. 3 Examples of Classical Fractals 3.1 The Cantor Set Because of the practicality of using fractal dimension to describe self-similar sets with intricate geometries, we will here further develop the concept of fractal dimension using the Cantor set as an example. Consider the construction of the Cantor set. Given [0,1] remove the open middle third segment ( 1 3, 2 3 ). Next, remove the middle thirds of the two remaining segments. Repeat this process n times. At the n th stage, there exists 2 n segments, each of length 1 n. 3

8 FRACTALS AND PHYSIOLOGY 8 Figure 2: Visual representation of the construction of the Cantor set. 1 3 At the n th stage of construction we can see that 2 n line segments of length n are needed in order to cover the set. Thus, we have N (3 ) r D = 2 n ( 1 3 )nd as the scaling relationship. Following this through we see: N (r) r D = 1 log 2 n ( 1 3 )nd = log 1 n(log 2 D log 3) = 0 D log 3 = log 2 D = log 2 log 3 The fractal dimension of the Cantor set then is log 2 log Sierpinski Gasket Using the Sierpinski Gasket as another example of a self-similar set we will again demonstrate the process of determining fractal dimensionality. In order

9 FRACTALS AND PHYSIOLOGY 9 to construct the Sierpinski Gasket start with a filled-in equilateral triangle. Next, perform a Cantor-like removal process by removing the interior of the middle equilateral triangle whose vertices are the midpoints of the three edges. Repeat this process on the three remaining filled-in triangle, and so on [2]. Figure 3: Visual representation of the construction of Sierpinski s Gasket. The Sierpinksi Gasket at the n th stage of construction is made up of 3 n equilateral triangles with sides of length 1 2 n. Calculating fractal dimension in the same manner as before we can see that: N (r) r D = 1 log 3 n ( 1 2 )nd = log 1 n(log 3 D log 2) = 0 D log 2 = log 3 D = log 3 log 2 The fractal dimension of the Sierpinski gasket then is log 3 log 2.

10 FRACTALS AND PHYSIOLOGY 10 4 Fractal Organization of the Pulmonary Tree Considering the usefulness of fractals in biology, one way in which fractals can be tremendously helpful is in the description of anatomical and physiological features. In particular, fractals are useful in describing structures that reveal finer and finer detail upon microscopic examination where the small-scale structure resembles the large-scale structure [6]. An example of such can be seen when analyzing the pulmonary tree and the way in which it appears to infinitely branch. Figure 4: The branching pattern of the bronchial tree is self-similar in nature. In the case of the pulmonary tree, measurements of bronchial dimensions

11 FRACTALS AND PHYSIOLOGY 11 exist and therefore it is possible to develop this idea more mathematically. In the past, bronchial scalings from one level of branching to the next have been represented by an exponential curve: d(z) = d(z 0 )e (az ) Where d(z) is the average diameter of tubes in the z th generation, d 0 is the tracheal diameter, and a is the characteristic scale factor [6]. This representation however, is not sufficient beyond the first ten bronchial generations. Fractal representation, as will be shown, is a better means by characterizing this anatomical structure. For a fractal tree like what is seen in the pulmonary tree, there will not be a single characteristic scale factor. Rather, multiplicity of scales is seen. These multiple scales of the fractal model will have different probabilities of occurrence. Modeling the pulmonary tree as a fractal then, it is expected that bronchial diameter from generation to generation should decrease in the form of an inverse power law. The bronchial scaling then, d(z), will be proportional to the following: d(z) = 1 z µ In this case, µ is the power-law index. Research shows that this fractal representation and power-law scaling is a good fit for over twenty generations of pulmonary tree branching. Further, this type of fractal model accounts for variability not only between generations, but within generations. In particular, the variation of bronchial diameter within a generation is a consequence of having multiplicity of scales with different probabilities of occurrence [6]. Overall, modeling the pulmonary tree with a fractal model not only explains the branching from generation to generation more completely than the traditionally used exponential model, but also accounts for variability both within and between branching generations.

12 FRACTALS AND PHYSIOLOGY 12 Obtaining measurement of the bronchial tree is unrealistic for the scope of this project, and therefore calculation of fractal dimension of the pulmonary tree by hand is not possible. Using the box counting method in conjunction with photographs of the human bronchial tree it becomes possible to obtain a value for fractional dimension. The box counting method is a sampling process that can be used to determine fractional dimension. Essentially, in this method a series of grids of decreasing calibre are laid over an image and the number of boxes are counted for each calibre. For each grid calibre, the boxes which have any part of the important detail in the image are counted. In the box counting method changing the size of the boxes in the grid is a manner of approximating scale. By observing the count for how detail changes with scale it becomes possible to find the slope of the logarithmic regression line, also known as the fractal dimension. FracLac, a program developed by the National Institutes of Health, utilizes the box counting method in order to assign fractional dimension values to images. FracLac was used to analyze the following image, and a dimension of was determined. Figure 5: Digital image of the bronchial tree used in conjunction with FracLac to determine a value for fractal dimension.

13 FRACTALS AND PHYSIOLOGY 13 Upon noticing that the bronchial tree displays this type of fractal organization one may question why this type of organization is relevant. The bronchial tree is a crucial component of the lungs in mammals responsible for inhalation and exhalation of oxygen. Due to the fact that the bronchial tree grows as a branching fractal, the surface area across which gaseous exchange can occur is increased without significantly increasing the size of the lungs themselves. A fractal dimension of 1 represents a locally flat structure, whereas a dimension of 2 represents a locally space filling structure. Hence, the image above with a fractal dimension of indicates that the structure is more locally space filling than flat. In terms of functionality, a more space filling structure allows for greater surface area for gas exchange to occur. Thus, this example of the bronchial tree displaying fractal organization depicts one of many instances where fractal geometry contributes to physiological structure and function. 5 Fractal Analysis of Heart Rate In addition to fractal geometry being useful in the description of various biological and physiological structures, fractals can be used to characterize certain processes. An excellent example of this can be seen when considering ECG patterns. Often, a signal that varies with time (such as an ECG signal) must be analyzed to diagnose various diseases. These types of temporal processes are usually analyzed using Fourier transform technique, chaos dynamics, and other complex mathematical techniques [8]. These methods are complex, not particularly easy to analyze, and often require pre-processing of data. Hence, a more simple method for analyzing time varying signals would be useful. Since ECG signals are self similar [8] in nature, fractal analysis can be used in order to gather information and potentially classify signals. To explore the idea of classification of ECG signals based on fractal dimension four signal images were used in conjunction with FracLac in order

14 FRACTALS AND PHYSIOLOGY 14 to determine a value for fractal dimension. Figure 6: Signals of four different patients retrieved from [5] Each of these signals were run through the FracLac program with identical settings and a value for fractal dimension was determined for each.

15 FRACTALS AND PHYSIOLOGY 15 Figure 7: Fractal dimension of patient A was determined to be Figure 8: Fractal dimension of patient B was determined to be Figure 9: Fractal dimension of patient C was determined to be Figure 10: Fractal dimension of patient D was determined to be These values for dimensionality represent the amount of structure that exists within the image. A fractal dimension of 1 represents a curve that is

16 FRACTALS AND PHYSIOLOGY 16 locally flat, and a fractal dimension of 2 represents a curve that is locally space filling. In these four signals, that of patient B is considered to be normal. Signals A and C are from patients in sinus rhythm with severe congestive heart failure and D is from a subject with atrial fibrillation, which produces an erratic heart rate [5]. From these three measurements it can clearly be seen that variability exists between the fractal dimension of normal and abnormal signals. These results suggest that in the future it may be possible to use fractal analysis in classifying signals. Further, it may become possible to use this technology as a type of preventative analysis. Through performing fractal analysis of ECG signals (or other quasi-self similar signals of physiological processes) a range of normal and abnormal fractal dimensions can be developed. With these ranges established it may be possible to determine the fractal dimension of a signal from a patient and make a conjecture as to whether or not that patient is at risk for a particular disease. Considering the analysis above using FracLac it is important to note that these images were analyzed without significant preprocessing and that in order to develop a clinically useful tool it is necessary to determine a scaling and resolution that produces consistently accurate data. The analysis above is intended to demonstrate the correlation between visual appearance and dimension rather than provide fractal dimension values corresponding to disease and health. In the future with more research it may become possible that fractal analysis of heart signals could serve as diagnostic tools for physicians in diagnosis of heart conditions. Further, similar analysis of brain waves may provide insight into conditions including Parkinson s and epilepsy [3].

17 FRACTALS AND PHYSIOLOGY 17 6 Conclusion The information presented above is only a basic introduction into the research that has been done regarding the utility of fractals in physiology. Within the past few decades it has become increasingly clear that fractal geometry is applicable in both physiological structures and functions. With regard to structure, fractal geometry can be used to describe features including the bronchial tree, blood vessels, and certain cardiac muscle bundles which all serve to rapidly and efficiently transport substances over a complex network. The idea of fractals can be applied beyond physiological form in many cases and used in analyzing complex processes that lack a single scale of time [5]. Considering these processes, perhaps the most important notion to take away is the idea that these physiological processes are part of complex feedback systems. The forces impacting processes such as heart rate or brain waves are hardly simple, hence the signals derived from these processes are among the most complex in nature. It is clear that mathematics relating to nonlinear mechanisms and complex systems can be useful in developing greater knowledge of these systems, however for these tools of analysis to become clinically useful collaboration between clinicians and mathematicians is necessary. Research in the field has revealed that scale-invariance appears to be a general mechanism underlying various structures and functions in physiology [4]. Given this, the development of new quantitative tools adapted from fractal mathematics to measure healthy variability of is plausible. Additionally, if we know that certain physiological structures and functions have this type of fractal central organization we can make potentially useful predictions regarding when these processes are disturbed. For instance, if a physiological system has a fractal-like organization, we can reasonably predict that disease will be associated with a breakdown of scale-free structure and dynamics. Paradoxically, although many diseases are coined as dis-orders, many of these maladies display more regular and periodic behavior than their healthy state counterparts [4]. This loss of complexity in disease implies that the infor-

18 FRACTALS AND PHYSIOLOGY 18 mation content of the physiological system becomes degraded, and therefore the systems and processes become less able to contend with the constantly evolving environment of the human body. As the foundation of knowledge regarding the utility of fractals and non-linear dynamics in physiology continues to be developed the prospect of physiological monitoring, diagnosis of disease, and perhaps even therapeutic intervention becomes conceivable.

19 FRACTALS AND PHYSIOLOGY 19 References [1] Casey SD. Using dimension theory to analyze and classify the generation of fractal sets. Computers and Graphics (5): [2] Casey SD, Reingold NF. Self-Similar Fractal Sets: Theory and Procedure. IEEE Computer Graphics and Applications (3): [3] Cipra BA. A Healthy Heart Is a Fractal Heart. SIAM News (7):1-2. [4] Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, Mark RG, Mietus JE, Moody GB, Peng C-K, Stanley HE. PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals. Circulation 101(23):e215-e220 [Circulation Electronic Pages; (June 13). [5] Goldberger AL, Amaral L, Stanley HE. Fractal Dynamics in Physiology: Alterations with disease and aging. PNAS : [6] Goldberger AL, West BJ. Fractals in Physiology and Medicine. The Yale Journal of Biology and Medicine : [7] Hemming E. Fractal Analysis of Heart Rate Variability in Patients Following Surgical Repair of Acute Type A Aortic Dissection. Web. CollegeDay2008/Award/hemmig jurdieu doc.pdf [8] Islam N, Hamid N, Mahmud A, Rahman S, Khan A. Detection of Some Major Heart Diseases Using Fractal Analysis. International Journal of Biometrics and Bioinformatics. 4(2): [9] Karperien A. FracLac for ImageJ. /fraclac/flhelp/introduction.htm [10] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York Print. [11] Rasband, W.S., ImageJ, U. S. National Institutes of Health, Bethesda, Maryland, USA,

20 FRACTALS AND PHYSIOLOGY 20 [12] West BJ. Fractal physiology and chaos in medicine. 2nd ed. Singapore: World Scientific, Print.