T HEOR ETICALBIO LOGY

F O RUM

105 · 2/2012

PISA · ROMA

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ISSN 0035-6050ISSN ELETTRONICO 1825-6538

CONTENTS

editorial

Paolo Freguglia, Mathematical, Computational and Physical Tools for Theo-retical Biology 11

articles

Matej Plankar, Emilio Del Giudice, Alberto Tedeschi, Igor Jerman,The Role of Coherence in a Systems View of Cancer Development 15

Gianluca Martelloni, Franco Bagnoli, Stefano Marsili Libelli, ADynamical Population Modeling of Invasive Species with Reference to the Cray-fish Procambarus Clarkii 47

Anna Maria Caroli, Flavia Pizzi, Livestock Biodiversity: from Genes to Ani-mal Products through Safeguard Actions 71

Sergio Pennazio, Metals Essential for Plants: the Nickel Case 83

Dušan Ristanović, Nebojša T. Milošević, Fractal Analysis: Methodologiesfor Biomedical Researchers 99

FRACTAL ANALYSIS: METHODOLOGIESFOR BIOMEDICAL RESEARCHERS

Dušan Ristanović · Nebojša T. MiloševićDepartment of Biophysics, Faculty of Medicine, University of Belgrade, 11122 Belgrade 102, Serbia

Corresponding author: Dušan Ristanović. E-mail: dusan@ristanovic.com

Contents: 1. Introduction. 2. Fractal geometry. 2. 1. Von Koch curve set. 2. 2. Geometricalself-similarity. 2.3. Statistical self-similarity. 3. Fractal analysis. 3. 1. Fractal analysis of anon-fractal geometrical object. 3. 2. Segment-counting method applied to an unformatted curve.3. 3. Space-filling curves. 3. 4. Box-counting method. 4. An experimental illustration. 4. 1. Ex-perimental material. 4.2. Image processing. 4.3. Illustration of our finding. 5. Discussion.5. 1. Fractal geometry and systems in the nature. 5. 2. Fractal geometry and fractal analysis. 5. 3.Fractal analysis methods. 6. Conclusion.

Keywords: Fractal dimension; Fractal geometry; von Koch curve set; Richardson’smethod; Self-similarity.

1. Introduction

ractal analysis has proven to be a useful tool in analysis of various phenome-na in numerous natural sciences [1]. It has been widely applied and used for

quantitative morphometric studies mainly in calculating the fractal dimensions ofobjects [2]. Most of the authors determine the fractal dimension of borders of ob-jects in flat plane [3]. It is conceivable since one can claim that the borders of somenatural objects could be fractal.

Although standard quantitative methods in science are based on classical Euclid-ean geometry [4], fractal geometry [5] is developed as a new geometry of nature [2,3, 6, 7]. It was conceived in 1975 by Benoît B. Mandelbrot, with the aim to describethe complexity of forms and processes met in nature [1, 5]. Early work on fractalgeometry showed that most commonly biological patterns were characterized by

Abstract: Fractal analysis has become a popular method in all branches of scientific investigations including biology and medicine.Although there is a growing interest in the application of fractal analysis in biological sciences, questions about the methodology offractal analysis have partly restricted its widerand comprehensible application. It is a notablefact that fractal analysis is derived from fractalgeometry, but there are some unresolved issuesthat need to be addressed. In this respect, we discuss several related underlying principles for

fractal analysis and establish the meaningful relationship between fractal analysis and fractalgeometry. Since some concepts in fractal analy-sis are determined descriptively and/or qualita-tively, this paper provides their exact mathemat-ical definitions or explanations. Another aim ofthis study is to show that nowadays fractalanalysis is an independent mathematical andexperimental method based on Mandelbrot’sfractal geometry, Euclidean traditional geome-try and Richardson’s coastline method.

F

100 dušan ristanović · nebojša t. milošević

fractal geometry [1, 8-12]. Up to now fractal geometry is being used in diverse re-search areas [13-16] and is proving to be an increasingly useful tool. However, thereare herewith some important unresolved issues that need to be addressed.

Neuronal and vascular structures are among the most complex morphologiesand fractal analysis has been consid ered as useful tool for assessing their complexi-ties [1-3]. This is particularly an appropriate method for quantifying neuronal ar-borization [12] and for studying the straightness of individual neuronal dendrites[17]. When the human is concerned, fractal analysis has been considered as a usefultool for assessing the complexity of the body structure and processes in time [1].This is particularly an appropriate method for quantifying neuronal arborization [2]and for describing how neurons fill their dendritic fields [4] as well as to study thestraightness of individual dendrites [18].

Today, fractal analysis stands a contemporary nontraditional mathematical andcomputational method of measuring complexity of patterns in geometry and na-ture using the fractal dimension. The complexity and fractal dimension are largerif the object’s border is more rugged, branching pattern more abundant and linesmore irregular. Most researchers use a suitable method of fractal analysis to meas-ure a pattern’s degree of fractality [3, 7].

At the time of writing this article, a search on ‘fractal analysis’ throughout WorldWide Web yielded more than 660 articles. This number is expected to grow rapid-ly. In spite of that, no consensus emerged so far about the all-inclusive definition offractal analysis and its relation to fractal geometry. The aim of the present study isto support the hypothesis that fractal analysis is an independent mathematical andexperimental discipline which uses methodologies and concepts of Mandelbrot’sfractal geometry, Euclidean traditional geometry and Richardson’s coastlinemethod to explore the structures, functions and other properties of real objects.Some related methodological considerations and underlying principles for fractalgeometry and fractal analysis are discussed and several important qualitative con-cepts quantitatively determined.

2. Fractal geometry

Fractal geometry, as a contemporary branch of pure and applied mathematics andthe basis of all fractal ideas, is developed as a new geometry of nature [1, 5]. It wasconceived in 1975 by Benoit B. Mandelbrot [5], after his extensive work describingthe complexity of forms and processes met in nature [1]. Nowadays fractal geome-try is being used in diverse research areas and is proving to be an increasingly use-ful tool for characterizing biological patterns [2, 7, 15, 19, 20].

Fractals are classified into geometrical and statistical [3, 5, 21, 22]. Each geometricalfractal should be considered as an infinite ordered set of fractal objects defined on ametric space. To determine a fractal set we need to specify four things [1, 11, 21]:

ii. the shape of a starting object – the initiator or initial template,ii. the recursion (generating, iterated) algorithm – enabling its iterative application on the

initiator and then, repeatedly, on all obtained objects (the results being the generators, gen-erations),

fractal analysis: methodologies for biomedical researchers 101

iii. the conditions – which these generators should satisfy, before all, the properties of self-similarity, scaling or scale-invariance, and

iv. the fractal dimension – as the main quantifier to measure complexity of the objects.

In that case, such objects (generators) are called prefractals [1, 3, 6]. The final resultof such infinite procedure is the limit fractal [5]. The initiator, prefractals and limitfractal represent the geometrical fractal set [11, 21].

Basic definitions and laws of fractal planimetry can be demonstrated on some clas-sical fractal models [5, 11]. For that purpose we chose the triadic (snowflake) von Kochcurve set.

2. 1. Von Koch curve set

The sequential construction of the von Koch curve set begins with the initiatorwhich is an equilateral triangle of the side length r0 (Fig. 1A). The iterated algorithmto generate the set of the von Koch curves (known as prefractals) is to recursively re-ducing the straight line segment (or the scale) by 1/3 exchanging repeatedly the mid-dle third of each side of the initiator, or a preceding generator, with two sides of asmaller, equilateral triangle whose side is one-third the length of a previous side.The result after the first iteration (the stage of construction z = 1) is shown in Fig. 1B,and those after the second (the stage of construction z = 2) and third iteration (z =3), in Fig. 1C and D. Continuation of this process results in the limit von Koch curve.

For the von Koch generators, the length of a segment at the zth stage of con-struction (rz) and the number of segments at the same stage (Nz) are, respectively,

rz = r0 , Nz = 3 · 4z, (1)3z

where r0 is the side length of the initiator. The length of the fractal curve, actuallythe perimeter since the curve is closed (Lz), is defined as a product of the numberof segments and the length of a segment, at the zth stage of construction,

Lz = Nz · rz = 3r0 · ⎧4⎫z

. (2)⎩3⎭

From Eq. (1) it is evident that the number of segments of the von Koch prefractalsdiverges as z approaches infinity. The same holds for the perimeter (Eq. (2)). Thelimit curve is a line of infinite length surrounding a finite area. For that reason thiscurve cannot be pictured. If the values of Lz and Nz are plotted against the length ofsegments (rz) the corresponding graphs exhibit a typical hyperbolic decrease (Figs.2A and B).

Indeed, the length for the von Koch curves (shown in Fig. 2A) is

Lz = 3 (3)rz0.262

with the coefficients of determination R2 = 1 and where r0 = 1 for visual clarity. Gen-erally, if we wish to express Lz as a function of rz for similar fractal set, two constantsof proportionality (· and ‚) should be used, thus the length can be written as

102 dušan ristanović · nebojša t. milošević

Fig. 1. Sequential construction of the von Koch curve set. (A) The initiator is the equilateraltriangle of the side length r0. (B) The first stage of construction (z = 1).

(C) The second stage of construction (z = 2). (D) The third stage of construction (z = 3).Details below the drawings B and C represent the generating elements

of the von Koch prefractals shown in B and C.

Fig. 2. Hyperbolic decrease of the perimeters and number of segments for the von Kochcurves set. (A) Perimeter (Lz) of the von Koch prefractals generated from an equilateral triangle

(initiator) with an edge length of 1 cm. (B) The number of segments (Nz) of the von Kochprefractals, both plotted against the segment length (rz).

fractal analysis: methodologies for biomedical researchers 103

Lz = ‚ . (4)rz

·

The value ‚ is the prefactor and · is the scaling exponent. From Eq. (3) it is seen thatfor the von Koch fractal set ‚ = 3 and · = 0.262. We say that the length of a curve(a prefractal) scales with the length of the corresponding fractal segment.

The simplest scaling relationship has the power law form [1]. The mathematicalform of such scaling is an inverse power law scaling (Eq. 4). It describes how a prop-erty L of the system depends on the scale r at which it is measured [1]. Thus, thisscaling relationship shows how the perimeter of a prefractal depends on the lengthof its segment: the smaller the length of the segment, the larger the perimeter. SinceLz = Nz · rz, from Eq. (4) it follows that

Nz = ‚ · rz–D (5)

where a positive number

D = 1 + · (6)

is the fractal dimension. For the von Koch fractal set the fractal dimension is D =1.262.

Equations (4) and (5) reveal as decreasing straight-line plots when the results of cal-culating the values of curve perimeters and these of counting the numbers of seg-ments of a fractal set (Eq. (5)) are plotted on log–log axes against the values of thesegment length (Fig. 3). Indeed, from Equations (4) and (5) it follows numerically

log (Lz) = – · · log (rz) + log (‚), log (Nz) = – D · log (rz) + log (‚). (7)

Both expressions represent the equations of straight lines; the scaling exponent (·)is given by the slope of the line of best fit (Fig. 3A), whereas the fractal dimension(D) can be obtained as a gradient of the straight line (Fig. 3B).

Fig. 3. Results of calculating the values of perimeters of the von Koch prefractals and numbersof segments, shown on log–log axes, against the length of a segment rz. (A) The perimeters Lz.

(B) The numbers of segments Nz. The graphs are obtained using equations inscribedon the pictures. R2 is the coefficient of determination.

104 dušan ristanović · nebojša t. milošević

2.2. Geometrical self-similarity

The object’s property known as self-similarity was first coined by Mandelbrot [5] andcan be geometrical or statistical [1, 11, 21]. A fractal pattern is said to be geometricallyself-similar if each small piece of it is an exact replica (i.e. ‘duplicate’) of the wholeobject [1, 3, 5]. Thus, self-similarity qualitatively means that every small piece of anobject resembles the whole object [1, 11]. This definition of the concept ‘geometri-cal self-similarity’ should be quantified since small pieces that constitute geometri-cal or natural objects are rarely identical copies of the whole object [19].

We have offered a more exact interpretation of this descriptive definition intro-ducing a generating element of a generator as a «small piece» [6]. A generator is usually made up of straight-line segments (for instance, see Fig. 1). A particular andlogical concatenation of some segments of a generator could be thought of as thegenerating element [11] of a generator if the whole object can be completely built withsuch elements by their translations and/or rotations.

In our example shown in Fig. 1 the generators of the von Koch prefractals, at thefirst and second stages of construction, have the generating elements made up offour equal segments each, as shown as details below the drawings in Fig. 1B and C.For example, the drawing in Fig. 1B can be subdivided into three generating ele-ments, that in Fig. 1C into 12 elements, and so on. The length (lz) of a generatingelement of the von Koch image at the zth stage of construction and the number(nz) of generating elements of the von Koch prefractal are, respectively,

lz = 4 · r0 , nz = 3 · 4z–1. (8)3z

Two generating elements of two generators of a fractal set can be geometricallysimilar or not. According to the definition of similarity in Euclidean planimetry, twogenerating elements of the generators at stages z and z + 1 (say, those in Fig. 1B andC) are similar to each other if (a) the ratio of the measure of a segment of the generating element at stage z + 1 and the measure of the corresponding segmentof the element at stage z is constant for all pairs of corresponding segments (e.g.,for the four pairs of segments of the two mentioned details in Fig. 1) and for all z,and (b) the angles between the pairs of corresponding segments of the two gener-ating elements are congruent. If the generating elements of the generators at stagesz and z + 1 are similar for every z, we say that the whole class of the generators isgeometrically self-similar, or, that this set has the property of geometrical self-simi-larity. We should mention in connection with it that generating elements can be defined on some other fractals [1], such as, Sierpinski’s triangle (generating elementis equilateral triangle), Cantor set (straight line segment), asymmetric Y-branch (astem with two branches) etc.

Scale invariance is a feature of objects or laws that they do not change if scales oflength or other variables are multiplied by a common factor. For example, this com-mon factor for the von Koch fractal set is 1/3. If r0 is 1 cm (Fig. 1A), the length ofthe segment in Fig. 1B is 1/3 cm, that in Fig. 1C is 1/9 cm, etc. This means that ifwe multiply the length of each segment of the generating element in Fig. 1B by 1/3,

fractal analysis: methodologies for biomedical researchers 105

the same but reduced generating element of the prefractal shown in Fig. 1C will beobtained.

2.3. Statistical self-similarity

The pieces of biological objects are rarely exact reduced copies of the whole objects[1]. Rather than being geometrically self-similar, they could be statistically self-sim-ilar. The natural object is statistically self-similar if a statistical property of everysmall piece of an object is not significantly different from same statistical propertymeasured on the whole object [1-3, 11].

It is instructive to see how similar definition has been used by Bernard et alii [23]in solving their experimental problem. They verified, in fact, the statistical self-sim-ilarity of rat cerebral oligodendrocytes in their culture conditions. Using statistics,this can be carried out by comparing the fractal dimension (as a cell property) cal-culated for an entire single cell, with the mean fractal dimension calculated fromthe fractal dimensions for many smaller parts of the same cell. The obtained resultscould be analyzed for significance by statistical tests. Statistical analysis could showno significant difference. The obvious similarity of the fractal dimension values ob-tained from the whole cell and from small parts of it confirmed the statistical self-similarity of oligodendrocytes, showing that this cell represents a statistical fractal.Similar result should be expected by comparing the density of the cell arbor withthe mean density of parts of the cell’s arbor [2].

A pattern in nature is hypothesized to be a fractal-like, i.e., statistically self-simi-lar across a range of scales [4]. Some authors tried to illustrate graphically the self-similarity of geometrical and natural fractals [1, 9-11, 14] showing that a detail of astructure or signal at smaller scales has a similar, but reduced, form to the entirestructure. Cameron et alii [24] claimed that most objects in the real world possessfinite level of detail, and are thus not fractal structures. In fact, a natural object canbe a statistical fractal if (a) it is statistically self-similar to a complex system whosethis object is a part, or (b) it consists of many small pieces being statistically self-sim-ilar to the whole object.

3. Fractal analysis

According to the numerous researchers, fractal analysis is derived from fractalgeometry [1-3, 7]. Previous work on fractal analysis represented traditionally a two-dimensional analysis [12] focused mainly to the analysis of border’s outlines of nat-ural objects [2]. Fractal analysis is a mathematical and experimental discipline whichuses methodology of Mandelbrot’s fractal geometry, Euclid’s traditional geometryand Richardson’s coastline method [6] to investigate the structure (mainly of the ob-ject’s border and its complexity), function and other properties of real objects. Theproblem is guided to calculating the fractal dimension and analyzing what the valueof this parameter tells us about the considered object.

By analogy with study of a geometrical fractal set in fractal geometry, the use offractal analysis signifies specifying four things:

106 dušan ristanović · nebojša t. milošević

i. the shape of a starting object – which corresponds to the initiator of a fractal set,ii. the algorithm enabling its repeated application to the starting object – which gives rise

to the generators,iii. the conditions – which these generators should satisfy, before all, the property of scaling,

andiv. the fractal dimension.

The limit generator is the same as the starting object, because it was not changed during repeated application of the algorithm.

In fractal analysis there are two basic approaches to measuring the fractal di-mension of objects in a plane [2]. The first and most commonly used is length-relat-ed method comprising the classical Richardson’s coastline method [5], box-countingmethod [4, 25-28] and dilation method [12, 26]. The second method is mass-relatedmethod [2].

The use of different methods of fractal analysis has made comparison of resultsdifficult because each method of determination of the fractal dimension givesslightly different results when uses to analyze the same structure [3, 12]. It shouldbe noted that the coastline method is equivalent to using the box-counting methodonly when one analyzes the border of a system [12].The problem to present, analyzeand compare different methods for measuring the fractal dimension is still open.

3. 1. Fractal analysis of a non-fractal geometrical object

Suppose we inscribe an equilateral triangle (Fig. 4A) in a smooth circle of radius 1cm (the starting object) and state the following algorithm: from the middle of each triangle side the normal is erected to the section with the circle. These sections areconnected to the nearest endpoints of the triangle sides forming a regular hexagon(Fig. 4B) inserted in the circle. The result of the next iteration using the same al-gorithm is a regular dodecagon (Fig. 4C) etc.

Fig. 4. A circle as an example of fractal analysis of a non-fractal geometrical object.(A) The equilateral triangle as the initiator inscribed in a circle of the radius 1 cm.

(B) The first-stage generator (the hexagon). (C) The second-stage generator (the dodecagon).Details below the drawings B and C represent the generating elements

of the polygons shown in B and C.

fractal analysis: methodologies for biomedical researchers 107

It is known that the side length of a regular polygon inscribed in a circle of unitradius can be calculated from the formula

r2n = ����2 – 2������1 – 1 rn2�� (9)4

where rn is the side length of a regular polygon with n sides (where n = 3, 6, 12, 24,…). It is seen that we applied here almost the same method as applied in con-structing the von Koch prefractals. By inspecting the details below the drawings inFigures 4B and C, which represent the generating elements for these polygons (gen-erators) inscribed in the circle, it is obvious that they are not geometrically similarsince the angles between the segments are different. It means that the condition ofself-similarity is not satisfied (Subsection 2.2) and the set of regular polygons in-scribed in the circle cannot be considered as a prefractal set [6].

As has already been mentioned, the perimeter of prefractals of the von Koch frac-tal set tends to infinity when the segment length of the curves decrease. This facthas developed in the inverse power law scaling (Eq. (4), Subsection 2. 1). In fractalanalysis, where one analyzes the fractal properties of real objects, this law shouldchange its form. For example, as the lengths of the polygonal segments decreaseand finer details of the structure are revealed, the measured lengths of the polygonsincrease converging to an asymptotically stable value. In previous example (Fig. 4)this stable value is 2 = 6.2831… (see Fig. 5A). Or, what is the same thing, if thenumber of iterations increases, the polygons tends to the starting circle. Since thefitting line can be determined by a second-degree equation (shown in Fig. 5A) thelaw scaling can be thought of as the inverse polynomial law scaling. Such law is char-acteristic of geometrical and real objects in fractal analysis.

In order to calculate the fractal dimension of the circle as a non-fractal object, wepresented in Fig. 5B the graph of relationship between the number of polygonal

Fig. 5. Fractal analysis of a non-fractal geometrical object (the circle shown in Fig. 4).(A) Length L of the inscribed polygons against the segment size r plotted on log–log axes.The second-degree polynomial fitting the calculated data points (open circles) is depicted

in the figure. (B) Number of the polygonal segments N against the segment length r plottedon log–log axes. Depicted formula is the equation of the straight line in this system.

R2 is the coefficient of determination.

108 dušan ristanović · nebojša t. milošević

segments and their sizes on log–log axes. It is seen that the obtained data can be fit-ted with a decreasing straight line with the fractal dimension D = 1.008. Althoughthe circle and the set of polygons are not a fractal set yet they have the correspon-ding fractal dimension. Measurement of the fractal dimension is usually not in-tended to indicate whether an object is a fractal object or not [3].

3.2. Segment-counting method applied to an unformatted curve

Lewis F. Richardson (1961) wondered if there are more wars between nations thatshare longer common borders [29]. Answering this question required measuringthe length of national borders. As he looked the borders closer and closer, he sawmore and more new details. To measure the length of the border drawn on a mapof Atlas, he used as a ruler a divider having sharp points at the end of both arms.The total length of the border was given by the number of divider steps multipliedby the distance between the ends of the divider. When he reduced the distance between the ends of the divider and repeated the entire measurement, the lengthdid not converge to a stable value but kept increasing. That is, the length scales withthe resolution of the instrument used to measure such a length. Richardson also measured the length of the west coast of Great Britain. The coastline paradox is thefinding that the measured length of a stretch of coastline depends on the scale ofmeasurement. Empirical evidence suggests that the smaller the increment of meas-urement, the longer the measured length becomes. It seems that the measuredlength increased without limit as the measurement scale decreased towards zero.

In order to apply segment-counting method to an unformatted curve (Fig. 6), ascale (e.g., a ruler) of various lengths (r) was used to measure curve lengths (Richard-

Fig. 6. An unformatted smooth curve subjected to fractal analysisusing Richardson’s ruler-counting method.

2 cm

fractal analysis: methodologies for biomedical researchers 109

son’s coastline or segment-counting method). For that purpose, we employed a dividerwhich is like a compass except that it had sharp points at the ends of both arms,keeping a fixed distance between the two divider’s ends by means of its adjustingscrew [6]. Since the measurement was performed in such a way that the dividermoved along the curve until the curve was completely traversed, the obtained ap-proximate (scaled) length (L) of the curve was given by the number of ruler steps(N) multiplied by the ruler length (r), or, more precisely, by the distance betweenthe ends of the divider. The measurement of the curve length was done around thelongitudinal axis (the skeleton) of the curve. The smallest span to enable reliable re-sult of measurement was 2 mm.

It is well known that L scales with r, i.e. for each ruler length there is a matchingcurve length. Usually, as the length of the divider decreases, the measured length ofthe object increases. The polygonal lines connecting the corresponding adjacentpoints on the line (Fig. 6) carried out on the drawing of the line by the divider’s tips,represent the generators. The algorithm could be a demand that the ruler lengthchanges from larger to smaller values (say, from 3 cm to 2 mm) but it does not mat-ter since the starting object does not change its shape during (iterative) measure-ment of its length (this algorithm is crucial in forming the geometrical fractal sets),so that the order of measuring the length is unconscious.

If the data obtained by segment-counting method are fitted by the second-degreefunction, for the relationship between the L and r (Fig. 7A), the following formulais obtained

L = – 0.465 · r2 – 1.721 · r + 30.963 (10)

where the coefficient of determination R2 = 0.994. The graph shown in Fig. 7A rep-resents the right tail of a convex-upward second-degree parabola, the critical valueof its maximum being rc = -1.85. But the rest of the parabola, being in negative (thesecond quadrant) area of the coordinate plane, has no physical sense (negative

Fig. 7. Fractal analysis of unformatted smooth line shown in Fig. 6. using ruler-counting(Richardson’s) method. (A) Curve length L measured using the divider, against the segment

(ruler) length r. (B) Number of segments (rulers) against the segment length r.The corresponding formulae of the fitting lines are shown in the text.

110 dušan ristanović · nebojša t. milošević

length does not make physical sense, like negative mass). For the relationship be-tween the N and r it is obtained

N = 26.264 · r –1.145 (11)

where R2 = 0.994 (Fig. 7B). The exponent in Eq. (11) represents the fractal dimen-sion of the curve. It is obvious that this parameter measures the degree of devia-tion of the smooth curve in Fig. 6 as compared to a straight line segment whoseD = 1. If a pattern branches, the approximate total length of all branches is the sumof lengths of each branch.

It should be noted that if r tends to zero (Eq. 10), the length of the curve shownin Fig. 7A tends to a stable value (being 30.693). This mathematical consequenceshould be corrected by the fact that every experimental measurement of the lengthmust be performed with the corresponding experimental error. Segment-countingmethod, in this example ruler-counting method, which uses the compass with two tipson its legs, can be successfully carried out if the ruler length is not smaller than r =0.2 cm. For that value of the ruler length, the acceptable value of the curve lengthin Fig. 6 should be 30.33 cm (from Eq. (10) with r = 0.2 cm). If r = 0, it follows 30.693cm representing the stable (asymptotic) value of the curve.

It should finally be noted that real finite objects, however complex, have finitelength. It means that the length of the object starts repeating if the measuringlength becomes small enough. For example, after the 10th iteration in example pre-sented in Subsection 3.1 (Fig. 5A), when the segment length of a polygon becomes2.0453·10-3 cm and inscribed polygon attains 3072 sides, the six decimals of the meas-ured length starts repeating and the stable (asymptotic) values of the perimeter becomes approximately 6.283186.

3.3. Space-filling curves

The topological (Euclidean) dimension (DT) of a space or object is defined as the min-imum number of coordinates needed to specify a position of any point within it.Thus a straight or irregular line has a dimension 1. Following the same deduction,a surface (such as a plane) has a topological dimension 2, while the inside of a cube,a cylinder or a sphere has dimension 3.

As we stated before (Section 3), the D is the basic parameter in fractal analysis. Itis a measure of pattern’s complexity [18], or more precisely, this parameter analyzeshow details of a pattern change with the scale at which they are measured. This pa-rameter estimates the degree of tortuosity and irregularity of smooth lines com-prising a pattern [3,6]. For example, it is noted that a highly irregular line filling uptwo-dimensional space would have a higher fractal dimension, closer to 2.

The topological dimension DT is always an integer but the D need not be [5]. Ithas also been used as a measure of the space-filling capacity of a pattern [4]. How-ever, in the last assertion there is an unresolved issue that needs to be focused.

In mathematical analysis, a space-filling (Peano’s) curve is a close packing curvewhose range contains the entire two-dimensional unit square. It is a self-intersect-ing curve that passes through every point of the unit square. Most well-known

fractal analysis: methodologies for biomedical researchers 111

space-filling curves are constructed iteratively. Space-filling curves are special casesof fractal constructions. The fractal dimension of the Peano’s curve is 1.67. For thatreason some authors also use the fractal dimension to measure the degree of thespace-filling of a pattern [4]. More precisely, there exists the attitude that the fractaldimension measures how completely a pattern fills the space. Indeed, a line seg-ment, that has topological dimension DT = 1, could be so wiggly that it nearly com-pletely fills a two-dimensional space and thus its fractal dimension D could be near-ly 2. We would like to notice that it is not always the case.

It is well known that the fractal dimension of a smooth straight line is 1. In Fig.8 it is demonstrated, using segment-counting (ruler-counting) method, that a col-lection of 5 smooth straight line segments (Fig. 8A) has also the fractal dimensionalmost 1 (Fig. 8B). Imagine the unit square in which it is not inscribed Peano’s curve,but instead a large number of straight line segments. This space is filled in a largedegree. But, from the present example it is obvious that the fractal dimension ofsuch system is again 1. On the other hand, the pattern which also contains 5 but wig-gly line segments (Fig. 9A), having nearly the same space-filling degree as perviousexample (Fig. 8), has larger fractal dimension than 1 (Fig. 9B). It seems to be mostlikely that the fractal dimension does not measure the space-filling of a pattern, butits complexity, before all tortuosity and irregularity of the pattern’s smooth curvesand ruggedness of the border of an object [26].

3.4. Box-counting method

Richardson’s (segment-counting) method of measuring the fractal dimension is ro-bust with very high correlation coefficients but it is, at the same time, tedious andtime consuming. Therefore, the need for more handsome methods for that meas-

Fig. 8. Measurement of the space-filling capacity of straight line segments.(A) Five straight line segments. (B) Log-log plot of the relationship between

the numbers of segments N measured using Richardson’s ruler-counting methodand the segment lengths r. R2 is the coefficient of determination.

112 dušan ristanović · nebojša t. milošević

urement emerges. The conventional [2] box-counting is one method suitable tomeasure fractal dimensions of real objects [2-4, 10, 11, 13, 22, 27-32]. This method isbased on the concept of ‘covering’ the image with rectangular coordinate grid [2]:the image (e.g. the curve) is overlaid with a grid, and the number of boxes inter-sected by the pattern counted. Each set of boxes is characterized by the square sider. The corresponding number of squares N necessary to cover the border (or anycurve) is presented as a function of r. Fractal dimension D of an image is determinedas the slope of the log–log relationship between N and r.

Although the method is not devoted to measuring the lengths and other featuresof patterns, this is the best technique to estimate the fractal dimension. Thereforethe box-counting method is commonly used among other fractal techniques [12, 18,20, 26, 27]. Unfortunately, it seems that the box-counting method demonstratessome mistakes. Using this method we showed that the fractal dimension of astraight line segment is 0.927, not 1. The fractal dimension of a circle of the radius1 cm (Fig. 5A), obtained using purely mathematical method, is 1.008, while thatmeasured using the box-counting method is 1.157.

4. An experimental illustration

4. 1. Experimental material

Golgi-impregnated neurons of the human spinal cord were traced using camera lucida. The images were obtained from 21 bodies. These images were previouslypublished (Schoenen [34] -with permission) and also used in our publication [27]. Adetailed description of the histological procedure can be found in these references.The images of all human neurons were divided into groups based on their laminarposition.

Fig. 9. Measurement of the space-filling capacity of irregular line segments.(A) Five irregular segments. (B) Log-log plot of the relationship between the numbersof segments N measured using Richardson’s ruler-counting method and the segment

lengths r. R2 is the coefficient of determination.

fractal analysis: methodologies for biomedical researchers 113

4.2. Image processing

The drawings of the neurons were converted into digitized images using a scanner‘Optic Pro 96301’, Mustek, with a resolution of 600 dpi. All images were analyzedas binary (black-and-white) and skeletonized tracings. Corresponding transforma-tions were carried out on a PC computer using the public Image J software(http://rsbweb.nih.gov/ij), developed at the US National Institute of Health. Allscanned images were imported into the software. Axons, spines and somata wereremoved digitally from the drawings. Since the mentioned study dealt only with thequantitative analysis of dendritic arborization patterns, removing the somata,spines and axons from digitized images of neurons was indispensable. These im-ages were analyzed applying two different methods of fractal analysis: Richardson’sruler counting and most popular box-counting method.

4.3. Illustration of our finding

It was noticed above that the fractal dimension obtained using ruler-countingmethod does not measure the space-filling of a pattern, but its complexity, beforeall tortuosity and irregularity of the pattern’s smooth curves and ruggedness of theborder of an object [26]. On the other hand, the box-counting method measuresmost probably only the space-filling. To illustrate this finding on the mentionedneurons, we applied both methods on each cell. In Fig. 10A a stellate cell in laminaII from a 34 week-old fetus, is expanded in this illustration. Using Image J we havefound that the fractal dimension of the binary image was 1.188, while that of theskeletonized image is a bit smaller and is given as 1.136. It is obvious that this pa-rameter reflects the ‘mass’ of the pixels in a given figure. On the other hand, usingRichardson’s method (Fig. 10B) we found that this dimension was nearly 1 (i.e.0.985) showing the value for a straight-line segment. It is expected since the den-drites of this neuron are mainly rectilinear.

5. Discussion

5. 1. Fractal geometry and systems in the nature

It is noticed before that geometrical self-similarity means that every small piece of anobject resembles the whole object. Many authors tried to demonstrate graphicallythis definition [1, 4, 14]. We showed that such objects can be self-similar if their gen-erating elements are geometrically similar (Subsection 2. 2). Any two prefractals ofthe von Koch set are not similar, contrary to their generating elements. This is ob-vious from Fig. 1B and C: these two images could only be visually alike, but notgeometrically similar.

The simplest scaling relationship between some properties of the system and thescale has the power law form [1], or, more precisely, an inverse power law scaling (Sub-section 2. 1, Eq. (4)). It should be noted that property inverse power law scaling givenby Eq. (4) is necessary but not necessary and sufficient condition for geometrical frac-tality of a set of objects. In order that a set of objects represents a fractal set, it is

114 dušan ristanović · nebojša t. milošević

(only) necessary that this set satisfies the inverse power law scaling (Eq. (4)). It meansthat the relationship L = f(r) represents a straight line on log–log axes (see Fig. 3).The opposite is rarely true: if a set of objects satisfy the condition ‘a straight line onlog–log axes’, this set may not be a fractal set (because this condition is only neces-sary). This is illustrated in Fig. 5: although the set of regular polygons gives rise tothe fractal dimension (Fig. 5B), the set of data for L is not fractal since it does notsatisfy the vital condition for fractality – the self-similarity (see Fig. 4B and C, wherethe generating elements are not geometrically similar to each other). Bassingth-waighte et alii [1] also noticed that the power law scaling is a result of self-similari-ty. Therefore it happens that some researchers – from satisfying the condition ‘astraight line on log–log axes’– too quickly conclude that an examined set of objectsis fractal. It can be fractal, but it may not be. The natural objects can satisfy the in-verse power low scaling only within an upper and lower limit of a restricted regionand be characterized, across this region, by the fitting parameter D [33]. To showthat the corresponding set of objects represents a fractal subset it should be essen-tial to prove out that they are self-similar, that is, geometrically similar to each oth-er. We have not found such a proof in the articles where such a problem is consid-ered. Therefore, it is not necessary to insist on fractality of these objects; it is

Fig. 10.

fractal analysis: methodologies for biomedical researchers 115

enough to analyze the value of the fractal dimension (and other fitting parameters)for an objective assessment of degree and kind of complexity of natural objects [2,3]. In fact, the fractal dimension is not intended to indicate whether the image is afractal object or not [3]. The fractal dimension in fractal analysis is only a useful descriptive parameter, like the dendritic field area or the size of the cell’s soma [3].Furthermore, obtaining a D for natural images by using fractal analysis does notnecessary imply that the image is fractal [4, 18].

The natural object is statistically self-similar if a statistical property of every smallpiece of an object is proportional to that property measured on the entire object [1,2]. It means that a property of a small piece of a growing object varies in a directrelation to this property of the entire object but the growing object is not alwaysthe subject of fractal analysis of object’s morphology. We find that this definition israther perplexed when applied to experimental dataset. Therefore, we replaced thenotion ‘proportional to’ with ‘is not significantly different from’. We find that ourdefinition of statistical self-similarity does not contradict the official definition, butit is more acceptable in research procedures and closer to the subject of statistics.

5.2. Fractal geometry and fractal analysis

The extension of the concepts of fractal geometry toward the life sciences has ledto significant progress in understanding complex functional properties and struc-tural features [1, 3, 8, 21, 22]. On the other hand, fractal analysis is assessing fractalcharacteristics of data [2, 3, 22]. It consists of several methods to measure a fractaldimension and other fractal characteristics of a dataset. According to the numerousauthors, fractal analysis is derived from fractal geometry [3, 6, 22] and used to de-scribe the organization of objects found in nature, quantifying their complexitywith a value of the fractal dimension [13-15, 19, 20, 23, 25, 26, 27, 30, 31].

Comparing the use of fractal analysis with that of fractal geometry it is our im-pression that these two important areas of science are mostly mutually independ-ent. For example, the initiator in fractal geometry is included into the fractal itera-tions as a part of prefractals (see also the Cantor fractal set and Sierpinski trianglein Bassingthwaighte et alii [1]), while the starting object in fractal analysis does notchange its integrity during iterations keeping the same shape as a limit object. Theself-similarity and scaling, being the basic principle in fractal geometry, have differ-ent meanings in fractal analysis (Section 3).

5.3. Fractal analysis methods

The essential idea of the fractal dimension (i.e. fractal analysis) has a long historyin mathematics [19], but the term itself was introduced by Mandelbrot based on hispaper on self-similarity [34] in which he discussed ‘fractional’ dimension. In that pa-per, Mandelbrot cited previous work by Richardson [29] and noted that the dia-grams presented in his paper lead to the conclusion that lengths of the coastlinesand frontiers between countries can be given by

L (r) ∝ F · r 1–D (12)

116 dušan ristanović · nebojša t. milošević

where the value of the exponent D seems to depend upon the coastline that is cho-sen. Topology fails to discriminate between different coastlines [5]. To Richardson,the D was a simple exponent of no particular significance. Having ‘unearthed’Richardson’s work in which he claimed that his lines’ slopes had no theoretical in-terpretation [5], Mandelbrot proposed that the exponent D should be interpreted asa fractal dimension [34]. It is our impression that the Richardson’s work had a pri-mary and decisive influence on development of Mandelbrot’s fractal geometry(Section 3). Also, we find that among many fractal analysis techniques, only theRichardson’s ruler-counting method enables calculation of the length of an object’sborder or irregular line (Subsections 3. 2 and 3. 3). The other techniques analyze theslopes of the straight lines representing the relationship between the number ofscale steps and scale length on log–log axes. However, it should be borne in mindthat the number of distinct scales of length of natural patterns is, for all practicalpurposes, always infinite [5].

The traditional Richardson’s segment-counting method is by no means basic formeasuring the fractal dimension within segment-related techniques. Unfortunate-ly, this measuring is tedious [3, 6, 31]. It would be best to prepare a suitable programfor that analysis. We have offered the circle-counting method [35]. This is a techniquefor measuring the length of irregular lines (like the cell dendrites). The analyticalimage was input into our proprietary software for drawing circles, written in Visu-al Basic (Microsoft) and the lengths of dendrites were successfully measured.

One of the main shortcomings of the box-counting method is that for large box-es the number of boxes intersected is sensitive to the location of the object uponthe grid [2], so it is necessary to average many trials (maybe 200 [4]) for each gridsize. It seems that dilation method is less sensitive to the location of the image in aframe [2]. Although time consuming, the ruler-counting method perhaps needs only one additional measurement, that is, from the end point of the curve to its beginning.

6. Conclusion

Fractal analysis is an independent mathematical and experimental method derivedfrom Fractal geometry and used to determine the fractal dimension of geomet-rical or natural objects. The fractal dimension describes the complexity of an object. The fractal dimension is larger if the complexity is higher, that is, the borders of objects more rugged, branching patterns more profuse, and linearstructures more wiggle. To calculate the fractal dimension of an object it is notnecessary to investigate its main fractal properties (such as, the self-similarity, scal-ing, scale-invariance and space-filling). Investigation of these properties is themain subject of fractal geometry. The chief aim of fractal analysis is to calculatethe fractal dimension of an object and ascertain what this value tells us about thecomplexity of the object. Finally, we should notice that the aim of the present article is not to discuss are fractals everywhere [36] or is the Geometry of naturefractal? [37].

fractal analysis: methodologies for biomedical researchers 117

Acknowledgement

The authors are grateful to the Ministry of Education and Science, Republic of Serbia, Research Project III41031.

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