JOURNAL OF PATHOLOGY, VOL. 169: 383-385 (1993)

TECHNICAL METHOD

A SIMPLE TECHNIQUE FOR THE MEASUREMENT OF FRACTAL DIMENSIONS IN

HISTOPATHOLOGICAL SPECIMENS HARRY SANDERS A N D JOHN CROCKER

Histopathology Department, East Birmingham Hospital, Bordesley Green East, Birmingham B9 5ST, U.K.

Received I7 August 1992 Accepted 20 October 1992

SUMMARY A computer program has been devised which enables a simple method for the fractal dimension analysis of

histological or cytological material to be performed rapidly with relatively inexpensive hardware. The fractal dimension of a structure refers to its self-similarity over a range of scales and appears to be well suited to histopathological preparations. Thus, the fine ‘roughness’ of object profiles can be assessed more appropriately by this technique than by means of conventional Euclidean geometry. Images of the object(s) to be investigated are firstly digitized after presen- tation to a high-performance computer via a video camera or a graphics tablet system and then analysed using the computer program, which has its basis in a standard perimeter estimate method. It is proposed that such techniques may find extensive use in investigative and diagnostic histopathology.

KEY WORDS-Fractals, chaos theory, morphometry.

INTRODUCTION

The potential value in pathology of fractal analysis has recently been promulgated in both the medicall-‘ and the computer literature.’ This is no doubt a response to the increasing popularity and understanding of ‘chaos theory’* in biological systems. Indeed, a large number of biological structures have shapes which are too irregular or ‘rough’ to be defined accurately by conventional or Euclidean mathematics o r geometry. The fractal properties of a structure reflect its self-scaling similarity; this will be true over a wide or even infinite range of scales. Examples of such fractal forms can be seen in the profiles of a coastline or a bracken frond. In view of the irregular surfaces of many structures in histopathology, they would seem ideally suited to non-Euclidean, fractal dimension analysis; accordingly, we have devised a relatively

Addressee for correspondence: Dr J. Crocker, Histopathology Department, East Birmingham Hospital, Bordesley Green East, Birmingham B9 5ST, U.K.

simple technique for the application of this potentially powerful approach to histological material.

MATERIALS AND METHODS

Image analysis Digitized images of the structures to be analysed

are produced by means of a Summagraphics Bit Pad 2 graphics tablet and cursor or by the use of a high-resolution video camera system linked to an Acorn Archimedes A5000 32-bit computer. The data are processed by means of the Digit software package (BPHayes Software, 3 1 Prior Gardens, Berkhamsted, Herts, HP4 2DS).

Fractal dimension analysis A computer program was devised to determine

the fractal dimension of structures of interest which had been digitized previously by means of the software described above. The fractal dimension

0022-341 7/93/03038343 $06.50 0 1993 by John Wiley & Sons, Ltd.

384 H. SANDERS AND J. CROCKER

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was calculated using a series of estimates of the perimeter of the structure. This method of calcu- lation of the fractal dimension uses the equation described by Falconer:'

lim log MdF) s=6+o -log6

where s is the fractal dimension; F is the plane curve (in this case, the series of points on the structure); M J F ) is the perimeter; and 6 is the scale of measurement (in this case, the y-coordinate separation).

The method is described by Kaye." A series of perimeter estimates of each structure was found using a number of polygons constructed on the digitized image of the structure. The polygons were formed by selecting coordinate pairs produced by the digitizing software using different y-coordinate spacings, i.e., using every y-coordinate, every second y-coordinate, every third y-coordinate, and so on (Fig. 1). Where the digitizing software had not generated a y-coordinate at the given spacing, the missing coordinate pair was derived by interpola- tion. Since the digitizing software produced integer coordinate pairs, the y-coordinate pairs were 1, 2, 3 , . . . , etc. Thus for a given y-coordinate spacing, an estimate of the perimeter of the structure was found by calculating the perimeter of the polygon based on the coordinate pairs which occur at the given spacing. Further, for a set of y-coordinate spacings, within a given range, a set of perimeter estimates was obtained.

The range of y-coordinate spacings can be regarded as a measure of the resolution of the

Fig. 1-Perimeter estimate based on unbiased step size. The perimeter estimate is given by the perimeter of a polygon formed by the intersection of the outline curve and a series of equally- spaced parallel lines. The perimeter is thus based on step size y , = ab + bc+ cd +. . + op+pa

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log of step sue (normahsed)

Fig. 2-A Richardson plot of a series of data for a relatively smooth-surfaced object; accordingly, the slope is quite gentle

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.I 5 .I 4 -1.3 - 1 2 -1 I 0.9

10s of step slze (normalised)

Fig. 3-A Richardson plot with a steeper slope than that in Fig. 2, indicating a relatively rougher surface

technique. Thus, a range of small y-coordinate spacings will give the fractal dimension, which relates to the roughness of the surface detail of the structure, whilst a range of large spacings will give a measure of the irregularity of the overall shape of the object. A regularly-shaped structure with a rough surface will have a low fractal dimension (smooth, regular) within a large range ofy-coordinate spacings (overall shape) but a high fractal dimen- sion (rough, irregular) within a range of small y-coordinate spacings (surface detail).

Each y-coordinate spacing and its corresponding perimeter estimate were 'normalized' with respect to the Feret diameter (maximum x- and y-diameter) of the structure, equivalent to the maximum between

FRACTAL DIMENSION ANALYSIS I N PATHOLOGY 385

any two points on the structure. The fractal dimen- sion can be found by plotting the normalized y- coordinate spacings and their respective normalized perimeter estimates on a Richardson (log-log) chart (Figs 2 and 3). The slope of the line through the data points on the plot gives the basis of the fractal dimension. In the case of the computer program used by us, the data pairs given by the logarithm of normalized y-coordinate spacings and the logarithm of the respective normalized perimeter estimates werecollected and regression was performed. When- ever a specimen is to be studied, the average fractal dimension can be found for the images stored and related to the range of y-coordinate spacings.

DISCUSSION

'Conventional' or Euclidean mathematics is able to describe structures by means of geometries based on shapes such as squares, triangles, or circles; how- ever, such forms are rarely features of biological entities. In these, irregularities are usually con- tinuous and may be internally repetitive; that is to say, the complexity remains regardless of the level of magnification of the object. The latter phenomenon is seen, for example, in the outline of bracken fronds, islands or continents, or, of course, in the familiar Mandelbrot or Julia set patterns. While the shapes of highly irregular (dendritic) or, for example, cloven structures can be measured with some accuracy by standard Euclidean parameters (such as the form factor"*I2), roughened surfaces, such as those often encountered in histological studies (especially as the ultrastructural level), are not at all well suited to such assessments.

It is for this reason that we have devised a com- puter program of the sort described to enable fractal dimension analysis. We made use of a standard polygon-based perimeter estimate technique, With an irregular structure, the smaller the step size, the more irregularities that will be detected, and this process can continue ad infiniturn, giving ever- greater measures of the total length of the periphery.

The fractal dimension of a structure such as may be encountered in histopathology would be expected to lie between 1 and 2, the fractal dimension of a straight line being 1 and that of a two-dimensional plane being 2. We suggest that this simple technique, run with a relatively inexpensive hardware system and using simple mathematical analysis, will find extensive use in histopathology, and that it should be investigated widely by histomorphometrists.

ACKNOWLEDGEMENTS

We are most grateful to the Shirley Monkspath Rotarian group for their support of this project. Stephen Crocker gave invaluable assistance with computing advice.

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