fractals in pathology

, . 182: 1–8 (1997)

REVIEW ARTICLE

FRACTALS IN PATHOLOGY

.

Department of Pathology, University of Sheffield Medical School, Sheffield, U.K.

SUMMARY

Many natural objects, including most objects studied in pathology, have complex structural characteristics and the complexity of theirstructures, for example the degree of branching of vessels or the irregularity of a tumour boundary, remains at a constant level over awide range of magnifications. These structures also have patterns that repeat themselves at different magnifications, a property knownas scaling self-similarity. This has important implications for measurement of parameters such as length and area, since Euclideanmeasurements of these may be invalid. The fractal system of geometry overcomes the limitations of the Euclidean geometry forsuch objects and measurement of the fractal dimension gives an index of their space-filling properties. The fractal dimension may bemeasured using image analysis systems and the box-counting, divider (perimeter-stepping) and pixel dilation methods have allbeen described in the published literature. Fractal analysis has found applications in the detection of coding regions in DNA andmeasurement of the space-filling properties of tumours, blood vessels and neurones. Fractal concepts have also been usefully incorporatedinto models of biological processes, including epithelial cell growth, blood vessel growth, periodontal disease and viral infections.? 1997 by John Wiley & Sons, Ltd.

J. Pathol. 182: 1–8, 1997.No. of Figures 3. No of Tables 0. No. of References 74.

KEY WORDS—fractals; fractal dimension; mathematical biology; mathematical modelling

INTRODUCTION

The concept of fractal geometry was formulated bythe mathematician Benoit Mandelbrot, building on thework of Poincaré, Cantor, Sierpinski, and others andwas comprehensively expressed in his definitive book in1982.1 Since then, the concept has had a considerableinfluence on pure and applied mathematics and has foundmany applications in the physical sciences. This articlepresents a brief introduction to the theory of fractalgeometry, considers the methods for measuring fractaldimensions in biological science, and reviews present andfuture applications of fractal geometry in pathology.

THE CONCEPTS OF FRACTAL GEOMETRY

The main geometrical system which is taught inschools and with which most of us are familiar isEuclidean geometry. This system deals with regularshapes which have parameters that can be expressed insimple algebraic formulae, e.g., ðr2 for the area of acircle. This system has three dimensions which areintegers; an idealized line with length but no width has adimension of one, a planar area has a dimension of two,and a solid volume-occupying object has a dimension ofthree. The Euclidean system has obvious applications insome areas, such as calculation of the spoke length for a

wheel of given circumference, but many deficiencies areapparent when it is applied to biological systems.Figure 1 shows a pure mathematical fractal object

(a Julia set). The Julia set has a complex boundary andthe level of complexity remains the same at increasingmagnification. Histopathologists will recognize thesimilarity between these images and the boundary of amalignant tumour, such as a breast carcinoma, whenviewed by light microscopy. From low to high magnifi-cation the level of complexity of the boundary of thetumour remains relatively constant down to the levelof individual cells, where the relatively smooth cellmembrane is the dominant feature. The deficiencies ofEuclidean geometry in describing the boundary of suchobjects become apparent when one considers measure-ment of the perimeter. If the perimeter of the Julia setwere to be measured at the magnification of Fig. 1a, thenthe complexities seen in Fig. 1d would not be included,because they are not visible at this magnification andso the estimate of the perimeter length would beartefactually reduced.The problem which concerned Mandelbrot was that

for pure mathematical structures the level of complexityremains the same at all magnifications, so an absolutemeasurement of parameters such as the perimeter orarea is not possible. Mandelbrot’s solution to this was toplot the measured perimeter at different magnificationson a log–log graph, which gives a straight line, and totake the gradient of that line as the fractal dimension.He did this for the coastline of Britain (as reproducedon maps) and found that the fractal dimension was1·25.2 This fractal dimension differs from Euclidean

*Correspondence to: Dr Simon S. Cross, Senior Lecturer, Depart-ment of Pathology, University of Sheffield Medical School, Beech HillRoad, Sheffield S10 2UL, U.K. E-mail: [email protected]

CCC 0022–3417/97/050001–08 $17.50 Received 4 October 1996? 1997 by John Wiley & Sons, Ltd. Revised 25 October 1996

Accepted 7 November 1996

dimensions in that it does not have to be an integer; itsvalue lies between the topological (Euclidean) dimensionof the object (in this case, a one-dimensional line) andthe object in which it is embedded (in this example, atwo-dimensional plane). The fractal dimension is anindex of the space-filling properties of an object so thatthe closer the dimension is to the topological dimensionin which the object is embedded, the greater its space-filling properties. Thus, the coastline of Britain, with afractal dimension of 1·25, is more space-filling than astraight line, because its dimension is greater than one;but it does not completely fill the plane in which it isembedded, because its dimension is less than two.2 Thefractal dimension also leads to the definition of a fractalobject, which is an object whose fractal dimension isgreater than its topological dimension.1 For mathemati-cal fractal objects this definition can be examined alge-braically, but for biological objects the fractal dimensionrequires to be empirically measured and statisticallycompared with the topological dimension.3

MEASUREMENT OF FRACTAL DIMENSIONS

A large number of different types of fractal dimen-sions have been described4,5 but many are only

applicable to pure mathematical fractal objects. Thethree most commonly used methods in biological scienceare the box-counting dimension,6–9 the perimeter-stepping (or divider) dimension,10,11 and the pixel dila-tion method.12–14 Of these, the box-counting dimensionis the easiest to implement and the methodology isillustrated in Fig. 2. In a typical implementation of thismethod on an image analysis system, an image will bedigitized and converted into a single pixel outline by anappropriate algorithm, boxes of varying sizes (preferablyover a range of at least two decades) are applied to theoutline, and the number of outline containing squares iscounted. The formula for the box-counting dimensionis given by

DB = limlog N (å)

, (1)å]0 log(1/å)

where DB is the box-counting fractal dimension of theobject, å is the side length of the box, and N(å) is thesmallest number of boxes of side length å required tocover completely the outline of the object beingmeasured. However, the limit zero cannot be appliedto biological objects, so an empirical method is used,given by

Fig. 1—Images of a pure mathematical structure, a Julia set, with the black outline box in (a) being the area shown in (b) and so through to (d).The total magnification from (a) to (d) is 600 times but it can be seen that the complexity of the boundary remains the same

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? 1997 by John Wiley & Sons, Ltd. J. Pathol. 182: 1–8 (1997)

DB=d, (2)

where d is the slope of the graph of log {N (å)} againstlog 1/å. Such a log–log graph for a colorectal carcinomais shown in Fig. 3. It can be seen that where the box sizeapproaches, or is equal to, the resolution of the image atthe right side of the graph, there is some curvature of thepoint distribution and there is scattering of the points atthe largest box sizes on the left side of the graph.Between the extremes of the range, there is a segment ofpoints lying on a straight line and the gradient of this istaken to be the box-counting fractal dimension. Thismethodology has been used in several applications inpathology which are described later.The perimeter-stepping (divider) dimension has a

similar principle to the box-counting dimension butinstead of boxes of varying sizes, steps of different sizesare used to measure the boundary length of an object.This is the method which Mandelbrot used to measurethe fractal dimension of the coastline of Britain.2 It hasbeen used in some biological studies,11 but it is less easyto implement since it usually requires the whole objectunder consideration to be contained within a single fieldof view. The pixel dilation method can be seen as a

variant of the perimeter-stepping method, but it is easierto implement on image analysis systems.12,13 An import-ant caution when using computerized image analysissystems is to avoid the use of processing techniquessuch as binary noise reduction, since these will causean artefactual reduction in the measured fractaldimension.15

APPLICATIONS OF FRACTAL GEOMETRY INPATHOLOGY

Molecular biologyThe concepts of fractal geometry have found several

applications in molecular biology. One of the biggestproblems that has arisen with the advent of totalsequencing of DNA in many organisms, includingHomo sapiens, is the analysis of such data, especiallythe identification of functional and ‘non-functional’sequences and division of the former into introns andexons. The nucleotide sequences, when presented in alinear letter encoded form, e.g., ACAGCCG, etc.,are not amenable to visual analysis by unaidedhuman observers. Many statistical methods have been

Fig. 2—Diagrammatic representation of measuring the box-counting fractal dimension. A digitized image of a tumour (a) is thresholded andconverted to a single pixel outline (b). Boxes of different side lengths are applied to the outline (c and d) and the number of outline-containing squaresis counted for each different size

3FRACTALS IN PATHOLOGY


developed for the detection of coding regions, based onthe frequencies of encounter of particular types ofnucleotide, or their combinations in particular positionsof the site. Such methods are often reliant on correctalignment of sequences and this information is notalways available for new sequences. A number of novelgraphical methods of representing DNA sequences havebeen described and fractal analysis has been applied tothese. Korolev et al.16 have used a method where eachbase is depicted as a point within a square, the apices ofthe squares corresponding to each of the four bases.Each nucleotide is displayed at a point at half thedistance between the preceding point and the corre-sponding apex, which produces a complex visual imagewith self-similar patterning. By applying fractal analysisto these patterns, Korolev et al. have produced a reliablesearch method for homologous sequences which doesnot rely on alignment. Other investigators have success-fully used related methods for embedding nucleotidesequences in two-dimensional space, with subsequentmeasurement of fractal dimensions,17–19 to differentiatebetween regions including exon and intron sequences.19Such analyses have also been applied to protein struc-ture, with fractal indices for the twistedness or extend-edness of partial protein segments.20 A morecontroversial idea derives from analysis of the frequencydistribution of bases through long lengths of DNA.Some authors report that this analysis shows that thereare long-range correlations between regions separatedby thousands of base pairs,21,22 but the biologicalmeaning of such observations is unclear.

Tumour pathology

The irregular boundaries of tumours can be examinedby fractal geometric analysis and many tumours havebeen shown to have a fractal dimension which is largerthan their topological dimension. Malignant melanomashave been investigated by two groups23,24 using photo-graphs of the lesions in vivo before excision and subse-quent histopathological confirmation of the diagnosis.Both groups found that the fractal dimension of theboundary of the lesions was significantly greater thanthe topological dimension, but that the values lay mainlyin the range 1·05–1·30, so the boundaries were onlyfilling up to a third of the area of the plane in which theywere embedded. One of the groups23 found that thefractal dimension was a useful diagnostic discriminantfor malignant melanoma from benign melanocyticnaevi, but this was not confirmed by the other group.24In the first study23 there were several cases of lentigomaligna, which had comparatively high fractal dimen-sions, whereas the samples in the second study didnot contain any such lesions.24 Colorectal polyps,particularly tubulovillous adenomas, have arborizingpatterns with some self-similarity and so subjectivelymight be expected to have a fractal element to theirstructure. The box-counting dimension of routinehaematoxylin and eosin-stained slides from 359 colo-rectal polyps (including tubulovillous adenomas, meta-plastic polyps, and inflammatory polyps) has beenanalysed and the measured fractal dimension exceededthe topological dimension in all cases.25 The mean value

Fig. 3—Log–log graph used to calculate the box-counting fractal dimension (these data are from the boundaryof a colorectal carcinoma). It can be seen that there is a central linear segment and the gradient of this (in thisexample 1·66) is taken to be the box-counting fractal dimension

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for tubulovillous adenomas (1·74, 25th and 75th centiles1·69–1·78) was significantly larger than that for meta-plastic polyps (1·61, 1·52–1·68) or inflammatory polyps(1·49, 1·40–1·66) but assignment of individual cases todiagnostic groupings was not precise enough for clinicaldiagnostic work (overall kappa statistic 0·60).25 Theirregular boundaries of chorionic villi in partial hyda-tidiform moles are often described as ‘geographical’ intextbooks of gynaecological pathology26 and the fractaldimensions of molar and non-molar pregnancies havebeen measured.27 Again, the fractal dimension exceededthe topological dimension in all cases and there weresignificant differences between non-molar and molarpregnancies, but prediction of the histopathologicaldiagnosis using the fractal dimension alone was poor(kappa statistic 0·26).27 Irinopoulou et al.28 have usedfractal analysis of nuclear dimension and texture inprostatic carcinoma as predictors of prognosis. Theiranalysis provided accurate prediction of metastasis in a3-year follow-up period, but larger studies are requiredto confirm their findings, as their study populationcomprised only 23 cases.28 The fractal ‘texture’ ofnuclear chromatin patterns has also been shown to be auseful discriminant function in differentiating betweenthe nuclei of benign and malignant cells.29Landini and Rippon30,31 have made extensive studies

of dysplastic and malignant epithelial lesions in the oralcavity using sophisticated methods of fractal analysis.Their initial study used the box-counting and perimeter-stepping methods to measure the overall fractal dimen-sion of the epithelial–stromal interface in the lesions.30This study showed that an increase in subjective irregu-larity of this interface (which was associated with severedysplasia and invasive carcinoma) was associated withan increase in the fractal dimension and the range offractal dimensions was wide (1·00 for normal epitheliumto 1·61 for invasive carcinoma). Landini has now devel-oped a method for measuring the local fractal dimen-sion31 which gives a distribution of frequencies of fractaldimensions across the whole lesion. This method wasmore discriminant than the overall fractal dimension inthe sample population examined and avoids the prob-lems of sampling or averaging out of clonal behaviourwithin the tumour. The surfaces of individual cells havefractal characteristics when examined at particularranges of magnification32 and there have been studies ofthe fractal dimension in cultured tumour cells.33 Thefractal dimension of cultured cells correlates with thesubjective increase in complexity as they grow and couldbe used to quantitate the effect of factors in cultureconditions.

Bone pathology

Histomorphometry on bone biopsies is often per-formed in the assessment of metabolic bone disease andEuclidean parameters, such as surface area and trabecu-lar thickness, are the usual measures. If trabecular bonehas a fractal element to its structure, then this couldinvalidate the Euclidean measures outside preciselydefined conditions of magnification and resolution.3 The

current literature in this area is divided on whethertrabecular bone has a fractal structure when viewed bylight microscopy. One group of investigators used iliaccrest biopsies from 62 subjects examined by lightmicroscopy at a total magnification of 25 and an imple-mentation of the box-counting dimension.34 The meanfractal dimension was 0·99 (95 per cent confidenceintervals 0·93–1·05) and this was not greater than thetopological dimension (one), so the authors concludedthat trabecular bone did not have a fractal structure inthe conditions of the study. A second group have usedhigh resolution nuclear magnetic resonance images(pixel size 50 ìm) of cancellous bone in human lumbarvertebrae and the box-counting method.35 They foundthe trabecular boundary to be a smooth surface relativeto the achievable resolution and thus non-fractal. Intheir study, the fractal dimension was undefined andvaried significantly as a function of image signal-to-noise ratio. They concluded that the apparent fractaldimension achieved by the box-counting method was afunction of marrow pore size. By contrast, two othergroups have used the box-counting method on trabecu-lar bone and have found measured fractal dimensionssignificantly greater than the topological dimension,concluding that there is a fractal element to trabecularstructure.36–38 In all of the studies on trabecular bone,the methods of measuring the fractal dimension werevalidated using images with known fractal dimensionsand the methods were shown to be accurate and repro-ducible. The apparent differences in the studies are thusnot readily explicable at present. In both of the studieswhich did show a fractal structure37,38 the measuredfractal dimension was highly correlated with conven-tional Euclidean measurements such as area and vol-ume, indicating that the usual methods of assessing bonebiopsies can still be used with confidence.

Vascular pathology

The branching self-similar patterns of blood vesselshave long attracted attention as potential fractal struc-tures.39 The retinal vessels have been extensively studiedsince they represent a virtually two-dimensional systemand are easily visualized. Landini et al. have used thebox-counting method to measure the fractal dimensionof vessels seen in fluorescein angiograms from normalsubjects and found an overall dimension of 1·63 (SD0·02) for the combined arterial and venous system.40Daxer has studied the vessels in proliferative diabeticretinopathy41,42 and has found that there is a signifi-cantly higher fractal dimension, 1·85 (SD 0·06), insubjects with neovascularization near the optic disc andthat this measure can be used to select patients whorequire panretinal laser treatment. Daxer has combinedthese data with the spatial correlation of vessel branch-ing sites in embryonic vessels, to produce models ofretinal vasculogenesis.43 He has also studied cornealvascularization following injury44 and another grouphas reported on models of this process using the tech-nique of diffusion limited aggregation.45 These studiesare obviously more directly relevant to ophthalmology,



but the methodologies described in them could beapplied to the quantitation of angiogenesis in otherpathological contexts.46 The renal arterial tree has beenassessed using the box-counting fractal dimension andthe normal adult arterial tree has been found to have adimension of 1·61 (SD 0·06).47 In this study there was nocorrelation between the fractal dimension and diseasewhich had developed in adult life, such as hypertension,but congenitally abnormal kidneys, such as those withhypoplastic renal dysplasia, had significantly reduceddimensions. The fractal dimension of the renal arterialtree appears constant from about 21 weeks’ gestation,48suggesting that the degree of branching, at least at alobar and lobular level, does not increase after this age.These studies47,48 were made using radiographs of wholekidneys, which involves the projection of a three-dimensional object onto a two-dimensional plane; fortu-nately there exists a proof to show that in almost allcircumstances, the fractal component of the dimensionwill be retained in the projection.49 There are manyother studies of the fractal geometry of vascular trees,most of which are found in the radiological literature.50

Neuropathology

The branching processes of neurons and astroglialcells again resemble self-similar structures which may befractal and several empirical studies have confirmedthis.13,51–53 There has recently been some controversy asto whether neurones are truly fractal and not simplyquasi-regular lattices,54,55 but the weight of publishedstudies is heavily in favour of the former. Reichenbachet al.51 have measured the fractal dimension of varioustypes of astroglial cells and have found that the proto-plasmic type has larger fractal dimensions than fibrousastrocytes. They relate these differences to the functionof buffering K+ currents, even through very long pro-cesses. On a larger scale, the whole brain can beregarded as a fractal object, with measurement of theconvoluted surface (with a theoretical range of fractaldimensions from 2 to 3) giving a fractal dimension of2·70 (SD 0·07). This has been related to the folding andcompartmentalization which appears to have occurredduring evolution.56 The arborizing patterns of per-ipheral nerves can also be quantified using fractalgeometry.57

Modelling biological processes using fractals

The images of mathematical fractal objects whichcan be created on computers58 have some subjectivesimilarity to natural objects, such as clouds and land-scapes.1,59–61 These images are generated by multipleiterations of relatively simple equations with graphing ofthe results1 and there have been several studies whichhave sought to model biological and pathological pro-cesses using such methods. Landini and Rippon62 havedescribed a cell growth model with two cell types,in which the cells are capable of displacing adjacentpopulations. The model gives rise to patches which arefractally distributed (fractal fragmentation) and they

suggest that fractal fragmentation is the natural out-come of multiple small perturbations in the spatialrearrangement of cells during multiplication. Landinihas also produced a fractal model of periodontal break-down in periodontal disease which is based on 1/f noisewith a fractal distribution.63 The model produces burstsand remissions with advancing age, which agrees withepidemiological data on this particular disease and couldbe useful in modelling other diseases with this typeof temporal progression. Fractal geometry has beenapplied to the cell ‘patches’ in the liver parenchyma ofrat chimeras, showing that the surface fractal dimensiondoes not change with the proportion of the two parentallineages in the chimera. A model has been produced toshow how a simple cell division process can account forthe generation of the liver parenchyma.64 A model ofcellular cytoplasm has been developed based on perco-lation clusters, a type of ‘random’ fractal structure, andthe effect of this on cytoplasmic enzymes and substrateshas been assessed.65 Diffusion-limited aggregation hasbeen used to model the vasculature of the retina45 andpercolation clusters have produced structures similar tobone.66,67 The challenge now lies in producing closerlinkage between these models and the biology, withidentification of mechanisms (e.g., feedback systems ofgrowth factors, expression of homeobox genes) whichcould provide the biological implementation of theseiterative algorithms.

Miscellaneous applications in pathology

The geometry of Herpes simplex corneal ulcers hasbeen quantified by fractal geometry and these propertieshave been related to the planar spread of viral infec-tion.68 Fractal techniques can be applied to manyaspects of image manipulation, such as compression ofimages, without significant loss of resolution,69 andautomatic segmentation of images29,70 including distinc-tion between heterochromatin and euchromatin inelectron micrographs.70

CONCLUSIONS

There are strong theoretical reasons for using fractalgeometry in measurements of biological systems andpublished studies have shown practical evidence ofthis.71,72 In any quantitative study of objects with anirregular shape whose complexity stays at a constantlevel over a range of magnifications, empirical measure-ment of the fractal dimension should be made. If this issignificantly greater than the topological dimension,then a fractal dimension will be the most valid methodof expressing the object’s space-filling properties; Eucli-dean measurements will only be valid at preciselydefined conditions of magnification and resolution andso are unlikely to be able to be directly comparable toother published values.3,5 The fractal dimension is not acomplete descriptor of the shape of an object, so insystems of category assignment (such as diagnosis orprognosis) other parameters will be included in a multi-variate system to provide the most accurate available

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prediction. Some of these other parameters may bederived from fractal geometry and of these, the measure-ment of lacunarity appears the most promising.73,74

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