The Fractal Properties of Retinal Vessels: Embryological and Clinical Implications
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Eye (1990) 4, 235-241 The Fractal Properties of Retinal Vessels: Embryological and Clinical Implications MARTIN A. MAINSTER Kansas City, Kansas, USA Summary The branching patterns of retinal arterial and venous systems have characteristics of a fractal, a geometrical pattern whose parts resemble the whole. Fluorescein angio gram collages were digitised and analysed, demonstrating that retinal arterial and venous patterns have fractal dimensions of 1.63 ± 0.05 and 1. 71 ± 0.07, respectively, consistent with the 1.68 ± 0.05 dimension of diffusion limited aggregation. This find ing prompts speculation that factors controlling retinal angiogenesis may obey Laplace's equation, with fluctuations in the distribution of embryonic cell-free spaces providing the randomness needed for fractal behaviour and for the unique ness of each individual's retinal vascular pattern. Since fractal dimensions charac terise how completely vascular patterns span the retina, they can provide insight into the relationship between vascular patterns and retinal disease. Fractal geometry offers a more accurate description of ocular anatomy and pathology than classical geometry, and provides a new language for posing questions about the complex geo metrical patterns that are seen in ophthalmic practice. Inspection and photography of ophthalmo ing patterns.4,5 Branching patterns are com scopic images are vital to modern ophthalmic mon in nature, occurring in such diverse practice, but analysis of those images remains phenomena as river tributary networks, light largely a subjective process. In otht:r disci ning discharge pathways, and erosion chan plines, images are routinely broken into a nels in porous media. Retinal arteries and large number of tiny picture elements, called veins have similar branching patterns. pixels, in a process known as digitisation. A branching pattern may have two impor Computers can digitise, enhance and analyse tant fractal properties. First, it may be self images. For example, computerised image similar over a limited range of length scales. enhancement can restore ophthalmoscopic That is, it may have similar detail (roughness, detail lost due to ocular media haziness.! branching, etc) when inspected at different Computers can also analyse images, measur magnifications, so that small parts of the pat ing structures2 or even quantitating spatial tern exhibit the pattern's overall structure. information in much the same way that con Second, the pattern may decrease in density trast sensitivity testing characterises an indi as it grows in size. Self-similarity and decreas vidual's visual performance. 3 ing density are two important characteristics Fractal analysis provides a different way of of fractals, shapes whose parts resemble their analysing digitised images, one that is particu whole.4.8 larly useful when they contain branch- Because of self-similarity, fractal properties Correspondence to: Martin A. Mainster, M.D., Ph.D., Department of Ophthalmology, Kansas University Medical Center, 39th and Rainbow Boulevard, Kansas City, Kansas 66103, USA. 236 MARTIN A. MAINSTER differ from those of the simple lines and produced by dielectric breakdown,to electro curves of ordinary Euclidean geometry. Since deposition,!! viscous fingering,5,!2.!5 and a magnifying a fractal only reveals more detail, variety of other physical phenomena.4-8 They length is not a useful parameter for describing also resemble human retinal arterial and a fractal. For example, a straight line has the venous patterns. Since DLA behaviour could same length regardless of the size of the scale offer insight into the relationship between ret used to measure it. Fractals exhibit more inal angiogenesis and the geometry of mature detail at higher magnification, however, so retinal vessels, retinal vessel patterns were their length increases as the size of the studied to determine if they have fractal measuring scale decreases. A coastline is a properties and a fractal dimension consistent simple example of this principle.4,5 An ant with DLA. crawling along the convoluted edge of a shoreline travels a much greater distance Methods between two locations than an automobile Arterial and venous patterns were identified driving between the locations. in the wide-field fluorescein angiogram col Although length is not a useful measure of a lages!6 of six individuals with early back fractal, other parameters such as the fractal ground diabetic retinopathy. A map of retinal dimension.(D) are valuable descriptors. The arteries and veins of each individual was pre fractal dimension describes a structure's con pared for vessels larger than capillaries. volutedness. Unlike ordinary Euclidean Arteries and veins were digitised separately, dimensions that are integers, fractal dimen with the optic disk centred on a two-dimen sions are fractions. For example, the Eucli sional 1000 x 1000 array of square pixels. dean dimension (E) of a straight line, an area Figures 1-3 show the retinal vessel map and (e.g. the retinal surface),. and a volume (e.g. digitised arterial and venous patterns of the the vitreous gel) are 1, 2, and 3, respectively. left eye of a 57 year old woman. Pixel size on An irregular but relatively smooth curve has a the retina was 20 !lm, assuming an average fractal dimension near one (e.g. 1.2), whilst a optic disc diameter of 1.5 mm in retinal highly tortuous curve has a fractal dimension photographs. near two (e.g. 1.8). Digitising an arterial or venous pattern con In a practical sense, a pattern's fractal sists of determining whether each pixel is dimension describes how thoroughly it fills occupied by any part of the vascular pattern. space. For example, squares or circles can Each black square or circle in Figure 2 or 3 completely cover a 2-dimensional space (sur represents a pixel occupied by part of the face), and their fractal dimension is two. arterial or venous pattern, respectively. The Branching linear structures span 2-dimen remaining pixels were vacant of vascular sional spaces less completely and their fractal detail and are not shown. dimension is less than two. As a 2-dimen There are several ways to determine the sional pattern's fractal dimension increases fractal dimension of a digitised retinal vas toward two, the structure covers its surface cular pattern.5.7,!7.20 The simplest method is to more completely, and its density drops off construct a series of circles of different radii more slowly as the pattern increases in size. centred about the optic nerve, and count the Decreasing fractal density may have a power 'mass' (number) of occupied pixels within a law dependence, with an exponent D-E, circle of a given radius. If the logarithm of the where D is the fractal dimension of the pat pixel mass within a circle of a given radius is tern, and E is the Euclidean dimension of the then plotted against the logarithm of the rad space containing it. ius of the circle, the slope of the log-log plot is The diffusion-limited aggregation (DLA) D, the fractal dimension of the pattern. A model of Witten and Sander has helped to fractal dimension can also be determined explain how growth and structure are related from pixel density, mass/area, where area is in fractal branching patterns.6.9 Computer altered to account for projection of the curved generated DLA patterns are similar in retina onto a flat ophthalmoscopic image. appearance and fractal dimension to patterns Mapping, digitisation and analysis were THE FRACTAL PROPERTIES OF RETINAL VESSELS 237 I , ' "'4. ,- ., .... " , -,.-r-\ ..., -f .....f � � \ \,'\ \ \, I 1------_ "- . ,( , \ ' , \ I \,t' ... , , ,. " , I I _./_ _ ., ,, I , I , I , Fig. 1. Retinal arterial (---) and venous (- ----) systems in the fluoresceinangiogram collage of the left eye of a 57 year old woman. The optic nerve is in the centre of the branching pattern, with the macula to its right. carried out manually, and required over 20 Results hours to analyse a single angiogram. In the Retinal vascular density decreases from the future, video retinal photography systems will central to peripheral retina, following a have the required 1024 x 1024 (or, optimally, power-law relationship with mean arterial and 2048 x 2048) pixel resolution needed for this venous fractal dimensions of 1.63±0,05 and type of analysis. Computer algorithms then 1.71±0,07, respectively (± std). The differ could be developed for comparing different ence between arterial and venous fractal angiogram phases to differentiate between dimensions was statistically significant arterial and venous systems and for compoun (P<0.05, two-tailed t test). Fractal dimen ding fractal dimensions by methods5-7,17-20 such sions change by less than 3% with a correction as the mass-radius relationship discussed for retinal curvature estimated from emme above. tropic model eye parameters, 21 Fractal dimen- 238 MARTIN A. MAINSTER Fig. 2. Retinal arterial patternfor the individual in Fig. 1, digitised on a 1000 x 1000 array of square 20 !J.I11pixels. The optic nerve is in the centre of the branching pattern, with the macula to its right. Each black square is a pixel occupied by part of the arterial system (squares may be contiguous with adjacent squares). The remaining pixels are vacant of vascular detail and not shown. The fractal dimension for the arterial pattern is 1.61. sions from fluorescein angiogram analysis density (D = 1.63). In other words, the retina should be more sensitive to vascular non-per is spanned most completely by the capillary fusion than narrowing. system, less completely by the venous system, It is interesting to note that earlier mor and least completely by the arterial system. phologic data22 on retinal capillary density can be re-expressed in a power-law relationship, Discussion yielding a fractal dimension of 1.82. Thus, Retinal arterial and venous patterns have a proceeding outward from the optic disc to the structure and fractal dimension similar to peripheral retina, capillary density drops off those of two-dimensional DLA and related more slowly than venous density (D = 1.71), phenomena.