Eye (1990) 4, 235-241

The 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 provides a different way of of , 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 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

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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. The fractal dimension of com­ which drops off more slowly than arterial puter-simulated two-dimensional DLA is THE FRACTAL PROPERTIES OF RETINAL VESSELS 239

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Fig. 3. Retinal venous pattern for the individual in Fig. 1, digitised on a 1000 x 1000 array of square 20 f-lmpixels. The optic nerve is in the centre of the branching pattern, with the macula to its right. Each black circle is a pixel occupied by part of the venous system. The remaining pixels are vacant of vascular detail and not shown. The fractal dimension for the venous pattern is 1.76. 2 1.68±0. 05, 3 similar to the fractal dimensions tions.6.8,12 In viscous fingering, for example, measured for physical phenomena such as threadlike branching fingersdevelop as water dielectric breakdown (=1.7),10 electrodeposi-. under pressure displaces a more viscous fluid 5. 2· 5 tion (1.66±0.03),1l and viscous fingering in a porous medium. 1 1 Pressure is governed (1.64±0.04).14 The retinal capillary system by Laplace's equation, whilst local fluctu­ has a larger fractal dimension than DLA ations in pore geometry of the porous structures. material introduce the randomness needed. 6·8, 2 DLA-like phenomena probably produce for fractal behaviour. 1 This process has a similar patterns because their non-equilib­ remarkable parallel in retinal angiogenesis. rium growth is governed by Laplace's Retinal vessels develop from primitive mes­ equation and appropriate boundary condi- enchymal cells. These cells differentiate into 240 MARTIN A. MAINSTER cords of endothelial cells that canalise to form ina due to vascular budding. an interconnecting polygonal (usually penta­ Neovascularisation produces polarised frac­ gonal) capillary network.24-26 An annular zone tal-canopy networks4 that have a smaller frac­ of capillary formation spreads radially out­ tal dimension than normal retinal vessel ward from the optic disc. Capillary remodell­ patterns. Differences between the vascular ing at the trailing edge of the centrifugal architecture of human and non-human ret­ angiogenic wave is directed by haemody­ inae or between retinal and non-retinal tissues namic factors such as arterial pressure and may arise from differences in a variety of chemical factors such as oxygen demand and parameters such as vascular growth rate; the diffusable growth modulators from the retinal number and location of arterial sources; local pigment epithelium. Capillary hypertrophy geometrical factors such as retinal thickness and retraction produce the larger arterial and and curvature; and the size, distribution and venous blood vessels of the mature retina. regularity of embryonic spaces available for Capillary and larger vessel formation prob­ vascular growth and development. ably occur in inner retinal cell-free spaces sep­ Laplacian patterns are common in the eye. arated by Muller cell cytoplasmic extensions. For example, the retinal nerve fibre layer has Thus, arterial and venous patterns may repre­ the form of lines of force in a two-dimensional sent adaptations within the constraints Laplacian field (e.g. an electric field), with a imposed by the lateral distribution of cell-free point sink (e.g. a negative charge) at the optic spaces.24-26 In view of the similarity between nerve, and a point source (e.g. a positive vascular patterns and DLA, it is tempting to charge) displaced temporally from the fovea. speculate that Laplace's equation describes Feynman ascribed the ubiquity of Laplacian the global haemodynamic and metabolic processes in nature to the smoothness of spacetime.27 The randomness needed for frac­ forces responsible for canalisation and expan­ tal vascular branching in the retina may arise sion of angioblastic cords, whilst local fluctu­ from the graininess of the milieu in which ations in cell-free space geometry provide the Laplacian forces act during angiogenesis. randomness needed for fractal geometry and Demonstration that the retinal vasculature the uniqueness of each individual's retinal has DLA-like characteristics does not identify vascular pattern. the formative Laplacian forces or the origin of It should be emphasised that a simple two­ local graininess, but it does suggest that these dimensional DLA model does not account for may be key factors in normal retinal angio­ the absence of vascularisation in the fovea or genesis and that fractal analysis may offer new outer retina. One factor that could limit vas­ insight into retinal vascular development. cularisation in the macular region is the spa­ Fractal analysis may also offer insight into tial hindrance of competition for cell-free retinal vascular dysfunction. If differences in spaces between developing vessels and gan­ how effectively vascular patterns span the ret­ glion cells.2 5 Although two-dimension D LA is ina are significant in determining individual consistent with the lateral branching patterns susceptibility to disorders such as diabetic ret­ of the retinal arterial and venous systems, the inopathy, fractal dimension measurements retinal capillary system's fractal dimension is could answer potentially significant clinical too large for DLA. Furthermore, a two­ questions. For example, are diabetics with dimensional model cannot account for axial retinal vessel patterns of relatively low fractal separation of capillary networks or the dimension at increased risk of diabetic ret­ absence of outer retinal vascularisation. This inopathy? Do venous fractal dimensions vertical vascular stratificationemphasises that decrease with progression of diabetic retino­ the determinants of evolving spatial vascular pathy, and if so, is this change a harbinger of patterns include not only spatial factors, but neovascularisation? Can fractal dimensions also hydrostatic, chemical and perhaps even help individualise grid laser photocoagulation electrical ones as well. patterns for treating diabetic macular Angiogenesis in the embryonic retina from oedema? capillary remodelling is fundamentally differ­ Similar qustions can be posed for other ent from neovascularisation in the mature ret- ophthalmic problems. 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