Services on Demand
On-line version ISSN 1414-431X
Braz J Med Biol Res vol.42 no.10 Ribeirão Preto Oct. 2009 Epub Sep 04, 2009
Can the fractal dimension be applied for the early diagnosis of non-proliferative diabetic retinopathy?
The fractal dimension has been employed as a useful parameter in the diagnosis of retinal disease. Avakian et al. (Curr Eye Res 2002; 24: 274-280), comparing the vascular pattern of normal patients with mild to moderate non-proliferative diabetic retinopathy (NPDR), found a significant difference between them only in the macular region. This significant difference in the box-counting fractal dimension of the macular region between normal and mild NPDR patients has been proposed as a method of precocious diagnosis of NPDR. The aim of the present study was to determine if fractal dimensions can really be used as a parameter for the early diagnosis of NPDR. Box-counting and information fractal dimensions were used to parameterize the vascular pattern of the human retina. The two methods were applied to the whole retina and to nine anatomical regions of the retina in 5 individuals with mild NPDR and in 28 diabetic but opthalmically normal individuals (controls), with age between 31 and 86 years. All images of retina were obtained from the Digital Retinal Images for Vessel Extraction (DRIVE) database. The results showed that the fractal dimension parameter was not sensitive enough to be of use for an early diagnosis of NPDR.
Key words: Box-counting; Fractal dimension; Information fractal dimension; Retina; Vascularization; Non-proliferative diabetic retinopathy
The retina captures and propagates images to the brain through receptor (cones and rods) and neural cells. Because of the high demand for oxygen due to elevated cellular activity, this tissue is highly vascularized, with a consequent complex vessel distribution (1-3). Therefore, a geometric description of this retinal vascularization is of great interest for the early diagnosis of some diseases that attack the vessels of the retina (4-6).
Complex geometric patterns similar to those formed by the retinal vessels can be described by fractal geometry. Fractal objects or processes have been described by Mandelbrot (7) and are characterized by the following properties: 1) self-similarity, which means that parts of an object or process resemble the whole object or process; 2) scaling, which means that the measured properties depend on the scale at which they are measured; 3) fractal dimension, which provides a quantitative description of self-similarity and scaling, and 4) the anomalous statistical properties of the fractals. From a practical point of view, the fractal dimension of a pattern describes how thoroughly the pattern fills the space in which it is embedded. For example, squares can completely cover a two-dimensional space and their fractal dimension is two. Retinal vessels span the two-dimensional space less completely, and their fractal dimension is less than two, but greater than one (8,9).
The fractal dimension has been employed as a useful parameter in the diagnosis of retinal diseases (10-15). Daxer (16) found a correlation fractal dimension for vascular pattern equal to 1.85 ± 0.06 (mean ± SD) in patients with proliferative diabetic retinopathy. This fractal dimension is significantly different from that found in normal individuals, in which this value is about 1.70 (17). Avakian et al. (13), comparing the vascular pattern of normal and mild to moderate non-proliferative diabetic retinopathy (NPDR) individuals, using box-counting (Df) and grid intersection (ρv) methods, found a significant difference between them only in the macular region. Significant differences in the fractal dimension of the vascular pattern were not observed for other retinal regions or for the whole retina. This significant difference in the box-counting fractal dimension of the macular region between normal and mild NPDR individuals can be used for an early diagnosis of NPDR. The possibility for the ophthalmologist to establish a parameter permitting the diagnosis of NPDR in the early phase is an important clinical goal.
The main objective of the present study was to answer the following question: is the fractal dimension a sufficiently sensitive parameter to be used for an early diagnosis of NPDR? A strategy developed in the present study to answer this question was to use a large sample for the control condition (28 diabetic but ophthalmically normal individuals) in order to reach consistent statistical conclusions.
We used 28 retinographies of diabetic, but ophthalmically normal patients (Figure 1A) and 5 retinographies (Figure 1D) of patients diagnosed with mild NPDR as shown elsewhere (18). All retinographies were obtained from the Digital Retinal Images for Vessel Extraction (DRIVE) database (18). The images were acquired from patients aged 31 to 86 years using a Canon CR5 non-mydriatic 3CCD camera with a 45-degree field of view. Each image has a size of 768 x 584 pixels, 8 bits per color channel and a field of view of approximately 540 pixels in diameter. The images were saved in JPEG-format (19).
Manually segmented images available in DRIVE were used for the calculation of the fractal dimensions (Figure 1B,E). The segmentation consists of extracting geometric information about the retinal vessels from the eye fundus images. The segmented images were saved in BMP-format. Figure 1 (panels B, C, E, and F) shows images of the segmentation of the retina with black fundus and white vessels, which do not permit the differentiation between control and mild NPDR images with the naked eye. Each image was divided into nine equal regions using the program Adobe Image Ready 7.0.1 (Figure 1C,F) and saved in BMP-format. After this procedure, box-counting and information fractal dimensions were automatically calculated for each retinal anatomical region and for the whole retina of control and NPDR individuals using the 1.3 Benoit™ Fractal Analysis System (TruSoft, USA) software.
Theoretical method (fractal dimension)
Box-counting fractal dimension. Box-counting is probably the method most frequently used to calculate fractal dimension. The procedure consists of covering the fractal object with N(r) boxes of different sizes, which contain at least one point of the fractal object. The Benoit™ software, depending on the size of the image, defines the side-length largest box and a coefficient of box size decrease that permit the determination of the size of the other boxes. Here, the side-length largest box was 146 pixels, the coefficient of box size decrease was 1.3 and the number of box sizes was 19. A plot of log N(r) versus log r (sides of the boxes) was then constructed. The slope of the line relating these two variables, with the inverted signal, is the dimension of box-counting that can be defined by the following equation (20):
The above equation is the derivative of log N in relation to log r (d log N/d log r) that represents the slope of the plot log N versus log r. It is easy to show that if the base of the logarithm is changed from Neperian to decimal the DBC value does not change. The sizes of the sides of the boxes were chosen by the Benoit™ software in a way that permitted us to obtain a graph of log N versus log r with a sufficient number of points to determine its slope with precision (or d log N/d log r). Nineteen different box sizes were determined for an image of 768 x 584 pixels. The program permits to increase the number of boxes, but the fractal dimension remains the same when the box sizes are changed.
Information fractal dimension
The information dimension is defined by the following equation:
is called Kolmogorov entropy, N is the number of boxes and mi = Mi / M, where Mi is the number of points in the ith box and M is the total number of points of the fractal object. The information dimension attributes weights to the boxes and those that contain a larger number of points weigh more than boxes with fewer points. The information dimension is calculated from the slope of the graph of the Kolmogorov entropy line and the logarithms of the sides of the boxes used to cover the fractal object (21). Similar to the box-counting, the value of the information fractal dimension is independent of the base of the logarithm used.
We first determined if the values of fractal dimensions of control and NPDR conditions from box-counting and information methods followed normal (Gaussian) distributions, in order to decide if parametric or nonparametric statistical analysis should be applied.
The Shapiro-Wilks test was used to test the normality of the distributions of the box-counting and information fractal dimension values of the vessel patterns in control and NPDR individuals. Data showing normal distribution were analyzed by a parametric test (t- or Z-test) and data showing non-normal distribution were analyzed by the nonparametric Mann-Whitney U-test to compare the average values of the fractal dimensions. The Z-test was applied to determine if the average value of fractal dimensions of the individuals with NPDR belongs to control Gaussian distribution.
Tables 1 and 2 show measurements of position and variability for the box-counting and information fractal dimension values for nine retinal anatomical regions and for the whole retina of 28 diabetic but ophthalmically normal individuals (controls) and of 5 patients with NPDR. Table 1 shows that the box-counting fractal dimensions of the retinal vascularization pattern of diabetic but ophthalmically normal individuals (controls) followed a Gaussian distribution in all retinal anatomical regions as well as in the whole retina, except for the optic disc and temporal regions. The information dimension of the vascular pattern for the optic disc, temporal, inferotemporal, and inferior regions did not follow a normal distribution, but for other retinal regions and for the whole retina the distributions were Gaussians (Table 1). Table 1 also shows that the mean values of the box-counting and information fractal dimensions of the control were significantly different for each region of the retina as well as for the whole retina.
The box-counting fractal dimensions of vessel patterns of controls and of patients with NPDR did not differ significantly regarding different anatomical regions of the retina or the whole retina (Table 2). The probability (P) of the normal and NPDR retinal fractal dimensions to be equal was 0.305 in the less probable case and it occurred in the macular region (Table 2). The information fractal dimension of vessel patterns of control and NPDR individuals did not differ significantly. In this case, the probability of the control and NPDR retinal fractal dimensions to be equal was in the least favorable case equal to 0.322, occurring in the inferior nasal region. A Z-distribution was used to determine these probabilities (Table 2).
For the box-counting fractal dimension in the optic disc (P = 0.687) and temporal (P = 0.393) regions and information fractal dimension in the optic disc (P = 0.580), temporal (P = 0.580), inferotemporal (P = 0.421), and inferior (P = 0.269) regions in which the fractal dimension values did not follow Gaussian distributions, the nonparametric Mann-Whitney U-test was used to determine this probability (Table 2).
These results show that the box-counting and information fractal dimensions are not adequate parameters to be used in order to distinguish between control and mild NPDR patients.
[View larger version of this image (187 K JPG file)]
Table 1. Box-counting (DBC) and information (DINF) methods for determining the fractal dimensions of each anatomical retinal region and of the whole retina of 28 control individuals (without non-proliferative diabetic retinopathy).
[View larger version of this table (136 K JPG file)]
|Table 2. Box-counting (DBC) and information (DINF) methods for determining the fractal dimension of each anatomical retinal region and of the whole retina of 5 patients with non-proliferative diabetic retinopathy (NPDR).|
[View larger version of this table (256 K JPG file)]
A geometric description of the retinal vascular pattern has been proposed to facilitate the diagnosis of retinal vascular diseases. Avakian et al. (13) compared the vascular pattern of ophthalmically normal and mild to moderate NPDR using the box-counting and grid intersection methods and observed a significant difference between normal and NPDR only for the macular region. For the whole retina, no difference was observed in the fractal dimension of the vascular pattern between normal and NPDR patients.
The first point of the present study was to answer the question: can fractal dimension be safely applied for an early diagnosis of mild NPDR? To answer the question, we compared the retinal vascular fractal patterns of the control and mild NPDR using box-counting and information methods. The results did not show significant differences between control and mild NPDR vascular patterns. In the present study, the vascular patterns were compared using a quantitative method of region-based fractal analysis, as proposed by Avakian et al. (13), but no significant differences were observed between control and mild NPDR vascular retinal patterns. The difference between our results and those of Avakian et al. (13) may have been due to the larger number of control images in our investigation. This fact permitted the establishment of a vascular fractal pattern for the control. In addition, we also analyzed a larger number of regions than Avakian et al. (13). Daxer (10), using the correlation method, studied proliferative diabetic retinopathy in patients with neovascularization near the optic disc and found a fractal dimension with a mean value (± SD) equal to 1.85 ± 0.06. This value differs from that found by Family et al. (17) for normal retinal vascular patterns (D ≈ 1.7). Therefore, when applied to patients with advanced stages of vascular retinal diseases, fractal analysis seems to be sensitive enough to disclose a difference between normal and sick retinas. Supporting this conclusion, Cheung et al. (5), when examining the relationship between retinal fractal dimension and the different stages of retinopathy in young people with type one diabetes, concluded that an increasing retinal vascular fractal dimension was significantly associated with an increased stage of retinopathy.
Another objective of the present study was to analyze the relevance of the method used for the identification of the retinal vascular pattern. Different methods have been used for the calculation of the fractal dimension (17,22-26). Family et al. (17) calculated the fractal dimension (D) of the vascular pattern of the human retina using mass radius (D = 1.71 ± 0.07) and correlation (D = 1.72 ± 0.03) methods, but did not observe a significant difference between them. Previous studies from our group have shown that the fractal dimension values depend on the method used for the calculation of the fractal structure (27,28). In the present study, we showed that box-counting and information fractal dimensions differ significantly in each region as well as in the whole retina. These results suggest that the human retinal vessel pattern is a multifractal process, as pointed out by Stosic and Stosic (29).
Our study showed that the fractal dimension was not a useful tool to distinguish between normal and mild NDPR, and reinforced our previous results showing that different values of fractal dimensions can be obtained depending on the method used to calculate the vascular retinal pattern.
1. Kretzer FL, Mehta RS, Johnson AT, Hunter DG, Brown ES, Hittner HM. Vitamin E protects against retinopathy of prematurity through action on spindle cells. Nature 1984; 309: 793-795. [ Links ]
2. Kretzer FL, McPherson AR, Hittner HM. An interpretation of retinopathy of prematurity in terms of spindle cells: relationship to vitamin E prophylaxis and cryotherapy. Graefes Arch Clin Exp Ophthalmol 1986; 224: 205-214. [ Links ]
3. Kretzer FL, Hittner HM. Retinopathy of prematurity: clinical implications of retinal development. Arch Dis Child 1988; 63: 1151-1167. [ Links ]
4. Liew G, Wang JJ, Cheung N, Zhang YP, Hsu W, Lee ML, et al. The retinal vasculature as a fractal: methodology, reliability, and relationship to blood pressure. Ophthalmology 2008; 115: 1951-1956. [ Links ]
5. Cheung N, Donaghue KC, Liew G, Rogers SL, Wang JJ, Lim SW, et al. Quantitative assessment of early diabetic retinopathy using fractal analysis. Diabetes Care 2009; 32: 106-110. [ Links ]
6. Shildkrot Y, Young JA, Lee CM, Glavas IP. Fractal dimensions of retinal vascular patterns seen on intravenous fluorescein angiogram photographs of normal eyes and eyes with background diabetic retinopathy. Invest Ophthalmol Vis Sci 2004; 45: E-Abstract 2800. [ Links ]
7. Mandelbrot BB. The fractal geometry of nature. Nova York: W.H. Freeman and Co.; 1983. [ Links ]
8. Bassingthwaighte JB, Liebovitch LS, West JB. Fractal physiology. New York: Oxford University Press; 1994. [ Links ]
9. Kamiya A, Takahashi T. Quantitative assessments of morphological and functional properties of biological trees based on their fractal nature. J Appl Physiol 2007; 102: 2315-2323. [ Links ]
10. Daxer A. The fractal geometry of proliferative diabetic retinopathy: implications for the diagnosis and the process of retinal vasculogenesis. Curr Eye Res 1993; 12: 1103-1109. [ Links ]
11. Hesse L. Fractal dimension in diabetic retinopathy. Graefes Arch Clin Exp Ophthalmol 1994; 232: 447-448. [ Links ]
12. Landini G, Murray PI, Misson GP. Local connected fractal dimensions and lacunarity analyses of 60 degrees fluorescein angiograms. Invest Ophthalmol Vis Sci 1995; 36: 2749-2755. [ Links ]
13. Avakian A, Kalina RE, Sage EH, Rambhia AH, Elliott KE, Chuang EL, et al. Fractal analysis of region-based vascular change in the normal and non-proliferative diabetic retina. Curr Eye Res 2002; 24: 274-280. [ Links ]
14. Cheng SC, Huang YM. A novel approach to diagnose diabetes based on the fractal characteristics of retinal images. IEEE Trans Inf Technol Biomed 2003; 7: 163-170. [ Links ]
15. Mainster MA. The fractal properties of retinal vessels: embryological and clinical implications. Eye 1990; 4 (Part 1): 235-241. [ Links ]
16. Daxer A. Characterisation of the neovascularisation process in diabetic retinopathy by means of fractal geometry: diagnostic implications. Graefes Arch Clin Exp Ophthalmol 1993; 231: 681-686. [ Links ]
17. Family F, Masters RB, Platt DE. Fractal pattern formation in human retinal vessels. Physica D 1989; 38: 98-103. [ Links ]
18. DRIVE: Digital retinal images for vessel extraction. http://www.isi.uu.nl/Research/Databases/DRIVE/index.php. Accessed June 25, 2006. [ Links ]
19. Staal J, Abramoff MD, Niemeijer M, Viergever MA, van Ginneken B. Ridge-based vessel segmentation in color images of the retina. IEEE Trans Med Imaging 2004; 23: 501-509. [ Links ]
20. Mandelbrot BB. Objetos fractais. Lisboa: Gradiva; 1991. [ Links ]
21. Vicsek T. Fractal growth phenomen. Singapore: World Scientific; 1991. [ Links ]
22. Landini G, Misson GP, Murray PI. Fractal analysis of the normal human retinal fluorescein angiogram. Curr Eye Res 1993; 12: 23-27. [ Links ]
23. Masters BR. Fractal analysis of normal human retinal blood vessels. Fractals 1994; 2: 103-110. [ Links ]
24. Masters BR. Fractal analysis of the vascular tree in the human retina. Annu Rev Biomed Eng 2004; 6: 427-452. [ Links ]
25. Misson GP, Landini G, Murray PI. Fractals and ophthalmology. Lancet 1992; 339: 872. [ Links ]
26. Masters BR, Platt DF. Development of human retinal vessels: a fractal analysis. Invest Ophthalmol Vis Sci 1989; 30: 391. [ Links ]
27. Kunicki AC, Araújo LS, Sá FB, Stosic B, Barbosa CT, Nogueira RA. Fractal analysis of retinal vessel patterns in ophthalmically normal dogs. Vet Ophthalmol 2008; 11: 2-6. [ Links ]
28. Mendonca MB, Amorim Garcia CA, Nogueira RA, Gomes MA, Valenca MM, Orefice F. Fractal analysis of retinal vascular tree: segmentation and estimation methods. Arq Bras Oftalmol 2007; 70: 413-422. [ Links ]
29. Stosic T, Stosic BD. Multifractal analysis of human retinal vessels. IEEE Trans Med Imaging 2006; 25: 1101-1107. [ Links ]
We acknowledge Dr. Wamberto A. Varanda (Faculdade de Medicina de Ribeirão Preto, USP, Ribeirão Preto, SP, Brazil) for review, comments and suggestions.
Address for correspondence: R.A. Nogueira, Departamento de Morfologia e Fisiologia Animal, Universidade Federal Rural de Pernambuco, Rua Dom Manuel de Medeiros, s/n, 52171-900 Recife, PE, Brasil. Fax: +55-81-3320-6057. E-mail: firstname.lastname@example.org
A.C.B. Kunicki was the recipient of a PIBIC/CNPq fellowship. Received November 21, 2008. Accepted July 31, 2009. Available Online September 4, 2009.
The Brazilian Journal of Medical and Biological Research is partially financed by