Fractal Dimension and Retinal Pathology: A Meta-analysis

Due to the fractal nature of retinal blood vessels, the retinal fractal dimension is a natural parameter for researchers to explore and has garnered interest as a potential diagnostic tool. This review aims to summarize the current scientific evidence regarding the relationship between fractal dimension and retinal pathology and thus assess the clinical value of retinal fractal dimension. Following the PRISMA guidelines, a literature search for research articles was conducted in several internet databases (Embase, PubMed, Web of Science, Scopus). This led to a result of 28 studies included in the final review, which were analyzed via meta-analysis to determine whether the fractal dimension changes significantly in retinal disease versus normal individuals


Retinal Vasculature and Fractal Dimension
The retina is of crucial importance to eye care professionals as retinal diseases are the leading cause of blindness worldwide. The retina is a thin, light-sensitive neural layer and is supplied by a sophisticated microvascular network that delivers nutrients and carries away waste. As part of the human circulatory system, the network's development tends to seek configurations which minimize operational energy expenditure, reflected by Murray's Law of Minimal Work which relates the radii of parent and daughter vessels, giving rise to the network's branching pattern [1]. Often diseases will have a vascular component that can manifest as abnormalities in this network and thus the network can be observed to acquire insight into the presence(or absence) of disease [2][3][4][5]. With advancements in non-invasive ocular imaging techniques such as optical coherence tomography angiography (OCTA) permitting the segmentation of the vasculature into well-defined layers [6], the retinal vasculature has become more accessible than ever for researchers. Hence, increasing attention has been paid to analyzing its quantitative characteristics as a potential diagnostic tool.
The retinal blood vessels form a complex branching pattern that has been shown to be fractal [5]. Therefore, a natural parameter for describing the retinal vasculature is the fractal dimension, first described by [7] and then introduced into ophthalmology by [4]. The fractal dimension is a real number that describes how an object's detail changes at different magnifications. It can be thought of as an extension of the familiar Euclidean dimensions to allow for intermediate states. The fractal dimension of the retinal vascular tree lies between 1 and 2 [5], indicating that its branching pattern fills space more thoroughly than a line, but less than a plane. Thus, the retinal fractal dimension provides a measure of the tree's global branching complexity, which can be altered by the rarefaction or proliferation of blood vessels in a disease scenario. In healthy human subjects, the retinal FD is around 1.7, which is similar to that of a 2D diffusion-limited aggregation process [4,5]. It has been postulated that this is because the retinal vasculature grows through diffusion of angiogenic factors in the retinal plane [8].

Common Methods for Measuring Retinal Fractal Dimension
After image acquisition, retinal images need to be processed first to extract the features of the vascular tree, a process known as vessel segmentation, before image binarization and then fractal analysis. Several algorithms exist for this purpose and the reader is referred to the literature [9][10][11]. We will now briefly discuss commonly used methods for calculating retinal fractal dimension.
Box-counting (capacity) dimension: The simplest and most common method used in the literature is the box-counting method [7] for fractal dimension. Given a binarized image of the retinal vascular tree, we overlay the image with a grid of boxes of side-length ε and count how many boxes contain a part of the tree. By decreasing ε, we capture more and more fine details of the tree from the covering. Taking N(ε) to be the box-count as a function of ε, the box-counting (capacity) dimension [12] is defined as It should be noted that the base of the logarithm does not affect the calculated value.
Information dimension: Similar to the method for determining the box-counting dimension, we overlay the retinal image with a grid of boxes of side-length ε. Instead of a box count however, we assign to each box a weight based on its contribution to the tree's information entropy and sum up the weights for each box, defining the information dimension [12,13] as where ((ε) is the total number of boxes that contain a part of the tree and / ' = 0 ' /1 is the proportion of retinal tree contained in the i-th box; 0 ' is the number of pixels in the i-th box and 1 is the total number of pixels in the tree.
Correlation dimension: We overlay the retinal image with a grid of side-length ε boxes and define the correlation integral 2(ε) [12] as where H is the Heaviside step function and counts the number of tree pixel pairs such that the distance between them is less than ε. The correlation dimension is then defined as Generalized dimensions: When discussing multifractal structures such as the retinal vasculature, they are more accurately described by an infinite hierarchy of fractal dimensions [14]. For any real q, the generalized fractal dimension is defined as where It can be seen and verified mathematically that ! & , ! . , ! 4 correspond to the capacity, information, and correlation dimensions respectively as previously discussed. If 9 . > 9 4 then ! 7 " ≤ ! 7 ! [15], so we have that ! & ≥ ! . ≥ ! 4 , which is a useful check when computing fractal dimensions in practice.

Objective
This objective of this review is to provide an overview of the current scientific evidence of the association between human retinal FD and common retinal disorders, such as diabetic retinopathy (DR), retinal detachment, glaucoma, etc..

Research question
Does the fractal dimension of the retinal vasculature change significantly due to retinal disease when compared to normals?

Search Strategy
The Preferred Items for Systematic Review and Meta-Analysis (PRISMA) guidelines [16] were followed for this review. A search was conducted in the databases: EMBASE, MEDLINE, Web of Science, and Scopus. Search queries were constructed by combining relevant subject headings and keywords with Boolean operators.

Inclusion/Exclusion Criteria
The goal of this review is to examine the scientific evidence comprehensively for a wide variety of retinal disorders rather than a select few. The eligibility criteria was determined using the PICO framework [17]. The study populations comprised of participants of any age, race, or gender that had been diagnosed with a retinal disorder. Interventions consisted of calculation of fractal dimensions on processed retinal vasculature images of subjects. Preferably, studies would state the model of the imaging device used, the region of interest which fractal analysis is conducted over, fractal analysis software packages used, and the types of fractal parameters that were calculated. Finally, studies had to compare fractal dimension in case subjects versus control subjects or contrast fractal dimension with disease progression from baseline to follow-ups.
Case-control studies, cross-sectional studies, cohort studies, and case series were included. Studies that employed the following imaging techniques were acceptable: Reviews and other types of articles without original research were not included. No restrictions on submission date or language were placed.

Screening Process
Search results from each of the databases were exported to Mendeley Reference Manager (Mendeley, London, UK) where duplicates were identified and removed. Subsequently, articles were screened based on title and abstract for relevance.

Meta-analysis
To answer the question of whether the retinal fractal dimension changes significantly due to retinal disease, a meta-analysis was conducted using the data from the included studies. For each disease category, sample mean (SD) fractal dimensions were collected from each study of that category and pooled together to compare normal with case subjects. If the study did not provide sample mean (SD) fractal dimension, then it was estimated from other statistics such as median (IQR) fractal dimension using methods by [18,19]. Otherwise, if there was too much discrepancy, then the study was excluded from the meta-analysis. Mean (SD) fractal dimensions were used to determine effect sizes for each study, which were then used to derive a summary effect size for each disease category using a random-effects model [20].

Results of Study Selection
A total of 280 records were found initially from the search, leaving 139 records after duplicates were removed. The titles and abstracts were then screened for relevance based off the PICO criteria described in the Methods section. This left 54 studies remaining to be considered, which were all read in-depth, resulting in 28 studies included in the final synthesis with reasons for exclusion for those that did not make the cut. The selection process is summarized in Figure 1. 27 of the selected studies exclusively focused on one disease: 10 on diabetic retinopathy (DR), 5 on myopia, 5 on diabetes mellitus (DM) in general, 2 on glaucoma, 2 on hypertension, 1 on macular telangiectasia (MacTel), 1 on retinal occlusions, and 1 on nonarteritic anterior ischemic optic neuropathy (NAION). Only one study examined multiple diseases: DM, hypertension, age-related macular degeneration (AMD), myopia, and glaucoma. We decided to include the results of an earlier investigation conducted by [8], where the box-counting fractal dimension was measured for RNF and FFA 2 images collected from patients at the University of Missouri School of Optometry. The results are summarized in Table 1 [21,22], 2 using the International Diabetic Retinopathy Severity Scale [23], and the remaining 3 not specifying a grading system.

9
Classification of myopia varied among the myopia studies. Some studies considered myopia to be defined as a spherical equivalent refraction (SER) worse than -6 10 diopters, while others set the boundary to be at -1.00 diopters. SER values between -1.00 and 1.00 diopters were generally considered emmetropic. Of the studies 11 which examined glaucoma, two used the International Society of Geographical and Epidemiological Ophthalmology scheme [24], while one used the Glaucoma

12
Hemifield Test [25], pattern standard deviation, and optic nerve damage as indicators but did not say whether it was part of some standardized procedure. Studies 13 on hypertension generally defined it as systolic blood pressure above 140 mmHg and diastolic blood pressure 80 mmHg. The clinical characteristics for each study 14 can be seen in Table 2. Most of the studies adjusted for potential confounding factors such as age, gender, cardiovascular risk factors, and optical disorders using exclusion and/or multivariate regression. Age ranges varied between studies from schoolchildren as young at ages 11-12 to elderly adults as old as 71.  (21) NPDR: 62 (14) PDR: 53 (14) Controls: 47 (18

42
FD and an association between diabetic retinopathy and FD was ruled out after multivariate 43 regression analysis [26]. For the studies that looked at myopia [26][27][28][29][30][31], hypertension [26,32,33], 44 glaucoma, and diabetes [34][35][36][37][38], all generally observed a mean decrease in fractal dimension 45 relative to the control group or a decreasing trend with respect to increasing disease severity. In 46 contrast, findings from the diabetic retinopathy studies were more mixed. Decreased retinal FD was found to be associated with presence of NPDR [39,40] and PDR [40]. Retinal FD also showed a 48 decreasing trend with respect to diabetic retinopathy severity ranging from mild NPDR to PDR 49 [37,38,41,42]. However, three studies found that greater retinal FD was associated with early 50 retinopathy signs in young type 1 diabetic patients [43], incidence of referable diabetic retinopathy 51 in urban Malay adults [44], and presence of proliferative diabetic retinopathy in type 2 diabetes [45].

52
For retinographies from the DRIVE database, no significant change in FD was reported for patients 53 with mild NPDR for the entire retina although a significant difference was found for the macular 54 region [46]. A cohort study [47] also reported no association found between retinal FD and incident 55 diabetic retinopathy in young diabetes patients after a mean (SD) follow-up period of 2.9 (2.0) years.
For the studies that examined less common retinal diseases, [48] found decreased FD for OCTA

Results of Meta-Analysis
The results of the meta-analysis are reported in Table 4. with the relationship remaining uncertain due the confidence intervals encompassing positive effect sizes as well.

-Limitations/Stability of Fractal Dimension
The outcomes of fractal analysis show decreased retinal FD in myopia, glaucoma, and hypertension but show inconsistent results for diabetic retinopathy. Other diseases that were considered in this review: AMD, MacTel, NAION, and retinal occlusions were not investigated extensively enough in the literature for meta-analysis or to form a conclusive consensus. It is difficult to compare fractal analysis outcomes due to heterogeneity in methods. Often studies considered different regions of interest and vascular layers, as well as ignoring certain types of blood vessels instead of calculating over the entire tree. Variation in imaging methods, image processing, and fractal analysis tools is also concerning when comparing research between studies. Previously, [12] studied the robustness of the FD parameter with respect to different methodological setups and concluded that retinal FD can be misleading in clinical applications due to its sensitivity to image quality and technique used. Similarly, [51] found a significant dependence of FD on the vessel segmentation and dimensional calculation methods used. Even lesser factors such as image brightness, contrast, and focus can have significant impact on the final FD estimate [52]. [53] conducted a review on the association between retinal FD and neurodegenerative diseases: Alzheimer's, cognitive impairment, and stroke. Although a general decrease in retinal FD for patients with neurodegenerative pathology was observed, [53] also expressed difficulties with study comparison and called for the uniformization and standardization of procedures related to calculating retinal FD before establishing clinical applications.

-Meta-analysis
To answer the question of whether retinal FD changes significantly in a quantitative manner, a meta-analysis was conducted for each retinal disease category to derive a summary effect size using the available studies. A negative summary effect size was found for diabetic retinopathy and myopia, suggesting decreased retinal fractal dimension for patients with those pathologies while the results for diabetes mellitus, glaucoma, hypertension are uncertain. Some limitations of this analysis include combining the retinal vascular layers into one group, estimation of mean (SD) FD, study heterogeneity, as well as low sample size for diabetes, glaucoma, and hypertension. Taking the combined average of mean (SD) FD calculations over different retinal vascular layers is a source of error as the fractal dimension over the whole retina can be greater than over its constituent layers. Estimating the mean (SD) FD for a study given median (IQR) FD can also lead to errors in the meta-analysis. Furthermore, the demographics of each study vary widely in age and ocular history. This combined with the many variables that go into each stage of data acquisition/processing when performing fractal analysis leads to large heterogeneity. Because of this, we reiterate that a standardized procedure for retinal fractal analysis should be developed to facilitate inter-study comparison and support future meta-analyses.

Conclusion
This review summarizes the current scientific literature on the association between FD and retinal disease. The nature of the association depends on the type of retinal disease in consideration. The results of the qualitative synthesis show decreased fractal dimension associated with presence of glaucoma, hypertension, and myopia. However, the results of the meta-analysis show that the decrease is strongest with diabetic retinopathy and myopia and weak for diabetes, glaucoma, hypertension. Due to the variances in methodological setups for retinal image processing and FD calculation, it is difficult to form a consensus on effect. Hence, before moving onto clinical applications of FD, it is necessary that a standardized protocol for image acquisition/processing be established first to facilitate inter-study comparison.