Fractal Analysis of the Vascular Tree in the Human Retina

Lens opacity and refractive influences on the measurement of retinal vascular fractal dimension

PURPOSE

To examine the influence of lens opacity and refraction on the measurement of retinal vascular fractal dimension (Df).

METHODS

  Optic disc photographs (right eyes) of 3654 baseline Blue Mountains Eye Study participants (aged 49-97) were digitized. Retinal vascular Df was quantified using a computer-based program. Summated severity scores for nuclear, cortical and posterior subcapsular (PSC) cataract were assessed from lens photographs. Refraction data were converted to spherical equivalent refraction (SER), as sum spherical plus 0.5 cylinder power. Axial length was measured at 10-year follow-up examinations using an IOL master.

RESULTS

  Mean Df of the retinal vasculature was 1.444±0.023 for 2859 eligible participants. Increasing lens opacity scores were associated with significant reduction in Df (β=-0.0030, p<0.0001). Both cortical and PSC cataract involving central lens area were associated with reduced Df, after controlling for confounding factors (p(trend) ≤0.0105). Increasing myopia severity was associated with reduced Df after adjusting for lens opacity scores and other confounders (p(trend) <0.0001). The slope of Df decrease per SER reduction was 0.0040 in eyes with SER≤-4D, compared to -0.0016 in eyes with SER>-4D. For axial length quintiles, there were no significant differences in mean Df in all groups except a reduction in the fifth quintile (axial length ≥24.15mm) (all p<0.05).

CONCLUSION

  Ocular media opacity independently influenced retinal vascular Df measurement, but we found no evidence supporting any refractive axial magnification effect on this measure. Myopic refraction ≤-4D was associated with a reduction in Df, suggesting rarefaction of retinal vasculature associated with high myopia.

Retinal fractals and acute lacunar stroke

This study aimed to determine whether retinal fractal dimension, a quantitative measure of microvascular branching complexity and density, is associated with lacunar stroke. A total of 392 patients presenting with acute ischemic stroke had retinal fractal dimension measured from digital photographs, and lacunar infarct ascertained from brain imaging. After adjusting for age, gender, and vascular risk factors, higher retinal fractal dimension (highest vs lowest quartile and per standard deviation increase) was independently and positively associated with lacunar stroke (odds ratio [OR], 4.27; 95% confidence interval [CI], 1.49-12.17 and OR, 1.85; 95% CI, 1.20-2.84, respectively). Increased retinal microvascular complexity and density is associated with lacunar stroke.

Fractal analysis of retinal vessels suggests that a distinct vasculopathy causes lacunar stroke

OBJECTIVES

Lacunar strokes account for 25% of all ischemic strokes and may represent the cerebral manifestation of a systemic small vessel vasculopathy of unknown etiology. Altered retinal vessel fractal dimensions may act as a surrogate marker for diseased cerebral vessels. We used a cross-sectional study to investigate fractal properties of retinal vessels in lacunar stroke.

METHODS

We recruited patients presenting with lacunar stroke and patients with minor cortical stroke as controls. All patients were examined by a stroke expert and had MRI at presentation. Digital retinal photographs were taken of both eyes. Monofractal and multifractal analyses were performed with custom-written semiautomated software.

RESULTS

We recruited 183 patients. Seventeen were excluded owing to poor photographic quality, leaving 166 patients (86 with lacunar and 80 with cortical stroke). The mean age was 67.3 years (SD 11.5 years). The patients with lacunar stroke were younger but the prevalence of diabetes, hypertension, and white matter hyperintensities did not differ between the groups. The mean Dbox (monofractal dimension) was 1.42 (SD 0.02), the mean D0 (multifractal dimension) 1.67 (SD 0.03). With multivariate analysis, decreased Dbox and D0 (both representing decreased branching complexity) were associated with increasing age and lacunar stroke subtype after correcting for hypertension, diabetes, stroke severity, and white matter hyperintensity scores.

CONCLUSIONS

Lacunar stroke subtype and increasing age are associated with decreased fractal dimensions, suggesting a loss of branching complexity. Further studies should concentrate on longitudinal associations with other manifestations of cerebral small vessel disease.

Age-related rarefaction in the fractal dimension of retinal vessel

Previous work suggests a general reduction in complexity with aging, referred to as the aging-complexity theory. Fractal dimension (FD) of the vessels in the retina is a global measure of the complexity of the vasculature. However, earlier works did not find any correlation between aging and FD of the retinal vasculature, in contrast to the findings of reduced complexity in other parts of the body. The authors tested the hypothesis that reduced complexity develops with advancing age in the structure of the retinal vasculature. To overcome the limitations of earlier works, a three-dimensional representation of the vasculature, together with Fourier fractal dimension (FFD) techniques, was used. Based on the analysis of 748 retinal images taken of persons aged 49-89 years, we observed a significant decrease in the FFD with aging (p < 0.0001). These data provide evidence supporting rarefaction (i.e. reduction) of the retinal vasculature with aging, consistent with observations from other human organ systems.

Retinal vascular fractal dimension and risk of early diabetic retinopathy: A prospective study of children and adolescents with type 1 diabetes

OBJECTIVE

To examine the prospective association of retinal vascular fractal dimension with diabetic retinopathy risk in young people with type 1 diabetes.

RESEARCH DESIGN AND METHODS

This was a hospital-based prospective study of 590 patients aged 12-20 years with type 1 diabetes free of retinopathy at baseline. All patients had seven-field retinal photographs taken of both eyes. Incident retinopathy was ascertained from retinal photographs taken at follow-up visits. Fractal dimension was measured from baseline photographs using a computer-based program following a standardized protocol.

RESULTS

Over a mean +/- SD follow-up period of 2.9 +/- 2.0 years, 262 participants developed mild nonproliferative diabetic retinopathy (15.0 per 100 person-years). After adjusting for age, sex, diabetes duration, A1C, and other risk factors, we found no association between retinal vascular fractal dimension and incident retinopathy.

CONCLUSIONS

Retinal vascular fractal dimension was not associated with incident early diabetic retinopathy in this sample of children and adolescents with type 1 diabetes.

Deterministic chaos and fractal complexity in the dynamics of cardiovascular behavior: perspectives on a new frontier

Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts.

Quantitative assessment of early diabetic retinopathy using fractal analysis

OBJECTIVE

Fractal analysis can quantify the geometric complexity of the retinal vascular branching pattern and may therefore offer a new method to quantify early diabetic microvascular damage. In this study, we examined the relationship between retinal fractal dimension and retinopathy in young individuals with type 1 diabetes.

RESEARCH DESIGN AND METHODS

We conducted a cross-sectional study of 729 patients with type 1 diabetes (aged 12-20 years) who had seven-field stereoscopic retinal photographs taken of both eyes. From these photographs, retinopathy was graded according to the modified Airlie House classification, and fractal dimension was quantified using a computer-based program following a standardized protocol.

RESULTS

In this study, 137 patients (18.8%) had diabetic retinopathy signs; of these, 105 had mild retinopathy. Median (interquartile range) retinal fractal dimension was 1.46214 (1.45023-1.47217). After adjustment for age, sex, diabetes duration, A1C, blood pressure, and total cholesterol, increasing retinal vascular fractal dimension was significantly associated with increasing odds of retinopathy (odds ratio 3.92 [95% CI 2.02-7.61] for fourth versus first quartile of fractal dimension). In multivariate analysis, each 0.01 increase in retinal vascular fractal dimension was associated with a nearly 40% increased odds of retinopathy (1.37 [1.21-1.56]). This association remained after additional adjustment for retinal vascular caliber.

CONCLUSIONS

Greater retinal fractal dimension, representing increased geometric complexity of the retinal vasculature, is independently associated with early diabetic retinopathy signs in type 1 diabetes. Fractal analysis of fundus photographs may allow quantitative measurement of early diabetic microvascular damage.

Modeling the variability of shapes of a human placenta

BACKGROUND

Placentas are generally round/oval in shape, but "irregular" shapes are common. In the Collaborative Perinatal Project data, irregular shapes were associated with lower birth weight for placental weight, suggesting variably shaped placentas have altered function.

METHODS

(I) Using a 3D one-parameter model of placental vascular growth based on Diffusion Limited Aggregation (an accepted model for generating highly branched fractals), models were run with a branching density growth parameter either fixed or perturbed at either 5-7% or 50% of model growth. (II) In a data set with detailed measures of 1207 placental perimeters, radial standard deviations of placental shapes were calculated from the umbilical cord insertion, and from the centroid of the shape (a biologically arbitrary point). These two were compared to the difference between the observed scaling exponent and the Kleiber scaling exponent (0.75), considered optimal for vascular fractal transport systems. Spearman's rank correlation considered p<0.05 significant.

RESULTS

(I) Unperturbed, random values of the growth parameter created round/oval fractal shapes. Perturbation at 5-7% of model growth created multilobate shapes, while perturbation at 50% of model growth created "star-shaped" fractals. (II) The radial standard deviation of the perimeter from the umbilical cord (but not from the centroid) was associated with differences from the Kleiber exponent (p=0.006).

CONCLUSIONS

A dynamical DLA model recapitulates multilobate and "star" placental shapes via changing fractal branching density. We suggest that (1) irregular placental outlines reflect deformation of the underlying placental fractal vascular network, (2) such irregularities in placental outline indicate sub-optimal branching structure of the vascular tree, and (3) this accounts for the lower birth weight observed in non-round/oval placentas in the Collaborative Perinatal Project.

Effects of old age on vascular complexity and dispersion of the hepatic sinusoidal network

OBJECTIVES

In old age, there are marked changes in both the structure of the liver sinusoidal endothelial cell and liver perfusion. The objective of this study was to determine whether there are also aging changes in the microvascular architecture and vascular dispersion of the liver that might influence liver function.

METHODS

Vascular corrosion casts and light micrographs of young (4 months) and old (24 months) rat livers were compared. Fractal and Fourier analyses and micro-computed tomography were used. Vascular dispersion was determined from the dispersion number for sucrose and 100-nm microspheres in impulse response experiments.

RESULTS

Age did not affect sinusoidal dimensions, sinusoidal density, or dispersion number. There were changes in the geometry and complexity of the sinusoidal network as determined by fractal dimension and degree of anisotropy.

CONCLUSIONS

There are small, age-related changes in the architecture of the liver sinusoidal network, which may influence hepatic function and reflect broader aging changes in the microcirculation. However, sinusoidal dimensions and hepatic vascular dispersion are not markedly influenced by old age.

Euclidean and fractal geometry of microvascular networks in normal and neoplastic pituitary tissue

In geometrical terms, tumour vascularity is an exemplary anatomical system that irregularly fills a three-dimensional Euclidean space. This physical characteristic and the highly variable shapes of the vessels lead to considerable spatial and temporal heterogeneity in the delivery of oxygen, nutrients and drugs, and the removal of metabolites. Although these biological characteristics are well known, quantitative analyses of newly formed vessels in two-dimensional histological sections still fail to view their architecture as a non-Euclidean geometrical entity, thus leading to errors in visual interpretation and discordant results from different laboratories concerning the same tumour. We here review the literature concerning microvessel density estimates (a Euclidean-based approach quantifying vascularity in normal and neoplastic pituitary tissues) and compare the results. We also discuss the limitations of Euclidean quantitative analyses of vascularity and the helpfulness of a fractal geometry-based approach as a better means of quantifying normal and neoplastic pituitary microvasculature.

Dynamical patterning modules: physico-genetic determinants of morphological development and evolution

The shapes and forms of multicellular organisms arise by the generation of new cell states and types and changes in the numbers and rearrangements of the various kinds of cells. While morphogenesis and pattern formation in all animal species are widely recognized to be mediated by the gene products of an evolutionarily conserved 'developmental-genetic toolkit', the link between these molecular players and the physics underlying these processes has been generally ignored. This paper introduces the concept of 'dynamical patterning modules' (DPMs), units consisting of one or more products of the 'toolkit' genes that mobilize physical processes characteristic of chemically and mechanically excitable meso- to macroscopic systems such as cell aggregates: cohesion, viscoelasticity, diffusion, spatiotemporal heterogeneity based on lateral inhibition and multistable and oscillatory dynamics. We suggest that ancient toolkit gene products, most predating the emergence of multicellularity, assumed novel morphogenetic functions due to change in the scale and context inherent to multicellularity. We show that DPMs, acting individually and in concert with each other, constitute a 'pattern language' capable of generating all metazoan body plans and organ forms. The physical dimension of developmental causation implies that multicellular forms during the explosive radiation of animal body plans in the middle Cambrian, approximately 530 million years ago, could have explored an extensive morphospace without concomitant genotypic change or selection for adaptation. The morphologically plastic body plans and organ forms generated by DPMs, and their ontogenetic trajectories, would subsequently have been stabilized and consolidated by natural selection and genetic drift. This perspective also solves the apparent 'molecular homology-analogy paradox', whereby widely divergent modern animal types utilize the same molecular toolkit during development by proposing, in contrast to the Neo-Darwinian principle, that phenotypic disparity early in evolution occurred in advance of, rather than closely tracked, genotypic change.

Fractal analysis of the retinal vascular network in fundus images

Complexity of the retinal vascular network is quantified through the measurement of fractal dimension. A computerized approach enhances and segments the retinal vasculature in digital fundus images with an accuracy of 94% in comparison to the gold standard of manual tracing. Fractal analysis was performed on skeletonized versions of the network in 40 images from a study of stroke. Mean fractal dimension was found to be 1.398 (with standard deviation 0.024) from 20 images of the hypertensives sub-group and 1.408 (with standard deviation 0.025) from 18 images of the non-hypertensives subgroup. No evidence of a significant difference in the results was found for this sample size. However, statistical analysis showed that to detect a significant difference at the level seen in the data would require a larger sample size of 88 per group.

A reliability study of fractal analysis of the skeletonised vascular network using the "box-counting" technique

Binary vascular networks of the human retina were obtained by computerized processing of digital fundus images. Fractal analysis was performed on skeletonised versions. The effect of three parameters in the vascular segmentation and skeletonization algorithm was quantified, with threshold level found to have the greatest influence. The results were compared to fractal analysis of skeletons derived by manual tracing. The mean difference in fractal dimension between 2 observers is 0.004 (0.3%) and the coefficient of repeatability is +/- 0.050 (3.4%). Between a single observer and the computerized approach the mean difference is 0.012 (0.8%) and the coefficient is +/- 0.038 (2.6%). The computerized approach demonstrated a superior reliability compared to manual segmentation.

Fractal-based automatic localization and segmentation of optic disc in retinal images

In this paper, we proposed a novel algorithm to detect optic disc location in retinal images. Optic disc is a bright disk area and all major blood vessels and nerves originate from it. With its high fractal dimension of blood vessel, optic disc can be easily differentiated from other bright regions such as hard exudates and artifacts. Compared with existing algorithms, ours has much lower computational cost and is more robust. With its location known, segmentation of optic disc can be done with simple local histogram analysis. The algorithm can be valuable for automated processing for early stage retinal disease.

From intrusive to oscillating thoughts

This paper focused on the possibility that intrusive thoughts (ITs) are a form of an evolutionary, adaptive, and complex strategy to prepare for and resolve stressful life events through schema formation. Intrusive thoughts have been studied in relation to individual conditions, such as traumatic stress disorder and obsessive-compulsive disorder. They have also been documented in the average person experiencing everyday stress. In many descriptions of thought intrusion, it is accompanied by thought suppression. Several theories have been put forth to describe ITs, although none provides a satisfactory explanation as to whether ITs are a normal process, a normal process gone astray, or a sign of pathology. There is also no consistent view of the role that thought suppression plays in the process. I propose that thought intrusion and thought suppression may be better understood by examining them together as a complex and adaptive mechanism capable of escalating in times of need. The ability of a biological mechanism to scale up in times of need is one hallmark of a complex and adaptive system. Other hallmarks of complexity, including self-similarity across scales, sensitivity to initial conditions, presence of feedback loops, and system oscillation, are also discussed in this article. Finally, I propose that thought intrusion and thought suppression are better described together as an oscillatory cycle.

The clinical assessment of retinal microvascular structure and therapeutic implications

Examination of the retinal microvasculature is widely used to assess diabetic eye disease and as an indicator of target organ damage in hypertension. The diagnostic value of grading of hypertensive retinopathy is dubious; however, many recent studies have demonstrated that hypertensive retinopathy is associated with a range of risk factors for cardiovascular disease and may predict cardiovascular events independently of blood pressure. Developments in digital imaging and computer-assisted analysis have facilitated the quantitative assessment of microvascular changes in cardiovascular disease. These approaches may be useful for assessing cardiovascular risk and targeting therapeutic intervention.

Automated segmentation of retinal blood vessels and identification of proliferative diabetic retinopathy

Proliferative diabetic retinopathy can lead to blindness. However, early recognition allows appropriate, timely intervention. Fluorescein-labeled retinal blood vessels of 27 digital images were automatically segmented using the Gabor wavelet transform and classified using traditional features such as area, perimeter, and an additional five morphological features based on the derivatives-of-Gaussian wavelet-derived data. Discriminant analysis indicated that traditional features do not detect early proliferative retinopathy. The best single feature for discrimination was the wavelet curvature with an area under the curve (AUC) of 0.76. Linear discriminant analysis with a selection of six features achieved an AUC of 0.90 (0.73-0.97, 95% confidence interval). The wavelet method was able to segment retinal blood vessels and classify the images according to the presence or absence of proliferative retinopathy.

Anatomy and flow in normal and ischemic microvasculature based on a novel temporal fractal dimension analysis algorithm using contrast enhanced ultrasound

Strategies for improvement of blood flow by promoting new vessel growth in ischemic tissue are being developed. Recently, contrast-enhanced ultrasound (CEU) imaging has been used to assess tissue perfusion in models of ischemia-related angiogenesis, growth-factor mediated angiogenesis, and tumor angiogenesis. In these studies, microvascular flow is measured in order to assess the total impact of adaptations at different vascular levels. High-resolution methods for imaging larger vessels have been developed in order to derive "angiograms" of arteries, veins, and medium to large microvessels. We describe a novel method of vascular bed (microvessel and arterial) characterization of vessel anatomy and flow simultaneously, using serial measurement of the fractal dimension (FD) of a temporal sequence of CEU images. This method is proposed as an experimental methodology to distinguish ischemic from nonischemic tissue. Moreover, an improved approach for extracting the FD unique to this application is introduced.

Multifractal analysis of human retinal vessels

In this paper, it is shown that vascular structures of the human retina represent geometrical multifractals, characterized by a hierarchy of exponents rather then a single fractal dimension. A number of retinal images from the STARE database are analyzed, corresponding to both normal and pathological states of the retina. In all studied cases, a clearly multifractal behavior is observed, where capacity dimension is always found to be larger then the information dimension, which is in turn always larger then the correlation dimension, all the three being significantly lower then the diffusion limited aggregation (DLA) fractal dimension. We also observe a tendency of images corresponding to the pathological states of the retina to have lower generalized dimensions and a shifted spectrum range, in comparison with the normal cases.

Architecture of infection thread networks in developing root nodules induced by the symbiotic bacterium Sinorhizobium meliloti on Medicago truncatula

During the course of the development of nitrogen-fixing root nodules induced by Sinorhizobium meliloti on the model plant Medicago truncatula, tubules called infection threads are cooperatively constructed to deliver the bacterial symbiont from the root surface to cells in the interior of the root and developing nodule. Three-dimensional reconstructions of infection threads inside M. truncatula nodules showed that the threads formed relatively simple, tree-like networks. Some characteristics of thread networks, such as branch length, branch density, and branch surface-to-volume ratios, were remarkably constant across nodules in different stages of development. The overall direction of growth of the networks changed as nodules developed. In 5-d-old nodules, the overall growth of the network was directed inward toward the root. However, well-defined regions of these young networks displayed an outward growth bias, indicating that they were likely in the process of repolarizing their direction of development in response to the formation of the outward-growing nodule meristem. In 10- and 30-d-old nodules, the branches of the network grew outward toward the meristem and away from the roots on which the nodules developed.

A Novel Three-Dimensional Computer-Assisted Method for a Quantitative Study of Microvascular Networks of the Human Cerebral Cortex

OBJECTIVE

Detailed information on microvascular network anatomy is a requirement for understanding several aspects of microcirculation, including oxygen transport, distributions of pressure, and wall shear stress in microvessels, regulation of blood flow, and interpretation of hemodynamically based functional imaging methods, but very few quantitative data on the human brain microcirculation are available. The main objective of this study is to propose a new method to analyze this microcirculation.

METHODS

From thick sections of india ink-injected human brain, using confocal laser microscopy, the authors developed algorithms adapted to very large data sets to automatically extract and analyze center lines together with diameters of thousands of brain microvessels within a large cortex area.

RESULTS

Direct comparison between the original data and the processed vascular skeletons demonstrated the high reliability of this method and its capability to manage a large amount of data, from which morphometry and topology of the cerebral microcirculation could be derived.

CONCLUSIONS

Among the many parameters that can be analyzed by this method, the capillary size, the frequency distributions of diameters and lengths, the fractal nature of these networks, and the depth-related density of vessels are all vital features for an adequate model of cerebral microcirculation.

Abnormal oral vascular network pattern geometry: a new clinical sign of down syndrome

Down syndrome is a leading genetic cause of mental retardation. Here, we show high fractal dimensions and Lempel-Ziv complexity and lower minimum path fractal dimension (P < or = .0006) for the oral vascular networks of patients (n = 14) and their unaffected parents. This newly recognized sign may provide a useful phenotypical marker for identifying couples potentially at risk for offspring with Down syndrome.

Retinal image analysis: concepts, applications and potential

As digital imaging and computing power increasingly develop, so too does the potential to use these technologies in ophthalmology. Image processing, analysis and computer vision techniques are increasing in prominence in all fields of medical science, and are especially pertinent to modern ophthalmology, as it is heavily dependent on visually oriented signs. The retinal microvasculature is unique in that it is the only part of the human circulation that can be directly visualised non-invasively in vivo, readily photographed and subject to digital image analysis. Exciting developments in image processing relevant to ophthalmology over the past 15 years includes the progress being made towards developing automated diagnostic systems for conditions, such as diabetic retinopathy, age-related macular degeneration and retinopathy of prematurity. These diagnostic systems offer the potential to be used in large-scale screening programs, with the potential for significant resource savings, as well as being free from observer bias and fatigue. In addition, quantitative measurements of retinal vascular topography using digital image analysis from retinal photography have been used as research tools to better understand the relationship between the retinal microvasculature and cardiovascular disease. Furthermore, advances in electronic media transmission increase the relevance of using image processing in 'teleophthalmology' as an aid in clinical decision-making, with particular relevance to large rural-based communities. In this review, we outline the principles upon which retinal digital image analysis is based. We discuss current techniques used to automatically detect landmark features of the fundus, such as the optic disc, fovea and blood vessels. We review the use of image analysis in the automated diagnosis of pathology (with particular reference to diabetic retinopathy). We also review its role in defining and performing quantitative measurements of vascular topography, how these entities are based on 'optimisation' principles and how they have helped to describe the relationship between systemic cardiovascular disease and retinal vascular changes. We also review the potential future use of fundal image analysis in telemedicine.

Hierarchical self-assembly of a coiled-coil peptide into fractal structure

Here we report the hierarchical self-assembly of a cross-linkable coiled-coil peptide containing an internal cysteine. Atomic force microscopy (AFM) experiments revealed the fractal structure of the assemblies, and molecular simulations showed that the peptides cross-linked to form clusters of coiled-coils, which further assembled to form globules of tens of nanometers in diameter. Such hierarchical organization was modulated by pH or thiol-reducing agent. Exploitation of the fractal structures through chemical methods may be valuable for the fabrication of materials spanning multiple length scales.