Fractal Analysis of the Vascular Tree in the Human Retina

Entropy of corneal nerve fibers distribution observed by laser scanning confocal microscopy: A noninvasive quantitative method to characterize the corneal innervation in Sjogren's syndrome patients

Sjogren's syndrome (SS) is a progressive autoimmune condition mainly affecting the salivary and lacrimal glands with an incidence of primary SS between 1/100 and 1/1,000. SS implies an alteration in the epithelium and subepithelium innervation, with consequent reduction of corneal sensitivity. It is necessary to have noninvasive quantitative methods to characterize the status of the corneal nerve fibers of the patients in order to choose and follow the best therapy. Entropy (information dimension) of the nerve corneal fibers distribution observed by confocal microscopy was evaluated in patients with primary SS (n = 30, 6 males, 24 females, 21-81 years), diagnosed by biopsy of salivary gland and blood tests and in sex- age-matched healthy subjects (n = 12). Corneal nerve fiber density, Langerhans cell count, and cell density in the nerve plexus images were also evaluated. In selected patients salivary gland atrophy degree was also evaluated. Nerve corneal distribution observed by confocal microscopy is fractal. Entropy of the corneal nerve distribution statistically distinguishes between SS patients and healthy subjects: patients present a lower value of information dimension of the corneal nerve fibers distribution than healthy individuals (P < 0.001). Percentage of grouped cases classified by entropy according to the subjects (selected patients vs. healthy) showed a 100% sensitivity and 96% specificity, P < 0.0001 with a low value of coefficient of variation among the individuals (6-7 times lower than the other morphometric indexes). Entropy correlated with the severity of the disease (salivary gland atrophy degree, P < 0.01). Evaluation of entropy of the corneal nerve distribution observed by a laser confocal microscopy appears to quantitatively and noninvasively characterize an aspect of the SS patients in relation to the recognition of an impairment of their ocular surface, giving us for the first time a method to objectively and precisely characterize the corneal innervation status in the SS patients.

Fractality of sensations and the brain health: the theory linking neurodegenerative disorder with distortion of spatial and temporal scale-invariance and fractal complexity of the visible world

The theory that ties normal functioning and pathology of the brain and visual system with the spatial-temporal structure of the visual and other sensory stimuli is described for the first time in the present study. The deficit of fractal complexity of environmental influences can lead to the distortion of fractal complexity in the visual pathways of the brain and abnormalities of development or aging. The use of fractal light stimuli and fractal stimuli of other modalities can help to restore the functions of the brain, particularly in the elderly and in patients with neurodegenerative disorders or amblyopia. Non-linear dynamics of these physiological processes have a strong base of evidence, which is seen in the impaired fractal regulation of rhythmic activity in aged and diseased brains. From birth to old age, we live in a non-linear world, in which objects and processes with the properties of fractality and non-linearity surround us. Against this background, the evolution of man took place and all periods of life unfolded. Works of art created by man may also have fractal properties. The positive influence of music on cognitive functions is well-known. Insufficiency of sensory experience is believed to play a crucial role in the pathogenesis of amblyopia and age-dependent diseases. The brain is very plastic in its early development, and the plasticity decreases throughout life. However, several studies showed the possibility to reactivate the adult's neuroplasticity in a variety of ways. We propose that a non-linear structure of sensory information on many spatial and temporal scales is crucial to the brain health and fractal regulation of physiological rhythms. Theoretical substantiation of the author's theory is presented. Possible applications and the future research that can experimentally confirm or refute the theoretical concept are considered.

Novel Method for Automated Analysis of Retinal Images: Results in Subjects with Hypertensive Retinopathy and CADASIL

Morphological analysis of the retinal vessels by fundoscopy provides noninvasive means for detecting and staging systemic microvascular damage. However, full exploitation of fundoscopy in clinical settings is limited by paucity of quantitative, objective information obtainable through the observer-driven evaluations currently employed in routine practice. Here, we report on the development of a semiautomated, computer-based method to assess retinal vessel morphology. The method allows simultaneous and operator-independent quantitative assessment of arteriole-to-venule ratio, tortuosity index, and mean fractal dimension. The method was implemented in two conditions known for being associated with retinal vessel changes: hypertensive retinopathy and Cerebral Autosomal Dominant Arteriopathy with Subcortical Infarcts and Leukoencephalopathy (CADASIL). The results showed that our approach is effective in detecting and quantifying the retinal vessel abnormalities. Arteriole-to-venule ratio, tortuosity index, and mean fractal dimension were altered in the subjects with hypertensive retinopathy or CADASIL with respect to age- and gender-matched controls. The interrater reliability was excellent for all the three indices (intraclass correlation coefficient ≥ 85%). The method represents simple and highly reproducible means for discriminating pathological conditions characterized by morphological changes of retinal vessels. The advantages of our method include simultaneous and operator-independent assessment of different parameters and improved reliability of the measurements.

Retinal vascular fractal dimension, childhood IQ, and cognitive ability in old age: the Lothian Birth Cohort Study 1936

Purpose

Cerebral microvascular disease is associated with dementia. Differences in the topography of the retinal vascular network may be a marker for cerebrovascular disease. The association between cerebral microvascular state and non-pathological cognitive ageing is less clear, particularly because studies are rarely able to adjust for pre-morbid cognitive ability level. We measured retinal vascular fractal dimension (D_f) as a potential marker of cerebral microvascular disease. We examined the extent to which it contributes to differences in non-pathological cognitive ability in old age, after adjusting for childhood mental ability.

Methods

Participants from the Lothian Birth Cohort 1936 Study (LBC1936) had cognitive ability assessments and retinal photographs taken of both eyes aged around 73 years (n = 648). IQ scores were available from childhood. Retinal vascular D_f was calculated with monofractal and multifractal analysis, performed on custom-written software. Multiple regression models were applied to determine associations between retinal vascular D_f and general cognitive ability (g), processing speed, and memory.

Results

Only three out of 24 comparisons (two eyes × four D_f parameters × three cognitive measures) were found to be significant. This is little more than would be expected by chance. No single association was verified by an equivalent association in the contralateral eye.

Conclusions

The results show little evidence that fractal measures of retinal vascular differences are associated with non-pathological cognitive ageing.

Early risk stratification in pediatric type 1 diabetes

In the late 1980s all Danish children with type 1 diabetes were invited for a nationwide evaluation of glycemic control. Approximately 75% (n = 720) participated and have later been referred to as The Danish Cohort of Pediatric Diabetes 1987 (DCPD1987). The results were surprisingly poor glycemic control among these young patients which lead to a great emphasis on glycemic control in the Danish Pediatric Departments. In 1995 the participants were invited for yet another evaluation but this time with main focus on early signs of microvascular complications - 339 participated. The mean HbA1c had remained at high levels (9.6%) and 60% of the participants had some level of Diabetic Retinopathy (DR). However, as the patients with DR mostly had the very milder forms it was believed that stricter glycemic control would reverse or at least stop progression of the disease in accordance with results from the large intervention study DCCT. This was investigated further at follow-up in 2011. The first study in the present thesis aimed to describe the 16-year incidence, progression and regression of DR in 185 participants from the DCPD1987 cohort. The 16-year incidence of proliferative retinopathy (PDR), 2-step progression and regression of DR was 31.0, 64.4, and 0.0%, respectively. As expected, the participants with PDR at follow-up had significantly higher HbA1c-values at both baseline and follow-up than those without PDR. However; a significantly larger decrease in HbA1c was also observed in the group with PDR over the study period, which in accordance with DCCT should have prevented the development of PDR to some extent. A surprisingly high incidence of proliferative retinopathy amongst young patients with type 1 diabetes in Denmark was found despite improvements in HbA1c over time. The improvement in HbA1c was either too small or happened too late. This study highlights that sight-threatening diabetic retinopathy remain a major concern in type 1 diabetes and the importance of early glycemic control. Identifying high-risk patients at a very early stage is not only desired for prevention of diabetic retinopathy - neuropathy and nephropathy similarly remain frequent in type 1 diabetes. Early risk stratification will allow for timely implementation of effective interventions and for individualized screening and diabetes care. The second and third studies of this thesis provide the longest prospective studies to date on both retinal vessel calibers and retinal fractal dimensions and their predictive value on diabetic microvascular complications. Semi-automated computer software has been developed to measure smaller changes in the retinal vessels on retinal photographs. Two of the first parameters to be reliably estimated by these programs were retinal vessel calibers and retinal vascular fractal dimensions (a quantitative measure on vascular complexity). There is very limited knowledge on their predictive value on diabetic complications thus far. In the second and third study, a consistent relation between narrower retinal arteriolar calibers, wider retinal venular calibers, lower fractal dimensions and the 16-year incidences of diabetic neuropathy, nephropathy and proliferative retinopathy was found. This has never been shown before. The results on vessel analyzes provides indications of a common pathogenic pathway for diabetic microvascular complications and therefore a possibility of universal risk estimation for development of neuropathy, nephropathy and retinopathy in type 1 diabetes.

Validation of the automatic image analyser to assess retinal vessel calibre (ALTAIR): a prospective study protocol

INTRODUCTION

The fundus examination is a non-invasive evaluation of the microcirculation of the retina. The aim of the present study is to develop and validate (reliability and validity) the ALTAIR software platform (Automatic image analyser to assess retinal vessel calibre) in order to analyse its utility in different clinical environments.

METHODS AND ANALYSIS

A cross-sectional study in the first phase and a prospective observational study in the second with 4 years of follow-up. The study will be performed in a primary care centre and will include 386 participants. The main measurements will include carotid intima-media thickness, pulse wave velocity by Sphygmocor, cardio-ankle vascular index through the VASERA VS-1500, cardiac evaluation by a digital ECG and renal injury by microalbuminuria and glomerular filtration. The retinal vascular evaluation will be performed using a TOPCON TRCNW200 non-mydriatic retinal camera to obtain digital images of the retina, and the developed software (ALTAIR) will be used to automatically calculate the calibre of the retinal vessels, the vascularised area and the branching pattern. For software validation, the intraobserver and interobserver reliability, the concurrent validity of the vascular structure and function, as well as the association between the estimated retinal parameters and the evolution or onset of new lesions in the target organs or cardiovascular diseases will be examined.

ETHICS AND DISSEMINATION

The study has been approved by the clinical research ethics committee of the healthcare area of Salamanca. All study participants will sign an informed consent to agree to participate in the study in compliance with the Declaration of Helsinki and the WHO standards for observational studies. Validation of this tool will provide greater reliability to the analysis of retinal vessels by decreasing the intervention of the observer and will result in increased validity through the use of additional information, especially in the areas of vascularisation and vessel branching patterns.

TRIAL REGISTRATION NUMBER

Clinical Trials.gov Identifier: NCT02087605.

Hematocrit dispersion in asymmetrically bifurcating vascular networks

Quantitative modeling of physiological processes in vasculatures requires an accurate representation of network topology, including vessel branching. We propose a new approach for reconstruction of vascular network, which determines how vessel bifurcations distribute red blood cells (RBC) in the microcirculation. Our method follows the foundational premise of Murray's law in postulating the existence of functional optimality of such networks. It accounts for the non-Newtonian behavior of blood by allowing the apparent blood viscosity to vary with discharge hematocrit and vessel radius. The optimality criterion adopted in our approach is the physiological cost of supplying oxygen to the tissue surrounding a blood vessel. Bifurcation asymmetry is expressed in terms of the amount of oxygen consumption associated with the respective tissue volumes being supplied by each daughter vessel. The vascular networks constructed with our approach capture a number of physiological characteristics observed in in vivo studies. These include the nonuniformity of wall shear stress in the microcirculation, the significant increase in pressure gradients in the terminal sections of the network, the nonuniformity of both the hematocrit partitioning at vessel bifurcations and hematocrit across the capillary bed, and the linear relationship between the RBC flux fraction and the blood flow fraction at bifurcations.

Hypoxia in vascular networks: a complex system approach to unravel the diabetic paradox

In this work we model the extent of hypoxia in the diabetic retina as a function of the area affected by vessel disruption. We find two regimes that differ on the ratio between the area of disrupted vasculature and the area of tissue in hypoxia. In the first regime the hypoxia is localized in the vicinity of the vascular disruption, while in the second regime there is a generalized hypoxia in the affected tissue. The transition between these two regimes occurs when the tissue area affected by individual sites of vessel damage is on the order of the square of the characteristic irrigation length in the tissue (the maximum distance that an irrigated point in the tissue is from an existing vessel). We observe that very high levels of hypoxia are correlated with the rupture of larger vessels in the retina, and with smaller radii of individual sites of vessel damage. Based on this property of vascular networks, we propose a novel mechanism for the transition between the nonproliferative and the proliferative stages in diabetic retinopathy.

Zone specific fractal dimension of retinal images as predictor of stroke incidence

Fractal dimensions (FDs) are frequently used for summarizing the complexity of retinal vascular. However, previous techniques on this topic were not zone specific. A new methodology to measure FD of a specific zone in retinal images has been developed and tested as a marker for stroke prediction. Higuchi's fractal dimension was measured in circumferential direction (FDC) with respect to optic disk (OD), in three concentric regions between OD boundary and 1.5 OD diameter from its margin. The significance of its association with future episode of stroke event was tested using the Blue Mountain Eye Study (BMES) database and compared against spectrum fractal dimension (SFD) and box-counting (BC) dimension. Kruskal-Wallis analysis revealed FDC as a better predictor of stroke (H = 5.80, P = 0.016, α = 0.05) compared with SFD (H = 0.51, P = 0.475, α = 0.05) and BC (H = 0.41, P = 0.520, α = 0.05) with overall lower median value for the cases compared to the control group. This work has shown that there is a significant association between zone specific FDC of eye fundus images with future episode of stroke while this difference is not significant when other FD methods are employed.

Influence of refractive condition on retinal vasculature complexity in younger subjects

OBJECTIVE

The aim of this study was to compare the retinal vasculature complexity between emmetropia, and myopia in younger subjects.

METHODS

A total of 82 patients (24.12 ± 1.25 years) with two types of refractive conditions, myopia and emmetropia were enrolled in this study. Refraction data were converted to spherical equivalent refraction. These retinal images (right eyes) were obtained from NAVIS Lite Image Filing System and the vasculature complexity was measured by fractal dimension (D f ), quantified using a computer software following a standardized protocol.

RESULTS

There was a significant difference (P < 0.05) in the value of D f between emmetropic (1.5666 ± 0.0160) and myopic (1.5588 ± 0.0142) groups. A positive correlation (rho = 0.260, P < 0.05) between the D f and the spherical equivalent refraction was detected in this study. Using a linear model, it was estimated that 6.7% of the variation in D f could be explained by spherical equivalent refraction.

CONCLUSIONS

This study provides valuable findings about the effect of moderate to high myopia on the fractal dimension of the retinal vasculature network. These results show that myopic refraction in younger subjects was associated with a decrease in D f , suggesting a loss of retinal vessel density with moderate to high myopia.

Retinal vascular fractals predict long-term microvascular complications in type 1 diabetes mellitus: the Danish Cohort of Pediatric Diabetes 1987 (DCPD1987)

AIMS/HYPOTHESIS

Fractal analysis of the retinal vasculature provides a global measure of the complexity and density of retinal vessels summarised as a single variable: the fractal dimension. We investigated fractal dimensions as long-term predictors of microvasculopathy in type 1 diabetes.

METHODS

We included 180 patients with type 1 diabetes in a 16 year follow-up study. In baseline retinal photographs (from 1995), all vessels in a zone 0.5-2.0 disc diameters from the disc margin were traced using Singapore Institute Vessel Assessment-Fractal image analysis software. Artefacts were removed by a certified grader, and fractal dimensions were calculated using the box-counting method. At follow-up (in 2011), diabetic neuropathy, nephropathy and proliferative retinopathy were assessed and related to baseline fractal dimensions in multiple regressions adjusted for sex and baseline age, diabetes duration, HbA1c, BP, BMI, vibration perception threshold, albuminuria, retinopathy and vessel diameters.

RESULTS

Mean baseline age and diabetes duration were 21.0 and 13.4 years, respectively, and of patients 50.0% were males. The mean fractal dimension was 1.3817. The 16 year incidences of neuropathy, nephropathy and proliferative retinopathy were 10.8%, 8.0% and 27.9%, respectively. Multiple regression analyses showed a lower fractal dimension to significantly predict incident neuropathy (OR 1.17 per 0.01 fractal dimension decrease [95% CI 1.01, 1.36]), nephropathy (OR 1.40 per 0.01 fractal dimension decrease [95% CI 1.10, 1.79]) and proliferative retinopathy (OR 1.22 per 0.01 fractal dimension decrease [95% CI 1.09, 1.37]).

CONCLUSIONS/INTERPRETATION

The retinal vascular fractal dimension is a shared biomarker of diabetic microvasculopathy, thus indicating a possible common pathogenic pathway. Retinal fractal analysis therefore is a potential tool for risk stratification in type 1 diabetes.

Fractal-based analysis of optical coherence tomography data to quantify retinal tissue damage

BACKGROUND

The sensitivity of Optical Coherence Tomography (OCT) images to identify retinal tissue morphology characterized by early neural loss from normal healthy eyes is tested by calculating structural information and fractal dimension. OCT data from 74 healthy eyes and 43 eyes with type 1 diabetes mellitus with mild diabetic retinopathy (MDR) on biomicroscopy was analyzed using a custom-built algorithm (OCTRIMA) to measure locally the intraretinal layer thickness. A power spectrum method was used to calculate the fractal dimension in intraretinal regions of interest identified in the images. ANOVA followed by Newman-Keuls post-hoc analyses were used to test for differences between pathological and normal groups. A modified p value of <0.001 was considered statistically significant. Receiver operating characteristic (ROC) curves were constructed to describe the ability of each parameter to discriminate between eyes of pathological patients and normal healthy eyes.

RESULTS

Fractal dimension was higher for all the layers (except the GCL + IPL and INL) in MDR eyes compared to normal healthy eyes. When comparing MDR with normal healthy eyes, the highest AUROC values estimated for the fractal dimension were observed for GCL + IPL and INL. The maximum discrimination value for fractal dimension of 0.96 (standard error =0.025) for the GCL + IPL complex was obtained at a FD ≤ 1.66 (cut off point, asymptotic 95% Confidence Interval: lower-upper bound = 0.905-1.002). Moreover, the highest AUROC values estimated for the thickness measurements were observed for the OPL, GCL + IPL and OS. Particularly, when comparing MDR eyes with control healthy eyes, we found that the fractal dimension of the GCL + IPL complex was significantly better at diagnosing early DR, compared to the standard thickness measurement.

CONCLUSIONS

Our results suggest that the GCL + IPL complex, OPL and OS are more susceptible to initial damage when comparing MDR with control healthy eyes. Fractal analysis provided a better sensitivity, offering a potential diagnostic predictor for detecting early neurodegeneration in the retina.

Crescimento vascular em membrana do saco vitelínico e desenvolvimento embrionário de codornas japonesas (Coturnix japonica) expostas a campo magnético de baixa frequência

O objetivo deste trabalho foi observar a influência do campo magnético (CM) de baixa frequência na membrana do saco vitelínico (MSV) e no desenvolvimento do embrião de codornas japonesas (Coturnix japonica) em 72 horas de incubação. Ovos fertilizados foram expostos a nove horas consecutivas de CM, sendo um grupo a partir das 48 horas e o outro a partir das 63 horas de incubação. A quantificação da vascularização da MSV foi determinada pela obtenção da dimensão fractal por meio dos métodos de box-counting e de dimensão de informação, enquanto o peso corporal e o percentual de comprimento cefálico dos embriões foram utilizados como parâmetros de desenvolvimento embrionário. O CM não causou diferenças significativas na densidade vascular da MSV nem no desenvolvimento embrionário, quando comparados ao grupo controle.

A prospective photographic study of the ocular fundus in obstructive sleep apnea

BACKGROUND

The prevalence of optic nerve and retinal vascular changes within the obstructive sleep apnea (OSA) population are not well-known, although it has been postulated that optic nerve ischemic changes and findings related to an elevated intracranial pressure may be more common in OSA patients. We prospectively evaluated the ocular fundus in unselected patients undergoing overnight diagnostic polysomnography (PSG).

METHODS

Demographic data, medical/ocular history, and nonmydriatic fundus photographs were prospectively collected in patients undergoing PSG at our institution and reviewed for the presence of optic disc edema for which our study was appropriately powered a priori. Retinal vascular changes were also evaluated. OSA was defined using the measures of both sleep-disordered breathing and hypoxia.

RESULTS

Of 250 patients evaluated in the sleep center, fundus photographs were performed on 215 patients, among whom 127 patients (59%) had an apnea/hypopnea index (AHI) ≥ 15 events per hour, including 36 with severe OSA. Those with AHI <15 served as the comparison group. None of the patients had optic disc edema (95% confidence interval [CI]: 0%-3%). There was no difference in rates of glaucomatous appearance or pallor of the optic disc among the groups. Retinal arteriolar changes were more common in severe OSA patients (odds ratio: 1.09 per 5 unit increase in AHI; 95% CI, 1.02-1.16; P = 0.01), even after controlling for mean arterial blood pressure.

CONCLUSIONS

We did not find an increased prevalence of optic disc edema or other optic neuropathies in our OSA population. However, retinal vascular changes were more common in patients with severe OSA, independent of blood pressure.

Gold(I) biscarbene complexes derived from vascular-disrupting combretastatin A-4 address different targets and show antimetastatic potential

Gold N-heterocyclic carbene (NHC) complexes are an emerging class of anticancer drugs. We present a series of gold(I) biscarbene complexes with NHC ligands derived from the plant metabolite combretastatin A-4 (CA-4) that retain its vascular-disrupting effect, yet address different cellular and protein targets. Unlike CA-4, these complexes did not interfere with tubulin, but with the actin cytoskeleton of endothelial and cancer cells. For the highly metastatic 518A2 melanoma cell line this effect was accompanied by a marked accumulation of cells in the G1 phase of the cell cycle and a suppression of active prometastatic matrix metalloproteinase-2. Despite these mechanistic differences the complexes were as strongly antivascular as CA-4 both in vitro in tube formation assays with human umbilical vein endothelial cells, and in vivo as to blood vessel disruption in the chorioallantoic membrane of chicken eggs. The antiproliferative effect of the new gold biscarbene complexes in a panel of six human cancer cell lines was impressive, with low sub-micromolar IC50 values (72 h) even against CA-4-refractory HT-29 colon and multidrug-resistant MCF-7 breast carcinoma cells. In preliminary studies with a mouse melanoma xenograft model the complexes led to significant decreases in tumor volume while being very well tolerated.

Quantification of tortuosity and fractal dimension of the lung vessels in pulmonary hypertension patients

Pulmonary hypertension (PH) can result in vascular pruning and increased tortuosity of the blood vessels. In this study we examined whether automatic extraction of lung vessels from contrast-enhanced thoracic computed tomography (CT) scans and calculation of tortuosity as well as 3D fractal dimension of the segmented lung vessels results in measures associated with PH.

In this pilot study, 24 patients (18 with and 6 without PH) were examined with thorax CT following their diagnostic or follow-up right-sided heart catheterisation (RHC). Images of the whole thorax were acquired with a 128-slice dual-energy CT scanner. After lung identification, a vessel enhancement filter was used to estimate the lung vessel centerlines. From these, the vascular trees were generated. For each vessel segment the tortuosity was calculated using distance metric. Fractal dimension was computed using 3D box counting. Hemodynamic data from RHC was used for correlation analysis.

Distance metric, the readout of vessel tortuosity, correlated with mean pulmonary arterial pressure (Spearman correlation coefficient: ρ = 0.60) and other relevant parameters, like pulmonary vascular resistance (ρ = 0.59), arterio-venous difference in oxygen (ρ = 0.54), arterial (ρ = −0.54) and venous oxygen saturation (ρ = −0.68). Moreover, distance metric increased with increase of WHO functional class. In contrast, 3D fractal dimension was only significantly correlated with arterial oxygen saturation (ρ = 0.47).

Automatic detection of the lung vascular tree can provide clinically relevant measures of blood vessel morphology. Non-invasive quantification of pulmonary vessel tortuosity may provide a tool to evaluate the severity of pulmonary hypertension.

Trial Registration

ClinicalTrials.gov NCT01607489

Age-related rarefaction in retinal vasculature is not linear

The fractal dimension is a global measure of complexity and is useful for quantifying anatomical structures, including the retinal vascular network. A previous study found a linear declining trend with aging on the retinal vascular fractal dimension (D_F); however, it was limited to the older population (49 years and older). This study aimed to investigate the possible models of the fractal dimension changes from young to old subjects (10-73 years). A total of 215 right-eye retinal samples, including those of 119 (55%) women and 96 (45%) men, were selected. The retinal vessels were segmented using computer-assisted software, and non-vessel fragments were deleted. The fractal dimension was measured based on the log-log plot of the number of grids versus the size. The retinal vascular D_F was analyzed to determine changes with increasing age. Finally, the data were fitted to three polynomial models. All three models are statistically significant (Linear: R² = 0.1270, 213 d.f., p < 0.001, Quadratic: R² = 0.1536, 212 d.f., p < 0.001, Cubic: R² = 0.1529, 211 d.f., p < 0.001). The quadratic regression is significantly better than the linear regression (p < 0.001); however, the increase in R² from the quadratic model to the cubic model is not significant (p = 0.97). These results suggest that the decreasing trend of the fractal dimension associated with aging is better explained by the quadratic model than by the linear and cubic models in a sample with a broader age spectrum.

Fractals in tissue engineering: toward biomimetic cell-culture matrices, microsystems and microstructured implants

Tissue engineering is a rapidly evolving field in which the complexity of biomaterials and biostructures, with typically non-Euclidean or fractal-like geometries, has to be adequately taken into account for the promotion of enhanced and even personalized diagnostic and therapeutic solutions. This study covers the main applications of fractals in the field of tissue engineering, including their advantages for modeling biological processes and cell-culture procedures, but specially focusing on their benefits for describing the complex geometries and structures of biomaterials (both natural and synthetic), many of which have potential uses for the development of cell culture microsystems, scaffolds for tissue repair and implants for tissue repair in general. We also explore the main supporting design, simulation and manufacturing technologies, as well as the most remarkable difficulties and limitations linked to the generalized use of fractals in engineering design, and also detail some current solution proposals and future directions.

Fractal analysis in radiological and nuclear medicine perfusion imaging: a systematic review

OBJECTIVES

To provide an overview of recent research in fractal analysis of tissue perfusion imaging, using standard radiological and nuclear medicine imaging techniques including computed tomography (CT), magnetic resonance imaging (MRI), ultrasound, positron emission tomography (PET) and single-photon emission computed tomography (SPECT) and to discuss implications for different fields of application.

METHODS

A systematic review of fractal analysis for tissue perfusion imaging was performed by searching the databases MEDLINE (via PubMed), EMBASE (via Ovid) and ISI Web of Science.

RESULTS

Thirty-seven eligible studies were identified. Fractal analysis was performed on perfusion imaging of tumours, lung, myocardium, kidney, skeletal muscle and cerebral diseases. Clinically, different aspects of tumour perfusion and cerebral diseases were successfully evaluated including detection and classification. In physiological settings, it was shown that perfusion under different conditions and in various organs can be properly described using fractal analysis.

CONCLUSIONS

Fractal analysis is a suitable method for quantifying heterogeneity from radiological and nuclear medicine perfusion images under a variety of conditions and in different organs. Further research is required to exploit physiologically proven fractal behaviour in the clinical setting.

KEY POINTS

• Fractal analysis of perfusion images can be successfully performed. • Tumour, pulmonary, myocardial, renal, skeletal muscle and cerebral perfusion have already been examined. • Clinical applications of fractal analysis include tumour and brain perfusion assessment. • Fractal analysis is a suitable method for quantifying perfusion heterogeneity. • Fractal analysis requires further research concerning the development of clinical applications.

Regional morphology and pathophysiology of retinal vascular disease

Disturbances in the retinal vascular supply are involved in the pathophysiology of the most frequent diseases causing visual impairment and blindness in the Western World. These diseases are diagnosed by noting how morphological lesions in the retina vary in shape, size, location and dynamics, and subsequently concluding the presence of a specific disease entity. This diagnostic approach can be used to identify the site of a retinal vascular occlusion, to assess whether retinal diseases are primarily due to changes in the larger retinal vessels or the microcirculation, and to differentiate the relative involvement of the choroidal and the retinal vascular systems. However, a number of morphological manifestations of retinal vascular disease cannot presently be related to the underlying pathophysiology. The review concludes that there is a need for developing new methods for assessing vascular structure and function in the ciliary vascular system supplying the choroid and the optic nerve head. Presently, the study of these structures relies on imaging techniques with limited penetration and resolution into the tissue. Secondly, there is a need for studying oscillations in retinal vascular function occurring within days to weeks, and for studying regional manifestations of retinal vascular disease. This may constitute the basis for future research in retinal vascular pathophysiology and for the development of new treatment modalities to reduce blindness secondary to retinal vascular disease.

Relationship between retinal fractal dimensions and retinal circulation in patients with type 2 diabetes mellitus

PURPOSE

To investigate the relationship between retinal fractal dimensions (Dfs) and retinal circulation, which is impaired in early-stage diabetic retinopathy (DR) in patients with type 2 diabetes mellitus (DM).

METHODS

Using a laser Doppler velocimetry system, we measured the retinal vessel diameter (D) and blood velocity (V) and calculated the retinal blood flow (RBF) in the retinal arterioles and venules in 106 eyes (106 patients, mean age ± standard deviation, 58.7 ± 9.8 years). Patients with type 2 DM had no (n = 86) or mild nonproliferative DR (n = 20). The Dfs were measured on the retinal photographs using a semiautomatic computer-based program.

RESULTS

The average D, V, and RBF in the retinal arterioles and venules were, respectively, 107.9 ± 13.3 and 139.4 ± 20.1 µm, 33.2 ± 7.0 and 22.6 ± 6.1 mm/s and 9.3 ± 2.9 and 10.4 ± 3.8 µl/min. The average Df was 1.4276 ± 0.0193. There was no association between the Df and any retinal circulatory parameters of the retinal arterioles. In contrast, we found significant correlations between the Df and the vessel D (r = 0.37, p = 0.0002) and RBF (r = 0.22, p = 0.026) in the retinal venules.

CONCLUSION

The Df might be associated with changes in the retinal circulation in patients with type 2 DM.

Multifractal geometry in analysis and processing of digital retinal photographs for early diagnosis of human diabetic macular edema

OBJECTIVE

The purpose of this paper is to determine a quantitative assessment of the human retinal vascular network architecture for patients with diabetic macular edema (DME). Multifractal geometry and lacunarity parameters are used in this study.

MATERIALS AND METHODS

A set of 10 segmented and skeletonized human retinal images, corresponding to both normal (five images) and DME states of the retina (five images), from the DRIVE database was analyzed using the Image J software. Statistical analyses were performed using Microsoft Office Excel 2003 and GraphPad InStat software.

RESULTS

The human retinal vascular network architecture has a multifractal geometry. The average of generalized dimensions (Dq) for q = 0, 1, 2 of the normal images (segmented versions), is similar to the DME cases (segmented versions). The average of generalized dimensions (Dq) for q = 0, 1 of the normal images (skeletonized versions), is slightly greater than the DME cases (skeletonized versions). However, the average of D2 for the normal images (skeletonized versions) is similar to the DME images. The average of lacunarity parameter, Λ, for the normal images (segmented and skeletonized versions) is slightly lower than the corresponding values for DME images (segmented and skeletonized versions).

CONCLUSION

The multifractal and lacunarity analysis provides a non-invasive predictive complementary tool for an early diagnosis of patients with DME.

The fractal spatial distribution of pancreatic islets in three dimensions: a self-avoiding growth model

The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, has not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension of 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with a fractal dimension of 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas.

The role of the circadian system in fractal neurophysiological control

Many neurophysiological variables such as heart rate, motor activity, and neural activity are known to exhibit intrinsic fractal fluctuations - similar temporal fluctuation patterns at different time scales. These fractal patterns contain information about health, as many pathological conditions are accompanied by their alteration or absence. In physical systems, such fluctuations are characteristic of critical states on the border between randomness and order, frequently arising from nonlinear feedback interactions between mechanisms operating on multiple scales. Thus, the existence of fractal fluctuations in physiology challenges traditional conceptions of health and disease, suggesting that high levels of integrity and adaptability are marked by complex variability, not constancy, and are properties of a neurophysiological network, not individual components. Despite the subject's theoretical and clinical interest, the neurophysiological mechanisms underlying fractal regulation remain largely unknown. The recent discovery that the circadian pacemaker (suprachiasmatic nucleus) plays a crucial role in generating fractal patterns in motor activity and heart rate sheds an entirely new light on both fractal control networks and the function of this master circadian clock, and builds a bridge between the fields of circadian biology and fractal physiology. In this review, we sketch the emerging picture of the developing interdisciplinary field of fractal neurophysiology by examining the circadian system's role in fractal regulation.

Uniform enhancement of optical micro-angiography images using Rayleigh contrast-limited adaptive histogram equalization

Optical microangiography is an imaging technology that is capable of providing detailed functional blood flow maps within microcirculatory tissue beds in vivo. Some practical issues however exist when displaying and quantifying the microcirculation that perfuses the scanned tissue volume. These issues include: (I) Probing light is subject to specular reflection when it shines onto sample. The unevenness of the tissue surface makes the light energy entering the tissue not uniform over the entire scanned tissue volume. (II) The biological tissue is heterogeneous in nature, meaning the scattering and absorption properties of tissue would attenuate the probe beam. These physical limitations can result in local contrast degradation and non-uniform micro-angiogram images. In this paper, we propose a post-processing method that uses Rayleigh contrast-limited adaptive histogram equalization to increase the contrast and improve the overall appearance and uniformity of optical micro-angiograms without saturating the vessel intensity and changing the physical meaning of the micro-angiograms. The qualitative and quantitative performance of the proposed method is compared with those of common histogram equalization and contrast enhancement methods. We demonstrate that the proposed method outperforms other existing approaches. The proposed method is not limited to optical microangiography and can be used in other image modalities such as photo-acoustic tomography and scanning laser confocal microscopy.

Fractal organization of the human T cell repertoire in health and after stem cell transplantation

T cell repertoire diversity is generated in part by recombination of variable (V), diversity (D), and joining (J) segments in the T cell receptor β (TCR) locus. T cell clonal frequency distribution determined by high-throughput sequencing of TCR β in 10 stem cell transplantation (SCT) donors revealed a fractal, self-similar frequency distribution of unique TCR bearing clones with respect to V, D, and J segment usage in the T cell repertoire of these individuals. Further, ranking of T cell clones by frequency of gene segment usage in the observed sequences revealed an ordered distribution of dominant clones conforming to a power law, with a fractal dimension of 1.6 and 1.8 in TCR β DJ and VDJ containing clones in healthy stem cell donors. This self-similar distribution was perturbed in the recipients after SCT, with patients demonstrating a lower level of complexity in their TCR repertoire at day 100 followed by a modest improvement by 1 year post-SCT. A large shift was observed in the frequency distribution of the dominant T cell clones compared to the donor, with fewer than one third of the VDJ-containing clones shared in the top 4 ranks. In conclusion, the normal T cell repertoire is highly ordered with a TCR gene segment usage that results in a fractal self-similar motif of pattern repetition across levels of organization. Fractal analysis of high-throughput TCR β sequencing data provides a comprehensive measure of immune reconstitution after SCT.

Multifractal analysis of microvasculature in health and disease

A growing body of evidence suggests that there is a strong association between neurodegenerative diseases such as Alzheimer's Diseases and the abnormality of the cerebral vasculature, in particular the microvessels/capillaries that are responsible for the exchange of nutrients across the blood-brain barrier [1]. Many microvessels are described as being kinked or distorted [2], implying that they are modified by some destructive process. Imaging devices such as microCT can achieve resolutions on the order of several µm, allowing imaging the three dimensional (3D) microvasculature down to the capillary level. However, the main weakness of using microCT for vascular research is considered to be the lack of software for 3D quantification of microvasculature and microvascular image databases for developing and testing algorithms. In this paper we describe a multifractal analysis method for the microvasculature automatically segmented from microCT images of the mouse brain. Due to the lack of a benchmark microCT image database, the method has been tested using a surrogate database--a publicly available retinal vessel database. The results are preliminary indication of the multifractal properties of mouse brain vasculature. A potential solution to automated classification of healthy and disease brains are discussed.

Retinal vascular fractal dimension and its relationship with cardiovascular and ocular risk factors

PURPOSE

To examine the influence of a range of cardiovascular risk factors and ocular conditions on retinal vascular fractal dimension in the Singapore Malay Eye Study.

DESIGN

Population-based cross-sectional study.

METHODS

Fractal analysis of the retinal vessels is a method to quantify the global geometric complexity of the retinal vasculature. Retinal vascular fractal dimension (D(f)) and caliber were measured from retinal photographs using a computer-assisted program. D(f) and arteriolar caliber were combined to form a retinal vascular optimality score (ranging from 0 to 3). Data on cardiovascular and ocular factors were collected from all participants based on a standardized protocol.

RESULTS

Two thousand nine hundred thirteen (88.8% of 3280 participants) persons had retinal photographs of sufficient quality for the measurement. The mean D(f) was 1.405 (standard deviation, 0.046; interquartile range, 1.243 to 1.542). In the multiple linear regression analysis, after controlling for gender, serum glucose, intraocular pressure, anterior chamber depth, and retinal vascular caliber, smaller D(f) was associated independently with older age (standardized regression coefficient [sβ] = -0.311; P < .001), higher mean arterial blood pressure (sβ = -0.085; P < .001), a more myopic spherical equivalent (sβ = 0.152; P < .001), and presence of cataract (sβ = -0.107; P < .001). Retinal vascular optimality score was associated significantly with higher mean arterial blood pressure (P > .001 for trend).

CONCLUSIONS

Age, blood pressure, refractive error, and lens opacity had significant influence on retinal vascular fractal measurements. A new score of retinal vascular optimality combining fractals and caliber showed strong association with blood pressure. Quantitative analysis of retinal vasculature therefore may provide additional information on microvascular architecture and optimality.

Retinal vascular geometry and glaucoma: the Singapore Malay Eye Study

PURPOSE

To determine the associations of geometric measurements (tortuosity, branching angle, and fractal dimension) of retinal vessels with glaucoma.

DESIGN

Population-based, cross-sectional study.

PARTICIPANTS

Persons aged 40 to 80 years who participated in the Singapore Malay Eye Study (n=3280; 78.7% response rate).

METHODS

Quantitative retinal vascular parameters (tortuosity, branching angle, and fractal dimension) were measured from digital retinal fundus photographs using a computer-assisted program following a standardized grading protocol. Glaucoma was diagnosed according to the International Society of Geographic and Epidemiological Ophthalmology classification system.

MAIN OUTCOME MEASURES

The associations among retinal vascular parameters with glaucoma, the main glaucoma subtype primary open-angle glaucoma (POAG), and ocular hypertension (OHT).

RESULTS

A total of 123 persons (4.4% of the 2789 participants) had glaucoma in the final analysis, 87 (70.7%) of whom were diagnosed with POAG. After adjusting for age, sex, body mass index, diabetes, hypertension, smoking, axial length, and intraocular pressure (IOP), decreased retinal arteriolar tortuosity (odds ratio [OR], 1.73; 95% confidence interval [CI], 1.38-2.18, comparing lowest vs. highest quartiles), decreased retinal venular tortuosity (OR, 1.59; 95% CI, 1.29-1.97), and narrower retinal venular branching angle (OR, 1.22; 95% CI, 1.00-1.48) were associated with glaucoma. Similar associations were found between these retinal vascular parameters and POAG. Decreased retinal vascular fractal dimension was associated with OHT (OR 1.37; 95% CI, 1.04-1.82).

CONCLUSIONS

Certain features of retinal vascular geometry are associated with glaucomatous optic neuropathy independently of vascular risk factors and IOP.

FINANCIAL DISCLOSURE(S)

The author(s) have no proprietary or commercial interest in any materials discussed in this article.

Quantifying optical microangiography images obtained from a spectral domain optical coherence tomography system

The blood vessel morphology is known to correlate with several diseases, such as cancer, and is important for describing several tissue physiological processes, like angiogenesis. Therefore, a quantitative method for characterizing the angiography obtained from medical images would have several clinical applications. Optical microangiography (OMAG) is a method for obtaining three-dimensional images of blood vessels within a volume of tissue. In this study we propose to quantify OMAG images obtained with a spectral domain optical coherence tomography system. A technique for determining three measureable parameters (the fractal dimension, the vessel length fraction, and the vessel area density) is proposed and validated. Finally, the repeatability for acquiring OMAG images is determined, and a new method for analyzing small areas from these images is proposed.

An automated computational framework for retinal vascular network labeling and branching order analysis

Changes in retinal vascular morphology are well known as predictive clinical signs of many diseases such as hypertension, diabetes and so on. Computer-aid image processing and analysis for retinal vessels in fundus images are effective and efficient in clinical diagnosis instead of tedious manual labeling and measurement. An automated computational framework for retinal vascular network labeling and analysis is presented in this work. The framework includes 1) detecting and locating the optic disc; 2) tracking the vessel centerline from detected seed points and linking the breaks after tracing; 3) extracting all the retinal vascular trees and identifying all the significant points; and 4) classifying terminal points into starting points and ending points based on the information of optic disc location, and finally assigning branch order for each extracted vascular tree in the image. All the modules in the framework are fully automated. Based on the results, morphological analysis is then applied to achieve geometrical and topological features based on branching order for one individual vascular tree or for the vascular network through the retinal vascular network in the images. Validation and experiments on the public DRIVE database have demonstrated that the proposed framework is a novel approach to analyze and study the vascular network pattern, and may offer new insights to the diagnosis of retinopathy.

Alveolar fractal box dimension inversely correlates with mean linear intercept in mice with elastase-induced emphysema

RATIONALE

A widely applicable model of emphysema that allows efficient and sensitive quantification of injury is needed to compare potential therapies.

OBJECTIVES

To establish such a model, we studied the relationship between elastase dose and the severity of emphysema in female C57BL/6J mice. We compared alveolar fractal box dimension (D(B)), a new measure which is an assessment of the complexity of the tissue, with mean linear intercept (L(m)), which is commonly used to estimate airspace size, for sensitivity and efficiency of measurement.

METHODS

Emphysema was induced in female C57BL/6J mice by administering increasing intratracheal doses of porcine pancreatic elastase (PPE). Changes in morphology and static lung compliance (C(L)) were examined 21 days later. Correlation of D(B) with L(m) was determined in histological sections of lungs exposed to PPE. The inverse relationship between D(B) and L(m) was supported by examining similar morphological sections from another experiment where the development of emphysema was studied 1 to 3 weeks after instillation of human neutrophil elastase (HNE).

RESULTS

L(m) increased with PPE dose in a sigmoidal curve. C(L) increased after 80 or 120 U/kg body weight (P < 0.05), but not after 40 U/kg, compared with the control. D(B) progressively declined from 1.66 ± 0.002 (standard error of the mean) in controls, to 1.47 ± 0.006 after 120 U PPE/kg (P < 0.0001). After PPE or HNE instillation, D(B) was inversely related to L(m) (R = -0.95, P < 0.0001 and R = -0.84, P = 0.01, respectively), with a more negative slope of the relationship using HNE (P < 0.0001).

CONCLUSION

Intratracheal instillation of increasing doses of PPE yields a scale of progression from mild to severe emphysema. D(B) correlates inversely with L(m) after instillation of either PPE or HNE and yields a rapid, sensitive measure of emphysema after elastase instillation.

Analysis and algorithmic generation of hepatic vascular systems

A proper geometric model of the vascular systems in the liver is crucial for modeling blood flow, the connection between the organ and the rest of the organism. In vivo imaging does not provide sufficient details, so an algorithmic concept for extending measured vascular tree data is needed such that geometrically realistic structures can be generated. We develop a quantification of similarity in terms of different geometric features. This involves topological Strahler ordering of the vascular trees, statistical testing, and averaging. Invariant features are identified in human clinical in vivo CT scans. Results of the existing "Constrained Constructive Optimization" algorithm are compared to real vascular tree data. To improve bifurcation angles in the algorithmic results, we implement a postprocessing step calibrated to the measured features. This framework is finally applied to generate realistic additional details in a patient-specific hepatic vascular tree data set.

Digital ocular fundus imaging: a review

Ocular fundus imaging plays a key role in monitoring the health status of the human eye. Currently, a large number of imaging modalities allow the assessment and/or quantification of ocular changes from a healthy status. This review focuses on the main digital fundus imaging modality, color fundus photography, with a brief overview of complementary techniques, such as fluorescein angiography. While focusing on two-dimensional color fundus photography, the authors address the evolution from nondigital to digital imaging and its impact on diagnosis. They also compare several studies performed along the transitional path of this technology. Retinal image processing and analysis, automated disease detection and identification of the stage of diabetic retinopathy (DR) are addressed as well. The authors emphasize the problems of image segmentation, focusing on the major landmark structures of the ocular fundus: the vascular network, optic disk and the fovea. Several proposed approaches for the automatic detection of signs of disease onset and progression, such as microaneurysms, are surveyed. A thorough comparison is conducted among different studies with regard to the number of eyes/subjects, imaging modality, fundus camera used, field of view and image resolution to identify the large variation in characteristics from one study to another. Similarly, the main features of the proposed classifications and algorithms for the automatic detection of DR are compared, thereby addressing computer-aided diagnosis and computer-aided detection for use in screening programs.

Automated detection of proliferative retinopathy in clinical practice

Timely intervention for diabetic retinopathy (DR) lessens the possibility of blindness and can save considerable costs to health systems. To ensure that interventions are timely and effective requires methods of screening and monitoring pathological changes, including assessing outcomes. Fractal analysis, one method that has been studied for assessing DR, is potentially relevant in today’s world of telemedicine because it provides objective indices from digital images of complex patterns such as are seen in retinal vasculature, which is affected in DR. We introduce here a protocol to distinguish between nonproliferative (NPDR) and proliferative (PDR) changes in retinal vasculature using a fractal analysis method known as local connected dimension (D_conn) analysis. The major finding is that compared to other fractal analysis methods, D_conn analysis better differentiates NPDR from PDR (p = 0.05). In addition, we are the first to show that fractal analysis can be used to differentiate between NPDR and PDR using automated vessel identification. Overall, our results suggest this protocol can complement existing methods by including an automated and objective measure obtainable at a lower level of expertise that experts can then use in screening for and monitoring DR.

Approximation of potential-driven flow dynamics in large-scale self-similar tree networks

Dynamic analysis of large-scale flow networks is made difficult by the large system of differential-algebraic equations resulting from its modelling. To simplify analysis, the mathematical model must be sufficiently reduced in complexity. For self-similar tree networks, this reduction can be made using the network’s structure in way that can allow simple, analytical solutions. For very large, but finite, networks, analytical solutions are more difficult to obtain. In the infinite limit, however, analysis is sometimes greatly simplified. It is shown that approximating large finite networks as infinite not only simplifies the analysis, but also provides an excellent approximate solution.

Fractal analysis reveals reduced complexity of retinal vessels in CADASIL

The Cerebral Autosomal Dominant Arteriopathy with Subcortical Infarcts and Leukoencephalopathy (CADASIL) affects mainly small cerebral arteries and leads to disability and dementia. The relationship between clinical expression of the disease and progression of the microvessel pathology is, however, uncertain as we lack tools for imaging brain vessels in vivo. Ophthalmoscopy is regarded as a window into the cerebral microcirculation. In this study we carried out an ophthalmoscopic examination in subjects with CADASIL. Specifically, we performed fractal analysis of digital retinal photographs. Data are expressed as mean fractal dimension (mean-D), a parameter that reflects complexity of the retinal vessel branching. Ten subjects with genetically confirmed diagnosis of CADASIL and 10 sex and age-matched control subjects were enrolled. Fractal analysis of retinal digital images was performed by means of a computer-based program, and the data expressed as mean-D. Brain MRI lesion volume in FLAIR and T1-weighted images was assessed using MIPAV software. Paired t-test was used to disclose differences in mean-D between CADASIL and control groups. Spearman rank analysis was performed to evaluate potential associations between mean-D values and both disease duration and disease severity, the latter expressed as brain MRI lesion volumes, in the subjects with CADASIL. The results showed that mean-D value of patients (1.42±0.05; mean±SD) was lower than control (1.50±0.04; p = 0.002). Mean-D did not correlate with disease duration nor with MRI lesion volumes of the subjects with CADASIL. The findings suggest that fractal analysis is a sensitive tool to assess changes of retinal vessel branching, likely reflecting early brain microvessel alterations, in CADASIL patients.

Fractal immunology and immune patterning: potential tools for immune protection and optimization

Fractals are self-similar geometric patterns that are inherently embedded throughout nature. Their discovery and application have produced significant benefits across a wide variety of biomedical applications. Recently, complex physiological systems (e.g., neurological, respiratory, cardiovascular) have been shown to exhibit fractal dimensions that are capable of distinguishing among physiologic function versus dysfunction and, in turn, health versus disease. Additionally, fractal data suggest that the immune system operates under similar patterned relationships, and this is in keeping with the recent findings that immune-based diseases are organized according to specific patterns. This review considers the potential benefits of using fractal analysis along with considerations of nonlinearity, scaling, and chaos as calibration tools to obtain holistic information on immune-environment interactions. The potential uses of both synthetic and artificial immune systems for improved protection of the biological immune system are also discussed. The addition of holistic measures of immune status to currently collected biomarkers of immunotoxicity has the potential to increase the effectiveness of health risk assessment. The objective of extending fractal physiology analyses to the immune system would be to promote immune optimization as a public health benefit, which would include improved: (1) immunotoxicity testing and effective health risk reduction and (2) measures of effective immune management for children, adults, and aged individuals.

Robust methodology for fractal analysis of the retinal vasculature

We have developed a robust method to perform retinal vascular fractal analysis from digital retina images. The technique preprocesses the green channel retina images with Gabor wavelet transforms to enhance the retinal images. Fourier Fractal dimension is computed on these preprocessed images and does not require any segmentation of the vessels. This novel technique requires human input only at a single step; the allocation of the optic disk center. We have tested this technique on 380 retina images from healthy individuals aged 50+ years, randomly selected from the Blue Mountains Eye Study population. To assess its reliability in assessing retinal vascular fractals from different allocation of optic center, we performed pair-wise Pearson correlation between the fractal dimension estimates with 100 simulated region of interest for each of the 380 images. There was Gaussian distribution variation in the optic center allocation in each simulation. The resulting mean correlation coefficient (standard deviation) was 0.93 (0.005). The repeatability of this method was found to be better than the earlier box-counting method. Using this method to assess retinal vascular fractals, we have also confirmed a reduction in the retinal vasculature complexity with aging, consistent with observations from other human organ systems.

Multifractal and lacunarity analysis of microvascular morphology and remodeling

OBJECTIVE

Classical measures of vessel morphology, including diameter and density, are employed to study microvasculature in endothelial membrane labeled mice. These measurements prove sufficient for some studies; however, they are less well suited for quantifying changes in microcirculatory networks lacking hierarchical structure. We demonstrate that automated multifractal analysis and lacunarity may be used with classical methods to quantify microvascular morphology.

METHODS

Using multifractal analysis and lacunarity, we present an automated extraction tool with a processing pipeline to characterize 2D representations of 3D microvasculature. We apply our analysis on four tissues and the hyaloid vasculature during remodeling.

RESULTS

We found that the vessel networks analyzed have multifractal geometries and that kidney microvasculature has the largest fractal dimension and the lowest lacunarity compared to microvasculature networks in the cortex, skin, and thigh muscle. Also, we found that, during hyaloid remodeling, there were differences in multifractal spectra reflecting the functional transition from a space filling vasculature which nurtures the lens to a less dense vasculature as it regresses, permitting unobstructed vision.

CONCLUSION

Multifractal analysis and lacunarity are valuable additions to classical measures of vascular morphology and will have utility in future studies of normal, developing, and pathological tissues.

Retinal vascular fractal dimension measurement and its influence from imaging variation: results of two segmentation methods

AIM

To assess the influences of imaging variation (different photographic angle) on the measurement of retinal vascular fractal dimension (D(f)), using two segmentation methods.

MATERIALS AND METHODS

Nonlinear orthogonal projection segmentation (International Retinal Imaging Software-Fractal, termed IRIS-Fractal) and curvature-based segmentation (Singapore Institute Vessel Assessment-Fractal, termed SIVA-Fractal) methods were used to measure D(f) and were assessed for their reproducibility in detecting retinal vessels of 30 stereoscopic pairs of optic disc color images. Each pair was taken from the same eye with slightly different angles of incidence. Each photograph of the pairs had subtle variations in brightness between areas temporal and nasal to the optic disc.

RESULTS

Intragrader reproducibility of D(f) measurement was similar (intraclass correlation 0.81 and 0.96, respectively) for IRIS-Fractal and SIVA-Fractal. Within-image pair Pearson's correlation coefficients (r) of D(f) measurements were moderate for both methods (0.57 and 0.48, respectively).

CONCLUSIONS

Both nonlinear orthogonal projection and curvature-based retinal vessel segmentation methods were found to be sensitive to variations in image brightness, resulting from iris shadowing associated with different angle of photographic incidence.

Effect of obesity on retinal vascular structure in pre-adolescent children

OBJECTIVES. In adults, obesity is linked to changes in the retinal microvasculature. Limited research has been conducted into this association in children. We examined in a cohort of pre-adolescents the relationship between body mass index (BMI) and retinal vascular structure, including retinal vessel diameter and retinal vessel fractals, a measure of geometric patterns reflecting vessel density.

METHODS

A population-based study among school children (2 353/3 144 [75.3%], median age, 12.7 years) recruited from a random cluster sample of 21 schools was conducted during 2004-5. Retinal images were taken and vessel diameter and fractal dimension were quantified using computer-based programs following standardized protocols. Anthropometric measures (weight, height, waist circumference, and BMI) were obtained and defined using standardized protocols.

RESULTS

Children with the highest quartile of BMI had narrower retinal arteriolar diameter (2.8 μm narrower, p<0.0001) and wider venular caliber (4.2 μm wider, p=0.001) than those with BMI in the lowest quartile, after adjusting for age, sex, ethnicity, mean arterial blood pressure, iris color, axial length, birth weight and fellow retinal vessel caliber. Obese children had narrower retinal arteriolar diameter (2.8 μm narrower, p=0.01) and wider venular diameter (4.5 μm wider, p=0.01) than in non-overweight/obese children. Retinal fractal dimension was not significantly associated with BMI. Increasing quartiles of waist circumference were also associated with retinal vessel diameter.

CONCLUSIONS

Our data provide evidence of adverse retinal microvascular caliber changes in obese pre-adolescent children, well before the onset of metabolic and cardiovascular disease in adulthood, and support an early, generalized adverse effect of adiposity on microvascular structure.