Fractal analysis reveals reduced complexity of retinal vessels in CADASIL

Intraoperative superb microvascular ultrasound imaging in glioma: novel quantitative analysis correlates with tumour grade

Accurate grading of gliomas is critical to guide therapy and predict prognosis. The presence of microvascular proliferation is a hallmark feature of high grade gliomas which traditionally requires targeted surgical biopsy of representative tissue. Superb microvascular imaging (SMI) is a novel high resolution Doppler ultrasound technique which can uniquely define the microvascular architecture of whole tumours. We examined both qualitative and quantitative vascular features of gliomas captured with SMI, analysing flow signal density, vessel number, branching points, curvature, vessel angle deviation, fractal dimension, and entropy. Results indicate that high-grade gliomas exhibit significantly greater vascular complexity and disorganisation, with increased fractal dimension and entropy, correlating with known histopathological markers of aggressive angiogenesis. The integrated ROC model achieved high accuracy (AUC = 0.95), highlighting SMI's potential as a non-invasive diagnostic and prognostic tool. While further validation with larger datasets is required, this study opens avenues for SMI in glioma management, supporting intraoperative decision-making and informing future prognosis.

The Morphology of Brain Neurons: The Box-Counting Method in the Quantitative Analysis of 2D Images

This chapter begins by showing the difference between fractal geometry and fractal analysis. The text shows the difference between mathematical and natural fractals and how they are best defined by explaining the concept of fractal analysis. Furthermore, the text presents the most famous technique of fractal analysis: the box-counting method. Defining this method and showing the methodology that leads to the precise value of the fractal (i.e., the box) dimension is done by demonstrating the images of human dentate neurons. A more detailed explanation of the methodology was presented in the previous version of this chapter.This version promotes the notion of monofractal analysis and shows how three types of the same neuronal images can quantify four image properties. The results showed that monofractal parameters successfully quantified four image properties in three nuclei of the cerebellum. Finally, the author discusses the results of this chapter and previously published conclusions. The results show how the monofractal parameters discriminate images of neurons from the three nuclei of the human cerebrum. These outcomes are discussed along with the results of previous studies.

ImageJ in Computational Fractal-Based Neuroscience: Pattern Extraction and Translational Research

To explore questions asked in neuroscience, neuroscientists rely heavily on the tools available. One such toolset is ImageJ, open-source, free, biological digital image analysis software. Open-source software has matured alongside of fractal analysis in neuroscience, and today ImageJ is not a niche but a foundation relied on by a substantial number of neuroscientists for work in diverse fields including fractal analysis. This is largely owing to two features of open-source software leveraged in ImageJ and vital to vigorous neuroscience: customizability and collaboration. With those notions in mind, this chapter's aim is threefold: (1) it introduces ImageJ, (2) it outlines ways this software tool has influenced fractal analysis in neuroscience and shaped the questions researchers devote time to, and (3) it reviews a few examples of ways investigators have developed and used ImageJ for pattern extraction in fractal analysis. Throughout this chapter, the focus is on fostering a collaborative and creative mindset for translating knowledge of the fractal geometry of the brain into clinical reality.

Box-Counting Fractal Analysis: A Primer for the Clinician

This chapter lays out the elementary principles of fractal geometry underpinning much of the rest of this book. It assumes a minimal mathematical background, defines the key principles and terms in context, and outlines the basics of a fractal analysis method known as box counting and how it is used to perform fractal, lacunarity, and multifractal analyses. As a standalone reference, this chapter grounds the reader to be able to understand, evaluate, and apply essential methods to appreciate and heal the exquisitely detailed fractal geometry of the brain.

Reduced cerebral vascular fractal dimension among asymptomatic individuals as a potential biomarker for cerebral small vessel disease

Cerebral small vessel disease is a neurological disease frequently found in the elderly and detected on neuroimaging, often as an incidental finding. White matter hyperintensity is one of the most commonly reported neuroimaging markers of CSVD and is linked with an increased risk of future stroke and vascular dementia. Recent attention has focused on the search of CSVD biomarkers. The objective of this study is to explore the potential of fractal dimension as a vascular neuroimaging marker in asymptomatic CSVD with low WMH burden. D_f is an index that measures the complexity of a self-similar and irregular structure such as circle of Willis and its tributaries. This exploratory cross-sectional study involved 22 neurologically asymptomatic adult subjects (42 ± 12 years old; 68% female) with low to moderate 10-year cardiovascular disease risk prediction score (QRISK2 score) who underwent magnetic resonance imaging/angiography (MRI/MRA) brain scan. Based on the MRI findings, subjects were divided into two groups: subjects with low WMH burden and no WMH burden, (WMH⁺; n = 8) and (WMH⁻; n = 14) respectively. Maximum intensity projection image was constructed from the 3D time-of-flight (TOF) MRA. The complexity of the CoW and its tributaries observed in the MIP image was characterised using D_f. The D_f of the CoW and its tributaries, i.e., D_f (w) was significantly lower in the WMH⁺ group (1.5172 ± 0.0248) as compared to WMH⁻ (1.5653 ± 0.0304, p = 0.001). There was a significant inverse relationship between the QRISK2 risk score and D_f (w), (r_s = − .656, p = 0.001). D_f (w) is a promising, non-invasive vascular neuroimaging marker for asymptomatic CSVD with WMH. Further study with multi-centre and long-term follow-up is warranted to explore its potential as a biomarker in CSVD and correlation with clinical sequalae of CSVD.

Retinal biomarkers of Cerebral Small Vessel Disease: A systematic review

INTRODUCTION

Cerebral Small Vessel Disease (CSVD), a progressive degenerative disorder of small caliber cerebral vessels, represents a major contributor to stroke and vascular dementia incidence worldwide. We sought to conduct a systematic review of the role of retinal biomarkers in diagnosis and characterization of CSVD.

METHODS

We conducted a systematic review of MEDLINE, PubMed, Scopus, the Cochrane Library Database, and Web of Science. We identified studies of sporadic CSVD (including CSVD not otherwise specified, Cerebral Amyloid Angiopathy, and Hypertensive Arteriopathy) and the most common familial CSVD disorders (including CADASIL, Fabry disease, and MELAS). Included studies used one or more of the following tools: visual fields assessment, fundus photography, Optical Coherence Tomography and OCT Angiography, Fluorescein Angiography, Electroretinography, and Visual Evoked Potentials.

RESULTS

We identified 48 studies of retinal biomarkers in CSVD, including 9147 cases and 12276 controls. Abnormalities in retinal vessel diameter (11 reports, n = 11391 participants), increased retinal vessel tortuosity (11 reports, n = 617 participants), decreased vessel fractal dimension (5 reports, n = 1597 participants) and decreased retinal nerve fiber layer thickness (5 reports, n = 4509 participants) were the biomarkers most frequently associated with CSVD. We identified no reports conducting longitudinal retinal evaluations of CSVD, or systematically evaluating diagnostic performance.

CONCLUSION

Multiple retinal biomarkers were associated with CSVD or its validated neuroimaging biomarkers. However, existing evidence is limited by several shortcomings, chiefly small sample size and unstandardized approaches to both biomarkers' capture and CSVD characterization. Additional larger studies will be required to definitively determine whether retinal biomarkers could be successfully incorporated in future research efforts and clinical practice.

Morphological characterization of Etv2 vascular explants using fractal analysis and atomic force microscopy

The rapid engraftment of vascular networks is critical for functional incorporation of tissue explants. However, existing methods for inducing angiogenesis utilize approaches that yield vasculature with poor temporal stability or inadequate mechanical integrity, which reduce their robustness in vivo. The transcription factor Ets variant 2 (Etv2) specifies embryonic hematopoietic and vascular endothelial cell (EC) development, and is transiently reactivated during postnatal vascular regeneration and tumor angiogenesis. This study investigates the role for Etv2 upregulation in forming stable vascular beds both in vitro and in vivo. Control and Etv2⁺ prototypical fetal-derived human umbilical vein ECs (HUVECs) and adult ECs were angiogenically grown into vascular beds. These vessel beds were characterized using fractal dimension and lacunarity, to quantify their branching complexity and space-filling homogeneity, respectively. Atomic force microscopy (AFM) was used to explore whether greater complexity and homogeneity lead to more mechanically stable vessels. Additionally, markers of EC integrity were used to probe for mechanistic clues. Etv2⁺ HUVECs exhibit greater branching, vessel density, and structural homogeneity, and decreased stiffness in vitro and in vivo, indicating a greater propensity for stable vessel formation. When co-cultured with colon tumor organoid tissue, Etv2⁺ HUVECs had decreased fractal dimension and lacunarity compared to Etv2⁺ HUVECs cultured alone, indicating that vessel density and homogeneity of vessel spacing increased due to the presence of Etv2. This study sets forth the novel concept that fractal dimension, lacunarity, and AFM are as informative as conventional angiogenic measurements, including vessel branching and density, to assess vascular perfusion and stability.

The Contribution of Ocular Biomarkers in the Differential Diagnosis of Alzheimer's Disease versus Other Types of Dementia and Future Prospects

With dementia becoming increasingly prevalent, there is a pressing need to become better equipped with accurate diagnostic tools that will favorably influence its course via prompt and specific intervention. The overlap in clinical manifestation, imaging, and even pathological findings between different dementia syndromes is one of the most prominent challenges today even for expert physicians. Since cerebral microvasculature and the retina share common characteristics, the idea of identifying potential ocular biomarkers to facilitate diagnosis is not a novel one. Initial efforts included studying less quantifiable parameters such as aspects of visual function, extraocular movements, and funduscopic findings. However, the really exciting prospect of a non-invasive, safe, fast, reproducible, and quantifiable method of pinpointing novel biomarkers has emerged with the advent of optical coherence tomography (OCT) and, more recently, OCT angiography (OCTA). The possibility of analyzing multiple parameters of retinal as well as retinal microvasculature variables in vivo represents a promising opportunity to investigate whether specific findings can be linked to certain subtypes of dementia and aid in their earlier diagnosis. The existing literature on the contribution of the eye in characterizing dementia, with a special interest in OCT and OCTA parameters will be reviewed and compared, and we will explicitly focus our effort in advancing our understanding and knowledge of relevant biomarkers to facilitate future research in the differential diagnosis between Alzheimer's disease and common forms of cognitive impairment, including vascular dementia, frontotemporal dementia, and dementia with Lewy bodies.

Remodeling mechanisms determine size distributions in developing retinal vasculature

The development of retinal blood vessels has extensively been used as a model to study vascular pattern formation. To date, various quantitative measurements, such as size distribution have been performed, but the relationship between pattern formation mechanisms and these measurements remains unclear. In the present study, we first focus on the islands (small regions subdivided by the capillary network). We quantitatively measured the island size distribution in the retinal vascular network and found that it tended to exhibit an exponential distribution. We were able to recapitulate this distribution pattern in a theoretical model by implementing the stochastic disappearance of vessel segments around arteries could reproduce the observed exponential distribution of islands. Second, we observed that the diameter distribution of the retinal artery segment obeyed a power law. We theoretically showed that an equal bifurcation branch pattern and Murray’s law could reproduce this pattern. This study demonstrates the utility of examining size distribution for understanding the mechanisms of vascular pattern formation.

Optimizing Our Patients' Entropy Production as Therapy? Hypotheses Originating from the Physics of Physiology

Understanding how nature drives entropy production offers novel insights regarding patient care. Whilst energy is always preserved and energy gradients irreversibly dissipate (thus producing entropy), increasing evidence suggests that they do so in the most optimal means possible. For living complex non-equilibrium systems to create a healthy internal emergent order, they must continuously produce entropy over time. The Maximum Entropy Production Principle (MEPP) highlights nature's drive for non-equilibrium systems to augment their entropy production if possible. This physical drive is hypothesized to be responsible for the spontaneous formation of fractal structures in space (e.g., multi-scale self-similar tree-like vascular structures that optimize delivery to and clearance from an organ system) and time (e.g., complex heart and respiratory rate variability); both are ubiquitous and essential for physiology and health. Second, human entropy production, measured by heat production divided by temperature, is hypothesized to relate to both metabolism and consciousness, dissipating oxidative energy gradients and reducing information into meaning and memory, respectively. Third, both MEPP and natural selection are hypothesized to drive enhanced functioning and adaptability, selecting states with robust basilar entropy production, as well as the capacity to enhance entropy production in response to exercise, heat stress, and illness. Finally, a targeted focus on optimizing our patients' entropy production has the potential to improve health and clinical outcomes. With the implications of developing a novel understanding of health, illness, and treatment strategies, further exploration of this uncharted ground will offer value.

Retinal vascular fractal dimension and cerebral blood flow, a pilot study

PURPOSE

Ocular and brain microcirculation share embryological and histological similarities. The retinal vascular fractal dimension (FD) is a marker of retinal vascular complexity of the vascular tree. The purpose of this study was to explore the relationship between cerebral blood flow (CBF), retinal vascular FD and other retinal vascular markers.

METHODS

Cross-sectional analysis comprising 26 individuals ≥65 years old from the Cognitive REServe and Clinical ENDOphenotype (CRESCENDO) cohort of relative healthy older adults. Retinal vascular FD was measured from fundus photographs by using the semi-automated Singapore Eye Vessel Assessment (SIVA) software. CBF was estimated using a 2D pulsed ASL MRI sequence. Associations between blood flow and retinal parameters were analysed using linear regression models adjusted for age and sex.

RESULTS

Cerebral blood flow was positively associated with venular FD (R²  = 0.32, p = 0.03). This association was stronger in the anterior versus posterior brain territories (R²  = 0.35 [p = 0.001] versus R²  = 0.16 [p = 0.07], respectively). Global CBF was correlated with arteriolar branching angle (R²  = 0.23, p = 0.01) and tortuosity (R²  = 0.20, p = 0.02). Global CBF was not correlated with other SIVA parameters.

CONCLUSIONS

Retinal venular complexity summarized by the FD was associated with cerebral blood flow as well as retinal arteriolar tortuosity and branching angle. Larger prospective clinical studies are needed to confirm these results.

Systematic Review on Fractal Dimension of the Retinal Vasculature in Neurodegeneration and Stroke: Assessment of a Potential Biomarker

Introduction: Ocular manifestations in several neurological pathologies accentuate the strong relationship between the eye and the brain. Retinal alterations in particular can serve as surrogates for cerebral changes. Offering a “window to the brain,” the transparent eye enables non-invasive imaging of these changes in retinal structure and vasculature. Fractal dimension (FD) reflects the overall complexity of the retinal vasculature. Changes in FD could reflect subtle changes in the cerebral vasculature that correspond to preclinical stages of neurodegenerative diseases. In this review, the potential of this retinal vessel metric to serve as a biomarker in neurodegeneration and stroke will be explored.

Methods: A literature search was conducted, following the PRISMA Statement 2009 criteria, in four large bibliographic databases (Pubmed, Embase, Web Of Science and Cochrane Library) up to 12 October 2019. Articles have been included based upon their relevance. Wherever possible, level of evidence (LOE) has been assessed by means of the Oxford Centre for Evidence-based Medicine Level of Evidence classification.

Results: Twenty-one studies were included for qualitative synthesis. We performed a narrative synthesis and produced summary tables of findings of included papers because methodological heterogeneity precluded a meta-analysis. A significant association was found between decreased FD and neurodegenerative disease, mainly addressing cognitive impairment (CI) and dementia. In acute, subacute as well as chronic settings, decreased FD seems to be associated with stroke. Differences in FD between subtypes of ischemic stroke remain unclear.

Conclusions: This review provides a summary of the scientific literature regarding the association between retinal FD and neurodegenerative disease and stroke. Central pathology is associated with a decreased FD, as a measure of microvascular network complexity. As retinal FD reflects the global integrity of the cerebral microvasculature, it is an attractive parameter to explore. Despite obvious concerns, mainly due to a lack of methodological standardization, retinal FD remains a promising non-invasive and low-cost diagnostic biomarker for neurodegenerative and cerebrovascular disease. Before FD can be implemented in clinic as a diagnostic biomarker, the research community should strive for uniformization and standardization.

NOTCH3 and CADASIL syndrome: a genetic and structural overview

CADASIL syndrome is a rare disease that belongs to a group of disorders called leukodystrophies. It is well established that NOTCH3 gene on chromosome 19 is primarily responsible for the development of the CADASIL syndrome. Herein, an attempt is made to shed light on the actual molecular mechanism underlying CADASIL syndrome, through insights extracted from comprehensive evolutionary studies and in silico modelling on Notch 3 protein. In particular, we suggest the use of optical coherence tomography angiography for the detection of early signs of small vessel diseases, which are the major precursors to a repertoire of neurodegenerative conditions, including CADASIL.

Retinal microvasculature changes in amyloid-negative subcortical vascular cognitive impairment compared to amyloid-positive Alzheimer's disease

BACKGROUND AND PURPOSE

To investigate small vessel abnormalities in patients with cognitive impairment, we compared retinal microvascular alterations between patients with cognitive impairment related to Alzheimer's disease (ADCI) and those with subcortical vascular cognitive impairment (SVCI).

METHODS

We prospectively recruited 29 amyloid-positive ADCI patients, 28 amyloid-negative SVCI patients that were confirmed by ¹¹C-PiB-PET scan and 34 individuals with normal cognition (NC). The three groups were compared in terms of retinal vascular variables (retinal fractal dimension, vascular caliber, tortuosity and branching angle) by using a semi-automated, computer-assisted analysis of digital fundus photographs. We also investigated the relationship between retinal variables and white matter hyperintensities (WMH) on MRI.

RESULTS

Compared to NC individuals, the SVCI patients had smaller total and arteriolar fractal dimensions, whereas there was no significant difference of fractal dimension between ADCI and NC. Other retinal variables did not differ among the three groups. A significant correlation existed between fractal dimension and WMH volume.

CONCLUSIONS

Retinal microvascular alterations, especially retinal fractal dimension, may be useful markers that reflect cerebral microvascular changes in patients with SVCI as opposed to ADCI, who had no definite difference in retinal variables compared to the NC group.

Retinal Vascular Imaging in Vascular Cognitive Impairment: Current and Future Perspectives

Vascular cognitive disorders are heterogeneous and increasingly recognized entities with intricate correlation to neurodegenerative conditions. Retinal vascular analysis is a noninvasive approach to study cerebrovascular pathology, with promise to assist particularly during early disease phases. In this article, we have systematically summarized the current understanding, potential applications, and inevitable limitations of retinal vascular imaging in patients with vascular cognitive impairment. In addition, future directions in the field with support from automated technology using deep learning methods and their existing challenges are emphasized.

A spectrum of retinal vasculature measures and coronary artery disease

BACKGROUND AND AIMS

We aimed to comprehensively describe a spectrum of retinal vessel measures including fractal dimension (D_f) and their associations with indices of coronary artery disease (CAD) extent and severity, as well as hypertension and diabetes.

METHODS

The Australian Heart Eye Study (AHES) is an observational study that surveyed 1680 participants presenting to a tertiary referral hospital for the evaluation of potential CAD by coronary angiography. A range of newer retinal vessel geometric measures (D_f, curvature tortuosity, and branching angle) were quantified from retinal photographs using semi-automated software, the Singapore 'I' Vessel Assessment (SIVA) tool. A combined retinal score was constructed, aiming to assess the joint effect of multiple retinal vessel parameters on CAD, comprising of those variables that were most strongly significant in multivariate analysis - D_f, arteriolar curvature tortuosity, and retinal arteriolar calibre. CAD was objectively quantified using a range of measures obtained from coronary angiography.

RESULTS

A total of 1187 participants had complete data on retinal vessel measurements and coronary vessel evaluation. Retinal vascular D_f and curvature tortuosity decreased with increasing age; women had significantly lower D_f than men (p<0.003). Straighter retinal vessels were associated with CAD extent and Gensini scores in multivariable analysis (p<0.02). Accounting for media opacity by sub-group analysis in pseudophakic patients, the combined retinal score was associated with stenosis greater than 50% in any coronary artery segment (vessel score) and obstructive coronary stenosis in all three main coronary arteries (segment score) (p = 0.01). Lower D_f and narrower arteriolar branching angle were associated with CAD vessel score (p<0.03). In sex-stratified multivariate analyses, straighter arterioles were associated with greater odds of CAD in men, and narrower venular branching angle was associated with CAD in women.

CONCLUSIONS

A range of retinal vessel measures were associated with CAD extent and severity. A sparser retinal microvascular network (smaller D_f) was associated with older age and female gender. After accounting for the impact of media opacity, retinal vessel measures were associated with more diffuse and severe CAD.

Association of the Fractal Dimension of Retinal Arteries and Veins with Quantitative Brain MRI Measures in HIV-Infected and Uninfected Women

Objective

The fractal dimension of retinal arteries and veins is a measure of the complexity of the vascular tree. We hypothesized that retinal fractal dimension would be associated with brain volume and white matter integrity in HIV-infected women.

Design

Nested case-control within longitudinal cohort study.

Methods

Women were recruited from the Brooklyn site of the Women’s Interagency HIV study (WIHS); 34 HIV-infected and 21 HIV-uninfected women with analyzable MRIs and retinal photographs were included. Fractal dimension was determined using the SIVA software program on skeletonized retinal images. The relationship between predictors (retinal vascular measures) and outcomes (quantitative MRI measures) were analyzed with linear regression models. All models included age, intracranial volume, and both arterial and venous fractal dimension. Some models were adjusted for blood pressure, race/ethnicity, and HIV-infection.

Results

The women were 45.6 ± 7.3 years of age. Higher arterial dimension was associated with larger cortical volumes, but higher venous dimension was associated with smaller cortical volumes. In fully adjusted models, venous dimension was significantly associated with fractional anisotropy (standardized β = -0.41, p = 0.009) and total gray matter volume (β = -0.24, p = 0.03), and arterial dimension with mean diffusivity (β = -0.33,.p = 0.04) and fractional anisotropy (β = 0.34, p = 0.03). HIV-infection was not associated with any retinal or MRI measure.

Conclusions

Higher venous fractal dimension was associated with smaller cortical volumes and lower fractional anisotropy, whereas higher arterial fractal dimension was associated with the opposite patterns. Longitudinal studies are needed to validate this finding.

Retinal Vascular Changes are a Marker for Cerebral Vascular Diseases

The retinal circulation is a potential marker of cerebral vascular disease because it shares origin and drainage with the intracranial circulation and because it can be directly visualized using ophthalmoscopy. Cross-sectional and cohort studies have demonstrated associations between chronic retinal and cerebral vascular disease, acute retinal and cerebral vascular disease, and chronic retinal vascular disease and acute cerebral vascular disease. In particular, certain qualitative features of retinopathy, retinal artery occlusion, and increased retinal vein caliber are associated with concurrent and future cerebrovascular events. These associations persist after accounting for confounding variables known to be disease-causing in both circulations, which supports the potential use of retinal vasculature findings to stratify individuals with regards to cerebral vascular disease risk.

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.

Fractals in the neurosciences, Part II: clinical applications and future perspectives

It has been ascertained that the human brain is a complex system studied at multiple scales, from neurons and microcircuits to macronetworks. The brain is characterized by a hierarchical organization that gives rise to its highly topological and functional complexity. Over the last decades, fractal geometry has been shown as a universal tool for the analysis and quantification of the geometric complexity of natural objects, including the brain. The fractal dimension has been identified as a quantitative parameter for the evaluation of the roughness of neural structures, the estimation of time series, and the description of patterns, thus able to discriminate different states of the brain in its entire physiopathological spectrum. Fractal-based computational analyses have been applied to the neurosciences, particularly in the field of clinical neurosciences including neuroimaging and neuroradiology, neurology and neurosurgery, psychiatry and psychology, and neuro-oncology and neuropathology. After a review of the basic concepts of fractal analysis and its main applications to the basic neurosciences in part I of this series, here, we review the main applications of fractals to the clinical neurosciences for a holistic approach towards a fractal geometry model of the brain.

Microvascular network alterations in the retina of patients with Alzheimer's disease

BACKGROUND

Although cerebral small-vessel disease has been implicated in the development of Alzheimer's disease (AD), the cerebral microcirculation is difficult to visualize directly in vivo. Because the retina provides a noninvasive window to assess the microcirculation, we determined whether quantitatively measured retinal microvascular parameters are associated with AD.

METHODS

We conducted a case-control study (case:control matching ≈ 1:2). Retinal photographs were analyzed using a computer program, and a spectrum of quantitative retinal microvascular parameters (caliber, fractal dimension, tortuosity, and bifurcation) were measured. Logistic regression models were used to compute the odds ratio (OR) and 95% confidence interval for AD adjusting for age, gender, ethnicity, smoking, hypertension, diabetes, hypercholesterolemia, and history of myocardial infarction.

RESULTS

We included 136 demented patients with AD and 290 age-gender-race-matched controls. Persons with narrower venular caliber (OR per standard deviation [SD] decrease, 2.01 [1.27-3.19]), decreased arteriolar and venular fractal dimension (OR per SD decrease 1.35 [1.08-1.68], 1.47 [1.17-1.84], respectively) and increased arteriolar and venular tortuosity (OR per SD increase, 1.84 [1.40-2.31], 1.94 [1.48-2.53], respectively) were more likely to have AD. These associations still persisted when only AD cases without a history of cerebrovascular disease were included.

CONCLUSIONS

Patients with AD have altered microvascular network in the retina (narrower retinal venules and a sparser and more tortuous retinal vessels) compared with matched nondemented controls. These changes in retinal microvasculature may reflect similar pathophysiological processes in cerebral microvasculature in the brains of patients with AD.

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.

Fractal dimension and vessel complexity in patients with cerebral arteriovenous malformations

The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs.