Can the fractal dimension be applied for the early diagnosis of non-proliferative diabetic retinopathy?

A new fractal index to classify forest fragmentation and disorder

Context

Forest loss and fragmentation pose extreme threats to biodiversity. Their efficient characterization from remotely sensed data therefore has strong practical implications. Data are often separately analyzed for spatial fragmentation and disorder, but no existing metric simultaneously quantifies both the shape and arrangement of fragments.

Objectives

We present a fractal fragmentation and disorder index (FFDI), which advances a previously developed fractal index by merging it with the Rényi information dimension. The FFDI is designed to work across spatial scales, and to efficiently report both the fragmentation of images and their spatial disorder.

Methods

We validate the FFDI with 12,600 synthetic hierarchically structured random map (HRM) multiscale images, as well as several other categories of fractal and non-fractal test images (4880 images). We then apply the FFDI to satellite imagery of forest cover for 10 distinct regions of the Romanian Carpathian Mountains from 2000–2021.

Results

The FFDI outperformed its two individual components (fractal fragmentation index and Rényi information dimension) in resolving spatial patterns of disorder and fragmentation when tested on HRM classes and other image types. The FFDI thus offers a clear advantage when compared to the individual use of fractal fragmentation index and the Information Dimension, and provided good classification performance in an application to real data.

Conclusions

This work improves on previous characterizations of landscape patterns. With the FFDI, scientists will be able to better monitor and understand forest fragmentation from satellite imagery. The FFDI may also find wider applicability in biology wherever image analysis is used.

The Longitudinal Assessment of Vascular Parameters of the Retina and Their Correlations with Systemic Characteristics in Type 2 Diabetes-A Pilot Study

The aim of the study was to assess various retinal vessel parameters of diabetes mellitus (DM) patients and their correlations with systemic factors in type 2 DM. A retrospective exploratory study in which 21 pairs of baseline and follow-up images of patients affected by DM were randomly chosen from the Sankara Nethralaya−Diabetic Retinopathy Study (SN DREAMS) I and II datasets. Patients’ fundus was photographed, and the diagnosis was made based on Klein classification. Vessel thickness parameters were generated using a web-based retinal vascular analysis platform called VASP. The thickness changes between the baseline and follow-up images were computed and normalized with the actual thicknesses of baseline images. The majority of parameters showed 10~20% changes over time. Vessel width in zone C for the second vein was significantly reduced from baseline to follow-up, which showed positive correlations with systolic blood pressure and serum high-density lipoproteins. Fractal dimension for all vessels in zones B and C and fractal dimension for vein in zones A, B and C showed a minimal increase from baseline to follow-up, which had a linear relationship with diastolic pressure, mean arterial pressure, serum triglycerides (p < 0.05). Lacunarity for all vessels and veins in zones A, B and C showed a minimal decrease from baseline to follow-up which had a negative correlation with pulse pressure and positive correlation with serum triglycerides (p < 0.05). The vessel widths for the first and second arteries significantly increased from baseline to follow-up and had an association with high-density lipoproteins, glycated haemoglobin A1C, serum low-density lipoproteins and total serum cholesterol. The central reflex intensity ratio for the second artery was significantly decreased from baseline to follow-up, and positive correlations were noted with serum triglyceride, serum low-density lipoproteins and total serum cholesterol. The coefficients for branches in zones B and C artery and the junctional exponent deviation for the artery in zone A decreased from baseline to follow-up showed positive correlations with serum triglycerides, serum low-density lipoproteins and total serum cholesterol. Identifying early microvascular changes in diabetic patients will allow for earlier intervention, improve visual outcomes and prevent vision loss.

Fractal signature of coronaviruses related to severe acute respiratory syndrome

Purpose

The aim of this work was to investigate the genome of SARS-CoV, MERS-CoV, and SARS-CoV-2 by the paradigm of chaos theory and fractal geometry. Coronavirus is the agent that causes the severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS) and emergent infections such as the present pandemic of COVID-19. Understanding its genome pattern is important for developing new and faster ways of testing for identifying the genome of the virus and also for better understanding of its origin and evolution.

Methods

For that, it was calculated the alpha coefficient by detrended fluctuation analysis (DFA) for the sequences of these genomes converted to binary numbers in order to determine if it is a chaotic or a random series of data. Also, it applied the random walking for obtaining a fractal map of the whole genome and calculated the fractal dimension (FD) by box-counting of this map by two different software.

Results

With this, it was found that the alpha coefficient of the first SARS viruses was > 0.5, indicating that the series is chaotic or fractal, and has a persistent long-range memory or self-similarity along its sequence. This is not the case for MERS virus, which showed to have a completely random sequence (α < 0.5). For the fractal dimension, SARS viruses presented a FD around 1.5, and for MERS the fractal dimension decreases (FD < 1.5).

Conclusion

The images generated by random walking of the entire RNA genome are by itself a fractal signature of the virus, which may be applied for studying its origin and for faster diagnostic of COVID-19.

Retinal vascular geometry detection as a biomarker in diabetes mellitus

OBJECTIVE

To compare the vessel geometry characteristics of color fundus photographs in normal control and diabetes mellitus (DM) patients and to find potential biomarkers for early diabetic retinopathy (DR) based on a neural network vessel segmentation system and automated vascular geometry parameter analysis software.

METHODS

A total of 102 consecutive patients with type 2 DM (T2DM) and 132 healthy controls were recruited. All participants underwent general ophthalmic examinations, and retinal fundus photographs were taken with a digital fundus camera without mydriasis. Color fundus photographs were input into a dense-block generative adversarial network (D-GAN)-assisted retinal vascular segmentation system (http://www.gdcerc.cn:8081/#/login) to obtain binary images. These images were then analyzed by customized software (ocular microvascular analysis system V2.9.1) for automatic processing of vessel geometry parameters, including the monofractal dimension (D_box), multifractal dimension (D₀), vessel area ratio (R), max vessel diameter (d_max), average vessel diameter (d_ave), arc-chord ratio (A/C), and tortuosity (τ_n). Geometric differences between the healthy subjects and DM patients were analyzed. Then, regression analysis and receiver operating characteristic (ROC) curve analysis were performed to evaluate the diagnostic efficiency of the vascular geometry parameters.

RESULTS

No significant differences were observed between the baseline characteristics of each group. DM patients had lower D_box and D₀ values (1.330 ± 0.041; 1.347 ± 0.038) than healthy subjects (1.343 ± 0.048, p < 0.05; 1.362 ± 0.042, p < 0.05) and showed increasing values of d_max, d_ave, A/C, and τ_n compared with normal controls, although only the differences in d_ave and τ_n between the groups were statistically significant. In the regression analysis, d_ave and τ_n showed a good correlation with diabetes (d_ave, OR 1.765, 95% CI 1.319-2.362, p < 0.001; τ_n, OR 9.323, 95% CI 1.492-58.262, p < 0.05).

CONCLUSIONS

We demonstrated the relationship between retinal vascular geometry and the process in DM patients, showing that D_box, D₀, d_ave, and τ_n may be indicators of morphological changes in retinal vessels in DM patients and can be early biomarkers of DR.

Chaotic model for COVID-19 growth factor

Purpose

The new COVID-19 pandemic caused by SARS-CoV-2 initiated in the world a large quarantine, due to the exponential capacity of the virus in spreading from human contact. In the present work, the dynamics of such spreading was evaluated by the analysis of the growth factor of the disease.

Method

It was applied the space phase of the time series, the detrended fluctuation analysis (DFA) of the series, and the fractal dimension of the space phase.

Results

It was possible to notice a strange attractor in the space phase of the growth factor indicating that the process is a deterministic chaos. The value of the alpha coefficient by DFA showed to be less than 0.5, characteristic of anti-persistent long-range memory, in which large events alternate with small events, and vice-versa. The fractal dimension of the phase space set was a fractal number, between 1 and 2, another indicator that the growth factor of the disease is not random.

Conclusion

This global analysis is pointing that the spread of COVID-19 is a deterministic chaos, with long-range memory. Understanding such dynamics may help the control of this disease.

Relationship Between Retinal Fractal Dimension and Nonperfusion in Diabetic Retinopathy on Ultrawide-Field Fluorescein Angiography

PURPOSE

To correlate fractal dimension (FD) of the retinal vasculature with the extent of retinal nonperfusion area in diabetic retinopathy (DR) on ultrawide-field fluorescein angiography (FA).

DESIGN

Cross-sectional study.

METHODS

Baseline Optos 200Tx ultrawide-field FA images of 80 eyes with DR from the DAVE (NCT01552408) and RECOVERY (NCT02863354) studies were stereographically projected at the Doheny Image Reading Center. The retinal vasculature was extracted from an early-phase FA frame by exploiting the elongated nature of the vessels and then skeletonized for calculation of FD using a box-counting method. The nonperfusion area was delineated by 2 independent, reading center-certified graders who were masked to the study groups and who were using a standardized protocol and then computed in millimeters squared.

RESULTS

While no difference in FD was observed for the entire retina in DR compared with normal control subjects, a significantly smaller FD was found in the far-periphery of the DR eyes (P < .001). FD for the entire retina was negatively associated with global nonperfusion area (R = -0.44; P < .001), and this relationship was also present within the 3 concentric retinal zones (posterior: R = -0.31, P = .016; midperiphery: R = -0.35, P = .007; and far periphery: R = -0.31, P = .015).

CONCLUSIONS

Peripheral FD on ultrawide-field FA is reduced in DR eyes compared with normal eyes and is correlated with severity of retinal nonperfusion. FD can be calculated automatically without the need for correction of peripheral distortion, and therefore it may prove to be a useful surrogate biomarker when precise quantification of nonperfusion is not feasible.

Quantifying Microvascular Density and Morphology in Diabetic Retinopathy Using Spectral-Domain Optical Coherence Tomography Angiography

Purpose

To quantify changes in retinal microvasculature in diabetic retinopathy (DR) by using spectral-domain optical coherence tomography angiography (SD-OCTA).

Methods

Retrospective, cross-sectional, observational study of healthy and diabetic adult subjects with and without DR. Retinal microvascular changes were assessed by using SD-OCTA images and an intensity-based optical microangiography algorithm. A semiautomated program was used to calculate indices of microvascular density and morphology in nonsegmented and segmented SD-OCTA images. Microvascular density was quantified by using skeleton density (SD) and vessel density (VD), while vessel morphology was quantified as fractal dimension (FD) and vessel diameter index (VDI). Statistical analyses were performed by using the Student's t-test or analysis of variance with post hoc Tukey honest significant difference tests for multiple comparisons.

Results

Eighty-four eyes with DR and 14 healthy eyes were studied. Spearman's rank test demonstrated a negative correlation between DR severity and SD, VD, and FD, and a positive correlation with VDI (ρ = −0.767, −0.7166, −0.768, and +0.5051, respectively; P < 0.0001). All parameters showed high reproducibility between graders (ICC = 0.971, 0.962, 0.937, and 0.994 for SD, VD, FD, and VDI, respectively). Repeatability (κ) was greater than 0.99 for SD, VD, FD, and VDI.

Conclusions

Vascular changes in DR can be objectively and reliably characterized with SD, VD, FD, and VDI. In general, decreasing capillary density (SD and VD), branching complexity (FD), and increasing average vascular caliber (VDI) were associated with worsening DR. Changes in capillary density and morphology were significantly correlated with diabetic macular edema.

Noninvasive Retinal Markers in Diabetic Retinopathy: Advancing from Bench towards Bedside

The retinal vascular system is the only part of the human body available for direct, in vivo inspection. Noninvasive retinal markers are important to identity patients in risk of sight-threatening diabetic retinopathy. Studies have correlated structural features like retinal vascular caliber and fractals with micro- and macrovascular dysfunction in diabetes. Likewise, the retinal metabolism can be evaluated by retinal oximetry, and higher retinal venular oxygen saturation has been demonstrated in patients with diabetic retinopathy. So far, most studies have been cross-sectional, but these can only disclose associations and are not able to separate cause from effect or to establish the predictive value of retinal vascular dysfunction with respect to long-term complications. Likewise, retinal markers have not been investigated as markers of treatment outcome in patients with proliferative diabetic retinopathy and diabetic macular edema. The Department of Ophthalmology at Odense University Hospital, Denmark, has a strong tradition of studying the retinal microvasculature in diabetic retinopathy. In the present paper, we demonstrate the importance of the retinal vasculature not only as predictors of long-term microvasculopathy but also as markers of treatment outcome in sight-threatening diabetic retinopathy in well-established population-based cohorts of patients with diabetes.

Sensitivity and Specificity of Fractal Analysis to Distinguish Between Healthy and Pathologic Rectal Mucosa Microvasculature Seen During Colonoscopy

PURPOSE

Conventional endoscopy is limited by human capability to recognize and to differentiate pathology. Fractal analysis of blood vessels has been used in other organs, such as the retina, but never before to supplement colonoscopy. The aim of this study was to assess whether it is possible to differentiate between healthy and pathologic rectal mucosa using fractal analysis of the mucosal microvascular architecture seen during colonoscopic examination (Supplemental Digital Content 1, http://links.lww.com/SLE/A145).

METHODS

A total of 300 consecutive patients, 133 females and 167 males with a mean age of 49.1 (±11.3) years, undergoing endoscopy were included in the prospective cohort study. Colonoscopy of the sigmorectal region was performed, and then analyzed with computer-assisted image fractal analysis.

RESULTS

Fractal analysis of mucosal vasculature allowed for differentiation between healthy and pathologic rectal mucosa, as well as different pathologies (P<0.0001). The sensitivity of fractal analysis to diagnose rectal neoplasia was 92.8% to 96.4%, while the specificity was 91.9% to 98.5% depending on the fractal parameter. The sensitivity of fractal analysis to diagnose rectal colitis was 84.2% to 92.1%, while the specificity was 95.0% to 96.0%, depending on the fractal parameter.

CONCLUSIONS

Computer-assisted fractal analysis allows for differentiation between healthy and pathologic rectal mucosa, as well as between different mucosal pathologies, seen during colonoscopy. Fractal analysis improves the sensitivity and specificity of colonoscopy to aid in the diagnosis of neoplasia or colitis, and should be included in the screening and surveillance of these pathologies.

Characterisation of human non-proliferative diabetic retinopathy using the fractal analysis

AIM

To investigate and quantify changes in the branching patterns of the retina vascular network in diabetes using the fractal analysis method.

METHODS

This was a clinic-based prospective study of 172 participants managed at the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and December 2013. A set of 172 segmented and skeletonized human retinal images, corresponding to both normal (24 images) and pathological (148 images) states of the retina were examined. An automatic unsupervised method for retinal vessel segmentation was applied before fractal analysis. The fractal analyses of the retinal digital images were performed using the fractal analysis software ImageJ. Statistical analyses were performed for these groups using Microsoft Office Excel 2003 and GraphPad InStat software.

RESULTS

It was found that subtle changes in the vascular network geometry of the human retina are influenced by diabetic retinopathy (DR) and can be estimated using the fractal geometry. The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is slightly lower than the corresponding values of mild non-proliferative DR (NPDR) images (segmented and skeletonized versions). The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is higher than the corresponding values of moderate NPDR images (segmented and skeletonized versions). The lowest values were found for the corresponding values of severe NPDR images (segmented and skeletonized versions).

CONCLUSION

The fractal analysis of fundus photographs may be used for a more complete undeTrstanding of the early and basic pathophysiological mechanisms of diabetes. The architecture of the retinal microvasculature in diabetes can be quantitative quantified by means of the fractal dimension. Microvascular abnormalities on retinal imaging may elucidate early mechanistic pathways for microvascular complications and distinguish patients with DR from healthy individuals.

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.

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.

Efeito dos ácidos graxos essenciais sobre lipidemia e vascularização da membrana vitelina de codornas japonesas

Verificou-se o efeito de diferentes fontes de ácidos graxos essenciais ômega-3 e ômega-6 sobre o perfil lipídico de codornas japonesas e sobre o crescimento vascular na membrana vitelina dos embriões de aves suplementadas com óleo de peixe, 2% e 4%, e óleo de soja, 2% e 4%, em relação à dieta-controle, sem suplementação lipídica. Foi usado o método enzimático com reação colorimétrica para estimar o perfil lipídico sérico de colesterol total, triglicerídeos e HDL das aves. A vascularização na membrana vitelina foi quantificada por meio da dimensão fractal, utilizando-se o método de box-counting. A concentração de colesterol dos grupos controle e tratados não diferiu entre si. Para o HDL, o grupo que recebeu maior proporção de óleo de peixe, 4%, diferiu dos outros grupos. Os níveis de triglicerídeos das codornas suplementadas com óleo de peixe foram superiores aos dos demais grupos. A dimensão fractal da vascularização da membrana vitelina dos embriões de codornas dos grupos que receberam 4% de óleo de peixe e 4% de óleo de soja foi significativamente menor que a dos demais grupos, indicando efeitos antiangiogênicos no processo de formação vascular.

Fractal analysis of extra-embryonic vascularization in Japanese quail embryos exposed to extremely low frequency magnetic fields

Magnetic fields (MF) can alter the dynamic behavior of vascular tissue and may have a stimulatory or inhibitory effect on blood vessel growth. Fractal geometry has been used in several studies as a tool to describe the development of blood vascular networks. Due to its self-similarity, irregularity, fractional dimension, and dependence on the scale of vessel dimensions, vascular networks can be taken as fractal objects. In this work, we calculated the fractal dimension by the methods of box counting (D(bc)) and information dimension (D(inf)) to evaluate the development of blood vessels of the yolk sac membrane (YSM) from quail embryos exposed to MF with a magnetic flux density of 1 mT and a frequency of 60 Hz. The obtained results showed that when the MF was applied to embryos aged between 48 and 72 h, in sessions of 2 h (6 h/day) and 3 h (9 h/day) with exposure intervals between 6 and 5 h, respectively, blood vascular formation was inhibited. Exposure sessions shorter than 2 h or longer than 3 h had no observable change on the vascular process. In contrast, the magnetic field had no observable change on the YSM vascular network for embryos aged between 72 and 96 h, irrespective of the exposure time. In conclusion, these results show a "window effect" regarding exposure time.