Fractal analysis of dendrite morphology using modified box-counting method

Morphology and Fractal-Based Classifications of Neurons and Microglia in Two and Three Dimensions

Microglia and neurons live physically intertwined, intimately related structurally and functionally in a dynamic relationship in which microglia change continuously over a much shorter timescale than do neurons. Although microglia may unwind and depart from the neurons they attend under certain circumstances, in general, together both contribute to the fractal topology of the brain that defines its computational capabilities. Both neuronal and microglial morphologies are well-described using fractal analysis complementary to more traditional measures. For neurons, the fractal dimension has proved valuable for classifying dendritic branching and other neuronal features relevant to pathology and development. For microglia, fractal geometry has substantially contributed to classifying functional categories, where, in general, the more pathological the biological status, the lower the fractal dimension for individual cells, with some exceptions, including hyper-ramification. This chapter provides a review of the intimate relationships between neurons and microglia, by introducing 2D and 3D fractal analysis methodology and its applications in neuron-microglia function in health and disease.

Fractal analysis of the macular region in healthy eyes using swept-source optical coherence tomography angiography

PURPOSE

This cross-sectional observational study evaluated the relationship between retinal vascular fractal dimension (FD) and age, as well as other vascular parameters in healthy eyes using swept-source optical coherence tomography angiography (SS-OCTA).

METHODS

The study cohort consisted of 222 eyes of 116 healthy participants with no ocular or systemic disease. SS-OCTA images were captured and analyzed using the Plex Elite 9000 and software tools available in the advanced retinal imaging (ARI) network hub. The retinal vascular layers were defined by the instrument's automatic retinal layer segmentation. The fractal analysis was performed on the superficial capillary plexus (SCP), deep capillary plexus (DCP), and the whole retina. Grayscale OCTA images were standardized and binarized using ImageJ and fractal box-counting analyses were performed using Fractalyse software. Pearson's correlation was used to analyze the correlation between FD and retinal vascular parameters.

RESULTS

The results showed that FD values were significantly higher in the 6 mm ring and the whole 6 × 6 scan region when compared to the 1 mm ETDRS central subfield. The correlation between age and FD was weak with a significant positive correlation between age and FD of the SCP in the 6 mm ring and between age and FD of the DCP in the 1 mm ring. Overall, differences in FD values in these healthy eyes were extremely small regardless of age or macular location.

CONCLUSION

FD values in normal eyes show little variation with age and are relatively stable across the macula. This suggests that FD values may not need adjustment for age or location when evaluated in the context of retinal disease.

Polymer Physics-Based Classification of Neurons

Recognizing that diverse morphologies of neurons are reminiscent of structures of branched polymers, we put forward a principled and systematic way of classifying neurons that employs the ideas of polymer physics. In particular, we use 3D coordinates of individual neurons, which are accessible in recent neuron reconstruction datasets from electron microscope images. We numerically calculate the form factor, F(q), a Fourier transform of the distance distribution of particles comprising an object of interest, which is routinely measured in scattering experiments to quantitatively characterize the structure of materials. For a polymer-like object consisting of n monomers spanning over a length scale of r, F(q) scales with the wavenumber [Formula: see text] as [Formula: see text] at an intermediate range of q, where [Formula: see text] is the fractal dimension or the inverse scaling exponent ([Formula: see text]) characterizing the geometrical feature ([Formula: see text]) of the object. F(q) can be used to describe a neuron morphology in terms of its size ([Formula: see text]) and the extent of branching quantified by [Formula: see text]. By defining the distance between F(q)s as a measure of similarity between two neuronal morphologies, we tackle the neuron classification problem. In comparison with other existing classification methods for neuronal morphologies, our F(q)-based classification rests solely on 3D coordinates of neurons with no prior knowledge of morphological features. When applied to publicly available neuron datasets from three different organisms, our method not only complements other methods but also offers a physical picture of how the dendritic and axonal branches of an individual neuron fill the space of dense neural networks inside the brain.

Controlled assembly of retinal cells on fractal and Euclidean electrodes

Controlled assembly of retinal cells on artificial surfaces is important for fundamental cell research and medical applications. We investigate fractal electrodes with branches of vertically-aligned carbon nanotubes and silicon dioxide gaps between the branches that form repeating patterns spanning from micro- to milli-meters, along with single-scaled Euclidean electrodes. Fluorescence and electron microscopy show neurons adhere in large numbers to branches while glial cells cover the gaps. This ensures neurons will be close to the electrodes’ stimulating electric fields in applications. Furthermore, glia won’t hinder neuron-branch interactions but will be sufficiently close for neurons to benefit from the glia’s life-supporting functions. This cell ‘herding’ is adjusted using the fractal electrode’s dimension and number of repeating levels. We explain how this tuning facilitates substantial glial coverage in the gaps which fuels neural networks with small-world structural characteristics. The large branch-gap interface then allows these networks to connect to the neuron-rich branches.

Aperiodic measures of neural excitability are associated with anticorrelated hemodynamic networks at rest: A combined EEG-fMRI study

The hallmark of resting EEG spectra are distinct rhythms emerging from a broadband, aperiodic background. This aperiodic neural signature accounts for most of total EEG power, although its significance and relation to functional neuroanatomy remains obscure. We hypothesized that aperiodic EEG reflects a significant metabolic expenditure and therefore might be associated with the default mode network while at rest. During eyes-open, resting-state recordings of simultaneous EEG-fMRI, we find that aperiodic and periodic components of EEG power are only minimally associated with activity in the default mode network. However, a whole-brain analysis identifies increases in aperiodic power correlated with hemodynamic activity in an auditory-salience-cerebellar network, and decreases in aperiodic power are correlated with hemodynamic activity in prefrontal regions. Desynchronization in residual alpha and beta power is associated with visual and sensorimotor hemodynamic activity, respectively. These findings suggest that resting-state EEG signals acquired in an fMRI scanner reflect a balance of top-down and bottom-up stimulus processing, even in the absence of an explicit task.

The analysis of F-actin structure of mesenchymal stem cells by quantification of fractal dimension

The actin cytoskeleton is indispensable for the motility and migration of all types of cells; therefore, it plays a crucial role in the ability of the tissues to repair. Mesenchymal stem cells are intensively used in regenerative medicine, but usually relatively low percent of transplanted cells reaches the injury. To overcome this evident limitation, researchers try to enhance the motility and migration rate of the cells. As one of the approaches, co-cultivation and preconditioning of stem cells with biologically active compounds, which can cause actin cytoskeleton rearrangements followed by an increase of migratory properties of the cells, could be applied. The observed changes in F-actin structure induced by the compounds require quantitative estimation, and measurement of fluorescence intensity of the F-actin image captured by various microscopic techniques is commonly used nowadays. However, this approach could not always accurately detect the observed changes in the shape and structure of actin cytoskeleton. At this time, the image of F-actin has an irregular geometric pattern, and thus could be considered and characterized as a fractal object. To quantify the re-organization of cellular F-actin in terms of fractal geometry Minkovsky's box-counting method is suitable, but it is not widely used nowadays. We modified and improved the previously described method for fractal dimension measurement, and successfully applied it for the quantification of the F-actin structures of human mesenchymal stem cells.

The Complexity and Fractal Geometry of Nuclear Medicine Images

Irregularity in shape and behavior is the main feature of every anatomical system, including human organs, tissues, cells, and sub-cellular entities. It has been shown that this property cannot be quantified by means of the classical Euclidean geometry, which is only able to describe regular geometrical objects. In contrast, fractal geometry has been widely applied in several scientific fields. This rapid growth has also produced substantial insights in the biomedical imaging. Consequently, particular attention has been given to the identification of pathognomonic patterns of "shape" in anatomical entities and their changes from natural to pathological states. Despite the advantages of fractal mathematics and several studies demonstrating its applicability to oncological research, many researchers and clinicians remain unaware of its potential. Therefore, this review aims to summarize the complexity and fractal geometry of nuclear medicine images.

Correlations between visual acuity and macular microvasculature quantified with optical coherence tomography angiography in diabetic macular oedema

PURPOSE

To explore the impact of macular ischaemia on vision in diabetic macular oedema (DMO) by analysing the correlations between visual acuity and macular microvascular parameters using optical coherence tomography angiography (OCTA).

METHODS

OCTA was performed in 81 eyes of 48 patients with DMO, and 3 × 3-mm² en face OCTA images of the superficial capillary plexus and deep capillary plexus in the central macula were retrospectively collected. Microvascular parameters including the number of microaneurysms, area of foveal avascular zone (FAZ), acircularity index of FAZ, vessel density, skeleton density, vessel density index and fractal dimension were measured. Central retinal thickness (CRT) and the presence of ellipsoid zone disruption at the fovea were also recorded. Linear mixed models were used to evaluate the correlations between best-corrected visual acuity (BCVA) and the microvascular parameters.

RESULTS

After adjustment for CRT and ellipsoid zone disruption at the fovea, lower skeleton density and lower fractal dimension in the deep capillary plexus were correlated with poorer BCVA (P = 0.030 and 0.024, respectively). None of the microvascular parameters of the superficial capillary plexus were correlated with BCVA after adjustment for CRT and ellipsoid zone disruption (all, P > 0.05).

CONCLUSIONS

For eyes with DMO, low skeleton density and low branching complexity in the deep capillary plexus of central macula were correlated with poor vision. OCTA could offer quantified parameters of macular microvasculature to measure the impact of macular ischaemia on visual acuity in DMO.

Fractal dimension in the evaluation of different treatments of muscular injury in rats

OBJECTIVES

To evaluate alterations from different therapies in muscular injury using the Fractal Dimension (FD) method.

METHODS

35 animals were allocated in Control Group (C), Injury Control Group (IC), Injury Low Level Laser Therapy Group (ILT), Injury Platelet Rich Plasma Group (IP), and Injury LLLT and PRP Group (ILP). The animals suffered a stretch injury in gastrocnemius muscle and after that IP and ILP groups received PRP application. The ILT and ILP groups received daily LLLT applications for seven days. After seven days the animals were euthanized and the gastrocnemius muscle removed and frozen. The muscles were stained with Hematoxylin and Eosin (HE) and Picrosirius Red, for observation of the morphology of the injury and semi-quantitative and quantitative analysis through the Fractal Dimension (FD) method.

RESULTS

In the qualitative and semi-quantitative analysis, in relation to IC group, the ILT presented a reduction in rounded fibers and the IP in angular fibers. The ILP group demonstrated a reduction in both polymorphic fibers and inflammatory infiltrate. The FD of the muscles stained with HE was higher in the groups that suffered the injury when compared to the C group (p < 0.05); the FD of the collagen demonstrated no statistical difference between the groups.

CONCLUSION

Both treatments were able to accelerate injury repair, and the association of both presented better results than the isolated applications. However, the FD method showed no sensitivity to differentiate the treatments, either in the histological aspects or the injury in collagen.

Reliability of Using Retinal Vascular Fractal Dimension as a Biomarker in the Diabetic Retinopathy Detection

The retinal fractal dimension (FD) is a measure of vasculature branching pattern complexity. FD has been considered as a potential biomarker for the detection of several diseases like diabetes and hypertension. However, conflicting findings were found in the reported literature regarding the association between this biomarker and diseases. In this paper, we examine the stability of the FD measurement with respect to (1) different vessel annotations obtained from human observers, (2) automatic segmentation methods, (3) various regions of interest, (4) accuracy of vessel segmentation methods, and (5) different imaging modalities. Our results demonstrate that the relative errors for the measurement of FD are significant and FD varies considerably according to the image quality, modality, and the technique used for measuring it. Automated and semiautomated methods for the measurement of FD are not stable enough, which makes FD a deceptive biomarker in quantitative clinical applications.

Fractal dimension of apical dendritic arborization differs in the superficial and the deep pyramidal neurons of the rat cerebral neocortex

Pyramidal neurons of the mammalian cerebral cortex have specific structure and pattern of organization that involves the presence of apical dendrite. Morphology of the apical dendrite is well-known, but quantification of its complexity still remains open. Fractal analysis has proved to be a valuable method for analyzing the complexity of dendrite morphology. The aim of this study was to establish the fractal dimension of apical dendrite arborization of pyramidal neurons in distinct neocortical laminae by using the modified box-counting method. A total of thirty, Golgi impregnated neurons from the rat brain were analyzed: 15 superficial (cell bodies located within lamina II-III), and 15 deep pyramidal neurons (cell bodies situated within lamina V-VI). Analysis of topological parameters of apical dendrite arborization showed no statistical differences except in total dendritic length (p=0.02), indicating considerable homogeneity between the two groups of neurons. On the other hand, average fractal dimension of apical dendrite was 1.33±0.06 for the superficial and 1.24±0.04 for the deep cortical neurons, showing statistically significant difference between these two groups (p<0.001). In conclusion, according to the fractal dimension values, apical dendrites of the superficial pyramidal neurons tend to show higher structural complexity compared to the deep ones.