Multi-scale fractal Fourier Ptychographic microscopy to assess the dose-dependent impact of copper pollution on living diatoms.

Accumulation of bioavailable heavy metals in aquatic environment poses a serious threat to marine communities and human health due to possible trophic transfers through the food chain of toxic, non-degradable, exogenous pollutants. Copper (Cu) is one of the most spread heavy metals in water, and can severely affect primary producers at high doses. Here we show a novel imaging test to assay the dose-dependent effects of Cu on live microalgae identifying stress conditions when they are still capable of sustaining a positive growth. The method relies on Fourier Ptychographic Microscopy (FPM), capable to image large field of view in label-free phase-contrast mode attaining submicron lateral resolution. We uniquely combine FPM with a new multi-scale analysis method based on fractal geometry. The system is able to provide ensemble measurements of thousands of diatoms in the liquid sample simultaneously, while ensuring at same time single-cell imaging and analysis for each diatom. Through new image descriptors, we demonstrate that fractal analysis is suitable for handling the complexity and informative power of such multiscale FPM modality. We successfully tested this new approach by measuring how different concentrations of Cu impact on Skeletonema pseudocostatum diatom populations isolated from the Sarno River mouth.

Main role of fractal-like nature of conformational space in subdiffusion in proteins.

Protein dynamics involves a myriad of mechanical movements happening at different time and space scales, which make it highly complex. One of the less understood features of protein dynamics is subdiffusivity, defined as sublinear dependence between displacement and time. Here, we use all-atoms molecular dynamics (MD) simulations to directly interrogate an already well-established theory and demonstrate that subdiffusivity arises from the fractal nature of the network of metastable conformations over which the dynamics, thought of as a diffusion process, takes place.

Retinal Fractal Dimension Is a Potential Biomarker for Systemic Health-Evidence From a Mixed-Age, Primary-Care Population.

Purpose: To investigate whether fractal dimension (FD), a retinal trait relating to vascular complexity and a potential "oculomics" biomarker for systemic disease, is applicable to a mixed-age, primary-care population. Methods: We used cross-sectional data (96 individuals; 183 eyes; ages 18-81 years) from a university-based optometry clinic in Glasgow, Scotland, to study the association between FD and systemic health. We computed FD from color fundus images using Deep Approximation of Retinal Traits (DART), an artificial intelligence-based method designed to be more robust to poor image quality. Results: Despite DART being designed to be more robust, a significant association (P < 0.001) between image quality and FD remained. Consistent with previous literature, age was associated with lower FD (P < 0.001 univariate and when adjusting for image quality). However, FD variance was higher in older patients, and some patients over 60 had FD comparable to those of patients in their 20s. Prevalent systemic conditions were significantly (P = 0.037) associated with lower FD when adjusting for image quality and age. Conclusions: Our work suggests that FD as a biomarker for systemic health extends to mixed-age, primary-care populations. FD decreases with age but might not substantially decrease in everyone. This should be further investigated using longitudinal data. Finally, image quality was associated with FD, but it is unclear whether this finding is measurement error caused by image quality or confounded by age and health. Future work should investigate this to clarify whether adjusting for image quality is appropriate. Translational Relevance: FD could potentially be used in regular screening settings, but questions around image quality remain.

Investigating the interpretability of schizophrenia EEG mechanism through a 3DCNN-based hidden layer features aggregation framework.

BACKGROUND AND OBJECTIVE: Electroencephalogram (EEG) signals record brain activity, with growing interest in quantifying neural activity through complexity analysis as a potential biological marker for schizophrenia. Presently, EEG complexity analysis primarily relies on manual feature extraction, which is subjective and yields varied findings in studies involving schizophrenia and healthy controls. METHODS: This study aims to leverage deep learning methods for enhanced EEG complexity exploration, aiding early schizophrenia screening and diagnosis. Our proposed approach utilizes a three-dimensional Convolutional Neural Network (3DCNN) to extract enhanced data features for early schizophrenia identification and subsequent complexity analysis. Leveraging the spatiotemporal capabilities of 3DCNN, we extract advanced latent features and employ knowledge distillation to reintegrate these features into the original channels, creating feature-enhanced data. RESULTS: We employ a 10-fold cross-validation strategy, achieving the average accuracies of 99.46% and 98.06% in subject-dependent experiments on Dataset 1(14SZ and 14HC) and Dataset 2 (45SZ and 39HC). The average accuracy for subject-independent is 96.04% and 92.67% on both datasets. Feature extraction and classification are conducted on both the re-aggregated data and the original data. Our results demonstrate that re-aggregated data exhibit superior classification performance and a more stable training process after feature extraction. In the complexity analysis of re-aggregated data, we observe lower entropy features in schizophrenic patients compared to healthy controls, with more pronounced differences in the temporal and frontal lobes. Analyzing Katz's Fractal Dimension (KFD) across three sub-bands of lobe channels reveals the lowest α band KFD value in schizophrenia patients. CONCLUSIONS: This emphasizes the ability of our method to enhance the discrimination and interpretability in schizophrenia detection and analysis. Our approach enhances the potential for EEG-based schizophrenia diagnosis by leveraging deep learning, offering superior discrimination capabilities and richer interpretive insights.

Analysis and comparative study of a deterministic mathematical model of SARS-COV-2 with fractal-fractional operators: a case study.

In this paper, we investigate a fractal-fractional-order mathematical model with the influence of hospitalized patients and the impact of vaccination with fractal-fractional operators. The respective derivatives are considered in the Caputo, Caputo Fabrizio, and Atangana-Baleanu senses of fractional order α and fractal dimension τ . For the proposed problem, some results regarding basic reproduction number and stability are given. Using the next-generation matrix approach, we have investigated the global and local stability of several types of equilibrium points. We provide a detailed analysis of the existence and uniqueness of the solution. Moreover, we fit the model with the real data of Pakistan from June 01, 2020, till March 24, 2021. Then, we use the fractal-fractional derivative to find a numerical solution for the model. MATLAB software is used for numerical illustration. Graphical presentations corresponding to different parameteric values are given as well.

Tenets and Methods of Fractal Analysis (1/f Noise).

This chapter deals with the methodical challenges confronting researchers of the fractal phenomenon known as pink or 1/f noise. This chapter introduces concepts and statistical techniques for identifying fractal patterns in empirical time series. It defines some basic statistical terms, describes two essential characteristics of pink noise (self-similarity and long memory), and outlines four parameters representing the theoretical properties of fractal processes: the Hurst coefficient (H), the scaling exponent (α), the power exponent (ß), and the fractional differencing parameter (d) of the ARFIMA (autoregressive fractionally integrated moving average) method. Then, it compares and evaluates different approaches to estimating fractal parameters from observed data and outlines the advantages, disadvantages, and constraints of some popular estimators. The final section of this chapter answers the questions: Which strategy is appropriate for the identification of fractal noise in empirical settings and how can it be applied to the data?

Multifractal Analysis in Neuroimaging.

The characteristics of biomedical signals are not captured by conventional measures like the average amplitude of the signal. The methodologies derived from fractal geometry have been a very useful approach to study the degree of irregularity of a signal. The monofractal analysis of a signal is defined by a single power-law exponent in assuming a scale invariance in time and space. However, temporal and spatial variation in the scale-invariant structure of the biomedical signal often appears. In this case, multifractal analysis is well-suited because it is defined by a multifractal spectrum of power-law exponents. There are several approaches to the implementation of this analysis, and there are numerous ways to present these.In this chapter, we review the use of multifractal analysis for the purpose of characterizing signals in neuroimaging. After describing the tenets of multifractal analysis, we present several approaches to estimating the multifractal spectrum. Finally, we describe the applications of this spectrum on biomedical signals in the characterization of several diseases in neurosciences.

Neuronal Fractal Dynamics.

Synapse formation is a unique biological phenomenon. The molecular biological perspective of this phenomenon is different from the fractal geometrical one. However, these perspectives are not mutually exclusive and supplement each other. The cornerstone of the first one is a chain of biochemical reactions with the Markov property, that is, a deterministic, conditional, memoryless process ordered in time and in space, in which the consecutive stages are determined by the expression of some regulatory proteins. The coordination of molecular and cellular events leading to synapse formation occurs in fractal time space, that is, the space that is not only the arena of events but also actively influences those events. This time space emerges owing to coupling of time and space through nonlinear dynamics. The process of synapse formation possesses fractal dynamics with non-Gaussian distribution of probability and a reduced number of molecular Markov chains ready for transfer of biologically relevant information.

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.

Fractality of Cranial Sutures.

It has long been known that skull suture has a typical fractal structure. Although the fractal dimension has been utilized to assess morphology, the mechanism of the fractal structure formation remains to be elucidated. Recent advances in the mathematical modeling of biological pattern formation provided useful frameworks for understanding this mechanism. This chapter describes how various proposed mechanisms tried to explain the formation of fractal structures in cranial sutures.

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.

The Fractal Geometry of the Human Brain: An Evolutionary Perspective.

The evolution of the brain in mammals is characterized by changes in size, architecture, and internal organization. Consequently, the geometry of the brain, and especially the size and shape of the cerebral cortex, has changed notably during evolution. Comparative studies of the cerebral cortex suggest that there are general architectural principles governing its growth and evolutionary development. In this chapter, some of the design principles and operational modes that underlie the fractal geometry and information processing capacity of the cerebral cortex in primates, including humans, will be explored. It is shown that the development of the cortex coordinates folding with connectivity in a way that produces smaller and faster brains.

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.

Clinical Sensitivity of Fractal Neurodynamics.

Among the significant advances in the understanding of the organization of the neuronal networks that coordinate the body and brain, their complex nature is increasingly important, resulting from the interaction between the very large number of constituents strongly organized hierarchically and at the same time with "self-emerging." This awareness drives us to identify the measures that best quantify the "complexity" that accompanies the continuous evolutionary dynamics of the brain. In this chapter, after an introductory section (Sect. 15.1), we examine how the Higuchi fractal dimension is able to perceive physiological processes (15.2), neurological (15.3) and psychiatric (15.4) disorders, and neuromodulation effects (15.5), giving a mention of other methods of measuring neuronal electrical activity in addition to electroencephalography, such as magnetoencephalography and functional magnetic resonance. Conscious that further progress will support a deeper understanding of the temporal course of neuronal activity because of continuous interaction with the environment, we conclude confident that the fractal dimension has begun to uncover important features of the physiology of brain activity and its alterations.

Fractals, Pattern Recognition, Memetics, and AI: A Personal Journal in the Computational Neurosurgery.

In this chapter, the personal journey of the author in many countries, including Italy, Germany, Austria, the United Kingdom, Switzerland, the United States, Canada, and Australia, is summarized, aimed to merge different translational fields (such as neurosurgery and the clinical neurosciences in general, biomedical engineering, mathematics, computer science, and cognitive sciences) and lay the foundations of a new field defined computational neurosurgery, with fractals, pattern recognition, memetics, and artificial intelligence as the common key words of the journey.

Fractal Analysis in Neurodegenerative Diseases.

Neurodegenerative diseases are defined by progressive nervous system dysfunction and death of neurons. The abnormal conformation and assembly of proteins is suggested to be the most probable cause for many of these neurodegenerative disorders, leading to the accumulation of abnormally aggregated proteins, for example, amyloid ß (Aß) (Alzheimer's disease and vascular dementia), tau protein (Alzheimer's disease and frontotemporal lobar degeneration), α-synuclein (Parkinson's disease and Lewy body dementia), polyglutamine expansion diseases (Huntington disease), or prion proteins (Creutzfeldt-Jakob disease). An aberrant gain-of-function mechanism toward excessive intraparenchymal accumulation thus represents a common pathogenic denominator in all these proteinopathies. Moreover, depending upon the predominant brain area involvement, these different neurodegenerative diseases lead to either movement disorders or dementia syndromes, although the underlying mechanism(s) can sometimes be very similar, and on other occasions, clinically similar syndromes can have quite distinct pathologies. Non-Euclidean image analysis approaches such as fractal dimension (FD) analysis have been applied extensively in quantifying highly variable morphopathological patterns, as well as many other connected biological processes; however, their application to understand and link abnormal proteinaceous depositions to other clinical and pathological features composing these syndromes is yet to be clarified. Thus, this short review aims to present the most important applications of FD in investigating the clinical-pathological spectrum of neurodegenerative diseases.

Fractals and Chaos in the Hemodynamics of Intracranial Aneurysms.

Computing the emerging flow in blood vessel sections by means of computational fluid dynamics is an often applied practice in hemodynamics research. One particular area for such investigations is related to the cerebral aneurysms, since their formation, pathogenesis, and the risk of a potential rupture may be flow-related. We present a study on the behavior of small advected particles in cerebral vessel sections in the presence of aneurysmal malformations. These malformations cause strong flow disturbances driving the system toward chaotic behavior. Within these flows, the particle trajectories can form a fractal structure, the properties of which are measurable by quantitative techniques. The measurable quantities are well established chaotic properties, such as the Lyapunov exponent, escape rate, and information dimension. Based on these findings, we propose that chaotic flow within blood vessels in the vicinity of the aneurysm might be relevant for the pathogenesis and development of this malformation.

Fractals in Neuroimaging.

Several natural phenomena can be described by studying their statistical scaling patterns, hence leading to simple geometrical interpretation. In this regard, fractal geometry is a powerful tool to describe the irregular or fragmented shape of natural features, using spatial or time-domain statistical scaling laws (power-law behavior) to characterize real-world physical systems. This chapter presents some works on the usefulness of fractal features, mainly the fractal dimension and the related Hurst exponent, in the characterization and identification of pathologies and radiological features in neuroimaging, mainly, magnetic resonance imaging.

Fractal Dimension Studies of the Brain Shape in Aging and Neurodegenerative Diseases.

The fractal dimension is a morphometric measure that has been used to investigate the changes of brain shape complexity in aging and neurodegenerative diseases. This chapter reviews fractal dimension studies in aging and neurodegenerative disorders in the literature. Research has shown that the fractal dimension of the left cerebral hemisphere increases until adolescence and then decreases with aging, while the fractal dimension of the right hemisphere continues to increase until adulthood. Studies in neurodegenerative diseases demonstrated a decline in the fractal dimension of the gray matter and white matter in Alzheimer's disease, amyotrophic lateral sclerosis, and spinocerebellar ataxia. In multiple sclerosis, the white matter fractal dimension decreases, but conversely, the fractal dimension of the gray matter increases at specific stages of disease. There is also a decline in the gray matter fractal dimension in frontotemporal dementia and multiple system atrophy of the cerebellar type and in the white matter fractal dimension in epilepsy and stroke. Region-specific changes in fractal dimension have also been found in Huntington's disease and Parkinson's disease. Associations were found between the fractal dimension and clinical scores, showing the potential of the fractal dimension as a marker to monitor brain shape changes in normal or pathological processes and predict cognitive or motor function.

Computational Fractal-Based Analysis of MR Susceptibility-Weighted Imaging (SWI) in Neuro-Oncology and Neurotraumatology.

Susceptibility-weighted imaging (SWI) is a magnetic resonance imaging (MRI) technique able to depict the magnetic susceptibility produced by different substances, such as deoxyhemoglobin, calcium, and iron. The main application of SWI in clinical neuroimaging is detecting microbleedings and venous vasculature. Quantitative analyses of SWI have been developed over the last few years, aimed to offer new parameters, which could be used as neuroimaging biomarkers. Each technique has shown pros and cons, but no gold standard exists yet. The fractal dimension (FD) has been investigated as a novel potential objective parameter for monitoring intratumoral space-filling properties of SWI patterns. We showed that SWI patterns found in different tumors or different glioma grades can be represented by a gradient in the fractal dimension, thereby enabling each tumor to be assigned a specific SWI fingerprint. Such results were especially relevant in the differentiation of low-grade versus high-grade gliomas, as well as from high-grade gliomas versus lymphomas.Therefore, FD has been suggested as a potential image biomarker to analyze intrinsic neoplastic architecture in order to improve the differential diagnosis within clinical neuroimaging, determine appropriate therapy, and improve outcome in patients.These promising preliminary findings could be extended into the field of neurotraumatology, by means of the application of computational fractal-based analysis for the qualitative and quantitative imaging of microbleedings in traumatic brain injury patients. In consideration of some evidences showing that SWI signals are correlated with trauma clinical severity, FD might offer some objective prognostic biomarkers.In conclusion, fractal-based morphometrics of SWI could be further investigated to be used in a complementary way with other techniques, in order to form a holistic understanding of the temporal evolution of brain tumors and follow-up response to treatment, with several further applications in other fields, such as neurotraumatology and cerebrovascular neurosurgery as well.