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.

Emergence of fractal geometries in the evolution of a metabolic enzyme.

Fractals are patterns that are self-similar across multiple length-scales1. Macroscopic fractals are common in nature2-4; however, so far, molecular assembly into fractals is restricted to synthetic systems5-12. Here we report the discovery of a natural protein, citrate synthase from the cyanobacterium Synechococcus elongatus, which self-assembles into Sierpinski triangles. Using cryo-electron microscopy, we reveal how the fractal assembles from a hexameric building block. Although different stimuli modulate the formation of fractal complexes and these complexes can regulate the enzymatic activity of citrate synthase in vitro, the fractal may not serve a physiological function in vivo. We use ancestral sequence reconstruction to retrace how the citrate synthase fractal evolved from non-fractal precursors, and the results suggest it may have emerged as a harmless evolutionary accident. Our findings expand the space of possible protein complexes and demonstrate that intricate and regulatable assemblies can evolve in a single substitution.

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.

Evaluation of the Correlation Between the Structural Parameters of Trabecular Bone in CBCT and the Primary Stability of Dental Implants.

PURPOSE: To investigate the relationship between the structural parameters of trabecular bone obtained from CBCT imaging and the primary stability of dental implants. MATERIALS AND METHODS: Sixty patients underwent implant placement followed by primary stability evaluation via measurement of the insertion torque (IT) and the implant stability quotient (ISQ). Gray values (GV) and the fractal dimension (FD) were also measured using pretreatment CBCT images. RESULTS: FD values showed a positive and significant relationship with ISQ and IT values (P = .017 and P = .004, respectively). Additionally, there was a positive and significant correlation between GV and IT (P = .004) as well as between GV and ISQ (P = .010). FD and GV showed a considerable difference between the maxillary and mandibular jaws and were higher in the mandible. Only FD was significantly different between men and women and was higher in men. In the two age groups (older and younger than 45 years), only GV was considerably higher in people older than 45 (P < .05). CONCLUSIONS: Both fractal dimension and gray values obtained from CBCT are efficient methods for predicting the primary stability of the implant due to their relationship with ISQ and IT values.

Improving SSVEP-BCI Performance Through Repetitive Anodal tDCS-Based Neuromodulation: Insights From Fractal EEG and Brain Functional Connectivity.

This study embarks on a comprehensive investigation of the effectiveness of repetitive transcranial direct current stimulation (tDCS)-based neuromodulation in augmenting steady-state visual evoked potential (SSVEP) brain-computer interfaces (BCIs), alongside exploring pertinent electroencephalography (EEG) biomarkers for assessing brain states and evaluating tDCS efficacy. EEG data were garnered across three distinct task modes (eyes open, eyes closed, and SSVEP stimulation) and two neuromodulation patterns (sham-tDCS and anodal-tDCS). Brain arousal and brain functional connectivity were measured by extracting features of fractal EEG and information flow gain, respectively. Anodal-tDCS led to diminished offsets and enhanced information flow gains, indicating improvements in both brain arousal and brain information transmission capacity. Additionally, anodal-tDCS markedly enhanced SSVEP-BCIs performance as evidenced by increased amplitudes and accuracies, whereas sham-tDCS exhibited lesser efficacy. This study proffers invaluable insights into the application of neuromodulation methods for bolstering BCI performance, and concurrently authenticates two potent electrophysiological markers for multifaceted characterization of brain states.

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.

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.

Fundamentals of sleep regulation: Model and benchmark values for fractal and oscillatory neurodynamics.

Homeostatic, circadian and ultradian mechanisms play crucial roles in the regulation of sleep. Evidence suggests that ratios of low-to-high frequency power in the electroencephalogram (EEG) spectrum indicate the instantaneous level of sleep pressure, influenced by factors such as individual sleep-wake history, current sleep stage, age-related differences and brain topography characteristics. These effects are well captured and reflected in the spectral exponent, a composite measure of the constant low-to-high frequency ratio in the periodogram, which is scale-free and exhibits lower interindividual variability compared to slow wave activity, potentially serving as a suitable standardization and reference measure. Here we propose an index of sleep homeostasis based on the spectral exponent, reflecting the level of membrane hyperpolarization and/or network bistability in the central nervous system in humans. In addition, we advance the idea that the U-shaped overnight deceleration of oscillatory slow and fast sleep spindle frequencies marks the biological night, providing somnologists with an EEG-index of circadian sleep regulation. Evidence supporting this assertion comes from studies based on sleep replacement, forced desynchrony protocols and high-resolution analyses of sleep spindles. Finally, ultradian sleep regulatory mechanisms are indicated by the recurrent, abrupt shifts in dominant oscillatory frequencies, with spindle ranges signifying non-rapid eye movement and non-spindle oscillations - rapid eye movement phases of the sleep cycles. Reconsidering the indicators of fundamental sleep regulatory processes in the framework of the new Fractal and Oscillatory Adjustment Model (FOAM) offers an appealing opportunity to bridge the gap between the two-process model of sleep regulation and clinical somnology.

Analysis of the quality of tunnel roof topography by automatic cutting control under the coupling of multiple factors.

The automatic cutting of coal and rock surface morphology modeling based on the actual geological environment of coal mine underground excavation and mining is of great significance for improving the surface quality of coal and rock after cutting and enhancing the safety and stability of advanced support. To this end, using the principle of coordinate transformation, the kinematic trajectory of the cutting head of the tunneling machine is established, and the contour morphology of the cutting head under variable cutting technology is obtained. Then, based on the regenerative vibration theory of the cutting head, a dynamic model of the cutting head coal wall is established, and the coordinate relationship of the cutting head in the tunnel coordinate system under vibration induction is analyzed. Based on fractal theory and Z-MAP method, a simulation method for the surface morphology of coal and rock after cutting is proposed, which is driven by the cutting trajectory Under the coupling effect of cutting vibration induction and random fragmentation of coal and rock, simulation of the surface morphology of comprehensive excavation tunnels was conducted, and relevant experiments were conducted to verify the results. A 1:3 similarity experimental model of EBZ160 tunneling machine was used to build a cutting head coal and rock system cutting experimental platform for comparative experiments of cutting morphology. Furthermore, statistical methods were used to compare and evaluate the simulated roof with the actual roof. The results show that the relative errors between the maximum range of peaks and valleys, the peak skewness coefficient of height standard deviation, and the kurtosis coefficient of the actual roof are 1.3%, 24.5%, 16%, and 2.9%, respectively. Overall, this indicates that the surface morphology distribution characteristics of the simulated roof and the actual roof are similar, verifying the effectiveness of the modeling and simulation method proposed in this paper, and providing theoretical support for the design and optimization of advanced support in the future.

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.

Mathematical modeling and control of lung cancer with IL2 cytokine and anti-PD-L1 inhibitor effects for low immune individuals.

Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The aim of this work is to examine that the Lung Cancer detection and treatment by introducing IL2 and anti-PD-L1 inhibitor for low immune individuals. Mathematical model is developed with the created hypothesis to increase immune system by antibody cell's and Fractal-Fractional operator (FFO) is used to turn the model into a fractional order model. A newly developed system TCDIL2Z is examined both qualitatively and quantitatively in order to determine its stable position. The boundedness, positivity and uniqueness of the developed system are examined to ensure reliable bounded findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions are employed to identify the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of IL2 and anti-PD-L1 inhibitor for low immune individuals. Fractal fractional operator is used to derive reliable solution using Mittag-Leffler kernel. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of lung cancer with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of Lung Cancer disease to verify the relationship of IL2, anti-PD-L1 inhibitor and immune system. Also identify the real situation of the control for lung cancer disease after detection and treatment by introducing IL2 cytokine and anti-PD-L1 inhibitor which helps to generate anti-cancer cells of the patients. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies from our justified outcomes.

Fractal Analysis in Clinical Neurosciences: An Overview.

Over the last years, fractals have entered into the realms of clinical neurosciences. The whole brain and its components (i.e., neurons and astrocytes) have been studied as fractal objects, and even more relevant, the fractal-based quantification of the geometrical complexity of histopathological and neuroradiological images as well as neurophysiopathological time series has suggested the existence of a gradient in the pattern representation of neurological diseases. Computational fractal-based parameters have been suggested as potential diagnostic and prognostic biomarkers in different brain diseases, including brain tumors, neurodegeneration, epilepsy, demyelinating diseases, cerebrovascular malformations, and psychiatric disorders as well. This chapter and the entire third section of this book are focused on practical applications of computational fractal-based analysis into the clinical neurosciences, namely, neurology and neuropsychiatry, neuroradiology and neurosurgery, neuropathology, neuro-oncology and neurorehabilitation, neuro-ophthalmology, and cognitive neurosciences, with special emphasis on the translation of the fractal dimension and other fractal parameters as clinical biomarkers useful from bench to bedside.

Fractal Dimension Analysis in Neurological Disorders: An Overview.

Fractal analysis has emerged as a powerful tool for characterizing irregular and complex patterns found in the nervous system. This characterization is typically applied by estimating the fractal dimension (FD), a scalar index that describes the topological complexity of the irregular components of the nervous system, both at the macroscopic and microscopic levels, that may be viewed as geometric fractals. Moreover, temporal properties of neurophysiological signals can also be interpreted as dynamic fractals. Given its sensitivity for detecting changes in brain morphology, FD has been explored as a clinically relevant marker of brain damage in several neuropsychiatric conditions as well as in normal and pathological cerebral aging. In this sense, evidence is accumulating for decreases in FD in Alzheimer's disease, frontotemporal dementia, Parkinson's disease, multiple sclerosis, and many other neurological disorders. In addition, it is becoming increasingly clear that fractal analysis in the field of clinical neurology opens the possibility of detecting structural alterations in the early stages of the disease, which highlights FD as a potential diagnostic and prognostic tool in clinical practice.

Point of Care Testing (POCT) in Psychopathology Using Fractal Analysis and Hilbert Huang Transform of Electroencephalogram (EEG).

Research has shown that relying only on self-reports for diagnosing psychiatric disorders does not yield accurate results at all times. The advances of technology as well as artificial intelligence and other machine learning algorithms have allowed the introduction of point of care testing (POCT) including EEG characterization and correlations with possible psychopathology. Nonlinear methods of EEG analysis have significant advantages over linear methods. Empirical mode decomposition (EMD) is a reliable nonlinear method of EEG pre-processing. In this chapter, we compare two existing EEG complexity measures - Higuchi fractal dimension (HFD) and sample entropy (SE), with our newly proposed method using Higuchi fractal dimension from the Hilbert Huang transform (HFD-HHT). We present an example using the three complexity measures on a 2-minute EEG recorded from a healthy 20-year-old male after signal pre-processing. Furthermore, we showed the usefulness of these complexity measures in the classification of major depressive disorder (MDD) with healthy controls. Our study is in line with previous research and has shown an increase in HFD and SE values in the full, alpha and beta frequency bands suggestive of an increase in EEG irregularity. Moreover, the HFD-HHT values decreased in those three bands for majority of electrodes which is suggestive of a decrease in irregularity in the frequency-time domain. We conclude that all three complexity measures can be vital features useful for EEG analysis which could be incorporated in POCT systems.

Fractals in Neuroanatomy and Basic Neurosciences: An Overview.

The introduction of fractal geometry to the neurosciences has been a major paradigm shift over the last decades as it has helped overcome approximations and limitations that occur when Euclidean and reductionist approaches are used to analyze neurons or the entire brain. Fractal geometry allows for quantitative analysis and description of the geometric complexity of the brain, from its single units to the neuronal networks.As illustrated in the second section of this book, fractal analysis provides a quantitative tool for the study of the morphology of brain cells (i.e., neurons and microglia) and its components (e.g., dendritic trees, synapses), as well as the brain structure itself (cortex, functional modules, neuronal networks). The self-similar logic which generates and shapes the different hierarchical systems of the brain and even some structures related to its "container," that is, the cranial sutures on the skull, is widely discussed in the following chapters, with a link between the applications of fractal analysis to the neuroanatomy and basic neurosciences to the clinical applications discussed in the third section.

Fractal Time Series: Background, Estimation Methods, and Performances.

Over the past 40 years, from its classical application in the characterization of geometrical objects, fractal analysis has been progressively applied to study time series in several different disciplines. In neuroscience, starting from identifying the fractal properties of neuronal and brain architecture, attention has shifted to evaluating brain signals in the time domain. Classical linear methods applied to analyzing neurophysiological signals can lead to classifying irregular components as noise, with a potential loss of information. Thus, characterizing fractal properties, namely, self-similarity, scale invariance, and fractal dimension (FD), can provide relevant information on these signals in physiological and pathological conditions. Several methods have been proposed to estimate the fractal properties of these neurophysiological signals. However, the effects of signal characteristics (e.g., its stationarity) and other signal parameters, such as sampling frequency, amplitude, and noise level, have partially been tested. In this chapter, we first outline the main properties of fractals in the domain of space (fractal geometry) and time (fractal time series). Then, after providing an overview of the available methods to estimate the FD, we test them on synthetic time series (STS) with different sampling frequencies, signal amplitudes, and noise levels. Finally, we describe and discuss the performances of each method and the effect of signal parameters on the accuracy of FD estimation.

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.