Fractals in the Neurosciences, Part II

Characterization of antiseizure medications effects on the EEG neurodynamic by fractal dimension

Objectives

An important challenge in epilepsy is to define biomarkers of response to treatment. Many electroencephalography (EEG) methods and indices have been developed mainly using linear methods, e.g., spectral power and individual alpha frequency peak (IAF). However, brain activity is complex and non-linear, hence there is a need to explore EEG neurodynamics using nonlinear approaches. Here, we use the Fractal Dimension (FD), a measure of whole brain signal complexity, to measure the response to anti-seizure therapy in patients with Focal Epilepsy (FE) and compare it with linear methods.

Materials

Twenty-five drug-responder (DR) patients with focal epilepsy were studied before (t1, named DR-t1) and after (t2, named DR-t2) the introduction of the anti-seizure medications (ASMs). DR-t1 and DR-t2 EEG results were compared against 40 age-matched healthy controls (HC).

Methods

EEG data were investigated from two different angles: frequency domain-spectral properties in δ, θ, α, β, and γ bands and the IAF peak, and time-domain-FD as a signature of the nonlinear complexity of the EEG signals. Those features were compared among the three groups.

Results

The δ power differed between DR patients pre and post-ASM and HC (DR-t1 vs. HC, p < 0.01 and DR-t2 vs. HC, p < 0.01). The θ power differed between DR-t1 and DR-t2 (p = 0.015) and between DR-t1 and HC (p = 0.01). The α power, similar to the δ, differed between DR patients pre and post-ASM and HC (DR-t1 vs. HC, p < 0.01 and DR-t2 vs. HC, p < 0.01). The IAF value was lower for DR-t1 than DR-t2 (p = 0.048) and HC (p = 0.042). The FD value was lower in DR-t1 than in DR-t2 (p = 0.015) and HC (p = 0.011). Finally, Bayes Factor analysis showed that FD was 195 times more likely to separate DR-t1 from DR-t2 than IAF and 231 times than θ.

Discussion

FD measured in baseline EEG signals is a non-linear brain measure of complexity more sensitive than EEG power or IAF in detecting a response to ASMs. This likely reflects the non-oscillatory nature of neural activity, which FD better describes.

Conclusion

Our work suggests that FD is a promising measure to monitor the response to ASMs in FE.

Structural brain morphometry differences and similarities between young patients with Crohn's disease in remission and healthy young and old controls

Introduction

Crohn's disease (CD), one of the main phenotypes of inflammatory bowel disease (IBD), can affect any part of the gastrointestinal tract. It can impact the function of gastrointestinal secretions, as well as increasing the intestinal permeability leading to an aberrant immunological response and subsequent intestinal inflammation. Studies have reported anatomical and functional brain changes in Crohn's Disease patients (CDs), possibly due to increased inflammatory markers and microglial cells that play key roles in communicating between the brain, gut, and systemic immune system. To date, no studies have demonstrated similarities between morphological brain changes seen in IBD and brain morphometry observed in older healthy controls..

Methods

For the present study, twelve young CDs in remission (M = 26.08 years, SD = 4.9 years, 7 male) were recruited from an IBD Clinic. Data from 12 young age-matched healthy controls (HCs) (24.5 years, SD = 3.6 years, 8 male) and 12 older HCs (59 years, SD = 8 years, 8 male), previously collected for a different study under a similar MR protocol, were analyzed as controls. T1 weighted images and structural image processing techniques were used to extract surface-based brain measures, to test our hypothesis that young CDs have different brain surface morphometry than their age-matched young HCs and furthermore, appear more similar to older HCs. The phonemic verbal fluency (VF) task (the Controlled Oral Word Association Test, COWAT) (Benton, 1976) was administered to test verbal cognitive ability and executive control.

Results/Discussion

On the whole, CDs had more brain regions with differences in brain morphometry measures when compared to the young HCs as compared to the old HCs, suggesting that CD has an effect on the brain that makes it appear more similar to old HCs. Additionally, our study demonstrates this atypical brain morphometry is associated with function on a cognitive task. These results suggest that even younger CDs may be showing some evidence of structural brain changes that demonstrate increased resemblance to older HC brains rather than their similarly aged healthy counterparts.

White matter integrity in bipolar disorder investigated with diffusion tensor magnetic resonance imaging and fractal geometry

BACKGROUND

Growing evidence suggests the presence of white matter (WM) alterations in bipolar disorder (BD). In this study we aimed to investigate the state of WM structures, in terms of tissue integrity and morphological complexity, in BD patients compared to healthy controls (HC), in an attempt to better elucidate the microstructural changes associated with BD.

METHODS

We collected a dataset of 399 Diffusion Tensor Magnetic Resonance Imaging (167 BD and 232 healthy controls) images, acquired at five different sites, which was processed with Tract-Based Spatial Statistics (TBSS) and fractal analysis.

RESULTS

The TBSS analysis demonstrated significantly lower FA values in the BD group. Diffusion abnormalities were primarily located in the temporo-parietal network. The Fractal Dimension (FD) analysis did not reveal consistent significant differences in the morphological complexity of WM structures between the groups. When the FD values of patients were considered individually, it is possible to notice some localized significant deviations from the healthy population.

LIMITATIONS

DTI sequences have not been harmonized before acquisition, samples' sizes are heterogeneous.

CONCLUSIONS

This study, by applying both TBSS and FD analyses, allows to evaluate diffusion and structural alterations of WM at the same time. The evaluation of WM integrity from these two different perspectives could be useful to better understand the pathophysiological and morphological changes underpinning bipolar disorder.

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.

On Multiscaling of Parkinsonian Rest Tremor Signals and Their Classification

Self-similar stochastic processes and broad probability distributions are ubiquitous in nature and in many man-made systems. The brain is a particularly interesting example of (natural) complex system where those features play a pivotal role. In fact, the controversial yet experimentally validated "criticality hypothesis" explaining the functioning of the brain implies the presence of scaling laws for correlations. Recently, we have analyzed a collection of rest tremor velocity signals recorded from patients affected by Parkinson's disease, with the aim of determining and hence exploiting the presence of scaling laws. Our results show that multiple scaling laws are required in order to describe the dynamics of such signals, stressing the complexity of the underlying generating mechanism. We successively extracted numeric features by using the multifractal detrended fluctuation analysis procedure. We found that such features can be effective for discriminating classes of signals recorded in different experimental conditions. Notably, we show that the use of medication (L-DOPA) can be recognized with high accuracy.

Fractal Geometry Meets Computational Intelligence: Future Perspectives

Characterizations in terms of fractals are typically employed for systems with complex and multiscale descriptions. A prominent example of such systems is provided by the human brain, which can be idealized as a complex dynamical system made of many interacting subunits. The human brain can be modeled in terms of observable variables together with their spatio-temporal-functional relations. Computational intelligence is a research field bridging many nature-inspired computational methods, such as artificial neural networks, fuzzy systems, and evolutionary and swarm intelligence optimization techniques. Typical problems faced by means of computational intelligence methods include those of recognition, such as classification and prediction. Although historically conceived to operate in some vector space, such methods have been recently extended to the so-called nongeometric spaces, considering labeled graphs as the most general example of such patterns. Here, we suggest that fractal analysis and computational intelligence methods can be exploited together in neuroscience research. Fractal characterizations can be used to (i) assess scale-invariant properties and (ii) offer numeric, feature-based representations to complement the usually more complex pattern structures encountered in neurosciences. Computational intelligence methods could be used to exploit such fractal characterizations, considering also the possibility to perform data-driven analysis of nongeometric input spaces, therby overcoming the intrinsic limits related to Euclidean geometry.

Analyzing Eye Paths Using Fractals

Visual patterns reflect the anatomical and cognitive background underlying process governing how we perceive information, influenced by stimulus characteristics and our own visual perception. These patterns are both spatially complex and display self-similarity seen in fractal geometry at different scales, making them challenging to measure using the traditional topological dimensions used in Euclidean geometry.However, methods for measuring eye gaze patterns using fractals have shown success in quantifying geometric complexity, matchability, and implementation into machine learning methods. This success is due to the inherent capabilities that fractals possess when reducing dimensionality using Hilbert curves, measuring temporal complexity using the Higuchi fractal dimension (HFD), and determining geometric complexity using the Minkowski-Bouligand dimension.Understanding the many applications of fractals when measuring and analyzing eye gaze patterns can extend the current growing body of knowledge by identifying markers tied to neurological pathology. Additionally, in future work, fractals can facilitate defining imaging modalities in eye tracking diagnostics by exploiting their capability to acquire multiscale information, including complementary functions, structures, and dynamics.

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.

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.

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 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.

Fractal Analysis in MATLAB: A Tutorial for Neuroscientists

MATLAB is one of the software platforms most widely used for scientific computation. MATLAB includes a large set of functions, packages, and toolboxes that make it simple and fast to obtain complex mathematical and statistical computations for many applications. In this chapter, we review some tools available in MATLAB for performing fractal analyses on typical neuroscientific data in a practical way. We provide detailed examples of how to calculate the fractal dimension of 1D, 2D, and 3D data in MATLAB. Furthermore, we review other software packages for fractal analysis.

The Fractal Geometry of the Brain: AnOverview

The first chapter of this book introduces some history, philosophy, and basic concepts of fractal geometry and discusses how the neurosciences can benefit from applying computational fractal-based analysis. Further, it compares fractal with Euclidean approaches to analyzing and quantifying the brain in its entire physiopathological spectrum and presents an overview of the first section of this book as well.

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 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 Phototherapy in Maximizing Retina and Brain Plasticity

The neuroplasticity potential is reduced with aging and impairs during neurodegenerative diseases and brain and visual system injuries. This limits the brain's capacity to repair the structure and dynamics of its activity after lesions. Maximization of neuroplasticity is necessary to provide the maximal CNS response to therapeutic intervention and adaptive reorganization of neuronal networks in patients with degenerative pathology and traumatic injury to restore the functional activity of the brain and retina.Considering the fractal geometry and dynamics of the healthy brain and the loss of fractality in neurodegenerative pathology, we suggest that the application of self-similar visual signals with a fractal temporal structure in the stimulation therapy can reactivate the adaptive neuroplasticity and enhance the effectiveness of neurorehabilitation. This proposition was tested in the recent studies. Patients with glaucoma had a statistically significant positive effect of fractal photic therapy on light sensitivity and the perimetric MD index, which shows that methods of fractal stimulation can be a novel nonpharmacological approach to neuroprotective therapy and neurorehabilitation. In healthy rabbits, it was demonstrated that a long-term course of photostimulation with fractal signals does not harm the electroretinogram (ERG) and retina structure. Rabbits with modeled retinal atrophy showed better dynamics of the ERG restoration during daily stimulation therapy for a week in comparison with the controls. Positive changes in the retinal function can indirectly suggest the activation of its adaptive plasticity and the high potential of stimulation therapy with fractal visual stimuli in a nonpharmacological neurorehabilitation, which requires further study.

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.

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.

Fractal-Based Analysis of Arteriovenous Malformations (AVMs)

Arteriovenous malformations (AVMs) are cerebrovascular lesions consisting of a pathologic tangle of the vessels characterized by a core termed the nidus, which is the "nest" where the fistulous connections occur. AVMs can cause headache, stroke, and/or seizures. Their treatment can be challenging requiring surgery, endovascular embolization, and/or radiosurgery as well. AVMs' morphology varies greatly among patients, and there is still a lack of standardization of angioarchitectural parameters, which can be used as morphometric parameters as well as potential clinical biomarkers (e.g., related to prognosis).In search of new diagnostic and prognostic neuroimaging biomarkers of AVMs, computational fractal-based models have been proposed for describing and quantifying the angioarchitecture of the nidus. In fact, the fractal dimension (FD) can be used to quantify AVMs' branching pattern. Higher FD values are related to AVMs characterized by an increased number and tortuosity of the intranidal vessels or to an increasing angioarchitectural complexity as a whole. Moreover, FD has been investigated in relation to the outcome after Gamma Knife radiosurgery, and an inverse relationship between FD and AVM obliteration was found.Taken altogether, FD is able to quantify in a single and objective value what neuroradiologists describe in qualitative and/or semiquantitative way, thus confirming FD as a reliable morphometric neuroimaging biomarker of AVMs and as a potential surrogate imaging biomarker. Moreover, computational fractal-based techniques are under investigation for the automatic segmentation and extraction of the edges of the nidus in neuroimaging, which can be relevant for surgery and/or radiosurgery planning.

Entropy and fractal analysis of brain-related neurophysiological signals in Alzheimer's and Parkinson's disease

Brain-related neuronal recordings, such as local field potential, electroencephalogram and magnetoencephalogram, offer the opportunity to study the complexity of the human brain at different spatial and temporal scales. The complex properties of neuronal signals are intrinsically related to the concept of 'scale-free' behavior and irregular dynamic, which cannot be fully described through standard linear methods, but can be measured by nonlinear indexes. A remarkable application of these analysis methods on electrophysiological recordings is the deep comprehension of the pathophysiology of neurodegenerative diseases, that has been shown to be associated to changes in brain activity complexity. In particular, a decrease of global complexity has been associated to Alzheimer's disease, while a local increase of brain signals complexity characterizes Parkinson's disease. Despite the recent proliferation of studies using fractal and entropy-based analysis, the application of these techniques is still far from clinical practice, due to the lack of an agreement about their correct estimation and a conclusive and shared interpretation. Along with the aim of helping towards the realization of a multidisciplinary audience to approach nonlinear methods based on the concepts of fractality and irregularity, this survey describes the implementation and proper employment of the mostly known and applied indexes in the context of Alzheimer's and Parkinson's diseases.

Examining electroencephalogram signatures of people with multiple sclerosis using a nonlinear dynamics approach: a systematic review and bibliographic analysis

Background

Considering that brain activity involves communication between millions of neurons in a complex network, nonlinear analysis is a viable tool for studying electroencephalography (EEG). The main objective of this review was to collate studies that utilized chaotic measures and nonlinear dynamical analysis in EEG of multiple sclerosis (MS) patients and to discuss the contributions of chaos theory techniques to understanding, diagnosing, and treating MS.

Methods

Using the preferred reporting items for systematic reviews and meta-analysis (PRISMA), the databases EbscoHost, IEEE, ProQuest, PubMed, Science Direct, Web of Science, and Google Scholar were searched for publications that applied chaos theory in EEG analysis of MS patients.

Results

A bibliographic analysis was performed using VOSviewer software keyword co-occurrence analysis indicated that MS was the focus of the research and that research on MS diagnosis has shifted from conventional methods, such as magnetic resonance imaging, to EEG techniques in recent years. A total of 17 studies were included in this review. Among the included articles, nine studies examined resting-state, and eight examined task-based conditions.

Conclusion

Although nonlinear EEG analysis of MS is a relatively novel area of research, the findings have been demonstrated to be informative and effective. The most frequently used nonlinear dynamics analyses were fractal dimension, recurrence quantification analysis, mutual information, and coherence. Each analysis selected provided a unique assessment to fulfill the objective of this review. While considering the limitations discussed, there is a promising path forward using nonlinear analyses with MS data.

Physics-informed neural entangled-ladder network for inhalation impedance of the respiratory system

BACKGROUND AND OBJECTIVES

The use of machine learning methods for modelling bio-systems is becoming prominent which can further improve bio-medical technologies. Physics-informed neural networks (PINNs) can embed the knowledge of physical laws that govern a system during the model training process. PINNs utilise differential equations in the model which traditionally used numerical methods that are computationally complex.

METHODS

We integrate PINNs with an entangled ladder network for modelling respiratory systems by considering a lungs conduction zone to evaluate the respiratory impedance for different initial conditions. We evaluate the respiratory impedance for the inhalation phase of breathing for a symmetric model of the human lungs using entanglement and continued fractions.

RESULTS

We obtain the impedance of the conduction zone of the lungs pulmonary airways using PINNs for nine different combinations of velocity and pressure of inhalation. We compare the results from PINNs with the finite element method using the mean absolute error and root mean square error. The results show that the impedance obtained with PINNs contrasts with the conventional forced oscillation test used for deducing the respiratory impedance. The results show similarity with the impedance plots for different respiratory diseases.

CONCLUSION

We find a decrease in impedance when the velocity of breathing is lowered gradually by 20%. Hence, the methodology can be used to design smart ventilators to the improve flow of breathing.

The Fractal Viewpoint of Tumors and Nanoparticles

Even though the promising therapies against cancer are rapidly improved, the oncology patients population has seen exponential growth, placing cancer in 5th place among the ten deadliest diseases. Efficient drug delivery systems must overcome multiple barriers and maximize drug delivery to the target tumors, simultaneously limiting side effects. Since the first observation of the quantum tunneling phenomenon, many multidisciplinary studies have offered quantum-inspired solutions to optimized tumor mapping and efficient nanodrug design. The property of a wave function to propagate through a potential barrier offer the capability of obtaining 3D surface profiles using imaging of individual atoms on the surface of a material. The application of quantum tunneling on a scanning tunneling microscope offers an exact surface roughness mapping of tumors and pharmaceutical particles. Critical elements to cancer nanotherapeutics apply the fractal theory and calculate the fractal dimension for efficient tumor surface imaging at the atomic level. This review study presents the latest biological approaches to cancer management based on fractal geometry.

Hierarchical multi-class Alzheimer’s disease diagnostic framework using imaging and clinical features

Due to the clinical continuum of Alzheimer’s disease (AD), the accuracy of early diagnostic remains unsatisfactory and warrants further research. The objectives of this study were: (1) to develop an effective hierarchical multi-class framework for clinical populations, namely, normal cognition (NC), early mild cognitive impairment (EMCI), late mild cognitive impairment (LMCI), and AD, and (2) to explore the geometric properties of cognition-related anatomical structures in the cerebral cortex. A total of 1,670 participants were enrolled in the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database, comprising 985 participants (314 NC, 208 EMCI, 258 LMCI, and 205 AD) in the model development set and 685 participants (417 NC, 110 EMCI, 83 LMCI, and 75 AD) after 2017 in the temporal validation set. Four cortical geometric properties for 148 anatomical structures were extracted, namely, cortical thickness (CTh), fractal dimension (FD), gyrification index (GI), and sulcus depth (SD). By integrating these imaging features with Mini-Mental State Examination (MMSE) scores at four-time points after the initial visit, we identified an optimal subset of 40 imaging features using the temporally constrained group sparse learning method. The combination of selected imaging features and clinical variables improved the multi-class performance using the AdaBoost algorithm, with overall accuracy rates of 0.877 in the temporal validation set. Clinical Dementia Rating (CDR) was the primary clinical variable associated with AD-related populations. The most discriminative imaging features included the bilateral CTh of the dorsal part of the posterior cingulate gyrus, parahippocampal gyrus (PHG), parahippocampal part of the medial occipito-temporal gyrus, and angular gyrus, the GI of the left inferior segment of the insula circular sulcus, and the CTh and SD of the left superior temporal sulcus (STS). Our hierarchical multi-class framework underscores the utility of combining cognitive variables with imaging features and the reliability of surface-based morphometry, facilitating more accurate early diagnosis of AD in clinical practice.

Sexual dimorphism in the relationship between brain complexity, volume and general intelligence (g): a cross-cohort study

Changes in brain morphology have been reported during development, ageing and in relation to different pathologies. Brain morphology described by the shape complexity of gyri and sulci can be captured and quantified using fractal dimension (FD). This measure of brain structural complexity, as well as brain volume, are associated with intelligence, but less is known about the sexual dimorphism of these relationships. In this paper, sex differences in the relationship between brain structural complexity and general intelligence (g) in two diverse geographic and cultural populations (UK and Indian) are investigated. 3D T1-weighted magnetic resonance imaging (MRI) data and a battery of cognitive tests were acquired from participants belonging to three different cohorts: Mysore Parthenon Cohort (MPC); Aberdeen Children of the 1950s (ACONF) and UK Biobank. We computed MRI derived structural brain complexity and g estimated from a battery of cognitive tests for each group. Brain complexity and volume were both positively corelated with intelligence, with the correlations being significant in women but not always in men. This relationship is seen across populations of differing ages and geographical locations and improves understanding of neurobiological sex-differences.

Emergence of lump-like solitonic waves in Heimburg-Jackson biomembranes and nerves fractal model

The aim of this study is to extend the soliton propagation model in biomembranes and nerves constructed by Heimburg and Jackson for the case of fractal dimensions. Our analyses are based on the product-like fractal measure concept introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media and electromagnetic fields. The mathematical model presented in the paper is formulated for only a part of a single nerve cell (an axon). The analytical and numerical envelop soliton of this equation are reported. The results obtained prove the emergence of lump-type solitonic waves in nerves and biomembranes. In particular, these waves decay algebraically to the background wave in space direction. This scenario is viewed as a particular class of rational localized waves which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev-Petviashvili I equation. The effects of fractal dimensions are discussed and were found to be significant to some extents.

Cortical Complexity Estimation Using Fractal Dimension: A Systematic Review of the Literature on Clinical and Nonclinical Samples

Fractal geometry has recently been proposed as a useful tool for characterizing the complexity of the brain cortex, which is likely to derive from the recurrence of sulci–gyri convolution patterns. The index used to describe the cortical complexity is called fractal dimensional (FD) and was employed by different research exploring the neurobiological correlates of distinct pathological and nonpathological conditions. This review aims to describe the literature on the application of this index, summarize the heterogeneities between studies and inform future research on this topic. Sixty‐two studies were included in the systematic review. The main research lines concern neurodevelopment, aging and the neurobiology of specific psychiatric and neurological disorders. Overall, the included papers indicate that cortical complexity is likely to reduce during aging and in various pathological processes affecting the brain. Nevertheless, the high heterogeneity between studies strongly prevents the possibility of drawing conclusions. Further research considering this index besides other morphological values is needed to better clarify the role of FD in characterizing the cortical structure.

Fractal dimension (FD) is a useful tool for the assessment of cortical complexity. In healthy controls, FD is associated with development, aging and cognition. Alterations in FD have been observed in different neurological and psychiatric disorders.

Region-specific complexity of the intracranial EEG in the sleeping human brain

As the brain is a complex system with occurrence of self-similarity at different levels, a dedicated analysis of the complexity of brain signals is of interest to elucidate the functional role of various brain regions across the various stages of vigilance. We exploited intracranial electroencephalogram data from 38 cortical regions using the Higuchi fractal dimension (HFD) as measure to assess brain complexity, on a dataset of 1772 electrode locations. HFD values depended on sleep stage and topography. HFD increased with higher levels of vigilance, being highest during wakefulness in the frontal lobe. HFD did not change from wake to stage N2 in temporo-occipital regions. The transverse temporal gyrus was the only area in which the HFD did not differ between any two vigilance stages. Interestingly, HFD of wakefulness and stage R were different mainly in the precentral gyrus, possibly reflecting motor inhibition in stage R. The fusiform and parahippocampal gyri were the only areas showing no difference between wakefulness and N2. Stages R and N2 were similar only for the postcentral gyrus. Topographical analysis of brain complexity revealed that sleep stages are clearly differentiated in fronto-central brain regions, but that temporo-occipital regions sleep differently.

BrainFD: Measuring the Intracranial Brain Volume With Fractal Dimension

A few methods and tools are available for the quantitative measurement of the brain volume targeting mainly brain volume loss. However, several factors, such as the clinical conditions, the time of the day, the type of MRI machine, the brain volume artifacts, the pseudoatrophy, and the variations among the protocols, produce extreme variations leading to misdiagnosis of brain atrophy. While brain white matter loss is a characteristic lesion during neurodegeneration, the main objective of this study was to create a computational tool for high precision measuring structural brain changes using the fractal dimension (FD) definition. The validation of the BrainFD software is based on T1-weighted MRI images from the Open Access Series of Imaging Studies (OASIS)-3 brain database, where each participant has multiple MRI scan sessions. The software is based on the Python and JAVA programming languages with the main functionality of the FD calculation using the box-counting algorithm, for different subjects on the same brain regions, with high accuracy and resolution, offering the ability to compare brain data regions from different subjects and on multiple sessions, creating different imaging profiles based on the Clinical Dementia Rating (CDR) scores of the participants. Two experiments were executed. The first was a cross-sectional study where the data were separated into two CDR classes. In the second experiment, a model on multiple heterogeneous data was trained, and the FD calculation for each participant of the OASIS-3 database through multiple sessions was evaluated. The results suggest that the FD variation efficiently describes the structural complexity of the brain and the related cognitive decline. Additionally, the FD efficiently discriminates the two classes achieving 100% accuracy. It is shown that this classification outperforms the currently existing methods in terms of accuracy and the size of the dataset. Therefore, the FD calculation for identifying intracranial brain volume loss could be applied as a potential low-cost personalized imaging biomarker. Furthermore, the possibilities measuring different brain areas and subregions could give robust evidence of the slightest variations to imaging data obtained from repetitive measurements to Physicians and Radiologists.

Foundations of Multiparametric Brain Tumour Imaging Characterisation Using Machine Learning

The heterogeneity of brain tumours at the molecular, metabolic and structural levels poses significant challenge for accurate tissue characterisation. Artificial intelligence and radiomics have emerged as valuable tools to analyse quantitative features extracted from medical images which capture the complex microenvironment of brain tumours. In particular, a number of computational tools including machine learning algorithms have been proposed for image preprocessing, tumour segmentation, feature extraction, classification, and prognostic stratifications as well. In this chapter, we explore the fundamentals of multiparametric brain tumour characterisation, as an understanding of the strengths, limitations and applications of these tools allows clinicians to better develop and evaluate models with improved diagnostic and prognostic value in brain tumour patients.

Spatial and time domain analysis of eye-tracking data during screening of brain magnetic resonance images

INTRODUCTION

Eye-tracking research has been widely used in radiology applications. Prior studies exclusively analysed either temporal or spatial eye-tracking features, both of which alone do not completely characterise the spatiotemporal dynamics of radiologists' gaze features.

PURPOSE

Our research aims to quantify human visual search dynamics in both domains during brain stimuli screening to explore the relationship between reader characteristics and stimuli complexity. The methodology can be used to discover strategies to aid trainee radiologists in identifying pathology, and to select regions of interest for machine vision applications.

METHOD

The study was performed using eye-tracking data 5 seconds in duration from 57 readers (15 Brain-experts, 11 Other-experts, 5 Registrars and 26 Naïves) for 40 neuroradiological images as stimuli (i.e., 20 normal and 20 pathological brain MRIs). The visual scanning patterns were analysed by calculating the fractal dimension (FD) and Hurst exponent (HE) using re-scaled range (R/S) and detrended fluctuation analysis (DFA) methods. The FD was used to measure the spatial geometrical complexity of the gaze patterns, and the HE analysis was used to measure participants' focusing skill. The focusing skill is referred to persistence/anti-persistence of the participants' gaze on the stimulus over time. Pathological and normal stimuli were analysed separately both at the "First Second" and full "Five Seconds" viewing duration.

RESULTS

All experts were more focused and a had higher visual search complexity compared to Registrars and Naïves. This was seen in both the pathological and normal stimuli in the first and five second analyses. The Brain-experts subgroup was shown to achieve better focusing skill than Other-experts due to their domain specific expertise. Indeed, the FDs found when viewing pathological stimuli were higher than those in normal ones. Viewing normal stimuli resulted in an increase of FD found in five second data, unlike pathological stimuli, which did not change. In contrast to the FDs, the scanpath HEs of pathological and normal stimuli were similar. However, participants' gaze was more focused for "Five Seconds" than "First Second" data.

CONCLUSIONS

The HE analysis of the scanpaths belonging to all experts showed that they have greater focus than Registrars and Naïves. This may be related to their higher visual search complexity than non-experts due to their training and expertise.

Spherical coordinates transformation pre-processing in Deep Convolution Neural Networks for brain tumor segmentation in MRI

Magnetic Resonance Imaging (MRI) is used in everyday clinical practice to assess brain tumors. Deep Convolutional Neural Networks (DCNN) have recently shown very promising results in brain tumor segmentation tasks; however, DCNN models fail the task when applied to volumes that are different from the training dataset. One of the reasons is due to the lack of data standardization to adjust for different models and MR machines. In this work, a 3D spherical coordinates transform during the pre-processing phase has been hypothesized to improve DCNN models' accuracy and to allow more generalizable results even when the model is trained on small and heterogeneous datasets and translated into different domains. Indeed, the spherical coordinate system avoids several standardization issues since it works independently of resolution and imaging settings. The model trained on spherical transform pre-processed inputs resulted in superior performance over the Cartesian-input trained model on predicting gliomas' segmentation on Tumor Core and Enhancing Tumor classes, achieving a further improvement in accuracy by merging the two models together. The proposed model is not resolution-dependent, thus improving segmentation accuracy and theoretically solving some transfer learning problems related to the domain shifting, at least in terms of image resolution in the datasets.

Investigating sexual dimorphism in human brain structure by combining multiple indexes of brain morphology and source-based morphometry

Computational morphometry of magnetic resonance images represents a powerful tool for studying macroscopic differences in human brains. In the present study (N participants = 829), we combined different techniques and measures of brain morphology to investigate one of the most compelling topics in neuroscience: sexual dimorphism in human brain structure. When accounting for overall larger male brains, results showed limited sex differences in gray matter volume (GMV) and surface area. On the other hand, we found larger differences in cortical thickness, favoring both males and females, arguably as a result of region-specific differences. We also observed higher values of fractal dimension, a measure of cortical complexity, for males versus females across the four lobes. In addition, we applied source-based morphometry, an alternative method for measuring GMV based on the independent component analysis. Analyses on independent components revealed higher GMV in fronto-parietal regions, thalamus and caudate nucleus for females, and in cerebellar- temporal cortices and putamen for males, a pattern that is largely consistent with previous findings.

Differential Effects of a Left Frontal Glioma on the Cortical Thickness and Complexity of Both Hemispheres

Glioma is a type of brain tumor that infiltrates and compresses the brain as it grows. Focal gliomas affect functional connectivity both in the local region of the lesion and the global network of the brain. Any anatomical changes associated with a glioma should thus be clarified. We examined the cortical structures of 15 patients with a glioma in the left lateral frontal cortex and compared them with those of 15 healthy controls by surface-based morphometry. Two regional parameters were measured with 3D-MRI: the cortical thickness (CT) and cortical fractal dimension (FD). The FD serves as an index of the topological complexity of a local cortical surface. Our comparative analyses of these parameters revealed that the left frontal gliomas had global effects on the cortical structures of both hemispheres. The structural changes in the right hemisphere were mainly characterized by a decrease in CT and mild concomitant decrease in FD, whereas those in the peripheral regions of the glioma (left hemisphere) were mainly characterized by a decrease in FD with relative preservation of CT. These differences were found irrespective of tumor volume, location, or grade. These results elucidate the structural effects of gliomas, which extend to the distant contralateral regions.

Assessment of eye-tracking scanpath outliers using fractal geometry

Outlier scanpaths identification is a crucial preliminary step in designing visual software, digital media analysis, radiology training and clustering participants in eye-tracking experiments. However, the task is challenging due to the visual irregularity of the scanpath shapes and the difficulty in dimensionality reduction due to geometric complexity. Conventional approaches have used heat maps to exclude scanpaths that lack a similarity pattern. However, the typically-used packages, such as ScanMatch and MultiMatch often generate discordant results when outlier identification is done empirically. This paper introduces a novel outlier evaluation approach by integrating the fractal dimension (FD), capturing the geometrical complexity of patterns, as an additional parameter with the heat map. This additional parameter is used to evaluate the degree of influence of a scanpath within a dataset. More specifically, the 2D Cartesian coordinates of a scanpath are fitted to a space filling 1D fractal curve to characterise its temporal FD. The FDs of the scanpaths are then compared to match their geometric complexity to one another. The findings indicate that the FD can be a beneficial additional parameter when evaluating the candidacy of poorly matching scanpaths as outliers and performs better at identifying unusual scanpaths than using other methods, including scanpath matching, Jaccard, or bounding box methods alone.

Novel Volumetric and Surface-Based Magnetic Resonance Indices of the Aging Brain - Does Male and Female Brain Age in the Same Way?

Introduction: Novel post-processing methods allow not only for assessment of brain volumetry or cortical thickness based on magnetic resonance imaging (MRI) but also for more detailed analysis of cortical shape and complexity using parameters such as sulcal depth, gyrification index, or fractal dimension. The aim of this study was to analyze changes in brain volumetry and other cortical indices during aging in men and women.

Material and Methods: Material consisted of 697 healthy volunteers (aged 38–80 years; M/F, 264/443) who underwent brain MRI using a 1.5-T scanner. Voxel-based volumetry of total gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF) was performed followed by assessment of cortical parameters [cortical thickness (CT), sulcal depth (SD), gyrification index (GI), and fractal dimension (FD)] in 150 atlas locations using surface-based morphometry with a region-based approach. All parameters were compared among seven age groups (grouped every 5 years) separately for men and women. Additionally, percentile curves for men and women were provided for total volumes of GM, WM, and CSF.

Results: In men and women, a decrease in GM and WM volumes and an increase in CSF volume seem to progress slowly since the age of 45. In men, significant GM and WM loss as well as CSF increase start above 55 years of age, while in women, significant GM loss starts above 50 and significant WM loss as well as CSF increase above 60. CT was found to significantly decrease with aging in 39% of locations in women and in 36% of locations in men, SD was found to increase in 13.5% of locations in women and in 1.3% of locations in men, GI was decreased in 3.4% of locations in women and in 2.0% of locations in men, and FD was changed in 2.7% of locations in women compared to 2.0% in men.

Conclusions: Male and female brains start aging at the similar age of 45. Compared to men, in women, the cortex is affected earlier and in the more complex pattern regarding not only cortical loss but also other alterations within the cortical shape, with relatively longer sparing of WM volume.

Brain volumetric and fractal analysis of synthetic MRI: A comparative study with conventional 3D T1-weighted images

PURPOSE

The estimation of brain volumetric measurements based on Synthetic MRI (SyMRI) is easy and fast, however, the consistency of brain volumetric and morphologic measurements based on SyMRI and 3D T1WI should be further addressed. The current study evaluated the impact of spatial resolution on brain volumetric and morphologic measurements using SyMRI, and test whether the brain measurements derived from SyMRI were consistent with those resulted from 3D T1WI.

METHOD

Brain volumetric and fractal analysis were applied to thirty healthy subjects, each underwent four SyMRI acquisitions with different spatial resolutions (1 × 1 × 2 mm, 1 × 1x3mm, 1 × 1 × 4 mm, 2 × 2 × 2 mm) and a 3D T1WI (1 × 1 × 1 mm isotropic). The consistency of the SyMRI measurements was tested using one-way non-parametric Kruskal-Wallis test and post hoc Dwass-Steel-Critchlow-Fligner test. The association between SyMRI and 3D T1WI derived measurements was evaluated using linear regression models.

RESULTS

Our results demonstrated that both in- and through-plane resolutions show an impact on brain volumetric measurements, while brain parenchymal volume showed high consistency across the SyMRI acquisitions, and high association with the measurements from 3D T1WI. In addition, SyMRI with 1 × 1 × 4 mm resolution showed the strongest association with 3D T1WI compared to other SyMRI acquisitions in both volumetric and fractal analyses. Moreover, substantial differences were found in fractal dimension of both gray and white matter between the SyMRI and 3D T1WI tissue segmentations.

CONCLUSIONS

Our results suggested that the measurements from SyMRI with relatively higher in-plane and lower through-plane resolution (1 × 1 × 4 mm) are much closer to 3D T1WI.

Cortical fractal dimension predicts disability worsening in Multiple Sclerosis patients

BACKGROUND

Fractal geometry measures the morphology of the brain and detects CNS damage. We aimed to assess the longitudinal changes on brain's fractal geometry and its predictive value for disease worsening in patients with Multiple Sclerosis (MS).

METHODS

We prospectively analyzed 146 consecutive patients with relapsing-remitting MS with up to 5 years of clinical and brain MRI (3 T) assessments. The fractal dimension and lacunarity were calculated for brain regions using box-counting methods. Longitudinal changes were analyzed in mixed-effect models and the risk of disability accumulation were assessed using Cox Proportional Hazard regression analysis.

RESULTS

There was a significant decrease in the fractal dimension and increases of lacunarity in different brain regions over the 5-year follow-up. Lower cortical fractal dimension increased the risk of disability accumulation for the Expanded Disability Status Scale [HR 0.9734, CI 0.8420-0.9125; Harrell C 0.59; Wald p 0.038], 9-hole peg test [HR 0.9734, CI 0.8420-0.9125; Harrell C 0.59; Wald p 0.0083], 2.5% low contrast vision [HR 0.4311, CI 0.2035-0.9133; Harrell C 0.58; Wald p 0.0403], symbol digit modality test [HR 2.215, CI 1.043-4.705; Harrell C 0.65; Wald p 0.0384] and MS Functional Composite-4 [HR 0.55, CI 0.317-0.955; Harrell C 0.59; Wald p 0.0029].

CONCLUSIONS

Fractal geometry analysis of brain MRI identified patients at risk of increasing their disability in the next five years.

Surface-based morphometry study of the brain in benign childhood epilepsy with centrotemporal spikes

Background

The study aimed to explore cortical morphology in benign childhood epilepsy with centrotemporal spikes (BECTS) and the relationship between cortical characteristics and age of onset and intelligence quotient (IQ).

Methods

Cortical morphometry with surface-based morphometry (SBM) was used to compare changes in cortical thickness, gyrification, sulcal depth, and fractal dimension of the cerebral cortex between 25 BECTS patients and 20 healthy controls (HCs) with two-sample t-tests [P<0.05, family-wise error (FWE) corrected]. Relationships between abnormal cortical morphological changes and age of onset and IQ, which included verbal intelligence quotient (VIQ), performance intelligence quotient (PIQ), and full-scale intelligence quotient (FIQ) were investigated with Spearman correlation analysis (P<0.05, uncorrected).

Results

The BECTS patients showed extensive cortical thinning predominantly in bilateral frontal, temporal regions, and limbic system. Cortical gyrification increased in the left hemisphere and partial right hemisphere, and the decreased cortical gyrification was only in the left hemisphere. The increased sulcal depth was the left fusiform gyrus. There are no statistically significant differences in the fractal dimension. Correlation analysis revealed the negative correlation between age of onset and cortical thickness in the right precentral gyrus. It also revealed the negative correlation between the age of onset and cortical gyrification in the left inferior parietal gyrus. Also, there was negative correlation between VIQ and cortical gyrification in the left supramarginal gyrus of BECTS patients.

Conclusions

This study reveals aberrant cortical thickness, cortical gyrification, and sulcal depth of BECTS in areas related to cognitive functions including language, attention and memory, and the correlation between some brain regions and VIQ and age of onset, providing a potential marker of early neurodevelopmental disturbance and cognitive dysfunction in BECTS.

Toward a more reliable characterization of fractal properties of the cerebral cortex of healthy subjects during the lifespan

The cerebral cortex manifests an inherent structural complexity of folding. The fractal geometry describes the complexity of structures which show self-similarity in a proper interval of spatial scales. In this study, we aimed at evaluating in-vivo the effect of different criteria for selecting the interval of spatial scales in the estimation of the fractal dimension (FD) of the cerebral cortex in T₁-weighted magnetic resonance imaging (MRI). We compared four different strategies, including two a priori selections of the interval of spatial scales, an automated selection of the spatial scales within which the cerebral cortex manifests the highest statistical self-similarity, and an improved approach, based on the search of the interval of spatial scales which presents the highest rounded R²_adj coefficient and, in case of equal rounded R²_adj coefficient, preferring the widest interval in the log–log plot. We employed two public and international datasets of in-vivo MRI scans for a total of 159 healthy subjects (age range 6–85 years). The improved approach showed strong associations of FD with age and yielded the most accurate machine learning models for individual age prediction in both datasets. Our results indicate that the selection of the interval of spatial scales of the cerebral cortex is thus critical in the estimation of FD.

Von Interaktionen zu Interaktionsnetzwerken: Zeitabhängige funktionelle Netzwerke am Beispiel der Epilepsie

Das menschliche Gehirn ist ein komplexes Netzwerk aus interagierenden nichtstationären Subsystemen (Netzwerk von Netzwerken), deren komplizierte räumlich-zeitliche Dynamiken bis heute nur unzureichend verstanden sind. Dabei versprechen aktuelle Entwicklungen im Bereich der Zeitreihenanalyse sowie der Theorie komplexer Netzwerke neue und verbesserte Einblicke in die Dynamiken von Hirnnetzwerken auf verschiedenen räumlich-zeitlichen Skalen. Wir geben einen Überblick über diese Entwicklungen und besprechen am Beispiel zeitabhängiger epileptischer Hirnnetzwerke Fortschritte im Verständnis von Hirndynamiken, die über multiple Skalen hinweg variieren.

Haemodynamic activity characterization of resting state networks by fractal analysis and thalamocortical morphofunctional integrity in chronic migraine

BACKGROUND

Chronic migraine (CM) can be associated with aberrant long-range connectivity of MRI-derived resting-state networks (RSNs). Here, we investigated how the fractal dimension (FD) of blood oxygenation level dependent (BOLD) activity may be used to estimate the complexity of RSNs, reflecting flexibility and/or efficiency in information processing in CM patients respect to healthy controls (HC).

METHODS

Resting-state MRI data were collected from 20 untreated CM without history of medication overuse and 20 HC. On both groups, we estimated the Higuchi's FD. On the same subjects, fractional anisotropy (FA) and mean diffusivity (MD) values of bilateral thalami were retrieved from diffusion tensor imaging and correlated with the FD values.

RESULTS

CM showed higher FD values within dorsal attention system (DAS) and the anterior part of default-mode network (DMN), and lower FD values within the posterior DMN compared to HC. Although FA and MD were within the range of normality, both correlated with the FD values of DAS.

CONCLUSIONS

FD of DAS and DMN may reflect disruption of cognitive control of pain in CM. Since the normal microstructure of the thalamus and its positive connectivity with the cortical networking found in our CM patients reminds similar results obtained assessing the same structures but with the methods of neurophysiology, in episodic migraine during an attack, this may be yet another evidence in supporting CM as a never-ending migraine attack.

Serotonergic psychedelics LSD & psilocybin increase the fractal dimension of cortical brain activity in spatial and temporal domains

Psychedelic drugs, such as psilocybin and LSD, represent unique tools for researchers investigating the neural origins of consciousness. Currently, the most compelling theories of how psychedelics exert their effects is by increasing the complexity of brain activity and moving the system towards a critical point between order and disorder, creating more dynamic and complex patterns of neural activity. While the concept of criticality is of central importance to this theory, few of the published studies on psychedelics investigate it directly, testing instead related measures such as algorithmic complexity or Shannon entropy. We propose using the fractal dimension of functional activity in the brain as a measure of complexity since findings from physics suggest that as a system organizes towards criticality, it tends to take on a fractal structure. We tested two different measures of fractal dimension, one spatial and one temporal, using fMRI data from volunteers under the influence of both LSD and psilocybin. The first was the fractal dimension of cortical functional connectivity networks and the second was the fractal dimension of BOLD time-series. In addition to the fractal measures, we used a well-established, non-fractal measure of signal complexity and show that they behave similarly. We were able to show that both psychedelic drugs significantly increased the fractal dimension of functional connectivity networks, and that LSD significantly increased the fractal dimension of BOLD signals, with psilocybin showing a non-significant trend in the same direction. With both LSD and psilocybin, we were able to localize changes in the fractal dimension of BOLD signals to brain areas assigned to the dorsal-attenion network. These results show that psychedelic drugs increase the fractal dimension of activity in the brain and we see this as an indicator that the changes in consciousness triggered by psychedelics are associated with evolution towards a critical zone.

Cortical thickness, local gyrification index and fractal dimensionality in people with acute and recovered Anorexia Nervosa and in people with Bulimia Nervosa

Eating disorders (EDs) have a possible neurodevelopmental pathogenesis. Our study aim was to assess regional cortical thickness (CT), local gyrification index (lGI) and fractal dimensionality (FD), as specific markers of cortical neurodevelopment in ED females. Twenty-two women with acute anorexia nervosa (acuAN), 10 with recovered anorexia nervosa (recAN), 24 with bulimia nervosa (BN) and 35 female healthy controls (HC) underwent a 3T MRI scan. All data were processed by FreeSurfer. Compared to recAN group women with acuAN showed a lower CT in multiple areas, while compared to HC they showed lower CT in temporal regions. BN group showed higher CT values in temporal and paracentral areas compared to HC. In multiple cortical areas, AcuAN group showed greater values of lGI compared to recAN group and lower values of lGI compared to HC. The BN group showed lower lGI in left medial orbitofrontal cortex compared to HC. No significant differences were found in FD among the groups. Present results provide evidence of CT and lGI alterations in patients with AN and, for the first time, in those with BN. Although these alterations could be state-dependent phenomena, they may underlie psychopathological aspects of EDs.

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.

Fractal dimension of cortical functional connectivity networks & severity of disorders of consciousness

Recent evidence suggests that the quantity and quality of conscious experience may be a function of the complexity of activity in the brain and that consciousness emerges in a critical zone between low and high-entropy states. We propose fractal shapes as a measure of proximity to this critical point, as fractal dimension encodes information about complexity beyond simple entropy or randomness, and fractal structures are known to emerge in systems nearing a critical point. To validate this, we tested several measures of fractal dimension on the brain activity from healthy volunteers and patients with disorders of consciousness of varying severity. We used a Compact Box Burning algorithm to compute the fractal dimension of cortical functional connectivity networks as well as computing the fractal dimension of the associated adjacency matrices using a 2D box-counting algorithm. To test whether brain activity is fractal in time as well as space, we used the Higuchi temporal fractal dimension on BOLD time-series. We found significant decreases in the fractal dimension between healthy volunteers (n = 15), patients in a minimally conscious state (n = 10), and patients in a vegetative state (n = 8), regardless of the mechanism of injury. We also found significant decreases in adjacency matrix fractal dimension and Higuchi temporal fractal dimension, which correlated with decreasing level of consciousness. These results suggest that cortical functional connectivity networks display fractal character and that this is associated with level of consciousness in a clinically relevant population, with higher fractal dimensions (i.e. more complex) networks being associated with higher levels of consciousness. This supports the hypothesis that level of consciousness and system complexity are positively associated, and is consistent with previous EEG, MEG, and fMRI studies.