Fractal Dimension Values Research Articles

Dynamic evolution of crack fractal of polypropylene fiber reinforced geopolymer during flexural process

Crack mouth opening displacement (CMOD) is the most widely used parameter to evaluate the crack width changing process in concrete fracture research. However, it is difficult to accurately and comprehensively describe the multi-crack cracking characteristics of high-toughness concrete with only crack width. This paper investigates the dynamic evolution of the fractal dimension (FD) of surface cracks on polypropylene fiber-reinforced geopolymer (PFRG) during flexural process. The box counting method was used to calculate the FD of cracks in the segmentation images that were generated by deep learning. Crack FD at the peak load and the fracture failure load could be used as indicators of deflection hardening behavior, with threshold values of 0.83 and 0.86, respectively. Crack development can be divided into four stages based on load, FD, CMOD, and energy capacity with time: elastic segment, internal crack development segment, surface crack extension segment, and crack destabilization segment. FD shows an obvious exponential increase in the surface crack extension segment and crack destabilization segment, and the changing trend is basically parallel with the load in the surface crack extension segment. The FD of peak load and fracture failure load showed a trend of increasing and then decreasing as the fiber factor was increased. The FD value peaks at 585 when W/B = 0.35 and at 252 when W/B = 0.38. To achieve the best stress diffusion and crack propagation process and achieve the toughening objective, it is necessary to reasonably match the fiber factor and matrix strength.

Global dynamics and computational modeling approach for analyzing and controlling of alcohol addiction using a novel fractional and fractal–fractional modeling approach

In recent years, alcohol addiction has become a major public health concern and a global threat due to its potential negative health and social impacts. Beyond the health consequences, the detrimental consumption of alcohol results in substantial social and economic burdens on both individuals and society as a whole. Therefore, a proper understanding and effective control of the spread of alcohol addictive behavior has become an appealing global issue to be solved. In this study, we develop a new mathematical model of alcohol addiction with treatment class. We analyze the dynamics of the alcohol addiction model for the first time using advanced operators known as fractal–fractional operators, which incorporate two distinct fractal and fractional orders with the well-known Caputo derivative based on power law kernels. The existence and uniqueness of the newly developed fractal–fractional alcohol addiction model are shown using the Picard–Lindelöf and fixed point theories. Initially, a comprehensive qualitative analysis of the alcohol addiction fractional model is presented. The possible equilibria of the model and the threshold parameter called the reproduction number are evaluated theoretically and numerically. The boundedness and biologically feasible region for the model are derived. To assess the stability of the proposed model, the Ulam–Hyers coupled with the Ulam–Hyers–Rassias stability criteria are employed. Moreover, utilizing effecting numerical schemes, the models are solved numerically and a detailed simulation and discussion are presented. The model global dynamics are shown graphically for various values of fractional and fractal dimensions. The present study aims to provide valuable insights for the understanding the dynamics and control of alcohol addiction within a community.

Determination of local fractal dimension of the catalyst support MgAl2O4 surface

In this paper, a procedure for estimating local fractal dimensionality values for greyscale images was developed. This procedure was tested on synthesised fractal and multifractal Brownian surfaces. The use of the method of detrended fluctuation analysis to determine the Hurst index for samples with a size not exceeding 2000 points was analysed. The optimal values of the sample size and the scale parameter have been obtained, allowing the Hurst index to be calculated with an error of less than 20%. The procedure was also tested on greyscale images, which are 8-bit representations of a synthesised fractal surface. The peculiarities of the application of the procedure for calculating the local fractal dimension for images obtained by scanning electron microscopy have been analysed. The values of the local fractal dimension of the oxide support MgAl2O4 are calculated using SEM images at different magnifications. It has been shown that, taking into account the fractal dimensionality calculation, different morphologies of the particle surface are observed at different scales of image magnification. At the highest magnification, the particle surface is found to consist of embedded smooth micron particles. The surface morphology of the MgAl2O4 particle depends on the micro- and mesopores between the embedded particles, this surface can be characterised by a fractal surface with FD2 = 2.3–2.4.

Measurement of 2D and 3D Fractal Features of Urban Morphology from an Architectural View and Its Influencing Factors

Urban morphology has been empirically demonstrated to be self-organized and can be quantified by fractal dimension. However, the spatial variation rule of fractal features at the sub-zone scale has yet to be uncovered, as well as the relationship between fractal dimension values and road network or land-use patterns. In this study, the urban area is partitioned into 158 grid units, with subsequent calculations conducted to determine the fractal dimensions (using 2D box-counting and 3D voxel-counting methods), road network characteristics, and land-use patterns within each individual unit. The pattern of how architectures fill into the 2D or 3D embedding space at the grid level is revealed. Moreover, the spatial relationship between the road network, land-use, and their impacts on the local architectural layout is elucidated by employing MGWR, a model that incorporates the principles of fitting localized spatial regression. The results are as follows: (1) urban morphology follows fractal laws at a sub-zone scale, both in a 2D plane and 3D volume; (2) the filling degree of architecture is high in the urban center but low in the periphery areas; (3) the selected variables fit well with the regression models; (4) there is spatial heterogeneity regarding the influence of each factor. The research findings provide valuable insights into the theoretical relationship between urban morphology and the composite structure of road networks and land use. This facilitates identifying crucial areas and priority directions for urban renewal construction, as well as optimizing architectural design to improve efficiency and functionality.

On the fractal dimension of carbon black particles in pyrolysis flow reactors

This study presents a numerical approach for simulating the internal morphology of carbon black particles formed in a pyrolysis reactor. The simulation process involves a two-step approach using population balance models (PBM) and detailed population balance models (DPBM), solved via sectional and stochastic methods, to simulate the arrangement of primary particles within aggregates to allow determination of their fractal dimension (FD). The outcome is a novel introduction of simulating real flow reactors that for the first time provides the fractal dimension of particles as an output, rather than an assumed input. The results of this study have practical implications for optimizing the synthesis processes in carbon black production. The effects of various production parameters, including aggregation efficiency, temperature, pressure, and acetylene concentrations, on the fractal dimension values of carbon black particles are examined. It is observed that higher temperatures lead to the formation of larger fractal shapes with lower fractal dimensions and larger primary particle diameters. Moreover, increased reactor pressure and higher aggregation efficiency enhance the formation of carbon black aggregates, but also have a time-based effect with higher compactness at longer residence times. The time-based effect reveals the importance of sintering, where high loads of small particles enhance the overall sintering of the aggregates. These findings provide insights into the interplay between temperature, pressure, and particle morphology, highlighting the dynamic nature of carbon black nanoparticles and their response to synthesis process conditions.

Comparison of periapical radiography, panoramic radiography, and CBCT in the evaluation of trabecular bone structure using fractal analysis.

The aim of this study is to compare imaging techniques to evaluate trabecular bone structure using Fractal Analysis (FA). Fifteen sheep hemimandibles were used for this study. Digital images were obtained using periapical radiography, panoramic radiography, and cone-beam computed tomography (CBCT). CBCT imaging was performed in standard (STD) and high-resolution (HR) modes. FA was conducted using ImageJ 1.3 software with the box-counting method on the images. The fractal dimension (FD) values were analyzed by the statistical software Jamovi 1.6.23. Statistical significance was accepted as p < 0.05. The highest mean FD value was the FD on digital periapical radiographs (PaFD) (1.28 ± 0.04), and the lowest mean FD value was the FD on standard resolution cone-beam computed tomography images (STD-CBCTFD) (1.12 ± 0.10). Although there was no statistically significant difference between the PaFD and the FD on digital panoramic radiographs (PanFD) (p = 0.485), the PaFD was found to be significantly higher than STD-CBCTFD (p < 0.001), and the FD on high-resolution cone-beam computed tomography images (HR-CBCTFD) (p = 0.007). The PanFD was found to be significantly higher than the STD-CBCTFD (p = 0.004). According to our results, in the evaluation of trabecular bone structure using FA, periapical radiographs and panoramic radiographs have similar image quality for assessment of the FD. On the other hand, CBCT results did not correlate with results from any of the other techniques in this study.

Bone Trabecular Pattern Analysis in Odontogenic Cysts Using Cone Beam Computed Tomography: A Clinical Retrospective Study.

Introduction The cysts of the maxillofacial region account for one of the most common pathologies of the head and neck region after the mucosal pathologies. Radiography provides an essential clue in early diagnosis and triaging, but it continues further as it is used to evaluate the post-treatment outcome. However, manual analysis is prone to errors. In this scenario, fractal analysis (FA) in radiographs uses mathematical methods to analyse the changes in grey scales in a given radiographic image. FA in odontogenic cysts is used to characterise their complexity, uncover hidden patterns, monitor treatment response, and potentially provide prognostic information. This paper aimed to assess the fractal characteristics of the radicular cyst (RC), dentigerous cyst (DC), and odontogenic keratocyst (OKC) using cone beam computed tomography (CBCT). The objective was to calculate fractal dimension (FD) values expressed in each of these cysts, which could prove to be a radiological adjunct in diagnosing the above cysts. Materials and methods As this is a retrospective study, the archives of CBCT images from June 2021 to December 2023 were obtained from patients diagnosed and confirmed with a histopathological diagnosis with RC, DC, and OKC. The FA was performed using Image J Software(Ver 1.51, National Institute ofHealth Bethesda, Fiji). The cortical and cancellous bones were segmented using thresholding techniques and converted to binary images. The mean FD of the three planes was then compared to establish the distinctive fractal characteristic for the specific odontogenic cysts. A one-way ANOVA was performed using the Statistical Product and Service Solutions (SPSS) (version 23.0; IBM SPSS Statistics for Windows, Armonk, NY) to determine the difference between FD values of RC, DC, and OKC with a significance level less than 0.05. Results The FD values of DC, RC, and OKC were 1.33 ± 0.17, 1.08 ± 0.16, and 1.65 ± 0.12, respectively. The results indicated that OKC had higher FD values than DC and RC, which means that OKC had lesser bone destruction compared to DC and RC. Inferential statistics showed that the one-way ANOVA was used to compare the means of the three groups of FD data. When calculated for the three groups, the F-statistic value was at 7.29, which yielded a P value of 0.03, making it statistically significant for a 95% confidence interval (p<0.05). Conclusion Our CBCT study on bone trabecular pattern analysis using FD and FAin odontogenic cysts reveals distinct alterations in bone parameters among different cyst types. The probability of higher FD values in OKC is because of lesser cortical bone destruction in OKC compared to the other cyst types. These findings have potential implications for diagnosing, treating, and prognosticating odontogenic cysts.

Fractal Changes of the Retinal Microvasculature in Syphilitic Uveitis

ABSTRACT Purpose To quantify chorioretinal microvascular damage and recovery post-treatment in patients with acute syphilitic posterior placoid chorioretinitis (ASPPC) using fractal dimension (FD). Methods Retrospective cohort study of patients with serologically confirmed syphilitic uveitis. We obtained optical coherence tomography angiography (OCTA) scans at baseline and follow-up after intravenous penicillin treatment and computed FD of the superficial capillary plexus (SCP), deep capillary plexus (DCP), and choriocapillaris (CC) using ImageJ. Results We enrolled seven patients with ASPPC (11 eyes), and 17 control subjects (34 eyes). Pre-treatment averages of FD-SCP, FD-DCP, and FD-CC were: 1.672 (±0.115), 1.638 (±0.097), and 1.72 (±0.137); post-treatment: 1.760 (±0.071), 1.764 (±0.043), and 1.898 (±0.047). After treatment FD-CC increased in all 11 eyes with an average of 0.163 (p = 0.003); FD-DCP increased in 10 (91%) eyes with an average of 0.126 (p = 0.003); and FD-SCP increased in seven (64%) eyes with an average of 0.089 (p = 0.059). Compared to the post-treatment FD values in the syphilitic group, controls had similar FD-SCP (p = 0.266), FD-DCP (p = 0.078), and FD-CC (p = 0.449). Conclusions CC and DCP are mostly affected in ASPPC with minimal changes in the SCP. All vascular layers FD recovered after completing antibiotic treatment.

Evaluation of Jaw Bone Change in Children and Adolescents with Rheumatic Heart Disease by Fractal Analysis.

Recognizing the oral needs of people living with rheumatic heart disease (RHD), particularly in relation to their jaws, is of paramount importance. This study aimed to evaluate potential jaw bone changes in children and adolescents diagnosed with RHD and compare them with a group of healthy children and adolescents. The dental panoramic radiographs (DPRs) of 32 children and adolescents diagnosed with RHD comprised the case group. In contrast, the control group consisted of DPRs from 32 children and adolescents without any known diseases. Two regions of interest (ROI1 and ROI2) of trabecular bone and a separate region of interest (ROI3) of cortical bone were selected for fractal analysis. The mandibular cortical index (MCI) was evaluated. The ROI1 fractal dimension (FD) value of the individuals in the case group was statistically higher in the right jaw than in the left jaw (P < 0.01). Significant differences were determined between the case and control groups regarding the left ROI1, and the left and the right ROI2 (P < 0.05). No significant difference was found between the patient and control groups regarding the MCI value (P > 0.05). The results of this study showed that bone metabolism was affected unilaterally in the condyle region and bilaterally in the gonial region of patients with RHD. It also revealed that the cortical layer was not affected.

Optical coherence tomography angiography macular biomarkers of peripheral retinal ischemia in diabetic macular edema: secondary endpoints from the clinical study "FOVEA".

To investigate the relationship between the macular values of fractal dimension (FD) and lacunarity (LAC) on optical coherence tomography angiography (OCTA) images and the presence of peripheral retina non-perfusion areas (NPAs) on fluorescein angiography (FA) in patients with treatment-naïve diabetic macular edema (DME). Fifty patients with treatment-naïve DME underwent a full ophthalmic examination, including best-corrected visual acuity measurement, FA, spectral-domain optical coherence tomography, and OCTA. Specifically, FA was performed to detect the presence of retinal NPAs, whereas fractal OCTA analysis was used to determine macular FD and LAC values at the level of thesuperficial and deep capillary plexus (SCP and DCP). FA montage frames of the posterior pole and peripheral retina, as well as macular OCTA slabs of the SCP and DCP, were obtained. Thirty (60%) eyes with FA evidence of peripheral retinal NPAs in at least one quadrant showed significantly lower FD and higher LAC in both SCP and DCP, when compared with eyes presenting a well-perfused peripheral retina. Furthermore, macular FD and LAC values were found to be significantly associated with the extent of retinal NPAs. Macular FD and LAC of both SCP and DCP seem to be strongly associated with the extent of peripheral retinal NPAs, thus suggesting that may be useful predictive biomarkers of peripheral ischemia in treatment-naïve DME eyes.

Kinematics, intensity, and spatial arrangement of extensional fractures in the Tuxtla-Malpaso fault system: Chiapas strike-slip fault province, southern Mexico

We describe the orientation, number, size, distribution, and fracture spatial arrangement in the central part of the kilometric Tuxtla-Malpaso fault system, one of the largest in the south of Mexico which cuts Cretaceous calcareous rocks. In this area, there are two sub-vertical extensional fracture sets (T1 and T2). The T1 fractures have ∼ N–S orientation and recorded 4.48 % of extension. The T2 fractures cut the T1 fractures, have NE-SW orientation, and accommodate an amount of extension of 3.87 %. These two fracture sets are parallel to two normal fault groups. The youngest normal faults (F2) have NE-SW strike and cut the N–S normal faults (F1). The F1 faults record a reactivation as strike-slip faults contemporaneous with F2.The T1 and T2 fractures have a fracture intensity between 9.08 and 9.39 (P10) on mesoscopic scale. In the case of the spacing, the value of this parameter (0.28–1) is related to fracture clustering. This behavior agrees with the values of fractal box dimension (0.72–0.81), correlation dimension (0.24–0.6), Lyapunov exponent (1.31–2.42), and coefficient of variation (0.94–1.85) on mesoscopic scale. The cumulative aperture vs. distance shows that T1 and T2 fractures have heterogeneous strain distribution. The fracture spatial arrangement is verified with the normalized correlation count analysis (NCC). The NCC graphs show a clustering behavior for T1 and T2 fractures, which are related to a regulary-spaced fractal clusters pattern. We suggest that this pattern is related to the mechanical stratigraphy and the strain location in the fault zones, while the fracture intensity depends on the degree of fracture clustering and rock type more than the amount of strain.Finally, the T1 fractures were associated with the right strike-slip kinematic for the NW-SE Tuxtla-Malpaso fault. This activity recorded a transtension behavior during the middle Miocene. The T2 fractures were formed during the reactivation (Pliocene) of the Tuxtla-Malpaso fault as the left strike-slip fault behind wrenching-transpression kinematics.

Prognostic value of right ventricular trabecular complexity in patients with arrhythmogenic cardiomyopathy.

The present study aimed to investigate the incremental prognostic value of the right ventricular fractal dimension (FD), a novel marker of myocardial trabecular complexity by cardiac magnetic resonance (CMR) in patients with arrhythmogenic cardiomyopathy (ACM). Consecutive patients with ACM undergoing CMR were followed up for major cardiac events, including sudden cardiac death, aborted cardiac arrest, and appropriate implantable cardioverter defibrillator intervention. Prognosis prediction was compared by Cox regression analysis. We established a multivariable model supplemented with RV FD and evaluated its discrimination by Harrell's C-statistic. We compared the category-free, continuous net reclassification improvement (cNRI) and integrated discrimination index (IDI) before and after the addition of FD. A total of 105 patients were prospectively included from three centers and followed up for a median of 60 (48, 66) months; experienced 36 major cardiac events were recorded. Trabecular FD displayed a strong unadjusted association with major cardiac events (p < 0.05). In the multivariable Cox regression analysis, RV maximal apical FD maintained an independent association with major cardiac events (hazard ratio, 1.31 (1.11-1.55), p < 0.002). The Hosmer-Lemeshow goodness of fit test displayed good fit (X2 = 0.68, p = 0.99). Diagnostic performance was significantly improved after the addition of RV maximal apical FD to the multivariable baseline model, and the continuous net reclassification improvement increased 21% (p = 0.001), and the integrated discrimination index improved 16% (p = 0.045). In patients with ACM, CMR-assessed myocardial trabecular complexity was independently correlated with adverse cardiovascular events and provided incremental prognostic value. The application of FD values for assessing RV myocardial trabeculae may become an accessible and promising parameter in monitoring and early diagnosis of risk factors for adverse cardiovascular events in patients with ACM. • Ventricular trabecular morphology, a novel quantitative marker by CMR, has been explored for the first time to determine the severity of ACM. • Patients with higher maximal apical fractal dimension of RV displayed significantly higher cumulative incidence of major cardiac events. • RV maximal apical FD was independently associated with major cardiac events and provided incremental prognostic value in patients with ACM.

Schramm-Loewner Evolution in 2D Rigidity Percolation.

Amorphous solids may resist external deformation such as shear or compression, while they do not present any long-range translational order or symmetry at the microscopic scale. Yet, it was recently discovered that, when they become rigid, such materials acquire a high degree of symmetry hidden in the disorder fluctuations: their microstructure becomes statistically conformally invariant. In this Letter, we exploit this finding to characterize the universality class of central-force rigidity percolation (RP), using Schramm-Loewner evolution (SLE) theory. We provide numerical evidence that the interfaces of the mechanically stable structures (rigid clusters), at the rigidification transition, are consistently described by SLE_{κ}, showing that this powerful framework can be applied to a mechanical percolation transition. Using well-known relations between different SLE observables and the universal diffusion constant κ, we obtain the estimation κ∼2.9 for central-force RP. This value is consistent, through relations coming from conformal field theory, with previously measured values for the clusters' fractal dimension D_{f} and correlation length exponent ν, providing new, nontrivial relations between critical exponents for RP. These findings open the way to a fine understanding of the microstructure in other important classes of rigidity and jamming transitions.

STUDY ON QUANTIFICATION OF ROCK FRACTURE NETWORK TO PROMOTE SHALE GAS DEVELOPMENT

Shale gas is produced by fracturing the reservoir to increase production. Quantifying the complex structure of the rock fracture network has a very important role in shale gas development. In order to quantify the complex structure of rock fracture network, in this paper, combined with the hierarchical structure of fractal figure, long tail distribution and fractal self-similarity, the current calculation method of fractal dimension is improved, and the basic fractal dimension calculation method is proposed based on the principle. By removing the limit calculation method which causes the deviation of numerical results, a reasonable and convenient calculation model for engineering application is proposed. A set of two-dimensional fracture pictures are used to test the model. In the 10 pictures, the more complex the figure is, the larger the new fractal dimension value is, which is in line with the characteristics of fractal dimension describing the complexity of graphics. Furthermore, lattice Boltzmann method (LBM) is used to simulate the velocity field, and the permeability of the fracture network is calculated according to Darcy’s law. The permeability and fractal dimension of the fracture network are compared and analyzed. The conclusion that the permeability is positively related to the fractal dimension of the fracture network is obtained, which provides theoretical support for the further study of shale mining problems such as hydraulic fracturing.

Determination of cutting modes when turning railway wheelsets used in heavy traffic

Increasing the durability of the cutting tool when restoring the profile of the rolling surface of railway wheels by controlling the vibrations of the machine-tool-part system. Vibrations arising during the turning of the rolling surface of wagon wheels under various cutting modes were studied. To do this, a matrix of experimental planning was built using the Box-Wilson method, according to which all factors change in turn (cutting depth, cutting speed, feed). After the experiment, a mathematical model is selected and the numerical values of the coefficients of this equation are estimated. In accordance with the power dependences accepted in the theory of cutting, a mathematical model was constructed for the experiments in the form of a logarithmic polynomial of the first degree series. Mathematical processing of measurement results using fractal dimension was carried out. The fractal dimension was determined by the normalized span method. A correspondence has been established between the durability of the he cutter and the intensity of its vibrations when turning railway wheels. A mathematical dependence of the fractal dimension on the cutting modes is obtained, which makes it possible to adjust the cutting speed or feed during the restoration of the rolling surface of the wheel in the presence of defects. Optimal cutting modes, which allow to achieve a given quality of the rolling surface of the wheel in the shortest possible time, give the highest value of fractal dimension. Defects in railway wheels such as sliders, bumps, and gouges lead to a decrease in fractal dimension. The determination of the fractal dimension in the process of turning wheelsets makes it possible to automatically adjust the cutting modes to ensure a given quality of the rolling surface of the wheel and the required durability of the cutting tool.

Prognostic value of left ventricular trabeculae fractal analysis in patients with dilated cardiomyopathy

The prognostic value of left ventricular (LV) myocardial trabecular complexity on cardiovascular magnetic resonance (CMR) in dilated cardiomyopathy (DCM) remains unknown. This study aimed to evaluate the prognostic value of LV myocardial trabecular complexity using fractal analysis in patients with DCM. Consecutive patients with DCM who underwent CMR between March 2017 and November 2021 at two hospitals were prospectively enrolled. The primary endpoints were defined as the combination of all-cause death and heart failure hospitalization. The events of cardiac death alone were defined as the secondary endpoints.LV trabeculae complexity was quantified by measuring the fractal dimension (FD) of the endocardial border based on fractal geometry on CMR. Cox proportional hazards regression and Kaplan-Meier survival analysis were used to examine the association between variables and outcomes. The incremental prognostic value of FD was assessed in nested models. A total of 403 patients with DCM (49.31±14.68years, 69% male) were recruited. After a median follow-up of 43months (interquartile range, 28-55months), 87 and 24 patients reached the primary and secondary endpoints, respectively. Age, heart rate, New York Heart Association functional class >II, N-terminal pro-B-type natriuretic peptide, LV ejection fraction, LV end-diastolic volume index, LV end-systolic volume index, LV mass index, presence of late gadolinium enhancement, global FD, LV mean apical FD, and LV maximal apical FD were univariably associated with the outcomes (all P<0.05). After multivariate adjustment, LV maximal apical FD remained a significant independent predictor of outcome [hazard ratio=1.179 (1.116, 1.246), P<0.001]. The addition of LV maximal apical FD in the nested models added incremental prognostic value to other common clinical and imaging risk factors (all <0.001; C-statistic: 0.84-0.88, P<0.001). LV maximal apical FD was an independent predictor of the adverse clinical outcomes in patients with DCM and provided incremental prognostic value over conventional clinical and imaging risk factors.

Morphological changes of cerebral gray matter in spinocerebellar ataxia type 3 using fractal dimension analysis.

Spinocerebellar ataxia type 3 (SCA3), or Machado-Joseph disease, presents as a cerebellar cognitive affective syndrome (CCAS) and represents the predominant SCA genotype in Taiwan. Beyond cerebellar involvement, SCA3 patients exhibit cerebral atrophy. While prior neurodegenerative disease studies relied on voxel-based morphometry (VBM) for brain atrophy assessment, its qualitative nature limits individual and region-specific evaluations. To address this, we employed fractal dimension (FD) analysis to quantify cortical complexity changes in SCA3 patients. We examined 50 SCA3 patients and 50 age- and sex-matched healthy controls (HC), dividing MRI cerebral gray matter (GM) into 68 auto-anatomical subregions. Using three-dimensional FD analysis, we identified GM atrophy manifestations in SCA3 patients. Results revealed lateral atrophy symptoms in the left frontal, parietal, and occipital lobes, and fewer symptoms in the right hemisphere's parietal and occipital lobes. Focal areas of atrophy included regions previously identified in SCA3 studies, alongside additional regions with decreased FD values. Bilateral postcentral gyrus and inferior parietal gyrus exhibited pronounced atrophy, correlating with Scale for the Assessment and Rating of Ataxia (SARA) scores and disease duration. Notably, the most notable focal areas were the bilateral postcentral gyrus and the left superior temporal gyrus, serving as imaging biomarkers for SCA3. Our study enhances understanding of regional brain atrophy in SCA3, corroborating known clinical features while offering new insights into disease progression.

Analysis of the difference between Alzheimer's disease, mild cognitive impairment and normal people by using fractal dimensions and small-world network.

This research examined the distinctions in brain network characteristics among individuals with Alzheimer's disease (AD), mild cognitive impairment (MCI), and a control group. Magnetic resonance imaging (MRI) and mini-mental state examination (MMSE) data were retrieved from the Alzheimer's Disease Neuroimaging Initiative (ANDI) database, comprising 40 subjects in each group. Correlation maps for evaluating brain network connectivity were generated using fractal dimension (FD) analysis, a method capable of quantifying morphological changes in cortical and cerebral regions. Employing graph theory, each parcellated brain region was represented as a node, and edges between nodes were utilized to compute small-world network properties for each group. In the comparison between control and AD demonstrated the significantly lower FD values (P<0.05) in temporal lobe, motor cortex, part of occipital and parietal, hippocampus, amygdala, and entorhinal cortex, which present the atrophy. Similarly, comparing control group to MCIs, regions closely associated with memory, such as the hippocampus, showed significantly lower FD values. Furthermore, both AD and MCI groups displayed diminished connectivity and decreased network efficiency. In conclusion, fractal dimension (FD) analysis illustrate the progression of structural declination from mild cognitive impairment (MCI) to Alzheimer's disease (AD). Additionally, structural small-world network analysis presents itself as a potential method for assessing network efficiency and the progression of AD. Moving forward, further clinical assessments are warranted to validate the findings observed in this study.

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.

Analisis Pola Harga Saham dengan Modifikasi Metode Eksponen Hurst dan Box Counting

A stock chart is a graphical representation of a stock's past performance. The dynamic pattern of stock prices is important to know because an investor wants to invest, expecting high returns with low risk. The dynamic pattern of stock prices can be known by fractal dimension analysis because the stock price graph is self-affine, which is one of the properties of fractal objects. In this study, a modification of the Hurst exponent method and box counting are used to analyze the fractal dimension. The calculated results are classified into three types, namely random, persistent and anti-persistent. Two data intervals are observed, namely January 2018-December 2021 stock prices (48 data) and January 2018-June 2022 stock prices (54 data). The resulting Hurst exponents of the two intervals are 0.043 and 0.003. Based on the resulting Hurst exponent value, the data is anti-persistent because the value 0&lt;H&lt;0.5. Then, the fractal dimension value obtained by applying the box-counting method is 1.547 and 1.562, meaning that Bank Rakyat Indonesia's price pattern is anti-persistent. The meaning of anti-persistent is that in certain months the stock has a high price and in the following months the stock has a low price to be traded.