Measurement Of Fractal Dimension Research Articles

Predictive and Explainable Artificial Intelligence for Neuroimaging Applications.

The aim of this review is to highlight the new advance of predictive and explainable artificial intelligence for neuroimaging applications. Data came from 30 original studies in PubMed with the following search terms: "neuroimaging" (title) together with "machine learning" (title) or "deep learning" (title). The 30 original studies were eligible according to the following criteria: the participants with the dependent variable of brain image or associated disease; the interventions/comparisons of artificial intelligence; the outcomes of accuracy, the area under the curve (AUC), and/or variable importance; the publication year of 2019 or later; and the publication language of English. The performance outcomes reported were within 58-96 for accuracy (%), 66-97 for sensitivity (%), 76-98 for specificity (%), and 70-98 for the AUC (%). The support vector machine and the convolutional neural network registered the best performance (AUC 98%) for the classifications of low- vs. high-grade glioma and brain conditions, respectively. Likewise, the random forest delivered the best performance (root mean square error 1) for the regression of brain conditions. The following factors were discovered to be major predictors of brain image or associated disease: (demographic) age, education, sex; (health-related) alpha desynchronization, Alzheimer's disease stage, CD4, depression, distress, mild behavioral impairment, RNA sequencing; (neuroimaging) abnormal amyloid-β, amplitude of low-frequency fluctuation, cortical thickness, functional connectivity, fractal dimension measure, gray matter volume, left amygdala activity, left hippocampal volume, plasma neurofilament light, right cerebellum, regional homogeneity, right middle occipital gyrus, surface area, sub-cortical volume. Predictive and explainable artificial intelligence provide an effective, non-invasive decision support system for neuroimaging applications.

Fractal dimension and lacunarity measures of glioma subcomponents are discriminative of the grade of gliomas and IDH status.

Since the overall glioma mass and its subcomponents-enhancing region (malignant part of the tumor), non-enhancing (less aggressive tumor cells), necrotic core (dead cells), and edema (water deposition)-are complex and irregular structures, non-Euclidean geometric measures such as fractal dimension (FD or "fractality") and lacunarity are needed to quantify their structural complexity. Fractality measures the extent of structural irregularity, while lacunarity measures the spatial distribution or gaps. The complex geometric patterns of the glioma subcomponents may be closely associated with the grade and molecular landscape. Therefore, we measured FD and lacunarity in the glioma subcomponents and developed machine learning models to discriminate between tumor grades and isocitrate dehydrogenase (IDH) gene status. 3D fractal dimension (FD3D) and lacunarity (Lac3D) were measured for the enhancing, non-enhancing plus necrotic core, and edema-subcomponents using preoperative structural-MRI obtained from the The Cancer Genome Atlas (TCGA) and University of California San Francisco Preoperative Diffuse Glioma MRI (UCSF-PDGM) glioma cohorts. The FD3D and Lac3D measures of the tumor-subcomponents were then compared across glioma grades (HGGs: high-grade gliomas vs. LGGs: low-grade gliomas) and IDH status (mutant vs. wild type). Using these measures, machine learning platforms discriminative of glioma grade and IDH status were developed. Kaplan-Meier survival analysis was used to assess the prognostic significance of FD3D and Lac3D measurements. HGG exhibited significantly higher fractality and lower lacunarity in the enhancing subcomponent, along with lower fractality in the non-enhancing subcomponent compared to LGG. This suggests that a highly irregular and complex geometry in the enhancing-subcomponent is a characteristic feature of HGGs. A comparison of FD3D and Lac3D between IDH-wild type and IDH-mutant gliomas revealed that mutant gliomas had ~2.5-fold lower FD3D in the enhancing-subcomponent and higher FD3D with lower Lac3D in the non-enhancing subcomponent, indicating a less complex and smooth enhancing subcomponent, and a more continuous non-enhancing subcomponent as features of IDH-mutant gliomas. Supervised ML models using FD3D from both the enhancing and non-enhancing subcomponents together demonstrated high-sensitivity in discriminating glioma grades (~97.9%) and IDH status (~94.4%). A combined fractal estimation of the enhancing and non-enhancing subcomponents using MR images could serve as a non-invasive, precise, and quantitative measure for discriminating glioma grade and IDH status. The combination of 2-hydroxyglutarate-magnetic resonance spectroscopy (2HG-MRS) with FD3D and Lac3D quantification may be established as a robust imaging signature for glioma subtyping.

Resting-state EEG spectral and fractal features in dementia with Lewy bodies with and without visual hallucinations

ObjectiveComplex visual hallucinations (VH) are a core feature of dementia with Lewy bodies (DLB), though they may not occur in all patients. Power spectral density (PSD) analysis of resting-state EEG (rs-EEG) shows associations between some frequency bands (e.g., θ), individual alpha frequency (IAF) and VH. However, new tools that improve early differential diagnosis and symptom-based stratification with higher sensitivity and specificity, even within the DLB population, are desirable. We aimed to assess differences in rs-EEG data between DLB patients with VH (DLB-VH+) and without VH (DLB-VH-), comparing innovative non-linear approaches with more traditional linear ones. MethodsWe retrospectively analyzed rs-EEG recordings of DLB-VH+, DLB-VH-, Alzheimer’s disease patients and age-matched healthy controls. EEG was analyzed using the nonlinear Higuchi’s Fractal Dimension (FD) measure, and the results were compared with those of entropy and standard linear methods based on PSD and IAF. ResultsOnly the FD measure could discriminate between DLB-VH+ and DLB-VH-. ConclusionsIn conclusion, rs-EEG differences between DLB-VH+ and DLB-VH- are better characterized by FD analysis than by a more traditional power spectrum approach. SignificanceThis suggests that the presence of complex VH is associated with less complex brain dynamics at rest, as reflected by the FD measure.

Exploring cortical morphology biomarkers of amnesic mild cognitive impairment using novel fractal dimension-based structural MRI analysis.

Amnestic mild cognitive impairment (aMCI) is considered as an intermediate stage of Alzheimer's disease, but no MRI biomarkers currently distinguish aMCI from healthy individuals effectively. Fractal dimension, a quantitative parameter, provides superior morphological information compared to conventional cortical thickness methods. Few studies have used cortical fractal dimension values to differentiate aMCI from healthy controls. In this study, we aim to build an automated discriminator for accurately distinguishing aMCI using fractal dimension measures of the cerebral cortex. Thirty aMCI patients and 30 health controls underwent structural MRI of the brain. First, the atrophy of participants' cortical sub-regions of Desikan-Killiany cortical atlas was assessed using fractal dimension and cortical thickness. The fractal dimension is more sensitive than cortical thickness in reducing dimensional effects and may accurately reflect morphological changes of the cortex in aMCI. The aMCI group had significantly lower fractal dimension values in the bilateral temporal lobes, right limbic lobe and right parietal lobe, whereas they showed significantly lower cortical thickness values only in the bilateral temporal lobes. Fractal dimension analysis was able to depict most of the significantly different focal regions detected by cortical thickness, but additionally with more regions. Second, applying the measured fractal dimensions (and cortical thickness) of both cerebral hemispheres, an unsupervised discriminator was built for the aMCI and healthy controls. The proposed fractal dimension-based method achieves 80.54% accuracy in discriminating aMCI from healthy controls. The fractal dimension appears to be a promising biomarker for cortical morphology changes that can discriminate patients with aMCI from healthy controls.

Investigating Cortical Complexity in Mixed Dementia through Nonlinear Dynamic Analyses: A Resting-State EEG Study.

Objective: Dementia is a broad term referring to a decline in problem-solving abilities, language skills, memory, and other cognitive functions to a degree that it significantly disrupts everyday activities. The underlying cause of dementia is the impairment or loss of nerve cells and their connections within the brain. The particular symptoms experienced are contingent upon specific regions of the brain affected by this damage. In this research, we aimed to investigate the nonlinear dynamics of the mixed demented brain compared to healthy subjects using electroencephalogram (EEG) analysis. Method : For this purpose, EEG was recorded from 66 patients with mixed dementia and 65 healthy subjects during rest. After signal preprocessing, sample entropy and Katz fractal dimension analyses were applied to the preprocessed EEG data. Analysis of variance with repeated measures was utilized to compare the nonlinear dynamics of brain activity between dementia and healthy states and partial correlation analysis was employed to explore the relationship between EEG complexity measures and cognitive and neuropsychiatric symptoms of patients. Results: Based on repeated measures ANOVA, there was a significant main effect between groups for both Katz fractal dimension (F = 4.10, P = 0.01) and sample entropy (F = 4.81, P = 0.009) measures. Post hoc comparisons revealed that EEG complexity was significantly reduced in dementia mainly in the occipitoparietal and temporal areas (P < 0.05). MMSE scores were positively correlated with EEG complexity measures, while NPI scores were negatively correlated with EEG complexity measures, mainly in the occipitoparietal and temporal areas (P < 0.05). Moreover, using a KNN classifier, all significant complexity measures yielded the best classification performance with an accuracy of 98.05%, sensitivity of 97.03% and specificity of 99.16% in detecting dementia. Conclusion: This study demonstrated a unique dynamic system within the brain impacted by dementia that results in more predictable patterns of cortical activity mainly in the occipitoparietal and temporal areas. These abnormal patterns were associated with patients' cognitive capacity and neuropsychiatric symptoms.

Evaluation of mandibular trabecular and cortical structure by fractal analysis in hyperlipidemia and hypertension patients

IntroductionThis study aimed to evaluate the mandibular trabecular and cortical changes in patients with hyperlipidemia (HL) and/or hypertension (HT) using fractal dimension (FD) analysis, mandibular cortical width (MCW), panoramic mandibular index (PMI) and mandibular cortical index (MCI). Materials and methodsPanoramic radiographs of 100 patients were evaluated. FD measurement of three region of interest (ROI) including the angulus, corpus and interdental bone area were made. MCW, PMI and MCI were also measured and noted. ResultsAngulus, corpus and interdental FD values were significantly lower in three disease groups than the control group. Angulus, corpus, and interdental FD values were significantly lower in the HL+HT group than in the HL group and HT group. MCW value was significantly lower in the HL group, HT group, and HL+HT group than the control group. The cortical index C1 was more common in the control group while C2 was more common in the HT, HL and HL+HT group. ConclusionThe fact that FD was significantly lower in the HL+HT group compared to the HL and HT groups indicates the positive effect of their association on bone loss and quality. FD measurements on images obtained using a direct digital panoramic system can be used for treatment planning and follow-up of patients with HL and/or HT.

Pattern formation by the drying of saline droplets on pillars

The application of saline solutions for surface coating is pertinent across multiple biomedical fields, various technological sectors, and industries, including agriculture. The drying of salt solution droplets is key to understanding and controlling morphological structures on surfaces. In this paper, we report the study of pattern formation from the evaporation of saline drops (NaCl, CsCl, and KCl) placed on pillars. Our findings indicate that regardless of saline concentration and type, the drying process can be categorized into three modes: stable, metastable, and unstable. In both the stable and metastable modes, the droplet fixes to the pillar during the drying process and ensuing pattern formation; however, the crucial distinction lies in the metastable mode, where the droplet additionally undergoes mass loss through fluid drainage from the walls of the pillars. Conversely, in the unstable drying mode, the droplet undergoes rapid collapse, leading to a substantial loss of mass. The distinct drying modes dictate the resulting patterns on the pillars. We employ measurements of configurational entropy and fractal dimension as quantitative metrics to assess the complexity, reproducibility, and similarity of patterns. The texture analysis reveals that, at low concentrations of both CsCl and KCl, significant differences emerge among the patterns produced by the different drying modes, especially highlighting differences in patterns of the stable drying mode. Finally, we analyze fertilizer deposition patterns to prove that all three drying modes can occur in complex fluids.

Applicability of Oculomics for Individual Risk Prediction: Repeatability and Robustness of Retinal Fractal Dimension Using DART and AutoMorph.

To investigate whether fractal dimension (FD)-based oculomics could be used for individual risk prediction by evaluating repeatability and robustness. We used two datasets: "Caledonia," healthy adults imaged multiple times in quick succession for research (26 subjects, 39 eyes, 377 color fundus images), and GRAPE, glaucoma patients with baseline and follow-up visits (106 subjects, 196 eyes, 392 images). Mean follow-up time was 18.3 months in GRAPE; thus it provides a pessimistic lower bound because vasculature could change. FD was computed with DART and AutoMorph. Image quality was assessed with QuickQual, but no images were initially excluded. Pearson, Spearman, and intraclass correlation (ICC) were used for population-level repeatability. For individual-level repeatability, we introduce measurement noise parameter λ, which is within-eye standard deviation (SD) of FD measurements in units of between-eyes SD. In Caledonia, ICC was 0.8153 for DART and 0.5779 for AutoMorph, Pearson/Spearman correlation (first and last image) 0.7857/0.7824 for DART, and 0.3933/0.6253 for AutoMorph. In GRAPE, Pearson/Spearman correlation (first and next visit) was 0.7479/0.7474 for DART, and 0.7109/0.7208 for AutoMorph (all P < 0.0001). Median λ in Caledonia without exclusions was 3.55% for DART and 12.65% for AutoMorph and improved to up to 1.67% and 6.64% with quality-based exclusions, respectively. Quality exclusions primarily mitigated large outliers. Worst quality in an eye correlated strongly with λ (Pearson 0.5350-0.7550, depending on dataset and method, all P < 0.0001). Repeatability was sufficient for individual-level predictions in heterogeneous populations. DART performed better on all metrics and might be able to detect small, longitudinal changes, highlighting the potential of robust methods.

Assessment of mandibular trabecular structure and cervical vertebral maturation using fractal analysis.

Changes in bone trabeculae occur during skeletal maturation. Fractal analysis is a technique for assessing changes in the internal structure of the bone. The aim of this study was to investigate correlation of bone trabeculation measured on panoramic and lateral cephalometric radiographs with cervical vertebral maturation stages (CVMS). CVMS was evaluated on lateral cephalometric radiographs of 120 individuals aged 6 to 18 years. The subjects were divided into 6 groups (n= 20 each) according to CVMS. On panoramic and lateral cephalometric radiographs, fractal dimension (FD) analyses were performed in mandibular corpus, gonial and condylar regions and cervical vertebral bodies (C2, C3, C4). FD measurements from the mandibular corpus region bilaterally and C2-C4 vertebral bodies showed significant positive correlations with CVMS (r= 0.418, r= 0.412, r= 0.324, r= 0.304, r= 0.263, respectively). Significant differences were found in the FD values of the right/left mandibular corpus and C2 and C3 vertebrae compared to CVMS (P< 0.001, P< 0.001, P= 0.005, P= 0.019, respectively). Significant positive moderate correlations between the FD values of the right and left mandibular corpus and CVMS suggest that fractal analysis may be useful in determining skeletal maturity stage. The significant FD values obtained from the right/left mandibular corpus, and C2 and C3 vertebra indicate that the complexity of the internal trabecular structure increases with maturation.

Fractal dimensions and machine learning for detection of Parkinson’s disease in resting-state electroencephalography

Parkinson’s disease (PD) is an incurable neurological disorder that degenerates the cerebrospinal nervous system and hinders motor functions. Electroencephalography (EEG) signal analysis can provide reliable information regarding PD conditions. However, EEG is a complex, multichannel, and nonlinear signal with noise that problematizes identifying PD symptoms. A few studies have employed fractal dimension (FD) to extract distinguishing PD features from EEG signals. However, no exploratory study exists, as per our knowledge, on the efficiency of the different FD measures. We aim to conduct a comparative analysis of the various FDs that, as feature extraction measures, can discriminate PD patients who are ON and OFF medication from healthy controls using ML architecture. This study has implemented and analyzed several techniques for segmentation, feature extraction, and ML models. The results show that k-nearest neighbors (KNN) classifier with Higuchi FD and 90% overlap for segmented window delivers the highest accuracies, yielding a mean accuracy of 99.65±0.15%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$99.65\\pm 0.15\\%$$\\end{document} for PD patients ON medication and 99.45±0.18%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$99.45\\pm 0.18\\%$$\\end{document} for PD patients OFF medication, respectively. The model accurately identifies the signs of the disease in resting-state EEG with almost equivalent accuracy in both OFF and ON medication patients. To enhance the interpretability in our study, we leveraged XGB’s feature importance to generate brain topographic plots. This integration of explainable AI (XAI) enhanced the transparency and comprehensibility of our model’s classifications. Additionally, a comparison between the performance of FD and a few entropy measures has also been drawn to validate the significance of FD as a superior feature extraction measure. This study contributes to the body of knowledge with an architectural pipeline for detecting PD in resting-state EEG while emphasizing fractal dimension as an effective way of extracting salient features from EEG signals.

3D Fractal Dimension Analysis: Prognostic Value of Right Ventricular Trabecular Complexity in Participants with Arrhythmogenic Cardiomyopathy.

Arrhythmogenic cardiomyopathy (ACM) is characterized by progressive myocardial fibro-fatty infiltration accompanied by trabecular disarray. Traditionally, two-dimensional (2D) instead of 3D fractal dimension (FD) analysis has been used to evaluate trabecular disarray. However, the prognostic value of trabecular disorder assessed by 3D FD measurement remains unclear. To investigate the prognostic value of right ventricular trabecular complexity in ACM patients using 3D FD analysis based on cardiac MR cine images. Retrospective. 85 ACM patients (mean age: 45 ± 17 years, 52 male). 3.0T/cine imaging, T2-short tau inversion recovery (T2-STIR), and late gadolinium enhancement (LGE). Using cine images, RV (right ventricular) volumetric and functional parameters were obtained. RV trabecular complexity was measured with 3D fractal analysis by box-counting method to calculate 3D-FD. Cox and logistic regression models were established to evaluate the prognostic value of 3D-FD for major adverse cardiac events (MACE). Cox regression and logistic regression to explore the prognostic value of 3D-FD. C-index, time-dependent receiver operating characteristic (ROC) curves and area under the ROC curve (AUC) to evaluate the incremental value of 3D-FD. Intraclass correlation coefficient for interobserver variability. P < 0.05 indicated statistical significance. 26 MACE were recorded during the 60 month follow-up (interquartile range: 48-67 months). RV 3D-FD significantly differed between ACM patients with MACE (2.67, interquartile range: 2.51 ~ 2.81) and without (2.52, interquartile range: 2.40 ~ 2.67) and was a significant independent risk factor for MACE (hazard ratio, 1.02; 95% confidence interval: 1.01, 1.04). In addition, prognostic model fitness was significantly improved after adding 3D-FD to RV global longitudinal strain, LV involvement, and 5-year risk score separately. The myocardial trabecular complexity assessed through 3D FD analysis was found associated with MACE and provided incremental prognostic value beyond conventional ACM risk factors. 4 TECHNICAL EFFICACY: Stage 1.

Fractal Neurodynamics.

The neuronal ongoing electrical activity in the brain network, the neurodynamics, reflects the structure and functionality of generating neuronal pools. The activity of neurons due to their excitatory and inhibitory projections is associated with specific brain functions. Here, the purpose was to investigate if the local ongoing electrical activity exhibits its characteristic spectral and fractal features in wakefulness and sleep across and within subjects. Moreover, we aimed to show that measures typical of complex systems catch physiological features missed by linear spectral analyses. For this study, we concentrated on the evaluation of the power spectral density (PSD) and Higuchi fractal dimension (HFD) measures. Relevant clinical impact of the specific features of neurodynamics identification stands primarily in the potential of classifying cortical parcels according to their neurodynamics as well as enhancing the effectiveness of neuromodulation interventions to cure symptoms secondary to neuronal activity unbalances.

Public Art Design Practice under Visual Communication Design

Abstract The expansion of visual communication design in public art makes the scope of visual communication wide. This new art form reflects the cross and integration of disciplines but also makes the form of public art in our life richer, and people can get more beautiful enjoyment. Research on the fractal algorithm in the escape time algorithm studied the Julia set in different function conditions of the fractal graph of the change and the function of different indices of the joint. The fractal graph obtained a colorful, more tense structure. Subsequently, it is applied to public art design, and after testing its performance, the fractal design of visual communication is combined with the IPA model to explore the practical effect of the fractal design in public art design. The results show that the improved fractal algorithm proposed in this paper increases the pattern generation rate from 61.5% to 92.6%. The fractal dimension measurement of 15 typical batik patterns shows that more than 85% of the batik patterns have an average value of more than 15000.

Fractal analysis and radiomorphometric indices: comparison of mandibular bone structure changes on digital panoramic radiographs of smokers

Purpose: The purpose of the study was to assess how smoking affected the mandibular bone structure using a variety of radiomorphometric indices and fractal dimension (FD) analysis.&#x0D; &#x0D; Materials and methods: 56 patients (28 smokers and 28 non-smokers) were included in this retrospective study. In the trabecular bone of the mandible, eight areas of interest of 45x45 pixels were chosen, and fractal dimension analysis was carried out. All digital panoramic radiographs were used to measure the mandibular index and mandibular cortical width. All values were compared between groups. Utilizing the Shapiro-Wilk and Mann-Whitney U tests, collected data were evaluated.&#x0D; &#x0D; Results: There was a statistically significant difference between the groups in ROI4 and ROI8, but not in the mean FD values of ROI1, ROI2, ROI3, ROI5, ROI6 and ROI7 in the mandibular trabecular bone. No statistically significant differences between groups were seen for PMI and MCW measures (p &gt; .05). The intraclass correlation coefficient (ICC) values for the FD, PMI and MCW measurements were excellent.&#x0D; &#x0D; Conclusion: The trabecular bone structures of the anterior of the mental foramen and the condyle were different in smokers. There was no difference in cortical bone structures.

Identifying and differentiating dementias with EEG fractal dimension distributions

AbstractBackgroundAlzheimer’s disease dementia (AD‐dementia) and non‐AD dementia have both been associated with decreases in fractal dimension and spectral slowing in electroencephalography (EEG). Here, we describe a set of EEG features ‐ Fractal Dimension Distributions (FDD) ‐ which quantifies the stability of fractal dimensions over time within specific frequency bands. We compare the performance of models trained using this FDD feature to models using age, spectral power, or overall fractal dimension.MethodWe analyzed resting‐state data from older adults (age M = 71, range = 55‐86) with subjective cognitive impairment (SCI; N = 97, 59 female) or dementia (N = 51, 32 female), including 38 with AD (N = 33). We calculated the Fractal Dimension Distributions (FDD) by first computing the Higuchi fractal dimension within moving windows, then summarizing the distribution of these fractal values using the mean and standard deviation within standard frequency bands. Finally, we trained regularized logistic regression models to classify participants as having SCI or dementia, and as having SCI or AD‐dementia. We compared the performance of models using FDD or standard measures of fractal dimension. Models were evaluated using balanced accuracy, weighted sensitivity, and weighted specificity calculated using 5‐fold cross‐validation.ResultFor SCI vs Dementia, the model using FDD (accuracy = 74%, sensitivity = 67%, specificity = 81%, p &lt; .001) had higher accuracy, sensitivity, and specificity than models trained on age, standard measures of fractal dimension, or spectral power. Similarly, the model trained to predict SCI vs AD‐Dementia was most accurate when using FDD (accuracy = 73%, sensitivity = 59%, specificity = 97%, p &lt; .005). The features useful for discriminating SCI from Dementia partially overlapped with the features useful for discriminating SCI from AD‐Dementia.ConclusionOur novel feature, FDD, outperforms standard measures of fractal dimensions or spectral band power at discriminating between SCI and dementia. Moreover, FDD may help to distinguish between AD and other varieties of dementia.

A novel approach to documenting water diffusion in ancient obsidian artifacts via the complexity analysis of microscope images

The diffusion of water through the surface of archaeological obsidian is mainly a temperature-dependent and concentration driven phenomenon that forms the basis of the hydration dating (OHD) method. For the first time, we apply a novel fractal approach, which explores the various attributes of images linked to the variation in water concentration and structural deficiencies within the hydrated layer. The microscopic images of hydrated layers on two obsidian specimens from Africa spanning thousands of years have been investigated. Seven complexity measures of fractal dimension (succolarity, lacunarity, correlation dimension, local connected fractal dimension (LCFD), fractal fragmented index (FFI), entropy and normalized Kolmogorov complexity (TIFF-LZW)), were applied to the hydration layer images. It was found that the effective determination of diffusion depth mainly comes from the application of succolarity, FFI and LCFD fractal indices, plus the structural deficiencies at the surface and the misconnected lacunas at the interface of the water diffusion layer and unaltered obsidian. The difference between fractal dimensions of the hydrated area and the non-hydrated images turned out to be a crucial quantitative parameter. Accordingly, we found that especially the succolarity, which is based on the capacity of a water fluid to percolate through the obsidian area in a given direction, was critical to determine the characteristics of the obsidian hydration layer. The FFI and LCFD estimate the compaction/fragmentation and connectivity, respectively. Therefore, the fractals play an important role in understanding the general diffusion in solids theory and the study aims at providing a new direction in the OHD and diffusion fields concerning the complex dynamic systems and structures which display a fractal pattern.

Assessment and Comparison of Nonlinear Measures in Resting-State Magnetoencephalograms in Alzheimer's Disease and Mild Cognitive Impairment.

Nonlinear dynamical measures, such as fractal dimension (FD), entropy, and Lempel-Ziv complexity (LZC), have been extensively investigated individually for detecting information content in magnetoencephalograms (MEGs) from patients with Alzheimer's disease (AD). To compare systematically the performance of twenty conventional and recently introduced nonlinear dynamical measures in studying AD versus mild cognitive impairment (MCI) and healthy control (HC) subjects using MEG. We compared twenty nonlinear measures to distinguish MEG recordings from 36 AD (mean age = 74.06±6.95 years), 18 MCI (mean age = 74.89±5.57 years), and 26 HC subjects (mean age = 71.77±6.38 years) in different brain regions and also evaluated the effect of the length of MEG epochs on their performance. We also studied the correlation between these measures and cognitive performance based on the Mini-Mental State Examination (MMSE). The results obtained by LZC, zero-crossing rate (ZCR), FD, and dispersion entropy (DispEn) measures showed significant differences among the three groups. There was no significant difference between HC and MCI. The highest Hedge's g effect sizes for HC versus AD and MCI versus AD were respectively obtained by Higuchi's FD (HFD) and fuzzy DispEn (FuzDispEn) in the whole brain and was most prominent in left lateral. The results obtained by HFD and FuzDispEn had a significant correlation with the MMSE scores. DispEn-based techniques, LZC, and ZCR, compared with HFD, were less sensitive to epoch length in distinguishing HC form AD. FuzDispEn was the most consistent technique to distinguish MEG dynamical patterns in AD compared with HC and MCI.

Assessment of correlation between hand-wrist maturation and cervical vertebral maturation: a fractal analysis study

BackgroundTo investigate whether fractal dimension (FD) measurements from hand-wrist radiographs and lateral cephalometric radiographs are correlated with each other and with skeletal maturation stages.MethodsIn this retrospective study conducted on hand-wrist and lateral cephalometric radiographs obtained from patients between 2017 and 2023, hand-wrist maturation stages (HWMS) and cervical vertebral maturation stages (CVMS) of 144 subjects (6 to 17 years of age) were assessed radiographically. The participants were divided into nine groups (n = 16 each) based on HWMS. Fractal analysis was performed on the radiographs of the radius, the middle finger phalanges (proximal, medial and distal), and the cervical vertebral bodies (C2, C3, C4). Mean and standard deviation values, Spearman’s and Pearson correlation analyses, one-way ANOVA, Kruskal-Wallis H tests and Mann-Whitney-U test were used to evaluate the data.ResultsPositive correlations were found between the FD values of the radius and HWMS or CVMS (r = .559, P = .001, r = .528 P = .001 respectively). The FD values of the radius were positively correlated with those of all cervical vertebrae (C2, C3, C4), proximal and medial phalanges as well as age. FD values measured from the proximal phalanx, medial phalanx and radius showed significant differences among both HWMS and CVMS (P < .05). HWMS was strongly correlated with CVMS (r = .929, P = .001). Age was strongly correlated with HWMS (r = .795, P = .001) and CVMS (r = .756, P = .001). There was a significant difference in terms of age distribution among HWMS and CVMS (P < .05).ConclusionsFD measurements on hand-wrist radiographs can provide useful information for the assessment of skeletal maturation stage. Especially, FD measurements from the radius are important and more reliable to predict skeletal maturation stage.

Dynamical fractal: Theory and case study

Urbanization is a phenomenon of concern for planning and public health: projections are difficult because of policy changes and natural events, and indicators are multiple. There are previous studies of development that used fractals, but none for this specific problem, nor extrapolating the future trend. In the first part of this paper, we construct a theoretical framework for analyzing dynamic (changing) fractals and extrapolating their future trends based on their fractal dimension—a measure of the complexity of the fractal. We believe this approach holds enormous potential for applications in analyzing changing fractals in the real world, such as urban growth, cells, cancers, etc., all of which are invaluable to research. This theoretical framework may shed light on a factor overlooked in past research: the trend of how fractals change. In the second part of this paper, we apply this theoretical framework to study the urbanization of Boston. We compare several maps and measurements of fractal dimensions, produce code11https://github.com/yinj66/fractal-dimension. that reads the maps and divides the city into subsections, and ultimately graph the fractal dimension over time using both differential and difference equations. Finally, we postulate the logistic equation as a model to fit the evolution of the fractal dimension, as well as the total population derived from census data, which serves as a component for comparing dynamical fractals.

(Invited) Using Molecular Dynamics Simulations to Relate Morphology to Water and Ion Dynamics in Hydrated Ion-Conducting Polymers

Hydrated ion-containing polymer membranes are used as electrolytes in a variety of applications, including in fuel cells and in alkaline batteries. Achieving high conductivity in hydrated, ion-conducting polymers typically requires the hydrophobic backbone of the polymers to be nano-phase separated from the hydrophilic domains so that the ions (protons or hydroxide ions) have well-defined water domains for efficient transport. The ion transport thus depends strongly on the hydration level and on the detailed morphology of the hydrophilic domains. It is often difficult to determine the morphology of these domains experimentally. I will present results from atomistic molecular dynamics (MD) simulations that reveal the domain morphology in various ionic polymers. Increasing water content generally leads to an increase in the characteristic spacing between hydrophilic domains. However, this swelling can be obscured in X-ray scattering due to a loss of scattering contrast between the hydrophobic and hydrophilic domains. The MD simulations reveal that the systems are nano-phase separated, even when X-ray scattering appears to indicate otherwise. Calculated and measured scattering profiles are in good qualitative agreement. I will discuss several methods to characterize the water domain morphology in more detail, including various measures of pore size distributions, surface areas, and fractal dimension. Water and ion diffusivities calculated from the MD simulations follow the experimentally-measured trends in conductivity and are consistent with changes in the nanoscale morphology with changing water content. I will discuss the implications of the results for the future design of ion-conducting polymers for use as membranes in electrochemical devices.This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE’s National Nuclear Security Administration under contract DE-NA-0003525. Figure 1