Multifractal Analysis of Brain Tumor Interface in Glioblastoma.

The dynamics of tumor growth is a very complex process, generally accompanied by numerous chromosomal aberrations that determine its genetic and dynamical heterogeneity. Consequently, the tumor interface exhibits a non-regular and heterogeneous behavior often described by a single fractal dimension. A more suitable approach is to consider the tumor interface as a multifractal object that can be described by a set of generalized fractal dimensions. In the present work, detrended fluctuation and multifractal analysis are used to characterize the complexity of glioblastoma.

Scale-Fractal Detrended Fluctuation Analysis for Fault Diagnosis of a Centrifugal Pump and a Reciprocating Compressor.

Reciprocating compressors and centrifugal pumps are rotating machines used in industry, where fault detection is crucial for avoiding unnecessary and costly downtime. A novel method for fault classification in reciprocating compressors and multi-stage centrifugal pumps is proposed. In the feature extraction stage, raw vibration signals are processed using multi-fractal detrended fluctuation analysis (MFDFA) to extract features indicative of different types of faults. Such MFDFA features enable the training of machine learning models for classifying faults. Several classical machine learning models and a deep learning model corresponding to the convolutional neural network (CNN) are compared with respect to their classification accuracy. The cross-validation results show that all models are highly accurate for classifying the 13 types of faults in the centrifugal pump, the 17 valve faults, and the 13 multi-faults in the reciprocating compressor. The random forest subspace discriminant (RFSD) and the CNN model achieved the best results using MFDFA features calculated with quadratic approximations. The proposed method is a promising approach for fault classification in reciprocating compressors and multi-stage centrifugal pumps.

Evaluating the roughness dynamics of kefir biofilms grown on Amazon cupuaçu juice: a monofractal and multifractal approach.

We conducted a comprehensive analysis of the surface microtexture of kefir biofilms grown on Theobroma grandiflorum Shum (cupuaçu) juice using atomic force microscopy. Our goal was to investigate the unique monofractal and multifractal spatial patterns of these biofilms to complement the existing limited literature. The biofilms were prepared dispersing four different concentrations of kefir grains in cupuaçu juice. Our morphological analysis showed that the surface of the obtained biofilms is essentially formed by the presence of cupuaçu fibers and microorganisms like lactobacilli and yeast. The topographic height-based parameter analysis reveals that there is a dependence between surface roughness and the concentration of kefir grains used. The strongly anisotropic well-centralized topographical height distribution of the biofilms also exhibited a quasi-symmetrical and platykurtic pattern. The biofilms exhibit comparable levels of spatial complexity, surface percolation and surface homogeneity, which can be attributed to their similar topographic uniformity. This aspect was further supported by the presence of similar multifractality in the biofilms, suggesting that despite their varying topographic roughness, their vertical growth dynamics follow a similar pattern. Our findings demonstrate that the surface roughness of kefir biofilms cultivated on cupuaçu juice is influenced by the concentration of kefir grains in the precursor solution. However, this dependence follows a consistent pattern across different concentrations. Graphical Abstract.

A new approach to assess the degree of contamination and determine sources and risks related to PTEs in an urban environment: the case study of Santiago (Chile).

In 2017, a geochemical survey was carried out across the Commune of Santiago, a local administrative unit located at the center of the namesake capital city of Chile, and the concentration of a number of major and trace elements (53 in total) was determined on 121 topsoil samples. Multifractal IDW (MIDW) interpolation method was applied to raw data to generate geochemical baseline maps of 15 potential toxic elements (PTEs); the concentration-area (C-A) plot was applied to MIDW grids to highlight the fractal distribution of geochemical data. Data of PTEs were elaborated to statistically determine local geochemical baselines and to assess the spatial variation of the degree of soil contamination by means of a new method taking into account both the severity of contamination and its complexity. Afterwards, to discriminate the sources of PTEs in soils, a robust Principal Component Analysis (PCA) was applied to data expressed in isometric log-ratio (ilr) coordinates. Based on PCA results, a Sequential Binary Partition (SBP) was also defined and balances were determined to generate contrasts among those elements considered as proxies of specific contamination sources (Urban traffic, productive settlements, etc.). A risk assessment was finally completed to potentially relate contamination sources to their potential effect on public health in the long term. A probabilistic approach, based on Monte Carlo method, was deemed more appropriate to include uncertainty due to spatial variation of geochemical data across the study area. Results showed how the integrated use of multivariate statistics and compositional data analysis gave the authors the chance to both discriminate between main contamination processes characterizing the soil of Santiago and to observe the existence of secondary phenomena that are normally difficult to constrain. Furthermore, it was demonstrated how a probabilistic approach in risk assessment could offer a more reliable view of the complexity of the process considering uncertainty as an integral part of the results.

Head-to-Tail and Head-to-Head Molecular Chains of Poly(p-Anisidine): Combined Experimental and Theoretical Evaluation.

Poly(p-anisidine) (PPA) is a polyaniline derivative presenting a methoxy (-OCH3) group at the para position of the phenyl ring. Considering the important role of conjugated polymers in novel technological applications, a systematic, combined experimental and theoretical investigation was performed to obtain more insight into the crystallization process of PPA. Conventional oxidative polymerization of p-anisidine monomer was based on a central composite rotational design (CCRD). The effects of the concentration of the monomer, ammonium persulfate (APS), and HCl on the percentage of crystallinity were considered. Several experimental techniques such as X-ray Diffraction (XRD), Scanning Electron Microscopy (SEM), multifractal analysis, Nuclear Magnetic Resonance (13C NMR), Fourier-transform Infrared spectroscopy (FTIR), and complex impedance spectroscopy analysis, in addition to Density Functional Theory (DFT), were employed to perform a systematic investigation of PPA. The experimental treatments resulted in different crystal structures with a percentage of crystallinity ranging from (29.2 ± 0.6)% (PPA1HT) to (55.1 ± 0.2)% (PPA16HT-HH). A broad halo in the PPA16HT-HH pattern from 2θ = 10.0-30.0° suggested a reduced crystallinity. Needle and globular-particle morphologies were observed in both samples; the needle morphology might have been related to the crystalline contribution. A multifractal analysis showed that the PPA surface became more complex when the crystallinity was reduced. The proposed molecular structures of PPA were supported by the high-resolution 13C NMR results, allowing us to access the percentage of head-to-tail (HT) and head-to-head (HH) molecular structures. When comparing the calculated and experimental FTIR spectra, the most pronounced changes were observed in ν(C-H), ν(N-H), ν(C-O), and ν(C-N-C) due to the influence of counterions on the polymer backbone as well as the different mechanisms of polymerization. Finally, a significant difference in the electrical conductivity was observed in the range of 1.00 × 10-9 S.cm-1 and 3.90 × 10-14 S.cm-1, respectively, for PPA1HT and PPA16HT-HH.

Nanoscale surface dynamics of spatial patterns of polymeric bilayered particles loaded with essential oil.

Gelatin/PCL bilayered particles loaded with Piper nigrum essential oil was synthesized aiming to access their morphological and surface dynamic patterns. Atomic force microscopy (AFM) was applied to investigate the 3D morphology and multifractal aspects of the particles surface. The AFM maps revealed spherical surfaces and well dispersed particles, besides a rougher surface on the loaded system. Minkowski functionals showed that shape of the rough peaks was similar in the unloaded and loaded systems; however, the presence of deep valleys on the loaded particles revealed their rougher pattern. Multifractal analysis revealed that unloaded and loaded particles presented multifractal behavior with different surface dynamics. The loaded surface presented a greater width of the multifractal spectrum and smaller difference of fractal dimensions, confirming their more vertically growing. These results can be useful in the development of novel polymeric-based particles loaded with essential oil. Their unique surface dynamics can provide enhanced physical properties and performance in emerging biotechnological applications.

Nonlinear Statistical Features of the Seismicity in the Subduction Zone of Tehuantepec Isthmus, Southern México.

After the M8.2 main-shock occurred on 7 September 2017 at the Isthmus of Tehuantepec, Mexico, the spatial distribution of seismicity has showed a clear clusterization of earthquakes along the collision region of the Tehuantepec Transform/Ridge with the Middle America Trench off Chiapas. Furthermore, nowadays, the temporal rate of occurrence in the number of earthquakes has also showed a pronounced increase. On the basis of this behavior, we studied the sequence of magnitudes of the earthquakes which occurred within the Isthmus of Tehuantepec in southern Mexico from 2010 to 2020. Since big earthquakes are considered as a phase transition, after the M8.2 main-shock, one must expect changes in the Tehuantepec ridge dynamics, which can be observed considering that the b-value in the Gutenberg-Richter law, has also showed changes in time. The goal of this paper is to characterize the behavior of the seismic activity by using the Gutenberg-Richter law, multifractal detrended fluctuation analysis, visibility graph and nowcasting method. Those methods have showed important parameters in order to assess risk, the multifractality and connectivity. Our findings indicate, first that b-value shows a dependency on time, which is clearly described by our analyses based on nowcasting method, multifractality and visibility graph.

A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect.

Stock markets can become inefficient due to calendar anomalies known as the day-of-the-week effect. Calendar anomalies are well known in the financial literature, but the phenomena remain to be explored in econophysics. This paper uses multifractal analysis to evaluate if the temporal dynamics of market returns also exhibit calendar anomalies such as day-of-the-week effects. We apply multifractal detrended fluctuation analysis (MF-DFA) to the daily returns of market indices worldwide for each day of the week. Our results indicate that distinct multifractal properties characterize individual days of the week. Monday returns tend to exhibit more persistent behavior and richer multifractal structures than other day-resolved returns. Shuffling the series reveals that multifractality arises from a broad probability density function and long-term correlations. The time-dependent multifractal analysis shows that the Monday returns' multifractal spectra are much wider than those of other days. This behavior is especially persistent during financial crises. The presence of day-of-the-week effects in multifractal dynamics of market returns motivates further research on calendar anomalies for distinct market regimes.

Scaling Analysis of an Image Encryption Scheme Based on Chaotic Dynamical Systems.

Despite that many image encryption systems based on chaotic or hyperchaotic systems have been proposed to protect different kinds of information, it has been crucial to achieve as much security as possible in such systems. In this sense, we numerically implement a known image encryption system with some variants, making special emphasis when two operations are considered in the scrambling stage. The variants of such an encryption system are based on some hyperchaotic systems, which generated some substitution boxes and the keys of the system. With the aim to have a more complete evaluation, some internal stages of the image encryption scheme have been evaluated by using common statistical tests, and also the scaling behavior of the encrypted images has been calculated by means of a two-dimensional detrended fluctuation analysis (2D-DFA). Our results show that the image encryption systems that include two operations or transformations in the scrambling stage present a better performance than those encryption systems that consider just one operation. In fact, the 2D-DFA approach was more sensitive than some common statistical tests to determine more clearly the impact of multiple operations in the scrambling process, confirming that this scaling method can be used as a perceptual security metric, and it may contribute to having better image encryption systems.

Multifractal Characteristics of Geomagnetic Field Fluctuations for the Northern and Southern Hemispheres at Swarm Altitude.

This paper explores the spatial variations of the statistical scaling features of low to high latitude geomagnetic field fluctuations at Swarm altitude. The data for this study comes from the vector field magnetometer onboard Swarm A satellite, measured at low resolution (1 Hz) for one year (from 9 March 2016, to 9 March 2017). We estimated the structure-function scaling exponents using the p-leaders discrete wavelet multifractal technique, from which we obtained the singularity spectrum related to the magnetic fluctuations in the North-East-Center (NEC) coordinate system. From this estimation, we retain just the maximal fractal subset, associated with the Hurst exponent H. Here we present thresholding for two levels of the Auroral Electrojet index and almost the whole northern and southern hemispheres, the Hurst exponent, the structure-function scaling exponent of order 2, and the multifractal p-exponent width for the geomagnetic fluctuations. The latter quantifies the relevance of the multifractal property. Sometimes, we found negative values of H, suggesting a behavior similar to wave breaking or shocklet-like propagating front. Furthermore, we found some asymmetries in the magnetic field turbulence between the northern and southern hemispheres. These estimations suggest that different turbulent regimes of the geomagnetic field fluctuations exist along the Swarm path.

Multifractal Spectrum Curvature of RR Tachograms of Healthy People and Patients with Congestive Heart Failure, a New Tool to Assess Health Conditions.

We calculate the multifractal spectra of heartbeat RR-interval time series (tachograms) of healthy subjects and patients with congestive heart failure (CHF). From these time series, we obtained new subseries of 6 h durations when healthy persons and patients were asleep and awake respectively. For each time series and subseries, we worked out the multifractal spectra with the Chhabra and Jensen method and found that their graphs have different shapes for CHF patients and healthy persons. We suggest to measure two parameters: the curvature around the maximum and the symmetry for all these multifractal spectra graphs, because these parameters were different for healthy and CHF subjects. Multifractal spectra of healthy subjects tend to be right skewed especially when the subjects are asleep and the curvature around the maximum is small compared with the curvature around the maximum of the CHF multifractal spectra; that is, the spectra of patients tend to be more pointed around the maximum. In CHF patients, we also have encountered differences in the curvature of the multifractal spectra depending on their respective New York Heart Association (NYHA) index.

Information Dynamics in Urban Crime.

Information production in both space and time has been highlighted as one of the elements that shapes the footprint of complexity in natural and socio-technical systems. However, information production in urban crime has barely been studied. This work copes with this problem by using multifractal analysis to characterize the spatial information scaling in urban crime reports and nonlinear processing tools to study the temporal behavior of this scaling. Our results suggest that information scaling in urban crime exhibits dynamics that evolve in low-dimensional chaotic attractors, and this can be observed in several spatio-temporal scales, although some of them are more favorable than others. This evidence has practical implications in terms of defining the characteristic scales to approach urban crime from available data and supporting theoretical perspectives about the complexity of urban crime.

Three-dimensional multifractal analysis of trabecular bone under clinical computed tomography.

PURPOSE: An adequate understanding of bone structural properties is critical for predicting fragility conditions caused by diseases such as osteoporosis, and in gauging the success of fracture prevention treatments. In this work we aim to develop multiresolution image analysis techniques to extrapolate high-resolution images predictive power to images taken in clinical conditions. METHODS: We performed multifractal analysis (MFA) on a set of 17 ex vivo human vertebrae clinical CT scans. The vertebrae failure loads (FFailure) were experimentally measured. We combined bone mineral density (BMD) with different multifractal dimensions, and BMD with multiresolution statistics (e.g., skewness, kurtosis) of MFA curves, to obtain linear models to predict FFailure. Furthermore we obtained short- and long-term precisions from simulated in vivo scans, using a clinical CT scanner. Ground-truth data - high-resolution images - were obtained with a High-Resolution Peripheral Quantitative Computed Tomography (HRpQCT) scanner. RESULTS: At the same level of detail, BMD combined with traditional multifractal descriptors (Lipschitz-Hölder exponents), and BMD with monofractal features showed similar prediction powers in predicting FFailure (87%, adj. R2 ). However, at different levels of details, the prediction power of BMD with multifractal features raises to 92% (adj. R2) of FFailure. Our main finding is that a simpler but slightly less accurate model, combining BMD and the skewness of the resulting multifractal curves, predicts 90% (adj. R2) of FFailure. CONCLUSIONS: Compared to monofractal and standard bone measures, multifractal analysis captured key insights in the conditions leading to FFailure. Instead of raw multifractal descriptors, the statistics of multifractal curves can be used in several other contexts, facilitating further research.

Multifractal spectrum and lacunarity as measures of complexity of osseointegration.

OBJECTIVES: The goal of this study is to contribute to a better quantitative description of the early stages of osseointegration, by application of fractal, multifractal, and lacunarity analysis. MATERIALS AND METHODS: Fractal, multifractal, and lacunarity analysis are performed on scanning electron microscopy (SEM) images of titanium implants that were first subjected to different treatment combinations of i) sand blasting, ii) acid etching, and iii) exposition to calcium phosphate, and were then submersed in a simulated body fluid (SBF) for 30 days. All the three numerical techniques are applied to the implant SEM images before and after SBF immersion, in order to provide a comprehensive set of common quantitative descriptors. RESULTS: It is found that implants subjected to different physicochemical treatments before submersion in SBF exhibit a rather similar level of complexity, while the great variety of crystal forms after SBF submersion reveals rather different quantitative measures (reflecting complexity), for different treatments. In particular, it is found that acid treatment, in most combinations with the other considered treatments, leads to a higher fractal dimension (more uniform distribution of crystals), lower lacunarity (lesser variation in gap sizes), and narrowing of the multifractal spectrum (smaller fluctuations on different scales). CONCLUSION: The current quantitative description has shown the capacity to capture the main features of complex images of implant surfaces, for several different treatments. Such quantitative description should provide a fundamental tool for future large scale systematic studies, considering the large variety of possible implant treatments and their combinations. CLINICAL RELEVANCE: Quantitative description of early stages of osseointegration on titanium implants with different treatments should help develop a better understanding of this phenomenon, in general, and provide basis for further systematic experimental studies. Clinical practice should benefit from such studies in the long term, by more ready access to implants of higher quality.

Multifractal dimension and lacunarity of yolk sac vasculature after exposure to magnetic field.

Several studies have reported about the effects of magnetic fields (MFs) on vascular tissue. Extremely low frequency magnetic fields (ELF-MFs) can promote either inhibition or stimulation of vasculogenesis and angiogenesis, depending upon the intensity and time of exposure to the MF. To investigate the possible effects of ELF-MF on vascular processes, it is necessary to employ methods that allow parameterization of the vascular network. Vascular network is a structure with fractal geometry; therefore, fractal methods have been used to evaluate its morphometric complexity. Here, we used the lacunarity parameter (complementary method of fractal analysis) and multifractal analyses to investigate angiogenesis and vasculogenesis in the embryonic yolk sac membrane (YSM) of Japanese quails (Coturnix japonica) with and without exposure to an external MF of 1 mT and 60 Hz. Lacunarity results showed that the vascular density was lower for the group exposed to the magnetic field for 9 h/day. In addition, multifractal analysis showed reduced vascularization in the experimental groups (6 h/day and 9 h/day of exposure to MF). Furthermore, multifractal analysis showed difference between the groups exposed for 12 and 24 h/day. Using multifractal methods (generalized dimensions and singularity spectrum), it was possible to characterize the vascular network of the quail embryo YSM as a multifractal object, therefore proving this method to be a more appropriate application than the traditional monofractal methods.