Fractal Dimension Research Articles

How does the direction of region of interest selection affect the fractal dimension?

How does the direction of region of interest selection affect the fractal dimension?

Relationship Between Salt Accumulation and Soil Structure Fractals in Cotton Fields in an Arid Inland Basin

The relationship between soil structure and salt accumulation is unclear; thus, experiments on salt accumulation under different soil structures were conducted in cotton fields in arid areas of northwest China. Thirty-nine sets of soil samples were collected from the 0 to 180 cm profile of three experimental areas. The total salt content of the soil extracts and the particle size distribution of the soil samples were determined using a JENCO TDS and a laser particle size analyzer, respectively, and the fractal dimension of the soil structure was obtained using fractal theory. Pearson’s correlation analysis and Tukey’s test (p < 0.01) were used to analyze the correlation between soil salinity, soil particle size distribution, and fractal dimensions in the three profiles. The results showed soil salinity accumulation was affected mutually by soil texture and soil structure, and soil salinity tended to accumulate in fine-grained soil. The soil fractal dimension (D) could indicate soil texture and quantify soil salinity content. When the sand content was more than 50%, there was a significant positive correlation between the soil fractal dimension and soil salinity (correlation coefficient R = 0.943). The results provide valuable insights into cotton production in arid areas.

Interfaces of the two-dimensional voter model in the context of SLE

Abstract This paper investigates various geometrical properties of interfaces of the two-dimensional voter model. Despite its simplicity, the model exhibits dual characteristics, resembling both a critical system with long-range correlations, while also showing a tendency towards order similar to the Ising–Glauber model at zero temperature. This duality is reflected in the geometrical properties of its interfaces, which are examined here from the perspective of Schramm–Loewner evolution. Recent studies have delved into the geometrical properties of these interfaces within different lattice geometries and boundary conditions. We revisit these findings, focusing on a system within a box of linear size L with Dobrushin boundary conditions, where values of the spins are fixed to either +1 or −1 on two distinct halves of the boundary, in order to enforce the presence of a pinned interface with fixed endpoints (or chordal interface). We also expand the study to compare the geometrical properties of the interfaces of the voter model with those of the critical Ising model and other related models. Scaling arguments and numerical studies suggest that, while locally the chordal interface of the voter model has fractal dimension d f = 3 / 2 , corresponding to a parameter κ = 4, it becomes straight at large scales, confirming a conjecture made by Holmes et al (2015 J. Stat. Phys. 159 937–57), and ruling out the possibility of describing the chordal interface of the voter model by SLE κ , for any non-zero value of κ. This contrasts with the critical Ising model, which is described by SLE3, and whose interface fluctuations remain of order L, and more generally with related critical models, which are in the same universality class.

Abstract 4113951: Calcification Alters Luminal Surface Complexity in Cadaveric Left Anterior Descending Arteries

Background: Understanding the intricacies of luminal surface topographic changes in pathologic coronary arteries may yield insight not only on the efficacy of existing clinical interventions (e.g. impact of surface changes on stent anchoring), but also on the subsequent hemodynamic changes and pathophysiology of thrombus formation. In this study, we aim to detect and quantify luminal surface alterations induced by the presence of calcification in cadaveric left anterior descending arteries (LADs) using a novel approach–surface metrology. Methods: LADs (n=10) were harvested from cadaveric hearts using a systematic dissection approach, scanned using the Bruker Skyscan 1173 microCT scanner, and then underwent threshold-based image segmentation on Dragonfly by Object Research Systems to quantify the total calcium volume present throughout the vessel. Each specimen was then splayed open and up to 15 scans were performed at 20X magnification using the Sensofar S Neox optical profiler. Each scan subsequently underwent surface metrologic scale-sensitive fractal analyses in SensoMap 10. Results: Extent of calcification was standardized and expressed using the following ratio: total calcium volume by total vessel length (TC/VL). We considered a TC/VL ratio of 0-0.25 as low calcification and 0.26-0.5 as high calcification. Interestingly, our findings demonstrated statistically significant increases in smooth-rough crossover (SRC) and scale of maximum complexity (Smfc) with positive correlation, but reductions in fractal complexity (Lsfc) and fractal dimension (Dls) with negative correlation, in highly calcified compared to lowly calcified LADs. Conclusions: Our data suggests that the presence of calcium in the walls of LADs may translate to increased roughness in luminal surfaces generally, but smoothness over calcified plaques, locally. Our proposed mechanism is increased endothelial tension directly over areas of calcification, thus reducing surface complexity. Future studies will involve modeling blood flow using smooth particle hemodynamics (SPH) based on complete luminal and surface topographical geometries to evaluate the impact of these surface changes on hemodynamics.

Fractal dimensions of fractal transformations and quantization dimensions for bi-Lipschitz mappings

In this paper, we study the fractal dimension of the graph of a fractal transformation and also determine the quantization dimension of a probability measure supported on the graph of the fractal transformation. Moreover, we estimate the quantization dimension of the invariant measure corresponding to a weighted iterated function system consisting of bi-Lipschitz mappings under the strong open set condition.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

Fractal Geometry in Vertebrate Respiratory Systems: A Comparative Analysis of Branching Patterns Across Species

This study investigates the application of fractal geometry to the analysis of vertebrate respiratory systems, focusing on the branching patterns of the bronchial tree across diverse species. Using a combination of mathematical modeling and computational visualization, we explore how fractal dimensions and branching characteristics reflect evolutionary adaptations to various habitats and physiological demands. The research encompasses a range of vertebrates, including humans, horses, dolphins, chickens, iguanas, and bullfrogs, representing diverse taxonomic groups and ecological niches. Our analysis reveals a clear gradient of complexity in respiratory structures, correlating strongly with metabolic rates and environmental adaptations. Mammals, particularly those adapted for high‐performance activities like horses, demonstrate the most complex fractal patterns, indicating highly efficient gas exchange systems. In contrast, ectothermic species such as iguanas and bullfrogs exhibit simpler structures, reflecting their lower metabolic demands and alternative respiratory strategies. The study employs fractal dimension calculations, L‐system modeling, and lacunarity analysis to quantify and visualize these differences. Our findings suggest that fractal analysis provides valuable insights into the relationship between form and function in vertebrate respiratory systems, offering a novel perspective on comparative physiology and evolutionary biology. This research not only enhances our understanding of respiratory system evolution but also has potential applications in biomimetic design, medical diagnostics, and ecological physiology. It demonstrates the power of interdisciplinary approaches, combining advanced mathematics with biological principles to uncover fundamental patterns in nature.

The impact of the type of playful massage movements on the perception of tactile stimulation in children: EEG study

Tactile play and playful massage are a normal interaction between adults and children in many cultures. Experimental data show that activation of the C-tactile system during such playful touches contributes to children’s socialization and the establishment of psychoemotional bonds in the parent-child dyad. However, comprehensive studies of the effects of different types of touch on perception of tactile stimulation in children have not been conducted before. Twenty-three children participated in the present study: 10 in the preschool age group and 13 in the elementary school age group. Children in both groups received play massage containing different types of touch with simultaneous recording of electroencephalogram (EEG) and heart rate. EEG processing included determination of spectral power, alpha rhythm peak frequency and fractal dimension. Analysis of the results showed that the relaxing effect of the procedure was more pronounced in children of primary school age, which can be explained both by the greater maturity of the CNS and the greater need for tactile contact, which is not realized in the school environment.

Effect of Thick–Hard Main Roof Fracturing on the Spatiotemporal Evolution of Overburden Fractures

Gas gushing and the co-mining of coal and gas outbursts are common disasters in coal mines. The weakening of thick–hard main roof fracturing can effectively alleviate the gas overrun in the initial mining stage. Four numerical simulation schemes were designed to determine the optimal fracturing treatment scheme for the gas overrun problem in the initial mining stage of the 6505# working face in the Huanghou Coal Mine. Compared with the scheme of the non-fracturing main roof, the weakening of the main roof fracturing could alleviate the concentration of advanced abutment stress. The fissure ratio of the overburden was solved by the binary method, and the dominant channels of gas migration and their morphological evolution under different fracturing schemes were obtained. On this basis, the permeability model of “permeability–fissure ratio–fractal dimension” was established. The results show that the overburden fracture network of the horizontal fracturing scheme has a larger fractal dimension, indicating that the overburden fracture under the scheme is more fully developed and more conducive to gas migration. The research results provide valuable reference for the design of the gas extraction scheme.

Anomalous scaling of branching tidal networks in global coastal wetlands and mudflats

Branching networks are key elements in natural landscapes and have attracted sustained research interest across the geosciences and numerous intersecting fields. The prevailing consensus has long held that branching networks are optimized and exhibit fractal properties adhering to power-law scaling relationships. However, tidal networks in coastal wetlands and mudflats exhibit scaling properties that defy conventional power-law descriptions, presenting a longstanding enigma. Here we show that the observed atypical scaling represents a universal deviation from an ideal fractal branching network capable of fully occupying the available space. Using satellite imagery of tidal networks from diverse global locations, we identified an inherent “laziness” in this deviation—where the increased ease of channel formation paradoxically decreases the space-filling efficiency of the network. We developed a theoretical model that reproduces the ideal fractal branching network and the laziness phenomenon. The model suggests that branching networks can emerge under a localized competition principle without adhering to conventionally assumed optimization-driven processes. These results reveal the dual nature of branching networks, where “laziness” complements the well-known optimization process. This property provides more flexible strategies for controlling tidal network morphogenesis, with implications for coastal management, wetland restoration, and studies in fluvial and planetary systems.

Retinal morphology across the menstrual cycle: insights from the UK Biobank

Oestradiol and progesterone levels are higher in menstruating women than men of the same age, and their receptors are present in their neurosensory retina and retinal pigment epithelium. However, the impact of this hormonal environment on retinal physiology in women remains unclear. Using self-reported menstrual cycle phases as a surrogate for fluctuating hormonal levels, we investigated associations with retinovascular indices on colour fundus photograph and retinal thickness in optical coherence tomography across regularly menstruating women in the UK Biobank. We found no differences in retinal thickness across the cycle; however, vessel density, arteriolar and venular, and fractal dimension were higher in the luteal phase than follicular. The calibre of the central retinal vessels did not differ. This study suggests that the menstrual cycle phase might be associated with retinal microvasculature density in non-invasive imaging. It raises awareness for this understudied area, providing insights into neuroscience fields and epidemiological studies.

Testing floc settling velocity models in rivers and freshwater wetlands

Abstract. Flocculation controls mud sedimentation and organic carbon burial rates by increasing mud settling velocity. However, calibration and validation of floc settling velocity models in freshwater are lacking. We used a camera, in situ laser diffraction particle sizing, and suspended sediment concentration–depth profiles to measure flocs in Wax Lake Delta, Louisiana. We developed a new workflow that combines our multiple floc data sources to distinguish between flocs and unflocculated sediment and measure floc attributes that were previously difficult to constrain. Sediment finer than ∼10 to 55 µm was flocculated with median floc diameter of 30 to 90 µm, bulk solid fraction of 0.05 to 0.3, fractal dimension of ∼2.1, and floc settling velocity of ∼0.1 to 1 mm s−1, with little variation along water depth. Results are consistent with a semi-empirical model indicating that sediment concentration and mineralogy, organics, water chemistry, and, above all, turbulence control floc settling velocity. Effective primary particle diameter is ∼2 µm, about 2 to 6 times smaller than the median primary particle diameter, and is better described using a fractal theory. Flow through the floc increases settling velocity by an average factor of 2 and up to a factor of 7 and can be described by a modified permeability model that accounts for the effect of many primary particle sizes on flow paths. These findings help explain discrepancies between observations and an explicit settling model based on Stokes' law that depends on floc diameter, permeability, and fractal properties.

SLOctolyzer: Fully Automatic Analysis Toolkit for Segmentation and Feature Extracting in Scanning Laser Ophthalmoscopy Images.

The purpose of this study was to introduce SLOctolyzer: an open-source analysis toolkit for en face retinal vessels in infrared reflectance scanning laser ophthalmoscopy (SLO) images. SLOctolyzer includes two main modules: segmentation and measurement. The segmentation module uses deep learning methods to delineate retinal anatomy, and detects the fovea and optic disc, whereas the measurement module quantifies the complexity, density, tortuosity, and caliber of the segmented retinal vessels. We evaluated the segmentation module using unseen data and measured its reproducibility. SLOctolyzer's segmentation module performed well against unseen internal test data (Dice for all-vessels = 0.91; arteries = 0.84; veins = 0.85; optic disc = 0.94; and fovea = 0.88). External validation against severe retinal pathology showed decreased performance (Dice for arteries = 0.72; veins = 0.75; and optic disc = 0.90). SLOctolyzer had good reproducibility (mean difference for fractal dimension = -0.001; density = -0.0003; caliber = -0.32 microns; and tortuosity density = 0.001). SLOctolyzer can process a 768 × 768 pixel macula-centered SLO image in under 20 seconds and a disc-centered SLO image in under 30 seconds using a laptop CPU. To our knowledge, SLOctolyzer is the first open-source tool to convert raw SLO images into reproducible and clinically meaningful retinal vascular parameters. It requires no specialist knowledge or proprietary software, and allows manual correction of segmentations and re-computing of vascular metrics. SLOctolyzer is freely available at https://github.com/jaburke166/SLOctolyzer. SLO images are captured simultaneous to optical coherence tomography (OCT), and we believe SLOctolyzer will be useful for extracting retinal vascular measurements from large OCT image sets and linking them to ocular or systemic diseases.

The Applicability and Reflection Characteristics of Coal Failure Events for External Monitoring-While-Drilling of Underground Pressure Relief Drilling

Previous research results preliminarily indicated that the Coal Failure Events (CFEs) that occurred during the process of Underground Pressure Relief Drilling (UPRD) represented the phenomenon of coal fracture and energy release. The research results had excellent value for the monitoring and response of pressure relief drilling while drilling, but there were still some special situations that needed to be analyzed and studied in actual on-site testing. So, through on-site testing and data statistical analysis, the study investigated the applicability of the innovative external Monitoring-While-Drilling (MWD) method for UPRD with more coal failure events and made a quantitative statistic of the CFEs and their relationship with abutment pressure to reveal the applicability of the external MWD method and characteristic of CFEs. The results showed that hundreds of CFEs were produced in the UPRD process, which must be removed to ensure the accuracy of the MWD method. Although CFEs bring recognition difficulties, they also provide conditions for studying their own distribution and characteristics. Results showed that more CFEs were produced in the depth of difficult drilling, which indicated that there was a positive correlation between the degree of difficulty in drilling and the number of CFEs. In addition, spectrum analysis showed that the depths with more CFE occurrence were more likely to produce high-frequency events. When the surrounding stress of drilling rocks is high, the occurrence of small fractures with a higher main frequency may become more frequent and consistent; more fractures with similar failure forms would occur, which may have a lower fractal dimension and promote the generation of more failure. The research results were of great significance for the MWD method for UPRD, a quantitative study of CFEs and their generation characteristics during UPRD construction.

A numerical simulation study on the characteristics of wrap shell lightweight aggregate concrete based on random aggregate model

It is a way to reuse solid waste by using dredged silt as raw material to prepare unburned wrap shell lightweight aggregates and replacing crushed stone aggregate to prepare unburned wrap shell lightweight aggregates concrete specimen. In order to understand the characteristics of wrap shell lightweight aggregates concrete, this paper adopts the method of combining laboratory test and numerical simulation to design the mix ratio test scheme of concrete specimens with different substitution rates of wrap shell aggregate (0 %, 25 %, 50 %, 75 %, 100 %), and conducts indoor mechanical properties test and numerical simulation simulation test on them, so as to obtain and analyze the mechanical properties and failure characteristics.The results show that:(1) The peak stress intensity of concrete specimens decreases linearly with the increase of the substitution rate of wrap shell aggregate.(2) The peak stress of concrete under different wrap shell round aggregate substitution rates is normally distributed, and the dispersion of peak strength decreases with the increase of wrap shell aggregate substitution rate. When the substitution rate is 0 %, the dispersion of peak strength is the largest.(3) Under the uniaxial compression simulation test, the failure mode of concrete specimens changes with the addition of the substitution rate of wrap shell aggregate. When the crack failure occurs, natural coarse aggregate with high strength is chosen to bypass, while when it comes to wrap shell aggregate with low strength, it is chosen to penetrate. With the continuous addition of round aggregate, the stress concentration phenomenon inside concrete can be effectively improved and the number of micro-cracks can be reduced.But at the same time, the penetration cracks gradually increase and gradually tend to be straight.The fractal dimension is negatively correlated with the compressive strength and positively correlated with the substitution rate of wrap shell aggregate.The above research can provide theoretical basis for the popularization of recycled concrete with wrap shell aggregate.

Long-term skid resistance evolution and influence mechanism of asphalt pavement based on self-developed wear equipment

Skid resistance decay is the critical functional distress of asphalt pavement, directly restricting its safety and service life. In order to grasp the characteristics and mechanism of long-term skid resistance evolution, this study developed a laboratory wheel-accelerated wear equipment (WAWE) and its testing methodology for asphalt pavement and coarse aggregate. Five coarse aggregates and five gradations combinations were selected to investigate the evolution. The accuracy of the evolution model and the loading conversion relationship were analyzed based on the Research Institute of Highway (RIOH) Track. Finally, the pavement anti-skid texture tester (PATT) was used to investigate the evolution characteristics of coarse aggregate microtexture. Results showed that the Asymptotic evolution model could characterize the skid resistance under the influence of coarse aggregate and gradation. RIOH Track verified the model's accuracy and showed that a single WAWE wear was equivalent to 265 loading times on actual pavement. The long-term skid resistance could be improved by increasing the nominal maximum particle size and the gradation fractal dimension indexes D and Df. The microtexture roughness of the coarse aggregate decreased after wear, then stabilized and fluctuated, which showed "regeneration" characteristics. The study can guide asphalt mixture design and maintenance timing decisions based on long-term skid resistance.

Complex shape markers can detect alterations in the spatial distribution of cell nuclei in human lung squamous cell carcinoma: a useful tool for automatic analysis?

Lung cancer is the leading cause of cancer-related death. The use of computational methods to quantify changes that are not perceptible to the human eye is increasing in digital pathology imaging and has quickly improved detection rates at a low cost. Therefore, the present study aims to use complex computational shape markers as tools for automated analysis of the spatial distribution of cells in microscopy images of squamous cell lung carcinoma (SqCC). Photomicrographs from pathology glass slides in the LC25000 dataset were used in this study. Compared with those of the control, the fractal dimension (28%) and lacunarity (41%) of the cell nuclei changed in SqCC. The multifractal analysis revealed a significant difference in parameters Dq, α, and f(α) for all values of q (-10 to + 10), with a greater increase for more positive q values. The values at q + 10 increased by 34% for Dq, 36% for α, and 53% for f(α) in the SqCC images. The circularity, area, and perimeter also changed in the SqCC images. However, the parameters of aspect ratio, roundness, and solidity did not significantly differ between SqCC and benign tissue. The complex shape markers with the greatest changes in this study were the f(α) values for multifractality (53%) and lacunarity (41%). In conclusion, automated quantification of the spatial distribution of cell nuclei can be a fast, low-cost tool for evaluating the microscopic characteristics of SqCC; therefore, complex shape markers could be useful tools for software and artificial intelligence to detect lung carcinoma.

Study on the Development Rule of Mudstone Cracks in Open-Pit Mine Dumps Improved with Xanthan Gum

The stability of open-pit mine slopes is crucial for safety, especially for spoil dump slopes, which are prone to cracks leading to landslides. This study investigates the use of xanthan gum (XG) to enhance the stability of mudstone in spoil dumps. Various concentrations of xanthan gum were mixed with mudstone and subjected to dry–wet cycle tests to assess the impact on crack development. Pore and crack analysis system (PCAS) was utilized for image recognition and crack analysis, comparing the efficiency of crack rate and length modification. The study found that xanthan gum addition significantly improved mudstone’s resistance to crack development post-drying shrinkage. A 2% xanthan gum content reduced the mudstone crack rate by 45% on average, while 1.5% xanthan gum reduced crack length by 46.2% and crack width by 26.3%. Xanthan gum also influenced the fractal dimension and water retention of mudstone cracks. The optimal xanthan gum content for mudstone modification was identified as between 1.5% and 2%. Scanning electron microscopy imaging and X-ray diffraction tests supported the findings, indicating that xanthan gum modifies mudstone by encapsulation and penetration in wet conditions and matrix concentration and connection in dry conditions. These results are expected to aid in the development of crack prevention methods and engineering applications for open-pit mine spoil dump slopes.

Chronic inhalation exposure to the herbicide 2,4-dichlorophenoxyacetic acid (2,4-D) alters cardiac collagen in Wistar rats

Abstract Introduction: 2,4-dichlorophenoxyacetic acid (2,4-D) is one of the most used in the world and exposure to herbicides can affect animals and humans, causing toxic effects that include cardiotoxicity. This is the first study to evaluate cardiac remodeling after experimental simulation of environmental exposure by chronic inhalation (6 months) to the herbicide 2,4-D. Methods: Thirty male Wistar rats were exposed to two different concentrations of the 2,4-D formulation (low – 187.17 mg/m3; and high – 467.93 mg/m3) and the control group exposed to nebulization of chloride solution 0.9% sodium. Inhalation exposure lasted 6 months. Mice hearts were collected for histology. Results: There was a difference between exposure concentrations in relation to the increase in cardiac collagen (P < 0.0001). In mice exposed to a low dose of 2,4-D and a decrease in the fractal dimension of cardiac collagen in the high dose of 2, 4-D (P = 0.010). Discussion: This study shows that chronic inhalation exposure to 2,4-D induces cardiotoxicity in rats, characterized by extracellular matrix reorganization and changes in collagen levels, highlighting the impact of the exposure route on cardiac outcomes. Conclusion: There was no difference in relation to anatomical parameters, cardiomyocyte area, collagen types I and III and analysis of arteriole thickness. Chronic exposure at different doses to the 2,4D herbicide had the potential to cause damage to cardiac remodeling by altering cardiac collagen in rats.

Quantifying the fractal complexity of nutrient transport channels in Escherichia coli biofilms under varying cell shape and growth environment.

Recent mesoscopic characterization of nutrient-transporting channels in Escherichia coli has allowed the identification and measurement of individual channels in whole mature colony biofilms. However, their complexity under different physiological and environmental conditions remains unknown. Analysis of confocal micrographs of colony biofilms formed by cell shape mutants of E. coli shows that channels have high fractal complexity, regardless of cell phenotype or growth medium. In particular, colony biofilms formed by the mutant strain ΔompR, which has a wide-cell phenotype, have a higher fractal dimension when grown on rich medium than when grown on minimal medium, with channel complexity affected by glucose and agar concentrations in the medium. Osmotic stress leads to a dramatic reduction in the ΔompR cell size but has a limited effect on channel morphology. This work shows that fractal image analysis is a powerful tool to quantify the effect of phenotypic mutations and growth environment on the morphological complexity of internal E. coli biofilm structures. If applied to a wider range of mutant strains, this approach could help elucidate the genetic determinants of channel formation in E. coli colony biofilms.