Multifractality approach of a generalized Shannon index in financial time series.

Multifractality is a concept that extends locally the usual ideas of fractality in a system. Nevertheless, the multifractal approaches used lack a multifractal dimension tied to an entropy index like the Shannon index. This paper introduces a generalized Shannon index (GSI) and demonstrates its application in understanding system fluctuations. To this end, traditional multifractality approaches are explained. Then, using the temporal Theil scaling and the diffusive trajectory algorithm, the GSI and its partition function are defined. Next, the multifractal exponent of the GSI is derived from the partition function, establishing a connection between the temporal Theil scaling exponent and the generalized Hurst exponent. Finally, this relationship is verified in a fractional Brownian motion and applied to financial time series. In fact, this leads us to proposing an approximation called local fractional Brownian motion approximation, where multifractal systems are viewed as a local superposition of distinct fractional Brownian motions with varying monofractal exponents. Also, we furnish an algorithm for identifying the optimal q-th moment of the probability distribution associated with an empirical time series to enhance the accuracy of generalized Hurst exponent estimation.

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

BACKGROUND: Changes in bone trabeculae occur during skeletal maturation. Fractal analysis is a technique for assessing changes in the internal structure of the bone. OBJECTIVE: 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). METHODS: 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). RESULTS: 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). CONCLUSION: 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.

Being fractal: Embodying antiracism values in course-based participatory action research.

In the winter and spring of 2021, I-a White, female, graduate student-taught a six-month course surrounding the theme: Disrupting Systemic Racism at our University Through Action Research. I was challenged to lead a meaningful course in a two-dimensional virtual space, amidst rising death tolls of the COVID-19 pandemic and the rhythmic beat of calls for racial justice pulsing through our Zoom class periods. This experience opened my eyes as an educator, budding community psychologist, and an antiracist White accomplice. In this critical autoethnographic case study, I recount my experience adapting the community organizing principle of fractals into a pedagogical framework that guided my instructional practices in a community psychology course. In doing so, I echo the call for community psychologists to connect our work more tightly to Black, Indigenous, and people of Color social justice organizers and movements to fortify the field's relevance in the struggle for racial justice.

Assessment of spatiotemporal characteristics of gait, trough the Phyphox® app: a case series.

BACKGROUND: Spatiotemporal characteristics from human locomotion can provide effective clinical metrics to assess motor control and brain function. This case report aims to assess the temporal structure of variability in stride-to-stride time and calculated the intrinsic fractal frame that is hidden below the repetitive structure of physiological gait through the "Phyphox" app. This is an innovative study from the perspective of analyzing gait variables through a mobile app. CASE PRESENTATION: Five older adults Caucasian (3 women; age = 73 ± 10,5 years; body mass = 62,2 ± 15,1 kg; height = 1,56 ± 0,1 m; 2 men; age = 75,5 ± 7,8 years; body mass = 86,3 ± 18,0 kg; height = 1,77 ± 0,1 m) participated in this study. Five participants were asked to walk with a natural cadence, two of the participants presented a value greater than 80 step's/minute (81.14 ± 0.01; 86.67 ± 0.02); and the others had values between 55 and 65 step's/minute (55.20 ± 0.02; 55.78 ± 0.05; 61.02 ± 0.05). Regarding the coefficient of variation, only one participant presented 10.08%. For the total number of steps, three of the participants had values greater than 1000 steps. The variability of these stride-to-stride time has been quantified through detrended fluctuation analysis; one participant presented a value above 1. CONCLUSIONS: This study provides evidence that a smartphone might provide a valid measure to assess the spatiotemporal characteristics of gait.

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

BACKGROUND: To investigate whether fractal dimension (FD) measurements from hand-wrist radiographs and lateral cephalometric radiographs are correlated with each other and with skeletal maturation stages. METHODS: In 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. RESULTS: Positive 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). CONCLUSIONS: FD 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.

Advancing dentistry: fractal assessment of bone health in pediatric patients with celiac disease using dental images.

OBJECTIVES: To assess the effects of a gluten-free diet on bone structure in children with celiac disease using fractal analysis on panoramic radiographs. METHOD AND MATERIALS: A total of 49 patients with celiac disease aged 6 to 13 years, separated into two groups as previously and newly diagnosed, and a control group of 32 healthy individuals were evaluated. In previously and newly diagnosed patients with celiac disease, body mass index Z-scores were calculated, calcium, alkaline phosphatase, vitamin D3, and parathormone levels were measured, and bone mineral density Z-scores were obtained from dual energy x-ray absorptiometry. In all patients, the fractal dimensions of the right and left temporomandibular condyles were evaluated with the fractal analysis method on panoramic radiographs. RESULTS: The mean values of serum biomarker levels and the body mass index and bone mineral density Z-scores for both celiac groups were within the normal reference range. No statistically significant difference was determined between right and left condyle fractal dimensions values in the three groups examined. In terms of both right and left condyle fractal dimensions values, there was a statistically significant difference between groups. The highest fractal dimensions values were determined in the previously diagnosed group. CONCLUSIONS: Differences in fractal dimensions values were observed among patients with celiac disease following the gluten-free diet. Utilizing fractal analysis on panoramic radiographs can prove valuable for dental practitioners in evaluating bone mineral density due to its cost-effectiveness, easy accessibility, and reduced radiation exposure for patients, enabling them to provide comprehensive oral health care and potential early interventions for patients with celiac disease.

Assessment of storm impact on coral reef structural complexity.

Extreme weather events are increasing in frequency and magnitude. Consequently, it is important to understand their effects and remediation. Resilience reflects the ability of an ecosystem to absorb change, which is important for understanding ecological dynamics and trajectories. To describe the impact of a powerful storm on coral reef structural complexity, we used novel computational tools and detailed 3D reconstructions captured at three time points over three years. Our data-set Reefs4D of 21 co-registered image-based models enabled us to calculate the differences at seven sites over time and is released with the paper. We employed six geometrical metrics, two of which are new algorithms for calculating fractal dimension of reefs in full 3D. We conducted a multivariate analysis to reveal which sites were affected the most and their relative recovery. We also explored the changes in fractal dimension per size category using our cube-counting algorithm. Three metrics showed a significant difference between time points, i.e., decline and subsequent recovery in structural complexity. The multivariate analysis and the results per size category showed a similar trend. Coral reef resilience has been the subject of seminal studies in ecology. We add important information to the discussion by focusing on 3D structure through image-based modeling. The full picture shows resilience in structural complexity, suggesting that the reef has not gone through a catastrophic phase shift. Our novel analysis framework is widely transferable and useful for research, monitoring, and management.

Economic Choice and Heart Rate Fractal Scaling Indicate That Cognitive Effort Is Reduced by Depression and Boosted by Sad Mood.

BACKGROUND: People with depression typically exhibit diminished cognitive control. Control is subjectively costly, prompting speculation that control deficits reflect reduced cognitive effort. Evidence that people with depression exert less cognitive effort is mixed, however, and motivation may depend on state affect. METHODS: We used a cognitive effort discounting task to measure propensity to expend cognitive effort and fractal structure in the temporal dynamics of interbeat intervals to assess on-task effort exertion for 49 healthy control subjects, 36 people with current depression, and 67 people with remitted depression. RESULTS: People with depression discounted more steeply, indicating that they were less willing to exert cognitive effort than people with remitted depression and never-depressed control subjects. Also, steeper discounting predicted worse functioning in daily life. Surprisingly, a sad mood induction selectively boosted motivation among participants with depression, erasing differences between them and control subjects. During task performance, depressed participants with the lowest cognitive motivation showed blunted autonomic reactivity as a function of load. CONCLUSIONS: Discounting patterns supported the hypothesis that people with current depression would be less willing to exert cognitive effort, and steeper discounting predicted lower global functioning in daily life. Heart rate fractal scaling proved to be a highly sensitive index of cognitive load, and data implied that people with lower motivation for cognitive effort had a diminished physiological capacity to respond to rising cognitive demands. State affect appeared to influence motivation among people with current depression given that they were more willing to exert cognitive effort following a sad mood induction.

Assessment of the relationship between fractal analysis of mandibular bone and orthodontic treatment duration : A retrospective study.

PURPOSE: This retrospective study aimed to determine whether a correlation exists between the fractal dimension value and overall orthodontic treatment duration in children and young adults. METHODS: The study included a total of 643 patients (age: 10-25 years) who received orthodontic treatment between January 2015 and March 2020. Patient records and pretreatment panoramic radiographs were evaluated. The regions of interest selected for calculating fractal dimension were the bilateral mental foramen regions of the mandible. Fractal dimension was set in relation to orthodontic treatment duration using a linear regression model which was also adjusted for potential confounding variables. Total treatment duration was the outcome variable of interest used as a continuous variable. The predictor variables of interest included age, gender, type of dental and skeletal malocclusion, vertical growth pattern, extraction type, and fractal dimension. RESULTS: The mean age, treatment duration, and fractal dimension were 14.56 years, 27.01 months, and 1.23 mm, respectively. Multiple linear regression analysis showed that the fractal dimension had a significant influence on overall treatment duration (P < 0.001). From the other variables, Angle class II malocclusion significantly influenced treatment duration (P < 0.01), age showed a significant negative correlation with treatment duration (P < 0.01), and treatment duration significantly increased for patients with tooth extractions (P < 0.001). CONCLUSION: There was a negative correlation between fractal dimensions at the mandibular mental region and total orthodontic treatment duration. Fractal dimension analysis may help to understand physiologic features of alveolar bone and predict orthodontic tooth movement.

Fractality of tics as a quantitative assessment tool for Tourette syndrome.

Tics manifest as brief, purposeless and unintentional movements or noises that, for many individuals, can be suppressed temporarily with effort. Previous work has hypothesized that the chaotic temporal nature of tics could possess an inherent fractality, that is, have neighbour-to-neighbour correlation at all levels of timescale. However, demonstrating this phenomenon has eluded researchers for more than two decades, primarily because of the challenges associated with estimating the scale-invariant, power law exponent-called the fractal dimension Df-from fractional Brownian noise. Here, we confirm this hypothesis and establish the fractality of tics by examining two tic time series datasets collected 6-12 months apart in children with tics, using random walk models and directional statistics. We find that Df is correlated with tic severity as measured by the YGTTS total tic score, and that Df is a sensitive parameter in examining the effect of several tic suppression conditions on the tic time series. Our findings pave the way for using the fractal nature of tics as a robust quantitative tool for estimating tic severity and treatment effectiveness, as well as a possible marker for differentiating typical from functional tics.

The Evaluation of Mean-Detrended Cross-Correlation Analysis Portfolio Strategy for Multiple risk Assets.

The fractal characteristics of the security market were considered in portfolio strategy optimization. First, the detrended cross-correlation analysis was adopted to measure the fractal correlation of different securities. Second, the fractal correlation was embedded into the mean-variance criterion of the modern portfolio theory. Third, the mean-detrended cross-correlation analysis portfolio strategy of multiple risk assets was constructed, and the analytic solution of the strategy was given. Finally, the evaluation results revealed that the constructed the mean-detrended cross-correlation analysis portfolio strategy clearly improved investment performance, thus achieving the goal of optimizing the multiple risk asset portfolio strategy.

Accuracy of fractal analysis and PI-RADS assessment of prostate magnetic resonance imaging for prediction of cancer grade groups: a clinical validation study.

OBJECTIVES: Multiparametric MRI with Prostate Imaging Reporting and Data System (PI-RADS) assessment is sensitive but not specific for detecting clinically significant prostate cancer. This study validates the diagnostic accuracy of the recently suggested fractal dimension (FD) of perfusion for detecting clinically significant cancer. MATERIALS AND METHODS: Routine clinical MR imaging data, acquired at 3 T without an endorectal coil including dynamic contrast-enhanced sequences, of 72 prostate cancer foci in 64 patients were analyzed. In-bore MRI-guided biopsy with International Society of Urological Pathology (ISUP) grading served as reference standard. Previously established FD cutoffs for predicting tumor grade were compared to measurements of the apparent diffusion coefficient (25th percentile, ADC25) and PI-RADS assessment with and without inclusion of the FD as separate criterion. RESULTS: Fractal analysis allowed prediction of ISUP grade groups 1 to 4 but not 5, with high agreement to the reference standard (κFD = 0.88 [CI: 0.79-0.98]). Integrating fractal analysis into PI-RADS allowed a strong improvement in specificity and overall accuracy while maintaining high sensitivity for significant cancer detection (ISUP > 1; PI-RADS alone: sensitivity = 96%, specificity = 20%, area under the receiver operating curve [AUC] = 0.65; versus PI-RADS with fractal analysis: sensitivity = 95%, specificity = 88%, AUC = 0.92, p < 0.001). ADC25 only differentiated low-grade group 1 from pooled higher-grade groups 2-5 (κADC = 0.36 [CI: 0.12-0.59]). Importantly, fractal analysis was significantly more reliable than ADC25 in predicting non-significant and clinically significant cancer (AUCFD = 0.96 versus AUCADC = 0.75, p < 0.001). Diagnostic accuracy was not significantly affected by zone location. CONCLUSIONS: Fractal analysis is accurate in noninvasively predicting tumor grades in prostate cancer and adds independent information when implemented into PI-RADS assessment. This opens the opportunity to individually adjust biopsy priority and method in individual patients. KEY POINTS: • Fractal analysis of perfusion is accurate in noninvasively predicting tumor grades in prostate cancer using dynamic contrast-enhanced sequences (κFD = 0.88). • Including the fractal dimension into PI-RADS as a separate criterion improved specificity (from 20 to 88%) and overall accuracy (AUC from 0.86 to 0.96) while maintaining high sensitivity (96% versus 95%) for predicting clinically significant cancer. • Fractal analysis was significantly more reliable than ADC25 in predicting clinically significant cancer (AUCFD = 0.96 versus AUCADC = 0.75).

Fractal Brownian Motion Assessment of the Center of Pressure Excursion During Impulse Phase on Standard Vertical Jump.

This study presents and applies fractal Brownian motion assessment of the center of pressure (COP) excursion during feet ground contact on standard vertical jump impulse phase with long and short countermovement (CM) in relation with lower limb muscle stretch-shortening cycle (SSC) comparing it with no CM and SSC. Fifty-four tests were performed by a group of six healthy male students of sports and physical education degree without previous injury, specific training, or fitness ability. Three repetitions were performed by each subject of a squat jump (SJ) without CM and SSC, countermovement jump (CMJ) with long CM and SSC, as well as drop jump (DJ) with short CM and SSC after depth jump from a 40 cm step. During trial tests ground reaction force and force moments were acquired with force platform and impulse phases were segmented for COP coordinates computation. Fractal Brownian motion analysis of COP excursion during impulse phases conduced to detection of differences between critical time and displacement as well as short and long-term diffusion coefficient (Ds, Dl) and Hurst index scale exponent (Hs, Hl), with Ds, Dl presenting statistical significative correlations -0.491, -0.559 and Hs, Hl non statistical significative correlations 0.266 and -0.424 with MVJ height (ht) at 5% significance for explaining underlying mechanisms on CM and SSC at MVJ.Clinical Relevance- This work contributes with new method for the study expansion of the center of pressure excursion and stability during feet ground contact from orthostatic standing position to the impulse phase during standard maximum vertical jump as the most adequate method for assessment of lower limb muscle stretch-shortening cycle.

Multiscale assessment of the degree of multifractality for physiological time series.

Recent advancements in detrended fluctuation analysis (DFA) allow evaluating multifractal coefficients scale-by-scale, a promising approach for assessing the complexity of biomedical signals. The multifractality degree is typically quantified by the singularity spectrum width (WSS), a method that is critically unstable in multiscale applications. Thus, we aim to propose a robust multiscale index of multifractality, compare it with WSS and illustrate its performance on real biosignals. The proposed index is the cumulative function of squared increments between consecutive DFA coefficients at each scale n: αCF(n). We compared it with WSS calculated scale-by-scale considering monofractal/monoscale, monofractal/multiscale, multifractal/monoscale and multifractal/multiscale random processes. The two indices provided qualitatively similar descriptions of multifractality, but αCF(n) differentiated better the multifractal components from artefacts due to crossovers or detrending overfitting. Applied on 24 h heart rate recordings of 14 participants, the singularity spectrum failed to always satisfy the concavity requirement for providing meaningful WSS, while αCF(n) demonstrated a statistically significant heart rate multifractality at night in the scale ranges 16-100 and 256-680 s. Furthermore, αCF(n) did not reject the hypothesis of monofractality at daytime, coherently with previous reports of lower nonlinearity and monoscale multifractality during the day. Thus, αCF(n) appears a robust index of multiscale multifractality that is useful for quantifying complexity alterations of physiological series. This article is part of the theme issue 'Advanced computation in cardiovascular physiology: new challenges and opportunities'.

Updated fractal assessment of Italian Covid-19 disease.

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Diagnostic accuracy of quantitative flow ratio for assessment of coronary stenosis significance from a single angiographic view: A novel method based on bifurcation fractal law.

OBJECTIVES: We aimed to evaluate the diagnostic accuracy of computation of fractional flow reserve (FFR) from a single angiographic view in patients with intermediate coronary stenosis. BACKGROUND: Computation of quantitative flow ratio (QFR) from a single angiographic view might increase the feasibility of routine use of computational FFR. In addition, current QFR solutions assume a linear tapering of the reference vessel size, which might decrease the diagnostic accuracy in the presence of the physiologically significant bifurcation lesions. METHODS: An artificial intelligence algorithm was proposed for automatic delineation of lumen contours of major epicardial coronary arteries including their side branches. A step-down reference diameter function was reconstructed based on the Murray bifurcation fractal law and used for QFR computation. Validation of this Murray law-based QFR (µQFR) was performed on the FAVOR II China study population. The µQFR was computed separately in two angiographic projections, starting with the one with optimal angiographic image quality. Hemodynamically significant coronary stenosis was defined by pressure wire-derived FFR ≤0.80. RESULTS: The µQFR was successfully computed in all 330 vessels of 306 patients. There was excellent correlation (r = 0.90, p < .001) and agreement (mean difference = 0.00 ± 0.05, p = .378) between µQFR and FFR. The vessel-level diagnostic accuracy for µQFR to identify hemodynamically significant stenosis was 93.0% (95% CI: 90.3 to 95.8%), with sensitivity, specificity, positive predictive value, negative predictive value, positive likelihood ratio, and negative likelihood ratio of 87.5% (95% CI: 80.2 to 92.8%), 96.2% (95% CI: 92.6 to 98.3%), 92.9% (95% CI: 86.5 to 96.9%), 93.1% (95% CI: 88.9 to 96.1%), 23.0 (95% CI: 11.6 to 45.5), 0.13 (95% CI: 0.08 to 0.20), respectively. Use of suboptimal angiographic image view slightly decreased the diagnostic accuracy of µQFR (AUC = 0.97 versus 0.92, difference = 0.05, p < .001). Intra- and inter-observer variability for µQFR computation was 0.00 ± 0.03, and 0.00 ± 0.03, respectively. Average analysis time for µQFR was 67 ± 22 s. CONCLUSIONS: Computation of µQFR from a single angiographic view has high feasibility and excellent diagnostic accuracy in identifying hemodynamically significant coronary stenosis. The short analysis time and good reproducibility of µQFR bear potential of wider adoption of physiological assessment in the catheterization laboratory.

A quasi-Monte Carlo based flocculation model for fine-grained cohesive sediments in aquatic environments.

The quasi-Monte Carlo (QMC) method was enhanced to solve the population balance model (PBM) including aggregation and fragmentation processes for simulating the temporal evolutions of characteristic sizes and floc size distributions (FSDs) of cohesive sediments. Ideal cases with analytical solutions were firstly adopted to validate this QMC model to illustrate selected pure aggregation, pure fragmentation, and combined aggregation and fragmentation systems. Two available laboratory data sets, one with suspended kaolinite and the other with a mixture of kaolinite and montmorillonite, were further used to monitor the FSDs of cohesive sediments in controlled shear conditions. The model results show reasonable agreements with both analytical solutions and laboratory experiments. Moreover, different QMC schemes were tested and compared with the standard Monte Carlo scheme and a Latin Hypercube Sampling scheme to optimize the model performance. It shows that all QMC schemes perform better in both accuracy and time consumption than standard Monte Carlo scheme. In particular, compared with other schemes, the QMC scheme using Halton sequence requires the least particle numbers in the simulated system to reach reasonable accuracy. In the sensitivity tests, we also show that the fractal dimension and the fragmentation distribution function have large impacts on the predicted FSDs. This study indicates a great advance in employing QMC schemes to solve PBM for simulating the flocculation of cohesive sediments.

Time-varying properties of asymmetric volatility and multifractality in Bitcoin.

This study investigates the volatility of daily Bitcoin returns and multifractal properties of the Bitcoin market by employing the rolling window method and examines relationships between the volatility asymmetry and market efficiency. Whilst we find an inverted asymmetry in the volatility of Bitcoin, its magnitude changes over time, and recently, it has become small. This asymmetric pattern of volatility also exists in higher frequency returns. Other measurements, such as kurtosis, skewness, average, serial correlation, and multifractal degree, also change over time. Thus, we argue that properties of the Bitcoin market are mostly time dependent. We examine efficiency-related measures: the Hurst exponent, multifractal degree, and kurtosis. We find that when these measures represent that the market is more efficient, the volatility asymmetry weakens. For the recent Bitcoin market, both efficiency-related measures and the volatility asymmetry prove that the market becomes more efficient.

A combined experimental and numerical approach to the assessment of floc settling velocity using fractal geometry.

Sedimentation processes are fundamental to solids/liquid separation in water and wastewater treatment, and therefore a robust understanding of the settlement characteristics of mass fractal aggregates (flocs) formed in the flocculation stage is fundamental to optimized settlement tank design and operation. However, the use of settling as a technique to determine aggregates' traits is limited by current understanding of permeability. In this paper, we combine experimental and numerical approaches to assess settling velocities of fractal aggregates. Using a non-intrusive in situ digital image-based method, three- and two-dimensional fractal dimensions were calculated for kaolin-based flocs. By considering shape and fractal dimension, the porosity, density and settling velocities of the flocs were calculated individually, and settling velocities compared with those of spheres of the same density using Stokes' law. Shape analysis shows that the settling velocities for fractal aggregates may be greater or less than those for perfect spheres. For example, fractal aggregates with floc fractal dimension, Df = 2.61, floc size, df > 320 µm and dp = 7.5 µm settle with lower velocities than those predicted by Stokes' law; whilst, for Df = 2.33, all aggregates of df > 70 µm and dp = 7.5 µm settled below the velocity calculated by Stokes' law for spheres. Conversely, fractal settling velocities were higher than spheres for all the range of sizes, when Df of 2.83 was simulated. The ratio of fractal aggregate to sphere settling velocity (the former being obtained from fractal porosity and density considerations), varied from 0.16 to 4.11 for aggregates in the range of 10 and 1,000 µm, primary particle size of 7.5 µm and a three-dimensional fractal dimension between 2.33 and 2.83. However, the ratio decreases to the range of 0.04-2.92 when primary particle size changes to 1.0 µm for the same fractal dimensions. Using the floc analysis technique developed here, the results demonstrate the difference in settlement behaviour between the approach developed here and the traditional Stokes' law approach using solid spheres. The technique and results demonstrate the improvements in understanding, and hence value to be derived, from an analysis based on fractal, rather than Euclidean, geometry when considering flocculation and subsequent clarification performance.

Chinese ASCVD risk equations rather than pooled cohort equations are better to identify macro- and microcirculation abnormalities.

BACKGROUND: We hypothesized that discriminating the early subclinical organ damage would serve as a great opportunity for prevention against atherosclerotic cardiovascular disease (ASCVD). Brachial-ankle pulse wave velocity (baPWV), low retinal vascular fractal dimension, and albuminuria are surrogates of subclinical vascular changes. METHODS: The aim of this study was to use Pooled Cohort Equations (PCE) and ASCVD risk equations derived from "Prediction for ASCVD Risk in China project (CHINA-PAR)" to observe the prevalence of macro- and microcirculation abnormalities. A total of 2166 subjects were involved. Characteristics were investigated using questionnaire and physical examinations. We calculated the urine albumin to creatinine ratio (UACR). The baPWV was measured using a fully automatic arteriosclerosis detector. The retinal vascular fractal dimension was measured by a semiautomated computer-based program. The 10-year ASCVD risk was estimated using the PCE and CHINA-PAR model. RESULTS: The cut-off values for the elevated baPWV were 2.82 and 2.92% in the PCE model and CHINA-PAR model, respectively, with nearly 85% sensitivity and an average specificity of 74%. For low retinal fractal dimension, at the cut-off point of 3.8%, we acquired an acceptable sensitivity of 66.27-68.24% and specificity of 62.57-67.45%. All the C-statistics presented a significant improvement from the PCE model to the CHINA-PAR model (P < 0.05). For all categories-net reclassification improvement (NRI) values were significant and clearly varied (0.329, 0.183, and 0.104, respectively) depending on the cut-off set at 3%. CONCLUSION: Our study demonstrated that the CHINA-PAR equations rather than PCE could provide better identification of macro- and microcirculation abnormalities. A lower cut-off point for the subclinical vascular changes may be selected in a population from southeast China.