Fractional Threshold Research Articles

The linear system for Sudoku and a fractional completion threshold

The linear system for Sudoku and a fractional completion threshold

Modelling Yeast Prion Dynamics: A Fractional Order Approach with Predictor–Corrector Algorithm

This work aims to comprehend the dynamics of neurodegenerative disease using a mathematical model of fractional-order yeast prions. In the context of the Caputo fractional derivative, we here study and examine the solution of this model using the Predictor–Corrector approach. An analysis has been conducted on the existence and uniqueness of the selected model. Also, we examined the model’s stability and the existence of equilibrium points. With the purpose of analyzing the dynamics of the Sup35 monomer and Sup35 prion population, we displayed the graphs to show the obtained solutions over time. Graphical simulations show that the behaviour of the populations can change based on fractional orders and threshold parameter values. This work may present a good example of how biological theories and data can be better understood via mathematical modelling.

A new denoising method of ship-radiated noise: Improved variational mode decomposition coupled with fractional order entropy double threshold criterion

A new denoising method of ship-radiated noise: Improved variational mode decomposition coupled with fractional order entropy double threshold criterion

Image Edge Detection by Global Thresholding Using Riemann–Liouville Fractional Integral Operator

It is difficult to give a fractional global threshold (FGT) that works well on all images as the image contents are totally different. This paper describes an interesting use of fractional calculus in the field of digital image processing. In the proposed method, the fractional global threshold-based edge detector (FGTED) is established using the Riemann–Liouville fractional integral operator. FGTED is used to find the microedges in minimum time for any input digital images. The results demonstrate that the FGTED outperforms conventional techniques for detecting microtype edges. The image with a higher entropy was produced by the FGT value-based approach. Tables and images are used to summarize the output performance analysis of various images using structural similarity index measure, F-score (F-measure), precision and recall, signal-to-noise ratio, peak signal-to-noise ratio, and computational time. The FGTED can be used to detect very thin or microtype edges more accurately in minimum time without training or prior knowledge.

Significance of navigated transcranial magnetic stimulation and tractography to preserve motor function in patients undergoing surgery for motor eloquent gliomas

Resection of gliomas in or close to motor areas is at high risk for morbidity and development of surgery-related deficits. Navigated transcranial magnetic stimulation (nTMS) including nTMS-based tractography is suitable for presurgical planning and risk assessment. The aim of this study was to investigate the association of postoperative motor status and the spatial relation to motor eloquent brain tissue in order to increase the understanding of postoperative motor deficits.Patient data, nTMS examinations and imaging studies were retrospectively reviewed, corticospinal tracts (CST) were reconstructed with two different approaches of nTMS-based seeding. Postoperative imaging and nTMS-augmented preoperative imaging were merged to identify the relation between motor positive cortical and subcortical areas and the resection cavity.38 tumor surgeries were performed in 36 glioma patients (28.9% female) aged 55.1 ± 13.8 years. Mean distance between the CST and the lesion was 6.9 ± 5.1 mm at 75% of the patient-individual fractional anisotropy threshold and median tumor volume reduction was 97.7 ± 11.6%. The positive predictive value for permanent deficits after resection of nTMS positive areas was 66.7% and the corresponding negative predictive value was 90.6%. Distances between the resection cavity and the CST were higher in patients with postoperative stable motor function. Extent of resection and distance between resection cavity and CST correlated well.The present study strongly supports preoperative nTMS as an important surgical tool for preserving motor function in glioma patients at risk.

A GENERALIZATION OF THRESHOLD-BASED AND PROBABILITY-BASED MODELS OF INFORMATION DIFFUSION

Diffusion of information through complex networks is of interest in studies such as propagation prediction and influence maximization, both of which have applications in viral marketing and rumor controlling. There are a variety of information diffusion models, all of which simulate the adoption and spread of information over time. However, there is a lack of understanding of whether, despite their conceptual differences, these models represent the same underlying generative structures. For instance, if two different models utilize different conceptual mechanisms, but generate the same results, does the choice of model matter? A classification of diffusion of information models is developed based on the neighbor knowledge of the model infection requirement and the stochasticity of the model. This classification allows for the identification of models that fall into each respective category. The study involves the analysis of the following agent-based models on directed scale-free networks: (1) a linear absolute threshold model (LATM), (2) a linear fractional threshold model (LTFM), (3) the independent cascade model (ICM), (4) Bass-Rand-Rust model (BRRM) (5) a stochastic linear absolute threshold model (SLATM) (6) a stochastic fractional threshold model (SLFTM), and (7) Dodds–Watts model (DWM). Through the execution of simulations and analysis of the experimental results, the distinctive properties of each model are identified. Our analysis reveals that similarity in conceptual design does not imply similarity in behavior concerning speed, final state of nodes and edges, and sensitivity to parameters. Therefore, we highlight the importance of considering the unique behavioral characteristics of each model when selecting a suitable information diffusion model for a particular application.

Efficiency enhancements of a Monte Carlo beamlet based treatment planning process: implementation and parameter study

Objective. The computational effort to perform beamlet calculation, plan optimization and final dose calculation of a treatment planning process (TPP) generating intensity modulated treatment plans is enormous, especially if Monte Carlo (MC) simulations are used for dose calculation. The goal of this work is to improve the computational efficiency of a fully MC based TPP for static and dynamic photon, electron and mixed photon-electron treatment techniques by implementing multiple methods and studying the influence of their parameters. Approach. A framework is implemented calculating MC beamlets efficiently in parallel on each available CPU core. The user can specify the desired statistical uncertainty of the beamlets, a fractional sparse dose threshold to save beamlets in a sparse format and minimal distances to the PTV surface from which 2 × 2 × 2 = 8 (medium) or even 4 × 4 × 4 = 64 (large) voxels are merged. The compromise between final plan quality and computational efficiency of beamlet calculation and optimization is studied for several parameter values to find a reasonable trade-off. For this purpose, four clinical and one academic case are considered with different treatment techniques. Main results. Setting the statistical uncertainty to 5% (photon beamlets) and 15% (electron beamlets), the fractional sparse dose threshold relative to the maximal beamlet dose to 0.1% and minimal distances for medium and large voxels to the PTV to 1 cm and 2 cm, respectively, does not lead to substantial degradation in final plan quality compared to using 2.5% (photon beamlets) and 5% (electron beamlets) statistical uncertainty and no sparse format nor voxel merging. Only OAR sparing is slightly degraded. Furthermore, computation times are reduced by about 58% (photon beamlets), 88% (electron beamlets) and 96% (optimization). Significance. Several methods are implemented improving computational efficiency of beamlet calculation and plan optimization of a fully MC based TPP without substantial degradation in final plan quality.

Influence maximization under limited network information: seeding high-degree neighbors

The diffusion of information, norms, and practices across a social network can be initiated by compelling a small number of seed individuals to adopt first. Strategies proposed in previous work either assume full network information or a large degree of control over what information is collected. However, privacy settings on the Internet and high non-response in surveys often severely limit available connectivity information. Here we propose a seeding strategy for scenarios with limited network information: Only the degrees and connections of some random nodes are known. This new strategy is a modification of ‘random neighbor sampling’ (or ‘one-hop’) and seeds the highest-degree neighbors of randomly selected nodes. Simulating a fractional threshold model, we find that this new strategy excels in networks with heavy tailed degree distributions such as scale-free networks and large online social networks. It outperforms the conventional one-hop strategy even though the latter can seed 50% more nodes, and other seeding possibilities including pure high-degree seeding and clustered seeding.

Evolution of a fractional-charge optical vortex upon free-space propagation

Previously, it has been demonstrated theoretically (J. Opt. 6, 259 (2004)) and experimentally (Opt Express 19, 5760 (2011)) that upon the free-space propagation, an initial fractional-charge optical vortex acquires an integer topological charge (TС) equal to the nearest smaller integer if the fractional part is less than 0.5, otherwise becoming equal to the nearest larger integer. In this work, we demonstrate by the numerical modeling that as an initial fractional-charge optical vortex propagates in free space, the TC changes to the nearest smaller or larger integer at a fractional part threshold of 0.12. The fact is that an additional singularity center is born on the beam periphery where the field intensity is near-zero (one millionth part of the maximum), meaning that it cannot be experimentally detected but can be numerically simulated.

Association between non-wire based computational angiography fractional flow reserve treatment threshold and major adverse cardiac events in patients with stable coronary artery disease

Abstract Introduction Despite class IA guideline recommendations, the use of fractional flow reserve (FFR) in guiding percutaneous coronary intervention (PCI) in stable coronary artery disease (CAD) patients remains low due to limitations including the need of guidewire placement and hyperaemic stimulus. A novel non-invasive index, computational pressure-flow dynamics derived FFR (caFFR), was developed for measuring functional myocardial ischemia and overcoming the limitations of FFR. However, the clinical relevance of caFFR remains to be investigated. In the present study, we aim at evaluating the prognostic value of caFFR among stable CAD patients. Methods We retrospectively included patients with stable CAD who underwent coronary angiography during 2014–2016 at our center. Based on the caFFR value, patients were considered to be ischemic (caFFR ≤0.8) and non-ischemic (caFFR >0.8). Further, we recombined the patients to form the adherence cohort, where patients were defined as adherent-to-caFFR if they were ischemic with PCI or non-ischemic without PCI, and nonadherent-to-caFFR if they were ischemic without PCI or non-ischemic with PCI. The primary endpoint was major adverse cardiac events (MACE), defined as a composite of all-cause mortality, non-fatal myocardial infarction, and any revascularization. Inverse probability of treatment weighting was used to account for treatment selection bias (PCI vs without PCI, or adherent vs non-adherent), and Cox proportional hazard model was used to evaluate the association with MACE. Results A total of 1322 patients, 782 patients in the ischemic cohort and 540 patients in the non-ischemic cohort respectively, were included in our analysis. PCI was associated with a lower risk of MACE in the ischemic cohort (hazard ratio [HR] 0.52; 95% confidence interval [CI], 0.34–0.80; P=0.002), but was not associated with MACE in the non-ischemic cohort. In the adherence cohort, adherent-to-caFFR group (n=803) had a lower risk of MACE compared with nonadherent-to-caFFR group (n=566) (HR, 0.61; 95% CI, 0.44–0.85; P=0.003). Conclusion Our study is the first to demonstrate the prognostic value of caFFR, a non-wire based assessment of myocardial ischemia, in patients with stable CAD undergoing PCI. These findings support the use of caFFR that bears the potential of a wider adoption compared with wire-based FFR through a reduction in procedure time, risk and costs. Funding Acknowledgement Type of funding sources: None. Weighted Kaplan-Meier curvesWeighted Cox proportional hazards model

Risk Assessment by Presurgical Tractography Using Navigated TMS Maps in Patients with Highly Motor- or Language-Eloquent Brain Tumors.

Patients with functionally eloquent brain lesions are at risk of functional decline in the course of resection. Given tumor-related plastic reshaping and reallocation of function, individual data are needed for patient counseling and risk assessment prior to surgery. This study evaluates the utility of mapping by navigated transcranial magnetic stimulation (nTMS) and nTMS-based diffusion tensor imaging fiber tracking (DTI FT) for individual risk evaluation of surgery-related decline of motor or language function in the clinical setting. In total, 250 preoperative nTMS mappings (100 language and 150 motor mappings) derived from 216 patients (mean age: 57.0 ± 15.5 years, 58.8% males; glioma World Health Organization (WHO) grade I & II: 4.2%, glioma WHO grade III & IV: 83.4%, arteriovenous malformations: 1.9%, cavernoma: 2.3%, metastasis: 8.2%) were included. Deterministic tractography based on nTMS motor or language maps as seed regions was performed with 25%, 50%, and 75% of the individual fractional anisotropy threshold (FAT). Lesion-to-tract distances (LTDs) were measured between the tumor mass and the corticospinal tract (CST), arcuate fascicle (AF), or other closest language-related tracts. LTDs were compared between patients and correlated to the functional status (no/transient/permanent surgery-related paresis or aphasia). Significant differences were found between patients with no or transient surgery-related deficits and patients with permanent surgery-related deficits regarding LTDs in relation to the CST (p < 0.0001), AF (p ≤ 0.0491), or other closest language-related tracts (p ≤ 0.0435). The cut-off values for surgery-related paresis or aphasia were ≤12 mm (LTD—CST) and ≤16 mm (LTD—AF) or ≤25 mm (LTD—other closest language-related tract), respectively. Moreover, there were significant associations between the status of surgery-related deficits and the LTD when considering the CST (range r: −0.3994 to −0.3910, p < 0.0001) or AF (range r: −0.2918 to −0.2592, p = 0.0135 and p = 0.0473 for 25% and 50% FAT). In conclusion, this is the largest study evaluating the application of both preoperative functional mapping and function-based tractography for motor and language function for risk stratification in patients with functionally eloquent tumors. The LTD may qualify as a viable marker that can be seamlessly assessed in the clinical neurooncological setup.

Overcoming challenges of the human spinal cord tractography for routine clinical use: a review.

The spinal cord (SC) is a dense network of billions of fibers in a small volume surrounded by bones that makes tractography difficult to perform. We aim to provide a review collecting all technical settings of SC tractography and propose the optimal set of parameters to perform a good SC tractography rendering. The MEDLINE database was searched for articles reporting "spinal cord" "tractography" in "humans". Studies were selected only when tractography rendering was displayed and MRI acquisition and tracking parameters detailed. From each study, clinical context, imaging acquisition settings, fiber tracking parameters, region of interest (ROI) design, and quality of the tractography rendering were extracted. Quality of tractography rendering was evaluated by several objective criteria proposed herein. According to the reported studies, to obtain a good tractography rendering, diffusion tensor imaging acquisition should be performed with 1.5 or 3Tesla MRI, in the axial plane, with >20 directions; b value: 1000smm-2; right-left phase-encoding direction for cervical SC; isotropic voxel size; and no slice gap. Concerning the tracking process, it should be performed with determinist approach, fractional anisotropy threshold between 0.15 and 0.2, and curvature threshold of 40°. ROI design is an essential step for providing good tractography rendering, and their placement has to consider partial volume effects, magnetic susceptibility effects, and motion artifacts. The review reported herein highlights that successful SC tractography depends on many factors (imaging acquisition settings, fiber tracking parameters, and ROI design) to obtain a good SC tractography rendering.

MRI Tumor Regression Grade and Circulating Tumor DNA as Complementary Tools to Assess Response and Guide Therapy Adaptation in Rectal Cancer.

Response to preoperative chemo-radiotherapy (CRT) varies. We assessed whether circulating tumor DNA (ctDNA) might be an early indicator of tumor response or progression to guide therapy adaptation in rectal cancer. A total of 243 serial plasma samples were analyzed from 47 patients with localized rectal cancer undergoing CRT. Up to three somatic variants were tracked in plasma using droplet digital PCR. RECIST and MRI tumor regression grade (mrTRG) evaluated response. Survival analyses applied Kaplan-Meier method and Cox regression. ctDNA detection rates were: 74% (n = 35/47) pretreatment, 21% (n = 10/47) mid CRT, 21% (n = 10/47) after completing CRT, and 13% (n = 3/23) after surgery. ctDNA status after CRT was associated with primary tumor response by mrTRG (P = 0.03). With a median follow-up of 26.4 months, metastases-free survival was shorter in patients with detectable ctDNA after completing CRT [HR 7.1; 95% confidence interval (CI), 2.4-21.5; P < 0.001], persistently detectable ctDNA pre and mid CRT (HR 3.8; 95% CI, 1.2-11.7; P = 0.02), and pre, mid, and after CRT (HR 11.5; 95% CI, 3.3-40.4; P < 0.001) compared with patients with undetectable or nonpersistent ctDNA. In patients with detectable ctDNA, a fractional abundance threshold of ≥0.07% mid CRT or ≥0.13% after completing CRT predicted for metastases with 100% sensitivity and 83.3% specificity for mid CRT and 66.7% for CRT completion. All 3 patients with detectable ctDNA post-surgery relapsed compared with none of the 20 patients with undetectable ctDNA (P = 0.001). ctDNA identified patients at risk of developing metastases during the neoadjuvant period and post-surgery, and could be used to tailor treatment.

Fractional Hölder mean-based image segmentation for mouse behavior analysis in conditional place preference test

Conditioned place preference (CPP) test is a common protocol where animals (usually rodents) are used to study the rewarding effects of addictive drugs and potential substances that may alleviate drug dependence. Automated CPP systems lead to increased throughput and reduced biasness; however, due to cost of such systems, most laboratories opt for manual testing. In this study, we present a new method for automatic analysis of CPP videos through a fractional Holder mean (FHM)-based method for segmenting mouse from the video frames, which can later be used for analyzing mouse movement and behavior (fractional segmentation method), and eventually understanding the effect of addictive drug. We show that FHM satisfies some basic properties, such as convergence criteria and contractive criteria, which are the most important properties in segmentation. The proposed FHM is based on local minima and maxima, which makes it robust to contrast variation that is found in the video frames, allowing it to successfully separate mouse from the background. The proposed segmentation results show that the use of fractional Holder mean-based threshold leads to a better pixel accuracy of object segmentation compared to other standard existing methods.

Association of Fractional Flow on 3D-TOF-MRA with Cerebral Perfusion in Patients with MCA Stenosis.

Fractional flow measured on 3D-TOF-MRA was proposed to quantify cerebral hemodynamic changes in patients with artery stenosis. We investigated the association between fractional flow and cerebral perfusion changes in patients with symptomatic MCA stenosis. This prospective study was approved by the institutional review board, and all participants provided written informed consent. From June 2015 to May 2018, four hundred twenty-nine patients with symptomatic intracranial arterial stenosis were consecutively recruited and underwent conventional brain MR imaging, 3D-TOF-MRA, and brain CTP. A total of 91 patients with unilateral M1 segment stenosis of the MCA and a stenosis degree of 50%∼99% were included in the analysis. Fractional flow was measured by comparing distal and proximal signal intensity changes across the stenosis on 3D-TOF-MRA. The cutoff value for fractional flow for discriminating between normal perfusion and hypoperfusion was obtained from the receiver operating characteristic curve. Associations between fractional flow and hypoperfusion were assessed using univariate and multivariate analyses. The receiver operating characteristic curve showed a significant fractional flow threshold value at 0.90 (sensitivity, 70.1%; 95% CI, 55.9%-81.2%; specificity, 69.6%; 95% CI, 47.6%-84.1%). Participants with a fractional flow of ≤0.90 were independently associated with cerebral hypoperfusion downstream from the stenosis site (adjusted OR, 3.68; 95% CI, 1.63-11.62; P = .027). Fractional flow measured on 3D-TOF-MRA may serve as a noninvasive and practical tool for determining the cerebral hypoperfusion in patents with symptomatic MCA stenosis.

Correlation between diffusion tensor imaging parameters of the distal femoral physis and adjacent metaphysis, and subsequent adolescent growth.

Diffusion-tensor imaging (DTI) depicts the movement of water through columns of cartilage and newly formed bone and provides information about velocity of growth and growth potential. To determine the correlation between DTI tractography parameters of the distal femoral physis and metaphysis and the height change after DTI in pubertal and post-pubertal children. We retrospectively analyzed DTI images of the knee in 47 children with a mean age of 14.1years in a 2-year period. In sagittal echoplanar DTI studies, regions of interest were placed in the femoral physis. Tractography was performed using a fractional anisotropy threshold of 0.15 and a maximum turning angle of 40°. The sample was divided to assess short-term and long-term growth after DTI. Short-term growth (n=25) was the height change between height at MRI and 1year later. Long-term growth (n=36) was the height gain between height at MRI and at the growth plateau. For the short-term group, subjects with larger tract volume (R2=0.40) and longer track lengths (R2=0.38) had larger height gains (P<0.01). For the long-term group, subjects with larger tract volume (R2=0.43) and longer track lengths (R2=0.32) had a larger height gain at the growth plateau (P<0.01). Intra- and inter-observer variability were good-excellent. Follow-up data of growth 1year after DTI evaluation and at skeletal maturity confirms that DTI parameters are associated with the amount of post-imaging growth.

Medical image segmentation method based on the improved artificial bee colony algorithm

Objectives: The aim is to study the application of artificial bee colony (ABC) algorithm in medical image threshold segmentation. Methods: A new image segmentation method based on the improved ABC and thresholding medical image threshold segmentation method is proposed, which is variable coefficient ABC (VCABC) optimization algorithm, which is used to determine n-1 optimal n level threshold on a given image. The proposed method is compared with the Particle Swarm Optimization fractional image threshold segmentation method and the ABC fractional medical image threshold segmentation method. Results: When considering a variety of conditions, the performance of this method is better than that of other methods. Conclusions: The improved method of combining ABC and fractional medical image threshold segmentation method is effective.

Optimizing Fractional Intensity Threshold for FSLBrain Extraction Tool (BET) and Comparing with FreeSurfer on 3D T1W MR Images

Objective: To find out the optimal Brain Extraction Tool (BET) parameter (fractional intensity threshold) for measuring the brain volume compared with the standard manual method in our institute and to compare with those of automated FreeSurfer software Methods: This retrospective study was performed in 10 healthy adult subjects with data of 3D-T1W on 3T MR machine. The manual gold standard brain volume measurements were done by two independent readers. The automated segmentations using BET with varied parameters and FreeSurfer software were also performed. Then, the two automated methods were compared with the manual tracing to make the optimal parameter by seeking for the highest intraclass correlation coefficient (ICC) using SPSS software. Results: The fractional intensity threshold for whole brain volume measurement of 0.1- 0.6 showed high ICC with the manual gold standard (ranging from 0.639 to 0.748). The best value was 0.1, showing highest ICC of 0.748 (p < 0.006) with confidence interval of 95% equal to (0.242; 0.932). There were no optimal parameters for right and left hippocampus volume measurement by BET due to very low ICC between BET and the reader (ICC ranging from 0.017 to 0.139 and from 0.012 to 0.110, for the right and left hippocampus volume, respectively). The ICC values of the automated FreeSurfer method with the manual tracing were also very low (0.063, 0.068, and 0.063 for right, left and bilateral hippocampi respectively). Conclusion: The optimal BET parameter (fractional intensity threshold) for automated brain volume measurement in our institute is 0.1 similar to the suggested value by prior study with high agreement (ICC=0.748) to the manual method.

Pilot study on renal magnetic resonance diffusion tensor imaging: are quantitative diffusion tensor imaging values useful in the evaluation of children with ureteropelvic junction obstruction?

Ureteropelvic junction (UPJ) obstruction is a common cause of renal injury in children. Indications for surgery are still controversial. Currently, there is no threshold to differentiate patients with suspected UPJ obstruction requiring surgery from the ones that do not, or to predict renal outcome after surgery. Several studies have demonstrated that diffusion tensor imaging (DTI) results may correlate with microstructural changes in the kidneys. To evaluate the feasibility of using DTI to identify UPJ obstruction kidneys. We analyzed functional MR urography (fMRU) with renal DTI (b=0 and b=400, 20 directions, 1.5Tesla, no respiratory triggering) in 26 kidneys of 19 children (mean age: 6.15years) by comparing 13 kidneys with UPJ obstruction configuration that underwent pyeloplasty following the fMRU, and 13 anatomically normal age- and gender-matched kidneys. DTI tractography was reconstructed using a fractional anisotropy threshold of 0.10 and an angle threshold of 55°. User-defined regions of interest (ROIs) of the renal parenchyma (excluding collecting system) were drawn to quantify DTI parameters: fractional anisotropy, apparent diffusion coefficient (ADC), track length and track volume. The failure rate was evaluated. All DTI parameters changed with age; fractional anisotropy decreased (P<0.032). Track volume and track length increased (P<0.05). ADC increased with age in normal kidneys (P<0.001) but not in UPJ obstruction kidneys (P=0.11). After controlling for age, the fractional anisotropy (UPJ obstruction mean: 0.18, normal kidney mean: 0.21; P=0.001) and track length (UPJ obstruction mean: 11.9mm, normal kidney mean: 15.4mm; P<0.001) were lower in UPJ obstruction vs. normal kidneys. There was a trend toward a higher ADC in UPJ obstruction kidneys vs. normal kidneys (P=0.062). The failure rate in UPJ obstruction kidneys due to technical limitations of DTI was 13/26 (50%). We demonstrated that fractional anisotropy is lower in UPJ obstruction than in normal kidneys. It is necessary to improve this technique to increase the success rate and to perform more studies to evaluate if a decrease in fractional anisotropy can differentiate UPJ obstruction kidneys from hydronephrotic kidneys without UPJ obstruction.

Fractional derivative approach to non-Darcian flow in porous media

Fractional Darcian model and fractional Swartzendruber model are proposed for non-Darcian flows with high and low velocity in the paper, respectively. The analytical solutions of the fractional derivative models are given and all parameters are determined based on test data. The dependence of fractional derivative models on parameters such as fractional derivative order and threshold hydraulic gradient are analyzed by sensitivity studies. The fractional derivative models provide better description of non-Darcian flow in porous media under the conditions of high and low velocity. Furthermore, the memory effectiveness of fractional calculus is discussed, showing that the fractional derivative order is an indicator of non-Darcian flow in porous media.