Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

The purpose of the current study was to evaluate the performance of a continuous-time random-walk (CTRW) diffusion model for differentiating malignant and benign breast lesions and to consider the potential association between CTRW parameters and the Ki-67 expression. Sixty-four patients (46.2 ± 11.4 years) with breast lesions (29 malignant and 35 benign) were evaluated with the CTRW model, intravoxel incoherent motion model, and diffusion-weighted imaging. Echo planar diffusion-weighted imaging was conducted using 13 b-values (0-3000 s/mm2 ). Three CTRW model parameters, including an anomalous diffusion coefficient Dm , and two parameters related to temporal and spatial diffusion heterogeneity, α and β, respectively, were obtained, and had MRI b-values of 0-3000 s/mm2 . Receiver operating characteristic (ROC) analysis was conducted to determine the sensitivity, specificity, and diagnostic accuracy of CTRW parameters for differentiating malignant from benign breast lesions. In malignant breast lesions, the CTRW parameters Dm , α, and β were significantly lower than the corresponding parameters of benign breast lesions. In the malignant breast lesion group, the CTRW parameter Dm was significantly lower in high Ki-67 expression than in low Ki-67 expression. In ROC analysis, the combination of CTRW parameters (Dm , α, β) demonstrated the highest area under the curve value (0.985) and diagnostic accuracy (94.23%) in differentiating malignant and benign breast lesions. The CTRW model effectively differentiated malignant from benign breast lesions. The CTRW diffusion model offers a new way for noninvasive assessment of breast malignancy and better understanding of the proliferation of malignant lesions.

Fluid flow and solute transport simulations in tight geologic formations: Discrete fracture network and continuous time random walk analyses

The transport of materials is of fundamental importance, with studies on diffusion being at the forefront. Diffusion in a simple matrix is typically considered Fickian. However, anomalous diffusion in various media is a dominant process. It is well documented that anomalous diffusion results from medium heterogeneity, extreme events, phase transitions, medium surface dynamics, and other factors. The present work demonstrated that the diffusion of a cationic dye (crystal violet, CV) in an agar gel medium was anomalous, as shown by spatial and time-series data of dye movement. We estimated the classical diffusion coefficients using the Einstein-Smoluchowski solution to Fick's law, but these did not yield consistent Gaussianity, stationarity, or non-seasonality. However, anomalous diffusion was confirmed by modelling the experimental data. The exponent values (α) ranged from 0.468 ± 0.027 to 0.883 ± 0.107, indicating anomalous sub-diffusion that depended on the concentration of the CV dye in the same medium. We then evaluated time- and ensemble-averaged mean squared displacement using the dye-spreading data via image processing. The discrepancy in the distribution functions over a long experimental period highlighted the non-ergodic nature of the stochastic process. The dimensionless ergodicity-breaking parameter was evaluated, confirming the non-ergodic nature of the process. We found that the continuous time random walk (CTRW) model was well-suited for describing the anomalous diffusion in this system. We attributed the sub-diffusion of the dye to violations of the assumptions of Brownian motion of the particles and the nature of the diffusing medium.

Fickian and non-Fickian behaviors were often detected for contaminant transport activity owed to the preferential flow and heterogeneity of soil media. Therefore, using diverse methods to measure such composite solute transport in soil media has become an important research topic for solute transport modeling in soil media. In this article, the continuous-time random walk (CTRW) model was applied to illustrate the relative concentration of transport in low-permeability homogeneous and saturated soil media. The solute transport development was also demonstrated with the convection-dispersion equation (CDE) and Two Region Model (TRM) for comparison. CXTFIT 2.1 software was used for CDE and TRM, and CTRW Matlab Toolbox v.3.1 for the CTRW simulation of the breakthrough curve. It was found that higher values of determination coefficient (R2) and lower values of root mean square error (RMSE) concerning the best fits of CDE, TRM, and CTRW. It was found that in the comparison of CDE, TRM, and CTRW, we tend to use CTRW to describe the transport behavior well because there are prevailing Fickian and non-Fickian transport. The CTRW gives better fitting results to the breakthrough curves (BTCs) when β has an increasing pattern towards 2.00. In this study, the variation of parameters in three methods was investigated and results showed that the CTRW modeling approach is more effective to determine non-reactive contaminants concentration in low-permeability soil media at small depths.

Continuous time random walks for non-local radial solute transport

On fractional diffusion and continuous time random walks

Non-Fickian diffusion has been increasingly documented in hydrology and modeled by promising time nonlocal transport models. While previous studies showed that most of the time nonlocal models are identical with correlated parameters, fundamental challenges remain in real-world applications regarding model selection and parameter definition. This study compared three popular time nonlocal transport models, including the multi-rate mass transfer (MRMT) model, the continuous time random walk (CTRW) framework, and the tempered time fractional advection–dispersion equation (tt-fADE), by focusing on their physical interpretation and feasibility in capturing non-Fickian transport. Mathematical comparison showed that these models have both related parameters defining the memory function and other basic-transport parameters (i.e., velocity v and dispersion coefficient D) with different hydrogeologic interpretations. Laboratory column transport experiments and field tracer tests were then conducted, providing data for model applicability evaluation. Laboratory and field experiments exhibited breakthrough curves with non-Fickian characteristics, which were better represented by the tt-fADE and CTRW models than the traditional advection–dispersion equation. The best-fit velocity and dispersion coefficient, however, differ significantly between the tt-fADE and CTRW. Fitting exercises further revealed that the observed late-time breakthrough curves were heavier than the MRMT solutions with no more than two mass-exchange rates and lighter than the MRMT solutions with power-law distributed mass-exchange rates. Therefore, the time nonlocal models, where some parameters are correlated and exchangeable and the others have different values, differ mainly in their quantification of pre-asymptotic transport dynamics. In all models tested above, the tt-fADE model is attractive, considering its small fitting error and the reasonable velocity close to the measured flow rate.

Non-Fickian transport ubiquitously occurs across all scales within fractured geological media. Detailed characterization of non-Fickian transport through single fractures is thus critical for predicting the fate of solutes and other fluid-borne entities through fractured media. Our direct numerical simulations of solute transport through two-dimensional rough-walled fractures showed early arrival and heavy tailing in breakthrough curves (BTCs), which are salient characteristics of non-Fickian transport. Analyses for dispersion coefficients (DADE) using the standard advection-dispersion equation (ADE) led to errors which increased linearly with fracture heterogeneity. Estimated Taylor dispersion coefficients deviated from estimated DADE even at higher Peclet numbers. Alternatively, we used continuous time random walk (CTRW) model with truncated power law transition rate probability to characterize the non-Fickian transport. CTRW modeling markedly and consistently improved fits to the BTCs relative to those fitted with ADE solutions. The degree of deviation of transport from Fickian to non-Fickian is captured by the parameter β of the truncated power law. We found that β is proportional to fracture heterogeneity. We also found that the CTRW transport velocity can be predicted based on the flow velocity. Along with the ability to predict β, this is a major step toward prediction of transport through CTRW using measurable physical properties.

ObjectivesTo demonstrate the feasibility of a novel fractional motion (FM) diffusion model for distinguishing low- versus high-grade pediatric brain tumors; and to investigate its possible advantage over apparent diffusion coefficient (ADC) and/or a previously reported continuous-time random-walk (CTRW) diffusion model.Materials and methodsWith approval from the institutional review board and written informed consents from the legal guardians of all participating patients, this study involved 70 children with histopathologically-proven brain tumors (30 low-grade and 40 high-grade). Multi-b-value diffusion images were acquired and analyzed using the FM, CTRW, and mono-exponential diffusion models. The FM parameters, Dfm, φ, ψ (non-Gaussian diffusion statistical measures), and the CTRW parameters, Dm, α, β (non-Gaussian temporal and spatial diffusion heterogeneity measures) were compared between the low- and high-grade tumor groups by using a Mann-Whitney-Wilcoxon U test. The performance of the FM model for differentiating between low- and high-grade tumors was evaluated and compared with that of the CTRW and the mono-exponential models using a receiver operating characteristic (ROC) analysis.ResultsThe FM parameters were significantly lower (p < 0.0001) in the high-grade (Dfm: 0.81 ± 0.26, φ: 1.40 ± 0.10, ψ: 0.42 ± 0.11) than in the low-grade (Dfm: 1.52 ± 0.52, φ: 1.64 ± 0.13, ψ: 0.67 ± 0.13) tumor groups. The ROC analysis showed that the FM parameters offered better specificity (88% versus 73%), sensitivity (90% versus 82%), accuracy (88% versus 78%), and area under the curve (AUC, 93% versus 80%) in discriminating tumor malignancy compared to the conventional ADC. The performance of the FM model was similar to that of the CTRW model.ConclusionsSimilar to the CTRW model, the FM model can improve differentiation between low- and high-grade pediatric brain tumors over ADC.

In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

Superdiffusion in decoupled continuous time random walks

Continuous time random walk model as a model of anomalous relaxation

The Portevin–Le Chatelier effect in the continuous time random walk framework

We study the upscaling of pore-scale solute transport in partially saturated porous media at different saturation degrees. The interaction between structural heterogeneity, phases distribution and small-scale flow dynamics induces complex flow patterns and broad probability distributions of flow. In turn, this spatial distribution of flow velocities, at the pore scale, induces irregular (non-Fickian) transport of dissolved substances (e.g., contaminants), causing an earlier arrival and longer tailing, which may have grave consequences in underestimating risk assessments and prolonged cleanup times of contaminated sites.Here, we suggest an integrated continuous time random walk (CTRW) modeling framework, which accounts for also the entrapping of particles in zones of low flow velocities, to estimate the resident times of solutes in the media. Furthermore, comparing the results of the CTRW model to a well-established numerical simulation method allows a phenomenological evaluation of the model's physical parameters for different conditions (i.e., volume of entrapped air, mean water flow rate, or solute molecular diffusion coefficient).&amp;#160; &amp;#160;In this study we show that entrapped air promotes preferential solute transport and solute trapping in low flow regions. Moreover, we demonstrate that the trapping frequency and trapping time depend on the interaction between advection and diffusion (i.e., the P&amp;#233;clet number). An integrative CTRW model captures the effects of trapping in stagnant regions and preferential transport on non-Fickian dispersion of solutes.

Motivation: Lymphovascular space invasion (LVSI) is associated with poor outcomes in non-small cell lung cancer (NSCLC). Goal(s): To investigate whether 18F-FDG PET, a continuous-time random walk (CTRW) model, amide proton transfer-weighted imaging (APTWI)-based parameters may contribute to the non-invasive assessment of LVSI in NSCLC. Approach: A total of 78 LVSI-positive and 52 LVSI-negative NSCLC patients underwent chest multiparametric 18F-FDG PET/MRI. Results: The SUVmax and MTRasym(3.5ppm) were higher, while the ADC, α, β, and Dm were lower in the LVSI-positive group than in the LVSI-negative group. SUVmax, α, β, Dm, and MTRasym(3.5ppm) were independently associated with LVSI positivity. Impact: The analysis of tumour glucose metabolism, water molecule diffusion, temporal / spatial heterogeneities, and mobile proteins / peptides measurement at multiparametric 18F-FDG PET/MRI enables the prediction of LVSI in NSCLC.