Continuous Time Random Walk Model Research Articles

Characterization of prostatic cancer lesion and gleason grade using a continuous-time random-walk diffusion model at high b-values

BackgroundDistinguishing between prostatic cancer (PCa) and chronic prostatitis (CP) is sometimes challenging, and Gleason grading is strongly associated with prognosis in PCa. The continuous-time random-walk diffusion (CTRW) model has shown potential in distinguishing between PCa and CP as well as predicting Gleason grading.PurposeThis study aimed to quantify the CTRW parameters (α, β & Dm) and apparent diffusion coefficient (ADC) of PCa and CP tissues; and then assess the diagnostic value of CTRW and ADC parameters in differentiating CP from PCa and low-grade PCa from high-grade PCa lesions.Study typeRetrospective (retrospective analysis using prospective designed data).PopulationThirty-one PCa patients undergoing prostatectomy (mean age 74 years, range 64–91 years), and thirty CP patients undergoing prostate needle biopsies (mean age 68 years, range 46–79 years).Field strength/SequenceMRI scans on a 3.0T scanner (uMR790, United Imaging Healthcare, Shanghai, China). DWI were acquired with 12 b-values (0, 50, 100, 150, 200, 500, 800, 1200, 1500, 2000, 2500, 3000 s/mm2).AssessmentCTRW parameters and ADC were quantified in PCa and CP lesions.Statistical testsThe Mann-Whitney U test was used to evaluate the differences in CTRW parameters and ADC between PCa and CP, high-grade PCa, and low-grade PCa. Spearman’s correlation of the pathologic grading group (GG) with CTRW parameters and ADC was evaluated. The usefulness of CTRW parameters, ADC, and their combinations (Dm, α and β; Dm, α, β, and ADC) to differentiate PCa from CP and high-grade PCa from low-grade PCa was determined by logistic regression and receiver operating characteristic curve (ROC) analysis. Delong test was used to compare the differences among AUCs.ResultsSignificant differences were found for the CTRW parameters (α, Dm) between CP and PCa (all P<0.001), high-grade PCa, and low-grade PCa (α:P=0.024, Dm:P=0.021). GG is correlated with certain CTRW parameters and ADC(α:P<0.001,r=-0.795; Dm:P<0.001,r=-0.762;ADC:P<0.001,r=-0.790). Moreover, CTRW parameters (α, β, Dm) combined with ADC showed the best diagnostic efficacy for distinguishing between PCa and CP as well as predicting Gleason grading. The differences among AUCs of ADC, CTRW parameters and their combinations were not statistically significant (P=0.051–0.526).ConclusionCTRW parameters α and Dm, as well as their combination were beneficial to distinguish between CA and PCa, low-grade PCa and high-grade PCa lesions, and CTRW parameters and ADC had comparable diagnostic performance.

Utilization of diffusion-weighted derived mathematical models to predict prognostic factors of resectable rectal cancer.

This study explored models of monoexponential diffusion-weighted imaging (DWI), diffusion kurtosis imaging (DKI), stretched exponential (SEM), fractional-order calculus (FROC), and continuous-time random-walk (CTRW) as diagnostic tools for assessing pathological prognostic factors in patients with resectable rectal cancer (RRC). RRC patients who underwent radical surgery were included. The apparent diffusion coefficient (ADC), the mean kurtosis (MK) and mean diffusion (MD) from the DKI model, the distributed diffusion coefficient (DDC) and α from the SEM model, D, β and u from the FROC model, and D, α and β from the CTRW model were assessed. There were a total of 181 patients. The area under the receiver operating characteristic (ROC) curve (AUC) of CTRW-α for predicting histology type was significantly higher than that of FROC-u (0.780 vs. 0.671, p = 0.043). The AUC of CTRW-α for predicting pT stage was significantly higher than that of FROC-u and ADC (0.786 vs.0.683, p = 0.043; 0.786 vs. 0.682, p = 0.030), the difference in predictive efficacy of FROC-u between ADC and MK was not statistically significant [0.683 vs. 0.682, p = 0.981; 0.683 vs. 0.703, p = 0.720]; the difference between the predictive efficacy of MK and ADC was not statistically significant (p = 0.696). The AUC of CTRW (α + β) (0.781) was significantly higher than that of FROC-u (0.781 vs. 0.625, p = 0.003) in predicting pN stage but not significantly different from that of MK (p = 0.108). The CTRW and DKI models may serve as imaging biomarkers to predict pathological prognostic factors in RRC patients before surgery.

Dynamics accelerate the kinetics of ion diffusion through channels: Continuous-time random walk models beyond the mean field approximation.

Ion channels are proteins that play a significant role in physiological processes, including neuronal excitability and signal transduction. However, the precise mechanisms by which these proteins facilitate ion diffusion through cell membranes are not well understood. This is because experimental techniques to characterize ion channel activity operate on a time scale too large to understand the role of the various protein conformations on diffusion. Meanwhile, computational approaches operate on a time scale too short to rationalize the observed behavior at the microscopic scale. In this paper, we present a continuous-time random walk model that aims to bridge the scales between the atomistic models of ion channels and the experimental measurement of their conductance. We show how diffusion slows down in complex systems by using 3D lattices that map out the pore geometry of two channels: Nav1.7 and gramicidin. We also introduce spatial and dynamic site disorder to account for system heterogeneity beyond the mean field approximation. Computed diffusion coefficients show that an increase in spatial disorder slows down diffusion kinetics, while dynamic disorder has the opposite effect. Our results imply that microscopic or phenomenological models based on the potential of mean force data overlook the functional importance of protein dynamics on ion diffusion through channels.

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

Understanding the effect of fracture network heterogeneity on flow and transport in tight formations with low permeability values is of great importance in a wide variety of applications, such as groundwater pollution and storage as well as water production. Although numerous studies have investigated solute transport in fractured media, investigation about scale effects on flow and transport in low-permeability aquifers is limited. More specifically, we are not aware of any study that has addressed how the continuous time random walk (CTRW) model parameters vary with scale and fracture network properties. To this end, we performed a set of numerical simulations in sparse fracture networks of sizes L = 20, 35, and 50 m under two different volumetric fracture counts (p30 = 0.05 and 0.1) using the discrete fracture network approach. The fractures in the networks were elliptical in shape, and their radii followed a truncated power-law distribution with exponents α = 1.5, 2, and 2.5. We simulated fluid flow using the Reynolds equation, and solute transport was simulated using particle tracking. The solute transport behavior was quantified by fitting the solution of the CTRW transport equation to the arrival time distributions. Results showed that as the truncated power law exponent α increased, the permeability of the networks decreased. In all the fracture networks studied, we found non-Fickian solute transport behavior deduced from small values of β(0.69≤β≤1.25), the exponent in the truncated power-law arrival time distribution in the CTRW model. In our fracture networks, the aperture size spanned less than one order of magnitude meaning that the fracture permeability distribution was relatively narrow. Therefore, the observed non-Fickian behavior should be attributed to sparse fracture networks and low connectivity. We found that the effective permeability, keff, at p30 = 0.05 was more scale-dependent compared to that at p30 = 0.1 because the fracture networks with p30 = 0.05 were less connected and, thus, more heterogeneous. Our simulation also disclosed the scale dependence of the CTRW model parameters. We conducted multiple linear regression analysis to link the CTRW model parameters to the network size as well as the geometrical and topological properties of the network. Results of regression analysis showed that all the CTRW model parameters except the average solute velocity were significantly dependent on the system size L (p-value < 0.05). Further investigations using a broader range of fracture network properties, such as broader aperture size distributions, smaller and larger network sizes and volumetric fracture counts are still required.

Staging liver fibrosis with various diffusion-weighted magnetic resonance imaging models.

Diffusion-weighted imaging (DWI) has been developed to stage liver fibrosis. However, its diagnostic performance is inconsistent among studies. Therefore, it is worth studying the diagnostic value of various diffusion models for liver fibrosis in one cohort. To evaluate the clinical potential of six diffusion-weighted models in liver fibrosis staging and compare their diagnostic performances. This prospective study enrolled 59 patients suspected of liver disease and scheduled for liver biopsy and 17 healthy participants. All participants underwent multi-b value DWI. The main DWI-derived parameters included Mono-apparent diffusion coefficient (ADC) from mono-exponential DWI, intravoxel incoherent motion model-derived true diffusion coefficient (IVIM-D), diffusion kurtosis imaging-derived apparent diffusivity (DKI-MD), stretched exponential model-derived distributed diffusion coefficient (SEM-DDC), fractional order calculus (FROC) model-derived diffusion coefficient (FROC-D) and FROC model-derived microstructural quantity (FROC-μ), and continuous-time random-walk (CTRW) model-derived anomalous diffusion coefficient (CTRW-D) and CTRW model-derived temporal diffusion heterogeneity index (CTRW-α). The correlations between DWI-derived parameters and fibrosis stages and the parameters' diagnostic efficacy in detecting significant fibrosis (SF) were assessed and compared. CTRW-D (r = -0.356), CTRW-α (r = -0.297), DKI-MD (r = -0.297), FROC-D (r = -0.350), FROC-μ (r = -0.321), IVIM-D (r = -0.251), Mono-ADC (r = -0.362), and SEM-DDC (r = -0.263) were significantly correlated with fibrosis stages. The areas under the ROC curves (AUCs) of the combined index of the six models for distinguishing SF (0.697-0.747) were higher than each of the parameters alone (0.524-0.719). The DWI models' ability to detect SF was similar. The combined index of CTRW model parameters had the highest AUC (0.747). The DWI models were similarly valuable in distinguishing SF in patients with liver disease. The combined index of CTRW parameters had the highest AUC.

Stochastic modeling of injection induced seismicity based on the continuous time random walk model

The spatiotemporal evolution of earthquakes induced by fluid injections into the subsurface can be erratic owing to the complexity of the physical process. To effectively mitigate the associated hazard and to draft appropriate regulatory strategies, a detailed understanding of how induced seismicity may evolve is needed. In this work, we build on the well-established continuous-time random walk (CTRW) theory to develop a purely stochastic framework that can delineate the essential characteristics of this process. We use data from the 2003 and 2012 hydraulic stimulations in the Cooper Basin geothermal field that induced thousands of microearthquakes to test and demonstrate the applicability of the model. Induced seismicity in the Cooper Basin shows all the characteristics of subdiffusion, as indicated by the fractional order power-law growth of the mean square displacement with time and broad waiting-time distributions with algebraic tails. We further use an appropriate master equation and the time-fractional diffusion equation to map the spatiotemporal evolution of seismicity. The results show good agreement between the model and the data regarding the peak earthquake concentration close to the two injection wells and the stretched exponential relaxation of seismicity with distance, suggesting that the CTRW model can be efficiently incorporated into induced seismicity forecasting.

Diffusion transitions induced by shear-thinning viscosity: application to laser-cooled atomic gases

We study the diffusive dynamics of a system in a nonlinear velocity-dependent frictional environment within a continuous time random walk model. In this model, the motion is governed by a shear-thinning frictional force, −γ0v/[1+(v2/vc2)]μ ( 0<μ⩽1 ), where γ 0 represents the coefficient of static friction and µ is the scaling index. Through analytical and numerical results, we construct a diffusion phase diagram that encompasses different regimes upon variations in parameters γ 0 and µ: normal diffusion; superdiffusion; and hyperdiffusion. These transitions occur because the induced weaker friction enhances the diffusion. With a decrease in the scaling index, we find that the γ 0-dependent exponent of diffusion converges towards the experimental findings for ultracold 87Rb atoms because the strong effective friction arises. The discrepancies between the fractional Lévy kinetics and the experimental findings may be potentially reconciled. We believe that these findings are helpful for analyzing experimental observations of cold atoms diffusing in optical lattices.

Can Geometric Parameters Enable Direct Prediction of Non‐Fickian Transport in Rock Fractures Across Diverse Flow Regimes?

AbstractAnomalous solute migrations in fractured rocks are governed by geometric characteristics and flow regimes. Although existing inverse models can describe this behavior, the underlying physics for quantifying key transport coefficients remains largely unexplored. Here, we investigate the quantitative impacts of geometric heterogeneity and flow regimes on solute transport in rock fractures. We conduct numerical experiments to simulate water flow and conservative solute transport in 3D fractures with varying geometric features and Reynolds numbers. Our results show that the non‐Fickian transport is prevalent across the entire flow regime, with Darcy flows attributed to geometric heterogeneity and non‐Darcian flows influenced by additional eddy zones. We employ the mobile‐immobile (MIM) domain model and continuous time random walk (CTRW) model to inversely model simulated breakthrough curves. Inverse analyses demonstrate that both models effectively characterize anomalous transport behaviors. The fitted transport coefficients of the MIM model exhibit stronger quantitative relationships with aperture and roughness parameters, as well as Reynolds number, compared to the CTRW model. By incorporating parameterized transport coefficients, we propose physics‐ and statistics‐based models to directly predict anomalous transport behaviors under different flow regimes. These prediction models accurately reproduce solute transport processes of all simulated cases with acceptable errors. The feasibility of directly predicting solute transport under varying flow regimes using geometric information is thus validated. Our study not only supports the study of substance migration based on geometric structure features, but also serves as a foundation for investigating geological activities based on substance migration information.

Fractional advection diffusion asymmetry equation, derivation, solution and application

The non-Markovian continuous-time random walk model, featuring fat-tailed waiting times and narrow distributed displacements with a non-zero mean, is a well studied model for anomalous diffusion. Using an analytical approach, we recently demonstrated how a fractional space advection diffusion asymmetry equation, usually associated with Markovian Lévy flights, describes the spreading of a packet of particles. Since we use Gaussian statistics for jump lengths though fat-tailed distribution of waiting times, the appearance of fractional space derivatives in the kinetic equation demands explanations provided in this manuscript. As applications we analyse the spreading of tracers in two dimensions, breakthrough curves investigated in the field of contamination spreading in hydrology and first passage time statistics. We present a subordination scheme valid for the case when the mean waiting time is finite and the variance diverges, which is related to Lévy statistics for the number of renewals in the process.

Coupled continuous time random walk with Lévy distribution jump length signifies anomalous diffusion?

The continuous time random walk (CTRW) models are highly valued for studying anomalous diffusion, a ubiquitous phenomenon in complex media whose mean squared displacement (MSD) exhibits power-law behavior. It is widely accepted that the Lévy distribution jump length of random particles with infinite second moment could be responsible for the superdiffusion phenomenon. In this paper, we study the coupled CTRW with exponentially truncated Lévy distribution jump length, deduce the corresponding integrodifferential diffusion equation, and give the propagator expression. It is interesting that we find the MSD for random particles is linearly related to time in the symmetric Lévy distribution case, and it reduces to a simple expression. Additionally, for a non-zero truncated exponent, the MSD also has a linear relationship with time, which reflects the special dependence between waiting time and jump length in this coupled CTRW model.

Fluctuation relation in continuous-time random walks driven by an external field

We study a fluctuation relation representing a non-equilibrium equality indicating that the ratio between the distribution of trajectories obtained by exchanging the initial and final positions is characterized by free energy differences for the duration of the trajectories. We examine the fluctuation relation for noninteracting charge carriers driven by an external electric field by using a continuous-time lattice random walk model with a general waiting-time distribution of transitions. The fluctuation relation is obtained regardless of the lattice structure factor or the form of the waiting-time distribution. However, the fluctuation relation is satisfied only after taking the continuum limit in the presence of a reflecting boundary. Moreover, in free space without boundary conditions, exchanging the initial and final positions is equivalent to exchanging the field (or drift) directions. However, we show that the exchanging field (or drift) directions is not relevant for studying the fluctuation relation under the reflecting boundary condition.

Staging liver fibrosis by a continuous-time random-walk diffusion model

PurposeNoninvasive assessment of liver fibrosis holds significant clinical importance. We aimed to evaluate the clinical potential of using a continuous-time random-walk diffusion model (CTRW) for staging liver fibrosis. MethodsThis prospective study included 52 patients suspected of liver disease and scheduled for liver biopsy. All patients underwent multi-b value diffusion-weighted imaging (DWI) using a 1.5 T MR scanner to derive the anomalous diffusion coefficient (D) and temporal (α) and spatial (β) diffusion heterogeneity indexes sourced from the CTRW. The mono-exponential DWI-derived apparent diffusion coefficient (ADC), transient elastography-derived liver stiffness measurement (LSM), aspartate aminotransferase-to-platelet ratio index (APRI), and fibrosis-4 (FIB-4) index were calculated. We assessed and compared the correlations of these parameters with fibrosis stages and their efficacy in staging liver fibrosis. ResultsSignificant correlations with fibrosis stages were found for APRI (r = 0.336), FIB-4 (r = 0.351), LSM (r = 0.523), D (r = −0.458), and ADC (r = −0.473). Significant differences were observed between APRI, LSM, D, and ADC of different fibrosis stages. The diagnostic performance of an index that combined D, α, β, ADC, and LSM was superior to that of ADC or LSM alone for fibrosis stage F ≥ 2 and better than the index that combined D, α, β for fibrosis stage F ≥ 4. ConclusionsAccurate liver fibrosis staging was achieved with a model that combined CTRW-derived parameters (D, α, and β), ADC, and LSM. The model could serve as a reliable tool for noninvasive fibrosis evaluation.

Renewal processes linked to fractional relaxation equations with variable order

We introduce and study here a renewal process defined by means of a time-fractional relaxation equation with derivative order α(t) varying with time t≥0. In particular, we use the operator introduced by Scarpi in the seventies [23] and later reformulated in the regularized Caputo sense in [5], inside the framework of the so-called general fractional calculus. The obtained model extends the well-known time-fractional Poisson process of fixed order α∈(0,1) and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memory degree of the inter-arrival times) with respect to time. The variable order renewal process is proved to fall outside the usual subordinated representation, since it can not be simply defined as a Poisson process with random time (as happens in the standard fractional case). Finally a related continuous-time random walk model is analyzed and its limiting behavior established.

Identifying potential imaging markers for diffusion property changes in a mouse model of amyotrophic lateral sclerosis: Application of the continuous time random walk model to ultrahigh b-value diffusion-weighted MR images of spinal cord tissue.

Diffusion MRI (dMRI) explores tissue microstructures by analyzing diffusion-weighted signal decay measured at different b-values. While relatively low b-values are used for most dMRI models, high b-value diffusion-weighted imaging (DWI) techniques have gained interest given that the non-Gaussian water diffusion behavior observed at high b-values can yield potentially valuable information. In this study, we investigated anomalous diffusion behaviors associated with degeneration of spinal cord tissue using a continuous time random walk (CTRW) model for DWI data acquired across an extensive range of ultrahigh b-values. The diffusion data were acquired in situ from the lumbar level of spinal cords of wild-type and age-matched transgenic SOD1G93A mice, a well-established animal model of amyotrophic lateral sclerosis (ALS) featuring progressive degeneration of axonal tracts in this tissue. Based on the diffusion decay behaviors at low and ultrahigh b-values, we applied the CTRW model using various combinations of b-values and compared diffusion metrics calculated from the CTRW model between the experimental groups. We found that diffusion-weighted signal decay curves measured with ultrahigh b-values (up to 858,022 s/mm2 in this study) were well represented by the CTRW model. The anomalous diffusion coefficient obtained from lumbar spinal cords was significantly higher in SOD1G93A mice compared with control mice (14.7 × 10-5 ± 5.54 × 10-5 vs. 7.87 × 10-5 ± 2.48 × 10-5 mm2 /s, p = 0.01). We believe this is the first study to illustrate the efficacy of the CTRW model for analyzing anomalous diffusion regimes at ultrahigh b-values. The CTRW modeling of ultrahigh b-value dMRI can potentially present a novel approach for noninvasively evaluating alterations in spinal cord tissue associated with ALS pathology.

Simulation Study on the Mechanism of Intermediate Subdiffusion of Diffusive Particles in Crowded Systems.

The intermediate subdiffusion of diffusive particles in crowded systems is studied for two model systems: the continuous time random walk (CTRW) model and the obstruction-binding model. For the CTRW model with an arbitrarily given longest waiting time τmax, we find that the diffusive particle exhibits subdiffusion below τmax and recovers normal diffusion above τmax. For the obstruction-binding model with randomly distributed attractive obstacles, the diffusion of the diffusive particle is dependent on the binding energy and the density of obstacles. Interestingly, diffusion curves for different binding strengths can be overlapped by rescaling the simulation time, indicating that the diffusive particle in the obstruction-binding model can change from the intermediate subdiffusion to the normal diffusion at a long-term simulation scale. The results of the two model systems show that the diffusive particles only exhibit intermediate subdiffusion below the longest waiting time. Therefore, long timescale subdiffusion would only be observed in the CTRW model with an infinitely long waiting time and in the obstruction-binding model with an infinitely large binding strength.

Subdiffusion with particle immobilization process described by a differential equation with Riemann-Liouville-type fractional time derivative.

An equationdescribing subdiffusion with possible immobilization of particles is derived by means of the continuous time random walk model. The equationcontains a fractional time derivative of Riemann-Liouville type which is a differential-integral operator with the kernel defined by the Laplace transform; the kernel controls the immobilization process. We propose a method for calculating the inverse Laplace transform providing the kernel in the time domain. In the long time limit the subdiffusion-immobilization process reaches a stationary state in which the probability density of a particle distribution is an exponential function.

Evaluation of breast cancer malignancy, prognostic factors and molecular subtypes using a continuous-time random-walk MR diffusion model

PurposeTo assess the continuous-time random-walk (CTRW) model’s diagnostic value in breast lesions and to explore the associations between the CTRW parameters and breast cancer pathologic factors. MethodThis retrospective study included 85 patients (70 malignant and 18 benign lesions) who underwent 3.0T MRI examinations. Diffusion-weighted images (DWI) were acquired with 16b-values to fit the CTRW model. Three parameters (Dm, α, and β) derived from CTRW and apparent diffusion coefficient (ADC) from DWI were compared among the benign/malignant lesions, molecular prognostic factors, and molecular subtypes by Mann-Whitney U test. Spearman correlation was used to evaluate the associations between the parameters and prognostic factors. The diagnostic performance was assessed by the area under the receiver operating characteristic curve (AUC) based on the diffusion parameters. ResultsAll parameters, ADC, Dm, α, and β were significantly lower in the malignant than benign lesions (P < 0.05). The combination of all the CTRW parameters (Dm, α, and β) provided the highest AUC (0.833) and the best sensitivity (94.3%) in differentiating malignant status. And the positive status of estrogen receptor (ER) and progesterone receptor (PR) showed significantly lower β compared with the negative counterparts (P < 0.05). The high Ki-67 expression produced significantly lower Dm and ADC values (P < 0.05). Additionally, combining multiple CTRW parameters improved the performance of diagnosing molecular subtypes of breast cancer. Moreover, Spearman correlations analysis showed that β produced significant correlations with ER, PR and Ki-67 expression (P < 0.05). ConclusionsThe CTRW parameters could be used as non-invasive quantitative imaging markers to evaluate breast lesions.

A concise continuous time random-walk diffusion model for characterization of non-exponential signal decay in magnetic resonance imaging

Diffusion-weighted magnetic resonance imaging (dMRI) is a method of capturing the signal of water molecules diffusing in heterogeneous materials. Gaussian diffusion is interrupted when water mobility is hampered by obstructions in complex structures, and the dMRI signal decay does not match the single exponential decay in Brownian motion. In this study, a concise continuous time random-walk diffusion model is derived with less parameters than the continuous time random walk (CTRW) model and used to characterize the attenuation signal of brain tissue. The fitting results are compared with the CTRW model and the mono-exponential model reflecting the sub-diffusion and the long tail phenomenon of signal decay. Three sample experiments on rat brain and human brain are chosen to evaluate the validity in explaining the anomalous diffusion of water molecules in biological tissues, particularly in brain tissues in diverse directions, which also extends the applications of the concise continuous time random-walk diffusion model. Furthermore, we note that the concise continuous time random-walk diffusion model has practical advantages over the classical exponential model from the perspective of computational accuracy especially in the case of large b values, and has less parameters and is comparable to the CTRW model.

Breast cancer diagnosis and prognosis using a high b-value non-Gaussian continuous-time random-walk model

To compare the diagnostic performance of mono-exponential model-derived apparent diffusion coefficient (ADC), continuous-time random-walk (CTRW) model-derived Dm, α, β and their combinations in discriminating malignancy of breast lesions, and investigate the association between model-derived parameters and prognosis-related immunohistochemical indices. A total of 85 patients with breast lesions (51 malignant, 34 benign) were analysed in this retrospective study. Clinical characteristics include oestrogen receptor (ER), progesterone receptor (PR), human epidermal receptor 2 (HER2), and Ki-67. The ADC was fitted using a mono-exponential model (b-values=0, 800 s/mm2), while Dm, α, and β were fitted using a CTRW model. Independent Student's t-test and the Mann-Whitney U-test were used for the comparison of parameters. Discrimination performance was accomplished by receiver operating characteristic (ROC) analysis, and Spearman's correlation analysis was used to explore the association between immunohistochemical indices and diffusion parameters, the statistical significance level was p<0.05. Dm and ADC demonstrated similar performance in differentiating malignant and benign lesions (AUC=0.928 versus 0.930), while the combination of Dm, α, and β could improve the AUC to 0.969. The combined parameter generated by ADC, Dm, α, and β was effective in identifying the ER+/ER- and PR+/PR- patients. Temporal heterogeneity parameter α correlated significantly with the expression of PR. Diffusion parameters derived from the CTRW model could effectively discriminate the malignancy of breast lesions. Meanwhile, the hormone receptor expression could be distinguished by combined diffusion parameters, and have the potential to reflect the prognosis.

Preoperative prediction of VETC in hepatocellular carcinoma using non-Gaussian diffusion-weighted imaging at high b values: a pilot study.

Vessels encapsulating tumor clusters (VETC) have been considered an important cause of hepatocellular carcinoma (HCC) metastasis. To compare the potential of various diffusion parameters derived from the monoexponential model and four non-Gaussian models (DKI, SEM, FROC, and CTRW) in preoperatively predicting the VETC of HCC. 86 HCC patients (40 VETC-positive and 46 VETC-negative) were prospectively enrolled. Diffusion-weighted images were acquired using six b-values (range from 0 to 3000 s/mm2). Various diffusion parameters derived from diffusion kurtosis (DK), stretched-exponential (SE), fractional-order calculus (FROC), and continuous-time random walk (CTRW) models, together with the conventional apparent diffusion coefficient (ADC) derived from the monoexponential model were calculated. All parameters were compared between VETC-positive and VETC-negative groups using an independent sample t-test or Mann-Whitney U test, and then the parameters with significant differences between the two groups were combined to establish a predictive model by binary logistic regression. Receiver operating characteristic (ROC) analyses were used to assess diagnostic performance. Among all studied diffusion parameters, only DKI_K and CTRW_α significantly differed between groups (P=0.002 and 0.004, respectively). For predicting the presence of VETC in HCC patients, the combination of DKI_K and CTRW_α had the larger area under the ROC curve (AUC) than the two parameters individually (AUC=0.747 vs. 0.678 and 0.672, respectively). DKI_K and CTRW_α outperformed traditional ADC for predicting the VETC of HCC.