Non-Gaussian water diffusion kurtosis imaging of prostate cancer

Purpose

To evaluate the non-Gaussian water diffusion properties of prostate cancer (PCa) and determine the diagnostic performance of diffusion kurtosis (DK) imaging for distinguishing PCa from benign tissues within the peripheral zone (PZ), and assessing tumor lesions with different Gleason scores.

Materials and Methods

Nineteen patients who underwent diffusion weighted (DW) magnetic resonance imaging using multiple b-values and were pathologically confirmed with PCa were enrolled in this study. Apparent diffusion coefficient (ADC) was derived using a monoexponential model, while diffusion coefficient (D) and kurtosis (K) were determined using a DK model. Differences between the ADC, D and K values of benign PZ and PCa, as well as those of tumor lesions with Gleason scores of 6, 7 and ≥ 8 were assessed. Correlations between parameters D and K in PCa were analyzed using Pearson’s correlation coefficient. ADC, D and K values were correlated with Gleason scores of 6, 7 and ≥ 8, respectively.

Results

ADC and D values were significantly (p < 0.001) lower in PCa (0.79 ± 0.14 μm²/ms and 1.56 ± 0.23 μm²/ms, respectively) compared to benign PZ (1.23 ± 0.19 μm²/ms and 2.54 ± 0.24 μm²/ms, respectively). K values were significantly (p < 0.001) greater in PCa (0.96 ± 0.20) compared to benign PZ (0.59 ± 0.08). D and K showed fewer overlapping values between benign PZ and PCa compared to ADC. There was a strong negative correlation between D and K values in PCa (Pearson correlation coefficient r = − 0.729; p < 0.001). ADC and K values differed significantly in tumor lesions with Gleason scores of 6, 7 and ≥ 8 (p < 0.001 and p = 0.001, respectively), although no significant difference was detected for D values (p = 0.325). Significant correlations were found between the ADC value and Gleason score (r = − 0.828; p < 0.001), as well as the K value and Gleason score (r = 0.729; p < 0.001).

Conclusion

DK model may add value in PCa detection and diagnosis. K potentially offers a new metric for assessment of PCa.     

Spatio-temporal anomalous diffusion imaging: results in controlled phantoms and in excised human meningiomas

Recently, we measured two anomalous diffusion (AD) parameters: the spatial and the temporal AD indices, called γ and α, respectively, by using spectroscopic pulse gradient field methods. We showed that γ quantifies pseudo-superdiffusion processes, while α quantifies subdiffusion processes. Here, we propose γ and α maps obtained in a controlled heterogeneous phantom, comprised of packed micro-beads in water and in excised human meningiomas. In few words, α maps represent the multi-scale spatial distribution of the disorder degree in the system, while γ maps are influenced by local internal gradients, thus highlighting the interface between compartments characterized by different magnetic susceptibility. γ maps were already obtained by means of AD stretched exponential imaging and α-type maps have been recently achieved for fixed rat brain with the aim of highlighting the fractal dimension of specific brain regions. However, to our knowledge, the maps representative of the spatial distribution of α and γ obtained on the same controlled sample and in the same excised tissue have never been compared. Moreover, we show here, for the first time, that α maps are representative of the spatial distribution of the disorder degree of the system.     

Solutions for a non-Markovian diffusion equation

Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.     

Entropy as a measure of diffusion

The time variation of entropy, as an alternative to the variance, is proposed as a measure of the diffusion rate. It is shown that for linear and time-translationally invariant systems having a large-time limit for the density, at large times the entropy tends exponentially to a constant. For systems with no stationary density, at large times the entropy is logarithmic with a coefficient specifying the speed of the diffusion. As an example, the large-time behaviors of the entropy and the variance are compared for various types of fractional-derivative diffusions.     

Diffusion kurtosis imaging of the human kidney: A feasibility study

Purpose

To assess the feasibility and to optimize imaging parameters of diffusion kurtosis imaging (DKI) in human kidneys.

Methods

The kidneys of ten healthy volunteers were examined on a clinical 3 T MR scanner. For DKI, respiratory triggered EPI sequences were acquired in the coronal plane (3 b-values: 0, 300, 600 s/mm², 30 diffusion directions). A goodness of fit analysis was performed and the influence of the signal-to-noise ratio (SNR) on the DKI results was evaluated. Region-of-interest (ROI) measurements were performed to determine apparent diffusion coefficient (ADC), fractional anisotropy (FA) and mean kurtosis (MK) of the cortex and the medulla of the kidneys. Intra-observer and inter-observer reproducibility using Bland-Altman plots as well as subjective image quality of DKI were examined and ADC, FA, and MK parameters were compared.

Results

The DKI model fitted better to the experimental data (r = 0.99) with p < 0.05 than the common mono-exponential ADC model (r = 0.96).Calculation of reliable kurtosis parameters in human kidneys requires a minimum SNR of 8.31 on b = 0 s/mm² images.Corticomedullary differentiation was possible on FA and MK maps. ADC, FA and MK revealed significant differences in medulla (ADC = 2.82 × 10^− 3 mm²/s ± 0.25, FA = 0.42 ± 0. 05, MK = 0.78 ± 0.07) and cortex (ADC = 3.60 × 10^− 3 mm²/s ± 0.28, FA = 0.18 ± 0.04, MK = 0.94 ± 0.07) with p < 0.001.

Conclusion

Our initial results indicate the feasibility of DKI in the human kidney presuming an adequate SNR. Future studies in patients with kidney diseases are required to determine the value of DKI for functional kidney imaging.     

A practical approach to in vivo high-resolution diffusion tensor imaging of rhesus monkeys on a 3-T human scanner

The nonhuman primate brain study provides important supplemental means for human brain exploration since the two species share close anatomical and functional similarities. MR diffusion tensor imaging (DTI) in human brain has revealed exquisite details of brain structures especially in the brain white matter. However, most previous monkey brain DTI results lack the spatial resolution in comparison to the conventional tracing and postmortem imaging methods, especially when it is acquired in commonly available human MRI scanners of field strength of 3 T or lower. To meet the increasing demands for nonhuman primate DTI studies, we proposed an in vivo high-resolution monkey DTI acquisition protocol that is practically feasible and combined it with an improved postprocessing procedure for a 3-T human scanner. The acquisition protocol, susceptibility distortion correction method with phase reversal acquisition, and postprocessing steps were proved to be effective in our study of rhesus monkeys. Results from diffusion tensor estimations and fiber tractography at 1 x 1 x 1 mm(3) resolution were found to be comparable to previous ex vivo DTI studies with much longer acquisition times. Effects of image resolution were evaluated and it was confirmed that the partial volume effect due to the larger voxel size in low-resolution data biased the diffusion tensor estimation and produced erroneous fiber tractography. Our results suggest that in vivo high-resolution monkey brain DTI can be achieved within practical time, which allows accurate diffusion tensor estimation and fiber tractography in monkey brains, so that the complex anatomical structures within many small but important anatomic structures can be delineated.     

Quantitative assessment of motion correction for high angular resolution diffusion imaging

Several methods have been proposed for motion correction of high angular resolution diffusion imaging (HARDI) data. There have been few comparisons of these methods, partly due to a lack of quantitative metrics of performance. We compare two motion correction strategies using two figures of merit: displacement introduced by the motion correction and the 95% confidence interval of the cone of uncertainty of voxels with prolate tensors. What follows is a general approach for assessing motion correction of HARDI data that may have broad application for quality assurance and optimization of postprocessing protocols. Our analysis demonstrates two important issues related to motion correction of HARDI data: (1) although neither method we tested was dramatically superior in performance, both were dramatically better than performing no motion correction, and (2) iteration of motion correction can improve the final results. Based on the results demonstrated here, iterative motion correction is strongly recommended for HARDI acquisitions.     

Elevations of diffusion anisotropy are associated with hyper-acute stroke: a serial imaging study

Diffusion tensor imaging (DTI) studies of human ischemic stroke within 24 h of symptom onset have reported variable findings of changes in diffusion anisotropy. Serial DTI within 24 h may clarify these heterogeneous results. We characterized longitudinal changes of diffusion anisotropy by analyzing discrete ischemic white matter (WM) and gray matter (GM) regions during the hyperacute (2.5-7 h) and acute (21.5-29 h) scanning phases of ischemic stroke onset in 13 patients. Mean diffusivity (MD), fractional anisotropy (FA) and T2-weighted signal intensity were measured for deep and subcortical WM and deep and cortical GM areas in lesions outlined by a > or =30% decrease in MD. Average reductions of approximately 40% in relative (r) MD were observed in all four brain regions during both the hyperacute and acute phases post stroke. Overall, 9 of 13 patients within 7 h post symptom onset showed elevated FA in at least one of the four tissues, and within the same cohort, 11 of 13 patients showed reduced FA in at least one of the ischemic WM and GM regions at 21.5-29 h after stroke. The fractional anisotropy in the lesion relative to the contralateral side (rFA, mean+/-S.D.) was significantly elevated in some patients in the deep WM (1.10+/-0.11, n=4), subcortical WM (1.13+/-0.14, n=4), deep GM (1.07+/-0.06, n=1) and cortical GM (1.22+/-0.13, n=5) hyperacutely (< or =7 h); however, reductions of rFA at approximately 24 h post stroke were more consistent (rFA= 0.85+/-0.12).     

Accurate measurement of translational diffusion coefficients: a practical method to account for nonlinear gradients

For NMR probes equipped with pulsed field gradient coils, which are not optimized for gradient linearity, the precision and accuracy of experimentally measured translational diffusion coefficients are limited by the linearity of the gradient pulses over the sample volume. This study shows that the accuracy and precision of measured diffusion coefficients by the Stejskal--Tanner spin-echo pulsed field gradient experiment can be significantly improved by mapping the gradient z-profile and by using the mapped calibration parameters in the data analysis. For practical applications the gradient distribution may be approximated by a truncated linear distribution defined by minimum and maximum values of the gradient. By including the truncated linear gradient distribution function in the Stejskal--Tanner equation, the systematic deviation between the fitted curve and the experimental attenuation curve decreases by an order of magnitude. The gradient distribution may be calibrated using an intense NMR signal from a sample with a known diffusion coefficient. The diffusion coefficient of an unknown sample may then be determined from a two-parameter fit, using the known gradient distribution function.     

Correlation function induced by a generalized diffusion equation with the presence of a harmonic potential

An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations, which includes short, intermediate and long-time memory effects described by the waiting time probability density function. Analytical expression for the correlation function is obtained and analyzed, which can be used to describe, for instance, internal motions of proteins. The result shows that the generalized diffusion equation has a broad application and it may be used to describe different kinds of systems.     

On the effects of dephasing due to local gradients in diffusion tensor imaging experiments: relevance for diffusion tensor imaging fiber phantoms

The effect of susceptibility differences between fluid and fibers on the properties of DTI fiber phantoms was investigated. Thereto, machine-made, easily producible and inexpensive DTI fiber phantoms were constructed by winding polyamide fibers of 15 microm diameter around a circular acrylic glass spindle. The achieved fractional anisotropy was 0.78+/-0.02. It is shown by phantom measurements and Monte Carlo simulations that the transversal relaxation time T(2) strongly depends on the angle between the fibers and the B(0) field if the susceptibilities of the fibers and fluid are not identical. In the phantoms, the measured T(2) time at 3 T decreased by 60% for fibers running perpendicular to B(0). Monte Carlo simulations confirmed this result and revealed that the exact relaxation time depends strongly on the exact packing of the fibers. In the phantoms, the measured diffusion was independent of fiber orientation. Monte Carlo simulations revealed that the measured diffusion strongly depends on the exact fiber packing and that field strength and -orientation dependencies of measured diffusion may be minimal for hexagonal packing while the diffusion can be underestimated by more than 50% for cubic packing at 3 T. To overcome these effects, the susceptibilities of fibers and fluid were matched using an aqueous sodium chloride solution (83 g NaCl per kilogram of water). This enables an orientation independent and reliable use of DTI phantoms for evaluation purposes.     

Non-Markovian diffusion equations and processes: Analysis and simulations

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.     

Diffusion tensor imaging to study anisotropy in a particular porous system: the trabecular bone network

During the last decade, considerable effort has been invested into the development of diffusion tensor imaging (DTI) mainly used to investigate cerebral morphology. The aim of this paper is to review and to discuss our recent results about high magnetic field DTI application to study spongy bone tissue. Due to its peculiar properties, spongy bone represents a particular porous system sample. Strategies to perform DTI on porous systems and issues linked to DTI outcome interpretation are presented on the basis of our results concerning trabecular bone network characterization.     

A review of chemical diffusion: Criticism and limits of simplified methods for diffusion coefficient calculation

A review of the physics and modelling of mass diffusion involving different gaseous chemical species is firstly proposed. Both accurate and simplified models for mass diffusion involve the calculation of individual species diffusion coefficients. Since these are computationally expensive, in CFD they are commonly estimated by assuming constant Lewis or Schmidt numbers for each chemical species. The constant Lewis number assumption is particularly used. As a matter of fact, these assumptions have never been theoretically justified nor verified in practical flames. The only published information are the first observations by Smooke and Giovangigli about the Lewis number against temperature distributions in methane–air premixed and counterflow diffusion one-dimensional flames. The aim of this work is to verify these assumptions. Functional dependences of molecular properties appearing in these numbers are made explicit to show that while Sc _i depends only on composition, Le _i depends also on temperature and therefore it certainly cannot be assumed constant in a flame. Then, accurately calculating molecular properties, distributions of these characteristic numbers against temperature are obtained a posteriori from numerical simulations of different flames, premixed and non-premixed, and burning different fuels. For non-premixed flames, individual species Lewis number distributions are broad for most of the species considered in this article, whilst they are tight for premixed flames. Some attention is focused on the particular shape of Lewis distributions in non-premixed flames: they are characterized by four or five (when extinction is experienced) branches associated to precise regions in the flame (basically, lean, rich and stoichiometric combusting zones). Instead, the Schmidt distributions are always tighter, also when extinctions take place: for many species they can be approximatively assumed constant. Finally, a simplified procedure to estimate individual species diffusion coefficients is suggested, assuming the median of non-premixed flame Schmidt distributions has a constant value for each chemical species.     

A framework for quality control and parameter optimization in diffusion tensor imaging: theoretical analysis and validation

In this communication, a theoretical framework for quality control and parameter optimization in diffusion tensor imaging (DTI) is presented and validated. The approach is based on the analytical error propagation of the mean diffusivity (D(av)) obtained directly from the diffusion-weighted data acquired using rotationally invariant and uniformly distributed icosahedral encoding schemes. The error propagation of a recently described and validated cylindrical tensor model is further extrapolated to the spherical tensor case (diffusion anisotropy approximately 0) to relate analytically the precision error in fractional tensor anisotropy (FA) with the mean diffusion-to-noise ratio (DNR). The approach provided simple analytical and empirical quality control measures for optimization of diffusion parameter space in an isotropic medium that can be tested using widely available water phantoms.     

Two-Higgs-doublet models in light of current experiments: a brief review

We briefly survey several typical CP-conserving two-Higgs-doublet models (2HDMs) in light of current experiments. First we derive the masses and couplings of the mass eigenstates from the Lagrangians. Then we analyze the constraints from theory and oblique electroweak parameters. Finally, we delineate the status of 2HDM in light of the LHC searches, the dark matter detections and the muon g − 2 measurement.     

数据采集方案对神经扩散模型影响的评估

基于单次数据采集的多种扩散模型联合应用已逐渐成为临床研究的热点,本研究比较了三种采集方案对于神经扩散模型定量计算的影响,包括Q空间笛卡尔网格（QGrid）、多壳层异向（Free）和多壳层同向（MDDW）采集方案,涉及的扩散模型包含扩散张量成像（DTI）;扩散峰度成像（DKI）;神经突方向分散度和密度成像（NODDI）;平均表观传播（MAP）模型.结果表明DTI和DKI模型对采集方案相对不敏感,而NODDI和MAP对采集方案和最大b值的设置相对较敏感,并且QGrid和Free方案一致性较高,因此在大样本和多中心研究中需要考虑采集方案的选择.此外,考虑到QGrid和Free方案分别在结合更多扩散模型和神经纤维束成像应用上更具优势,因此推荐使用.     

Validation of practical diffusion approximation for virtual near infrared spectroscopy using a digital head phantom

Light propagation in the digital head phantom for virtual near infrared spectroscopy and imaging is calculated by diffusion theory. In theory, diffusion approximation is not valid in a low-scattering cerebrospinal fluid (CSF) layer around the brain. The optical path length and spatial sensitivity profile predicted by the finite element method based upon the diffusion theory are compared with those predicted by the Monte Carlo method to validate a practical implementation of diffusion approximation to light propagation in an adult head. The transport scattering coefficient of the CSF layer is varied from 0.01 to 1.0 mm⁻¹ to evaluate the influence of that layer on the error caused by diffusion approximation. The error is practically ignored and the geometry of the brain surface such as the sulcus structure in the digital head phantom scarcely affects the error when the transport scattering coefficient of the CSF layer is greater than 0.3 mm⁻¹.     

Crossover from normal diffusion to single-file diffusion of particles in a one-dimensional channel: LJ particles in zeolite zsm-22

The crossover from normal Fickian diffusion, where mean-squared displacement goes as t to single-file diffusion, where <z²> t?^0.5 is studied as function of particle size confined in zeolite zsm-22 using molecular dynamics simulations. The simulation results indicate that the crossover is smooth as the particle size increases and the diffusion reaches single file through a series of subdiffusion processes. A small number of mutual passage of particles can destroy the correlated single-file diffusion. The density dependence on single-file mobility obtained from molecular dynamics simulations are in very good agreement with theoretical predictions.