Diffusion tensor parameters and principal eigenvector coherence: Relation to b-value intervals and field strength

Diffusion-weighted MRI images acquired at b-value greater than 1000 s mm^− 2 measure the diffusion of a restricted pool of water molecules. High b-value images are accompanied by a reduction in signal-to-noise ratio (SNR) due to the application of large diffusion gradients. By fitting the diffusion tensor model to data acquired at incremental b-value intervals, we determined the effect of SNR on tensor parameters in normal human brains, in vivo. In addition, we also investigated the impact of field strength on the diffusion tensor model. Data were acquired at 1.5 and 3 T, at b-values 0, 1000, 2000 and 3000 s mm^− 2 in twenty diffusion-sensitised directions. Fractional anisotropy (FA), mean diffusivity (MD) and principal eigenvector coherence (κ) were calculated from diffusion tensors fitted between datasets with b-values 0–1000, 0–2000, 0–3000, 1000–2000 and 2000–3000 s mm^− 2. Field strength and b-value effects on diffusion parameters were analysed in white and grey matter regions of interest. Decreases in FA, κ and MD were found with increasing b-value in white matter. Univariate analysis showed a significant increase in FA with increasing field strength in highly organised white matter. These results suggest there are significant differences in diffusion parameters at 1.5 and 3 T and that the optimal results, in terms of the highest values of FA in white matter, are obtained at 3 T with a maximum b = 1000 s mm^− 2.     

Geometric analysis of the b-dependent effects of Rician signal noise on diffusion tensor imaging estimates and determining an optimal b value

The optimal diffusion weighting (DW) factor, b, for use in diffusion tensor imaging (DTI) studies remains uncertain. In this study, the geometric relations of DW quantities are examined, in particular, the effects of Rician noise in the measured magnetic resonance signal. This geometric analysis is used to make theoretical predictions for selecting a b value to reduce the influence of noise. It is shown that the optimal b value for DTI studies in healthy human parenchyma is approximately b=1200 s mm⁻², with a simple relation given as well for a given expected apparent diffusion coefficient. Monte-Carlo simulations on sets of realistic DTI measures are then performed, verifying the optimal DW for minimizing estimate errors. The effects of noise on various DTI parameters such as anisotropy indices (fractional anisotropy and scaled relative anisotropy), mean diffusivity, radial diffusivity, eigenvalues and the direction of the first eigenvector are investigated as well.     

Magnetic resonance diffusion-weighted imaging: quantitative evaluation of age-related changes in healthy liver parenchyma

The purpose of this study was to verify in healthy liver parenchyma the possible influence of age on DwI-related parameters: apparent diffusion coefficient (ADC), perfusion fraction (PF), diffusion and pseudodiffusion coefficient (D and D^?). Forty healthy adult volunteers (age range 26-86 years), divided into four age groups, were prospectively submitted to a breath-hold magnetic resonance diffusion imaging (MR-DwI) (two b values, 0-300 and 0-1000 s/mm²). A smaller cohort of 16 subjects underwent a free-breath multi-b acquisition (16 b values, 0-750 s/mm²). Quantitative analysis was performed by two observers with manually defined regions of interest, on the most homogeneous portion of the right liver lobe. Individual and group statistical analysis of data was performed: ANOVA to establish differences between groups and Pearson correlation coefficient to investigate the association between DwI parameters and age. The mean, S.D. and 95% limits of agreement of ADC values for each age-defined group are reported. ANOVA showed no significant differences between group means (P always >.05). No significant correlation between subjects' age and DwI parameters was established, both in breath-hold and free-breath acquisitions, on the whole range of adopted b values. Our study conducted on healthy liver parenchyma shows that there are no significant differences in ADC, PF, D and D^? of younger or older subjects.     

Diffusion-weighted magnetic resonance imaging of human skeletal muscles: gender-, age- and muscle-related differences in apparent diffusion coefficient

Purpose

To evaluate the apparent diffusion coefficient (ADC) of skeletal muscle based on signal intensity (SI) attenuation vs. increasing b values and to determine ADC differences in skeletal muscles between genders, age groups and muscles.

Materials and Methods

Diffusion-weighted images (b values in the range of 0–750 s/mm² at increments of 50 s/mm²) of the ankle dorsiflexors (116 subjects) and the erector spinae muscles (86 subjects) were acquired with a 1.5-T MR device. From the two different slopes obtained in SI vs. b-value logarithmic plots, ADC^b0–50 (b values=0 and 50 s/mm²) reflected diffusion and perfusion, while ADC^b50–750 (b values in the range of 50–750 s/mm² at increments of 50 s/mm²) approximated the true diffusion coefficient. Moreover, to evaluate whether this b-value combination is appropriate for assessing the flow component within muscles, diffusion-weighted images of the ankle dorsiflexors (10 subjects) were obtained before and during temporal arterial occlusion.

Results

ADC^b0–50 and ADC^b50–750 were found to be 2.64×10^–3 and 1.44×10^–3 mm²/s in the ankle dorsiflexors, and 3.02×10^–3 and 1.49×10^–3 mm²/s in the erector spinae muscles, respectively. ADC^b0–50 was significantly higher than ADC^b50–750 in each muscle (P<.01). The erector spinae muscles showed significantly higher ADC values than the ankle dorsiflexors (P<.01). However, for each muscle, there were few significant gender- and age-related ADC differences. Following temporal occlusion, ADC^b0–50 of the ankle dorsiflexors decreased significantly from 2.49 to 1.6×10^–3 mm²/s (P<.01); however, ADC^b50–750 showed no significant change.

Conclusion

Based on the SI attenuation pattern, muscle ADC could be divided into ADC that reflects both diffusion and perfusion, and ADC that approximates a true diffusion coefficient. There were significant differences in ADC of functionally distinct muscles. However, we barely found any gender- or age-related ADC differences for each muscle.     

Biexponential apparent diffusion coefficients in prostate cancer

Purpose

The purpose of this study was to investigate the need for biexponential signal decay modeling for prostate cancer diffusion signal decays with b-factor over an extended b-factor range.

Materials and Methods

Ten healthy volunteers and 12 patients with a bulky prostate cancer underwent line scan diffusion-weighted MR imaging in which b-factors from 0 to 3000 s/mm² in 16 steps were sampled. The acquired signal decay curves were fit with both monoexponential and biexponential signal decay functions and a statistical comparison between the two fits was performed.

Results

The biexponential model provided a statistically better fit over the monoexponential model on the peripheral zone (PZ), transitional zone (TZ) and prostate cancer. The fast and slow apparent diffusion coefficients (ADCs) in the PZ, TZ and cancer were 2.9±0.2, 0.7±0.2×10⁻³ mm²/ms (PZ); 2.9±0.4, 0.7±0.2×10⁻³ mm²/ms (TZ); and 1.7±0.4, 0.3±0.1×10⁻³ mm²/ms (cancer), respectively. The apparent fractions of the fast diffusion component in the PZ, TZ and cancer were 70±10%, 60±10% and 50±10%, respectively. The fast and slow ADCs of cancer were significantly lower than those of TZ and PZ, and the apparent fraction of the fast diffusion component was significantly smaller in cancer than in PZ.

Conclusions

Biexponential diffusion decay functions are required for prostate cancer diffusion signal decay curves when sampled over an extended b-factor range, providing additional, unique tissue characterization parameters for prostate cancer.     

In vivo lipid diffusion coefficient measurements in rat bone marrow

The diffusion coefficient of lipids, D_l, within bone marrow, fat deposits and metabolically active intracellular lipids in vivo will depend on several factors including the precise chemical composition of the lipid distribution (chain lengths, degree of unsaturation, etc.) as well as the temperature. As such, D_l may ultimately prove of value in assessing abnormal fatty acid distributions linked to diseases such as cystic fibrosis, diabetes and coronary heart disease. A sensitive temperature dependence of D_l may also prove of value for MR-guided thermal therapies for bone tumors or disease within other fatty tissues like the breast. Measuring diffusion coefficients of high molecular weight lipids in vivo is, however, technically difficult for a number of reasons. For instance, due to the much lower diffusion coefficients compared to water, much higher b factors than those used for central nervous system applications are needed. In addition, the pulse sequence design must incorporate, as much as possible, immunity to motion, susceptibility and chemical shift effects present whenever body imaging is performed. In this work, high b-factor line scan diffusion imaging sequences were designed, implemented and tested for D_l measurement using a 4.7-T horizontal bore animal scanner. The gradient set available allowed for b factors as high as 0.03 μs/nm² (30,000 s/mm²) at echo times as short as 42 ms. The methods were used to measure lipid diffusion coefficients within the marrow of rat paws in vivo, yielding lipid diffusion coefficients approximately two orders of magnitude smaller than typical tissue water diffusion coefficients. Phantom experiments that demonstrate the sensitivity of lipid diffusion coefficients to chain length and temperature were also performed.     

Evaluation of noncirrhotic hepatic parenchyma with and without significant portal vein stenosis using diffusion-weighted echo-planar MR on the basis of multiple-perfusion-components theory

To determine whether diffusion-weighted echo-planar MR images are sensitive to liver perfusion difference.Noncirrhotic livers of 71 patients (43 males, 28 females; age range, 22-87 years; mean, 61 years) without (n=51) and with (n=20) significant (>70%) portal vein stenosis (accompanying proximal hepatic arterial stenosis and/or biliary tract obstruction in 10) by tumors were examined with diffusion-weighted echo-planar sequences (modified for b factors of 1, 28, 66, 288 and 600 s/mm²). On the basis of multiple-perfusion-components theory, i.e., assuming logarithm of signal intensity for liver perfusion is linearly attenuated versus logarithm of a smaller b factor, we defined the slope of the line as the perfusion-related D′ value. The D′ values of these livers were calculated from images with b factors of 1, 28, and 66 s/mm². The livers' apparent diffusion coefficient values for diffusion (ADC_d values) were calculated from images with b factors of 288 and 600 s/mm².The livers with significant portal vein stenosis had statistically lower mean D′ values than the livers without portal vein stenosis (P<.001 on the Mann-Whitney U test). However, there was no significant difference in ADC_d values between these liver types (P>.05).The D′ value calculated from diffusion-weighted echo-planar sequences with plural smaller b factors may be sensitive to liver perfusion difference.     

Diffusion tensor magnetic resonance imaging of the normal breast

Purpose

The objective of this study was to evaluate diffusion anisotropy of the breast parenchyma and assess the range and repeatability of diffusion tensor imaging (DTI) parameters in normal breast tissue.

Materials and Methods

The study was approved by our institutional review board and included 12 healthy females (median age, 36 years). Diffusion tensor imaging was performed at 1.5 T using a diffusion-weighted echo planar imaging sequence. Diffusion tensor imaging parameters including tensor eigenvalues (λ₁, λ₂, λ₃), fractional anisotropy (FA) and apparent diffusion coefficient (ADC) were measured for anterior, central and posterior breast regions.

Results

Mean normal breast DTI measures were λ₁=2.51×10⁻³ mm²/s, λ₂=1.89×10⁻³ mm²/s, λ₃=1.39×10⁻³ mm²/s, ADC=1.95±0.24×10⁻³ mm²/s and FA=0.29±0.05 for b=600 s/mm². Significant regional differences were observed for both FA and ADC (P<.05), with higher ADC in the central breast and higher FA in the posterior breast. Comparison of DTI values calculated using b=0, 600 s/mm² vs. b=0, 1000 s/mm², showed significant differences in ADC (P<.001), but not FA. Repeatability assessment produced within-subject coefficient of variations of 4.5% for ADC and 11.4% for FA measures.

Conclusion

This study demonstrates anisotropy of water diffusion in normal breast tissue and establishes a normative range of breast FA values. Attention to the influence of breast region and b value on breast DTI measurements may be important for clinical interpretation and standardization of techniques.     

DTI at 7 and 3 T: systematic comparison of SNR and its influence on quantitative metrics

Diffusion tensor imaging (DTI) and advanced related methods such as diffusion spectrum and kurtosis imaging are limited by low signal-to-noise ratio (SNR) at conventional field strengths. DTI at 7 T can provide increased SNR; however, B0 and B1 inhomogeneity and shorter T2? still pose formidable challenges. The purpose of this study was to quantify and compare SNR at 7 and 3 T for different parallel imaging reduction factors, R, and TE, and to evaluate SNRs influences on fractional anisotropy (FA) and apparent diffusion coefficient (ADC). We found that R>4 at 7 T and R≥2 at 3 T were needed to reduce geometric distortions due to B0 inhomogeneity. For these R at 7 T, SNR was 70-90 for b=0 s/mm² and 22-28 for b=1000s/mm² in central brain regions. SNR was lower at 3 T (40 for b=0 s/mm² and 15 for b=1000 s/mm²) and in lateral brain regions at 7 T due to B1 inhomogeneity. FA and ADC did not change with MRI field strength, SENSE factor or TE in the tested range. However, the coefficient of variation for FA increased for SNR <15 and for SNR <10 in ADC, consistent with published theoretical studies. Our study demonstrates that 7 T is advantageous for DTI and lays the groundwork for further development. Foremost, future work should further address challenges with B0 and B1 inhomogeneity to take full advantage for the increased SNR at 7 T.     

Parton densities beyond perturbation theory

I compute non-perturbative corrections to the kernel governing the evolution of non-singlet parton densities. The model used is QED in the limit of many charged particles. I find an infrared renormalon at b₀t = −1, where t is the Borel variable and b₀ is the first coefficient of the β-function. This term has a non-trivial dependence on the variable x = -q²/(2p · q) and its coefficient scales as x³/(1 − x)² (p is the momentum of the hadron and q is the momentum transfer). An extrapolation of my results to QCD implies a breakdown of the parton model near the elastic region.     

Some estimates for magnetic fluctuations in a turbulent medium

A new integral relationship between the fluctuations b(r, t) of a magnetic field and its mean B ₀(r, t) is derived for the steady-state magnetic field in a turbulent medium. This formula provides the estimate 〈b?curlb〉=?B ₀?curlB ₀. Simultaneously, the coefficient of amplification of the mean magnetic field α effect) is obtained: α=(η+β)B ₀? curlB ₀/B ₀ ² . The formula for α allows for a decrease in this coefficient owing to the back action of the magnetic field on the turbulent velocity field. It is shown that the Zel’dovich’s estimate 〈 b ²〉?β/η B ₀ ² for two-dimensional turbulence holds for magnetic fields at the instant the fluctuations 〈a ²〉 of the vector potential, rather than 〈b ²〉, reach a maximum. Here, η and β are the ohmic (molecular) and turbulent diffusion coefficients, respectively. This estimate is refined with allowance made for the fact that the condition for diffusion approximation itself relates the β, b, and B ₀ quantities to each other.     

Optimised diffusion-weighting for measurement of apparent diffusion coefficient (ADC) in human brain

This work studies the effect of diffusion-weighting on the precision of measurements of the apparent diffusion coefficient (ADC, or D) by diffusion-weighted magnetic resonance imaging. The precision in the value of the ADC was described in terms of a diffusion-to-noise ratio (DNR) which was calculated as the signal-to-noise ratio in the resultant ADC. A theoretical analysis decomposed the DNR into the signal-to-noise ratio in the diffusionweighted image and the sensitivity of diffusion-weighting, “κ_D”. The latter reflects the effect of the sampling strategy in the diffusion-weighting domain on the DNR. The theoretical analysis demonstrated that optimal two-point diffusion-weighting could be achieved in the vicinity of ξ = D(b₂−b₁) = 1.1, where ξ is a nondimensional parameter of diffusion-weighting, and b₁ and b₂ are the diffusion-weighting factors for the two-point diffusion-weighting. This approach also derived an optimised signal averaging scheme. The limitations and restrictions of the two-point scheme for in vivo ADC measurement were also considered; these included a detailed discussion on partial volume effects. The theory was verified by experiments on phantoms and on the brain of a healthy volunteer using a diffusion-weighted echo-planar imaging protocol. This led to an optimal two-point diffusion-weighting for ADC measurement in human brain using b₁ = 300, and b₂ = 1550 ± 100 s/mm². Such a two-point scheme successfully measured values of the ADC in gray matter, white matter and cerebrospinal fluid in human brain. It thus offers an alternative to the commonly used multiple-point schemes and has the advantage of requiring significantly shorter imaging times.     

On the effects of a varied diffusion time in vivo: is the diffusion in white matter restricted?

The aim of this work was to study the diffusion-related signal attenuation curves (signal-vs.-b curves) measured perpendicular and parallel to the neuronal fibers of the corticospinal tract in vivo and to determine whether effects of restricted diffusion could be observed when varying the diffusion time (T_D). A biexponential model and a two-compartment model including exchange according to the Kärger formalism were employed to analyze the signal-vs.-b curves. To validate the two-compartment model, restricted diffusion with exchange was simulated for uniformly sized cylinders, using different diameters and exchange times. The model was shown to retrieve the simulated parameters well, also when the short gradient pulse approximation was not met. The in vivo measurements performed perpendicular to the tracts, using b values up to 28000 s/mm² and T_D values between 64 and 256 ms, did not show the effects of restricted diffusion as expected from previous ex vivo studies. The applied two-compartment model yielded an average axonal diameter of about 4 μm and an intracellular exchange time of about 300 ms, but did not fit statistically well to the data. In conclusion, this study indicates that if the diffusion is modeled as two compartments, of which one is restricted, exchange must be included in the model.     

On the connection between the macroscopical and microscopical evolution in an exactly soluble hopping model: I. Neutral particles

The hopping rate equation for neutral particles, on an arbitrary periodical lattice, can be solved exactly. It is shown that if one scales the time t and the distances x(t→λ²t, x→λx) then, in the λ→∞ limit, the particle density tends to the solution of the diffusion equation faster than λ^?3. The diffusion coefficient is the same as obtained from both Kubo and Miller-Abrahams theory via the Einstein relation.     

Usefulness of diffusion derived vessel density computed from a simplified IVIM imaging protocol: An experimental study with rat biliary duct blockage induced liver fibrosis

ObjectivesLiver vessel density can be evaluated by DDVD (diffusion derived vessel density): DDVD_(b0b1) = Sb0/ROIarea0 – Sb1/ROIarea1, where Sb0 and Sb1 refer to the liver signal when b is 0 or 1 s/mm². Sb1 and ROIarea1 may be replaced by other b-values. With a rat biliary duct ligation (BDL) model, this study assesses the usefulness of liver DDVD computed from a simplified IVIM imaging protocol using b = 25 and b = 50 to replace b = 1 s/mm², alone and in combination with other IVIM parameters.MethodsMale Sprague-Dawley rats were used. The rat number was 5, 5, 5, and 3 respectively, for the timepoints of 7, 14, 21, 28 days post-BDL surgery. 12 rats had partial biliary duct recanalization performed after the rats had BDL for 7 days and then again followed-up for a mean of 14 days. Liver diffusion MRIs were acquired at 3.0 T with a b-value distribution of 0, 25, 50, 75, 100, 150, 300, 700, 1000 s/mm². DDVD_mean (control rats n = 6) was the mean of DDVD_(b0b25) and DDVD_(b0b50). IVIM fitting started from b = 0 s/mm² with segmented fitting and a threshold b of 50 s/mm² (n = 5 for control rats). Three 3-D spaces were constructed using a combination of the four diffusion parameters.ResultsThe control rats and BDL rats (n = 18) had a liver DDVD_mean of 84.0 ± 26.2 and 44.7 ± 14.4 au/pixel (p < 0.001). All 3-D spaces totally separated healthy livers and all fibrotic livers (n = 30, BDL rats and recanalization rats). The mean relative distance between healthy liver cluster and fibrotic liver cluster was 0.331 for PF, D_slow, and D_fast; 0.381 for PF, D_fast, and DDVD_mean; and 0.384 for PF, D_slow, and DDVD_mean.ConclusionA combination of PF, D_slow, and D_fast allows total separation of healthy livers and fibrotic livers and the integration of DDVD improved the separation.     

Superposition model (SPM) analysis of zero-field splitting parameters of Mn doped ZnAl2S4 crystal

In this study, superposition model (SPM) is employed to investigate the local environment around the different Mn²⁺ centers in ZnAl₂S₄ spinel. Using SPM and crystallographic data, the zero-field splitting (ZFS) parameters b₂⁰, b₄⁰, and b₄³ are calculated for Mn²⁺ at the B-site (with local symmetry D_3d), whereas b₄⁰ and b₄⁴ for the A-site (T_d). The lattice relaxation due to Mn²⁺ impurities is analyzed in terms of the bonding lengths and the angles between the Mn-S bond and the crystallographic axis [1 1 1]. Our SPM analysis of ZFS parameters indicates that satisfactory agreement can be achieved between the theoretical and the experimental results. Additional structural information about Mn²⁺ impurity centers is also obtained.     

Effect of diffusion on gain saturation in gas lasers

A theoretical model is developed to account for the effect of diffusion on gain saturation in gas lasers. It is shown that the effect can be characterized by the functionK(k ₂ a, k ₂ b), which depends on pressure, temperature, radius of the beama, and the radius of the cylindrical cavityb. It is concluded that diffusion can be very effective whenk ₂ a is small andb is several times larger thana.     

Growth mechanism of Al2O3 scales developed on Fe Cr Al alloys

By combining classical techniques used for oxidation studies, electrochemical measurements on the oxide scale and ¹⁸O profilings by using the ¹⁸O(p, α) ¹⁵N reaction, it was possible to obtain reliable data on the growth mechanisms of alumina scales developed at 1082°C on three Fe Cr Al alloys differing by their impurity amount. It was shown that aluminum diffusion through the scale is the major material transport mechanism during oxidation, resulting in an outside scale growth. Nevertheless, carbon and other impurities enhance the oxygen inward diffusion and improve the scale adherence. Alumina scales behave as a mixed ionic and electronic conductor, but the incorporated impurities decrease the values of t_i, σ_i and kp, thus accounting for a slower oxidation rate in an industrial alloy than in a high purity one.     

Quantifying hepatic fibrosis using a biexponential model of diffusion weighted imaging in ex vivo liver specimens

The purpose of this study was to evaluate the non-Gaussian behavior of diffusion related signal decay of the ex vivo murine liver tissues from a dietary model of hepatic fibrosis. To this end, a biexponential formalism was used to model high b-value diffusion imaging (up to 3500 s/mm²), the findings of which were correlated with liver histopathology and compared to a simple monoexponential model. The presence of a major, fast diffusing component and a minor, slow diffusing component was demonstrated. With increasing hepatic fibrosis, the fractional contribution of the fast diffusing component decreased, as did the diffusion coefficient of the fast diffusing component. Strong correlation between the degrees of liver fibrosis and a two-predictor regression model incorporating parameters of the biexponential model was found. Using Akaike's Information Criterion analyses, the biexponential model resulted in an improved fit of the high b-value diffusion data when compared to the monoexponential model.     

The far-infrared spectrum of tetrahydrothiophene: The bend/twist vibrations and associated barriers

The far-infrared spectrum of tetrahydrothiophene is reinvestigated with a resolution of 0.12 cm^?1 in the region of 50–350 cm^?1. In addition to the bend transitions (ν_b) below 120 cm^?1 reported previously, a number of sequences revealed by the improved resolution are observed for the first time and assigned to transitions of 2ν_b, of the twist (ν_t), and of difference combinations (ν_t - ν_b). Simple one-dimensional modeling of the twist sequence, which is derived from a self-consistent bend-twist energy level diagram, with a quadratic-quartic Hamiltonian suggests a barrier to planarity on the order of 4250 cm^?1. A two-dimensional potential function in the dimensionless coordinates is found to be V(q_b, q_t) = ?249.6q_b² + 4.48q_b⁴ ? 215.5q_t² + 2.73q_t⁴ + 7.00q_b²q_t².