Root mean squared error in the training data as a function of the SNR in the measurements. For high SNR the regression method needs a higher order than linear to do an optimal fit. Since the model is nonlinear and the analytical inversion requires a logarithm, as we increase the SNR, polynomials become suboptimal. For low SNR all regressions perform equally well, and their behaviour is represented by the theory.

Standard deviation of three regression methods for each parameter in the SM. As Eq. 2 shows, for high SNR the variance reaches the values in the training while for low SNR it tends to 0. Maps for f estimated for the high SNR data (right) and different levels of artificially decreased SNR (values are an average over a WM ROI). Even though the noise increases from right to left, the standard deviation of the values decreases (WM ROI). The corresponding SNR values are highlighted, the trend observed is similar. Additionally, the mean values get closer to the expected value of the prior used.

Figure 3 - A water phantom with a folded surface demonstrates the effect of stationary background gradients. In addition to image distortions, background gradients cause signal variation when waveforms are rotated, causing a positive bias in FA (true value is zero in pure water). Direction encoded color-maps, brightness scaled by FA$$$\in[0.0~0.4]$$$, show prominent coross-term effects for conventional waveforms (high FA), compared to cross-term compensated waveforms, which appear more robust to background gradients (low FA). Violin plots show FA distributions in the slice.

Figure 3 - A water phantom with a folded surface demonstrates the effect of stationary background gradients. In addition to image distortions, background gradients cause signal variation when waveforms are rotated, causing a positive bias in FA (true value is zero in pure water). Direction encoded color-maps, brightness scaled by FA$$$\in[0.0~0.4]$$$, show prominent coross-term effects for conventional waveforms (high FA), compared to cross-term compensated waveforms, which appear more robust to background gradients (low FA). Violin plots show FA distributions in the slice.

Figure 3: IVIM parameter maps (D, f, D^*) of the least-squares, Bayesian and IVIM-NET_optim approaches to IVIM fitting of a PDAC patient before receiving CRT. The red ROI represents the PDAC. The least-squares and Bayesian approaches appear noisy, whereas IVIM-NET_optim shows less noisy and more detailed parameter maps.

Figure 1: Representation of the PI-DNN with different hyperparameter options. Here, the input signal, consisting of the measured DWI signal, is fed forward either through (A) a parallel network design or (B) the original single fully-connected network design. The blue crossed circles indicate random dropout. The output layer is constrained with either absolute or sigmoid activation functions. Subsequently, the network predicts the IVIM signal which is used to compute the loss function.

Figure 2. Panel (A): Different training data distribution strategies: (i) uniform distribution, (ii) in-vivo parameter combinations from traditional model fitting, (iii) in-vivo white matter parameter combinations and (iv) in-vivo grey matter parameter combinations. Panel (B): RMS errors of v_cyl estimates (top) and λ_cyl estimates (bottom) at different parameter combinations. Panel (C): Parameter maps estimated from each training data distribution. Panel (D): Difference between parameter maps in panel (C) and parameter maps from traditional model fitting in Figure 1.

Figure 4. The bias in parameter estimates for (i) the uniform training data distribution and for (ii) the in-vivo parameter distribution. The arrows start from the ground truth of each parameter combination and end at the estimates. Red arrows mark the tissue abnormalities shown in Figure 3. In (ii), the red contours show the training data density. Multiple regions of the parameter space act as “sinks” towards which estimates of nearby parameters are biased.

Figure 2. Deploying trained networks on previously unseen synthetic and in vivo data provides anatomically plausible parameter maps in under 10 s. The parameter maps corresponding to the synthetic dataset (second column) are compared to the corresponding ground-truth labels (first column), with difference maps being shown in the third column. Parameter maps obtained from applying a trained network on in vivo brain data are displayed in the fourth column.

Figure 5. Sensitivity of different acquisition protocols to 10% parameter modulations. The matrices display the relation between an induced parameter change and the observed response. When a single parameter on the y-axis is modulated by 10%, the response can be read in all other parameters along the x-axis. An ideal network would report a diagonal matrix with the value 10% on the diagonal, and zero otherwise. Protocol A appears sensitive in all parameters, whereas Protocols B and C lack sensitivity to D_Δ;Z and D_I;S, respectively.

Figure 5: Representative parametric maps of D (first row), V_b (second row), and F (third row) for the monopolar and ten-point b-M₁-optimized data samplings for two subjects. Reduced variability in the estimates in the right lobe of the liver is observed for the estimates obtained using the b-M₁-optimized sampling in comparison to the monopolar sampling. For the monopolar sampling, Subject 2 demonstrates areas of instability in the fitting (yellow arrows), whereas the b-M₁-optimized sampling demonstrates stability.

Figure 2: b-M₁-optimized waveforms associated with the Cramer-Rao lower bound (CRLB) b-M₁-optimized data sampling (Figure 1b). Waveforms were designed by combining a monopolar gradient waveform with the M₁-nulled gradient waveform, which allows for the simultaneous control of b and M₁.⁹ Of these ten waveforms or data points in the b-M₁-optimized sampling, numbers 1, 2, 5, 8, and 9 provide the optimal solution to Eq. (2) for a five-point b-M₁-optimized sampling, and the addition of number 6 provides a six-point b-M₁-optimized sampling.

Fig.2 – Ex vivo CTI kurtosis measures for a representative slice of brain specimens from the 3 h post-stroke (A) and sham (B) animal groups. From left to right: panels A and B show the total kurtosis (K_T), anisotropic kurtosis (K_aniso), isotropic kurtosis (K_iso) and intracompartmental kurtosis (K_intra). K_T and K_intra reveal to have elevated contribution in both white and gray matter in the ipsilesional hemisphere, where the infarct core is located.

Fig.3 – Specificity analysis after ANOVA (multiple comparisons) across different regions of interest (ROIs). Different kurtosis sources are plotted along rows and each plot contains the respective kurtosis source measure for the L1, R1 and counter control animal group ipsilateral hemisphere, consecutively. Each column refers to only GM and only WM, respectively (p < 0.05).

Figure 3. Scalar maps obtained by employing the QTI+ estimation on the full, p81, p56, and p39 protocols. Note how a large part of the information available in the full and p81 protocols is retained, when the p56 and p39 protocols were employed. The bars on the last row show the mean absolute deviation of the maps from their ground truth values (maps computed on the full dataset) for the three protocols. These plots highlight the resilience to downsampling achieved by imposing positivity constraints.

Figure 5. (a) $$$\mu FA$$$ maps obtained by fitting the QTI model with WLLS(ss) and emplying QTI+ on the four protocols. The fits performed with both methods on the full protocol are used as reference. (b) Difference between the reference $$$ \mu FA$$$ maps and those estimated with both methods on the three protocols. (c) Histograms showing the distribution of $$$ \mu FA$$$ values for the three protocols.

Coefficients of Variation over an ROI of the posterior limb of the internal capsule for rotational invariants, DTI parameters, and DKI parameters, averaged over all subjects. P1 v. P2 and S1 v. S2 measure scan-rescan variation. TE=92 v. TE=128 measure the effect of varying the echo time, CV for Prisma 1 and Prisma 2 scan vs. Prisma 3 scan were averaged. P v. S measure inter-scan variation, where scans Skyra 1 and Skyra 2 scan vs. Prisma 3 scan (with matched echo times) were averaged. Rescaling to the voxel level by $$$1/\sqrt{N voxels}$$$ yields CV ~ 1%.

Voxel-wiseVoxelwise coefficients of variation are shown for MD, MK, and FA for scan-rescan data on the Prisma and the Skyra. Generally, theThe CVs are at 5-10% level on a voxelwise basis in whiter matter and higher in gray matter.

Figure 5. Diffusion dispersion rate (Λ) maps of a single slice from each subject (different rows) for all b-values (different columns), for OGSE shapes without flow compensation. All sampled frequencies for each respective b-value are accounted for in computation. Voxels with negative Λ are masked. The underlying contrast is similar for all b-values but stabilizes as the b-value increases (going from left to right), as fewer pixels exhibit spurious intensities.

Figure 1. Comparison of OGSE gradient waveforms and power spectra. All plots are in arbitrary units, and the gradient plots show effective shapes including the inversion by the refocusing RF pulse. Power spectra are pictured for a standard OGSE shape with a half-period gap (A), a reduced gap OGSE shape² (B), and a further modification to the latter producing FC (C). Note that the power spectra of (B) and (C) have higher peaks and smaller side lobes than that of (A). Also, the zero frequency peak of power spectrum (B) is eliminated in (C) by extending the innermost gradient lobes of (C) by T_FC.

Figure 1. Scheme of the neural network multi-parametric mapping pipeline. Real and imaginary parts of the 12-point bSSFP phase-cycling data are fed voxelwise as input into the NN (green box). For each voxel, a 3x3 window of nearest neighbors in the axial plane is extracted leading to 216 input features. WM asymmetries in the magnitude profile are shown for a representative ROI in corpus callosum (orange box). The trained NNs enable voxelwise predictions of four diffusion metrics: MD, FA, azimuth (Φ) and inclination (ϴ) (blue box).

Figure 3. NN predictions (middle) are depicted for a representative axial slice of a testing subject not included into NN training in comparison to reference data (left) obtained with standard SE-EPI-based DTI fitting. On the right, the corresponding relative uncertainty maps of the NN are shown.

Segmentation of multiple-sclerosis lesions in six scans from the validation set. (a) Ground truth of one example slice per scan, (b) predictions for that slice using our equivariant model, (c) predictions for that slice using the non-rotation-equivariant reference model with 3D convolutional layers, (e) ROC curves of all models on the full scans, (f) precision-recall curves of all models on the full scans. While our equivariant model is very certain (yellow areas) at most positions, the non-rotation-equivariant model has large areas of high uncertainty (purple areas).

Segmentation of multiple-sclerosis lesions in a scan from the validation set (not used for training) using our equivariant (top) and the non-rotation-equivariant (bottom) model, both trained on reduced subsets of the training set (from left to right) with the ground truth segmentation in the left column. While our equivariant model achieves quite accurate segmentations when trained on 26.3% of the training scans, the segmentations of the non-rotation-equivariant model only start getting accurate when trained on 65.8% of the set, indicating better generalization of our model.

Figure 1: Diffusion weighted images for an example healthy volunteer showing improved SNR (separability of gray and white matter compartments in b-values >=2000 s/mm²) without spatial smoothing across multiple intensities of DL-based denoising.

Figure 3: DL-based denoising improved differentiation across signal compartments while preserving quantitative tensor-based metrics. (A-C) Histogram analysis demonstrating distribution of FA values in whole brain CSF, Gray matter (GM) and WM. Quantitative diffusion metrics were compared across two level of DL denoising (50% and 75%). Results were consistent across both volunteers and across acquisitions. (D) FA values across 50 WM regions remain unchanged demonstrating that DL-based denoising maintains tensor-based quantitative metrics.

Figure 5: First column shows high b-value prospective DWI with slice thickness 2 mm, NEX=1, 2. More noise in test images can be observed with reduction in slice thickness. Last column shows their denoised counterparts. Noisy & denoised parts are magnified (highlighted with red and yellow squares) in second & third columns. Significant noise reduction and increase in entropy measures can be noted.

Figure 2: Visualization of Openneuro high b-value DWI denoising: First column (second row onwards) shows input images with Rician noise at σ =0.01,0.03 and 0.05 respectively. Last column shows denoised images. Noisy & denoised parts are magnified (highlighted with red and yellow squares respectively) in second & third columns. Significant reduction in noise can be observed in the denoised version.

Example of an unseen slice fed into the ensemble model. (a) Normalized Axial FA Slice. (b) Z-Scoring label. (c – e) Model predictions for 2D-UNets with Resnet101, Vgg19, InceptionV4 Encoders with Dice scores 0.69, 0.69, 0.70 respectively. (f) Ensemble Predictive Uncertainty. (g) Ensemble Prediction. (h) Ensemble Prediction with “Optimal” Threshold and a Dice Score of 0.71.

Dice-Score vs Threshold for all models with and without TTA. The ensemble performs most reliably over the threshold range.

Figure 1. Outline of the neural network architecture.

The output value has small risk to become an outlier because it is generated by the combination of the weighted averages for the neighboring pixels in the original image. The loss to be minimized consists of both the mean absolute error between the output and the target images and the Euclid distance between the derived diffusion tensors for efficient optimization.

Figure 4. The results of the ROI-based analysis.

dNR was closer to NEX8 than NEX1 in most regions and parameter maps, and some significant differences between NEX1 and NEX8 were resolved in between dNR and NEX8. However, dNR was far from NEX8 than NEX1 in some combinations of the region and the parameter. Especially, in the regions of deep white matter and periventricular white matter in ODI, the difference compared to NEX8 was significant in dNR and not in NEX1.

Figure 3. Illustration of DTI images, ADC maps, FA maps and colored FA maps of a selected slice, from top to bottom row. Each column from left to right are reference images reconstructed from full k-space data, images reconstructed from zero-filled under-sampled k-space data, images reconstructed from under-sampled k-space data using U-net, PD-net and Cascade-net.

Figure 1.a. Structure of U-net.^1,4 The number of filters in each layer is a quarter of the original structure. Zero-padding is adapted to maintain the size of images in convolutions. 1.b. Structure of PD-net.⁵ In this work, the activation function PReLU non-linearity in the original structure was substituted by ReLU non-linearity for consistency with U-net and Cascade-net.¹ 1.c. Structure of Cascade-net.² The number of filters was set as 48 in this work, which was 64 in the original structure.¹

Figure1: (a) In conventional DTI, acquired DWIs are first used to estimate the diffusion tensor. The resulting tensor is used to compute the DTI metrics. (b) In proposed DL-DTI, the acquired DWIs are first used to predict DWIs that were not acquired. The acquired and predicted DWIs are used together for estimating the diffusion tensor. The diffusion tensor is then used to compute the DTI metrics.

Figure 3: Comparison of DTI metrics. Directionally Encoded Color (DEC), Fractional Anisotropy (FA), Mean Diffusivity (MD), Axial Diffusivity (AD), and Radial Diffusivity (RD) maps are shown for a reference (N=64) dataset together with the same metrics obtained using the conventional DTI (Figure 1(a)) and DL-DTI (Figure 1(b)) approaches using a varying number of input DWIs. The metrics obtained from DL-DTI correspond well to those obtained using the reference dataset even at high acceleration rates.

Figure 5: The percentage accuracy of the classifier on test data for each of the 11 cross-validation iterations is shown, for both the network trained on DWI slices with intensity values windowed between 0-25, and for the network trained without windowing. The mean accuracy and corresponding standard deviation across all 11 iterations is also shown for both cases.

Figure 5: The percentage of slices containing Nyquist ghosts is shown for each pair of subjects. For each subject pair, the percentage for the constituent subjects are also shown, labelled with the subject number. The mean (44.2%) across all subject pairs is given, along with the standard deviation (5.3%) which is much lower than that of slices containing Nyquist Ghosting in individual subjects, as shown in Figure 2.

Figure 3. Direction-encoded fractional anisotropy maps. Fractional anisotropy maps color encoded by the primary eigenvector (red: left–right; green: anterior–posterior; blue: superior–inferior) derived from the diffusion tensors fitted using interpolated data (a, b), SRDTI super-resolution data (c), and ground-truth high-resolution data (d) along axial, coronal and sagittal directions, showing regions-of-interest in the internal capsule (i, ii), cerebral cortex (iii, iv) and pons (v, vi), respectively.

Figure 2. Image results. b=0 images (row a) and diffusion-weighted images (DWIs) (row c) along one direction of the six optimized diffusion-encoding directions (i.e., [0.91, 0.416, 0]) from interpolated data (i, ii) (interpolated data using cubic spline is the input to SRDTI), SRDTI output data (iii), ground-truth high-resolution data (iv), and their residuals comparing to the ground-truth high-resolution images (rows b, d). A region-of-interest in the deep white matter (yellow boxes) is displayed in enlarged views (rows, b, d, column iv).

Figure 1: Architecture design of the proposed residual CNN. Input is 144 stacked diffusion-weighted magnitude images, normalized to the mean LV intensity of the first image. The final output 6 channels represents the tensor entries. Convolutional layers (blue) have a $$$3\times3\times N$$$ kernel and ReLU activation. The residual blocks (green) consist of the parallel paths with different kernel sizes. The outputs are concatenated and reduced 1 convolution layer to match the input channels. The output of a residual block is added to the skip connection and fed to the next block.

Figure 2: Illustration of the simulated training data and visual comparison to real in vivo data. (a) Animation of single averages for each linear diffusion-weighting and (b) example of the correspondingly simulated and LV-masked training dataset. The red contours show a dilated LV-mask in which the inference on the unregistered data is performed. (c-d) Maps of mean MD, FA as well as an artificial lesion map resulting serving as ground truth of the simulated dataset.

Figure 1: Deep learning framework aiming at predicting the microstructural features from raw diffusion MRI (dMRI) data using histology images as ground truth. Example of prediction of main fiber orientation from previously acquired data on human thalamus.

Figure 2: Summary of the whole pipeline that enables to predict a single simulated fiber orientation using deep learning. A) Normalized dMRI signal from an B) infinitely long rotating cylinder as a function of the C) PGSE sequence b values for different gradient orientation. D) This signal is then fed to a neural network to predict the E) orientation of the cylinder given by the spherical angles theta and phi.

Figure 2) Top are the FA weighted colour maps of the primary direction of diffusion. The motor tract, highlighted in yellow, is enlarged at the bottom where the primary directions of diffusion are illustrated as colour encoded sticks. The sticks are masked such that only WM voxels remain, determined by FA>0.2. Estimations from Patch-CNN are visually more similar to the GT for both RGB colourmap and sticks.

Figure 4) Boxplots computed over the medians of errors for each of the 4 testing subjects. For (a,c,d) the median error is computed across all of the brain voxels at each subject. For (b) median error is computed for each subject across voxels for which the primary direction of diffusion is well defined, where the linearity coefficient⁹>0.6.

Figure 1. Schematic diagram of the nonselective STEAM prepared multislice echoplanar sequence.

Figure 2: Example multi-slice b=1000 diffusion weighted images after correction for differential slice delays by geometric averaging.

Figure 2 g-factor calculated for all accelerated acquisitions, red line indicates g-factor of 1, which represents the SNR of the unaccelerated (S1P1) dMRI acquisiton. p-values indicated by asterisks and were calculated using a paired t-test, in this figure, all p < 0.0001 (****).

Figure 3. FA, MD, MK, noise SD (σ) maps and their corresponding error maps of the M¹NCM-based methods, using the simulated data with unstationary noise level of 0.02. The error maps show the absolute difference between the reference parameters and the estimated parameters. The RMSE of each parameter map is shown in the right bottom of its error map. The unit of the MD is ×10^-3 mm²/s.

Figure 1. RMSE comparisons of FA, MD, MK maps estimated with the UVNLM-NLS, UVNLM-CWLLS, M¹NCM, M¹NCM-NSS, M¹NCM-LSS, M¹NCM-NSS-LSS, and M¹NCM-NSS-LSS-PR algorithms, using the simulated datasets with stationary and fixed noise levels of 0.02-0.09.

Figure 1: Schematic representation of a filter exchange imaging (FEXI) sequence using two pulsed gradient spin echo (PGSE) blocks. The first gradient pair used as the FEXI filter is followed by a varying mixing time during which the magnetization is longitudinally stored while transversal components are dephased. Before and after the second and third radiofrequency pulse, respectively, gradients to choose the right echo path are applied. The second gradient pair is a standard diffusion weighting. This block is followed by an echo planar imaging (EPI) readout.

Figure 2: Comparison of the standard deviations σ resulting from the different trigger experiments. No diffusion weighting was applied (b = 0 s/mm²). The three used orthogonal diffusion encoding directions (2/3, 2/3, 1/3), (1/3, 2/3, 2/3) and (2/3, 1/3, 2/3) are depicted in blue, red and black, respectively.

Fig. 4. f, D*, Dapp, Kapp maps and their corresponding error maps of the proposed UVNLM-NLS method. The error maps show the absolute difference between the reference parameters and the estimated parameters.

Fig. 1. RMSE comparisons of f, D*, Dapp, Kapp maps estimated with the conventional NLS, the PCA-NLS, the proposed UVNLM-NLS method.

Figure 1. Bland–Altman plots of four diffusion metrics for 3 different ROIs, namely whole spinal cord area, white matter, and grey matter. The x and y-axes represent, respectively, average, and mean differences between pre- and post-upgrade. The blue lines show the mean of the differences, while the pairs of dotted red lines denote the 95% confidence intervals.

Figure 2. Parametric maps for fractional anisotropy (FA), mean diffusity (MD), mean kurtosis (MK) and neurite density index (NDI) from a single subject obtained pre- and post-upgrade.

Figure 3. Derived microstructural maps and parameters. From top to bottom: Mean diffusivity (MD) from diffusion kurtosis fit, fractional anisotropy (FA) calculated from diffusion kurtosis fit, microscopic fractional anisotropy (μFA) from QTI, normalised size variance (C_MD) from QTI. STD=standard deviation.

Figure 4. (A) Mean diffusion-weighted LTE image at b-value of 1000 s/mm² shows contrast within the hippocampal subregions. Subregions were manually segmented for subiculum (blue), Cornu Ammonis (CA) regions CA1-3 (red) and CA4 with dentate gyrus (DG, yellow). (B) Microstructural parameters were calculated in each subregion; Subiculum, CA1-3 and CA4/DG.

Figure 2: Asymmetric motion compensated (M₁=M₂=0) arbitrary (ARB) waveform optimally designed with GrOpt (blue) and the corresponding trapezoidal (TRAP) implementation before (red) and after (green) balancing. Temporal evolution of the zero (M₀), first (M₁) and second (M₂) order gradient moments are represented for each waveform. ARB has the lowest residual moments, followed by balanced TRAP, then unbalanced TRAP.

Figure 4: Symmetric motion compensated trapezoidal (TRAP) waveforms (M₁=M₂=0) with SR_limit=50mT/m/s (blue) and SR_max=200mT/m/s (red) are compared to an arbitrary (ARB) cubic ramp-up waveform (green). For this scanner setup the best cubic ramp-up shape parameter was obtained for a=1.5x10^-3. The temporal evolution of the b-value as well as the peripheral nerve stimulation (PNS) plots are represented for each waveform. As marked by the red dashed line, a PNS ≥1 indicates potential stimulation.

Figure 2. Experimental (markers) and fitted (lines) “powder-averaged”¹⁸ signal S/S₀ vs. b-value. (a) Two-compartment phantom with pure water and a concentrated solution of magnesium nitrate give rise to two isotropic Gaussian (ω-independent) components. (b) Polydomain lamellar liquid crystal giving Gaussian axial and radial diffusivities, D_A and D_R, as estimated with a “Pake” model fit¹². (c) Sediment of yeast cells with intra- and extracellular compartments, the latter exhibiting restricted (ω-dependent) diffusion.

Figure 1. Gradient waveforms g_i(t) with duration τ for comprehensive sampling of the 2D space of root-mean-square frequency ω_rms and normalized anisotropy b_Δ (“shape”) of the tensor-valued diffusion encoding spectrum b(ω) with elements b_ij(ω). The q-vector trajectories shown for the b_Δ= 0 case are derived from magic-angle spinning (MAS) and double rotation (DORn). Superquadric tensor glyphs²⁹ along the vertical axis indicate the special values b_Δ = –1/2, 0, 1/2, and 1. The waveforms are scaled to give identical b-values.

Figure 1: Tuned ellipsoidal tensor encodings with varying spectral anisotropy featuring more encoding power at low frequencies (LF-ETE) or high frequencies (HF-ETE) along z-axis. While the trace spectra is similar for LF-ETE and HF-ETE, the power distribution across different directions is markedly different. Color coding is based on the RGB-weights given by projections of the encoding power in the three bands with crossover frequencies determined by $$$D(\omega)$$$ for a sphere with $$$D_0^2R^{-4}=8000 \,\mathrm{s}^{-2}$$$ reaching 1/3 and 2/3 of the asymptotic value.

Figure 3: Directional spread of signals (normalized STD, $$$\sigma_\mathrm{S}/\bar{S}$$$) as a function of compartment size for cylindrical, ellipsoidal and stick-like restrictions. HF-ETE (solid blue lines) and LF-ETE (dashed red lines) for Watson orientation distributions with varying order parameters (OP). Location of the minima is independent of OP. For cylindrical restrictions, the minima occur only with the HF-ETE, while for stick-like restrictions, the minima occur only with the LF-ETE. For ellipsoidal restrictions we have minima with both LF-ETE and HF-ETE.

Figure 1. A comparison between a high quality reference mean axial diffusivity map (left column) and mean axial diffusivity maps obtained using the REF and FAST methods. The label in the upper left corner shows the estimation method used and the numbers indicate number of b-tensors / number of averaged estimates. The number in the lower left corner shows the normalized mean absolute error between the image in the left column and the current image.

Figure 3. The time required to estimate one sample for the REF and FAST methods. Note that this time scales linearly with the number of samples. I.e. if 25 estimates are averaged as for some of the maps in Figure 1 and 2 the time in the table needs to be multiplied by 25.

Fig. 3. Summary statistical maps for each participant averaged across all sessions. μFA. As expected, the bright voxels are highest in dense white matter regions. Bright voxels in SD and COV maps show more variability in regions contaminated with cerebrospinal fluid such as the ventricles, and low variability in white matter regions. MKi. Bright voxels located in areas of isotropic diffusion such as CSF in the ventricles. MKa Bright voxels demonstrate regions of strong anisotropic diffusion.

Fig. 2A. Voxel-wise whole brain white matter Pearson correlations are presented between individual time points for μFA white matter skeletons. Voxels pooled across all subjects for each map, r = Pearson correlation coefficient, p<.0001 for all plots. Univariate histograms on the diagonal show distributions of voxels of μFA across all voxels. Calculation of metrics in session 2 of one subject was not robust and thus excluded.

The scatter plots of relative signal intensities in the enhanced lesions in brain tumors, the normal thalamus, the normal white matters, and the lateral ventricle. The horizontal and vertical axes are relative signal intensities from the parallel directions and from the anti-parallel directions, respectively. There were significantly correlated in all areas (p < 0.001).

The scatter plots of relative signal intensities in the enhanced lesions, the normal thalamus, the normal white matters, and the lateral ventricle. The horizontal and vertical axes are relative signal intensities from the collinear directions which are the averages of the parallel and anti-parallel directions, and from the orthogonal directions, respectively. There were significantly correlated in all areas (p < 0.001). The slopes of the thalamus and white matter seemed to be different from those in figure 2.

Fig.4 – Kurtosis estimates of a representative rat brain coronal slice: A) CTI kurtosis estimates; B) Tensor-valued kurtosis estimates obtained by fitting Eq. 2 to all acquired data (TV(all)); C) Tensor-valued kurtosis estimates obtained by fitting Eq. 2 to only the data acquired with b₁=b₂ and parallel/perpendicular gradient directions (TV(sel)).

Fig.5 – Scatter plots of tensor-valued vs CTI kurtosis estimates: A) Tensor-valued kurtosis estimates obtained by fitting Eq. 2 to all acquired data (TV(all)) vs CTI estimates; B) Tensor-valued kurtosis estimates obtained by fitting Eq. 2 to DDE data acquired with b₁=b₂ (TV(sel)) vs CTI estimates. The points in the scatter plots are color-coded according to CTI’s K_intra estimates.

Figure 4: The $$$\bar{K}$$$ maps for the various data set are shown for the ordinary and regularized NLS in the top and middle row, respectively. Moreover, we show the map of the predicted mean kurtosis $$$\hat{K}$$$ (bottom row) to illustrate the similarity in contrast.

Figure 2: The tract-averaged $$$\bar{K}$$$ using various fitting strategies and $$$\hat{K}$$$ (bottom row) for the test (filled marker) and retest data (open marker) each subject (labeled by marker shape). The graphs on the right column show the same data, but windowed differently for enhanced contrast. Seven major white matter tracts were evaluated: genu and splenium of the corpus callosum (GCC and SCC), corticospinal tract (CST), arcuate fasciculus (AF), inferior fronto-occipital fasciculus (IFO), Superior Longitudinal Fasciculus (SLF) and optic radiation (OR).

Figure 4: Time-scaling MRI parameters derived from in vivo MAP-MRI data with different diffusion times quantify anomalous diffusion in live brain tissues. The random walk dimension, d_w, and the spectral dimension, d_s, are dynamic exponents that describe the diffusion process in a fractal-like medium, while the fractal dimension, d_f, describes the scaling of the mass of the environment with distance.

Figure 1: Diffusion time dependence of MAP-MRI scalar parameters. Generally, the tissue contrast in Propagator Anisotropy (PA) and Non-Gaussianity (NG) images increases with diffusion time; meanwhile, both the Return-to-axis probability (RTAP) and the Return-to-origin-probability (RTOP) decrease with diffusion time throughout the parenchyma, especially in gray matter.

Figure 4: example of interpolation of the experimental D0(D,K) and L(D,K) observations for a SNR at b = 0 of 20. Top: observations for Δ = 50 ms and b_max = 1500 s mm^–2 (A, to the left: D0(D,K); B, to the right: L(D,K)). Bottom: radial basis function interpolation and extrapolation to the whole (D,K) domain (C, to the left: D0(D,K); D, to the right: L(D,K)).

Figure 1: example of synthetic hepatic cells considered in this study. From left to right: different cell realisations obtained by independent perturbations of regular prisms with the same base shape. From top to bottom: different base shapes (square, pentagonal and hexagonal). L corresponds to the inter-base distance.

Figure 1. Examples of placement of regions of interest (ROIs) on MD map(a), MK map(b), D map (d), PD map (e), f map(f)

Figure 2. The Pearson rank correlation analysis shows a negative correlation between the MD values and the Fibroscan (r=-0.452 , P=0.016)

SG-TIETA decays and inverted $$$\textbf{X}$$$ for D6, yeast, and water. (a) Decays analyzed as described in the text. Err. bars = $$$\pm1$$$ SD for, in legend order, $$$38, 3, 4, 25$$$ repetitions truncated at $$$n = 34, 17, 17, 15$$$, respectively. (b) $$$\textbf{X}$$$ solutions. Inversion parameters were identical to Fig. 2 other than $$$\Delta t(k)$$$ for D6. Again, $$$D_0$$$ and $$$D_\infty$$$ (dashed lines) were given as initial $$$\textbf{X}$$$ guesses. Zoomed plot compares experimental results to the presented theoretical short-time $$$D_{\mathrm{inst}}(t)$$$.

Example SG-TIETA sequence. (a) Timings: $$$m_j = \{1, 3, 1, 2, 1\}$$$, $$$\tau = 4\delta$$$, and $$$\delta = 14\; \mu$$$s $$$= 1$$$ dash. $$$\pi$$$-pulses occur at $$$t_n$$$ and echoes form at $$$T_n$$$. Magenta line indicates timing behavior: $$$T_n = t_{n} + h_n$$$, where $$$h_n$$$ is the $$$|F(t)|/\gamma g$$$ "height" at $$$t_n$$$, given recursively by $$$h_1 = \tau$$$ and $$$h_{n} = 2\tau + m_n\delta -h_{n-1}$$$ for $$$n > 1$$$. (b) Direct echo $$$F(t)$$$ drawn along with other coherence pathways that refocus (red, dash-dot) or do not refocus (gray, dotted).

Figure 3. Deep grey matter segmentations derived from workflow in Figure 2 displayed for three subjects spanning a large age range. Segmentations are displayed in 3D as well as on a single axial slice of the mean b1000 diffusion weighted image and FA map. Even with substantial subject variability in brain shape, reasonably accurate cortical segmentations were generated for each region

Figure 2. Segmentation workflow of deep GM structure based solely on DTI. (A) Initial segmentations registered from an atlas are (B) converted into surfaces. (C) The caudate and thalamus are deformed to an edge on the FA map or nearest ventricle edge (purple). (D) The globus pallidus (GP) is deformed on the mean b1000 image. The putamen is deformed on the FA map preventing deformation into the GP. A correction to the GP is applied on the FA map. (E) Visualization of final deep GM segmentations.

Microscopic fractional anisotropy (µFA) calculated from pairs of STE and tuned LTE for the long τ and TE (top left), short τ and long TE (top center) and short τ and short TE. ROI mean and standard deviation from the 3 maps are shown below.

Mean diffusivity (MD) maps calculated form the initial slope of the different LTE and TE configurations. Maps are ordered with increasing high frequency spectral content (see figure 1) towards the right.

(a): Example of a measured image of the monkey brain. (b): The signal components, W, from the msNMF. (c): The associated normalized mixture maps, H, labelled by the frame colors. A logarithmic color scale is used for the yellow component. Maps are given for both data sets and the right column shows the difference between the two (Short diffusion time subtracted from long diffusion time). The thin and thick black arrows mark the conspicuous visual cortex and cerebellum, respectively.

Result of the msNMF for the relaxometry data of rat spinal cord. (a): The signal components, W. (b): The associated mixture maps, H, indicated by the frame and label colors. Each column of images shows the maps for a specific rat (one from each group).

Fig.5: The relative error (as a percentage) between the total signal from the resulting corrected phantom obtained from both methods (Raffelt et al.^[7] and the log-domain method in this work) and the original phantom (unaffected by bias fields). The relative error is averaged across results for all 10 bias fields and overlaid with a WM image of the phantom for reference. The method by Raffelt et al.^[7] struggles mostly at the extremities (low and high) of the bias field value range; i.e., deep in the brain and at the edge. At the bottom of the mask, errors exceed 20% (up to 35% in some voxels).

Fig.4: The relative error (expressed as a percentage) between the total signal from the resulting corrected phantom obtained from both methods (Raffelt et al.^[7] and the log-domain method in this work) and the original phantom (unaffected by bias fields). The relative error is averaged over the entire brain mask and shown separately for each of the 10 "ground truth" bias fields. The log-domain method proposed in this work appears to consistently outperform the method by Raffelt et al.^[7], as evidenced by lower relative errors after intensity and inhomogeneity normalisation.

A reliable pipeline for the preprocessing of dMRI.

dMRIPrep produces visual reports that allow quality-control of the results while serving as a scaffold for understanding the pipeline.

Evaluation of diffusion strategy deriving the diffusion feature (x-axis) with highest Cohen’s d (Equation1, y-axis). The mean and standard deviation of each diffusion parameter, derived by either 3T-DTI or 3T-DKI or 7T-DTI, were examined and denoted by the letters “M” and “S” respectively (e.g. MFA for FA). The LPCA and Gibbs and Eddy pipeline was applied in each case. No smoothing was applied. The box-and-whisker plot captures variation among tracts.

Evaluation of tracts' segmentation. The six tracts (left and right fornix, left and right inferior cerebellar peduncle, left and right superior cerebellar peduncle) that exceeded the value of 0.19, were excluded for further analysis. A list with all abbreviations for the studied nerve tracts can be found at the github page of TractSeg ( https://github.com/MIC-DKFZ/TractSeg ).

Fig. 1. Effects on Diffusion-Weighted Images. Images for different b-values from the the original noisy dataset and after denoising with dwidenoise, VST-Mag, and OS-SVD.

Fig.3. Effects on Orientation Estimation and Tractography. OS-SVD improves estimation of the fiber orientation distribution function (fODF) and eventually tractography.

Figure 1. Examples of diffusion-weighted images with and without denoising, highlighting that signal components can be individually separated with improved conspicuity, and without spatial regularization or blurring. All images are scaled to the non-denoised b=500 s/mm² data.

Figure 2. Qualitative assessment of denoising indicates retention of spatial features such as striatal cell bodies connecting the caudate and putamen through the internal capsule (red arrow).