Fig. 1. a. By applying strong gradients G (Larmor gradient g=γG), magnetization far from the boundaries (cell membranes) vanishes, and that near the boundaries within a thickness of localization length l_g contributes to measured signal. This so-called localization regime emerges when l_g << diffusion length and cell dimension. b. Numerical simulation of diffusion inside an impermeable sphere (D₀=3 μm²/ms) demonstrates the localization regime at strong gradients (dashed line). The signal decay in localization regime is much slower than the free diffusion (dotted-dashed line).

Fig. 3. dMRI measurements by using wide pulse sequence of strong gradients at 3 different diffusion times in 2 healthy subjects demonstrates the localization regime in the cortical brain gray matter, where diffusion signals coincide with the theory in Eq. (2) and yield a soma size estimation (Fig. 4). In contrast, diffusion signals in the brain white matter are not compatible to theory of localization regime since the axon size ~1 μm is smaller than the localization length l_g = 3.4-6.0 μm in experiments.

Fig.4: GRAMMI parametric maps calculated with NLLS (A) & DL (B) from a multi-shell multi-td dataset. The maps are overall homogeneous, with good differentiation between GM & WM. C: Median & IQR of model parameters in the cortex ROI across the 4 datasets. Experimental trends agree with the simulations. For D_e and f, NLLS and DL results are consistent, with better precision for DL Regarding t_ex the two methods agree very well for Dataset #4, which had the highest SNR (larger voxels), but the specific t_ex estimate may be biased due to only 3 diffusion times available instead of 4 (Fig. 1).

Fig.3: Simulation results fitting multi-shell multi-t_d data jointly for random (A) and fixed (B) GT. A: Displayed are the medians & IQR in each bin. Black lines: ideal estimation ±10 % error. Without noise (A1) DL and NLLS fit all parameters with high accuracy and precision. At SNR=100 (A2) some sensitivity to D_i and high t_ex values is lost but still better than single t_d fits (Fig.2). DL has better precision than NLLS. B: At SNR=100, good accuracy is achived for t_ex, De and f with both NLLS and DL. For D_i the precision is poor with NLLS while DL biases the outcome

Visual abstract. On the top right we illustrate a multi-shell dMRI acquisition. Based on the proposed 3-compartment model, we then extract summary statistics from it. Applying a neural density estimator we can both estimate the tissue parameters and their full posterior distribution.

Microstructural measurements averaged over 31 HCP MGH subjects. We deemed stable measurements with a z-score larger than 2, where the standard deviation on the posterior estimates was estimated through our LFI fitting approach. In comparing with Nissl-stained cytoarchitectural studies we can qualitatively evaluate our parameter C_s: Broadmann area 44 (A) has smaller soma size in average than area 45 (B)¹³; large von Economo neurons predominate the superior anterior insula (C)¹²; precentral gyrus (E) shows very small somas while post-central (D) larger ones¹⁴.

Fig.2 Illustration of the morphological features investigated for an exemplar cell. We estimated general features of the whole structure and separated soma from neurites, processing them individually to estimate a set of other relevant features (see Methods). Additionally, we display the Gaussian curvature of the soma surface to show that it is a non-spherical geometry (always positive but not constant). A limitation of the current approach (and the majority of existing tools^23,24) is the slightly inaccurate definition of the soma surface, as shown in the top right corner (arrows).

Tab.2 Reference values for all the morphological features of neuronal and glial cells. The ranges and mean values obtained from the whole dataset investigated (N = 3,448) are reported, together with mean values for only neurons and glia. The ‘≥’ and ‘≤’ are used when the estimated value of the corresponding feature may be slightly (on average <20%) under- or over-estimated, respectively, given the known limitations of the approach used (e.g., see Fig.2).

Figure 1. a) ROI’s used for computing cortex and corpus callosum voxel-averaged signals. The ROI’s are depicted on an estimate of the b=0 signal (left) and the fractional anisotropy (right), both from a DKI fit to the data fulfilling b≤3ms/µm² for acquisition 1. Equivalent ROI’s were used for acquisition 2. b) The powder averaged signals as a function of $$$1/\sqrt{b}$$$ for acquisition 1. The black lines show linear fits to data in the interval [0.1; 0.2].

Figure 2. Cortex signal (markers) as a function of $$$1/\sqrt{b}$$$ for acquisition 1. Fits of SANDI (left panel) and the Kärger SM (right panel) are shown by black lines. The dashed lines depict the signal prediction at half the pulse separation using the estimated parameters from the fits. SANDI estimates are: extra-cellular f_ec = 59%, D_ec = 0.63µm²/ms, neurite f_in = 49%, D_in = 2.32µm²/ms, soma R = 5.6µm, offset < 1%. Kärger SM estimates are: extra-cellular f_ec = 51%, D_ec = 0.57µm²/ms, neurite D_in = 1.73µm²/ms, mean intra-neurite residence time τ_in = 7.9ms, offset = 1%.

Fig. 4. SM parameter estimates (columns) in each ROI (rows) as a function of t_d. Colors: b_max retained for the fit. Time-dependence is pronounced for f and c₂ while b-dependence is not. Conversely, for diffusivities, the b-dependence is pronounced while the time-dependence is neither systematic nor consistent. D_a decreases with time for all ROIs and b_max but most significantly in the cortex at b_max ≥ 6. Extra-axonal diffusivities show a less systematic behavior with some trend for D_e,|| to decrease and D_e,ꓕ to increase, but none of them significant.

Fig.1: Schematic of the SM of diffusion in WM, accounting for two non-exchanging compartments: the intra-axonal space (with a relative weight f) and the extra-axonal space. D_a: intra-axonal diffusivity, D_e,|| / D_e,ꓕ: extra-axonal axial / radial diffusivities. These unitary fascicles are distributed in the voxel given an orientation distribution function P(n), for which the first few rotationally-invariant spherical harmonics coefficients can be estimated – along with the 4 scalar parameters – by the RotInv framework. [Image taken from Ref. 1].

Figure 1. 3D reconstructions of A) 54 splenium axons (segmented at 75 nm resolution) and B) 58 crossing fiber axons (segmented at 500 nm resolution) in their respective XNH volumes. C) Combined 3D AD distributions over all measured diameters in the splenium (yellow) and crossing fiber region (blue).

Figure 3. AD estimation in real axons. Estimated diameter from the SMT approach ($$$d_{SMT-multi})$$$ and PL approach ($$$D_{PL}$$$) vs the geometrical volume-weighted AD, $$$d$$$ for A) infinite SNR (ignoring the inherent MC noise), B) SNR = 100, C) SNR = 50 and D) SNR = 20, where fitting fails for the population of splenium axons. Square marker: volume-weighted AD of splenium axon population, cross marker: volume-weighted AD of crossing fiber population.

Fig. 2 (a) The results of fitting the sphere radius in stick + ball + sphere model for different sphere signal fractions (GT = Ground Truth and E = Estimated). The figure also shows the p-value of the F-test in the presence of Gaussian, Rician, and corrected Rician noise respectively. (b) Estimated sphere and cylinder radii versus the ground truth sphere radius values for cylinder + ball + sphere model without noise (SNR = 200). The third row in (b) shows the reduced chi-square values for two scenarios where the sphere radius is fixed to 5 and 8 μm, blue and red curves respectively.

Fig. 4 Estimated stick (f_stick), ball (f_ball), and sphere (f_sphere) signal fractions, intra-axonal parallel diffusivity ($$$D_{\rm{in}}^{\mid\mid} (\mu m^2/ms)$$$), extra-cellular diffusivity ($$$D_{\rm{ec}} (\mu m^2/ms)$$$), sphere radius ($$$R_{\rm{sphere}} (\mu m)$$$), and standard deviation of the noise ($$$\sigma$$$) on axial, sagittal, and coronal views of the smoothed brain image (A 3D Gaussian kernel with standard deviation of 0.5 is used for smoothing).

Fig. 1. Intra-axonal axial diffusivity estimated using the proposed approach. Values were lowest in the anterior white matter (green), intermediate in the posterior white matter (red), and highest in the corticospinal tract (blue).

Fig. 2. Maximum-intensity projections of the intra-axonal axial diffusivity across multiple slices illustrating the elevated values of intra-axonal axial diffusivity found in the corticospinal tract and the brainstem.

ADC and strain-rate maps over the cardiac cycle for two subjects. Although slices were acquired with sagittal orientation and FH encoding, we present the fitted images for a transverse orientation. On the left, the T1-weighted image and the average ADC map are shown. Ten cardiac phases were reconstructed; the upper represents the relative difference of ADC over the cardiac cycle, the lower row represents the strain rates. Largest change was observed at peak systole (20-30% cardiac interval).

Relation between ADC and strain rate over the cardiac cycle, averaged over the conservative white matter mask, avoiding blood and CSF signals. Top: strain-rate (absolute values, used to avoid cancellation from averaging). Middle: measured ADC as percentage of the average ADC. Bottom: calculated ADC fluctuation, based on negative strain only (representing the ‘worst’ case). Negative strain was obtained by multiplying absolute strain-rate (top trace) by -1 times the mixing time (100ms).

Fig. 2: Left: The reference tissue structure (RTS) consists of 30 monkey glial cells in a VOI of 150x150x150μm³. To create an isotropic tissue environment 15 different cells were placed twice, each randomly rotated, in the VOI. The Blender physics engine was used to avoid cell collision and overlap. Right: Overview of the 15 different glial cells [NeuroMorpho.org IDs are provided].

Fig. 3: Left: swelling of a glial cell (ID: 73137) with a volume increase by 10%, 50% or 100% (zoomed insets to highlight differences). Right: property changes of the RTS in volume (V), surface area (A) and surface area to volume ratio (A/V) upon cell-swelling for the entire VOI presented in Fig. 2.

Figure 3. (a) Examples of realistic substrates generated to simulate axon loss and demyelination. Colors label different EAS fractions. (b) D(t) was obtained from MC simulations and Equation (1) was fitted. (c) $$$D_{\infty}$$$, and $$$D_0/D_{\infty}$$$ show similar behavior than for the substrates with cylinders, $$$t_c$$$ is more noisy and shows similar behavior for both conditions, $$$A$$$ differentiates between axon loss and demyelination, but shows a flat-like behavior for larger EAS. The effect of axon shape in $$$D(t)$$$ needs to be further investigated.

Figure 1. (a) A substrate containing only the centers, weighted by the area of the cylinders is created to compute the power-spectrum (Fourier transform of the density-correlation-function) $$$\Gamma(k)$$$, free of shape effects. (b) For small $$$k$$$, $$$\Gamma(k)\sim c_0+l_c^2k^2$$$. (c) Pearson correlation coefficient, between $$$l_c$$$ computed from $$$\Gamma(k)$$$ and $$$l_c$$$ computed from MC, indicate high and significant correlation. This indicates that structural information can be obtained from $$$D(t)$$$.

A) Our test substrate is a straight cylinder with mean radius $$$R_m$$$ and a variation in radius in axial direction with the wavelength $$$L$$$ and relative amplitude $$$k_a$$$. B) A 3D rendering. C) Scaling of step probability in relation to area. Only a portion relative to the overlapping area can move into the smaller i+1 section (green) whereas the excess walkers (red area) remain stationary. D) The corresponding propagator for a time step from the central section, where the restricted direction (right direction) is scaled by the ratio of cross-sectional areas.

Time dependent diffusivity vs. time (units in µm2/ms and ms) over a range of varicosity wavelengths and amplitudes. The 1D model is shown in red and the 3D model is shown in different shades of gray for different radii.

Fig. 4. MF-SMSI Indices. $$$v_{\text{IA}}$$$, $$$v_{\text{EC}}$$$, $$$v_{\text{FW}}$$$, $$$v_{\text{IS}}$$$, MAI, OCI, axonal radius $$$r_{\text{axon}}$$$ (μm), unbiased axonal radius $$$r_{\text{axon}}^\dagger$$$ (μm), axonaxonal permeability $$$\kappa_{\text{axon}}$$$ (10⁻⁶ μm μs⁻¹), unbiased membrane permeability $$$\kappa_{\text{axon}}^\dagger$$$ (10⁻⁶ μm μs⁻¹), soma radius $$$r_{\text{soma}}$$$ (μm), and unbiased soma radius $$$r_{\text{soma}}^\dagger$$$ (μm). Radius and permeability maps are overlaid on the T1w image.

Fig. 1. Examples of SpinDoctor geometric configurations. From left to right: a sphere representing a soma with impermeable membrane a cylinder representing an axon with impermeable membrane, a cylinder tightly wrapped with extra-cellular space, and a cylinder in a boxed extra-cellular space representing an axon with perme- able membrane allowing water exchange with extra-cellular space.

Figure 1. Overview of DIFFnet. Two DIFFnets, one for DTI and the other for NODDI were developed. (a) For DTI, DIFFnet_DTI was constructed to generate DTI parameters (FA, MD, AD, and RD) from a signal set with diffusion gradient directions and a single b-value of $$$b_0$$$ s/mm². (b) For NODDI, DIFFnet_NODDI was designed to generate NODDI model parameters (ICVF, ISOVF, and ODI) from a three-shell signals. Different from previously proposed networks, DIFFnet does not specify gradient directions and b-values for its input dataset.

Figure 2. Generalized approach for input gradient directions and b-values via “q-space projection and quantization”, producing “Qmatrix”. (a) A signal set was placed in a q-space with q-vectors. (b) Qmatrix was designed via projection and quantization. (c) In DTI, a signal set was projected on the three planes. Three projected signal sets were quantized and concatenated, producing a $$$q_n × q_n × 3$$$ matrix. (d) For NODDI, the projection was performed on each shell. Nine sets of projected signals were generated, producing a $$$q_n × q_n × 9$$$ matrix. For $$$q_n$$$, 5 to 25 were tested.

Illustration of simulated MR images with various animated artifacts (a bit excessive for illustration purposes): eddy current distortions (a), intensity drift (b), head motion and spike (c), head motion, eddy currents and noise (d), Gibbs ringing (e), inhomogeneity distortions (f).

Comparison between the B0 image contrast of a simulated (a) and the corresponding real reference image (b). This figure illustrates that using the automatic parameter optimization it was possible to closely approximate the contrasts of a real image.

Figure 1. Mesh of the substrate used for the Monte-Carlo simulations of spins dynamics to generate the DiSCo DW-MRI dataset. The mesh is composed of 12,196 tubular fibers, with gamma-distributed outer diameters ranging from 2um to 6um, connecting 16 ROIs. The 16 ROIs were placed on the surface of a sphere of 1mm in diameter.

Figure 2. Three dimensional (left) and cross-sectional (middle) views of the trajectories of the 12,196 tubular fibers. Trajectories sharing both endpoints are shown with the same color. They were used to compute the ground truth connectivity matrix (right), with weights corresponding to the sum of the cross-sectional area of the tubular fibers connecting each pair of ROIs.

Figure 2. Variability in fibre responses within a voxel at b=2000 s/mm² along with geometrical variation in fibres responsible for median, 10^th and 90^th percentile response. Notably, the 10^th percentile fibres tend to be more stick-like while 90^th percentile have more beading. Solid green line in FRF figures represents median response, shaded green area represents 10^th and 90^th percentile response. Same values for representative cylinders are in blue to demonstrate the variability due to noise.

Figure 3. fODFs estimated using different FRFs showing how assuming a different FRF can result in vastly different fODFs. Green outline in rightmost plots shows gold standard fODF for comparison. All fODFs are normalised to integrate to 1 for comparison.

Figure 3: ODFs in ROIs of Taxon bundles crossing at 90°, 45° and 30° (rows) as reconstructed by Multi-shell QBI (qODF), CSD (fODF), GQI (dODF), DSI (dODF) and RDSI (dODF) (columns). Gray bands represent ground truth fiber directions.

Figure 1: The anisotropic diffusion phantom contains single and crossing bundles of Taxons™ (textile water filled tubes [9] with 0.8 10-6 microtubes). a) T1-weighted image of the phantom layout. The ROIs used in this work are indicated along with crossing fiber angles and taxon densities. b) Taxon cross section schematic and SEM of Taxon detail.

Figure 2: A-D: ADC values of all phantoms. The inter-fiber spaces in (A/B) and (C/D) were respectively filled with the 0% and 20% solution. The phantom ADC values with different fiber permeabilities are shown separately. Each solid line connects values measured from three phantoms with different volume fractions of intra-fiber spaces. E-H: Representative ADC maps. For each subplot, the volume fractions of the intra-fiber spaces increase from left to right, and the PVP concentration of the intra-fiber spaces increase from top to bottom.

Figure 1: SEM images and phantom photographs. A: SEM images of the hollow PES fibers with small pore diameter (first row) and large pore diameter (second row). The images of the inner surfaces, outer surfaces, and cross sections are shown from left to right. B: Pore diameter distribution on the inner surface of two fibers. C: Photograph of the entire phantom. The red arrows point to the positions for the two openings of the inter-fiber spaces, and the blue arrows point to the positions of the intra-fiber spaces. D: Photograph of the potting layer.

Figure 3. Percent difference between ADC values measured at 3T and model predicted values at 20°C for all measurements (left) and individual measured temperatures per measurement (right).

Figure 1. Example trace (left) and ADC (right) images of the NIST / QIBA diffusion phantom acquired at 1.5T.

Figure 2

Examples of intraventricular temperature maps for the two DTI techniques with fractional anisotropy (FA) processing. The red arrows indicate some pixels around the foramen of Monro removed by FA processing.

Figure1

The second order motion compensation (2nd-MC) can be achieved by using suitable second moment nulling gradient waveforms. (a) conventional DTI and (b) 2nd-MC DTI．

Figure 1: PVP phantom structure a) 1H image showing tube positions b) ADC map c) signal decay vs increasing b values for water tube at the center of phantom exhibits noise-floor bias for b>1000 s/mm² (excluded from ADC fit)

Figure 2: ADC values for each PVP concentration at three measured temperatures. Doted lines display the linear fits to the data. Using linear regression, fit coefficients C1 and C2 are extracted for each line and tabulated.

Figure 3: a) FEXSY signal before and after cell wash. The spectrum peaks were used for the calculation of the ADCs for water and isopropanol, and AXR for water. (b, c) AXR relaxation curves fits for cell pellets before (b) and after the cell wash (c) for several isopropanol treatment intervals. (d, e, f, g) FEXSY measurements of the cell pellet before (yellow, red) and after the cell wash (blue) over time. (e, f, g) ADC values calculated from the FEXSY signal with diffusion filter off.

Figure 4: (a) AXR relaxation curves with AXR fit of the control and two yeast cell suspensions treated with 0.4% Triton X-100 concentrations for 15 minutes and 12.50h. The AXR curves were calculated from the water peak of the FEXSY signal spectrum. (b) Percentage of trypan blue stained cells of each suspension.

Figure 4: Parametric maps resulting from fitting equation 1 on data obtained on a meningioma patient. The tumour (located in the left temporal lobe) is marked with a red arrow and the oedema with a blue arrow. $$$E_D, E_R, V_D, C_{D,R}$$$ and $$$V_R$$$ denote mean diffusivity, mean restriction coefficient, variance of diffusivities, covariance of $$$D$$$ and $$$R$$$ and the variance of $$$R$$$, respectively. In addition to noise, there is a pronounced fat shift artefact at the bottom of the maps.

Figure 5: (a): Slice through the brain of a meningioma patient showing three ROIs placed in normal tissue and the tumour as well as the oedema around it. (b): Variation of the restriction coefficient and exchange rate in the three regions. The former increases in the tumour relative to normal tissue, while the latter decreases. (c): Quantification of the correlation between restriction and exchange in the three regions. The plot shows lack of correlation, meaning the parameters convey independent information.

Figure 2: Comparison between the newly proposed method and an acquisition without filter. While the newly proposed method is in good agreement with the simulation, the measurement without the filter strongly deviates at the lowest two q-values.

Figure 1: Schematic representation of the sequence used in this study. The long gradient of the long-narrow approach was split into a CPMG-like gradient train and an additional diffusion weighting, acting as a filter, was added.

Figure 2. Parameter maps derived from the per-voxel D-distributions for the 5 mm Hex, Mic, and Lam phantoms for microimaging and the 0.5 L Lam phantom on the whole-body scanner. (A) Image segmentation by binning in the 2D D_iso-D_A/D_R projection to capture tensor components characteristic of Hex (prolate), Mic (spherical), and Lam (oblate). (B) Per-voxel statistical measures E[x], Var[x] and Cov[x,y] of the D_iso and D_Δ² projections of the distributions. (C) Bin-resolved signal fractions (brightness) and per-bin means (color)_. Ideal lamellar gives rise to f_oblate = 1 and D_Δ² = 0.25.

Figure 3. Comparison of the information gain for estimating restriction-size distributions with NOGSE trapezoidal and sinusoidal modulations. (a) Difference between the information gain (sensitivity) of the trapezoidal and sinusoidal modulations as a function of the mean and standard deviation values for a Lognormal distribution. Colored squares represent the measured size distributions of the prepared phantoms. (b) Restriction-size distribution reconstruction using NOGSE of the used phantoms. They correspond to the colored squares in (a).

Figure 1. Transverse plane MRI of a phantom of aramid fibers bundles immersed in water in a 50 ml plastic tube. Different packing densities were designed for producing different size distributions.

Figure 2. The estimated parameters of the cylinder + ball model.

Figure 3. Mean and the median value of the estimated parameters in each tube.

Figure 1 Normalized NMR signal intensity of free (green, blue) and restricted (red) water molecules at different orientation of the diffusion gradient

Figure 2 Normalized NMR signal intensity of restricted water molecules at different orientation of diffusion gradient

Figure. 3. Effect of prior distribution on parameter estimation of synthetic data. The slope $$$\partial \mu_{x}^{e}/\partial\mu_{x}^{p}$$$ of linear regression of parameter estimate mean $$$\mu_{x}^{e}$$$ with respect to the mean of prior distribution $$$\mu_{x}^{p}$$$ is plotted against SNR for each SM parameter. We see that the diffusivities are notably more sensitive to prior (and less to measurement) than $$$f$$$ and $$$p_2$$$.

Figure. 4. Histogram of estimated parameters with different prior distributions for in-vivo data. Histograms of each parameter estimated by polynomial regression trained by different prior distributions are plotted for in-vivo data. Mean of the prior distribution $$$\mu_{x}^{p}$$$ and the estimate mean $$$\mu_{x}^{e}$$$ of the same parameter are provided in the legend for every SM parameter, while the slope $$$\partial\mu_{x}^{e}/\partial\mu_{x}^{p}$$$ is written under each subfigure.

Parametric maps of f shown for all protocols and scanners.

Standard model parametric maps for one of the subjects. A WM ROI extracted from the JHU WM atlas is drawn on top of the map. Bottom row shows the corresponding parametric histograms for the plotted ROI.

Fig.1 Soma and neurite densities (SD and ND) estimation from diffusion MRI metrics. Left: Diffusion weight images registration and DTI, DKI fitting. Cortical segmentation and overlay on fitted parameter maps. axial kurtosis (AK), radial kurtosis (RK), mean kurtosis (MK), fractional anisotropy (FA), axial diffusivity (AD), radial diffusivity (RD), mean diffusivity (MD). Middle: Each voxel in the cortex served as a training sample in random forest regressor. Right: Estimated SD and ND values. Estimation evaluation and extraction of feature importance from model.

Fig.4 Soma and neurite densities (SD and ND) estimation results and feature importance. A: Significant linear correlation between ground truth SD and estimated SD on validation set. (P<0.0001, mean absolute error (MAE)=0.017, Pearson correlation coefficient r=0.53). B: Significant linear correlation between ground truth ND and estimated ND on validation set. (P<0.0001, MAE= 0.038, Pearson correlation coefficient r=0.53). Panel C: Feature importance for each metric. MK ranks firsts for both SD and ND estimation.

Figure 1a) b0 images and powder averaged data for different b-values (increasing from left to right) from one representative mouse. The SNR of b0 images is 15-25 in WM and 20-50 in GM. b) SANDI parameter maps revealing high soma signal fraction in GM, high neurite signal fraction in WM and high extracellular diffusivity (and signal fraction, not shown) in CSF. Apparent soma radius and intra neurite diffusivity show less contrast between WM and GM. c) 3D surface rendering of soma and neurite fraction obtained in ImageJ using a visualisation threshold of 0.4 and 0.2, respectively.

Figure 3 Histograms of a) SANDI and b) DKI parameters in different ROIs (left to right: cortex, striatum, thalamus, hippocampus, corpus callosum, internal capsule and CSF). Different rows represent different model parameters, and the colours reflect the 3 mice. An overlap of the parameter histograms is observed for most parameters and ROIs.

Figure 1. Upper: Histogram of difference between conventional and tissue-weighted means for NDI (left) and ODI (right). The heavy tails of the distributions demonstrate how the conventional mean was in general a biased estimate of the tissue mean, under- and over-estimating NDI and ODI, respectively. Lower: Bland-Altmann style plot of the difference between the conventional and tissue-weighted means across all ROIs and subjects (2448 ROIs). The difference tends to decrease and increase as a function of tissue-weighted mean for NDI and ODI, respectively.

Figure 2. Percentage bias in the ROI between-subject averaged conventional means, for NDI (left) and ODI (right). Each boxplot shows the distribution across ROIs of the percentage difference in the between-subject average conventional and tissue-weighted mean for PV and non-PV ROIs in control and YOAD subjects. NDI tended to be under-estimated, apart from control subjects non-PV ROIs, which were over-estimated. ODI was over-estimated in PV and non-PV ROIs for control and YOAD subjects.

Figure 2. Scatter plots of NODDI indexes obtained within the normal-appearing cortex (NA-cortex) and cortical lesions (CLs) of MS patients and healthy controls.

Figure 1. The postprocessing is shown for a representative MS patient: after the segmentation of cortical lesions and gray matter, masks are overlapped on the maps of intracellular volume fraction (ICV_f) and orientation dispersion (ODI).

Figure 2: A) HSV (Hue, Saturation, Value) maps generated with f_ISO, ODI and NDI in respective channels- H, S and V shows single- and multi-shell maps generated with DLpN and NODDI. B) shows HSV channels and C) shows colormap for f_ISO and overall HSV space.

Figure 5: Comparison of single-shell DLpN with P2 (DLpN_P2) and multi-shell NODDI (NODDI_Pall) that reported minimum error, shows noisy estimation from multi-shell could be addressed with DLpN single-shell reconstructed NDI. White arrows show some regions with noisy overestimation in NODDI_Pall NDI.

Figure 1: Diffusion coefficients (A) and perfusion fraction (B) for effusion, synovial membrane and muscle tissue calculated with 12 different b combinations and 4 b-thresholds

Figure 2: Relative difference of the calculated diffusion (A) and perfusion fraction (B) values between b-value combination #1 and all other b-combinations.

Fig. 3: An axial image of the fractional anisotropy (FA) number of fibers (NuFO) and f_IVIM obtained with MRI data of a human brain containing both LTE and STE. f_IVIM[LTE] exhibits higher values than f_IVIM[STE], especially in correspondence of NuFO values above 1. f_IVIM[STE] and f_IVIM[STE-short] are remarkably similar showing higher value in gray matter than white matter, as expected from known physiology.

Fig. 4: Boxplots showing difference between f_IVIM and D_T estimated with LTE and STE data as a function of FA (top) and of the number of fibers (NuFO, bottom). The red dots represent the median value. The red bars with asterisks indicate a significant difference in distribution (sign test). The difference in f_IVIM and D_T between LTE and STE is significantly larger in voxels with intermediate to large FA, or containing 2 or 3 fibers, and have considerable non-zero magnitude. Significant differences are also detected when comparing STE to STE-short, but these have almost zero magnitude.

Figure 1. The optimal model maps with different NAs.

Figure 3. The influence of SNR (represented by number of averages, NA) on the percentage optimal model territory, f_p, and D^*.

Figure 1: IVIM processing scheme. Gaussian curve associated with the diffusion peak (blue), pseudo-diffusion peak (yellow), and a third peak obtained with the Gaussian Mixture (green).

Figure 3: Comparison of the Gaussian Mixture with the traditional method (find peaks) for two patients with glioma. The green arrow shows the region that we can see a contrast of the tumor in the pseudo-diffusion image.

Figure 3: IVIM parameter maps of the liver, obtained with model-based reconstruction on the left (a and c) and the conventional fit on the right-hand side (b and d). The conventional fit seems to overestimate the diffusion coefficient in the left liver lobe and perfusion fraction in the bottom of the liver (red arrows). This behavior is not observed in the model-based reconstructed maps.

Figure 1: Schematic view of the conventional (top) and model-based (bottom) reconstruction and fitting process. Conventionally, magnitude-only images are reconstructed prior to the model fitting. However, the modulus operation transforms the noise from a complex Gaussian to a Rician distribution. In the model-based reconstruction the quantitative model is included in the reconstruction process, thus the Gaussian noise assumption is valid. iFFT: inverse fast Fourier transform.

Proposed method estimated parameters maps compared to Barbieri's, with the ground truth maps on the bottom row.

Proposed method architecture, where the acquisition parameters are added to the network's input.

Figure 1. A network structure of the proposed method for a physics guided IVIM parameter estimation with a self supervised U-net architecture. An input consists of a 3 dimensional array, which is a concatenation of a 2D slice acquired at 7 b-values. The input is passed through a U-net to produce 4 IVIM parameter estimates at each pixel. In the second stage, the parameter estimates are used in the IVIM equation to reconstruct the original input image array. L2 loss between the output and the original input is used to propagate gradients backwards through the network.

Figure 2. An example of IVIM parameter estimates with the conventional voxelwise IVIM fitting method, DeepIVIM and the proposed self-supervised approach. Difference image between the proposed method and each of the alternative methods is shown in the bottom two rows. The proposed method shows strong agreement with estimates of the conventional fitting method for the estimates of D and f, but a more coherent visual graph of D*.

Fig 1. In-vivo comparison of estimated parameters using conventional IVIM (upper row) and alternative multi orientations method (bottom row).

Table 1 Correlation coefficient between ground truth and estimated parameters (D and D*) for the conventional method and proposed method.

Figure 2. MD, PD, RD and FA maps at different OGSE frequencies and different TEs.

Figure 3. Measured PD, MD and RD of white matter parcels in the splenium (SCC), body (BCC) and genu (GCC) of the corpus callosum at different frequencies and different TEs.

Axial Slice of intra and extra-axonal T₂. Higher values in the internal capsule show highly myelinated axons.

Axial Slice of intra-axonal volume fraction. Estimated intra-axonal volume shows a clear segmentation of white matter tissue that is consistent with SMT model.

Signal S vs. acquisition number n_acq (circles: measured, points: fit) and nonparametric D-R₂ distributions for representative voxels of an in vivo rat brain at 21.1 T. The distributions are shown as projections onto the 2D planes D_iso-D_Δ², D_iso-R₂, and D_Δ²-R₂, where D_iso is the isotropic diffusivity, D_Δ² the squared normalized anisotropy (41), and R₂ the transverse relaxation rate. Binning in the D_iso-D_Δ² plane allows calculation of nominally tissue-specific signal fractions f_bin1, f_bin2, and f_bin3 and associated diffusion-relaxation metrics.

Parameter maps derived from the per-voxel D-R₂ distributions. (a) Synthesized S(b=0,TE=0) image, bin definition in the D_iso-DΔ² plane, and RGB map color-coded by the signal fractions [f_bin1,f_bin2,f_bin3]. (b) Bin-resolved signal fractions and means E[x] of the D-R₂ metrics coded into image brightness and color (see quantitative scale bars). Direction-encoded colors derive from the lab-frame diagonal values [D_xx,D_yy,D_zz] and maximum eigenvalue D₃₃. (c) Per-voxel means E[x], variances V[x], and covariances C[x,y] of D_iso, D_Δ², and R₂.

Figure 3: Estimated diffusion and relaxometry parameters. (a) $$$T_1$$$ and $$$T_2$$$ relaxation times and apparent diffusion coefficient (ADC) and fractional anisotropy (FA) from a single slice of md-MRF scan using linear tensor encoding (LTE) (b) results of md-MRF scan using spherical tensor encoding (STE). M0 shows the proton density map.

Figure 1: Sequence diagram of an acquisition unit of mdMRF scan. (a) Each acquisition unit consists of multiple inversion, $$$T_2$$$ and diffusion preparation modules with various timing and b-values. The table lists the key parameters of the modules. (b) and (c) are LTE and STE diffusion gradients used in the diffusion preparation modules.

Fig. 4. maps of summed-weights: (A) in the region ‘a.s’ (S_w,a.s) ; (B) in the region ‘a.l’ (S_w,a.l)

Fig. 3. animal results: (A) averaged weights with standard deviations from the L₁-norm method in WT group (blue) and SOD group (red). Three regions are labeled as ‘a’, ‘b’ and ‘c’, and the region ‘a’ is divided into sub-region ‘a.s’ and ‘a.l’. The ‘*’ denotes a significant difference (P<0.05) is found in this region. (B) a representative fitting results from WT group (blue) and SOD group (red)