Evidence for microscopic kurtosis in neural tissue revealed by Correlation Tensor MRI
11 Evidence for microscopic kurtosis in neural tissue revealed by Correlation Tensor MRI
Rafael Neto Henriques , Sune N Jespersen and Noam Shemesh Champalimaud Research, Champalimaud Centre for the Unknown, Lisbon, Portugal Center of Functionally Integrative Neuroscience (CFIN) and MINDLab, Clinical Institute, Aarhus University, Aarhus, Denmark. Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark *Corresponding author: Dr. Noam Shemesh, Champalimaud Research, Champalimaud Centre for the Unknown, Av. Brasilia 1400-038, Lisbon, Portugal E-mail: [email protected]; Phone number: +351 210 480 000 ext.
Abstract
Purpose: The impact of microscopic diffusional kurtosis (µK) – arising from restricted diffusion and/or structural disorder – remains a controversial issue in contemporary diffusion MRI (dMRI). Recently, Correlation Tensor Imaging (CTI) was introduced to disentangle the sources contributing to diffusional kurtosis, without relying on a-priori assumptions. Here, we aimed to investigate µK in in vivo rat brains and assess its impact on state-of-the-art methods ignoring µK. Methods: CTI harnesses double diffusion encoding (DDE) experiments, which were here improved for speed and minimal bias using four different sets of acquisition parameters. The robustness of CTI estimates from the improved protocol is assessed in simulations. The in vivo CTI acquisitions were performed in healthy rat brains using a 9.4T pre-clinical scanner equipped with a cryogenic coil, and targeted the estimation of µK, anisotropic kurtosis, and isotropic kurtosis. Results: The improved CTI acquisition scheme substantially reduces scan time and importantly, also minimizes higher-order-term biases, thus enabling robust µK estimation, alongside K aniso and K iso metrics. Our CTI experiments revealed positive µK both in white and grey matter of the rat brain in vivo ; µK is the dominant kurtosis source in healthy grey matter tissue. The non-negligible µK substantially biases prior state-of-the-art analyses of K iso and K aniso . Conclusion: Correlation Tensor MRI offers a more accurate and robust characterization of kurtosis sources than its predecessors. µK is non-negligible in vivo in healthy white and grey matter tissues and could be an important biomarker for future studies. Our findings thus have both theoretical and practical implications for future experiments.
1. Introduction
Diffusion MRI (dMRI) has become one of the most important methods for non-invasively probing microstructural features on different microscopic and mesoscopic scales in health and in disease (1 – – – – – – – – ’s and Alzheimer ’s diseases (37,38). However, even though DKI is sensitive to microstructural alterations in health and disease, the biological interpretation of its measures is limited because non-Gaussian diffusion may arise from different sources (39 – – – – diffusion encoding probes diffusion correlations across different dimensions by (a) including additional pairs of pulse gradients (54 – continuous gradient waveforms with 3D trajectories (63–69). Under the strict assumption of multiple Gaussian components (no time-dependent diffusion and no kurtosis arising from restricted diffusion or structural disorder) , Westin et al. showed that different multidimensional diffusion encoding sequences can be generally described by its tensor-valued encoding information (70) . By combining multidimensional diffusion encoding experiments with different b-tensor shapes, one can resolve anisotropic and isotropic kurtosis sources ( 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 ) which are, respectively, related to the shape and size variances of tissue components which are represented by diffusion tensors (39,68,71,72). Such diffusion tensor variance (DTV) analyses can, however, be prone to bias due to diffusion time-dependence (73,74) and non-Gaussian diffusion effects of restricted diffusion and tissue structural disorder (i.e. microscopic kurtosis 𝜇𝐾 ) (40,41,49,67,73), as will be shown also in this work. Double diffusion encoding (DDE, (5,54 – 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 based on DTV analyses. Although prior studies have attempted to measure µK , the angular DDE experiments were strongly conflated with orientation dispersion (41,75). Going beyond the DTV framework, recent studies show that DDE can be used to measure 𝜇𝐾 via the Correlation Tensor Imaging (CTI) approach (40). The CTI framework allows the simultaneous decoupling of 𝐾 𝑎𝑛𝑖𝑠𝑜 , 𝐾 𝑖𝑠𝑜 from 𝜇𝐾 effects without resorting to the multi-Gaussian assumption (40) or without conflation with other mesoscopic effects (41). However, the CTI approach proposed previously (40) can suffer from inaccurate 𝜇𝐾 measurements due to higher-order effects, and furthermore was quite time-consuming to acquire (40). Here, we aimed to investigate the existence of 𝜇𝐾 in neural tissues and their impact on the increasingly popular DTV approaches. We first develop a framework for improving CTI acquisitions towards more accurate 𝜇𝐾 estimates, with minimized high-order-term biases, and highly accelerated acquisitions. We then use this framework to reveal the degree of 𝜇𝐾 on in vivo rat brains and compare the impact of 𝜇𝐾 on previous DTV analysis that ignore 𝜇𝐾 . Our results provide a new marker for future studies aiming to quantify microstructure in health and disease, and show that 𝜇𝐾 must be considered in future dMRI studies.
2. Theory
The SDE signal attenuation ( 𝐸 𝑆𝐷𝐸 ) can be expressed (with Einstein summation convention) as the following 2 nd order cumulant expansion (19): log(𝐸 𝑆𝐷𝐸 (𝑏, 𝒏)) = −𝑛 𝑖 𝑛 𝑗 𝑏𝐷 𝑖𝑗 + 𝑛 𝑖 𝑛 𝑗 𝑛 𝑘 𝑛 𝑙 𝑏 𝐷̅ 𝑊 𝑖𝑗𝑘𝑙 + 𝑂(𝑏 ) (1) where 𝑏 is the b-value defined by 𝑏 = (𝛾𝛿𝑔) (∆ − 𝛿/3) , 𝒏 is the diffusion gradient direction, 𝐷 𝑖𝑗 and 𝑊 𝑖𝑗𝑘𝑙 are the diffusion and excess-kurtosis tensors, and 𝐷̅ is the mean diffusivity ( 𝐷̅ =𝑡𝑟𝑎𝑐𝑒(𝐷 𝑖𝑗 ) ). To quantify non-Gaussian diffusion decoupled from confounding effects of tissue dispersion, it is also useful to consider the cumulant expansion of powder-averaged SDE signal decays (i.e. signals averaged across multiple gradient directions) (29,48,76): log(𝐸̅ 𝑆𝐷𝐸 (𝑏)) = −𝑏𝐷 𝑇 + 𝑏 𝐷 𝑇2 𝐾 𝑇 + 𝑂(𝑏 ) (2) where 𝐸̅ 𝑆𝐷𝐸 is the powder-averaged SDE signal decay, 𝐷 𝑇 and 𝐾 𝑇 are the isotropic diffusivity ( 𝐷 𝑇 = 𝐷̅ ) and isotropic excess-kurtosis of powder-averaged signals ( 𝐾 𝑇 = (𝑊 𝑖𝑖𝑗𝑗 + 2(𝐷 𝑖𝑗 𝐷 𝑖𝑗 /𝐷̅ − 6)/5 ) (40). In the absence of exchange, the total kurtosis 𝐾 𝑇 can be desbribed by the sum of three different sources (40): 𝐾 𝑇 = 𝐾 𝑎𝑛𝑖𝑠𝑜 + 𝐾 𝑖𝑠𝑜 + 𝜇𝐾 (3) where 𝐾 𝑎𝑛𝑖𝑠𝑜 is related to tissue miscrocospic anisotropy 𝜇𝐴 ( 𝐾 𝑎𝑛𝑖𝑠𝑜 =
65 𝜇𝐴 𝐷 ) (39,65,76), and 𝐾 𝑖𝑠𝑜 is related to the variance of tissue components ’ apparent mean diffusivities 𝐷 𝑖 ( 𝐾 𝑖𝑠𝑜 =3 𝑉(𝐷 𝑖 )𝐷 , with 𝑉(𝐷 𝑖 ) representing the variance across the mean diffusivities of tissue components) (39,51). Microscopic kurtosis 𝜇𝐾 , which was previously referred to as intra-compartmental kurtosis (40), is a weighted sum of different microscopic sources of non-Gaussian diffusion 𝜇𝐾 𝑖 𝜇𝐾 = 〈𝐷 𝑖2 𝜇𝐾 𝑖 〉𝐷 , (4) with 〈∙〉 representing the average over tissue components). Here 𝜇𝐾 𝑖 can be related to non-Gaussian diffusion arising from restricted diffusion (41,77,78) or tissue disorder due to the presence of microscopic hindrances to water molecules, e.g. membranes, organelles, axonal caliber variations etc. (49,79 – 𝐾 𝑇 can be estimated by fitting Eq. 2 to data acquired with at least two non-zero b-values, it is important to note that the kurtosis sources in Eq. 3 cannot be decoupled from SDE experiments in a model free manner. Recently, the Correlation Tensor Imaging (CTI) methodology was proposed to resolve different kurtosis sources from DDE signals (40). Fig. 1B shows an illustration of the DDE sequence which probes diffusion using two pairs of pulsed gradients with magnitudes 𝑔 and 𝑔 , widths 𝛿 and 𝛿 , separations time ∆ and ∆ , and mixing time 𝜏 𝑚 (Fig. 1B). Note that the DDE pairs can also be applied along different directions, 𝒏 and 𝒏 . To probe kurtosis for fixed timing parameters, CTI uses 𝛿 = 𝛿 = 𝛿 and ∆ = ∆ = ∆ . In the long mixing time regime and up to 2 nd order in b , the DDE signal attenuation ( 𝐸 𝐷𝐷𝐸 ) can be expressed as (40,60,76,83): log(𝐸 𝐷𝐷𝐸 (𝑏 , 𝑏 , 𝒏 , 𝒏 )) = −(𝑛 𝑛 𝑏 + 𝑛 𝑛 𝑏 )𝐷 𝑖𝑗 + (𝑛 𝑛 𝑛 𝑛 𝑏 + 𝑛 𝑛 𝑛 𝑛 𝑏 )𝐷̅ 𝑊 𝑖𝑗𝑘𝑙 + δ3 ) 𝑛 𝑛 𝑛 𝑛 𝑏 𝑏 𝑍 𝑖𝑗𝑘𝑙 + 𝑂(𝑏 ) (5) where 𝑏 = (𝛾𝛿𝑔 ) (∆ − 𝛿/3) and 𝑏 = (𝛾𝛿𝑔 ) (∆ − 𝛿/3) are the b-values associated with the two DDE gradients, and 𝑍 𝑖𝑗𝑘𝑙 is a tensor that approaches the covariance tensor ( 𝑍 𝑖𝑗𝑘𝑙 →4𝐶 𝑖𝑗𝑘𝑙 (∆ − δ3 ) ) at long mixing times. We previously showed that 𝐾 𝑎𝑛𝑖𝑠𝑜 , 𝐾 𝑖𝑠𝑜 , and 𝜇𝐾 can in theory be extracted from the tensors of Eq. 5 (40). However, our preliminary validation showed that high-order-terms 𝑂(𝑏 ) can introduce biases on the different kurtosis estimates that depend on dispersion levels. To suppress this dependency, here we consider an adapted version of CTI for DDE powder-averaged signals ( 𝐸̅ 𝐷𝐷𝐸 ) : log(𝐸̅ 𝐷𝐷𝐸 (𝑏 , 𝑏 , 𝜃))= −(𝑏 + 𝑏 )𝐷 + 16 (𝑏 + 𝑏 )𝐷 𝐾 𝑇 + 12 𝑏 𝑏 cos 𝜃 𝐷 𝐾 𝑎𝑛𝑖𝑠𝑜 + 16 𝑏 𝑏 𝐷 (2𝐾 𝑖𝑠𝑜 − 𝐾 𝑎𝑛𝑖𝑠𝑜 ) + 𝑂(𝑏 ) (6) Here 𝜃 is defined as the angle between the gradient directions 𝒏 and 𝒏 : note that several different pairs of 𝒏 and 𝒏 with constant 𝜃 are required for powder-averaging (vide infra). From the parameters of Eq. 6, microscopic kurtosis can be estimated by 𝜇𝐾 = 𝐾 𝑇 − 𝐾 𝑎𝑛𝑖𝑠𝑜 −𝐾 𝑖𝑠𝑜 (cf. Eq. 3). The prior CTI diffusion and kurtosis quantities were extracted based on extensive DDE acquisitions (40); one central aim of the current work is to accelerate the acquisition. In fact, CTI metrics can be extracted for powder-averaged 𝐸̅ 𝐷𝐷𝐸 using only four different sets of DDE experiments illustrated in Fig. 1C, in addition to acquisitions without diffusion sensitization, i.e. 𝑏 = 𝑏 = 0 . The 4 sets are as follows: 1) Powder-averaged signals with 𝑏 = 𝑏 a and 𝑏 = 0 . Note that these experiments are equivalent to SDE experiments (Fig. 1C1); 2) Powder-averaged symmetric DDE with diffusion weighting 𝑏 = 𝑏 = 𝑏 a /2 and parallel gradient directions ( 𝜃 = 0 o ).
3) Powder-averaged symmetric DDE with diffusion weighting 𝑏 = 𝑏 = 𝑏 a /2 and perpendicular gradient directions ( 𝜃 = 90 o ); 4) Powder-averaged symmetric and parallel DDE as 3) but with a different total b-value 𝑏 𝑡 = 𝑏 + 𝑏 = 𝑏 𝑏 < 𝑏 𝑎 . Note that all previous sets (1-3) have the same total b-value 𝑏 t =𝑏 + 𝑏 equal to 𝑏 a To ensure homoscedastic 𝐸̅ 𝐷𝐷𝐸 signals, all four experiment sets should be acquired with a similar number of gradient direction pairs for powder-averaging.
The analysis to resolve different kurtosis sources proceeds in the following way: a) 𝜇𝐾 can be extracted from the log difference of powder-averaged signals from the experiments’ set log( 𝐸 ̅ 𝐷𝐷𝐸 (𝑏 𝑎 , 0, 0°)) − log ( 𝐸 ̅ 𝐷𝐷𝐸 ( 𝑏 𝑎 , 𝑏 𝑎 , 0 o )) = 𝑏 𝑎2 𝐷 𝜇𝐾 (7) b) as pointed in previous studies (e.g., (76,84,85)), 𝐾 𝑎𝑛𝑖𝑠𝑜 can be extracted from the log difference of powder- averaged signal from the experiments’ set log (𝐸̅ 𝐷𝐷𝐸 ( 𝑏 𝑎 , 𝑏 𝑎 , 0 o )) − log (𝐸̅ 𝐷𝐷𝐸 ( 𝑏 𝑎 , 𝑏 𝑎 , 90 o )) = 𝑏 𝑎2 𝐷 𝐾 𝑎𝑛𝑖𝑠𝑜 (8) c) to decouple 𝐷 , 𝐾 𝑇 , and 𝐾 𝑖𝑠𝑜 , powder-averaged signals have also to be acquired for at least two non-zero total b-values 𝑏 𝑡 . Therefore, DDE experiments with symmetric intensities and parallel directions for a lower total b-value 𝑏 𝑡 = 𝑏 𝑎 are also considered (set Figure 1 – Experiments for kurtosis source estimation. A)
Parameters of a standard SDE pulse sequence, where ∆ is the diffusion time, 𝛿 is the gradient pulse duration, and 𝑔 is the gradient pulse direction (this sequence can also be described by a gradient direction 𝒏 and b-value 𝑏 = (𝛾𝑔𝛿) ∆ ); B) Parameters of a standard DDE pulse sequence, where ∆ and ∆ are the diffusion encoding blocks ’ diffusion times, 𝛿 and 𝛿 are their gradient pulse durations, 𝑔 and 𝑔 are their gradient intensities and 𝜏 𝑚 is the mixing time (i.e. the time between the two diffusion encoding bocks). At the long mixing time, DDE experiments for powder-averaged systems can be fully described by two b-values ( 𝑏 and 𝑏 ) and the angle 𝜃 between the directions of the two diffusion encoding modules 𝒏 (blue) and 𝒏 (red); C) Parameters for the different data sets required to estimate different kurtosis sources. The diffusion encoding profiles for all data sets are shown in panels C1.1, C2.1, C3.1, and C4.1 (note that all acquisitions in this work use ∆ = ∆ = ∆ and δ = δ = δ ). The gradient directions used for signal powder-average calculation are shown in panels C1.2, C2.2, C3.2, and C4.2. The equivalent b-tensor shapes for each experiment are shown in panels C1.3, C2.3, C3.3, C3.4. Under the Gaussian diffusion assumption, previous studies showed that 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 can be estimated from signals measured using any multidimensional encoding sequence that probes different b-tensor magnitudes 𝑏 𝑡 and shapes. For the sake of simplicity, here we only consider axial tensor-valued experiments (39,51,72), where the b-tensor shape is characterized by a single parameter 𝑏 ∆ ∈ [− , 1] . Thus, the powder-averaged signal is log(𝐸̅ 𝐷𝑇𝑉 (𝑏, 𝑏 ∆ )) = −𝑏 𝑡 𝐷 + 𝑏 𝑡2 𝐷 𝐾 𝑖𝑠𝑜 + 𝑏 𝑡2 𝑏 ∆2 𝐷 𝐾 𝑎𝑛𝑖𝑠𝑜 + 𝑂(𝑏 ) (9) Note that 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 can be estimated from Eq. 8, using the same dMRI experiments as for CTI, since these correspond to data acquired with at least two b-tensor shapes ( 𝑏 ∆ = 1 for sets 𝑏 ∆ = -1/2 for set 𝑏 𝑡 = 𝑏 𝑎 for sets 𝑏 𝑡 = 𝑏 𝑏 for set
3. Methods
The robustness of the different kurtosis source estimation strategies (DTV and CTI) was first assessed for synthetic signals simulated for five different models with known ground truth kurtosis sources: 1)
Sum of multiple isotropic Gaussian diffusion components with mean diffusivities 𝐷 𝑖 sampled from a Gaussian distribution with mean 〈𝐷 𝑖 〉 = 0.65𝜇𝑚 /𝑚𝑠 and standard deviation √𝑉(𝐷 𝑖 ) = 0.21𝜇𝑚 /𝑚𝑠 (Fig. 2A). Ground truth parameters for this model are: 𝐷 = 0.65𝜇𝑚 /𝑚𝑠 , 𝐾 𝑡 = 𝐾 𝑖𝑠𝑜 = 0.31 , 𝐾 𝑎𝑛𝑖𝑠𝑜 = 𝜇𝐾 = 0. Sum of multiple uniformly oriented anisotropic Gaussian diffusion components all with the same axial and radial diffusivities of 𝐴𝐷 𝑖 = 1.45𝜇𝑚 /𝑚𝑠 and 𝑅𝐷 𝑖 =0.25𝜇𝑚 /𝑚𝑠 (Fig. 2B). Ground truth parameters for this model are: 𝐷 = 0.65𝜇𝑚 /𝑚𝑠 , 𝐾 𝑡 =𝐾 𝑎𝑛𝑖𝑠𝑜 = 0.91 , 𝐾 𝑖𝑠𝑜 = 𝜇𝐾 = 0. Spherical restricted compartments with radius 𝑟 = 3𝜇𝑚 and intrinsic diffusivity 𝐷 = 2𝜇𝑚 /𝑚𝑠 (Fig. 2C). The signals for this system were produced using the MISST package (86,87). With the acquisition parameter values ∆ and 𝛿 corresponding to experiments ( vide infra ), ground truth parameters are: 𝐷 = 0.17𝜇𝑚 /𝑚𝑠 , 𝐾 𝑡 = 𝜇𝐾 = −0.48 , 𝐾 𝑖𝑠𝑜 =𝐾 𝑎𝑛𝑖𝑠𝑜 = 0 . 4) Single compartment with positive source of microscopic kurtosis. According to the effective medium theory (79), compartments with microscopic disorder can exhibit non-Gaussian diffusion with a positive microscopic kurtosis contribution 𝜇𝐾 𝑖 (79,80)). Although in Fig. 2D a single component with positive 𝜇𝐾 is sketched in a medium that encompasses randomly oriented anisotropic compartments, microstructural disorder in both intra- and extra- “ cellular ” components arising, e.g., due to the presence of small sub-structures (spines, boutons, axons, dendrites) with spatially varying diameter sizes and packing degree (49,80 – 𝜇𝐾 value for a medium represented in Fig. 2D will depend on the volume fraction, anisotropy, size, and packing of the anisotropic compartments as well as on the acquisition parameters (82). To simplify, the DDE signal decay for the single isotropic compartment with positive microscopic kurtosis is here numerically computed using the signal representation 𝐸(𝑏 , 𝑏 ) = exp (−(𝑏 + 𝑏 )𝐷 + (𝑏 + 𝑏 )𝐷 𝜇𝐾) with 𝐷 and 𝜇𝐾 ground truth set to an arbitrary value of /𝑚𝑠 and 1, respectively. 5) A system comprising different components and with non-zero contributions for all different kurtosis sources (Fig. 3A). For this system, we consider a sum of the compartments types used for simulations 1, 2, and 4 with equal weights. The signal for this model is first computed with the diffusivity values specified above, resulting in the following ground truth parameters:
𝐷 = 0.65𝜇𝑚 /𝑚𝑠 , 𝐾 𝑡 = 0.911 , 𝐾 𝑎𝑛𝑖𝑠𝑜 = 0.473 , 𝐾 𝑖𝑠𝑜 = 0.104 , and 𝜇𝐾 = 0.333 . As the mean diffusivities of simulations 1, 2 and 4 are equal, this ensemble model can assess the robustness of estimates for different kurtosis sources individually by varying concrete model parameters. In particular, different ground truth 𝐾 𝑖𝑠𝑜 were generated by changing the mean diffusivity variance of Gaussian isotropic components 𝐾 𝑖𝑠𝑜(𝑔.𝑡.) = 3𝑓 𝑉(𝐷 𝑖(1) )/𝐷 ; different ground truth 𝐾 𝑎𝑛𝑖𝑠𝑜 were created by changing the difference between the axial and radial diffusivities ( 𝛼 = 𝐴𝐷 𝑖(2) − 𝑅𝐷 𝑖(2) ) of uniformly oriented anisotropic Gaussian components 𝐾 𝑎𝑛𝑖𝑠𝑜(𝑔.𝑡.) = 4𝑓 𝛼 /15𝐷 . In the latter case, 𝐴𝐷 𝑖(2) and 𝑅𝐷 𝑖(2) were computed as 𝐷 +2𝛼/3 and
𝐷 − 𝛼/3 for 𝛼 values sampled from 0 to /𝑚𝑠 to keep the mean diffusivity constant. Finally, different ground truth 𝜇𝐾 values were generated by changing the microscopic kurtosis contribution of the non-Gaussian diffusion compartment 𝜇𝐾 (𝑔.𝑡.) = (1 − 𝑓 −𝑓 )𝜇𝐾 (3) . For all five models, powder-averaged signals were generated according to the four different sets of DDE acquisition parameters (c.f. Fig. 1) for total b-values 𝑏 𝑎 = 2.5𝑚𝑠/𝜇𝑚 (sets 𝑏 𝑏 = 1𝑚𝑠/𝜇𝑚 (set ∆ = 𝜏 𝑚 = 12𝑚𝑠 and 𝛿 = 3.5𝑚𝑠 for all experiments). For all four sets, the 45 directions of a 3D spherical 8-design (88) were used for the single encoding of set old ) DDE protocol suggested for CTI (40). The 𝑏 , 𝑏 and 𝜃 values, as well as gradient directions schemes used for the improved ( new ) and old CTI protocols (CTI new and CTI old ) are summarized in Table 1 (for more information on the old protocol, c.f. supporting material, section A). In addition to the diffusion-weighted signals for the different CTI sets, 135 signal replicas for 𝑏 = 𝑏 = 0 are incorporated in both protocols – this data is treated as an independent 𝑏 t = 0 set. To assess the robustness of estimates towards noise, all synthetic signals were corrupted by Rician noise with a nominal SNR of 40 before powder-averaging. Table 1 – Summary of the DDE parameter combination used for the “ new ” CTI protocol (CTI new ) and the reference “old” CTI protocol (CTI old ). Parameters b , b , b t are expressed in ms/µm New CTI protocol – CTI new set b b b t θ b Δ direction scheme o
1 45 directions of the 8-design (x3 repetitions) o
1 45 directions of the 8-design (x3 repetitions) for both diffusion encodings o -1/2 45 directions of the 8-design for the 1 st encoding, repeated for 3 orthogonal directions for the 2 nd encoding o
1 45 directions of spherical 8-design (x3 repetitions) for both diffusion encodings
Old CTI protocol – CTI old set b b b t θ b Δ direction scheme o
1 45 directions of the 8-design (x3 repetitions) o
1 45 directions of the 8-design (x3 repetitions) for both diffusion encodings o -1/2 45 directions of the 8-design for the 1 st encoding, repeated for 3 orthogonal directions for the 2 nd encoding
1 0 1 0 o
1 45 directions of the 8-design (x3 repetitions)
1 1 2 0 o
1 45 directions of the 8-design (x3 repetitions) for both diffusion encodings
1 1 2 90 o -1/2 45 directions of the 8-design for the 1 st encoding, repeated for 3 orthogonal directions for the 2 nd encoding Data processing: CTI estimates were obtained by fitting Eq. 6 to the log of the powder-averaged signals of both new and old protocols using an ordinary linear-least-squares (OLLS) procedure. 𝐾 𝑎𝑛𝑖𝑠𝑜 , 𝐾 𝑖𝑠𝑜 , and 𝐾 𝑡 were also estimated from the DTV approach by fitting Eq. 9 to the powder- average of CTI’s new protocol using an OLLS procedure. Mean and standard deviation of each kurtosis estimates were computed by repeating simulations for 1000 iterations. All animal experiments were preapproved by the institutional and national authorities and carried out according to European Directive 2010/63. Data was acquired from N=3 female Long Evans rats (ages = 22/19/22 weeks old, weights = 354.6/260.4/334.5g in a 12 h/12 h light/dark cycle with ad libitum access to food and water) under anesthesia (~2.5% Isoflurane in 28% oxygen) using a 9.4T Bruker Biospec scanner equipped with an 86 mm quadrature transmission coil and 4-element array reception cryocoil. Sagittal T -weighted images are first acquired using a RARE sequence with the following parameters : TR = 2000 ms, effective TE = 36 ms, RARE factor = 8, Field of View = 24 × 16.1 mm , matrix size 160 × 107, leading to an in-plane voxel resolution = 150 × 150 μm , slice thickness = 500 μm (21 slices), number of averages = 8. Diffusion MRI datasets were then acquired using an EPI pulse sequence written in-house (acquisition bandwidth = 400 kHz, number of shots = 1, partial Fourier = 1.25) for 3 evenly spaced coronal slices (Fig. 3A). Per-slice respiratory gating was applied. Acquisitions followed the first CTI protocol reported in Table 1 along with 24 𝑏 t = 0 acquisitions per DDE set. Other acquisition parameters included: = 𝜏 𝑚 = 12 ms, = 3.5 ms, TR/TE=3000/50.9 ms, in-plane resolution = 200 m , slice thickness = 0.9 mm. For each rat, the acquisition time for the diffusion-weighted data was around 40 mins. Data-processing: Thermal noise of each diffusion-weighted data set and for each cryocoil channel was suppressed using a threshold-based Marchenko‐Pastur
PCA denoising (89) in which signal components for eigenvalues 𝜆 < 𝜎 (1 + √𝑁/𝑀) were removed - 𝜎 is noise variance (computed from the repeated 𝑏 t = 0 acquisitions), N and M are the number of gradient directions pairs and image voxels. Denoised data was then Gibbs ringing corrected using a sub-voxel shift algorithm (29,90). Processed diffusion-weighted signals for the four channels were then combined using sum-of-squares. Combined data for different gradient direction pairs were then aligned using a sub-pixel registration technique (91). Kurtosis estimates using both CTI and DTV were obtained by voxelwise fitting of equations 6 and 8 to the set powder-averaged data (masked to avoid regions distorted due to b0 inhomogeneities (Fig. 3B) using an OLLS procedure. Ten regions of interest (ROIs) were also manually define d on 𝑏 t = 0 images, including the left and right cortical grey matter (GM), the right and left corpus callosum genu (CCg), the right and left corpus callosum body (CCb), the right and left corpus callosum splenium (CCs), and the right and left internal capsule (IC).
4. Results
Figure 2 shows the results from the synthetic diffusion-weighted signals generated for systems containing single components types (models 1-4). When considering a system consisting of isotropic Gaussian diffusion components with different mean diffusivities (Fig. 2A1), all DDE synthetic signal sets present identical log-signal dependencies with 𝑏 𝑡 (Fig. 2A2). Thus, all kurtosis estimation strategies correctly characterized the non-linear log-signal decays of these experiments as an effect of non-zero 𝐾 𝑖𝑠𝑜 (Fig. 2A3-6). For systems comprising isotropically distributed anisotropic Gaussian components (Fig. 2B1), perpendicular DDE synthetic signals present higher diffusion-weighted attenuations, as expected (Fig. 2B2). The old CTI scheme estimates a biased, non-zero 𝜇𝐾 (Fig. 2B5). On the other hand, the new CTI scheme (as well as the DTV approach) correctly attributes the DDE signal difference to 𝐾 𝑎𝑛𝑖𝑠𝑜 (Fig. 2B4 and Fig. 2B6). For systems characterized by restricted diffusion (Fig. 2C1, negative µK ) or systems with microscopic disorder (Fig. 2D1, positive µK ), asymmetric DDE signals (i.e., 𝐸̅ 𝐷𝐷𝐸 (𝑏 𝑡 , 0, 0°) ) differ from their symmetric DDE counterparts (i.e. 𝐸̅ 𝐷𝐷𝐸 (𝑏 𝑡 /2, 𝑏 𝑡 /2, 0 o ) and 𝐸̅ 𝐷𝐷𝐸 (𝑏 𝑡 /2, 𝑏 𝑡 /2, 90 o ) , Fig. 2C2 and Fig. 2D2). The finite 𝜇𝐾 strongly biases both 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 from DTV (Fig. 2C4 and Fig. 2D4). On the other hand, CTI – both old and new protocols – correctly estimate the finite 𝜇𝐾 . Figure 2 – Results for synthetic diffusion-weighted signals generated according to four systems containing single compartment types: A)
Isotropic Gaussian diffusion components with different mean diffusivities sampled from a Gaussian distribution with mean /𝑚𝑠 and standard deviation /𝑚𝑠 ; B) Isotropically oriented anisotropic Gaussian diffusion components with axial and radial diffusivities of /𝑚𝑠 and /𝑚𝑠 ; C) Non-Gaussian diffusion inside restricted spheres with radius of and intrinsic diffusivity of /𝑚𝑠 ; D) Non-Gaussian diffusion due to microscopic disorder with ground truth 𝜇𝐾 =1 and
𝐷 = 0.65𝜇𝑚 /𝑚𝑠 . From left to right, each panel shows: a schematic representation of the models (A1, B1, C1, D1); the signal decays for three different DDE experiment types in which signals of the improved CTI protocol are marked by the blue triangles (A2, B2, C2, D2); the kurtosis ground truth values (A3, B3, C3, D3); the kurtosis estimates obtained from DTV (A4, B4, C4, D4); the kurtosis estimates obtained from the CTI using its previous “old” protocol (A5, B5, C5, D5); the kurtosis estimates obtained from the CTI using i ts improved “new” protocol (A6, B6, C6, D6). In a realistic voxel, all kurtosis sources could be simultaneously present, and it is therefore instructive to assess whether the CTI framework can disentangle the kurtosis sources with specificity. Figure 3 shows the results from the synthetic signals for a model with non-zero contributions from all different kurtosis sources. For this system, log-signal DDE decays are different for the three conditions (Fig. 3A2). The new improved CTI protocol successfully estimates the kurtosis sources, as observed by the close correspondence with ground truth values (Fig. 3A3-6), while the old protocol overestimates 𝜇𝐾 . We then used the same model to investigate how changes in ground truth kurtosis sources would impact the different source estimates (see methods). Particularly, when changing the ground truth 𝐾 𝑖𝑠𝑜 (Fig. 3B) or 𝐾 𝑎𝑛𝑖𝑠𝑜 (Fig. 3C), the improved CTI and DTV approaches correctly show larger changes on 𝐾 𝑖𝑠𝑜 and 𝐾 𝑎𝑛𝑖𝑠𝑜 respectively. Note that 𝜇𝐾 from the old CTI protocol have limited specificity as varying 𝐾 𝑎𝑛𝑖𝑠𝑜 results in varying 𝜇𝐾 . 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 for the DTV approach also show limited specificity when the ground truth 𝜇𝐾 is varied (Fig. 3D). Interestingly, we find that CTI estimates from the new protocol correctly tracked the specific ground truth sources, i.e. major changes in improved CTI 𝐾 𝑖𝑠𝑜 , 𝐾 𝑎𝑛𝑖𝑠𝑜 , and 𝜇𝐾 estimates are only observed when 𝐾 𝑖𝑠𝑜 , 𝐾 𝑎𝑛𝑖𝑠𝑜 , and 𝜇𝐾 ground truths are varied, respectively (Fig. 3B3, Fig. 3C2, and Fig. 3D4), even if some offsets are also observed. Figure 3 – Results for synthetic diffusion-weighted signals generated according to a system containing a sum of different compartment types (isotropic Gaussian + anisotropic Gaussian + isotropic non-Gaussian components): A)
Results from the model ’s initial guess, i.e., isotropic components ’ mean diffusivities sampled from a Gaussian distribution with mean /𝑚𝑠 and standard deviation /𝑚𝑠 , anisotropic Gaussian diffusion components with axial and radial diffusivities of /𝑚𝑠 and /𝑚𝑠 ; and non-Gaussian diffusion with 𝐾 𝑖 = 1 and 𝐷 𝑖 = 0.65𝜇𝑚 /𝑚𝑠 . B) CTI and DTV kurtosis estimates ’ sensitivity to ground truth 𝐾 𝑖𝑠𝑜 alterations by changing the mean diffusivity variation of isotropic Gaussian components. C) CTI and DTV kurtosis estimates sensitivity to ground truth 𝐾 𝑎𝑛𝑖𝑠𝑜 alterations by changing the axial and radial diffusivities of anisotropic Gaussian compartments. D) CTI and DTV estimates sensitivity to ground truth 𝜇𝐾 alterations by changing the microscopic kurtosis level of the non-Gaussian component. Representative raw sagittal T2-weighted images and coronal 𝑏 t = 0 images for all three rats are shown in Fig. 3A-B. Nominal SNR (Fig. 3C) was ~30 for the raw data. Fig. 3D shows a representative diffusion-weighted image for a dMRI experiment acquired for the maximum b-value used on this study before and after PCA denoising. Denoised data presented an SNR gain of ~1.3 (nominal SNR of the denoised data was ~40). Figure 4 – Raw T2-weighted and diffusion-weighted data: A)
Midsagittal T2-weighted images of all three animals. These images were used as a reference for placing the 3 coronal slices for diffusion MRI acquisition – positions of the coronal slices are marked in red. B) Representative 𝑏 t = 0 image for all animals and coronal slices which are used to delineate the brain mask (blue line) and the regions of interest in the cortical grey matter (GM), the right and left corpus callosum genu (CCg), the right and left corpus callosum body (CCb), the right and left corpus callosum splenium (CCs), and the right and left internal capsule (IC). C) Representative nominal SNR map that was computed from all 𝑏 t = 0 acquisitions of Rat 𝑏 t =2.5 𝜇𝑚 /𝑚𝑠 (for 𝑏 = 2.5 𝜇𝑚 /𝑚𝑠 and 𝑏 = 0 ) before and after PCA denoising (right and left respectively) – these diffusion-weighted images were computed for Rat Fig. 5A shows the powder-averaged signal decays for the four sets of the improved CTI protocol. The log difference between the powder-averaged data from set sensitivity to 𝜇𝐾 (c.f. Eq. 4), while the log difference between the powder-averaged data from set 𝐾 𝑎𝑛𝑖𝑠𝑜 (c.f. Eq. 5). Figure 5 – Results from powder-averaged data for all slices of a representative animal (Rat
Powder-averaged signal decays for all four sets DDE experiments for CTI: set B) Sensitivity to microscopic kurtosis ( 𝜇𝐾 ) quantified by the log difference between powder-averaged signals of set C) Sensitivity to anisotropic kurtosis ( 𝐾 𝑎𝑛𝑖𝑠𝑜 ) quantified by the log difference between powder-averaged signals from set 𝑏 𝑎 = 2.5 𝜇𝑚 /𝑚𝑠 and 𝑏 𝑏 = 1 𝜇𝑚 /𝑚𝑠 . The CTI kurtosis estimates for all slices of rat 𝐾 𝑡 and 𝐾 𝑎𝑛𝑖𝑠𝑜 were higher in white matter regions (Fig. 6A-B). 𝐾 𝑖𝑠𝑜 and 𝜇𝐾 maps show noisier spatial profiles than 𝐾 𝑡 and 𝐾 𝑎𝑛𝑖𝑠𝑜 maps (Fig. 6C-D). Nevertheless, 𝜇𝐾 is lower in white matter brain regions (Fig. 6D, red arrows). The ROI analysis supported the trends observed in the kurtosis maps (Fig. 7A) – in general, higher values of 𝐾 𝑡 , 𝐾 𝑎𝑛𝑖𝑠𝑜 , and 𝐾 𝑖𝑠𝑜 and lower 𝜇𝐾 values were observed in white matter ROIs compared with grey matter ROIs. Fig. 7B shows the histograms of the different CTI-driven kurtosis estimates for combined grey and combined white matter ROIs. The mean values of grey and white matter values are significantly different (two-sample t-test with unequal variances, p<0.001 for kurtosis estimates and all three animals).
Figure 6 – Correlation Tensor Imaging kurtosis estimates. A)
Total Kurtosis 𝐾 𝑡 ; B) Anisotropic Kurtosis 𝐾 𝑎𝑛𝑖𝑠𝑜 ; C) Isotropic Kurtosis 𝐾 𝑖𝑠𝑜 ; D) Microscopic Kurtosis 𝜇𝐾 . The red arrow in panel D points to lower 𝜇𝐾 estimates observed particularly in white matter brain regions. On each panel, maps are presented for all slices of rat Figure 7 – Correlation Tensor Imaging kurtosis estimates in various brain regions. A)
Mean and standard deviation of 𝐾 𝑡 (A1), 𝐾 𝑎𝑛𝑖𝑠𝑜 (A2), 𝐾 𝑖𝑠𝑜 (A3), and 𝜇𝐾 (A4) estimates for different regions of interest - black bars shows the mean and standard deviations for the individual rats, while the coloured bars show the mean values across all three animals. Regions of interest for these panels include: the right and left cortical grey matter (GM r and GM l ); the right and left corpus callosum splenium (CCs r and CCs l ); the right and left corpus callosum body (CCd r and CCd l ); the right and left corpus callosum genu (CCg r and CCg l ); and the right and left internal capsule (IC r and IC l ). B) Histograms of the 𝐾 𝑡 (B1), 𝐾 𝑎𝑛𝑖𝑠𝑜 (B2), 𝐾 𝑖𝑠𝑜 (B3), and 𝜇𝐾 (B4) estimates for grey matter (histograms in green) and white matter (histograms in red) regions of interest - in each panel, the differences between the mean values from grey and white matter ROIs are statistically tested using a two-sampled t-test with unequal variances (* for p<0.05, ** for p < 0.01, and *** for p<0.001 ***). We then turned to assess the impact of the finite 𝜇𝐾 on analysis using only diffusion tensor variance (DTV). Figure 8A-C shows the kurtosis maps obtained from the DTV approach. In comparison to their CTI counterparts, DTV-derived 𝐾 𝑡 was lower in both grey and white matter regions (Fig. 8A), while DTV 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 values were higher. The mean and standard deviation of the different DTV kurtosis estimates for different ROIs are shown in Fig. 8D; the histograms of kurtosis values from white and grey matter ROIs are shown in Fig. 8E. As for CTI, DTV mean 𝐾 𝑡 and 𝐾 𝑎𝑛𝑖𝑠𝑜 appear higher in white matter ROIs (Fig. 8D1-2). However, non-significant differences between the white and grey matter voxels were observed for the DTV 𝐾 𝑖𝑠𝑜 estimates (Fig. 8E3). Figure 8 – Kurtosis estimates assuming only diffusion tensor variance (DTV). A)
DTV total kurtosis ( 𝐾 𝑡𝑇𝑒𝑛 ) maps. B) DTV anisotropic kurtosis 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 maps. C) DTV isotropic kurtosis 𝐾 𝑖𝑠𝑜𝐷𝑇𝑉 maps. D) Mean and standard deviation of 𝐾 𝑡 (D1), 𝐾 𝑎𝑛𝑖𝑠𝑜 (D2), and 𝐾 𝑖𝑠𝑜 (D3) DTV estimates for different regions of interest - black bars shows the mean and standard deviations for the individual rats, while the coloured bars show the mean values across all three animals. E) Histograms of the 𝐾 𝑡 (E1), 𝐾 𝑎𝑛𝑖𝑠𝑜 (E2), and 𝐾 𝑖𝑠𝑜 (E3) DTV estimates for all grey matter (histograms in green) and white matter (histograms in red) regions of interest – in each panel, the differences between the mean values of grey and white matter ROIs are statistically tested using a two-sampled t-test with unequal variances (* for p<0.05, ** for p < 0.01, and *** for p<0.001 ***). To further investigate the correlation between these metrics, scatter plots of DTV and CTI kurtosis estimates are shown in Fig. 9. Points in the scatter plots are color-coded according to CTI’s microscopic kurtosis ( 𝜇𝐾 ) estimates, showing that higher differences between CTI and DTV estimates are associated with higher degrees of 𝜇𝐾 . Figure 9 – Scatter plots between DTV and CTI estimates. A)
Scatter plots between DTV total kurtosis ( 𝐾 𝑡𝐷𝑇𝑉 ) and CTI total kurtosis ( 𝐾 𝑡𝐶𝑇𝐼 ) estimates; B) Scatter plots between DTV anisotropic kurtosis ( 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 ) and CTI anisotropic kurtosis ( 𝐾 𝑎𝑛𝑖𝑠𝑜𝐶𝑇𝐼 ) estimates; B) Scatter plots between DTV isotropic kurtosis ( 𝐾 𝑖𝑠𝑜𝐷𝑇𝑉 ) and CTI isotropic kurtosis ( 𝐾 𝑖𝑠𝑜𝐶𝑇𝐼 ) estimates. Points in the scatter plots are colour coded according to CTI’s microscopic kurtosi s ( 𝜇𝐾 ) estimates.
5. Discussion
The conflation of underlying kurtosis sources in SDE was a major motivation in the development of multidimensional diffusion encoding approaches which provide 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 , and has shown great promise for e.g. distinguishing different tumor types and grades (39,72), depicting healthy and pathological age-related microstructural alterations (92), mapping multiple sclerosis lesions (93,94), and characterizing body organs (69,95). However, approaches based only on the tensor-valued information of multidimensional diffusion encoding are analyzed assuming only diffusion tensor variance (DTV) and hence implicitly ignoring diffusion-time dependence and considering that 𝜇𝐾 is identically zero. Recent studies demonstrated that diffusion-time dependence can introduce directional dependent biases on these approaches (73,74). However, 𝜇𝐾 effects were hitherto not investigated in living tissues. We therefore sought in this study to characterize 𝜇𝐾 in vivo and assess its impact on the more conventional DTV approaches. Correlation Tensor Imaging was recently introduced for 𝜇𝐾 mapping. CTI goes beyond the tensor-value framework and simultaneously estimates 𝐾 𝑎𝑛𝑖𝑠𝑜 , 𝐾 𝑖𝑠𝑜 , and 𝜇𝐾 – albeit at the expense of a larger number of acquisitions (40). In addition, the original CTI framework included measurements that could bias 𝜇𝐾 in some scenarios. To alleviate these drawbacks, a sparse set of DDE acquisitions for robustly resolving kurtosis sources was developed. In the long mixing time regime (e.g. (54,55,57,76,85)), non-Gaussian diffusion effects due to microscopic diffusion anisotropy ( 𝐾 𝑎𝑛𝑖𝑠𝑜 ) can be estimated using the log-signal differences of DDE experiments acquired with parallel and perpendicular wave-vectors (c.f. Eq. 8). One central finding of this study is that 𝜇𝐾 can also be estimated from the log signal differences of two different DDE experiments (c.f. Eq. 7), specifically DDE signals acquired using 1) parallel symmetric DDE gradient intensities and 2) signals acquired for SDE-analogous experiments with the same total b-value. In addition to DDE experiments for a different total b-value to resolve the 𝐾 𝑇 and 𝐾 𝑖𝑠𝑜 , we showed that the four CTI quantities ( 𝐷 , 𝐾 𝑎𝑛𝑖𝑠𝑜 , 𝐾 𝑖𝑠𝑜 , 𝜇𝐾 ) can be fully resolved from only four different combinations of DDE parameters ( 𝑏 , 𝑏 , 𝜃 ) which we refer to as DDE acquisition sets (c.f. Fig. 1). Previous CTI kurtosis estimates were extracted from DDE data that took more than 2h to acquire, but in this study, CTI kurtosis estimates were obtained under 40 mins using the improved protocol. This significant acceleration opens the way to much more relevant applications for in vivo preclinical and even clinical mapping. In addition to the accelerated acquisition, we improved the acquisition scheme towards mitigating biases from higher-order terms afflicting the previous CTI estimates. Indeed, as pointed out in our previous study (40,85,96), higher-order biases are a common issue for techniques based on the truncation of the signal cumulant expansion. For example, non-zero 𝜇𝐾 estimates can be obtained even for synthetic systems characterized by multiple Gaussian components (e.g., Fig. 2A5, Fig. 2B5). While these biases were significant for 𝜇𝐾 estimation in the previous CTI implementation, we have here shown that these biases can be dramatically diminished for the improved CTI acquisition scheme by balancing the total b-values used for different DDE sets (c.f. Section A of the supporting material). Diminishing higher-order effects was shown to be important to promote CTI specificity (Fig. 3). Indeed, our simulations showed that from the improved CTI acquisitions, estimated kurtosis source alterations agreed with ground truth values. In (40), positive 𝜇𝐾 estimates were observed in both grey and white matter brain regions in the in vivo rat brain. Our new results with the improved CTI protocol confirmed the overall non-vanishing positive 𝜇𝐾 effects in both white and grey matter (Fig. 6, Fig. 7); however, significant 𝜇𝐾 differences were observed between grey and white matter brain regions (Fig. 7). These positive 𝜇𝐾 estimates are consistent with the expected positive non-Gaussian diffusion effects due to intra-cellular cross-section size variance (18,49,81) and/or the presence of obstacles in tortuous extra-cellular environments (79,82). Therefore, the 𝜇𝐾 differences between grey and white matter could perhaps be explained by differences in both intra- and extra-cellular microstructural configurations (e.g. different compartmental cross sectional variance, different degree of cellular packing, etc.), or due to the presence of a more negative 𝜇𝐾 contributions from restricted diffusion in white matter. Although 𝜇𝐾 may depend on all above-mentioned effects, positive 𝜇𝐾 contributions are expected to prevail as negative 𝜇𝐾 contributions from completely restricted diffusion are typically associated with low apparent diffusivities – note that the total 𝜇𝐾 measured by CTI is a weighted average of all its contributions, where the weights depend on the squared apparent diffusivities of each contribution ( 𝜇𝐾 = 〈𝐷 𝑖2 𝜇𝐾 𝑖 〉𝐷 ). Another significant result in this study, is that this non-vanishing 𝜇𝐾 can have a dramatic effect on kurtosis sources computed from multidimensional diffusion encoding DTV framework (Fig. 8). In general, both 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 derived from the DTV approach were biased towards higher values compared with their more accurate CTI counterparts. The color-coded scatter plots between DTV and CTI estimates revealed that differences can be fully explained by 𝜇𝐾 biases on DTV estimates (Fig. 9). In is important to note, that the influence of 𝜇𝐾 on 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 can obscure microstructural differences, such as the 𝐾 𝑖𝑠𝑜 differences between grey and white matter brain regions (Fig. 9). Although it does not rely on the Gaussian diffusion assumption, CTI is still a cumulant expansion of DDE signals, which induces some implicit assumptions. Namely, disregarding higher-order cumulant terms; assuming the long mixing time regime (which can be empirically evaluated); and ignores exchange. Higher-order-term biases were here minimized by the new CTI protocol. The long mixing time regime effectively suppresses unwanted time-dependent diffusion correlations from the Q and S-tensors (60,83) and guarantees that the Z-tensor in Eq. 5 approaches the covariance tensor (40,60,76,83). This regime was empirically verified in acquired signals by measuring parallel and antiparallel DDE experiments and showing that they produce identical signal decays (55,57,59,97) (supporting material, section B). Lastly, exchange effects can only decrease the kurtosis values (19,42,98,99), and thus cannot explain the non-zero kurtosis sources. In this study, simple models were used to investigate the origin of CTI-driven kurtosis sources, and to illustrate the impact of finite µK on previous DTV approaches. These simple models are, however, insufficient to accurately represent the complexity of biological tissues. Future studies should expand such in-silico experiments toward more complex simulations allowing the assessment of the relationships between different kurtosis sources and concrete (sub)cellular features (e.g., cellular cross-sectional variance, cellular packing, exchange, etc.) (81,98,100 – 𝜇𝐾 effects. This finding is also in line with Lundell et al. (2019), who showed that continuous diffusion gradient waveforms experiments probing identical b-tensors but different power spectra can provide different information about diffusion time-dependence and microscopic kurtosis (67). One could argue that biases of the DTV analysis could be reduced by adjusting the diffusion parameters of our acquisition protocol, or by entirely excluding some acquisitions. In section C of supporting material, we report DTV kurtosis estimates obtained by fitting Eq. 9 to only DDE sets 𝐾 𝑎𝑛𝑖𝑠𝑜 and CTI 𝐾 𝑎𝑛𝑖𝑠𝑜 are identical. However, the DTV 𝐾 𝑖𝑠𝑜 estimates from this modified protocol are still a combination of isotropic and microscopic kurtosis effects. In a future study, we expect to expand the comparison between CTI and DTV estimates to define the regimes in which multidimensional DTV estimates are accurate. This should also include the DTV estimates from faster continuous waveform acquisitions (63–69) . Here, we managed to accelerate the CTI scan times to about 40 mins. Although this is still not (yet) sufficiently rapid for clinical translation, we note taht the objective was only to identify the minimal acquisition set requirements for the extraction of all CTI quantities. Indeed, we acquired a large number of directions (135 per experiment set) to enhance the precision of our kurtosis estimates. In future studies, further acceleration could be obtained by reducing the number of directions acquired for powder-averaging (94,104). We note that microscopic kurtosis can also be estimated directly from only 2 sets of this minimal protocol – albeit at the expense of not resolving 𝐾 𝑎𝑛𝑖𝑠𝑜 and 𝐾 𝑖𝑠𝑜 – as was done for the raw 𝜇𝐾 sensitivity analysis in Fig. 5B. Future studies aiming to measure 𝜇𝐾 should also consider the desired estimation precision/accuracy when designing their experiments – some considerations on the relationship between 𝜇𝐾 precision and acquisition parameters are described in section D of the supporting material.
6. Conclusion
We developed and applied an improved Correlation Tensor Imaging approach for accurately resolving kurtosis sources, including the commonly neglected 𝜇𝐾 source. Significant 𝜇𝐾 is robustly found in in vivo rat brains, both in grey matter and in white matter; in fact, µK is the dominant kurtosis source in grey matter. Ignoring µK leads to significant bias in DTV approaches. Our findings suggest promising new biomarkers in health and disease, and also underscore the importance of accounting for µK in multidimensional diffusion encoding approaches. Acknowledgments
This study was funded by the European Research Council (ERC) (agreement No. 679058). The authors acknowledge the vivarium of the Champalimaud Centre for the Unknown, a facility of CONGENTO which is a research infrastructure co-financed by Lisboa Regional Operational Programme (Lisboa 2020), under the PORTUGAL 2020 Partnership Agreement through the European Regional Development Fund (ERDF) and Fundação para a Ciência e Tecnologia (Portugal), project LISBOA-01-0145-FEDER-022170. Supporting Material
Section A
In our previous work (Henriques et al., 2020), CTI estimates were produced for six pairs of 𝑏 and 𝑏 combinations, repeated for 2 sets of parallel/perpendicular directions. These correspond to the 12 sets of experiments summarized in Table A1. Table A1 – Summary of the complete DDE parameter combination used for the CTI estimates in Henriques et al. (2020). The 12 parallel directions of the 5-design and the 60 perpendicular directions of Jespersen’s scheme are reported by (Jespersen et al., 2
Complete old CTI protocol sets b b b t θ b Δ direction scheme b a b a a o
1 Parallel directions from 5-design and 8-design (12+45 directions) b a b a a o -1/2 Perpendicular directions of Jespersen’s DDE perpendicular scheme (60 directions) b a
0 b a o
1 Directions from 5-design and 8-design (12+45 directions) b a
0 b a o
1 1 st encoding directions of Jespersen’s perpendicular directions scheme (60 directions)
0 b a b a o
1 Directions from 5-design and 8-design (12+45 directions)
0 b a b a o
1 2 nd encoding directions of Jespersen’s perpendicular directions scheme (60 directions) b b b b b o
1 Parallel directions from 5-design and 8-design (12+45 directions) b b b b b o -1/2 Perpendicular directions of Jespersen’s DDE perpendicular scheme (60 directions) b b
0 b b o
1 Directions from 5-design and 8-design (12+45 directions) b b
0 b b o
1 1 st encoding directions of Jespersen’s perpendicular directions scheme (60 directions)
0 b b b b o
1 Directions from 5-design and 8-design (12+45 directions)
0 b b b b o
1 2 nd encoding directions of Jespersen’s perpendicular directions scheme (60 directions) It is important to note that in Henriques et al. (2020), experiment sets 𝒏 inverted directions to evaluate the long mixing time regime approximations. These inverted directions acquisitions are excluded in table A1 for the sake of simplicity. One can note that the protocol reported in Table A1 contains experiments for redundant powder-averaged signals. Particularly, experiment sets 𝐸̅ 𝐷𝐷𝐸 (𝑏 a , 0, 0 o ) = 𝐸̅ 𝐷𝐷𝐸 (0, 𝑏 a , 0 o ) and 𝐸̅ 𝐷𝐷𝐸 (𝑏 b , 0, 0 o ) = 𝐸̅ 𝐷𝐷𝐸 (0, 𝑏 b , 0 o ) respectively. For the reference old CTI protocol of our current study, these redundant experiment sets are removed, and the direction schemes of each 𝑏 , 𝑏 and 𝜃 are adapted to ensure homoscedasticity (Table A2). Table A2 – Summary of the reference non-optimized CTI protocol used on the current study.
Complete non-optimized CTI protocol sets b b b t θ b Δ direction scheme b a
0 b a o
1 45 directions of the 8-design (x3 repetitions) b a b a a o
1 45 directions of the 8-design (x3 repetitions) for both diffusion encodings b a b a a o -1/2 45 directions of the 8-design for the 1 st encoding, repeated for 3 orthogonal directions for the 2 nd encoding b b
0 1 0 o
1 45 directions of the 8-design (x3 repetitions) b b b b a o
1 45 directions of the 8-design (x3 repetitions) for both diffusion encodings b b b b b o -1/2 45 directions of the 8-design for the 1 st encoding, repeated for 3 orthogonal directions for the 2 nd encoding In contract to the improved protocol of the current study, one can note that the old protocol presents set of experiments with unbalanced total b-values, i.e. symmetric DDE experiments (i.e. sets 𝑏 a and 𝑏 b in Table A2 were set to and for the old protocol, which leaded to a high total b-value of for experiment sets 𝜇𝐾 observed on our previous study is a consequence of the unbalanced b-values used. To support this claim, bellow we show the simulation results for the old protocol for the models 2 and 5 applied with lower total b-values (i.e. 𝑏 𝑎 = 1.25𝑚𝑠/𝜇𝑚 and 𝑏 𝑏 = 0.5𝑚𝑠/𝜇𝑚 ). This figure shows that the unbalanced protocol can still overestimate 𝜇𝐾 even when the total b-value is decreased to half. Figure A1 – Kurtosis estimates of non-optimized CTI protocol with parameters 𝒃 𝒂 = 𝟏. 𝟐𝟓𝒎𝒔/𝝁𝒎 and 𝒃 𝒃 = 𝟎. 𝟓𝒎𝒔/𝝁𝒎 for two model simulations ; A) Evenly oriented anisotropic Gaussian diffusion components; B) Sum of different compartment types (isotropic Gaussian + anisotropic Gaussian + isotropic non-Gaussian components). From left to right, each panel shows: a schematic representation of the models (A1, B1); the kurtosis ground truth values (A2, B2); and the kurtosis estimates obtained from the CTI using its “old” protocol (A3, B3). Section B
To empirically check the long mixing time regime, the following two extra set of DDE experiments were performed for Rat • (set 𝑏 = 𝑏 = 1𝑚𝑠/𝜇𝑚 and parallel DDE directions ( 𝑆 =𝐸̅ 𝐷𝐷𝐸 (𝑏 , 𝑏 , 0 o ) ). For powder-averaging, 45 directions of the 8-design (repeated 3 times) were acquired. • (set 𝑏 = 𝑏 = 1𝑚𝑠/𝜇𝑚 and antiparallel DDE directions ( 𝑆 =𝐸̅ 𝐷𝐷𝐸 (𝑏 , 𝑏 , 180 o ) ). For powder-averaging 45 directions of the 8-design (repeated 3 times) were acquired. Note that these directions are inverted for the 2 nd encoding. Other acquisition parameters were equal to the main experiments of these study. Each experiment set was acquired together with 24 𝑏 t = 0 acquisitions for signal decay normalization. Experiment set 𝑆 and 𝑆 ). Figure B1 shows the powder-averaged signal decays for the two repetitions of experiment set Figure B1 – Powder-averaged data for the extra experiments performed on Rat
Powder-averaged signal decays for DDE parallel acquisitions (repetition B) Powder-averaged signal decays for DDE parallel acquisitions (repetition C) Powder-averaged signal decays for DDE antiparallel acquisitions. D) Histograms of the log differences between powder-averaged signals of experiment set 𝑏 𝑐 was set to (i.e total b-value = ). Section C
Here we report the results of kurtosis estimates from the diffusion tensor variance (DTV) framework obtained by fitting Eq. 9 to only DDE experiment sets 𝐾 𝑎𝑛𝑖𝑠𝑜 ( 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 ) and CTI 𝐾 𝑎𝑛𝑖𝑠𝑜 are identical (Fig. C1), while DTV 𝐾 𝑖𝑠𝑜 ( 𝐾 𝑖𝑠𝑜𝐷𝑇𝑉 ) are still a combination of isotropic and microscopic kurtosis effects. Note that these observations can be theoretically derived from Eqs. 6 and 9 as shown below. From Eq. 9, one can note that the 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 is resolved by: log 𝐸̅ 𝐷𝑇𝑉 (𝑏, 1) − log 𝐸̅
𝐷𝑇𝑉 (𝑏, − ) = 𝑏 𝐷 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 , (C1) Replacing 𝐸̅ 𝑇𝑒𝑛 (𝑏, 1) by 𝐸̅ 𝐷𝐷𝐸 ( 𝑏2 , 𝑏2 , 0 o ) (experiment set 𝐸̅ 𝐷𝑇𝑉 (𝑏, 1) by 𝐸̅ 𝐷𝐷𝐸 ( 𝑏2 , 𝑏2 , 90 o ) (experiment set 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 = 𝐾 𝑎𝑛𝑖𝑠𝑜𝐶𝑇𝐼 (note that this would not be the case if 𝐸̅ 𝐷𝑇𝑉 (𝑏, 1) was replaced by 𝐸̅ 𝐷𝐷𝐸 (𝑏, 0, 0 o ) ( experiment set 𝐾 𝑖𝑠𝑜𝐷𝑇𝑉 can be extracted by subtracting 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 from the total apparent kurtosis 𝐾 𝑡𝑎𝑝𝑝 of the b-value dependency of 𝐸̅ 𝐷𝑇𝑉 (𝑏, 1) . Considering that 𝐾 𝑡𝑎𝑝𝑝 from the multiple b-value information of experiment set 𝐾 𝑡𝑎𝑝𝑝 = 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 + 𝐾 𝑖𝑠𝑜𝑐𝑡𝑖 + 𝜇𝐾2 (c.f. Eq. 6), one can derived that 𝐾 𝑖𝑠𝑜𝐷𝑇𝑉 = 𝐾 𝑖𝑠𝑜𝑐𝑡𝑖 + 𝜇𝐾2 . Figure C1 – Diffusion Tensor Variance (DTV) kurtosis estimates using only data from experiment sets
DTV total kurtosis ( 𝐾 𝑡𝐷𝑇𝑉 ) maps. B) DTV anisotropic kurtosis 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 maps. C) DTV isotropic kurtosis 𝐾 𝑖𝑠𝑜𝐷𝑇𝑉 maps. D) Scatter plots between DTV anisotropic kurtosis ( 𝐾 𝑎𝑛𝑖𝑠𝑜𝐷𝑇𝑉 ) and CTI anisotropic kurtosis ( 𝐾 𝑎𝑛𝑖𝑠𝑜𝐶𝑇𝐼 ) estimates; E) Scatter plots between DTV isotropic kurtosis ( 𝐾 𝑖𝑠𝑜𝐷𝑇𝑉 ) and CTI isotropic kurtosis ( 𝐾 𝑖𝑠𝑜𝐶𝑇𝐼 ) estimates. Section D
Considering the propagation of uncertainty, the expected error of 𝜇𝐾 (𝜎 𝜇𝐾 ) can be calculated from Eq. 6 (similar calculations for microscopic diffusion anisotropy were done by Kerkelä et al. MRM 2019), which gives: 𝜎 𝜇𝐾 = 𝑎2 𝐷 √ 𝜎 𝑆 𝑁 + 𝜎 𝑆 𝑁 (D1) where 𝑆 = 𝐸̅ 𝐷𝐷𝐸 (𝑏 𝑎 , 0, 0 o ) , 𝑆 = 𝐸̅ 𝐷𝐷𝐸 ( 𝑏 𝑎 , 𝑏 𝑎 , 0 o ) , 𝜎 is the signal noise standard deviation, and N is the number of pairs of directions used to compute the powder-averages (here we assumed that the error of 𝐷 is negligible). From equation D1 one can note that 𝜎 𝜇𝐾 does not only depend on acquisition parameters (b-value and number of pairs of gradient directions) but on the expected diffusivity of the studied system. Particularly, higher b-values and number of pairs of gradient directions should be required to measure 𝜇𝐾 for systems with lower diffusivities. References
1. Jones DK. Diffusion MRI : theory, methods, and applications. (Jones D, editor.) Oxford
University Press; 2010. 2. Le Bihan D, Johansen-Berg H. Diffusion MRI at 25: exploring brain tissue structure and function. Neuroimage 2012;61:324 –
41 doi: 10.1016/j.neuroimage.2011.11.006. 3. Kiselev VG. Microstructure with diffusion MRI: what scale we are sensitive to? J. Neurosci. Methods 2021;347:108910 doi: 10.1016/j.jneumeth.2020.108910. 4. Stejskal EO, Tanner JE. Spin Diffusion Measurements: Spin Echoes in the Presence of a
Time‐Dependent Field Gradient. American Institute of Physics; 1965 pp. 288– –
87 doi: 10.1002/mrm.25901. 6. Moseley ME, Kucharczyk J, Mintorovitch J, et al. Diffusion-weighted MR imaging of acute stroke: correlation with T2-weighted and magnetic susceptibility-enhanced MR imaging in cats. AJNR. Am. J. Neuroradiol. 1990;11:423 –
9. 7. Moseley ME, Cohen Y, Kucharczyk J, et al. Diffusion-weighted MR imaging of anisotropic water diffusion in cat central nervous system. Radiology 1990;176:439 –
445 doi: 10.1148/radiology.176.2.2367658. 8. Le Bihan D, Breton E, Lallemand D, Grenier P, Cabanis E, Laval-Jeantet M. MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. Radiology 1986;161:401 –
407 doi: 10.1148/radiology.161.2.3763909. 9. Callaghan PT, Eccles CD, Xia Y. NMR microscopy of dynamic displacements: k-space and q-space imaging. J. Phys. E. 1988;21:820 –
822 doi: 10.1088/0022-3735/21/8/017.
10. Tuch DS, Reese TG, Wiegell MR, Van J. Wedeen. Diffusion MRI of Complex Neural Architecture. Neuron 2003;40:885 –
895 doi: 10.1016/S0896-6273(03)00758-X. 11. Wedeen VJ, Hagmann P, Tseng W-YI, Reese TG, Weisskoff RM. Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. Magn. Reson. Med. 2005;54:1377 – –
32 doi: 10.1016/j.neuroimage.2013.04.016. 13. Basser PJ, Mattiello J, LeBihan D. MR diffusion tensor spectroscopy and imaging. Biophys. J. 1994;66:259 –
267 doi: 10.1016/S0006-3495(94)80775-1. 14. Pierpaoli C, Basser PJ. Toward a quantitative assessment of diffusion anisotropy. Magn. Reson. Med. 1996;36:893 –
906 doi: 10.1002/mrm.1910360612. 15. Basser PJ, Pierpaoli C. Microstructural and Physiological Features of Tissues Elucidated by Quantitative-Diffusion-Tensor MRI. J. Magn. Reson. Ser. B 1996;111:209 –
219 doi: 10.1006/jmrb.1996.0086.
16. Stepišnik J. Time -dependent self-diffusion by NMR spin-echo. Phys. B Phys. Condens. Matter 1993;183:343 –
350 doi: 10.1016/0921-4526(93)90124-O. 17. Gore JC, Xu J, Colvin DC, Yankeelov TE, Parsons EC, Does MD. Characterization of tissue structure at varying length scales using temporal diffusion spectroscopy. NMR Biomed. 2010;23:745 –
756 doi: 10.1002/nbm.1531. 18. Fieremans E, Burcaw LM, Lee H-H, Lemberskiy G, Veraart J, Novikov DS. In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter. doi: 10.1016/j.neuroimage.2016.01.018. 19. Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K. Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging. Magn. Reson. Imaging 2005;53 doi: 10.1002/mrm.20508. 20. Assaf Y, Basser PJ. Composite hindered and restricted model of diffusion (CHARMED) MR imaging of the human brain. Neuroimage 2005;27:48 –
58 doi: 10.1016/j.neuroimage.2005.03.042. 21. Jespersen SN, Kroenke CD, Østergaard L, Ackerman JJH, Yablonskiy DA. Modeling dendrite density from magnetic resonance diffusion measurements. Neuroimage 2007;34:1473 – – – – – – – –
373 doi: 10.1016/j.neuroimage.2014.04.013. 31. Hui ES, Fieremans E, Jensen JH, et al. Stroke Assessment With Diffusional Kurtosis Imaging. Stroke 2012;43:2968 – –
881 doi: 10.1148/radiol.09090819. 33. Van Cauter S, Veraart J, Sijbers J, et al. Gliomas: Diffusion kurtosis MR imaging in grading. Radiology 2012;263:492 –
501 doi: 10.1148/radiol.12110927. 34. Abdalla G, Dixon L, Sanverdi E, et al. The diagnostic role of diffusional kurtosis imaging in glioma grading and differentiation of gliomas from other intra-axial brain tumours: a systematic review with critical appraisal and meta-analysis. Neuroradiology 2020;62:791 –
802 doi: 10.1007/s00234-020-02425-9. 35. Grossman EJ, Ge Y, Jensen JH, et al. Thalamus and cognitive impairment in mild traumatic brain injury: A diffusional kurtosis imaging study. J. Neurotrauma 2012;29:2318 – –
477 doi: 10.1016/j.neuroimage.2011.07.050.
37. Wang J-J, Lin W-Y, Lu C-S, et al. Parkinson Disease: Diagnostic Utility of Diffusion Kurtosis Imaging. Radiology 2011;261:210 –
217 doi: 10.1148/radiol.11102277. 38. Fieremans E, Benitez A, Jensen JH, et al. Novel White Matter Tract Integrity Metrics Sensitive to Alzheimer Disease Progression. Am. J. Neuroradiol. 2013;34:2105 – –
532 doi: 10.1016/J.NEUROIMAGE.2016.07.038. 40. Henriques RN, Jespersen SN, Shemesh N. Correlation tensor magnetic resonance imaging. Neuroimage 2020;211:116605 doi: 10.1016/J.NEUROIMAGE.2020.116605. 41. Paulsen JL, Özarslan E, Komlosh ME, Basser PJ, Song Y-Q. Detecting compartmental non-Gaussian diffusion with symmetrized double-PFG MRI. NMR Biomed. 2015;28:1550 – –
710 doi: 10.1002/nbm.1518. 43. Fieremans E, Jensen JH, Helpern JA. White matter characterization with diffusional kurtosis imaging. Neuroimage 2011;58:177 –
188 doi: 10.1016/j.neuroimage.2011.06.006. 44. Jespersen SN. White matter biomarkers from diffusion MRI. J. Magn. Reson. 2018;291:127 –
140 doi: 10.1016/j.jmr.2018.03.001. 45. Novikov DS, Veraart J, Jelescu IO, Fieremans E. Rotationally-invariant mapping of scalar and orientational metrics of neuronal microstructure with diffusion MRI. Neuroimage 2018;174:518 –
538 doi: 10.1016/j.neuroimage.2018.03.006. 46. Lampinen B, Szczepankiewicz F, Mårtensson J, van Westen D, Sundgren PC, Nilsson M. Neurite density imaging versus imaging of microscopic anisotropy in diffusion MRI: A model comparison using spherical tensor encoding. Neuroimage 2017;147:517 –
531 doi: 10.1016/j.neuroimage.2016.11.053. 47. Lampinen B, Szczepankiewicz F, Novén M, et al. Searching for the neurite density with diffusion MRI: Challenges for biophysical modeling. Hum. Brain Mapp. 2019;40:2529 – – –
406 doi: 10.1016/j.neuroimage.2017.10.051. 50. Jelescu IO, Palombo M, Bagnato F, Schilling KG. Challenges for biophysical modeling of microstructure. J. Neurosci. Methods 2020;344:108861 doi: 10.1016/j.jneumeth.2020.108861. 51. Topgaard D. Multidimensional diffusion MRI. J. Magn. Reson. 2017;275:98 –
113 doi: 10.1016/J.JMR.2016.12.007. 52. Szczepankiewicz F, Westin CF, Nilsson M. Gradient waveform design for tensor-valued encoding in diffusion MRI. J. Neurosci. Methods 2020;348:109007 doi: 10.1016/j.jneumeth.2020.109007. 53. Henriqu es RN, Palombo M, Jespersen SN, Shemesh N, Lundell H, Ianuş A. Double diffusion encoding and applications for biomedical imaging. J. Neurosci. Methods 2020 doi: 10.1016/j.jneumeth.2020.108989. 54. Cory DG, Garroway AN, Miller JB. Applications of spin transport as a probe of local geometry. Polym Prepr 1990;31:149. 55. Mitra PP. Multiple wave-vector extensions of the NMR pulsed-field-gradient spin-echo diffusion measurement. Phys. Rev. B 1995;51:15074 – –
143 doi: 10.1002/mrm.1910340202. 57. Ozarslan E. Compartment shape anisotropy (CSA) revealed by double pulsed field gradient MR. J. Magn. Reson. 2009;199:56 –
67 doi: 10.1016/j.jmr.2009.04.002. 58. Shemesh N, Özarslan E, Bar-Shir A, Basser PJ, Cohen Y. Observation of restricted diffusion in the presence of a free diffusion compartment: Single- and double-PFG experiments. J. Magn. Reson. 2009;200:214 –
225 doi: 10.1016/j.jmr.2009.07.005. 59. Shemesh N, Özarslan E, Basser PJ, Cohen Y. Measuring small compartmental dimensions with low-q angular double-PGSE NMR: The effect of experimental parameters on signal decay. J. Magn. Reson. 2009;198:15 –
23 doi: 10.1016/j.jmr.2009.01.004. 60. Jespersen SN, Buhl N. The displacement correlation tensor: Microstructure, ensemble anisotropy and curving fibers. J. Magn. Reson. 2011;208:34 –
43 doi: 10.1016/j.jmr.2010.10.003. 61. Lawrenz M, Koch MA, Finsterbusch J. A tensor model and measures of microscopic anisotropy for double-wave-vector diffusion-weighting experiments with long mixing times. J. Magn. Reson. 2010;202:43 –
56 doi: 10.1016/j.jmr.2009.09.015. 62. Hui ES, Jensen JH. Double-pulsed diffusional kurtosis imaging for the in vivo assessment of human brain microstructure. Neuroimage 2015;120:371 –
381 doi: doi: 10.1016/j.neuroimage.2015.07.013. 63. Topgaard D. Isotropic diffusion weighting in PGSE NMR: Numerical optimization of the q-MAS PGSE sequence. Microporous Mesoporous Mater. 2013;178:60 –
63 doi: 10.1016/j.micromeso.2013.03.009. 64. Eriksson S, Lasic S, Topgaard D. Isotropic diffusion weighting in PGSE NMR by magic-angle spinning of the q-vector. J. Magn. Reson. 2013;226:13 –
18 doi:
65. Lasič S, Szczepankiewicz F, Eriksson S, Nilsson M, Topgaard D. Microanisotropy imaging: quantification of microscopic diffusion anisotropy and orientational order parameter by diffusion MRI with magic-angle spinning of the q-vector. Front. Phys. 2014;2:11 doi: 10.3389/fphy.2014.00011. 66. Sjölund J, Szczepankiewicz F, Nilsson M, Topgaard D, Westin C-F, Knutsson H. Constrained optimization of gradient waveforms for generalized diffusion encoding. J. Magn. Reson. 2015;261 doi: 10.1016/j.jmr.2015.10.012. 67. Lundell H, Nilsson M, Dyrby TB, et al. Multidimensional diffusion MRI with spectrally modulated gradients reveals unprecedented microstructural detail. Sci. Rep. 2019;9:1 –
12 doi: 10.1038/s41598-019-45235-7. 68. Szczepankiewicz F, Sjö Lund J, Ståhlberg F, Lä Tt J, Nilsson M. Tensor-valued diffusion encoding for diffusional variance decomposition (DIVIDE): Technical feasibility in clinical MRI systems. PLoS One 2019;14:e0214238 doi: 10.1371/journal.pone.0214238.
69. Lasič S, Szczepankiewicz F, Dall’Armellina E, et al. Motion -compensated b-tensor encoding for in vivo cardiac diffusion-weighted imaging. NMR Biomed. 2020;33 doi: 10.1002/nbm.4213. 70. Westin CF, Szczepankiewicz F, Pasternak O, et al. Measurement tensors in diffusion MRI: Generalizing the concept of diffusion encoding. In: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 8675 LNCS. Springer Verlag; 2014. pp. 209 –
71. Szczepankiewicz F, Lasič S, van Westen D, et al. Quantification of microscopic diffusion anisotropy disentangles effects of orientation dispersion from microstructure: applications in healthy volunteers and in brain tumors. Neuroimage 2015;104:241 –
52 doi: –
73. Jespersen SN, Olesen JL, Ianuş A, Shemesh N. Effects of nongaussian diffusio n on “isotropic diffusion” measurements: An ex -vivo microimaging and simulation study. J. Magn. Reson. 2019;300:84 –
94 doi: 10.1016/J.JMR.2019.01.007. 74. Szczepankiewicz Filip, Lasic Samo, Nilsson Markus, Lundell Henrik, Westin Carl-Fredrik, Topgaard Daniel. Is spherical diffusion encoding rotation invariant? An investigation of diffusion time-dependence in the healthy brain. In: ISMRM 27th Annual Meeting & Exhibition. ; 2019. 75. Ji Y, Paulsen J, Zhou IY, et al. In vivo microscopic diffusional kurtosis imaging with symmetrized double diffusion encoding EPI. Magn. Reson. Med. 2019;81:533 –
541 doi: 10.1002/mrm.27419. 76. Jespersen SN, Lundell H, Sønderby CK, Dyrby TB. Orientationally invariant metrics of apparent compartment eccentricity from double pulsed field gradient diffusion experiments. NMR Biomed. 2013;26:1647 –
62 doi: 10.1002/nbm.2999. 77. Callaghan PT, Coy A, MacGowan D, Packer KJ, Zelaya FO. Diffraction-like effects in NMR diffusion studies of fluids in porous solids. Nature 1991;351:467 –
469 doi: 10.1038/351467a0. 78. Callaghan PT. Pulsed-Gradient Spin-Echo NMR for Planar, Cylindrical, and Spherical Pores under Conditions of Wall Relaxation. J. Magn. Reson. Ser. A 1995;113:53 –
59 doi: 10.1006/JMRA.1995.1055. 79. Novikov DS, Kiselev VG. Effective medium theory of a diffusion-weighted signal. NMR Biomed. 2010;23:682 –
697 doi: 10.1002/nbm.1584.
80. Burcaw LM, Fieremans E, Novikov DS. Mesoscopic structure of neuronal tracts from time-dependent diffusion. Neuroimage 2015;114:18 –
37 doi: 10.1016/j.neuroimage.2015.03.061. 81. Lee HH, Papaioannou A, Kim SL, Novikov DS, Fieremans E. A time-dependent diffusion MRI signature of axon caliber variations and beading. Commun. Biol. 2020;3:1 –
13 doi: 10.1038/s42003-020-1050-x. 82. Lee HH, Papaioannou A, Novikov DS, Fieremans E. In vivo observation and biophysical interpretation of time-dependent diffusion in human cortical gray matter. Neuroimage 2020;222:117054 doi: 10.1016/j.neuroimage.2020.117054. 83. Jespersen SN. Equivalence of double and single wave vector diffusion contrast at low diffusion weighting. NMR Biomed. 2012;25:813 –
818 doi: 10.1002/nbm.1808. 84. Cheng Y, Cory DG. Multiple Scattering by NMR. J. Am. Chem. Soc. 1999;121:7935 –
85. Ianuş A, Jespersen SN, Serradas Duarte T,
Alexander DC, Drobnjak I, Shemesh N. Accurate estimation of microscopic diffusion anisotropy and its time dependence in the mouse brain. Neuroimage 2018;183:934 –
949 doi: 10.1016/j.neuroimage.2018.08.034. 86. Drobnjak I, Zhang H, Hall MG, Alexander DC. The matrix formalism for generalised gradients with time-varying orientation in diffusion NMR. J. Magn. Reson. 2011;210:151 –
157 doi: 10.1016/j.jmr.2011.02.022.
87. Ianuş A, Alexander DC, Drobnjak I. Microstructure imaging sequence simulation toolbox. In: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9968 LNCS. Springer Verlag; 2016. pp. 34 –
44. doi: 10.1007/978-3-319-46630-9_4.
88. Hardin RH, Sloane NJAA. McLaren’s impro ved snub cube and other new spherical designs in three dimensions. Discret. Comput. Geom. 1996;15:429 –
441 doi: –
406 doi: 10.1016/j.neuroimage.2016.08.016. 90. Kellner E, Dhital B, Kiselev VG, Reisert M. Gibbs-ringing artifact removal based on local subvoxel-shifts. Magn. Reson. Med. 2016;76:1574 – –
8. 92. Kamiya K, Kamagata K, Ogaki K, et al. Brain White-Matter Degeneration Due to Aging and Parkinson Disease as Revealed by Double Diffusion Encoding. Front. Neurosci. 2020;14:1091 doi: 10.3389/fnins.2020.584510.
93. Andersen KW, Lasič S, Lundell H, et al. Disentangling white -matter damage from physiological fibre orientation dispersion in multiple sclerosis. Brain Commun. 2020;2 doi: 10.1093/braincomms/fcaa077. 94. Yang G, Tian Q, Leuze C, Wintermark M, McNab JA. Double diffusion encoding MRI for the clinic. Magn. Reson. Med. 2018;80:507 –
520 doi: 10.1002/mrm.27043. 95. Nery F, Szczepankiewicz F, Kerkelä L, et al. In vivo demonstration of microscopic anisotropy in the human kidney using multidimensional diffusion MRI. Magn. Reson. Med. 2019;82:2160 – –
182 doi: 10.1016/j.jmr.2009.02.003. 98. Fieremans E, Novikov DS, Jensen JH, Helpern JA. Monte Carlo study of a two- compartment exchange model of diffusion. NMR Biomed. 2010;23:711 –
24 doi: 10.1002/nbm.1577.
99. Ning L, Nilsson M, Lasič S, Westin C -F, Rathi Y. Cumulant expansions for measuring water exchange using diffusion MRI. J. Chem. Phys. 2018;148:074109 doi: 10.1063/1.5014044. 100. Skinner NP, Kurpad SN, Schmit BD, Budde MD. Detection of acute nervous system injury with advanced diffusion-weighted MRI: A simulation and sensitivity analysis. NMR Biomed. 2015;28:1489 – –
61 doi: 10.1016/j.neuroimage.2018.06.046. 102. Palombo M, Alexander DC, Zhang H. A generative model of realistic brain cells with application to numerical simulation of the diffusion-weighted MR signal. Neuroimage 2019;188:391 ––