Diffusion magnetic resonance imaging (dMRI) sensitizes the signal to the random motion of the water molecules in the tissue [1]. By probing the water motion in the tissue, one can infer information about the underlying microstructure [2–6]. Some basic features of the tissue, such as fiber orientation or anisotropy can be captured using the diffusion weighted signal. In tissue that is highly ordered on the micron-scale, water molecules experience fewer boundaries along one direction and travel further per unit time than along other directions [7,8]. Altered microstructure is the hallmark of many diseases, which manifests itself in altered diffusion properties. [9] showed the reduction in the apparent diffusivity by increase in cell density in tumors. The first clinical application of diffusion MRI was on detection of early stage cerebral ischemia [10,11], which at that time could not be depicted with computed tomography (CT) or other MRI contrasts. Since then, diffusion MRI has been used in diagnosis of other diseases, such as epilepsy, stroke, tumors in central nervous system, breast and prostate, as well as surgical planning [12–22]. Diffusion MRI has also been invaluable in the study of brain development [23], learning [24,25], and connectivity [26,27]. More recently, diffusion MRI of the heart has regained some significant interest, enabled by advances in MR scanner hardware and experimental design [28–31]. Diffusion MRI has been also used in the imaging of other organs with skeletal muscle such as breast [32,33], kidney [34,35], and prostate [36,37]. In this review, we briefly explain different diffusion encoding schemes and the advantages of using advanced diffusion encoding in brain and body imaging are discussed.

In this section, we briefly explain single, double, and triple diffusion encoding (SDE, DDE, and TDE), as well as free gradient waveforms and b-tensor encoding with the special cases of linear, planar, and spherical tensor encoding (LTE, PTE, and STE).

Most diffusion MRI studies in the literature are based on conventional Stejskal–Tanner acquisitions [38], which has one pair of pulsed field gradients that encode diffusion along a single axis. In the nomenclature proposed by [39] this is referred to as Single Diffusion Encoding (SDE). A drawback of this technique is that the effect of microscopic anisotropy, orientation dispersion, and isotropic variance are entangled. This means different combinations of these factors lead to the same signal attenuation with SDE-so one may need to change the signal attenuation properties to separate them [40–47].

Double Diffusion Encoding (DDE) which contains two pairs of pulsed-field gradients that are separated from each other with a mixing time τ [39,48] has been used to disentangle the effect of microscopic anisotropy from orientation dispersion [46,47,49–55]. The encoding direction of each pair can be controlled independently and therefore facilitates measuring the diffusivity along two directions using a single preparation of the signal. The principles of DDE-based approaches have been described in several studies [45,56–59].

Varying the relative gradient directions of the two SDE blocks, one can estimate microscopic diffusion anisotropy [44,46,52,60,61] whereas varying the gradients’ strengths while keeping them orthogonal to each other reveals compartmental kurtosis [62,63]. To estimate exchange, e.g., through the membrane between extra-cellular and intra-cellular spaces, parallel gradients with variable mixing time can be used [64–69]. Another application of DDE is the estimation of compartment size using parallel and antiparallel gradients with a short mixing time [61,70].

Triple Diffusion Encoding (TDE) allows for disentangling microscopic anisotropy from isotropic diffusion, which is not feasible using SDE alone and also the advantage of TDE over DDE is that the isotropic diffusivity can be obtained from TDE using a single measurement [40,71–74].

Isotropic diffusion encoding was introduced by [75] and [71] for fast measurement of mean diffusivity. [76] used the combination of SDE and spherical/isotropic diffusion encoding to probe microscopic anisotropy, while [77] developed the method to quantify it (See Figure 1 as an example). The difference between the signals from SDE and isotropic diffusion encoding is related to the microscopic anisotropy. The non-monoexponential decay of the diffusion weighted signal as a function of b-value from isotropic diffusion encoding can show the presence of multiple compartments within a voxel [42,43,77,78].

Although SDE, DDE, and TDE are the most common gradient waveforms there is no reason to limit the shape of the gradient to a rectangular/trapezoidal waveform. Free gradient waveforms may be more useful than the trapezoidal ones, as explained below [79–81].

[82] proposed a general framework to describe diffusion encoding for arbitrary gradient waveforms. In this framework, the b-value and encoding direction were replaced by the “b-tensor”, which includes the shape of the diffusion encoding [74,82–85]. In this framework, SDE is just a special realization of linear tensor encoding (LTE) where the b-tensor has only one non-zero eigenvalue as all gradients are in the same orientations. DDE can yield encoding with up to two non-zero eigenvalues and can be designed to be Planar Tensor Encoding (PTE), some asymmetric rank-2 b-tensor or LTE. In spherical tensor encoding (STE) the gradients point in all directions at some time giving rise to a rank-3 b-tensor.

Optimization of gradient waveforms in terms of echo time (providing the maximum b-value in a given echo time) has allowed for b-tensor encoding to be used across many clinical systems [86,87]. It has been used to study the tissue microstructure in the healthy brain [43,88–91], brain tumors [78,92], multiple sclerosis [93,94], epilepsy [95], as well as other organs such as breast [33], heart [96], kidney [35], and prostate [36,37]. It has shown the improvement of parameter estimates in biophysical models [91,97–100] and fiber dispersion quantification [101]. The extra dimensionality provided by b-tensor encoding helps to improve model fitting in situations where the analysis based on LTE alone has resulted in ambiguities in model parameters.

Each imaging voxel contains an ensemble of microenvironments (over a million cells for brain tissue). The diffusion within each microenvironment can be modeled by a microscopic diffusion tensor (assuming R1.1 multiple Gaussian components (MGC), i.e. no time dependence and microscopic kurtosis, μK = 0 [62,102–104]) and therefore the whole voxel has a distribution of diffusion tensors [46,77,83,105,106]. Single diffusion tensor [3] from a voxel is equivalent to the average of the microscopic tensors. Although the voxel level diffusion tensor has a lot of applications [107], it does not provide information about the underlying distribution of microscopic diffusion tensors. To obtain such information, the distribution of the microscopic diffusion tensors can be parametrized in terms of mean diffusivity (MD) and two components of diffusional variance; anisotropic and isotropic variance [43,77]. Isotropic and anisotropic mean kurtosis (MK_I and MK_A) are proportional to isotropic and anisotropic variances respectively (for more details see [78,106]). Fractional anisotropy (FA) reflects the average anisotropy of the voxel [108] whereas microscopic fractional anisotropy (μFA) is not influenced by the orientational order of the tissue [43,46,49,77]. Apparent diffusion coefficient (ADC) can show the macro heterogeneity (across many voxels) or the local average (in one voxel) [109,110], however, it cannot capture microheterogeneity within a voxel.

Figures 1A–F provides an overview of various diffusion encoding schemes. As shown in this figure, b-tensor encoding allows for the data to be acquired in a shorter echo time compared to DDE and TDE. Figures 1G–I shows the plots of the MR signal versus b-value measured in three synthetic environments using linear, planar, and spherical b-tensors. The three synthetic cases represent three distinct scenarios with different distributions of microenvironments (fast and slow isotropic compartments, randomly oriented anisotropic compartments, and randomly oriented anisotropic compartments with different anisotropies). Fitting a diffusion tensor to the signal from these examples will lead to a spherical diffusion tensor on the macroscopic level for all of them, while the difference between the LTE, PTE, and STE signal shows the difference in the underlying microstructure [74,77]. Figure 1 (j-R1.4n) show examples of brain [92], prostate [37], cardiac [96], breast [33], and kidney [35] images (The images (j-n) are taken with permission from [33,35,92,96,98]).

Tensor-valued diffusion encoding has been used to study the tissue microstructure in healthy brain [43,88,91,97,98,111–116]. Because of fiber crossings and the orientation dispersion, the FA measure extracted from conventional diffusion MRI is not able to show the changes in the microscopic level properly. Therefore microscopic anisotropy can be used to show the changes in the underlying microstructure independent of fiber architecture. In the normal brain, microscopic anisotropy is high in white matter and low in cortex [41,99,113,117,118].

[83] used tensor-valued diffusion encoding and extracted the scalar maps representing the mean and variance of the diffusion tensor distribution, to study the changes in schizophrenia compared to normal brains. An increase in the variance of mean diffusivity (V _MD, the variance in mean diffusivities between local microenvironments) was observed. This cannot be explained by a homogeneous increase in the local mean diffusivity but it shows a higher fraction of free water (water molecules that diffuse freely, only likely to be found in the extracellular space). This indicated the elevated extracellular water content due to the neuro-inflammatory process, which is the porposed primary mechanism to explain the changes in the white matter diffusion in schizophrenia [119]. Reduction in the microscopic anisotropy in schizophrenia patients could indicate axonal degeneration at the microscopic level. The advantage of using tensor-valued diffusion encoding for the study of Schizophrenia is that the changes in the microstructure of the tissue, such as axonal degeneration can be reflected in the microscopic anisotropy while this was not necessarily clear in the macroscopic anisotropy.

[78] used the combination of LTE and STE to investigate the link between diffusional variance and tissue heterogeneity in meningiomas and gliomas. The eccentric cells in meningiomas lead to high structural anisotropy which can be captured by anisotropic mean kurtosis (MK_A) [78]. These structures are not present in gliomas. Normal white matter has high microscopic anisotropy and low tissue heterogeneity, while tumours have low to intermediate microscopic anisotropy and low to high tissue heterogeneity (Meningioma contains microscopically anisotropic tissue [78]). High tissue heterogeneity can be captured by the variation of the diffusivity (MK_I) within the voxel. This can be explained by partial necrosis within the voxel which means in some parts of the voxel there is high cell density and low apparent diffusivity while other parts are necrotic with high diffusivity. [92] extended the exploration to other tumour types and with better waveforms and a shorter acquisition scheme.

One of the main causes of drug-resistant epilepsy is malformations of cortical development (MCD) [120]. It can produce seizures that are mostly treated through surgical resection. [95] used tensor-valued dMRI to obtain information about tissue microstructure on MCD. In MCD, the variation in microscopic anisotropy is consistent with variations in axonal content reported in the previous studies [121–125].

[94] and [93] showed that microscopic fractional anisotropy (μFA) improves the microstructural imaging of cerebral white matter in multiple sclerosis (MS) compared to standard diffusion tensor imaging. MS lesions are areas with demyelination and axonal degeneration. A considerable reduction in μFA was reported by [94] in the MS patients compared to healthy controls. In the presence of crossing fibers, the degeneration in one set of fibers may cause an increase in the FA value [126] while the anisotropy is decreased microscopically. Reduced μFA suggests a change in the volume fraction of the cellular spaces due to demyelination or axonal degeneration. In addition, more supporting cells such as glial cells in the microstructural environment may cause a decrease in μFA [127] (however, if we have glial processes, these will be picked up as microscopic anisotropic domains).

[128] used DDE to investigate white matter degeneration in Parkinson disease (PD). In PD, mean diffusivity (MD) increases, while FA, mean kurtosis (MK), anisotropic mean kurtosis (MK_A) and μFA decrease [128–134]. Some features of the neurodegeneration in PD include neuroinflammation, degeneration of myelin sheath, axonal swelling/beading, and axonal loss [135,136]. The analysis of kurtosis in [128] shows that the reductions of MK in PD are likely from the reduction in microscopic anisotropy. The increase of isotropic mean kurtosis (MK_I) and decrease of MK_A have different time trajectories during PD progression. [137] suggest that the increase in MK_I is related to early neuroinflammation and the decrease of MK_A is associated with the subsequent degeneration, so MK may have a non-monotonical trajectory, increasing in the beginning followed by a decrease. A large free-water fraction reported by [137] can explain the decrease in microscopic anisotropy although this is not the only reason and other factors such as axonal loss and demyelination may have the same effect on μFA.

Diffusion weighted imaging is increasingly used in breast cancer imaging [14,33]. In the presence of pathology, microstructural features of tissue such as cellular density, membrane permeability, shape and orientation may change. These alterations are reflected in the diffusion weighted signal that is obtained from tissue. [33] studied the feasibility of non-invasive microstructural characterization of normal and neoplastic breast tissue using b-tensor encoding. They aimed for potential use of b-tensor encoding in the clinic to disentangle the fibroglandular breast tissue (FGT) from breast cancer. Their findings show that the breast cancer tissue has low isotropic diffusivity and high anisotropy, while normal FGT exhibited a low amount of anisotropy and high isotropic diffusivity which means the normal breast tissue has non-hindered isotropic environment where the water molecules can diffuse fast.The average of isotropic diffusivities in a voxel is equivalent to some conventional imaging biomarkers such as ADC that is useful to disentangle healthy tissue from benign and malignant lesions [138]. Previous studies in breast lesions showed that the tissue cellularity is inversely correlated with MD. [33] showed that isotropic diffusivity in FGT ( ∼ 2 × 1 0 − 3 m m 2 / s ) [139–141] were significantly higher than cancers which is in agreement with previous findings on MD [142–145].[33] reported that the fractional anisotropy and microscopic anisotropy values in FGT were significantly lower than tumors, in line with the previous literature [139,146].In healthy breast tissue, there are elongated structures such as lobules, ducts, and stroma in FGT that have large diameters compared to the mean displacement of water molecules during the diffusion time [139]. This may lead to low microscopic anisotropy and FA values in healthy breast tissue. The limitations of this work are the low resolution of images that may affect the delineation of small lesions and the contrast injection which may cause bias in the estimated diffusivity values.

ADC and FA have been used as common biomarkers in detection of prostate cancer [147,148]. However, there is contradiction in the reported results by different groups as some found higher and others reported lower FA in normal glandular tissue compared to the cancerous one [148–150]. This can be explained by different factors, such as echo time, diffusion time or the spatial resolution in different studies which all may affect the estimated FA. Especially the low resolution causes each voxel of image to include cells with different orientations and leads to lower FA value due to orientation dispersion [43,108]. This is common in prostate images because of high orientation dispersion [151]. [37] and [36] used tensor-valued diffusion encoding to scan the prostate in patients with cancer. They showed that the tissue with more elongated cell structures has higher microscopic diffusion anisotropy (microscopic anisotropic kurtosis (MK_A)) and isotropic heterogeneity (microscopic isotropic kurtosis (MK_I)) compared to normal tissue.In the prostate, regions with stromal smooth muscle have high microscopic anisotropy [151]. This can be detected as high FA if the image resolution is high enough to avoid the orientation dispersion inside a voxel which is not usually feasible in in vivo clinical scans. As cancer progresses from Gleason pattern 3 to pattern 4 the well-formed glands are replaced by fused glands [152]. This leads to a disorganized and heterogeneous tissue that has high MK_I. Low resolution of the imaging protocol may prevent the accurate delineation of the lesions. In addition, it may cause partial volume effect. There is a lack of voxel-to-voxel histology to match with each voxel of MRI data [153].

FA has been used in the kidney as a measure of tubular integrity [154,155]. Several studies have shown higher FA in the kidney medulla compared to the kidney cortex [156–158]. Comparing FA in patients and healthy controls showed that FA is reduced in kidney disease patients [159]. However, FA is not able to disentangle different pathophysiological features that cause renal dysfunction [154]. Therefore, more specific biomarkers of renal microstructure are desirable. [35] used the combination of LTE and STE to extract the microscopic FA in the human kidney in vivo. The lower bound for the b-value range that is required to provide microstructural information about kidney tissue is around 500 s/mm ² [35]. Clear divergence between LTE and STE curves by increasing the b-value (due to microscopic anisotropy) in the cortex and medulla of the kidney was observed without the need for any model fitting [35].

Cardiac diffusion weighted imaging is one of the most challenging medical imaging techniques because of the macroscopic motion of the beating heart and of respiration, which are several orders of magnitude larger than the length scale of displacement of water molecules during the diffusion time. Motion-compensated diffusion encoding overcomes this limitation [160–163]. Most of the cardiac dMRI studies are based on single diffusion encoding, which has already led to interesting insights in the healthy [28,90,96,161,162] and diseased heart, including myocardial infarction [29], hypertrophic and dilated cardiomyopathy [164], amyloidosis [165] and athlete’s heart [166]. Isotropic diffusion encoding can be used to estimate mean diffusivity (MD) in a shorter time compared to conventional single diffusion encoding [71,75]. First order nulling of isotropic encoding was proposed by [71]. [96] proposed b-tensor encoding with arbitrary order nulling to compensate the higher order motion in cardiac dMRI [28,90,167]. The nulling of concomitant field was also considered, this is done in numerical optimization.

Table 1 represents a summary of the application of tensor-valued diffusion encoding in the neuro and non-neuro applications.

To use b-tensor encoding, optimized waveforms in terms of echo time are necessary [86]. However, some hardware limits such as slew rate, maximum gradient amplitude and Peripheral Nerve Stimulation (PNS) are the limiting factors, especially in designing the motion compensated waveforms [79,96]. In addition, the effect of Maxwell terms should be considered. These may cause an extra gradient term, proportional to the G ² (G–gradient strength), which can lead to a signal loss and bias in the metrics of interest [87,168,169]. Timing of the linear and spherical encodings are sometimes different (in the design of the waveforms) which may cause differences in the effective diffusion time for the two b-tensor encoding schemes, and therefore confound the measurements [79,103,104,113,170,171,172,173] and introduce parameter bias.

There are approaches other than tensor valued diffusion encoding for quantifying microstructure such as SDE with different diffusion times, correlation tensor imaging (CTI) [62], oscillationg gradient spin echo (OGSE) [174]. Using CTI, one can disentangle three sources of kurtosis; isotropic, anisotropic, and intra-component kurtosis [62]. OGSE is useful to investigate small sizes in the tissue.