Combined diffusion-relaxometry microstructure imaging: Current status and future prospects

2.1.1 Intrinsic relaxation in diffusion MRI experiments

The standard technique for dMRI is single-shot echo planar imaging (ssEPI). In ssEPI the diffusion preparation is followed by an EPI read-out train as shown in Figure 3. In this illustrated example, the diffusion preparation, described by the gradient waveform, is a standard pulsed gradient experiment (ie, pulsed gradient spin echo [PGSE]) with two gradient lobes of equal length , equal polarity, and equal gradient strength . One lobe lies between the excitation and refocusing pulse, with the other between the refocusing pulse and the start of the EPI read-out. The spacing between the gradient lobes () determines the diffusion time. In this classic setup, the b-value is determined by , a special case of the more generic formulation

(1)

where

(2)

describes the gradient waveform, and denotes the field-strength-dependent gyromagnetic ratio over 2. Typically the echo time (TE) is minimized, which achieves the highest possible SNR. Therefore, the key limitations are the length of EPI read-out train before the TE ( in Figure 3), and the available gradient strength . Figure 3 also illustrates that the diffusion acquisition does not happen in isolation, but in parallel to the / decay. It follows that the measured ssEPI signal is influenced both by decay and by the effect of diffusion gradients and can be written, assuming monoexponential relaxation and diffusion decays

(3)

where D is the apparent diffusion coefficient and is the relaxation time. It follows that the choice of the highest b-value in a dMRI acquisition, and hence the minimum TE possible due to the restrictions outlined above (assuming fixed TE across all volumes), has a direct effect on the signal attenuation for all b-values in the acquisition. Therefore, these choices influence quantitative diffusion-related metrics derived from dMRI experiments. Two artifacts arising from the influence of the transverse relaxation are shine-through and blackout, where variations in the time (eg, in hematomas or myelin water) influence the dMRI signal intensity.²⁷

2.1.2 Basic combined diffusion-relaxometry acquisition

The inherent relationship between diffusion and relaxation acquisition sequences, alongside the opportunity to probe tissue microstruture in more detail, motivates MRI acquisition sequences that probe diffusion and relaxation properties and their correlations. We will now introduce and review such acquisition sequences. For clarity, here and throughout we avoid using the terms “multicontrast” and “multimodal”, which can refer to any scan (or series of individual scans) that measures multiple MR contrasts. Instead we use “combined diffusion-relaxometry,” to mean an experiment where diffusion and relaxation encoding parameters are varied in a 2D (or higher) acquisition space (eg, a diffusion-prepared sequence repeated at multiple TEs), and “simultaneous combined diffusion-relaxometry” for the special case where multiple diffusion and relaxation encoding parameters are varied within a single repetition time (TR) (eg, diffusion-prepared sequence where multiple TEs are measured in a single TR).

Consecutive acquisitions

The most common way to acquire combined diffusion-relaxometry data is by repeating a diffusion-weighted scan with varying relaxation-encoding parameters. For example, since relaxation time can be estimated by acquiring multiple spin echoes with different TEs,²⁸ typical -diffusion experiments comprise acquiring multiple diffusion prepared spin echoes with varying TEs (eg, Figure 4A). With this approach, each sample in the high-dimensional acquisition space (eg, Figures 1 and 4) is obtained with its own acquisition block of length TR. This is inefficient, as the time per sample is limited by the time required to achieve the specific diffusion preparation. This can lead to large idle times compared to typical dMRI—where TE is minimized subject to the constraints of the required b-value and imaging gradients (eg, Figure 5A)—as data are acquired at a range of TEs with some longer than the minimal TE (eg, Figure 5B). Consequently, typical consecutive combined diffusion-relaxometry -diffusion acquisitions, such as in Figure 4A, have taken 1 hour to scan the whole brain with around 500 volumes at 2.5 mm isotropic resolution.^{26, 29} As well as being slow and inefficient, this approach adds to the risk of inconsistencies and bias through motion and modifying the diffusion time.

The effect for -diffusion experiments is even more drastic. A basic -diffusion acquisition comprises spin echo acquisitions preceded with a global inversion pulse, as in a typical inversion recovery sequence,³⁰ with the time between the global inversion pulse and the first spin echo pulse (the inversion time, TI) varied to yield sensitivity. This introduces significant delays compared to typical dMRI sequences, as the time between inversion and excitation, which sets the achieved TI (see Figure 6B), is typically in the range of 0-3000 ms. Moreover, the inversion recovery sequence is typically performed slice-by-slice, with the global inversion pulse followed by reading out a single slice. An early in vivo -diffusion combined diffusion-relaxometry experiment gave whole brain -diffusion coverage with 2 mm isotropic voxels in 1 hour.³¹

This intrinsic inefficiency in sequences, together with the huge number of possible data points—two previous 2D diffusion-relaxometry studies using uniform sampling utilized 1024³² and 225 000³³ data points—hampers multidimensional scanning, and in the past prevented translation of such techniques from NMR to imaging. It is therefore clear that methods for faster and more efficient combined diffusion-relaxometry acquisition protocols are of paramount importance. Two commonly employed strategies for working toward this goal are:

1. modifying the pulse sequence to acquire data points more efficiently (ie, efficient sampling schemes)
2. selecting acquisition parameters that maximize the information content for subsequent analysis and interpretation (ie, optimized sampling schemes)

2.2.1 Efficient sampling schemes

There are multiple techniques that apply the aforementioned first strategy, often inspired by efficient simultaneous diffusion and relaxation acquisitions pioneered in the NMR domain.^34-36 These bespoke MR sequences typically improve acquisition efficiency by combining multiple contrast preparations and multiple read-outs within a single TR. In particular, for transverse relaxometry, time can be saved by acquiring multiple echoes (either spin or gradient) after the initial diffusion preparation and spin echo readout (eg, Figure 4B). This is further illustrated for the multiple spin echo case, which captures information, in Figure 5C. Similarly, additional gradient echoes would capture information, akin to a typical multi-echo gradient echo scan for measuring relaxation.³⁷ In both spin and gradient echo cases, this approach means that a single diffusion preparation is shared across multiple echoes, rather than being repeated for each echo, with the benefits of a more efficient scan and a consistent diffusion preparation across echoes. This results in a more complete sampling of the TE-diffusion preparation space within a fixed scan time. The exact speed up depends on the number of TEs chosen, and the required extension of the shot length and the duty cycle requirements set by the heating of the gradients. For large b-values, the TR might be extended due to this requirement of cooling periods, potentially offsetting some time gains when acquiring multiple echoes.

For relaxometry experiments where a global inversion pulse is inserted before the first excitation (ie, Figure 6B), again there are NMR-inspired techniques that more efficiently sample multiple TIs over the imaging volume. Typical inversion recovery experiments acquire multiple spatial slices sequentially, meaning that each slice has a different effective TI value, allowing the measurement of many points on the inversion recovery curve, albeit at different spatial locations, within a single TR. The addition of slice-shuffling as originally proposed in 1990 and recently done in a number of studies,^38-40 where the acquisition order of spatial slices is changed in subsequent TRs, allows the efficient sampling of each TI at each spatial slice location. Efficient slice shuffling has recently been utilized to yield whole brain T1-diffusion coverage in 13 minutes for 2.6 mm isotropic voxels.⁴⁰ An alternative approach avoids inversion recovery altogether by only varying TR to yield T1 sensitivity.⁴¹

Further acquisition flexibility comes from applying the slice shuffling principle to the diffusion preparation, as demonstrated recently by Hutter et al⁴² who, instead of acquiring all spatial slices with the same diffusion preparation in a single TR, changed the diffusion preparation on a slice level. The ZEBRA (Z-location shuffling, multiple echoes and B-interleaving for relaxometry-diffusion acquisitions) technique combines three strategies^{42, 43}—multiple subsequent gradient echoes are combined with slice-shuffling and interleaved slice-level diffusion encoding. This allows the acquisition parameter spaces required for , , and diffusion contrast to be efficiently sampled in a significantly reduced acquisition time—a ZEBRA whole-brain --diffusion scan with 1344 volumes at 2.5 mm isotropic resolution was demonstrated in a total scan time of 52 minutes.⁴⁴ However, there are remaining challenges and constraints for such pulse sequences: acquisition times are still too long for clinical translation, the number of subsequent echoes that can be acquired is constrained by the length of the EPI read-out train and crusher gradients, and the acquired data cannot be easily split into shells—as each gradient direction will have different TIs due to slice shuffling—preventing the use of standard analysis techniques.

Stimulated echo (STEAM) techniques, albeit commonly employed to alter the diffusion time, also have an inherent sensitivity,⁴⁵ although 50% of the signal is lost compared to a spin echo in the conventional setup (see Figure 6C for a representative STEAM sequence). However, a recent example exploiting stimulated echo pathways, MESMERISED (Multiplexed Echo Shifted Multiband Excited and Recalled Imaging of STEAM Encoded Diffusion), uses echo-shifting to remove the dead time in STEAM acquisitions and hence allows more efficient -diffusion acquisition.⁴⁶

Similarly, there are multiple techniques proposed to acquire transverse relaxometry data more efficiently. These include echo planar time-resolved imaging (EPTI),⁴⁷ which uses novel sampling strategies to exploit correlations in k-space and time to enable faster and relaxometry. This approach was recently demonstrated for diffusion-relaxometry with PEPTIDE (PROPELLER EPTI with dynamic encoding).⁴⁸ Also related are high strength gradient systems, as they can enable lower TEs for the same diffusion weighting ⁴⁹ although it does remain the case that lower minimum TEs are possible in 1D relaxometry experiments than combined diffusion-relaxometry acquisitions. This may inhibit the ability of diffusion-relaxometry to measure structures with low / relaxation times, such as myelin water.² magnetic resonance fingerprinting (MRF) is another relevant technique. This allows simultaneous quantification of multiple tissue properties by combining a highly accelerated acquisition that varies all relevant parameters, with dictionary matching to prior computed signal curves. The simulation of the MRF dictionary⁵⁰ with additional parameters constitutes a large computational effort, limiting diffusion MRF so far largely to variation of the b-value to allow calculation of the ADC value, for example, Ref. [51]. However, recent efforts have started to overcome this limitation by supporting this step with deep learning techniques, resulting in combined diffusion-relaxometry MRF⁵² including variation of b-value and gradient direction. Another proposed alternative is MR multitasking, a continuous acquisition that uses using real-time low-rank modeling to account for motion, relaxation and other dynamics and hence efficiently quantify MR properties,⁵³ and was recently demonstrated for simultaneous , , and ADC mapping.⁵⁴

2.2.2 Optimized sampling schemes

In addition to sequence efficiency, the choice of acquisition parameters is of key importance. Sampling the acquisition parameter space (ie, Figures 1 and 4) with a uniform grid is the obvious place to start. However, similar to nonuniform k-space sampling strategies, this is not required for analysis and not necessarily the most efficient approach. For example, the Cramer-Rao lower bound (CRLB) is an estimation theoretic tool that can be used to quantify the efficiency of different encoding schemes,⁵⁵ and CRLB-based analysis suggests that there can be many sampling locations which provide little additional information within a uniform diffusion-relaxometry sampling grid.⁵⁶ Samples that do not provide much information can be skipped to enable high-quality data from a very small number of samples. This approach has enabled estimation of multidimensional spectra from as few as 12 samples for 2D diffusion-relaxation,⁵⁶ and fitting of a two compartment -diffusion model from a 15 minutes whole brain scan at 2.5 mm isotropic resolution.⁵⁷

The MADCO (marginal distributions constrained optimization) framework is another example of the reduced-sampling approach. It exploits the fact that lower dimensional distributions, or spectra, of MR parameters estimated from the data are essentially projections of the corresponding multidimensional spectra. MADCO utilizes a hierarchical encoding scheme: first, fully sampled 1D data is acquired for all sampling dimensions; then 2D (or higher) datapoints are very sparsely sampled (see Figure 4C). This, when combined with analysis techniques detailed in later sections, supports estimation of multidimensional spectra using many less data points that would typically be required—MADCO was shown to achieve an acceleration factor of up to 50.^{58, 59} MADCO also has the significant advantage that acquisition times do not dramatically increase for higher dimensions, as demonstrated for the 3D case.⁶⁰

Further acceleration techniques that reduce the number of required samples, all unified by the assumption that the data is sparse, have been proposed. While strategies like MADCO leverage the marginalized axes to constrain where peaks can appear, compressed sensing approaches—traditionally applied in k-space—impose a presumed basis set in which the data is sparse. These methods have been used to recover 2D spectra using significantly less data in NMR⁶¹ and MRI⁶² contexts. Similarly, a framework based on PCA-like optimization has been proposed⁶³ and aims to retrieve the sparse basis from the data. Finally, recent frameworks using machine learning techniques such as SARDU-net⁶⁴ have shown excellent ability to retrieve information based on fewer samples.⁴⁴

2.3.1 Tensor-valued diffusion encoding schemes

Arbitrary diffusion-weighting gradient waveforms (eg, Figure 6A) can be described in terms of a symmetric second-order b-tensor defined as⁷⁰

(4)

where q is defined in Equation (2).

The tensor-valued diffusion encoding framework describes protocols where diffusion encoding is executed in more than one direction per signal preparation to enable the measurement of b-tensors with arbitrary traces (ie, b-values), shapes and orientations.⁷⁰ In contrast, diffusion encoding schemes based on the Stejskal–Tanner design can only modulate the trace and orientation of the b-tensor. Consequently, the benefit of tensor-valued encoding is that it unlocks the “shape of the b-tensor,” a new encoding dimension that can be modulated to control the sensitivity of the detected signal to microscopic diffusion anisotropy, as reviewed in detail in Ref. [71].

The parallels between novel diffusion gradient waveforms and combined diffusion-relaxometry are clear—both are acquisition techniques that can precisely quantify and disentangle distinct tissue microenvironments and, excitingly, both approaches are complementary. It follows that modified gradient waveforms are necessary in order to maximize the utility of combined diffusion-relaxometry acquisitions (and vice versa). For example, a sequence combining tensor-valued diffusion encoding with multiple TR weighting was introduced as a means to establish correlations between and diffusion properties and thus characterize complex fiber orientation layouts.⁴¹ However, the combined relaxation and b-tensor protocols developed thus far have utilized a rather straightforward design, particularly from the relaxation encoding perspective, with suboptimal echo and recovery time samplings. In this regard, there is potential for improvement through combining arbitrary b-tensor shape with acquisition strategies that encode relaxation information more efficiently. We also note that, particularly when combined with relaxometry, the additional degrees of freedom describing an advanced diffusion gradient waveform put further emphasis on efficiently selecting the acquisition parameters.

2.3.2 Varying diffusion time

As well as modifying diffusion gradient shape, we can also modify the time scale at which diffusion is observed. In the conventional PGSE approach, , the spacing between diffusion encoding pulsed gradients, can be varied for this purpose. Importantly, is closely linked to a change in TE as described above, due to the intertwining between the structure used for the spin echo and the placement of the diffusion gradients after the excitation and after the refocusing pulse.

There are a number of advanced acquisition techniques which probe the diffusion time. Higher gradients with faster slew rates can be used to allow MRI systems to access short and may be critical for measurement of white matter features, such as the axonal diameter.⁷² Alternative sequences, such as the oscillating gradient spin echo (OGSE) are associated with significantly shorter effective diffusion times.⁷³ For structures with larger radii (prostate lumen,⁷⁴ muscle tissue⁷⁵), PGSE cannot offer long enough diffusion times, due to signal loss from -weighting.

The aforementioned STEAM diffusion preparations (Figure 6C) can maintain high SNR for diffusion times above 50 ms. Although STEAM conflates and diffusion time dependencies, this may be overcome with a twice-refocused STEAM preparation.⁷⁶ In combination with the appropriate models (as discussed in later sections) this approach can investigate the correlations between the time-dependence of diffusivity and relaxation times.

Modifying the diffusion time yields particular sensitivity to porous structures, such as cells.⁷⁷ Such structures also affect relaxation properties, for example, differences in magnetic susceptibility between the pores and surrounding material cause spatial variations in the field.⁷⁸ Song et al demonstrated that this can be exploited using sequences that vary the TE and diffusion time to measure the decay due to diffusion in the internal field (DDIF), and hence determine the characteristic length-scales of a porous structure.⁷⁹ Alvarez et al developed a related sequence sensitive to diffusion and relaxation properties to measure internal gradient distribution tensors, which probe local magnetic susceptibility properties.⁸⁰

These experiments show that incorporating diffusion time dependence into combined diffusion-relaxometry models is an exciting prospect for quantifying microscopic structures. Alongside magnetic susceptibility, future acquisition methods could also probe and relaxation. These methods can potentially detect compartments with small (1 m) and impermeable features, which are not experimentally accesible with diffusion-only sequences due to the limited diffusion times available on clinical MR scanners (20-50 ms).

2.3.3 field strength

Higher B0 field strength offers significant advantages for combined diffusion-relaxometry, including increased SNR, contrast, and spatial and temporal resolution.⁸¹ However, relaxation times vary strongly with B0, and the specific relationship often depends on the tissue type, such as with in the brain.⁸² Since they are affected by relaxation, diffusion metrics may also depend on ,⁸³ particularly at ultra-high field (ie, 7T and above).^{84, 85} In some cases, it may be important to account for these dependencies in the clinic, such as when setting cut-off values between healthy and diseased tissue. There are also likely opportunities to exploit this dependency—in some applications distinguishing between tissue types may be easier at a certain . Ultra-high-field MRI seems most promising for revealing and exploiting dependencies of diffusion metrics. In particular, the reported increase of micro-FA at ultra-high-field deserves further investigation.^{85, 86} Another promising avenue is the B0 dependency of the intravoxel incoherent motion (IVIM) signal curve at ultra-high-field,^{87, 88} where the weight of the arterial pool should further increase, combining this with additional relaxometry information can potentially make “arterial-pool-weighted” IVIM feasible.⁸⁹

2.3.4 Exchange

During acquisitions, some spins will inevitably move between microenvironments that have different MR properties. Traditional diffusion-relaxometry acquisition techniques do not explicitly account for exchange or provide a means to modulate its effects, but exchange will affect the signal and hence derived diffusion and relaxation parameters. A class of related and relevant techniques can detect exchange between compartments using sequences incorporating variable mixing times. These include diffusion exchange spectroscopy (DEXSY)⁶⁹ and filter exchange imaging (FEXI)^{90, 91} for diffusion, and relaxation exchange spectroscopy (REXSY) for ⁹² and ⁹³ relaxation. Such techniques can only measure the apparent exchange rate, as differences in relaxation times (for diffusion exchange) or diffusivities (for relaxation exchange) between compartments will affect the measured exchange rate.⁹⁴ Techniques that acquire combined diffusion-relaxometry data with variable mixing times have the potential to account for these differences, and hence measure exchange rates more closely related to underlying values. However, naively merging these techniques would lead to prohibitively long acquisition times, so novel pulse sequences, such as methods that rapidly measure exchange,⁹⁵ are desirable.

2.3.5 Final thoughts

Further improvements will come from combining specific multidimensional data acquisition techniques with important parallel developments in MRI acceleration. These include multiband imaging^{96, 97} and compressed sensing approaches^{62, 98} among others. In summary, the efficient acquisition of combined diffusion-relaxometry data is an exciting and ongoing field which will continue to be driven and influenced by a multitude of developments in both MRI and NMR. Such accelerated acquisition techniques are crucial to provide data for the appropriate analysis method while avoiding prohibitively long acquisition times.

3.2.1 Mapping voxelwise spectra

Unlike modeling methods that calculate a single, or small number of, values per voxel (as discussed later), the continuum modeling approach we describe calculates the spectrum, f, in each voxel. The typical way to derive meaningful maps from such voxelwise spectra is known as spectral integration.^{60, 100, 124} In summary, spectral intergration comprises first manually identifying prominent regions of the spectrum—typically by examining a spectrum derived from the signal averaged over a large representative ROI. The proportion of each voxelwise spectra that lies within each of these prominent regions is then calculated (hence the name spectral integration), yielding scalar indices often termed apparent spectral volume fractions. The data-driven regularization methods described above^{117, 118} provide an alternative approach for deriving maps.

In contrast to continuum modeling, a useful and widely used constraint in single-contrast diffusion and relaxation experiments is to assume that the number of components M in Equation (1) is known in advance and is very small, while also assuming simple/parsimonious parametric models for the signal observed from each component. This approach has been used with combined diffusion-relaxometry data,^{26, 74, 125} and uses modeling assumptions that are substantially more restrictive than those previously discussed. However, when the modeling assumptions are accurate, they lead to a simpler inverse problem that requires substantially lower SNR data to solve than the more general formulations described previously (calculating multidimensional spectra, particularly for single voxels, requires very high SNR). We discuss two examples of this approach in the next sections.

4.1.1 Background and motivation: Brain applications

There are a wide range of diffusion MRI brain microstructure imaging techniques, many of which show promise in the clinic.¹⁵⁴ Compared to relaxometry and more conventional DTI, the main advantage offered by these techniques is a more direct link between quantitative parametric maps and the underlying tissue microstructure. For example, simple diffusion models have shown remarkable value for imaging acute ischemic stroke,¹⁵⁵ but the exact microstructural changes driving the contrast in ADC and DTI maps remain unclear. Moreover, unaccounted for changes in relaxation properties of brain tissue due to pathological conditions (eg, increase of relaxation time in ischemic lesion¹⁵⁶) may bias parameter estimates and lead to misleading or wrong interpretation of the ongoing pathological mechanisms. Incorporating relaxation into diffusion MRI techniques can lead to more specific measures of brain microstructure, leading to improved understanding of disease and new clinical applications.

4.1.2 Current state-of-the-art: Brain applications

There are a wide range of brain microstructure models, spanning signal representations and biophysical models.¹³⁸ The most common brain microstructure model encompasses a large set of white matter models consisting of Gaussian compartments, is the so-called “Standard model”.¹³⁵ This describes the measured dMRI signal as the convolution of a fiber orientation distribution function (ODF), , and a response kernel, from a straight fiber oriented along , where is the unit gradient direction:

(14)

while the Standard model does not fix individual parameters,^{26, 57, 157} there are numerous particular cases of this model that involve prior assumptions or bounds on the scalar parameters of the kernel (diffusivities, volume fractions), inclusion of free water compartment, and assumptions relating to the functional form of the ODF, most generally represented by an expansion of spherical harmonic coefficients. Fixing compartment diffusion coefficients and the ODF shape introduces an unknown a priori bias¹⁴¹ on the remaining parameters, which is particularly problematic in regions with diffusivity changes and no change in the volume fractions of tissue compartments.¹⁵⁸ On the other hand, without any priors or constraints the standard model suffers from degeneracy, that is, multiple combinations of the model parameters can provide the same indistinguishable signals.¹⁴¹ Simultaneously measuring relaxation and diffusion properties can help address this issue and relax a number of the priors and constraints. This was recently demonstrated in the brain, where an optimized 5D protocol comprising multiple TE and b-tensor shapes was shown to enable the fit of a minimally constrained white matter model.⁵⁷

Both and have been utilized to resolve inherent ambiguities in diffusion measurements. De Santis et al used combined -diffusion acquisition to resolve crossing fiber populations^{31, 125} (see Figure 8A). More recently, Leppert et al investigated tract specific mapping.⁴⁰ Veraart et al used combined -diffusion to improve separation of presumed intra- and extra-axonal compartments²⁶ (Figure 8B), and a similar strategy was later adopted within the NODDI model framework²⁹ (Figure 8C). An interesting different multidimensional approach has been recently proposed to map intra-axonal values using the extra-axonal water suppression provided by high diffusion-weighting. Using a diffusion-filtered asymmetric spin echo, Kleban et al jointly estimated diffusion and susceptibility effects in major brain white matter tracts, showing that intra-axonal is lower in the corticospinal tract than in corpus callosum and cingulum.¹⁵⁹

Moving away from biophysical model-based approaches, more data-driven methods to multidimensional diffusion-relaxation analysis of the brain tissue have been recently proposed, carrying very exciting perspectives (for a more comprehensive review of the most recent methods, we refer the reader to Refs. [55, 160]). For example, de Almeida Martins et al proposed an MRI framework to quantify the microscopic heterogeneity of the living human brain by spatially resolving five-dimensional relaxation-diffusion distributions from combined 5D relaxation acquisition,^{161, 162} with experiments on a healthy volunteer demonstrating that the retrieved 5D -D distributions can resolve distinct microscopic tissue environments within the living human brain (see Figure 8D). It is worthwhile to mention here that these data-driven multidimensional diffusion-relaxation approaches can be extremely useful to characterize parts of the central nervous system other than the brain, such as the spinal cord^{100, 163} (eg, Figure 8E). Recently, in a comprehensive postmortem multidimensional MRI and histopathology combined study that included multiple human subjects ranging in axonal injury severities, Benjamini et al showed that axonal injury has a multidimensional spectral signature,¹⁶⁴ which can be used to generate injury biomarker images that closely follow amyloid precursor protein (APP, axonal injury marker) histopathology, with strong correlation (see Figure 8F).

Of course, more general data-driven multidimensional approaches come together with new challenges,¹⁶⁰ for example, how to improve the robustness (in terms of accuracy and precision) of the statistical descriptors that are estimated directly from the data,¹⁶⁵ or how to summarize and visualize in an effective and simple way the large amount of information that can be retrieved.^{161, 166} A particularly interesting model-free application of multidimensional relaxation-diffusion approach to the same problem previously tackled by De Santis et al using a model-based approach has been recently proposed for the nonparametric estimation of fiber-specific diffusion- features.⁴¹ However, for spinal cord, the application of such methods to in vivo clinical settings is hampered by challenges involving low SNR and high motion corruption. Nevertheless, recently proposed strategies to reduce the number of samples¹⁶⁷ may provide useful tools to address some of these limitations, opening exciting possibilities for in vivo clinical applications.

4.1.3 Future promise: Brain applications

Combined diffusion-relaxometry MRI is quickly evolving and moving into more and more clinically feasible applications.¹⁶⁰ As discussed in the previous sections, this new approach comes with new challenges, but several recent works have already highlighted interesting avenues toward the resolution of some of them.^{161, 165, 166} Notwithstanding the current challenges, the possibilities offered by combined diffusion-relaxometry for brain microstructure characterization are exciting, especially for pathological conditions. In particular, the rich data available from combined diffusion-relaxometry lend itself to data-driven analysis (eg, continuum modeling), which in principle can better generalize across diseases than typical brain microstructure models that make a priori assumptions about tissue structure.

In addition, new acquisition techniques have the ability to probe very short TEs,⁴⁹ paving the way toward very interesting applications, for example, the quantification of combined diffusion-relaxation properties of myelin, whose is very short (typically <10 ms), raising in turns new challenges for the acquisition. Furthermore, advanced MRI hardware, utilizing stronger diffusion gradients such as those available in the CONNECTOM scanner,⁴⁹ may allow the exploration of an additional dimension, the diffusion time, that may carry unique information about microstructural features such as membrane permeability, spatial disorder and more (see further the earlier Diffusion Time Section). Promising future applications may involve the multidimensional exploration of sub-cellular membrane structures¹⁶⁸ or the combined diffusion-relaxation of brain metabolites.^{169, 170}

4.2.1 Background and motivation: Prostate applications

The multiparametric MRI (mp-MRI) examination for diagnosis, staging, and risk stratification of prostate cancer relies on -weighted and diffusion MRI images.¹⁷¹ The synergy between modalities is abundantly clear to clinicians; moreover, decoupling signals from multiple tissue compartments promises biomarkers with greater diagnostic value.¹⁷² This is a monumental modeling challenge, as the prostate is heterogeneous at multiple length scales, where each voxel presents unknown mixtures of stroma (20 m), epithelium (10 m), vasculature (10 m), and lumen (100 m). The observation^{173, 174} of a nonmonotonic diffusion time dependence, necessarily means that at least 1 compartment falls within the short or long-time diffusion limits. For this reason, it is incredibly difficult to estimate compartment fractions via diffusion, as this would require definition of higher order cumulants of the Taylor expansion of for each compartment. A potential avenue to resolve the model complexity of diffusion is to rely on relaxometry for compartment estimation, where separation of signal into cellular (stromal + epithelium) and luminal compartments using alone has been performed reliably in prostate tissue as early as 1987¹⁷⁵ and has been reproduced on numerous occasions.^{145, 176-178} The practical challenge remains on how to describe the prostate signal through diffusion-relaxometry, without biasing parameter estimates or over complicating the model.

4.2.2 Current state-of-the-art: Prostate applications

The model of Chatterjee et al¹⁷⁹ incorporates relaxometry and hypothesizes that prostate diffusion can be described as a sum of 3 Gaussian compartments: epithelium, stroma, and lumen (see Figure 9A). The model of Lemberskiy et al⁷⁴ circumvents the issue of higher order cumulants, by modeling DWI at low-b (where diffusion remains sufficiently Gaussian) and performing the assumed compartmental decomposition of the diffusion tensor entirely through , thereby preserving the full diffusion tensor for each compartment (see Figure 9B). This approach allowed for the functional validation of diffusion time dependence within putative cellular (epithelium and stroma), which was found to agree with random permeable barriers, and luminal compartments were found to agree with the short-time limit. Though high b-values were not considered, exchange/permeability could be measured for the cellular compartment via a random permeable barrier model.^{75, 180} This approach did not account for vascularity, which while small, may be a critical microstructural biomarker. VERDICT does account for vascularity,¹³⁷ it originally did not include relaxometry but was recently expanded to account for and dependence.¹⁴³ Zhang et al recently applied continuum modeling with spatial regularization to quantify epithelium, stroma, and lumen compartments, and showed that these correlated well with results from histopathlogy¹⁸¹ (see Figure 9C).

4.2.3 Future promise: Prostate applications

The complexity of individual compartments and the potential for diffusion time-dependence present a challenge to modeling prostate microstructure. Unlike the mapping of axonal diameter in the brain,⁷² MR gradient systems are sufficiently strong to measure various microstructural features. With the hardware challenge resolved, the primary challenge is to determine the optimal combined diffusion-relaxometry acquisition and its subsequent interpretation through appropriate biophysical modeling.

4.3.1 Background and motivation: Placenta applications

Placental MRI is emerging as a sensitive tool which can supplement antenatal ultrasound in the clinic. There are a number of MR contrasts which show promise as markers of placenta-related pregnancy complications. relaxometry—which relates to oxygenation levels—reflects placental dysfunction in fetal growth restriction (FGR)^{182, 183} and pre-eclampsia.¹⁸⁴ Diffusion MRI is also sensitive to these changes, with ADC¹⁸⁵ and IVIM-derived perfusion fraction^{186, 187} decreased in FGR cases.

The placenta is unique in that it contains two distinct and nonmixing circulations: fetal and maternal blood. Fetal blood circulates in vasculature within convoluted villous tree structures. On the other hand, maternal blood, uniquely, does not flow within vasculature, but resides in intervillous space surrounding fetal villous trees. This one-of-a-kind structure is a leading motivation behind applying combined diffusion-relaxometry in the placenta. Melbourne et al make the following initial speculations about placental tissue environments¹⁸⁸: fetal and maternal blood have similar relaxation times, and hence their relative fractions cannot be determined with relaxometry; However, they can likely be differentiated by diffusivity properties, since fetal blood perfusing in vasculature travels much faster than maternal blood. Moreover, maternal blood cannot be separated from water in tissue by diffusivity, but can be separated by relaxation time. It follows that diffusion-relaxometry is necessary to adequately disentangle these respective compartments.

4.3.2 Current state-of-the-art: Placenta applications

The first example of combined diffusion-relaxometry in the placenta was presented by Melbourne et al¹⁸⁸ (see Figure 10A). The DECIDE model contains three putative tissue compartments: fetal blood (characterized by high diffusivity and long ), maternal blood (low diffusivity, long ), and tissue (low diffusivity, short ). This enables estimation of fetal and maternal blood values, and the maternal to fetal blood volume ratio. These are potential biomarkers of placental dysfunction that are unavailable through single contrast measurements.

Slator et al¹⁸⁹ simultaneously probed and diffusivity in the placenta (see Figure 10B). The -ADC spectrum was calculated with an ILT, and showed multiple separated peaks potentially reflecting multiple tissue microenvironments. Encouragingly, although the sample size is small, -ADC spectra clearly separated normal and dysfunction placentas, demonstrating the potential of this approach to quantify and predict pregnancy complications.

4.3.3 Future promise: Placenta applications

A promising avenue for future work is to account for anisotropy in analysis techniques. It has been shown that models incorporating anisotropy—in both perfusion and diffusion regimes—explain the placental dMRI signal well ¹⁹⁰. Incorporating this into diffusion-relaxometry analysis techniques has the potential to increase sensitivity to pregnancy complications since, for example, fetal vascular tree morphology is significantly altered in fetal growth restriction ¹⁹¹. The ultimate aim is to develop targeted imaging tools that can supplement antenatal ultrasound in the clinic. Such a tool should predict and precisely diagnose pregnancy complications, and hence be beneficial for potential future therapies. The extent to which such a tool is sensitive to the fetal to maternal blood ratio, oxygenation levels, and placental microstructure is an open question.