## Introduction

This works exploits these unique xSPEN capabilities, to map diffusion in areas that have hitherto proved inaccessible to single-shot studies. This mapping is complicated by the constant gradient employed by xSPEN MRI, which while capable to provide T2 *-free images for a large range of ΔB o inhomogeneities, also imparts a heavy diffusion weighting of its own. Such weightings can overwhelm the effects that the ±G d have to impose for monitoring motions along orthogonal axes, making it hard to obtain full anisotropic details –or even isotropically-weighted information– about water’s diffusivity. In order to perform such full tensorial mapping this study relies on a local b-matrix analysis that unifies the time- and space-dependent effects imposed by both the diffusion and the imaging gradients, and uses it to devise xSPEN-based pulsed-gradient spin-echo (PGSE) schemes that can sample diffusivity over a sufficiently large range of directions. The usefulness of these new tools is demonstrated in a series of preclinical and clinical imaging tests providing diffusivity information in human head and brain regions that are often unreachable by single-shot methods.

## Results

### xSPEN MRI

To better understand the challenges that xSPEN imaging poses to diffusion measurements, it is convenient to briefly review the features that distinguish this methodology from its SPEN predecessors. SPEN relies on a progressive excitation/inversion and refocusing of the spins, achieved by applying a frequency-swept radio frequency (RF) pulse lasting for a time T e and acting whilst in the presence of an encoding gradient G e 32,33. Whether this RF pulse is used for an excitation or inversion, the result is a spatially parabolic phase profile. Assuming for concreteness swept 180° inversion pulses applied while in the presence of a y-axis gradient, this phase can be written, within an unimportant constant, as

$${\phi }_{e}(y)=-\frac{{(\gamma {G}_{e})}^{2}}{R}{y}^{2}$$
(1)

where R is a sweep rate defined by γG y FOV/T e , and FOV defines the targeted field-of-view along the y-axis being encoded. The quadratic coefficient of the parabolic phase in Eq. [1] defines the spatial extent of the spins emitting at any given moment, as signal emission will be dominated by spins positioned at the apex of this phase parabola. To probe the full FOV this stationary point is displaced, by applying an additional acquisition gradient G a over an acquisition time T a . Extending this 1D rasterization into a single-shot 2D MRI experiment requires encoding a second (e.g., x-) axis, something that is usually achieved by a conventional oscillating readout (RO) gradient. Fourier transform (FT) along the RO axis followed by a magnitude or super-resolved calculation, then delivers the final 2D image20,34,35.

The xSPEN pulse sequence takes Eq. [1] one step further, by melding into it the kind of encoding used in so-called “ultrafast” single-shot 2D NMR spectroscopy36,37. In this experiment the initial spin excitation is combined with two frequency-swept inversion pulses, acting in unison with a pair of bipolar gradients. This replaces the parabolic phase introduced in Eq. [1] by a bilinear phase encoding, proportional to both the chemical shifts Ωi of the targeted sites as well as on the spins’ positions along the axis of the bipolar gradient: $${\phi }_{e}=C{{\rm{\Omega }}}_{i}y+{\phi }_{0}$$, with C = 4T e /FOV a spatiotemporal constant under control and ϕ 0 a position-independent phase. Having imposed such encoding, the application of G a during acquisition leads to the generation of site-specific gradient echoes, enabling the acquisition of arbitrary nD NMR correlations in a single scan. In xSPEN, this spectroscopy-oriented approach is converted onto an imaging one by replacing the chemical shifts Ωis by a spatially-dependent frequency. This inhomogeneous frequency broadening can be imparted, for instance, by activating a constant G z along the slice-select z-axis (Fig. 1a). Considering an unknown term δω(r) coming from field heterogeneities and chemical shifts that adds to this z gradient, leads to an encoding phase profile

$${\phi }_{e}(y,z)=-Cy\cdot [\gamma {G}_{z}z+\delta \omega (y,z)].$$
(2)

$$C=\frac{4{T}_{e}\gamma {G}_{y}}{2\pi BW}$$ is now defined as a function of the bandwidth BW characterizing the swept RF that is applied while in the presence of the two encoding gradients: $$BW=(\gamma {G}_{y}FO{V}_{y}+\gamma {G}_{z}{L}_{z})/2\pi$$, with L z a nominal slice thickness. The hyperbolic encoding phase y.z dominating Eq. [2] has the unique feature that it allows the sum of the “encoding” frequencies $$\gamma {G}_{z}z+\delta \omega (y,z)$$ to eventually decode a distortion-less y image. In other words, G z and δω become both the encoding mechanism of the experiment, as well as the decoding tool revealing the undistorted positions of the spins along the y-axis –regardless of the size of the field distortion30,31. Figure 1c exemplifies this by presenting results collected on a phantom incorporating a titanium screw and a Lego piece; clearly xSPEN provides more faithful representations of these objects than any of the remaining single-scan counterparts.

### Alternative diffusion-monitoring strategies

The application of a continuous G z over the course of both the encoding and the decoding processes, imparts a heavy diffusion weighting into xSPEN MRI. This weighting is further complicated by xSPEN’s progressive spatial inversion/observation of the spins’ throughout the encoding and acquisition processes, which make these diffusion losses dependent on the y position being decoded. Understanding these effects requires a framework capable of computing the diffusion-derived signal attenuation produced by gradients such as G z /δω and G y , as well as accounting for the frequency (i.e., spatially) progressive nature of the RF encoding and readout processes25,26,27,38. Supporting Information A describes the formalism developed to estimate these effects, and to calculate diffusivity maps in single-shot xSPEN MRI. Based on those analytical and numerical derivations, Fig. 2 describes in further detail the challenges that the usual, variable-G d PGSE scheme, faces in the retrieval of xSPEN diffusivity data. To visualize these challenges the upper row in Fig. 2 depicts the $$\bar{\bar{b}}$$ -tensors arising in various DTI schemes, summarizing them in terms of the strength and direction that the eigenvector associated with the largest eigenvalue (b max ), exhibits within an imaging-based frame. Notice that whereas in techniques like EPI the weak effects of the imaging gradients allow one to explore the full range of necessary b-values and directions by suitably varying the strengths and directions of a conventional PGSE block, the continuous G z gradient employed in xSPEN strongly biases the range of b-elements and directions that can be sampled. For instance, when considering bipolar diffusion modules whose directions are uniformly spread over the x-y plane, the dominant b-matrix eigenvector ends up subtending a cone in the (x, y, z) sub-space –even though components solely in the first two of these axes should have differed from zero (Fig. 2b, gray). Details of this behavior are further illustrated in Extended Data Figure S1, which describes how these principal eigenvectors –as well as other aspects of the full $$\bar{\bar{b}}$$ -tensor– vary for different positions along the imaged axis over the course of an xSPEN acquisition. These strong b-modulations may not impede diffusivity measurements based on xSPEN PGSE schemes if dealing with preclinical scanners, where strong gradients capable of spreading the b-eigenvectors over a sufficiently large range of directions are available. Figure 3 illustrates this with axial slices recorded on an ex-vivo rat head using xSPEN and SE-EPI pulse sequences using the standard PGSE scheme in Fig. 1a. Both SE-EPI and xSPEN methods provide similar Apparent Diffusion Coefficient (ADC) maps for the brains, even if distortions for other regions in the head are noticeable in the EPI (yellow arrows, Fig. 3b). Still, this flexibility will not be available for the ca. ten-fold weaker pulsed gradients available in conventional clinical scanners. Not even the sequence shown in Fig. 1b, incorporating a second bipolar diffusion-weighting block39 placed on the far ends of both inversion pulses and furnishing higher b-values for a given maximal G d strength, will provide a sufficient range of accessible b-values under clinical scanning conditions.

Given these circumstances, we investigated alternative strategies for enabling reliable human diffusivity measurements using xSPEN. To this end an imaging scheme was devised, whereby the low-bandwidth xSPEN dimension was decoded twice, in two independent single-shot experiments that alternated the gradients employed along the RO and SS axes (Fig. 1b, purple labels). As a result of this the dominating b-weight associated with the SS axis also alternated among different (e.g., axial and sagittal) orientations. Figures 2c and 2d illustrate how this leads to two sets of orthogonal conic shapes in the sampling of the $$\bar{\bar{b}}$$-tensor space, rotated by 90° about y. While not involving the comprehensive exploration of the tensorial space that is possible in SE-EPI40, these two orthogonal xSPEN acquisitions lead to a sufficiently wide sampling to reliably measure the diffusion parameters. The lower panels in Fig. 2 demonstrate this with a series of fractional anisotropy (FA) and ADC plots employed to assess the reliability of the different $$\bar{\bar{b}}$$-space sampling schemes. These plots summarize a series of simulations where various 3D media (2601 “tissues”) were assumed, each of them having a random proton density and realistic FA and ADC values (see Supporting Information and figure captions for further details). The signals arising from these synthetic “tissues” under the action of the different gradient schemes and pulse sequences were then calculated, and employed to estimate FA and ADC values on the basis of $$S(\bar{\bar{b}})/S(0)=exp(-{\sum }_{i,j}{D}_{ij}{b}_{ij})$$, where {D ij }i,j=1-3 represent the diffusion tensor elements41. r 2-values were then calculated against the ground truth FAs and ADCs known for the various “tissues”, and from these the reliability of the various approaches was estimated. The most reliable ADC/FA assessments arose in all cases from SE-EPI (Fig. 2a); by contrast, a similar PGSE-based strategy gave unreliable results when incorporated into the original xSPEN sequence (Fig. 2b). Switching to two orthogonal acquisitions where the roles of the RO and SS axes in the xSPEN process are swapped and their outcomes processed in a combined analysis increased this considerably (Fig. 2c); the reliability of this double-cone $$\bar{\bar{b}}$$-space acquisition could be further enhanced by incorporating into the fits, independent b o -samplings (i.e., samplings with all G d s set to null) for each of the imaging schemes (Fig. 2d). The Extended Data Figure S2 illustrates an experimental validation of the resulting b o-including “double-cone” approach, conducted on a water phantom in a clinical 3 T MRI machine, showing essentially the same reliability as EPI-based maps. This latter gradient scheme was adopted for the human ADC and DTI xSPEN mapping.

### Diffusion in humans by xSPEN MRI

Figure 4 compares SE-EPI and xSPEN results obtained on two representative human head slices, showing the b o magnitude images, ADC, FA and cFA (color-coded FA) maps to which each experiment leads. For an upper, relatively homogeneous brain slice (Fig. 4a), both methods show similar tissue contrasts and diffusivity information. Still, the FA maps show a slightly stronger contrast for SE-EPI vis-à-vis xSPEN –perhaps reflecting the different ADC mapping accuracies of the two methods as discussed in the context of Fig. 2. For a lower brain slice (Fig. 4b), however, the SE-EPI data suffers from evident inhomogeneity distortions, clearly seen in the sinus regions and in the shapes of the eyes (yellow arrows). Interestingly, these are not only reflected in the SE-EPI b o images but also translate in diffusion map artifacts –for instance, in false anisotropies that SE-EPI describes in the center of the eyes’s vitreous cFA maps.

Figure 5 compares another set of axial, coronal and sagittal human head images, showing SE-EPI and xSPEN ADC maps collected at 4 mm isotropic resolution to probe diffusivity on head regions challenged by field inhomogeneity effects. Some of these regions are highlighted in the Figure by yellow arrows and include (a) the eyes, (b, c and e) the nose and nasal cavity, (d and g) the cerebellum, (f) the brain stem and (h) the tongue area. In SE-EPI these regions are partially or fully distorted and the ensuing ADC maps convey limited useful information, whereas xSPEN makes them clearly accessible. Worth comparing among these sets are the results observed for the vitreous humor, which in the center of the eyes include average ADCs of 3.7 ± 0.2 × 10−3 mm2/s and 3.6 ± 0.4 × 10−3 mm2/s for xSPEN and SE-EPI, respectively. These values are in good agreement with each other, as well as with literature reports42.

Figure 6 shows an additional example collected at a higher, 3 mm isotropic resolution, showing zoomed-in DTI maps of the head’s anterior region. This folding-free zoom-in is possible at no cost in the sequence’s complexity, and it helps to highlight features associated with the frontal lobes and eyes (Fig. 6a). It also shows how xSPEN could help to target the diffusivity of the optical nerves, for which xSPEN FA and ADC maps were collected at a slight tilt (Fig. 6b) that allowed us to observe regions with distinct diffusion values in the center of these nerves. These features include a more mobile core with average ADC and FA values of 1.5 ± 0.3 × 10−3 mm2/s and 0.25 ± 0.09 respectively, and a more slowly diffusing periphery with average ADC and FA values of 1.1 ± 0.3 × 10−3 mm2/s and 0.28 ± 0.10 respectively. Judging by the corresponding T1-weighted images, these data are affected by minimal artifacts despite the strong air-interface ΔB o s associated to these tissue regions. These ADC values are in agreement with literature reports based on fast spin-echo and on EPI measurements (0.8–1.4 × 10−3 mm2/s), even if reported FA values for optic nerves have been usually higher (0.39–0.64)43,44. The origin of this variation is under investigation.

## Discussion and Conclusions

Different diffusion-monitoring variants have been explored over the years as diffusion-monitoring alternatives to EPI, particularly within the context of overcoming heterogeneous field environments. These include multi-shot schemes liable to motion-derived errors, as well as single-shot acquisitions based on radial samplings or on the use of multiple rf-driven spin echoes15,16,17,18,19. Recently introduced single-shot 2D SPEN methods can achieve many of these goals20,35,45,46,47, yet if overwhelmed by inhomogeneities even these methods are liable to yield corrupted images reflecting ΔB o displacements. xSPEN by contrast shows unprecedented resilience to in-plane field inhomogeneities, with the sole distortions identified in their single-scan 2D images arising from limitations in the slice selection process31. xSPEN’s immunity to field inhomogeneities stems from its reliance on a constantly active frequency broadening mechanism that incorporates background inhomogeneities into the image-formation process. This use of a continuous gradient throughout the encoding and acquisition, however, results in a strong intrinsic b-weighting that needs to be accounted for when attempting quantitative diffusion measurements. In order to enable such accounting, the sequence’s diffusion effects were quantitatively evaluated using a customized, spatially-localized b-matrix analysis. With the aid of this formalism and of numerical simulations on “synthetic tissues”, a diffusion-weighting scheme was devised that overcomes xSPEN’s original limitations. Such scheme operates by exchanging, in independent measurements, the roles that orthogonal gradients play as the broadening mechanism enabling the xSPEN image formation. Phantom experiments validated the quantitativeness of the resulting “double-cone” b-sampling approach; when the diffusion gradients were not sufficiently intense, a double PGSE block could be readily incorporated into the sequence. When implemented on human volunteers (Figs 46) the ensuing double-PGSE, double-cone scheme improved the ADC and FA depiction afforded by SE-EPI, providing reliable maps of isotropic and anisotropic diffusivity for the brain stem, cerebellum, mouth and the ocular regions. Thanks to its reliance on frequency swept pulses xSPEN also provided a built-in “zooming” capability that allowed us to focus on the frontal brain region, where ADC mapping of the optical nerve was demonstrated in all three orientations –always evidencing a good, rounded shape and faithful mapping of the nerves’ diffusivity characteristics to their original locations.

Despite these features, xSPEN still exhibits a number of limitations that remain to be overcome. Foremost among these is its SNR limitation; while we have managed to overcome these for the original SPEN experiment using super-resolution procedures48,49, similar procedures remain to be devised for xSPEN. In their absence, we could not explore higher human spatial resolutions than 3 mm isotropic. An additional limitation of xSPEN compared to other single-shot methods –included its SPEN predecessors– is its higher SAR. This results from xSPEN’s use of two frequency-swept inversion pulses, and it led to SAR reaching ca. 90% of its maximum allowed (normal mode) value when performing the multi-slice experiments illustrated above. Approaches to minimize this problem’s impact are also under investigation. Despite these two main limitations we believe that xSPEN’s ability to target challenging inhomogeneity-dominated regions, of the kind that are usually inaccessible by traditional single-shot methods, can open new opportunities in basic and in clinical investigations. This immunity to field heterogeneities was here demonstrated for various head regions of healthy volunteers; we are exploring what new avenues can be opened when imaging other body regions, as well as tissues proximate to metal implants including spine, mouth and dental MRI.

## Methods

### Preclinical scans

Animal experiments were preapproved by Weizmann’s Animal Care and Use Committee (protocol 01500312-3) and all procedures were carried in accordance with the guidelines of this IACUC. Weizmann’s animal program is accredited by the Association for Assessment and Accreditation of Laboratory Animal Care (AAALAC), by the US NIH’s Office of Laboratory Animal Welfare, and by Israel’s Ministry of Health. Phantom and ex-vivo whole-head rat experiments were performed on a DD2 7 T/110 mm horizontal magnet scanner (Agilent Technologies, Santa Clara, CA) using a quadrature-coil probe. For these scans the xSPEN pulse sequence shown in Fig. 1a was used and compared against a SE-EPI sequence with similarly structured timing and gradient strengths. These sequences ran in and were processed within Agilent’s VNMRJ 3.2 software environment. For the preclinical anatomical references, scanner-provided fast spin-echo multi-shot (SEMS) sequences were used. Scanning parameters for the Titanium-Lego phantom included TE ≈ 50 ms, T a  = 22.0 ms, FOV = 32 × 32 mm2, resolution = 0.5 × 0.5 mm2, 3.0 mm slice. Number of averages: SEMS = 1, SE-EPI = 1 SPEN = 1 and xSPEN = 2. TR: SEMS = 2 s, SE-EPI = 8 s, SPEN = 8 s, xSPEN = 8 s. Additional parameters: SPEN’s encoding bandwidth BW = 18.0 kHz, G y  = 1.33 G/cm; xSPEN’s encoding bandwidth BW = 5.8 kHz, G y  = 0.21 G/cm, G z  = 2.27 G/cm. The DWI parameters for all phantom experiments were δ = 3 ms, ∆ = 12 ms, maximum diffusion gradients G d  = 35 G/cm applied in three orthogonal directions leading to a b max  = 870 s/mm2, six b max -scaling values (0, 0.25, 0.63, 0.77, 0.89 and 1.0). For the ex vivo DWI experiments, xSPEN and SE-EPI parameters were δ = 5 ms, ∆ = 13 ms, maximum diffusion gradients G d  = 26 G/cm applied in three orthogonal directions leading to a b max  = 1370 s/mm2, six b max -scaling values (0, 0.25, 0.63, 0.77, 0.89 and 1.0). Common scanning parameters included TR = 5 s, FOV = 32 × 32 mm2, resolution = 0.5 × 0.5 mm2, 3.0 mm slice thickness. Number of averages: SEMS = 2, SE-EPI = 4, xSPEN = 4. Additional parameters: for SE-EPI TE = 40 ms and T a  = 22 ms; for xSPEN middle TE = 62 ms, T p  = T a /2 = 11.0 ms, encoding bandwidth BW = 3.6 kHz, G y  = 0.13 G/cm, G z  = 1.42 G/cm.

### Clinical scans

Phantom and in vivo human head diffusion maps were acquired at 3 T using a Siemens TrioTIM scanner (Erlangen, Germany) equipped with a 32-channels head coil. Three subjects (males aged 26, 28 and 30) were scanned for improving the optic nerve and ocular globe diffusional statistics. All these experiments were approved by the Internal Review Board WOMC-0091-11 of the Wolfson Medical Center (Holon, Israel) and by Weizmann’s IRB 141-1 protocol; all experiments were performed in accordance with these institutions relevant guidelines and regulations, and images were collected after obtaining suitable informed consents from all the participants. The phantom used consisted of a plastic bottle containing 1.9 L of water with 3.75 g NiSO4*6H2O + 5 g NaCl. For these scans the diffusion xSPEN pulse sequence in Fig. 1b was used, and compared against a double bipolar diffusion SE-EPI sequence39. For the simulations (Figs 2 and S1), the water phantom experiments (Fig. S2) and the lower-resolution human head scans (Figs 4 and 5), the b max used for both xSPEN and SE-EPI had a nominal 1000 s/mm2 value. Additional xSPEN diffusion parameters included δ = 16 ms, ∆ = 57 ms, G d  = 3.26 G/cm. xSPEN scanning parameters for these data were: 48 slices, one nominal |b| = b max value, T p  = T a /2 = 10.9 ms, G y  = 0.027 G/cm, G z  = 1.24 G/cm, chirp bandwidth BW = 2.2 kHz, FOV = 185 × 193 mm2 (PExRO), resolution = 4.0 mm3 isotropic, TR = 25 s, SAR = 90%. Number of averages: SE-EPI = 1, xSPEN = 2. Other SE-EPI scanning parameters: TE = 80 ms, TR = 10 s, FOV = 184 × 256 mm2 (PExRO), resolution = 4.0 mm3 isotropic. For the higher, 3.0 mm3 isotropic resolution head scans (Fig. 6) the b max -value for both xSPEN and SE-EPI was 800 s/mm2. Additional xSPEN parameters: δ = 17.2 ms, ∆ = 49.1 ms, G d  = 2.95 G/cm, 32 slices, one b-value, TR = 15 s, FOV = 90 × 96 mm2 (PExRO), T p  = T a /2 = 7.75 ms, G y  = 0.0504 G/cm, G z  = 1.52 G/cm, chirp bandwidth BW = 2 kHz, SAR = 90%. Additional scanning parameters: Number of averages: SE-EPI = 1, xSPEN = 4. Other SE-EPI scanning parameters: TE = 84 ms, TR = 10 s, FOV = 180 × 252 mm2 (PExRO), resolution = 3.0 mm3 isotropic. T1- and T2-weighted scans are also included as anatomical references.