#### Resolving Chemical Shift Information in SPEN Imaging: Principles and an Extended SR Formulation

To summarize the way by which SPEN can deliver, at the same time, both spatial and spectroscopic aspects about the spins' evolution, we consider the simplest 1D form of quadratic encoding illustrated in Figure 1a. This involves an RF-driven excitation of the spins based on a chirped 90° pulse acting in the presence of an excitation gradient *G*_{exc}, followed by a signal readout under the action of a decoding acquisition gradient *G*_{acq}. The range of the sweep γ*G*_{exc}FOV defines the maximum observed field of view (FOV), and the excitation/acquisition times are tuned to enable the full read-out of the encoded information by fulfilling *G*_{exc}*T*_{exc} = −*G*_{acq}*T*_{acq}. Assuming the encoding occurs along the *y* dimension on a site with chemical shift offset ω_{CS}, the initial excitation will impart the spins with a quadratic phase

- (1)

Subsequent acquisition under a wavenumber *k*(*t*) = γ∫_{0}^{t} *G*_{acq} (*t*^{′})d*t*^{′} acting over the acquisition time *t*, results on a FID signal

- (2)

Here Δ*y* is a nominal spatial resolution defined during the encoding by , and ρω_{cs} is the spatial density image being sought for this particular chemical site. The focal, highest-sensitivity point of the resulting acquisition is then unraveled on a *t-*dependent basis according to the decoding condition

- (3)

The linear ω_{CS}·*t* term modulating the FID in Eq. (2) implies that the sample's chemical shift spectrum can arise from a FT of the signal, provided that the remaining terms are accounted for. As all these remaining exponential terms are given by a series of parameters that are both known a priori and common to all chemically shifted sites in the sample, they can be removed by suitable postprocessing of the FID. In fact multiplying the time-domain signal *S*(*t*) on a point-by-point basis by a suitable gradient-dependent conjugate phasor

- (4)

leads to a usual-looking spectroscopic FID, described in Eq. (4)'s right-hand side under the assumption of multiple CS-shifted sites. With the acquired signal thus modified, Tal and Frydman [21] described a simple series that could yield, for each chemically shifted site, its corresponding spin density image as follows: (i) calculate the modified signal *S*′(*t*); (ii) obtain by Fourier-transform of it the 1D sample's nuclear magnetic resonance spectrum resolving every chemical shifted peak; (iii) separate the individual contributions arising from each ensuing chemical-shifted site by applying a suitable spectral filtering, and (iv) inverse FT each of these filtered peaks to reconstruct—by calculating the magnitude mode of the resulting signals—each chemical site's SPEN image (the phasor multiplication leading from *S*(*t*) to *S′*(*t*) having no effect upon doing such calculation). Notice that unlike what happens with the EPSI sequence (Fig. 1b), such procedure can result on an array of spectrally resolved spatial images without the need of oscillating the imaging gradient.

A main drawback of the original SPEN formulation vis-à-vis normal k-based imaging, rested in the former's lower spatial resolving power per unit gradient strength [20]. For a given set of imaging acquisition parameters *T*_{acq}, Δ*y*, and FOV, SPEN's gradient demands would be √(FOV/Δ*y*) times larger than those of its EPSI counterpart. In actuality, however, a large redundancy of digitized information would then characterize the signal. As discussed in Ref. [22], it is possible to exploit this redundancy using SR formalisms, which go beyond retrieving the sought image from a simple magnitude calculation. In the original, single ω_{CS} = 0 site implementation of this algorithm, this was exploited by recasting the discrete digitized form of Eq. (2) as a system of linear equations

- (5a)

- (5b)

and subsequently solving for by standard minimization methods. Herein the matrix contains the phase encoding arising from the SPEN manipulation, and relates the *N* number of points acquired in the time-domain, with the *M* number of the points desired along the spatial dimension. As *M* is usually large enough to fulfill *M* > (FOV/Δ*y*), an improved spatial resolution could be achieved in this fashion at no extra penalty. To exploit SPEN's full spectroscopic imaging potential, this SR formulation is here extended to the reconstruction of spatially resolved images from multiple chemically shifted sites. To do so, we assume that the acquired signal (Eq. (2)) is proportional to the sum of spin densities arising from all *Q* chemical sites potentially contributing to the sample's spectrum, and its phase modulation reflects both the quadratic *y*^{2} dependence coming from the spatial encoding and an additional linear chemical shift phase modulation. Taking into account the discrete nature of the digitized SPEN signal, of the spatial image being sought and of the chemical shift spectrum, enables one to recast this problem algebraically as

- (6)

which describes each chemical site's spin densities (ρ^{1}…ρ^{Q}) as a (potentially super-resolved) spatially dependent vector of its own. Combining these *Q* individual contributions in a single column vector of dimension *Q*·*M*, it is possible to recast Eq. (6) into a multisite system of linear equations

- (7)

where (1…1) is an all-ones vector of dimension *Q*·*M*, and the *A*_{1}…*A*_{Q} are matrices akin to those in Eq. (5a)—possessing identical spatial point spread functions but differing by their chemical shift evolution phasors, as given in the right-hand side of Eq. (4). Figure 2 illustrates the kind of properties adopted by the resulting “extended” *A*_{ext} block-diagonal matrix for two chemical sites (e.g., fat and water at 3 T) in terms of the magnitude and phase of its components. The latter linear phase modulations along *y*, which were absent in the original SPEN treatment dealing solely with a single on-resonance chemical site, encompass the effects of shifts introduced by the presence of multiple inequivalent chemical sites.

Equation (7) can often be inverted to get a description of the spatial densities ρ_{ext}(*y*) = [ρ^{1}(*y*)…ρ^{Q}(*y*)], characterizing the spatial distribution of every chemical component in the sample. For instance the least squares criterion

- (8)

based on finding *A*_{ext}'s pseudoinverse matrix *A*^{+}_{ext} with the aid of an iterative regularization procedure, can yield a well-defined image description provided that the number of digitized signal points *N* is equal or larger than the number of points *M* × *Q* sought for the combined super-resolved vector [ρ_{1}…ρ_{Q}]. Given, however, that the conditioning of the inversion problem associated to the passage from Eq. (7) to (8) is fairly well behaved, a simpler, noniterative solution using the conjugate gradient method suggested in Ref. [22], can also be used. This involves applying a gaussian weighting on the *A*_{ext} matrix, and performing a single iteration of the form

- (9)

This was found to give good results also in the present spectroscopically resolved case—even if it meant that given a fixed number of sampled data points *N*, a best fit reconstruction of the spatial images arising from each chemical site had to proceed at the expense of reducing the number of points in each image vector by a factor *Q*. This is similar to what was shown to be the case in Refs. [21] and [22] using the filtering method. Yet this will not necessarily always be the limiting case when processing spectroscopic imaging data using the SR approach; for example, when the signal is sparse and/or the difference between the relevant shifts is a priori known (as is often the case when dealing with fat and water), resolution can be improved without sacrifices in the stability of the inversion problem. We consequently found super-resolved solutions stemming from Eq. (9)'s minimization to yield equal or better spectral and spatial results than their filtering-derived counterparts. Results obtained using the previously suggested filtering/magnitude calculation method versus the new suggestion given herein, are further compared later.

#### Pulse Sequencing Considerations

Previous SR-oriented SPEN studies were often based on applying a quadratic spatial encoding on the spins via a sequential excitation imparted by a frequency-chirped 90° pulse. To facilitate multislice acquisitions, however, this study relied on sequences that started with a fixed-frequency 90° slice-selective excitation pulse, and imparted the quadratic spatial encoding sought with a subsequent 180° inversion chirp applied in the presence of an encoding gradient (Fig. 1a, center). While such combination can impart on the targeted slice a parabolic phase profile, the use of an inversion pulse will take as well all remaining spins in the sample away from equilibrium. This effect was counteracted by the introduction of a second, nonselective 180° inversion pulse, akin to that described in Ref. [23] but of a “hard” rather than a “chirp” rewinding character. Although several, equally valid choices could be selected for placing such 180° pulse, Figure 3 shows the version that ended up giving an optimum experimental performance in this study. The remaining of the sequence shown in this figure is of the typical “hybrid” character, where SPEN replaces what would usually be EPI's phase-encoding dimension (“*y*”), and a readout direction (“*x*”) is encoded as is usual in magnetic resonance imaging in the corresponding *k*-domain. Features worth noticing about the resulting sequence include

- a reliance on 180° inversions, which will change the actual chemical shift modulation from the form given in Eq. (4) to
*S*′(*t*) - the sequence's introduction of additional delays between the pulses, capable of defining the exact echoing timing for arbitrary chirped pulse and acquisition durations,
*T*_{180} and *T*_{acq}, respectively - the sequence's need for a sufficiently long acquisition time to resolve among chemically inequivalent peaks as given by (where
*T*_{acq} is a physical evolution time that also takes into account the period taken by the readout decoding). - the benefits that arise by ensuring that the remaining gradient, sweep and timing parameters are chosen so as to achieve a full excitation of the targeted FOV despite of the shielding offsets, that the shift scaling factors is not too small, that appropriate choices are made for the amplitude and the central frequency of the chirped pulses, and that any a priori information available (like the chemical sites' relative displacements) is used in the reconstruction data processing.
- the limitation of the sequence's site-separation abilities to Δ
*B*_{o} field inhomogeneity distortions, which by imposing a corresponding γΔ*B*_{o} line broadening will prevent the resolution of inequivalent sites unless their similarly scaled Δν shift differences exceeds this value.

#### Materials

Besides numerical corroborations of the new procedures here introduced, experiments were conducted to test the method's ability to provide multislice 2D spatial images of chemically distinct species in a single shot. Among the conditions assayed were measurements on phantoms containing various chemical sites at differing concentrations. These experiments were carried out at 7 T on a Varian VNMRS vertical microimaging system. The feasibility and advantages associated with the SPEN-based chemical shift imaging were also examined with a series of in vivo experiments, using two different platforms. Experiments on mice were conducted at 7 T on a Varian VNMRS vertical imaging system using a quadrature-coil Millipede® probe with FOVs of 30 × 30 × 46 mm^{3}. These experiments were performed on the abdominal region, which contains a relative high fat content. These in vivo experiments, as well as all associated animal handling procedures, were done in accordance with protocols approved by the Weizmann Institute's Animal Care and Use Committee. A second set of experiments focused on female human volunteers, were conducted on a 3T Siemens TIM TRIO clinical system using a 4-channels breast coil. This set of experiments sought to verify our new method's ability to separate fat- from water-based (i.e., connective tissue) images in breast and were performed according to procedures approved by the Internal Review Board of the Wolfson Medical Center (Holon, Israel) after obtaining informed suitable written consents.

All the SPEN pulse sequences used in this work were custom written. For the Varian-based experiments RF pulses and gradient shapes were designed in MATLAB® (The MathWorks, Inc., Natick, MA) and uploaded onto the scanner; in the Siemens-based experiments, RF pulses and gradient waveforms were mostly based on available Siemens software. Images were reconstructed in all instances using custom-written MATLAB packages, which included the possibility to process hybrid SPEN-/k-space data with/without SR along the spatiotemporal dimension, and Fourier transformation along the *k*-dimension. Following Ref. [22], manipulations in the SR data processing included the alignment of positive and negative readout echoes; zero-filling, weighting and other conventional procedures were included in the procedure described earlier, as needed.