Natural Linewidth Chemical Shift Imaging (NL-CSI)

Abstract

The discrete Fourier transform (FT) is a conventional method for spatial reconstruction of chemical shifting imaging (CSI) data. Due to point spread function (PSF) effects, FT reconstruction leads to intervoxel signal leakage (Gibbs ringing). Spectral localization by imaging (SLIM) reconstruction was previously proposed to overcome this intervoxel signal contamination. However, the existence of magnetic field inhomogeneities creates an additional source of intervoxel signal leakage. It is demonstrated herein that even small field inhomogeneities substantially amplify intervoxel signal leakage in both FT and SLIM reconstruction approaches. A new CSI data acquisition strategy and reconstruction algorithm (natural linewidth (NL) CSI) is presented that eliminates effects of magnetic field inhomogeneity-induced intervoxel signal leakage and intravoxel phase dispersion on acquired data. The approach is based on acquired CSI data, high-resolution images, and magnetic field maps. The data are reconstructed based on the imaged object structure (as in the SLIM approach) and a reconstruction matrix that takes into account the inhomogeneous field distribution inside anatomically homogeneous compartments. Phantom and in vivo results show that the new method allows field inhomogeneity effects from the acquired MR signal to be removed so that the signal decay is determined only by the “natural”R₂ relaxation rate constant (hence the term “natural linewidth” CSI).

Magnetic resonance spectroscopy (MRS) has proved to be of considerable value in the detection and management of several diseases, including cancer, multiple sclerosis, and psychiatric disorders (1-5). The direct biochemical information that is provided by MRS augments our understanding of the underlying disease mechanisms. There are two major factors that limit the usefulness of in vivo spectroscopy: 1) the signal-to-noise ratio (SNR) and 2) the quality and reliability of spectral localization. In the chemical shift imaging (CSI) method, a set of free induction decay (FID) signals is acquired and phase encoding is performed in the relevant spatial dimensions (6). Two issues limit the spatial resolution in CSI experiments: First, the clinically relevant metabolites occur in low concentrations in vivo and cannot be discriminated from noise when a conventional MR image resolution is used. Second, the need to phase encode in two dimensions makes the data acquisition time excessively long if data are obtained at high spatial resolution (7). Therefore, to improve SNR and keep experimental time within reasonable limits, MR spectroscopic images are often collected and reconstructed from a small set of low spatial frequencies, e.g., a typical 2D (two spatial dimensions and one spectral dimension) CSI experiment can have a spatial matrix of 16 × 16 or 32 × 32.

The most common approach to spatially reconstruct phase-encoded CSI data is the discrete Fourier transform (FT); however, it is well known that the FT is inadequate for truncated data (8). FT reconstruction of spectroscopic imaging data leads to a sinc-like point spread function (PSF) that causes the signal to “ring” away from the main central lobe and contaminate pixels quite distant from the actual source. One way to reduce the ringing caused by truncation is to apodize the k-space data prior to FT. This approach causes blurring, degrades the spatial resolution, and reduces the SNR (per unit time) from optimally achievable levels. Another strategy is to employ a FT series windowing approach whereby a varying number of excitations are acquired based on a filter function, usually Hanning or Blackman functions, and the resulting PSF is free of side lobes (9-12). These windowing functions also change the effective spatial resolution and blur the image, i.e., the spectrum assigned to each voxel is a weighted average of the spectra from neighboring voxels.

Several alternatives to filtering have been reported to cope with cross-voxel contamination in MR spectroscopic imaging (MRSI). One such technique, spectral localization by imaging (SLIM), proposed by Hu et al. (13), is used to improve scanning efficiency and eliminate intervoxel signal leakage. This technique uses model-based reconstruction whereby the spatial coordinates of the boundaries of several compartments of interest with arbitrary shape are determined based on a high-resolution MR image. This information is used to reconstruct the compartmental spectra from the spectroscopic signals obtained with a phase-encoding pulse sequence. A modified version, called generalized SLIM (GSLIM), was introduced to reduce the spectral leakage arising from compartmental inhomogeneity errors in SLIM (14). In a similar vein, Von Kienlin and Mejia (15) developed a spatial localization method, termed spectral localization with optimal point spread function (SLOOP), which like SLIM reconstruction also uses prior information available on high-resolution anatomical MR images to reduce spectral leakage. Similarly, Plevritis and Macovski (16,17) showed that by using the Papoulis-Gerchberg algorithm and anatomic information derived from a proton image, one can produce a spectroscopic image that is better resolved than a zero-filled FT image.

Various other groups have also tried to address the problem of spectral leakage. For example, Hu et al. (18,19) developed a method in which phase-encoded data are acquired with various resolutions and degrees of signal averaging, i.e., a standard MRS data set is acquired with low spatial resolution and high SNR (many averages), and high-spatial-resolution data are acquired with low SNR (few averages). They developed algorithms to combine these two data sets to improve spectral localization and preserve the SNR in the MRS data sets. Using these approaches, Hu et al. (18,19) showed that they could eliminate the leakage of extracranial lipid signal into brain voxels. However, since the final spectroscopic image is generated by FT, this method and other similar approaches do not address the issue of signal leakage between more closely adjacent brain voxels. Non-Fourier encoding methods, such as wavelet encoding (20), have also been suggested to reduce the ringing artifact associated with Fourier encoding, but their application and use have been limited to date.

While the above-mentioned techniques have significantly contributed to improving the quality of CSI results, they do not address another source of spectral leakage and spectral distortion in CSI experiments that, as we will show later, is associated with the magnetic field inhomogeneities present in any MRI or MRS experiment. If a subject is placed in the scanner, magnetic susceptibility variations within tissues and at air/tissue interfaces cause distortions in the magnetic field and introduce inhomogeneities. Some of these inhomogeneities can be removed by shimming the magnetic field, which involves applying small corrective field gradients using a set of specially designed coils for that purpose (21,22). While these coils can partially correct for relatively coarse, low-spatial-frequency field inhomogeneities, they cannot completely eliminate magnetic field inhomogeneities at the interfaces of air, tissue, and bone, where the susceptibility changes rapidly over short distances. MRS studies are increasingly being performed on high-field magnets to improve the amount, quality, and/or specificity of the information content resulting from the increased SNR and chemical shift dispersion of the signals. However, if the increased static magnetic field inhomogeneities experienced at higher fields are not taken into account, they can offset or cancel the SNR and chemical shift dispersion advantages.

Given the inherent low resolution (large voxel size) of spectroscopic imaging, it should be recognized that even after careful shimming these magnetic field variations may be substantial over the extent of the voxel dimensions. In this paper we show that the presence of magnetic field inhomogeneities can cause severe spectral leakage and distortion in a CSI experiment. We also demonstrate a new method that eliminates both these artifacts. First, we outline the theory and demonstrate by simulations the spectral leakage effect due to the field inhomogeneities in a CSI experiment. Importantly, we demonstrate that such spectral leakage is present in both FT reconstruction and non-Fourier-based SLIM reconstruction. We then introduce a new method to eliminate the effect of magnetic field inhomogeneity-induced spectral leakage on spectra from all selected voxels. We show through simulations using this new approach that ideal (noncontaminated) spectra can be reconstructed in all selected voxels. Phantom and in vivo human experiments are also described that demonstrate the effect of spectral leakage and distortion of spectra due to magnetic field inhomogeneities, and confirm the feasibility of this newly proposed method to reduce this effect. To simplify the presentation and analysis, all simulations and results are presented for 1D CSI (one spatial dimension); however, the technique can be easily extended to higher dimensions.

THEORY

MR signal acquired in a uniform external magnetic field from a 1D phase-encoded spectroscopic experiment is given by

$$S(k_n,t)={\int }_{-\infty }^{\infty }{\int }_{\Re }\rho (\Re ,f)e^{-\mathrm{i}2\pi (k_{n}x-ft)}d\Re df$$ [1]

where ρ(ℜ, f) is the spin density function representing the spatial distribution of the spectral information of the excited region R, and k_n is the reciprocal spatial term corresponding to the applied gradients. In MRS experiments only N discrete pieces of information for S(k_n, t) are available at k_n = n/FOV for n = -N/2, ···, N/2 - 1. In a conventional CSI experiment, truncated Fourier series reconstruction is used to reconstruct the approximate solution to ρ(ℜ, f) from S(k_n, t). Since only a small number of central k-space points are acquired (typically 16-32 in spectroscopic imaging), a Fourier series reconstruction leads to severe Gibbs ringing artifact. This ringing causes intervoxel signal leakage and contaminates the signals from individual voxels.

As noted above, Hu et al. (13) developed the SLIM reconstruction technique to reduce scanning time and avoid spectral leakage caused by truncated Fourier series reconstruction. The technique is based on the principle that if the desired object is first segmented into a set of generalized homogeneous compartments, one can then derive the signal from each compartment by fitting the compartment model to the measured phase-encoded data. It is assumed that these homogeneous compartments can be identified on a high-resolution proton image obtained from the same spatial coordinates as the spectroscopic imaging data. Hu et al. (13) showed that, in principle, a set of only M (or more) phase-encoding data is sufficient to obtain spectra from M compartments of arbitrary shape. In an experiment with N phase-encoding steps and M(M ≤ N) homogeneous compartments in the object, they showed that mathematically this problem can be stated as a set of linear equations:

$$s_n\left(t\right)=\sum _m{\mathrm{g}}_{\mathrm{nm}}{\rho }_m\left(t\right),m=1,\cdots ,M;n=1,\cdots ,N,$$ [2]

where s_n(t) is the nth phase-encoded spectroscopic signal, ${\rho }_m\left(t\right)={\int }_{-\infty }^{\infty }\rho (\mathfrak{R},f)e^{i2\pi ft}df$ is an FID signal from the m-th compartment in the absence of phase-encoding gradients, and g_nm is the geometric encoding matrix

$${\mathrm{g}}_{nm}={\int }_{{\Re }_m}{\mathrm{e}}^{-i2\pi k_{n}x}d\Re ,$$ [3]

where the integral is taken over the m-th compartment volume ℜ_m. After calculating the geometric encoding matrix, one can obtain the FID signal from each homogeneous region by solving the set of linear equations given by Eq. [2] in a least-squares sense.

Both discrete FT reconstruction and SLIM reconstruction assume that Eq. [1] represents the acquired phase-encoded spectroscopic imaging data, and ignore the fact that there are always static susceptibility-induced magnetic field inhomogeneities present in an MR experiment. The MR signal in the presence of field inhomogeneities is given by

$$S(k_n,t)={\int }_{-\infty }^{\infty }{\int }_{\Re }\rho (\Re ,f)e^{-i2\pi (k_{n}x-ft)}{e}^{-i\varphi (\Re ,t)}d\Re df,$$ [4]

where

$$\varphi (\Re ,t)={\varphi }_0\left(\Re \right)+\gamma \Delta B\left(\Re \right)t$$ [5]

is the additional phase due to the background magnetic field inhomogeneities, where ϕ₀(ℜ) is due to B₁ (RF) inhomogeneities, and ΔB(ℜ) is the main field inhomogeneity. These magnetic field inhomogeneities lead to well known artifacts in routine clinical imaging; however they are not as severe as in CSI, for two major reasons. First, imaging is usually done at higher resolution and therefore the phase dispersion induced by the magnetic field inhomogeneities over dimensions of a voxel is smaller compared to that encountered in low-resolution CSI. Second, many imaging applications employ either a spin echo, which refocuses the magnetic field gradients at the echo time (TE) when the data are acquired, or short-TE gradient-echo imaging in which this effect is minimized. However, the FID in spectroscopic imaging is usually obtained over a data acquisition time of 0.5-1 s. As phase dispersion across the voxel linearly increases with time, the signal leakage increases as acquisition time increases. As we demonstrate in this paper, even small magnetic field inhomogeneities can lead to severe contamination of the spectral signal in a CSI sequence if they are not accounted for.

Herein we propose a new method, termed natural line-width (NL) CSI, which takes these magnetic field inhomogeneities into account and provides for elimination of their deleterious effects on voxel spectra. This method is a generalization of the SLIM compartmental modeling method, in which we explicitly acknowledge 1) the presence of susceptibility-induced field inhomogeneities in the spatial encoding matrix in addition to the magnitude and direction of the phase-encoding gradients, and 2) the size, shape, and location of the compartments. The new spatial encoding matrix is now given by

$$g_{nm}\left(t\right)={\int }_{{\Re }_m}{e}^{-i2\pi k_{n}x}{e}^{-i\varphi (\Re ,t)}d\Re ,$$ [6]

where ϕ(ℜ, f) given by Eq. [5] is the phase introduced by magnetic field inhomogeneities. Instead of Eq. [2], a new equation describing CSI signal can be written in matrix notation as

$$S\left(t\right)=\mathrm{G}\left(t\right)\rho \left(t\right),$$ [7]

where the encoding matrix G(t) is given by Eq. [6]. The FID signal can now be obtained for each defined compartment by solving the set of linear equations given in Eq. [7]. The important aspect is that the encoding matrix g_nm(t) is now a function of time and takes into account the presence of magnetic field inhomogeneities. This method allows complete removal of field inhomogeneity effects from the acquired MR signal so that the signal decay is determined only by the R₂ relaxation rate constant (hence the term “natural linewidth” CSI).

MATERIALS AND METHODS

Simulations

To demonstrate the effect of magnetic field inhomogeneities on data obtained by means of CSI, we conducted several 1D simulations. The purpose of these simulations is to demonstrate that even small magnetic field inhomogeneities can lead to severe spectral leakage of voxel signals in both FT and SLIM reconstruction, and can be removed by the proposed NL-CSI method.

We used 1D simulations with the following parameters: field of view (FOV) = 256 mm, number of phase-encoding steps = 16, vector size = 1024 (in the time domain), and spectral bandwidth = 2000 Hz, resulting in an FID duration of 0.512 s. We assumed that the phase-encoding gradient duration was 750 μs, which would give a maximum phase-encoding gradient strength of 0.979 mT/m. The choice of these parameters is within the practical ranges that are generally used for clinical spectroscopic imaging. We also introduced a background magnetic field gradient that was 1% of the maximum phase-encoding gradient strength, i.e., 9.78 × 10^-3 mT/m. This is a very small gradient, and stronger susceptibility-induced magnetic field inhomogeneities are expected in vivo, especially at the interfaces of air, tissue, and bone. The direction of the background magnetic field gradient was chosen to be the same as the phase-encoding gradient. This is to include both effects of intervoxel signal leakage and intravoxel signal dispersion. A choice of gradient perpendicular to the phase-encoding gradient direction would cause only intravoxel signal dephasing. We considered two kinds of simulated objects, as shown in Fig. 1a and 2a. In the first case the object perfectly occupies a single voxel (16 mm) in the FOV (an ideal case when discrete FT reconstruction results in minimal leakage). In the second case the object occupies six voxels (96 mm, approximately on the order of the size of human brain) in the FOV. Simulated data were generated by numerical integration over the object, and for SLIM and NL-CSI reconstruction the object was considered as discrete with a pixel resolution of 0.5 mm.

FIG. 1.

a: Location and size of the object (shown in gray) used in the simulations. The vertical lines represent the CSI voxel locations. Only voxels 9 and 10 were used as regions for SLIM and NL-CSI reconstruction. b: FID signal from voxel 9. c: FID signal from voxel 10 using three different spatial reconstruction algorithms. Discrete FT reconstruction leads to an approximately 10% signal loss at time zero and a sinc-like signal decay of the FID in voxel 9. The signal in the voxel 10 exists solely due to the leakage effect and initially grows with time. SLIM reconstruction recovers the FID signal at time zero in voxel 9 but resembles FT reconstructed signal for times greater than zero. NL-CSI reconstruction fully recovers the “inhomogeneity-free” FID signal and there is no leakage in neighboring voxels.

FIG. 2.

a: Location and size of the object used in the simulations. The size of the object is 96 mm and the FOV is 256 mm. The vertical lines show the location of voxels when the discrete FT is used for spatial reconstruction. For SLIM and NL-CSI reconstruction voxels 6-12 were chosen as regions (total = 7). Only half regions of voxels 6 and 12 were selected (i.e., where the object is present). b and c: Reconstructed signals from voxels 6 and 9 with T₂ = ∞. FT and SLIM reconstructions show substantially nonlinear behavior. In voxel 6 the signal peaks around 70 ms and then decays rapidly. NL-CSI reconstruction shows constant linear signal with time, as expected. d and e: Reconstructed signals from voxels 6 and 9 with T₂ = 50 ms. The plots show that FT and SLIM reconstruction lead to severely distorted FID signal, whereas NL-CSI reconstruction shows true exponential decay of the signal.

In the first set of simulations T₂ was chosen to be infinite. This was done to emphasize the role of magnetic field inhomogeneities in signal leakage and dispersion. In the second set of simulations we assumed T₂ = 50ms, which is typical for in vivo brain spectroscopy at 1.5-3T. The simulated data were spatially reconstructed using discrete FT, SLIM reconstruction, and the proposed NL-CSI method.

Phantom Study

A spherical phantom (∼10 cm diameter) containing 100 mM NAA, 50 mM KH₂PO₄, and 56 mM NaOH in deionized water was prepared and 1% agarose was added to the mixture to make it gelatinous. All experiments were done on Siemens 3T Magnetom Allegra system (Siemens Medical Systems, Erlangen, Germany) and the data were collected using a circularly polarized head coil. A 1D spin-echo CSI sequence was developed with an adiabatic refocusing pulse (23) to improve the slice profile of the excited region. A 16 mm × 16 mm wide region was excited and spectroscopic imaging experiments were performed using FOV = 256 mm, TR/TE = 1000/40 ms. Non-water-suppressed spectra were acquired with 16 and 32 phase-encoding steps and eight and four averages, respectively, giving a Fourier voxel resolution of 16 × 16 × 16 mm³ and 16 × 16 × 8 mm³. A vector size of 1024 with 2000 Hz bandwidth was acquired with a data collection time of 0.512 s. To demonstrate the effect of magnetic field inhomogeneities on CSI, only magnet default values were used for shimming.

3D high-resolution images from the same spatial coordinates used for the CSI data were also collected using a multi-gradient-echo pulse sequence with isotropic 1 mm³ resolution at two different TEs of 3 and 20 ms. The modified sequence acquired k-space data at both TEs with positive readout gradients to reduce the effect of image distortions and eddy currents on the phase images. Shim settings were kept the same as the 1D CSI experiments to evaluate field inhomogeneities needed for NL-CSI reconstruction of CSI data. Other imaging parameters were as follows: TR = 25 ms, FOV = 256 × 256 × 40 mm³, and matrix size = 256 × 256 × 40, for a total data acquisition time of 4.5 min. Using phase images, background magnetic field inhomogeneities were calculated as described in the Data Processing section below.

A localized time domain MR FID signal was also collected using a point-resolved spectroscopy sequence (PRESS) and used as a standard for comparison with the reconstructed data. The size of the voxel was 10 × 10 × 10 mm³ and it was placed in the center of the phantom to reduce magnetic field inhomogeneities originating from the air/phantom interface. The magnet was shimmed manually to achieve a linewidth of <10 Hz. Other parameters for the single-voxel experiment were TR = 1500 ms, TE = 40 ms, BW = 2000, vector size = 1024, and number of averages = 64.

In Vivo Study

1D CSI was performed on healthy volunteers after they gave informed consent and the experimental protocol was approved by the Washington University Medical School IRB. Non-water-suppressed 1D CSI data were acquired with an excited region of 16 mm × 16 mm, FOV = 256 mm, TR/TE = 1500/40 ms, vector size = 1024, bandwidth = 2000 Hz, and data collection time = 0.512 s. Sixteen phase-encoding steps were acquired with 16 averages giving a Fourier voxel resolution of 16 × 16 × 16 mm³. Shimming was performed over the excited volume. 3D high-resolution images for field mapping were also acquired from the same location with parameters as described above for the phantom study.

Data Processing

All data were processed by software developed in MAT-LAB (The Mathworks, Natick, MA, USA). 3D magnitude and phase images were generated from the raw data set of the 3D imaging experiment and phase images were unwrapped using custom-developed 3D unwrapping algorithms based on mask cut and quality-guided maps (24). Magnitude images were used as references to segment the object into homogeneous compartments of approximately 16 × 16 × 16 mm³ regions. In order to compare the results of SLIM and NL-CSI reconstruction with discrete FT reconstruction, we chose the size and location of the regions to closely match with the voxels defined when the discrete FT was used to spatially reconstruct the CSI data. 3D phase maps of each voxel were fitted to second-order polynomials to generate noise-filtered phase maps in order to improve the SNR. These filtered phase maps were then used to generate the ϕ_o and ΔB (Eq. [5]) distributions in the voxels to be used in calculating the spatial encoding matrix as given in Eq. [6]. This encoding matrix could then be used with the phase-encoded 1D CSI data to calculate the natural linewidth FID signal from each voxel using Eq. [2] in a least-squares sense by evaluating

$$\underset{\rho }{\min }{\parallel \mathrm{G}\left(t\right)\rho \left(t\right)-S\left(t\right)\parallel }^2$$ [8]

We note that the encoding matrix G(t) may become ill-conditioned such that its true rank cannot be determined and a least-squares solution may not be an option (25). Several regularization methods have been developed to overcome such ill-conditioned problems (26-28). Of these methods, Tikhonov regularization is the most widely accepted (26,29). Tikhonov regularization replaces the inverse least-squares problem with a constrained least-squares problem, and the idea is to find a regularized solution that minimizes the weighted combination of the residual norm and the constraint as follows:

$$\underset{\rho }{\min }[{\parallel G\left(t\right)\rho \left(t\right)-S\left(t\right)\parallel }^2+\lambda {\parallel {\mathrm{L}}_{\rho }\left(t\right)\parallel }^2]$$ [9]

where L approximates the first derivative operator (Tikhonov first-order regularization) and the regularization parameter λ controls the weight given to minimization of constraint relative to minimization of the residual norm. The optimal regularization parameter λ is usually calculated by using the L-curve method (30). For this study we explored two approaches of choosing the regularization parameter: first λ was calculated using the L-curve method, and second the regularization parameter was made a function of time: λ(t) = λ × logspace(-1,1, vector size), i.e., spanning a range from 0.1λ to 10λ. The advantage of this approach is that for the initial parts of the FID signal where the signal dominates the noise, λ is small (much less than that determined by L-curve) and least-squares part dominate in Eq. [9]. Conversely, for the later parts of the FID where noise dominates the signal and little useful information is present, the side constraint dominates the solution. This regularization approach not only helps ensure the stability of the solution, it also helps prevent magnification of the noise.

Noise Comparison

In previous sections we compared the signal achieved with the three different reconstruction approaches. The effect of these reconstruction approaches on the noise was also compared for both simulation and in vivo studies. For simulation studies multivoxel data were generated by adding random noise to the simulated signal. The standard deviation (SD) of the noise was calculated for the reconstructed time domain FID signals in each voxel. For simulation studies the noise was calculated separately for the first and last 25 ms of the reconstructed signal. For in vivo experiments the noise was calculated only for the last 25 ms of the FID signals, since the lipid and metabolite signal oscillations have a higher amplitude than the noise during the first 25 ms.

RESULTS

Simulation Results

Figures 1a and 2a show the location of objects and voxels for 1D simulation experiments. The gray area defines the extent of the objects, and the white lines represent the boundaries of voxels as defined when the discrete FT is used to spatially reconstruct the spectroscopic imaging data. Figure 1b shows the reconstructed FID of voxel 9 from the simulated data when the object occupies only one voxel in the FOV and T₂ = ∞. With the discrete FT and SLIM reconstructions, the resulting FID signal intensity shows a sinc-like decay with time and passes through a null at approximately 160 ms. At each time point the FT reconstruction also shows decreased signal compared to SLIM reconstruction; for example, at time point zero the relative FT-reconstructed FID signal is ∼0.9 as compared to 1.0 for SLIM reconstruction. This difference is due to Gibbs ringing, which is inherent to FT reconstruction and is present at every time point of the FID signal. This Gibbs ringing is removed by SLIM reconstruction at time point zero; however, background magnetic field (inhomogeneity) gradients lead to signal decay with increasing time in SLIM reconstruction. This signal decay is usually interpreted as $T_2^{\ast }$ decay due to intravoxel signal phase dispersion in the presence of field inhomogeneities. In fact, this interpretation is not quite correct, because signal spreadout to the surrounding voxels substantially contributes to the signal formation if FT or SLIM reconstruction is employed (see Discussion). Using NL-CSI reconstruction, which takes into account the background magnetic field inhomogeneities, the ideal FID, which is unity for all time, is recovered. Figure 1c shows a reconstructed signal in voxel 10 of Fig. 1a where no object is present. This plot demonstrates the spectral leakage of signal from voxel 9 when FT or SLIM reconstructions are used. When NL-CSI reconstruction is used there is no leakage signal in neighboring voxels. SLIM reconstruction has this property only at t = 0, when the contribution of field inhomogeneity is not present.

When the object occupies several voxels of the FOV, the background magnetic field inhomogeneities can lead to erratic signal leakage (“contamination”) with both SLIM and FT reconstruction. Figure 2b and c show representative FID signals from two voxels (6 and 9 in Fig. 2a), with T₂ = ∞. Contrary to the expected $T_2^{\ast }$-like decay, both SLIM and FT reconstruction show “unusual” oscillatory intensity behavior for short times and rapidly decaying intensity at longer times. For example, in voxel 6 the signal actually demonstrates a twofold intensity increase over the ideal signal at ∼70 ms, followed by an almost exponential intensity decrease with time. NL-CSI reconstruction reproduces the ideal signal, is not affected by field inhomogeneities, and is constant with time, having an intensity of 0.5 in voxel 6 and 1.0 in voxel 9.

The results obtained with the same simulated object but with T₂ = 50 ms are shown for voxels 6 and 9 in Fig. 2d and e. In both voxels the signal shows substantial deviations from true exponential decay when discrete FT or SLIM reconstruction is used for spatial reconstruction. When NL-CSI reconstruction is used with the same simulated data, the results produce true exponential decay. Phased absorption mode spectrum of the simulated signal from voxel 9 reconstructed with all three methods is shown in Fig. 3. With FT and SLIM reconstruction the spectrum is distorted, and these distortions are removed when NL-CSI reconstruction is used.

FIG. 3.

Absorption mode spectra from voxel 9 in Fig. 2a of the multivoxel simulation experiment using (a) FT, (b) SLIM, and (c) NL-CSI reconstruction.

Experimental Results

Figure 4 shows on a semi-log scale the magnitude of the FID signal vs. acquisition time. When 16 phase-encoding steps are used for data acquisition, both FT (Fig. 4a) and SLIM (Fig. 4b) reconstruction show that the magnitude of the FID signal decays in a non-monoexponential fashion (i.e, it is distorted) even though the voxel (voxel 1, Fig. 4a inset) is near the center of the phantom. The FID signal after NL-CSI reconstruction (Fig. 4c) shows that most of the distortions are removed and the signal compares favorably with the reference spectrum (Fig. 4d) obtained using a PRESS sequence. This confirms that the distortions seen in the FID signal with the FT and SLIM reconstructions are due to the background magnetic field inhomogeneities and can be mostly eliminated using the NL-CSI approach.

FIG. 4.

Images of the spherical phantom showing the 1D CSI grid and FT voxel locations with 16 and 32 phase encodes are shown as insets. For SLIM and NL-CSI reconstruction, multiple regions were selected only in the phantom, and the regions closely matched the FT voxel locations and size. The magnitude of the FID signal from voxels 1 and 2 plotted on a semi-log scale is shown: (a and e) FT reconstruction, (b and f) SLIM reconstruction, (c and g) NL-CSI reconstruction, and (d and h) PRESS FID signal from a well-shimmed single-voxel spectroscopy experiment. FT and SLIM reconstructions show nonlinear behavior of the FID signal. NL-CSI-reconstructed signal shows linear behavior, which agrees very well with the reference single-voxel spectroscopy signal.

When 32 phase-encoding steps are used for a CSI experiment, the resultant voxel dimension is reduced by half, and hence the magnetic field variation across the voxel is also reduced. Even in this case the FT (Fig. 4e) and SLIM (Fig. 4f) reconstructions show distortions of the FID signal from a selected voxel (voxel 2, Fig. 4e inset). However, this distortion is significantly reduced compared to the experiment with 16 phase-encoding steps. NL-CSI (Fig. 4g) reconstruction in this case leads to almost perfect reconstruction of the signal, and the results exactly match the reference single-voxel spectroscopy data (Fig. 4h).

Figure 5 (left column) shows the phased absorption mode water spectrum of voxel 1 indicated in Fig. 4a. Spectral distortions are visible with the FT (Fig. 5a) and SLIM (Fig. 5b) methods of spatial reconstruction, whereas NL-CSI (Fig. 5c) almost completely removes these distortions and agrees with the reference spectrum from the PRESS sequence (Fig. 5d). Figure 5e (FT), f (SLIM), g (NL-CSI), and h (PRESS) show the spectrum after water peak was removed using the SVD filter in the JMRUI (31) package, and the scale is expanded to show the spectrum from the metabolite resonances. NL-CSI reconstruction demonstrates marked improvement in the lineshape of the metabolite spectra and improved discrimination of the two peaks at ∼2.7 ppm.

FIG. 5.

The left column shows the absorption mode spectra (water peak only) from voxel 1 shown in Fig. 4. (a) FT and (b) SLIM spatial reconstructions of the phase-encoded data show distortions of the spectrum. (c) NL-CSI reconstruction leads to removal of this distortion, as shown by improvement of the lineshape, and agrees with the (d) PRESS-acquired spectrum. In the right column the scale is expanded to show the metabolite peaks in the spectrum: (e) FT, (f) SLIM, (g) NL-CSI, and (h) PRESS. NL-CSI reconstruction show significantly improved lineshape over FT and SLIM reconstructions.

In Vivo Results

Figure 6a shows a gradient-echo scout image overlaid with the location of the 16 mm × 16 mm 1D interrogated CSI region and resultant FT voxels. For SLIM and NL-CSI reconstruction, the skull and scalp were chosen as two compartments and the brain was segmented into compartments to closely match the FT voxels. Figure 6b-d show the magnitude of FIDs plotted on a semi-log scale from voxels marked in Fig. 6a. FT- and SLIM-reconstructed signals from three representative voxels clearly show signal distortions as compared to the NL-CSI-reconstructed signal. In all three voxels the NL-CSI approach effectively removes the distorting effects of field inhomogeneities, leading to almost pure monoexponential signal decay, which is seen as linear decay on the semi-log scale. High-frequency oscillations are visible on the FT-reconstructed signal at short times due to the extracranial lipid signal that leaks into the signal from brain voxels when FT reconstruction is used. This extracranial lipid signal leakage is also eliminated with NL-CSI reconstruction, resulting in uncontaminated FID signal from the voxels.

FIG. 6.

(a) Gradient-echo reference image of a head, showing the location of 16 mm × 16 mm 1D excited region and FT voxel locations. Parts b-d show on a semi-log scale the magnitude of the FID signals from voxels marked in a. The FT-reconstructed signals from three representative voxels show nonlinear signal decay due to field inhomogeneities. The NL-CSI reconstructed signal in all three voxels demonstrates the effectiveness of the technique in removing the field inhomogeneity. High-frequency oscillations are clearly visible on the FT-reconstructed signal due to the extracranial lipid signal that leaks into the signal of brain voxels when FT reconstruction is used.

Figure 7 shows the representative spectrum from the voxel marked as “c” in Fig. 6a. The water peak was removed and the scale was expanded to show the spectrum from the metabolite resonances. No line-broadening was applied to the signal. FT reconstruction (Fig. 7a) shows that even halfway into the brain there is extracranial lipid leakage. It can be seen that both SLIM (Fig. 7b) and NL-CSI (Fig. 7c) are very effective in removing this extracranial lipid spectral leakage, but the lineshape is improved with NL-CSI.

FIG. 7.

Absorption mode spectra after (a) FT, (b) SLIM, and (c) NL-CSI spatial reconstruction of CSI data from voxel “c” shown in Fig. 6a. No line-broadening was applied. FT reconstruction leads to significant extracranial lipid signal leakage even into a voxel near the center of the brain. SLIM and NL-CSI are very effective in removing the lipid leakage (see Discussion), and NL-CSI provides better lineshape than SLIM reconstruction.

Noise Comparison

Table 1 shows the noise calculated for all three reconstruction approaches. Simulation studies show that both SLIM and NL-CSI have similar noise for the first 25 ms of the FID, which is slightly higher than the noise in the FT-reconstructed data. For the last 25 ms of the FID (when the signal has decayed to zero and only the noise remains), the NL-CSI shows significantly increased noise levels over FT and SLIM reconstruction, but regularization appreciably reduces this noise. The in vivo data also show similar noise performance, and NL-CSI reconstruction with regularization has noise values comparable to those of the discrete FT and SLIM reconstructions.

Table 1.

Noise Comparison^*

	Fourier	SLIM	NL-CSI
			a	b	c
Simulation
First 25 ms	0.49 ± 0.06	0.62 ± 0.09	0.62 ± 0.05	0.33 ± 0.03	0.60 ± 0.04
Last 25 ms	0.45 ± 0.04	0.57 ± 0.05	15.9 ± 6.8	2.4 ± 0.35	2.5 ± 0.01
In vivo
Last 25 ms	0.5 ± 0.35	0.51 ± 0.34	4.5 ± 3.7	0.6 ± 0.12	0.45 ± 0.12

^*Noise is shown as mean ± SD for all voxels and all the noise values are scaled by 10⁴. a = NL-CSI solution with no regularization, b = NL-CSI with regularization coefficient λ, c = NL-CSI with regularization parameter λ(t) as described in Methods.

DISCUSSION

The presence of a person or an object in the MR scanner causes magnetic field inhomogeneities even in an ideal magnet. Careful shimming can reduce these field inhomogeneities but cannot fully eliminate them. These field inhomogeneities are significant at the interfaces of air, tissue, blood, and bone, where the susceptibility changes rapidly over short distances. In this paper we have shown both theoretically and experimentally that even very small background magnetic field inhomogeneities can lead to severe spectral leakage between voxels and contaminate the quantitative results when FT is used for spatial reconstruction of CSI data. As in vivo MRS moves to higher-field-strength magnets to take advantage of greater SNR and chemical shift dispersion, these higher field strengths are also accompanied by increased magnetic field inhomogeneities. Our results show that if these background field inhomogeneities are not taken into account, the advantages of moving to higher field may be compromised. We have introduced a non-FT-based reconstruction method (NL-CSI) that eliminates or reduces signal distortions resulting from these magnetic field inhomogeneities, and thus produces near ideal (by content and lineshape) spectra from each voxel.

In a CSI experiment, phase encoding is performed for spatial localization and the FID signal is recorded in the absence of any externally applied field gradients. In a spin-echo CSI sequence (the most commonly used CSI sequence), the data acquisition usually starts at the middle of the spin echo and continues for typically 0.5-1 s after the echo. At the spin echo, the phase accumulated due to background static field inhomogeneities is refocused and the contribution of γΔB(Rt term in Eq. [5] to the measured signal is nulled. However, this contribution increases linearly with time. The effect of this term is twofold: First, it produces phase dispersion inside each voxel, leading to a well-known effect of signal decay in each voxel ($T_2^{\ast }$-type phenomena). Second, if the magnetic field gradient has a component along a phase-encoding direction, it introduces an additional time-dependent phase. When FT is performed to spatially localize the phase-encoded data, this additional phase dispersion can lead to a situation in which spins in a particular voxel are mapped to neighboring voxels, resulting in contamination of their signal and vice versa. This contamination effect increases as signal acquisition progresses and is in addition to the previously well-documented sinc-like PSF behavior inherent to discrete FT.

Simulations show that even when a small linear background field gradient is present, FT reconstruction leads to a sinc-like signal decay with time. When the object occupies only one voxel in the FOV, the effect is easily predictable from Eq. [1]. However, objects that occupy a single voxel in the FOV are not generally encountered in real-life experiments. For an object occupying multiple voxels, even the presence of a small linear background field gradient leads to a complicated signal behavior. When FT reconstruction is used (Fig. 3), signals from different voxels severely contaminate each other. The background field inhomogeneities in this case can cause additional signal leakage, leading to near doubling of the actual signal from a given voxel at a specific time. This result is rather unexpected because it is usually assumed that magnetic field inhomogeneities lead to increased signal decay, not amplification.

In real experiments, heterogeneous objects cause background field inhomogeneities that are nonlinear and the strength of these inhomogeneities can be much larger than what we chose for our simulations, especially near tissue, air, and bone interfaces. In this scenario there can be severe signal contamination in spectroscopic imaging experiments if these field inhomogeneities are not taken into account while the phase-encoded data are reconstructed, and the resulting quantitative estimates of the metabolite signals may be erroneous. We have already discussed and demonstrated how this effect manifests itself in phantom studies. Our in vivo results also clearly demonstrate this point. The signal decay obtained with FT reconstruction is highly nonlinear on a semi-log scale, demonstrating that it is contaminated and distorted due to field inhomogeneities. For example, the signal can decrease more rapidly (Fig. 6c and d) or increase above (Fig. 6b) the true brain signal. The signal in a given voxel can only increase if the signals from neighboring voxels bleed into this given voxel, an effect that can be seen from the simulation studies. When FT reconstruction is used, static field inhomogeneities often lead to increased rates of signal decay. The signal from voxel “c,” for example, is decaying very rapidly due to the presence of a large blood vessel, seen in the scout image (Fig. 6a), which causes marked field inhomogeneities. The NL-CSI method is very effective in removing effects of such field inhomogeneities and recovers the “natural” signal from the voxel. When the JMRUI software package (31) was used to quantify the spectroscopy data using the Hankel Lanczos singular values decomposition filter (HLSVD) signal quantitation algorithm (32), water and NAA decay rates showed a decrease of 23% and 52%, respectively, compared to the FT-reconstructed data.

Another advantage of the proposed technique is the removal of extracranial lipid signal that leaks into brain voxels when FT reconstruction is used, as shown in Fig. 7a. There are two sources of intervoxel signal leakage in FT reconstruction: a sinc-like PSF and field inhomogeneities. Even for a voxel in the center of the brain, there are significant amounts of lipid spectral leakage, which can dominate the signals from metabolites. The most common approach to reduce lipid leakage is to use outer volume suppression pulses to saturate the lipid magnetization. Although this approach may eliminate the lipid spectral leakage, it cannot prevent intervoxel signal contamination from water and metabolite signals, which will compromise quantitative results. Both SLIM and NL-CSI are very effective in removing the lipid spectral leakage. SLIM is effective in removing the lipid leakage because $T_2^{\ast }$ of lipids is significantly shorter than that of water and metabolites. The source of spectral leakage in SLIM reconstruction arises only from field inhomogeneities, since most of the lipid signal decays before the signal distortion due to field inhomogeneities starts to dominate. However, SLIM still allows spectral leakage of water and metabolite signals due to field inhomogeneities, as can be seen by the nonlinear FID signal decay on semi-log plots (Fig. 6). The NL-CSI approach is very effective in removing both these sources of signal leakage.

It is important to note that spectral distortions are not only present in voxels near the phantom or tissue/air interfaces, where field inhomogeneities are greatest; significant signal distortions also occur in voxels in the center of a spherical phantom or the brain. With NL-CSI reconstruction the ideal signal is nearly restored as compared to a single-voxel spectroscopy experiment, which is used as a benchmark in this study. Following application of NL-CSI, only very small signal deviations from the reference FID signal are visible on plots (Fig. 4c and g). This may be due to errors in the phase maps or incorrect localization of the region of interests (ROIs) on the phase maps. With better phase-mapping sequences and localization techniques, these small distortions can be further reduced.

For any technique involving spectroscopic imaging, the SNR is always of great concern. Our data show that direct NL-CSI reconstruction can lead to higher noise at the end of the reconstructed FID signal (i.e., where the signal has decayed to the noise level). The reason for this increased noise is a decrease of the matrix coefficients of the NL-CSI encoding matrix (Eq. [6]) with time. Thus a least-squares solution to (Eq. [7]) leads to a higher noise in the reconstructed FID signal. It is also demonstrated that using regularization the noise is appreciable reduced and a more stable solution to Eq. [7] is obtained. It is important to note that this increased noise is observed when the signal has decayed to the level of noise and there is not much useful information present in the FID signal. For the initial part of the FID signal (where the SNR is high), the noise in NL-CSI reconstruction is similar to that in SLIM reconstruction and comparable to that in FT reconstruction.

Also, it should be noted that in this study we used a simple regularization approach. A variety of different regularization approaches are available to solve such systems of linear equations, and they need to be evaluated to determine the best possible approach that will maximize the SNR and reduce errors introduced by reconstruction. We propose to use the Twomey regularization method that minimizes the difference between the solution and some estimate of the solution (33). A simple Lorentzian model can be used for this estimated signal in Twomey regularization. Using this approach and a good choice for the modeled signal, high SNR can be achieved without sacrificing the accuracy of the results. Another elegant approach was proposed by Webb et al. (34), who used phase information over a voxel to estimate the $R_2^{\ast }$ signal decay rate, and a model-based approach to estimate the amplitudes of the peaks. There are two main advantages of this approach: First, they were able to estimate the decay rate and used this information to account for the phase dispersion in the voxel. Second, since all of the data are used to evaluate a small number of model parameters, the method is highly effective in cases with low SNR. An extension of this approach coupled with NL-CSI can be used to directly evaluate the resonances in a spectroscopic imaging sequence. With this approach, complete phase-encoded data and a time-dependent geometric encoding matrix (Eq. [6]) can be used to evaluate the model parameters. This could present a robust and efficient means of analyzing the data.

One approach to further improve these results is to use higher-spatial-resolution acquisition strategies previously used with FT reconstruction (35). A smaller voxel size results in reduced field inhomogeneity-induced phase dispersion across the voxel, and therefore improved spectral quality for MRSI. Increased resolution also leads to reduced leakage and contamination of the FID signal as compared to low-spatial-resolution spectroscopic images. However, significant leakage/contamination can still be present in a high-resolution CSI experiment, as can clearly be seen as nonlinear signal behavior on the semi-log plots in Fig. 4e (FT) and 4f (SLIM). For absolute quantification of resulting metabolite signals, this high-resolution approach coupled with discrete FT reconstruction of phase-encoded spectroscopic imaging data may still lead to unacceptable signal contamination. However, this high-spatial-resolution strategy coupled with NL-CSI reconstruction eliminates intervoxel signal contamination, resulting in near-idealized voxel spectra (Fig. 4g). A potential disadvantage of using this high-spatial-resolution acquisition strategy is a reduced SNR ratio per unit time, but the accuracy of the results and improved spectral lines can more than compensate for this. As demonstrated by Li et al. (36), reducing the voxel dimensions may increase $T_2^{\ast }$ and partially compensate for signal loss with decreased voxel volume. However, this high-resolution approach cannot be extended indefinitely, as eventually the reduced SNR will make identification and quantification of metabolite resonances impractical.

Hu et al. (13) proposed SLIM reconstruction as an alternative to FT reconstruction to overcome the limitations of the FT approach and to speed up imaging time by reducing the number of phase-encoding steps. The SLIM approach exploits the structural information available from a high-resolution MR image to reconstruct the spectroscopic imaging data. As shown in Figs. 2 and 3, at time zero (when field inhomogeneities are refocused in a spin-echo experiment), SLIM reconstruction results in the true signal from each voxel. Thus, at time zero, SLIM reconstruction is effective in eliminating the Gibbs ringing artifact inherent to the FT approach. However, in the presence of background field inhomogeneities, SLIM reconstruction also leads to severe signal contamination due to the intervoxel signal leakage. This contamination can be even more severe than that encountered with the FT reconstruction, as shown in Fig. 2. This effect is due to violation of the primary assumption in the SLIM approach, specifically that the voxel is homogeneous. Even if the original voxel has a homogeneous spin density, the presence of magnetic field inhomogeneities compromises this homogeneity in a time-dependent manner. NL-CSI uses a modified spatial encoding matrix that incorporates information on background field inhomogeneities to significantly reduce or even eliminate signal contamination, and thus produces near-ideal signal from selected voxels or ROIs.

Although we have shown that NL-CSI technique is very effective in producing noncontaminated voxel spectra, it should be noted that it is based on certain assumptions and can produce erroneous results if the assumptions are violated. One important constraint for the NL-CSI approach to be effective is the requirement that segmented compartments should be homogeneous. Liang and Lauterbur (37) previously showed that compartmental inhomogeneity can lead to intervoxel signal leakage when SLIM reconstruction is used. To reduce the effects of intervoxel signal contamination due to compartmental inhomogeneities in SLIM reconstruction, Liang and Lauterbur (14) introduced GSLIM reconstruction approach for spectroscopic imaging data. They showed that with the use of GSLIM, intervoxel signal leakage due to compartmental inhomogeneities can be significantly reduced. In this study we did not address the issue of compartmental inhomogeneity, since our focus was to demonstrate the effect of field inhomogeneity on intervoxel signal leakage. Compartmental inhomogeneity was not an issue for the simulation and phantom studies because the compartments were truly homogeneous. For the in vivo study, the compartments were chosen to be aligned with FT voxels to facilitate direct comparison. Although this violates the condition that the compartments should be homogeneous, the results demonstrate that even with this selection of compartments, NL-CSI resulted in significantly improved lineshapes and an almost complete removal of extracranial lipid leakage as compared to FT reconstruction. Further improvements can be achieved by careful automatic segmentation of the high-resolution images. Also, the NL-CSI technique can be incorporated with GSLIM reconstruction to improve the results presented herein and reduce the effect of compartmental inhomogeneity.

Obviously, this approach can be used in addition to, but not as a substitute for a good shimming of the magnetic field. Strong magnetic field inhomogeneities can cause geometric distortions of the gradient-echo images used to obtain the high-resolution base images and to determine the field maps used for data reconstruction. This may lead to inaccurate spatial and phase information in the encoding matrix, leading to erroneous results. In addition, a highly inhomogeneous field will cause the FID signal to decay below the noise level very quickly, and no reconstruction approach will be able to recover it. It should also be noted that field mapping adds additional time to the study; in this study approximately five additional minutes were needed to acquire complete data sets for NL-CSI. However, we believe that this additional time is certainly not prohibitive in clinical settings and can easily be incorporated into the protocol.

CONCLUSIONS

Even the small magnetic field inhomogeneities that are present during any spectroscopic imaging experiment can lead to severe contamination of spectroscopic data if the discrete FT is used for spatial reconstruction. A new method, NL-CSI, takes into account magnetic field inhomogeneities and produces spectral maps that are nearly free of intra- and intervoxel contamination, and have significantly improved lineshape.