Localization Errors in MR Spectroscopic Imaging due to the Drift of the Main Magnetic Field and their Correction

Abstract

PURPOSE

To analyze the effect of B₀ field drift on multi voxel MR spectroscopic imaging and to propose an approach for its correction.

THEORY AND METHODS

It is shown, both theoretically and in a phantom, that for ~30 minute acquisitions a linear B₀ drift (~0.1 ppm/hour) will cause localization errors that can reach several voxels (centimeters) in the slower varying phase encoding directions. An efficient and unbiased estimator is proposed for tracking the drift by interleaving short (~T₂^*), non-localized acquisitions on the non-suppressed water each TR, as shown in 10 volunteers at 1.5 and 3 T.

RESULTS

The drift is shown to be predominantly linear in both the phantom and the volunteers at both fields. The localization errors are observed and quantified in the phantom. The unbiased estimator is shown to reliably track the instantaneous frequency in-vivo despite only using a small portion of the FID.

CONCLUSION

Contrary to single-voxel MR spectroscopy, where it leads to line broadening, field drift can lead to localization errors in the longer chemical shift imaging experiments. Fortunately, this drift can be obtained at a negligible cost to sequence timing, and corrected for in post processing.

Introduction

Temporal instabilities of the spins’ resonant frequency have long been recognized as sources of artifacts in various applications ranging from MR spectroscopy (MRS) to thermometry and fMRI (1–3). Over sufficiently long (many minutes) scans, the resonance frequency undergoes a predominantly linear shift, referred to as field “drift”, due to tiny dissipative losses in the superconducting coil. These are specified in the magnet’s data sheets at under 0.1 ppm/hour, compared with ~10⁻³ ppm/day of frequency synthesizers (see programmedtest.com/pts310.html). Heating of the passive shims by currents in the active ones is another source of an approximately linear drift (4). Other sources of instability are sequence dependent, such as heating of the magnet’s heat shield by fast-switching gradients (5), and physiological, such as respiration, which introduces a periodic component, often on the order of a sequence’s repetition time, TR (4).

The specific effects of the drift vary, depending on the sequence and total acquisition time, with longer protocols, e.g., fMRI and MR spectroscopic imaging, being the most vulnerable. For EPI based fMRI, geometric image distortions and ways of correcting them have been reported (6,7). Spatial misregistration that can lead to localization errors can also be incurred with spectrally and spatially selective pulses (8). In single-voxel MRS, field drift leads to line broadening which can affect spectral resolution and quantification (9,10). In phase-encoded chemical shift imaging (CSI), however, it can lead to localization errors as previously mentioned but without detailed analysis of its origins and quantification of its consequences (1,11).

The drift’s effects can be removed if the instantaneous frequency is known in each scan, assuming it remains constant throughout a single TR. Several suggestions have been made for monitoring the instantaneous frequency. Some are based on specialized external hardware (12). Others combine drift and motion correction using navigator echoes (6). In fast imaging, the drift has often been estimated from the central k-space point (6,13). Since no spatial localization is required, a simple approach involves exciting the water with a (possibly frequency-selective) small tip angle pulse, acquiring a free induction decay (FID), and computing the position of the its peak (4,14,15). However, this requires acquisition a full FID, which requires considerable available “dead time” in the sequence. Unless limited by SAR, such dead time is best kept to a minimum to reduce total scan time or, alternatively, to improve spatial coverage and SNR per unit time (16). 3D protocols which involve time-multiplexing or multi-slab acquisitions, and which have little “dead-time” during which a full interleaved FID can be acquired, can be problematic as well (16,17).

In this paper we analyze the effects of a predominantly linear field drift, showing that, unlike single-voxel MRS, it does not necessarily broaden spectral lines, but rather introduces localization errors, equivalent to spatial blurring due to a moving filter. To address its effects we also present a method to obtain the instantaneous resonant frequency at negligible cost in time, by acquiring just the first few milliseconds of the non-localized, non-water-suppressed FID every TR. Then, using a weighted least-squares (WLS) fitting procedure we obtain an unbiased and efficient estimator for the water frequency: its expectation value matches the true value and it attains the Cramer Rao lower bound, i.e., the smallest possible variance of all possible unbiased estimators. We demonstrate its effectiveness (few tenths of a Hz precision) and the validity of the linear drift approximation in vivo in 10 volunteers at 1.5 and at 3 T.

Theory

B₀ Drift Leads to Localization Errors

As shown experimentally below, in-vivo drifts are characterized by a dominant linear component, whose effect on the localization of phase-encoded CSI data is demonstrated first in one dimension. The analysis will assume the resonant frequency changes negligibly during the acquisition of a single FID. Although rapid changes within a TR are possible, e.g., due to breathing and blood flow, these are periodic and occur on a fast scale (~ seconds) and, therefore, we assume their effect is decoupled from that of the slow, linear decline component of interest.

To obtain N voxels at a given field-of-view (FOV), N scans are carried out to sample k-space at N equidistant points: k_n = −k_max/2 + nΔk (n=0,1,…,N−1), with Δk=FOV⁻¹ and k_max=N·Δk. If a drift of Δν_scan Hz accrues between subsequent scans, spaced TR apart, and if the receiver is initially centered on the chemical species in question, then the signal acquired during the n^th scan is:

$${\text{s}}_{\text{n}}(\text{t})={\text{e}}^{2\pi \text{in}\mathrm{\Delta }{\nu }_{\text{scan}}\text{t}}{\int }_{-\infty }^{\infty }{\text{M}}_0(\text{x}){\text{e}}^{2\pi {\text{ik}}_{\text{n}}\text{x}}\text{dx},$$ [1]

where M₀(x) is the spatial spin density and t is the intra-scan acquisition time which varies between 0 and T_acq, on the order of several hundred milliseconds. The signal from the m^th voxel is obtained with an inverse discrete Fourier transform taken over Eq. [1]:

$${\widehat{s}}_m=\sum _{n=0}^{N-1}{s}_n{e}^{-\frac{2\pi \mathit{imn}}{N}}={\int }_{-\infty }^{\infty }{M}_0(x)\left\{\sum _{n=0}^{N-1}{e}^{2\pi in\mathrm{\Delta }{\nu }_{\text{scan}}t}{e}^{2\pi i{k}_{n}x}{e}^{-\frac{2\pi \mathit{imn}}{N}}\right\}dx.$$ [2]

Carrying out the summation within the curly brackets, we obtain:

$${\widehat{s}}_m(t)={\int }_{-\infty }^{\infty }{M}_0(x)\text{PSF}(x+\mathrm{\Delta }{\nu }_{\text{scan}}·\mathit{FOV}·t-m\mathrm{\Delta }x)dx,$$ [3]

where Δx=FOV/N is the voxel size and PSF(x) is the point spread function associated with the phase encoding process (18):

$$\text{PSF}(\text{x})={\text{e}}^{\frac{-\pi \text{ix}}{\text{FOV}}}\frac{sin\left[\frac{\pi \text{x}}{\mathrm{\Delta }\text{x}}\right]}{sin\left[\frac{\pi \text{x}}{\text{FOV}}\right]},$$ [4]

i.e., the signal from the m^th voxel is given by a convolution of a PSF and the spin distribution, M₀(x). The PSF, however, also shifts during the acquisition, picking up signal from the center of the voxel at t=0 and shifting linearly up to:

$${\delta }_{\text{x}}=\mathrm{\Delta }{\nu }_{\text{scan}}·\text{FOV}·{\text{T}}_{\text{acq}}=\mathrm{\Delta }{\nu }_{\text{scan}}·{\text{N}}_{\text{scans}}·\mathrm{\Delta }\text{x}·{\text{T}}_{\text{acq}}.$$ [5]

Typical parameters (TR=1 s, 0.1 ppm/hour drift at 3 T, FOV= 8 cm, N_scans=8, T_acq=0.5 sec) yield δ≈0.1 mm, which is small compared to the typical (~ 1 cm)³ CSI voxel.

Although negligible in 1D, the effective frequency drift between successive points in k-space becomes non-negligible in 2D and 3D. Indeed, if one acquires k-space rectilinearly,

$$\begin{array}{l}{\mathbf{k}}_{\text{pqm}}=-2^{-1}({\text{k}}_{max,\text{x}},{\text{k}}_{max,\text{y}},{\text{k}}_{max,\text{z}})+(\text{p}\mathrm{\Delta }{\text{k}}_{\text{x}},\text{q}\mathrm{\Delta }{\text{k}}_{\text{y}},\text{m}\mathrm{\Delta }{\text{k}}_{\text{z}})\\ (\text{p},\text{q},\text{m})=(0,0,0),(1,0,0),\dots ,({\text{N}}_{\text{x}}-1,0,0),(0,1,0),(1,1,0),\dots ,({\text{N}}_{\text{x}}-1,{\text{N}}_{\text{y}}-1,{\text{N}}_{\text{z}}-1),\end{array}$$ [6]

where Δk_q=FOV_q⁻¹ (q=x,y,z), the signal from the k_pqm coordinate in k-space is given by:

$${\text{s}}_{\text{pqm}}(\text{t})={\text{e}}^{2\pi \text{i}(\text{p}+{\text{qN}}_{\text{x}}+{\text{mN}}_{\text{x}}{\text{N}}_{\text{y}})\mathrm{\Delta }{\nu }_{\text{scan}}·\text{t}}\int {\text{M}}_0(\mathbf{r}){\text{e}}^{2\pi \text{i}{\mathbf{k}}_{\text{pqm}}·\mathbf{r}}\text{d}\mathbf{r}.$$ [7]

Reconstruction by an inverse 3D discrete Fourier transform over all three axes yields:

$${\widehat{s}}_{\mathit{pqm}}(t)=\int \left[\begin{array}{l}{M}_0(\mathbf{r})·\text{PSF}(x+\mathrm{\Delta }{\nu }_{\text{scan}}·{\mathit{FOV}}_x·t-p\mathrm{\Delta }x)\\ ·\text{PSF}(y+\mathrm{\Delta }{\nu }_{\text{scan}}·{\mathit{FOV}}_y·N_x·t-q\mathrm{\Delta }y)\\ ·\text{PSF}(z+\mathrm{\Delta }{\nu }_{\text{scan}}·{\mathit{FOV}}_z·N_x·N_y·t-m\mathrm{\Delta }z)\end{array}\right]d\mathbf{r}.$$ [8]

As before, M₀(x) is convolved with a moving PSF. Note, however, that the shift increases N_x-fold in the 2^nd dimension, and N_x·N_y-fold in the third, due to longer delays between consecutive k-space points along these axes. Taking N_x=N_y=16, FOV_z =40 mm, the shift along the z-axis is δ_z=Δν_scan·FOV_z·N_x·N_y·T_acq≈17 mm, which approaches FOV_z itself. Note that Eq. [3] and its 2 and 3D analogues hold even if spatial filtering is used to smooth the PSF.

Criterion for Neglecting Drift

The drift’s effects can be neglected when the shift it induces is negligible compared to the voxel size. For a 3D CSI, the greatest shift is shown to occur along the slowest z-axis, δ_z=Δν_scan·FOV_z·N_x·N_y·T_acq, and the criterion becomes δ_z«Δz. Since FOV_z=Δz·N_z, and N_scans=N_x·N_y·N_z, this becomesΔν_scan·N_scans·T_acq«1. Since T_acq is typically of the order of T₂^* in a voxel, approximately the inverse of the linewidth Δν_FWHM - the criterion becomes:

$$\mathrm{\Delta }{\nu }_{\text{scan}}·{\text{N}}_{\text{scans}}\ll \mathrm{\Delta }{\nu }_{\text{FWHM}},$$ [9]

where Δν_scan·N_scans is simply the total drift accrued throughout the experiment. Eq. [9] can be derived for 1D and 2D acquisitions as well.

Frequency Estimation using Weighted Least Squares

The spins’ instantaneous frequency can be monitored by interleaving the CSI scans with a pulse-acquire module. Such a module using a small tip angle - so as not to disturb the magnetization appreciably - will yield an FID dominated by the intense water signal, with some lipids contribution. A single dominant frequency component can be modeled as a decaying exponential embedded in normally distributed noise with a fixed variance, σ^(xy):

$${\text{s}}_{\text{n}}={\text{s}}_0·{\text{e}}^{2\pi \text{i}{\nu }_0{\text{t}}_{\text{n}}-\frac{{\text{t}}_{\text{n}}}{{\text{T}}_2^{\ast }}}+{\epsilon }_{\text{R}}+\text{i}{\epsilon }_{\text{I}},\text{n}=0,1,\dots ,\text{N}-1$$ [10]

where s₀ is the signal-to-noise-ratio (SNR) at the initial point of the FID, t_n=nΔt and Δt is the acquisition dwell time. In the presence of intermediate-to-high SNR≥3, it is well known that the phase of s_n, ${\phi }_{\text{n}}=arctan\left(\frac{Im({\text{s}}_{\text{n}})}{Re({\text{s}}_{\text{n}})}\right)$, follows a normal distribution with mean 2πν₀t_n and standard deviation ${\sigma }_{\text{n}}={\sigma }^{(\text{xy})}·{\text{s}}_0^{-1}·exp({\text{t}}_{\text{n}}/{\text{T}}_2^{\ast })$ (19); that is, as t_n increases, so does the phase’s uncertainty. The instantaneous frequency,ν₀, can be estimated from the slope of the phase data φ_n. It is known that the optimal estimator for this slope is obtained via weighted least squares (WLS), where time point is assigned a weight proportional to its SNR (20):

$$\widehat{\nu }=\frac{1}{2\pi }\left[\left(\sum _{\text{n}=0}^{\text{N}-1}\frac{{\text{t}}_{\text{n}}}{{\sigma }_{\text{n}}^2}\right)\left(\sum _{\text{n}=0}^{\text{N}-1}\frac{{\phi }_{\text{n}}}{{\sigma }_{\text{n}}^2}\right)-\sum _{\text{n}=0}^{\text{N}-1}\left(\frac{{\text{t}}_{\text{n}}{\phi }_{\text{n}}}{{\sigma }_{\text{n}}^2}\right)\left(\sum _{\text{n}=0}^{\text{N}-1}\frac{1}{{\sigma }_{\text{n}}^2}\right)\right]/\left[\sum _{\text{n}=0}^{\text{N}-1}{\left(\frac{{\text{t}}_{\text{n}}}{{\sigma }_{\text{n}}}\right)}^2-\sum _{\text{n}=0}^{\text{N}-1}\left(\frac{{\text{t}}_{\text{n}}^2}{{\sigma }_{\text{n}}^2}\right)\sum _{\text{n}=0}^{\text{N}-1}\left(\frac{1}{{\sigma }_{\text{n}}^2}\right)\right]$$ [11]

which is valid even for non-equispaced time points. It is unbiased, efficient and computationally fast, requiring only addition and multiplication of O(N) terms. Its standard deviation can be calculated explicitly (20):

$${\text{SD}}_{\widehat{\nu }}=\sqrt{\text{Var}(\widehat{\nu })}=\frac{1}{2\pi }·\sqrt{\sum _{\text{n}=0}^{\text{N}-1}\frac{1}{{\sigma }_{\text{n}}^2}/\sum _{\text{n}=0}^{\text{N}-1}\left(\frac{{\text{t}}_{\text{n}}^2}{{\sigma }_{\text{n}}^2}\right)\sum _{\text{n}=0}^{\text{N}-1}\left(\frac{1}{{\sigma }_{\text{n}}^2}\right)-\sum _{\text{n}=0}^{\text{N}-1}{\left(\frac{{\text{t}}_{\text{n}}}{{\sigma }_{\text{n}}}\right)}^2}$$ [12]

Materials and Methods

The experiments were done in a 3 T TIM-Trio and a 1.5 T Avanto full body MR imagers using their standard transmit-receive head-coils, capable of delivering 22μT (0.9 kHz) of B₁ (Siemens AG, Erlangen Germany). In vivo experiments were done on 10 (8 males, 2 females) healthy 22–39 year old volunteers. Each provided institutional review board (IRB)-approved written consent.

3D CSI in a Phantom

To observe the localization artifacts described in Eq. [3], a 3D CSI experiment was conducted at 3 T, on a 1.6 cm diameter sphere containing water. A 3D MPRAGE image was acquired at 1 mm isotropic resolution for image guidance of the CSI grid. The 3D CSI sequence consisted of a 90° excitation, followed by quick (250 μsec) gradient phase encoding along all 3 axes and a subsequent 1-second FID acquisition of 1024 points, as shown in Fig. 1a. Following each acquisition, a small, 1°, flip angle pulse was applied and a second FID (1024 points for 1024 ms) was acquired without any localization to track the water’s frequency drift. A total of 10×10×10 phase encoding steps were applied along the x, y, and z axes at 200×200×200 mm³ FOV, leading to 2×2×2 cm³ voxels. The phantom was contained entirely within the central voxel of the FOV (Fig. 2a). k-space was acquired rectilinearly (Eq. [6]), with the z-axis being the “slowest” to change. The TR, which includes the second drift-tracking acquisition, was 2.2 seconds, leading to a total acquisition time of 10×10×10×2.2 seconds ≈ 36.6 minutes. Strong, random gradients (~18 mT/m) at the end of each TR spoiled any remaining transverse magnetization.

Fig. 1.

Pulse sequences used for (a) phantom experiments and (b) in-vivo experiments. (a) is 3D CSI, while (b) is 2D CSI PRESS. In both sequences, k-space was encoded in this order: k_x-k_y-k_z, i.e., the k_x coordinate changed every TR, k_y every N_x·TR, and k_z every N_x·N_y·TR, as described by Eq. [6]. Both sequences feature an additional interleaved small, 1°, tip-angle pulse-acquire module to track the water signal’s instantaneous frequency. The durations of all elements, pulses’ bandwidths (at FWHM) and gradients’ strengths are marked. Dark gradients represent spoilers/crushers and light-grayed once represent slice/volume selection. The phase encoding gradients, which follow the refocusing gradient in (b), are omitted for brevity.

Fig. 2.

The 3D CSI spectra acquired at 3 T from a homogeneous spherical phantom occupying only the center voxel. (a) A schematic diagram of the (yz) plane containing the phantom and of the placement of the FOV. (b) The instantaneous frequency as a function of the experiment’s duration, as estimated from the pulse-acquire module interleaved with the CSI acquisition. A linear fit yielded a drift of −18.4 Hz/hour. (c) Spectra from the set of voxels containing the phantom along the y-axis (k_y is incremented every N_x·TR=30 seconds). The voxel containing the phantom is marked by a small sphere. (d) Spectra from the set of voxels containing the phantom along the “slow” z-axis (k_z is incremented every N_x·N_y·TR=300 seconds), showing the pronounced effects of drift. (e) Following correction of the drift, the spectra shown in (d) appear much more symmetrical and well proportioned, and signal becomes mostly confined to the central voxel as expected.

The acquired 3D CSI FIDs were apodized with a Gaussian and zero-filled to 4096 points in the time domain and Fourier transformed along all four (k_x-k_y-k_z-time) axes to yield a localized spectrum from each voxel. Automatic zero order phase correction was applied in all voxels.

To estimate the instantaneous frequency, each interleaved non-localized FID was zero-filled 16-fold to increase its spectral resolution and then Fourier transformed. The spectral position of the water peak was used as an estimator for the instantaneous frequency which was then plotted as a function of total scan time (36.6 minutes) and inspected visually for linearity. It was linearly fit,

$$\nu (\text{t})={\alpha }_1\text{t}+{\alpha }_0.$$ [13]

The drift, α₁, in Hz/Hour, was then corrected for by multiplying each k^th FID by exp(2πi·α₁·τ_k·t), where τ_k=k·TR is the time at which it was collected and 0≤t≤1024 ms the ADC time variable.

The correction quality was assessed by visual inspection of the drift-corrected spectra and by calculating the total spatial leakage outside the voxel containing the phantom. To do so, the area under the real part of the spectrum was obtained by integrating from ±20 Hz about the phased water peak. This yielded a positive number for each voxel, L_ijk (i,j,k=1, …,10). The total leakage was defined as the percentage of signal outside the voxel containing the phantom:

$${\text{L}}_{\text{total}}=\sum _{\text{i},\text{j},\text{k}\ne \text{phantom}}{\text{L}}_{\text{ijk}}/\sum _{\text{i},\text{j},\text{k}=1}^{10}{\text{L}}_{\text{ijk}}·100\%.$$ [14]

L_total will be greater than 0 even in a perfect, driftless 3D CSI experiment, due to the non-local nature of the PSF (Eq. [4]), and can be calculated by simulating the Bloch equations numerically using the same parameters used in the experiment. The simulation yields L_total=38% in the absence of any drift. Calculating L_total before and after drift-correction would quantify the effect of the drift and the quality of its correction.

2D PRESS-CSI and Drift Correction In-Vivo

The frequency drift was tracked in 10 healthy volunteers at 1.5 and 3.0 T to quantify its magnitude in-vivo. First, a localizer was run, followed by a 5:26 minute 3D MPRAGE (1×1×1 mm³ resolution and 256×256×192 mm³ FOV). Our in-house whole-head shimming routine was then used to perform B₀ shimming in each volunteer in about 5 minutes (21). This was followed by a 2D PRESS CSI acquisition of a 12 mm thick axial 2D slice in the plane of the superior lateral ventricles and an 8×8 cm² VOI was prescribed in a 24×24 cm² FOV, with a N_x×N_y=24×24 acquisition matrix in its plane. With a TR/TE=3000/60 ms the total acquisition time was 29 minutes (including 4 dummy scans to establish a dynamic equilibrium). The CSI was interleaved with the same 1° tip angle pulse-acquire module described above for the phantom. The instantaneous water frequency was estimated using two approaches: (i) by zero-filling 16-fold (for smoothing), Fourier-transforming and noting its peak position, and: (ii) by using the WLS estimator (Eq. [11]) on the initial points of the FID, up until T=10 ms. The choice of T=10 ms is made using Eq. [12] as follows: taking T₂^*=10 ms for a well shimmed whole-head, a conservative s₀/σ^(xy)=50 and a dwell time of 0.5 ms (implying N=21 points acquired during T=10 ms), Eq. [12] yields a standard deviation of SD_ν=0.25 Hz. This is much smaller than the overall drift-induced change in frequency we would expect during the entire scan and hence acceptable. A larger portion of the FID, longer than 10 ms, can be used to further minimize the standard deviation.

Results

3D CSI in a Phantom

The positioning of the sample and spectroscopic imaging grid is shown in Fig. 2a for the (yz) slice containing the phantom. The shimmed linewidth was 3±1Hz. The instantaneous frequency, extracted from the pulse-acquire module interleaved with each CSI scan, is shown in Fig. 2b, along with the fitted linear line, which yielded a drift of α₁=−18.4 Hz/hour=−0.15 ppm/hour. The spectra from the set of voxels containing the phantom along the “slowest” varying z-axis, highlighted in Fig. 2a, are shown in Fig. 2d. Significant signals outside the main voxel are evident. For comparison, the spectra from the set of voxels containing the phantom along the y-axis are shown in Fig. 2c. These show significantly less artifacts since the time between consecutive points along the k_y-axis is much shorter (by a factor of N_z=10) than along the k_z- axis. This is expected since the time difference between adjacent points along k_y is shorter than between adjacent points along k_z by a factor of N_z=10.

The leakage (Eq. [14]) before drift-correction was found to be L_total=75%, in contrast to the expected 38% in the absence of drift. The same set of 10 voxels in Fig. 2d are shown in Fig. 2e after applying the drift correction, using the calculated drift coefficient (−18.4 Hz/hour). The corrected spectra display significantly improved localization and spectral quality. The leakage was found to be L_total=41%, in better agreement with the theoretically predicted 38%.

2D CSI In-Vivo

Due to the temporal proximity of the small tip angle excitation and the FID readout, the first 10 points of each interleaved FID were distorted and were therefore discarded prior to carrying out any frequency estimation. The instantaneous frequency recorded from all 10 volunteers is shown in Fig. 3 for both the Fourier-transformed (blue) and WLS (red) estimation, with the WLS estimator using only the first 21 points (10 ms) of the FID. Both estimators show good agreement. Both frequency curves were linearly fitted, with the results displayed in Fig. 3 in ppm/hour. The average±standard deviation taken over all volunteers of the difference between the two fits is 0.001±0.11 ppm/hour. A sample interleaved FID from a single scan from one of the volunteers is shown in Fig. 4, alongside the fitted linear curve calculated from the WLS estimator.

Fig. 3.

In-vivo drift-tracking in 10 healthy volunteers at 1.5 and 3 T, showing for each volunteer: (i) the instantaneous frequency as a function of scan time, as estimated from the position of the water peak from the interleaved scans (blue, solid); (ii) the weighted-least squares estimation (Eq. [11]) from just the first 21 points (10 ms) in each interleaved FID (red, solid); (iii) the linear fit (black, dashed) to the Fourier transformed (blue solid) data; (iv) The drift, in ppm/hour, as calculated from both estimators (blue: Fourier transformed. red: weighted least squares). Note that all drifts are, to a good approximation, linear, and that the two estimators show excellent agreement.

Fig. 4.

A representative interleaved non-localized FID (out of 24×24=576) from one of the volunteers. (a) The real part of the FID, which decays quickly due to the short whole-head T₂^*. (b) The whole-head spectrum in magnitude mode. Note the dominant water peak and the much smaller lipid peak. (c) The phase of the FID (black, solid), superimposed on the fitted weighted least squares line (red, dashed). The slope of the line equals the instantaneous frequency (multiplied by 2π) during that particular scan. Only the first 10 ms (shaded) are used for fitting the line using Eq. [11]. Note how, for low SNR (≤3) around ca. 350–400 ms, the phase unwrapping algorithm begins to fail as the noise in the phase domain becomes too large.

Sample in vivo spectra from a volunteer (Fig. 3d) are shown in Fig. 5 before and after drift correction. Correction was applied after smoothing the instantaneous frequency curve in Fig. 3d and multiplying each PRESS FID by exp(2πiν_mt), where 0≤t≤1024 ms is the ADC time variable and ν_m (m=1,2, …,576) is the instantaneous frequency during the m^th scan. Voxels adjacent to discontinuities in the spin density along the “slow” y-axis (Figs. 5c-e) show distorted lineshapes, and benefit considerably from the drift correction. This is similar to the effects observed in the phantom before (Fig. 2d) and after (Fig. 2e) drift correction. In particular, the drift correction eliminated the spurious signal in the ventricles (grey arrow in Fig. 5e). On the other hand, no striking improvements are observed when the voxel is surrounded by other voxels having a similar spin density (Fig. 5f). Here, the drift-induced spatial averaging implied by Eqs. [3] and [8] merely averages the voxel with similar adjacent voxels which do not modulate the signal appreciably; i.e., “the spatial average of a homogeneous medium is a homogeneous medium”.

Fig. 5.

Sample in vivo spectra from a volunteer. The corresponding drift curve is shown in Fig. 3d. (a, b) VOI (red box) and FOV (white grid) placement on anatomical MPRAGE images. Spectra before and after drift correction are shown in (c–f). Note that the voxels in (c–e) are positioned close to discontinuities in the spin density and, consequently, visibly benefit from the drift correction. In particular, note the elimination of the spurious NAA signal in the ventricles in (e). Much less pronounced artifacts are observed in (f) due to the voxel’s homogeneous surroundings.

Discussion

The spins’ resonant frequency always declines over time due to small resistive losses in any superconducting magnet. This drift is often disclosed in the magnet’s specification sheets at under 0.1 ppm/hour, but this is often measured under no load and no gradient and shim activity. The drifts measured herein were mostly in the 0.1–0.2 ppm/hour range for both 1.5 and 3.0 T magnets. Literature reports, summarized in Table 1, range from 0.03 up to 6.5 ppm/hour and vary with magnet, field strength and sequence used, but most are approximately a few tenths of a ppm/hour. A 0.1 ppm/hour drift at 3 T would accrue about 6 Hz over 30 minutes, on the order of the voxels’ spectral linewidth, and hence lead to non-negligible localization errors according to Eq. [9].

Table 1.

Literature review of typical drift magnitudes, spanning a wide range of manufacturers, magnets, fields and subjects.

Ref.	B0	Manufacturer	Drift (ppm/hr)	Subject	Sequence
(27)	1.5 T	GE Signa	0.55	Phantom	Thermometry using spoiled gradient echo imaging
(4)	3 T	Bruker Avance	0.5	Human	PRESS interleaved with FID acquisition
(25), (7)	1.9 T	GE/Elscint Prestige	Not reported1	Human	EPI fMRI
(11)	1.5 T	Siemens Magnetom	0.05	Phantom	Multiple pulse-acquire before and after acquiring a CSI dataset
(6)	1.5 T	GE Signa Horizon	Sinusoidal2	Phantom	EPI fMRI
(26)	7 T	Bruker Biospec	0.03	Rat	PRESS interleaved with navigator scan
(13)	1.5 T	Siemens Sonata	1.49	Phantom	FLASH and SSFP
(14)	3 T	Siemens Trio	0.35	Human	EPSI interleaved with FID acquisition
(5)	4 T	Varian INOVA	0.85	Human	EPI fMRI
(29)	3 T	GE Signa	0.48	Human	Spiral EPI fMRI
(8)	1.5 T	Siemens Avanto	2.34	Human	DTI
(28)	3 T	Philips Achieva	6.5	Phantom	Multi gradient echo thermometry
(15)	3 T	Siemens Trio	0.033	Phantom	PRESS with interleaved FID acquisition

¹The drift was shown to induce geometrical distortions in EPI of 2–3 voxels.

²A sinusoidal ±5 Hz variation in B₀ with a period of ca. 2 minutes was reported, attributed by the authors to a power line in the room.

³These drifts were observed following an intensive diffusion weighted EPI sequence. Prior to that, even smaller drifts (not exceeding 0.01 ppm/hour) were observed.

In contrast to single-voxel measurements, drift does not induce line broadening in phase-encoded CSI. Rather, the drift-induced phase and frequency uncertainties result in localization errors. Line broadening, however, can be observed as a consequence of the spatial averaging when the shim, spin density or relaxation change appreciably on the order of the spatial shift experienced by the PSF (e.g., Eq. [5] in 1D). For example, the lineshape in the voxel containing the spherical phantom appears broadened before correction (before: Fig. 2c vs. after: Fig 2e): as the PSF shifts outside the voxel containing the phantom, the signal decays to zero on a time scale faster than the voxel’s T₂^*, thus broadening (and slightly distorting) the lineshape. The exact effects will depend on the local environment of each voxel.

Several 3D CSI variants incorporate echo planar spectroscopic imaging (EPSI) modules to speed up the acquisition. The most common approach uses EPSI along one axis and phase-encoding along others (22), while others extend it to all three axes (23). When EPSI is used for one direction, the drift effects appear only along the two phase-encoded axes, in a similar manner as described so far (Eq. [3]), For example, for a 2D acquisition in the k_x-k_y plane in which the k_x-axis is phase-encoded and EPSI is used for the k_y-axis, is the signal from the n^th acquisition is:

$${\text{s}}_{\text{n}}(\text{t})={\text{e}}^{2\pi \text{in}\mathrm{\Delta }{\nu }_{\text{scan}}\text{t}}{\int }_{-\infty }^{\infty }{\text{e}}^{2\pi {\text{ik}}_{\text{n}}\text{x}}\left[{\int }_{-\infty }^{\infty }{\text{M}}_0(\text{x},\text{y}){\text{e}}^{2\pi {\text{ik}}_{\text{y}}(\text{t})\text{y}}\text{dy}\right]\text{dx}$$ [15]

whereΔν_scan is the change in the spins’ resonance frequency between consecutive phase-encoded scans, k_n = −k_max/2 + nΔk (n=0,1, …,N−1), Δk=FOV⁻¹ and k_max=N·Δk. Eq. [15] is similar in form to Eq. [1], and the conclusions are identical. Due to the low in vivo SNR encountered, EPSI sequences need the same time as ordinary phase-encoded CSI (24), making them just as susceptible to drift effects. The effects of field drifts on a full volumetric 3D EPSI acquisition, in which all of k-space is acquired in a very short window of time, are of a different nature, and resemble those encountered in single-voxel MRS (4): Each scan acquires the entire spectrum, which drifts throughout acquisition. Averaging the different spectra will then result in line broadening but no localization errors since localization is achieved in a single, fast (~1 sec) volumetric scan of k-space, during which the effects of drift are negligible.

Drift Correction

The effects of any drift slow enough to be considered constant within a single scan can be corrected if the instantaneous spins’ resonant frequency is known. The question, therefore, becomes one of instantaneous frequency estimation. The correction can then be done on the fly or retrospectively via post-processing. On-the-fly approaches are preferred since they address the drift-induced mismatch between the magnet’s and RF’s center frequencies, which can decrease the performance of water suppression and volume selection pulses. Henry et. al. (4) use interleaved FIDs to adjust B₀ in real time via a Z0 shim coil, not commonly found in human imagers. In contrast, Benner et. al.(8) circumvent the need for specialized hardware by adjusting the receiver and transmitter center frequencies directly during an EPI-based DTI acquisition. To track the instantaneous frequency, the authors use the phase of the central point of k-space, which has a high signal to noise ratio and is acquired every TR. While this works well for repeated single-scan acquisitions, such as fMRI (7,25) and DTI (8), which visit the central k-space point each TR, it is inappropriate for CSI in which the center of k-space is visited only once or at most a small number of times. Thus, regardless of whether a post-processing or real-time approach is taken, sequence and/or hardware probes are required to track the instantaneous frequency during CSI.

Hardware probes offer an excellent means for observing not only temporal but also spatial field variations (12), but require specialized hardware. The alternative is to interleave the pulse sequence with some form of reference scan, which is the approach taken by the majority of other publications dealing with field drift during CSI (4,11,14,15). For example, Ebel et. al. used an EPSI-module for their reference scan on non-suppressed water, which afforded additional B₀ correction capabilities, but requires fast-switching gradients and specialized post-processing expertise to implement. In (26) a localized navigator echo was incorporated for correcting for both field drift and motion artifacts, albeit at the cost of additional volume-selection RF pulses. The much simpler acquisition of a non-localized FID, used in the current paper, was also used in (4). Although this method does not correct spatial field variations, its appeal lies in its simplicity, robustness and minimal sequence modifications required.

The interleaved scans should be kept as short as possible for two reasons: (i) not to perturb TR or impose unnecessary restrictions on each scan’s duration; (ii) to provide some immunity from the instantaneous frequency changes even on short, ~1 sec, time scales due to patient motion, respiration and blood flow. Fig. 3 shows that, for a well shimmed sample (~20–30 Hz whole-head at 3 T), even a 10 ms acquisition is sufficient to obtain excellent (few tenths of a Hz) accuracy if the WLS estimator is used (Eq. [11]). Although this computation requires the SNR at each time point of the interleaved scans’ FIDs (σ_n in Eq. [11]), it is highly tolerant to estimation errors - any errors will merely give slightly different weight to different time points, increasing the standard deviation proportionally, but never beyond that of a non-weighted least squares procedure.

Drift Linearity

On the time scales of a CSI experiment, the drifts encountered were predominantly linear (Fig. 3). This linearity is not necessary for correcting the drift’s effect. However, it does serve to validate the theoretical effects analyzed in this paper. The drift’s linearity is supported by the literature: Peters et. al. (27) reports a linear drift in a phantom, measured with a simple pulse-acquire over 25 minutes, modulated only by variations in the phantom’s temperature due to air conditioning variations, that are not expected in vivo. A ~0.5 ppm/hour drift was observed over 10 minutes due to heating of the passive shims in a PRESS sequence (4). An almost-linear drift of about 0.35 ppm/hour, with a small quadratic component, was recorded in an EPSI acquisition (14). Therefore, when it is impossible to modify the sequence or hardware, the linear approximation is a useful first order correction; the user can measure the water’s resonant frequency before and after the CSI experiment with (possibly) several acquisitions to average out physiological variations.

Non linear drift components can be observed depending on magnet quality and whether a high duty cycle sequence was run on the magnet prior to CSI as described in (15); such nonlinearities can persist for half an hour, depending on the amount of passive shim elements in the system. For a complete correction, the spins’ instantaneous frequency must be tracked. Several suggestions have been made for tracking the temporal frequency changes, including interleaving reference scans and adjusting the Z₀ gradient (4) or correcting in post-processing (15), using navigator echoes and centroids (7,26), introducing specialized field probes into the scanner (12), or using reference-less approaches which rely on multiple gradient echoes (28).

As evident from Fig. 3, the instantaneous frequency displays a noisy component which cannot be attributed solely to the estimator’s variance; this “noise”, the source of which is most probably physiological, is characterized by frequency variations on the order of TR. Before correction is attempted it is advised that a polynomial fit or similar smoothing procedure be applied, to minimize the effects of frequency shifts within a single TR.

Localization errors are particularly insidious in vivo in the brain, where small changes in metabolite levels are sought, as they are difficult to detect by visual inspection: since the effect of a linear drift is to spatially average adjacent voxels (Eq. [3]), and since the brain’s metabolite levels are fairly spatially homogeneous, the drift’s effects can be subtle (Fig. 5f). However, that is not to say it is unimportant: drift will average spin density/shim/linewidth variations between, e.g., healthy and diseased tissue, thus reducing both specificity and sensitivity. Voxels close to the ventricles and the VOI’s edges also benefit from proper drift correction (Figs. 5c-e).

Conclusions

All superconducting magnets suffer a drift. In real-world scenarios, this drift has a strong linear component (ca. 0.1 ppm/hour in vivo, as shown herein) and introduces localization errors in chemical shift imaging protocols, which can become substantial for long acquisition times. Correcting for it is straightforward if the instantaneous resonant frequency can be estimated and smoothed. Interleaved FIDs offer a simple, easily implemented and low SAR solution for doing so; fitting the FID’s phase using weighted least squares provides high accuracy and precision, while keeping the duration of the interleaved FIDs to a minimum (ca. 10 ms).