Beat phenomena of oscillating readouts

To demonstrate slowly varying, erroneous magnetic field gradients for oscillating readouts due to the mechanically resonant behavior of gradient systems.


INTRODUCTION
To perform faithful imaging, high-fidelity gradients are required.However, the MRI gradient chain including amplifiers and coils exhibit low-pass and resonant behavior caused by the amplifier, 1 eddy currents arising in conductive structures, 2 and mechanical resonances of the gradient coils and their support structures. 3 Assuming a linear time-invariant (LTI) system, the gradient modulation transfer function (GMTF) describes the relationship between actual and desired gradient waveforms.To this end, a GMTF measurement permits, for example, the correction of k-space trajectories of echo planar, 4,5 spiral, [6][7][8][9] or radial imaging 10 sequences.However, the validity of the LTI assumption can be challenged if nonlinearity effects (e.g., due to strong diffusion gradients 11 ) or temperature changes 12 in the gradient chain are no longer negligible.
In practice, pre-emphasis is used to pre-distort demand gradient waveforms to counteract eddy current (EC) fields.Apart from the contribution of vibrational ECs at ultrahigh fields, 13 ECs arise in conductive parts according to Lenz's rule.They can be modeled by L-R series circuits. 2ccordingly, their magnetic-field contributions are modeled as exponentially decaying signals 14 in the impulse response function or, equivalently, as Lorentzian functions centered at zero frequency in the transfer function of the gradient system.Therefore, the effects on the magnetic-field distribution due to ECs monotonically increase with the frequency of the gradient waveform.Earlier works have included the term "oscillating eddy currents" 15 ; this may, however, be misleading.Without any pre-emphasis applied, the gradient-system response exclusively due to ECs for a nonoscillating input is also nonoscillating.The signal can be interpreted as low-pass-filtered in the temporal domain, yet no oscillations are introduced by EC decay.As the level of EC compensation using pre-emphasis in MRI systems is limited by several factors, 16 residual ECs may persist even with pre-emphasis applied.
In addition to ECs, mechanical resonance effects exist.These are caused by Lorentz forces resulting in the deformation of the gradient coils and its support structures, 17 thereby contributing to acoustic noise emission and oscillating magnetic-field deviations.In the temporal domain (spectral domain), the mechanical resonance effects result in magnetic-field oscillations (peak/dips).Their frequencies are identified in the GMTF, forming antiresonance/resonance pairs.Pre-emphasis including oscillating filters may reduce effects due to mechanical resonances which however may change their frequency due to heating. 18Contrary to ECs, mechanical resonance effects are most pronounced at their corresponding resonance frequency, while being negligible at higher frequencies for example.This corresponds to a non-monotonical behavior with respect to the spectral components contained in a gradient waveform.As mechanical resonances can be a concern in MRI, 19 measures to avoid their excitation including prefiltering of the waveform 7,20 or sequence optimization [21][22][23] have been proposed.However, residual oscillating magnetic-field deviations are still detectable. 23,24revious work has investigated the influence of mechanical resonances in MR spectroscopy, where "loud" sequences have been associated with increased linewidths. 23In addition, the relation between residual background image phase and mechanical resonances has been revealed for phase-contrast flow measurements. 25n the present work, we hypothesize that the interplay of mechanical resonances with oscillating readouts can cause a slowly varying phase at a frequency given by the difference of the readout's oscillation frequency and the mechanical resonance frequency of the activated gradient axis.The effect is referred to as a beat phenomenon.[28][29][30] 2 THEORY

Beat phenomenon
The equation of motion for a driven, damped oscillator is where ẍ, , ̇x,  0 , x, X 0 , and  denote acceleration, damping constant, velocity, undamped oscillation (or resonance) angular frequency, position, driving amplitude, and angular driving frequency, respectively.Assuming initial conditions x(0) = ̇x(0) = 0 and an angular driving frequency near resonance, the solution of Eq. ( 1) is given by ] . ( The general solution (which includes Eq. [2] as special case) consists of the steady-state solution x ss and transient solution x tr .The latter decays exponentially over time (on the order of  −1 ), leaving the system oscillating solely at the angular driving frequency after a sufficient time duration has passed (Figure 1).
In the case of vanishing damping and driven close to the resonance, Eq. (2) simplifies to where A, B, and C denote an amplitude factor, the beat amplitude modulation, and the "mean" angular frequency oscillation, respectively.The derivations of Eqs. ( 2) and ( 3) are given in the Supporting Information.

F I G U R E 1
Illustration of the beat phenomenon.An EPI gradient waveform consisting of a fundamental frequency f EPI (left) played out on the MRI gradient system may result in mechanical deflections and in a change in the position of the coil.Evaluation of Eq. ( 2) with  = 2f EPI = 2 × 1390 Hz,  0 = 2f res = 2 × 1298 Hz, and  = 100 rad∕s results in a beat modulation (middle) observed during the transient phase (right).The duration of the transient phase is dependent on the damping  and on the order of  −1 .During the steady phase, the position oscillates with  = 2f EPI .Depending on the value of  for the MRI gradient system, the steady phase may not be reached within TR.
The MRI gradient system, on the other hand, experiences mechanical Lorentz forces due to rapidly changing currents in the gradient coils situated in the main magnetic field.These cause mechanical deflections that give rise to mechanical resonance frequencies.Therefore, the gradient system may be treated approximately as a driven, damped oscillator, where the driving amplitude and the position correspond to the gradient waveforms being played and the position of the gradient coil, respectively.The ith mechanical resonance frequency f res,{x,y,z},i of the x, y, or z gradient axis relates to  0 = 2f res,{x,y,z},i in Eqs. ( 1) and (3).Oscillating readouts, such as used in EPI, consist of a fundamental frequency f EPI that corresponds to the monochromatic angular driving frequency  = 2f EPI in Eqs. ( 2) and (3).
For MRI gradient coils, it is desirable to shift the mechanical eigenmodes toward higher frequencies by adding mass and/or stiffness to avoid overlap with frequencies contained in typical gradient waveforms. 13Analogously, these approaches would decrease the damping constant in Eq. ( 2), and therefore prolong the decay of the transient solution.
Assuming very low damping in an MRI gradient system and small positional deviations from its mechanical equilibrium, the system may be approximated as an undamped oscillator during a limited time duration of the transient solution (Figure 1, middle).Subsequently, exciting it near its resonance frequency by playing oscillating gradient waveforms may result in a slowly oscillating, erroneous magnetic-field modulation e(t) described by the beat phenomenon in Eq. (3) as follows: where A ′ refers to an amplitude factor that is dependent on the gradient amplitude G 0 and the frequency difference between f res and f EPI .Here, B ′ and C ′ denote the beat amplitude modulation by the frequency f beat = (f res − f EPI )∕2 and the "mean" frequency defined by (f res + f EPI )∕2.Obviously, increased positional deviations of the MRI gradient system from its equilibrium are limited by the structure itself, resulting in increased mechanical damping to prevent damage of the coils.Figure 2 illustrates the relationship of the MRI gradient waveforms, the beat phenomenon, and the spatial encoding bandwidths involved.The EPSI sequence used in the current manuscript uses two RF pulses forming a spin echo, whereas gradient echoes after the second RF pulse are acquired starting at the TE.During the readout, the signal dephases according to a T 2 * decay.For illustration purposes, five echoes corresponding to the five central k-space lines along k y are marked in green.The acquisition-time duration per echo, including sampling during gradient ramping, is shown in pink together with the time points around which individual k-samples are centered (same time scale).
Assuming a simple phantom geometry and structure, the main portion of the signal energy of the central k-space line is contained within a few samples (red) around the k-space center as shown in a logarithmic plot of the k-space magnitude data of an EPSI measurement used later in the manuscript.It is found that the signal energy of only five central k-space samples (Index j) corresponds to almost 90% of the signal energy of the whole k-space line (Index k).A comparison of the fundamental frequency of the gradient waveform (1∕(2f EPI )) and its third harmonic with the bandwidth per pixel BW x demonstrates that the change of the gradient waveform during the acquisition of the central k-space samples is negligible for this particular setup.Therefore, for simplicity, instantaneous readout at the echo time points could be assumed.Subsequently, Eq. ( 4) (A) Illustration of the echo-planar spectroscopic imaging (EPSI) sequence (gradient amplitudes not to scale; time scales are consistent within the separate plots), the beat phenomenon during the EPI train, and justification for assuming instantaneous encoding.For the one-dimensional Fourier transform sequence, phase encoding is omitted.The data sampling during the EPSI train (top) is related to the fundamental frequency f EPI and higher harmonic 3f EPI of the EPI gradient waveform (lower left).Due to the simple geometry and structure of the phantom, 90% of the k-space energy of the central k y line (data from real EPSI scan, log scale) is contained within five samples around the center (lower right).Therefore, and as the bandwidth in frequency-encoding direction (BW x ) is much higher than f EPI and 3f EPI , instantaneous encoding at the TE can be assumed, allowing for monitoring the beat phenomenon over time at the respective TEs.
where n ∈ N 0 and the echo spacing is ES = 1∕(2f EPI ).Assuming that (f res + f EPI )∕2 ≈ 1∕(2ES) results in C ′ = const.∀t  {ES∕2, … }, leading to a constant contribution to the erroneous magnetic-field modulation e(t) at instantaneous readout times t.Subsequently, when changing f EPI , the erroneous contribution due to the beat phenomenon is dependent on the difference frequency f res − f EPI .The term consisting of f res + f EPI in Eq. ( 4) yields a constant value referred to as  ES in Section 3 and is subtracted from the data before evaluation.
In addition to the temporal deviations, the magnetic-field gradient may vary spatially.If spherical harmonics are fitted, 31 zeroth-order (GMTF0; constant magnetic field offset), first-order GMTF (GMTF1; magnetic field gradient offset constant in space), and higher-order GMTFs are obtained.The errors in the MRI signal due to zeroth-order effects are spatially invariant, whereas first-order effects increase linearly with the distance from the isocenter.In addition, the sign of effects due to first-order effects depends on the sign of the distance from the isocenter.For gradient-recalled-echo sequences using high readout bandwidths in frequency-encoding direction (such as EPSI used in the current manuscript), zeroth-order and first-order effects result in a spatial shift and stretching/compression, respectively.This is exploited later in the article.

Phantom experiments
Measurements were performed on two 3T MRI systems (Philips Healthcare, Best, the Netherlands) with the same system specifications including magnetic-field gradients of 30 mT/m at a slew rate of 200 T/m/s.The GMTFs were measured previously 5,32 and resulted in the identification of mechanical resonance frequencies equal for both systems.The GMTF0 and GMTF1 along with the list of mechanical resonance frequencies for the systems used in this work are provided in Figure 3.The presented GMTF data were acquired using a spatially resolved thin-slice method 33 with a frequency resolution of 1/128 ms = 7.81 Hz (inverse of the acquisition duration) 32

F I G U R E 3
First-order (A) and (B) zeroth-order (B) gradient modulation transfer functions (GMTFs).(C) Mechanical resonance frequencies on the x-, y-, and z-axis are given in the table.Selected mechanical resonance frequencies are marked by dashed lines in (A) and (B).
with eddy current compensation activated.In the main manuscript, results of only one system are shown, while others can be found in the Supporting Information.Raw data were stored using additional metadata and processed in MATLAB 2022a (MathWorks, Natick, MA, USA) using MRecon (Gyrotools LLC, Winterthur, Switzerland).Neither EPI phase correction nor ringing filter were applied.
A spherical phantom (GE Healthcare, Waukesha, WI, USA) was positioned at the isocenter.First-order shimming was performed.B 0 maps were acquired to ensure acceptable shimming (see Figure S1).To evaluate mechanical resonance effects, a single-shot EPI sequence with its phase blips set to zero was used, resulting in one-dimensional (1D) projections of the phantom along the phase-encoding direction over time (Figure 4).After Fourier transform along the k x -direction, data were available with coordinates where k y = { 1, 2, … , n ky } (n ky total number of k y profiles) and Here, we differentiate between time coordinate t ′ and "real world" time t, where the step size of the former corresponds to the inverse of the bandwidth in the phase-encoding direction (= time for acquiring one k-space line along the frequency encoding direction).The TE and TR were set to 25 ms and 1000 ms, respectively, while varying the frequency of the EPI train between f EPI = [1121, 1580] Hz.
From the data of the erroneous phase, where Δk 1st and ΔB0 0th denote the erroneous k-space position offset due to first-order GMTF terms and the erroneous B 0 offset due to zeroth-order GMTF terms, respectively.Of note, Δ ( t ′ ) in Eq. ( 6) is induced exclusively by the switching magnetic-field gradients as opposed to  B0 ( t ′ ) due to field inhomogeneity.To eliminate the latter from the data, acquisitions with inverted readout gradient sign (opposite water fat shift direction [termed WFS {y,z} for the frequency encoding direction], or equivalently, the EPI train, being aligned with axis {z, y}) were acquired for every EPI frequency step, and the phase images were subsequently subtracted.
Illustration of EPI phase discrepancies and EPI phase-correction approaches for eliminating erroneous echo phase difference Δ ES .Without B 0 inhomogeneity or beating, the phase discrepancy between odd and even echoes is constant and can easily be removed (left).With B 0 inhomogeneity, a linear model can be fitted (middle).With beating, however, linear correction approaches fail due to the nonlinear trend over encoding time and results in an erroneous phase Δ.magnetic-field gradients in slice direction due to first-order and higher-order GMTF terms would have canceled out due to the positioning of the phantom at the isocenter.
Of note, Δ ( x, t ′ ) along the image coordinates [ x, t ′ ] is approximately constant along x as the bandwidth per pixel in frequency-encoding direction is much larger than the beat frequency (Figure 2).Due to the interferometric nature of the method, the signal obtained depends on the ratio of (1) the temporal change of the magnetic field due to mechanically resonating gradient coils and (2) the echo spacing (ES).Therefore, if f EPI = 1∕(2ES) and f res are sufficiently far from each other, a difference frequency can be determined for the estimation of f res .However, if f EPI is very close to f res , due to the subtraction of even and odd echoes and assuming instantaneous readout at the echo time points, the effects are not visible to their full extent (Figure 6).
To evaluate the effect of erroneous first-order and zeroth-order GMTF effects on imaging, an echo-planar spectroscopic imaging (EPSI) sequence 34 was used for scanning a spherical gel phantom placed at the isocenter while covering a FOV of 200 × 200 mm 2 with a slice thickness of 10 mm.The EPSI encoding strategy resulted in one image per echo along the readout train (63 echoes in total) and enabled the reconstruction of images over time t ′ .This is similar to the 1D-EPI mode used previously, while additionally using phase encodings at the beginning of each readout train (Figure 4).Data coordinates were hence For EPSI imaging, the erroneous phase  ( t ′ ) still satisfies Eq. ( 5); however, elimination of B 0 effects from  B0 ( t ′ ) is not feasible using the current WFS approach.A possible solution would be shifting of the echo train by 1∕(2f EPI ) 28,29 ; however, this has not been implemented for the current work.Instead, optimized shimming was used to minimize the influence of B 0 .The decay due to intravoxel phase dispersion at boundary pixels and T 2 * was fitted along the magnitude data over time t ′ using a Illustration of the beat phenomenon observable in the phase-difference data for evaluation of Eq. ( 2) while assuming instantaneous readout at the respective TEs for a hypothetical experiment.A mechanical resonance of f res = 1298 Hz and two different fundamental EPI frequencies f EPI,1 = 1299 Hz and f EPI,2 = 1390 Hz are assumed.For f EPI,1 ≈ f res , the position difference of even and odd echoes does not necessarily reflect the position.Of note, it is assumed that spatial encoding happens instantaneously; hence, imaging effects are not considered.mono-exponential model.Given the very long T 2 * of one evaluated voxel, a linear model was fitted with acceptable agreement.For all fits, the first eight echoes were omitted to allow for stabilization of the signal.Remaining signal variations in the image magnitude were attributed to gradient-induced effects and evaluated in frequency and phase-encoding direction for different EPI frequencies.Due to the even/odd echo discrepancy of the magnitude signal being centered around zero after subtraction of the fitted curve, the sum of the magnitude data for even and odd echoes was used, respectively.For the EPSI data, erroneous zeroth-order gradients have no spatial dependency, therefore resulting in an erroneous offset phase that is equal for any voxel.According to the Fourier shift theorem, this leads to spatial shifting of the imaged object without spatial distortion, assuming that no other erroneous effects are at play.Therefore, if the signal magnitude of opposing voxels (e.g., top and bottom) follows an opposing temporal trend (out of phase), zeroth-order effects are identified as of erroneous phase.Erroneous first-order gradients, however, have linear spatial dependence, therefore resulting in an erroneous offset phase that is dependent on the distance of the voxel to the isocenter.According to the Fourier spatial-encoding model, this leads to spatial stretching/compression originating at the center of the imaged object if it was positioned at the isocenter.Therefore, if the signal magnitude of opposing voxels follows a similar temporal trend (in phase), first-order effects are identified as cause of the erroneous phase.This is used for the EPSI data analysis.

Data analysis
Given the image coordinates , the beat period was calculated by visual identification of the coordinate distance t ′ beat = t ′ (max 2 ) − t ′ (max 1 ) of two maxima of the oscillation of either the image phase (1D EPI) or the image magnitude (EPSI).For the 1D EPI, image coordinates Using Eq. ( 4) enables identification of the mechanical resonance frequency, causing the observed beat frequency by where the absolute value of the beat frequency

RESULTS
For the 1D Fourier transform data, Figure 7 shows the image phase difference Δ ( x, k y ) for different EPI frequencies.A slowly varying beat pattern is found, with its frequency and amplitude depending on the EPI frequency f EPI .The frequency of the beat pattern was found to fulfill the relationship S1) for the z-axis with a single resonance.Additional data for EPI along the y-axis and phase plots without subtraction of even/odd echoes are shown in Figures S2-S11.Figure 8 presents the evaluation along the central voxel, yielding a maximum phase deviation of 0.8 rad for f EPI = 1305 Hz.The maximum amplitudes over all data were found to be 1 rad (Figures S8 and S9).
In addition, Figures 7 and 8 and Supporting Information Figures S8 and S9 confirm the non-monotonic relationship between |f beat | and f EPI,k when f EPI,k > f res,k,i or f EPI,k < f res,k,i .The amplitude of the beating scales with |f beat | as predicted in Eq. ( 4). Figure 9 depicts the evaluation of Eq. ( 7) for different f EPI,z (A-C) and for f EPI,y (D).For the latter, the EPI train (WFS direction) is aligned along the y-axis, which consists of several f res,y,i .In the plot, a superposition of beat patterns is found, which corresponds to f res,y,2 and f res,y,3 .

F I G U R E 7
Phase plots with image coordinates [ x, t ′ ] for all one-dimensional Fourier transform measurements across different f EPI with the EPI train aligned along the z-axis.The mechanical resonance frequency of the corresponding axis is f res,z,1 = 1298 Hz.

F G U R E
Phase plots for x = 30 along time t ′ for all one-dimensional Fourier transform measurements across different f EPI with the EPI train aligned along the z-axis.The mechanical resonance frequency of the corresponding axis is f res,z,1 = 1298 Hz.
For the EPSI data, Figure 10 presents the magnitude over time for selected voxels (line plots in Figure 10A; colored voxels and emphasized by rectangles in Figure 10B) after subtraction of T 2 * and intravoxel phase-dispersion fitting results.The WFS direction is altered every other row to align with either the y-axis or z-axis.A beat pattern is visible for every other row (marked by black and white triangles) and most pronounced on the axis Phase plots with image coordinates showing a slowly, sinusoidally varying phase component.The relationship in Eq. ( 7) reveals the underlying mechanical resonance, which causes the beat pattern to be identified using Eq. ( 8).(A) For f EPI,z < f res,z,1 (EPI train along z-direction), the beat frequency is 86 Hz.Evaluation of Eq. ( 8) yields f ′ res = 1284 Hz as the possible mechanical resonance frequency.(B) For f EPI,z ≈ f res,z,1 , no beating is observable (Figure 6).comprising the EPSI train.Its frequency . For a single WFS direction, | f ′ beat | changes with f EPI , and the non-monotonic trend is confirmed (marked by brackets and asterisks).The asterisks correspond to a frequency f EPI,k where a zero-crossing of | f ′ beat | occurs.Evaluation of the center voxel (gray curves) demonstrates the validity of the T 2 * and intravoxel phase-dispersion fit over time.Figure 10C compares the magnitude over time for spatially opposing voxels for different WFS directions.For f EPI,k ≈ f res,k,i , first-order effects are found corresponding to time-varying stretching of the FOV.For f EPI,k ≈ f res,j,i where k ≠ j (different axis), zeroth-order effects occur that relate to time-varying spatial shifts of the isocenter.Figure S12 shows the validity of T 2 * and intravoxel phase-dispersion fitting of the magnitude data for selected voxels and f EPI .Of note, a linear fit for the top voxel (red), and mono-exponential fits for the other voxels, were used.Figure S13 shows exemplary magnitude data before summation of even/odd echoes.

DISCUSSION
In this work, the mechanically oscillating behavior of a gradient system and its interplay with oscillating readouts have been studied in detail.A slowly varying phase was observed with a frequency given by the difference of the oscillating readout frequency and the mechanical resonance frequency of the activated gradient axis.Accordingly, a gradient system be seen a mechanical oscillator.A 1D-EPI sequence with fundamental frequencies below above the mechanical resonance frequency of the gradient system allowed us to demonstrate the occurrence of the beat phenomenon and to confirm its non-monotonic behavior a function of EPI frequency.In Figure 9A, Eq. ( 8) yielded 1284 Hz as a possible frequency, where indeed the main mechanical resonance of corresponding axis f res,z,1 = Hz.same held for Figure 9C. Figure 9D revealed a of beating phenomena for an axis with several mechanical Evaluation of the voxels' magnitude sum for even/odd echoes over time after subtraction of T 2 * decay while using the echo-planar spectroscopic imaging (EPSI) sequence.(A) Temporal evolution of the magnitude for different f EPI and water-fat-shift (WFS) directions aligned with the y-axis and z-axis for voxels and WFS directions shown in (B), selected at voxel positions at the phantom's boundary.WFS direction corresponds to the phase-encoding direction (orthogonal to the axis comprising the EPSI train).(A) The beat effect is found on the axis on which the EPI train is played (black/white triangles).(B) Red, blue, green, and black voxels lie on the same axis.(C) Comparison of the phase of the beat effect for opposing voxels enables a differentiation between first-order (similar variation; stretching/compression) and zeroth-order effects (opposing variation; spatial shift).Of note, first-order effects are found on the axis on which the EPI train is played (e.g., for red/blue curve, f EPI,z + f beat = 1336 Hz, where f res,z,1 = 1298 Hz), whereas zeroth-order effects arise on the axis different from the one with the EPI train (for red/blue curve, f EPI,y + f beat = 1041 Hz, where f res,y,2 = 1050 Hz).
In this study, the maximum erroneous phase observed due to the beat effect was 1 rad (see Figures S8 and  S9).This would relate to a FOV stretching/compression of 1 pixel during phase encoding.Therefore, a possible implication of the beat phenomenon for oscillating readouts in imaging is blurring.However, as illustrated in Figure 6, the actual erroneous phase may be larger for EPI frequencies close to mechanical resonance frequencies, resulting in more severe image artifacts.As the erroneous beat phase oscillates with low frequency, the resulting k-space data are inconsistent, as the encoded FOV in the frequency-encoding direction is stretched/compressed in an oscillatory fashion during the phase encoding of k y lines.This has been demonstrated in a simulation for a slotted phantom in another work. 35The maximum erroneous phase at the mechanical resonance frequency could be determined by a magnetic-field camera measurement.
Figure 10 evaluated the influence of the beat effect on imaging using an EPSI sequence.The data confirmed several hypotheses: First, the beat phenomenon dominated on the axis comprising the EPSI train.This is in line with the assumption that the oscillating readout gradient causes the erroneous beat.Second, the non-monotonic behavior predicted by Eq. ( 4) was evident while varying f EPI to be smaller and greater than f res,z,1 = 1298 Hz (blue line plots) and f res,y,2 = 1237 Hz and f res,y,3 = 1237 Hz (black line plots).Moreover, for this axis consisting of several f res,y,i , the superposition of beats is confirmed.Third, first-order effects are found on the axis on which the EPI train is played (e.g., for red/blue curve: f EPI,z + | f beat |= 1336 Hz where f res,z,1 = 1298 Hz), whereas zeroth-order effects arise on the axis different from the one with the EPI train (for red/blue curve: f EPI,y + |f beat | = 1041 Hz where f res,y,2 = 1050 Hz).This may be related to zeroth-order effects on an axis due to mechanical resonances on other axes (Figure 3).However, this interpretation remains vague and shall not be continued here.Summarizing, it was demonstrated that the beat effect cannot simply be avoided by choosing f EPI away from f res (Figure 10A) and that effects of first order occur in the spatial direction aligned with the EPI train, while zeroth-order effects may arise along the other direction.
Current correction approaches for mitigating even/odd echo shifts in EPI (usually referred to as EPI phase correction) assume the temporal evolution of the phase difference of even/odd echoes to be of low order (e.g., linear in time; Figure 5, left).This corresponds to a phase accumulation in time due to B 0 inhomogeneities, for example (Figure 5, mid).However, if the beat phenomenon is present, linear EPI phase corrections would fail to predict the phase over time in the presence of oscillating erroneous phase due to mechanical resonances as soon as sin 5, right).Prescans for EPI phase correction are applied by default before acquiring EPI images and consist of a 1D Fourier transform scan similar to the one used in the current manuscript.Therefore, the beat phenomenon may be visible in the available correction data without overhead and allowed for fitting an extended phase-correction model as in Eq. ( 4).
Low-frequency magnetic-field fluctuations while playing EPI sequences have been previously reported. 24The authors stated that given the low number of principal components, correction of the fluctuations may be conceivable.Referring to Eq. ( 4), the two unknowns for characterization of the magnetic-field fluctuations are the factors A ′ and C ′ , given that the mechanical resonances of the MRI gradient system are known.In fact, for long TRs, allowing for the gradient system to settle to equilibrium, the physical properties of the beat phenomenon require it to have zero amplitude at the beginning.For the data available, the profile at k y = 1 was not filled with data due to unknown reasons.The value at k y = 1 in Figure 9A, for example, can be extrapolated and would yield approximately zero beat amplitude, therefore fulfilling Eq. ( 4).However, for other f EPI , this relation does not hold (Figure 7).
In the current manuscript, solely the fundamental frequency f EPI has been considered, while higher harmonics 3f EPI or 5f EPI have been omitted.Previously identified mechanical resonance frequencies for the MRI system used were < 1780 Hz, as reported by Vannesjo et al. 3 For smaller, dedicated gradient head coils, mechanical resonances up to 2907 Hz are possible. 36However, it is to be expected that the damping would increase rapidly for mechanical resonances at higher frequencies.Due to amplifier nonlinearities and the used chirp input function, the previously measured transfer function does not allow for interpretation of characteristics > 2000 Hz.Nevertheless, for the identified mechanical resonance frequencies, it holds that f res < 3f EPI for all combinations of f res and f EPI on all axes.Subsequently, excitation of mechanical resonances by higher-order harmonics is not likely for the EPI frequencies and the MRI system used.
For other oscillating readout techniques, such as spiral imaging or balanced SSFP, the gradient waveform spectra hold information about the possibility to excite mechanical resonances.To this end, combined spectral and temporal analysis methods such as the short-time Fourier transform could be used.In the current manuscript, however, the beat effect due to the fundamental frequency of EPI readouts (which was constant during spatial encoding) was evaluated.
Concurrent field monitoring 31,37 mitigates errors due to imperfections of the MRI gradient chain at reconstruction time. 31However, it requires specialized and expensive hardware.A one-time calibration using the GMTF 3 allows for reduction of the effects and would be favorable; however, due to heating effects, the mechanical resonance frequencies may change during measurement time. 12,18,24 possible alternative, simple method for the identification of mechanical resonances using audio recordings has been presented previously, 38 which could be used concurrently to measurements and allow monitoring mechanical resonance frequencies online.However, this method may be limited to modes that also couple acoustically. 39pproaches to mitigate gradient imperfections based on the gradient system transfer function (GMTF/GIRF) include prospective or retrospective correction of gradient waveforms by convolution with the impulse response of the corresponding gradient axis.If an oscillatory gradient waveform is used, the output will also reflect the beat phenomenon, given that the GMTF resolves the mechanical resonance.However, given that the LTI assumption may not hold for strong gradients as used in diffusion MRI, 11 the validity of the gradient correction may be limited as well.In addition, the time horizon for the impulse response (and therefore for the validity of the convolution) is limited by the time duration of the GMTF acquisition.The latter is limited, in turn, by the signal decay of the measurement (which includes the worsening of SNR throughout acquisition of the GMTF).
The current study confirms previous works identifying mechanical resonances as a source for erroneous background phases, as they related the echo time point to the arising effect and demonstrated that it is non-monotonic. 25,40,41To the best of our knowledge, however, the current work is the first to explain the underlying cause in detail.
The data shown act as a proof-of-concept for the theory arising from physical principles.It is expected to hold for other MRI systems as well.GMTFs of other systems by other vendors published in literature 4,7 exhibit similar dips and peaks due to mechanical resonances that support the claim.The GMTFs have been acquired at an earlier timepoint than the data shown in the manuscript.However, due to their long-term stability, 5 the mechanical resonance frequencies are not expected to differ.

CONCLUSION
Oscillating readouts such as used in EPI can result in low-frequency, erroneous phase contributions, which are explained by the beat phenomenon.Therefore, EPI phase-correction approaches may need to include beat effects for accurate image reconstruction.
. One-dimensional Fourier transform (1DFT) phase-difference plots evaluated at x = 30 for different EPI bandwidths played on the y-axis for System S1.The number in the rectangles corresponds to the scan ID given in Supporting Information Table S1.Some scan IDs have been acquired several times and are shown here for the purpose of demonstrating the repeatability of the results.Figure S6.One-dimensional Fourier transform (1DFT) phase images for different EPI bandwidths played on the y-axis for System S1 (even/odd echoes have not been subtracted).Some scans have been acquired several times and are shown here for the purpose of demonstrating the repeatability of the results.Figure S7.One-dimensional Fourier transform (1DFT) phase plots evaluated at x = 30 for different EPI bandwidths played on the y-axis for System S1 (even/odd echoes have not been subtracted).Some scans have been acquired several times and are shown here for the purpose of demonstrating the repeatability of the results.Figure S8.One-dimensional Fourier transform (1DFT) phase-difference images for different EPI bandwidths played on the z-axis for System S2.Some scans have been acquired several times and are shown here for the purpose of demonstrating the repeatability of the results.Compared with Figure 7, the phase is not constant in the x-direction.This may be related to time-varying magnetic field gradients at around the TE or because the phantom's position was offset from isocenter.However, this effect is not the focus of the current manuscript.
Acquisition schemes.One-dimensional Fourier transform (1DFT) acquisition refers to an EPI scan with phase-encoding blips deactivated and results in multiple acquisitions of the central k y line (=1D projection along y) over time (left).Standard EPI trajectory uses phase blips and allows for reconstruction of a single image (middle).Echo-planar spectroscopic imaging (EPSI) uses phase blips before the EPI train and allows for the reconstruction of 2D images over time (right).Data ordering in EPSI ensures consistency of k-space data within single images in contrast to EPI.
[ x, y, t ′ ].An acquisition resolution of 5 × 5 mm 2 was used and reconstructed to a 96 × 96-pixel matrix by zero-padding the data.A total of 50 averages was acquired per EPI frequency step.The TE and TR were fixed to TE = 10.2 ms and TR = 100 ms, respectively, while varying the frequency of the EPSI train between 1016 Hz and 1369 Hz.The readout train alignment with gradient axes y and z resulted in data with WFS directions (i.e., phase-encoding direction) WFS z and WFS y.
(C) For f EPI,z > f res,z,1 , the phase oscillation changes sign and yields f ′ res = 1274 Hz. (D) For f EPI,y , a superposition of oscillations is found corresponding to resonances of the y-axis yielding f ′ res,1 = 1236 Hz and f ′ res,2 = 1042 Hz.Gradient modulation transfer function-based identification yielded f res,z,1 = 1298 Hz, f res,y,2 = 1050 Hz, and f res,y,3 = 1237 Hz.

Figure S9 .
Figure S5.One-dimensional Fourier transform (1DFT) phase-difference plots evaluated at x = 30 for different EPI bandwidths played on the y-axis for System S1.The number in the rectangles corresponds to the scan ID given in Supporting Information TableS1.Some scan IDs have been acquired several times and are shown here for the purpose of demonstrating the repeatability of the results.FigureS6.One-dimensional Fourier transform (1DFT) phase images for different EPI bandwidths played on the y-axis for System S1 (even/odd echoes have not been subtracted).Some scans have been acquired several times and are shown here for the purpose of demonstrating the repeatability of the results.FigureS7.One-dimensional Fourier transform (1DFT) phase plots evaluated at x = 30 for different EPI bandwidths played on the y-axis for System S1 (even/odd echoes have not been subtracted).Some scans have been acquired several times and are shown here for the purpose of demonstrating the repeatability of the results.FigureS8.One-dimensional Fourier transform (1DFT) phase-difference images for different EPI bandwidths played on the z-axis for System S2.Some scans have been acquired several times and are shown here for the purpose of demonstrating the repeatability of the results.Compared with Figure7, the phase is not constant in the x-direction.This may be related to time-varying magnetic field gradients at around the TE or because the phantom's position was offset from isocenter.However, this effect is not the focus of the current manuscript.Figure S9.One-dimensional Fourier transform (1DFT) phase-difference plots evaluated at x = 30 for different EPI bandwidths played on the z-axis for System S2.Some scans have been acquired several times and are shown here for the purpose of demonstrating the repeatability of the results.Figure S10.One-dimensional Fourier transform (1DFT) phase images for different EPI bandwidths played on the

Figure S12. T 2 *
and intravoxel phase-dispersion fit (black dashed line) for the magnitude data (points and crosses) over time for voxels colored according to their color in Figure10, even though the first eight data points shown here have not been used for the fitting.Of note, due to the long T 2 * of the phantom used in the experiment and varying intravoxel phase dispersion depending on the voxels' positions, mono-exponential fitting is used for the bottom, left, right, and center voxels (blue, green, black, and gray), whereas for the top voxel (red), a polynomial of first order was used.Water-fat-shift (WFS) direction Z (A), WFS Z (B), WFS Y (C), WFS Z (D), WFS Y (E), and WFS Z (F).

Figure S13 .
Bottom voxel's magnitude (blue voxel in Figure10) for f EPI = 1369 Hz (WFSZ) after subtraction of the mono-exponential decay fit.Left: before sum of even/odd echoes' magnitude; right: after summing even/odd echoes.The right plot corresponds to the processing of the data shown in the magnitude plots in Figure10for all voxels.