Radiation damping at clinical field strength: Characterization and compensation in quantitative measurements

In any MR experiment, the bulk magnetization acts on itself, caused by the induced current in the RF receiver circuit that generates an oscillating damping field. This effect, known as “radiation damping” (RD), is usually weak and, therefore, unconsidered in MRI, but can affect quantitative studies performed with dedicated coils that provide a high SNR. The current work examined RD in a setup for investigations of small tissue specimens including a quantitative characterization of the spin‐coil system.


INTRODUCTION
Achieving a high SNR is challenging in MR of specimens that are small compared to the sensitive volume of standard receiver coils. 1,2In such cases, dedicated coils with an increased "filling factor"  and quality factor Q are often employed.However, this may concomitantly result in relevant radiation damping (RD) [3][4][5][6][7][8][9][10][11][12] caused by inductive coupling of the bulk magnetization and the coil, that is, the processing magnetization interacts with itself mediated by the detection circuit.According to Lenz's law, the induced current generates an oscillating damping field that rotates the magnetization without altering its length.A well-known RD effect is an increased linewidth of strong signals (e.g., solvent peaks) in high-resolution NMR. 11,13,14espite a common assumption, RD is not unique to high magnetic fields, 9 but has been observed in NMR at lower fields [3][4][5] and occasionally MRI at clinical field strength. 15,16revious works have described RD in various experiments, 12,[16][17][18][19][20][21][22][23][24] including relaxation and magnetization-transfer (MT) measurements, perfusion MRI, spectroscopy, and more.Of note, RD is not only present during free precession (with or without signal detection), but also during RF transmission. 7,25Here, RD is more difficult to characterize and may interfere with the desired magnetization trajectory during the pulse, thereby altering the effective flip angle θ eff .Damping during transmission is more prominent for long, low-power pulses with durations  p of several milliseconds.
Various techniques have been proposed for mitigating, suppressing or even utilizing 26 RD, including a reduced sample region contributing to the signal, 21 small flip-angle pulse trains to counterbalance RD, 27 coils with switchable Q, 28 or active electronic feedback. 291][32] If this is not applicable (e.g., during RF pulses), alternative solutions are required. 7,25Approaches to obtain RD-insensitive RF pulses were based on theoretical considerations, 33 composite pulses and gradient optimization, 7 or optimal-control theory. 34urrently, little is known how RD might confound the accuracy of quantitative MRI, which is increasingly used to study tissue microstructure or composition (e.g., myelin or iron content). 35As more prominent perturbations result with high  and Q, unconsidered damping-related bias may be of particular concern when scanning small specimens with bespoke RF coils.This is a typical approach for correlating results from MRI and other modalities. 36,37For example, investigations in fixed marmoset brain yielded cortical T 1 values ranging from 371 ms at the gray-white matter boundary to 359 ms in the stria of Gennari, corresponding to a variation of only 3.2%. 37This underscores the degree of precision required to identify the gross signature of cortical layers using relaxation measurements.Small effect sizes are also characteristic of other MRI experiments, for example, arterial spin labeling 16 or MT 22,37 and relaxation anisotropy measurements in white matter, 38,39 indicating that their performance may be impacted by RD.Damping effects were also observed in reference measurements of T 1 of blood with a standard transmit-receive (Tx/Rx) head coil. 40The focus of the current work was, therefore, on a comprehensive characterization of RD in simple phantoms employing a setup for small samples on a clinical scanner.The degree of damping was externally adjusted through a hardware modification.Finally, RD was independently quantified for free precession and transmission.

THEORY
To account for RD, the evolution of the magnetization is described by augmenting the Bloch equations by: an additional back-action field B RD . 5Its amplitude is proportional to the (complex) transverse magnetization M + = M x + M y : where  is the gyromagnetic ratio. is the damping rate, which depends on the characteristics of the RF circuit: with and  = arctan  LC (5)   where  0 is the vacuum permeability, L the coil's inductivity,  0 the Larmor frequency, and  LC the circuit's resonance frequency.The RD field lags behind the transverse magnetization by an angle π∕2 − ψ, that is, both are in quadrature for a perfectly tuned coil with Ω ≡  0 −  LC = 0.The (complex) nutation frequency  RD = −B RD has, therefore, components This notation is slightly different from recent work 10,12 but follows Vlassenbroek et al. 6,14 With  1 = −B 1 , the combined spin-coil system is, hence, described by the Bloch-Maxwell equations in the rotating frame as: Often, an effective damping time constant  RD is used as an indicator of the RD strength.It is related to  by: To account for static field inhomogeneities, T 2 is commonly replaced by T * 2 . 13However, this simplification may not capture transient RD effects, 9,41 and an approach based on isochromats may be preferable in some cases. 6,9,14,41

METHODS
All experiments were performed at 3 T on a MAGNETOM Skyra fit (Siemens Healthineers, Erlangen, Germany) operated under the software baseline syngo MR VE11E.Various tailored pulse sequences were developed to ensure sufficient experimental flexibility. 42They provide a variety of RF pulse types for preparation, excitation and refocusing, whereby every pulse may be spatially selective or non-selective.Different preparation schemes (excitation, saturation, inversion) are supported followed by an adjustable relaxation period including variable crusher gradients.Selectable readout types included a simple FID, a spin-echo train, or multiple-echo (ME) gradient-recalled echoes (GREs).Two spherical phantoms were used for the measurements with inner diameters of 19 and 24 mm.They were filled with aqueous MnCl 2 solution from the same batch.The concentration (0.135 mM) was adjusted to produce a T 1 of approximately 770 ms at room temperature. 43

Helmholtz coil
A linear Tx/Rx Helmholtz coil was used in all experiments (Figure 1A). 44It was optimized for small ex-vivo specimens to obtain a high SNR and B 1 homogeneity and short RF pulse durations (≈20 μs for θ=90  For initial tests, the loaded coil was connected to a standard Tx/Rx switch via a 50 Ω coaxial cable of random length.Initial experiments indicated characteristic RD effects with potential impact on the precision of quantitative experiments.They included an echo-like signal build-up after flip angles >90 • (Figure S1) and flip-angle dependent line broadening, which was not improved by shimming.Hardware-based methods to minimize RD, such as a small , active electronic feedback or detuning, were discarded due to reduced SNR or high circuit complexity.Alternatively, the very low input impedance of GaAs field-effect transistor (FET)-based preamplifiers at optimal noise matching was utilized in a new Tx/Rx switch (Figures 1B and S2). 45The basic idea for this switch was modified by (i) adding an actively biased PIN-diode switch and (ii) supplementing the LC transformer for impedance reduction in the Tx branch by a resistive voltage divider.Instead of a trimmer, fixed-value capacitors were employed to match the loaded coil to 50 Ω.The GaAs-based preamplifier and the voltage divider (resistors R 1 = 2 Ω and R 2 = 1 Ω) yielded a strong impedance mismatch ensuring that most of the induced signals in the coil were reflected back.An alternative setup was also realized with R 1 = 49 Ω and R 2 = 1 Ω.Here, the LC transformer was not necessary; therefore, C s1 , C s2 , and L p were omitted.This resulted in a higher voltage needed for a 180 • rectangular 1 ms pulse (referred to as "reference voltage" in the following), which was undesirable for many intended applications but useful for demonstrations.
The adapted Tx/Rx switch, a suitable cable length and the impedance mismatch guaranteed that the induced current in the coil, i in , and the current due to reflected power, i re , were almost perfectly out of phase, leading to destructive interference.Therefore, the loop currents cancel out, and RD is minimized without SNR degradation.The required phase shift is easily adjusted by the cable length.Note that the same principle is employed for preamplifier decoupling. 46If the cable length is extended by a quarter of the wavelength (/4), the loop currents reach a maximum (in-phase interference of i in and i re ). 47hus, a coaxial cable of appropriate length between the coil and the Tx/Rx switch achieves changing from minimum to maximum RD with nearly identical coil characteristics.This permits convenient checks if pulse sequences are Custom-built linear Helmholtz coil with a spherical water phantom.The sample holder is designed for either standard 5 mm NMR tubes or 3D-printed spherical containers up to 25 mm diameter (as shown here).For studies of orientation dependence of MR parameters (not used in the current study), it can be tilted about the x-axis and z-axes of the magnet, together with the self-supporting coil elements.Angle indications support the adjustment of a desired tilt angle.The center of the sample and the coil always remain in the magnet's isocenter upon tilting.(B) Simplified schematic of the Helmholtz coil and the Tx/Rx switch.The trimmer capacitor C t is integrated for tuning.The required cable length for maximum radiation damping (RD) reduction depends on the choice of the (fixed) matching capacitors C m1 and C m2 .The LC transformer (C s1 , C s2 , L p ) matches the transmitter output (50 Ω) to the input impedance of the voltage divider (approximately R 1 + R 2 ).Further details are presented in Figure S1.
affected by RD or if RD suppression works.The following cables were utilized: 1 Optimized length for minimal RD ("setup 1"). 2 Extended length by λ/6 for moderate RD (phase-shifted interference of i in and i re ; "setup 2"). 3 Extended length by λ/4 for maximal RD ("setup 3").
Coil tuning and matching were performed in Tx mode, using a vector network analyzer (ZVT 8; Rohde & Schwartz, München, Germany).Presumably, the Rx mode was not perfectly tuned, due to the two distinct electronic branches (Figure 1B). 48,49The RD-modifying cables were attached inside the scanner without retuning.
All samples were shimmed using the scanner's manual adjustment, which proved challenging due to the simultaneous presence of RD and field inhomogeneities.Therefore, shimming was performed once for setup 1 and not readjusted after changing the cables to setups 2 and 3.An FWHM of 9.5 and 11 Hz was achieved for the 19 and 24 mm sample, respectively.

Characterization of RD
Previous studies highlighted specific RD characteristics during free precession and transmission, 10,48 suggesting a need for different experimental strategies for both coil operating modes.Therefore, individual parameters  RD,Rx ,  RD,Tx as well as  Rx , ψ Rx and  Tx , ψ Tx are employed for the Rx and Tx mode, respectively.

Pulse calibration and pulse sequences
According to Keifer, 50 an array of spectra acquired with an increasing pulse duration from scan to scan offers detailed characterization of the MR system, including off-resonance effects, insufficient relaxation delays, probe arcing, and so forth.Besides accurate 90 • -pulse calibration, RD effects can also be visualized.In the current work, it was used to evaluate the circuit's stability and Tx-field homogeneity.Data were collected following non-selective excitation with a rectangular pulse ( p = 0.02-4.42ms).The total array consisted of 111 measurements (TR = 4.5 s).The spectra were phase-corrected, and a region of ±1.5 ppm around the resonance position was extracted and concatenated.The array was normalized by setting the amplitude of the highest positive peak to 1.

RD during free precession
RAdiation Damping Difference SpectroscopY (RADDSY) compares relaxation after a short (0.4 ms) rectangular 90 • pulse in the absence and presence of RD followed by another 90 • readout pulse. 32,52A shift of a trapezoidal crusher gradient (amplitude G = 14 mT/m; duration 3 ms) yields two sequence variants: (i) 90 • -G--90 • -ADC (undamped reference) and (ii) 90 • --G-90 • -ADC (damped case).The ADC (analog-to-digital converter) period indicates the time of signal detection;  is an evolution time (between 5 and 6000 ms).If the crusher is applied immediately after the first 90 • pulse, no coherent transverse magnetization is present during the evolution period, and RD is suppressed.In contrast, damping is maximal in the second sequence version, and magnetization is returned faster to the z-axis during the time  to be detected following the second 90 • pulse.In additional experiments, a 1D GRE train (48 echoes; ΔTE = 6 ms, TR = 7 s) was utilized to measure T * 2 .The readout gradient was applied along the x-, y-, and z-axes in consecutive experiments, and the data from the three acquisitions was averaged for further analysis.
RADDSY only achieves quantitation of RD in terms of  RD,Rx without yielding information about  Rx and ψ Rx (Eq.8).Experiments that are sensitive to ψ Rx exploit the resonance shift caused by RD, that is, the imperfect quadrature alignment of the transverse magnetization and the damping field ( ψ Rx ≠ 0 ) if the coil is not perfectly tuned. 11,53,54Two approaches were implemented: (i) A flip-angle array to examine the dependence of the proton frequency on θ (variation between 0 • and 360 • in increments of 10 • ; adiabatic 5 ms BIR4-pulse 55 ). 54ii) Saturation-recovery experiments (train of six 90 • pulses and crusher gradients with alternating direction, x, y, z, x, y, z, and decreasing amplitude) with different saturation levels, assuming that a weaker initial magnetization induces less RD.19 This leads to a corresponding line broadening depending on M z accompanied by a frequency shift.Because of the long scan time (>10 min), B 0 drifts might be misinterpreted as coil detuning.Therefore, every second acquisition was replaced by a reference scan to correct for frequency shifts unrelated to RD.

RD during transmission
A comprehensive characterization of RD during transmission was performed with four distinct preparation schemes consisting of two types of composite pulses.One type was designed to compensate only for inhomogeneity of the transmit field amplitude B + 1 and off-resonance effects. 56e other type, 7 not only accounts for these imperfections but also compensates for RD effects.After preparation, a crusher gradient was applied, and the magnetization was read out by a 200 μs) 90 • pulse.We assume that differences observed with RD-compensated and uncompensated preparations primarily reflect magnitude and phase variations of the damping field during transmission and refer to this approach as RAdiation Damping Difference EXcitation (RADDEX).The following composite pulses were used in successive experiments: (i) 90 x (damped 180 • rotation).For each scheme, 36 measurements (TR = 5 s) were performed with increasing  p .The duration for a (rectangular) 90 • subpulse in the inversion composite pulse was varied between 0.2 and 5.0 ms yielding total durations of 0.9-19.3ms, 1.8-38.6ms, 1.0-25.0ms, and 0.8-20.0ms for schemes (i), (ii), (iii), and (iv), respectively.For the 90 • -excitation experiment, the maximum duration of the 315 • subpulse was 15 ms (maximum duration of 4.3 ms for the 90 • subpulse) in scheme (i).
To render RF pulses immune to RD, we adapted a method to modify the complex RF field describing the pulse shape in order to counterbalance the induced damping field. 25This involves an iterative procedure in which the Bloch equations are initially solved to obtain M + (t) without relaxation under the assumption of an undamped RF pulse.With this result, the complex reaction field is computed with Eqs. ( 1), (6a), (6b) and can be canceled out by changing the nutation frequency of the amplitude and phase-modulated pulse.This procedure requires the knowledge of the RD characteristics of the investigated system.To support online application on the scanner, an interface was created on the user interface's "special card" to allow input of pre-determined RD parameters.During runtime, the Bloch-Maxwell equations were solved numerically without restriction to a perfectly tuned coil, and the rectangular pulse shape was adapted to compensate for RD. Tx and ψ Rx were provided via the user interface, and the nutation frequencies were corrected accordingly.The RD parameters were not iteratively optimized as suggested previously 25 but determined by RADDEX.Instead of "pulse-width arrays," "pulse-amplitude arrays" were employed for the investigation of RD-related distortions of θ.Hence, FIDs were recorded following a 7.5 ms pulse, whose amplitude was incremented to achieve nominal flip angles varying from 0 • to 360 • in steps of 5 • .The online compensation method improved pulse performance, which is relevant for long, low-amplitude RF pulses, and served as an indicator of the reliability of RD quantification.

Simulations and parameter fitting
Simulations and non-linear least squares (NLLS) fitting were based on calculations of the evolution of the magnetization during the entire pulse sequence using numerical solutions of Eqs.(7a)-(7c) implemented in Matlab (R2020b; MathWorks, Natick, MA, USA).The algorithm included an ordinary differential equation solver (ode45) based on explicit Runge-Kutta formulas of orders 4 and 5.The (digitized) RF pulse shapes were directly obtained as played out during runtime from the scanner's sequence programming software (IDEA; Siemens Healthineers). 38,57,58The NLLS fitting used the function nlinfit (or also lsqnonlin).Confidence intervals were calculated using the nlparci algorithms of Matlab's Statistics and Machine Learning Toolbox (R2020).Errors are given as the 95% confidence intervals for the parameter estimates (approximately two SDs for each parameter as obtained from the covariance matrix).
A frequently used T 1 measurement is the variable flip angle (VFA) method. 59,60Due to RF-pulse application for each k-space line, it may be influenced by RD during transmission.2][63] The longitudinal magnetization in a 3D spoiled GRE acquisition was calculated by solving Eqs.(7a)-(7c) for a series of 256 repetitions (TR = 30 ms) with a rectangular RF pulse.
These simulations were performed for 11 different excitation angles (between 4 • and 60 • ) for the damped ( RD,Tx = 17 ms) and undamped case ( RD,Tx → ∞) considering two values of  p (0.5 and 5 ms).Otherwise ideal conditions (ideal pulse shapes, perfect B 0 and B + 1 homogeneity, perfect spoiling, no noise) were assumed to simplify the identification of RD effects.A typical assumption in VFA T 1 measurements is the establishment of a (periodic) steady state after a sufficiently large number of pulses (here, N = 256) according to the Ernst equation: 64 S 0 is the signal intensity obtained with TR ≫ T 1 and θ = 90 • .Consistently, estimates of T 1 and deviation from ground-truth input values were obtained from NNLS fits of the simulated signal intensity after the final RF pulse to Eq. ( 9).

Pulse-width arrays
Pulse-width arrays acquired with all coil setups and two spherical samples are shown in Figure 2. The results obtained with setup 1 (Figure 2A,D) showed undistorted sinusoidal amplitude oscillations as a function of  p ,

F I G U R E 2
Pulse-width arrays measured with coil setup 1 (undamped; A, D), 2 (moderately damped; B, E), and 3 (maximally damped; C, F).With all setups, the width of a rectangular pulse was stepwise increased from 0.02 to 4.42 ms at constant amplitude.(A-F) Results obtained with the spherical 19 mm sample and the 24 mm sample, respectively.Broken lines indicate the ratio of the second-highest and highest (positive) peak amplitude.Expected damped sine oscillations of the peak amplitudes for an ideal configuration are indicated by solid yellow lines.Significant deviations due to radiation damping (RD) are evident for coil setup 2 and 3.The amplitude was adjusted to obtain a pulse length of 1 ms for a 180 • pulse under ideal conditions.

T A B L E 1
Summary of the reference voltages obtained by adjusting a 360 • pulse with a pulse-width array for all coil setups and samples.verifying the absence of relevant RD for optimized power mismatch.These oscillations were slightly damped due to B + 1 inhomogeneity, as evident from the amplitudes obtained with the 450 • (i.e., 360 • + 90 • ) and 90 • pulse.Corresponding ratios >0.96 for the 19 mm and >0.9 for the 24 mm sample indicate excellent homogeneity. 50n contrast to setup 1, substantial RD is evident for setup 2 (Figure 2B,E) and setup 3 (Figure 2C,F) with extended cable lengths.The oscillations are skewed, shifting the maximum positive and negative signals to flip angles >90 • and < 270 • , respectively.These sawtooth-like profiles result from line broadening for θ < 90 • and distorted lineshapes between 90 • and 270 • . 50The distortions are more pronounced with the 24 mm sample with increased  and magnetic moment.The 360 • pulse is largely unaffected by RD 7 and, therefore, provides a robust means to adjust the reference voltage.Corresponding results are summarized in Table 1.

RD during free precession
Results from RADDSY experiments are shown in Figure 3.
Damping effects are easily identified as deviations from an exponential recovery (blue triangles) and, correspondingly, as a difference between the damped and undamped sequence versions (black circles).Subtle RD effects are even evident in the data obtained with setup 1.
For quantitative analysis, magnitudes of the numerically integrated spectra were rescaled to account for small deviations from a 90 • flip angle caused by relaxation or RD and residual B + 1 inhomogeneities.This scaling factor was determined by extrapolating the undamped recovery curve to t = 0.This approach reflects the z-magnetization amplitude during free evolution, whose time course was fitted to Eqs. (7a)-(7c) to estimate T 1 , T * 2 , and  RD,Rx .Further important is the interplay between RD and transverse relaxation because fast dephasing may prevent an efficient rotation of the magnetization by the damping field.This is reflected in large off-diagonal elements in the covariance matrix and limits estimations of T * 2 and  RD,Rx (Pearson correlation coefficient r > 0.96).Therefore, additional ME-GRE acquisitions, which are sensitive to T 2 * but not to RD (due to the gradient pulses), were simultaneously fitted (combined analysis).Figure 3 demonstrates a remarkable agreement of the fits (solid lines) and experimental data, which also benefited from the high SNR.
Fitted parameters and T * 2 ∕ RD,Rx ratios are included in Table 1.A comparison of the samples shows that the  RD,Rx -values scale with the magnetic moments (i.e., their ratio corresponds to that of the sample volumes, 243/193 ≈ 2.02).This is expected if the increased  obtained with the larger sphere is approximately compensated by a lower Q (Eq.8).The small T 1 difference for both samples (3%-4%) is probably due to temperature differences as the measurements were performed on different days.To avoid ambiguity, further analysis is restricted to the 19 mm sample.
To go beyond the assumption of a perfectly tuned coil, flip-angle arrays were employed to analyze the phase relation (i.e., ψ Rx ) between the damping field and the transverse magnetization (Figure 4), combined with ME-GRE data for estimating T * 2 .Characteristic features of the degree Radiation damping difference spectroscopy (RADDSY) experiments with coil setup 1 (undamped; A, D), 2 (moderately damped; B, E), and 3 (maximally damped; C, F). (A-F) Results obtained with the spherical 19 mm sample and the 24 mm sample, respectively.The peak amplitude after full recovery (evolution time of 6 s) was set to 1 for normalization of the signals.Note that only results for evolution times below 950 ms are shown for better visualization of the effects.Blue and orange triangles show data for the two sequence variants corresponding to the damped case and undamped reference, respectively.Black circles show the difference of the two measurements.An accelerated recovery of longitudinal magnetization due to radiation damping (RD) is clearly evident for coil setup 2 and 3, whereas the good agreement between both sequence versions obtained with coil setup 1 verifies conditions that are mostly free from RD (consistent with the pulse-width array results in Figure 2A,D).

(A) (B) (C) F I G U R E 4
Overview of the results obtained with flip-angle arrays and the 19 mm sample with coil setup 1 (undamped; A), 2 (moderately damped; B), and 3 (maximally damped; C).The top row shows color-coded contour plots of stacked spectra (phase-corrected real part, individually normalized to an amplitude of 1) as a function of the flip angle illustrating the changes in the peak with and position due to radiation damping (RD).The bottom row shows experimental data (colored circles) and corresponding fits (solid lines) of the FWHM (orange) and the peak position (blue). of RD and detuning (i.e.,  0 ≠  LC ) were a dependence of the linewidth and peak position on .For the undamped setup 1, an invariant FWHM and insignificant frequency shifts verify negligible RD, consistent with the above results.By contrast, prominent RD for setups 2 and 3 produce distinct FWHM variations and frequency shifts, the latter being more pronounced for the maximally damped setup 3.
Overview of the results obtained with saturation recovery (evolution times 10 ms ≤  ≤ 6 ms) and the 19 mm sample with coil setup 1 (undamped; A), 2 (moderately damped; B), and 3 (maximally damped; C).The results are plotted in dependence of M z (τ)/M 0 instead of τ.The top row shows color-coded contour plots of stacked spectra (phase-corrected real part, individually normalized to an amplitude of 1) as a function of the flip angle illustrating the changes in the peak with and position due to radiation damping (RD).Note that the evolution time was first increased from 10 ms to 6 s in sequential measurements and then decreased from 6 s to 10 ms.Thus, the entire series appears twice, with M z /M 0 (on the abscissa) first increasing from 0.02 to 1 and then decreasing back to 0.02.This presentation allows a better assessment of the reproducibility of the results.The bottom row shows experimental data (colored circles) and corresponding fits of the FWHM (orange) and the peak position (blue).Shaded areas illustrate 95% confidence intervals of the fits.
For quantification, FIDs were simulated, and the FWHM and peak position were fitted considering four adjustable parameters:  Rx , ψ Rx , T * 2 and a frequency term f accounting for B 0 drift.T 1 = 770 was a fixed parameter because it varied by ≤0.6% in different measurements (Table 1).The results suggest that the phase term may be neglected for setup 3 but not for setup 2 ( ψ Rx = -25.9• ⇒ cos ψ Rx = 0.900 ) .Therefore, the larger phase offset associated with a detuned setup 2 (cos[5.1 • ]/cos[−25.9• ] ≈ 1.107) explains the increased of  RD compared to setup 3 ( RD [3]/ RD [2]  = 26.5/23.8≈ 1.113).The alternative saturation-recovery approach leads to consistent findings (Figure 5 and Table 2; cos[4.7 • ]/cos[−28.3• ] ≈ 1.132).Despite this agreement, different limitations affect the accuracy of both experiments: Saturation-recovery acquisitions may suffer from system instabilities (e.g., B 0 drifts, eddy currents), isochromat simulations (data not shown) indicate that the single-vector model may fail for θ > 90 • , in line with previous results. 9With improved field homogeneity, such deviations become smaller and are negligible.

RD during transmission
Results of the RADDEX measurements are shown in Figure 6A,B.Improved stability of the fits was achieved by fitting all data from the four excitation schemes simultaneously.Damping during signal acquisition was neglected as only the magnitude of the summed spectra was considered in the fits yielding the following parameters: ζ Tx , ψ Tx , T 2 ,  , and c B1 .As in the free-precession case, the f was used to correct for B 0 drifts, whereas the scaling factor c B1 corrects for B + 1 deviations.Of note, relaxation during transmission is better approximated by T 2 instead of T * 2 used in the free-precession experiments. 65,66Results are also in Table 2.
The compensation of RD-related flip-angle deviations during runtime of a simple pulse-and-acquire sequence is demonstrated in Figure 6C,D showing the normalized signal magnitude as a function of the amplitude of a rectangular pulse of constant duration.The different version of the Tx/Rx switch (R 1 = 49 Ω) was used in these experiments, leading to a higher reference voltage (≈5×) to avoid discretization effects at long  p .Results of corresponding RADDEX measurements are provided as Figure S3, Table S1.Similar to Figure 2, the performance of the 360 • pulse was not affected.Without correction, oscillations were distorted, especially around a 180 • pulse, where RD effects are enhanced.Efficient mitigation is evident if  RD,Tx and ψ Tx are properly considered in the amplitude-and pulse-modulated pulse.Overestimation of RD leads to opposite bias, whereas a subtle underestimation appeared benign.This suggests a Summary of the results obtained with the flip angle-array and saturation-recovery measurements during free precession as well as with RADDEX during transmission.certain tolerance in practical applications of the correction.

RD in relaxation measurements
Simulations of VFA experiments and fits to Eq. ( 9) are presented in Figure 7. Remarkably, T 1 was slightly overestimated even in the absence of RD, despite the assumption of noise-free acquisitions.Absolute deviations (2 and 22 ms for the 0.5 ms and 5-ms pulse, respectively) did not vary significantly with intrinsic T 1 but increased with pulse duration.This is explained by relaxation during the pulse, which is considered in Eqs.(7a)-(7c) (i.e., in the simulations), but not in the fit function, Eq. ( 9).Therefore, θ eff is reduced, and the T 1 overestimation increases with  p due to increasing relaxation effects.Simulated VFA experiments with shorter pulses ( p = 5 μs; data not shown) confirmed this interpretation, yielding deviations <0.1 ms from ground truth.In contrast to T 1 overestimation due to unaccounted relaxation, RD caused T 1 underestimation by 0.7% and 7% for the 0.5 ms and the 5 ms pulse, respectively.Notably, both relaxation and RD lead to reduced θ eff .However, relaxation causes rather uniform re-scaling, whereas the impact from RD is flip-angle dependent.Figure 8 presents T 2 measurements (19 mm sample) with different echo spacings.Apparent deviations from mono-exponential echo decays were not evident (Figure 8B,C).For the undamped case (setup 1), estimated T 2 times decreased monotonically from 60.5 ms (ΔTE = 9 ms) to 59.7 ms (ΔTE = 20 ms).This may be explained by increasing diffusion-related echo attenuation with longer ΔTE.A varying influence of pulse imperfections may also contribute to this observation. 51Similar results were obtained for the moderately damped case (setup 2), whereas deviations at longer ΔTE for the maximally damped setup 3 suggested RD.Results for the 24 mm sample are shown as Figure S4, indicating larger T 2 deviations (≈2%) for setups 2 and 3, consistent with enhanced RD due to increased .Overall, while the high SNR supported the detection of RD effects, related T 2 deviations were smaller than those typically expected in repeated measurements due to subtle variations of the experimental conditions (e.g., temperature changes).

DISCUSSION
This work demonstrates RD at a clinical field strength of 3T.Assuming that such effects are more common in experiments with dedicated coils for small samples, a Helmholtz coil was built that achieves a high η and Q, which are crucial requirements under such conditions.It also supports the generation of well-defined, short RF pulses ( p ≥ 20 μs), allowing to investigate RD or other nuisance effects over a wide range of pulse durations.Passive feedback integrated in the coil setup largely suppressed RD following the established concept of preamplifier decoupling.Alternatively, maximization or, more generally, modulation of RD was achieved by integrating coaxial cables of suitable length between the coil and the Tx/Rx switch.In combination, this allowed a comprehensive characterization of the spin-coil system and RD effects during excitation and free precession.
Results obtained with pulse-width arrays demonstrate substantial lineshape distortions if RD is not suppressed in the setup.This may compromise flip-angle calibration procedures, which are typically based on an automated adjustment of a reference voltage for a 180 • pulse on clinical scanners.Resulting flip-angle errors are non-linear, and maximal errors are expected for a 180 • pulse, whereas a 360 • pulse is almost unaffected.Consequently, they are not well corrected by rescaling employing B + 1 mapping.Residual RD may, therefore, impact quantitative MR, such as VFA-based T 1 measurements.Our simulations indicate only small errors for  p ≤ 1 ms.Significantly longer pulses of potentially high amplitude are often used in MT imaging, 37,67,68 or CEST. 58,69Integration of RD suppression approaches may, hence, be beneficial in such experiments.
T 2 measurements appeared to be relatively immune to RD-related perturbations.This robustness of spin-echo sequences can be explained by the absence of RD when  nuclear spins are out of phase in an inhomogeneous field as already discussed by Bloembergen and Pound. 4nly experiments with ΔTE ≥ 20 ms indicated small T 2 errors, being more pronounced for larger .This ΔTE is of the order of  RD,Rx (23.8 and 12.0 ms for setup 3 and the 19 and 24 mm sample, respectively; Table 1) supporting previous assumptions 19 that RD effects in CMPG sequences may be ignored if ΔTE∕ RD,Rx ≪ 1.Given current gradient limitations on clinical scanners, TE is typically long in diffusion-weighted experiments.While diffusion-sensitizing gradients would suppress RD, they are not applied in acquisitions with b = 0, which are, hence, more susceptible if RD is not mitigated in the coil setup.
Despite agreement with earlier observations, it remains unclear to what extent T 2 discrepancies can be attributed solely to RD.For a comprehensive analysis, an extension of the single-vector model to a distribution of isochromats would be required. 41Alternatively, we demonstrate experimental quantification of RD, allowing an assessment of whether perturbations must be accounted for in a specific measurement or how their impacts can be mitigated.For a general evaluation, it is recommended to use ratios T 2 ∕ RD or T * 2 ∕ RD comparing the damping-field duration to the lifetime of coherence of the spin system.The latter ratio also appears to be more robust against erroneous T * 2 values if RADDSY experiments are analyzed without further ME-GRE data, as correlation between T * 2 and  RD limits accurate measurements of both parameters.Our results suggest that the damping rate and phase relation during free precession and transmission may deviate (Table 2), which has not been thoroughly studied before.Previous experiments with a small dual-loop coil indicated a higher coil current in the Tx mode due to a reduced fraction of reflected power. 70A correspondingly increased mismatch of i in and i re leads to less efficient cancellation and, hence, reduced RD suppression, which is consistent with our results of  Tx M 0 >  Rx M 0 (58.7 vs. 45.9 s −1 , respectively, for coil setup 3; Table 2).Finally, the assumption of a perfectly tuned coil ( = 0) may not hold, which can lead to errors in the estimated damping time constant if unconsidered.

CONCLUSIONS
Radiation damping may impact MR investigations of small tissue samples with dedicated RF coils and should be considered, especially in cases when precise quantification of MR parameters, such as relaxation rates, is required.Difference experiments comparing results with maximum and minimum RD impact provide a comprehensive quantitative characterization.Efficient mitigation can be achieved hardware-based with a preamplifier decoupling approach or sequence-based with suitable RF pulse or gradient schemes.Simple VFA or multi-echo experiments are relatively robust against RD as long as pulse durations or echo spacings are short compared to the damping time constant.
B) Radiation damping difference excitation (RADDEX) experiments with the 19 mm sample and coil setups 2 (moderately damped; A) and 3 (maximally damped; B).The signal magnitude (normalized to the signal after preparation with a 180 • composite pulse of minimal duration) is plotted as a function of the composite pulse duration.Orange, red, light blue, and dark blue symbols refer to preparations by an "undamped" 90 • , a "damped" 90 • , an "undamped" 180 • , and a "damped" 180 • composite pulse, respectively.Corresponding solid lines indicate simultaneous NNLS fits to the combined data.Considerable radiation damping (RD) effects, which increase with pulse duration, lead to deviations from a normalized signal of 0 (after 90 • preparation) or 1 (after 180 • preparation) in all experiments, but are more pronounced for the "damped" preparations.An undesired effect of the discretization of the transmitter voltage is visible as a step function of data points acquired with longer pulse durations due to the relatively small reference voltage.(C, D) Demonstration of the online correction of a 7.5 ms rectangular RF pulse during runtime.Normalized signal intensities obtained with "pulse amplitude arrays" and coil setups 2 (C) and 3 (D) are shown.As in Figure2, the 360 • pulse is minimally affected by RD.The uncorrected pulse (red squares) yields a shift of the minimum toward higher nominal flip angles.Results obtained with a corrected pulse shape based on a RADDEX acquisition (green triangles) follow an undistorted sinusoidal oscillation (indicated by the yellow line) and a minimum at half the amplitude of the 360 • pulse.An overcorrection (use of a too short τ RD,Tx due to overestimation of RD; dark blue triangles in C) produces a minimum shift toward the opposite direction, whereas a moderate underestimation (dark blue circles in D) still yields an acceptable correction.

7
Simulations of variable flip angle (VFA) acquisitions with short (τ p = 0.5 ms; A, C) and long (τ p = 5 ms; B, D) rectangular pulses and typical relaxation times for gray matter (T 1 = 1500 ms, T 2 = 100 ms; A,B) and white matter (T 1 = 900 ms, T 2 = 70 ms; C, D).Blue crosses and red circles indicate calculated flip angle-dependent steady-state signals without (τ RD , Tx → ∞) and with relaxation damping (τ RD , Tx = 17 ms), respectively.Blue solid and red dotted lines show corresponding fits to Eq. (9).Subtle overestimation of T1 results even in the absence of RD due to relaxation during pulse application, which is enhanced for longer τ p (A: 1502 ms; B: 1522 ms; C: 902 ms; D: 922 ms).By contrast, radiation damping (RD) leads to an increasing underestimation of T 1 with increasing τ p (A: 1492 ms; B: 1424 ms; C: 895 ms; D: 855 ms).

F
I G U R E 8 (A) Experimental estimations of T 2 in the 19 mm sample as a function of the echo-spacing in a Carr-Purcell-Meiboom-Gill (CPMG) sequence.Green circles, orange squares, and blue triangles show the results obtained with coil setups 1, 2, and 3, respectively.Error bars indicate ±1 SD.The green solid line shows an exponential fit to the undamped data (coil setup 1).A subtle deviation from the undamped result is visible at the longest echo spacing for the maximally damped case.Exemplary fits to the normalized echo amplitudes recorded with coil setup 1 (B; ΔTE = 9 ms) and coil setup 2 (C; ΔTE = 11 ms) do not indicate relevant deviations from a mono-exponential decay or the occurrence of stimulated echoes.The insets of (B, C) show the data with a scaled y-axis.
).Most mechanical parts were designed with CAD software and 3D-printed (Objet Eden260VS; Stratasys, Eden Prairie, MN, USA) using Objet MED610 Biocompatible Clear material (Stratasys).The final coil consisted of two loops of 2 mm diameter silver-plated copper wire connected in series.Radius and spacing of the loops were 16 mm.Bench-top measurements yielded Q ≈ 470 of the empty coil.
The 19 mm sample was used for all acquisitions.Abbreviation: RADDEX, radiation damping difference excitation. Note: