Confounding factors in breast magnetic resonance fingerprinting: B1+ , slice profile, and diffusion effects

Magnetic resonance fingerprinting (MRF) offers rapid quantitative imaging but may be subject to confounding effects (CE) if these are not included in the model‐based reconstruction. This study characterizes the influence of in‐plane B1+ , slice profile and diffusion effects on T1 and T2 estimation in the female breast at 1.5T.


| INTRODUCTION
Quantitative magnetic resonance imaging (qMRI) aims for the quantitative evaluation of the tissue relaxation times T 1 and T 2 , 1 and its clinical value as has been demonstrated, for example, in cardiac 2 and muscoskeletal 3 MRI. Moreover, qMRI promises vendor-independent data, which is of advantage during multi-center clinical trials and for deep learning-based analysis methods. In the female breast, qMRI has previously been explored for response assessment of neoadjuvant chemotherapy [4][5][6] and for distinction between invasive ductal carcinoma from healthy tissue 7 or between different types of lesions. 8,9 Adversely, gold standard qMRI sequences measuring either T 1 or T 2 are of slow acquisition speed, which complicates their routine clinical use and therefore limits qMRI investigations of breast malignancies. Recently, magnetic resonance fingerprinting (MRF) has been presented as a rapid quantitative sequence, 10,11 and qMRI in the breast within a clinical feasible scan time using MRF has been demonstrated. 7 MRF extracts multiple quantitative parameters from a time series of undersampled images with varying acquisition parameters. By varying the flip angle and/or the sequence timings from image to image, MRF encodes a tissue's spin history over the image series without leaving the transient state, thereby yielding distinct signal evolutions for different tissues. Typically, the quantitative parameters of interest measured by MRF include the tissue relaxation times T 1 and T 2 , although MRF, by design, is not limited to the relaxation times only. Parameter maps are reconstructed by matching the measured temporal signal evolution in every voxel to a pre-calculated dictionary of possible signal evolutions, from which the best-matching entry yields the quantitative parameters for that voxel. Hence, the accuracy of the parameter maps crucially depends on the extent to which the signal model replicates the physical reality behind the MRF measurement. In other words, MRF can only be accurate if all influences on the magnetization dynamics are taken into account. For unbalanced gradient-echo MRF sequences, extended phase graph (EPG) simulations are used for dictionary calculation. 12,13 The signal model commonly includes instantaneous RF excitations defined by the nominal MRF flip angle train, T 1 and T 2 relaxation between excitations, as well as re-and dephasing of magnetization induced by the unbalanced spoiling gradient moments.
However, other physical effects may influence the signal evolution and therefore act as confounding effects (CE). On the one hand, deviations from the nominal sequence parameters may lead to bias on the MRF results. B + 1 effects (ie, both in-plane B + 1 inhomogeneity and a non-rectangular slice excitation profile [SP]), were observed to affect the outcome of MRF measurements [14][15][16][17] and are often included into the matching reconstruction, either as prior information 15,16 or as additional free dictionary parameters. 14,18,19 However, different MRF sequences and anatomies may be more or less susceptible to a CE. A dedicated analysis of CEs on the accuracy of MRF has been conducted for cardiac MRF sequences, addressing bias because of a non-rectangular SP, in-plane B + 1 inhomogeneity as well as preparation pulse efficiency. 20 On the other hand, tissue properties not included in the EPG signal model may lead to deviations of measured from modeled signal evolutions. Examples are diffusion or magnetization transfer (MT). Depending on the spoiling gradient strength, diffusion was found to explain bias on MRF T 2 estimates, most pronounced when spoiling gradient moment, T 2 value and ADC were high. 21 In the presence of a semi-solid pool, MT introduced bias on the T 1 and T 2 relaxation times that increased when additional off-resonant pulses were included in the MRF sequence design. 22 Along with physical effects, the presence of noise may have an impact on both accuracy and precision of MRF results. Whereas electronic noise of constant amplitude is present in all MRI data, aliasing artefacts additionally overlay undersampled spiral MRF data. Up to a certain amplitude, the inner product matching used during reconstruction can filter out noise and even the aliasing artefacts; however, they may also act as an additional confounder. 23,24 Because strong dielectric effects are present in the female breast, 25 especially in-plane B + 1 inhomogeneity is expected to induce bias on the MRF results. In fibroglandular tissue (FGT), moreover, diffusion may influence the measured MRF signal evolutions depending on the employed spoiling gradient moment, because an ADC of (1.95 ± 0.24) × 10 −3 mm 2 /s is reported for healthy FGT, 26 being rather high in comparison to other tissues. 27,28 Concerning MT in the breast, the (qualitative) MT ratio has been assessed in different breast tissues including tumors. 29,30 There is also a first report on (quantitative) pool size ratios in the healthy breast. 31 We have described breast MRF measurements on 6 healthy volunteers before, 32 in which MRF data were acquired at 1.5T using a 2D MRF-Dixon sequence. Adding to these results, the current study aims to theoretically investigate the potential bias because of in-plane B + 1 inhomogeneity, SP, and diffusion effects by means of simulation and to evaluate their | 1867 importance with respect to the experimental MRF estimates in the breast. Additionally, the impact of both electronic noise and aliasing artefacts will be included in the analysis.

| MRF-Dixon measurements
MRF measurements in the female breast were performed on a 1.5T Achieva System (Philips, Best, The Netherlands) with a 4-channel breast coil (InVivo, Gainesville, FL), using a 2D single-slice MRF-Dixon sequence with spiral readout. 32 Data were acquired in 6 healthy female volunteers, age range 24-31 years, in accordance with the institutional ethics board. Spiral imaging leads to off-resonance blurring, being more severe for fatty tissue (FT) than for FGT because of the fat chemical shift. Therefore, the MRF-Dixon sequence acquires 3 MRF trains separated by a delay time (t d ) and differing solely in their echo time (TE). Here, the 1st/2nd/3rd flip angle (FA) train exhibited a constant TE = (4.6/6.9/9.2) ms and a TR of 20 ms. Likewise, water-fat separation and subsequent deblurring of spiral off-resonance artefacts were conducted before the MRF matching reconstruction. As depicted in Figure 1A, the MRF sequence used a FA train of 500 TR intervals preceded by a 180° inversion pulse that has shown good encoding capability. 33 The present study re-evaluates the 6 undersampled MRF measurements (ie, 1 data set per volunteer), which were realized with one rotating spiral interleave of 7 ms acquisition time and 20-fold undersampling. N int = 20 spiral interleaves would be needed to fully sample k-space.

| Theoretic bias analysis including noise and aliasing artefacts
To analyze the theoretic amount of bias that a CE adds to the MRF dictionary matching reconstruction, a standard dictionary (StdDict) was calculated using EPG simulations. Signal evolutions within the StdDict take into account the tissue relaxation times T 1 and T 2 , dephasing because of spoiler gradients (G), the flip angle series (FA(t)) and the timings TR and TE of the MRF sequence; however, all of the investigated CEs are excluded. Signals in the StdDict covered a T 1 range of [5,2000] ms in steps of 5 ms and a T 2 range of [2,200] in steps of 2 ms. Unphysical signals with T 1 < T 2 were not included. Subsequently, test signals were calculated over a grid of values T test 1 and T test 2 with spacing ΔT test 1 = 40 ms and ΔT test 2 = 20 ms, respectively, including either B + 1 , SP, or diffusion effects, explained below. Each test signal was matched to the StdDict based on maximum inner product matching. 11 By plotting the deviations of the test signal's nominal T 1 (T 2 ) value from the matched T 1 (T 2 ) value, ie, (T matched 1 −T test 1 ) and (T matched 2 −T test 2 ), respectively, we assessed the expected theoretic matching error that results from omitting a CE. To test different noise conditions, the above analysis was conducted three times. First, the test signals were computed without adding noise. Electronic noise or spiral aliasing artefacts were then added to the test signals before matching. A noisy signal m was simulated by adding a noise vector n to a simulated "true" signal swhere (m, n, s) ∈ ℂ 1×N are defined in image space, with N the number of TR intervals of the MRF sequence: In turn, n may be extracted from a measured signal m by subtracting the best-matching signal s. Following, 34 we define SNR for a noisy MRF signal as with the complex norm | | 2 . Before calculating r, m and s are normalized to 1. Given a noise vector n � ∈ ℂ 1×N of arbitrary magnitude, we obtain a noisy test signal m with adjustable r as 34 To investigate the effects of electronic noise, we set n ′ to be complex white Gaussian noise (WGN). Test signals with electronic noise levels between r = 10 and r = 50 were simulated. For each test signal, adding noise and matching was repeated 200 times, so that a mean matched T 1 (T 2 ) value and SD were obtained to predict accuracy and precision.
To investigate the effects of spiral aliasing artefacts that resemble noise-like fluctuations around the true signal evolutions, 11 a different "noise" model was chosen for n ′ . In the literature, such aliasing artefacts have been mimicked by adding complex WGN scaling with the signal strength to simulated MRF signals. 23,35 We further adapted this artefact model to our undersampled MRF data: because the artefact patterns exhibit a periodicity of N int = 20, caused by the rotation of the spiral interleaf with (360/N int )° between successive TR intervals, we defined periodic WGN vectors before scaling them with the signal strength. More specifically, we composed a periodic noise vector n p ∈ ℂ 1×N in image space of 25 replicates of a shorter complex WGN pattern of length 20. Then, we set where ⊙ denotes the Hadamard product. In the context of spiral aliasing artefacts, we will refer to r as the "signal-to-artefact ratio." Test signals with signal-to-artefact ratios between r = 1 and r = 5 were simulated. Again, adding aliasing artefacts and matching was repeated 200 times for each test signal to predict accuracy and precision.

| Experimental bias analysis
The experimental bias analysis on in vivo breast MRF data was conducted by means of difference maps as summarized in Figure 1D. In addition to the StdDict described above, we calculated one dictionary per CE, ie, an in-plane B + 1 dictionary (B + 1 Dict), SP dictionary (SPDict), and diffusion dictionary (DiffDict), using the same range and resolution as StdDict, but including the respective CE into the signal model as described below. In addition, a dictionary combining SP and in-plane B + 1 effects (SP&B + 1 Dict) was calculated. To evaluate the experimentally observed bias, the breast MRF data sets were matched to the StdDict as well as to the B +
(4) n � = s ⊙ n p , Dict, SPDict, SP&B + 1 Dict, and DiffDict, respectively. T 1 and T 2 difference maps comparing the 2 MRF reconstructions were computed, based on which mean difference and SD in regions of interest (ROIs) of FGT and FT were evaluated for all 6 volunteers. To define the ROIs, maps of the mean water (W) and fat signal (F) over the MRF train were retrieved and the mean fat signal fraction FSF = F∕ W + F was computed. Voxels with a FSF of <15% were counted as FGT and voxels with a FSF > 85% were counted as FT. For the inplane B + 1 analysis, mean T 1 and T 2 values and SDs were separately calculated for the left and right breast. Left-right differences were compared between StdDict and B + 1 Dict reconstructions.
To compare the experimentally observed bias with the predictions from simulation, we calculated a second set of difference maps. Potential differences between the experimentally observed bias and the predicted bias may point out an influence of noise or the simultaneous presence of another CE. To predict the in-plane B + 1 difference map, for example, the respective (T 1 , T 2 , B + 1 ) combination resulting from the reconstruction with the B + 1 Dict was used to compute a noise-free test signal for each voxel, which was subsequently matched back to the StdDict to obtain the predicted bias for that voxel (and analogously for the other CEs). To further investigate whether any observed discrepancies are attributable to spiral aliasing artefacts, a third set of difference maps was calculated by adding, once for each voxel, simulated aliasing artefacts to the test signal before matching it back to the StdDict. Test signals including aliasing artefacts were simulated using periodic complex scaled WGN as described by Equations (3) and (4). The signal-to-artefact ratio in every voxel was estimated beforehand from the data using Equation (2).

| EPG-simulations including confounding effects
In the following, we outline briefly how each of the CEs was included into the EPG signal calculations.

inhomogeneity
In-plane B + 1 inhomogeneity causes deviations from the nominal FA train ie, it acts as a multiplicative factor f B + 1 on the nominal flip angle train. The correction factor in a voxel (x, y) may be measured using a B + 1 mapping technique. Along with the MRF scans, we acquired B + 1 maps of the MRF slice using 3D actual flip angle imaging (AFI) with FOV, slice thickness, and resolution matching the MRF slice (scan duration = 3:40 minutes). 36 The B + 1 Dict included correction factors f B + 1 between 0.7 and 1.3 with a step size of 0.025. In comparison to the StdDict, its size and computation time therefore increased by a factor of 25. Matching the measured MRF signal of a voxel to the B + 1 Dict was restricted to the subset of simulated signal evolutions with f B + 1 closest to the measured correction factor in that voxel.

| Slice profile effects
A non-rectangular SP results in a flip angle distribution along the slice direction z for each RF excitation. To minimize this effect during the in vivo breast MRF measurements, an RF pulse shape of high time-bandwidth product (TBW = 10.2) was used ( Figure 1B). For each FA, the SP was calculated by Shinnar-Le Roux transformation 37 of the RF pulse shape. Figure 1C shows the resulting excitation profile for a FA of 60°, which is indeed close to, but not perfectly block-shaped. To account for the SP in the MRF dictionary, the z-axis was subdivided into n b = 77 discrete bins. The bin size was chosen larger in regions of constant or slowly varying SP amplitude to reduce computation time. One MRF signal was first calculated for each bin. Subsequently, a weighted sum of signals over the slice was performed, with weighting factors taking into account the individual bin sizes. The SP&B + 1 Dict additionally included the correction factor f B + 1 on the FA train in each bin. Although the SPDict (SP&B + 1 Dict) is of equal size than the StdDict (B + 1 ), the respective computation time increased by n b , (or by (n b − 1)/2 + 1 for a symmetric SP), which required parallel computing.

| Diffusion effects
The diffusion sensitivity of any MRI sequence increases when adding de-and rephasing gradients, in between which microscopic motion of nuclear spins leads to an additional signal decay. 38 For isotropic diffusion in 1D, the associated signal attenuation S/S 0 is commonly expressed as 38,39 with the isotropic diffusion constant D b takes the effect of all diffusion gradients over 1 TR into account Here, k(t) = ∫ t 0 G( )d is the gradient moment and γ is the gyromagnetic ratio in rad/(s × T). Gradient-spoiled MRF sequences commonly exhibit 1 unbalanced spoiler gradient per TR interval. Much like the diffusion-sensitizing gradients in conventional diffusion-weighted sequences, the consecutive spoiler gradients within the MRF sequence de-and rephase the spin system and thereby add diffusion weighting to MRF signals. Numeric equivalents of Equations (6) and (7) can be included in the EPG framework as a diffusion operator Here, i: order of de-/rephasing state (with i > 0 for dephasing and i < 0 for rephasing states), Δk = GT: amount of de-and rephrasing introduced by the application of one spoiler gradient, D: gradient strength, T: gradient duration, gr on : 1 if gradient is being played out, 0 otherwise. 13,40 The spoiler gradients of our breast MRF sequence exhibited a gradient moment of Δk = 4 × 2π over the 5 mm slice per TR. We assumed D = 1.95 × 10 −3 mm 2 /s for FGT. 26 We neglected diffusion in FT, because diffusion constants are reported to be ~2 orders of magnitude smaller than in aqueous tissues. 41 The DiffDict is of equal size and similar computation time as the StdDict. resulting from matching the test signals to the StdDict. Hence, these plots predict the impact of a CE on MRF accuracy. The analysis was conducted (1) without noise, (2) with electronic noise of SNR r = 50 that we selected as a representative value for a fully sampled MRF scan, and (3) with aliasing artefacts of r = 1.5, resembling the experimentally observed signal-to-artefact ratios. The control plots comprise test signals without any CE, but with added noise or aliasing artefacts, and therefore show bias introduced by noise/aliasing artefacts alone. Supporting Information Figure S1 presents the SD of T 1 and T 2 over the 200 matching steps per test signal as a measure of the predicted precision when including electronic noise or spiral aliasing artefacts along with a CE. Supporting Information Figure S2 presents the predicted accuracy and precision for more signal-to-noise/signal-toartefact ratios.

| Theoretic bias analysis
For 2 example correction factors of f B + 1 = 0.9 and f B + 1 = 1.1, corresponding to actual FAs smaller and larger than the nominal ones, it is visible that omitting in-plane B + 1 effects in the MRF dictionary leads to T 1 under-/T 2 overestimation if f B + 1 < 1 and to T 1 over-/T 2 underestimation if f B + 1 >1. Omitting the SP during dictionary calculation results in T 1 under-/T 2 overestimation. For the employed spoiling gradient moment of 4 × 2π over the slice, omitting diffusion in the MRF dictionary causes T 1 underestimation toward longer T 1 and T 2 , as well as T 2 underestimation toward longer T 2 . Yet, for the here-investigated spoiling moment, diffusion constant D and dictionary resolution, an effect of diffusion becomes visible from T 2 > 100 ms onward only. Simulation results for more values of f B + 1 and D are presented in Supporting Information Figure S3.
Adding WGN with an SNR of r = 50 to the test signals has a smoothing effect on the T 1 and T 2 bias prediction, but overall leaves the mean predicted T 1 and T 2 values unchanged. As visible from the control plots, matching accuracy is not affected. Adding aliasing artefacts with r = 1.5 to the test signals has a slightly stronger smoothing effect on both T 1 and T 2 and, moreover, adds some granularity to the T 1 bias predictions. However, the corresponding control plots in Figure 2 reveal no systematic effect, but rather a fluctuation ~0. It should be noted that the already small bias caused by diffusion is hardly noticeable anymore after aliasing artefacts were added. As shown in Supporting Information Figures S1 and S2, precision is more strongly affected for T 1 than for T 2 . For spiral aliasing artefacts with r = 1.5, predicted T 1 SDs are larger than for electronic noise with r = 50.
Supporting Information Figure S4 presents simulated MRF signal evolutions of an example test signal for the different CEs. In line with the previous observations, the signal evolutions differ hardly when including diffusion, but notably when including in-plane B + 1 inhomogeneity or the SP.

| Experimental bias analysis
Next, we analyzed the experimental bias on the in vivo MRF results. Figure 4A,B show, for volunteer 5, the MRF T 1 and T 2 map after matching to the StdDict (ie, without accounting for any of the CEs during reconstruction). Figure 4C shows the measured B + 1 map. A strong left-right inhomogeneity is visible between both breasts, which visibly transmits into the T 2 map in Figure 4B, as marked by arrows indicating regions of f B + 1 smaller/larger than 1 and the corresponding areas of higher/lower T 2 values. Figure 4D,E summarize the effect of a combined correction for in-plane B + 1 inhomogeneity and the and T test 2 . The first and second row present the expected bias on T 1 because of in-plane B + 1 effects, for 2 example correction factors ( f B + 1 = 0.9 and f B + 1 = 1.1), corresponding to under-or overestimation of the nominal flip angle train, respectively. The third row shows the expected bias on T 1 because of slice profile effects. In the fourth row, the expected bias on T 1 because of diffusion effects is shown for a diffusion coefficient of D = 1.95 × 10 −3 mm 2 /s. The 3 columns correspond to different noise conditions. While the test signals are left noise-free in the first column, white Gaussian noise with an SNR of r = 50 was superimposed for the second column, and spiral aliasing artefacts with a signal-to-artefact ratio of r = 1.5 were superimposed for the third column before matching the test signals to the StdDict. Adding electronic noise/spiral aliasing artefacts was repeated 200 times. The plots therefore show the mean bias. For the control plots in the fifth row, no confounding effects have been included to test the effect of electronic noise or spiral aliasing artefacts alone on the accuracy of the matching procedure  Figure 2, the first and second row present the expected bias on T 2 because of in-plane B + 1 effects, for 2 example correction factors ( f B + 1 = 0.9 and f B + 1 = 1.1), corresponding to under-or overestimation of the nominal flip angle train, respectively. The third row shows the expected bias on T 2 because of slice profile effects. In the fourth row, the expected bias on T 2 because of diffusion effects is shown for a diffusion coefficient of D = 1.95 × 10 −3 mm 2 /s. Again, the 3 columns correspond to different noise conditions. Although the test signals are left noise-free in the first column, white Gaussian noise with an SNR of r = 50 was superimposed for the second column, and spiral aliasing artefacts with a signal-to-artefact ratio of r = 1.5 were superimposed for the third column before matching the test signals to the StdDict. Adding electronic noise/spiral aliasing artefacts was repeated 200 times. The plots therefore show the mean bias. In analogy to Figure 2, no confounding effects have been included for the control plots in the fifth row, to test the effect of electronic noise or spiral aliasing artefacts alone on the accuracy of the matching procedure | 1873

F I G U R E 3 Simulated bias on T 2 for different
non-rectangular SP, which permits to compare the corrected T 1 and T 2 map with their uncorrected counterparts in Figure  4A,B. Next to the improved in-plane B + 1 homogeneity, overall lower T 2 values are visible in the corrected T 2 map ( Figure  4D). T 1 maps before and after correction look similar. Figure  4F shows the ROI masks for segmentation of FGT and FT. Table 2 presents mean inner product (IP) values and SDs in FGT and FT regions that result from matching the data set to the different dictionaries. IPs closer to 1 indicate a better matching quality. Although changes in mean IP values between different dictionaries are subtle, they increase when including in-plane B + 1 or the SP into reconstruction. The SP&B + 1 Dict fits the measured signal evolutions best. Difference maps between MRF results with and without including a CE "x" in the MRF dictionary, ie, (T 1 x -T 1 std ) and (T 2 x -T 2 std ), are shown in the first column of Figures 5 and  6 for T 1 and T 2 , respectively. The second column predicts, based on the observed values T 1 x and T 2 x , the difference map expected from the theoretic simulations. For the third column, the prediction additionally includes spiral aliasing artefacts.

inhomogeneity
As predicted by simulations, factors f B + 1 < 1 result in an overestimation of T 2 and vice versa when reconstructing with the StdDict instead of the B + 1 Dict, whereas the inverse is seen to a less strong extent for T 1 . When comparing the measured T 1 difference map to the 2 predicted ones, the predicted bias without aliasing artefacts is similar, but slightly less important than the measured bias. Adding aliasing artefacts results in a more noisy appearance. For T 2 , including aliasing artefacts has no visually obvious effect. Local minima in the right breast of the experimental difference map ( Figure 6 marked by white arrows) are not reproduced by any of the predicted difference maps and are therefore not attributable to the presence of spiral aliasing artefacts.  Figure 5. Averaged over all 6 volunteers, the absolute difference |T L 1 − T R 1 | changes from 42 to 40 ms in FGT (mean relative difference stays at 4%) and from 13 to 18 ms in FT (mean relative difference changes from 4% to 6%), if reconstructing with the B + 1 Dict instead of the StdDict. Here, we calculated the mean relative difference for each volunteer as ∕T both 1 , (ie, the absolute left-right difference was normalized to the mean value calculated over both breasts). Averaged over all 6 volunteers, the absolute is reduced from 29 to 5 ms in FGT (mean relative difference changes from 38% to 8%) and from 29 to 9 ms in FT (mean relative difference changes from 21% to 7%). A separately conducted phantom study, presented in Supporting Information Figure S5, showed that reconstructing with the B + 1 Dict improved both the accuracy of the MRF T 2 estimates with respect to reference values and the repeatability of the MRF experiment on repositioning. MRF T 1 estimates were barely affected by the in-plane B + 1 correction. These findings support the in vivo analysis presented here.

| Slice profile effects
Matching to the SPDict resulted in increased T 1 values and smaller T 2 values than matching to the StdDict. The predicted difference maps without aliasing artefacts look more homogeneously structured than the measured ones. Adding aliasing artefacts results in more noisy difference maps of improved resemblance with the experimental ones. Table 3 lists the mean deviations in T 1 and T 2 in areas of FGT and of FT and their SD for all volunteers. With average relative deviations of (2, −22, 2, −6)% for (T 1,FGT , T 2, FGT , T 1,FT , T 2,FGT ), SP corrections affected T 2 more strongly than T 1 and FGT more strongly than FT. This is in line with the theoretic analysis, which predicted a stronger SP effect on T 2 for tissues with higher T 1 (Figure 3). For comparison, difference maps visualizing the combined effect of in-plane B + 1 inhomogeneity and the SP are shown in the third row. Qualitatively, the two CEs seem to add up.

| Diffusion effects
Apart from some noisy pixels that are visible in the experimentally observed T 1 difference map and, more strongly, also in its predicted counterpart including aliasing artefacts, no difference between matching to the DiffDict and StdDict is visible. From Table 3, matching including diffusion resulted in slightly higher T 2 in areas of FGT than matching to the StdDict, corresponding to average relative deviations of 0.1% for T 1,FGT and 1% for T 2,FGT . Yet, the observed mean differences are within the SDs, which is in line with the simulations.

| DISCUSSION
This study explored the extent to which bias from different CEs affected the outcome of MRF measurements in the female breast. In vivo, including in-plane B + 1 inhomogeneity into the MRF dictionary reduced the left-right asymmetry in T 2 maps of the breast, whereas correcting for slice profile effects yielded decreased T 2 values especially in FGT. For the spoiling moment of 4 × 2π over the slice used here, diffusion effects in FGT proved not to be significant, corresponding well to the theoretic predictions.
Our analysis underlines the need to consider CEs during MRF reconstruction. Some effects, such as the shape of the SP, are independent of the scanned subject and can, therefore, readily be included into the dictionary, albeit at an increase in computation time. Other effects like in-plane B + 1 inhomogeneity commonly need a calibration measurement (ie, the B + 1 map). The same would have held true for the ADC, if our MRF sequence were sensitive to this effect. Such prescans add to the scan time of MRF, and compromise its main promise: to offer rapid quantitative imaging. Ideally, MRF sequences would allow for inclusion of a CE as a free parameter in the dictionary and result in unambiguous matching. For example, adding rapid variations in the end of the flip angle series reduced matching errors because of in-plane B + 1 inhomogeneity, 18 and including different coil modes into the MRF framework permitted to estimate in-plane B + 1 during matching reconstruction at high field strength. 19 However, undersampling may pose a challenge to robust parameter estimation when the number of free parameters augments.
In breast MRI, the potential of the ADC as a native biomarker for the distinction of malignant from benign breast lesions and therapy response [42][43][44][45][46] is increasingly investigated. This encourages the incorporation of the ADC as a free parameter into MRF reconstruction; yet, few MRF studies so far showed successful matching of T 1 , T 2 , and ADC. 47,48 This challenge is probably present because diffusion acts very similarly to T 2 because of its signal-attenuating nature. 21 Hence, as long as an MRF sequence is not able to distinguish F I G U R E 5 Experimental bias analysis for T 1 by means of T 1 difference maps. Presented from top to bottom, the CEs under investigation are in-plane B + 1 inhomogeneity, the non-rectangular slice profile, the combination thereof, as well as diffusion. The difference maps presented in the first column show the experimentally observed bias. For their calculation, the T 1 map obtained by matching the experimental data to the StdDict was subtracted from the T 1 map obtained by matching the same data set to the dictionary including a CE. The second column shows the bias predicted by simulations. To calculate this second set of difference maps, one test signal per voxel including a CE was generated, with T 1 and T 2 as obtained by matching to the corresponding dictionary, and matching back to the StdDict. The resulting matched T 1 value was subtracted from the test signal's T 1 value to calculate the predicted difference map. For the third column, aliasing artefacts with a signal-to-artefact ratio of r = 1.5 were added to the test signals before matching them to the StdDict F I G U R E 6 Experimental bias analysis for T 2 by means of T 2 difference maps. Presented from top to bottom, the CEs under investigation are in-plane B + 1 inhomogeneity, SP effects, the combination thereof as well as diffusion effects. The difference maps presented in the first column show the experimentally observed bias. For their calculation, the T 2 map obtained by matching the experimental data to the StdDict was subtracted from the T 2 map obtained by matching the same data set to the dictionary including a CE. The second column shows the bias prediction by simulations. To calculate this second set of difference maps, 1 test signal per voxel including a CE was generated, with T 1 and T 2 , as obtained by matching to the corresponding dictionary, and matched back to the StdDict. The resulting matched T 2 value was subtracted from the test signal's T 2 value to calculate the predicted difference map. For the third column, aliasing artefacts with a signal-to-artefact ratio of r = 1.5 were added to the test signals before matching them to the StdDict. White arrows indicate minima in the experimental difference maps that are not reproduced by the predicted difference maps

T A B L E 3 Experimental bias because
of slice profile and diffusion effects between diffusion and relaxation effects, little diffusion bias is, at first, a positive attribute for the MRF sequence. Repeated signal simulations including spiral aliasing artefacts, implemented as periodic complex WGN patterns scaled to the signal strength, suggest that spiral aliasing artefacts have a larger impact on precision than on accuracy of the matched T 1 and T 2 values, while affecting T 1 precision more strongly than T 2 . Likewise, the aliasing artefacts themselves may act as a confounder, at least if low signal-to-artefact ratios are present such as in highly undersampled MRF scans. It should be emphasized that our periodic WGN simulations do not equal a full simulation of the k-space encoding based on a given object shape, sampling trajectory and MRF sequence. 24 Using pWGN patterns to simulate spiral aliasing artefacts may, however, give an estimate on the possible strength of spiral undersampling artefacts for a given measurement without knowledge about the exact trajectory design or the object shape. Importantly, the predicted artefacts patterns in our difference maps are spatially uncorrelated, as the noise patterns are generated independently for each voxel. Therefore, approximating spiral aliasing artefacts by pWGN will fail to capture any global spatial patterns that may be occur at high undersampling factors, which is an important limitation. An additional simulation study, comparing the pWGN approximation of spiral aliasing artefacts to a full simulation of k-space encoding in the breast, is available to the interested reader in Supporting Information Figure S6.
The in vivo bias analysis of left-right inhomogeneity caused by in-plane B + 1 effects was based on mean T 1 and T 2 values, calculated over ROIs containing mostly FGT or FT. Left and right breast were assigned separate ROIs; yet, they enclosed a range of different B + 1 values and, in the case of FGT, also heterogeneous tissue. Therefore, the calculated mean values comprised voxels with different amounts of bias. This explains why the bar plots in Figure 7 show large SD. Tissue heterogeneity may further be the reason for any residual differences observed between left and right breast FGT. However, we would not have expected leftright discrepancies to remain at least for FT after including inplane B + 1 effects into reconstruction, as FT should be relatively homogeneous throughout the breasts. The residual discrepancies might indicate inaccuracies in the AFI B + 1 mapping method. We have not analyzed magnetization transfer (MT) effects as a fourth, potentially CE, although there is evidence for MT to be present in FGT. [29][30][31] In FT, no MT effects are expected. 49 The missing analysis of bias because of MT in this work is because of a lack of quantitative MT literature values for the breast, as next to the fractional pool size also T 1 and T 2 of the semisolid pool as well as the exchange rate between the 2 pools are required for EPG simulations including MT. 50 Importantly, a recent study on MRF and MT has shown that MT is expected to affect MRF signal evolutions. 22 Although our MRF sequence did not include additional off-resonant pulses, the RF excitation pulses are nevertheless of a finite bandwidth and power,. Therefore, we cannot exclude the occurrence of residual MT effects at present and recommend a future bias analysis of MT for MRF in the breast.
In conclusion, we have shown that it is important to correct MRF measurements in the breast for 2 B + 1 -related effects, ie, for in-plane inhomogeneity and the SP shape, while the employed MRF sequence was not sensitive to diffusion. The analysis underlines that MRF has the power to overcome system imperfections, because their effect on the signal evolution can be understood, modeled, and hence, incorporated into MRF reconstruction. Yet, at the same time, it points out a major challenge for MRF: if an MRF sequence is sensitive to a CE, efforts need to be taken to desensitize the sequence-or to optimize it in such a way that the effect is no longer a confounding one, but becomes separable from other dictionary parameters during reconstruction.