Water diffusion and T2 quantification in transient‐state MRI: the effect of RF pulse sequence

In quantitative measurement of the T2 value of tissues, the diffusion of water molecules has been recognized as a confounder. This is most notably so for transient‐state quantitative mapping techniques, which allow simultaneous estimation of T1 and T2 . In prior work, apparently conflicting conclusions are presented on the level of diffusion‐induced bias on the T2 estimate. So far there is a lack of studies on the effect of the RF pulse angle sequence on the level of diffusion‐induced bias. In this work, we show that the specific transient‐state RF pulse sequence has a large effect on this level of bias. In particular, the bias level is strongly influenced by the mean value of the RF pulse angles. Also, for realistic values of the spoiling gradient area, we infer that the diffusion‐induced bias is negligible for non‐liquid human tissues; yet, for phantoms, the effect can be substantial (15% of the true T2 value) for some RF pulse sequences. This should be taken into account in validation procedures.

the acquired signal is simply ignored.The review by Poorman et al. 18 does not even mention diffusion-related confounders amongst the many challenges for MRF.
Meanwhile, Kobayashi and Terada 20 show that diffusion can cause substantial T 2 underestimation in non-balanced MRF; however, Nolte et al. 21show that diffusion causes negligible T 2 bias in non-balanced MRF.This looks like a contradiction, although the influence of the spoiling gradient has been clearly identified. 20Section 6 of this article will elaborate on this in more detail.
None of the aforementioned publications considers the influence of the RF pulse sequence, despite the fact that, in methods such as MRF, MR-STAT or QTI, sequences of time-varying RF pulses are an essential characteristic of the method.It is an active field of research to optimize such RF pulse sequences on the precision of the outcome [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] -but not on its sensitivity to (diffusion-induced) bias.Our aim is therefore to quantify the diffusion-induced bias on the reconstructed T 2 map (in brief, "DI-bias") and the dependence thereof on the RF pulse sequence.We do this within the MR-STAT framework, but we expect the effects to be similar for other transient state techniques such as MRF or QTI.For this purpose, we apply simulations to study the dependence of the DI-bias on various values of the diffusion constant D (which we assume to be isotropic), on T 1 , on T 2 , and on the RF pulse sequence.We then validate the simulations using experiments on gel phantoms.We show that the DI-bias strongly depends on the RF pulse sequence.We apply phantom experiments because confounding factors are more controlled than with in vivo scans; moreover, we did not manage to clearly measure the effect in vivo, where the simulations indicate less than 2% DI-bias for the worst of the sequences tested.Yet, for phantoms-where D and T 2 values can be relatively high-an evaluation of the DI-bias is relevant, since phantoms are important for the technical validation of quantitative methods. 38

| SIMULATION: METHODS
We simulate data acquisitions for a set of sequences, described in detail below.This simulation is done using the extended phase graph (EPG) model 39 which uses T 1 , T 2 and (isotropic) diffusion D as parameters.The coil sensitivity is assumed to be constant, as well as the proton density; the slice profile is assumed to be perfectly rectangular.Subsequently, the raw data is used to reconstruct T 1 , T 2 and proton-density values.
Diffusion is purposely not part of the reconstruction model.This is obviously a model error, which we expect to lead to DI-bias.
The simulated sequences consist of applying different patterns of RF pulse sequences on a base MR-STAT sequence.The base sequence (see Figure 1) consists of an unbalanced single-slice Cartesian pseudo-SSFP (steady state free precession) sequence on a 3 T scanner (Philips Elition), with T R = 10 ms, T E = 5 ms, voxel size 1 mm Â 1 mm, slice thickness 5 mm, and an FOV of 224 mm; this requires 224 phaseencoding steps for Nyquist sampling.The set of phase-encoding steps is repeated six times, allowing MR-STAT reconstruction of proton density, T 1 , and T 2 maps.In total, 1344 readout-lines are simulated to be acquired in 13.4 s.See Figure 2A-F.Subtracting the T 2 -simulation input from the obtained T 2 maps results in DI-bias ¼ T 2,true À T 2,simulated .Note that, according to this definition, a positive value of the DI-bias by the institution that the simulated (or measured) value of T 2 is lower than the true value.
To investigate the dependence of the DI-bias on the RF pulse sequence, the simulations are applied for six different time-varying RF pulse sequences (Figure 2).These RF pulse sequences were previously optimized with the BLAKJac framework 37 (Block Analysis of K-Space View on the Jacobian), using a variety of constraints and criteria, as shown in Table 1.The sequence optimization focused on the predicted noise level of the reconstructed T 1 and T 2 values; the values of the DI-bias were not included in the design, which led to a relatively high spread of these functions.In Figure 2, the orange line represents the second derivative of the phase of these RF pulses. 40,41A non-zero phase derivative means that RF phase cycling is applied.Taking A and C as an example, this second derivative is equal to À2 during the first half of the sequence and +2 during the second half, such that ϕ n ¼ Àn 2 during the first half and Data is simulated by varying one parameter around a reference value of T 1 and T 2 .This reference value is chosen to be comparable to the values actually encountered in a real-world phantom (see Section 4).The setpoints for the relaxation parameters are chosen to be T 1 ¼ 1:2 s, For the diffusion, the simulation depends on the product Here, Δk is the dephasing that the unbalanced sequence applies within each T R period; Δk is assumed to be constant throughout a sequence; it is expressed in rad per unit length, that is, Δk 2π is the number of cycles inflicted during each T R period, which in our case is exactly 1000 m À1 , applied in the frequency-encoding direction.D is the diffusion constant in this direction; taken to be 2.1 μm 2 /ms for gel vials (see Section 4).The parameter in our phantom, we assume T D to be 2:1 ≈ 12:06 s, so T R =T D ≈ 0:000829.

| SIMULATIONS: RESULTS
Figure 3A shows the dependence of the DI-bias on T R =T D at the setpoint of T 1 ¼ 1:2 s, T 2 ¼ 0:2 s, for the six different sequences.Figure 3B,C shows, respectively, the dependence of the DI-bias on T 1 and on T 2 .Figure 3B suggests that the DI-bias can be reasonably modelled as independent of T 1 ; this T 1 -independent modelling incurs an error in DI-bias of approximately 1.6 ms (root mean square (RMS) over the range of T 1 values and over the six sequences).We thus model the DI-bias as a function of only T R =T D and T 2 , that is, The subscript S makes this model sequence dependent.
Figure 3C suggests that, for a fixed T R =T D , the T 2 dependence can be modelled by a polynomial model.For this purpose, we introduce the . From all possible polynomial models, we discard the zeroth-order term, since this would lead to an unphysical non-zero bias at T 2 ¼ 0. Allowing orders beyond the quadratic (i.e., containing a 3 T 3 2 or higher) is also counter-indicated: each of these models leads, for at least one of the tested sequences, to non-monotonic bias curves, eventually descending to negative bias values at relatively high values of T 2 ; this indicates overfitting.So, our empirical bias model reduces to where a 1 and a 2 are sequence-dependent parameters.After fitting the simulated data to this quadratic model over each sequence and over the range T 2 0:07 s,0:55 s ½ , we obtain an RMS error of just 1.2 ms (see Figure 4).

| MEASUREMENTS: METHODS
Here, our aim is to validate the model identified in the simulation using actual measurements on phantoms.For this purpose, we want to establish the bias function f S T 2 ð Þ for a given RF sequence.Even if we know T 2,true , it is difficult to reliably measure the DI-bias directly, since T 2,measured contains many causes of bias. 18To resolve this difficulty, we define a pair of scans that differ only in spoiling gradient area, that is, that differ only in Δk.The first element of the pair is called "full gradient spoiling" (FS).The net gradient area in between RF pulses consists almost exclusively of Schematically drawn basic element of the sequence.The amplitude (and possibly phase) of RF pulses varies from excitation to excitation, as does the phase encode value.The post-echo readout gradient plus the spoiling gradient constitute the net gradient area A (shaded).This area is constant over all repetitions of the basic element of the sequence.
the frequency-encoding gradient (see shaded area in Figure 1); for FS, the area is approximately 23:5 ms mT m .Strictly speaking, the slice-select gradient also has a non-zero net gradient area, but its contribution is negligible in terms of Δk 2 .The second element of the pair, "half gradient spoiling" (HS), has been obtained by the same sequence timings and flip angles, but with a doubled field of view and doubled voxel size; in effect, this halves the strength of the frequency-encoding gradient, and the spoiling gradient strength also scales proportionally, bringing the total net gradient area to approximately 11:8 ms mT m .The reconstructed T 2 maps according to the FS and HS sequences are denoted as T 2,FS and T 2,HS respectively.Using the model from Equation (1) and using Δk HS ¼ 1 2 Δk FS (thus Δk 2 HS ¼ 1 4 Δk 2 FS ), we can write The six different RF pulse sequences employed in this study.In blue is the amplitude (in degrees); the increment of the phaseincrement from RF pulse to RF pulse is shown in orange, also in degrees. 40,41The RF pulse sequences are optimized on SNR of the resulting T 1 and T 2 maps under varying circumstances, not related to this study (see Table 1). 37A B L E 1 The six different RF pulse sequences are optimized for presence/absence of a leading inversion pulse, for excitation pulses with or without phase, differing levels of the allowed RMS of the RF pulse angles, and different granularities (degrees of freedom) of the RF pulse angle pattern.
, we can then estimate the DI-bias as where T 2,HS À T 2,FS ð Þcan be measured for various reconstructed values of T 2 , and Q S T 2 ð Þ is a function of the true T 2 value, which can be estimated from simulations.See Figure 5 for plots of Q S T 2 ð Þ.In the absence of simulations, one could naively assume Q T 2 ð Þ¼0:75 (assuming ; this "naive" value of Q would lead to slightly less accurate end-results (results not shown).Figure 5 shows that this naïve approximation is most accurate for the Q values of sequences A, C, and D.
In our experiments, we use the Eurospin phantom, 42 which consists of 12 gel vials with different relaxation properties.The base sequence has been described in Section 2. We used the same six sequences as in the simulations (Figure 2).
For performance analysis, this base sequence is acquired 10 times (with 5 s pause in between to allow for full relaxation), which leads to 10 separate reconstructions; this enables the calculation of the noise level at each voxel and boosts the precision of the outcome by ffiffiffiffiffiffi 10 p .
For further analysis, we also need an estimate of the diffusion-coefficient D in the gel vials.We measured this value using six-direction diffusion tensor imaging (DTI) data with B = 1000 s/mm 2 and T E ¼ 86 ms.The native system calculation provides an apparent diffusion coefficient map, which is used for estimating D.

F I G U R E 4
For the six different RF pulse sequences, the simulated DI-bias as a function of T 2 (blue dots).For each sequence, the fitted model a 1,S T 2 þ a 2,S T 2 2 is also plotted (black line).The six graphs are displayed at the same scale to emphasize the difference between sequences.
To quantify the DI-bias, we measured T 2,FS À T 2,HS ð Þfor different sequences and for different values of T 2 .As an example, Figure 6 shows the reconstructed T 2 maps of RF Pulse Sequence A for the two spoiling levels (HS and FS, respectively) as well as their difference.Note the difference in T 2 in some of the 12 gel vials.The difference in measured T 1 values was not significant.
Figure 7 shows, for the six different RF pulse sequences, the difference The "gold standard" value of T 2 was obtained using single-echo spin-echo experiments with echo times of [8, 28, 48, 88, 138, 188] ms, for which we assume the diffusion effects to be negligible.The orange lines in Figure 7 represent the best fit of the second-order model, derived from simulations (note that in our experiments Example of reconstructed relaxation maps (RF Sequence A) for the full (FS) and half (HS) spoiling and the mutual difference (righthand column).For analysis, we focus on the 12 gel vials, not on the surrounding liquid.Most prominent is the difference indicated by the arrow, which occurs in the gel vial with the highest T 2 .
Using this model, we derived the polynomial parameters shown in Table 2.The table also shows the mean DI-bias value evaluated at Given 10 repetitions of the same measurement, we computed f S T 2,ref ð Þ for each individual dataset.The standard deviation over these 10 bias outcomes is reported in the table as σ.
The results from Table 2 can be used to compare the simulation results with the measurement results.This comparison is shown in Figure 8.
It is clear that simulation and measurements correlate (correlation coefficient of 0.92); the best fitting slope is 0.81.
The measurements and the simulation both lead to the observation that there is a large difference in DI-bias between sequences with different RF excitation sequences; both simulation and measurement suggest a factor of almost 8 difference in DI-bias when comparing sequences C and E.

| DISCUSSION
Diffusion causes bias in reconstructed T 2 maps, which we denote as DI-bias.This effect has been discussed for many types of quantitative MR sequence, including MRF.In this study we focused on the dependence on the RF pulse variation, which is an essential element of transient-state MRI.For the six different RF pulse sequences employed in our study, we found that the DI-bias can be as high as 15% of the true value; this holds for gel vials with a T 2 value of around 375 ms.
Furthermore, we observe a large difference in DI-bias over different RF pulse sequences.As shown in Figure 9, an important factor in this difference is the average RF power of the sequence.
For the six different RF pulse sequences, the estimate of the DI-bias (i.e., the value of , plotted against T 2 , for the 12 vials (blue circles; error bar calculated from the standard deviation over 10 reconstructions).For each sequence, the fitted model 2 is also plotted (orange line).The six graphs are displayed at the same scale to emphasize the difference between sequences.

T A B L E 2
The DI-bias model obtained from the phantom experiment.We should add that the set of sequences was not selected a priori on the value of the DI-bias; hence there might be sequences with even larger or smaller bias than observed for the six sequences at hand.Future work may aim to specifically optimize sequences on minimizing (or possibly maximizing) the DI-bias.
Looking at Figure 3, we could argue that the bias h S could be modelled as approximately proportional to Such a model deviates in terms of RMS from simulation results by 7.6 ms, over a range of D Δk ð Þ 2 T R 0,0:0028 ½ , T 2 0:06 s,0:44 s ½ ; the RMS value of the DI-bias is 37 ms, so the approximation mis-estimates the DI-bias by approximately 20%.Such approximation closely resembles the estimate derived by Freed et al., 43 although that was established for a stopped-SSFP experiment and its validity holds for T R > T 2 or T R > T D .
The validity of the simulations is shown by their similarity to experimental results, although the simulations indicate a 20% higher DI-bias than we infer from the measurements.Given the results of Figure 9, a cause thereof may be an error in actual B 1 , despite a correction for B 1 in the reconstruction process.Another cause may be that the diffusion in gel phantoms is slightly restricted, such that the D as seen by MR-STAT is slightly lower than the D ¼ 2:1 Â 10 À9 m 2 s measured at T E ¼ 84 ms.In this work, we focus on phantom experiments because confounding factors are more controlled than with in vivo scans and the measured effect is stronger than in vivo.In vivo, it is difficult to clearly measure the DI-bias given the much lower values of D and T 2 of human tissue with respect to the phantom we used.In particular, when considering brain tissue (grey or white) instead of gel vials, we can assume a value of D of around 0:65 Â 10 À9 m 2 s (Reference 44) and a T 2 of around 70 ms (Reference 45).For this combination of D and T 2 , the simulations indicate a DI-bias of less than 2% for the worst-case sequences B and C.This explains the difficulty in applying the FS/HS method on brain tissue: the precision and the accuracy of the measuring techniques are on the same order as the effect we would like to study.F I G U R E 9 Relation between the RMS value of the RF pulses and the simulated DI-bias.
Our study has been conducted on a single slice.Most transient-state MRI work consists either of multiple-single-slice sequences-for which our experiments should be representative-or of 3D sequences; in the latter case, the slice-select gradient is replaced by phase-encoding gradients, which are usually balanced.In effect, the gradient-area issue is similar to our experiments.
The insights from our analysis also allow comparison of the conclusions of Nolte et al. 21against those of Kobayashi and Terada. 20The latter work states "Without [correcting for] diffusion, T 2 from MRF-FISP was greatly under-estimated for large gradient moments," which can be quantified using referenced Figure 7C-E, 20 suggesting about 20% DI-bias for tissues such as grey matter, liver, and skeletal muscle.Nolte et al.
estimate the DI-bias in breast tissue to be less than 1%, which is indeed not observable experimentally.The explanation is certainly not due to the value of D in breast tissue (D ¼ 2:0 Â 10 À9 m 2 s ), 46,47 which is higher than the D in muscle (1:8 Â 10 À9 m 2 s ), 48 liver (1:1 Â 10 À9 m 2 s ), 49 or brain (0:65 Â 10 À9 m 2 s ).Yet, the value of Δk differs considerably: Δk 2π ¼ 800=m (Reference 21) against Δk 2π of up to 6400=m (Reference 20).Since, in coarse approximation (see Figure 3A), the DI-bias scales with Δk 2 , this explains a difference of almost two orders of magnitude (factor of 64) between the two approaches.In addition, we show that the RF pulse sequence itself has a large influence (Figure 8): both from simulations and from measurements, we observe a difference factor of 8 between the two extreme sequences from the set of six.This probably also contributes to the apparent discrepancy of conclusions between the referred studies.With our choice of Δk 2π ¼ 1000=m, which we consider sufficient for voxel sizes of 1 mm, we can simulate that the DI-bias is small (<2%) for most non-liquid human tissues, regardless of the choice of the sequence.

| CONCLUSION
We have shown that the DI-bias on the reconstructed T 2 is strongly dependent on the RF sequence applied, and particularly on the mean value of the RF pulses.
We can infer that for non-liquid human tissue, the DI-bias is generally negligible, that is, less than 2%, even for the sequences with a high sensitivity to DI-bias.For phantoms, our study shows that the DI-bias cannot be neglected: even if we establish it to be small for one specific RF pulse sequence, the effect can be non-negligible (up to 15%) for other sequences, even if the values of Δk and T R do not change.Since phantoms are used for validation of MR methods, the DI-bias should not be neglected.

F I G U R E 3
Simulation of the DI-bias at or around T 1 ¼ 1:2 s, T 2 ¼ 0:2 s, T R =T D ¼ 0:000829 on varying T R =T D (A), varying T 1 (B), and varying T 2 (C).

F I G U R E 8
Correspondence between simulated DI-bias and measured DI-bias, evaluated at T 2 ¼ 0:2 s.The full line represents the identity line; the dashed line is the best fitting slope.