Robust T2 estimation with balanced steady state free precession

To develop a novel signal representation for balanced steady state free precession (bSSFP) displaying its T2 independence on B1 and on magnetization transfer (MT) effects.

2][3] Rapid T 2 mapping using coherent SSFP, however, is generally more challenging than T 1 estimation, because both longitudinal and transverse magnetization components contribute to the signal (e.g., see Handbook of MRI Pulse Sequences). 4 Disentangling the mixed T 2 /T 1 contributions, therefore, either requires a priori knowledge of T 1 (e.g. using a global estimate) 5 or T 1 and T 2 must be simultaneously retrieved (e.g. using coherent and incoherent SSFP acquisitions) 6,7 using multiple phase-cycled balanced steady state free precession (pc-bSSFP) scans, [8][9][10] or using MR fingerprinting (MRF). 11In addition, all signal models depend critically on the choice of the flip angle and, therefore, typically show a more or less pronounced sensitivity on B 1 . 6,12,13Similarly, for reasons of simplicity, typically a single compartment tissue with a single T 2 is presumed, thereby neglecting magnetization transfer (MT) effects, which may also lead to pronounced bias in the parameter estimates. 14,15In summary, it appears evident that all rapid relaxometry methods may suffer from such an extrinsic or intrinsic bias (or a combination thereof) (i.e., B 1 , MT).
There was, however, given experimental evidence that T 2 estimation with triple-echo steady state (TESS) 16 or using multiple pc-bSSFP scans 9 is inert to B 1 field inhomogeneities.This rather surprising experimental finding was only recently corroborated theoretically by an approximate scaling for steady state sequences. 17In this work, we further elaborate on this recent theoretical finding to show that balanced steady state free precession (bSSFP) theory predicts that T 2 estimation should not only be insensitive to B 1 , but also to MT.All sources of bias, as introduced to the steady state bSSFP signal by B 1 field inhomogeneities and MT effects are absorbed into a large bias for T 1 .As a result, T 2 estimation with bSSFP in the steady state is expected to become inert to B 1 and to all protocol parameters that affect MT, such as the flip angle, the RF pulse characteristics, and the TR.

Pure tissue
We consider a train of instantaneous and equidistant RF excitation pulses (with TR, flip angle , and constant RF phase increment ) in the absence of motion and diffusion (as usual).It was shown by Zur et al. 18 that the steady state of bSSFP immediately after excitation can be expressed as a series which includes all unbalanced configuration orders (or modes) m (n) and where  denotes the accumulated phase within one repetition time interval (that is the off-resonance  0 minus the RF phase increment ).
The configuration orders can be expressed as 17 with the definitions In Eq. ( 2), the two lowest order configuration modes m (0) (corresponding to the FID) and m (−1) (corresponding to the ECHO) take the form where m E denotes the Ernst signal.Because quantitative imaging relies on signal ratios, let us define the leading order mode ratio as follows Using Eq. ( 2), we can explicitly calculate the series in Eq. ( 1) to yield and in combination with Eq. ( 7) we, therefore, find Equation (9) shows that, apart from a common scaling factor m E (which, as we will see more explicitly below, may be ignored), the magnetization only depends on the accumulated phase , T 2 relaxation (E 2 ), and on a single parameter c, which, according to Eq. ( 9), completely describes any dependence on the actual flip angle  and the relaxation time T 1 .In other words, in the quantitative analysis of bSSFP experiments with variable phase increments   , we may neglect any explicit dependencies on  and T 1 (or E 1 ), but rather treat c as a fitting parameter, the value of which does not matter in the context of T 2 relaxometry.

MT effects
The binary spin bath model 19 takes into account MT effects between a restricted (subscript r) and a free (subscript f ) proton pool of fractional size F (≔ M 0r ∕M 0f , where M 0r and M 0f denote the corresponding equilibrium magnetizations, respectively).The two pools have relaxation times T a∈{1r,1f ,2r,2f } and the exchange of magnetization between the two pools is reflected by two pseudo-first order rate constants k f (≔ RM 0r , where R is the fundamental rate constant between the two pools) and k r (≔ RM 0f ).The effect of pulsed RF irradiation on the restricted pool protons is calculated as a function of the flip angle  and at on-resonance (Δ → 0) taking into account the absorption line shape of the restricted pool protons (G(Δ)). 20It was shown that the signal of bSSFP in the presence of MT effects can be written in elliptical form 21 (note that we used the substitution  → − to be consistent with Eq. ( 1)), and using the definition 1∕T ′ 1f ≔ 1∕T 1f − 1∕TR ⋅ ln().The factor  depends on the details of the binary spin bath model and its exact form can be found in Eq. ( 3) in Wood et al. 21(in the notation of Wood et al.:  ≔ B∕A, but note that A and B have a different meaning as compared to our definitions in Eqs. ( 4) and ( 7)). 21emarkably, the magnetization of the pure tissue ( → 1) takes exactly the same elliptical form as Eqs.( 10) and ( 11) using the substitutions As a result, bSSFP in the presence of MT effects will, therefore, follow exactly the same line of equations as given for the pure tissue because the factor b MT (and, therefore, the nominator and denominator in Eq. ( 10)) has exactly the same functional form with and without the presence of MT effects.As a result, also in the presence of MT, bSSFP will neither show an explicit dependence on the flip angle  nor on E 1 (i.e., T 1 ).
In summary, an analysis of the frequency profile of bSSFP to retrieve the apparent T 1 and T 2 relaxation times using a configuration-based non-linear least squares fitting procedure will essentially lead to the same apparent T 2 independent of B 1 field inhomogeneities and independent of the presence of MT-related signal modulations (and, therefore independent of the RF pulse characteristics and the TR used).T 2 estimation with bSSFP should, therefore, be exceptionally robust and free of any MT-related effects.

Parameter estimation
For bSSFP experiments, conducted with variable phase increments   , the signal of bSSFP as a function of the RF phase increment (  ) is of form (see Eq. ( 9)) with two linear coefficients (f ± ) and a non-linear factor z ≔ e i 0 A that includes the off-resonance.
For N-phase cycled signal acquisitions (y ≔ [y 1 , y 2 , … , y N ] T ), Eq. ( 12) will take the form from which the least-squares estimate ẑ can be found using a variable projection (VARPRO) 22 with the projector and from which the least squares estimates f± are found using ( F ≡ F(ẑ)) In summary, VARPRO will yield the least squares-estimators θ0 (= arg(ẑ)) and Â (=| ẑ |) from Eq. ( 14), as well as the leading order configuration mode ratio ( B = f− ∕ f+ ) from Eq. ( 16).Finally, from Â and B the corresponding estimators for ĉ and Ê2 are found using (for details, see Appendix A) Generally, the local off-resonance  0 might be estimated independently from VARPRO (see Eq. ( 14)) and directly using a discrete Fourier transform (DFT) of y, as described in related work. 23In addition, the DFT also allows to yield an estimate of A either from the decay of the modes using a matrix pencil approach 24 or directly from m (1) /m (0) .As a result, the DFT-related estimate of θ0 and Â was used as a meaningful starting value of ẑ = e i θ0 Â for the VARPRO minimization.

MRI acquisition
One healthy volunteer (female; 50 years old) was scanned at 3 T (MAGNETOM Prisma, Siemens Healthcare) with a standard 20-channel receive head/neck coil.Informed written consent was obtained from the volunteer before the scans, which were approved by the local ethics board (Ethikkomission Nordwestschweiz).Brain MRI with bSSFP was performed in 3D in sagittal orientation with an isotropic resolution of 1.3 mm using a matrix size of 192 × 144 × 144, a GRAPPA acceleration factor of 2, partial Fourier factor 7/8 in slice and phase encoding direction, and elliptical scanning.A sinc-shaped slab selective excitation pulse (time-bandwidth-product of 1.0) was used and a preparation period (340 dummy TR of 4 ms, 100 dummy TR of 8 ms) before each pc-bSSFP scan to mitigate transient effects.For N pc-bSSFP scans, the total scan time was fixed to 7 min and the phase cycling condition was Centered echo times were used.Variable protocol settings for pc-bSSFP were explored within three different scan sessions resulting in a range of different pc-bSSFP sets: • Set 1: N = 12, TR of 4.0 ms, bandwidth of 744 Hz/pixel, and flip angle of 15 • (T RF of 320 μs).
• Set 8: as set 7, but with a flip angle of 20 • (T RF of 320 μs).
• Set 10: subsets of set 1, using N = 4 (Δ = −180 No image registration was performed between pc-bSSFP scans and between the acquired pc-bSSFP sets.If motion in-between pc-bSSFP scans is an issue, image registration can be performed.Skull stripping of the brain data was performed with FSL (Analysis Group, FMRIB).All other data processing and visualizations were performed in MATLAB (R2019a, MathWorks).

Numerical simulations
Pc-bSSFP data was generated from Eq. ( 9) for a range of flip angles ( = 1 … 30 [deg]), for various numbers of phase cycles (N = 3 … 24), and for a range of TRs (TR = 1 … 30 [ms]) for a fixed T 1 /T 2 of 1000 ms/100 ms and at on-resonance ( 0 = 0 [deg]).Complex Gaussian white noise was added to the simulated signal.A total of 10 000 Monte-Carlo (MC) runs were performed and each run analyzed using the proposed VARPRO method.For the resulting T 2 estimates, SNR was calculated from its mean divided by its SD.For better comparison and to be independent on the noise-level, relative SNR values (rSNR) are reported.To this end, the SNR is normalized by the SNR predicted for a set of N = 12 cycles, a TR of 4 ms, and a flip angle of 15 • (corresponding to set 1).

RESULTS
The results of the MC simulation are shown in Figure 1.
For a flip angle of about 7 • , the rSNR reaches its maximum, but drops rapidly for lower flip angles.As a result, a flip angle of about 7 • should be seen as the lower limit for the actual flip angle.At 3 T, because of B 1 field inhomogeneities, actual flip angles can vary considerably (e.g., between 0.7-1.3times the nominal one).This gives a lower limit for the optimal flip angle of about 10 • .In contrast, there is no optimal TR, any increase in the TR increases the rSNR (being corrected by the acquisition time).Interestingly, however, the rSNR (again corrected by the acquisition time) as a function of the number of phase cycles remains constant for N ≥ 5, but drops rapidly for smaller values.Therefore, the recommended minimum number of phase cycles is N = 5.
Example maps for T 2 , the parameter c (Eq. ( 3)) and the local off-resonance  0 retrieved from pc-bSSFP of dataset 1 are shown in Figure 2 for two axial positions.In dataset 1, an RF pulse duration of 320 μs was used, corresponding to Relative SNR for T 2 estimation for the suggested variable projection (VARPRO) method using Monte-Carlo simulations as a function of the flip angle (A), TR (B), and the number N of phase cycled balanced steady state free precession (pc-bSSFP) acquisitions (C).The circle in the plots represents the relative SNR (set to 1) for a scan using a flip angle of 15 • , a TR of 4 ms, and 12 pc-bSSFP acquisitions (i.e., of the parameters from set 1).

F I G U R E 2
Parameter estimation of T 2 , off-resonance  0 , and c using variable projection (VARPRO) showing two axial brain slices (for phase cycled balanced steady state free precession [pc-bSSFP] imaging with a flip angle of 15 • , a T RF of 320 μs, a TR of 4 ms, and 12 pc-bSSFP acquisitions).a certain amount of MT effects inherent to the bSSFP signal.A shortening of in the RF pulse duration will increase MT effects, whereas its prolongation will decrease these effects.Overall, a change in the RF pulse duration has a strong effect on the bSSFP signal which formally enters as a scaling of T ′ 1f for the bSSFP two-pool model signal equation (see Theory section).As a result, a modulation of the RF pulse duration must result in a corresponding change of the parameter c only.An increase of MT effects (by a shortening of the RF pulse) will decrease T ′ 1f , being reflected by an increase in c (note that c-for a fixed flip angle-is monotonically decreasing with increasing T 1 ), whereas a decrease of MT effects will result in a decrease of c.This is demonstrated in Figure 3.
Generally, for the estimation of the complex parameter ẑ (Eq.( 12)), the theoretical minimum number of pc-bSSFP scans required is N = 2, but MC simulations revealed that the SNR will be compromised for N < 5 (see Figure 1C).In Figure 4, the effect of decreasing the number N of pc-bSSFP on T 2 and c estimation is shown.Parameter maps show no apparent considerable loss of quality using six cycles instead of 12 (notably corresponding to half of Effect of RF pulse variation T RF and, therefore, of magnetization transfer-related signal variations on T 2 and c, shown in two axial brain slices for T RF of 80 and 1280 μs.In the left column of the T 2 maps, the positions of the regions of interest (ROIs) for the white matter (WM) values in Table 1 are depicted.In the left lower c map, the position of the ROI for the gray matter (GM) values in Table 1 is depicted.

F I G U R E 4
Dependence of T 2 and c parameter estimation on the number of phase cycles used.the acquisition time), but noise clearly becomes visible for four cycles.As a result, and as indicated by MC simulations, rather than performing a high number of phase cycles with short TR (e.g., 12 cycles with a TR of 4 ms) the use of a lower number of phase cycles with a longer TR (e.g., six phase cycles with a TR of 8 ms) seems advisable.The results of such a scan are collected in Figure 5.
Finally, for manually drawn regions-of-interest (ROIs) in the frontal and the parietal white matter (for definition of ROIs see Figure 3) corresponding T 2 values (mean ± SD) and corresponding c values (mean ± SD) are collected in Table 1.For all presented data, T 2 difference maps can be found in the Supporting Information.Overall, our limited experimental data suggests that T 2 estimation is indeed independent on the experimental conditions (measurement protocol), as expected and predicted from theory all the measurement variations are collected within the parameter c (see Table 1).

DISCUSSION
From MR theory, T 2 estimation with bSSFP is predicted to be inert to B 1 -field inhomogeneities, as well as to protocol Effect of TR and flip angle α variation on T 2 and c, shown for α = 10 • and α = 20 • and TR of 8 ms (using six phase cycled balanced steady state free precession acquisitions).Note the different scaling for the c maps.

T A B L E 1
T 2 and c values measured in the brain of a 50-year-old female healthy volunteer with the phase cycled bSSFP method using various RF pulse lengths (T RF ), pulse angles (α), TR, and numbers of cycles (N).variations, such as the RF pulse power, that may lead to MT-related signal modulations.This peculiar finding relates to the special representation of bSSFP variables in configuration space (Eq.( 2)) and its corresponding signal form in real space (Eq.( 9)).Moreover, the results as derived in this work generally require steady state conditions.Therefore, non-steady state conditions, such as with MRF, will essentially not feature the same properties.As pointed out above, we make use of a special and novel representation of bSSFP's signal (up to a constant factor) that depends-besides, the phase accrual -only on the two parameters E 2 and c (where c has an implicit dependence on the (actual) flip angle  and E 1 ) (Eq. ( 3)).This is in contrast to the common bSSFP signal representation with explicit dependence on , E 1 , E 2 , and the (actual) flip angle .Nevertheless, non-linear least squares-fitting should result in the same least-squares estimators and, therefore, reveal the same independency (as experimentally observed in previous studies). 9,10nitially, it might be rather surprising that bSSFP theory predicts that T 2 does not depend on MT, but this finding relates to the time evolution of the transverse magnetization, which is virtually absent for the restricted pool protons.As a result, the accumulated phase within any TR interval depends only on the phase accrual of the free pool protons.Therefore, in both cases (pure tissue and the two-compartment model for MT) the phase of bSSFP is solely and uniquely determined by the T 2 of the free pool protons.For other multi-compartment tissues, such as myelin-water or water-fat, shifts, this will no longer be valid, if not properly accounted for (e.g., by an extension of the signal model).Although myelin is present in our selected ROIs in the white matter, our example brain MRI experiments at 3 T did not show a significant change of T 2 estimates if TR was increased from 4 ms to 8 ms or if the nominal flip angle was increased from 10 • to 20 • .
In this work, we propose a VARPRO method to retrieve T 2 least squares estimates using Eq. ( 9) (i.e., using a novel representation of bSSFP's signal in real space).Obviously, the same independency of T 2 holds in configuration space (i.e., using Eq. ( 2)).Therefore, one might be tempted to work directly in configuration space rather than in real space using a fast DFT of pc-bSSFP data and using the lowest order configuration modes (as suggested in related work). 9,16In most practical cases, there will be no noticeable difference, but it should be kept in mind that for low number of sampling points N (typically less than approximately N = 8) or for low flip angles aliasing of modes can occur.Aliasing of modes is not so much an issue for tissues (other than fat), which generally show a rapid decay of modes; however, the signal of fluids is preserved even to high configuration orders.As a result, accurate T 2 estimation of fluids, such as for CSF, can be erroneous with bSSFP using a configuration analysis, unless a high flip angle (not recommended because of a prominent SNR loss) or a high number of pc-bSSFP scans are recorded (which is not advisable because of constant SNR per unit time) (see Figure 1C).The proposed VARPRO method in this work, however, does not suffer from this limitation and is always exact independently on the number N of phase cycles used.Nevertheless, a DFT of pc-bSSFP data is of interest here, in combination with our suggested method (e.g., serving as starting values).

CONCLUSION
In summary, bSSFP theory predicts that T 2 estimation in the steady state becomes independent on typical variations of the MR protocol settings, as well as on scanner imperfections, as supported by our example brain data, but given the rich complexity of tissues, a more rigorous testing is required to validate this finding in practice.