Getting the phase consistent: The importance of phase description in balanced steady-state free precession MRI of multicompartment systems

Purpose: Determine the correct mathematical phase description for balanced steady-state free precession (bSSFP) signals in multicompartment systems. Theory and Methods: Based on published bSSFP signal models, two distinct phase descriptions can be formulated: one predicting the presence and the other predicting the absence of destructive interference effects in multicompartment systems. Numerical simulations of bSSFP signals of water and acetone were performed to evaluate the predictions of these two distinct phase descriptions. For experimental validation, bSSFP profiles were measured at 3T using phase-cycled bSSFP acquisitions performed in a phantom containing mixtures of water and acetone, which replicates a system with two signal components. Localized single voxel MRS was performed at 7T to determine the relative chemical-shift of the acetone-water mixtures. Results: Based on the choice of phase description, the simulated bSSFP profiles of water-acetone mixtures varied significantly, either displaying or lacking destructive interference effects, as predicted theoretically. In phantom experiments, destructive interference was consistently observed in the measured bSSFP profiles of water-acetone mixtures, an observation which excludes the phase description that predicts an absence of destructive interference. The connection between the choice of phase description and predicted observation enables an unambiguous experimental identification of the correct phase description for multicompartment bSSFP profiles, which is consistent with Bloch equations. Conclusion: The study emphasizes that consistent phase descriptions are crucial for accurately describing multi-compartment bSSFP signals, as incorrect phase descriptions result in erroneous predictions.

Besides previous studies on water-fat fraction mapping studies 15 , most quantitative mapping methods using bSSFP profiles [9][10][11][12][13] assume that a voxel has a single compartment with a unique resonance frequency.In these scenarios, the bSSFP profile appears elliptical on a complex plane and symmetric when plotted against the RF phase increment.
The complex bSSFP profile was described for single compartment systems 2,9-12,16- 20 , in which the profile has a symmetric and elliptic shape.However, asymmetries in the bSSFP profiles were observed in the presence of gray matter, white matter and muscle 9,21,22 , and in voxels containing multiple resonance frequencies such as water and fat 14 .Descriptions for bSSFP profiles in multi-compartments were reported 11 but not thoroughly investigated nor validated for asymmetric signal behavior.
Because certain elements of bSSFP signal phase descriptions may affect how two single component profiles are superimposed, the study aimed to determine the correct mathematical phase description for multi-compartment bSSFP profiles.An intuitive derivation and experimental validation are provided using water acetone mixtures.
Acetone was selected over fat due to its improved solubility, and to mimic a scenario with two distinct signal components.

Theory
In bSSFP, for single component systems, a near zero signal is observed when the RF phase increment matches the phase accumulated between two successive RF excitation pulses due to the local off-resonance frequency 1 .This near-zero signal is responsible for the commonly observed banding artefacts in bSSFP.In multicompartment systems, asymmetric profiles were suggested to originate from convolution between the local frequency distribution and the respective bSSFP profiles 21,22 .Therefore, the final combined profile is an integration of all off-resonant profiles within a voxel, which can be understood using the superposition principle.It can be argued that a near-zero signal obtained with a specific RF phase increment might be the consequence of superposition rather than off-resonance, arising from destructive interference of signals from different tissue types, such as water and fat, within the same imaging voxel.This section aims to describe the destructive interference behavior of bSSFP in multicompartment systems with distinct signal components.

Phase-cycled bSSFP signal model
The signal equation at time  = 0 + directly after the RF excitation pulse of a bSSFP sequence can be written 2,18,20 in the rotating coordinate frame as ) , [1]     with  the RF phase increment and  the accumulated phase as defined in 18 , which is a phase directly proportional to local off-resonance i.e., chemical shift   and magnetic field inhomogeneity  0 .Unless specified, right-handed coordinate systems are used with The constants of Eq. [1] are defined 2,18,20 as  =  0 (1 −  ) is the relaxation term describing the longitudinal (T1) and transverse (T2) relaxation time ( = 1,2) with repetition time , RF excitation angle  and the thermal equilibrium magnetization  0 .Eq. [2] generates the phase factors of Eq. [1] as cos() ± i • sin() = ⅇ ±i which inverts phase signs in a left-handed coordinate system ⅇ ±i → ⅇ ∓i .Switching between left-and right-handed coordinate systems can be achieved by complex conjugation.

Definition of accumulated phase and phase evolution
Using the accumulated phase definition , the phase evolution for bSSFP signals at timepoint  can be described as: To identify the accumulated phase, Eq. [3] can be compared with the solution of the Bloch equation 23 with the main magnetic field along the z-direction () =  0   and for positive gyromagnetic ratios  24 .In rotating coordinate frames, only residual magnetic field inhomogeneity will remain.Hence, the exponent of Eq. [4] can be replaced by  0 →  0 , for isochromatic spins systems.For more than one type of isochromatic spins in a system corresponding to different chemical shifts   , the rotating coordinate frame corresponds to  0 → ( 0 +    0 ) 17,25 .By comparing Eqs.[3,4] the accumulated phase yields  = −( 0 +    0 ) [5].
A higher   value indicates a lower shielding effect of the respective chemical species for their respective nuclei against the  0 field 25 .
At this point, a first look at phases and their respective signs can be performed: • While the minus sign in Eq. [5] follows from the Bloch equations, reconstructed MRI data not necessarily rely on that convention and sometimes ignore it.This merely reflects a choice regarding the handedness of the underlying coordinate system.
• Regardless of the chosen coordinate system, the steady state [1] must depend on the difference  −  of accumulated phase  and RF phase increment , since changing both by the same amount must not modulate the steady-state amplitude.Getting this wrong may, depending on the situation, also affect quantitative analysis.In this Note, the possible implications of this kind of error will not be further pursued with the assumption that the difference is used.

2
. Apart from banding artifacts occurring at  ≈  +  ⋅ 2 (independent of the echo time), this reflects some of the fundamental properties of the bSSFP sequences at  =  2 ⁄ : a) The steady state is 4-periodic with respect to .
b) As can also be seen from the multicompartment solution Eq. [7], below, the signal phase, as a function of  − , is essentially constant (≈  2 ⁄ +  ⋅ ) within any bSSFP plateau (numbered by ) between two subsequent banding artifacts.This is also consistent with the well-known spin-echo like behavior of bSSFP 26 for moderate field inhomogeneity.
Properties (a) and (b) no longer hold if the phase evolution in Eq. [3] is replaced by its complex conjugate ⅇ − i    , since at  =  2 ⁄ this would correspond to a multiplication with ⅇ − i  2 instead of ⅇ i  2 , such that the phase evolution will become . But for a single isochromat with some fixed (and unknown) , this erroneous combination of steady state and phase evolution should not be noticeable, since the factor ⅇ −i would be absorbed in the unknown factor ⅇ i in Eq. [1].As a consequence, quantitative relaxometry, based on several bSSFP acquisitions with different RF phase increments   , should not be affected by conjugating the phase evolution in Eq. [3], as long as the tissue is pure 27 .
In the following, it will be investigated whether this still holds, when the tissue of interest is composed of several components with different   .To this end, Eq. [3] is multiplied with ⅇ i   𝜗 , such that a comparison can be made between the correct ( = 0) and the erroneous ( = −1) evolution.
Unlike pure tissues considered before, the outcome depends qualitatively on whether the correct ( = 0) phase evolution is used or not ( = −1): For  = 0, the sum in [7] is real and the total phase can only take two opposed values:  +  2 ⁄ , if the sum is positive and  +  2 ⁄ +  otherwise.Due to the periodicity of each addend, the sum at any  and  + 2 must be of equal magnitude but of opposite sign.Therefore, there must be values , where the signal  tot (, ), becomes zero.Note that this is not caused by banding artifacts associated with each component, but rather due to destructive interference.This cancellation must also be observable in more complex tissue compositions, consisting of three or more components.

II.
For  = −1, the addends become complex due to the factors ⅇ −i  , which will usually vary from component to component.Unlike the pure case, these factors can no longer be dragged out of the sum and absorbed in the global phase factor ⅇ i and are therefore observable: The sum will decompose into a real and imaginary component, each of which will exhibit destructive interference, but in general not for the same .Therefore, the magnitude of  tot (, ) cannot be expected to vanish for any .Also, the phase of  tot (, ) will show a more complicated behavior than the linear dependence on , which was obtained for  = 0.
Profiles were simulated for a 36%, 60% and 100% respectively.For visual comparison the simulated profiles were rotated using the negative angle of the complex sum of the profiles 10 .
0 values and ⅇ i factors (cf.Eqs.[4,5]) where randomized to check profile shapes for invariances to  0 values and global phase factors.

MRI experiments in an acetone-water phantom at 3T
The bSSFP profiles were measured at 3T (MAGNETOM Prisma, Siemens Healthineers) using 18 phase-cycled bSSFP acquisitions with  = were then used to calculate proton densities, as described in 28 .
Because data were acquired in a left-handed coordinate system 29,30 , bSSFP profiles were complex conjugated and then plotted in the (right-handed) complex plane to allow comparison with simulated data.To improve visual comparison, the bSSFP profiles were rotated using the negative angle of their complex sum before plotting.

MRS experiments at 7T
Localized single voxel MRS was performed on the phantom vials containing   = 36% and   = 60% acetone to verify the chemical shifts.For MRS, a STEAM 31 sequence was used at 7T (MAGNETOM Terra, Siemens Healthineers) with a // = 5/20/13, 32 averages, a voxel of 10x10x10mm3 , a spectral width of 5000Hz and 4096 data points.Spectra were processed with JMRUI 32 and fitted using AMARES 33 to obtain the chemical shifts.

Simulation results:
For single components, the elliptical shape of the bSSFP profile is independent of the phase definition of  = 0 and  = −1 and for randomized  0 values and global phase factors ⅇ i (Figure 1).Different ⅇ i factors led to a rotation of the profile in the complex plane.Note that this rotation is not visible because the profiles were rotated during post-processing.Changing the  0 values rotate the sampled points on the elliptical trajectories, while leaving the trajectories themselves unchanged.
Superposition was applied to obtain two-component profiles of water and acetone (Figure 2).For TR=3.62ms the phase difference of the water (Figure 2D) and acetone (Figure 2E) profile was 180°, resulting in a combined profile that is symmetric, appearing as a single component profile.
Two-component simulations demonstrate that different phase definitions ( = 0 or  = −1) lead to significant different profile shapes (Figure 3 and 4), appearing either ribbon shaped, or heart shaped.Complex conjugation leads to mirroring of the profile shape (Figure 3 and 4, columns 1 and 2), which implies that a 60% acetone fraction profile with  = 0 (Figure 3M,P, dashed box) would appear similar to a ~40% acetone fraction profile that is complex conjugated (Figure 4J, dotted box).
For the  = −1 case, destructive interference was observed for TR values close to 4.84ms (Figure 4S), and also for TR=3.62ms(Figure 3G).For other TR choices, the asymmetric profiles are appearing heart-shaped centered around origin, without destructive interference (Figure 3 and 4 third column), confirming theory.
For  = 0, destructive interference is present for all TR values (Figure 3 and 4 column 1 and 2), as expected according to theory.

MRI experiments
In the 100% acetone vial, which mimics a single component system, the bSSFP profiles are symmetric and elliptical (Figure 1).In a few of these bSSFP profiles, a gap can be observed within the ellipse, indicating B0-drift (Figure 1T and 3T, red arrow).
In vials containing acetone-water fractions of 36% and 60%, the bSSFP profiles appear ribbon-shaped (Figure 3 and 4, last column) and are similar to profiles simulated using the  = 0 case.For this case, for every TR, the observed destructive interference matches predictions (Figure 3 and 4, first column).In these experimental bSSFP profiles, the absence of destructive interference was never observed, which contradicts with the predicted profiles in the  = −1 case.
Interestingly, the bSSFP profiles obtained in the 60% fraction (Figure 3P) and the 36% fraction (Figure 4L), appear similar, but mirrored from one another.These results show that knowledge of the coordinate handedness is mandatory for a correct evaluation of tissue fractions based on experimental bSSFP profiles (cf. Figure 3 and

Figure 1.
Simulated and experimental bSSFP signal profiles obtained for a 100% acetone solution.All profiles remain elliptical for different repetition times.The red arrow indicates B0 drift effects.Banding is observed at the origin of the complex plots where the signal magnitude is near zero.
. acetone-water mixture (green) using the correct phase sign case of  = 0. Simulations were performed for different TR values.The RF phase increments are indicated in the plots in degrees (°).For TR=3.62, the acetone and water profiles are shifted 180 degrees, and the combined profile appears as a single component profile.Banding is observed at the origin of the complex plane of the single component profiles where the signal magnitude is near zero.Destructive interference is observed at the origin of the complex plane of the superimposed profile, where the signal magnitude is near zero.

Discussion
This work demonstrated the importance of correct phase descriptions (referring to  = 0 in Eq. [7]) for bSSFP profiles in multicompartment systems, which was shown theoretically and confirmed by simulations and experiments.In contrast to symmetric profiles, which are typically observed in single component systems, an incorrect phase description can lead to erroneous predictions of asymmetric profiles for multicomponent and compartment systems, which was tested and validated using acetonewater mixtures.Furthermore, to quantify component fractions correctly, knowledge about coordinate handedness of the data acquisition, which typically depends on the vendor, is crucial.
In single component systems, using either phase description has no effect on the bSSFP profile shapes, because such profiles are not subject to superposition and are thus symmetric.The shape is not affected by the changing coordinate handedness.
However, for two-compartment systems there is a clear correct ( = 0) and incorrect ( = −1) way to combine the steady state signals and subsequent phase evolution.
The combination of two single-component profiles, using superposition, leads to significantly different bSSFP profile shapes depending on the chosen phase description in Eq. [7].For the case where  = 0, destructive interferences were consistently observed, and for the case  = −1, they were mostly absent.Therefore, being inconsistent with the phase leads to wrong bSSFP profile shapes and is therefore likely to cause systematic errors.
In addition, the choice of the coordinate handedness has an effect on the "chirality" of the predicted profile, which is crucial for the correct estimation of tissue fractions.
The use of right-or left-handed coordinate systems is not interchangeable because it causes a mirroring of the asymmetric profile and therefore corresponds to a different acetone-water fraction.In essence, this could be viewed as multiplying the underlying chemical shifts with -1, or as a straightforward tissue swap.For example, a complex conjugated profile of 60% acetone-water fraction corresponds to a non-complex conjugated profile of a 40% acetone-water fraction.
In experimental data, the bSSFP profiles of 60% acetone-water appear as single component profiles for TR=3.62ms.This is expected because for certain TR values and chemical shifts, the individual bSSFP profiles of the two components may be shifted by 180 ° or 0 ° relative to each other, leading to two-component profiles which are visually perceived as elliptical single-component profiles.Therefore, it would be possible to estimate the chemical shift difference between water and acetone using bSSFP profile measurements.A rough chemical shift estimation based on bSSFP profiles agreed well with chemical shifts obtained with MRS.The observation that the chemical shift difference between acetone and water depends on the acetone-water fractions suggests that these types of mixtures are not well suited to validate tissue fraction quantification techniques such as those performed with Dixon techniques 34,35 , unless verified with spectroscopy.
In summary, working with a consistent phase description ( = 0) is crucial to correctly describe multicomponent and multicompartment bSSFP profiles, specifically to avoid possible systematic errors in the analysis of bSSFP profile asymmetries e.g. at higher field strengths 27 , for T1 and T2 quantification in gray matter, white matter and in muscles 9,11,21,22 , fat-water separation techniques 4,5,36 and fat quantification techniques using phase-cycled bSSFP 14 .

Conclusion
The study demonstrates the vital importance of a consistent and correct treatment of phases in bSSFP signal descriptions of multi-compartment systems.The use of incorrect phase descriptions leads to erroneous bSSFP profiles as well as erroneous quantitative analysis of multi-compartment systems, which has been demonstrated both theoretically and experimentally.

Figure 3 .
Figure 3. Simulated and experimental bSSFP signal profiles obtained for a 60%