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Hybrid-State Free Precession for Measuring Magnetic Resonance Relaxation Times in the Presence of B0 Inhomogeneities

Vladimir A. Kobzar¹, Carlos Fernandez-Granda², and Jakob Asslaender³

¹New York University Center for Data Science, New York University, New York, NY, United States, ²Courant Institute of Mathematical Sciences and New York University Center for Data Science, New York University, New York, NY, United States, ³Center for Biomedical Imaging, Dept. of Radiology, and Center for Advanced Imaging Innovation and Research, New York University, New York, NY, United States

Synopsis

Magnetic resonance fingerprinting is a methodology for the quantitative estimation of the relaxation times T_1,2. An important challenge is to make estimation robust to inhomogeneities of the main magnetic field B₀. Free precession sequences with smoothly varying parameters, such as balanced hybrid-state free precession (bHSFP) sequence, can be optimized for T_1,2-encoding performance. Previously, magnetic field deviations were assumed to be determined by a separate experiment. Here we develop a numerically optimized bHSFP sequence that takes into account variations in B₀ with the aim of mitigating bias due to B₀ inhomogeneities. Our numerical results indicate that this approach yields accurate T_1,2 estimates when B₀ inhomogeneities are unknown.

Introduction

In magnetic resonance fingerprinting (MRF) [1], the relaxation times T₁ and T₂ are determined quantitatively by matching the evolution of magnetization signal to a precomputed dictionary of patterns or "fingerprints" of tissues. However, inhomogeneities of the magnetic fields corrupt these estimates, producing for example so-called banding artifacts in the case of sequences with balanced gradient moments [1],[2]. The purpose of this work is to develop a numerically optimized fingerprinting sequence based on the hybrid state framework [2,3] that incorporates B₀ estimation and mitigates bias due to B₀ inhomogeneities. In particular, the approach succeeds in suppressing banding artifacts.

Theory

In reference [2], it was shown that slow flip angle variations lead to a so-called "hybrid state", which allows us to solve the Bloch equation analytically in spherical coordinates. In this state, the entire spin dynamics is captured by the radial component

$r$ of the magnetization, which is controlled by the polar angle

$\vartheta$ . The polar angle

$\vartheta$ can be approximated [2],[4] by:

$\sin^2\vartheta=\frac{\sin^2\frac{\alpha}{2}}{\sin^2\frac{\phi}{2}\cdot\cos^2\frac{\alpha}{2}+\sin^2\frac{\alpha}{2}}$ where

$\alpha$ denotes the flip angle (except in the vicinity of so-called stop bands, which are given by

$|\sin \phi | \ll 1$ ). The phase

$\phi = \phi^{nom} + \gamma\Delta B_0 T_R$ is composed of the phase increment of the radio-frequency (RF) pulses

$\phi^{nom}$ and the phase accumulated during the time

$T_R$ between pulses due to inhomogeneities in the main magnetic field. The gyromagnetic ratio is denoted by

$\gamma$ . Expressing the magnetization in terms of the control parameters and field inhomogeneities makes it possible to optimize the sequence with respect to the T₁ and T₂ parameters, even if

$\Delta B_0$ is unknown.

Methods

The Cramer-Rao bound [5],[6] is a lower bound on the error of any unbiased estimator of a parameter of interest. It is, therefore, a useful proxy for the sensitivity of the data with respect to the parameter, which can be minimized to optimize the measurements [7],[8],[9]. Here we follow such an approach to search for an efficient balanced hybrid-state free precession (bHSFP) sequence with anti-periodic boundary conditions (defined by $r(0)=-r(T_C)$ , where $T_C$ is the duration of one cycle of the pulse sequence [2]). We use the relative Cramer-Rao bound ( $rCRB$ ), normalized by the duration of the experiment, as figure of merit and assume that the signal depends on unknown $\Delta B_0$ variation parameters (as well as the magnetization and the relaxation times). We optimize an $\alpha$ pattern using the Broyden-Fletcher-Goldfarb-Shanno algorithm. In the most general approach, one would optimize $\phi^{nom}$ along with the flip angle pattern. However, we found this approach to show poor convergence behavior. Instead, we implicitly enforce equivalent encoding at all $\Delta B_0$ offsets (within the limits of the RF-hardware) by sweeping through $\gamma\Delta B_0 T_R \in [0, 2\pi]$ while repeating the same $\alpha$ pattern. For consistency with hybrid state conditions [2], the changes in the flip angles in consecutive RF pulses $\Delta \alpha$ were constrained by

$| \Delta \alpha | \leq \max \left \{\sin^2 \frac {\alpha}{2} - \frac {5}{2} \left(1 - \sqrt E_2 \right)^2,0 \right \} \label {eq:dalphaconstr}$

where $E_2 = \exp (-T_R/T_2)$ .

Results and Discussion

The optimized bHSFP sequences exhibit smooth

$\vartheta$ and

$\alpha$ patterns (Figure 1), and the

$\vartheta$ pattern is similar to one found when neglecting field inhomogeneities (

$\Delta B_0 = 0$ ) [2],[3]. Also when plotting the optimized

$rCRB$ as a function of

$T_C$ (Figure 2), we observe that the optimal duration

$T_C = 3.8$ s is also comparable to the ones found in literature for the idealized case. Lastly, Figure 3 indicates that the sequences optimized for unknown

$B_0$ have similar

$rCRB$ as the sequences optimized for known

$B_0$ variations. (Using the latter sequence in a setting when

$B_0$ variations are unknown would have resulted, however, in a significantly higher

$rCRB$ .) Moreover, the horizontal lines disappear in Figure 3, which indicates that

$T_1$ and

$T_2$ can be estimated without banding artifacts.

Conclusion and Outlook

Our results indicate that incorporating

$B_0$ variations while designing bHSFP sequences for magnetic resonance fingerprinting is a promising avenue to achieve robust estimates of

$T_1$ and

$T_2$ in the presence of

$B_0$ inhomogeneities. Future work will include validation on phantom and in vivo scans. In addition, we will extend our methodology to account for variations in the RF field

$B_1$ .

Acknowledgements

V.A.K. is supported by the Moore-Sloan Data Science Environment at New York University. C.F. is supported by NSF award DMS-1616340. J.A. is supported by the research grants NIH/NIBIB R21 EB020096 and NIH/NIAMS R01 AR070297, and this work was performed under the rubric of the Center for Advanced Imaging Innovation and Research (CAI2R, www.cai2r.net), a NIBIB Biomedical Technology Resource Center (NIH P41 EB017183).

References

[1] Dan Ma, Vikas Gulani, Nicole Seiberlich, Kecheng Liu, Jeffrey L. Sunshine, Jeffrey L. Duerk, and Mark A. Griswold. Magnetic resonance fingerprinting. Nature, 495(7440):187–192, 2013.

[2] Jakob Asslaender, Riccardo Lattanzi, Daniel K Sodickson, and Martijn A. Cloos. Hybrid state free precession in nuclear magnetic resonance. arXiv:1807.03424 [physics.med-ph], 2018.

[3] Jakob Asslaender, Daniel K Sodickson, Riccardo Lattanzi, and Martijn A Cloos. Hybrid State Free Precession for Measuring Magnetic Resonance Relaxation Times. In Proc. Intl. Soc. Mag. Reson. Med, 2018.

[4] Ray Freeman and H. D W Hill. Phase and intensity anomalies in fourier transform NMR. J. Magn. Reson., 4(3):366–383, 1971.

[5] Calyampudi Radhakrishna Rao. Information and the Accuracy Attainable in the Estimation of Statistical Parameters. Bull. Calcutta Math. Soc., 37(3):81–91, 1945.

[6] Harald Cramer. Methods of mathematical statistics. Princeton University Press, Princeton, NJ, 1946.

[7] J.A. Jones, P. Hodgkinson, A.L. Barker, and P.J. Hore. Optimal Sampling Strategies for the Measurement of Spin–Spin Relaxation Times. J. Magn. Reson. Ser. B, 113(1):25–34, 1996.

[8] J. A. Jones. Optimal sampling strategies for the measurement of relaxation times in proteins. J. Magn. Reson., 126(126):283–286, 1997.

[9] Bo Zhao, Justin P Haldar, Kawin Setsompop, and Lawrence L Wald. Optimal Experiment Design for Magnetic Resonance Fingerprinting. In Eng. Med. Biol. Soc. (EMBC), IEEE 38th Annu. Int. Conf., number 1, pages 453–456, 2016.

Figures

Fig. 1: The dynamics in (a) result from optimizing an α sequence of length T_C = 3.8 s where Δ B₀ is assumed to be 0 (cf. Fig 4 in [2] ), subject to 0 ≤ϑ ≤π/4. The dynamics in (b) result from optimizing the same sequence concatenated 96 times assuming Δ B₀ variations are unknown. In each period of length T_C a different φ^nomis introduced, with φ^nomuniformly distributed between -π and π. α was constrained to 0 ≤α ≤π/2, and its changes were limited according with (1). The blue ellipse indicates the steady state of balanced steady-state free precession sequences.

Fig. 2: The bHSFP trajectories described by the relaxation of the Bloch Equations are optimized for different lengths (T_C) of the flip angle α sequences. The rCRB values in the scenario where B₀ variations are unknown indicate that T_C = 3.8 s is optimal, as marked by the vertical bar. (These bounds result from a nonconvex optimization where the parameter space grows with T_C and convergence issues become apparent for larger T_C.) The rCRB in the scenario where B₀ variations are known is also depicted and the optimal T_C = 3.8 s is consistent with prior work (cf. Fig. 3 in [2]).

Fig. 3: The performance of the optimized experiments is illustrated through plots of the rCRB values, which provide a lower bound for the noise in the retrieved relaxation times. The performance is illustrated as a function of the magnetic field, which is parameterized by the main magnetic field variation Δ B₀, and by the RF field variation B₁. During the optimization, the nominal values of the inhomogeneities were fixed, as shown by the red squares / red lines. The experiments have T_C = 3.8 s (cf. Fig. 2 above). (The rCRB color bar is in log scale.)

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)

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Synopsis

p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica; color: #000000} Introduction

Theory

Methods

p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica; color: #000000} Results and Discussion

p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica; color: #000000} Conclusion and Outlook

Acknowledgements

References

Figures

Introduction

Results and Discussion

Conclusion and Outlook