Bahman Tahayori^{1,2}, Zhaolin Chen^{2}, Gary Egan^{2}, and N. Jon Shah^{1,2,3}

An approximate analytic expression is calculated for the phase of the transverse magnetisation during the excitation period. A spherical Bloch equation is used to find this expression. Simulation results are in agreement with the analytic solution. This analytic solution can be used where the phase information is required and, therefore, may be used to improve the performance of imaging through optimal pulse sequence design.

Assuming the Radio Frequency (RF) field is applied in the x'-direction, the Bloch equation during excitation for short pulses in the spherical coordinates for azimuth and colatitude components of the magnetisation vector in the rotating frame of reference can be written as^{5}

$$ \left\{ \begin{array}{l l} \dot{\theta} = -\Delta \omega_0+\omega_1(t) \cot \phi \cos \theta\\ \dot{\phi} = \omega_1(t) \sin\theta\\ \end{array} \right.$$

where $$$\Delta \omega_0$$$ is the filed inhomogeneities including gradient fields, $$$\omega_1(t) $$$ is the excitation field, $$$\theta = \tan^{-1}\frac{M_{y'}}{M_{x'}}$$$ and $$$\phi = \cos^{-1}M_{z'}$$$ in which $$$M_{x'}$$$, $$$M_{y'}$$$ and $$$M_{z'}$$$ are the magnetisation components in the Cartesian coordinates. We have used available tools in nonlinear dynamical systems^{6,7} to find the phase of the magnetisation as follows

$$\theta (t) \approx \frac{\pi}{2} - \Delta\omega_0\,t +\tan^{-1}\left(\cot\int_0^t\omega_1(\tau) \cos(\Delta\omega_0\tau)\, d\tau \,\,\tanh \int_0^t\omega_1(\tau) \sin(\Delta\omega_0 \tau)\,d\tau \right).$$

The above expression is valid for hard as well as soft RF pulse excitations. Furthermore, if field inhomogeneities are time varying, e.g. time varying gradient fields, the above equation applies by replacing $$$\Delta\omega_0 \tau$$$ with $$$\int_0^\tau \Delta\omega_0(s) ds$$$.

1. D.I. Holt, "The solution of the Bloch equations in the presence of a varying B1 field - An approach to selective pulse analysis," Journal of Magnetic Resonance, vol. 35, pp. 69-86, 1979.

2. J. Pauly, D. Nishimura, and A. Macovski, "A k-space analysis of small tip-angle excitation," Journal of Magnetic Resonance, vol. 81, pp. 43-56, 1989.

3. N. Boulant and D.I.Hoult, "High Tip Angle Approximation Based on a Modified Bloch-Riccati Equation," Magnetic Resonance in Medicine, vol. 67, pp. 339-343, 2012.

4. J. Zhang, M. Garwood, and J.Y. Park, "Full Analytical Solution of the Bloch Equation when Using a Hyperbolic-Secant Driving Function," Magnetic Resonance in Medicine, DOI: 10.1002/mrm.26252, 2016.

5. B. Tahayori, L.A. Johnston, I.M.Y. Mareels, and P.M. Farrell, "Novel insight into magnetic resonance through a spherical coordinate framework for the Bloch equation," in Proceedings of the SPIE Conference on Medical Imaging, Physics of Medical Imaging, pp. 72580Y1-72580Y12, Orlando, USA, Feb. 2009.

6. J. Sanders, F. Verhulst, and J. Murdock, *Averaging Methods in Nonlinear
Dynamical Systems.*
Springer-Verlag, 2007.

7. H. Khalil, *Nonlinear Systems, *3rd
ed. Prentice Hall, 2002

8. K. Shmueli , J.A.de Zwart, P. van Gelderen , T.Q. Li, S.J. Dodd, J.H. Duyn, "Magnetic susceptibility mapping of brain tissue in vivo using MRI phase data," Magnetic Resonance in Medicine, vol. 62, pp. 1510–1522, 2009.

9. E.M. Haacke, S. Liu, S. Buch, W. Zheng, D. Wu, and Y. Ye, "Quantitative susceptibility mapping: current status and future directions," Magnetic Resonance Imaging, vol. 33, pp. 1–25, 2015.

10. S.D. Robinson, B. Dymerska, W. Bogner, M. Barth, O. Zaric, S. Goluch, G. Grabner, X. Deligianni, O. Bieri, and S. Trattnig, "Combining phase images from array coils using a short echo time reference scan (COMPOSER)," Magnetic Resonance Imaging, DOI: 10.1002/mrm.26093, 2015.

11. M. Garwood, "MRI of fast-relaxing spins," *Journal of Magnetic Resonance, * vol. 229, pp. 49-54, 2013.

Fig1. Phase of magnetisation during a $$$\pi/4 $$$ soft pulse excitation for (a) $$$\Delta\omega_0 = 10^2$$$rad/s, (b) $$$\Delta\omega_0 = 10^3$$$rad/s, (c) $$$\Delta\omega_0 = 10^4$$$rad/s and (d) $$$\Delta\omega_0 = 10^5$$$rad/s.,

Fig2. Phase of magnetisation during a $$$\pi/2 $$$ soft pulse excitation for (a) $$$\Delta\omega_0 = 10^2$$$rad/s, (b) $$$\Delta\omega_0 = 10^3$$$rad/s, (c) $$$\Delta\omega_0 = 10^4$$$rad/s and (d) $$$\Delta\omega_0 = 10^5$$$rad/s.

Fig3. Spherical components of the magnetisation vector for a $$$\pi/4$$$ selective pulse, (a) azimuth component, (b) colatitude component calculated analytically and numerically.

Fig4. Spherical components of the magnetisation vector for a $$$\pi/2$$$ selective pulse, (a) azimuth component, (b) colatitude component calculated analytically and numerically.

Fig5. The error between the analytic and numerical solution of the azimuth component of the Bloch equation for (a) a $$$\pi/4$$$ selective pulse and (b) a $$$\pi/2$$$ selective pulse.