Bahman Tahayori^{1,2,3}, Zhaolin Chen^{2}, Gary Egan^{2}, and N. Jon Shah ^{4}

We have applied Volterra series expansion to the Bloch equation and have calculated the kernels for an arbitrary initial condition. We have shown that small tip angle approximation can be extended to the non-equilibrium initial condition. Simulation results illustrated the validity of the extended small tip angle approximation.

A linear system can be described completely by its impulse response. Volterra series extends the impulse response to nonlinear systems through an infinite sum of higher order convolution integrals using Volterra kernels [4,5]. Bilinear systems have convergent Volterra series expansion and the Volterra kernels can be calculated using matrix exponentials [5]. The Bloch equation, neglecting the relaxation effects, when the excitation is applied in the $$$x$$$-direction is written as

$$\dot{ \textbf{M}}= \left[\begin{array}{c}\dot{M}_{x'}\\\dot{M}_{y'}\\\dot{M}_{z'} \end{array}\right] = \left[ \begin{array}{ccc} 0& \Delta \omega & 0 \\ -\Delta
\omega & 0 & \omega_{1}(t) \\ 0 & -\omega_{1}(t)
& 0 \end{array} \right] \left[\begin{array}{c} {M}_{x'}\\{M}_{y'}\\{M}_{z'}
\end{array}\right],$$

where $$$\dot{ \textbf{M}}$$$ is the magnetisation vector, $$$\omega_1$$$ is the excitation and $$$\Delta\omega$$$ represents the off-resonance frequencies that can be generated by gradient fields. Assuming the magnetisation is initially at $$$\textbf{M}^0 = [M_x^0 \quad M_y^0 \quad M_z^0]^T$$$, the zeroth and the first kernels for the Bloch equation are

$$h_0 (t) = \left(\begin{array}{c} {M_x^0}\, \cos\!\left(\Delta{}\omega{}\, t\right) +{M_y^0}\, \sin\!\left(\Delta{}\omega{}\, t\right)\\ {M_y^0}\, \cos\!\left(\Delta{}\omega{}\, t\right) - {M_x^0}\, \sin\!\left(\Delta{}\omega{}\, t\right)\\ {M_z^0} \end{array}\right),$$

$$h_1 (t,\tau_1)= \left(\begin{array}{c} {M_z^0}\, \sin\!\left(\Delta{}\omega{}\, \left(t - {\tau{}}_{1} \right)\right)\\ {M_z^0}\, \cos\!\left(\Delta{}\omega{}\, \left(t-{\tau{}}_{1} \right)\right)\\ {M_x^0}\, \sin\!\left(\Delta{}\omega{}\, {\tau{}}_{1}\right) - {M_y^0}\, \cos\!\left(\Delta{}\omega{}\, {\tau{}}_{1}\right) \end{array}\right).$$

To illustrate the validity of the theoretical results, we compared the approximation with the numerical solution of the Bloch equation for the following cases

**Forward problem: **For a rectangular pulse excitation, we have calculated the $$$M_z$$$ profile when $$$\textbf{M}^0 = [ 1 \quad 0 \quad 0]^T$$$ and $$$\textbf{M}^0 = [ 0 \quad 1\quad 0]^T$$$. The resultant $$$M_z$$$ profiles for the above initial conditions using the Volterra series kernels are

$$M_z(\Delta\omega, \tau_p) = \mathcal{F}_s\{\omega_1(t)\} = \frac{\omega_1(1-\cos \Delta\omega\tau_p)}{\Delta\omega},$$

and

$$M_z(\Delta\omega, \tau_p) = \mathcal{F}_c\{\omega_1(t)\} = -\frac{\omega_1 \sin \Delta\omega\tau_p}{\Delta\omega},$$

repectively. In the above equations, $$$\omega_1$$$ is the pulse amplitude and $$$\mathcal{F}_s$$$ and $$$\mathcal{F}_c$$$ are Fourier sine and cosine transforms, respectively. The numerical solution of the Bloch equation as well as the presented approximate solutions are shown in Figure 1. These results show that the approximate solution is in agreement with the numerical solution.

**Backward problem:** Assuming the magnetisation is initially at $$$\textbf{M}^0 = [ 0 \quad 1\quad 0]^T$$$, we designed a slice selective pulse to rotate the magnetisation for $$$\pi/6$$$ about the $$$x$$$-axis. Given that the Fourier cosine transform of a rectangular function is a half a sinc function, the excitation that tips the bulk magnetisation from the $$$+y$$$-axis is a truncated half a sinc. The excitation pattern, as well as the resultant magnetisation profile are represented in Figure 2. Similar to the conventional Fourier method, to obtain a refocused slice, the gradient was reversed for half the time of the pulse duration with the same magnitude.

1. J. Pauly, D. Nishimura, and A. Macovski, "A k-space analysis of Small-tip-angle excitation," *Magnetic Resonance in Medicine, *vol. 81, pp. 43-56, 1989.

2. D. Nishimura, *Principles of Magnetic Resonance Imaging, *Stanford University, 1996.

3. M. Bernstein, K. King, and X. Zhou, *Handbook of MRI pulse sequences, * Elsevier, 2004.

4. L. Carassale and A. Kareem, "Modeling nonlinear systems by Volterra series,"* Journal of Engineering Mechanics, *vol. 136, 2010.

5. R. Brockett, "The early days of geometric nonlinear control," *Automatica, *vol. 50, pp. 2203-2224, 2014.

Figure 1. $$$M_z$$$ profile generated by a rectangular pulse excitation of a 12.8ms duration, (a) when the initial magnetisation was at $$$[1\quad 0 \quad 0]^T$$$ and (b) $$$[0 \quad 1 \quad 0 ]^T$$$. The STA approximation solution is represented by the solid line and the numerical solution of the Bloch equation at the end of the pulse duration is shown by the dashed line.

Figure 2. (a) The designed pulse to tip the bulk magetisation from $$$[0 \quad 1 \quad 0 ]^T$$$ for $$$\pi/6$$$ and (b) the resultant slice profile. The gradient was switched on post excitation in the reverse direction to achieve a refocused slice.