Victor Han^{1}, Jianshu Chi^{1}, and Chunlei Liu^{1,2}

^{1}Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, United States, ^{2}Helen Wills Neuroscience Institute, University of California, Berkeley, Berkeley, CA, United States

When multiple RF fields are applied at different frequencies, multiphoton excitation can occur when the sums or differences of integer multiples of these frequencies equal the Larmor frequency. No RF at the Larmor frequency is required. In this work, we describe the general principles of multiphoton pulsed selective excitation, providing a formalized treatment with design examples and implementations on a 3T scanner with an additional homemade z-direction (B_{z}) coil. With the additional B_{z} coil, we demonstrate additional flexibility, where the same excitation can be accomplished in several different ways with the same pulse duration.

In this work, we describe the general principles of multiphoton pulsed selective excitation, providing a formalized treatment with design examples and implementations on a 3T scanner with an additional homemade z-direction ($$$B_z$$$) coil. With the additional $$$B_z$$$ coil, we demonstrate additional flexibility, where the same excitation can be accomplished in several different ways with the same pulse duration.

$$\begin{bmatrix}\dot{M_x}\\\dot{M_y}\end{bmatrix}=\gamma\begin{bmatrix}0&B_z(t)\\-B_z(t)&0\end{bmatrix}\begin{bmatrix}M_x\\M_y\end{bmatrix}+\gamma\begin{bmatrix}-B_y(t)M_0\\B_x(t)M_0\end{bmatrix}.\:\:\:[1]$$

Denote the transverse magnetization and magnetic field as

$$m_{xy}=M_x+iM_y,\:\:\:[2]$$

$$B_{xy}=B_x+iB_y.\:\:\:[3]$$

For pulsed B fields over the time period from 0 to T, the solution to Eq. [1] is given by

$$m_{xy}(\textbf{r},T)=i\gamma M_0\int_0^TB_{xy}(\textbf{r},t)e^{-i\gamma\int_t^TB_z(\textbf{r},\tau)d\tau}dt.\:\:\:[4]$$

In the Larmor frequency rotating frame, let us define

$$B_z=B_{1,z}(t)\cos{(\omega_zt+\phi)}+\textbf{G}\cdot\textbf{r},\:\:\:[5]$$

$$B_{xy}=B_{1,xy}(t)e^{-i((\omega_{xy}-\omega_0)t+\theta(t))}.\:\:\:[6]$$

These are the typical fields in MRI, except with the addition of a uniform RF field in the z-direction for multiphoton excitation. When the frequency of the xy-RF is far off from the Larmor frequency, but satisfies the multiphoton resonance condition $$$\omega_{xy}-\omega_0=n\omega_z$$$

$$m_{xy}(\textbf{r},T)=i\gamma M_0\int_0^TB_{1,xy}(t)e^{-i(n\omega_zt+\theta(t))}e^{-i\gamma\int_t^TB_{1,z}(\tau)\cos{(\omega_z\tau+\phi)}d\tau}e^{i\textbf{k}(t)\cdot\textbf{r}}dt,\:\:\:[7]$$

where $$$-\gamma\int_t^T\textbf{G}(\tau)d\tau=\textbf{k}(t)$$$ as in excitation k-space

If $$$B_{1,z}(\tau)$$$ is slowly varying compared to $$$\cos{(\omega_{1,z}\tau+\phi)}$$$, then the integral of their product is approximately the product of $$$B_{1,z}(\tau)$$$ and the integral of $$$\cos{(\omega_{1,z}\tau+\phi)}$$$. With this assumption, and using the Jacobi-Anger expansion shown below, where $$$J_m(-)$$$ is the Bessel function of the first kind of order m,

$$e^{i\frac{\gamma B_{1,z}}{\omega_z}\sin{(\omega_zt+\phi)}}=\Sigma_{m=-\infty}^\infty J_m\left(\frac{\gamma B_{1,z}}{\omega_z}\right)e^{im(\omega_zt+\phi)},\:\:\:[8]$$

Eq. [7] can be rewritten as

$$m_{xy}(\textbf{r},T)\approx i\gamma M_0\int_0^TB_{1,xy}(t)e^{-i(n\omega_zt+\theta(t))}\left(\Sigma_{m=-\infty}^\infty J_m\left(\frac{\gamma B_{1,z}(t)}{\omega_z}\right)e^{im(\omega_zt+\phi)}\right)e^{-i\frac{\gamma B_{1,z}(t)}{\omega_z}\sin{(\omega_zT+\phi)}}e^{i\textbf{k}(t)\cdot\textbf{r}}dt.\:\:\:[9]$$

Only the term with $$$m=n$$$ contributes significantly to the integral, giving

$$m_{xy}(\textbf{r},T)\approx i\gamma M_0\int_0^TB_{1,xy}(t)e^{-i\theta(t)}J_n\left(\frac{\gamma B_{1,z}(t)}{\omega_z}\right)e^{-i(\frac{\gamma B_{1,z}(t)}{\omega_z}\sin{(\omega_zT+\phi)}-n\phi)}e^{i\textbf{k}(t)\cdot\textbf{r}}dt.\:\:\:[10]$$

Eq. [10] shows that $$$B_{xy}$$$ and $$$B_z$$$ contribute to the excitation profile in a similar way with $$$B_{1,xy}(t)$$$ and $$$J_n\left(\frac{\gamma B_{1,z}(t)}{\omega_z}\right)$$$ available for amplitude modulation and $$$e^{-i\theta(t)}$$$ and $$$ e^{-i(\frac{\gamma B_{1,z}(t)}{\omega_z}\sin{(\omega_zT+\phi)}-n\phi)}$$$ available for phase modulation. This contrasts with the standard one-photon case where we would just have $$$m_{xy}(\textbf{r},T)\approx i\gamma M_0\int_0^TB_{1,xy}(t)e^{-i\theta(t)} e^{i\textbf{k}(t)\cdot\textbf{r}}dt$$$.

1. Generate a prototype pulse using a conventional method like the SLR algorithm

2. If designing a standard one-photon pulse, directly set $$$B_{xy}$$$ to the prototype pulse and finish.

3. Else if designing a multiphoton pulse, choose $$$\omega_z$$$.

- Based on Eq. [10], choose values such that $$$B_{1,xy}(t)J_n\left(\frac{\gamma B_{1,z}(t)}{\omega_z}\right)$$$ equals the amplitude modulation of the prototype pulse, and $$$e^{-i\theta(t)}e^{-i(\frac{\gamma B_{1,z}(t)}{\omega_z}\sin{(\omega_zT+\phi)}-n\phi)}$$$ equals the frequency modulation of the prototype pulse. For multiphoton excitation, we have more variables to choose which can together achieve the same effects as in the one-photon case.
- Shift the center frequency of the $$$B_{xy}$$$ pulse by $$$n\omega_z$$$.

Base SLR prototype pulses were generated using SigPy.RF

Using the same one-photon and two-photon pulses as in Fig. 2, Fig. 5 shows the in-plane results of the one-photon and two-photon pulses in vivo under our institution's IRB approval. No significant differences between the images are observed for this set of parameters.

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DOI: https://doi.org/10.58530/2022/0451