To extend the Shinnar-Le-Roux (SLR) algorithm for designing joint $T_1$, $T_2$, and frequency selective RF pulses.Designing radio frequency pulses is in general a non-linear problem. The SLR transform^1,2 introduces a one-to-one mapping between frequency selective pulses to polynomials and hence allows us to design frequency selective pulses using linear filter design tools. Recently, Logi et al.³ further extends the SLR transform to $T_2$ selective pulses. In this work, we propose a generalization of the SLR transform and map a joint $T_1$, $T_2$ and frequency selective pulse to multi-dimensional polynomials. This enables us to create $T_1$, $T_2$ and frequency selective pulses using convex optimization filter design tools.

For illustration, we focus on the case where $T_1$ can be neglected. Derivation with $T_1$ is shown in figures.

To account for $T_1$ and $T_2$, we consider the standard representation of magnetization $(M_{xy}, M_{z})$ instead of spinors. We assume the hard pulse model (Figure 1), where the RF pulse is parametrized by hard pulse flip angle magnitude $\theta_n$ and phase $\phi_n$ with spacing $\Delta t$ for free precession, $T_2$ decay, and $T_1$ recovery. We define $z_2$ to be $\exp(\Delta t/T_2+i2\pi f\Delta t )$. Then, expanding the forward equation in Figure 1 recursively, we can express the magnetization after the $n$th hard pulse as: $$M_{xy,n}(z_2,z_2^*)=\sum_{i+j\le n}b_n{[i][j]}~z_2^{-i}{z^*_2}^{-j}\\M_{z,n}(z_2, z_2^*)=\sum_{i + j\le n}a_n{[i][j]}~z_2^{-i}{z^*_2}^{-j}\\$$ Moreover, these polynomials satisfy constraints that are invariant to the RF pulse: Since $M_{z,n}$ is real, $a_n[i][j]=a^*_n[j][i]$. The polynomials can also be verified to satisfy: $$\Re[ M_{xy,n}(z_2, z_2^*)M^*_{xy,n} (1/z_2^*, 1/z_2)]+M_{z,n}(z_2, z_2^*)M_{z,n}(1/z_2^*, 1/z_2)=1$$ which reduces to the original SLR energy conservation constraint $(|M_{xy,n}|^2 +|M_{z,n}|^2=1)$ when $T_2$ is neglected.

More importantly, any pair of two-dimensional polynomials that satisfies the invariant constraints can be mapped back to a $T_2$ and frequency selective pulse. In particular, if we go though the inverse equation in Figure 2, each polynomial must reduce its degree by one. By matching with the invariant constraints, one unique hard pulse angle can be found to reduce the polynomial degrees: $$\begin{pmatrix}\theta_n\\ \phi_n\end{pmatrix}=\begin{pmatrix}\tan^{-1}\left(\frac{|b_n{[0][0]}|}{a_n{[0][0]}}\right)\\ \angle\left(-ib_n{[0][0]}a_n{[0][0]}\right)\end{pmatrix}=\begin{pmatrix}\tan^{-1}\left(\frac{|2a_n{[0][n]}|}{|b_n[n][0]|-|b_n{[0][n]|}}\right)\\ \angle\left(ia_n{[0][n]}b_n[n][0]\right)\end{pmatrix}$$

Because of limited space, we omit the detailed derivation. By iterating through the inverse equations, we can show that there is a one-to-one mapping between a $T_2$ and frequency selective pulse and a pair of two-dimensional polynomials. A similar procedure can show that a $T_1$, $T_2$ and frequency selective pulse can be mapped one-to-one to three-dimensional polynomials, which are summarized in Figure 3.

We have now reduced the general pulse design to multi-dimensional polynomial design. However, one challenge is that multi-dimensional minimum phase decomposition does not exist and the invariant constraint becomes difficult to incorporate within the filter design. In this abstract, we consider a convex relaxation approach$^{4,5}$ as shown in Figure 4. However, more work is still needed to consistently design multi-dimensional polynomials.

We also consider the small-tip angle region, where $\sin^2( \theta_n / 2) \approx 0$. By propagating through the forward equation, the polynomial coefficients can be found to be mostly zeros, except $b_n[i][0]$, $a_n[i][0]$ and $a_n[0][j]$. Hence, the two-dimensional polynomials can be approximated as one-dimensional polynomials and the small-tip $T_2$ and frequency polynomial can be easily designed using the z-transform as shown in the results.

Because of the complexity of designing multi-dimensional polynomials, we have only implemented our method successfully with small-tip $T_2$ and frequency pulses, where the design reduces to a one-dimensional z-transform. Figure 2a shows minimum phase TBW=6 pulses designed to excite a bandwidth of 600 Hz for $T_2$ = 10000 ms and $T_2$ = 1ms separately, with $\Delta t = 50 \mu s$ and $n=20$. Note that the z-transform of the short $T_2$ pulse shows how the polynomial compensates for the short $T_2$ by shifting the zeros to the left, which results in a sharp frequency selective profile. Figure 2b shows a minimum phase pulse designed to null $T_2 = 1 ms$ with frequency $= 0Hz$ and $=-100Hz$, using the proposed method with $\Delta T = 250 \mu s$ and $n=20$. Note that the z-transform of the polynomial shows that the zeros are placed exactly at the corresponding $z_2$ null points.