## 2 SSFP/CWFP Theory  
The SSFP sequence uses a train of radiofrequency pulses (rf) with phase (\(\Phi\)), flip angle (\(\theta\)), and time between pulses (T\({}_{\text{p}}\)) shorter than the relaxation time T\({}_{2}\) (T\({}_{\text{p}}\) < T\({}_{2}\)). SSFP sequences have been used to improve the SNR in pulsed NMR spectroscopy since 1958 [1]. In 1966, [PERSON] and [PERSON] [2] derived the analytical solution for the SSFP regime. They demonstrated that the SSFP signal is composed of FID and echo signals. The FID component (M+) follows the rf pulses, and the echo component (M-) precedes the pulses, as given by the following equations:  
\[ \begin{array}{l}M_{x}^{-}=M_{0}(1-E_{1})[E_{2}\ sin\theta\ sin\Phi]/D\\ M_{y}^{-}=M_{0}(1-E_{1})[E_{2}\ sin\theta\ cos\Phi-E_{2}^{2}sin\theta]/D\\ M_{z}^{-}=M_{0}(1-E_{1})[1-E_{2}\ cos\Phi-E_{2}cos\theta(cos\theta-E_{2})]/D\\ M_{x}^{+}=M_{x}^{-}\\ M_{y}^{+}=M_{0}(1-E_{1})[\ (1-E_{2}\ cos\Phi)sin\theta]/D\\ M_{z}^{+}=M_{0}(1-E_{1})[E_{2}(E_{2}cos\Phi)+(1-E_{2}cos\Phi)cos\theta\ ]/D \end{array}  \tag{1}\]  
where \(D=[(1-E_{1}cos\theta)(1-E_{2}cos\Phi)]-[(E_{1}-cos\theta)(E_{2}-cos\Phi)E_{2}]\) with the precession angle \(\Phi=\Omega T_{p}\), offset frequency \(\Omega=\omega_{ref}-\omega_{0}\), and relaxation components \(E_{1}=exp\big{(}-T_{p}/T_{1}\big{)}\), and \(E_{2}=exp\big{(}-T_{p}/T_{2}\big{)}\). These equations allow us to calculate the magnitude of the magnetization in the \(xy\) plane after the \(n\)th rf pulse in the SSFP condition [4; 30]:  
\[|M|=\frac{M_{0}|sin\theta|\sqrt{2-2 cos\Phi}}{(1+cos\theta)(1-cos\Phi)+(1-cos \theta)2T_{1}/T_{2}} \tag{2}\]  
Therefore, the amplitude of the signal has a strong dependence on the ratio of the relaxation times T\({}_{1}\) and T\({}_{2}\), the flip angle \(\theta\), and the precession angle \(\Phi=\Omega\)\(T_{p}\). Equation (2) shows that the amplitude of the magnetization is null when \(\Phi=2n\pi\) and is the maximum when \(\Phi=(2n+1)\pi\), where \(n\) is an integer. These nulls and maximums of amplitude occur due to the destructive or constructive interference between the FID and echo components in the SSFP sequences [12]. These complex interferences between FID and echo signals depend on the time interval between pulses T\({}_{\text{P}}\), precession angle \(\Phi\), frequency offset \(\Omega\), flip angle \(\theta\), and phase alternation [23; 24]. The transition from regular pulse sequence to SSFP/CWFP regimes was demonstrated by [PERSON] et al. in 2000 [12] and is summarized in Figure 1. The NMR signals of Figure 1A-E were obtained by a numerical simulation using a train of 90\({}^{\circ}\) pulses, T\({}_{1}\) = 200 ms, T\({}_{2}\) = 150 ms, T\({}_{2}\)* = 0.5 ms and T\({}_{\text{P}}\) = 5T\({}_{1}\) (Figure 1A); T\({}_{\text{P}}\) = T\({}_{2}\) (Figure 1B); T\({}_{\text{P}}\) = 2T\({}_{2}\)* (Figure 1C); T\({}_{\text{P}}\) = 0.3 ms \(<\) T\({}_{2}\)* and \(\Phi=(2n+1)\pi\) (Figure 1D); and \(\Phi=n2\pi\) (Figure 1E). In Figure 1A,B, the pulses (gray bar) are shown at the center of the window of \(\pm 2\) ms to better present the FID and echo signals [12]. Figure 1C-E show other pulses (gray bars) outside the center. Figure 1A depicts the standard pulse sequence with 5T\({}_{1}\)\(\geq\) T\({}_{\text{P}}\) > 3T\({}_{2}\), where the amplitude of the echo is very small (normally not seen in experiments), and most of the magnetization has returned to the Boltzmann equilibrium. Figure 1B shows the typical SSFP signal after the \(n\)th pulse with T\({}_{2}\)* \(<\) T\({}_{\text{P}}\) = T\({}_{2}\). The FID signal has a lower amplitude than in Figure 1A, and a small echo signal forms preceding the pulse. In this regime, the FID/echo does not overlap because of the short T\({}_{2}\)*. Figure 1C, with T\({}_{2}\)\(>\) T\({}_{\text{P}}\) > T\({}_{2}\)*, shows the SSFP regime with partial FID and echo overlap. A special SSFP regime occurs when T\({}_{2}\)\(>>\) T\({}_{\text{P}}\) < T\({}_{2}\)* (Figure 1D,E).