Fully-balanced steady-state free precession (bSSFP)¹ is clinically useful because of its high SNR efficiency, imaging speed and unique T₂/T₁ contrast. A geometric solution to bSSFP has been derived and provides a simple and intuitive understanding of how the complex steady-state signals are formed². However, T₁ and T₂ relaxation has generally been ignored. In this work, we present an exact geometric solution to the bSSFP signals by including the T₁ and T₂ relaxation effects.

Following convention, the equilibrium magnetization, flip angle and the precession angle accumulated over a given TR are denoted as M₀, θ and β, respectively. Further, E₁ and E₂ are defined as $$$e^{-TR/T_1}$$$ and $$$e^{-TR/T_2}$$$, respectively. Assuming that the RF pulse is applied along the X-axis and alternates in phase every other TR, Fig. 1 shows the bSSFP signal evolution during two consecutive TRs in the Y-Z and X-Y planes, where A, B, C, D are the time points corresponding to 0⁺, TR^-, 0⁺, TR^-. Assuming that M_xy, β₁ and M_z are the transverse magnetization’s magnitude, transverse magnetization’s phase, and longitudinal magnetization’s magnitude of the bSSFP signal at time-point A. The corresponding parameters at time-point B can be written as M_xyE₂, β-β₁, and M_zE₁+M₀ (1-E₁), respectively.

Referring to Fig. 1, three signal equations can be written based on the geometry of the magnetizations. Eq. [1] stipulates that the magnitudes of the magnetization at time-points A and B are the same. Since RF pulses do not change the magnitude of the magnetization, the magnitude of the magnetization should also be same at time-point C.

$$M_{xy}^2 \frac{1-E_2^2}{1-E_1^2} + \left(M_z - \frac{E_1}{1+E_1}M_0\right)^2 = \frac{M_0^2}{(1+E_1)^2} \hspace{13mm} [1]$$

Eq. [2] relates to the initial phase β₁ of the transverse magnetization immediately after the RF pulse. Since the RF pulse is applied along the X-axis, the magnitude of M_x at time-points A and B are the same:

$$\tan(\beta_1) = \frac{E_2\sin\beta}{1+E_2\cos\beta} \hspace{66mm} [2]$$

Eq. [3] establishes the relationship between the RF flip angle and the signals at time-points A and B. Using geometry and recognizing that RF pulses are applied along the X-axis, the |ΔMz| over |ΔM_y| ratio should be the tangent of half the RF flip angle:

$$\tan(\frac{\theta}{2}) = \frac{(M_0-M_z)(1-E_1)} {M_{xy}(\cos\beta_1-E_2\cos(\beta-\beta_1))} \hspace{33mm} [3]$$

Eqs. [1-3] can be used to solve for the bSSFP signals and the results are presented as follows:

\begin{eqnarray}M_x &=& M_0 (1-E_1)\frac{-E_2\sin\theta\sin\beta}{d} \hspace{40mm}&[4]&\\M_y &=& M_0 (1-E_1)\frac{\sin\theta(1+E_2\cos\beta)}{d} &[5]&\\M_z &=& M_0 (1-E_1)\frac{[E_2(E_2+\cos\beta)+(1+E_2\cos\beta)\cos\theta]}{d}&[6]&\end{eqnarray}

where

$$d = (1-E_1\cos\theta)(1+E_2\cos\beta)-E_2(E_1-\cos\theta)(E_2+\cos\beta)\hspace{10mm}[7]$$

The above solutions are identical to those given in Ref. [3], except for a change from β to β + π. The difference can be attributed to the fact that we assumed that RF pulses in alternating phases, whereas Ref. [3] assumed non-alternating RF pulses.

The solutions given in Eqs. [4-7] are helpful in understanding different aspects of the bSSFP signals. It is well-known that the bSSFP signals display the spin-echo behavior at TE=TR/2 over a wide range of the precession angles⁴. Using the solutions above and Fig. 1, such behavior can be quantified using the difference between: a) the actual echo time relative to TR, as defined by β₁/β, and b) 0.5, which is the echo time at half TR. The complete result is presented in Fig. 2 as a function of E₂ and precession angles, where the negative contours indicate that the actual echo occurs before TR/2, and the positive contours indicate that the actual echo occurs after TR/2. Clearly, spins are always perfectly refocused at TR/2, if E₂ is 1 and regardless of β. However, as E₂ becomes less than one due to T₂-relaxation, spins will refocus either earlier or later than the TR/2 time point. The deviation is more pronounced for spins with a precession angle of approximately ±π, as compared to 0 or 2π.

The exact bSSFP solution including the T₁ and T₂ relaxation effects is derived based on the geometry of the magnetizations. The results are consistent with those based on matrix calculations. The geometric solutions are helpful in understanding different aspects of the bSSFP signal behavior.