Introduction

Balanced steady-state free precession (bSSFP) is an SNR-efficient imaging sequence that is regularly employed in various applications including cardiac, low field, and inner ear imaging. The development of the elliptical signal model (ESM) has enabled the potential realization of bSSFP quantitative parameter mapping, furthering its clinical value¹. However, achieving effective banding artifact removal and parametric quantitation has thus far required the acquisition of at least four phase-cycled bSSFP images¹, presenting a challenge for time-sensitive applications. Previously, a procedure to obtain an analytical solution that requires only three phase-cycled images has been described², promising to further improve time efficiency. However, the final solution was not explicitly presented, and its effectiveness has only been demonstrated under noiseless scenarios using an equally-spaced phase-cycled angle setup $$$(0^\circ,120^\circ,240^\circ)$$$. Here we present an explicit closed-form solution to the bSSFP system with 3 general phase-cycled acquisitions. Our method is demonstrated for phase-cycled images acquired at angles of $$$(10^\circ,118^\circ,258^\circ)$$$, although potentially any three angles may be chosen due to the solution’s flexibility to the arbitrary selection of phase cycling increments.

Methods

Theory:
The ESM for bSSFP is described in Eq.(1) and depicted in Figure 1, with the tissue and imaging parameters $$$M_0,T_1,T_2,\alpha,$$$ and $$$TR$$$ grouped into the ESM parameters $$$M,a,b$$$:$$\begin{aligned}I_j&=M\frac{1-a\,e^{i(\theta+\psi_j)}}{1-b\cos(\theta+\psi_j)}e^{i\phi},\quad\,j=1,2,3\quad(1)\\E_1:&=\exp(-TR/T_1)\\E_2:&=\exp(-TR/T_2)\\a:&=E_2\\M:&=\frac{M_0(1-E_1)\sin\alpha}{1-E_1\cos\alpha-E_2^2(E_1-\cos\alpha)}\\b:&=\frac{E_2(1-E_1)(1+\cos\alpha)}{1-E_1\cos\alpha-E_2^2(E_1-\cos\alpha)}\end{aligned}$$
Here $$$\psi$$$ denotes arbitrary RF phase cycling increments, where $$$\psi_1,\psi_2,\psi_3$$$ represent three bSSFP acquisitions, and $$$\theta$$$ and $$$\phi$$$ respectively denote the off-resonant phase accumulation at TR and TE.
Previous work² demonstrated that the demodulated magnetization $$$M$$$ may be determined by transformation of the signal ellipse to a J-circle formulation, which is defined by its center $$$M\,e^{i\phi}$$$, radius $$$Ma$$$ and is obtained through multiplication of $$$(1-b\cos(\theta+\psi_j))$$$ onto the elliptical bSSFP signal $$$I$$$:$$J_j=I_j(1-b\cos(\theta+\psi_j)),\quad\,j=1,2,3\quad(2)$$
Given that the relation between the three transformed points $$$J_1,J_2,J_3$$$ is guaranteed by the circular property:$$(J_3-J_2)E=(J_1-J_2)\quad(3)$$$$E=\frac{e^{i\psi_1}-e^{i\psi_2}}{e^{i\psi_3}-e^{i\psi_2}}\quad(4)$$
a linear system with 2 equations and 2 unknowns $$$(b\cos\theta,b\sin\theta)$$$ is deduced.
Solving the linear system then yields the following closed form solution for $$$(b\cos\theta,b\sin\theta)$$$:
$$\begin{aligned}b\cos\theta&=\frac{\mathrm{Im}\left[\cos\frac{\psi_1+\psi_2}{2}I_1^*I_2e^{i\frac{\psi_1-\psi_2}{2}}+\cos\frac{\psi_1+\psi_3}{2}I_3^*I_1e^{i\frac{\psi_3-\psi_1}{2}}+\cos\frac{\psi_2+\psi_3}{2}I_2^*I_3e^{i\frac{\psi_2-\psi_3}{2}}\right]}{\mathrm{Im}\left[\cos\frac{\psi_1-\psi_2}{2}I_1^*I_2e^{i\frac{\psi_1-\psi_2}{2}}+\cos\frac{\psi_3-\psi_1}{2}I_3^*I_1e^{i\frac{\psi_3-\psi_1}{2}}+\cos\frac{\psi_2-\psi_3}{2}I_2^*I_3e^{i\frac{\psi_2-\psi_3}{2}}\right]}\\b\sin\theta&=-\frac{\mathrm{Im}\left[\sin\frac{\psi_1+\psi_2}{2}I_1^*I_2e^{i\frac{\psi_1-\psi_2}{2}}+\sin\frac{\psi_1+\psi_3}{2}I_3^*I_1e^{i\frac{\psi_3-\psi_1}{2}}+\sin\frac{\psi_2+\psi_3}{2}I_2^*I_3e^{i\frac{\psi_2-\psi_3}{2}}\right]}{\mathrm{Im}\left[\cos\frac{\psi_1-\psi_2}{2}I_1^*I_2e^{i\frac{\psi_1-\psi_2}{2}}+\cos\frac{\psi_3-\psi_1}{2}I_3^*I_1e^{i\frac{\psi_3-\psi_1}{2}}+\cos\frac{\psi_2-\psi_3}{2}I_2^*I_3e^{i\frac{\psi_2-\psi_3}{2}}\right]}\quad(5)\end{aligned}$$
Once the transformed $$$J$$$-circle points $$$J_1,J_2,J_3$$$ are obtained, arithmetic manipulation allows solution for the point $$$M$$$ at the $$$J$$$-circle center. Notice that the point $$$M$$$ can also be interpreted as the circumcircle center of triangle $$$\triangle\,J_1J_2J_3$$$, suggesting that the coordinates of $$$M$$$ can be expressed through linear combination of $$$J_1,J_2,J_3$$$. The solution may then be found using barycentric coordinates of the triangle $$$\triangle\,J_1J_2J_3$$$ as follows:$$\begin{aligned}M\,e^{i\phi}&=\frac {W_{J1}J_1+W_{J2}J_2+W_{J3}J_3}{W_{J1}+W_{J2}+W_{J3}}\\W_{J1}&=\sin(\psi_3-\psi_2),\quad\,W_{J2}=\sin(\psi_1-\psi_3),\quad\,W_{J3}=\sin(\psi_2-\psi_1)\quad(6)\end{aligned}$$
This gives us the implicit solution for $$$M$$$ with arbitrary phase-cycled angles $$$\psi_1,\psi_2,\psi_3$$$. Using Eq.(2), (5) and (6), the closed-form solution for $$$M$$$ in terms of $$$I_1, I_2, I_3$$$ is expressed as follows:$$\begin{aligned}M\,e^{i\phi}&=\frac{W_{I1}I_1+W_{I2}I_2+W_{I3}I_3}{W_{I1}+W_{I2}+W_{I3}}\\W_{I1}&=\cos\left(\frac{\psi_2-\psi_3}{2}\right)\left(I_2^*I_3e^{i\frac{\psi_2-\psi_3}{2}}-I_2I_3^*e^{-i\frac{\psi_2-\psi_3}{2}}\right)\\W_{I2}&=\cos\left(\frac{\psi_3-\psi_1}{2}\right)\left(I_3^*I_1e^{i\frac{\psi_3-\psi_1}{2}}-I_3I_1^*e^{-i\frac{\psi_3-\psi_1}{2}}\right)\\W_{I3}&=\cos\left(\frac{\psi_1-\psi_2}{2}\right)\left(I_1^*I_2e^{i\frac{\psi_1-\psi_2}{2}}-I_1I_2^*e^{-i\frac{\psi_1-\psi_2}{2}}\right)\quad(7)\end{aligned}$$
Validation:
Three phase-cycled bSSFP images with arbitrary phase-cycling increment $$$\psi=10^\circ,118^\circ,$$$ and $$$258^\circ$$$ respectively were generated for simulations. All possible tissue properties and imaging conditions were covered by ESM parameters $$$a,b$$$ both within the realizable range of [0,1] and the real-world limitation defined by $$$b<\frac{2a}{1+a^2}$$$. Bivariate Gaussian noise at 2% of the mean signal intensity was added. The solution for the demodulated magnetization $$$M$$$ was then computed pixel-wise, compared with the complex sum (CS) solution and evaluated with total relative error (TRE) analysis³.

Results

Figure 2 depicts the three simulated bSSFP signal images. Figure 3 shows the simulated ground truth M, the complex sum (CS), and the proposed direct analytical solution (DAS) magnitude and phase images, followed by residual maps. Figure 3 shows that both DAS magnitude and phase TRE values are lower than those of the CS solution. The CS suffers from band sensitivity while the DAS exhibits higher accuracy and resilience near banding locations in the source images.

Discussion

The closed-form DAS to the bSSFP system has been demonstrated with only three phase-cycled acquisitions at arbitrary-angles. The simulation shows that DAS is exact with noiseless data. With noise introduced, DAS manifests good accuracy over most possible tissue and imaging parameters. The DAS phase faithfully recovers the contrast of the ground truth, indicating potential for field mapping with only 3 bSSFP acquisitions. Future work will entail algorithmic improvements and refinements to extend the application to real data, including effective regularization and optimal selective design of phase cycle angles.

Conclusion

An explicit closed form analytical solution to demodulate bSSFP images with arbitrary 3 phase-cycled angles is developed and regularized with a linearization approach using regional information, inspiring novel future qualitative and quantitative applications of rapid bSSFP imaging.

Acknowledgements

No acknowledgement found.