Introduction

Balanced steady state free precession (bSSFP) imaging is an SNR-efficient modality with high fluid-tissue contrast. These properties ensure its diagnostic capacity for imaging in regions such as cardiac and the internal auditory canal (IAC)¹. bSSFP would be even more valuable clinically if it provided quantitative tissue and environment parameter maps². Previously, a geometric solution (GS)³ based on the elliptic signal model (ESM) using 4 phase-cycled bSSFP images demonstrated the capacity to generate artifact-free images with an off-resonance field map^4,5. When coupled with complementary ESM computations such as the algebraic solution (AS)⁶, it is apparent that quantitative MR parameters may be realized analytically using the bSSFP ESM. While numerical methods such as the PLANET⁷ and DESPOT2⁸ exist, exact bSSFP parametric maps have yet to be calculated. Here the first analytical computation of $T_2$ relaxation and source ESM parameters using bSSFP is formulated via an exact geometric methodology.

Theory

Eq.(1) describes the ESM. $\theta=\Delta\omega\cdot{TR}+\psi$ is the off-resonant phase accumulation per repetition time $TR$ for precession frequency off-resonance $\Delta\omega$, $\varphi=-\Delta\omega\cdot{TE}$ represents the phase accumulation at echo time $t=TE$ and $\psi$ is the phase cycling increment.
$$I(\theta,\psi)=M\frac{1-ae^{i(\theta+\psi)}}{b\,\mathrm{cos}(\theta+\psi)}e^{i\varphi}\tag{1}$$
Eq.(2) defines the ESM parameters $|M|,a,b$, in terms of the equilibrium magnetization $M_0$, flip angle $\alpha$, $TR$, and relaxation times $T_1$ and $T_2$.
$$\begin{aligned}E_1&=\mathrm{exp}(-TR/T_1)\\a&=E_2=\mathrm{exp}(-TR/T_2)\\|M|&=\frac{M_0(1-E_1)\,\mathrm{sin}\alpha}{1-E_1\,\mathrm{cos}\alpha-E_2^2(E_1-\mathrm{cos}\alpha)}\\b&=\frac{E_2(1-E_1)(1+\mathrm{cos}\alpha)}{1-E_1\,\mathrm{cos}\alpha-E_2^2(E_1-\mathrm{cos}\alpha)}\end{aligned}\tag{2}$$
Four unique datasets are formed at phase cycling increments $\psi_1=0^{\circ},\psi_2=90^{\circ},\psi_3=180^{\circ},\psi_4=270^{\circ}$ respectively.
As shown by the orange signal ellipse curve in Fig. 1, point $P_i$ represents the signal $I_i$ with the phase cycling increment $\psi_i$. The blue circle with center at $M$ obtained by transforming points $P_i$ on the signal ellipse to points $Q_i$ is define by $J(\theta,\psi)$:
$$J(\theta,\psi)=I(\theta,\psi)(1-b\,\mathrm{cos}(\theta+\psi))=|M|e^{i\varphi}(1-ae^{i\theta})\tag{3}$$
The geometric cross-point $M=(M_x,M_y)$ is calculated via the geometric solution as described by Xiang and Hoff³.
The ratio of the distances $|MP_i|$ from signal points $P_i$ to the cross-point $M$ permits a geometric calculation of $b$ (and $\theta$):
$$b\,\mathrm{cos}\theta=\frac{|MP_1|-|MP_3|}{|MP_1|+|MP_3|},b\,\mathrm{sin}\theta=\frac{|MP_4|-|MP_2|}{|MP_2|+|MP_4|}\\b=\sqrt{\left(\frac{|MP_1|-|MP_3|}{|MP_1|+|MP_3|}\right)^2+\left(\frac{|MP_4|-|MP_2|}{|MP_2|+|MP_4|}\right)^2}\tag{4}$$
Coordinates $Q_i$ may then be computed by multiplying the signal coordinates of point $P_i$ by $(1\pm{b}\,\mathrm{cos}\theta)$ or $(1\pm{b}\,\mathrm{sin}\theta)$ from Eq.(4). The intersection of the circle and ellipse indicates that the circle $J$ radius $|MQ_i|=|M|a$. Therefore, $a$ may be computed once for each of the four transformed phase-cycled points $Q_i$:
$$\begin{aligned}a_i&=|MQ_i|/|M|\\\overline{a_{rms}}&=\sqrt{(a_1^2+a_2^2+a_3^2+a_4^2)/4}\end{aligned}\tag{5}$$
An $a$ map can be obtained by calculating the root mean square of $a_i$, and a $T_2$ map using Eq.(2) $a=\mathrm{exp}(-TR/T_2)$.

Methods

Four phase-cycled images are simulated using phase cycling increments $\psi=0^{\circ},90^{\circ},180^{\circ}\,\text{and}\,270^{\circ}$, flip angle $\alpha=30^{\circ}$ and $TR=10ms$. $T_1$ is held constant at $1000ms$. $T_2$ is varied from $47ms$ to $995ms$ such that $a$ varies from $0.81$ to $0.99$ and $b$ varies from $0.25$ to $0.87$ vertically. The off-resonant angle $\theta$ varies from $-2\pi$ to $2\pi$ horizontally. Bivariate gaussian noise set to $2\%$ of the mean intensity is also added.
The solution for $a,b$ and $T_2$ was computed pixel-by-pixel with signal real parts $x_i$, imaginary parts $y_i$ and the geometric solution cross-point $M=(M_x,M_y)$. The Total Relative Error (TRE) of each analytical solution $f_i(x,y)=a,b\,\text{and}\,\theta$ over all pixel values $x,y$ is computed by
$$TRE(f)=\frac{\sqrt{\sum_{x,y}[f_i(x,y)-f_g(x,y)]^2}}{\sum_{x,y}f_g(x,y)}\tag{6}$$
with respect to the gold standard $f_g$. Extended-size datasets were employed to permit computation of TRE vs. $a,T_2$ and $\theta$. $11$ datasets of similar simulated images were generated with Gaussian noise percentage ranging from $0\%$ to $10\%$ for the TRE vs. noise computation.

Results

Fig.3 shows good correspondence between the $a,b$ and $T_2$ gold standard and calculated analytical solution maps. Relative error is calculated and shown in the right column, indicating accurate contrast with some increased $\theta$-dependent variance.
Fig.4a and Fig.4b depict the TRE of $a$ and $b$ parameters calculated with varying $T_2$, and are plotted respectively against $T_2$ itself and the $a(T_2)$ value. TRE of the solutions gradually decrease as $T_2$ value increases.
Fig.4c shows a periodic dependency of the $a$ and $b$ TRE on off-resonant angle $\theta$. Fig.4d shows increased TRE as added noise increases.

Discussion

A 4-point geometric solution to $a,b\,\text{and}\,T_2$ is formulated, which realizes the first analytical computation of the relaxation parameter $T_2$ using the ESM of bSSFP. The solution to $a,b\,\text{and}\,T_2$ is exact and without error in noiseless scenarios. With noise present, the $a$ solution demonstrates relatively little error, whereas the calculated $b$ shows a stronger periodic variation from the gold standard at $\theta=N\pi/2$. The ESM indicates that when $b\mathrm{cos}\theta$ or $b\mathrm{sin}\theta$ approach zero at $N\pi/2$, added noise may relatively dominate signal terms. This is exacerbated by squaring the terms in the calculation of $b$. Note that it is the $b$-solution variance (and not contrast/level) that depends on $\theta$, and that the slight $\theta$-dependence of the GS solution variance⁶ may also contribute.
The calculated $T_2$ map is accurate when $T_2$ is small ($<200ms$). But the inverse logarithm computation $a=\mathrm{exp}(-TR/T_2)$ causes instability when $a$ is close to 1. Optimized processing of the averaged $a$ value should improve $T_2$ map accuracy.

Conclusion

The first analytic solution for $T_2$ using the ESM of bSSFP is shown to be relatively robust for calculations of ESM parameters $a$ and $b$ under variation of noise levels, $T_2$ values and band proximity. This inspires clinical application of this novel GS-scaled method to add bSSFP parametric maps to artifact-free images.

Acknowledgements

No acknowledgement found.