Kübra Keskin^{1,2} and Tolga Çukur^{1,2,3}

There is growing interest in use of balanced steady-state free precession (bSSFP) imaging for simultaneous mapping of $$$T_1$$$, $$$T_2$$$ and off-resonance. An elegant ellipse fitting approach in the complex plane was recently proposed for parameter estimation from multiple phase-cycled acquisitions. Since this approach requires at least six phase-cycles, it can limit scan efficiency. Here, we propose a new technique that integrates a geometric solution with constrained ellipse fitting to enable mapping with only four phase-cycled acquisitions. The proposed method yields accurate $$$T_1$$$, $$$T_2$$$ and off-resonance maps while significantly improving scan efficiency.

Balanced steady-state free precession (bSSFP) offers high imaging speed, and thereby holds great promise for parametric mapping^{1}. Yet, its elevated sensitivity to main field inhomogeneity and nonstandard $$$T_2$$$/$$$T_1$$$ contrast limit its applicability. Multiple-acquisition phase-cycled acquisitions are commonly used to alleviate banding artifacts^{2}. A powerful technique observes that magnetization values for multiple phase-cycles lie along an ellipse to suppress artifacts^{3}. This elliptical model was recently used to quantify $$$T_1$$$_{ }and $$$T_2$$$ values via direct least-squares fitting to at least six phase-cycled acquisitions^{4}. While powerful, this previous approach can limit scan efficiency.

Here, we propose a new method to improve efficiency of parametric mapping via multiple-acquisition bSSFP imaging. Specifically, we incorporate prior knowledge about the elliptical geometry formed by multiple phase-cycles to constrain the ellipse fitting problem. This approach enables simultaneous mapping of $$$T_1$$$, $$$T_2$$$ and off-resonance using only four acquisitions.

The phase-cycled bSSFP signal can be expressed in complex domain as:

$$S_n=M \frac{1-ae^{i\theta_n}}{1-b\cos{\theta_n}}e^{i\phi},\quad n\in[1,N]\qquad\qquad\qquad [1]$$

where $$a=E_2,\quad b=\tfrac{E_2(1-E_1)(1+\cos{\alpha})}{1-E_1\cos{\alpha}-E_2^2(E_1-\cos{\alpha})},\quad M=\tfrac{M_0(1-E_1)\sin{\alpha}}{1-E_1\cos{\alpha}-E_2^2(E_1-\cos{\alpha})}e^{-TE/T_{2}},$$ $$E_1=exp(-TR/T_1),\quad E_2=exp(-TR/T_2)$$.

Here, $$$N$$$ is the total number of acquisitions. $$$TR$$$ and $$$TE$$$ are repetition and echo times, $$$M_0$$$ is the equilibrium magnetization, and $$$\alpha$$$ is the flip angle. $$$\theta_n=\theta_0+\Delta\theta_n$$$ denotes phase acquired due to both off-resonance ($$$\theta_0=2\pi\Delta f_0TR$$$) and phase-cycling ($$$\Delta\theta_n=2\pi (n-1)/N$$$). The phase-cycling that varies across acquisitions causes the complex signal with phase $$$\phi=2\pi\Delta f_0TE+\phi_{acc}$$$ to rotate around the origin along an ellipse, so each $$$S_n$$$ in Equation 1 is a point on the ellipse^{3}. Therefore, the shape and positioning of the ellipse carry information about the bSSFP signals.

The proposed approach follows three steps to estimate $$$T_1$$$, $$$T_2$$$ and off-resonance (Figure 1):

First, we find the geometric solution (GS) of the ellipse, which is the cross-point of all signal pairs separated with $$$\pi$$$ phase-cycling angle^{3}.(For $$$N=4$$$, the following pairs would be used $$$\{0$$$-$$$\pi$$$; $$$\frac{\pi}{2}$$$-$$$\frac{3\pi}{2}\}$$$). This corresponds to the parameter $$$M$$$ in Equation 1. Here, we generalize the GS solution proposed in ^{3} for 4 phase cycles to N cycles. The cross-point is obtained by solving the following system of equations where $$$(x,y)_{m1}$$$ and $$$(x,y)_{m2}$$$ are $$$\pi$$$-separated signal pairs, $$$(x_0,y_0)$$$ is the cross-point, and $$$m=\{1,2,..,N/2\}$$$:

$$\begin{bmatrix}y_{12}-y_{11}&x_{11}-x_{12}\\y_{22}-y_{21}&x_{21}-x_{22}\\\vdots&\vdots\\y_{m2}-y_{m1}&x_{m1}-x_{m2}\end{bmatrix}\begin{bmatrix}x_0\\y_0\end{bmatrix}=\begin{bmatrix}x_{11}y_{12}-x_{12}y_{11}\\x_{21}y_{22}-x_{22}y_{21}\\\vdots\\x_{m1}y_{m2}-x_{m2}y_{m1}\end{bmatrix}\qquad\qquad [2]$$

Next, we use the cross-point as prior knowledge in constraining the ellipse fitting. We leverage the notion that prior information about the line on which the ellipse’s center lies enables accurate ellipse fitting with as few as four data points^{5}. To estimate this central line, we observe that the line connecting the origin to the cross-point has to pass through the center of the ellipse. Thus, we use the parameters of this line to a constrained ellipse-fitting procedure^{5} in Equation 3.

$$\min_{\gamma,\,\nu,\,h}{\left\Vert(\begin{bmatrix}D_0&1_N\end{bmatrix}-\gamma\begin{bmatrix}D_1&0_N\end{bmatrix})\begin{bmatrix}\nu\\h\end{bmatrix}\right\Vert^2_2}\quad\text{s.t.}\quad\nu^TB\nu=1\qquad\qquad [3]$$

Here, $$$\gamma$$$ is a parameter used for determining the ellipse center, vector $$$\nu$$$ and scalar $$$h$$$ are ellipse-related parameters, and $$$B$$$ is a matrix used to ensure that the fitting result is an ellipse. $$$D_0$$$ and $$$D_1$$$ are data matrices that prior information is integrated.

After the ellipse parameters are fit, we estimate the center and semi-axes of the ellipse. Based on these estimates, we can find the values of variables $$$a$$$ and $$$b$$$ in Equation 1. Lastly, $$$T_1$$$, $$$T_2$$$ and off-resonance values are computed based on the analytical solution proposed in ^{4}.

To demonstrate the proposed approach, we performed MRI experiments on a 3T scanner (Siemens Magnetom Trio). Phantom and in vivo results were obtained with N=4 and N=8 phase cycles, respectively. $$$B_1$$$ correction was performed prior to mapping.

This work was supported in part by a TUBITAK 1001 Grant (117E171), by a IEEE Research Encouragement Award 2017, by a TUBA GEBIP 2015 fellowship, by a BAGEP 2017 fellowship, and by a EMBO Installation Grant (IG 3028).

1. Björk M, Ingle R, Gudmundson E, Stoica P, Nishimura D, Barral J. Parameter estimation approach to banding artifact reduction in balanced steady-state free precession. Magn Reson Med. 2013;72(3):880-892. doi:10.1002/mrm.24986

2. Bangerter N, Hargreaves B, Vasanawala S, Pauly J, Gold G, Nishimura D. Analysis of multiple-acquisition SSFP. Magn Reson Med. 2004;51(5):1038-1047. doi:10.1002/mrm.20052

3. Xiang Q, Hoff M. Banding artifact removal for bSSFP imaging with an elliptical signal model. Magn Reson Med. 2014;71(3):927-933. doi:10.1002/mrm.25098

4. Shcherbakova Y, van den Berg C, Moonen C, Bartels L. PLANET: An ellipse fitting approach for simultaneous T1 and T2 mapping using phase-cycled balanced steady-state free precession. Magn Reson Med. 2017;79(2):711-722. doi:10.1002/mrm.26717

5. Waibel P, Matthes J, Gröll L. Constrained Ellipse Fitting with Center on a Line. J Math Imaging Vis. 2015;53(3):364-382. doi:10.1007/s10851-015-0584-x

Illustration of the proposed approach's steps. After the acquisition of N phase-cycled bSSFP data, first step is finding the cross-point of the data pairs separated with $$$\pi$$$ phase-cycling angle. Second step is fitting an ellipse to the data points by using the cross-point as a prior information to the constrained ellipse fitting. Third step is finding the center and semi-axes of the ellipse. Lastly, $$$T_1$$$ and $$$T_2$$$ values are estimated.

Representative $$$T_1$$$, $$$T_2$$$ and off-resonance maps recovered from a cylindrical phantom using phase-cycled bSSFP acquisitions are displayed for the proposed constrained ellipse fitting (CEF) and the direct ellipse fitting (DEF) approach. CEF allows $$$T_1$$$-$$$T_2$$$ mapping with only four phase-cycles, moreover it produces very similar maps to the ones recovered from eight phase-cycles, which reduces the scan time by half.

Comparisons of in vivo $$$T_1$$$ and $$$T_2$$$ maps recovered via the proposed constrained ellipse fitting (CEF) approach and the direct ellipse fitting (DEF) approach implemented with eight phase-cycles are shown. While both demonstrate similar anatomical features, CEF offers less oscillatory maps (e.g. lower part of the brain in T1 maps). This suggests that constrained ellipse fitting approach combining the prior information with the fitting problem can leverage mapping results.