Accurate T1 and T2 mapping by direct least-squares ellipse fitting to phase-cycled bSSFP data
Yulia Shcherbakova1, Cornelis A.T. van den Berg2, Jan J.W. Lagendijk3, Chrit T.W. Moonen4, and Lambertus W. Bartels5

1Center for Imaging Sciences/Imaging Division, UMC Utrecht, Utrecht, Netherlands, 2Dept. of Radiotherapy/Imaging Division, UMC Utrecht, Utrecht, Netherlands, 3Department of Radiotherapy/Centre for Image Sciences, UMC Utrecht, Utrecht, Netherlands, 4Center for Imaging Sciences/ Imaging Division, UMC Utrecht, Utrecht, Netherlands, 5Image Sciences Institute/Department of Radiology, UMC Utrecht, Utrecht, Netherlands

Synopsis

Björk et al. proposed to use the balanced steady-state free precession (bSSFP) pulse sequence with multiple phase-cycled acquisitions to estimate the values of T1 and T2 using a non-linear fitting approach. Unfortunately, they found that this non-linear approach would face large uncertainties for realistic SNRs. The purpose of our work was to demonstrate that by reformulating the signal model in the complex plane, an elliptical model can be fitted to the data points using a linear least squares method , allowing for more robust, accurate and simultaneous estimation of T1 and T2 values.

Target audience

Scientists who are interested in quantitative MRI.

Purpose

The balanced steady-state free precession (bSSFP) pulse sequence with multiple phase-cycled acquisitions was introduced to minimize banding artifacts (1). Björk et al. (2) extended this work by exploring the possibility to estimate T1 and T2 from phase-cycled acquisitions using non-linear fitting. They concluded that the technique would face large uncertainties for realistic SNRs.

Our purpose was to demonstrate that by reformulating the signal model in the complex plane, an elliptical model (3) can be fitted to the data points using a linear least squares method (4) and that a subsequent analytical solution for T1 and T2 exists. This transforms the reconstruction into a convex problem allowing for more robust, accurate and simultaneous estimation of T1 and T2 values.

Methods

The complex bSSFP signal S can be represented as:

$$$S=M\frac{1-ae^{i\theta}}{1-b\cos\theta}$$$ [1],

where $$$M=\frac{M_{0,eff}(1-E_{1})\sin\alpha}{1-E_{1}\cos\alpha-E_2^2(E_{1}-\cos\alpha)}$$$;$$$a=E_{2}$$$; $$$b=\frac{E_{2}(1-E_{1})(1+\cos\alpha)}{1-E_{1}\cos\alpha-E_2^2(E_{1}-\cos\alpha)}$$$;$$$E_{1}=\exp(\frac{-TR}{T_{1}})$$$;$$$E_{2}=\exp(\frac{-TR}{T_{2}})$$$

M0,eff is the effective equilibrium magnetization, α the flip angle, TR the repetition time, θ the resonance offset angle, θ = θ0+∆θ, θ0 is proportional to off-resonance frequency, ∆θ is user-controlled RF phase increment.

Equation 1 describes an ellipse in the complex plane. By fitting an ellipse to the set of data points acquired with different settings of the phase increment, T1 and T2 can in principle be retrieved. We propose the following approach:

First, the general second order polynomial representation of an ellipse, $$$Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$$$, is fitted to the measured data using the direct linear least-squares fitting method, specific to ellipses (4). Since there are 6 unknowns, we need at least 6 data points, which corresponds to acquiring data with at least 6 different phase increment settings. From the polynomial representation we have derived the conic equation of the ellipse and found the geometric center and semi-axes, which are related to parameters M, a, b through a system of nonlinear equations, which we have solved analytically. The T1 and T2 estimates can be found from a and b using:

$$$E_{1}=\frac{a(1+\cos\alpha-ab\cos\alpha)-b}{a(1+\cos\alpha-ab)-b\cos\alpha}$$$; $$$E_{2}=a$$$; $$$T_{1}=\frac{-TR}{\ln(E_{1})}$$$; $$$T_{2}=\frac{-TR}{\ln(E_{2})}$$$

Unlike T2 estimates, T1 estimates depend on α. Therefore the obtained T1 maps were calculated after a correction for spatial flip angle variations using an additionally acquired B1-field map.

To investigate the performance of the proposed approach, MRI experiments were performed on a clinical 1.5T MR scanner (Philips Achieva, Best, The Netherlands) using a phantom consisting of gel tubes with calibrated T1 and T2 values (TO5, Eurospin II test system, Scotland). T1 and T2 values of the gels were corrected for scanner room temperatures. Tubes were chosen with relaxation times in the range of human body tissues. An 8-channel head coil was used as a receive coil. The mean T1 and T2 values of 4 tubes placed close to the center of the phantom were calculated by averaging over an ROI inside the tubes on the T1 and T2 maps, after B1-field correction, for 3 slices in the center of the phantom.

Sequences parameter settings: 3D bSSFP, FOV 200x200x100mm3, voxel size 1.5x1.5x2mm3, TR 6.9 ms, TE 3.45 ms, NSA 2, α 45°, 10 phase cycles, ∆θ π/5.

Reference T1 and T2 values were calculated using a standard combined inversion recovery (for T1) and multi echo SE (for T2) sequence provided by the scanner manufacturer (5).

Results

Figure 1 provides an example of ellipses fitted to data points from two voxels inside tubes with different T1 and T2 combinations, acquired using 10 phase increment steps.

Figure 2 presents T1 and T2 maps, calculated for the phantom.

Figure 3 shows a comparison between tabulated relaxation times, those measured using reference sequence, and those calculated from our elliptical model for 4 different tubes near the center of the phantom. Standard deviations are presented in error bars.

Discussion and conclusion

We have demonstrated that direct least-squares ellipse fitting to phase-cycled bSSFP data allows the simultaneous estimation of relaxation parameters T1 and T2 from the same scan in a robust and fast manner. This is a clear advantage of this method compared to standard DESPOT1&2 methods, which require scanning using two different sequences. Reformulating the signal model in the complex plane allowed us to fit the ellipse directly using a linear least-squares method, which makes reconstruction time-efficient and avoids problems related to local minima associated with iterative fitting methods in the presence of noise. Contrary to Bjork et al., we foresee good potential for phase-cycled bSSFP imaging for simultaneous T1 and T2 mapping. Our future work will include a more detailed study of the precision and accuracy of the method in relation to the SNR. Furthermore we will optimize scan parameters to allow applications in vivo.

Acknowledgements

No acknowledgement found.

References

[1] N.K. Bangerter, B.A. Hargreaves et al. MRM 51 (2004); [2] M.Björk et al. MRM 72 (2014); [3] Q-S Xiang, M.N. Hoff MRM 71 (2014); [4] A.W. Fitzgibbon et al. TPAMI, Vol 21, 5 (1999); [5] J.J In den Kleef, J.J Cuppen MRM 5(6) (1987)

Figures

Figure 1. Two ellipses fitted to data points in the complex plane

Figure 2. Quantitative maps for the phantom

Figure 3. Tabulated and calculated T1 and T2 values for the phantom tubes



Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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