Scientists who are interested in quantitative MRI.

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.

The complex bSSFP signal S can be represented as:

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

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

M_0,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).

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.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.