Introduction

In cardiac T1 Mapping, the signal intensity is commonly modelled as a non-linear function of inversion time, parameterized by a set of variables. Estimation of these parameters involves a non-linear parametric fitting procedure. The minimum mean square error (MMSE) estimator is most widely used and can be interpreted as a maximum likelihood estimator under the assumption that the additive measurement noise is a zero-mean stationary Gaussian process. However, in the MR T1 mapping data, the noise may not follow a Gaussian model, especially as cardiac motion (or imperfect motion correction) can give rise to outliers (i.e. misalignment in structure) for fitting.

To achieve a robust T1 estimate with improved robustness, we propose to employ M-estimators¹ for non-linear parameter estimation of cardiac T1 mapping. The loss function of an M-estimator is designed in a way that outliers will have a reduced impact on the final estimate. In this work, we evaluate three M-estimator variants on two types of MR T1 mapping sequences: Modified Look-Locker Inversion Recovery (MOLLI) ² and Saturation Pulse Prepared Heart rate independent Inversion-Recovery (SAPPHIRE) sequence ³.

Methods

An M-estimator replaces the $$$l_2$$$-norm in MMSE with a robust loss function $$$\rho$$$:
$$\theta^*= \text{argmin}_\theta\ \sum_k \rho \left ( \vert y_k - f_\theta (t_k) \vert ^ 2\right ) = \text{argmin}_\theta\ \sum_k \rho(r_k^2) $$
where $$$y_k$$$ is the measurement at $$$t_k$$$, $$$f$$$ is the nonlinear signal intensity model parameterized by $$$\theta$$$, and $$$r_k$$$ denotes the $$$k$$$-th fitting residual. We evaluated three M-estimator variants, namely the Huber, Cauchy and Welsch losses:
$$\rho_{Huber}(r_k^2) = r_k^2 \cdot \mathbb{I}(\vert r_k \vert<S) + (2S \cdot \vert r_k \vert - S^2 ) \cdot \mathbb{I}(\vert r_k \vert \geq S)$$
$$\rho_{Cauchy} (r_k^2) = S^2 \cdot ln(1 + r_k^2/S^2) $$
$$\rho_{Welsch} (r_k^2) = S^2 \cdot (1 - exp(-r_k^2/S^2) ) $$
where $$$\mathbb{I}(\cdot)$$$ is the indicator function, and $$$S$$$ is a user-defined parameter to adjust the tolerance to the residual error. In this work, we set $$$S$$$ adaptively: $$$ S = r_{mid}\cdot \mathbb{I}(r_{mid} < r_{th}) + S_{min} \cdot (r_{mid} \geq r_{th}) $$$, where $$$r_{mid}$$$ is the absolute MMSE residual error median. The error median $$$r_{mid}$$$ reflects the inlier residual distribution if a moderate number of outliers are present. However, a high proportion of outliers may cause severe failure of MMSE, and consequently an elevated $$$r_{mid}$$$ much higher than a predefined threshold $$$r_{th}$$$. In such cases, we set $$$S$$$ as a limited value $$$S_{min}$$$ to curb the estimator's tolerance to residual errors. The Huber loss is a combination of $$$l_1$$$- and $$$l_2$$$- norm losses and suppresses the influence of outliers if $$$r_k$$$ is large. In contrast, the other two loss functions, the redescending functions, ignore outliers completely as the residual error approaches infinity. The Welsch loss tends to ignore more outliers than the Cauchy loss at a fixed $$$S$$$.

In our experiments, we solved the optimization problem with the non-derivative-based simplex-search solver, with the initial value set as the MMSE estimate. The MMSE and M-estimator variants were evaluated on 50 MOLLI scans (3.0T Ingenia, Philips Healthcare, Best, The Netherlands), in total 131 short-axis slices, with or without motion correction (MOCO) ⁴, and on 12 SAPPHIRE scans (3.0T Siemens Magnetom Prisma, Siemens Healthineers, Erlangen, Germany) without MOCO, in total 119 short-axis slices. The SD error map was estimated for each T1 map using the definition in ⁵. The myocardium regions were manually delineated as the region of interest (ROI).

Results and Discussions

Figure 1-a) illustrates an example of parametric fitting with motion-induced outliers, where the Cauchy M estimator yields a robust fitting to the majority of measurements. The statistics of the mean SD error in the myocardium ROI are shown in Figure 1-b) and 1-c), respectively. For the MOLLI scans without MOCO, all three M-estimator variants reduced the mean myocardial SD error significantly. The largest reduction is observed for the redescending variants, the Cauchy and Welsch losses, from 49.3 ± 16.6 ms (MMSE) to 41.6±16.6 ms (Cauchy), p < 0.001 by paired T-test. On the SAPPHIRE scans, the mean myocardial SD decreased slightly from 72.6±8.3 ms to 70±6.5 ms with the Cauchy loss, with p=0.02. In comparison to the MMSE estimator, the Cauchy estimator had a bias of -12.9 ± 14.3 ms on motion-corrected MOLLI scans and 20 ± 12.2 ms on SAPPHIRE scans, in terms of the estimated myocardial T1 values. Figure 3 shows the estimated T1 and SD maps of an exemplar MOLLI and SAPPHIRE acquisition, both without MOCO. From the MOLLI images, it can be observed that the sharpness of the myocardial border (lower septum) slightly improved when using the M-estimators, with reduced SD error in the same region. However, the Welsch variant can potentially introduce noise susceptible voxels in the estimated T1 map, as the fit may become instable if too many measurement points are discounted.

Conclusion

Nonlinear parameter estimation in quantitative cardiac T1 mapping can be realized by the M-estimators, which are more robust to non-Gaussian noises and motion-induced outliers than the conventional MMSE. Our study showed that M-estimators were able to significantly reduce the SD error in T1 map estimation for both MOLLI and SAPPHIRE sequences, in comparison to the MMSE estimator. The Cauchy variant outperformed other variants in terms of SD error reduction and solution stability.

Acknowledgements

No acknowledgement found.