The elimination of field contributions that are not generated by sources inside the region of interest (ROI), so called background fields, is an essential step in phase MRI and QSM. Background fields are perturbations caused primarily by the susceptibility difference between tissue and air, often resulting in field perturbations one to two orders of magnitude stronger than tissue-related fields.

A variety of different background elimination methods has been published with different underlying assumptions. Unfortunately, it is unclear how the various methods perform relative to reach other and what their respective strengths and weaknesses are, because a multi-center quantitative comparison has not yet been carried out.

In this work, we quantitatively compare inverse Laplace filtering (iLF)¹, SHARP², V-SHARP^3, iSMV⁴, LBV⁵, HARPERELLA⁶, iHARPERELLA⁷, PDF^8,9, and RE-SHARP¹⁰ in a collaborative effort.

To assess the various methods we applied them in a realistic numerical human torso model (1mm isotropic; Fig. 1, top-middle). The numerical model allowed us to compare the background corrected fields against a ground truth (Fig. 1, top-right) obtained by embedding the brain of the susceptibility model into a homogeneous susceptibility equal to the average of that of the brain.

To achieve the optimal possible result with each technique, the authors of the respective original publication of each algorithm applied their own implementation. Algorithmic parameters (where they existed) were optimized in a standardized way by calculating for each parameter setting the normalized root-mean-squared error (NRMSE) between the ground truth and the reconstructed internal fields in 32 brain-shaped shells of 1 mm thickness located within 1 to 32 mm from the brain’s surface. The parameter setting with the smallest sum of all 32 NRMSE values was considered as optimal. To ensure comparability, we used the same binary ROI mask for all techniques (Fig. 1, top-left).

For RESHARP the optimal regularization parameter, $$$\lambda$$$, was zero, which can be explained by the absence of non-harmonic background fields in the model, effectively turning RESHARP into a minimum norm solution.

Figure 1 shows exemplary slices of the background-corrected fields (middle) along with the respective differences to the ground truth (bottom). Figure 2 shows a plot of the NMRSE values as a function of the distance from the brain’s surface.

All techniques yielded comparable results with relatively smooth error patterns (Fig. 1) and decreasing error with increasing distance from the brain's surface (Fig. 2). Only iLF showed strong deviations from the ground truth, which can be attributed to the involved regularization. iSMV and LBV outperformed all other approaches with respect to nominal NMRSE values (Fig. 2-black), followed by V-SHARP, iHARPERELLA, PDF, and RESHARP. SHARP, and HARPERELLA resulted in the largest errors. The error patterns and NMRSE values of iSMV and LBV were similar and both methods did not suffer from the localized inhomogeneity at the sagittal sinus (arrows), reflecting that they solve the exact same mathematical problem.

Deviations are smaller in SHARP (R=15mm) compared to iLF (Figs. 1 and 2-red) because the more extended kernel of SHARP requires less regularization. V-SHARP extends the spatial support of the corrected field map compared to SHARP and its error pattern looks similar to that of the other methods (Fig. 1-bottom-right), but it produces inaccurate values close to the boundary (arrow). However, while these errors are discernible on the difference image, they do not result in a substantial deviation from other methods in the quantitative analysis (Fig. 2), where V-SHARP was on par with iHARPERELLA, PDF, and RESHARP (blue).

This is the first major multi-center effort to systematically compare all major background elimination techniques.

The collaborative design of our study and the systematic optimization of algorithmic parameters ensured that all algorithms were implemented correctly, yielded optimal results, and results are representative. While this ensured comparability of the results, it has yet to be clarified to what degree residual background fields or (regularization) artifacts actually propagate to susceptibility maps. Some QSM algorithms may be able to attribute certain residual background fields to susceptibility variations outside the ROI, not affecting the obtained susceptibility values inside the ROI. A thorough investigation of the propagation of residual background fields into susceptibility maps and practical aspects of finding the optimal algorithmic parameters for each technique will be subject to future research.