Quantitative susceptibility mapping (QSM) allows quantification of magnetic biomarkers such as iron, calcium and gadolinium in the brain, blood or liver ^[1]. Traditional QSM is generated from gradient echo (GRE) data using steps of field estimation, field/phase unwrapping, background removal and local dipole inversion ^[1] as shown in the left of Fig.1. Traditional QSM has a major drawback that an error in field unwrapping, often occurring at brain boundary due to low SNR, can cause severe artifacts in the final susceptibility map. Here we propose a novel QSM method directly from the complex GRE data (dcQSM) that requires neither field mapping nor phase unwrapping, by fitting both the background field and the local susceptibility distribution directly to original complex GRE data.

The optimization model for nonlinear background removal is:

$$\chi_B^*=arg\min_{\chi_B}\sum_j^{N_B}\parallel I_j - \mid I_j \mid e^{-i2\pi TE_j(d\star\chi_B)} \parallel_2^2 (1)$$

with $$$I_{j}$$$ the complex GRE image for the j-th echo, $$$TE_{j}$$$ the j-th echo time, $$$d$$$ the dipole kernel and $$$\chi_{B}$$$ the background susceptibility distribution responsible for background field inside the region of interest $$$M$$$. This is a nonlinear version of projection onto dipole field (PDF) ^[2] model but, additionally, fields are no longer explicitly obtained by instead fitting the susceptibility directly to complex images $$$I_{j}$$$, hence bypassing the field estimation as well as phase unwrapping. From the estimated $$$\chi_B^*$$$, the effect of the background inhomogeneity can be removed from the original complex GRE data:

$$ I_{Lj}=I_je^{i2\pi TE_j(d\star\chi_B^*)} (2) $$

Here the modified GRE data $$$I_{Lj}$$$ are complex images whose phase is induced by local field only. The local susceptibility $$$\chi_L^*$$$ in $$$M$$$ can now be directly estimated from this complex signal:

$$ \chi_L^{*}=arg\min_{\chi_L}\sum_j^{N_B}\parallel I_{Lj} - \mid I_{Lj} \mid e^{-i2\pi TE_j(d\star\chi_L)} \parallel_2^2+\lambda\parallel M_G\triangledown\chi_L \parallel_1 (3) $$

Here the binary mask $$$M_{G}$$$ is derived from anatomic magnitude image as in traditional QSM ^[1]. This dcQSM differs from traditional QSM ^[1] in that we are fitting to multi-echo complex signal rather than the local field. Both minimization problems (1) and (3) are solved using the iterative Gauss-Newton linearization and Conjugate Gradient search ^[1].

We applied the proposed dcQSM on an agarose phantom in which 4 balloons filled with Gadolinium solutions at prepared susceptibilities of 0.4, 0.8, 1.6 and 3.2ppm. The phantom was scanned at 3T (GE Healthcare, Waukesha, WI) with FA=15, FOV=18cm, 0.8$$$\times$$$0.8$$$\times$$$0.8 mm³ and echo spacing 3.5ms. We estimated mean susceptibility inside each balloon from the QSM. We also applied our proposed dcQSM on 5 human brain data to test its robustness against phase unwrapping problem. Subjects were scanned at 3T (Siemens Skyra), FA=15, FOV=24cm, 0.9$$$\times$$$0.9$$$\times$$$2 mm³ and echo spacing 5ms. For both phantom and in vivo experiment, dcQSM was compared with traditional QSM.

In the phantom experiment, as shown in Fig. 1, both traditional QSM and proposed dcQSM achieved comparable estimation accuracy for balloon susceptibilities (traditional QSM: 0.48, 0.84, 1.61, 3.13 ppm; Proposed dcQSM: 0.45, 0.84, 1.62, 3.25 ppm). In an example from in vivo experiments, the traditional method didn’t generate a correct local field due to phase unwrapping failure, resulting in a shadow artifact in QSM in Fig. 2. In contrast, the proposed dcQSM method reconstructed a susceptibility map without the artifact.Our proposed dcQSM removes the need for phase unwrapping by directly fitting the background field to the complex raw GRE data using nonlinear optimization (1). This dcQSM method achieved the same accuracy as the traditional method in phantom susceptibility estimation. In clinical MRI data, dcQSM is robust against the phase unwrapping errors sometimes seen with the traditional method. In the future work, we will focus on developing faster algorithm in solving our nonlinear background removal (1) and local susceptibility inversion problem (3).Our proposed dcQSM algorithm eliminates the need for phase unwrapping in traditional QSM.