Currently, QSM methods suffer from errors in the regularization assumptions and tissue boundary erosion ^[1] used for background field removal and from errors of shadow artifact caused by slow convergence to large local susceptibilities, such as brain hemorrhage ^[2]. Here we propose a novel approach to address these problems by incorporating background as a region of large susceptibility and by introducing preconditioning to accelerate convergence. Accordingly, the QSM over the entire field of view with a large range of susceptibilities is determined directly from the total field estimated from MRI data.

The proposed total field inversion problem is:

$$\chi^*=arg\min_{\chi}\frac{1}{2}\parallel w\left(Mf-M(d\star\chi)\right) \parallel_2^2 + \lambda \parallel M_G \triangledown\chi\parallel_1(1)$$

with $$$\chi$$$ the susceptibility map (both inside and outside the region of interest-ROI), $$$\star$$$ the convolution with the dipole kernel $$$d$$$, $$$M$$$ the mask for ROI, $$$f$$$ the total field estimated from multi-echo GRE images, $$$w$$$ the noise weighting, $$$\triangledown$$$ the gradient operator and $$$M_G$$$ the binary edge mask derived from magnitude image ^[3]. This is the combination of projection onto dipole field (PDF) ^[4] which is a background field removal method, and MEDI ^[3] which is for dipole inversion. The problem (1) is solved using Gauss Newton Conjugate Gradient ^[3]. In order to improve its convergence, we introduce preconditioner $$$P$$$ for (1):

$$y^*=arg\min_{y}\frac{1}{2}\parallel w\left(Mf-Md\star(P\cdot y)\right) \parallel_2^2 + \lambda \parallel M_G \triangledown(P\cdot y)\parallel_1(2)$$

and susceptibility is recovered with element-wise multiplication $$$\chi^*=P\cdot y^*$$$. $$$P$$$ is constructed as:

$$P_i=\begin{cases}1 & (i\in M\ \ and\ \ i\notin M_{HL}) \\30 & (i\in M_{HL}\ \ or\ \ i\notin M)\end{cases}(3)$$

The mask $$$M_{HL}$$$ is chosen by thresholding $$$R2^*$$$ map $$$(M_{HL}:=R2^*>50s^{-1})$$$, carving out large susceptibility regions (such as hemorrhage). The weight is constructed to reflect the difference in magnitude of susceptibility between regions, and values (30 and 1) are empirically chosen to improve convergence for brain QSM.

We applied the proposed method on a numerical brain phantom. True susceptibility $$$\chi_T$$$ was set to 9ppm outside ROI $$$M$$$, and 0 inside except for a single point source 0.1 ppm (Fig. 1). Both current QSM (PDF+MEDI) and unpreconditioned total field inversion (1) were applied on the total field $$$f$$$ generated by $$$\chi_T$$$, and the estimated susceptibility for the single point was compared with truth (0.1 ppm). This experiment was repeated with different point source locations, generating an error map indicating susceptibility accuracy for each method. We also performed total field inversion on human brains of 5 healthy subjects and 19 patients with brain hemorrhage. The in vivo data were scanned at 3T (16.0 GE, Waukesha, WI) with FA=15, FOV=24cm, 1$$$\times$$$1$$$\times$$$1 mm³ and echo spaceing 3.5 ms. For healthy subjects, both (1) and (2) were applied to show the effect of preconditioning. For all in vivo experiments, our current QSM routine, PDF+MEDI, was performed for comparison. Additionally, we applied the preconditioning scheme (3) without skull stripping, thereby generating whole head QSM.

In the numerical phantom simulations (Fig. 1), the proposed total field inversion produced less errors in susceptibility estimation than current method. For healthy subject (Fig. 2), preconditioning (2) accelerated convergence, reducing the required number of CG iterations in order to converge to reference QSM map (MEDI result) from 300 to 50. In hemorrhage patients (Fig. 3), preconditioned total field inversion (2) suppressed the large hemorrhagic shadow artifact appearing on PDF+MEDI. Fig. 4 showed that our preconditioned method (2) generated susceptibility maps for skull, air-filled sinuses, subcutaneous fat and skin in the head, as well as brain QSM comparable with MEDI.The proposed preconditioned method (2) reduced susceptibility inaccuracy/indeterminacy along tissue boundaries in standard QSM methods ^[1] by directly fitting the total field without partitioning the FOV into background and tissue. The preconditioner $$$P$$$ is consistent with the disparity between different tissue susceptibility levels and accelerated the CG convergence to a QSM solution free of shadow artifact in brain hemorrhage patients. Moreover, since the total field preserves the field generated by tissue outside the brain, we are able to render bone, air and all soft tissues in the head, including brain parenchyma using total field inversion.Our proposed preconditioned QSM method (2) handles background and local susceptibility in a single total field inversion, eliminating errors associated with background field removal and slow inversion convergence. This preconditioned QSM method (2) enables high quality susceptibility maps for challenging cases, e.g. intracerebral hemorrhage, and for the whole head rendering including skin, skull and air-filled sinuses.