Researchers interested in quantitative susceptibility mapping and white matterQuantitative susceptibility mapping (QSM) has been demonstrated to offer iron and myelin related contrast yet, some of the effects observed are dependent on the microstructural orientation of white matter (WM). The aim of this study is to combine the fiber orientation information retrieved from the DTI acquisition to compute susceptibility maps with a Lorentzian correction (LC)^[1]. This effect has been recently shown in ex-vivo studies to be dominant over the anisotropy of the susceptibility of myelin^[2,3]. The impact of the Lorentzian correction on the susceptibility values of major WM fiber bundles was studied.

Six subjects were scanned on a 3T scanner (Siemens,Germany) with a 32 channel head coil (NovaMedical) using the following sequences:

1.MP2RAGE:TR/TI1/TI2/TE=6000/700/2000/2340ms,res=1mm,Tacq=7min32sec

2.DWI-EPI:TR/TE=3490/74.6msec,res=1.5mm,matrix=150x150x90,b-value=1000s/mm2,Diffusion encoding directions=137,Taq=8min47sec

3.3D-GRE:TR/TE1-TE5=63/6.15-57.18ms,res=1.5 mm,iPAT=2x2,Tacq=2min39sec. This protocol was repeated up to 8 times with the subject’s head oriented along different orientations.

Relative head positions were computed by co-registration using FSL-FLIRT (www.fmrib.ox.ac.uk/fsl) as in^[4]. $$$R_2^{*}$$$ and field maps were computed as in^[4]. Rotation matrices from the previous co-registration were combined with DWI information to calculate orientation of fibres in respect to the static magnetic field $$$\vartheta_i$$$ for each of the head positions $$$i$$$. Hereby the primary and secondary fibre orientations were retrieved from DTI data by FSL-BEDPOSTX. Probabilistic tractography (FSL-PROBTRACKX) was used to extract regions of interest (ROI) based on the major WM fibres (cst-corticospinal tract, cg-cingulum, ilf-inferior longitudinal fasciculus, fmj-forceps major, fmi-forceps minor). Reference QSM maps were calculated with multiple orientation acquisitions (COSMOS)^[5]: $$\sum_i^R\parallel \delta B_{i}(\vec{r}) - FT^{-1}(DK_{i}\cdot FT(\chi_{COSMOS}(\vec{r}))) \parallel^{2}_{2}$$. Where $$$R$$$ is the number of acquired orientations, $$$\delta B_i$$$ the measured field map and $$$DK_i$$$ the dipole kernel for orientation $$$i$$$. $$$FT$$$ and $$$FT^{-1}$$$ denotes the Fourier and inverse Fourier transform, respectively. A WM mask was obtained using FSL-FAST on the MP2RAGE image. This mask was further refined by excluding deep grey matter (GM) regions based on QSM, $$$\chi_{COSMOS}$$$, and $$$R_2^{*}$$$ maps. In order to correct for the microstructural anisotropy effect, the fibre orientation was included using the Lorentzian correction^[1,2] in the following manner: $$\sum_{i}^R\parallel F_{ISO}(\chi^f_{ISO})+F_{LC}\chi^{f}_{LC,f} \parallel^{2}_{2}$$ $$$F_{ISO}=\delta B_{i}(\vec{r})-FT^{-1}(DK_{i}\cdot FT(\chi^F_{ISO}(\vec{r}))$$$ is equivalent to the QSM COSMOS formulation and $$$F_{LC}=M_{WM}\sum_{f}^F(3\cdot\sin^2(\vartheta^f_i)-2)\cdot\chi^{F}_{LC,f}$$$ accounts for the phase correction in presence of microstructural compartmentalization susceptibility in WM^[3]. $$$f$$$ refers to the number of WM fibre populations within each pixel. When LC was applied to the primary fibre orientation only, $$$f=1$$$, the method outputs an isotropic susceptibility $$$\chi^1_{ISO}$$$ and a susceptibility of the inclusions associated with the primary fibre orientation, PF, $$$\chi^1_{LC,1}$$$. When a secondary fibre orientation was used, $$$f=2$$$, a third susceptibility map is reconstructed associated with the secondary fibre orientation.

Isotropic susceptibility components for the three methods were similar in the cortical and deep GM, see Figure1(a,b,d-black arrows). The visible contrast in the $$$\chi_{iso}$$$ of white matter tracts, like fmj, on the COSMOS method was decreased when correction of the PF was applied. It is important to note that $$$\chi^1_{LC}$$$ has the same polarity throughout (physically meaningful). Also, a much sharper contrast is now present on $$$\chi^1_{LC}$$$ separating the large optic radiation from neighbouring fibres, see Figure1(c,e-white arrows). Figure2 shows that the correlation of the measured QSM inside deep GM remains high after first LC ($$$corr_{f=0,f=1}=0.97,r^2_{f=0,f=1}=0.91$$$ and decreases when the second LC correction is applied ($$$corr_{f=0,f=2}=0.91,r^2_{f=0,f=2}=0.80$$$). Mean values of QSM across subjects for $$$\chi_{iso}$$$ decrease for all WM fibres with increasing number of applied LC, Figure3(a,b,d). As for the PF components when the SF was added the main change was visible on cst while the other WM fibre components remained similar. The change of polarity in the susceptibility when doing SF correction (which is not physically meaningful) suggests that that term is fitting noise. This is specially the case in large fibre bundles as the cst, optic radiation and fmj, see Figure1(f-gray arrows). One potential improvement would be to create a different white matter mask for each of the LC. The SF will be only allowed if this population is significant and its orientation is different from the primary population by more than a given threshold. Although the isotropic susceptibility component changed with the number of applied corrections in WM, the correlation between the original method and the LC in deep GM structures remained high. By introducing the Lorentzian correction for the main fibre orientation, physically meaningful susceptibility maps were obtained with improved contrast between known fibre bundles. While it is known that various fibre populations exist in each pixel, trying to fit more than one population on our data and with the current methodology gave results with increased artefacts.