Projects:QuantitativeSusceptibilityMapping

Quantifying magnetic susceptibility in the brain from the phase of the MR signal provides a non-invasive means for measuring the accumulation of iron believed to occur with aging and neurodegenerative disease. Phase observations from local susceptibility distributions, however, are corrupted by external biasfields, which may be identical to the sources of interest. Furthermore, limited observations of the phase makes the inversion ill-posed. We describe a variational approach to susceptibility estimation that incorporates a tissue-air atlas to resolve ambiguity in the forward model, while eliminating additional biasfields through application of the Laplacian. Results show qualitative improvement over two methods commonly used to infer underlying susceptibility values, and quantitative susceptibility estimates show better correlation with postmortem iron concentrations than competing methods.

Description

There is increasing evidence that excessive iron deposition in specific regions of the brain is associated with neurodegenerative disorders such as Alzheimer's and Parkinson's disease [1]. The role of iron in the pathogenesis of these diseases remains unknown and is difficult to determine without a non-invasive method to quantify its concentration in-vivo. Since iron is a ferromagnetic substance, changes in iron concentration result in local changes in the magnetic susceptibility of tissue. In magnetic resonance imaging (MRI) experiments, differences in magnetic susceptibility cause perturbations in the local magnetic field, which can be computed from the phase of the MR signal (in a gradient echo sequence, the observed field is proportional to the MR phase).

The field perturbations caused by magnetic susceptibility differences can be modeled as the convolution of a dipole-like kernel with the spatial susceptibility distribution. In the Fourier domain, the kernel exhibits zeros at the magic angle, preventing direct inversion of the fieldmap [3]. Critically, limited observations of the field make the problem ill-posed. The observed data is also corrupted by confounding biasfields (ie. those from tissue-air interfaces, mis-set shims, and other non-local sources). Eliminating these fields is critical for accurate susceptibility estimation since they corrupt the phase contributions from local susceptibility sources.

In general, methods that rely heavily on agreement between observed and predicted field values computed using kernel-based forward models [2, 3, 6] are inherently limited since they cannot distinguish between low frequency biasfields and susceptibility distributions that are eigenfunctions of the model. Examples of such distributions include constant, linear, and quadratic functions of susceptibility along the main field (ie. 'z') direction. Applying the forward model to these distributions results in predicted fields that are proportional to the local susceptibility sources, but also identical in form to non-local biasfields (ie. those produced by a z-shim). Therefore, removing all low frequency fields prior to susceptibility estimation will eliminate the biasfield as well as fields due to the sources of interest, potentially preventing accurate calculation of the underlying susceptibility values. In contrast, inadequate removal of the biasfield may result in the estimation of artifactual susceptibility eigenfunctions in areas where the biasfield is strong, such as regions adjacent to tissue-air interfaces. This suggests that additional information such as boundary conditions or priors may be necessary to regularize an incomplete forward model and prevent the mis-estimation of low frequency biasfields.

We present a variational approach for Atlas-based Susceptibility Mapping (ASM) that performs simultaneous susceptibility estimation and biasfield removal using the Laplacian operator and a tissue-air susceptibility atlas. In [7, 8, 6] it was shown that applying the Laplacian to the observed field eliminates non-local biasfields due to mis-set shims and remote susceptibility distributions (ie. the neck/chest). In this method, large deviations from the susceptibility atlas are penalized, discouraging the estimation of artifactual susceptibility eigenfunctions in regions near tissue-air boundaries where the Laplacian may not be sufficient to eliminate the contribution of non-local sources and substantial signal loss corrupts the observed field. Agreement of predicted and observed fields within the brain is also enforced, but deviations in estimated susceptibility values outside the brain are not penalized, allowing values at the boundary to vary from the atlas-based prior to account for unmodeled external field sources (ie. shims).

Results

The method is evaluated by comparison of susceptibility maps estimated using ASM to results from Susceptibility Weighted Imaging (SWI) and Field Dependent Relaxation Imaging (FDRI). In SWI, a filtered phase map is obtained by applying a high-pass filter to the phase data, and the resulting SWI map is commonly used as a proxy for susceptibility. While SWI has shown some correlation with magnetic susceptibility differences due to iron and other sources, the phase maps it yields are only an indirect measure of susceptibility due to the non-local effects of the convolution kernel. In addition, the filtering process may remove some low frequency fields due to sources inside the brain. In FDRI, R2 maps are acquired at two different field strengths (ie. 1.5 and 3 Tesla) and the difference in R2 divided by the difference in field strength gives the FDRI. The mean FDRI in several regions of interest was previously compared to the mean iron concentration obtained from postmortem analysis and showed stronger correlation with iron content than the SWI maps computed for the same subjects. Obtaining FDRI measurements would be impractical for most studies, however, since it requires images to be collected on two separate scanners. In this work, quantitative results are obtained by comparison of mean susceptibility values in the thalamus (TH), caudate (CD), putamen (PT) and globus pallidus (GP) to corresponding results from SWI, FDRI and postmortem data.

ASM results for a young subject are shown in Fig. 1. Column 1 shows the T1 structural (row 1) and acquired fieldmap (row 2). Application of the Laplacian to the field map (row 2, column 2) removes substantial B0 inhomogeneities that bias the observed field. The susceptibility atlas is shown in row 1, column 2 and estimated external sources are shown in row 1, column 3. The estimated susceptibility map (row2, column 3) shares high frequency structure with the Laplacian of the observed field, while low frequency structure is preserved by enforcing agreement with additional information provided by the atlas-based prior and observed field.

Fig. 1: ASM Results. The first column shows the T1 structural image (row 1) and field map (row 2) with substantial inhomogeneity that was obtained from a young subject. Column 2 shows the susceptibility atlas (row 1), in which voxels take continuous values between [0,1] corresponding to susceptibility values between air and tissue. Taking the Laplacian of the fieldmap successfully eliminates biasfields (row 2, column 2). Estimates of external sources are shown in row 1, column 3. The estimated susceptibility map (row 2, column 3) shares similar high frequency structure with the Laplacian of the observed field while low frequency structure is preserved by enforcing agreement with the atlas and observed field.

Fig. 2 shows results from FDRI (row 1, column 2), SWI (row 1,column 3), ASM for a young subject (row 2, column 3), and ASM results for 2 elderly subjects (row 2, columns 1,2). The FDRI shows strong constrast between the ROIs and surrounding tissue, but less high frequency structure than the SWI. The SWI retains high frequency phase effects, but indiscriminately removes low order fields from both internal and external sources, resulting in artifactual low frequency structure. The ASM method accurately preserves the high frequency phase effects seen in SWI while showing improved estimation of low order susceptibility distributions. In addition, ASM provides direct estimates of susceptibility values rather than filtered phase values that serve as proxies for susceptibility.

Fig. 2: Comparison of Results. Row 1 shows the T1 structural image (column 1), FDRI (column 2) and SWI (column 3) results for a young subject. ASM results are shown in row 2 for young (column 3) and elderly (columns 1,2) subjects. The FDRI shows strong constrast between ROIs and adjacent tissue, but less high frequency structure than the SWI. The SWI retains high frequency phase effects, but indiscriminately removes low order fields from both internal and external sources, resulting in artifactual low frequency structure. ASM accurately preserves the high frequency structure seen in SWI while showing improved estimation of low order susceptibility distributions.

Quantitative results from ASM and previously reported results from FDRI and SWI for the same elderly subjects are shown in Fig. 3. The mean susceptibility values (relative to tissue susceptibility) in each ROI from all elderly subjects are plotted against the corresponding iron concentrations from postmortem analysis (only the mean and SD in each ROI was reported in [17]). ASM shows a high correlation with postmortem values, which is comparable to that seen in FDRI and substantially better than the correlation between phase and iron concentration obtained with SWI. In addition, for the structures that we analyzed, ASM results compare favorably to the correlation between postmortem iron and susceptibility estimates in corresponding ROIs computed from multi-angle acquisitions [6].

Fig. 3: Quantitative Results. The Mean +/- SD iron concentration (mg/100g fresh weight) in each ROI determined from postmortem analysis [17] is plotted on the x-axis. The y- axes show the Mean +/- SD FDRI (s^{-1}/Tesla), Mean +/- SD SWI (radians), and Mean +/- SD ASM susceptibility (ppm). Mean susceptibility values from ASM show a high correlation with the postmortem data, which agrees well with previous results from FDRI and shows improvement over SWI values reported for the same data [5].

Key Investigators

• MIT: Clare Poynton, Elfar Adalsteinsson, Polina Golland
• BWH/Harvard: William Wells
• Stanford: Adolf Pfefferbaum, Edith Sullivan