This lecture for physicists and engineers will introduce the principles of susceptibility in MRI and the effect of susceptibility on image contrast and phase. We will discuss algorithms and image acquisition requirements necessary for acquiring quantitative maps of magnetic susceptibility.
The lecture is designed for physicists and engineers with an interest in:
Upon completion of this course, participants should be able to:
Any material that is exposed to a static magnetic field, such as that in an MRI scanner, is magnetized.3 The magnetic susceptibility $$$\chi$$$ is a dimensionless macroscopic physical property that characterizes the dependence of this (induced) magnetization on the externally applied field. Materials with negative susceptibility are called diamagnetic materials; whereas the susceptibilities of paramagnetic and ferromagnetic materials are positive. Most biomaterials are diamagnetic or weakly paramagnetic with susceptibilities between $$$\pm 1$$$ parts per million (ppm) and $$$\pm 10$$$ ppm ($$$\pm 10^{−6}$$$ to $$$\pm 10^{−5}$$$; $$$|\chi|\ll 1$$$).7 Susceptibilities of ferromagnetic materials are several orders of magnitude higher ($$$\chi > 1$$$); these materials are considered as MRI unsafe.
For non-ferromagnetic materials the total field $$$B$$$ can be written in a first-order approximation as a convolution integral:2-4,6
$$B(\boldsymbol r) = B_0 \cdot \int \chi(\boldsymbol r')\cdot b_\chi(\boldsymbol r - \boldsymbol r') \mathrm d^3 \boldsymbol r',\quad\mathrm{(1),}$$
where $$$ b_\chi(\boldsymbol r)=\frac{3cos^2\theta-1}{4\pi\cdot ||\boldsymbol r||^3} \quad (\boldsymbol r\neq 0)$$$ is the macroscopic unit-dipole function3 ($$$\theta$$$ is the angle between $$$\boldsymbol r$$$ and the applied magnetic field $$$\boldsymbol B_0$$$). Due to the Larmor relation, the MR signal frequency directly depends on the magnetic field at the site of the nucleus:1
$$ f_\mathrm L (\boldsymbol r) = -\frac{\gamma}{2\pi}\cdot B(\boldsymbol r), \quad \mathrm{(2)}$$
where $$$\gamma$$$ is the gyromagnetic ratio. The goal of QSM is the measurement of $$$B(\boldsymbol r)$$$ (via Eq. 2) and solution of Eq. (1) with respect to $$$ \chi$$$.
QSM relies on the same pulse sequence as the widely available Susceptibility Weighted Imaging (SWI) technique, a gradient recalled echo (GRE) sequence. GRE phase images reflect the local Larmor frequency variation and, hence, provide a very sensitive tool to map the magnetic field (Eq. 2). Raw (unfiltered) phase images acquired with a conventional clinical SWI protocol may in principle also be used for QSM. However, SWI usually employs highly anisotropic voxels (0.5x0.5x2mm3) whereas more isotropic voxels are preferred for QSM.
The optimal phase contrast-to-noise ratio (CNR) is obtained when the echo time (TE) equals T2*, which translates to optimal echo times between 20 and 50ms in the human brain at 3 Tesla. Since phase contrast is comparable at different field strength when $$$B\cdot\mathrm{TE}$$$ remains constant, higher field strengths allow choosing lower TEs and, hence, reduce the repetition time (TR) as well as the total acquisition time correspondingly. Consequently, a higher field strength is preferable and should be used if the choice exists.
Measured phase is always aliased into an interval of range $$$2\pi$$$. This phase aliasing needs to be resolved to calculate a field map from phase measurements; a procedure also referred to as phase unwrapping.5 A variety of different algorithms has been proposed for phase unwrapping.5
The Fourier convolution theorem allows a straightforward calculation of the field perturbation from a known susceptibility distribution in the Fourier domain (the “forward problem”) based on Eq. 1. To this end, one simply has to Fourier transform both the 3D unit dipole response and the susceptibility distribution and to point-wise multiply their 3D Fourier spectra:
$$\mathrm{FT}(\Delta B) = \mathrm{FT}(d_\chi) \cdot \mathrm{FT}(\chi).\quad \mathrm{(3)}$$
Using Fast Fourier Transform (FFT) algorithms, Eq. 3 may be evaluated within seconds even for very large matrix sizes. All current QSM algorithms rely on Eq. 3 to solve for the unknown $$$\chi$$$ based on measurements of $$$\Delta B$$$. Solution strategies vary in different aspects of their numerical implementation and mathematical properties of the obtained solution.
A major challenge of solving Eq. 3 is that for spatial frequencies along the magic angle ($$$\theta = 54.74^\circ$$$) the magnitude of $$$\mathrm{FT}(d_\chi)$$$ is relatively small and can even become zero, implying that corresponding spatial frequencies of the susceptibility distribution (right-hand side of Eq. 3) are attenuated in the observed field perturbation (left-hand side of Eq. 3). Since measurements are always affected by noise, it is practically impossible to reconstruct the heavily attenuated frequency components of the susceptibility distribution without additional assumptions or a priori knowledge on the (true) susceptibility distribution. This issue is also referred to as ill-conditioning of the inverse problem.
Multi-orientation QSM techniques9,10 rely on multiple field maps acquired with the object (e.g. the head in a brain examination) in different orientations relative to the main magnetic field. If the orientations are chosen properly, the susceptibility distribution may be reconstructed solely based on measured field data. However, the requirement for repositioning of the subject in the scanner and the associated prolonged acquisition times make it unlikely that such techniques will be translated to the clinics.
Reconstruction of susceptibility maps from clinical GRE scans requires that the ill-conditioning is numerically addressed. In other words, over-fitting of the susceptibility distribution to measurement noise needs to be prevented, because it would otherwise result in excessive amplification of the noise in the calculated susceptibility map. This is commonly achieved by introducing additional information on the susceptibility distribution; a technique that is also known as regularization.
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[2] SN Hwang and FW Wehrli. The Calculation of the Susceptibility-Induced Magnetic Field from 3D NMR Images with Applications to Trabecular Bone. J Magn Reson B, 109(2):126–145, 1995.
[3] LZ Li and JS Leigh. Quantifying arbitrary magnetic susceptibility distributions with MR. Magn Reson Med, 51(5):1077–1082, 2004.
[4] JP Marques and RW Bowtell. Application of a Fourier-based method for rapid calculation of field inhomogeneity due to spatial variation of magnetic susceptibility. Concepts Magn Reson Part B Magn Reson Eng, 25B(1):65–78, 2005.
[5] SD Robinson, K Bredies, D Khabipova, B Dymerska, JP Marques, and F Schweser. An illustrated comparison of processing methods for MR phase imaging and QSM: combining array coil signals and phase unwrapping. NMR Biomed (epub).
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[7] JF Schenck. Health and physiological effects of human exposure to whole-body four-tesla magnetic fields during MRI. Ann N Y Acad Sci, 649:285–301, 1992.
[8] F Schweser, SD Robinson, L de Rochefort, W Li, and K Bredies. An illustrated comparison of processing methods for phase MRI and QSM: removal of background field contributions from sources outside the region of interest. NMR Biomed (epub).
[9] T Liu, P Spincemaille, L de Rochefort, B Kressler, and Y Wang. Calculation of susceptibility through multiple orientation sampling (COSMOS): a method for conditioning the inverse problem from measured magnetic field map to susceptibility source image in MRI. Magn Reson Med, 61(1), 196–204, 2009.
[10] C Liu. Susceptibility tensor imaging. Magn Reson Med, 63(6), 1471–7, 2010.