This talk will describe: what magnetic susceptibility is; how we measure magnetic susceptibility using MRI; and some ways of validating methods for measuring magnetic susceptibility.
Magnetic susceptibility, χ, is the property of a material that describes how easily it can be magnetized when exposed to an applied magnetic field (1). In most materials, any induced magnetization disappears when the magnetic field is removed (the exception being ferromagnetic materials that are not so relevant for biomedical MRI), and the strength of the magnetization, M, is proportional to the applied magnetic field, H. The induced magnetisation produces an additional magnetic field that affects the evolution of the magnetic resonance signals. This provides a way of making MR signals sensitive to the magnetic susceptibility (2).
Most materials, including water, have an intrinsic diamagnetism, which means that induced magnetisation opposes the applied magnetic field and reduces its strength. Some materials, which generally contain constituents with unpaired electrons, also display magnetisation that is parallel to the applied field (3). Endogenous iron-containing compounds, such as deoxy-haemoglobin in the blood, and ferritin, which is the storage form of iron, are the biggest source of paramagnetism in the body (4). Haemosiderin and other blood-breakdown products formed following a bleed are also paramagnetic, as are MRI contrast agents containing gadolinium ions or iron oxide particles. The strength of diamagnetism depends on a material’s molecular structure, but is also linked to its density. This means, for example, that cancellous bone is more diamagnetic than water and fat is less diamagnetic. To add to the complexity, the magnetic susceptibility can also be anisotropic in microstructurally-ordered materials, meaning that the induced magnetization and the related field perturbation depends on the orientation of the microstructure with respect to the applied field (5). This effect is most strongly manifested in vivo in white matter tracts, as a result of the ordered arrangement of lipid chains in the myelin sheaths (6).
In biological tissues the magnitude of the magnetic
susceptibility lies in a range of a few ppm around the -9 ppm susceptibility of
water (4). The magnetic
field perturbations produced by the magnetic susceptibility are consequently
very small. The largest field variations arise near interfaces between air,
which has a susceptibility of ~ 0 ppm, and tissue.
How do we measure susceptibility?
Measurement of magnetic susceptibility using MRI relies on characterising the field perturbations generated by the induced magnetisation, and knowing the mathematical relationship between the spatial pattern of field perturbation and the distribution of magnetic susceptibility. Fortunately, the phase of the MR signal, φ, is extremely sensitive to magnetic field perturbations, so magnetic susceptibility measurements usually start from a measurement of the phase of a gradient echo signal: at echo time, TE, the phase accumulated due to a field offset, ΔB is given by γ ΔB TE (7). Unfortunately, there are many other sources of field variation which can easily produce effects that are much larger than those generated by the magnetic susceptibility of the tissue of interest. These sources include imperfections in the magnet, and effects of body structures (particularly air/tissue interfaces) outside the region of interest, and also some sources of phase variation that do not depend on TE, but which can vary with spatial position. Sources of non-TE-dependent phase offsets, include receive and transmit RF phase effects, which become more significant at high field, as well as errors in sequence timings. In addition, the measured phase is only defined modulo 2π, so to reveal the true range of phase variation that is present in an image, the GE phase images have to be unwrapped (8). The unwrapped phase is needed to make inferences about patterns of field perturbation. Non-TE-dependent phase variation can be eliminated by using multi-echo gradient sequences, which allow the variation of phase with TE to be calculated, and the phase variation due to sources outside the region of interest can be eliminated by a process called “background field removal” (9). There are a number of different ways of carrying out background field removal, all generally relying on the different characteristics of the fields produced by sources inside and outside the region of interest (ROI): in particular the field from a source outside the ROI must be harmonic (i.e. it satisfies Laplace’s equation), whereas that from a source inside the ROI does not satisfy this condition throughout the ROI. The background field removal process generally requires selection of an ROI – e.g. the brain – and the separation of fields from internal and external sources is problematic in voxels at the periphery of the ROI (9).
Having unwrapped the phase and removed the background field,
the remaining field perturbation can be related directly to the magnetic
susceptibility distribution inside the ROI (1). This is done by assuming that each
voxel behaves like a small piece of magnetization that is aligned with the
applied field: the field perturbation at position r outside the voxel then has the characteristic dipolar
form, with an amplitude that depends on the amount of magnetization in the voxel (which is given by χΔVB0/μ0 when a tissue voxel of volume ΔV and susceptibility χ is exposed to a magnetic field of strength B0, i.e.
$$ \Delta B (\textbf{r}) =\frac{B_0 \chi \Delta V}{4\pi}\frac{(3cos^2\theta -1)}{r^3} \hspace{1cm} [1]$$
where, r and θ are the usual spherical polar co-ordinates.
The total field in the sample is given by convolving the susceptibility distribution with the dipolar field (plus an additional Lorentz-sphere correction term which takes account of the fact that the field within a voxel due to its own magnetization is not well characterized by Eq. [1]). The convolution simplifies on Fourier transformation, giving rise to a simple invertible relationship between the field and susceptibility in the Fourier domain (10):
$$\frac{ \Delta B (\textbf{k})}{B_0}=\left( \frac{1}{3}-\frac{ {k_z}^2} {k^2} \right)\chi(\textbf{k}) = D(\textbf{k})\chi(\textbf{k}).\hspace{1cm}[2]$$
Equation [2] forms the basis for calculating the susceptibility distribution from the measured field variation, but simple inversion of this equation is not so straightforward because the Fourier-domain dipolar kernel (D(k)) goes to zero on two conical surfaces which are oriented at the “magic-angle” (54.7o) to the positive and negative z-axes. This leads to noise amplification and artefacts in the calculated susceptibility map. These artefacts can be reduced by artificially limiting the minimum value of the dipolar kernel, in a process known as thresholded k-space division (TKD) (11). An alternative approach involves acquiring multiple images with the object oriented at different angles with respect to the applied magnetic field, B0. Rotating the object with respect to the field, rotates the Fourier transform (FT) of the dipole kernel (and hence the conical surfaces) with respect to the FT of the object. The susceptibility distribution can consequently be found at all locations in k-space using the field data acquired at multiple orientations. This COSMOS (Calculation of susceptibility through multiple orientation sampling) approach (12) is sometimes considered as the gold-standard in QSM, but does require that the anatomical region of interest for imaging (usually the head) can be rotated and held at different orientations to the field.
An additional issue that arises when simply inverting Eq. [2] to find χ(k), is that the field variation is only known in the region of interest, so the value of ΔB(k) derived from the measured field perturbation is an approximation. This issue can be dealt with by formulating the susceptibility calculation as a minimization problem, in which the deviation between the field variation measured in the ROI and that calculated using Eq. [2] is minimized by adjusting the susceptibility distribution inside the ROI (1). Posed in this way, the problem can be addressed using a range of numerical techniques, and it is also possible to incorporate different forms of regularization to stabilize the calculation of susceptibility from phase images acquired at a single orientation. The regularization can be based on information from other images: for example, it can involve penalising spatial variations in magnetic susceptibility which do not correlate with changes in the intensity of the magnitude GE image that is acquired simultaneously with the phase data (13). A range of different methods for phase unwrapping, background field removal and susceptibility calculation have been developed, and new methods (e.g. those based on deep learning) are still emerging. At this stage a consensus on the best approach to QSM data processing has not yet been formed.
All QSM methods do share a common characteristic, that the calculated susceptibility values are relative rather than absolute. This behaviour derives from the indeterminate form of Eq. [2] at k = 0 and also the somewhat arbitrary setting of the absolute resonance frequency during image acquisition. It means that when quoting susceptibility values it is important to indicate to what they are referenced, or to focus on differences in susceptibility between regions in the same susceptibility map. Common choices for the reference include the average susceptibility in the whole brain or the susceptibility in large regions of CSF in the ventricles. In addition, the QSM methods that are in common use assume that the susceptibility is isotropic at all locations, and so may be subject to errors in the presence of significant anisotropic susceptibility (e.g. close to large myelinated tracts in the brain) (14). An additional issue is the effect of microstructure in which there are ordered structures of different susceptibility than their surroundings which generate no (or a weak) NMR signal (15, 16). When these inclusions have a size that is larger than the diffusion length-scale their presence can break the simple link between the local field perturbation and the measured phase .
How to validate methods for measuring magnetic susceptibility?
QSM methods can be validated at several different levels. Processing methods for phase-unwrapping, background field removal and susceptibility calculation can be tested by using simulated data. For these, field maps can be calculated from model structures using Eq. [2], followed by the addition of background fields and noise. The whole processing pipeline can be tested using phantoms containing structures with a known susceptibility difference from water (1). Solutions containing gadolinium chelates or other paramagnetic species can be used for this purpose, taking advantage of their well characterised molar magnetic susceptibilities. Working with aqueous solutions requires confinement of the paramagnetic solution inside a container whose walls will also generate field perturbations (and no NMR signal), complicating matters. Use of agarose gel which has been loaded with different amounts of superparamagnetic iron oxide nanoparticles allows the production of phantom with a well controlled susceptibility variation and no boundaries between regions (11), although it is important to bear in mind the saturation of magnetization in these nanoparticles. While allowing the evaluation of the full process of formation of a susceptibility map from measured phase images, such phantoms do not reflect the complex microstructure of biological tissue. This can be accessed by making susceptibility measurements via QSM on post mortem tissue and then making comparison to measurements of susceptibility using other techniques. However, other susceptometry techniques, generally provide only a single bulk value of susceptibility in a small sample, and require extraction of a small sub-sample for testing. This requirement, and the strong sensitivity to the mass of the extracted sample, make comparisons to QSM difficult. When the focus is in on variation of susceptibility across tissue regions due to variation in the concentration of iron (or other elements), comparison of QSM measurements on post mortem tissue with measurements of elemental concentration made using laser ablation inductively coupled plasma mass spectrometry (17) or proton induced x-ray emission have been more productive (18). In the case of in vivo measurements, there is no non-MRI based gold standard susceptibility measurement technique with which to make comparisons, but comparisons of QSM measurements made on the same living subjects using different scanners and/or processing techniques are useful for testing the efficacy and robustness of different methods (19-22).
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