- Magnetic susceptibility is a material property that describes the degree of magnetization of a material when placed into an external magnetic field

- Magnetic materials may be classified as diamagnetic, paramagnetic, or ferromagnetic on the basis of their susceptibilities

- Tissue magnetic susceptibility can be anisotropic (e.g. white matter)

- Tissue susceptibility depends on the composition and microstructure of the tissue

- Susceptibility differences between tissues can be utilized to enhance vascular and other features due to modulation of the MRI signal induced by local phase shifts.

Scientists and clinicians interested in understanding the fundamentals of magnetic susceptibility and how it manifests in tissues.

Magnetic susceptibility is the physical quantity that describes the change of magnetization of a material in response to an applied magnetic field [1]. For isotropic, non-ferromagnetic (linear) materials, the magnetic volume susceptibility, χ, is defined by the linear relation

$$\overrightarrow{M}=\chi\cdot\overrightarrow{H}$$

where $$$\overrightarrow{M}$$$ is the magnetization or magnetic dipole moment per unit volume, $$$\overrightarrow{H}$$$ is the applied magnetic field intensity and χ is a scalar quantity. Whereas in free space, where there is no magnetization, the magnetic field can be described equally well by the vector fields$$$\overrightarrow{B}$$$ or $$$\overrightarrow{H}$$$, which are linearly related by $$$\overrightarrow{B}=\mu_{0}\cdot\overrightarrow{H}$$$ and thus scaled versions of each other, in magnetized matter the magnetic induction or magnetic flux density $$$\overrightarrow{B}$$$, and $$$\overrightarrow{H}$$$ are related by the equation

$$\overrightarrow{B}=\mu_{0}\cdot(\overrightarrow{H}+\overrightarrow{M})=\mu_{0}\cdot(1+\chi)\cdot\overrightarrow{H}$$

with $$$\mu_{0}$$$ being the magnetic permeability of free space. Since both $$$\overrightarrow{M}$$$ and $$$\overrightarrow{H}$$$ have the same units (A/m), the volume magnetic susceptibility is dimensionless. Introducing the relative permeability, $$$\mu_{r}=1+\chi$$$, the above relation can also be written as $$$\overrightarrow{B}=\mu_{0}\cdot\mu_{r}\cdot\overrightarrow{H}$$$.

In the general case, magnetic susceptibility is orientation dependent with respect to an external magnetic field and is then described by a symmetric second rank tensor $$$\widehat{\chi}$$$. Thus, the magnetization may depart from the direction of the magnetic field. Anisotropic susceptibility, which arises from asymmetry of orbitals or bonds has been observed in different biological structures of various sizes including e.g., helical proteins [2], lipid bilayers [3], retinal rods [4], or muscle fibers [5]. From the tensor $$$\widehat{\chi}$$$ different parameters can be extracted, such as the mean magnetic susceptibility $$$\overline{\chi}=\frac{1}{3}Tr(\widehat{\chi})$$$ or the axial magnetic susceptibility anisotropy $$$\overline{\chi_{aa}}=\chi_{11}-(\chi_{22}+\chi_{33})/2$$$.

In diamagnetic materials there are no magnetic moments in the absence of an external magnetic field. Only in the presence of an external field magnetic moments will be induced in the material and will create a finite magnetization. The induced magnetic moments are opposite to the inducing magnetic field (Lenz’s law), which means that the magnetic susceptibility of diamagnetic material is negative (χ < 0). Diamagnetic susceptibility is a generalized response of the orbiting paired electrons in a sample to the magnetic field. Most tissues are diamagnetic with very small susceptibility values. Due to the fact that diamagnetic susceptibility arises from the reaction of electrons in their orbitals to the presence of the magnetic field, this quantity is usually independent of temperature.Molecules may possess a permanent magnetic dipole moment, which typically arises from the presence of unpaired electron spin and the associated magnetic moments. In an external magnetic field these moments align in the direction of the applied field. The magnetic susceptibility is thus positive (χ > 0) and the paramagnetic response usually masks the weaker diamagnetic response of the material. Pure diamagnetic tissue response can thus only be observed if there are no magnetic moments present which is only possible, if the sum of spin moments and orbitals moments vanishes, i.e. is zero. The paramagnetic response of molecules to an external field is temperature dependent and the response decreases with increasing temperature.In ferromagnetism spontaneous magnetization occurs below a material specific temperature even in the absence of an external field. This is due to the quantum mechanical exchange interaction between permanent magnetic moment, which leads to a spatial orientation of the moments. Exchange interaction cannot be understood in classical terms, but is a consequence of the Pauli principle and the Coulomb interaction of electrons. The order of magnitude of this exchange interaction is typically in the range of 10 to 100 meV, which is much larger than classical dipole interactions between permanent magnetic moments that are on the order of 0.1 meV.

Magnetic susceptibilities of brain tissue are diamagnetic and rather small, typically only ±20 % of the susceptibility of water ($$$\chi_{H_{2}O}=-9.05\cdot10^{-6}$$$ in SI units) due to the abundance of the latter in the human body. The most important biophysical sources determining tissue magnetic susceptibility in vivo are water, lipid content, myelin, iron, and calcium. Calcium in tissues (calcifications) has substantially lower magnetic susceptibility than water. The susceptibility of hydroxyapatite (Ca²⁺), for instance, has been measured with NMR and found to be -14.83 ppm (6).

Iron

Iron can be a dominant paramagnetic contributor to tissue magnetic susceptibility in vivo (1,7). Iron in tissue is either present as storage iron (non-heme iron), such as ferritin and hemosiderin, or is bound to hemeproteins, such as in hemoglobin (heme iron) (1,7,8,9,10). In the healthy brain the total concentration ranges between 0 and approximately 200 µg/g tissue [11,12]. Correlation studies of non-heme iron content in deep gray matter nuclei using quantitative susceptibility mapping (QSM) have demonstrated strong correlations between magnetic susceptibility assessed by QSM and known iron concentrations in the brain [13-16]. The magnetic properties of hemoglobin, i.e. being paramagnetic in the deoxygenated and diamagnetic in its oxygenated state with a susceptibility difference between completely deoxygenated and completely oxygenated erythrocytes of $$$\triangle\chi_{do}=4\cdot\pi\cdot0.264 ppm$$$ [17] offers unique opportunities to probe oxygen metabolism in blood and tissues. Since the magnetic susceptibility χ of blood scales linearly with blood oxygen saturation Y, this enables non-invasive estimation of Y based on magnetic susceptibility in the venous vasculature [18-21].

Myelin

Myelin forms a compact multi-layered membrane by wrapping itself tightly around the axon and is predominantly present in white matter. It is a significant contributor to tissue magnetic susceptibility having a special membrane structure with high abundance of lipids (e.g., phospholipids [$$$\overline{\chi}=-9.68 ppm$$$, $$$\overline{\chi_{aa}}= -1.13 ppm$$$] [22], sphingolipids consisting of extremely long acids ranging from C₁₈ stearic acid [ $$$\overline{\chi}=-10.03 ppm$$$, $$$\overline{\chi_{aa}}= -1.22 ppm$$$] [23] to C₂₄ lignoceric acid), and proteins [24] that overall render myelin more diamagnetic than water. The contribution of myelin to the measured bulk susceptibility and its anisotropy has been demonstrated in a number of studies [25,26] and the latter property has been exploited in susceptibility tensor imaging to reveal white matter structure [27]. Reported values of susceptibility anisotropy in major fiber bundles determined from differences between parallel and perpendicular oriented white matter fibers ranged between 0.012 ppm and 0.022 ppm [26,28,29]. Recently, the anisotropy of the magnetic susceptibility of WM fiber bundles has been directly demonstrated experimentally by studying and analyzing the torque that tends to orient human spinal cord samples in a magnetic field in a direction that maximizes its magnetization [30].

Susceptibility variations in tissue or susceptibility differences between tissues due to different contributing constituents lead to spatial variations in the magnetic field which affects both phase (or frequency) and signal relaxation (R₂*) that can both be sensitively measured with gradient echo (GRE) imaging. The resonance frequency shifts occurring in response to these magnetic field variations offer then the possibility to extract information about the magnetic properties of tissue and potentially its structure from the MRI phase although this can become rather challenging due to the complexity of biological or human tissue. In particular, tissue composition as well as tissue microstructure and its orientation relative to the magnetic field will affect the MRI signal [31,32]. So-called structural anisotropy resulting from, e.g., orientation dependent frequency (or phase) shifts in elongated structures as well as susceptibility anisotropy of molecular tissue constituents (e.g. lipids in myelin) need to be considered and possibly taken into account when applying MR techniques, such as quantitative susceptibility mapping [13,14;19] or susceptibility tensor imaging [27], that aim to extract and quantify tissue magnetic susceptibility.