The interaction of electro-magnetic fields with tissues is mediated by Maxwell Equations and inherently related to the existing dielectric and magnetic tissue properties.
In this presentation we will cover some of the aspects known regarding: the physical mechanisms behind of magnetic susceptibility; the conductivity and electric permittivity at the frequencies of interest in MRI (MHz in the case of the resonating radio-frequency waves, and KHz in the case of the switching of encoding gradients).
The way these interactions influence not only the images acquired in MRI but also the comfort of subjects will be addressed.
Target audience:
Scientists and Clinicians interested in understanding the means of interaction of electromagnetic fields with tissue in the context of MRI, and its implications in image contrast and subject comfort.The motto of this abstract could be: “It’ s the Maxwell Equations, Stupid!". It is up to them to describe the interactions of electro-magnetic fields with Tissues. In this presentation we will focus on the effects associated with 3 of the Maxwell Equations:
$$ \nabla × \bf{H} -\epsilon_r \epsilon_0 \frac{\partial D}{\partial t}=\sigma E $$ [Eq. 1]
$$\nabla×E+\frac{\partial B}{\partial t}=0$$ [Eq. 2]
$$\nabla∙((1+\chi )\bf{H})=0$$ [Eq. 3]
The equations were written to explicitly highlight the role of both dielectric properties (relative permittivity, $$$\epsilon_r$$$ and conductivity,$$$ \sigma$$$) and magnetic properties (susceptibility, $$$\ch$$$i).
The first two equations (Ampére’s and Faraday’s Law(1)) are relevant to understand the interaction of RF waves with tissues. In the particular case where the dielectric properties are considered to be homogeneous in space, this equations can be used to derive the Helmohltz equation which is the workhorse of most electric property mapping methods presented to date (2).
In the absence of time varying electrical fields and in non-conductive environments, Ampèrs’s law, implies that a scalar potential field, $$$\Phi$$$, exists so that $$$ H=-\nabla \Phi $$$. In the case of a strong magnetic field aligned along z, the scalar potential field is given by $$$~-\frac{B_0}{\mu_0}z$$$. This information combined with Gauss’s law of Magnetism (Eq. 3) allows the computation of the static magnetic field induced by magnetic susceptibility in the presence of an external magnetic field (3).
Magnetic Susceptibility
Magnetic susceptibility, $$$\chi$$$, is the physical quantity that describes the change of magnetization (M) of a material in response to an applied magnetic field (H)
$$ \bf{M} = \chi \bf{H}$$ [Eq. 4]
For isotropic or amorphous, non-ferromagnetic (linear) materials susceptibility is a scalar implying that the induced magnetization be either aligned or anti-aligned depending on the material having a positive (paramagnetic) or negative (diamagnetic) susceptibility. In the general case, magnetic susceptibility is orientation dependent with respect to an external magnetic field and is described by a symmetric second rank tensor. Thus, the resulting magnetization may depart from the direction of the magnetic field.
Diamagnetism is present in all materials and arises from the effect of the external magnetic field on the electron orbits in molecules and atoms. Similar to Lenz Law, this induces a counteracting induced magnetic moment (negative susceptibility). Langevian theory of diamagnetism (valid for systems with closed shells, J=0) can be used to compute the diamagnetic moment. An electron in the presence of external field will precess at Larmor frequency, $$$ \omega=e B / (2 m_)$$$e . Magnetic moment can then be calculated as the product between the of current associated with the precessing electron and the orbital area perpendicular to the externally applied magnetic field. This simplistic image allows understanding why anisotropic susceptibility such as found in biological materials (helical proteins, lipid bilayers or muscle fibers (4,5)) is associated with diamagnetism. 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.
Paramagnetism
Molecules may possess a permanent magnetic dipole moment, which typically arises from the presence of unpaired electron spins (J \diff 0) and its associated net angular moment. Note that the magnetic moment of an electron is ~1836 times greater that of protons, making the effect of nuclear magnetization (used in NMR) negligible. In an external magnetic field there is a separation of the energy levels and these are populated in accordance to the Boltzmann distribution. The resulting moment is aligned in the direction of the applied field (resulting in minimum energy and $$$\chi > 0$$$), proportional to the magnetic field and temperature dependent (increasing with decreasing temperature).
Dielectric Properties
Electrical conductivity ($$$ \sigma$$$), reflects the ability to conduct charge and is thus the physical property that characterizes the current density (j) induced in the presence of an electric field, E. The electric permittivity ($$$ \epsilon $$$), measures how an electric field is affected by a dielectric medium. In electromagnetism, the electric displacement field D represents how an electric field E influences the organization of electric charges in a given medium, including charge migration and electric dipole reorientation:
$$ \mathbf {D} =\varepsilon \mathbf {E} =\varepsilon_0 \varepsilon_r \mathbf {E} $$ [Eq.5]
where the permittivity ε is a scalar for isotropic materials. If the medium is anisotropic, the permittivity is a second rank tensor. Both these dielectric properties are highly frequency dependent with the relative permittivity decreasing with increasing frequency while the conductivity increases with increasing frequency. The dielectric spectrum of a tissue is characterized by three main relaxation regions $$$\alpha$$$, $$$\beta$$$, and $$$\gamma$$$ at low (<1KHz), medium (100KHz-10MHz) and high frequencies (>1GHz),and other minor dispersions such as $$$\delta$$$ (~100MHz) (6,7). While in the lower frequencies dispersion is associated with interfacial polarisations with electric double layers and surface ionic conduction effects at membrane boundaries. For frequencies >100MHz, the capacitive shorting out effects of cell membranes take place, resulting in a more homogeneous electric structure. In this regime the dielectric characteristics of tissues reflect the properties of intra- and extra-cellular electrolytes and water concentration (8). In its simplest form, each of these relaxation regions is the manifestation of a polarization mechanism characterized by a single time constant, \tau which, to a first order approximation is described by a Debye expression (6). Thus, to describe the full frequency spectrum, superposition of broadened versions of the Debye expression, known as the Cole-Cole equation, have been used to characterize dielectric properties of biological tissues using data from ex vivo samples probed with open ended co-axial dielectric probes.
In the brain, the relative permittivity is expected to lie in the vicinity of 52, 73, and 84 for GM, WM, and CSF, respectively while the conductivity is 0.34, 0.59, and 2.14 S/m (9).
Interactions of the External Magnetic Field with Magnetic Susceptibility
The induced magnetization due to the dia- and paramagnetic materials present in tissues are very small compared to the external magnetic field applied in MRI (<1ppm). As a result, it is possible to derive from Eq. 3 that the magnetic field perturbation is simply a convolution of the field generated by a magnetic dipole with that of the magnetic susceptibility distribution. In MRI we are only sensitive to static field perturbations parallel to the externally applied magnetic field and the magnetic dipole projection in that direction can be analytically described in k-space (3,10) allowing fast calculations of the perturbation of magnetic field induced by magnetic susceptibility. Similar – or equivalent - descriptions of the magnetic field perturbation of an arbitrary (or not) shape of magnetic susceptibility have been extensively used to understand image distortions in MRI (11), T2* decay (12), understand phase contrast in gradient echo imaging and perform quantitative susceptibility mapping (13). While it is expected that the physical model relating susceptibility and field perturbation holds for biological tissues, it is increasingly clear that the sampling of magnetic field perturbations as done by MRI is a biased towards compartments rich in water. This bias breaks some of the assumptions inherent to the model used in QSM (14,15).
Interactions with RF waves with Dielectric properties
The dielectric properties at the Larmor frequency play an important role in the propagation of radiofrequency waves into the subject’s body. To a first order approximation, dielectric properties can be seen as dampening and inducing changes in the wave lengths of the rf. One of the main consequences of tissue dielectric properties, as we move to higher magnetic field strengths, is the observation of more inhomogeneous transmit field resulting from constructive and destructive interferences of the propagating rf waves. This increase in inhomogeneity increases the sensitivity electrical property mapping methods that explore the spatial variation of the transmit (or receive) fields (16).
Other Interactions: Vertigo, peripheral nerve stimulation and SAR
There are a variety of potential mechanisms for interaction between the human body and strong, static or slowly-varying magnetic fields, including those based upon electromagnetic induction, magneto-mechanical effects and electron spin interactions (17). However, to date none of these have shown to have clinically significant, long-term or short-term, health effects at currently accessible field strengths (18).
The most commonly reported effect by subjects is dizziness. This magnetic field induced vertigo (MFIV), results from interaction between the magnetic field, and the vestibular system of the inner ear that is responsible for balance. Suggested causes of this effect include: currents induced in the vestibular system due to movements that change the flux (e.g. translation through the region of large field gradient at the end of the magnet or head rotation in the uniform field at the centre of the magnet); forces on structures in the inner ear that have different magnetic susceptibility from the surrounding in the presence of large gradients in the static field magnitude; and magneto-hydrodynamic forces on fluid moving in the inner ear (19). Recently been shown that the most probable cause of MFIV is the Lorentz force on the ionic currents that flow continuously in the endolymph fluid within the semi-circular canals (20). In the presence of a strong field the Lorentz force produces small deflections of the cupulae mimicking the effect of head rotation, thus leading to feelings of vertigo. RF power deposition increases with field strength (due to the increase in radiofrequency) and at high field, because of the dielectric properties, the RF power deposition becomes less uniform, leading to local SAR hotspots (21,22). SAR is currently one of the biggest limitations of ultra-high field imaging.
One last form of interaction not mentioned up to now is that of the fast switching gradients and tissues resulting in peripheral nerve stimulation (PNS). PNS results from the induction of electric fields in the tissue by temporally varying magnetic fields. When this electric fields exceed the polarization threshold of a nerve, nerve stimulation occurs (23). This effect is peripheral because charges accumulate at the surface of conductors. PNS poses constraints in the design of gradients coils (concomitant magnetic field components along x and y contribute to PNS, favoring head gradient inserts) and limit fast imaging by putting upper bound on the switching rate of a given gradient coil (24).
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