Chapter 15
# Principles of radiology modalities: Magnetic Resonance Imaging

## Basics of Magnetic Resonance Imaging

## Instrumentation

## Complementary MRI and PET approaches for Medical Applications

Atoms with an odd number of protons and/or odd number of neutrons, possess a spin angular momentum which is simply referred to as spin. Qualitatively, these spins can be visualized as spinning charged spheres giving rise to small magnetic moments, a stack of compass needles randomly pointing along all spatial directions. When the spins are exposed to a static magnetic field, they orient themselves along the magnetic field lines, similar to compass needles, and create a net magnetic moment. As soon as the net magnetic moment is tipped away from the direction of the static magnetic field, which we denote B0, the effective magnetization vector precesses in the transverse plane orthogonal to the direction of the static magnetic field. This is the nuclear magnetic resonance (NMR) phenomenon discovered by Felix Bloch and Edward Purcell in 1946.

Magnetic resonance imaging (MRI) adds spatial dimensions to a 1D-NMR experiment and allows for spatially resolved NMR signals. Due to its soft tissue contrast and non-invasiveness, MRI has become a standard method in medical diagnostics. The basic principle of MRI is that the precession frequency of the nuclear spins in a static magnetic field, also denoted as Larmor frequency, can be made a function of space by the application of linear magnetic field gradients in three orthogonal directions. This idea was published by Sir Peter Mansfield and Paul C. Lauterbur in 1973, and they received the Nobel Prize in Physiology or Medicine for this concept in 2002[124,125]. The relationship between the Larmor frequency of spins and their spatial coordinates in the presence of a magnetic gradient is the basic equation of image formation:

*ω*(r)= *γ*B_{0} + *γ*G·r *(Eq. 1)*

It is worth noting that the Larmor frequency, ω, is effectively only modified by additional gradient field components parallel to B_{0}, since the gradient fields are several orders of magnitude smaller than B_{0} (γG << B_{0}). Horizontal gradient fields just slightly tilt the net field direction but do not influence the Larmor frequency in standard applications.

Considering nuclear spins within an external gradient field G at spatial position r with a sample volume dV and local spin density ρ(r), the MR signal dS from the spins can be expressed by:

*d*S(G,*t*) ~* **ρ*(r)*d*V exp(i*ω*(r)*t*) *(Eq. 2)*

This approximation neglects any relaxation effects. It holds for strong gradients ensuring that the phase spread γG·r is much more rapid than the relaxation decay. Taking the spatial dependence of the Larmor frequency by external field gradients into account, equation 2 can be written as:

*d*S(G,*t*) ~* ρ*(r)*d*V exp(i[*γ*B_{0} + *γ*G · r]*t*) *(Eq. 3)*

In the case of the reference frequency of the receiver being set to the Larmor frequency ω0, the integrated MRI signal equation becomes:

*S*(*t*)= *ρ*(r)exp(i*γ*G · r*t*)dr *(Eq. 4)*

This signal equation has the form of a Fourier transformation which becomes clearer when the concept of the reciprocal space vector, denoted as k, is introduced. k is the conjugate variable of the real space vector r:

k = *γ*G*t (Eq. 5)*

Therefore, we get a Fourier relationship between the signal intensity *S*(k) and the spin density *ρ*(r):

*S*(k)=∭*ρ*(r) exp(i2πkr)dr

*ρ*(r)=∭*S*(k) exp(-i2πkr)dk *(Eq. 6)*

The first step to get a spin density image in real space is the mapping of the inverse space, also denoted as k-space. Moving through k-space can happen by varying either the time t after a gradient was turned on or by changing the gradient amplitude G. The gradient direction determines the direction of the trajectory through k-space. After the mapping of k-space is completed, a Fourier transformation reveals the image in real space. This principle is shown in Fig. 1 for a linear gradient along the x-direction, which spatially encodes the spin distribution in x-direction during the acquisition of an FID. Because the spatial encoding of the spins is achieved through modulations of the Larmor frequency, this principle is also known as frequency encoding.

*Figure 1:* Frequency encoding. a) Spherical and cubical sample containing protons; a linear gradient Gz is applied in z-direction which locally varies the strength of the Bz field and causes the spins at different z-positions to precess at different Larmor frequencies during FID acquisition. b) Time domain signal S(t) in the receiver coil. c) Fourier transformation of b) shows the frequency encoding of the spin distribution projected along the z-axis.

An external magnetic field B_{1} from a radiofrequency (RF) coil tuned to the Larmor frequency can tip the effective magnetization vector of the spins to the transverse plane with their precession being detectable as a signal in the radiofrequency range. If an additional gradient perpendicular to the frequency encoding gradient is applied, the B_{1} pulse is merely exciting spins matching its frequency bandwidth. This procedure is called selective excitation - in contrast to non-selective excitation without an additional gradient. Slice selection is primarily used to image single slices within the object under investigation. For selectively exciting a slice of thickness ∆x, a pulse bandwidth of γG_{x}∆x is required. The RF pulse frequency bandwidth is inversely proportional to the pulse duration. To achieve a slice selection with a nearly rectangular frequency excitation profile, the B_{1} pulse form of choice is a sinc-function, since the Fourier transform of a sinc-function is close to a rectangle function.

After a selective excitation of a slab of spins (e.g. in the z-direction), the reconstruction of an image in real space is achieved by a two-dimensional Fourier transform. Frequency encoding of the FID signal can provide one dimension in k-space, which we will now denote as the x-direction for convenience. The gradient G_{x}, which is present for a readout period tro during the sampling of the FID, is called readout gradient. The spacing between successively mapped k_{x} points is determined by a combination of the so-called dwell-time t_{d}, which is the time-difference between two acquired data points of the FID, and the strength of the readout gradient. The magnitude of kx for a specific point in k-space is determined by the index number n ranging from −N/2 to N/2 and N being the maximum number of points sampled in one direction in k-space:

*(Eq.7)*

Further dimensions in k-space can be mapped by phase encoding. By applying another gradient G_{y} in _{y}-direction for a fixed time before the sampling of the FID begins, a phase modulation is imposed on the spins along the y-axis. Usually the gradient is turned on for a fixed time tph so that the phase shift due to the gradient becomes:

∆Φ = *γ*t *ph* ∆G_{y} y *(Eq.8)*

The mapping of k-space in ky direction is then accomplished by varying the gradient strength Gy:

*(Eq.9)*

With this combination of frequency and phase encoding, each FID during a given phase encoding step provides one line in k-space. By varying the gradient strength Gy we build up a 2D Cartesian grid in k-space. Once the whole k-space is mapped, a two-dimensional Fourier transform reconstructs the real space image ρ(r). Fig. 2 illustrates a standard 2D-imaging sequence and the principle of k-space sampling.

*Figure 2:* Basic 2D-imaging sequence with slice selection. The left diagram depicts the pulse sequence timing, while the right drawing shows the sampling of k-space. First, a sinc-shaped RF pulse is applied while the slice select gradient Gz is turned on to selectively excite a slice in the x-y plane. After the slice selection gradient is turned off, a reversed compensation gradient is turned on to rephase the spins. At the same time a phase encoding gradient with a specific gradient strength Gy is turned on for a fixed time tph. After a short delay, the signal is acquired with a sampling rate 1/td with td being the dwell time. During the readout, a gradient of constant strength Gx and fixed readout time tro is turned on. The sampling rate of kx and ky determines the field-of-view (FOV) of the image in real space.

A high soft-tissue contrast can be achieved in MRI, because the MRI signal depends on the timing and the tip angles of the excitation used for signal generation as well as on the relaxation properties of longitudinal and transversal magnetization, with their respective relaxation constants T1 and T2.

An MRI scanner consists of three major components: the magnet, the gradient coils, and the RF coils for transmission and reception of the magnetic resonance signals. Three types of magnets are currently used: permanent magnets, resistive magnets, and superconducting magnets. Clinical scanners usually use a superconducting magnet in the range of 1.5-3 T and maintain a B_{0} field homogeneity below 4 ppm within a spherical region of 50 cm diameter[126]. Linear magnetic field gradients along the x-, y-, and z-direction of the magnet are mounted concentrically within the magnet bore, allow spatial encoding of magnetic resonance signals, and are used for shimming. Every time, the bed position is moved for a whole body MR, the shim currents are automatically adjusted to improve B_{0} homogeneity. Typically, nuclear spins are excited using a whole body transmit volume coil mounted within the magnet bore creating a time-varying magnetic field B_{1} (on the order of several mT) on resonance with the Larmor frequency. Typically, such a volume coil provides uniform B_{1} across the entire homogeneous region of the magnet. The sensitivity of an MRI acquisition depends on the distance between the coil and the region being examined and the coil geometry. The closer the coil can be brought to the measured object and the better the coil dimensions match the region-of-interest, the higher the sensitivity. Therefore, dedicated radiofrequency coils have been developed for a variety of clinical questions. Coils designed for simultaneous PET/MR differ from MR-only coils in their design in two key elements: first, they are designed to minimize radiofrequency interferences with the PET electronics and second, the amount of material is reduced as much as possible, in order to minimize γ-photon scatter and attenuation for the PET image quantification. There are specific coils for breast, carotid artery, lungs, spine, prostate, knee, neonatal, and multi-nuclear applications among others. Receiver coils can also be used as phased arrays allowing acceleration of large volume image acquisitions.

There are an important number of applications in oncology, neurology, cardiology, and especially paediatric imaging where MRI in combination with simultaneous or subsequent PET could provide significant benefits over PET/CT and individual PET and MRI scans. Compared to PET/CT, PET/MR exposes the patient to less radiation, allows for high soft tissue contrast and multi-parametric information that can be exploited in different ways. Compared to individual MRI and PET scans, it is possible to perform simultaneous MR-PET in a single session, whereas, using individual scanners, usually two examinations need to be coordinated, which has an additional impact on overall logistics and also patient comfort. Quasi-perfect co-registration between anatomical and functional information is always a benefit of simultaneous PET/MR compared to sequential PET/MR, for which re-sampling or registration between the PET and MRI images may potentially introduce errors. In the case of sensitive patients, such as in paediatric imaging or for patients of reproductive potential, the avoidance of the CT radiation exposure is a major benefit of PET/MR.