Basic MRI Physics

Topic editor: Connie Tsao, M.D. (Beth Israel Deaconess Medical Center, Harvard Medical School)

Introduction
Cardiovascular MRI is a non-invasive imaging method that allows assessment of anatomy and function. MRI is an application of the principles of nuclear magnetic resonance (NMR), which was described by Felix Block and Edward Purcell in 1946. These investigators, who won the Nobel Prize in 1952, discovered that when charged nuclei were subjected to a magnetic field, they would absorb energy, which was emitted when transferred back to their original state. The detection of the energy emitted when protons change energy state is translated into images. In the 1973, Paul Lauterbur and Peter Mansfield used this principle to create two-dimensional images to image the human body, and also were awarded the Nobel Prize in Physiology or Medicine in 2003 for their discoveries. As described below, it relies on the following principles: 1) signal may be detected from manipulating the magnetization of protons, and 2) the body is largely composed of water which contains two hydrogen atoms, or protons.

The Hydrogen Atom
Hydrogen atoms (and other atoms with an odd number of protons and neutrons) possess a property called spin. Spin is visualized as an atom rotating about its own axis. Because atoms are charged particles, they generate magnetic field in the direction of the axis (Figure 1). Normally, without magnetization, the atoms are spinning in their own spin states randomly (Figure 2, Left). However, when they are placed in a magnetic field B0, their spins will largely align along or against the field, depending on the energy state (Figure 2, Right). This results in a net weak magnetization of the tissue, the magnitude of which is proportional to the magnitude of the external magnetic field.



Figure 1: An atom spinning about its own axis (right) creates a magnetic charge similar to a bar magnet (left).



Figure 2: Left: Protons spin with their axes in random directions in the absence of a magnetic field. Right. In the presence of an external magnetic field B0, the atoms largely align along or against the gradient.

In the magnetic field as well, the protons are not static but rather spin about the axis of B0 at a frequency ω0 (Figure 3).



Figure 3: The atom precesses about the axis of the magnetic field gradient B0, in the path of a cone.

The central relationship in MRI is described by the Larmor equation, which expresses the relationship between the strength of a magnetic field, B0, and the precessional frequency, ω0, of an individual spin: ω0 = γ∙B0. The gyromagnetic ratio, γ, is a constant, which for hydrogen is 4257 Hz/T. The net sum of the magnetization of all of the spinning protons is described as M, which is proportional to the strength of B0.

Effect of Radiofrequency Pulses
The net magnetization M can be changed by application, via a coil, of a radiofrequency (RF) pulse applied at the Larmour frequency ω0. The degree of rotation of the axis M is called the flip angle. This angle is determined by the strength and duration of the RF pulse. A 90˚ pulse moves the direction of M from the z plane into the transverse (xy) plane as shown in Figure 4.



Figure 4: A 90˚ pulse moves the spin into the transverse plane with the flip angle which M has rotated from the z axis.

After transmission of the RF pulse cease, the effective magnetic field returns to B0. M lies in the xy plane as shown, and precesses about the z axis (Figure 5, Left). This precession induces a current in a receiving coil, which the scanner uses to translate an image. As soon as the RF pulse ceases, however, the signal begins to diminish as Mxy (where the signal is at its maxiumum) returns to Mz. This free induction decay (FID) is shown in Figure 5, Right.



Figure 5: Left: After the pulse knocks M into the transverse plane, it rotates about the z axis. Right: The signal, which decreases over time, is detected by the receiver coil.

Coils
A coil is composed of loops of wire and is used for two main purposes: 1) transmit or detection of signal and 2) inducing changes in the magnetic field. A transmit or receiver coil either sends or receives an RF pulse, or performs both functions. The body coil is part of the large magnet that can be used for both actions. Surface coils, which are placed over the region of interest on the body, receive signal and increase the signal to noise ratio (SNR). Gradient coils are used to localize the object in space. This is done by inducing changes in magnetization through sending small radiofrequency pulses (on the order of 40 mT) on top of the existing large external field (1.5T), creating a perturbation in the magnetic field B0. Three gradient coils are positioned perpendicular to each other, and create the slice-select, phase-encoding, and frequency-encoding gradients, corresponding to the z, y, and x axes, respectively.

Relaxation
After the RF pulse ends, the hydrogen atoms dissipate their excess energy as heat and begin to realign along the existing magnetic field B0 as previously. Relaxation is the process of the magnetization of the hydrogen atoms returning to equilibrium. The difference in time for this regrowth of equilibrium allows for differentiation of various tissue types in the body.

The two different types of relaxation are the following:

T1 relaxation: T1, or longitudinal, relaxation describes the regrowth of the magnetization component in the z direction towards M. It is a relaxation time constant which is an intrinsic property of each tissue. After a 90˚ pulse, when all the z component is tipped into the transverse plane, T1 is the number of milliseconds it takes for Mz to grow to 63% of the original M. The relationship is described by the equation: Mz (t) = M∙(1-e(-t/T1)). T1 relaxation is depicted in Figure 6A.

T2 relaxation: T2, or transverse, relaxation describes the decay of the signal in the xy plane. It occurs due to the interactions between atoms as energy is released followed an RF pulse. T2 decay is the number of milliseconds for 37% of the magnetization in the xy plane (Mxy) to remain. It is described by the equation:

Mxy(t) = M∙e(-t/T2). T2 relaxation is depicted in Figure 6B.



Figure 6: Left: T1 relaxation: Recovery of signal in the Mz direction. T1 is number of ms for 63% of the original M to regrow. Right: T2 relaxation: Decrease in signal in the xy plane. T2 is the number of ms for 63% of the transverse magnetization to be lost.

T1 and T2 depend on the water content of each tissue. Tissues with low water content, such as bone, have fast relaxation constants whereas soft tissues have long relaxation constants. For example, fat has both a shorter T1 and T2 than myocardium, so it both regrows its longitudinal (z) component and decays its transverse (xy) component faster than myocardium. Thus, imaging at the moment of maximal difference between the T1 and T2 of tissues allows for contrast differentiation between them.

Image formation
MRI can obtain many different image orientations. Three methods are essential to create the images: slice encoding, frequency encoding, and phase encoding. The intensity of signal at each position is dependent upon the proton densities in each location.

Slice Encoding
As stated previously, in the presence of the external magnetic field, the spins in the sample align their axes along the field. In non-selective excitation, a RF excitation will tip all the spins into the transverse plane. However, in slice selective excitation, a gradient along the longitudinal (z) axis, is first applied, which causes the protons to experience different magnetic field strengths depending on their position along the gradient. The RF pulse will only excite a small slice of protons whose Larmor frequency is the same as the RF pulse transmitted. Both increased gradient strength and a narrower frequency bandwidth result in selection of a thinner slice. Once a particular slice is selected in the z direction, resolution must be made in the plane itself. This is accomplished through frequency and phase encoding.

Frequency Encoding
Frequency encoding is the process of determining spatial position in the x direction. A linear spatial gradient Bx is placed on the existing magnetic field B0, which results in different Lamor frequencies of spins in different positions along the x direction. The signal detected is the sum of the different frequencies. Fast Fourier Transform, a technique which is not elaborated upon here, allows separation of the signals and amplitudes. Figure 7 shows how Fourier Transforms convert the signal vs. time into signal as a function of frequency, ie, signal as a function of location along the axis, since frequency varies with position. Frequency encoding is also called the read-out gradient, as it is left on while the signal is being obtained, or read-out.



Figure 7: Examples of Fourier Transforms (bottom) of signals obtained in frequency encoding; bottom graphs depict signal as a function of frequency, which localizes the signal in the x direction.

Phase Encoding
Similarly to the previous temporary gradients applied for image localization, during phase encoding, a gradient is applied in the y-direction, with different spins resonating at different frequencies depending on their location. When this gradient is removed, the protons will return to spinning at the same frequency, but now they are out of phase with each other. The Gy gradient is applied right after the RF pulse, or prior to the Gx gradient. This way, the phase shift induced by Gy allows for differentiation between the areas of one slice when Gx is then applied, as different areas along each slice will have both different phases and frequencies.



Figure 8: A pulse sequence. The repeated application of these gradients results in complete data imaging. TR, or Repetition Time, is the time interval between one 90˚ RF pulse and the next. TE is the echo delay time, the time from the RF pulse to data measurement.

k-space
Data from each Repetition Time (TR) fills one line in a set of rows of k-space, which is a spatial-frequency map of the image in the x-y plane. k-space is a digitized version of the image which is obtained using a technique called Analog-to-digital conversion (ADC), which converts a time-varying signal to a digit-form one. The position in k-space is determined by the place in the pulse sequence and the areas under the gradient curves in Figure 8. When k-space is fully sampled, the true image is reconstructed using a Fourier Transform.