Introduction to MRI Physics

The Magnetic Resonance Effect

Around 70% of our bodily tissues, including the brain, contain water molecules composed of hydrogen atoms. A single hydrogen atom has a nucleus with a single proton that freely rotates around its axis. In general, their axes point randomly but when placed inside a magnetic field B, they become aligned with it. What happens next? There are protons pointing with and against the direction of B the antiparallel spins. The net polarity under such equilibrium is measured as M, the net magnetization. The direction of M is always along B. 

While in equilibrium, the protons are also behaving like a spinning top with a particular angular frequency. This phenomenon is called precession. The frequency is called Larmor frequency, the value of which is proportional to B. Ideally, B is always constant and homogeneous, so it is called the static magnetic field. By convention, the direction of B is in the longitudinal axis. Its strength, measured in Tesla (T), defines the MRI scanner specification, for example 1.5 T and 3.0 T.

Suppose there is this second magnetic field B₁ applied to these realigned protons, two things will happen. First, they get more energy and begin to wobble. As a result, the precession axes start to point further away from the longitudinal axis B. Look at the diagram below. Second, the protons then become in-phase and this yields to a growing net magnetization in the transverse plane, 

M_xy perpendicular to the original M. The longer we apply this field, the higher the precession angle so much so that the new M' is now fully flipped 90º to the transverse plane.

Fig-1: An excitation pulse changes the net magnetization that yields to detectable RF signals used in MRI. (a) In the body, each hydrogen atom or proton has a random orientation. (b) If the body is under the influence of static magnetic field B, the protons will try to align themselves with B, producing a net M. (c) The application of external excitation field will destabilize the alignment, the resulting new M will be perpendicular to the original M. (d) Once this external field is switched off, we can measure the relaxation process through an RF detector. 

The field B₁ is also called the excitation pulse and is switched on only for a certain period. What happens when we switch it off suddenly? The protons try to align back with the static magnetic field, the process of which is called relaxation. At the same time, they emit RF signals with Larmor frequency matched with B₁. If we put an RF detector or receiver coil nearby, we are able to capture this signal. Hence, the word ‘resonance’ makes sense because what we measure is the resonance effect.  Following Faraday’s Law, the change in RF pulses induces an electric current that we can measure externally.  This is the basic principle of Magnetic Resonance Imaging (MRI) as discovered independently by Bloch and Purcell in 1946. This technology was originally called NMR or Nuclear Magnetic Resonance.

Gradient Field for Spatial Localization
Before being able to image an object, we need to have a coordinate system for spatial localization. Why? We want to know the “composition” of protons at a different location in the brain in order to produce the 3D brain structure. Without this feature, our RF detector is not able to know where in the brain produces what signal. The very first idea of spatial localization came from the 1D imaging experiment by P. Lauterbur in 1973. 

Spatial localization is achieved by using a gradient field. This means that instead of giving a constant external magnetic field, we vary the intensity slightly according to the distance in a certain direction. Because Larmor frequency depends on the magnetic field strength, there will be varying frequency components at different positions captured by the RF detector. The number of protons is reflected as the signal intensity. Because this gradient field yields to different frequencies at a different location, this particular external pulse is also called frequency encoding. It is usually denoted as G_x. 

To create a full volumetric or 3D brain image, we require the second and third gradient fields. By convention, the Z-axis is parallel to the direction of the static magnetic field and is called the longitudinal axis. The scanner bore is along the longitudinal axis. The X and Y axes form a perpendicular plane called the transverse plane w.r.t the longitudinal axis. Obviously, the remaining gradient fields are set along the X and Y-axis. Because a gradient field in one axis will modulate the spin precision frequency on that particular axis, the three cannot be switched on at the same time.

Fig-2: Creating a full 3D brain image requires 3 different gradient pulses on top of the external RF pulse B₁ (from [4]).

The slice selection field G_z  along the Z-axis is switched on simultaneously with the excitation pulses. ‘Slice’ here literally refers to brain slice. We now acquire MRI signals for that particular slice and then resolve them based on the X and Y position on the transverse plane using 2 additional gradient fields. The X- gradient field can resolve position based on frequency components as mentioned earlier on, i.e. frequency encoding. The Y-gradient G_y allows us to resolve position based on the phase difference, that is, the phase encoding. By convention, the frequency encoding is also known as the readout gradient.

Refer to the diagram above. The excitation pulses and the three gradient pulses during MRI acquisition are collectively called the pulse sequence.

Relaxation Time and Image Contrast
As mentioned earlier, a brief excitation pulse B₁ will cause the protons to spin along the transverse plane; sometimes called the 90º excitation. Spinning protons return back to the longitudinal baseline position in the direction of the static magnetic field B once the pulse B₁ is switched off. This process is called relaxation. Bear in mind that different tissues have a different composition, and thus, different relaxation times. In the early 1970s, R. Damadian attempted the first full-body NMR image using different relaxation times.

Fig-3: Different relaxation time constants observed once the external pulse is switched off. Taken from [4].

There are two types of relaxation. The first relaxation is caused by spin-spin interaction that results in the gradual dephasing effect in the transverse plane. T₂ represents the rate of relaxation in the transverse plane M_xy and T₂-value decay over time. A variant of T₂, called T₂*, accounts for local field inhomogeneity due to tissue properties. In addition, the magnetization M_z along the longitudinal axis is restored over time and the recovery time is represented by T₁. The process is also called spin-lattice relaxation because there is a transfer of energy to the surrounding tissue, causing less antiparallel and more parallel spins.

In MRI, tissues with different relaxation times yield different image contrast. So a contrast can be created by exploiting T₁, T₂ , or T₂*. As such, there are two important time parameters in MRI: 

1. Repetition time (TR), i.e. the time interval between two consecutive excitation pulses. 
2. Echo time (TE), i.e. the interval between the start of an excitation pulse and data acquisition during the peak of the signal. This is akin to a reply of an echo that is captured by the receiver.

The T₁-weighted image makes use of the T₁ relaxation effect on different tissues. This effect is clearly seen by providing a short repetition time TR. T₂-weighted images, on the other hand, can be clearly seen with long TE, allowing more time for the spins to dephase completely. Another important parameter is the "How" which determines the sequence to produce this echo, e.g. gradient echo and spin echo. What does it mean by echo here?

k-space and Fourier Transform

The introduction of k-space is useful to elucidate the process of image reconstruction from mere nuclear magnetic equations. The k-value is an integral operation of the gradient curve across time, i.e. the area under the gradient curve. If the gradient point is below the axis, its value is negative and its integration over time t yields to a negative value too. Changes in k-values over time are represented by the k-space trajectory. In our example below, the frequency and phase-encoding gradient are shown as orange and brown color respectively. The k-space trajectory is shown in green on the right-hand side panel.
  

Fig-4: A simplified version of how to generate 2D k-space trajectory using two gradient pulses, Gx and Gy.

Suppose we are acquiring one brain slice. The k-space contains k_x and k_y coordinates. Before we are able to reconstruct the slice image, we have to fill up the entire coordinate with relevant data. Which data? The data acquired or sampled by the RF detector, each for each plane. How do we create the trajectory to plot the data at different k_x and k_y coordinates? By varying the frequency and phase encoding gradient fields across time. When there is no gradient field, the coordinate remains the same over time as no trajectory is formed.

Data points are measured in a discrete manner. The number of k-space measurements we make per slice determines how good the spatial resolution of the slice is. After the whole k-space contains relevant data, Inverse Fourier Transform operation can subsequently be done to produce an image.

Echo Planar Imaging

If we turn the excitation pulse longer, it is possible to flip M all the way to 180º in the opposite direction. This result is interesting as at this position, there is no resonance effect captured by the detector. We can also do something different. First, we apply 90º excitation pulse so that the precision is on the transverse plane, then we switch it off. Transverse magnetization M_xy experiences gradual dephasing on the transverse plane. Now half-way through we apply a strong pulse called the 180º refocusing pulse. It turns out that we are able to flip the direction of rotation (or phase) around the X-axis. The effect is interesting. The M_xy will increase again once more like an 'echo' before gradually dying off. This sequence is called the spin echo. As shown in Fig-5, the period from the 90º excitation pulse up to the appearance of the maximum detectable signal is called the echo time (TE), i.e. the whole panel (a) to (d). 

Almost similar to the spin echo is a sequence called the gradient echo that makes use of gradient fields to introduce 180º flip instead of using refocusing pulses. There are several techniques to produce MRI echos and to allow a very rapid acquisition, but the topic is out of scope at the moment. A type of pulses called gradient recalled echo (GRE) forms the basis of most angiographic MRI.

Fig-5: In the first instance (a), the magnetization has been flipped onto the XY plane. Afterward, a spin-echo can be produced by introducing a refocusing RF pulse in (b) that flips the net magnetization vector along the X-axis. See panel (c) that depicts the moment just after the 180-deg flip. After the flip in (d) the vector position shown becomes similar to that in (a) when the 90-deg excitation pulse was just briefly introduced. At this point, the resonance signal is at its maximum, and it is the best time to acquire or "Read". Adapted from [1].

Further development of gradient echo is the Echo Planar Imaging (EPI). An EPI sequence forms the basis for functional MRI and diffusion-weighted imaging. In the EPI sequence, multiple refocusing pulses are applied in the phase encoding G_y continuously. Rather than returning to (0,0) EPI sequence allows a transition to the next level simply by continuing the trajectory controlled by positive and negative frequency encoding gradient G_x. This allows for the continuous streaming of readout. What happens in k-space? We are able to cover k-space as quickly as possible by filling up one axis within a single shot of spin-echo, resulting in much faster image acquisition. Refer to Fig-6 below and see the arrow upward at either side denoting the image reconstruction process (compared it with Fig-4). The drawback of the EPI sequence is its poor resolution and contrast, e.g. 3 mm is very common. The EPI method was proposed by Peter Mansfield. More on EPI can be found here.

Fig-6: The pulse diagram for EPI sequence mainly used in functional MRI (fMRI). Taken from [1]. Although EPI sequence is attractive, it invites some issues with respect to local distortion, image ghosting, and high requirement in data sampling rate. For these and other tradeoffs, EPI yields to a poorer resolution.

Epilogue
There are three main imaging modalities popular for brain imaging I would say. The first two, CT scan and PET scan, have been popular since the early '90s. CT scanning stands for computerized tomography and it makes use of X-ray principles to produce high-resolution brain images. CT images are static. PET or positron emission tomography makes use of radioactive tracer injected into the body to produce functional brain images. PET is able to detect biochemical processes in the brain and thus dynamic. The third imaging tool, MRI, makes use of the nuclear magnetic resonance effect of the spinning protons. MRI is attractive because it provides better spatial resolution and hydrogen atoms are already abundant in the body.

The basic physics of NMR has previously been explained. In brief, MRI technology makes use of the magnetic properties of hydrogen atoms or protons. When placed under the influence of changing magnetic fields, different tissues in the brain have different properties of spinning protons. These differences can be observed in terms of relaxation time (T₁ or T₂). To produce a full 3D image we utilize three different gradient fields. Lastly, different imaging quality and quantity are influenced by different pulse sequence parameters. In practice, MRI physicists often use a dummy model called phantom to test the performance of a new pulse sequence.

In the subsequent post, functional MRI or fMRI will be discussed. This type of functional imaging is derived from MRI technology that is able to capture changes related to neuronal function in successive images. Unlike MRI, fMRI is not static. And unlike PET, there is no need to administer any radioactive tracer to produce image contrast. Rather, we can make use of our blood as the natural contrast agent. The method is called blood-oxygen level-dependent fMRI or BOLD-fMRI and this invention was discovered by Seiji Ogawa in 1990 using a gradient-echo imaging sequence. 

Most recently, the introduction of the multiband (MB) or simultaneous multi-slice (SMS) EPI technique effectively shortens acquisition time without decreasing TE and maintaining the same SNR by simultaneous acquisition of multiple slices at one go. For a review, refer to Feinberg and Setsompop (2013).

References
[1]  Jezzard, P., Mathhews, P. M., and Smith, S. M. (2001). Functional MRI: An Introduction to Methods. Oxford University Press.    
[2]  Missimo, Filippi. (2009). fMRI Techniques and Protocols, 1st ed.. Humana Press - Springer. 
[3]  Basic MRI principles (Youtube)
[4]  Box 19-3 of Chapter 19 (pp.370-374). In Kandel E.R. et. al. (2000). Principles of Neural Science 4e, McGraw-Hill.