CHAPTER 2: MAGNETIC RESONANCE IMAGING AND
2.7. Spatial Encoding, K-Space and Image Formation
The NMR signal has no inherent spatial information, and therefore cannot be used to produce an image. In order to spatially encode the spins, three sets of gradient coils are built into MRI machines to spatially-vary the magnetic field, thereby altering the Larmor precession frequency in a spatially dependent fashion. The magnetic field can be represented as:
𝐵\(𝒓) = 𝐵7+ 𝐺_𝑥a + 𝐺b𝑦a + 𝐺\𝑧̂ [2.10]
In the presence of the gradient fields, the transverse magnetization precesses with a spatially dependent Larmor frequency, accruing a different amount of phase that depends on the location of the spins as well as the intensity and duration of the gradient field:
𝜑(𝒓, 𝑡) = h𝜔jkklkkmn 𝑡7+ 𝛾𝑮??⃗. 𝒓
𝜔(𝒓) [2.11]
Omitting the w0 term, we can define the following parameters for a two-
dimensional image in the xy plane:
𝑘_(𝑡) = 𝛾
2𝜋𝐺_𝑡 , 𝑘bo𝑡bp = 𝛾
Given this, we can mathematically describe the NMR signal obtained for each acquisition with the gradients on for a total duration of t9:
𝑆Kro𝑘_, 𝑘bp = s 𝑚_b(𝑥, 𝑦)𝑒(Kvwoxy(z)_&x{(z)bp𝑑𝑥 𝑑𝑦
|}~•
[2.13]
where mxy(x,y) is the spatial distribution of the magnetization in the xy plane. Equation
[2.13] suggests that Fourier transform of this distribution in space describes the NMR signal represented in k-space. The duality property of the Fourier transform implies that the two-dimensional inverse Fourier transform of the k-space yields the spatial distribution of the spins, which is the image:
𝑚_b(𝑥, 𝑦) = s 𝑆Kro𝑘_, 𝑘bp𝑒Kvwoxy_&x{bp𝑑𝑘
_ 𝑑𝑘b €(•‚~ƒO
[2.14]
This equation implies that MR image acquisition involves obtaining enough information in the k-space and calculating its inverse Fourier transform, as shown in Figure 2.13. The range of k-space covered by the acquisition method controls the resolution of the image along the x and y axes:
Dx = 1
2𝑘_,…†_ , Dy = 1
2𝑘b,…†_ [2.15]
The sampling interval k-space, Dkx and Dky,determine the field-of-view (FOV) of
the image along the x and y axes:
FOV_ = 1
D𝑘_ , FOVb =
1
D𝑘b [2.16]
As data in k-space are obtained in a discrete fashion, the Nyquist criterion must be taken into account when selecting the sampling interval. This dictates that the selected FOV must
be equal to or larger than the desired object, otherwise the signal outside of the FOV folds back onto itself and creates an artifact known as aliasing. The relationship between the sampling interval and the FOV along the x and y axes is:
D𝑘 ≤ 1
FOVŒ•Ž [2.17]
Figure 2.13. MRI data acquired in the k-space represents the Fourier transform of the object. The actual image (right) can be obtained by applying inverse Fourier transform to the k-space data.
2.7.1. Acquisition Methods and MRI Pulse Sequences
The order by which an acquisition method covers the k-space is called the k-space trajectory. A few commonly used k-space trajectories are shown in Figure 2.14. The method by which the k-space data is acquired is described by an MRI pulse sequence, which describes the timing, shape and strength of the gradient fields, the shape and the power of the RF pulses with pulse sequence diagram, an example of which is shown in Figure 2.15.
The majority of MRI pulse sequences have a number of components in common including slice selection, phase encoding, frequency encoding and data acquisition.
Figure 2.14. (A) gradient-echo and (B) echo-planar readout trajectories are basic cartesian trajectories commonly used in both preclinical and clinical settings. Commonly used non-cartesian trajectories include
(C) radial and (D) spiral trajectories. The imaging methods in this thesis obtain the data in a Cartesian grid.
To obtain data to form a two-dimensional image, the first step is to excite the spins selectively in a given slice by turning the gradient on in the direction that slices are arranged. The thickness of the slice is determined by the bandwidth of the RF pulse, its shape and the strength of the gradient. Data is then spatially encoding using phase encoding
or frequency encoding gradients. Phase encoding occurs between excitation and data acquisition, and spatially alters the phase of the transverse magnetization. Frequency encoding, on the other hand, occurs during the data readout and spatially alters the
frequency of the transverse magnetization. This acquisition method, called a gradient-echo
(GRE) acquisition, is repeated every TR until the desired section of the k-space is traversed
and is the most common method for collecting MRI data.
Figure 2.15. Basic gradient-echo pulse sequence with a cartesian read-out trajectory. The components shown by the pulse sequence diagram are repeated regularly at the repetition time (TR)interval until the desired portion of the k-space is covered.
The time between the center of the excitation pulse and the point at which the maximum signal is formed during spatial encoding is called the echo time (TE) and is crucial for generating images with different types of tissue contrast. For instance, the signal from tissues with a short T2* time constant, such as the lungs, decay much more rapidly,