6
k-Space
After reading this chapter, you will be able to:
- Describe the characteristics of k-space.
- Explain different ways in which k-space is filled.
- Understand how pulse sequences determine how and when k-space is filled with data.
- Apply this understanding when altering parameters in the scan protocol.
INTRODUCTION
In Chapter 5, we discovered that spatial encoding selects an individual slice and produces a frequency shift of magnetic moments of spins along one axis of the slice and a phase shift along the other. The system now has a way of locating an individual signal within the image by measuring the number of times the magnetic moments of spins cross the receiver coil (frequency) and their position around their precessional path (phase).
The phase shift caused by the phase-encoding gradient creates a spatial frequency. This is because a waveform is derived from plotting the change of phase of magnetic moments of hydrogen nuclei located at different locations along the gradient. Spatial frequencies are also obtained from slice-selection and frequency encoding because they too cause frequencies that are dependent on spatial position. Data from these spatial frequencies are used by Fast Fourier Transform (FFT) mathematics to produce an image. During the scan, data are acquired and stored in k-space. k-Space is a spatial frequency domain, i.e. where information is stored about the frequency of a signal and where it comes from.
Every parameter we select in the scan protocol changes how k-space is filled with data. k-Space is therefore a very important concept. This chapter is broken down into the following five parts, and scan tips are used to link the theory of k-space to practice.
- What is k-space?
- How are data acquired and how are images created from these data?
- Some important facts about k-space!
- How do pulse sequences fill k-space?
- What are the different ways in which k-space is filled?
The following discussion is a jigsaw puzzle made up of these five parts. You need the whole puzzle to get the whole picture!
PART 1: WHAT IS k-SPACE?
k-Space is a storage device. It stores digitized data produced from spatial frequencies created from spatial encoding (see Chapter 5). Figure 6.1 illustrates k-space for one slice. k-Space is rectangular and has two axes perpendicular to each other. The frequency axis of k-space is horizontal and is centered in the middle of several horizontal lines. Data from frequency encoding are positioned in k-space along this axis. The phase axis of k-space is vertical and is centered in the middle of k-space perpendicular to the frequency axis. Data from phase encoding are positioned in k-space along this axis. In the simplest method of k-space filling, data are stored in horizontal lines that are parallel to the frequency axis and perpendicular to the phase axis of k-space.
Let’s for a moment imagine a room with a very high ceiling. Inside this room there are 20 chests of drawers, that extend from floor to ceiling. In each chest of drawers, there are 100 drawers and in the middle of the room is a large pile of clothes of different types – all jumbled up together. Imagine that you are tasked with sorting these clothes into different drawers and that the aim is to ensure that each drawer, in each of the chests of drawers, contains the same type and quantity of clothes. For example, the top drawer of every chest of drawers in the room contains 50 pairs of green socks. The next drawer down of every chest of drawers contains 50 red shirts, and the next, 50 yellow pairs of trousers, and so on. The task is complete when the pile of clothes in the center of the room is sorted into all the correct drawers of every chest of drawers in the room.
The information that you would probably need to complete this task (now called the clothes-sorting exercise) is as follows:
- The total number of chests of drawers in the room
- The total number of drawers in each chest of drawers
- Numbered drawers so that each is quickly identifiable
- The most efficient system to fill the drawers with the required quantity and type of clothing.
The MRI computer system needs the same information to fill k-space as you would need to complete the clothes sorting exercise. A single chest of drawers is analogous to k-space for a single slice. All the chests of drawers represent all the selected slices. The drawers inside a chest of drawers are equivalent to the lines of k-space filled with data from echoes from that slice. The large pile of jumbled-up clothes represents all the spatial frequencies, from all the echoes, from all the slices produced from spatial encoding during the scan. The task is to ensure that the data from these spatial frequencies are sorted out and located correctly in each drawer or line of k-space. So, what are the answers to the questions above?
- The number of chests of drawers in total equals the number of k-space areas, and this equals the total number of selected slices. This is a parameter selected in the scan protocol (20 in the example above).
- The number of drawers or lines of data equals the phase matrix selected in the scan protocol (100 in the example above). If a phase matrix of 256 is selected, then 256 lines or drawers are filled with data to complete the scan. If a phase matrix of 128 is selected, then 128 lines or drawers are filled with data to complete the scan (Figure 6.3).
- The lines of k-space are numbered so that the system always knows where it is in k-space. The following convention is used. The lines are numbered with the lowest number near to the central axis (e.g. lines ±1, 2, 3, …) and the highest numbers toward the outer edges (e.g. ±128, 127, 126, …) (Figure 6.4). The lines in the top half of k-space are called positive lines, and those in the bottom half are called negative lines. This is because the line filled with data is determined by the polarity of the phase-encoding gradient. Positive polarity phase-encoding gradients are associated with lines in the top half of k-space, whereas negative polarity phase-encoding gradients are associated with lines in the bottom half of k-space.
- There are many different k-space filling methods. The computer uses different systems depending on parameters we select in the scan protocol. The simplest method is called Cartesian filling. In this method, k-space is filled in a linear manner from top to bottom or bottom to top.
Using the Cartesian method and the chest of drawers analogy, look at Figure 6.5, which illustrates a conventional spin-echo pulse sequence. The top half of the diagram illustrates when gradients are applied during the pulse sequence. The bottom half shows k-space for a single slice, drawn as a chest of drawers. It may also help at this point to refer to the end of Chapter 5 when we explored the timing of different elements within a pulse sequence.
The slice-select gradient is applied during the RF excitation and rephasing pulses to selectively excite and rephase a slice. The slice-select gradient determines which slice is excited or which chest of drawers is selected. Each slice has its own area of k-space or chest of drawers. Note that although three chests of drawers are shown in Figure 6.5, they do not represent k-space for three separate slices in this diagram. In Figure 6.5, each chest of drawers represents the same slice at three different times in the sequence when each of the three gradients is switched on.
The phase-encoding gradient is then applied. This determines which line or drawer is filled with data. In the example shown in Figure 6.5, the third drawer down in the chest of drawers is opened. Let us say that this is equivalent to line +126 in k-space. To do this, the phase- encoding gradient is applied positively and steeply. Application of this gradient selects line +126 in k-space. The frequency-encoding gradient is then switched on. During its application, frequencies in the echo are digitized, and data are acquired and located in line +126 in k-space. These data are arranged in data points, and they are equivalent to pairs of green socks in the clothes sorting exercise (or red shirts, or yellow pairs of trousers!). The number of data points is determined by the frequency matrix (more on this later). After data points are collected and located in line +126, the frequency-encoding gradient switches off, and the slice-select gradient is applied again to excite and rephase the next slice (slice 2). This is equivalent to walking up to another chest of drawers (not shown in Figure 6.5). The phase-encoding gradient is applied again to the same polarity and amplitude as for the first slice, filling line +126 in k-space for slice 2. The process is repeated for slice 3 and all the other slices, with the same line filled for each k-space area.
Once this line is filled in every chest of drawers, the TR is repeated. The slice gradient selects slice 1 (or chest of drawers 1) again, but this time a different line of k-space is filled to that in the previous TR. If the Cartesian k-space filling method is used, either the line above or the line below is filled. In our example, let’s assume this is the line below that is labeled +125. To do this, the phase-encoding gradient is switched on positively but to a shallower slope than in the previous TR. This opens the next drawer down in the chest of drawers, or selects line +125. When the frequency-encoding gradient is switched on, data points are laid out in this line. When this is complete, the slice-select gradient is applied again to select slice 2. The same amplitude and polarity of phase-encoding gradient are applied to open the same drawer or pick the same line (+125) in k-space for slice 2.
This process is repeated for all the slices. As the pulse sequence continues, every TR the phase encoding amplitude is gradually decreased to step down through the lines of k-space until the center line is filled. To fill the bottom lines, the phase-encoding gradient is switched negatively with gradually increased amplitude every TR to progressively fill the outer lines. The central line (line 0) is always filled with data. To do this, the phase-encoding gradient is not switched on. In Cartesian k-space filling, if, for example, a 256 phase matrix is selected, the system fills 128 lines in the top half of k-space, line 0 and 127 lines in the bottom half of k-space (total 256). This is written as (+128, 0, −127). The system can, however, just as easily start at the bottom and work its way up through k-space, in which case it is written as (−128, 0, +127). This is the most common k-space filling method, although there are many others, and some of these are discussed in Part 5.
Table 6.1 Things to remember – k-space.
k-Space stores information about where frequencies within the slice are located |
Data points acquired over time are laid out in k-space during the scan and mathematically converted into information related to amplitude via FFT |
k-Space is analogous to a chest of drawers where the number of lines filled is the number of drawers in the chest of drawers |
Each gradient determines when and how the chest of drawers is filled |
In a standard sequence, the same drawer is filled for each of the chest of drawers in a single TR period |
The number of drawers equals the phase matrix |
The number of socks or data points in each drawer equals the frequency matrix |
Refer to animation 6.1 on the supporting companion website for this book: www.wiley.com/go/westbrook/mriinpractice
PART 2: HOW ARE DATA ACQUIRED AND HOW ARE IMAGES CREATED FROM THESE DATA?
This section specifically explores how spatial frequencies are converted into data points stored in k-space and how they create images. What are the socks in our clothes-sorting exercise? This is a difficult subject and one you may need some time to learn. However, it is important, as it affects several parameters we always select in the scan protocol.
As we learned earlier, a waveform is created by plotting the change of phase of magnetic moments either over time or over distance. Figure 6.6 shows three waveforms. These waveforms represent the change of phase over time of the magnetic moments of three spins precessing at three different frequencies (1, 2, and 4 Hz). The echo received by the receiver coil at time TE contains hundreds of different frequencies, and, unlike those shown in Figure 6.6, they also have different amplitudes. These frequencies are dependent on where signal is coming from in the slice during the scan. This is complicated because there are hundreds of different frequencies and many different amplitudes.
A few preliminary steps are needed to ensure that the data are in a format required for FFT mathematics to create an image of each slice. The first step is to simplify the frequencies and amplitudes present in the echo. This step is called frequency and amplitude modulation. The multiple waveforms in the echo are simplified into one waveform that represents the average amplitude of all the different frequencies in the echo at different time points (see bottom of Figure 6.6). This modulation is still a waveform (although it does not perhaps look like one). The echo that occurs at time TE is this modulation. The echo is a symmetrical representation of signal, and the peak of the echo usually occurs in the middle of the application of the frequency-encoding gradient (see Chapter 5). The next step is digitizing the modulation (called the echo from now on). This process is called analog-to-digital conversion (ADC). Analog is a term used for information about a variable illustrated as a waveform. Digitization displays the same information but in binary numbers. On modern scanners, digitization of analog signal is done inside either the receiver coil assembly or the body of the MRI scanner (see Chapter 9).
The frequency-encoding gradient is switched on while the system reads the echo and digitizes it. It is therefore sometimes called the readout or measurement gradient. The duration of the frequency-encoding gradient is called the sampling time, sampling window, or acquisition window (called the sampling window from now on). During the sampling window, the system samples or measures the echo at certain time points. Every time a sample is taken, this is stored as a data point in k-space. These are the pairs of socks used in the clothes-sorting exercise.
During the sampling window, several data points are acquired as the system samples the echo, and, in the simplest method, these are laid out in a line of k-space from left to right. The number of data points is determined by the frequency matrix. If the frequency matrix is 256, then 256 data points are acquired during the sampling window. If the frequency matrix is 512, then 512 data points are acquired. Every TR, a different line of k-space is filled with the same number of data points determined by the frequency matrix. This continues until all the lines are full of data points when the scan ends.
The process of data acquisition therefore results in a grid of data points:
- The number of data points horizontally in each line equals the frequency matrix, e.g. 512, 256, 1024, etc.
- The number of data points vertically in each column corresponds to the phase matrix, e.g. 128, 256, 384, 512, etc. (Figure 6.7).
Sampling
During the sampling window, the system samples or measures spatial frequencies in the echo. Every sample results in a data point that is placed in a line of k-space. Each data point contains information about spatial frequencies in the echo at different time points during the sampling window.
The rate at which sampling occurs is called the digital sampling rate or digital sampling frequency. It is the rate at which data points are acquired per second during the sampling window and has the unit Hz. If one data point is acquired per second, the digital sampling frequency is 1 Hz, 100 data points per second is 100 Hz, 1000 per second is 1000 Hz or 1 KHz, and so on. The digital sampling frequency is not directly selected in the scan protocol, but it affects several other parameters that are selected, so read on!
The digital sampling frequency determines the time interval between each data point. This time interval is called the sampling interval and is calculated by dividing the digital sampling frequency by 1 (Equation (6.1)). This relationship means that if the digital sampling frequency increases, then the sampling interval decreases and vice versa. For example:
- If the digital sampling frequency is 32 000 Hz (32 KHz), the sampling interval is 0.031 ms (1 ÷ 32 000).
- If the digital sampling frequency halves to 16 000 Hz (16 KHz), the sampling interval doubles to 0.062 ms (1 ÷ 16 000).
Equation 6.1 ω_{sampling} _{=} 1/ΔT_{s}
ω_{sampling} is the digital sampling frequency (KHz)
ΔT_{s} is the interval between each data point or sampling interval (ms)
This equation shows that if the digital sampling frequency increases, the sampling interval decreases
The digital sampling frequency is important. If it is too low, then there may not be enough data points in k-space to create an accurate image. If it is too high, then the resultant files might be large and unmanageable, and sampling might take too long. In addition, some of the frequencies sampled in the echo are unwanted noise frequencies. As the digital sampling frequency increases, more noise data are acquired, and this affects image quality (see Chapter 7).
The most optimum digital sampling frequency is determined by the Nyquist theorem. (The full name is Whittaker–Kotelnikov–Shannon–Raabe–Someya–Nyquist theorem [1]. Thankfully, it is abbreviated to the Nyquist theorem!) This calculates the minimum digital sampling frequency needed to acquire enough data points to create an accurate image. The Nyquist theorem states that when digitizing a modulation of several frequencies, the highest frequency present in the modulation must be sampled at least twice as frequently to accurately digitize or represent it.
Look at Figure 6.8. Sampling once per cycle, or at the same frequency as the frequency we are trying to digitize, results in a representation of a straight line or an absent frequency in the data (middle diagram). Sampling at less than once per cycle represents a completely incorrect frequency that leads to an artifact called aliasing (bottom diagram) (see Chapter 8). Sampling twice per cycle, or at twice the frequency we are trying to digitize, results in the correct representation of that frequency in the data (top diagram).
As long as the highest frequency present is sampled twice, it is represented correctly in the data. Lower frequencies are sampled more often at the same digital sampling frequency and are also represented accurately in the data. Digital sampling frequencies higher than this produce more data and therefore a more accurate representation of the original analog frequencies. However, due to time constraints, the digital sampling frequency is sometimes limited. High digital sampling frequencies also result in more noise data. The digital sampling frequency is therefore usually kept at just twice the highest frequency in the modulation (termed the Nyquist frequency) to avoid aliasing while still sampling in the most time-efficient manner. The digital sampling frequency is usually limited, therefore, to twice the Nyquist frequency (Equations (6.2)–(6.4)).
Equations 6.2–6.4: Sampling equations. | ||
ω_{sampling =} 2 × ω_{Nyquist} | ω_{sampling} is the digital sampling frequency (KHz) ω_{Nyquist} is the Nyquist frequency (KHz) – the highest frequency in the echo that can be sampled |
If Nyquist is obeyed, the highest frequency is sampled twice as fast as the Nyquist frequency, and this determines the digital sampling frequency |
RBW = 2 × ω_{Nyquist} therefore |
RBW is the receive bandwidth (KHz) ω_{Nyquist} is the Nyquist frequency (KHz) |
The receive bandwidth is the range of frequencies sampled on either side of center frequency. The RBW is therefore twice the highest frequency sampled |
ω_{sampling =} RBW | ω_{sampling} is the digital sampling frequency (KHz) RBW is the receive bandwidth (KHz) |
Combining the first two equations shows that when Nyquist is obeyed, the receive bandwidth has the same numerical value as the digital sampling frequency |
Refer to animation 6.2 on the supporting companion website for this book: www.wiley.com/go/westbrook/mriinpractice
The digital sampling frequency is not a parameter directly selected in the scan protocol. However, when Nyquist principles are obeyed, there is a selectable parameter that has the same numerical value as the digital sampling frequency. This is called the receive bandwidth.
Receive bandwidth
The bandwidth (or range of frequencies) used in the RF excitation pulse to excite a slice is called the transmit bandwidth (see Chapter 5). The transmit bandwidth permits slices of a certain thickness. The receive bandwidth is the range of frequencies accurately sampled or digitized during the sampling window. The receive bandwidth is determined by applying a filter to the frequencies encoded by the frequency-encoding gradient. This is achieved by selecting the center frequency of the echo and defining the upper and lower limits of frequencies that are accurately digitized on either side of this center frequency.
The receive bandwidth is equivalent to representing frequencies of half the receive bandwidth above the center frequency to half the receive bandwidth below the center frequency. For example, if the receive bandwidth is 32 KHz across the whole echo, then this represents 16 KHz above the center frequency of the echo to 16 KHz below. These frequencies are mapped across the field of view (FOV) after FFT.
Using the example above, the highest frequency in the echo is 16 KHz above the center frequency of the echo. This is the Nyquist frequency. The receive bandwidth is therefore twice the Nyquist frequency (2 × 16 = 32 KHz). As we have just learned, the digital sampling frequency is also twice the Nyquist frequency. Therefore, although the receive bandwidth and the digital sampling frequency are different parameters, they have the same numerical value (Equations (6.2)–(6.4)) when Nyquist principles are applied.
As the receive bandwidth is a selectable parameter in the scan protocol, it is used to determine the digital sampling frequency. When the receive bandwidth increases, the highest frequency sampled in the echo also increases. To accurately sample this higher frequency, the digital sampling frequency also increases (aliasing results if this does not occur). If the receive bandwidth increases from 32 to 64 KHz, for example, the Nyquist frequency is 32 KHz and the sampling frequency is twice this, i.e. 64 KHz which is the same as the receive bandwidth. Using a digital sampling frequency of 64 KHz, means that 64 000 data points are acquired per second so that the sampling interval halves (1 ÷ 64 000 is half 1 ÷ 32 000). Hence, the required number of data points (as determined by the frequency matrix) is acquired in half the sampling window. The opposite is true if the receive bandwidth decreases.
Sampling window (sampling time)
The sampling window is not a user-selectable parameter in the scan protocol. However, as the echo is usually centered in the middle of this time (i.e. the peak of the echo corresponds to the middle of the application of the frequency-encoding gradient), the duration of the sampling window indirectly affects the TE (which is selectable in the scan protocol). For example, if the frequency-encoding gradient is switched on for 8 ms (i.e. the sampling window is 8 ms), then the peak of the echo occurs after 4 ms. If the sampling window increases, the frequency-encoding gradient switches on for longer. Hence, the peak of the echo occurs later, increasing the time from the peak of the echo to the RF excitation pulse that created it (i.e. TE increases). The opposite is true if the sampling window decreases.