CHAPTER 1: Introduction to Magnetic Resonance Imaging

Nuclear magnetic resonance (NMR) has expanded rapidly from its initial discovery in 1946 into one of the most important analytical techniques in modern science, with applications in both physics and chemistry. It is NMR that forms the basis of magnetic resonance imaging (MRI), which has revolutionised diagnostic medicine since the late seventies. Indeed, even today the NMR techniques invented decades ago are finding application in MRI as new imaging sequences.

The introduction of linear magnetic field gradients to impose spatial information on the measured signal, in combination with the ubiquitous Fourier transform, led to an enormous research effort in the field of medical diagnostic imaging. In the early 1980s, there was already a growing feeling within the research community that NMR imaging would be used in other applications, and not solely for the study of the morphology of the body. Their prediction has indeed come to pass with the advent of functional MRI (fMRI), non-proton MRI to investigate metabolism and so on. In fact, it has been anecdotally reported that even the early NMR pioneers self-experimented. Edward Purcell placed his head in an NMR spectrometer to investigate differences in NMR signal shape dependent upon when he either concentrated hard on a specific task or freed his mind. Although no differences were measured at that time, it is apparent that these pioneers were already convinced about the possibility of applying NMR to study biological systems. A few decades later, MRI came of age.

In this chapter, even though some reference to quantum mechanics is briefly touched upon, the basics of MRI are described using a classical model, which suffices to explain standard proton-based imaging. Following on from the description of spin dynamics and the acquisition of the signal, image reconstruction methods are described, including up-to-date techniques for the acceleration of image acquisition employing parallel receive and multi-band excitation. The final section of this chapter concludes with a brief description of how sequences can be classified and, conceptually, understood.

Nuclei, composed of protons and neutrons, possess a basic property referred to as spin angular momentum, I⃑, which originates from the intrinsic spin of the nucleons. Nucleons are fermions and have a spin of ½ in units of ħ, where ħ is the reduced Planck's constant . Thus, a given nucleus has a net spin, which depends on its mass number Z (number of nucleons), atomic number A (number of protons) and neutron number N. In particular, a nucleus has a half-integer spin when the mass number Z is odd and an integer spin when the mass number Z is even and A is odd. In the case that Z and A are even, a nucleus has a zero nuclear spin. From here on, it is assumed that the nucleus under discussion has non-zero nuclear spin. In most cases, MRI is concerned with a spin ½ nucleus, namely the proton of hydrogen nuclei. Nevertheless, other MR-active nuclei, such as sodium, possess a non-integer spin and are also of interest to imaging. These will be discussed in later chapters.

Spin value determines the magnetic dipole moment  = γI⃑, with gyromagnetic ratio¹  given by γ = ge/2m_p. The gyromagnetic ratio depends on the nuclear g factor, the elementary charge, e, and the mass of a nucleon, m_p.¹  For reasons imposed by the uncertainty principle, the magnetic moment cannot be well defined in all spatial directions simultaneously and thus it is common to choose one axis along which the component of magnetic dipole moment is to be considered. Ordinarily, in NMR and MRI, one is interested in the component of the magnetic dipole moment along the axis of an externally applied magnetic field, which is normally labelled z. Nevertheless, this is based solely on convention, and in principle, the choice of this axis is arbitrary.²  The dipole moment is a quantised variable, with quantification defined through the magnetic number m_l (eqn (1.1))

The macroscopic manifestation of spin is referred to as magnetisation, M⃑. The magnetisation is the vector sum of the individual magnetic dipole moments of the nuclear spin ensemble in the volume of interest. In the absence of a magnetic field, the nuclear spins are oriented randomly in all directions of space, with the distribution determined by the thermal conditions, yielding a zero-net magnetisation (see Figure 1.1).

When an external magnetic field is applied to a spin ensemble, the distribution of nuclear spins becomes skewed towards the magnetic north pole, thus yielding a net magnetisation in the direction of the external field.³  The potential energy stored in a magnetic dipole is given by the scalar product of the magnetic dipole moment and the magnetic field vector, and thus depends on the angle between the dipole and the magnetic field θ. Assuming that the temperature of the system is high enough to apply the Boltzmann distribution, the angular distribution, P, is given by eqn (1.2)

here k_B is the Boltzmann constant and Z is the canonical partition function, which is given by eqn (1.3)

The expectation value of the z-component of the dipole moment (eqn (1.4))

can be approximated to (eqn (1.5))

This is a significant result for magnetic resonance methods, as not only does it show that there is a net magnetisation in the presence of the magnetic field, but it also implies that the available magnetisation depends linearly on the magnetic field, thus explaining the on-going endeavour to increase magnetic fields for MR imaging applications. It should be noted at this point that, for MR applications, higher equilibrium magnetisation translates to better image quality or reduced acquisition times. This should become clear to the reader during the course of this book.

Each spin state of the nucleus has an associated energy level. In the absence of a magnetic field, these levels are degenerate, but this degeneracy is lifted once an external field is applied.²  An energy level splitting, referred to as the Zeeman effect, occurs where the energy shift is proportional to the magnetic field strength. In the classical model, which can be applied here for a better understanding, this energy splitting manifests itself in a precessional motion of the magnetisation about the magnetic field vector at a characteristic frequency, ω₀. This is referred to as the Larmor frequency and depends on the strength of the magnetic field (eqn (1.6))

This equation is of fundamental importance since it relates the frequency of precession to the total magnetic field. This can be shown by deliberately changing the static field. When the static field is changed, an alteration in the frequency can be observed. The associated difference between the energy states is ΔE = ħγB₀ (see Figure 1.2).

In equilibrium, the net magnetisation of a system in the presence of an external field is oriented along the z-axis only, i.e. only longitudinal magnetisation exists. The transverse magnetisation of the system averages to zero. If the system is perturbed from equilibrium, this may no longer be the case. Transverse magnetisation might be present, and the longitudinal magnetisation might be changed or inverted. Over time, if the system is left to evolve under the influence of the static, external field, the magnetisation will return to its equilibrium state. The reason for this is that energy is transferred between individual spins and their environment.³  In the case of protons, the dynamics follow monoexponential decays and their characteristic time constants. The longitudinal relaxation (or spin–lattice relaxation) time T₁ represents the convergence of the z magnetisation to its equilibrium magnetisation M₀ and is caused by the interaction of spins with their environment. Transverse magnetisation decreases with the time constant T₂ and is an interaction between the individual spins, thus representing an intrinsic irreversible loss of coherence. This so-called spin–spin relaxation is commonly superimposed with other effects, such as field inhomogeneity. The effective time constant T$2*$ is then given by the following reciprocal relationship (eqn (1.7))

with T′₂ as the relaxation time due to field inhomogeneity. Although often undesirable, changes in T$2*$ relaxation can be used in imaging as a marker of brain activity (see Chapter 3, Section 3.1.2) or to map effects from paramagnetic entities, such as iron (see Chapter 4, Section 4.5.1). In general, 2T₁ is always longer than T₂, which must be longer than T$2*$.³  However, under most circumstances, T₁ > T₂ > T$2*$ in biological tissue.

The spin dynamics of ½-spin systems can easily be described by a set of three coupled first order differential equations, derived from classical physics. Nevertheless, one should keep in mind that this is an approximation, and the underlying principles are of quantum mechanical origin. Classically, at thermal equilibrium, M⃑ and B⃑ are aligned. If M⃑ is caused to point in a different direction from B⃑, then it is evident that M⃑ will precess about B⃑. Following this line of argument, the torque resulting on the net magnetisation in the magnetic field, . Noting that torque is the rate of change of angular momentum and, in the most basic form, ignoring the effects of relaxation and assuming a constant net external magnetic field B⃑, this set of equations takes the form of a simple vector rotation equation (eqn (1.8))

with magnetisation M⃑ = (M_x M_y M_z)^T. The solution of this equation can conveniently be expressed for the case B⃑₀ = (0 0 B₀)^Tas a rotation about the z-axis with rotation matrix such that (eqn (1.9))

thus, the magnetisation will precess about the z-axis at the Larmor frequency ω₀ = γB₀. In terms of the individual components of M⃑, the solution takes the form of eqn (1.10)

Following the tipping of the net magnetisation away from equilibrium (excitation), the transverse component decays to zero, whilst the longitudinal component returns to equilibrium. This behaviour is characterised by introducing additional phenomenological terms to represent the on-going relaxation processes. The coupled first order differential equations, commonly referred to as Bloch equations, can be obtained⁴  where the two additional terms describe the spin–lattice and spin–spin relaxation processes (eqn (1.11))

where M_eq is the equilibrium magnetisation. In terms of the individual components of M⃑, the equation can conveniently be expressed in one line: , with unit vectors x̂, ŷ, ẑ pointing in the three spatial directions in the laboratory frame.

Precession about the net magnetic field is not limited to the static external magnetic field alone, but can, in fact, include effects from the temporally and spatially varying magnetic fields created by the gradient system, as well as the radiofrequency (RF) field and effects due to (usually undesired) field inhomogeneity. Understanding these fields is necessary in order to fully appreciate the imaging process, which can be conveniently described by the Bloch equations. If necessary, further terms can be added to the Bloch equation to address effects such as diffusion (see Chapter 3, Section 3.3).

Solving Bloch equations can be an elaborate process in the laboratory reference frame, in which the observer is positioned in a stationary location and the spin system evolves and precesses accordingly. It is often convenient and useful to utilise the concept of the rotating frame of reference to solve the Bloch equations and understand spin dynamics. The rotating frame of reference is chosen such that it revolves about the z-axis at a given frequency, ω_r.

For now, it is assumed that the magnetisation is subject to a magnetic field, which is made up of two parts: the static field, B₀; and the effect of an oscillating linear field, B⃑₁ = 2B₂ cos(ω_rft)î , perpendicular to the B₀ field. The B₁ field can be expressed as a sum of left- and right-handed rotating magnetic fields. If observed relative to the precessing magnetisation, one component is rotating at a reduced relative angular frequency in the same direction as the magnetisation and the other component is rotating in the opposite direction, causing dynamic effects on a timescale not relevant to the measurement process. Omitting contributions from the latter and, for now, also the effects of relaxation, the Bloch equations reduce to a form in which they describe simple precession about the net magnetic field (eqn (1.12))

In order to solve this set of differential equations, it is convenient to transform to the rotating frame of reference, which rotates at the same angular frequency as the RF wave. Its unit vectors are determined by eqn (1.13)

such that (eqn (1.14))

Here, the effective magnetic field is introduced as eqn (1.15)

which represents a field that determines the motion of the magnetisation in the rotating frame of reference. It should be noted, that if the resonance condition, ω_RF = ω₀, is met, the effective magnetic field is determined by the transverse B₁ component only, i.e. in this case the magnetisation only evolves around the applied B₁ field. This is easy to comprehend if one considers an observer rotating about the same axis, watching an object precessing with the same angular frequency: the motion of the object would seem frozen to the observer (see Figure 1.3).

Sometimes it is beneficial to note that it is mathematically convenient to rewrite the Bloch equations in an axial representation,⁵  such that the transverse magnetisation is expressed as a complex quantity M_xy = M_x + iM_y, rather than using elements of the three-dimensional magnetisation vector M⃑. By adopting a similar notation for the transverse magnetic field B_xy = B_x + iB_y, the Bloch equations (see eqn (1.10)) can be expressed by the following set of coupled differential equations (eqn (1.16))

In the axial representation, transformation to the rotating frame of reference is a simple multiplication of transverse magnetisation with the exponential M_xy,r = e^−iω₀tM_xy. The longitudinal magnetisation remains equal in both the laboratory frame and in the rotating frame.

In order to understand the effects of RF pulses on the magnetisation, it is convenient to use the rotating frame of reference as introduced above. Assuming a resonant RF pulse generates a magnetic field B_xy = B₁e^{iφ−iω₀t}, it is observed as a constant field in the rotating frame of reference B_xy,r = B₁e^iφ. Considering the pulse duration τ to be sufficiently short, such that relaxation effects do not contribute to any significance, the dynamics are described by eqn (1.17)

and eqn (1.18)

where and denote the complex conjugates of M_xy,r and B_xy,r, respectively.

Solving these equations for the longitudinal magnetisation yields a solution of the form (eqn (1.19))

with constants of integration C_± to be determined by the initial values. The transverse magnetisation can then be found by integration (eqn (1.20))

Assuming that the system started from equilibrium, M_z(t = 0) = M_eq and M_xy,r(t = 0) = 0 yields for the constants of integration

and (eqn (1.22))

In the rotating frame, the magnetisation experiences a rotation about the RF field (B₁), which means that magnetisation is transferred from the z-axis to the xy-plane and vice versa periodically for the duration of the RF pulse.⁶  The axis of rotation lies in the xy-plane, and the RF phase φ determines the orientation of the rotation axis on that plane, i.e. for the short RF pulse, the effective field acts as if the magnetisation experiences a constant external field in the direction of B⃑₁, about which it precesses for the duration of the pulse (see Figure 1.4 for an example of a y pulse). The angle swept out by the magnetisation during the duration of the pulse is called the flip-angle, which can be determined by α = γB₁τ. It should be noted that although the exponential term, including the RF phase φ, determines the direction of the motion in the rotating frame, it does not influence the flip-angle, as its magnitude is 1. Two special cases are of particular interest in MRI: the or 90° pulse and the π or 180° pulse. The former rotates the magnetisation by 90°, i.e. all longitudinal magnetisation is transferred to the transverse plane. The latter inverts the magnetisation and has the effect of inverting relative phases thus acting as an inversion and refocusing pulse.

In the previous section, a perfectly resonant excitation pulse was considered; however, this condition is often not met. This is either deliberate, e.g. to exclude portions of the sample when excitation is only required in a slice of interest (see Section 1.3.1.1), or because resonance is simply not achieved technically. If the rotating frame of reference is assumed to rotate at the same frequency as the RF field B⃑₁, the Bloch equations can be rewritten as eqn (1.23)

and eqn (1.24)

These equations do not, however, have an analytical general solution, but can be solved for special cases. The off-resonance term in reflects the fact that the precession frequencies of the RF pulse and magnetisation are not equal. This manifests itself as a frequency shift and a persistent slow precession in the rotating frame. The consequence of this is that excitation occurs at a reduced efficiency, thus the achieved flip-angles are generally lower and, furthermore, a non-zero phase is introduced to the transverse magnetisation. If the off-resonance is significant, no excitation takes place at all, which may be detrimental but can also be desirable in some imaging applications.

Spatial magnetic field gradients are of utmost importance to magnetic resonance imaging. They are used to select imaging slices, to manipulate the phase of the processing spins and are essential for spatial encoding. The magnetic field is generated to vary linearly with position r⃑ = (x, y, z)^T depending on the magnetic field gradient G⃑ = (G_x,G_y,G_z)such that B(r⃑)ẑ = (G⃑(t)·r⃑ + B₀)ẑ. This causes the spin dynamics to become spatially dependent. In the rotating frame of reference with frequency ω_r = γ(G⃑·r⃑ + B₀), the transverse magnetisation evolves into eqn (1.25)

The solution of this equation takes the form (eqn (1.26))⁷ 

Decomposing G⃑·r⃑ into its components yields eqn (1.27)

which can be rewritten in terms of spatial frequencies , with n = x,y,z (eqn (1.28))

Thus, the application of a gradient creates a phase difference according to the value of k⃑. With this property, image information will be recovered. As far as the outcome of the application of the gradient is concerned, the shape of the gradient is of minor importance; it is rather the time integral (called gradient moment), which constitutes the net effect of the gradient.

It should be noted that in the context of MRI, gradients exist in three orthogonal spatial dimensions, where they describe a change of the magnetic field component in the z direction, i.e. an “x gradient” G⃑_x = (∂B/∂x, 0, 0)^Tgenerates a change in B₀ dependent on the position on the x axis. Similarly, a “y gradient” generates a change dependent on the position on the y axis and so on (see Figure 1.5).

MR devices are set up in such a way that the presence of rotating transverse magnetisation can be picked up by a radiofrequency coil. This coil couples inductively to the precessing magnetisation. Thus, if the system is in equilibrium, no transverse magnetisation is present and thus no signal is received. Assuming this equilibrium is disturbed, for example by the application of an RF pulse, and transverse magnetisation is created, the receiver coil picks up a signal. This signal possesses a carrier frequency, which is equal to the frequency at which the magnetisation precesses, i.e. the Larmor frequency. The envelope of the signal is given by exponential T$2*$ decay as the transverse magnetisation undergoes relaxation (see Figure 1.6). This signal, S, is commonly referred to as the free induction decay (FID). Starting from equilibrium, the highest FID signal occurs after a 90° pulse, which transfers all longitudinal magnetisation into the transverse plane.

Imaging is rarely carried out on the FID signal itself (see Section 4.3), this is due to the fact that the time required for spatial encoding will usually cause most of the FID signal to be lost before an image can be acquired. Instead, the FID is refocused to create echoes, which are then used for imaging. The time after which an echo occurs is often abbreviated to TE, for echo time. This time is usually measured from the centre of the RF pulse to the time at which the peak echo amplitude occurs.

Spin echoes occur when existing, but dephased, transverse magnetisation is recovered by applying a refocusing RF pulse.³  After coherent transverse magnetisation is created, the magnetisation is left to evolve for the time ½ TE, before the refocusing pulse is applied. During this time the magnetisation undergoes both intrinsic non-reversible relaxation, as well as dephasing, due to field inhomogeneity. The pulse causes the magnetisation to be mirrored along the axis of the pulse and, following the subsequent ½ TE, the reversible effects have been undone and a signal maximum occurs. Without further influences, an echo is seen which has an envelope similar to that of two FIDs facing each other, with the peak amplitude at the time TE given by the signal loss due to true T₂ (see Figure 1.7).

If the effect of a gradient is used to artificially promote dephasing along a spatial direction, an echo might be recovered if another gradient of opposite polarity, but with the same gradient moment as the first one, is applied.⁸  This would cause the spins, which were initially precessing at an increased rate due to the gradient, to be slowed down, whereas those originally precessing more slowly would be sped up accordingly. Eventually, once the effect of the first gradient has been undone, an echo would occur (see Figure 1.8). This type of echo does not undo the effects of B₀ field inhomogeneity.

Stimulated echoes occur after the application of a minimum of three consecutive RF pulses.⁹  The first pulse creates coherent transverse magnetisation, which then evolves for a time, t₁. At this point, the second pulse transforms the transverse magnetisation to longitudinal magnetisation. After a time, t₂ from the second pulse, a third pulse recalls this magnetisation from the longitudinal direction into the transverse plane. Here the magnetisation refocuses, and an echo is formed. Such echo is referred to as a stimulated echo and is found at a time 2t₁ + t₂ after the first RF pulse.

The imaging apparatus is set up in such a way that the receiver coil detects magnetic flux changes in the transverse direction only. The signal can be expressed in terms of the transverse magnetisation that creates the signal in the receiver coil. The total signal is therefore proportional to the induced voltage from all contributing spins in the receptive volume of the coil,⁷  assuming the coil has a uniform sensitivity in this region (eqn (1.29))

By substituting the appropriate form of the Bloch equations, one can predict the signal expected at any time during an MR experiment (eqn (1.30))

Since the external field is assumed to be homogeneous in the region-of-interest (ROI), the Larmor frequency is not dependent on position and can be taken out of the integral, and since the detected signal is demodulated anyway, this term can be omitted entirely (eqn (1.31))

i.e. demodulating the signal is as if one is to measure directly from within the rotating frame of reference.

MRI images are commonly acquired in all three spatial dimensions. Therefore, encoding of image information also needs to be performed in three directions. MRI uses its gradient system to achieve this. Gradients are used during pulses, during the readout and on their own to manipulate the phase of the spins and imprint information relating to the spatial location. For the purposes of illustration, simpler 2D MRI sequences can be broken down into three consecutive modules: (1) slice selection, (2) phase encoding and (3) frequency encoding. More complex methods use these modules but not necessarily in the order above and they can sometimes occur simultaneously.

If a linear gradient field is applied, the Larmor frequency of the nuclei becomes spatially dependent and varies linearly with the position along the gradient. If a narrow-band RF pulse is applied whilst one encoding gradient is present, the resonance condition is only met in a subset of voxels, that is, a slice perpendicular to the applied gradient. If one considers the rotating frame of reference and a local off-resonance due to a gradient G⃑_r, the frequency offset is given by ω_r − ω₀ = γG⃑_r · r⃑(see Figure 1.9). An RF pulse of the form B_xy,r = B₁e^{−i(ω_rf − ω_r)t + iφ} , with the carrier frequency ω_RF, acts on the transverse magnetisation such that (eqn (1.32))

If the flip angle is sufficiently small and the longitudinal magnetisation is not significantly impacted by the pulse, then the longitudinal and transverse magnetisations do not couple, and the solution becomes eqn (1.33)

and for a pulse of duration τ (eqn (1.34))

The integral is an FT of the envelop function of the RF pulse, which relates the time evolution of the RF pulse to the slice profile it excites in the spin system, which is under the influence of the gradient. From this, it becomes apparent that the slice has a width, which is dependent on the bandwidth of the RF pulse and the slope of the gradient.⁶  In order to excite a rectangular slice, a sinc-shaped envelope function is required, since the Fourier transform of a sinc profile is rectangular. Such an envelope would require an infinitely long pulse duration, which cannot exist in reality. In other words, the pulse is truncated. This causes the slice profile to extend outside the desired slice. To prevent image artefacts, slices are not usually sampled consecutively or directly adjacent to each other. The centre of the slice is given by , the thickness depends on the bandwidth of the RF pulse Δω_RF, and the strength of the applied gradient G_r = |G⃑_r|(see Figure 1.9).

In the case of arbitrarily shaped pulses, the bandwidth is usually defined as the full-width-at-half-maximum (FWHM). Further attention should be drawn to the phase term before the FT, as it causes the transverse magnetisation to be dephased after the RF pulse has been applied. In order to recover the maximum signal from the selected slice, this dephasing needs to be undone. This can be accomplished by manually inducing a phase shift in the opposite direction by applying a gradient with half the gradient moment of the slice selection gradient.^6,10 

Switching on a magnetic field gradient for a fixed amount of time, δt, causes the Larmor frequency to vary linearly with position along the applied gradient. Since the phase angle Δφ is the time integral of the Larmor frequency, Δφ also varies linearly with position^7,10 : with a solution M_xy,r(t + δt) = M_xy,r (t)e^{iγG⃑·r⃑δt} . The phase angle after time δt is given by Δφ = γG⃑·r⃑δt. Phase encoding is best understood in terms of sampling the object for periodicity: for a completely homogeneous object, the signal from the whole object after application of the phase encode gradient will be very small. This is because the induced phase shifts disperse the magnetisation, leading to a small net transverse magnetisation. Now, assuming the object has a certain periodicity along the gradient direction, a gradient strength exists at which the periodicity of the object is the same as that of the phase term induced by the gradient. This causes all spins in the periodic object to acquire a phase, which is a multiple of 2π, and since the transverse magnetisation is aligned, a large signal is to be expected (see Figure 1.10). If one is to test for periodicity by assuming a different gradient strength each time and repeating the steps often enough; it is possible to detect these spatial frequencies. After each phase encoding step, it is then possible to reconstruct the measured signal and to create the corresponding MR image via FT. Phase encoding can be carried out in 1-, 2- and 3-dimensions, the latter is only rarely carried out, as phase encoding is demanding on measurement time.

The final dimension can be determined by switching a gradient on while acquiring spectral data. This will cause the Larmor frequency of the nuclei to differ along the gradient, thus emitting signals with different carrier frequencies. Fourier transformation allows the spectral information to be recovered and converted into spatial coordinates.⁸ 

Two major modes of encoding exist in MRI⁸ : 2D and 3D. In 2D, imaging is on a per slice basis. That is, in a slice excited via slice selection, one dimension is encoded via phase encoding and the other dimension is obtained from frequency encoding. 3D methods, however, consist of phase encoding in 2D and frequency encoding in the other. Both acquisition schemes have advantages and disadvantages. 2D methods are usually faster and are not prone to aliasing artefacts, nevertheless, slices cannot be put arbitrarily close together as the slice profiles might overlap and crosstalk between slices might introduce artefacts. 3D methods need a longer scan time due to the requirement of more encoding steps, but since no slice profile needs to be accounted for, voxels can be located directly adjacent to each other in all spatial dimensions. In some circumstances, it is beneficial to prevent excitation outside the region-of-interest. One way to achieve this is to excite a very thick slice, covering the region-of-interest, referred to as a slab. The difference between this and slice selection is that in the slab method, individual phase encoding steps are carried out in the same direction to finally yield thin slices. This method is an alternative to an oversampling in the phase encodes direction (see Section 1.3.4.6).

The signal equation depends on the magnetisation of all contributing voxels at position r⃑ = (x, y, z)^T. Each voxel has a specific Larmor frequency ω(r⃑,t) and a phase offset φ, which are dependent on position (eqn (1.35) and (1.36))

where M_i (r⃑,t) is the magnetisation at r⃑ before spatial encoding commences. Λ (r⃑) is the slice selection function, which describes the selected slice profile. In the case of an ideal, rectangular slice in a 2D sequence (or for slab selection in 3D) with a width Δz along the z direction, this function is . In the case of a 3D sequence, it is simply Λ (r⃑) = 1. Considering the 2D case, in which the direction of frequency encoding is x, such that , phase encoding is applied for a duration τ along y, , and z is the direction of slice selection (eqn (1.37))

The time integral of the gradients , n = {x,y,z} are the spatial frequencies. In this case and . They form the basis of the acquired MRI data and are commonly referred to as k-space. The choice of k-space trajectory, i.e. the choice of how and when gradients are applied, is the key element in MR imaging. In order to reconstruct an image from the raw MR data acquired, the correct trajectory needs to be chosen in such a way that it contains sufficient information relating to the final image.

It should be noted that, particularly when phase manipulation via gradients is executed, the signal equation attains the form of a FT in 2D or 3D, where the components are mapped out by the choice of gradient pulses (eqn (1.38))

The signal gives information on the transverse magnetisation in the region-of-interest. The aim is to acquire sufficient knowledge about the underlying magnetisation from which this signal originates to be able to reconstruct an image containing the NMR parameters⁷  such as the density of the sample nucleus and relaxation times (T₁, T₂).

It should be noted that, for the FT, it is necessary to fill the matrix step-by-step, which creates the signal S, i.e. multiple iterations with different phase encoding gradients are necessary to populate S. In the case of 3D spatial encoding, the signal equation becomes a 3D FT. Acquisition commences stepwise for each combination of k_y, k_z (eqn (1.39))

The signal matrix S = S(k_x, k_y, k_z, t) is commonly referred to as k-space data.

Since the relationship between image space and k-space is that of an FT, simple inverse relationships between the field-of-view (FOV) and k-space line spacing Δk, as well as voxel size δ and the maximum k-space value k_max, can be used (eqn (1.40))

Here, N_k is the matrix size. It should be noted that in order to increase resolution, i.e. reduce voxel size, it is necessary to achieve greater k-space values. Which, for a set encoding time, requires an increase in gradient strength, thus the maximum gradient strength is a limiting factor for the detectable special resolution of the image.⁸  Another limit to the achievable resolution is imposed by the receiver bandwidth, which determines the maximum detectable frequency.

In MRI, spatial encoding for an imaging object is accomplished by means of gradient fields, which create location-dependent spatial frequencies in x, y or z directions. The spatial frequencies are physically denoted by ‘k’ and hence, the space where the acquired MR data initially exist is called ‘k-space’. The k-space represents how much each spatial frequency component contributes to the image and therefore, the k-space data do not directly show anatomical structures of the imaging object (see Figure 1.11(a)). The anatomical image can be obtained by transforming the data space from the frequency domain (k_x, k_y and k_z) to the spatial domain (x, y and z). This can be achieved by FT (see Section 1.3.1.5). The corresponding spatial domain image of the example k-space data (Figure 1.11(a)) is depicted in Figure 1.11(b).

For most anatomical images, the signals in the spatial domain change relatively gradually in each spatial direction. Therefore, a large fraction of their frequency components are distributed around the low-frequency region. This can also be observed in the example k-space data above; signals in central k-space are much brighter than those in peripheral k-space. Here, the central k-space contains most of the signal-to-noise ratio (SNR) and overall contrast of the object, whilst peripheral k-space plays an important role in determining the detailed resolution of the image. This feature can be simply verified by nulling the peripheral k-space data while maintaining the central k-space data. Figure 1.11(c) depicts the case where only a quarter of the original k-space remains in the k_y direction. Although the corresponding reconstructed image generally looks blurry (see Figure 1.11(d)), it can be seen that the image still preserves most of the SNR and the overall shape of the image. This fact can be effectively used for the signal compression or more efficient sampling schemes.^11–13 

As described in the previous section, an MR image is reconstructed by a linear combination of a set of linearly independent spatial frequency components. If high-frequency components are not sufficiently sampled, the detailed structures of the MR image cannot be well represented with the given Fourier basis set. This is more pronounced in the regions where signal discontinuities exist (e.g. edges)¹⁴  as those regions require a greater number of frequency components in representing the shape than the regions where signals change smoothly. The problem results in oscillating structures and is known as the so-called “Gibbs ringing artefact”. As an example, oscillating structures can be observed near the edge region of Figure 1.11(d), which is, however, an exaggerated case where many of the high-frequency components are missing.

A straightforward way to solve the problem is to simply increase the k-space sampling range. However, since this strategy requires an increased scanning time, there are several alternative approaches to minimise the artefacts. One simple solution is low-pass filtering. Although this is quite effective in alleviating the oscillating structures, it also causes blurring of the sharp structures. There are also more sophisticated schemes to remove the problem while maintaining the shape of the sharp structures.^14,15 

A typical and widely used coordinate system in sampling k-space is the Cartesian grid. Its main advantage is that an imaging sequence for this grid can be easily implemented and the corresponding image can be simply reconstructed by applying the FT. However, since standard Cartesian imaging allocates an identical number of sampling points per k-space area regardless of k-space density of natural images, its sampling strategy is not efficient in terms of acquisition time and SNR. One solution to overcome this issue is to employ non-Cartesian sampling using radial or spiral trajectories.¹⁶ 

Figure 1.12(a) depicts an example of a radial trajectory. It consists of a number of radial lines for a set of rotation angles (0° < θ < 180°), each of which crosses the central k-space point. According to the projection theorem, the k-space data on a radial line with rotation angle θ are equal to the FT of the projection data of the image space at the same rotation angle θ.¹⁷  Therefore, the reconstructed image can be obtained by first performing an FT to the radial k-space data and then applying the back-projection algorithm. In order to account for dense sampling around the central k-space region, additional compensation filtering is required. This reconstruction procedure is the so-called filtered-back projection.¹⁸  However, it is nevertheless possible to reconstruct the radial data with more sophisticated approaches such as regridding or using an inverse model. Although the approaches may require increased reconstruction time, they do yield enhanced image qualities.^18–20 

Figure 1.12(b) shows another non-Cartesian k-space trajectory for a conventional single-shot spiral out sequence.²  The trajectory starts from the origin of the k-space and spirals outward to the peripheral k-space. In this way, k-space can be more rapidly and efficiently sampled than Cartesian sampling and it has been proven to be robust against motion and distortions.¹²  Depending on the purposes of the study, several variants of the spiral trajectory have been proposed. For instance, a ‘spiral-in/out’ trajectory was presented to improve BOLD contrasts in the regions where signal-drop typically exists due to the susceptibility differences.²¹  Spiral data can also be reconstructed by using regridding or establishing an inverse model. One major issue in spiral imaging is that image blurring, caused by the off-resonance effects, often degrades the image quality. In order to correct image blurring several methods have been proposed using a field map acquisition^22,23  or an automatic approach without any field map calibration.^24,25 

Ever since the first days of NMR, leaving aside the continuous wave methods, pulse sequences have been the de facto measurement technique even if the term was invented somewhat later. And, in essence since imaging is ‘only’ NMR spectroscopy with gradients, it is no surprise that pulse sequences have a rich and varied history and still proliferate today.

In order to understand the concepts employed by modern pulse sequence designers, it is important to understand a few basic elements and to understand that the purpose of a pulse sequence in imaging is to generate image contrast between two neighbouring tissue types, for example, a tumour surrounded by healthy tissue or flowing blood in vessels. One of the fundamental considerations that determines the details of a pulse sequence is the manner and speed with which k-space is traversed and this will determine contrast, resolution and field-of-view, for example. It is important to note that each k-space trajectory brings with it a different set of advantages and disadvantages. The plethora of sequences that exist in MRI arise from the fact the modules alluded to below, and the order in which they are played out, has no strict modular meaning. It is this fact that fundamentally underpins the creativity of the sequence and leads to the manifold applications and richness of MRI.

Imaging pulse sequences can be broken down very conveniently by dimensionality:

• 1D (a spectrum of profile)

• 2D (a slice or many single slices)

• 3D (selection of a ‘3D slab’ which is then ‘sliced’ in the reconstruction)

• 4D (3 spatial dimensions and, e.g., a frequency, resulting in spectra for each voxel in the 3D image)

• and higher dimensionalities are also possible in principle but rarely used.

Another convenient classification results from asking how is the signal, that is to be spatially-encoded, is generated and here again a number of different methods can be used:

• free induction decay (FID); usually used in conjunction with non-Cartesian k-space trajectories²⁶ 

• gradient echo; gradient echo sequences lack refocusing RF pulses and generate T₂^* contrast

• spin echo; make use of a refocusing pulse to generate T₂ contrast

• stimulated echo; used in spectroscopic sequences such as STEAM.²⁷ 

The manner in which k-space is traversed is also a classification parameter: Cartesian or non-Cartesian (spirals, cones, helices). As noted above, each trajectory has pros and cons.

A pulse sequence can be conceptually modularised in the following way: contrast-generation; slice selection; phase encoding; frequency encoding; and reconstruction.

As an example, at the start of a pulse sequence, the application of a 180° inversion pulse (an inversion recovery module) can be applied to tip all spins to point along the −z direction from where they will begin to recover through T₁ relaxation, resulting in T₁ contrast. The application of a 90° pulse followed by crusher gradients and a time delay (saturation recovery module) will also produce T₁ contrast, albeit with a reduced dynamic range. Different tissues relax with different rates and this results in image contrast. Generally, though, it is not necessary to have a separate contrast generation module. Contrast can be generated through just the interplay of flip-angle, TE, and TR. A good example of a T₁-weighted sequence is the 3D, MP-RAGE method.²⁸  Other conceptual examples of contrast generation include diffusion-weighted imaging,²⁹  perfusion-weighted imaging,³⁰  flow-encoded imaging,³¹  etc.

In its simplest form, a 2D slice can be selected by applying an RF pulse in the presence of a gradient (see Section 1.3.1.1). For a given RF pulse, a stronger gradient selects a thinner slice and a weaker gradient selects a thicker slice, also known as a slab; in 3D imaging, this slab is then ‘sliced’ computationally by virtue of the fact that the third direction is phase encoded. Repeating the 2D selection module for different spatial locations results in the so-called 2D multi-slice method. For many years, this simple method—an RF pulse in the presence of a gradient—remained practically the only method for slice selection and, indeed, it is still the only method used in the vast majority of pulse sequences.

At ultra-high field, the use of parallel transmit is now standard (parallel transmit is analogous to parallel receive, see Section 1.3.4.2). The use of parallel transmit brings with it the possibility of using more complex RF pulses (of discrete phase and amplitude modulation) in the presence of multiple gradient lobes to select complex volumes and not just rectilinear slices.³² 

Further, in the pursuit of reduced image acquisition time, acceleration in the slice direction using so-called multi-band pulses has also become common. Multi-band excite multiple, superimposing, 2D slices simultaneously, which are then separated in image reconstruction.

Having selected a slice, the imaging task now reduces to an encoding of the two, orthogonal in-slice directions. For one of these directions, phase encoding is used (see Section 1.3.1.2). This typically involves the application of a blipped gradient of short duration along the chosen phase-encode axis. Phase encoding imparts a phase offset to the spins dependent upon their spatial location. The FOV in the phase direction is determined by the step size in k-space in this direction, and spatial resolution is determined by the extent of k-space (see Section 1.3).

The frequency encoding direction is synonymously called the readout direction and encoding in this direction involves reading the signal in the presence of a gradient, the readout gradient. The FOV in the readout direction is determined by the step size in k-space in this direction and spatial resolution is determined by the extent of k-space (see Section 1.3.1.3).

It is noted here that, for non-Cartesian trajectories, phase and frequency encoding take place simultaneously as the k-space trajectory is swept out.³³ 

Currently, the most common reconstruction method is the application of an inverse FT, of appropriate dimensionality, to k-space data acquired on a Cartesian grid. Non-Cartesian k-space data can be resampled onto a Cartesian grid and then Fourier transformed.

In the search for imaging speed, a number of other methods have been introduced that incompletely sample Cartesian k-space, reconstruct the missing data by various means before applying an FT; primary exponents in this class are GRAPPA (k-space-based) and SENSE (image-based). Even more recently, methods that deliberately sample k-space incoherently have been used in conjunction with compressed sampling (CS) to achieve higher imaging speeds. Here, clearly, it is the reconstruction scheme to be employed that dictates the pulse sequence to be used.

The two main introductory classes of sequence are: (1) gradient echo sequences, lacking refocusing pulses, in which low-flip angle pulses are repeated at short intervals, each followed with read gradients applied while data are acquired; and (2) spin echo sequences, which typically use higher-flip angle excitation pulses, with refocusing pulses to recover the spin magnetisation dephased by local magnetic field gradients.

Gradient echo sequences lack refocusing RF pulses and typically use low-flip angle pulses that are repeated with short TR intervals. Usually one readout per TR takes place during which data are acquired. There are three main contenders for high SNR and good grey matter/white matter contrast: (a) MP-RAGE sequence,³⁴  (b) a modified version using a second inversion pulse, called MP2-RAGE³⁵  and (c) the Modified Driven Equilibrium Fourier Transform (MDEFT) sequence, optimised by Deichmann.³⁶  Acquisition parameters for the MP-RAGE sequence have recently been optimised by Stöcker and Shah³⁷  and Bock et al.,³⁸  giving a substantial improvement in CNR per unit time.

In its simplest form, the spectroscopic spin echo (SE) sequence can be described as follows

This sequence can easily be turned into an imaging variant by the judicious application of gradients. The SE sequence has the advantage that it refocuses dephasing due to magnetic field inhomogeneities and is T₂-weighted. However, because of the use of 90° pulses, the relaxation delay is long, making the overall acquisition time long too.

An imaging variant of the CPMG sequence,^39,40  which uses multiple 180° pulses, was invented by Hennig in 1986 and called RARE.⁴¹  The RARE sequence can be represented as

The string of refocusing pulses following a single excitation pulse form the essence of the RARE (also known as turbo-spin-echo (TSE) or fast spin echo) greatly increased the efficiency of the spin echo sequence. Unfortunately, the increased number of 180° pulses also greatly increases SAR, making the standard TSE sequences difficult to use at field strength above 3 T. The disadvantage can be overcome by using the GRASE sequence.⁴² 

The echo planar imaging (EPI) sequence, invented by Mansfield in 1977 ⁴³  can be conceptually thought of as a gradient echo sequence in which, following a single 90° pulse, the polarity of the readout gradient is continually alternated, and a train of signals is thus read. At the time point of readout gradient reversal, a blipped gradient is applied to enable traversal of k-space in one shot following the application of only one excitation pulse. Since gradient alternation can be done in a millisecond or so, the EPI sequence is very rapid.

The introduction of a 180° between excitation and the readout train will result in a spin-echo EPI sequence giving T₂ contrast. Further, traversal of k-space can be segmented to produce multi-shot EPI⁴⁴  or combined with a keyhole to merge the advantages of single-shot/multi-shot EPI.⁴⁵  EPI is the mainstay of fMRI and DWI/DTI methodology.

In order to achieve high spatial resolution in MRI, it is necessary to increase the k-space sampling range while maintaining a sufficiently adequate sampling rate. High-resolution MRI requires a greater number of sampling points than low-resolution MRI and hence, a longer acquisition time is required. In order to address this and achieve a reduction in acquisition time, numerous acceleration methods have been presented.

Depending on the applied sequence scheme, that is, whether the sequence is based on a Cartesian or non-Cartesian trajectory, or whether the sequence is 2D or 3D imaging, some of the acceleration methods may not be possible. Here, for simplicity, each acceleration technique will be described under the condition of 2D multi-slice Cartesian imaging. For this case, the acceleration methods can be further divided into two categories: in-plane acceleration and through-plane acceleration. The in-plane acceleration technique reduces the scanning time required for encoding each slice and the through-plane acceleration technique shortens the scanning time by simultaneously exciting multiple imaging slices. These two techniques can even be combined to achieve both benefits. In the following sub-sections, several widely used methods in the community will be introduced along with their fundamental features.

For a conventional 2D imaging sequence such as GRE or SE, the required acquisition time for encoding each slice is mainly determined by the number of phase encoding lines. The partial Fourier technique shortens the acquisition time by skipping sampling for a part of the phase encoding lines. Figure 1.13(a) depicts an example where a quarter of the top k-space region is excluded for sampling. This strategy is established based on the assumption that the final natural image only contains real values (i.e. no imaginary signals) and hence, its corresponding k-space has a conjugate symmetry (Hermitian). In theory, the missing k-space part can be reconstructed by simply taking the complex conjugate of the acquired part. However, in practical situations, spatial or time domain effects caused by flow, T₂ decay, B₀ inhomogeneity or eddy currents may induce imaginary components in the MR image. As a consequence, the constraint of conjugate symmetry cannot be perfectly applied for reconstruction.⁴⁶  Instead, a more sophisticated reconstruction scheme needs to be employed. Figure 1.13(b) shows the reconstructed image obtained by filling the missing k-space region with the projection onto convex sets (POCS) method.⁴⁷  It is observed that detailed structures of the brain are more clearly delineated in the POCS reconstructed image than in the zero-padding reconstructed image. As shown in the example figure, the partial Fourier technique is effective in reducing the scanning time whilst maintaining good image quality. However, due to reduced sampling, partial Fourier leads to an SNR reduction by a square-root of the acceleration factor. In this example, the SNR of the partial Fourier image (SNR^PF) can be expressed as eqn (1.41)

where SNR^Full indicates the SNR when full k-space sampling is used.

To avoid aliasing artefacts when sampling k-space, it is important to keep the sampling interval small enough so that it satisfies the Nyquist criterion. For the given imaging FOV, the Nyquist sampling interval is given by Δk^Ny = 1/FOV. In conventional Cartesian imaging, this interval is maintained for the acquisition of each phase encoding line. Here, the k-space coverage (k^cov) is determined by an integer multiple of the interval (i.e. k^cov = Δk^Ny·N) and the acquisition time increases in proportion to the total number of phase encodings (N). Similar to partial Fourier imaging, parallel imaging also accelerates the acquisition by reducing the number of phase encoding steps (e.g. N′ → N/2). However, instead of cutting the k-space coverage, parallel imaging accelerates the acquisition time while keeping the k-space coverage by increasing the sampling distance (e.g. Δk^PI = 2Δk^Ny). This approach allows one to maintain the image resolution but leads to aliasing artefacts in the reconstructed images. The aliasing artefacts need to be eliminated by additional processing.

For the elimination of the aliasing artefacts, several methods have been presented.^48–51  The detailed procedure is different depending on each method, but all the methods require multi-channel data to retrieve the original voxels from the aliased voxels. The multi-channel data can be obtained using a multi-channel receiver coil. Each channel has its own distinct sensitivity profile, which is presented as a multiplication factor in the image in each channel. Figure 1.14 depicts an example k-space trajectory for a parallel imaging scheme where the acceleration factor is two (Δk^PI = 2Δk^Ny). The figure also exhibits the corresponding multi-channel images reconstructed by Fourier transform (see Figure 1.14(c)), each of which shows two-fold structures with its own individual sensitivity profile. From the figure, it can be seen that a voxel on the folded regions is the result of the superimposition of the original two voxels, each of which is weighted by the corresponding sensitivity profile. This can be expressed as the following formula (eqn (1.42))

where C_i^j, V^j and S_i denote the coil sensitivity profile of the i-th channel at the voxel location j, original (unfolded) signal intensity at the voxel location j and the obtained (folded) signal intensity of the i-th channel. The coil sensitivity profiles can be estimated from an additional reference scan obtained with full k-space sampling (see Figure 1.14(b)). Therefore, the original voxel intensity (V^a and V^b) can be computed by taking the pseudoinverse of the matrix equation. The reconstruction procedure described above is the scheme employed by SENSE⁴⁹  and its resultant image is shown in Figure 1.14(d). As shown, the method is quite effective in eliminating the aliasing artefacts, however, depending on the accuracy of the sensitivity profile estimation, the reconstructed image quality may be affected (see the region marked by the white arrow). In order to overcome this issue, another reconstruction scheme, GRAPPA, has been presented.⁵¹  GRAPPA also requires a reference scan for calibration, but it does not need a complicated estimation procedure to obtain coil sensitivity profiles. Figure 1.14(d) shows a GRAPPA reconstructed image where the signal drop-out visible in the SENSE reconstructed image has disappeared.

Due to the reduced sampling, parallel imaging acceleration also induces an SNR reduction by a square root of its acceleration factor (R). In addition, to the spatially dependent term, the coil geometry factor (g) also constitutes a further SNR reduction which describes the ability of the multi-channel coil to separate the superimposed voxels.⁴⁹  Overall, the SNR of the parallel imaging reconstructed image (SNR^PI) can be written as: SNR^PI = SNR^Full/(g√R).

This sub-section introduces acceleration techniques, which are particularly associated with/used in EPI techniques. Single-shot EPI is one of the fastest MRI techniques and it acquires all phase encoding lines following a single RF excitation. When the number of phase encoding lines is N, the length of the sampling window of EPI is N-times as large as that of the conventional GRE or SE sequences (assuming equal bandwidth in the readout direction). As a result, the effects of T₂/T₂^* decay or accumulated phase errors develop significantly, which in turn causes blurring artefacts and geometric distortions in the reconstructed images. Furthermore, the increased sampling window of EPI requires a relatively long minimum TE. The acceleration techniques in EPI are, thus, of great help in not only reducing the scanning time but also in alleviating the aforementioned issues. The above acceleration techniques, such as partial Fourier or parallel imaging techniques, are also applicable to EPI. Here, segmentation-based approaches (e.g. multi-shot or readout segmentation) particularly designed for EPI will additionally be described.

The basic concept behind the multi-shot approach is to split the long EPI readout into several segments, each of which is acquired with an individual RF excitation.⁵²  As shown in Figure 1.15(b), multi-shot imaging performs segmentation along the phase encoding direction; to aid comparison, a single-shot EPI trajectory is also depicted in Figure 1.15(a). This example shows the case of three-shot EPI where only every third line is sampled at every frame and the missing lines are acquired in an interleaved way in the following two consecutive scans. Figure 1.15(c) shows another possible way to achieve k-space segmentation, which is termed as ‘readout-segmented EPI’ where segmentation is applied along the readout direction.⁵³ 

The multi-shot EPI and readout segmentation techniques can effectively reduce the long readout duration of EPI. However, due to the increased number of excitations, both techniques require more time to completely acquire k-space than the single-shot technique. An alternative reconstruction scheme to overcome this issue is a sliding window reconstruction⁵⁴  where each shot is considered as a single temporal frame and the missing k-space data are filled by the data from neighbouring scans. Although this approach reduces scanning time, it employs the repeated use of the same k-space data in several temporal frames (autocorrelation).

Another major drawback of multi-shot EPI is that as shot-to-shot instabilities cause a substantial magnification of the physiological noise, its use for dynamic MRI studies such as fMRI may be limited unless a proper correction procedure is applied.^55,56  The difficulties associated with using the multi-shot scheme for fMRI can be overcome with, for example, EPIK (EPI with keyhole).^56–59  As depicted in Figure 1.15(d), the EPIK acquisition scheme resembles three-shot EPI (Δky′ = 3/FOV) in terms of employing segmented acquisition, but differs from it in that the segmented acquisition is applied only to peripheral k-space. For the central k-space, full Nyquist rate sampling (Δky = 1/FOV) is instead applied, which plays the role of the ‘keyhole’⁶⁰  for every single time frame. The missing k-space lines in the peripheral region are completed with those from two adjacent scans. Here, it is important to note that a sliding window technique⁵⁴  is used to reconstruct the images, which ensures that the keyhole and the periphery of k-space are continually updated, albeit at different rates.

Figure 1.16 shows an example EPIK reconstructed image. Due to the reduced readout duration, the EPIK image shows reduced geometric distortions around the frontal area when compared to the single-shot EPI image. The reduction of geometric distortions can be also observed in the multi-shot EPI image. However, in EPIK, due to full Nyquist sampling for the keyhole region at every temporal frame, the performance of EPIK in detecting dynamic functional signals is comparable to that of community-standard, single-shot EPI method^56,59  and better than that of multi-shot EPI; the performance of multi-shot EPI has been shown to be significantly degraded due to undersampling of the keyhole region.

The acceleration methods described above are in-plane (e.g. x- and y-direction) acceleration techniques. The multi-band technique is a different type of acceleration, which reduces the number of multiple slice loops (e.g. z-direction) in the acquisition of volume data (i.e. through-plane acceleration). Multi-band utilises a specially designed RF pulse to excite multiple slices simultaneously; a conventional 2D multi-slice sequence typically uses a sinc RF pulse to excite only a single slice per slice loop. Simultaneous excitation is realised by applying sinusoidal modulation to a sinc RF pulse, which yields multiple clones of the original excitation band.⁶¹  An example RF pulse and its corresponding excitation bands are shown in Figure 1.17. Here, the sinusoidally modulated RF pulse produces three excitation bands, whereas the sinc RF pulse yields only a single excitation band (marked by white lines in each figure).

However, simultaneous excitation results in the overlapping of signals from multiple slices. Each individual slice can be reconstructed by applying a signal unfolding equation, as in the parallel imaging technique.⁶²  For this reason, the multi-band data also need to be acquired with a multi-channel receiver coil. Here, in order to avoid imposing similar sensitivity profiles on the simultaneously excited slices, an additional imaging gradient is applied in the slice direction so that each slice has a different distance offset in the phase encoding direction. This scheme assists a more effective signal separation from the superimposed slice signals.⁶²  Figure 1.17(c) displays the superimposed slices with different distance offsets and corresponding reconstructed individual slices.

High-resolution MRI is of great interest not only for clinical diagnosis but also for research activities. However, in general, high-resolution MRI requires an increased acquisition time. Furthermore, depending on particular MRI applications, the acquisition time that can be applied is often limited by the regulations, the subject or by the purpose of the study. In order to achieve an accelerated acquisition even in such circumstances, there have been many attempts to combine two or more acceleration techniques. This subsection introduces a high-resolution imaging method which achieved half-millimetre in-plane resolution with a whole-brain coverage for fMRI using the EPIK technique.⁶³  The proposed scheme is illustrative in that it combines four in-plane acceleration techniques (i.e. parallel imaging, partial Fourier, EPIK and readout-segmentation techniques) and achieves a minimum required TR of around 97 ms for a single slice with an in-plane resolution of 0.5 × 0.5 mm². The result indicates that if further combined with a multi-band acceleration with a factor of three, the proposed method can provide whole brain coverage in a typical fMRI setting (TE/TR = 35/3000 ms). Figure 1.18 shows examples of reconstructed images where a detailed spatial representation of the anatomical structures such as gyri or sulci is observed.

The use of EPIK for high-resolution fMRI at 7 T brings a number of advantages over EPI given that the T₂^* at 7 T is reduced making of an increased number of k-space lines concomitantly more difficult to acquire with standard methods. The ultra-high fields allow one to acquire MR data with an increased SNR. This makes it possible to scan human subjects with a high spatial resolution whilst maintaining a sufficient SNR. Figure 1.19 depicts an example of sub-millimetre (0.73 × 0.73 mm²) resolution scans obtained using EPIK at 7 T, where parallel imaging acceleration of 2 and partial Fourier factor of 5/8 were used. In Figure 1.19(a) four representative slices from a whole-brain acquisition, shown in Figure 1.19(b), are shown. The same imaging protocol was applied to a finger-tapping fMRI study using a block-based paradigm. Prior to fMRI data analysis, the standard pre-processing procedure including realignment, coregistration, normalisation and spatial smoothing was carried out for the initial fMRI time series. The first-level analysis using a generalised linear model (GLM) model was then performed to obtain functional activation as shown in Figure 1.20; a more detailed explanation of fMRI paradigm design and analysis can be found in Section 3.1. Motor activation can be clearly seen, and it elegantly follows the cortical ribbon. Details of the imaging experiments are given in the respective figure captions.

The acceleration and reconstruction methods introduced in this chapter have been widely used and verified in a number of previous works. There are also other types of imaging methods such as random sampling to avoid coherent sampling patterns.¹¹  Depending on the given imaging sequence, more complicated reconstruction methods may need to be required, for instance, using an inverse model with a proper regularisation technique.⁶⁴  However, as describing these methods is beyond the scope of the purpose of this chapter, the reader is referred to the related works.^65,66