2.5 2D and 3D Imaging
2.7 MRI Pulse Sequences
2.7.1 Gradient Recalled Echo (GRE) Sequence
Formation of a Gradient Echo
The formation of a gradient echo (GRE) is explained using a one-dimensional experiment along the x-axis shown in figure 2.8. The extension to higher dimensions is straightfor- ward. After the RF excitation, a constant gradient along the x-axis with the amplitude -GDeph(x, t) is switched on for a duration τ . Local magnetization vectors with differ-
ent x-locations precess at different frequencies ω(x, t) during this time. The precession frequencies are described as:
ω(x, t) =−γ · Gdeph(x, t)· x. (2.42)
Assuming the initial phase to be 0◦, the phases accumulated at time τ can be written as:
φ(x, τ ) =−γ
τ
0 GDeph(x, t)· x · dt = −γGDeph(x, t)· x · τ. (2.43)
At time τ a second gradient GReph(x, t) of same strength and duration but different
polarity is switched on. The accumulated phase is now:
φ(x, τ ) =−γGReph(x, t)· x · τ + γ
2τ
τ GReph(x, t)· x · t. (2.44)
At time t = 2τ the local magnetization vectors are rephased and a signal echo is mea- sured. The time between the RF pulse and the occurrance of the gradient echo is known as the echo time T E.
In general, GRE are formed when the 0th gradient moment is zero for all components.
B0-inhomogeneities and susceptibility effects are not eliminated by gradient echoes. Con- sequently, the GRE signal still decays with T2∗.
Basic 2D GRE sequence
The pulse sequence diagram of a basic 2D gradient echo sequence is displayed in figure 2.9. During an RF pulse of flip angle α a slice selection gradient Gz is switched on
to excite only a specific slice along the z-axis. To compensate for phase dispersion throughout the selected slice a refocusing gradient of opposite polarity and half its area succeeds the slice selection gradient (A).
In (B) the Gy gradient is applied. For each of the presented Gy gradients the slice has
to be excited by a new RF pulse. The effect of the Gy gradient in k-space is to move the
k-space position along the ky- axis. Positive and negative gradients lead to positive and
negative ky-locations. In general, both polarities are needed since the Fourier integral
given in equation 2.25 requires both, negative and positive values. In practice, this process can be abbreviated for example using half-Fourier imaging methods [Noll1991].
Figure 2.8: The formation of a gradient echo: The gradient GDeph dephases the local
magnetization vectors, the gradient GRephof opposite polarity rephases them
again. When the areas of both gradients add up to zero, a gradient echo is formed. The measured echo signal still decays with T2∗-relaxation (adapted from [Guenther1999]).
Frequency encoding using a Gx gradient and data acquisition with an ADC are shown in
(C) and (D). Prior to data acquisition a Gx prephaser gradient of opposite polarity and
half area of the readout gradient is turned on. This is needed for full k-space coverage from positive to negative values. In k-space it serves as a ‘prewinder’, moving the kx-
position to the negative maximum -kx,max. The frequency encoding Gx gradient results
in the trajectory of a straight line from -kx,max to kx,max at a given ky location. During
signal acquisition a gradient-recalled echo is formed when the area under the prephaser gradient time curve equals the area under the readout gradient at kx = 0. With equal
gradient magnitudes of prephaser and readout gradient the echo occurs at half of the frequency encoding duration. After data acquisition a spoiler gradient (E) is applied to dephase all of the residual transversal magnetization. The spoiling process will be described more detailed later in this section.
2.7 MRI Pulse Sequences 21
Figure 2.9: Pulse sequence diagram of a basic 2D GRE sequence: (A) RF pulse, slice selection and slice refocusing gradient, (B) Gy gradient, (C) Gx-rephaser and
Gx readout gradient, (D) Data acquisition with an ADC, (E) spoiler gradient
(adapted from [Guenther1999]).
adjacent excitations of the same slice is known as repetition time (TR). For Ny ky-lines,
the total measurement duration ttot of the sequence becomes:
ttot= Ny· T R. (2.45)
Spoiled GRE sequence in the steady-state
To encode multiple phase encoding steps a series of identical RF pulses with flip angle
α, evenly spaced over time with period TR, is applied to the spin system:
α− T R − α − T R − α − T R − ... (2.46)
After a certain number of repetitions the transverse and longitudinal components of the magnetization Mxy and Mz reach a dynamic equilibrium, known as steady-state. The
steady-state describes a periodic behavior of the magnetization with period TR.
GRE sequences can be classified by the value of the transverse magnetization before each new RF pulse. If Mtr = 0 the sequence is called a spoiled GRE sequence. If Mtr = 0
the sequence is said to be a steady-state free precession (SSFP) sequence. In this thesis a spoiled GRE sequence is used, therefore it will be the focus of the remainder of this section.
The condition Mtr = 0 can be achieved by choosing TR to be at least 5T2∗. In
this case, the transverse magnetization automatically decays nearly to zero due to T2∗ decay. However, the disadvantage is, according to equation 2.45, that with a long TR the total measurement time of the sequence becomes large. A more practical way is to dephase or spoil Mxy before each new RF pulse. Spoiling can be achieved either with
spoiler gradients or using RF spoiling, or a combination of both. Spoiler gradients are
large gradients applied at the end of each sequence block. They dephase the transverse magnetization prior to each new RF pulse. With RF spoiling, the residual magnetization is killed by cycling the phase of each RF excitation pulse according to a predetermined schedule. The mechanisms of spoiling will be described in more detail at the end of this section. For now, it will be assumed that the transverse magnetization is zero before each new RF excitation.
The steady-state is reached when a counterbalance between the loss of longitudinal magnetization due to tipping and regrowth due to T1-relaxation is established. The steady-state of Mz of a spoiled GRE sequence is said to be incoherent. Mathematically,
the created transverse magnetization at each data acquisition and therefore the received signal of a spoiled GRE sequence can be determined as follows. The following notation will be used: The indices ‘1, 2, 3, ...’ describe the number of the RF pulse, the indices ‘-’ and ‘+’ stand for ‘just before the RF pulse’ and ‘just after the RF pulse’. Mz0 describes
the equilibrium value of the magnetization.
Each RF pulse tips the longitudinal magnetization by flip angle α. For example for the first RF pulse Mz,1 after tipping has the value:
Mz,+1 = Mz,−1cosα. (2.47)
During the TR interval between the two RF pulses at time points t+1 and t−2 Mz regrows
due to T1-relaxation. This is given according to the solutions of the Bloch equation by:
Mz,−2 = Mz,+1e− T R
T1 + Mz0(1− e−T RT1) = Mz,−1cosαE1+ Mz0(1− E1), (2.48)
with E1 := e−T RT1 .
For a system in steady-state the longitudinal magnetization just before two RF excita- tions is the same. Therefore, the condition for the steady-state is:
Mz,−1= Mz,−2. (2.49) The elimination of Mz,−2 in equation 2.48 and rewriting yields:
Mz,−1= Mz0
(1− E1)
1− E1cosα. (2.50)
The measured signal S, which is proportional to the transverse component after flipping
Mz into the transverse plane by α, is caused by the rephasing of the magnetization vec-
tors at echo time TE:
S = Mz0sinα(1− E1)
1− cosαE1 e
−T ET ∗
2.7 MRI Pulse Sequences 23
T R»T1, α«θE T R»T1, α≈ θE T R T1, α«θE T R T1, α≈ θE
T E << T2∗ ρ-weighting ρ-weighting ρ-weighting T1-weighting
T E≈ T2∗ ρ-T2∗-weighting ρ-T2∗-weighting ρ-T2∗-weighting T1-T2∗-weighting Table 2.1: Relationship between TR, TE, α and the image contrast in a spoiled GRE
sequence
The flip angle α that maximizes the signal S is called the Ernst angle θE.
θE = arccos(E1) = arccos(e− T R
T1). (2.52)
The Ernst angle monotonically increases as the ratio T RT
1 increases. Mathematically, the
approach to steady-state can be described as:
Sj = M0sinθ[ 1− E1 1− cosθE1 + (cosθE1) j−1 (1− 1− E1 1− cosθE1)]e −T E T ∗2 , (2.53)
where j indicates the jth RF pulse.
Contrast of spoiled GRE sequences
As equation 2.51 indicates, spoiled GRE images are weighted by a factor of e−
t
T ∗2. This
makes them prone to signal loss, especially in regions with high B0- or susceptibility- inhomogeneities, for example near metallic implants. If T E is chosen to be short relative to T2∗, then the term e−(T ET∗
2 ) tends to 1 and no weighting by T
∗
2 remains.
For small flip angles α the term cosα in equation 2.51 approaches the value 1 and the E1-dependence cancels out. In the case of short echo times T E and small flip angles the signal is only dependent on M0, which is proportional to the proton spin density ρ0. The signal is said to be proton density weighted. If a small echo time T E and a larger flip angle, preferably around the Ernst angle, are employed, the E1-factor becomes dominant and the signal is weighted by the factor e−T RT1. The signal is said to
be T1-weighted. The relationship between the choice of TR, TE and α and the resulting
contrast is summarized in table 2.1.
Fast imaging with spoiled GRE sequences
An advantage of spoiled gradient echo sequences is that they can be used for fast imag- ing, especially as needed for the coverage of large 3D volumes. Fast GRE sequences employ small flip angles between α = 2◦ and 70◦. Small flip angles have the advantage that most longitudinal magnetization is undisturbed while there is still an appreciable amount of transversal magnetization. Short TR times can be used because there is no long T1-relaxation.
Mechanisms of Spoiling Gradient Spoiling
At the end of a TR period residual transverse magnetization can remain. In a sequence where TR is repeated multiple times, this residual magnetization can interfere with the desired magnetization in the subsequent data acquisition and lead to imaging artifacts.
Spoiler gradients are used to kill the unwanted remaining magnetization. They are
usually gradients of large 0th gradient moments applied at the end of a sequence to dephase the residual transverse magnetization whilst leaving the longitudinal magneti- zation undisturbed.
Considering and arbitrary voxel in which the transverse residual magnetization Mtr(r)
remains at the end of an TR period. Within this voxel the magnetization is the sum of many local magnetization vectors. When a spoiler gradient Gsp is applied, these vectors
fan out with frequencies dependent on their location along the spoiler gradient direction. The acquired phase φ(r) of the spoiler gradient Gsp at time t after the starting time t = 0
of the spoiler is given by the product of the gradient moment and the voxel dimension along the gradient direction:
φ(r, τ ) = γ τ 0 Gsp(r, t)rdt = γ τ 0 Gsp(r, t)rdt = γrAsp, (2.54)
where Asp is the 0th moment of the spoiler gradient. The phase dispersion across the
whole voxel Δφ is:
Δφ(r) = γΔrAsp, (2.55)
where Δr is the voxel dimension in direction of Gsp. The minimal moment needed to spoil
the unwanted magnetization is normally determined experimentally. The spoiler gradient moment is incremented and the resulting images are monitored for the occurrence of artifacts. For most applications, the minimal phase dispersion across the voxel must be at least 2π [Bernstein2004]. Since the polarity of the spoiler gradients can be chosen flexibly, they should be selected such that they do not counteract the preceding gradients. For example if the spoiler is applied along the same direction as a readout gradient, it should be of the same polarity. The spoiler gradient is preferably applied along the axis with the largest voxel dimension, since here the largest phase dispersion can be achieved for the same gradient moment. The magnetization is killed to a large extent if the spoiler moment is large enough. However, if only small amount of transverse magnetization is left undestroyed, it can build up to a steady-state value after multiple repetitions. Therefore, it is recommended to vary the moment of the spoiler gradient with each excitation to prevent this build up.
RF Spoiling
RF spoiling is done by applying a phase offset to each new RF pulse. This prevents a
2.8 Noise in MR Images 25
cycled according to a predetermined schedule. A commonly used schedule is:
φj = φj−1+ jφ0, (2.56)
where j is the index of the jth RF pulse, φj is the phase of the jth RF pulse and the
starting value φ0 can be chosen freely. φ0 = 117◦ has been shown to lead to efficient spoiling [Bernstein2004]. In general, efficient spoiling can be achieved by combining both, RF and gradient spoiling.