Concepts of the 3D Golden Angle Radial Sequence

In the following, the concepts of the work by Chan et al [Chan2009], on which the sequence used in this chapter is based, are summarized. For conventional 3D radial sequences the vector tips of the radial profiles follow a spiral on the surface of a sphere.

A predefined number of profiles is needed to cover the whole 3D k-space. However, if the polar and azimuthal angles are chosen to be thegolden angles ΦGA and ΘGA, it can be achieved that the angular distribution of profiles is always approximately uniform, independent of the number of profiles. This is the 3D analogy of the 2D golden angle method using Θ₂D,GA ≈ 111^◦ [Winkelmann2007]. For both, 2D and 3D, the spatial distribution of the radial profiles for a varying number of profilesNprusing golden angles

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Figure 8.2: a) Using the two-dimensional golden means ϕ₁ and ϕ₂, points can be dis-tributed relatively uniformly on a unit sphere. b) By a change in topology, the 2D golden means can be used to uniformly distribute points across the surface of a sphere. c) Resulting vector tips of the radial profiles. (adapted from [Chan2009])

is shown in figure 8.1. The multidimensional golden angles for 3D radial imaging were derived by Chan et al from a modified Fibonacci sequence. They found the resulting two-dimensional golden means ϕ₁ and ϕ₂ to be:

ϕ₁ = 0.4656 ,ϕ₂= 0.6823. (8.1)

Using ϕ₁ and ϕ₂, points can be uniformly distributed on a unit square, as shown in figure 8.2 a). The coordinates of the points are placed according the following iterative process: Thex-coordinate of the first point is set to beϕ₁. Thex-coordinate of the next point is placed by incrementing the previous value byϕ₁and taking the modulus 1. This is iteratively repeated for all following points. The correspondingy-coordinates are cal-culated analogously usingϕ₂. This process can be denoted by (x, y) = ({mϕ₁},{mϕ₂}) withm= 1,2, ..., Npr.

By a change in topology, the result of the unit square is used to distribute the tips of the projections of the 3D radial k-space uniformly across the surface of the sphere surrounding the data. This is done by distributing the points along thekz-axis using the one-dimensional golden mean ϕ₁. The second golden meanϕ₂ is used to distribute the polar angle uniformly within the slices. This is illustrated in figure 8.2 b). The resulting distribution of the tips of the profiles is shown in figure 8.2 c).

The resulting golden azimuthal and polar angles ΘGA and ΦGA are given by:

ΘGA=sin⁻¹({mϕ₁}) , ΦGA = 2π({mϕ₂}). (8.2) 8.2.3 Sequence Design

The design of the implemented 3D GA radial sequence implemented in this work will be addressed in the following. The sequence parameters used are listed in table 8.1.

Figure 8.3: a) Sequence diagram of the first acquired profile and correspondingk-space trajectory. RF pulse and slab selection gradients have constant orientations, whilst the readout and spoiler gradients are rotated. b) Readout and spoiler gradients and correspondingk-space trajectories at three different repetition times, indicated by index m. The gradients of the first spoke are rotated with the current rotation matrixR to yield the subsequent profiles.

RF Pulse and Excitation Volume

A slab selection gradient is used to excite a large slab. The slab thickness is set to the same length as the FOV in in-plane direction. Additionally, a slab selection rephaser gradient is applied. For excitation, asinc-shaped RF pulse is used. RF spoiling is used according to equation 2.56 with phase increment φ₀ = 117^◦. The RF pulse and slab selection gradients are shown in the sequence diagram in figure 8.3 a).

Readout Gradients

The readout (RO) gradients are chosen such that data are sampled along a 3D GA radial k-space trajectory. Prior to each RO gradient, a prephaser gradient is played to start the readout at thek-space periphery.

In the sequence environment, a 3D rotation matrixR can be defined for each sequence element. It determines the orientation of the sequence element within the physical

scan-8.2 Methods 125

ner coordinate system. The MR scanner acquires images in the logical coordinate system, defined by the ‘frequency-encoding’ (FE), ‘phase-encoding’ (PE) and ‘slice-selection’ (SS) direction. To avoid specifying the amplitude for each of the gradientsGx,Gy,Gz in the logical coordinate system for every profile anew, R is exploited. The amplitude of the RO prephaser and the RO gradient is set for the first profile, which is aligned along the physical x-axis. All other profiles are determined by multiplication of the gradients of the initial profile with R:

R(m) =

⎛

⎜⎝

cosΦm −sinΦmcosΘm sinΦmsinΘm

sinΦm cosΦmcosΘm −cosΦmsinΘm

0 sinΘm cosΘm

⎞

⎟⎠, (8.3)

where the indexm indicates the profile number (m= 1, ..., Npr). This process is visual-ized for three profiles in figure 8.3 b).

Flip angleα 15^◦

BWpixel [Hz/px] 500

Spoiler momentAsp[mTs/m] 0.0352 spoiler phaseΔφ[rad] 3π

TR [ms] 5.19

TE [ms] 2.25

FOV [mm] 300

Table 8.1: Parameters used for the 3D GA radial sequence.

Data Acquisition and Pixel Bandwidth

Data acquisition is performed during the plateau of the trapezoidal RO gradients. The pixel bandwidthBWpixelis the adjustable parameter in the sequence which characterizes the receiver bandwidth, defined as:

BWpixel:= BWread

Nx

= 1 Ts

, (8.4)

whereBWread is the receiver bandwidth andTsis the total data acquisition time during one readout period. Nx is the number of sampled data during readout, including an oversampling factor of 2 performed automatically by the scanner. The unit of BWpixel

used is Hertz/pixel (Hz/px). The chosen bandwidth, being a compromise of imaging speed and SNR, is listed in table 8.1.

Spoiler Gradients

Spoiler gradients, as described in section 2.7.1, are gradients with a large area used to spoil the residual magnetization from previous excitations.

Figure 8.4: Diagram of sequence to measure gradient delays along the x-axis and cor-responding k-space trajectory. Opposing lines along the axes in k-space are repeatedly acquired. To prevent stimulated echoes, the spoiler gradients are rotated with the 3D golden angles.

Since for each profile a RO gradient is played without a rephaser gradient, there is already dephasing of the spins present at the end of each RO gradient. To take advantage of this fact, the polarity of the spoiler gradient is chosen to be equal to that of the RO gradient. Since the RO gradient in a 3D radial golden angle sequence is changing direction with every acquired profile, the spoiler direction changes as well with every profile. This has the wanted effect to prevent artifacts due to the build-up of a residual steady-state magnetization. The chosen area Asp and resulting phase Δφ across each voxel is listed in table 8.1.