Stabilization of Alternating TR Steady State Free Precession Sequences

Abstract

Alternating TR steady-state free precession (ATR SSFP) has been proposed as a method to achieve a favorable frequency response compared to that of conventional balanced SSFP. ATR SSFP, much like conventional SSFP, exhibits oscillatory transient signal behavior that can degrade image quality. Thus an efficient preparation scheme is desired in order to actively reduce this initial signal fluctuation. Using an approach similar to that of Le Roux (JMR 163:23–37), we construct a mathematical model for ATR SSFP sequences and show a Fourier relation between the separated odd and even terms of the RF flip angle increment sequence during an initial preparation, and the resulting oscillatory residues. A weighted Kaiser-Bessel windowed ramp can be used to design preparation schemes for arbitrary TR₁, TR₂, and RF phase cycling combinations. Optimized Kaiser-Bessel windowed ramp preparations for wideband SSFP and fat-suppressed ATR SSFP imaging are tested in phantoms. The results show substantially reduced transient signal oscillation with this new initial preparation method.

1. Introduction

In balanced steady-state free precession (SSFP) MR imaging [1], both the magnetization and the received signal oscillates during the approach to steady state. The fluctuating signal creates a nonsmooth weighting in k-space that results in substantial image artifact [2,3]. The duration of these transient oscillations is on the order of the tissue T₁ relaxation time. It is possible to simply wait for the steady-state to be reached, however this compromises the effectiveness of magnetization preparation schemes (e.g. fat saturation or inversion recovery) that are essential in many SSFP applications. Efficient initial preparation schemes are critical for the broad use of SSFP sequences. Simple methods such as α/2-TR/2 preparation [2] can quickly align the magnetization in the direction of its steady-state, but only for a limited off-resonance range.

To achieve good transient oscillation suppression for a wider range of frequencies, more elaborate methods were proposed. For conventional balanced SSFP, Le Roux [4] used the SU₂ formalism [5–7] to develop a simplified model for transient signal behavior, and to demonstrate a Fourier relation between RF flip angle increments and the resulting oscillatory residues. A Kaiser-Bessel windowed ramp preparation method was proposed to minimize oscillatory transients [4]. This approach has been widely adopted, and is the current method-of-choice for reducing transient artifacts in conventional balanced SSFP imaging. Initial preparation schemes based on the Shinnar-Le Roux (SLR) algorithm have also been proposed [8,9], which rely on sequences of RF pulses that accurately generate the steady-state frequency response by magnitude-scaling and direction-selection. In addition to its design complexity, this method is sensitive to B₁ variation.

Alternating TR SSFP has been recently proposed as a way to achieve a favorable frequency response compared to conventional SSFP. In the case of fat-suppressed ATR (FS-ATR) SSFP [10], a wide stopband is achieved capable of suppressing fat in the steady-state. In the case of wideband SSFP [11], the spacing between nulls in the frequency response can be increased which mitigates banding artifacts for a given TR. Single-tip preparation was previously used for alternating TR SSFP sequences [10]. Paired Kaiser-Bessel design has been recently shown to have a performance comparable to optimum SLR design when there is a sufficient number of preparation cycles [12].

In this work, we adapt the SU₂ formalism approach to build a mathematical model for alternating TR SSFP signal behavior. This model provides justification for the use of Kaiser-Bessel windowed ramps, and a method for optimizing these ramps. Optimized initial preparation sequences for fat suppressed ATR SSFP and wideband SSFP are validated in simulation and phantom studies. We also describe a preparation scheme that can be applied to any alternating TR sequence with any RF phase cycling scheme.

2. Theory

In alternating TR SSFP (Figure 1), two repetition times TR₁ and TR₂ are used, and the phases of RF excitation in the two TRs are 0 and (π + φ), respectively [10,11]. A full cycle of the steady state contains two excitations and two periods of free precession and relaxation. Unlike conventional SSFP, the steady-state magnetization is alternating between two positions instead of returning to the same place after each excitation.

Fig. 1.

An alternating TR SSFP sequence. Black arrows represent RF excitation pulses.

In the following section we set φ = 0 for simplicity, and later will show that the result can be extended to arbitrary φ when 0 ≤ φ < π. Relaxation is neglected since its effects on signal oscillation are not as significant as the direction of the magnetization [4].

2.1. Characterization of the cycle rotation

In order to determine the resulting magnetization after multiple cycles of ATR SSFP sequence, first we need to characterize its cycle-rotation matrix. We define R₀ to be the transition matrix for one full cycle of the alternating TR sequence. When R₀ rotates a magnetization from its position at TR₁/2 (M₁) to TR₂/2 (M₂) and back to M₁ at the next TR₁/2 (M₁ = R₀M₁, see Figure 2), we find M₁ to be the steady-state magnetization, which is identical to the axis of R₀ rotation. R₀ can be written as the product of a series of 3 × 3 orthogonal rotation matrices (all representing right-hand rotations):

$$\begin{array}{cc}{\mathbf{R}}_0\hfill & ={\mathbf{R}}_1·{\mathbf{R}}_2\hfill \\ \hfill & =[{\mathbf{R}}_z({\theta }_1){\mathbf{R}}_x(\alpha ){\mathbf{R}}_z({\theta }_2)]·[{\mathbf{R}}_z({\theta }_2){\mathbf{R}}_x(-\alpha ){\mathbf{R}}_z({\theta }_1)]\hfill \end{array}$$ (1)

where α is the flip angle, θ₁(= −Δf · π · TR₁) and θ₂(= −Δf · π · TR₂) are the magnetization phase offsets due to off-resonance during TR₁/2 and TR₂/2, respectively. R₁ and R₂ are the rotation matrices for the first and second halfcycles, that M₂ = R₂M₁ and M₁ = R₁M₂. By re-writing the above matrices using SU₂ formalism (see Appendix A), we find the rotation axis n₁(= n_1x î + n_1y ĵ + n_1z k̂, without normalization) and rotation angle Ω₁ of R₁ matrix to be

$$\begin{array}{cc}{n}_{1x}\hfill & =\text{sin}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{R\theta }{2}\right)\hfill \\ n_{1y}\hfill & =\sin \left(\frac{\alpha }{2}\right)\text{sin}\left(\frac{R\theta }{2}\right)\hfill \\ n_{1z}\hfill & =\text{cos}\left(\frac{\alpha }{2}\right)\text{sin}\left(\frac{\theta }{2}\right)\hfill \\ {\Omega }_1\hfill & =2{\text{cos}}^{-1}\left[\text{cos}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{\theta }{2}\right)\right]\hfill \end{array}$$ (2)

where θ = θ₁ + θ₂ and R = (θ₁ − θ₂)/(θ₁ + θ₂). Similarly, R₂ has its rotation axis n₂(= n_2xî + n_2yĵ + n_2zk̂, without normalization) and rotation angle Ω₂ as

$$\begin{array}{cc}{n}_{2x}\hfill & =-\text{sin}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{R\theta }{2}\right)\hfill \\ n_{2y}\hfill & =\text{sin}\left(\frac{\alpha }{2}\right)\text{sin}\left(\frac{R\theta }{2}\right)\hfill \\ n_{2z}\hfill & =\text{cos}\left(\frac{\alpha }{2}\right)\text{sin}\left(\frac{\theta }{2}\right)\hfill \\ {\mathrm{\Omega }}_2\hfill & =2{\text{cos}}^{-1}\left[\text{cos}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{\theta }{2}\right)\right]\hfill \end{array}$$ (3)

We then define Ω ≡ Ω₂ = Ω₁, where Ω ≈ 2 cos⁻¹[cos(θ/2)] = θ for small α values. n₁, n₂, M₁, and M₂ are plotted in Figure 3. Note that n₁ and n₂ are situated in the bisector plane of $\widehat{{\mathbf{M}}_1{\mathbf{M}}_2}$ and are symmetric about the y-z plane, with the norms

$$\begin{array}{cc}\Vert {\mathit{n}}_1\Vert =\Vert {\mathit{n}}_2\Vert \hfill & ={[1-{\text{cos}}^2(\alpha /2){\text{cos}}^2(\theta /2)]}^{\frac{1}{2}}\hfill \\ \hfill & ={[1-{\text{cos}}^2\mathrm{\Omega }]}^{\frac{1}{2}}\hfill \\ \hfill & =\text{sin}\mathrm{\Omega }\text{.}\hfill \end{array}$$ (4)

Fig. 2.

The steady-state magnetization path of an alternating TR SSFP sequence with (0, π) phase-cycling. A full-cycle rotation of a magnetization with resonance offset Δf starts at M₁. It experiences phase offset of θ₁(= −Δf · 2π · TR₁/2) $({\mathbf{M}}_1\to {\mathbf{M}}_1^{+})$, RF excitation of angle −α $({\mathbf{M}}_1^{+}\to {\mathbf{M}}_2^{-})$, phase offset θ₂(= −Δf · 2π · TR₂/2) and becomes M₂. Then it experiences another phase offset θ₂ $({\mathbf{M}}_2\to {\mathbf{M}}_2^{+})$, excitation α $({\mathbf{M}}_2^{+}\to {\mathbf{M}}_1^{-})$, phase offset θ₁ and it returns to M₁.

Fig. 3.

The relation between M₁, M₂, n₁, and n₂. (a) 2D plot in x-y plane. Dotted line represents the magnetization path as in Fig. 2. (b) 2D plot in y-z plane. ϕ₁ is the angle between z-axis and n (the projection of n₁ and n₂ on y-z plane), ϕ₂ is the angle between n and both M₁ and M₂. Vector n_⊥ is in the y-z plane and perpendicular to n. (c) 3D plot with three axes being (x, n, n_⊥). ϕ_x is the angle between n and n₁ in (x, n) plane.

2.2. Approach to Steady-State from Thermal Equilibrium

Figures 3a and 3b show the projections of the vectors in x-y and y-z planes, respectively, and Figure 3c contains the three-dimensional depiction. We define φ₁ as the zenith angle between z-axis and n (the projection of n₁ and n₂ on y-z plane), and φ₂ as the angle between n and M₁ (which is equal the angle between n and M₂). The values of φ₁ and φ₂ are determined by the RF flip angles of the current R₁ and R₂ rotation. These two variables indicate the direction of the steady-state magnetizations, hence are essential for the calculation of transient oscillating residues.

Figure 3 along side with the derivation in Appendix B describes the relation between φ₁, φ₂, and RF flip angle. Starting from thermal equilibrium (when magnetizations are aligned with z-axis), an initial varying RF sequence {α_k} is applied. From Figure 3b we find that after the k^th excitation, when the flip angle α_k is small, the change in ϕ₁ value during this half-cycle (denoted as Δϕ₁(k)) is proportional to the RF flip angle increment Δα_k(= α_k − α_k−1) that

$$\mathrm{\Delta }{\varphi }_1(k)=\mathrm{\Delta }{\alpha }_k·\frac{\text{sin}(R\theta /2)}{2\text{sin}(\theta /2)}$$ (5)

From Figure 3c we also find Δϕ₂(k) to be proportional to Δα_k:

$$\mathrm{\Delta }{\varphi }_2(k)=\mathrm{\Delta }{\alpha }_k·\frac{\text{cos}(R\theta /2)}{2\text{cos}(\theta /2)}$$ (6)

Please see Appendix B for complete derivations of Eq. (5) and (6).

2.3. Transient oscillatory residues

We define the oscillatory residue ε to be the magnetization component perpendicular to its steady-state position. It lies in the (u₁, x) plane (perpendicular to M₁) at TR₁/2 and in (u₂, x) plane (perpendicular to M₂) at TR₂/2 (see Fig. 3b). After the k^th RF excitation, the new oscillatory residue ε_k can be calculated by first adding the shift in the steady-state magnetization vector(= the origin of (u, x) plane) to ε_k−1, then apply the Ω rotation. The relation between the oscillatory residues in two consecutive echoes can thus be written as

$${\epsilon }_k=e^{-i\mathrm{\Omega }}[{\epsilon }_{k-1}+\mathrm{\Delta }{\varphi }_1(k)\left|\mathbf{M}\right|+e^{-i\pi k}\mathrm{\Delta }{\varphi }_2(k)\left|\mathbf{M}\right|]$$ (7)

The e^−iπk term represents the alternating sign of Δϕ₂ during TR₁ and TR₂.

Defining the phase-shifted oscillatory residue ε_k = ε_k · e^iΩk and setting the magnetization magnitude |M| = 1, with Eq. (5) and Eq. (6), we find the oscillatory residue ε_p after p excitations (p is an even number) to be

$$\begin{array}{cc}{\epsilon }_p=\hfill & \frac{1}{2}[\frac{\text{sin}(R\theta /2)}{\text{sin}(\theta /2)}{\displaystyle \sum _{k=1}^p(e^{i\theta (k-1)}\mathrm{\Delta }{\alpha }_k)}\hfill \\ \hfill & -\frac{\text{cos}(R\theta /2)}{\text{cos}(\theta /2)}{\displaystyle \sum _{k=1}^p(e^{i(\theta -\pi )(k-1)}\mathrm{\Delta }{\alpha }_k)}]\hfill \end{array}$$ (8)

which shows that the oscillatory residue function is a linear combination of the Fourier transforms of the flip angle increment sequence {Δα_k} (see Appendix C for the complete derivation). Note that when TR₁ = TR₂, this simplifies to balanced SSFP, and is equivalent to Eq.(24) in reference [4]. The preparation sequence for a steady-state sequence with imaging flip angle α′ has two constraints, which are α₀ = 0 and α_p = α′.

A windowing function {w_k} is then chosen as the base flip angle increment sequence for initial preparation. We split {w_k} into two separate series of its odd and even elements, multiply them with scalars b₁ and b₂ respectively (b₁ + b₂ = 1 in order to meet the α_p = α′ constraint). The new flip angle increment sequence becomes {b_mw_k} (m = 1 when k is even, m = 2 when k is odd), and the oscillatory residue can be re-written as

$$\begin{array}{ccc}{\epsilon }_p=\hfill & \frac{\text{sin}(R\theta /2)}{\text{sin}(\theta /2)}\hfill & [b_1{\displaystyle \sum _{k=1}^{p/2}(e^{i\theta (2k-1)}{w}_{2k})}\hfill \\ \hfill & \hfill & +b_2{\displaystyle \sum _{k=1}^{p/2}(e^{i\theta (2k-2)}{w}_{2k-1})}]\hfill \\ \hfill & +\frac{\cos (R\theta /2)}{\cos (\theta /2)}\hfill & [b_1{\displaystyle \sum _{k=1}^{p/2}(e^{i\theta (2k-1)}{w}_{2k})}\hfill \\ \hfill & \hfill & -b_2{\displaystyle \sum _{k=1}^{p/2}(e^{i\theta (2k-2)}{w}_{2k-1})}]\hfill \end{array}$$ (9)

We can then optimize b₁ and b₂ for any resonance offset by minimizing |ε_p| at θ ~ θ₀ = −Δf · π · (TR₁ + TR₂). From Eq.(9) we know that |ε_p|_θ~θ0 has its minimum when

$$\begin{array}{ccc}|{\epsilon }_p{|}_{\theta \sim {\theta }_0}\hfill & =\hfill & [\frac{\text{sin}(R\theta /2)}{\text{sin}(\theta /2)}(b_1+b_2)\hfill \\ \hfill & \hfill & +\frac{\text{cos}(R\theta /2)}{\text{cos}(\theta /2)}(b_1-b_2)]·ℱ\left\{w_{2k-1}\right\}=0\hfill \\ \hfill & =\hfill & \frac{\text{tan}(R\theta /2)}{\text{tan}(\theta /2)}(b_1+b_2)+(b_1-b_2)\hfill \end{array}$$ (10)

Considering a wideband SSFP sequence (alternating TR SSFP with (0, π) phase-cycling) [11], for the center of passband θ ~ 0, the optimized b₁/b₂ ≈ (1 − R)/(1 + R) = TR₂/TR₁. Figure 4a shows an optimized eight-step Kaiser-Bessel windowed flip angle increment sequence {Δα_k}, that its elements form an alternatively scaled Kaiser-Bessel windowing function. The actual flip angle ramp for the initial preparation is shown in Figure 4b.

Fig. 4.

Wideband SSFP sequence: (a) an RF flip angle increment sequence {Δα_k} designed using Kaiser-Bessel windowing functions with optimized ratios. (b) the actual RF ramp sequence calculated using values in (a), that ${\alpha }_k={\displaystyle {\sum }_{n=1}^k\mathrm{\Delta }{\alpha }_n}$. (c) The absolute value of simulated oscillatory residues after an optimized 8-step linear ramp preparation. Grey solid line represents the true residue values obtained using matrix rotation, black dashed line represents the result given by Eq.(9). (d) true residue values after 8-step linear ramp and Kaiser-Bessel windowed ramp preparation. Black solid line: optimized Kaiser-Bessel window (β = 3), grey dashed line: Kaiser-Bessel window (β = 3) with b₁ = b₂ = 0.5, grey solid line: linear ramp.

Numerical simulations of different choices of window function {w_k} and scalars (b₁, b₂) were performed for a wideband SSFP sequence with p = 8 (four full-cycles). Figure 4c shows the oscillatory residues after linear ramps (ramps based on a rectangular window, i.e. the flip angle increments are constant) with optimized b₁, b₂. The solid line represents the true oscillatory residue values calculated using four R₀ matrix rotations, and dotted line represents the approximated values obtained from Eq.(9). The approximated and true |ε_p| functions are in close agreement within the passband. Fig 4d shows true oscillatory residues after a linear ramp (optimized b₁, b₂) and Kaiser-Bessel windowed [13] ramps (flip angle increments forms a Kaiser-Bessel window) with both optimized (b₁, b₂) and b₁ = b₂ = 0.5. The Kaiser-Bessel window was chosen because its shape can be flexibly modified by adjusting one control parameter β, and because it provides a near-optimal solution [14]. The black solid line represents the Kaiser-Bessel windowed ramp with optimized b₁ and b₂, which shows the best oscillatory residue attenuation among the three methods.

2.4. General Model for ATR SSFP sequences

This approach can be generalized to alternating TR sequences with an RF phase-cycling of (0, π + φ), 0 ≤ φ < π by writing down the new full-cycle rotation matrix ${\mathbf{R}}_0^{\prime }$ as

$$\begin{array}{ccc}{\mathbf{R}}_0^{\prime }\hfill & =\hfill & {\mathbf{R}}_1^{\prime }·{\mathbf{R}}_2^{\prime }\hfill \\ \hfill & =\hfill & \left[{\mathbf{R}}_z({\theta }_1){\mathbf{R}}_x(\alpha ){\mathbf{R}}_z({\theta }_2)\right]\hfill \\ \hfill & \hfill & ·\left[{\mathbf{R}}_z({\theta }_2){\mathbf{R}}_z(\phi ){\mathbf{R}}_x(-\alpha ){\mathbf{R}}_z(-\phi ){\mathbf{R}}_z({\theta }_1)\right]\hfill \\ \hfill & =\hfill & [{\mathbf{R}}_z({\theta }_1^{\prime }){\mathbf{R}}_x(\alpha ){\mathbf{R}}_z({\theta }_2^{\prime })]·[{\mathbf{R}}_z({\theta }_2^{\prime }){\mathbf{R}}_x(-\alpha ){\mathbf{R}}_z({\theta }_1^{\prime })]\hfill \end{array}$$ (11)

which has the same form as R₀ in Eq.(1), with ${\theta }_1^{\prime }={\theta }_1-\phi /2,{\theta }_2^{\prime }={\theta }_2+\phi /2$. Therefore the oscillatory residue can be obtained by substituting Rθ in Eq.(9) with (Rθ − φ):

$$\begin{array}{ccc}{\epsilon }_p\hfill & =\frac{\text{sin}((R\theta -\phi /2)}{\text{sin}(\theta /2)}\hfill & [b_1{\displaystyle \sum _{k=1}^{p/2}(e^{i\theta (2k-1)}{w}_{2k})}\hfill \\ \hfill & \hfill & +b_2{\displaystyle \sum _{k=1}^{p/2}(e^{i\theta (2k-2)}{w}_{2k-1})}]\hfill \\ \hfill & +\frac{\text{cos}((R\theta -\phi /2)}{\text{cos}(\theta /2)}\hfill & [b_1{\displaystyle \sum _{k=1}^{p/2}(e^{i\theta (2k-1)}{w}_{2k})}\hfill \\ \hfill & \hfill & -b_2{\displaystyle \sum _{k=1}^{p/2}(e^{i\theta (2k-2)}{w}_{2k-1})}]\hfill \end{array}$$ (12)

and the relation between optimized scalars b₁ and b₂ becomes

$$\begin{array}{c}\left(1+\frac{\text{tan}\left[(R{\theta }_0-\phi )/2\right]}{\tan ({\theta }_0/2)}\right)\cdot b_1\\ =\left(1-\frac{\text{tan}\left[(R{\theta }_0-\phi )/2\right]}{\tan ({\theta }_0/2)}\right)\cdot b_2\end{array}$$ (13)

For example, an FS-ATR SSFP sequence [10] with TR₂ = TR₁/3 and $(0,\frac{3}{4}\pi )$ RF phase-cycling has center of passband (water frequency) at θ₀ = −π/2 (omitting the RF phase increment to shift water to θ = 0, for simplicity). Using Eq.(13), the optimized b₁ and b₂ for water frequency will be

$$b_1=b_2=0.5$$

We can also optimize b₁ and b₂ for the center of the stopband (fat frequency), which is at θ₀ = π/2, and obtain

$$b_1=0,b_2=1$$

Flip angle increments and amplitudes in the ATR SSFP preparation scheme designed using Kaiser-Bessel window and the optimized (b₁, b₂) values for the water band are shown in Figure 5a and b.

Fig. 5.

Fat-suppressed ATR SSFP sequence: (a) an RF flip angle increment sequence {Δα_k} designed using Kaiser-Bessel windowing functions with (b₁, b₂) optimized for water band. (b) the actual RF ramp sequence calculated using values in (a), that ${\alpha }_k={\displaystyle {\sum }_{n=1}^k\mathrm{\Delta }{\alpha }_n}$. (c) The absolute value of simulated oscillatory residues after an optimized eight-step linear ramp preparation. Grey solid line represents the true residue values obtained using matrix rotation, black dashed line represents the result given by Eq.(9). (d) true residue values after 8-step linear ramp and Kaiser-Bessel windowed ramp preparation. Black solid line: Kaiser-Bessel window (β = 3) optimized for water band, grey dotted line: Kaiser-Bessel window (β = 3) optimized for fat band, grey solid line: linear ramp optimized for water band.

Numerical simulations of different {w_k} and (b₁, b₂) were also performed for this sequence with p = 8 (four full-cycles), and the results are shown in Figure 5. In Figure 5c, solid and dotted lines represent the true oscillatory residue values after four ${\mathbf{R}}_0^{\prime }$ rotations and the approximated values from Eq.(12), respectively. The approximated and true |ε_p| functions are in close agreement except for frequencies around the null bands of the ATR SSFP. Figure 5d shows true oscillatory residues after a linear ramp (b₁ = b₂ = 0.5, optimized for water) and Kaiser-Bessel windowed [13] ramps (b₁ = b₂ = 0.5, optimized for water and b₁ = 0, b₂ = 1, optimized for fat). The black solid line represents the Kaiser-Bessel windowed ramp with b₁ and b₂ optimized for water, which shows the best oscillatory residue attenuation within the water passband. Grey dotted line represents Kaiser-Bessel windowed ramp with b₁ and b₂ optimized for fat, and this sequence suppresses oscillatory residues better inside the fat-band.

3. Experimental Methods

Experiments were performed on a Signa Excite HD 3T scanner (GE Healthcare, Waukesha, WI) with a single-channel head coil. To measure the transient signal of different resonant frequencies, a linear shim was used to generate a frequency gradient in a uniform spherical ball phantom (T₁/T₂ = 150/35). A pulse sequence was designed to wait for thermal equilibrium, apply an eight-step initial preparation sequence, and then acquire the same phase encode for 64 full-cycles. The process repeats for all the phase encoding steps, so that a complete image can be reconstructed for each imaging TR after starting from thermal equilibrium. Finally, we extract one line from each image to plot the spectral profile evolution during the approach to steady-state.

Experimentally measured transient signals were compared with Bloch simulations in MATLAB (Mathworks, Inc., South Natick, MA). Three types of initial preparation were considered: dummycycles, linear ramp and Kaiser-Bessel ramp preparations (b₁, b₂ optimized for water). All preparations consisted of eight pulses (four full-cycles) and were 28ms in length. Two alternating TR SSFP sequences were considered: wideband SSFP and FS-ATR SSFP. The prescribed flip angle was 45° in all scans.

Phantom imaging with eight-step initial preparations and centric phase-encoding ordering was also performed for wideband SSFP and FS-ATR SSFP sequences to observe the artifacts caused by transient signal oscillation. Three ramp types were used: dummy cycles, linear ramp, and Kaiser-Bessel windowed ramp. Shim gradients were applied to create a ±25 Hz off-resonance span in the uniform ball phantoms. In wideband SSFP experiments the ramps were optimized for the center of passband (Δf = 0). Ramps optimized for different frequencies were tested in FS-ATR SSFP scans. Two phantoms were imaged with their resonant frequencies centered at water (pass band) and fat (stop band).

4. Results

Figure 6 contains the simulated and measured transient signals from a uniform phantom, as functions of number of cycles and resonant frequency. The experimental measurements show good agreement with the simulation. The measured spectral profiles are smoother compared to simulated ones because the real phantom reflects a continuous intra-voxel frequency distribution, whereas the simulation is discretely sampled in frequency. This frequency distribution also facilitates the decay of signal oscillation in experiments compared to simulations. Grey arrows indicate the point when central dips [11] become visible. The formation of a central dip in the spectral profile is related to magnetization relaxation, and it begins to form after the sequence reaches a constant flip angle. Hence, the two ramp-up preparation schemes result in a delayed appearance of the dip compared to dummy-cycle preparation.

Fig. 6.

Simulated and measured transient signals of a ball phantom after different preparation methods. (a) wideband SSFP, grey arrows indicate when the central dips appear. (b) FS-ATR SSFP. The experimental measurements show good agreement with the simulation. Kaiser-Bessel windowed ramp significantly reduced the transient signal fluctuation.

Initial preparations consisted of eight RF pulses (four full cycles), were 28 ms in length, and consisted of either dummy cycles, linear ramp, or Kaiser-Bessel ramp optimized for the center of the water band. Table 1 and Table 2 contains the averaged measured magnitude of oscillation over half of the passband and stopband in the first ten imaging cycles after different preparation schemes. Table values reflect the magnitude of oscillation relative to the steady-state signal amplitude. In wideband SSFP (Table 1), linear ramp reduced the oscillation from 18.7% to 4.6%, and Kaiser-Bessel ramp preparation further reduced it to 1.8% (the percentage represents the amount of oscillation compared to steady-state magnitude). In FS-ATR SSFP (Table 2), linear and Kaiser-Bessel ramps optimized for water resonant frequency reduced the amount of oscillation in water band from 41.3% to 1.0% and 0.4%, respectively. The oscillation reduction in fat band was from 48.6% to 9.4% (linear ramp) and 8.3% (Kaiser-Bessel ramp).

Table 1.

Measured signal oscillation magnitude during the first ten cycles of wideband SSFP, for three initial preparation schemes. Values are averaged over half the passband.

	Dummy Cycles / Linear / Kaiser-Bessel
center passband	18.7%	4.6%	1.8%

Table 2.

Measured signal oscillation magnitude during the first ten cycles of FS-ATR SSFP, for three initial preparation schemes. Values are averaged over half the water band and fat band.

	No Prep / Linear / Kaiser-Bessel
water band	41.3%	1.0%	0.4%
fat band	48.6%	9.4%	8.3%

Figure 7 shows the actual images obtained after different eight-step initial preparation schemes. Images obtained with Kaiser-Bessel windowed ramp have significantly reduced artifacts in both sequences. In FS-ATR SSFP images (Figure 7b), initial ramps with different (b₁, b₂) scalars show slightly more uniform signal and less artifact along the phase-encoding (vertical) direction, around the frequency bands they are optimized for. However the visible differences are subtle.

Fig. 7.

Alternating TR SSFP phantom images after different preparation methods. Eight-step preparation was used for all the scans. Images obtained with Kaiser-Bessel windowed ramp show better image uniformity and reduced artifacts. (a) Wideband SSFP images. Ramps were optimized for the center of passband where Δf = 0. (b) FS-ATR SSFP images. Resonant frequencies of the two phantoms were centered at water band (pass band) and fat band (stop band), with a ±25 Hz span. Ramps optimized for water (left column) and fat (right column) bands were both tested.

5. Discussion

Simulations and phantom experiments demonstrate ATR SSFP imaging with reduced signal oscillation using optimized Kaiser-Bessel windowed ramp preparation, compared to dummy-cycles and linear ramp preparation. The preparation scheme can be easily modified to minimize oscillatory residue at the desired resonant frequency using Eq.(12), and only a simple recalculation is needed when changing the length of preparation. This method is expected to be robust in the presence of $B_1^{+}$ variation because it only relies on relative flip angle amplitudes.

The Fourier relation between RF amplitude increments and the amount of oscillatory residue is based on the assumption that half of the imaging flip angle α/2 is small. As the flip angle becomes larger, actual oscillatory residues start to deviate from Eq.(8). Nevertheless, note that the small angle approximation (tan(α/2) ≈ α/2) used in Eq.(5) and Eq.(6) only has a 10% error even when the value of α is as high as 60°. This more than covers the flip angle range typically used in ATR SSFP imaging.

Relaxation effects are neglected in our model, due to the fact that T₁ and T₂ has little effect on the magnetization direction during initial preparation [4]. In reality, relaxation would result in a decreasing magnetization length which would introduce an additional smooth transient signal weighting in k-space, which is not expected to cause substantial deviation from this model.

In this work the coefficients b₁ and b₂ are optimized solely with respect to the center of passband, independent of the chosen windowing function. Therefore, the ramps may become slightly less effective as the amount of off-resonance increases. For this reason a Kaiser-Bessel windowing function is preferred since it has peak concentration around its pass band center and good attenuation in the side-lobes [14], thus the oscillatory residue can remain suppressed for a wide range of frequencies. As Figure 7 shows, Kaiser-Bessel windowed ramps optimized for water frequency still perform well in the fat band, and the same for the opposite case. Further optimization using numerical calculation over a frequency band is possible, in which case the spectral response of the windowing function and subject off-resonance characteristic will also become factors in the decision of b₁ and b₂.

6. Conclusions

We have developed a model for transient magnetization in alternating TR SSFP. Initial preparation sequences can be designed using a two-step process: first choose a windowing function (preferably a Kaiser-Bessel window), then optimize the (b₁, b₂) parameters by minimizing the oscillatory residue given in Eq. (12). Kaiser-Bessel windowed ramps with scaling factors optimized for wideband SSFP and FS-ATR SSFP were tested in phantom experiments, and the results showed signifiant reduction of transient signal oscillation in both cases. The proposed design can be applied to any combination of repetition times and RF phase-cycling. It is easy to implement, and actively reduces oscillatory residues during the transient approach to steady-state.

Appendix A

Using SU₂ formalism, R₁ (representing the rotation from M₂ to M₁) can be re-written as a 2 × 2 spinor matrix Q₁:

$$\begin{array}{ccc}{\mathbf{Q}}_1\hfill & =\hfill & {\mathbf{Q}}_z({\theta }_1){\mathbf{Q}}_x(\alpha ){\mathbf{Q}}_z({\theta }_2)\hfill \\ \hfill & =\hfill & \left[\begin{array}{cc}{e}^{-i{\theta }_1/2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & e^{i{\theta }_1/2}\hfill \end{array}\right]·\left[\begin{array}{cc}\hfill \text{cos}(\frac{\alpha }{2})& -i\text{sin}(\frac{\alpha }{2})\hfill \\ -i\text{sin}(\frac{\alpha }{2})\hfill & \hfill \text{cos}(\frac{\alpha }{2})\end{array}\right]\hfill \\ \hfill & \hfill & ·\left[\begin{array}{cc}{e}^{-i{\theta }_2/2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & e^{i{\theta }_2/2}\hfill \end{array}\right]\hfill \\ \hfill & =\hfill & \text{cos}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{\theta }{2}\right)·\mathbf{I}\hfill \\ \hfill & \hfill & -i\left[\begin{array}{cc}\text{cos}(\frac{\alpha }{2})\text{sin}(\frac{\theta }{2})\hfill & \text{sin}(\frac{\alpha }{2})e^{-iR\theta /2}\\ \text{sin}(\frac{\alpha }{2})e^{iR\theta /2}\hfill & -\text{cos}(\frac{\alpha }{2})\text{sin}(\frac{\theta }{2})\hfill \end{array}\right]\hfill \end{array}$$

where θ = θ₁ + θ₂ and R = (θ₁ −θ₂)/(θ₁ +θ₂). As described in Ref. [8], an SU₂ matrix Q can have its rotation axis n(= n_xî + n_yĵ + n_zk̂) and rotation angle Ω extracted directly from the matrix elements, that

$$\mathbf{Q}=\text{cos}\left(\frac{\mathrm{\Omega }}{2}\right)·\mathbf{I}-i·\text{sin}\left(\frac{\mathrm{\Omega }}{2}\right)·\left[\begin{array}{cc}\hfill n_z\hfill & n_x-i{n}_y\hfill \\ n_x+i{n}_y\hfill & \hfill -n_z\hfill \end{array}\right]$$

I is the identity matrix. Thus for matrix Q₁ we find

$$\begin{array}{cc}{n}_{1x}\hfill & =\text{sin}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{R\theta }{2}\right)\hfill \\ n_{1y}\hfill & =\text{sin}\left(\frac{\alpha }{2}\right)\text{sin}\left(\frac{R\theta }{2}\right)\hfill \\ n_{1z}\hfill & =\text{cos}\left(\frac{\alpha }{2}\right)\text{sin}\left(\frac{\theta }{2}\right)\hfill \\ {\mathrm{\Omega }}_1\hfill & =2{\text{cos}}^{-1}\left[\text{cos}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{\theta }{2}\right)\right]\hfill \end{array}$$

The same approach can be applied to R₂ rotation to get the another 2 × 2 matrix Q₂:

$$\begin{array}{ccc}{\mathbf{Q}}_2\hfill & =\hfill & {\mathbf{Q}}_z({\theta }_2){\mathbf{Q}}_x(-\alpha ){\mathbf{Q}}_z({\theta }_1)\hfill \\ \hfill & =\hfill & \left[\begin{array}{cc}{e}^{-i{\theta }_2/2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & e^{i{\theta }_2/2}\hfill \end{array}\right]·\left[\begin{array}{cc}\hfill \text{cos}(\frac{\alpha }{2})& i\text{sin}(\frac{\alpha }{2})\hfill \\ i\text{sin}(\frac{\alpha }{2})\hfill & \hfill \text{cos}(\frac{\alpha }{2})\end{array}\right]\hfill \\ \hfill & \hfill & ·\left[\begin{array}{cc}{e}^{-i{\theta }_1/2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & e^{i{\theta }_1/2}\hfill \end{array}\right]\hfill \\ \hfill & =\hfill & \text{cos}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{\theta }{2}\right)\hfill \\ \hfill & \hfill & -i\left[\begin{array}{cc}\text{cos}(\frac{\alpha }{2})\text{sin}(\frac{\theta }{2})\hfill & -\text{sin}(\frac{\alpha }{2})e^{iR\theta /2}\\ -\text{sin}(\frac{\alpha }{2})e^{-iR\theta /2}\hfill & -\text{cos}(\frac{\alpha }{2})\text{sin}(\frac{\theta }{2})\hfill \end{array}\right]\hfill \end{array}$$

and its rotation axis n₂(= n_2xî + n_2yĵ + n_2zk̂) and rotation angle Ω₂ are

$$\begin{array}{cc}{n}_{2x}\hfill & =-\text{sin}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{R\theta }{2}\right)\hfill \\ n_{2y}\hfill & =\text{sin}\left(\frac{\alpha }{2}\right)\text{sin}\left(\frac{R\theta }{2}\right)\hfill \\ n_{2z}\hfill & =\text{cos}\left(\frac{\alpha }{2}\right)\text{sin}\left(\frac{\theta }{2}\right)\hfill \\ {\mathrm{\Omega }}_2\hfill & =2{\text{cos}}^{-1}\left[\text{cos}\left(\frac{\alpha }{2}\right)\text{cos}\left(\frac{\theta }{2}\right)\right]={\mathrm{\Omega }}_1\hfill \end{array}$$

Appendix B

In Figure 3b we define ϕ₁ as the angle between z-axis and n, and observe that

$$\begin{array}{cc}\text{tan}{\varphi }_1\hfill & =\frac{n_{1y}}{n_{1z}}=\frac{n_{2y}}{n_{2z}}=\frac{\text{sin}(\alpha /2)\text{sin}(R\theta /2)}{\text{cos}(\alpha /2)\text{sin}(\theta /2)}\\ \hfill & =\text{tan}\left(\frac{\alpha }{2}\right)·\frac{\text{sin}(R\theta /2)}{\text{sin}(\theta /2)}\hfill \end{array}$$

For small α/2 values we have tan(α/2) ≈ α/2, therefore

$${\varphi }_1\approx \alpha ·\frac{\text{sin}(R\theta /2)}{2\text{sin}(\theta /2)}$$

Considering a sequence of excitations with RF flip angles {α_k}, the change in ϕ₁ value after the k^th excitation can be expressed as

$$\begin{array}{cc}\mathrm{\Delta }{\varphi }_1(k)\hfill & ={\varphi }_1(k)-{\varphi }_1(k-1)\hfill \\ \hfill & =\left[{\alpha }_k-{\alpha }_{k-1}\right]·\frac{\text{sin}(R\theta /2)}{2\text{sin}(\theta /2)}\hfill \\ \hfill & =\mathrm{\Delta }{\alpha }_k·\frac{\text{sin}(R\theta /2)}{2\text{sin}(\theta /2)}.\hfill \end{array}$$

Defining ϕ₂ as the angle between n and bothM₁ and M₂, and plotting M₁, M₂, n₁ and n₂ in 3D space (Fig.3c), we find

$${\mathbf{M}}_2·\text{cos}{\varphi }_2·\text{sin}{\varphi }_x·\text{tan}\left(\frac{\mathrm{\Omega }}{2}\right)={\mathbf{M}}_2·\text{sin}{\varphi }_2$$

Solving for ϕ₂, we have

$$\begin{array}{cc}\text{tan}{\varphi }_2\hfill & =\text{sin}{\varphi }_x·\text{tan}\left(\frac{\mathrm{\Omega }}{2}\right)=\frac{\left|n_{1x}\right|}{\Vert {\mathit{n}}_1\Vert }·\frac{\text{sin}(\mathrm{\Omega }/2)}{\text{cos}(\mathrm{\Omega }/2)}\\ \hfill & =\frac{\text{sin}(\alpha /2)}{\text{cos}(\alpha /2)}·\frac{\text{cos}(R\theta /2)}{\text{cos}(\theta /2)}\hfill \\ \hfill & =\text{tan}\left(\frac{\alpha }{2}\right)·\frac{\text{cos}(R\theta /2)}{\text{cos}(\theta /2)}\hfill \end{array}$$

Again we apply small-α approximation and obtain

$${\varphi }_2\approx \alpha ·\frac{\text{cos}(R\theta /2)}{2\text{cos}(\theta /2)}$$

Hence the change in ϕ₂ value after the k^th excitation is

$$\begin{array}{cc}\mathrm{\Delta }{\varphi }_2(k)\hfill & ={\varphi }_2(k)-{\varphi }_2(k-1)\hfill \\ \hfill & =\left[{\alpha }_k-{\alpha }_{k-1}\right]·\frac{\text{cos}(R\theta /2)}{2\text{cos}(\theta /2)}\hfill \\ \hfill & =\mathrm{\Delta }{\alpha }_k·\frac{\text{cos}(R\theta /2)}{2\text{cos}(\theta /2)}\hfill \end{array}$$

The above equations indicate that for spins of a certain resonant frequency, the amounts of increment in ϕ₁ and ϕ₂ is proportional to the amount of RF flip angle increment Δα.

Appendix C

Multiplying e^iΩk to both side of Eq.(7), we obtain a phase-shifted form (ε_k) of the oscillatory residue that

$$\begin{array}{ccc}{\epsilon }_k\hfill & =\hfill & e^{i\mathrm{\Omega }k}{\epsilon }_k\hfill \\ \hfill & =\hfill & e^{i\mathrm{\Omega }(k-1)}{\epsilon }_{k-1}+e^{i\mathrm{\Omega }(k-1)}[\mathrm{\Delta }{\varphi }_1(k)+e^{-i\pi k}\mathrm{\Delta }{\varphi }_2(k)]\hfill \\ \hfill & =\hfill & {\epsilon }_{k-1}+e^{i\mathrm{\Omega }(k-1)}\frac{\text{sin}(R\theta /2)}{2\text{sin}(\theta /2)}·\mathrm{\Delta }{\alpha }_k\hfill \\ \hfill & \hfill & -e^{i(\mathrm{\Omega }-\pi )(k-1)}\frac{\text{cos}(R\theta /2)}{2\text{cos}(\theta /2)}·\mathrm{\Delta }{\alpha }_k\hfill \end{array}$$ (C.1)

Using the fact that when the sequence first starts from thermal equilibrium, ε₀ = ε₀ = 0 and Ω ≈ θ, the oscillatory residue ε_p after p excitations (p is an even number) can be calculated by summing up the last two terms in Eq.(C.1):

$$\begin{array}{cc}{\epsilon }_p=\frac{1}{2}\hfill & [\frac{\text{sin}(R\theta /2)}{\text{sin}(\theta /2)}{\displaystyle \sum _{k=1}^p(e^{i\theta (k-1)}\mathrm{\Delta }{\alpha }_k)}\hfill \\ \hfill & -\frac{\text{cos}(R\theta /2)}{\text{cos}(\theta /2)}{\displaystyle \sum _{k=1}^p(e^{i(\theta -\pi )(k-1)}\mathrm{\Delta }{\alpha }_k)}]\hfill \end{array}$$

which suggests the Fourier relation between ε_p and {Δα_k}: the first term is the Fourier transform of {Δα_k} multiplied by a function of θ; the second term is the same Fourier transform shifted by π and multiplied by another function of θ.