As the flip angle is increased, the bSSFP signal as a function of off-resonance frequency begins to approximate a sinusoid for many biological tissues (). (Note that this approximation strictly speaking requires echo time TE = TR/2. At other TE, a simple linear phase is introduced across the spectral profile and the signal can be modeled by a complex exponential.) This spectral profile can be arbitrarily shifted in frequency by incrementing the phase of the RF pulse by some Δϕ from excitation to excitation [

3,

4,

7,

8]. When two large-flip-angle (~50° or greater) bSSFP acquisitions are performed with Δϕ = 0° and Δϕ =180°, the spectral profiles of the acquisitions approximate a sine and cosine, respectively, with period 2/TR Hz. A data set with spectral profile shifted by a desired spectral shift Δf can then be synthesized from the two acquisitions, using the relationship

as illustrated in .

Choice of TR such that TR = n / (2*CS

_{f}), where n = [1, 3, 5, …] and CS

_{f} is the absolute chemical shift of fat relative to water (in Hz), will place fat in a signal null whenever water is at a signal maximum. Choice of TR to adjust the relative phase of fat and water is also exploited in [

25]. If the appropriate frequency shift Δf needed to place water on a signal maximum is then identified on a voxel-by-voxel basis, a fat-suppressed image without banding can theoretically be formed from just two phase-cycled acquisitions. This forms the basis for our large-angle multiple-acquisition (LAMA) bSSFP technique.

The required frequency shift for each voxel can be determined either through acquisition of a field map, or through an intelligent search of voxel intensities over a range of Δf values (eliminating the need for field map acquisition and the associated scan time penalty). In the latter case, a field map is essentially inferred by examining what values of Δf yield the maximum water and minimum fat signal for each voxel. When large jumps in these Δf values are observed in adjacent voxels, the voxels likely lie at a fat/water boundary. Region-growing algorithms can then be devised to identify voxels as predominantly fat or predominantly water, and a smooth map of Δf values synthesized.

Once a smooth Δf map is generated, water-only or fat-only images can be reconstructed. Let S1_{n} be the complex signal of the n^{th} voxel from the first phase-cycled acquisition (with Δϕ = 0°), S2_{n} be the complex signal of the n^{th} voxel from the second phase-cycled acquisition (with Δϕ = 180°), and Δf_{n} be the frequency shift required to put water in the center of the passband (and hence fat in the center of the null) for the n^{th} voxel. The reconstructed signal intensity of the n^{th} voxel in the water-only image (W_{n}) is then given by:

Similarly, the reconstructed signal intensity of the n^{th} voxel in the fat-only image (F_{n}) is given by:

The above relations are clearly only valid if the off-resonant spectral profiles of the tissues of interest are identically sinusoidal. While the assumption of sinusoidal spectral profiles is an approximation, we can expect reasonable results across a range of tissues even at flip angles down to approximately 50°. In the spectral profiles of several tissues (fat, arterial blood, venous blood, and muscle) at flip angles of 30°, 60°, and 90° are shown (first column). The second through fourth columns of show spectral profiles synthesized from two phase-cycled acquisitions at Δf values of 1/(12TR), 1/(6TR), and 1/(4TR) respectively. Note that this range of Δf values spans the full range of distortion in the spectral profiles: the distortion at Δf =1/(3TR) is equivalent to that at 1/(6TR), and so forth. While the profiles are distorted enough at a flip angle of 30° to suggest that the LAMA bSSFP method is impractical at these lower flip angles, the distortion is relatively mild at 60°, as illustrated in columns two through four of . The distortion continues to decrease for flip angles approaching 180° (irrespective of T1/T2), although signal attenuation becomes significant at flip angles approaching 180°.

The warping of synthesized profiles at different Δf values has an effect on signal contrast. An analysis of arterial/venous, arterial/muscle, and arterial/fat contrast (defined as the absolute signal difference between two tissues) is shown in as a function of both flip angle and resonant shift Δf. As in , graphs are shown for Δf = 0, 1/(12TR), 1/(6TR), and 1/(4TR). Interestingly, the distortion tends to improve contrast between the relevant tissues for our flow-independent peripheral angiography application. T1 and T2 values assumed in all simulations for fat, muscle, arterial blood, and venous blood are listed in .

| **Table 1**T1 and T2 values assumed in simulations for muscle, fat, arterial blood, and venous blood. Values were informed by parameters found in [22,26-27]. |