### Abstract

- Top of page
- Abstract
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

Variable-density *k*-space sampling using a stack-of-spirals trajectory is proposed for ultra fast 3D imaging. Since most of the energy of an image is concentrated near the *k*-space origin, a variable-density *k*-space sampling method can be used to reduce the sampling density in the outer portion of *k*-space. This significantly reduces scan time while introducing only minor aliasing artifacts from the low-energy, high-spatial-frequency components. A stack-of-spirals trajectory allows control over the density variations in both the *k*_{x}–*k*_{y} plane and the *k*_{z} direction while fast *k*-space coverage is provided by spiral trajectories in the *k*_{x}–*k*_{y} plane. A variable-density stack-of-spirals trajectory consists of variable-density spirals in each *k*_{x}–*k*_{y} plane that are located in varying density in the *k*_{z} direction. Phantom experiments demonstrate that reasonable image quality is preserved with approximately half the scan time. This technique was then applied to first-pass perfusion imaging of the lower extremities which demands very rapid volume coverage. Using a variable-density stack-of-spirals trajectory, 3D images were acquired at a temporal resolution of 2.8 sec over a large volume with a 2.5 × 2.5 × 8 mm^{3} spatial resolution. These images were used to resolve the time-course of muscle intensity following contrast injection. Magn Reson Med 50:1276–1285, 2003. © 2003 Wiley-Liss, Inc.

In MRI, for a given gradient system constraint and acquisition bandwidth, scan time becomes proportional to the number of points sampled in *k*-space. Conventionally, the number of points sampled is determined by the Nyquist sampling rate given the field of view (FOV) and spatial resolution requirements. This gives a lower limit on scan time, and in many MRI applications, particularly those requiring 3D imaging, scan time is a limiting factor.

One way to reduce the scan time beyond the scan time limitation imposed by the Nyquist criterion is to undersample in *k*-space. However, undersampling inevitably results in either aliasing or lower resolution. Different approaches have been used to overcome this problem using a priori knowledge. They include parallel imaging methods (1, 2), unaliasing by Fourier-encoding the overlaps using the temporal dimension (UNFOLD) (3), partial *k*-space reconstruction methods (4–6), keyhole imaging (7, 8) and variable-density *k*-space sampling. Variable-density sampling uses the knowledge that most of the energy of an image is concentrated at low spatial frequencies. Therefore, by allocating more of the sampling points near the *k*-space origin, reasonable image quality can be achieved with fewer sampling points compared to the case in which *k*-space is sampled uniformly.

Variable-density *k*-space trajectories have been used primarily for fast imaging (9–12), reduction of aliasing artifacts (13–15), motion artifacts (16), off-resonance artifacts (17), and chemical shift imaging artifacts (18). Marseille et al. (9) employed nonuniform phase encodes to reduce imaging time. Scheffler and Hennig (10) proposed the use of undersampled PR trajectories, which are inherently sampled with variable-density, to reduce scan time. Peters et al. (11) used the undersampled PR trajectories to reduce scan time for angiography applications. Spielman et al. (12) used variable-density spiral trajectories to increase the temporal resolution of fluoroscopy. Nayak and Nishimura (13) tried randomizing the *k*-space trajectory to reduce artifacts by making the artifacts noncoherent. Tsai and Nishimura (14) studied variable-density spiral trajectories to reduce aliasing artifacts from objects outside the FOV. Cline et al. (15) designed logarithmic *k*-space trajectories to reduce aliasing artifacts. Liao et al. (16) described the use of variable-density spiral trajectories to reduce motion artifacts by oversampling in the low-spatial-frequency region. Stochastic *k*-space trajectories were used by Scheffler et al. (17) to reduce the off-resonance artifacts by randomizing the distribution of the off-resonance signal. Variable-density spirals were also used by Adalsteinsson et al. (18) for spatial side lobe reduction in chemical-shift imaging.

Compared to other imaging modalities, variable-density sampling is more practical in MRI because sampling points are arbitrarily determined by the gradient waveforms. Therefore, no additional hardware is required to enable variable-density sampling. The gradient waveforms can be modified to choose sampling points in *k*-space as long as the trajectory is still smooth. Consequently, undersampling can be done selectively in a very flexible manner.

Here we propose the use of variable-density sampling applied to the stack-of-spirals trajectory to significantly reduce 3D scan time while maintaining reasonable image quality. The stack-of-spirals trajectory (19, 20) consists of spiral trajectories in the *k*_{x}–*k*_{y} plane and phase encoding in the *k*_{z} direction. Since the combination of the spiral trajectory and the phase encoding allows flexible control over the density variation, it is particularly suitable for variable-density *k*-space sampling. The proposed technique was demonstrated with phantom images and applied in vivo to the problem of assessing limb perfusion.

MRI, with its excellent spatial resolution, is an ideal tool for assessing peripheral perfusion. Methods such as BOLD and *T*_{1} measurement of perfusion (21, 22) have been developed, but a more robust and quantifiable method is needed. First-pass perfusion imaging using bolus tracking is a validated, potentially quantifiable technique (23, 24), but it requires sufficient volumetric coverage with high temporal resolution. To cover the lower leg adequately during a single contrast injection, a temporal resolution of 2–3 sec is required with a 2–3 mm spatial resolution while covering a large volume. In this work, we demonstrate that such imaging goals can be achieved with the use of our fast imaging technique.

### THEORY

- Top of page
- Abstract
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

The variable-density sampling concept allows the sampling density to be a function of the *k*-space location. This enables scan time to be flexibly allocated based on the signal characteristics, the scan time constraint, and the image quality that is acceptable for the application. Since most of the energy of an image is concentrated near the *k*-space origin, it is more important to accurately resolve the signals at low spatial frquencies than signals at higher spatial frequencies. Therefore, in cases in which the scan time must be reduced beyond the Nyquist criterion, more scan time can be allocated to sampling the inner *k*-space region while the outer *k*-space region is undersampled.

For a variable-density *k*-space trajectory with a variable sampling rate, sampling density becomes a design variable. Therefore, we need an accurate definition of the sampling density. Here we define the sampling density to be the inverse of the sampling interval in each *k*_{x}*, k*_{y}*, k*_{z} direction. With the definition above, the sampling density is equivalent to the FOV in the uniform-density sampling case. Therefore, with variable-density sampling, the FOV becomes a variable. This varying FOV, which equals the sampling density, is called the effective FOV. It should be noted that this effective FOV is different from the nominal FOV that defines the FOV of interest, which is a constant. The effective FOV in each sampling direction can be expressed as follows:

- (1)

- (2)

- (3)

We use this effective FOV as the design parameter for the variable-density *k*-space sampled trajectories.

The stack-of-spirals trajectory consists of spirals in the *k*_{x}–*k*_{y} plane and phase encodes in the *k*_{z} direction. Since both the spiral trajectory and the phase encoding allow arbitrary density selection, it is well suited for variable-density sampling. Fast *k*-space coverage in the *k*_{x}–*k*_{y} plane using spirals makes this trajectory suitable for fast imaging applications. The uniform-density stack-of-spiral trajectory is shown in Fig. 1a.

The goal of variable-density *k*-space sampling is to undersample in the low-energy, high-spatial-frequency region. Without any prior knowledge about the object being scanned, it can be generally assumed that the power spectral density of a 3D object is roughly radially symmetric. Therefore, it would be reasonable to design the trajectory to have a radially symmetric density.

In this work, the *k*_{z} dependence of *FOV*_{x}*, FOV*_{y} was not utilized; in other words, the same set of spirals was used for all *k*_{z} locations. Figure 1b shows this variable-density stack-of-spirals trajectory. Since the *k*_{z} dependence in the *FOV*_{x}*, FOV*_{y} is not exploited, less undersampling is done in the outer *k*-space region.

Given the assumptions above, the effective FOV in the *x,y* direction can be replaced using a single-parameter *FOV*_{r}, where *r* is the distance from the *k*_{z} axis. Since the *k*_{z} dependence was removed from *FOV*_{x}*, FOV*_{y}, the new variable *FOV*_{r} is only a function of which again can be replaced by a new variable *k*_{r}*.* The effective FOV expression can now be simplified as follows:

- (6)

- (7)

With this new expression, the 3D trajectory design problem is reduced to designing a spiral trajectory and a phase encode. For the spiral and phase-encode design, functions *g*_{1} and *g*_{2} were each chosen to be monotonically decreasing functions of *k*_{r} and *k*_{z}, respectively. Once *g*_{1} and *g*_{2} are specified, the spiral trajectory in the *k*_{x}–*k*_{y} plane, and the phase-encode location in the *k*_{z} direction can be determined.

There are several methods that can be used to design the spiral trajectory (25–27) in the *k*_{x}–*k*_{y} plane. Here we employ a simple method that uses fixed limits on the gradient amplitudes and slew rate. The constraints involved in designing the spiral trajectories are as follows: First, hardware constraints limit the maximum gradient amplitude and slew rate. Since the starting angle of the trajectory is a variable for interleaved spiral imaging, the gradient *vector* magnitude and slew rate *vector* magnitude are constrained. Second, FOV requirements place a limit on the rate of increase in the radial direction compared with the rate of turning. The FOV also places a limit on the maximum gradient *vector* magnitude.

The desired Archimedian spiral trajectory can be described as

- (8)

where *k* = *k*_{x} + *ik*_{y}. Note that and θ = tan ^{−1}k_{y}/k_{x}. The effective FOV in the *x,y* direction gives the following relationship that determines the rate of increase in the radial direction compared with the rate of turning:

- (9)

where *N* is the number of interleaves. Other constraints that are imposed by the hardware and the effective FOV are

- (10)

- (11)

where *G* is the gradient, *S* is the slew rate, and *T*_{s} is the sampling interval. The trajectory that satisfies the above constraints can be solved numerically. Unlike the case for uniform-density spirals, the FOV in the equation above is a variable. However, the problem can be solved using a numerical method similar to the uniform-density case (26–28). The numerical routine starts at *k* = 0 and *G* = 0, with the initial condition of θ = 0 and dθ/dt = 0. Since *g*_{1} is specified to be a monotonically decreasing function of *k*_{r}, the spirals will have increased spacing in the outer *k*-space region.

The phase-encoding locations in the *k*_{z} direction are determined by specifying the spacing between phase encodes. The spacings between the phase encodes are designed to be 1/*FOV*_{z}(*k*_{x}), since the *FOV*_{z} was defined as the inverse of the sample spacing in the *z* direction. With *g*_{2} designed to be a monotonically decreasing function, the spacing increases in the outer *k*-space regions.

Image reconstruction was performed using a gridding reconstruction algorithm (29, 30). Since the trajectory was designed to be separable in the *k*_{x}–*k*_{y} plane and *k*_{z} direction, the gridding was done assuming the separability of the gridding routine; that is, the raw data was first gridded in the *k*_{x}–*k*_{y} direction and then gridded in the *k*_{z} direction separately. A cone-shaped function with three times the grid space width was used as the convolution kernel. A 2D cone function was used for gridding on the *k*_{x}–*k*_{y} direction, and a 1D cone function was used for gridding in the *k*_{z} direction. The grid size was chosen to be the smallest power of 2 that is bigger than the data size determined by the FOV and the image resolution in each direction.