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Keywords:

  • variable-density;
  • spiral;
  • 3D;
  • perfusion;
  • first-pass

Abstract

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. 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 kxky plane and the kz direction while fast k-space coverage is provided by spiral trajectories in the kxky plane. A variable-density stack-of-spirals trajectory consists of variable-density spirals in each kxky plane that are located in varying density in the kz 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 mm3 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 kxky plane and phase encoding in the kz 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 T1 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

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. 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 kx, ky, kz 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:

  • equation image(1)
  • equation image(2)
  • equation image(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 kxky plane and phase encodes in the kz 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 kxky plane using spirals makes this trajectory suitable for fast imaging applications. The uniform-density stack-of-spiral trajectory is shown in Fig. 1a.

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Figure 1. a: Uniform-density stack-of-spirals trajectory. It consists of uniform-density spiral trajectories in the kx-ky plane and uniform density phase encoding in the kz direction. b: Variable-density stack-of-spirals trajectory. It consists of variable-density spirals in the kx-ky plane and variable-density phase encodes in the kz direction. Different variable-density spirals can also be used in each kx-ky plane so that the sampling density can be decreased to better approximate radial symmetry. This will further reduce scan time. However, for our experiments, the same set of spirals was used for each phase-encode location, as shown in the figure.

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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.

However, the stack-of-spirals trajectory imposes the following constraints on the density variation: For spirals used in the kxky plane, the effective FOV must be the same in the xy direction, and they can only be a function of the distance from the kz axis (kmath image + kmath image) and the distance from the kxky plane (kz). The effective FOV in the z direction can only depend on the distance from the kxky plane (kz). These constraints are expressed in the equations below:

  • equation image(4)
  • equation image(5)

Therefore, under these trajectory constraints, radial symmetry can be best approximated by using spirals that are not evenly spaced in kz, and by having spirals that vary in density within each kxky plane. Utilizing the kz dependence in the FOVx, FOVy expression, the spirals will be sparser for phase-encode locations that are farther away from the k-space origin.

In this work, the kz dependence of FOVx, FOVy was not utilized; in other words, the same set of spirals was used for all kz locations. Figure 1b shows this variable-density stack-of-spirals trajectory. Since the kz dependence in the FOVx, FOVy 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 FOVr, where r is the distance from the kz axis. Since the kz dependence was removed from FOVx, FOVy, the new variable FOVr is only a function of equation image which again can be replaced by a new variable kr. The effective FOV expression can now be simplified as follows:

  • equation image(6)
  • equation image(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 g1 and g2 were each chosen to be monotonically decreasing functions of kr and kz, respectively. Once g1 and g2 are specified, the spiral trajectory in the kxky plane, and the phase-encode location in the kz direction can be determined.

There are several methods that can be used to design the spiral trajectory (25–27) in the kxky 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

  • equation image(8)

where k = kx + iky. Note that equation image and θ = tan −1ky/kx. 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:

  • equation image(9)

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

  • equation image(10)
  • equation image(11)

where G is the gradient, S is the slew rate, and Ts 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 g1 is specified to be a monotonically decreasing function of kr, the spirals will have increased spacing in the outer k-space region.

The phase-encoding locations in the kz direction are determined by specifying the spacing between phase encodes. The spacings between the phase encodes are designed to be 1/FOVz(kx), since the FOVz was defined as the inverse of the sample spacing in the z direction. With g2 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 kxky plane and kz direction, the gridding was done assuming the separability of the gridding routine; that is, the raw data was first gridded in the kxky direction and then gridded in the kz 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 kxky direction, and a 1D cone function was used for gridding in the kz 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.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

All of the experiments were conducted on a GE 1.5 T CV/i whole-body scanner equipped with gradients capable of a 40 mT/m magnitude and a 150 T/m/s slew rate, and a receiver capable of 4 μs sampling (±125 kHz).

Phantom Experiment

A phantom experiment was first done to verify the properties of variable-density k-space sampling combined with the stack-of-spirals trajectory. For the phantom experiment, the stack-of-spirals trajectory was implemented using a gradient-recalled echo (GRE) sequence with a flip angle of 90°. A large flip angle was used since the phantom had very short T1. The nominal FOV was chosen to be 26 × 26 × 26 cm3 and the resolution was chosen to be 1.25 × 1.25 × 1.25 mm3. The TR was 14 ms and the readout time was fixed at 6 ms. A head coil was used for transmitting and receiving.

Four different sampling schemes were used for the phantom experiment. First, a uniform-density, sufficiently sampled trajectory was used to acquire an image that could be used as a reference to determine the image quality of the undersampled images. Then three different sampling schemes, each with 52% of the scan time, were used to compare different undersampling schemes. The three undersampled trajectories used the same number of interleaves and the same number of phase encodes in the kz direction, resulting in the same number of excitations and scan time. The three undersampled trajectories were designed to have uniform, linear, and quadratic density variations in the kr and kz directions.

Image quality resulting from different sampling schemes can be understood through point spread functions (PSFs), as shown in Figs. 2 and 3. Since the trajectories were designed to be separable in the kxky and kz directions, the PSF becomes separable the xy and z directions. In other words, the 3D PSF becomes the the multiplication of the 2D PSF in the xy direction and the 1D PSF in the z direction. Therefore, the 3D PSF can be fully described by the two PSFs in the x,y and z directions. Figure 2 shows the PSF in the xy direction, while Fig. 3 shows the PSF in the z direction. In Fig. 2, the effective FOV in the xy direction, determined by function g1 as in Eq. [6], is shown with the corresponding PSF in the xy direction. The first column shows the effective FOV in the xy direction as a function of kr. The second column shows the PSF, and the third column shows the cross section of the PSF along the center. The effective FOV in the z direction, determined by function g2 as in Eq. [7], is shown in Fig. 3 with the corresponding PSF in the z direction. It can be seen from the PSFs that the uniform-density undersampled case has a large, significant aliasing peak within the FOV, which will result in severe aliasing artifacts, while the variable-density undersampled cases have aliasing signals that are smoothed out with a smaller peak value. The width of the central peaks are similar in all four cases.

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Figure 2. Effective FOV in the x-y direction along kr, and PSF in the x-y direction for (a) uniform-density sufficiently-sampled, (b) uniform-density undersampled, (c)linear variable-density undersampled, and (d) quadratic variable-density undersampled trajectories. Each column shows the effective FOV, PSF in the x-y direction and the cross section of the PSF. These sampling densities were used for the phantom experiment. The variable-density undersampled trajectories show PSFs with a smaller peak aliasing signal inside the FOV compared to the uniform-density undersampled case.

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Figure 3. Effective FOV in the z direction along kz, and PSF in the z direction for (a) uniform-density sufficiently-sampled, (b) uniform-density undersampled, (c) linear variable-density undersampled, and (d) quadratic variable-density undersampled trajectories. The first column shows the effective FOV and the second column shows the PSF. These sampling densities were used for the phantom experiment. The variable-density undersampled trajectories show PSFs with a smaller peak aliasing signal inside the FOV compared to the uniform-density undersampled case.

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First-Pass Perfusion Study

The first-pass perfusion study was performed as an in vivo study that would benefit from the technique proposed in this work. The study was performed by scanning the infrapopliteal region of both legs repeatedly to obtain temporal information over a large volume. The procedure was as follows: The right leg was occluded by inflating a blood-pressure cuff to above the subject's systolic blood pressure level. About 20 s after the scan started, 10 cc of Magnevist (contrast agent) was injected over 3 s. Approximately 50–80 s after the injection, the cuff was deflated. Occlusion of one leg was done to demonstrate the differential perfusion of the ischemic vs. normal leg. The release gives a hyperemic response, which is potentially a better, more sensitive measure of ischemia.

For this study, the stack-of-spirals trajectory was implemented using a spoiled-gradient-recalled echo (SPGR) sequence with a flip angle of 90°. The flip angle was chosen to suppress the muscle signal. Since the infrapoplitial regions of both legs have to be covered for this application, the nominal FOV was chosen to be 27 × 27 × 28 cm3. The resolution was designed to be 2.5 × 2.5 × 8 mm3. TR was 22 ms. A head coil was used for both transmitting and receiving.

For this application, the quadratic undersampling scheme was chosen. In the xy direction the effective FOV was decreased to 15 cm, and in the z direction it was decreased to 18 cm. The readout duration was chosen to be 8.1 ms. This resulted in four-interleave spirals with 32 phase encodes in the z direction. The temporal resolution achieved was 2.8 s. In comparison, the use of a uniform-density stack-of-spirals trajectory would have resulted in a temporal resolution of 4.5 s. For our experiment, 70 3D data sets were acquired continuously over 3 min 17 s.

Spectral-spatial excitation pulses (31) were used to suppress fat and limit the FOV in the z direction. The rapid changes in the gradient for the spectral-spatial excitation pulse cause eddy currents, which in turn distort the readout gradient waveform. This results in artifacts in the image. This is currently overcome by delaying the data acquisition by 2 ms after the RF excitation.

We analyzed the 4D data (3D spatial, 1D temporal) acquired by observing the fractional muscle signal intensity change along the temporal dimension. The first five temporal data sets were discarded to avoid the use of data collected before the steady state was reached. Since there was a slight gap between the 3D data acquisitions due to data transfer time in the scanner memory, some of the data sets collected in the beginning may have been collected before the steady state was reached. The fractional change was calculated by taking the difference with respect to the first image and dividing the difference value with the maximum signal intensity of the first image. The maximum signal intensity of the first image represents the muscle signal intensity before contrast injection. The muscle signal intensity was used for normalization between different subjects to compensate for different imaging conditions. Twenty-five (5 × 5) pixel values, which were carefully chosen to be away from the arteries, were averaged for the analysis.

RESULTS

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

Phantom Experiment

The phantom experiment results are shown in Fig. 4. Axial and sagittal images from the 3D data set are shown for the four different sampling schemes chosen for the experiment. Figure 4a shows the uniform-density sampled case, which serves as a reference for image quality. Figure 4b shows the uniformly undersampled case with clearly visible aliasing artifacts. The axial image shows aliasing in both the x, y and z directions. The spiral artifact is seen in the outside edge of the image, and the z direction aliasing is clearly visible inside the resolution phantom, with an elevated signal level in the normally null signal area. Figure 4c and d show the linearly undersampled and quadratically undersampled case. The artifacts are not as significant as in the uniform-density undersampled case. The image quality for the variable-density undersampled cases is quite comparable to the uniform-density, sufficiently-sampled case. In the axial images, only minor aliasing can be seen at the edge of the bottle phantom. In the sagittal images, mild artifacts are visible at the top and bottom of the figure in the dark area. The two variable-density sampling schemes appear to provide similiar image quality for the undersampling factor chosen.

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Figure 4. Phantom images acquired using (a) uniform-density sufficiently-sampled, (b) uniform-density undersampled, (c) linear variable-density undersampled, and (d) quadratic variable-density undersampled trajectories. The uniform-density undersampled image shows significant aliasing, while the variable-density undersampled images show little aliasing artifacts. The arrows point to the aliasing artifacts.

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First-Pass Perfusion Study

Figure 5 shows the results from one of the normal subjects. For this subject, the cuff was deflated approximately 50 s after the injection. The four figures on the left column are four different slices from one of the 70 3D volumetric data sets. The blue boxes and red boxes on the images show the region of the unoccluded and occluded leg chosen for the signal intensity plot on the right-hand side of the images. The plots show the fractional change of the muscle signal intensity in each of the four slices. The bold line represents the muscle signal intensity of the unoccluded leg, and the regular line represents the muscle signal intensity of the occluded leg. The delay in the signal intensity rise is clearly visible in the occluded leg. The hyperemic response represented by the steep signal rise to a higher signal intensity is also clearly depicted. Note how the high temporal resolution enables good tracking of the signal rise.

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Figure 5. Normal volunteer first-pass perfusion study results. The delay in the signal rise for the occluded leg can be seen. The occluded leg shows a hyperemic response with a much steeper slope and higher peak signal value.

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The results shown in Fig. 6 are from a patient known to have a peripheral vascular disease (Ankle Brachial Index < 0.5) on the right leg. The ischemic leg was occluded to compare the hyperemic response with that of normal subjects. The cuff was deflated approximately 80 s after the injection for this patient. The results clearly depict the delay in the signal intensity rise of the occluded leg. The hyperemic response of the occluded leg shows a much smaller slope in the increase of the signal intensity compared to that in the normal volunteer.

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Figure 6. Patient first-pass perfusion study results. The delay in the signal rise for the occluded leg can be seen. The signal rise is generally smaller than in the normal volunteer, and the hyperemic response shows a significantly smaller slope.

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Figure 7 shows an oblique slice in the z direction of one of the 70 volumetric data sets for a normal volunteer and a patient. The color overlay represents the slope of the signal intensity rise for each pixel. Color was overlaid on pixels with a slope that is higher than 0.28%/s. The maximum slope was 1.32 %/s. The images were filtered with a low-pass filter before the slope was calculated, to reduce noise in the signal intensity curve. The slope of the signal intensity rise was estimated by taking the maximum value of the difference between signals that were 10 temporal samples apart. It can be clearly seen in the occluded leg (right leg) that the normal volunteer has a higher slope in most of the region. This demonstrates that patients can be distinguished from normals using this method, while regional information over a large volume is also obtained.

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Figure 7. An oblique slice in the z direction of one of 70 volumetric data sets is shown for both a normal volunteer (a) and a patient (b). The slope of the signal intensity rise in each pixel is overlaid with color. Color was overlaid only on pixels with a slope greater than a chosen threshold. The unoccluded leg does not show much color overlaid because the slope was smaller for the unoccluded leg. The occluded leg of the normal volunteer shows a broader region with a slope above the threshold. The occluded leg of the patient shows a much smaller region with a slope over the threshold. This preliminary result shows the feasibility of this method for showing regional perfusion over a substantial region of the lower limb.

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DISCUSSION

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

The effectiveness of a variable-density k-space sampling method applied to the stack-of-spiral trajectories for fast 3D imaging has been demonstrated. Phantom experiments show that reasonable image quality is preserved, while approximately half the scan time required without variable-density sampling is used. The first-pass perfusion study demonstrates that this fast-imaging technique facilitates studies that are otherwise difficult to perform because of scan time limitations.

To further define the properties of the proposed technique, the signal-to-noise-ratio (SNR) and the off-resonance properties are discussed in the following sections. We also discuss issues with respect to optimal sampling schemes, scan time reduction, combinations with other fast-imaging techniques, and other applications.

SNR

The SNR of a variable-density undersampled image is only slightly degraded because of the nonuniformity of the trajectory (16) itself. The SNR efficiency decrease due to the nonuniformity of the trajectory can be calculated as shown by Tsai and Nishimura (14). In Table 1, the value η representing the SNR efficiency degradation factor is calculated for the sampling schemes that were used for the phantom experiments. This factor takes into account only the SNR efficiency decrease due to the nonuniformity of the trajectory compared to a trajectory with the same scan time and uniform k-space coverage. As can be seen in Table 2, this factor is usually not very significant.

Table 1. SNR Efficiency Degradation Factor for Different Methods
MethodSampling ratioη
3DFT11
Uniform-density stack-of-spirals trajectory10.98
Under-sampled uniform-density stack-of-spirals trajectory0.520.97
Linear variable-density stack-of-spirals trajectory0.520.96
Quadratic variable-density stack-of-spirals trajectory0.520.94

It should be noted that there are aliasing artifacts in an undersampled variable-density sampled image that are not included in the noise analysis. In a variable-density k-space-sampled image, the amount of aliasing varies throughout the image and the aliasing effect is object-dependent, which makes it difficult to quantify.

Off-Resonance Effects

In spiral imaging, off-resonance results in image blurring (32). This can be demonstrated using PSFs. To understand the different off-resonance properties of the different sampling schemes used, PSFs corresponding to being –220 Hz off-resonance were calculated for the four different sampling schemes used in the phantom experiment, for which the readout time was 6 ms. A rather extreme off-resonance value was chosen so that the off-resonance effect could be clearly seen in the PSFs.

Since the trajectory was chosen to be cylindrical, with the same spirals for each kz location, points along the kz direction with the same kxky location are sampled at the same time from the excitation. Therefore, the off-resonance PSF in the z direction is the same as the PSF without off-resonance. The trajectory is not sensitive to off-resonance in the z direction.

Figure 8 shows the four off-resonance PSFs in the first column, and the cross section of the PSF through the center in the second column. Off-resonance in the uniform-density-sampled case (Fig. 8a and b) results in coherent rings around the central peak, but the variable-density-sampled case (Fig. 8c and d) shows a much smoother, incoherent blur. The central peak itself does not change much, but a ring occurs very close to the central peak. The amplitude of this ring is slightly lower in the variable-density-sampled cases.

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Figure 8. Off-resonance PSF in x-y for (a) uniform-density sufficiently-sampled, (b) uniform-density undersampled, (c) linear variable-density undersampled, and (d) quadratic variable-density undersampled trajectories. The first column shows the off-resonance PSFs, and the second column shows the cross section of the PSFs along the center. These off-resonance PSFs were generated for the four sampling densities used in the phantom experiment. The rings that cause blurring are smoother in the variable-density-sampled cases (c and d) compared to the uniform-density-sampled cases (a and b).

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Optimal Sampling Scheme

In this work, two different undersampled variable-density sampling schemes are demonstrated for comparison. For the phantom object and imaging parameters chosen, both sampling schemes provided images of similar quality. However, on close inspection of the aliasing artifacts, it can be seen that the quadratic undersampling scheme gives a smoother aliasing artifact. In the PSFs, it can also be observed that the aliasing signal peak is also smaller for the quadratically undersampled case. The optimum sampling scheme has not yet been determined.

Scan Time Reduction

The scan time reduction factor achievable using the variable-density sampling scheme can be subjective. As demonstrated in this study, a scan time reduction factor of approximately 0.5 still provided reasonable image quality. A further reduction down to 0.3 gave a reasonable image for the phantom configuration, FOV, and resolution used for our demonstration. It should be noted that the acceptable undersampling ratio will depend on the object power spectral density and imaging parameters. Since the achievable scan time reduction rate is not a fixed value, it can be flexibly adjusted to give the scan time needed for a certain application as long as the image quality achieved with such an undersampling ratio is acceptable. This provides a way to optimally trade off image quality with scan time.

Combination With Other Techniques

The variable-density sampling method samples the central k-space region more densely. This can be nicely combined with parallel imaging techniques (33, 34) because the sensitivity map can be calculated from the low-resolution image obtainable from the central portion of the k-space data. Partial k-space reconstruction methods can also be combined (35), since the phase maps can be obtained from the central portion of the k-space data.

Applications

Perfusion scanning was used as a demonstration of one possible use of variable-density 3D scanning. Our experiments clearly show promise in quantifying peripheral perfusion in patients with peripheral vascular disease. In this study, a fractional signal intensity rise in muscle was shown. A more accurate quantitative model should be used to provide accurate quantification (36, 37). The application of this fast imaging technique is not limited to first-pass perfusion studies of the lower limbs. It can also be used for whole-organ first-pass perfusion studies in organs such as the breast, brain, and kidney. Other time-constrained imaging such as angiography and cardiac imaging, can also benefit from this technique.

CONCLUSIONS

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

We have demonstrated that variable-density k-space sampling can achieve a significant scan time reduction for 3D imaging. Applied to the stack-of-spirals trajectory, this sampling scheme can provide a very fast 3D scan that enables imaging for time-constrained applications, such as first-pass perfusion studies. Normal volunteer and patient studies demonstrated the promise of this method for first-pass perfusion imaging.

Acknowledgements

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

Jin Hyung Lee acknowledges Steven Conolly, John Pauly, Phillip Yang, Krishna Nayak, Ann Shimakawa, and the Korea Foundation for Advanced Studies for their help with this research.

REFERENCES

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES