During the last decade, diffusion-weighted (DW) imaging (DWI) has become an established technique for the diagnosis of ischemia (1) and investigations of the anatomical connectivity of the human brain (2). Presently, no manufacturer delivers any phantoms dedicated to diffusion imaging, due to the complexity of their design. However, diffusion phantoms have numerous applications. They include calibration, validation of tractography algorithms, and validation of diffusion models. The phantom design should comply with the concrete application. For example, calibration requires a large region of interest (ROI) with a specific apparent diffusion coefficient (ADC), fractional anisotropy (FA), and principal orientation(s) to reduce the impact of acquisition noise on the measurements. On the other hand, to validate tractography, one would typically use a phantom made up of long bundles, similar to those found in brain white matter. To circumvent the intrinsic limitations of diffusion tensor imaging (DTI; i.e., the inability to resolve multiple fiber populations), a number of high-angular-resolution diffusion imaging (HARDI) models were introduced (3–12). They were conceived with the aim of providing an unbiased estimate of the probability density function (PDF) describing the displacements of the water molecules during a predefined time interval. Some models deliver only the radial projections of the PDF, known as the orientation distribution function (ODF). The phantoms employed in the studies of HARDI models could be adjusted to different fiber configurations (crossing, kissing, merging, and splitting), and angular distribution. Several diffusion phantom designs were proposed, based on fibrous vegetables (13), biological tissues (14), plastic capillaries (15–17), or textile fibers (18–20). In this work, we present a novel diffusion phantom dedicated to the validation of HARDI models. We developed two versions of this phantom corresponding to 45° and 90° fiber crossings, and used them to test the analytical Q-ball model.
We present new diffusion phantoms dedicated to the study and validation of high-angular-resolution diffusion imaging (HARDI) models. The phantom design permits the application of imaging parameters that are typically employed in studies of the human brain. The phantoms were made of small-diameter acrylic fibers, chosen for their high hydrophobicity and flexibility that ensured good control of the phantom geometry. The polyurethane medium was filled under vacuum with an aqueous solution that was previously degassed, doped with gadolinium-tetraazacyclododecanetetraacetic acid (Gd-DOTA), and treated by ultrasonic waves. Two versions of such phantoms were manufactured and tested. The phantom's applicability was demonstrated on an analytical Q-ball model. Numerical simulations were performed to assess the accuracy of the phantom. The phantom data will be made accessible to the community with the objective of analyzing various HARDI models. Magn Reson Med 60:1276–1283, 2008. © 2008 Wiley-Liss, Inc.
MATERIALS AND METHODS
The design of diffusion phantoms dedicated to study HARDI models depends on two key points related to mechanical and NMR properties. There are three main mechanical properties: the nature of fibers, their geometry, and the nature of the water solution. The use of an echo-planar DW sequence also poses the constraints on the possible ranges for the proton density, relaxation times, and diffusion indices (ADC and FA).
Type of Adequate Fibers
The purpose of diffusion phantoms is to understand the impact of the diffusion process on the MR signal, knowing exactly the geometry of the fibers and membranes. To fulfill the latter, the phantom design needs to conform to the following: 1) the fiber diameter should be close to the axon diameter (∼10 μm); 2) the fiber geometry should be tubular; 3) the fibers should be flexible but keep a low curvature; 4) the MR characteristics of a bundle of fibers should be similar to those of white matter bundles; and 5) the fiber layout should be rectilinear.
An important characteristic of fibers is their capability to capture or reject the water molecules. Most natural fibers are known to be hydrophilic and susceptible to biological degradation, while synthetic fibers are hydrophobic and very stable over time. For the purpose of this study, we decided to use acrylic fibers, sold as tufts of parallel filaments, with a diameter equal to dacrylic = 20 μm (Fig. 1a). However, to our knowledge, none of the available textile fibers are tubular, which reduces the number of compartments to a single, “extracellular” compartment.
Fiber-Crossing Manufacturing and Geometry Control
In contrast to plastic capillaries, which are more resilient, acrylic fibers are flexible and their geometry is difficult to control accurately. Fieremans et al. (20) used heat-shrinking tubes to hold the fibers tightly together and to create different geometries. This complicates the control of the exact fiber density, and the angular distribution of fibers cannot be fully regulated.
The use of a manufacturing support consisting of a negative impression of the fiber bundle geometry can significantly improve the control over the weaving operation. These supports are built from polyurethane. Layers of fibers are laid in the negative impression and stacked on each other (Fig. 1b).
Finally, the two parts of the support are assembled with screws controlling the tightening of the crossing. The obtained fiber density of 1900 fibers/mm2 implied an FA up to 0.4 in the main branches of the bundles (see the cross-section in Fig. 1c). The proton density was equal to one-third of the density measured inside the aqueous solution. The dimensions of the final crossing area of interest were 30 mm × 30 mm × 14 mm, resulting in very high signal-to-noise ratio (SNR).
Nature of the Solution and Filling of the Container
The format of the medium was such that it fitted strictly a plastic container of cylindrical shape (length 160 mm, diameter 130 mm). The nature of the solution and the process of filling the container may have a great impact on image quality when the single-shot DW EPI pulse sequence is employed. The spin-spin T2 relaxation time of the phantom immersed in the solution must stay greater or equal to the minimum echo time (TE) feasible on the MR system. The spin-lattice T1 relaxation time must stay within reasonable limits, reducing the global acquisition time. Lastly, the ADC of the phantom must have the same order of magnitude as the ADC of the human brain, and the solution must be stable over time.
Several papers have proposed the use of agarose gel or alkanes (21, 22). However, the former is susceptible to biological attacks, and the latter depends strongly on the storage conditions. In order to reduce the T1 relaxation time from 2.4 s to 1.0 s at 1.5T, we decided to use a solution of distilled water doped with gadolinium of concentration equal to 0.15 mM (Gd-DOTA with longitudinal and transverse relaxivities r1 = 3.5 s−1mM−1 and r2 = 4.5 s−1mM−1) (23). The original T2 relaxation time was only slightly reduced (from 120 ms to 100 ms).
The aqueous solution was previously degassed during 1 day. Then, the cylindrical container was filled in a vacuum chamber in order to remove the air bubbles present in the solution to avoid susceptibility artifacts. Finally, the solution was treated by ultrasonic waves, mechanically splitting air bubbles into smaller ones able to get out of the fiber bundles.
Evaluation of MR Characteristics
Two phantoms have been manufactured as described above, simulating two fiber crossings at 90° (Fig. 2a) and 45° (Fig. 2b). In order to assess the impact of gadolinium on T1 and T2 times, we decided to add gadolinium-tetraazacyclododecanetetraacetic acid (Gd-DOTA) only to the second phantom. The relative proton density, ρ/ρsolution, inside the bundles, as well as the T1 and T2 constants, were calculated from MR data acquired with a conventional spin-echo sequence. Acquisition parameters were as follows: readout bandwidth (RBW) = 6.58 kHz, field of view (FOV) = 24 cm, matrix size = 64 × 64, slice thickness (TH) = 15 mm, TE varying from 25 ms to 500 ms, and TR varying from 500 ms to 12 s.
Acquisition of HARDI Data Sets
The data were acquired on a 1.5T Signa MR system (GE Healthcare, Milwaukee, MN, USA), equipped with a whole-body gradient (40 mT/m, 150T/m/s) and an eight-channel head coil. A single-shot echo-planar pulse sequence (DW-SS-EPI) was employed using the twice refocusing technique proposed by Reese et al. (24), compensating for the Eddy currents to the first order.
These phantoms are adapted to very high b-values (up to 8000 s.mm−2). In order to improve the SNR obtained on clinical scanners, the diffusion phantoms were designed to use large voxel dimensions. As depicted in Fig. 1c, the size of the usable ROI inside the crossing area is equal to 30 mm × 30 mm × 14 mm. The RBW was fixed to 62.5 kHz, and the matrix size to 32 × 32. The SNR remained greater than 4 along the most attenuated direction (i.e., the direction of one of the two branches) for the six prescribed b-values, 0/2000/4000/6000/8000/10,000 s.mm−2, when the voxel volume was greater than 1400 mm3. This SNR limit enables the approximation of Rician noise by Gaussian. Consequently, the parameters were set as follows: FOV = 32 cm, matrix = 32 × 32, TH = 14 mm. The k-space was fully sampled with the minimum TE (130 ms) corresponding to the maximal b-value. According to the T1 relaxation times, the TRs were set to 4.5 s and 12.0 s for the 45° and 90° crossing phantoms, respectively.
The number of orientations was 4000, distributed uniformly over the unit sphere, according to the electrostatic repulsion model that takes into account the symmetry of the diffusion process, as described by Jones et al. (25). The large number of orientations enabled us to build almost uniform orientation subsets to study the relation between the number of orientations and the angular resolution of HARDI models. The average/minimum/maximum/standard deviation (SD) angles between neighboring orientations of the initial set were 2.28°/1.58°/4.68°/0.30°.
Analytical Q-Ball Model
We demonstrate the applicability of the described phantoms on the example of the Q-ball imaging (QBI) model introduced by Tuch (9). The QBI derives from the q-space methods that establish a Fourier relationship between the diffusion propagator and the normalized signal measured in the wave vector space q = (γ/2π)δG (where δ represents the width of the applied diffusion gradient pulse, and G is the gradient strength). The QBI reconstruction is based on the spherical Funk Radon transform that, for a given orientation, involves integrating the diffusion signal over the corresponding equator on a sphere of predetermined radius. We employ the recently proposed analytical QBI reconstruction scheme based on spherical harmonics, as described in Descoteaux et al. (26). For ODF visualization, we use the standard minimum-maximum (min-max) normalization operator.
Monte Carlo Simulation
In order to evaluate the goodness of the ODFs processed from real diffusion data with respect to the theoretically derived data, using the Bloch-Torrey equations, we implemented a numerical phantom as in Ref.27, sharing the same geometry as the physical phantoms that was made up of thin layers of nonpermeable cylindrical capillaries, 20 μm in diameter.
The interactions between the water molecules and the membranes were assumed to be elastic. The capillaries were (numerically) juxtaposed in thin interleaved layers. Water molecules within the capillaries accounted for 22% of the total water volume. The displacement of the water molecules was calculated using a free random walk process, with a step of 82.5 μs (which corresponds to a diffusion distance of 1 μm given the diffusion coefficient of water D = 2 × 10−9 m2.s−1 at 25°C). This large time interval was chosen to decrease the computation time. The MR signal was calculated under the assumption of a Stejskal-Tanner pair of diffusion gradients used for diffusion encoding, summing the phase contribution of the spin population. A detailed description of the numerical simulator can be found in Ref.28.
MR Characteristics of the Acrylic Fiber Bundles
The T1 relaxation times were 2.4 s and 1.0 s for the 45° and 90° crossing phantoms, respectively. The estimated T2 relaxation time was 100 ms, which is slightly greater than the T2 relaxation time of the human brain white matter, resulting in a higher SNR than in the brain for the same b-value. The relative proton density ρ/ρsolution of approximately 0.3 was close to the theoretical value of 0.22, in the case of the ideal geometry control. It would have been preferable to get tubular fibers in order to maximize the relative proton density. However, the value obtained is still much greater than what can be obtained with plastic capillaries.
The FA of the diffusion phantoms is directly linked to the fiber density. The higher the density, the greater the FA (Fig. 2c and d). At the current density of 1900 fibers/mm2 and b = 2000 s.mm−2, the FA varies from 0.3 to 0.4 in the branches of the phantoms, and from 0.2 to 0.3 inside the crossings. The plastic medium has a key role in the FA control. The FA in the brain white matter can reach higher values because the water molecules are more restricted: the average diameter of axons is on the order of 10 μm as the average diameter of the extracellular space of the phantoms. At b = 2000 s.mm−2, the ADC varies from 1.3 × 10−9 to 1.4 × 10−9m2.s−1. This value is between the value for white matter equal to 0.7 × 10−9m2.s−1, at 37°C, and the value for pure water equal to 2 × 10−9 m2.s−1, at 25°C.
Signal Magnitude on Equators
It is important to understand the raw MR signal before performing any processing. A graphical tool was developed to visualize the equators of the signal magnitude with a typical “peanut” shape (29). For an equator perpendicular to a given direction, we define a number of discrete equidistant points along it, and assign the values by nearest-neighbor interpolation of the measurements. Figure 3 depicts the equators corresponding to the three main axes of symmetry of the voxel inside the 90° crossing. The blue equator is characterized by two branches: the vertical branch corresponds to the signal coming from the fiber bundle aligned with the x-axis (red bundle), whereas the horizontal branch corresponds to the signal associated with the fiber bundle aligned with the y-axis (green bundle). The red and green equators are useful to define the signal E0 corresponding to the maximum ADC as well as the signal EMAX corresponding to the minimum ADC. E0 is contaminated by Rician noise inferior to 25% of E0.
Investigation of the Q-Ball Model
Figure 4 presents the results of the analytical QBI reconstruction for the 90° and 45° crossing phantoms obtained from the real MR data acquired with the DW-SS-EPI sequence whose parameters were previously described. The Q-ball ODFs of the 90° phantom corresponding to the voxels containing the single fiber population depict monomodal ODFs with the principal axis perfectly aligned with the underlying fibers. The Q-ball ODFs corresponding to the crossing area are characterized by two perpendicular lobes with axes parallel to the orientations of the two fiber populations. The width of the lobes decreases with increasing b-value, which is related to the improvement of the angular resolution of the model (9). In order to distinguish the crossing area in the 45° crossing phantom, the b-value must be superior or equal to 6000 s.mm−2 (in the absence of any sharpening postprocessing (26)). However, the manufacturing of these phantoms can already be assessed by the quality and accuracy of the ODF fields.
In addition, Fig. 4 depicts the ODFs resulting from the analytical QBI reconstruction on the simulated data, corresponding to the two geometries of the phantom. The shapes of the simulated ODFs are quite similar to the shapes obtained with the phantom.
In this study we have presented a novel diffusion-dedicated phantom and demonstrated its applicability to the study of HARDI models. Also, we made a comparison between the ODFs reconstructed from the phantom data and the ODFs obtained from the numerically simulated data. We believe that the presented phantom can be quite useful for the validation of HARDI models. The analysis herein conducted was mostly qualitative, and more quantitative study will be included in future work. Nonetheless, several important issues have been addressed, including the impact of the b-value, phantom geometry, Rician noise, and number of orientations on the angular resolution of the ODFs.
Role of b-Value on the Signal and Angular Resolution
To understand the effect of b-value on the angular resolution of the reconstructed Q-balls, it is important to investigate the 3D representation of raw MR signal. Figure 5 depicts the spherical meshes deformed according to the measured diffusion signal along each direction, in the case of the 45° crossing phantom. The second and third columns show the meshes for low and high b-values, respectively. At b = 2000 s.mm−2, two diffusion lobes corresponding to two single populations are relatively broad. The mesh observed for a voxel located inside the crossing ROI is unimodal, which does not allow the two underlying diffusion peaks to be distinguished. Its maximum corresponds to the bisecting line of the mixture. At b = 8000 s.mm−2, two diffusion lobes are much sharper, and the signal mesh corresponding to the crossing area becomes bimodal.
Geometry of the Extracellular Compartments
Tubular fiber structure is probably the first building element one thinks of concerning the creation of diffusion-dedicated phantoms. In this study, the acrylic fibers used to build the phantom were not tubular, which led to a diffusion process only outside the fibers. Therefore, the anisotropy is not due to the restriction phenomenon that intrinsically implies very high anisotropy, but rather to a mixture of restriction and hindering, which leads to less anisotropic diffusion. This observation corresponds well to the FA measured in the phantoms that remains limited to 0.3. Figure 6 shows the 3D renderings of two ROIs: ROI1, corresponding to the phantom branch, and ROI2, corresponding to the fiber crossing. The fibers inside ROI1 can be considered parallel with the superquadric cross-section of average diameter equal to (π/4) × dacrylic (∼78.5% of the diameter of the fiber). Therefore, it is more restrictive than would be the complementary intracellular compartment. In order to test this hypothesis, a new simulation was performed assuming tubular structure of the fibers, yielding an intracellular and an extracellular compartment. As depicted in Fig. 6d and e, the unimodal Q-balls of ROI1, reconstructed at b = 8000 s.mm−2, are sharper inside the extracellular compartment than inside the intracellular compartment. On the contrary, the geometry of the ROI2 extracellular compartment is much more complicated than the intracellular compartment, presenting more degrees of anisotropy. Consequently, its anisotropy is lower than the anisotropy of the extracellular compartment. It can also be assessed by the shape of the reconstructed Q-balls with lobes that are broader inside the extracellular compartment than inside the intracellular compartment. To conclude, the phantoms built from acrylic fibers require a higher b-value than those made of tubular fibers. The phantom design permits the use of large voxels to counterbalance the SNR decrease at high b-value.
Comparison Between Phantoms and Monte-Carlo Simulations
The shapes of the estimated ODFs are quite similar to the shapes obtained with the real MR data (Fig. 4). The lobes are somewhat sharper in the ODFs corresponding to the simulations for two putative reasons: First, the level of noise was deliberately set to zero for the simulations. Second, while we can have a full control over the geometry of the fibers in the Monte-Carlo simulations, it is not the case for the real data. The control of the geometry in real data depends mostly on how well we can tighten the fibers.
Influence of the Rician Noise
As explained in the Results section, the acquisition parameters were tuned to maintain an SNR greater than 4 along the direction of maximal diffusion. E0 and EMAX were defined as the magnitude corresponding to the direction of maximal and minimal diffusion, respectively. The noise average (η) is equal to 0.25 E0, at the maximum b-value of 8000 s.mm−2. This average that results from the norm of the Gaussian noise with null mean and SD σ, on both real and imaginary channels is known to follow Rician distribution. Since σ and η are linked by the relation , we obtain σ ≈ 0.2 E0 for the phantom data at 8000 s.mm−2. We performed a numerical simulation adding this Gaussian noise to both the real and imaginary channels of the simulated signal. The obtained Q-ball ODFs were only slightly inflated, demonstrating the robustness of the manufactured phantoms to noise.
Influence of the Number of Measurement Orientations
We deliberately acquired the DW data along a huge number of orientations in order to study the effect of the number of the sensitization directions on the angular resolution of the model. As previously described, the average angular separation between two neighboring orientations was equal to 2.28°, which enabled almost any uniform subset to be built with smaller number of orientations. Figure 7 shows the mean-square difference between the current Q-ball ODF (calculated from a subset of the full orientation set) and the reference ODF (calculated from the complete orientation set) with respect to the number of orientations in the current subset. Such a set of L-curves for a large range of crossing angles (0–90°) would provide a powerful tool for automatically deciding on the minimum number of orientations needed given the constraint on the minimum angular resolution, and the SNR along the direction of maximal diffusion.
In this work, a new diffusion phantom dedicated to the study and validation of HARDI models is presented. Acrylic fibers were chosen to build fiber crossings because of their diameter, hydrophobicity, and flexibility. A polyurethane medium was filled under vacuum with an aqueous solution that was previously degassed, doped with Gd-DOTA, and treated by ultrasonic waves. The phantom was applied to an analytical Q-ball model and tested by a numerical simulator. The acquisition data will be provided to the diffusion community for validation of various HARDI models.
We thank Dr. Patrick Le Roux (GE HealthCare, Advanced Science Laboratory, Buc, France) for sharing his expertise in MR physics and pulse sequences during this work. The data corresponding to this work, and dedicated to the validation of HARDI models, can be freely obtained on demand from firstname.lastname@example.org.