NOTE Magnetic Resonance in Medicine 73:1171–1176 (2015)
Diffusion Weighted Imaging with Circularly Polarized Oscillating Gradients Henrik Lundell,* Casper Kaae Sïnderby, and Tim B. Dyrby Purpose: The short diffusion time regime provides an interesting probe for tissue microstructure and can be investigated with oscillating gradient spin echo (OGSE) experiments. Several studies report new contrasts in preclinical settings and the first in vivo human experiments have recently been presented. One major hurdle in practical implementation is the low effective diffusion weighting provided at high frequency with limited gradient strength. Theory: As a solution to the low diffusion weighting of OGSE, circularly polarized OGSE (CP-OGSE) is introduced. CP-OGSE gives a twofold increase in diffusion weighting with encoding in a plane rather than in one direction. CP-OGSE can be used for rotationally invariant acquisitions on anisotropic tissues. Methods: Experiments with a 4.7 T preclinical scanner on a postmortem monkey brain as well as simulations were performed using conventional OGSE and CP-OGSE. Results: Simulations and experiments show that CP-OGSE provides the same microstructural information as OGSE but provides more robust parameter estimates with limited gradient strength. Conclusions: CP-OGSE can be an important contribution for making OGSE imaging more effective in clinical imaging settings with limited gradient strength. Furthermore, the improved diffusion weighting can also be used to expand the investigated frequency range. Magn Reson Med 73:1171–1176, C 2014 Wiley Periodicals, Inc. 2015. V Key words: diffusion tensor imaging; short diffusion time; oscillating gradient spin echo; circularly polarized
INTRODUCTION Diffusion weighted imaging uses water diffusion as a probe to study tissue microstructure (1). Conventional pulsed field gradient experiments, such as pulsed gradient spin echo (PGSE) or stimulated echo (STEAM) diffusion sequences, are normally probing diffusion processes on time scales above 10 ms. Water displacement probed on this time scale is on the size or above the size of most
cells and diffusion time in this range is a weak contrast parameter (2,3). To probe diffusion on a shorter time scale, oscillating gradient spin echo (OGSE) experiments can be used to investigate the frequency dependent diffusion coefficient (4). The rate of increase in diffusivity varies in different restricted geometries with increasing oscillation frequency, i.e., decreasing effective diffusion time. This technique provides a new contrast in brain regions with densely packed cells like in the hippocampus and in the cerebellum (5–8), or as a marker of cell morphology in detection and treatment of tumors (9,10). Unfortunately, the effective diffusion weighting decreases rapidly at high frequencies with an f 2 dependence making acquisitions on clinical scanners and other systems with moderate gradient performance (11) difficult due to insufficient signal attenuation at realistic signal to noise (SNR). Increased diffusion weighting can be used to shorten scan time, increase resolution, expand the investigated frequency range, or improve parameter estimates, all of which are important factors for clinical implementations. We introduce circularly polarized OGSE (CP-OGSE) gradients to further improve the diffusion weighting of OGSE when the maximum gradient strength is limited. We combine two simultaneous but orthogonally oriented oscillating gradients played out with a quarter wave phase shift. This creates a planar sensitivity and increases the effective diffusion weighting with a factor of two compared to the corresponding linearly polarized gradient shape with the same length and amplitude. The same increase in diffusion weighting is reached with two orthogonal directions in phase but that can only be applied at maximum gradient strength in fixed directions along the vertices of a cube. The circular polarization is not restricted to these directions and can be applied in any arbitrary plane and we can with multiple acquisitions extract rotationally invariant data for imaging anisotropic tissues.
THEORY Danish Research Centre for Magnetic Resonance, Centre for Functional and Diagnostic Imaging and Research, Copenhagen University Hospital Hvidovre, Denmark Grant sponsor: Future and Emerging Technologies (FET) program within the Seventh Framework Program for Research of the European Commission. *Correspondence to: Henrik Lundell, Ph.D., Danish Research Centre for Magnetic Resonance, Centre for Functional and Diagnostic Imaging and 30, Research, Copenhagen University Hospital Hvidovre, Kettegaards Alle DK 2650, Denmark. E-mail: [email protected] Received 17 July 2013; revised 30 January 2014; accepted 17 February 2014 DOI 10.1002/mrm.25211 Published online 17 March 2014 in Wiley Online Library (wileyonlinelibrary. com). C 2014 Wiley Periodicals, Inc. V
The temporal diffusion spectrum, D(f), describes the frequency dependent diffusion coefficient and is in an NMR experiment sensitized with a filter given by the squared spectral density of a time varying gradientinduced phase modulation (12). A practical imaging setup is limited by gradient strength and T2-decay, and OGSE MRI is often carried out by the few oscillation frequencies allowed by an even number of periods over a fixed echo time. For a time modulated gradient dominated by a well defined and isolated frequency peak, we approximate the accumulated spectral power to be
1171
1172
Lundell et al.
FIG. 1. a: The CP-OGSE diffusion weighting is constructed by two orthogonally oriented oscillating gradients, g1 and g2, shown in red and blue. The first gradient is initiated with a trapezoidal pulse with the same area as a quarter wave (shaded in red) to approach a cosine in k(t) shown in (b). Both gradient trains are balanced in phase space and has the frequency specific qualities of a cosine modulated gradient. c: The encoding spectra of the two individual waves at 200 Hz (red and blue lines) and the collected weighting of the two waves played out simultaneously as in CP-OGSE (black line).
assigned to the oscillation frequency. This holds for cosine or apodized trapezoidal gradients. For such experiment, we can calculate the conventional B-matrix as in PGSE diffusion tensor imaging (DTI) experiments: Z t kðtÞ ¼ g g ðt0 Þdt 0 0 Z t kðtÞkT ðtÞdt B¼
[1] [2]
0
where g ðtÞ is the effective diffusion encoding gradient and kðtÞ is the phase modulation. We consider a cosine modulated gradient gðtÞ ¼ Gcos ð2pftÞ with the phase gG modulation kðtÞ ¼ 2pf sin ð2pftÞ. With circular polarizagG tion in the xy-plane, we get kcp ðtÞ ¼ 2pf ½sin ð2pftÞ; cos ð2p ftÞ; 0 and inserted into Eq. [2], we get for a full number of periods during the gradient duration s:
Bcp ¼
1 gG 2 2pf
2
2
t
6 6 40
0 t
0 0
0
3
7 07 5
[3]
0
This matrix clearly has two nonzero eigenvalues and the measured signal will thus be the product of the diffusivities along two directions or the trace of the planar diffusion tensor spanned by the plane of the two eigenvectors. The term cosine modulated can be confusing but refers to gradients with no DC offset in the phase space which would induce sensitivity to the zero-frequency of the diffusion spectrum. Cosine gradients fulfill this requirement but other bipolar waves with an appropriate apodization can also be used (4,8,13). Figure 1 shows our realization of a CP-OGSE design with two independent orthogonally oriented gradients g1 and g2 with a quarter wave phase shift creating a circular trajectory in phase space. Note that kðtÞ has the quality of a cosine modulated gradient in both directions as there is no DCoffset. In the first direction, g1, of our implementation, we use an initial trapezoidal pulse at maximum gradient amplitude with an area corresponding to a quarter wave oscillation to approach a circular trajectory in phase space as quickly as possible. The other direction, g2,
uses a linear ramp to a cosine wave adjusted to a total area corresponding to a quarter wave. The same initiation could also be used in g1, but is impractical at low frequencies when a quarter wavelength has a significant duration compared to the length of the whole gradient train. Unwanted additional diffusion weighting in one direction is generated when entering and leaving the circular trajectory. This skews the circular polarization and gives two unequal nonzero eigenvalues in Bcp . To circumvent this, the second gradient train after the refocusing pulse is phase shifted by a quarter wave creating two identical nonzero eigenvalues in Bcp . Just as in a conventional diffusion tensor imaging, the calculated Bcp describes the sensitivity of the six tensor elements for a given acquisition. The diffusion tensor can be fitted to a normalized dataset with a minimum of six noncoplanar CP-OGSE measurements. The spectral encoding is shown in Figure 1c. One should note that the optimal polarity of the second half of the gradient trains in Figure 1a should be matched to the frequency and the length of the gradient silent period during the 180 -pulse to maintain the phase in k(t). This prevents splitted peaks in the encoding spectrum as described earlier (8). The calculation includes the 3-ms 180 -pulse used in the experiment but the contribution from imaging gradients is neglected. METHODS Gradient Waveform Design Two protocols were constructed. The first protocol tested the difference between conventional OGSE and CPOGSE. Gradient amplitude G ¼ 0.5 T/m was used for the highest frequency set to f ¼ 200 Hz giving b-values 220 and 440 s/mm2 for the OGSE and CP-OGSE, respectively, with a gradient train length of 21 ms on each side of the refocusing pulse. The gradient strengths of the lower frequencies f ¼ [50, 100, 150] Hz were adjusted to G ¼ [0.14 0.26 0.38] T/m to match the same b-values of the f ¼ 200 Hz gradients. The frequencies f ¼ [50, 100, 150, 200] Hz correspond to effective diffusion times 5, 2.5, 1.7, and 1.3 ms. The trapezoidal prepulse was G ¼ 0.5 T/m, which
Circularly Polarized Diffusion Encoding
was the same for all frequencies. CP-OGSE trajectories were oriented in planes with normal vectors along 20 uniformly distributed directions from an electrostatic simulation (14). A conventional OGSE experiment was realized as gradient g2 only repeated along the same 20 uniformly distributed normal vectors. The second protocol was constructed to investigate higher diffusion weighting at lower frequencies. The parameters were identical to the first protocol except: (1) only CP-OGSE gradients at f ¼ [50, 150] Hz and (2) G ¼ [0.18 0.50] T/m resulting in a higher diffusion weighting of b ¼ 775 s/ mm2. Brain Tissue Sample An excised cerebellum from a 3.5-year-old perfusion fixed Vervet monkey was used and prepared following an optimized procedure for postmortem diffusion weighted imaging (15). The monkey was handeled and cared for on the island of St. Kitts (The Caribbean Primate Center of St. Kitts) according to a protocol approved by the local ethics commitee (16). Imaging Data was acquired on a Varian 4.7 T preclinical scanner with 600 mT/m gradients. Both protocols were acquired with single line readout TR/TE ¼ 2500/68 ms, 0.16 0.16 mm2 in plane resolution, and 2-mm slice thickness. The image plane was centered sagittally over vermis of the cerebellum. The same scan time and number of excitations were used for the OGSE and CP-OGSE protocols and the protocols were interleaved with b ¼ 0 references for every fifth acquisition. The total scan time was 27 h for the first protocol and 6 h and 45 min for the second protocol. The first protocol was acquired with a 20-mm diameter surface coil and the second acquisition was acquired with a quadrature volume coil. A 4-h gradient heavy dummy scan preceded the experiments to reduce short term mechanic and thermal instabilities (15). Imaging datasets can be downloaded from our homepage http://dig.drcmr.dk/. Simulation Monte Carlo simulations were performed in model substrates of restricted diffusion in cylinders with radii r ¼ 1 and 3 mm and the free diffusion coefficient set to D ¼ 2 109 m2 =s comparable to water at room temperature. We implemented the tissue model in the Camino software with 10,000 walkers and time steps of 4.2 ms (17). The walkers were uniformly distributed in the intracellular compartment only. Ideal linearly and circularly polarized cosine modulated gradients were constructed for a range of frequencies, 50–800 Hz, and gradient strengths, 200–1000 mT/m, at a fixed total gradient train length of 40 ms. Numerical simulations were compared to analytical expressions for the OGSE signal in cylindrical geometries (18). Noise propagation through the diffusion tensor model was evaluated by calculating the diffusion tensor signal S ¼ exp ððbxx Dxx þ byy Dyy þ bzz Dzz þ 2bxy Dxy þ 2 bxz Dxz þ 2byz Dyz ÞÞ with the B-matrices from the imaging study scaled from b ¼ 50–1000 s/mm2 for OGSE and b ¼ 100–2000 s/mm2 for the CP-OGSE. This simulates
1173
diffusion weightings possible at different systems in the contrast rich frequency range around 100 Hz (5,8). An anisotropic test tensor with the eigenvalues [0.5 0.7 1.8] 109 m2/s was used. The impact of SNR was investigated by contaminating 10,000 repetitions of the datasets with Gaussian noise in the complex plane and then calculating the magnitude to model the Rician noise distribution of MRI data (19). Data Analysis Image data was registered and resliced in plane to one b ¼ 0 image and normalized to the mean of all b ¼ 0 images to account for eddy current effects with SPM8 (http://www.fil.ion-.ucl.ac.uk/spm). The B-matrices were calculated by trapezoidal integration of the gradient amplifier input sampled at a 4-ms time step. Diffusion tensors were estimated using linear least square fitting and we calculated mean diffusivity (MD) as the mean of the eigenvalues and radial diffusivity (RD) as the mean of the two minor eigenvalues. The increase in diffusivity as a function of frequency in restricted geometries is sinc-shaped but DMD, the linear increase in MD over frequency, can be used for visualization in narrow frequency ranges and has been shown to have a contrast similar to histological Nissl-stains (5). We calculated DMD using least squares fitting to MD over the four frequency measurements. All diffusion analyses were performed using in-house scripts written in MATLAB (Mathworks Inc., Navick, MA, USA). RESULTS Simulation For diffusion tensor imaging, we assume that the CPOGSE experiment probes diffusivity along two orthogonal directions independently and should thus be equivalent to the product of two conventional OGSE experiments. Figure 2a shows the frequency dependence of the signal at different gradient strengths in cylindrical restrictions for the CP-OGSE and the product of the two corresponding OGSE experiments applied in the plane perpendicular to the length axis of the cylinder. The CP-OGSE is in very good correspondence with both simulated and analytical estimates of the OGSE signal. The components of restricted diffusivity in MD increase with frequency as shown in Figure 2b. In the narrow frequency range possible in our experimental settings, this effect can be described by the linear increase DMD. Figure 2c shows estimates of the individual eigenvalues, MD and RD from the two techniques over a range of b-values at SNR ¼ 20. The repulsion of the individual eigenvalues at low b-values in both OGSE and CP-OGSE is due to the sorting bias inherent in diffusion tensor analysis (20). CP-OGSE excel OGSE in terms of lower standard deviation in estimates at low b-values. The benefits remain up to moderate b-values in the RD and MD estimates but eigenvalues become more biased and noisy. Imaging Imaging results from the first protocol are shown in Figure 3. Figure 3a shows a b ¼ 0 image with three ROIs
1174
Lundell et al.
FIG. 2. a: Monte Carlo simulations were performed inside cylindrical geometries with radii r ¼ 3 mm (top) and r ¼ 1 mm (bottom) and shows good correspondence between CP-OGSE and OGSE. The signal attenuation as a function of frequency for fixed gradient strengths in the range 0.2–1 T/m is shown. The CP-OGSE correspond well to the product of two individual orthogonal OGSE gradients from simulations (red dotted line) as well as from an analytical description of the OGSE in restricted geometries as derived by Xu et al. (18) (green thin line). b: The frequency dependence on MD in the 3 and 1 mm cylinders at SNR ¼ 10 (top) from a Monte Carlo simulation corresponding to the experimental settings. Better separation between sizes is achieved at higher frequencies. Below a schematic illustration of the effect with random walks in restrictions with different dimensions. At higher frequencies (shorter diffusion time), diffusivity in larger restrictions is less affected by barriers and is thus higher. c: The influence of noise on MD and RD (top) and the individual eigenvalues (l1;2;3 ; bottom), over a range of b-values at SNR ¼ 20. The mean over 10,000 repetition are plotted with thick lines and the standard deviations are shown as thin lines. The blue lines indicate the true value. CP-OGSE performs better than OGSE at low b-values but a larger bias and noise sensitivity in the individual eigenvalues is introduced at increasing b. The benefits remain up to moderate b-values in RD and in MD. Strong bias in the individual eigenvalues are seen both in OGSE and CP-OGSE and are due to the general sorting bias problem in diffusion tensor analysis prominent at low b-values or SNR.
drawn in white matter (green), in the inner granular layer of the cerebellar cortex (red), and in the outer molecular layer (blue). Differentiation between the three regions increased at higher oscillation frequency with the largest MD increases in the densely packed granular layer (Fig. 3b), the colors correspond to the colors of the ROIs in Figure 3a. The MD estimates were less affected by noise in our CP-OGSE measurement compared to the conventional OGSE (solid vs. dashed lines). Due to low SNR, MD could only be reconstructed in white matter using the CP-OGSE. DMD-maps, shown in Figure 4a,b for OGSE and CP-OGSE, respectively, had large contrast between the granular and molecular layers not visible in T2W or PGSE MD images. Voxels with negative eigenvalues, i.e., insufficient contrast to noise, were excluded from the analysis and are colored green in Figure 4c,d. The overall visual appearances of CP-OGSE results were similar to OGSE but less noisy. Figure 5 shows MD (a) and color coded FA (b) from the second protocol with lower maximum frequency but higher diffusion weighting. Contrast is enhanced in MD maps at higher frequencies but FA has similar appearance at both frequencies as observed earlier with conventional OGSE (5). Note
FIG. 3. a: A T2W reference image with three ROIs covering white matter (green), granular layer (red) and the molecular layer (blue). b: MD values extracted from the OGSE (dashed lines) and CPOGSE (solid lines). The colors correspond to the ROIs shown in (a). Due to different microstructure, the contrast increase at higher frequencies. MD was not resolved in white matter with OGSE due to low SNR.
Circularly Polarized Diffusion Encoding
1175
FIG. 4. DMD-maps from the first protocol acquired at b ¼ 220/440 s/mm2, showing the linear increase in MD as a function of frequency for the (a) OGSE and the (b) CP-OGSE acquisitions from the first gradient protocol. Due to the increased diffusion weighting with CP-OGSE, parameter estimates are improved in (b) compared to (a) despite the same gradient limitations. Voxels with insufficient SNR are excluded and colored green.
that directional information is correctly and consistently resolved with CP-OGSE in the color coded maps in Figure 5b.
DISCUSSION In this study, we propose the use of CP-OGSE to increase the diffusion weighting of OGSE experiments for a given set of hardware limitations. In contrast to linearly polarized harmonic or trapezoidal gradients, CP-OGSE always run on maximum gradient strength and can still be rotated in any arbitrary orientation. We suggest that data can be analyzed using conventional diffusion tensor techniques and show in simulations and experiments that CP-OGSE is comparable to OGSE but provides more robust parameter estimates at low b-values. This can thus be an important contribution for applications on systems with limited gradient performance, like clinical scanners. Simulations confirmed that CP-OGSE is equivalent to two OGSE experiments at the moderate diffusion weightings applied in this study and within the limits reasonable on clinical imaging systems. Similarly, Portnoy et al. recently reported a resemblance between OGSE and PGSE experiments at b < 1000 s/mm2 (7). This indicates that the effect of interaction between spin displacements during subsequent waves of the OGSE is negligible and OGSE can more be considered as the accumulated attenuation from multiple PGSE experiments as initially proposed by Gross and Kosfeld (21). It is noteworthy to mention that CP-OGSE yields a natural contrast for anisotropy and fiber direction similar to double wave experiments (22), or the Funk–Radon transform used in Q-ball imaging and previously also explored with planar PGSE encoding (23,24). The choice between OGSE and CP-OGSE must be considered in relation to gradient limitations, maximum frequency (and thereby the maximal b-value), the underlying diffusion tensor, and the parameters of inter-
FIG. 5. a: MD maps from the second protocol acquired with CPOGSE only at b ¼ 775 s/mm2 for f ¼ 50 Hz (left) and f ¼ 150 Hz (right). b: Corresponding color coded FA maps for f ¼ 50 Hz (left) and f ¼ 150 Hz (right). Anisotropy pattern is maintained and CPOGSE renders directional information correctly with colors following the main white matter branches. The cerebellar cortex is red presumably due to the unmyelinated but densely packet parallel fibers running transversally over the midsagittal plane.
est. CP-OGSE primary improves estimates of isotropic features of the diffusion tensor, like MD and RD, but also to a lesser degree the individual eigenvalues in the b D 1 regime (Fig. 2c). Care must always be taken when inferring anisotropy based on subtle differences in eigenvalues (20,25,26) Our OGSE experiment in Figure 3 as well as other recent OGSE studies operate around b MD 0:1 to 0.3 (5,6,8). The contrast between different gradient orientations does not improve with CP-OGSE and will thus not improve eigenvalue estimates at moderate b-values. But sufficient signal attenuation, given by b D, compared to the noise is essential for robust estimates, which is why CP-OGSE still provides a benefit at low b. The noise distribution across the different eigenvalues depends on their sizes in relation to each other as each measurement will be the interaction of two directions. The benefits of trapezoidal compared to harmonic gradient shapes has also been reported (8,27,28). For a square wave inserted in Eq. [2] the increase in b-value is by a factor ð4=pÞ2 1:62 compared to a conventional OGSE, but the effect will be less with finite slew rate limitations. Besides from the higher twofold increase in diffusion weighting, the CP-OGSE also benefits from lower slew rates and is thus less prone to peripheral nerve stimulation in in vivo settings. CONCLUSIONS CP-OGSE provides twice the diffusion weighting compared to conventional OGSE which is a large experimental benefit on MRI systems with limited gradient
1176
strength. We demonstrated increased robustness in parameter estimates at low b in simulations and experiments on perfusion fixed brain tissue. Simulations showed that results are equivalent to conventional OGSE. CP-OGSE can be a significant contribution for making OGSE possible for studies of the human brain in vivo on clinical systems with moderate gradient strengths.
REFERENCES 1. Nilsson M, van Westen D, Sta˚hlberg F, Sundgren PC, L€ att J. The role of tissue microstructure and water exchange in biophysical modelling of diffusion in white matter. MAGMA 2013;26:345–370. 2. Clark CA, Hedehus M, Moseley ME. Diffusion time dependence of the apparent diffusion tensor in healthy human brain and white matter disease. Magn Reson Med 2001;45:1126–1129. 3. Pfeuffer J, Fl€ ogel U, Dreher W, Leibfritz D. Restricted diffusion and exchange of intracellular water: theoretical modelling and diffusion time dependence of 1H NMR measurements on perfused glial cells. NMR Biomed 1998;11:19–31. 4. Gore J, Xu J, Colvin D, Yankeelov TE, Parsons EC, Does MD. Characterization of tissue structure at varying length scales using temporal diffusion spectroscopy. NMR Biomed 2010;23:745–756. 5. Aggarwal M, Jones MV, Calabresi PA, Mori S, Zhang J. Probing mouse brain microstructure using oscillating gradient diffusion MRI. Magn Reson Med 2012; 1:98–109. 6. Kershaw J, Leuze C, Aoki I, Obata T, Kanno I, Ito H, Yamaguchi Y, Handa H. Systematic changes to the apparent diffusion tensor of in vivo rat brain measured with an oscillating-gradient spin-echo sequence. NeuroImage 2013;70:10–20. 7. Portnoy S, Flint JJ, Blackband SJ, Stanisz GJ. Oscillating and pulsed gradient diffusion magnetic resonance microscopy over an extended b-value range: implications for the characterization of tissue microstructure. Magn Reson Med 2013;69:1131–1145. 8. Van AT, Holdsworth SJ, Bammer R. In vivo investigation of restricted diffusion in the human brain with optimized oscillating diffusion gradient encoding. Magn Reson Med 2014;71:83–94. 9. Xu J, Li K, Smith RA, Waterton JC, Zhao P, Chen H, Does MD, Manning HC, Gore JC. Characterizing tumor response to chemotherapy at various length scales using temporal diffusion spectroscopy. PLoS One 2012;7:1–8. 10. Colvin DC, Yankeelov TE, Does MD, Yue Z, Quarles C, Gore JC. New insights into tumor microstructure using temporal diffusion spectroscopy. Cancer Res 2008;68:5941–5947.
Lundell et al. 11. Dyrby TB, Sïgaard LV, Hall MG, Ptito M, Alexander DC. Contrast and stability of the axon diameter index from microstructure imaging with diffusion MRI. Magn Reson Med 2013;70:711–721. 12. Stepisnik J. NMR measurement and Brownian movement in the short-time limit. Physica B 1994;198:299–306. 13. Stepisnik J, Mohoric A, Duh A. Diffusion and flow in a porous structure by the gradient spin echo spectral analysis. Physica B 2001;307: 158–168. 14. Jones DK, Horsfield MA, Simmons A. Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging. Magn Reson Med 1999;42:515–525. 15. Dyrby TB, Baar e WFC, Alexander DC, Jelsing J, Garde E, Sïgaard LV. An ex vivo imaging pipeline for producing high-quality and highresolution diffusion-weighted imaging datasets. Hum Brain Mapp 2010;32:544–563. 16. Burke MW, Zangenehpour S, Ptito M. Brain banking: making the most of your research specimens. J Vis Exp 2009;1260. 17. Hall MG, Alexander DC. Convergence and parameter choice for Monte-Carlo simulations of diffusion MRI. IEEE Trans Med Imaging 2009;28:1354–1364. 18. Xu J, Does MD, Gore JC. Quantitative characterization of tissue microstructure with temporal diffusion spectroscopy. J Magn Reson 2009; 200:189–197. 19. Gudbjartsson H, Patz S. The Rician distribution of noisy MRI data. Magn Reson Med 1995;34:910–914. 20. Pierpaoli C, Basser PJ. Toward a quantitative assessment of diffusion anisotropy. Magn Reson Med 1996;36:893–906. 21. Gross B, Kosfeld R. Anwendung der Spin-Echo-Methode bei der Messung der Selbstdiffusion. Messtechnik 1969;7:171–177. 22. Jespersen SN, Lundell H, Sïnderby CK, Dyrby TB. Orientationally invariant metrics of apparent compartment eccentricity from double pulsed field gradient diffusion experiments. NMR Biomed 2013;26: 1647–1662. 23. Tuch DS. Q-ball imaging. Magn Reson Med 2004;52:1358–1372. 24. Wedeen VJ, Dai G, Tseng WYI, Benner T. Diffusion encoding with 2D gradient trajectories yields natural contrast for 3D fiber orientation. In Proceedings of the 14th Annual Meeting of ISMRM, Seattle, Washington, USA, 2006. p. 851. 25. Jones DK, Basser PJ. Squashing peanuts and smashing pumpkins: how noise distorts diffusion-weighted MR data. Magn Reson Med 2004;52:979–993. 26. Lundell H, Nielsen JB, Ptito M, Dyrby TB. Distribution of collateral fibers in the monkey cervical spinal cord detected with diffusionweighted magnetic resonance imaging. NeuroImage 2011;56:923–929. 27. Lundell H, Sïnderby CK, Dyrby TB. DTI using modulated gradients at short effective diffusion times. In Proceedings of the 20th Annual Meeting of ISMRM, Melbourne, Australia, 2012. p. 355. 28. Ianus¸ A, Siow B, Drobnjak I, Zhang H, Alexander DC. Gaussian phase distribution approximations for oscillating gradient spin echo diffusion MRI. J Magn Reson 2013;227:25–34.
Diffusion Weighted Imaging with Circularly Polarized Oscillating Gradients Henrik Lundell,* Casper Kaae Sïnderby, and Tim B. Dyrby Purpose: The short diffusion time regime provides an interesting probe for tissue microstructure and can be investigated with oscillating gradient spin echo (OGSE) experiments. Several studies report new contrasts in preclinical settings and the first in vivo human experiments have recently been presented. One major hurdle in practical implementation is the low effective diffusion weighting provided at high frequency with limited gradient strength. Theory: As a solution to the low diffusion weighting of OGSE, circularly polarized OGSE (CP-OGSE) is introduced. CP-OGSE gives a twofold increase in diffusion weighting with encoding in a plane rather than in one direction. CP-OGSE can be used for rotationally invariant acquisitions on anisotropic tissues. Methods: Experiments with a 4.7 T preclinical scanner on a postmortem monkey brain as well as simulations were performed using conventional OGSE and CP-OGSE. Results: Simulations and experiments show that CP-OGSE provides the same microstructural information as OGSE but provides more robust parameter estimates with limited gradient strength. Conclusions: CP-OGSE can be an important contribution for making OGSE imaging more effective in clinical imaging settings with limited gradient strength. Furthermore, the improved diffusion weighting can also be used to expand the investigated frequency range. Magn Reson Med 73:1171–1176, C 2014 Wiley Periodicals, Inc. 2015. V Key words: diffusion tensor imaging; short diffusion time; oscillating gradient spin echo; circularly polarized
INTRODUCTION Diffusion weighted imaging uses water diffusion as a probe to study tissue microstructure (1). Conventional pulsed field gradient experiments, such as pulsed gradient spin echo (PGSE) or stimulated echo (STEAM) diffusion sequences, are normally probing diffusion processes on time scales above 10 ms. Water displacement probed on this time scale is on the size or above the size of most
cells and diffusion time in this range is a weak contrast parameter (2,3). To probe diffusion on a shorter time scale, oscillating gradient spin echo (OGSE) experiments can be used to investigate the frequency dependent diffusion coefficient (4). The rate of increase in diffusivity varies in different restricted geometries with increasing oscillation frequency, i.e., decreasing effective diffusion time. This technique provides a new contrast in brain regions with densely packed cells like in the hippocampus and in the cerebellum (5–8), or as a marker of cell morphology in detection and treatment of tumors (9,10). Unfortunately, the effective diffusion weighting decreases rapidly at high frequencies with an f 2 dependence making acquisitions on clinical scanners and other systems with moderate gradient performance (11) difficult due to insufficient signal attenuation at realistic signal to noise (SNR). Increased diffusion weighting can be used to shorten scan time, increase resolution, expand the investigated frequency range, or improve parameter estimates, all of which are important factors for clinical implementations. We introduce circularly polarized OGSE (CP-OGSE) gradients to further improve the diffusion weighting of OGSE when the maximum gradient strength is limited. We combine two simultaneous but orthogonally oriented oscillating gradients played out with a quarter wave phase shift. This creates a planar sensitivity and increases the effective diffusion weighting with a factor of two compared to the corresponding linearly polarized gradient shape with the same length and amplitude. The same increase in diffusion weighting is reached with two orthogonal directions in phase but that can only be applied at maximum gradient strength in fixed directions along the vertices of a cube. The circular polarization is not restricted to these directions and can be applied in any arbitrary plane and we can with multiple acquisitions extract rotationally invariant data for imaging anisotropic tissues.
THEORY Danish Research Centre for Magnetic Resonance, Centre for Functional and Diagnostic Imaging and Research, Copenhagen University Hospital Hvidovre, Denmark Grant sponsor: Future and Emerging Technologies (FET) program within the Seventh Framework Program for Research of the European Commission. *Correspondence to: Henrik Lundell, Ph.D., Danish Research Centre for Magnetic Resonance, Centre for Functional and Diagnostic Imaging and 30, Research, Copenhagen University Hospital Hvidovre, Kettegaards Alle DK 2650, Denmark. E-mail: [email protected] Received 17 July 2013; revised 30 January 2014; accepted 17 February 2014 DOI 10.1002/mrm.25211 Published online 17 March 2014 in Wiley Online Library (wileyonlinelibrary. com). C 2014 Wiley Periodicals, Inc. V
The temporal diffusion spectrum, D(f), describes the frequency dependent diffusion coefficient and is in an NMR experiment sensitized with a filter given by the squared spectral density of a time varying gradientinduced phase modulation (12). A practical imaging setup is limited by gradient strength and T2-decay, and OGSE MRI is often carried out by the few oscillation frequencies allowed by an even number of periods over a fixed echo time. For a time modulated gradient dominated by a well defined and isolated frequency peak, we approximate the accumulated spectral power to be
1171
1172
Lundell et al.
FIG. 1. a: The CP-OGSE diffusion weighting is constructed by two orthogonally oriented oscillating gradients, g1 and g2, shown in red and blue. The first gradient is initiated with a trapezoidal pulse with the same area as a quarter wave (shaded in red) to approach a cosine in k(t) shown in (b). Both gradient trains are balanced in phase space and has the frequency specific qualities of a cosine modulated gradient. c: The encoding spectra of the two individual waves at 200 Hz (red and blue lines) and the collected weighting of the two waves played out simultaneously as in CP-OGSE (black line).
assigned to the oscillation frequency. This holds for cosine or apodized trapezoidal gradients. For such experiment, we can calculate the conventional B-matrix as in PGSE diffusion tensor imaging (DTI) experiments: Z t kðtÞ ¼ g g ðt0 Þdt 0 0 Z t kðtÞkT ðtÞdt B¼
[1] [2]
0
where g ðtÞ is the effective diffusion encoding gradient and kðtÞ is the phase modulation. We consider a cosine modulated gradient gðtÞ ¼ Gcos ð2pftÞ with the phase gG modulation kðtÞ ¼ 2pf sin ð2pftÞ. With circular polarizagG tion in the xy-plane, we get kcp ðtÞ ¼ 2pf ½sin ð2pftÞ; cos ð2p ftÞ; 0 and inserted into Eq. [2], we get for a full number of periods during the gradient duration s:
Bcp ¼
1 gG 2 2pf
2
2
t
6 6 40
0 t
0 0
0
3
7 07 5
[3]
0
This matrix clearly has two nonzero eigenvalues and the measured signal will thus be the product of the diffusivities along two directions or the trace of the planar diffusion tensor spanned by the plane of the two eigenvectors. The term cosine modulated can be confusing but refers to gradients with no DC offset in the phase space which would induce sensitivity to the zero-frequency of the diffusion spectrum. Cosine gradients fulfill this requirement but other bipolar waves with an appropriate apodization can also be used (4,8,13). Figure 1 shows our realization of a CP-OGSE design with two independent orthogonally oriented gradients g1 and g2 with a quarter wave phase shift creating a circular trajectory in phase space. Note that kðtÞ has the quality of a cosine modulated gradient in both directions as there is no DCoffset. In the first direction, g1, of our implementation, we use an initial trapezoidal pulse at maximum gradient amplitude with an area corresponding to a quarter wave oscillation to approach a circular trajectory in phase space as quickly as possible. The other direction, g2,
uses a linear ramp to a cosine wave adjusted to a total area corresponding to a quarter wave. The same initiation could also be used in g1, but is impractical at low frequencies when a quarter wavelength has a significant duration compared to the length of the whole gradient train. Unwanted additional diffusion weighting in one direction is generated when entering and leaving the circular trajectory. This skews the circular polarization and gives two unequal nonzero eigenvalues in Bcp . To circumvent this, the second gradient train after the refocusing pulse is phase shifted by a quarter wave creating two identical nonzero eigenvalues in Bcp . Just as in a conventional diffusion tensor imaging, the calculated Bcp describes the sensitivity of the six tensor elements for a given acquisition. The diffusion tensor can be fitted to a normalized dataset with a minimum of six noncoplanar CP-OGSE measurements. The spectral encoding is shown in Figure 1c. One should note that the optimal polarity of the second half of the gradient trains in Figure 1a should be matched to the frequency and the length of the gradient silent period during the 180 -pulse to maintain the phase in k(t). This prevents splitted peaks in the encoding spectrum as described earlier (8). The calculation includes the 3-ms 180 -pulse used in the experiment but the contribution from imaging gradients is neglected. METHODS Gradient Waveform Design Two protocols were constructed. The first protocol tested the difference between conventional OGSE and CPOGSE. Gradient amplitude G ¼ 0.5 T/m was used for the highest frequency set to f ¼ 200 Hz giving b-values 220 and 440 s/mm2 for the OGSE and CP-OGSE, respectively, with a gradient train length of 21 ms on each side of the refocusing pulse. The gradient strengths of the lower frequencies f ¼ [50, 100, 150] Hz were adjusted to G ¼ [0.14 0.26 0.38] T/m to match the same b-values of the f ¼ 200 Hz gradients. The frequencies f ¼ [50, 100, 150, 200] Hz correspond to effective diffusion times 5, 2.5, 1.7, and 1.3 ms. The trapezoidal prepulse was G ¼ 0.5 T/m, which
Circularly Polarized Diffusion Encoding
was the same for all frequencies. CP-OGSE trajectories were oriented in planes with normal vectors along 20 uniformly distributed directions from an electrostatic simulation (14). A conventional OGSE experiment was realized as gradient g2 only repeated along the same 20 uniformly distributed normal vectors. The second protocol was constructed to investigate higher diffusion weighting at lower frequencies. The parameters were identical to the first protocol except: (1) only CP-OGSE gradients at f ¼ [50, 150] Hz and (2) G ¼ [0.18 0.50] T/m resulting in a higher diffusion weighting of b ¼ 775 s/ mm2. Brain Tissue Sample An excised cerebellum from a 3.5-year-old perfusion fixed Vervet monkey was used and prepared following an optimized procedure for postmortem diffusion weighted imaging (15). The monkey was handeled and cared for on the island of St. Kitts (The Caribbean Primate Center of St. Kitts) according to a protocol approved by the local ethics commitee (16). Imaging Data was acquired on a Varian 4.7 T preclinical scanner with 600 mT/m gradients. Both protocols were acquired with single line readout TR/TE ¼ 2500/68 ms, 0.16 0.16 mm2 in plane resolution, and 2-mm slice thickness. The image plane was centered sagittally over vermis of the cerebellum. The same scan time and number of excitations were used for the OGSE and CP-OGSE protocols and the protocols were interleaved with b ¼ 0 references for every fifth acquisition. The total scan time was 27 h for the first protocol and 6 h and 45 min for the second protocol. The first protocol was acquired with a 20-mm diameter surface coil and the second acquisition was acquired with a quadrature volume coil. A 4-h gradient heavy dummy scan preceded the experiments to reduce short term mechanic and thermal instabilities (15). Imaging datasets can be downloaded from our homepage http://dig.drcmr.dk/. Simulation Monte Carlo simulations were performed in model substrates of restricted diffusion in cylinders with radii r ¼ 1 and 3 mm and the free diffusion coefficient set to D ¼ 2 109 m2 =s comparable to water at room temperature. We implemented the tissue model in the Camino software with 10,000 walkers and time steps of 4.2 ms (17). The walkers were uniformly distributed in the intracellular compartment only. Ideal linearly and circularly polarized cosine modulated gradients were constructed for a range of frequencies, 50–800 Hz, and gradient strengths, 200–1000 mT/m, at a fixed total gradient train length of 40 ms. Numerical simulations were compared to analytical expressions for the OGSE signal in cylindrical geometries (18). Noise propagation through the diffusion tensor model was evaluated by calculating the diffusion tensor signal S ¼ exp ððbxx Dxx þ byy Dyy þ bzz Dzz þ 2bxy Dxy þ 2 bxz Dxz þ 2byz Dyz ÞÞ with the B-matrices from the imaging study scaled from b ¼ 50–1000 s/mm2 for OGSE and b ¼ 100–2000 s/mm2 for the CP-OGSE. This simulates
1173
diffusion weightings possible at different systems in the contrast rich frequency range around 100 Hz (5,8). An anisotropic test tensor with the eigenvalues [0.5 0.7 1.8] 109 m2/s was used. The impact of SNR was investigated by contaminating 10,000 repetitions of the datasets with Gaussian noise in the complex plane and then calculating the magnitude to model the Rician noise distribution of MRI data (19). Data Analysis Image data was registered and resliced in plane to one b ¼ 0 image and normalized to the mean of all b ¼ 0 images to account for eddy current effects with SPM8 (http://www.fil.ion-.ucl.ac.uk/spm). The B-matrices were calculated by trapezoidal integration of the gradient amplifier input sampled at a 4-ms time step. Diffusion tensors were estimated using linear least square fitting and we calculated mean diffusivity (MD) as the mean of the eigenvalues and radial diffusivity (RD) as the mean of the two minor eigenvalues. The increase in diffusivity as a function of frequency in restricted geometries is sinc-shaped but DMD, the linear increase in MD over frequency, can be used for visualization in narrow frequency ranges and has been shown to have a contrast similar to histological Nissl-stains (5). We calculated DMD using least squares fitting to MD over the four frequency measurements. All diffusion analyses were performed using in-house scripts written in MATLAB (Mathworks Inc., Navick, MA, USA). RESULTS Simulation For diffusion tensor imaging, we assume that the CPOGSE experiment probes diffusivity along two orthogonal directions independently and should thus be equivalent to the product of two conventional OGSE experiments. Figure 2a shows the frequency dependence of the signal at different gradient strengths in cylindrical restrictions for the CP-OGSE and the product of the two corresponding OGSE experiments applied in the plane perpendicular to the length axis of the cylinder. The CP-OGSE is in very good correspondence with both simulated and analytical estimates of the OGSE signal. The components of restricted diffusivity in MD increase with frequency as shown in Figure 2b. In the narrow frequency range possible in our experimental settings, this effect can be described by the linear increase DMD. Figure 2c shows estimates of the individual eigenvalues, MD and RD from the two techniques over a range of b-values at SNR ¼ 20. The repulsion of the individual eigenvalues at low b-values in both OGSE and CP-OGSE is due to the sorting bias inherent in diffusion tensor analysis (20). CP-OGSE excel OGSE in terms of lower standard deviation in estimates at low b-values. The benefits remain up to moderate b-values in the RD and MD estimates but eigenvalues become more biased and noisy. Imaging Imaging results from the first protocol are shown in Figure 3. Figure 3a shows a b ¼ 0 image with three ROIs
1174
Lundell et al.
FIG. 2. a: Monte Carlo simulations were performed inside cylindrical geometries with radii r ¼ 3 mm (top) and r ¼ 1 mm (bottom) and shows good correspondence between CP-OGSE and OGSE. The signal attenuation as a function of frequency for fixed gradient strengths in the range 0.2–1 T/m is shown. The CP-OGSE correspond well to the product of two individual orthogonal OGSE gradients from simulations (red dotted line) as well as from an analytical description of the OGSE in restricted geometries as derived by Xu et al. (18) (green thin line). b: The frequency dependence on MD in the 3 and 1 mm cylinders at SNR ¼ 10 (top) from a Monte Carlo simulation corresponding to the experimental settings. Better separation between sizes is achieved at higher frequencies. Below a schematic illustration of the effect with random walks in restrictions with different dimensions. At higher frequencies (shorter diffusion time), diffusivity in larger restrictions is less affected by barriers and is thus higher. c: The influence of noise on MD and RD (top) and the individual eigenvalues (l1;2;3 ; bottom), over a range of b-values at SNR ¼ 20. The mean over 10,000 repetition are plotted with thick lines and the standard deviations are shown as thin lines. The blue lines indicate the true value. CP-OGSE performs better than OGSE at low b-values but a larger bias and noise sensitivity in the individual eigenvalues is introduced at increasing b. The benefits remain up to moderate b-values in RD and in MD. Strong bias in the individual eigenvalues are seen both in OGSE and CP-OGSE and are due to the general sorting bias problem in diffusion tensor analysis prominent at low b-values or SNR.
drawn in white matter (green), in the inner granular layer of the cerebellar cortex (red), and in the outer molecular layer (blue). Differentiation between the three regions increased at higher oscillation frequency with the largest MD increases in the densely packed granular layer (Fig. 3b), the colors correspond to the colors of the ROIs in Figure 3a. The MD estimates were less affected by noise in our CP-OGSE measurement compared to the conventional OGSE (solid vs. dashed lines). Due to low SNR, MD could only be reconstructed in white matter using the CP-OGSE. DMD-maps, shown in Figure 4a,b for OGSE and CP-OGSE, respectively, had large contrast between the granular and molecular layers not visible in T2W or PGSE MD images. Voxels with negative eigenvalues, i.e., insufficient contrast to noise, were excluded from the analysis and are colored green in Figure 4c,d. The overall visual appearances of CP-OGSE results were similar to OGSE but less noisy. Figure 5 shows MD (a) and color coded FA (b) from the second protocol with lower maximum frequency but higher diffusion weighting. Contrast is enhanced in MD maps at higher frequencies but FA has similar appearance at both frequencies as observed earlier with conventional OGSE (5). Note
FIG. 3. a: A T2W reference image with three ROIs covering white matter (green), granular layer (red) and the molecular layer (blue). b: MD values extracted from the OGSE (dashed lines) and CPOGSE (solid lines). The colors correspond to the ROIs shown in (a). Due to different microstructure, the contrast increase at higher frequencies. MD was not resolved in white matter with OGSE due to low SNR.
Circularly Polarized Diffusion Encoding
1175
FIG. 4. DMD-maps from the first protocol acquired at b ¼ 220/440 s/mm2, showing the linear increase in MD as a function of frequency for the (a) OGSE and the (b) CP-OGSE acquisitions from the first gradient protocol. Due to the increased diffusion weighting with CP-OGSE, parameter estimates are improved in (b) compared to (a) despite the same gradient limitations. Voxels with insufficient SNR are excluded and colored green.
that directional information is correctly and consistently resolved with CP-OGSE in the color coded maps in Figure 5b.
DISCUSSION In this study, we propose the use of CP-OGSE to increase the diffusion weighting of OGSE experiments for a given set of hardware limitations. In contrast to linearly polarized harmonic or trapezoidal gradients, CP-OGSE always run on maximum gradient strength and can still be rotated in any arbitrary orientation. We suggest that data can be analyzed using conventional diffusion tensor techniques and show in simulations and experiments that CP-OGSE is comparable to OGSE but provides more robust parameter estimates at low b-values. This can thus be an important contribution for applications on systems with limited gradient performance, like clinical scanners. Simulations confirmed that CP-OGSE is equivalent to two OGSE experiments at the moderate diffusion weightings applied in this study and within the limits reasonable on clinical imaging systems. Similarly, Portnoy et al. recently reported a resemblance between OGSE and PGSE experiments at b < 1000 s/mm2 (7). This indicates that the effect of interaction between spin displacements during subsequent waves of the OGSE is negligible and OGSE can more be considered as the accumulated attenuation from multiple PGSE experiments as initially proposed by Gross and Kosfeld (21). It is noteworthy to mention that CP-OGSE yields a natural contrast for anisotropy and fiber direction similar to double wave experiments (22), or the Funk–Radon transform used in Q-ball imaging and previously also explored with planar PGSE encoding (23,24). The choice between OGSE and CP-OGSE must be considered in relation to gradient limitations, maximum frequency (and thereby the maximal b-value), the underlying diffusion tensor, and the parameters of inter-
FIG. 5. a: MD maps from the second protocol acquired with CPOGSE only at b ¼ 775 s/mm2 for f ¼ 50 Hz (left) and f ¼ 150 Hz (right). b: Corresponding color coded FA maps for f ¼ 50 Hz (left) and f ¼ 150 Hz (right). Anisotropy pattern is maintained and CPOGSE renders directional information correctly with colors following the main white matter branches. The cerebellar cortex is red presumably due to the unmyelinated but densely packet parallel fibers running transversally over the midsagittal plane.
est. CP-OGSE primary improves estimates of isotropic features of the diffusion tensor, like MD and RD, but also to a lesser degree the individual eigenvalues in the b D 1 regime (Fig. 2c). Care must always be taken when inferring anisotropy based on subtle differences in eigenvalues (20,25,26) Our OGSE experiment in Figure 3 as well as other recent OGSE studies operate around b MD 0:1 to 0.3 (5,6,8). The contrast between different gradient orientations does not improve with CP-OGSE and will thus not improve eigenvalue estimates at moderate b-values. But sufficient signal attenuation, given by b D, compared to the noise is essential for robust estimates, which is why CP-OGSE still provides a benefit at low b. The noise distribution across the different eigenvalues depends on their sizes in relation to each other as each measurement will be the interaction of two directions. The benefits of trapezoidal compared to harmonic gradient shapes has also been reported (8,27,28). For a square wave inserted in Eq. [2] the increase in b-value is by a factor ð4=pÞ2 1:62 compared to a conventional OGSE, but the effect will be less with finite slew rate limitations. Besides from the higher twofold increase in diffusion weighting, the CP-OGSE also benefits from lower slew rates and is thus less prone to peripheral nerve stimulation in in vivo settings. CONCLUSIONS CP-OGSE provides twice the diffusion weighting compared to conventional OGSE which is a large experimental benefit on MRI systems with limited gradient
1176
strength. We demonstrated increased robustness in parameter estimates at low b in simulations and experiments on perfusion fixed brain tissue. Simulations showed that results are equivalent to conventional OGSE. CP-OGSE can be a significant contribution for making OGSE possible for studies of the human brain in vivo on clinical systems with moderate gradient strengths.
REFERENCES 1. Nilsson M, van Westen D, Sta˚hlberg F, Sundgren PC, L€ att J. The role of tissue microstructure and water exchange in biophysical modelling of diffusion in white matter. MAGMA 2013;26:345–370. 2. Clark CA, Hedehus M, Moseley ME. Diffusion time dependence of the apparent diffusion tensor in healthy human brain and white matter disease. Magn Reson Med 2001;45:1126–1129. 3. Pfeuffer J, Fl€ ogel U, Dreher W, Leibfritz D. Restricted diffusion and exchange of intracellular water: theoretical modelling and diffusion time dependence of 1H NMR measurements on perfused glial cells. NMR Biomed 1998;11:19–31. 4. Gore J, Xu J, Colvin D, Yankeelov TE, Parsons EC, Does MD. Characterization of tissue structure at varying length scales using temporal diffusion spectroscopy. NMR Biomed 2010;23:745–756. 5. Aggarwal M, Jones MV, Calabresi PA, Mori S, Zhang J. Probing mouse brain microstructure using oscillating gradient diffusion MRI. Magn Reson Med 2012; 1:98–109. 6. Kershaw J, Leuze C, Aoki I, Obata T, Kanno I, Ito H, Yamaguchi Y, Handa H. Systematic changes to the apparent diffusion tensor of in vivo rat brain measured with an oscillating-gradient spin-echo sequence. NeuroImage 2013;70:10–20. 7. Portnoy S, Flint JJ, Blackband SJ, Stanisz GJ. Oscillating and pulsed gradient diffusion magnetic resonance microscopy over an extended b-value range: implications for the characterization of tissue microstructure. Magn Reson Med 2013;69:1131–1145. 8. Van AT, Holdsworth SJ, Bammer R. In vivo investigation of restricted diffusion in the human brain with optimized oscillating diffusion gradient encoding. Magn Reson Med 2014;71:83–94. 9. Xu J, Li K, Smith RA, Waterton JC, Zhao P, Chen H, Does MD, Manning HC, Gore JC. Characterizing tumor response to chemotherapy at various length scales using temporal diffusion spectroscopy. PLoS One 2012;7:1–8. 10. Colvin DC, Yankeelov TE, Does MD, Yue Z, Quarles C, Gore JC. New insights into tumor microstructure using temporal diffusion spectroscopy. Cancer Res 2008;68:5941–5947.
Lundell et al. 11. Dyrby TB, Sïgaard LV, Hall MG, Ptito M, Alexander DC. Contrast and stability of the axon diameter index from microstructure imaging with diffusion MRI. Magn Reson Med 2013;70:711–721. 12. Stepisnik J. NMR measurement and Brownian movement in the short-time limit. Physica B 1994;198:299–306. 13. Stepisnik J, Mohoric A, Duh A. Diffusion and flow in a porous structure by the gradient spin echo spectral analysis. Physica B 2001;307: 158–168. 14. Jones DK, Horsfield MA, Simmons A. Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging. Magn Reson Med 1999;42:515–525. 15. Dyrby TB, Baar e WFC, Alexander DC, Jelsing J, Garde E, Sïgaard LV. An ex vivo imaging pipeline for producing high-quality and highresolution diffusion-weighted imaging datasets. Hum Brain Mapp 2010;32:544–563. 16. Burke MW, Zangenehpour S, Ptito M. Brain banking: making the most of your research specimens. J Vis Exp 2009;1260. 17. Hall MG, Alexander DC. Convergence and parameter choice for Monte-Carlo simulations of diffusion MRI. IEEE Trans Med Imaging 2009;28:1354–1364. 18. Xu J, Does MD, Gore JC. Quantitative characterization of tissue microstructure with temporal diffusion spectroscopy. J Magn Reson 2009; 200:189–197. 19. Gudbjartsson H, Patz S. The Rician distribution of noisy MRI data. Magn Reson Med 1995;34:910–914. 20. Pierpaoli C, Basser PJ. Toward a quantitative assessment of diffusion anisotropy. Magn Reson Med 1996;36:893–906. 21. Gross B, Kosfeld R. Anwendung der Spin-Echo-Methode bei der Messung der Selbstdiffusion. Messtechnik 1969;7:171–177. 22. Jespersen SN, Lundell H, Sïnderby CK, Dyrby TB. Orientationally invariant metrics of apparent compartment eccentricity from double pulsed field gradient diffusion experiments. NMR Biomed 2013;26: 1647–1662. 23. Tuch DS. Q-ball imaging. Magn Reson Med 2004;52:1358–1372. 24. Wedeen VJ, Dai G, Tseng WYI, Benner T. Diffusion encoding with 2D gradient trajectories yields natural contrast for 3D fiber orientation. In Proceedings of the 14th Annual Meeting of ISMRM, Seattle, Washington, USA, 2006. p. 851. 25. Jones DK, Basser PJ. Squashing peanuts and smashing pumpkins: how noise distorts diffusion-weighted MR data. Magn Reson Med 2004;52:979–993. 26. Lundell H, Nielsen JB, Ptito M, Dyrby TB. Distribution of collateral fibers in the monkey cervical spinal cord detected with diffusionweighted magnetic resonance imaging. NeuroImage 2011;56:923–929. 27. Lundell H, Sïnderby CK, Dyrby TB. DTI using modulated gradients at short effective diffusion times. In Proceedings of the 20th Annual Meeting of ISMRM, Melbourne, Australia, 2012. p. 355. 28. Ianus¸ A, Siow B, Drobnjak I, Zhang H, Alexander DC. Gaussian phase distribution approximations for oscillating gradient spin echo diffusion MRI. J Magn Reson 2013;227:25–34.
