Diffusion Propagator Imaging: Using Laplace s Equation and Multiple Shell Acquisitions to Reconstruct the Diffusion Propagator

Transcription

1 Diffusion Propagator Imaging: Using Laplace s Equation and Multiple Shell Acquisitions to Reconstruct the Diffusion Propagator Maxime Descoteaux 1, Rachid Deriche 2,DenisLeBihan 1, Jean-François Mangin 1,andCyrilPoupon 1 1 NeuroSpin, IFR 49 CEA Saclay, France 2 INRIA Sophia Antipolis - Méditerranée, France Abstract. Many recent single-shell high angular resolution diffusion imaging reconstruction techniques have been introduced to reconstruct orientation distribution functions (ODF) that only capture angular information contained in the diffusion process of water molecules. By also considering the radial part of the diffusion signal, the reconstruction of the ensemble average diffusion propagator (EAP) of water molecules can provide much richer information about complex tissue microstructure than the ODF. In this paper, we present diffusion propagator imaging (DPI), a novel technique to reconstruct the EAP from multiple shell acquisitions. The DPI solution is analytical and linear because it is based on a Laplace equation modeling of the diffusion signal. DPI is validated with ex vivo phantoms and also illustrated on an in vivo human brain dataset. DPI is shown to reconstruct EAP from only two b-value shells and approximately 100 diffusion measurements. 1 Introduction One of the quest of diffusion-weighted (DW) imaging is the reconstruction of the full three-dimensional (3D) ensemble average propagator (EAP) describing the diffusion process of water molecules in biological tissues. Many recent high angular resolution diffusion imaging (HARDI) techniques have been proposed to recover complex diffusion orientation distribution functions (ODF) of the white matter geometry. However, these orientation functions derived from HARDI only capture the angular structure of the diffusion process on a single shell and are therefore mostly useful for fiber tractography applications. The EAP can capture richer information by considering both radial and angular information part of the q-space diffusion signal. Thus, the EAP might provide means to infer axonal diameter and also be sensitive to white matter anomalies [1]. In order to relate the observed diffusion signal to the underlying tissue microstructure, we need to understand how the diffusion signal is influenced by the tissue geometry and its properties. Under the narrow pulse assumption, the relationship between the diffusion signal, E(q), in q-space and the EAP, P (R), in real space, is given by an inverse Fourier transform (IFT) [3] as J.L. Prince, D.L. Pham, and K.J. Myers (Eds.): IPMI 2009, LNCS 5636, pp. 1 13, c Springer-Verlag Berlin Heidelberg 2009

2 2 M. Descoteaux et al. P (R) = E(q)e 2πiq R dq, (1) q R 3 where E(q) =S(q)/S 0 is the diffusion signal measured at position q in q-space and S 0 is the baseline image acquired without any diffusion sensitization (q =0). We denote q = q and q = qu, R = R 0 r,whereu and r are 3D unit vectors, and q, R 0 R. The wave vector q is q = γδg/2π, withγ the nuclear gyromagnetic ratio and G the applied diffusion gradient vector. Various methods already exist to reconstruct the EAP [4,5,6,7,8,9,10,11,12]. Among the most commonly used methods, diffusion tensor imaging (DTI) [4] is limited by the Gaussian assumption of the free diffusion model, which excludes observedin vivo phenomena such as restriction, heterogeneity, anomalous diffusion, and finite boundary permeability. Diffusion spectrum imaging (DSI) [5] can account for DTI limitations. The technique has the advantage of being model-free but requires hundreds of DW measurements sampled on a dense Cartesian grid, which requires strong gradient fields, in order to evaluate the Fourier transform of Eq. 1. DSI was also shown to be possible on a non-cartesian grid in [11] using less measurements on multiple spherical shells. More recently, inspired by computed tomography, another technique was proposed to perform measurements along many radial lines before computing 1D tomographic projections to reconstruct the 3D EAP [10]. This technique also requires hundreds of samples on a few radial lines of q-space to recover the EAP. The results are promising but have not yet been appliedonanin vivo brain. Other techniques suggest using multiple spherical shell acquisitions in order to reconstruct the features of the EAP, such as generalized high order tensors [6] based on cumulant expansions; or the composite and hindered restricted model of diffusion (CHARMED) [7]; or the diffusion kurtosis [8]; or the diffusion orientation transform (DOT) [9]; or hybrid diffusion properties of the EAP [11]; or a fourth order Cartesian tensor representation of the probability profile [12]. Unfortunately, for most of these methods, many DW measurements are needed. Moreover, it remains unclear what is the right number of spherical shells needed and most of the results lack validation. Of all the mentioned techniques, DOT is in closer spirit to our approach and will be revisited later. In this paper, we develop diffusion propagator imaging (DPI), a novel technique for analytical EAP reconstruction from multiple shell acquisitions. Our solution is simple, linear and compact. DPI is based on a 3D Laplace equation modeling of the q-space diffusion signal, which greatly simplifies the solution to Eq. 1 and allows one to obtain an analytical solution. An important part of this paper is dedicated to validate DPI, both the signal fitting with Laplace equation and the EAP reconstruction, on real datasets from ex vivo phantoms [13]. We also illustrate DPI on a real in vivo human brain. 2 Diffusion Propagator Imaging (DPI) q-space Signal Approximation with Laplace s Equation. Modeling the 3D q-space diffusion signal to recover the EAP and capture complex fiber crossing configurations was proposed before in CHARMED [7] and DOT [9], where the

3 Diffusion Propagator Imaging: Using Laplace s Equation 3 diffusion signal was modeled with multiple fiber compartments; in CHARMED, with a mixture of restricted and hindered compartements and in DOT, with a mixture of exponential decay functions (mono-, bi or tri-exponential). In our approach, we do not want to assume any mixture models apriori. We seek a simpler representation of the diffusion signal that will naturally capture multiple shell measurements and allow for an analytical EAP solution. Our main assumption is that the diffusion signal attenuation can be estimated using the 3D Laplace Equation. Under this assumption, we express the q-space diffusion signal E(q) =S(q)/S 0 in terms of any radius using the general or total solution of the Laplace equation in spherical coordinates, which gives E(q) =E(qu) = [ j=0 c j q l(j)+1 + d jq l(j) ] Y j (u) for q>0, (2) where l(j) is the order associated with element j of the spherical harmonic (SH) basis Y j, which is defined to be real and symmetric, and c j and d j are the unknown SH coefficients describing the signal. Also, for q = 0, E(q) = 1. The Laplace equation requires boundary conditions. In our problem, we need at least two shell measurements and more diffusion measurements N than unknown coefficients to properly constrain Eq. 2. Intuitively, our Laplace equation modeling can be seen as the heat equation between each given shell measurements, i.e. the solution is obtained when the heat does not change between the temperature measurements given at each shell. Analytical Diffusion Propagator Reconstruction. Under this Laplace equation modeling assumption, we prove, in Appendix A, that the EAP can be reconstructed as P (R 0 r)=2 j=0 ( 1) l(j)/2 2π l(j) 1 R l(j) 2 0 c j Y j (r) for R 0 > 0, (3) (2l(j) 1)!! where (n 1)!! = (n 1) (n 3) This expression is analytical and quite simple to compute. Note also that the EAP expression is linear and only depends on the c j coefficients, but the d j coefficients are nonetheless important in the diffusion signal fitting/modeling procedure of Eq. 2. We finally note that if R 0 =0,P (0) = q R 3 E(q)dq, the average diffusion signal in q-space. 3 Methods As a starting point, we are given n HARDI shell datasets with the same number of diffusion measurements N per shell. It is now standard in HARDI processing techniques to use a modified real and symmetric SH basis of order l with elements Y j,wherej := j(k, m) =(k 2 + k +2)/2 +m is defined from the order l and phase m standard SH Yl m (see [14,15,16,17]). We can then generate the linear system associated with the Laplace equation signal estimation given in Eq. 2. We let S n be the N x 1 vector representing the

4 4 M. Descoteaux et al. diffusion signal of shell number n at each of the N diffusion encoding gradient direction. We also let C and D represent the R x 1 vectors of unknown SH coefficients, c j and d j in Eq. 2, where R =1/2(l+1)(l+2). Next, we let B represent the N x R matrix constructed with the modified SH basis (as ine [14,15,16,17]). Finally, we define R l (q) =r l and I l (q) =r l 1 to capture the regular and irregular radial part of the total Laplace equation. We can then construct the matrices of coefficients for each shell, F n and G n,astwor x R diagonal square matrices with diagonal entries R l(j) (q n )andi l(j) (q n ) respectively. As before, q n is the q-value of shell n and l(j) is the order of the jth SH coefficient 1. Lastly, for each of the n shell, we have the linear system S n = BG n C+BF n D. Combining all n systems, we obtain the general linear system representing Eq. 2, S = AX. This system of over-determined equations is solved with a standard least-square solution yielding the vector X =[C D ] T,givenby X =(A T A) 1 A T S. Therefore, the estimated signal, S, can be recovered simply with AX. We can then report the mean and standard deviation (std) of the Euclidean error S n S n percentage between the original and estimated diffusion signal for each of the n shell and over all N diffusion measurements. Finally, taking the first part of the estimated vector X, we can extract the C coefficients needed to compute the EAP. The spherical function P R0 representing the EAP for given R 0, can be obtained with a simple matrix multiplication P R0... = B 2( 1) l(j)/2 2π l(j) 1 R l(j) 2 0 /(2l(j) 1)!!... P R0 can then be visualized for different values of R 0. C, (4) Data Acquisition. DPI was used to estimate the diffusion signal and the associated EAP on ex vivo phantoms with fibers crossing at 90 and 45 designed in [13] with parameters: FOV=32cm, matrix 32x32, TH=14mm, TE=130ms, TR=4.5s,12.0s (45,90 ), BW=200KHz and b-values of 2000, 4000, 6000, 8000 s/mm 2 and 4000 uniformly distributed orientations. The number of directions were also resampled to N =15, 25 and 60 to test DPI under lower and more realistic sampling schemes. DPI was also applied on data acquired from a 3T Trio MR Siemens system, equipped with a whole body gradient (40 mt/m and 200 T/m/s) and an 32 channel head coil. The acquisition parameters were TE/TR = 147ms/11.5s, BW=1680Hz/pixel, 96x96 matrix, isotropic 2mm resolution, and 60 axial slices. We acquired a b=0 diffusion image followed by four b-values acquisitions with 64 uniform directions, at b = 1000, 2000, 4000 and 6000 s/mm 2. 4 Results Diffusion Propagator Imaging of the Ex Vivo Phantom. DPI is applied on the 90 and 45 degree phantoms shown in Figure 1. We pick the center 1 For j = {1, 2, 3, 4, 5, 6, 7, 8,...}, l j = {0, 2, 2, 2, 2, 2, 4, 4,...}.

5 Diffusion Propagator Imaging: Using Laplace s Equation 5 (a) 90 2 (b) 45 2 Fig. 1. Physical phantoms designed in [13]. (a) 90 (photograph), and (b) 45 (fast spin-echo map and ROI)) phantoms. We also show the original diffusion signal equators perpendicular to the z (in blue, axial view) and x (in red, sagittal view) axes, from b = 2000, 4000, 6000, and 8000 s/mm 2, showing the diffusion signal decay. Table 1. Mean and standard deviation of the percentage error in the multi-shell signal fit depending on the order l and number of measurements N used (sampling scheme) (a) Estimation order l, multi-shell experiment with N = 4000 shell 90 crossing 45 crossing s/mm 2 l =2 l =4 l =16 l =2 l =6 l = ±2.5% 2.6±0.3% 2.5±0.3% 13±3.3% 2.0±0.2% 2.0±0.3% ±3.9% 4.3±1.0% 4.3±1.1% 13±3.4% 3.9±0.7% 3.7±0.9% ±4.4% 5.1±1.6% 4.5±1.1% 14±3.8% 5.2±1.2% 5.2±1.4% ±6.5% 8.5±2.8% 8.5±3.0% 15±6.6% 8.5±2.7% 8.4±3.1% (b) Sampling scheme N, multi-shell experiment shell 90 crossing, l =4 45 crossing, l =6 s/mm 2 N = 4000 N =60 N =15 N = 4000 N =60 N = ±0.3% 2.7±0.3% 2.8±0.4% 2.0±0.2% 2.1±0.2% 2.3±1.0% ±1.0% 4.3±1.0% 5.1±1.4% 3.9±0.7% 4.0±0.8% 4.0±1.1% ±1.6% 5.2±1.8% 6.2±2.6% 5.2±1.2% 5.3±1.3% 6.3±2.0% ±2.8% 8.5±2.7% 8.6±3.2% 8.5±2.7% 8.5±2.5% 8.5±2.8% voxel of the phantom, which contains approximately equal proportion of the two fiber branches of the crossing. The diffusion signal attenuation is also shown in Figure 1. In this visualization, red and blue lines illustrate equators of the original diffusion signal perpendicular to the x and z plan respectively, whereas later, black lines illustrate the associated estimated signal in the signal fit experiments. Can Laplace s Equation be used for diffusion signal estimation? Table 1 quantitatively shows that the diffusion signal can be modeled using Laplace s equation, on both ex vivo phantoms. This is qualitatively confirmed in Figure 2. From Table 1, we first see that estimation is accurate and that there is less than 10% mean signal fitting error with a small standard deviation at every shell, for every estimation order l and every sampling scheme N. Wealsosee that the mean percentage error is increasing with increasing b-value. The most significant error systematically occurs at shell b = 8000 s/mm 2. This is due to the intrinsic smoothness of the diffusion signal fit with Laplace s equation. As

6 6 M. Descoteaux et al. 90 multi-shell, =4andN =15 45 multi-shell, =6andN =60 b = 2000 b = 4000 b = 6000 b = 8000 b = 2000 b = 4000 b = 6000 b = 8000 Fig. 2. Multi-shell signal fit experiment on the 90 and 45 degree crossing phantoms. We show original (colored) and estimated (black) diffusion signal equators perpendicular to the z (in blue) and x (in red) axes respectively. seen in curves of Figure 2, a desired smoothing occurs in the diffusion signal estimation, which for high b-values, increases the mean error between the more jagged original signal curves in color and the smooth estimated curves in black. What is the right estimation SH order? From Table 1, we note that the signal estimation accuracy is more or less equivalent for SH order l 4. Under the simple geometrical phantom configuration, increasing estimation order does not significantly reduce the mean percentage error nor the standard deviation. What is the right number of diffusion measurements? We do not need hundreds of diffusion measurements for accurate diffusion signal fit. From Table 1, we note that the diffusion signal estimation accuracy remains relatively stable for all sampling schemes N. No significant error increase is observed as one decreases the number of measurements, down to N =15and28fororder l = 4 and 6 respectively. Can we recover fiber crossings? Because we can accurately estimate the diffusion signal, we can obtain EAP reconstructions that recover fiber crossing information. First, we note in Figure 3a) that a SH order of 2 is insufficient to discriminate the fiber crossing. This is in fact reflected in the first column (l = 2) of Table 1 with large percentage errors in the diffusion signal estimation. A DPI reconstruction of order 2 is similar to a DTI reconstruction of the Gaussian diffusion propagator. Next, we also note in Figure 3b)-f), that a SH order 4 is sufficient to discriminate the 90 degree crossing but insufficient for the 45 degree crossing. One must use SH order l 6 or higher to discriminate the crossing fibers in the 45 degree crossing example, at the cost of having some small spurious peaks appearance for higher radii R 0. Hence, even though increasing SH order does not significantly improve the signal fitting accuracy (seen in Table 1), increasing SH order does improve the angular resolution of the EAP reconstruction. However, higher SH order EAP reconstruction have higher frequency coefficients that are more perturbed by noise and thus, they can have spurious peaks. We therefore choose l = 4 in the 90 crossing phantom and l =6 in the 45 degree crossing phantom in the rest of our experiments. What is the right number of shells? Figure 4 shows that two-shell diffusion signal fitting is also possible using Laplace s equation. We see that the diffusion signal estimation remains relatively accurate even if one uses only

7 Diffusion Propagator Imaging: Using Laplace s Equation 90, 7 = 2, N = 4000, multiple shells (b = 2000, 4000, 6000, 8000 s/mm2 ) a) = 4, N = 4000, multiple shells (b = 2000, 4000, 6000, 8000 s/mm2 ) b) = 4, N = 15, two outermost shells (b = 2000, 8000 s/mm2 ) c) R0 = 0.1 μm 0.25 μm 0.5 μm 1 μm 2 μm 3 μm 5 μm 10 μm 45, = 4, N = 4000, multiple shells (b = 2000, 4000, 6000, 8000 s/mm2 ) d) = 8, N = 4000, multiple shells (b = 2000, 4000, 6000, 8000 s/mm2 ) e) = 6, N = 60, two outermost shells (b = 2000, 8000 s/mm2 ) f) R0 = 0.1 μm 0.25 μm 0.5 μm 1 μm 5 μm 10 μm 20 μm 50 μm Fig. 3. EAP reconstruction on the 90 and 45 crossing phantom for different SH order, sampling scheme N, number of shells, and radius R0 90 two-shell, = 4 and N = two-shell, = 6 and N = 60 b = 2000 b = 4000 b = 6000 b = 8000 b = 2000 b = 4000 b = 6000 b = 8000 Fig. 4. Two-shell signal fit experiment on the 90 and 45 crossing phantoms using shells b = 2000 and 8000 s/mm2

8 8 M. Descoteaux et al. GFA map RGB map from principal direction of the q-ball ODF b = 1000 s/mm2 b = 2000 s/mm2 b = 4000 s/mm2 b = 6000 s/mm2 Fig. 5. Multiple shell real data HARDI acquisition with 4 shells. We show a coronal slice of the GFA map and RGB map with a region of interest. two-shell measurements. We also see that the best fit occurs for the shells used in the estimation. On the other hand, as expected, the largest errors occur for the signal associated to the extrapolated shells. Although not shown here, these results are confirmed quantitatively with mean percentage errors and standard deviations as computed for the multi-shell experiment. Even though the error is higher for the unused shell s signal, the extrapolated curves are smooth and are still able to capture the angular contrast of the signal. As a result, one can reliably reconstruct the diffusion EAP from two shell measurements. As seen in Figure 3c) and f), the reconstructed EAP from two-shell are qualitatively similar to the full multi-shell reconstruction. Diffusion Propagator Imaging of the Human Brain. We now apply DPI on the in vivo brain dataset illustrated in Figure 5. Based on the previous section, we choose SH order = 4 and use all measurements, either in a full multi-shell DPI reconstruction, using all four shells, or in a two-shell DPI reconstruction using the two outermost shells (b = 1000 and 6000 s/mm2 ). Figure 6 first shows that the EAP reconstructions agree with the underlying anatomy in the ROI shown in Figure 5. The extracted spherical functions of the EAP, P (R0 r), emphasize directionality of the EAP and we see that the CC in red, CST in blue and SLF in green are well identified. Most importantly, we also see that the crossing fiber configurations are recovered and well discriminated in the EAP, especially for high radii Ro. In fact, as the radius R0 increases, the angular resolution improves, at the cost of having slightly more noisy profiles with some spherical function that become spiky. Figure 6 also shows that twoshell DPI closely agrees with the full multi-shell DPI. This confirms the ex vivo phantom results that DPI can be done reliably with only two different shells.

9 Diffusion Propagator Imaging: Using Laplace s Equation 9 Two-shell DPI Multi-shell DPI EAP, R 0 =2μm EAP, R 0 =5μm Fig. 6. DPI on the human brain dataset in the same crossing region as Figure 5. In the first row, a multi-shell DPI reconstruction and in the second row, a two-shell DPI was done using the two outermost shells b = 1000 and 6000 s/mm 2. 5 Discussion and Conclusions The Laplace equation was successfully used to model the diffusion signal and reconstruct the EAP but apriori, there is no physical reason why this should be. There might exist other possible model and family of functions to do so. In fact, [16] and [18] recently proposed new orthogonal bases to estimate the diffusion signal. In [16], the Spherical Polar Fourier (SFP) basis is used to extract features of the EAP. The basis is composed of an angular part using spherical harmonics and a radial part using an orthonormal radial basis function called the Gaussian-Laguerre polynomials. However, [16] do not have the full analytical EAP reconstruction itself as we do. Only EAP features such as the ODF or other angular functions of the EAP can be computed. Separately, [18] proposed

10 10 M. Descoteaux et al. to use Hermite polynomials to estimate the 1D q-space signal from spectroscopy imaging. While forming a complete orthogonal basis, the Hermite polynomials also have the important property that their Fourier transform can be expressed in terms of themselves. This property could potentially be very powerful to evaluate Eq. 1. Although the Hermite polynomials were used for 1D q-space imaging, an extension to the 3D problem seems conceivable. One now has to think about the best way to sample q-space (with Cartesian points, spherical shells, or radial lines) in order to have sufficient measurements to robustly reconstruct the EAP and avoid the necessity for hundreds of measurements. Different sampling schemes might be better adapted depending on the chosen basis functions. Our Laplace equation assumption also has several positive aspects that make DPI an appealing technique. The advantages of DPI are threefold: i) The solution is analytical. ii) The signal modeling performs some level of smoothing. iii) The solution is linear and compact. First, having an analytical expression for the EAP allows one to estimate probability values outside the actual q-space range prescribed by the acquisition. In comparison, the DSI technique is limited by the boundaries of q-space and thus, the reconstructed EAP is also bounded. Our approach is in closer spirit in with the DOT technique, where an expression of P (R 0 r) is also obtained based on an exponential assumption of the signal decay. A systematic comparison of DSI, DOT and DPI thus seems important to highlight the limitations of the modeling and numerical versus analytical reconstructions. Next, as seen in the Result section, DPI performs some level of smoothing in the estimation. We believe this intrinsic smoothing of the Laplace modeling is actually desired. Again, in comparison, DSI [5] uses a Hanning window to smooth the signal before Fourier transform computation. No such lowpass filtering is needed for robust diffusion estimation and EAP reconstruction in DPI. Finally, the DPI solution is linear and quite compact, which makes the EAP reconstruction extremely fast; as fast as a simple least-square DTI or QBI reconstruction. DPI reconstruction runs in less than 2 minutes on a standard PC and its solution can be expressed in a small number of SH coefficients. To conclude, we have introduced diffusion propagator imaging, a novel technique for reconstructing the diffusion propagator from multiple shell acquisitions. We have shown that DPI can provide a new and efficient framework to study characteristics of the diffusion EAP, which opens interesting perspectives for tissue microstructure investigation. We now need to exploit DPI in clinical acquisitions and clinical settings. Future work will focus on new measures that integrate radial part of the diffusion signal, to lead to the development of new biomarkers sensitive to white-matter anomalies, brain development and aging, and might also provide means to infer axonal diameter. Acknowledgments. The authors are thankful to A. Ghosh and the diffusion MRI group of NeuroSpin for insightful discussion on the diffusion propagator estimation. Moreover, thanks to K-H. Cho, Y-P. Chao and Pr. C-P. Lin for their participation in he data acquisition. This work was supported by Programme Hubert Curien Orchid 2008 franco-taiwanais, l Ecole des Neurosciences Paris-Ile de France (ENP), and l Association France Parkinson for the NucleiPark project.

11 Diffusion Propagator Imaging: Using Laplace s Equation 11 References 1. Cohen, Y., Assaf, Y.: High b-value q-space analyzed diffusion-weighted mrs and mri in neuronal tissues - a technical review. NMR Biomed. 15, (2002) 2. Tuch, D.S.: Diffusion MRI of Complex Tissue Structure. PhD thesis, Massachusetts Institute of Technology (2002) 3. Callaghan, P.T.: Principles of nuclear magnetic resonance microscopy. Oxford University Press, Oxford (1991) 4. Basser, P., Mattiello, J., LeBihan, D.: Estimation of the effective self-diffusion tensor from the nmr spin echo. J. Magn. Reson. B 103(3), (1994) 5. Wedeen, V.J., Hagmann, P., Tseng, W.Y.I., Reese, T.G., Weisskoff, R.M.: Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. Magn. Reson. Med. 54(6), (2005) 6. Liu, C., Bammer, R., Acar, B., Moseley, M.E.: Characterizing non-gaussian diffusion by using generalized diffusion tensors. Magn. Reson. Med. 51, (2004) 7. Assaf, Y., Freidlin, R.Z., Rohde, G.K., Basser, P.J.: New modeling and experimental framework to characterize hindered and restrcited water diffusion in brain white matter. Magn. Reson. Med. 52, (2004) 8. Jensen, J.H., Helpern, J.A., Ramani, A., Lu, H., Kaczynski, K.: Diffusional kurtosis imaging: The quantification of non- gaussian water diffusion by means of magnetic resonance imaging. Magn. Reson. Med. 53, (2005) 9. Özarslan, E., Shepherd, T., Vemuri, B., Blackband, S., Mareci, T.: Resolution of complex tissue microarchitecture using the diffusion orientation transform (dot). NeuroImage 31(3), (2006) 10. Pickalov, V., Basser, P.: 3-D tomographic reconstruction of the average propagator from MRI data. In: IEEE ISBI, pp (2006) 11. Wu, Y.C., Alexander, A.L.: Hybrid diffusion imaging. NeuroImage 36, (2007) 12. Barmpoutis, A., Vemuri, B.C., Forder, J.R.: Fast displacement probability profile approximation from hardi using 4th-order tensors. In: IEEE ISBI, pp (2008) 13. Poupon, C., Rieul, B., Kezele, I., Perrin, M., Poupon, F., cois Mangin, J.F.: New diffusion phantoms dedicated to the study and validation of hardi models. Magn. Reson. Med. 60, (2008) 14. Hess, C., Mukherjee, P., Han, E., Xu, D., Vigneron, D.: Q-ball reconstruction of multimodal fiber orientations using the spherical harmonic basis. Magn. Reson. Med. 56, (2006) 15. Tournier, J.D., Calamante, F., Connelly, A.: Robust determination of the fibre orientation distribution in diffusion mri: Non-negativity constrained super-resolved spherical deconvolution. NeuroImage 35(4), (2007) 16. Assemlal, H.E., Tschumperlé, D., Brun, L.: Efficient computation of pdf-based characteristics from diffusion mr signal. In: Metaxas, D., Axel, L., Fichtinger, G., Székely, G. (eds.) MICCAI 2008, Part II. LNCS, vol. 5242, pp Springer, Heidelberg (2008) 17. Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Regularized, fast, and robust analytical q-ball imaging. Magn. Reson. Med. 58(3), (2007) 18. Özarslan, E., Koay, C.G., Basser, P.J.: Simple harmonic oscillator based estimation and reconstruction for one-dimensional q-space mr. In: ISMRM, p. 35 (2008)

12 12 M. Descoteaux et al. A Proof of the Analytical Diffusion Propagator Solution We want to prove that, under the Laplace equation assumption (Eq. 2), P (R 0 r) is given by Eq. 3. We need the following four identities: (1) The pointwise convergent expansion of the plane wave in spherical coordinates is given by e ±2πiq R =4π (±i) l(j) j l(j) (2πqR 0 )Y j (u)y j (r), j=0 where j n is the spherical Bessel function (also used in the DOT derivation [9]). (2) We also need the following definite integral involving the Bessel function x m J 0 n (x)dx =2 m (Γ (n/2+m/2+1/2))/(γ (n/2 m/2+1/2)). Note however that this integral blows up when the denominator is undefined, which occurs in the specific case m = n + 1 because Γ (0) is undefined. (3) Hence, we need to solve property (2) in the special case that we have 0 x n J n 1 (x)dx. To do so, we need the following recurrence relation J n 1 (x) = 2n/xJ n (x) J n+1 (x). It is straightforward to show that the integral is zero. (4) Finally, Γ (n +1/2) = π(2n 1)!!/2 n for n =1, 2, 3,..., where(2n 1)!! = (2n 1). For n =0,Γ (1/2) = π. Now, we first write the 3D Fourier integral (Eq. 1) in spherical coordinates P (R 0 r)= E(q)e 2πiR0q r dq = q 2 E(qu)e 2πiR0qu r dqdu. (5) q=0 u =1 Using the pointwise convergent expansion of property (1) above, it implies that P (R 0 r)=4π ( i) l(j) Y j (r) j=0 q=0 u =1 q 2 E(qu)j l(j) (2πqR 0 )Y j (u)dqdu (6) Next, we replace E(q) by the Laplacian equation given in Eq. 2 to obtain P (R 0 r) 4π ( i) l(j) Y j (r) j=0 q 2 k=0 [ ] ck q l k+1 + d kq l k Y k (u)y j (u)j l(j) (2πqR 0 )dqdu Because the SH basis is orthonormal, Y k (u)y j (u)du = δ kj.also,sincel(j) is even in our basis, ( i) l(j) =( 1) l(j)/2. Finally, we use j n (x) = π/2xj n+1/2 (x). P (R 0 r)= 2π R0 ( 1) l(j) j=0 2 Yj (r) ( cj 0 q l(j) d j q l(j)+ 3 2 ) J l(j)+ 1 2 (2πqR 0)dq } {{ } I l(j) (7) I l(j) = c j q 1/2 l(j) J l(j)+1/2 (2πqR 0 )dq + d j q l(j)+3/2 J l(j)+1/2 (2πqR 0 )dq 0 } {{ } 0 } {{ } P 1 P 2

13 Diffusion Propagator Imaging: Using Laplace s Equation 13 From property (3), we see that P 2 is zero. For P 1, we do the change of variable x =2πqR 0,dx=2πR 0 dq, and use properties (2) and (4) to obtain P 1=c j (2πR 0 ) l(j) l(j) Γ (l(j) l(j) ) Γ ( l(j) l(j) ) = c j 2 (2πR 0 ) l(j) 3 2 π (2l(j) 1)!! We insert P 1andI l(j) back into Eq. 7 and we obtain the desired result of Eq. 3 with some algebra. This completes the proof.