Magnetic Resonance in Medicine 51:924 937 (2004) Characterizing Non-Gaussian Diffusion by Using Generalized Diffusion Tensors Chunlei Liu, 1,2 Roland Bammer, 1 Burak Acar, 3 and Michael E. Moseley 1 * Diffusion tensor imaging (DTI) is known to have a limited capability of resolving multiple fiber orientations within one voxel. This is mainly because the probability density function (PDF) for random spin displacement is non-gaussian in the confining environment of biological tissues and, thus, the modeling of self-diffusion by a second-order tensor breaks down. The statistical property of a non-gaussian diffusion process is characterized via the higher-order tensor (HOT) coefficients by reconstructing the PDF of the random spin displacement. Those HOT coefficients can be determined by combining a series of complex diffusion-weighted measurements. The signal equation for an MR diffusion experiment was investigated theoretically by generalizing Fick s law to a higher-order partial differential equation (PDE) obtained via Kramers-Moyal expansion. A relationship has been derived between the HOT coefficients of the PDE and the higher-order cumulants of the random spin displacement. Monte-Carlo simulations of diffusion in a restricted environment with different geometrical shapes were performed, and the strengths and weaknesses of both HOT and established diffusion analysis techniques were investigated. The generalized diffusion tensor formalism is capable of accurately resolving the underlying spin displacement for complex geometrical structures, of which neither conventional DTI nor diffusion-weighted imaging at high angular resolution (HARD) is capable. The HOT method helps illuminate some of the restrictions that are characteristic of these other methods. Furthermore, a direct relationship between HOT and q-space is also established. Magn Reson Med 51:924 937, 2004. 2004 Wiley-Liss, Inc. Key words: magnetic resonance imaging; diffusion; diffusion tensor imaging; high angular resolution; fiber; probability density function; cumulants Diffusion anisotropy in biological tissues has been the subject of extensive studies following the discovery of anisotropic water diffusion in the cat central nervous system by Moseley et al. in early 1990 (1,2). The diffusion tensor formalism for biological tissues introduced by Basser et al. (3 5) provides a method for characterizing diffusion anisotropy. However, a second-order diffusion 1 Lucas MRS/I Center, Department of Radiology, Stanford University, Stanford, California. 2 Department of Electrical Engineering, Stanford University, Stanford, California. 3 Electrical-Electronics Engineering Department, Bogazici University, Istanbul, Turkey. Grant sponsor: National Institutes of Health; Grant numbers: NIH- 1R01NS35959; NIH-1R01EB2711; Grant sponsors: Center of Advanced MR Technology of Stanford, the Lucas Foundation. *Correspondence to: Michael E. Moseley, Ph.D., Radiological Science Laboratory at the Richard Lucas MRS/I Center, Department of Radiology, Stanford University, 1201 Welch Road, Stanford, CA 94305-5488. E-mail: moseley@stanford.edu Received 25 November 2002; Revised 5 December 2003; Accepted 8 December 2003. DOI 10.1002/mrm.20071 Published online in Wiley InterScience (www.interscience.wiley.com). 2004 Wiley-Liss, Inc. 924 tensor, as used for diffusion tensor imaging (DTI), is based on the assumption that water molecules obey Gaussian diffusion in biological tissues (e.g., gray matter and white matter). This assumption limits the information that can be gained from DTI. When DTI is used to study the connectivity of white matter tracts, difficulty is often encountered in regions where the fibers cross or merge (6), and with current MR resolution, voxel averaging of different fiber tracts is frequent and unavoidable (7). New methods such as q-space spectral imaging (8 13) and more recently high angular resolution diffusionweighted imaging (HARD) (14 17) have been proposed to overcome this problem. Both methods offer higher resolution of the directionally dependent spin diffusion, and therefore provide better connectivity information for white matter tracts in regions where multiple fiber orientations exist. Theoretically, q-space imaging is a correct way to image the probability density function (PDF) of spin displacement in general, because it does not, in contrast to DTI, assume a particular diffusion model (10,18). Unfortunately, the gradient strength and RF pulse duration requirements for q-space imaging (i.e., both need to approximate a Dirac function) cannot currently be satisfied on a whole-body MR system, or they are precluded by patient safety guidelines that limit the maximum time rate of change of magnetic fields (19). In this study, we investigated the signal equation of an MR diffusion experiment by treating the diffusion process as a first-order Markov process. We derived the relationship between the complex MR signal and the higher-order statistics of the spin displacement based on statistical methods. This relationship can be used to reconstruct the PDF of the spin displacement. To evaluate this method, Monte-Carlo simulations were performed for diffusion in a bounded space with different geometrical shapes. The theoretical framework and the results from the Monte-Carlo simulations lead to a better understanding of the DTI and HARD approaches. Moreover, the generalized diffusion tensor formalism clearly reveals the relationship between diffusion tensor and q-space, and the fact that a secondorder approximation to this method is DTI. Finally, the extension of this work to in vivo applications, a generalized diffusion tensor imaging (GDTI) method (20), will be addressed. THEORY Generalized Diffusion Equation In 3D real space, the particle flux vector (a tensor of order 1) is denoted as: F F k e k F 1 e 1 F 2 e 2 F 3 e 3. [1]
Characterizing Non-Gaussian Diffusion 925 Here e k s are a set of orthogonal unit vectors, and F k (k 1, 2, 3) is the component of F along the direction of e k. In Eq. [1] and throughout this discussion, index notation follows Einstein s summation rule: if an index appears twice in an expression, then summation over that index is implied. The macroscopic theory of Gaussian diffusion is based on the hypothesis of Fick s first law: the flux, F, ofthe diffusing substances is proportional to the concentration gradient, i.e.: F k D kl C x l, k, l 1, 2, 3. [2] Here C is the concentration, x l is the l-th spatial coordinate, and D kl is an element of the self-diffusion tensor. Fick s first law assumes a linear relationship between the flux and the concentration gradient for an anisotropic diffusion. More generally, one can write the relationship as: F k D C 2 3 kl D x klm l 2 C 4 D x l x klmn m 3 C x l x m x n [3] to an arbitrarily high-order partial differentiation of the concentration without opposing this linear constraint. Here the coefficient D i1 i 2 i n is an n-th order tensor, where the superscript n in parentheses indicates the order of the tensor, and the subscript indicates the coordinate. Eq. [3] reduces to Fick s first law in the case that the higher-order coefficients are all zero, i.e.: n D i1i 2i n 0, for n 3. By using the standard mathematical notation: Eq. [3] can be written as: n i1i 2i n. x i1 x i2 x in F k D 2 kl l C D 3 4 klm lm C D klmn lmn C. [4] Based on the law of conservation of mass, one can apply the continuity theorem and obtain the following generalized partial differential equation for diffusion: C t kf k k D 2 kl l C D 3 4 klm lm C D klmn lmn C D 2 kl kl C D 3 4 klm klm C D klmn klmn C, [5] which is a generalization of Fick s second law. In obtaining the last equality, the D i1 i 2 i n tensors are assumed to be spatially invariant for a given voxel of interest. Equation [5] is better known as the Kramers-Moyal expansion of the linearized 3D Boltzmann equation (21,22). Consider a 3D stochastic process x(t), where x(t) is a 3D vector. Let C(x,t) be the PDF of the random variable at time t and let C(x,tx 0,t 0 ) denote the transition probability density function, i.e., the conditional PDF of x(t) at time t given that x(t 0 ) x 0. The linearized Boltzmann equation can be written as: Cx, t lim t t30 1 t Cx, tcx, t tx, t Cx, tcx, t tx, t dx. [6] Expanding the right-hand side of this equation using Kramers-Moyal expansion, one obtains Eq. [5]. It can be shown that the D i1 i 2...i n tensors are related to the moments of x x (23). Let x x x, then: n D i1i 2i n 1n 1 lim n! t30 t x i1 x i2 x in Cx, t tx, t dx. [7] The physical significance of the higher-order tensors hereafter becomes clear: they characterize the higher-order moments of the PDF. For a pure Gaussian diffusion, it will be shown later that D i1 i 2 i n 0 for n 3. For non-gaussian diffusion, however, these higher-order tensors become significant, and it is important to recognize that the higherorder terms shown in Eq. [5] have to be considered in such situations. MR Signal Equation The macroscopic nuclear magnetization density vector M(x,t) at time t is proportional to the spin concentration. Considering the spin diffusion, one can therefore write the Bloch equation in the rotating frame as (24): Mx, t t Mx, t Bx, t M 1e 1 M 2 e 2 M 3 M0 e T 2 T 3 1 D 2 kl kl M D 3 4 klm klm M D klmn klmn M. [8] Here, M k (k 1, 2, 3) is the component of M(x,t) along the direction of e k, is the gyromagnetic ratio, B(x,t) is the magnetic field vector, and T 1 and T 2 are the spin lattice and spin spin relaxation time constants, respectively. Following a derivation similar to Torrey s (24), Eq. [8] can be solved for a spin echo sequence (Fig. 1) and the transverse magnetization evaluates to (see Appendix A): mb m0exp n2 j n n D i1i 2i n n b i1i 2i n, [9] where m(0) and m(b) are the transverse magnetization measured at the echo time (TE) in the absence and pres-
926 Liu et al. q G. [15] Then the signal intensity as a function of q can be written as (18): mq m0 Pr, expjq r dr. [16] FIG. 1. Time diagram for a spin echo sequence with diffusion encoding gradients. ence of diffusion gradients, respectively. j is the square root of 1. The elements of tensor b are defined as: Here P(r,) is the PDF of the spin displacement r during the gradient separation time interval. m(q)/m(0) is the characteristic function, i.e., the Fourier transformation of the PDF of the random displacement vector r. In general, this characteristic function can be expanded as an exponential function of a series of cumulants (25): t 2 n b i1i 2i n k i1 k i2 k in d t1 TE 1 nt2 where k(t) and (t) are defined as: kt 0 i1 i2 in d, [10] t G d, [11] t kt 2kt 1. [12] The G() in Eq. [11] is the time-varying magnetic field gradient vector. If the diffusion gradients are constant over time, it can be shown that (see Appendix A): n b i1i 2i n n G i1 G i2 G in n n 1 n 1. [13] Here, is the separation time of the two diffusion gradients, and is the duration of each gradient lobe (see Fig. 1). Note that for n 2, Eq. [13] becomes the familiar b matrix: b kl 2 G k G l 2 1 3. [14] Because of the j n factor, Eq. [9] reveals an important property of the D i1 i 2 i n tensors: even-order tensors only affect the magnitude of the signal, whereas odd-order tensors only affect the phase of the signal. Relationship Between D Tensors and q-space The physical and statistical significances of the D i1 i 2 i n tensors are evident in the Kramers-Moyal expansion, as seen in Eq. [7]. The moments of the displacement are linear functions of the D i1 i 2 i n tensors. This result can be further explored by considering a spin echo sequence with narrow diffusion gradient pulses, such as used in the q- space approach. Define q as (18): mq m0 expjq i1 1! 1 q i1 j2 Q 2 i1i 2 q i1 q i2 2! jn n Q i1i 2i n q i1 q i2 q in n!. [17] The expansion coefficients Q i1 i 2 i n are called the n-th order cumulants of the random variable r. As shown in Eq. [17], cumulants are the coefficients of the Taylor expansion of the logarithm of the characteristic function. Cumulants are functions of statistical moments, and they are well defined if the corresponding moments are finite. By comparing Eq. [17] to Eq. [9], it is evident that the D i1 i 2 i n tensors are the equivalent cumulants of r. In fact, Q i1 i 2 i n is proportional to D i1 i 2 i n (see Appendix B for a more detailed derivation): n Q i1i 2i n 1 n n n!d i1i 2i n n 1 n 1 1 n n n!d i1i 2i n. [18] In the case that r is a zero-mean Gaussian random variable (i.e., assuming unrestricted Brownian motion), all but the second-order cumulants are zero (25), because the Fourier transformation of a Gaussian function remains Gaussian (26). Under this assumption, Eq. [9] reduces to the familiar ordinary signal equation in DTI (3,4,5): mte m0 expb kld kl. [19] The PDF of the spin displacement r can be reconstructed via the Gram-Charlier series (25,27): Pr N0, Q 2 kl 1 Q 3 klm 3! H r Qklmn 4 klm 4! H klmn r, [20]
Characterizing Non-Gaussian Diffusion 927 FIG. 2. A cross-sectional view in the XY plane of 3D synthetic phantoms. The black dots are uniformly distributed initial positions of the spins. a: Phantom 1, isotropic sphere. b: Phantom 2, single tube. c: Phantom 3, crossing tube. d: Phantom 4, Y-shaped tube. Spins are allowed to diffuse freely inside the phantom and they are reflected elastically at the boundary indicated by the solid lines. The box in the middle drawn with dotted line is the voxel of interest. where N(0, Q (2) kl ) is the normal distribution with zeromean and covariance matrix Q (2) kl, and H... i1 i in 2 (r) isthe n-th order Hermite tensor (25,27). Although cumulants Q i1 i 2 i n are not directly MR-measurable quantities, they can be computed using Eq. [18] from MR-measurable quantities, i.e., the D i1 i 2 i n tensors. As shown in Eq. [4], the D i1 i 2 i n tensors relate the rate of change of the spin density to the n-th order derivatives of the density. In an anisotropic medium, the spin density propagates with different velocities in different directions, resulting in an irregularly shaped wave front. The underlying model for fiber tracking is that the rate of diffusion along the axis of the fiber is higher than that in directions perpendicular to it, and therefore the shape of the reconstructed PDF can be used to infer the structure of the fiber. The advantage of the generalized diffusion tensor formalism is that, in foregoing the assumption of Brownian motion, the random displacement r for any given particle is not constrained to have a Gaussian distribution, and the PDF can assume more complex shapes, i.e., can have nonzero higher-order cumulants. GDTI vs. DTI As mentioned above, DTI is a second-order approximation of the generalized diffusion tensor imaging method (GDTI), i.e., all the D i1 i 2 i n tensors of orders higher than two are zero. Therefore, the estimated PDF is Gaussian and is completely characterized by the covariance matrix associated with the second-order diffusion tensor. This approximation is exact in the case of an unrestricted self-diffusion, for which the PDF of the spin displacement is Gaussian and all cumulants vanish except for the second-order cumulant. However, if the self-diffusion is restricted by boundaries (which is usually the case in biological tissues), the PDF is constrained and might be asymmetric. Such a situation is certainly not well represented by a Gaussian function. Moreover, if the diffusing medium is heterogeneous over the spin diffusing range, then, in general, the resultant PDF also will not be a Gaussian function (18). Even if boundary effects are negligible, the spatial variation in the viscosity of the diffusing medium can cause a non- Gaussian PDF. MATERIALS AND METHODS Simulation In order to assess the usefulness and accuracy of the generalized diffusion tensor theory, Monte-Carlo simulations were conducted using four computer-synthesized phantoms with simple geometric structures. The four constructed phantoms (Fig. 2) are: an isotropic sphere (Phantom 1), a single tube (Phantom 2), two perpendicularly crossing tubes (Phantom 3), and a Y- shaped tube (Phantom 4). All the tubes have a square cross section with a width of 20 m, which is roughly on the order of the diameter of the axon of a neuron (28). The ends of the tubes are closed. The boundaries of the tubes were assumed to be impermeable. The particle motion is represented as a stochastic process, denoted by x(t) as before. The process behaves in the
928 Liu et al. interior of the phantoms like standard Brownian motion (uncorrelated components with zero drift). At the boundary x(t) reflects elastically, which happens instantaneously and without loss of energy. A common way to simulate self-diffusion is to treat the diffusion as a random walk process. The particle motion is represented as a sequence of small random displacement: xt t xt x, [21] where x is the random displacement vector of the particle in the time interval t. We have chosen to generate random walks in the continuous space. The particle may jump in any direction in the 3D space and the length of the jump varies. Specifically, the particle displacement x is distributed according to the Gaussian distribution with the following PDF: Px 1 3 4Dt ex2 /4Dt. [22] The particle trajectory is obtained by using Eq. [21] with the random displacement generated according to Eq. [22]. The trajectory is corrected with consideration of the boundary condition. For perfectly reflecting walls located at x i a and x i a (here i indicates the i-th coordinate), the boundary effect is taken into account by replacing x i with 2a x i when x i a, or by replacing x i with 2a x i when x i a. If there are multiple boundaries adjacent, the nearest one is chosen as the reflecting boundary. For a sufficiently small t, the probability that the resultant particle position is outside the phantom goes to zero. For the sequence depicted in Fig. 1, the simulated spin phase at TE for a single spin n is given by: t 2 n G tt 2 t 1 x n t x n tt 1 tt, [23] where x n (t) is the position vector of spin n at time t. In this simulation, we consider only the central voxel of size 80 80 80 m 3 (Fig. 2). This better simulates the real physical situation where spins are moving in and out of each voxel. This also excludes the effect from the edge of the phantoms, since the edges are distant from the central voxel. The complex signal of the spins located in this voxel at TE is given by: S 1 N 0 e jn, [24] ni where I is the set of all trajectories ending within the voxel of interest at TE: I {n; x n (TE) voxel}, and N 0 is the total number of spins ending within the voxel in the absence of diffusion gradients. Note that the signal attenuation due to T 1 and T 2 decay has been neglected. For each phantom, the signal was obtained by simulating 2.5 10 5 spin trajectories with uniformly distributed starting positions (Fig. 2). As a result, N 0 was around 1.0 10 5. The simulation time step was t 0.2 ms. The diffusion encoding parameters for all simulations were chosen to be comparable to those used for in vivo experiments. The duration,, of the diffusion sensitizing gradients was 30.0 ms, their separation time,, was 40.0 ms, and the TE was 80.0 ms. With these values, the effective diffusion time was 30.0 ms according to Eq. [14]. The maximum diffusion gradient strength was 40 mt/m, which is roughly what is currently offered by most whole-body systems. For an unrestricted diffusion, the diffusion coefficient D was set to be 2.02 x 10-3 mm 2 /s, which is equal to the diffusion coefficient of water at about 20 C (29). In a separate set of simulations, Gaussian noise with different variance is added to the signal in order to assess the effect of noise. The simulations are conducted with the same set of parameters as described in the previous paragraphs. Tensor Estimation To reconstruct the PDF of the spin displacement, the D i1 i 2 i n tensors are estimated by combining a series experiments with different diffusion gradient vectors and solving the set of linear equations defined by Eq. [9]. This becomes more evident by rewriting Eq. [9] in the following form: lns/s 0 b i1i 2 2 2 D i1i 2 1 n 2n b i1i 2i 2n 5 b i1i 2i 3i 4i 5 5 D i1i 2i 3i 4i 5 4 4 b i1i 2i 3i 4 D i1i 2i 3i 4 2n D i1i 2i 2n 1 n 2n1 b i1i 2i 2n1 3 3 jb i1i 2i 3 D i1i 2i 3 2n1 D i1i 2i 2n1. [25] The real part of the logarithmic signal is solely determined by the even-order tensors, while the imaginary part is completely governed by odd-order tensors. The b i1 i 2 i n tensors are functions of the direction, the magnitude, and the timing of the diffusion-encoding gradients. With each set of the above parameters, there is a unique corresponding linear equation in the form of Eq. [25]. In principle, the D i1 i 2 i n tensors can then bedetermined to an arbitrary order by truncating the series summation as desired (see Eq. [9]). The resulting set of linear equations can then be solved by using matrix computation (30,31). Suppose m sets of measurements are acquired using different directions and strengths of the diffusion gradient and S k (k 1,2 m) is the complex signal of the k-th measurement. We also introduce the following matrix notations:
Characterizing Non-Gaussian Diffusion 929 Y r reallns 1 /S 0 reallns 2 /S 0 reallns m /S 0 T, 2 X r D 11 2 D 12 2 D 13 2 D 33 4 D 1111 4 D 1112 4 D 1113 4 D 3333 T, b11 B r 2,1 2,2 b 11 2,1 2,1 2b 12 b 33 2,2 2b 12 2,2 b 33 4,1 b 1111 4,2 b 1111 4,1 4,1 4b 1112 b 3333 4,2 4,2 4b 1112 b 3333, b 11 2,m 2b 12 b 33 b 1111 4,m 4b 1112 4,m b 3333 Y i imaglns 1 /S 0 imaglns 2 /S 0 imaglns m /S 0 T, 3 X i D 111 3 D 112 3 D 113 3 D 333 5 D 11111 T, b111 B i 3,1 3,2 b 111 3,1 3,1 3b 112 b 333 5,1 b 11111 3,2 3b 112 3,2 b 333 5,2 b 11111 b 111 3,m 3b 112 b 333 5,m b 11111 5,1 5,1 5b 11112 b 33333 5,2 5,2 5b 11112 b 33333 5,m 5,m 5b 11112 b 33333. Here, real indicates the real part and imag represents the imaginary part. The symbol T represents the matrix transpose operation, and b,m i1 i 2 i n denotes the element of the b tensor from the m-th measurement. The number in front of the b tensor element is the number of permutations of the subscript, which accounts for the symmetry of b and D tensor as discussed in the following paragraph. According to Eq. [25], one finally obtains Y r,i B r,i X r,i for both the real and imaginary parts. The D i1 i 2 i n tensors possess a very useful symmetry property that greatly reduces the complexity of the estimation problem. By its definition in Eq. [10], b i1 i 2 i n is clearly a symmetric tensor, which means that the order of the subscripts will not affect the values of the tensor elements. From the following relationships: 2n D i1i 2i 2n 2n1 D i1i 2i 2n1 1 lns/s 0 n 2n, [26] b i1i 2i 2n j1 lns/s 0 n 2n1, [27] b i1i 2i 2n1 it is evident that D i1 i 2 i n is a symmetric tensor as well. In general, a symmetric tensor of order n has (n1)(n2)/2 independent elements out of the total 3 n elements (27). By utilizing this symmetry property, the number of tensor elements to be estimated is greatly reduced. For example, to determine a tensor of order 4, only 15 of 81 elements need to be computed. Without a priori knowledge of the shape of the PDF, the best directional sampling scheme is a uniform sampling on the surface of a sphere (15,16). In this simulation, 200 sampling directions were evenly distributed on the surface of a sphere. At each orientation the gradient strength was varied from 0 to 40 mt/m in 10 uniform steps and a total of 2,000 linear equations were obtained. The D i1 i 2 i n tensors up to an order 4 were estimated using the singular valve decomposition (SVD) method (30,31) that decomposes a matrix B into B UV T. Here, is a diagonal matrix whose entries are the singular values of B; U is a matrix whose column vectors are the left singular vectors of B; and V is a matrix whose column vectors are the right singular vectors of B. To avoid noise amplification and oscillation of the solution vector due to the ill-conditioning of the coefficient matrix, all singular values less than 10-8 of the largest singular value were set to zero. The solutions for X r and X i are given by: X r,i V r,i 1 r,i U T r,i Y r,i [28] Besides the generalized diffusion tensor representation, the simulation data served to calculate the conventional second-order diffusion tensor used in DTI and to compute relevant data to perform HARD. Visualization Using the estimated D i1 i 2 i n tensors, the PDF of spin displacement was reconstructed via Eq. [20]. For visualization purposes, two rendering techniques were used: an isosurface plot of the PDF, and an isosurface plot of the deviation of the PDF from Gaussian PDF. The latter is called the skewness and kurtosis map (skewness map for simplicity), which is determined via the following equation: Pr Pr N0, Q 2 i1i 2 N0, Q 2 i1i 2 Q 4 i 1i 2i 3i 4 4! Q 3 i 1i 2i 3 3! H i1i 2i 3 r H i1i 2i 3i 4 r. [29]
930 Liu et al. Phantoms 2, 3, and 4, an obvious deviation from a simple exponential decay was observed for the magnitude of the signal because of the restricted diffusion (Fig. 3a). This is very similar to what has been observed in human white matter (32). More importantly, the phase of the signal for Phantom 4 is significant (Fig. 3b), indicating the existence of odd-order tensors. FIG. 3. Simulated signal attenuation as a function of the b-value with the diffusion gradient applied in the direction of ( 0.228, 0.171, 0.281). The b-value is calculated according to Eq. [14]. a: The magnitude of the signal. b: Phase of the signal. As explained in the Theory section, for a given observation interval in which enough spins encounter the confining environment, the restricting geometric structure of the tube used in these simulations determines the resulting shape of the PDF. Along the direction of the tube, the spin diffusion is not restricted; therefore, the PDF along this direction has relatively larger values compared to the corresponding Gaussian diffusion. This is demonstrated by the positive values on the skewness map. Along the directions perpendicular to the tube, the values of the PDF are reduced and smaller than Gaussian diffusion, and negative values would appear in the skewness map. For easy recognition of the positive and negative values, color-coding was introduced: green represents positive values and red represents negative values. RESULTS In Fig. 3, a representative signal attenuation curve in the absence of noise is plotted as a function of gradient strength along the direction of ( 0.228, 0.171, 0.281) for each phantom. Since the diffusion is not restricted in Phantom 1, the signal decay for the magnitude is purely exponential (Fig. 3a), whereas the phase of the signal is negligible, but nonzero. There are two reasons for this observation: the limited numerical precision of the calculation and the fact that a pseudorandom number generator is used with a finite number of samples. However, for Phantom 1 For each phantom, the estimated elements of the D i1 i 2 i n tensors are listed in Table 1. For Phantom 1, only the second-order tensor is significant. The estimated value of the diagonal elements of the second-order tensor is also close to the predetermined diffusion coefficient in the simulation. For the other three phantoms, because of their structural characteristics, diffusion anisotropy is clearly evident in the second-order tensors, with the value of tensor elements similar to what is observed in human brain white matter (3,4,33). The higher-order tensor elements are several orders larger compared to Phantom 1. Although these values are still numerically much smaller than the second-order tensor elements, they are significant in determining the shape of the PDF. The resultant 3D PDF isosurface plots and the skewness maps obtained from the four phantoms in the absence of noise are plotted in Fig. 4. The diffusion ellipsoid determined by DTI and the angular distribution of apparent diffusion coefficient (ADC) determined by HARD are also shown for comparison. To better represent the 3D distribution, Fig. 5 contains several different projections of the 3D results. For the isotropic Phantom 1, all three methods (i.e., DTI, HARD, and GDTI) produced almost identical results, which is to be expected since the diffusion is orientation independent. This is illustrated in GDTI by the spherical PDF isosurface, which means that the spin had the same probability density to diffuse in every direction. Phantom 2 Both the skewness map and PDF isosurface plot have the shape of rectangular tubes and, thus, convey the true shape of the phantom. The accuracy of the GDTI method is well demonstrated by the square cross-section of PDF isosurface. While DTI and HARD are both capable of revealing faster diffusion along the tube, some subtleties have been lost because only the second-order diffusion tensor is utilized (as shown in Fig. 4). Phantom 3 Both the skewness map and PDF isosurface plot are able to determine the shape of the phantom. Although the rectangular shape of the tube and the symmetry of the structure are not reproduced perfectly, the two maps successfully represent the primary geometrical structure of the phantom. The imperfection is the result of the intrinsic randomness of the diffusion process, the insufficient diffusion exposure time, and the increasing complexity of the structure. Note that averaging over a finite number of spins will not produce exactly the true expectation value of the signal. Furthermore, for a complex structure such as the one
Characterizing Non-Gaussian Diffusion 931 Table 1 Estimated Values for Tensor Elements D i1 i 2 i n in the Absence of Noise Index of D i1 i 2 i n D i1 i 2 i n Isotropic D i1 i 2 i n Single tube D i1 i 2 i n Crossing tube D i1 i 2 i n Y-tube (1, 1) 2.02 10 3 2.00 10 3 1.41 10 3 1.36 10 3 (1, 2) 3.24 10 7 6.88 10 7 1.18 10 6 4.63 10 7 (1, 3) 2.00 10 7 2.06 10 6 1.37 10 6 2.74 10 6 (2, 2) 2.02 10 3 0.231 10 3 1.41 10 3 1.28 10 3 (2, 3) 3.09 10 7 1.43 10 6 3.97 10 7 1.98 10 7 (3, 3) 2.02 10 3 0.231 10 3 0.230 10 3 0.232 10 3 (1, 1, 1) 10 9 6.03 10 9 10 9 2.36 10 8 (1, 1, 2) 10 9 3.21 10 9 3.23 10 9 4.79 10 7 (1, 1, 3) 10 9 1.57 10 9 5.57 10 9 10 9 (1, 2, 2) 10 9 2.26 10 9 1.17 10 8 7.73 10 9 (1, 2, 3) 10 9 2.53 10 9 4.39 10 9 5.75 10 9 (1, 3, 3) 10 9 10 9 2.36 10 9 2.82 10 9 (2, 2, 2) 10 9 10 9 7.04 10 9 2.87 10 7 (2, 2, 3) 10 9 10 9 6.78 10 9 10 9 (2, 3, 3) 10 9 10 9 4.98 10 9 3.61 10 8 (3, 3, 3) 10 9 2.57 10 9 2.18 10 9 4.88 10 9 (1, 1, 1, 1) 10 9 7.25 10 9 7.41 10 9 4.19 10 9 (1, 1, 1, 2) 10 9 10 9 10 9 10 9 (1, 1, 1, 3) 10 9 10 9 10 9 10 9 (1, 1, 2, 2) 10 9 10 9 10 9 1.68 10 9 (1, 1, 2, 3) 10 9 10 9 10 9 10 9 (1, 1, 3, 3) 10 9 10 9 10 9 10 9 (1, 2, 2, 2) 10 9 10 9 10 9 10 9 (1, 2, 2, 3) 10 9 10 9 10 9 10 9 (1, 2, 3, 3) 10 9 10 9 10 9 10 9 (1, 3, 3, 3) 10 9 10 9 10 9 10 9 (2, 2, 2, 2) 10 9 10 9 7.28 10 9 3.80 10 9 (2, 2, 2, 3) 10 9 10 9 10 9 10 9 (2, 2, 3, 3) 10 9 10 9 10 9 10 9 (2, 3, 3, 3) 10 9 10 9 10 9 10 9 (3, 3, 3, 3) 10 9 1.06 10 9 10 9 10 9 Here, (i 1, i 2, i 3, i 4 ) represent all possible permutations of the indexes i 1, i 2, i 3, and i 4. For 2 nd order tensor, the unit is mm 2 /s; for 3 rd order tensor, the unit is mm 3 /s; and for 4 th order tensor, the unit is mm 4 /s. The accuracy of the estimation is about 10 11 with a respective unit for each tensor element. represented by Phantom 3, a fourth-order approximation of the GDTI method (as used in this study) might be insufficient and may be improved by the inclusion of higherorder terms. It is important to note that DTI and HARD do not have the fidelity to correctly represent the true diffusion property possessed by the spins inside this phantom. Although the HARD measurement demonstrates a crossing structure, the azimuthal ADC maxima deviate by 45 from the direction of the tube (17,34). This result can be verified theoretically via Eq. [17]. For this specific case, it can be shown that the signal intensity decays fastest when the diffusion gradient is applied along the 45 line. Therefore, the direct one-to-one correspondence between the geometry of the fiber structure and the geometry of the angular distribution of the ADC is no longer valid. A simple ADC plot is not sufficient for determining the structure of the fibers without further analysis (16). This point is even clearer for an asymmetrical fiber system such as the one modeled by Phantom 4. Phantom 4 The Y-shaped tube is asymmetric about the origin. The GDTI method is capable of reproducing this structure as seen in both the skewness map and the PDF isosurface plot. DTI, as a second-order approximation, fails to identify the underlying structures. Interestingly, the HARD plot remains symmetric about the origin even though the tube does not have this symmetry property. Again, the six branches shown in the HARD plot do not correspond to the principle orientations of the tube. Figure 6 illustrates the effect of Gaussian noise on the PDF skewness map and PDF isosurface map for Phantom 4. Figure 6a shows the result in the absence of noise; Fig. 6b,c is the result obtained in the noisy situation with different SNR (ratio of signal amplitude to noise standard deviation). In Fig. 6b, SNR 31.5 when b 313 s/mm 2 ; and SNR 4.6 when b 3126 s/mm 2 which is the largest b-value used. In Fig. 6c, SNR 15.9 when b 313 s/mm 2 ; and SNR 2.4 when b 3126 s/mm 2. For each b-value, the SNR was computed as the average SNR over all diffusion directions. DISCUSSION Diffusion tensor imaging (DTI) is known to have a limited capability of resolving multiple fiber tracts within a voxel for spatial resolutions currently possible on whole-body MRI systems. Recently, it has been demonstrated that in cases where fibers intersect, a second-order diffusion ten-
932 Liu et al. FIG. 4. Comparison of GDTI, DTI, and HARD. For Phantom 1, the higher-order tensor elements are several orders smaller than those of the other three phantoms. The skewness is negligible and not plotted here. In the PDF and diffusion ellipsoid plots, the blue lines are the x and y axis; in the HARD image, red line is the y axis and green is the x axis. In the PDF skewness maps red represents negative values and is displayed either as opaque or semiopaque isosurface. sor is not the correct model to represent the diffusion process (14,17). New techniques, such as high angular resolution measurements (14 16) and multiple tensors (17) have been suggested to improve MRIs ability to correctly reveal the underlying fiber structure. As a potential adjunct to these approaches, the generalized diffusion tensor formalism might be able to considerably increase the angular resolution of the PDF of the spin s random displacement and, thus, could potentially improve the fiber tractography (35,36) results. In this study, we have shown that the MR signal in a diffusion experiment is determined by a set of higherorder diffusion tensors D i1 i 2 i n (Eq. [9]). Assuming that the underlying diffusion process is a Markov process, the D i1 i 2 i n tensors can be related to the higher-order statistics of the spin s random displacement. The higher-order moments of the random displacement result from various physical conditions, such as nonuniform viscosity and geometrical restrictions. Specifically, the higher-order cumulants of the random spin displacements are linearly related to the D i1 i 2 i n tensors. It has also been shown that the generalized diffusion tensor formalism is related to the signal equation resulting from the q-space formalism. The generalized diffusion tensor approach reduces to standard DTI and HARD methods by appropriate nulling of higher-order tensor elements. For n 2, the exponential decay function is obtained, which is the classic result of exponential signal decay caused by Brownian motion, and Eq. [18] reduces to the following familiar form: Q 2 ij 2D 2 ij. [30] For an isotropic diffusion, this leads to Einstein s equation for the mean square displacements of Brownian particles (37): r i 2 2D. [31] Here, r i 2 is the mean square displacement along the direction of e i. The importance of including higher-order tensors is evident from the simulation results of Phantom 4. In this case, the PDF of the random spin displacement is not an even function. Generally, every function can be written as the combination of its even part and odd part (26). The
Characterizing Non-Gaussian Diffusion 933 FIG. 5. Different perspective views of the 3D skewness maps. Silver indicates positive skewness and dark represents negative skewness. Left images are for Phantom 3; right images are for Phantom 4; (a) front view; (b) top view; (c) side view. even part has a real and even Fourier transformation, whereas the odd part has an imaginary and odd Fourier transformation (26), which makes the signal complex. By computing the ADC along a certain direction, a secondorder model is also unintentionally assumed. Methods based on simple ADC evaluation (including HARD) will have the same limitations caused by order reduction as DTI and will suffer from unrecoverable information loss. More specifically, the computation of the directionally dependent ADC relies on the assumption that the underlying PDF is Gaussian and is symmetric about its mean value (i.e., zero for pure self-diffusion). Due to this symmetry, reversing the direction of diffusion gradient will not affect the signal intensity and, therefore, the angular distribution of the ADC will be symmetrical about the origin, regardless of whether the underlying structure is asymmetric (see Phantom 4). Further analysis of the HARD data, for example, by spherical harmonic decomposition, can help obtain more structural information (16); however, the ADC measured in this way will be biased when the signal is not an exponential function of the b-value. The random movement of water molecules is completely represented in the infinite time limit by its PDF. Therefore, by reconstructing the true PDF of spin displacement the generalized diffusion tensor approach can potentially improve the accuracy of fiber tractography results. It can be shown that the PDF reconstructed from the higherorder tensors is equivalent to the PDF of the random displacement averaged over all starting positions as characterized by the q-space method. Neither higher-order tensors nor the q-space approach can avoid problems resulting from voxel-averaging, but they do have the capability to resolve multiple orientations of the PDF and, therefore, can improve the angular resolution of the fiber structure. It is important to note that even-order tensors only affect the magnitude of the signal and odd-order tensors only affect the phase of the signal. Nevertheless, the accuracy of the estimation for tensors of even and odd orders is theoretically the same because the full complex signal can be utilized. Since magnitude and phase information are both present in the real and imaginary parts of the signal, the error of estimation is spread and phase wrapping is not a problem because real and imaginary parts are treated separately (see Eq. [28]). One motivation for this study was to examine the behavior of the MR signal in a diffusion experiment on a theoretical basis and reveal the limitations of current data interpretation methods, such as DTI and HARD. The simulations served to demonstrate that the diffusion MR signal is correctly represented by Eq. [9], and the PDF of the spin displacement can be obtained by utilizing its higherorder statistics. However, currently there are still experimental difficulties that need to be overcome in order to implement the GDTI method in vivo. The main difficulty is phase perturbations due to bulk physiologic motion. The previous paragraph mentioned that the higher-order tensor information is buried in the complex MR signal; a FIG. 6. The effect of Gaussian noise on the reconstructed PDF for Phantom 4. The left column is the PDF skewness map and the right column is the PDF isosurface. a: Without noise. b: SNR 31.5 with b 313 s/mm 2, and SNR 4.6 with b 3126 s/mm 2. c: SNR 15.9 with b 313 s/mm 2, and SNR 2.4 with b 3126 s/mm 2.
934 Liu et al. full characterization of the underlying diffusion process requires total information of the complex signal. However, the phase of in vivo signals can be easily corrupted by physiologic and involuntary patient motion. In such situations, one can elect to ignore the phase of the signal. By doing so, useful information regarding the diffusion process is discarded and only partial characterization of the diffusion process can be achieved. With the magnitude of the signal, only tensors of even order can be estimated and the PDF is restricted to be symmetric. So far there is no experimental demonstration of the odd-order tensors. The observation of odd-order terms in the Y- phantom simulation is of theoretical interest only, and is unlikely to be seen in vivo. It is important to realize that similar limitations also exist for q-space spectrum imaging, in which phase information currently is usually discarded, even though one cannot assume that the q-space signal is generally real. Another potential problem is the signal noise. When the deviation from a Gaussian distribution is not large, the magnitude of the higher-order statistics of the PDF will be small. As a result, the noise can affect the accuracy of the estimation of the higher-order tensors and result in a distorted PDF reconstruction. Nevertheless, in our phantom simulations it was possible to reconstruct the PDF reasonably well in the noisy situation, as illustrated by Fig. 6. It is also observed that the PDF isosurface is less sensitive to the noise than the PDF skewness map. CONCLUSIONS A new formalism has been developed to characterize the diffusion processes. In this formalism, the statistical property of a diffusion process is represented by an infinite series of higher-order diffusion tensors. These are symmetrical tensors and can be measured by MR. It was shown that the cumulants of the random spin displacement can be calculated from the generalized diffusion tensors for an MR spin echo sequence. The generalized diffusion tensors can be theoretically determined to an arbitrarily high order; the actual order, however, is determined by the maximum number of measurements that can be acquired with different b tensors within a reasonable scan time. The simulations that were performed using four computer-synthesized phantoms demonstrated the feasibility and accuracy of this method. Most importantly, while the generalized tensor approach successfully revealed the geometric structure for all four phantoms, DTI and HARD had limited success in determining the complex geometry. Conventional DTI is a second-order approximation of the generalized diffusion tensor formalism. In summary, the generalized diffusion tensor method is capable of mapping the PDF to a high accuracy and provides a novel perspective for interpreting MR diffusion experiments and for understanding the limitations of conventional methods for processing diffusion data. Further work should be conducted to make both q-space and GDTI sufficiently robust to reveal the full PDF for in vivo applications. ACKNOWLEDGMENTS The authors thank Professor Persi Diaconis of the Statistics Department of Stanford University for thoughtful discussions on cumulants, and Karen Y. Chen, MS, Sharon Mason, MS, and David Clayton, PhD, for proofreading the manuscript. APPENDIX A Solution to the Generalized Diffusion Equation for a Spin Echo Sequence In a rotating reference frame, the transverse magnetization m(t) is governed by the following equation: m t jg xm m D 2 3 T i1i 2 i1i 2 m D i1i 2i 3 m 2 By writing m(t) in the form of: m Mtexp j 0 Eq. [A.1] reduces to: dmt dt where: 4 D i1i 2i 3i 4 i1i 2i 3i 4 m. t G x d expt/t 2. j 2 2 k i1 k i2 D i1i 2 j 3 3 k i1 k i2 k i3 D i1i 2i 3 j 4 4 k i1 k i2 k i3 k i4 D i1i 2i 3i 4 Mt, k i t 0 t G i d. [A.1] [A.2] [A.3] [A.4] When there is no discontinuity in the function of G(t), a direct integration of both sides yields: t Mt Mt 1 exp j2 2 D i1i 2 k i1 k i2 d t1 j 4 4 D i1i 2i 3i 4 j 3 3 D i1i 2i 3 t t1 t t1 To simplify the notation, let: k i1 k i2 k i3 d k i1 k i2 k i3 k i4 d. n k i1i 2i n t k i1 tk i2 t k in t. [A.5] [A.6] Then following Einstein s summation rule, Eq. [A.5] can be written as: Mt Mt 1 exp jn n D i1i 2i n t t1 n k i1i 2i n d. [A.7]
Characterizing Non-Gaussian Diffusion 935 Here, n ranges from 2 to infinity. Because the diffusion gradient is not always present, to solve the above equation for a spin echo sequence (Fig. 1), one has to solve the equation piece by piece and consider the continuity boundary condition that needs to be satisfied. By applying a technique similar to Torrey s (24), and considering the phase inversion (caused by 180 pulse) at TE/2, one obtains the magnetization at time TE: Hence, the b tensor is given by: n b i1i 2i n n G i1 G i2 G in n n 1 n 1. [A.16] When n 2, it reduces to the familiar Stejskal-Tanner equation (38): TE mx, TE Mt 1 exp 2 T expjx t 2 t 2 exp jn D nt1 k n d 1 nt2 d TE n. Note in this case: b i1i 2 2 G i1 G i2 2 1 3. n t 2 0. [A.17] [A.18] Define an n-th order tensor b : t 2 n b i1i 2i n t1 n k i1i 2i n TE d 1 nt2 [A.8] n i1i 2i n d, [A.9] where () isann-th order tensor. Its elements are defined as: n i1i 2i n i1 i2 in, i t k i t 2k i t 1. [A.10] [A.11] If G(t) G(Gis a constant gradient) over the time period (t 1, t 1 ) and (t 2, t 2 ), where: t 1 t 1 t 2 t 2, t 2 t 1, [A.12] [A.13] then the integration in Eq. [A.9] can be carried out explicitly as follows: t 2 t1 n k i1i 2i n t 2 1 nt2 t dt t1 t 1 t 1 t1 n G i1 G i1 G in t t 1 n dt n G i1 G i1 G in n dt n G i1 G i1 G in n n n 1 n1, n i1i 2i n t dt [A14] By applying Eq. [A.9] and Eq. [A.18] to Eq. [A.8], one finally obtains: mb m0expj n n D i1i 2i n where: n b i1i 2i n m0 Mt 1 exp TE T 2 M0exp TE T 2,, n 2, 3,, [A.19] if t 1 0. [A.20] Here, m(0) is the magnetization measured in the absence of a diffusion gradient. APPENDIX B Reconstruction of the Probability Density Function (PDF) of the Nuclear Spin Displacement The PDF of spin displacement can be reconstructed by its higher-order cumulants via the Gram-Charlier series. Generally, the PDF may not be a Gaussian. To reveal the relationship between the experimentally measurable quantities (i.e., generalized diffusion tensors) and the higher-order cumulants, it requires a statistical analysis of the diffusion imaging experiment. If x(t) is the position of a given spin at time t, then the phase shift introduced by the diffusion gradient over a short time interval dt is: d Gt xt dt. [B.1] The accumulated phase shift at time TE will be (for a spin echo sequence shown in Fig. 1): TE 0 TE/2 TE Gt xt dt TE/2 Gt xt dt. t 1 1 nt1 n G i1 G i2 G in t t 1 n dt n G i1 G i2 G in 1 n 1 n1. [A15] Consequently, the magnetization at time TE is: mte m0expjte. [B.2] [B.3]
936 Liu et al. In general, neither m(te) nor (TE) is a Gaussian random variable; its ensemble average is: Q 1 G i1 Q 1 r,i1, mte m0expjte. [B.4] Q 2 2 2 2 G i1 G i2 Q r,i1i 2, The right-hand side can be expanded using the cumulants: mte m0exp jq 1 1! j2q 2 2! j3q 3 3! j4 4 Q, 4! [B.5] where Q is the n-th order cumulant of the random variable (TE). Cumulants are the coefficients of the Taylor expansion of the generating function defined as follows: Q n n n n G i1 G i2 G in Q r,i1i 2i n. [B.10] Combine Eq. [B.10], [B.8], and [A.16]: 1 n n G i1 G i2 G in n n Q r,i1i 2i n n! n G i1 G i2 G in n n 1 n 1 n D i1i 2i n, [B.11] logexpjte jq 1 1! j2q 2 2! j3q 3, 3! [B.6] where is the coordinate in the reciprocal space. Per definition, the magnetization given by Eq. [A.19] and Eq. [B.5] should equal each other, i.e.: m0expj n n D i1i 2i n n b i1i 2i n m0exp jn n Q n!. [B.7] and one obtains: n Q r,i1i 2i n 1 n n n!d i1i 2i n n 1 n 1 For n 2, Eq. [B.12] evaluates to: 2 Q r,i1i 2 2D 2 i1i 2. 1 n n n!d i1i 2i n, [B.12] [B.13] A comparison of the coefficients yields: n D i1i 2i n D 1 i1 b 1 i1 Q 1 1!, D 2 i1i 2 b 2 i1i 2 Q 2 2!, n b i1i 2i n 1 Q n n n!. [B.8] Hence, given a set of diffusion tensors of order n, D, the cumulants Q of the random phase (TE) can be calculated. The cumulants of the spin displacement can be determined by the cumulants of the random phase (TE) as shown next. If the diffusion gradient s duration is short or the gradient separation is much longer than the duration, then the spins could be considered to be static during this short time period. Hence, Eq. [B.2] reduces to: TE Gt 1 xt 1 Gt 2 xt 2 G r, [B.9] where r() x(t 2 ) x(t 1 ) is the spin displacement during the time interval. From Eq. [B.9], the relationship between the cumulants of (TE) and r() can be derived as given below: This is essentially Einstein s formula for a homogeneous medium: r i 2 2D. [B.14] Given the cumulants of r obtained through Eq. [B.12], one can readily approximate its PDF via the Gram-Charlier series as: 2 Pr N0, Q r,i1i 2 1 Q 3 r,i 1i 2i 3 3! H i1i 2i 3 r Q 4 r,i 1i 2i 3i 4 H 4! i1i 2i 3i 4 r, [B.15] (2) where N(0, Q r,i1 i 2 ) is the normal distribution with zero (2) mean and covariance matrix Q r,i1 i 2. H i1 i 2...i n (r) is the Hermite polynomial (27). REFERENCES 1. Moseley ME, Cohen Y, Kucharczyk J. Diffusion-weighted MR imaging of anisotropic water diffusion in cat central nervous system. Radiology 1990;176:439 446. 2. Moseley ME, Cohen Y, Kucharczyk J. Early detection of regional cerebral ischemic injury in cats: evaluation of diffusion and T2-weighted MRI and spectroscopy. Magn Reson Med 1990;14:330 346. 3. Basser PJ, Mattiello J, Turner R, Le Bihan D. Diffusion tensor echoplanar imaging of human brain. In: Proc 1st Annual Meeting ISMRM, Dallas, 1993. p 584. 4. Basser PJ, Mattiello J, Le Bihan D. Estimation of the effective selfdiffusion tensor from the NMR spin echo. J Magn Reson 1994;103:247 254. 5. Basser PJ, Mattiello J, Le Bihan D. MR diffusion tensor spectroscopy and imaging. Biophys J 1994;66:259 267.