NeuroImage xxx (2012) xxx xxx. Contents lists available at SciVerse ScienceDirect. NeuroImage. journal homepage:

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1 YNIMG-9753; No. of pages: ; 4C: NeuroImage xxx () xxx xxx Contents lists available at SciVerse ScienceDirect NeuroImage ournal homepage: Bessel Fourier Orientation Reconstruction (): An analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices Q3 A. Pasha Hosseinbor a,b,, Moo K. Chung b,c, Yu-Chien Wu b,d,e, Andrew L. Alexander a,b,f a Department of Medical Physics, University of Wisconsin-Madison, Madison, WI, USA b Waisman Laboratory for Brain Imaging and Behavior, University of Wisconsin-Madison, Madison, WI, USA c Department of Biostatistics and Medical Informatics, University of Wisconsin-Madison, Madison, WI, USA d Dartmouth Brain Imaging Center, Dartmouth College, Hanover, NH, USA e Department of Radialogy, University of Wisconsin-Madison, Madison, WI, USA f Department of Psychiatry, University of Wisconsin-Madison, Madison, WI, USA article info Article history: 3 Accepted August 4 Available online xxxx Keywords: 9 Ensemble average propagator Bi-Gaussian model Extrapolation q-space indices 3 High angular resolution diffusion imaging 4 Heat diffusion smoothing Introduction abstract 45 The aim of diffusion-weighted imaging (DWI) is to non-invasively 46 recover information about the diffusion of water molecules in bio- 47 logical tissues. The most common form of DWI is diffusion tensor im- 48 aging (DTI) (Basser et al., 994), which is a good model of diffusion- 49 weighted signal behavior at low levels of diffusion weighting. 5 However, DTI is limited by the Gaussian assumption, which is invalid 5 at higher levels of diffusion weighting (b> s/mm ) and its inabil- 5 ity to resolve multiple fiber orientations within a voxel (Alexander et al., 53 ; Frank, ; Wiegell et al., ). In order to recover complex 54 white matter (WM) geometry, high angular resolution diffusion imag- 55 ing (HARDI) (Tuch et al., ), which reduces the diffusion signal 56 sampling to a single sphere (i.e. single level of diffusion weighting) 57 within q-space, was proposed. Many HARDI techniques (Agan et al., 58 ; Canales-Rodriguez et al., 9; Descoteaux et al., 7; Hess et 59 al., 6; Tuch, 4) seek to extract the orientation distribution func- 6 tion (ODF), a probability density function describing the angular The ensemble average propagator (EAP) describes the 3D average diffusion process of water molecules, cap- 5 turing both its radial and angular contents. The EAP can thus provide richer information about complex tissue 6 microstructure properties than the orientation distribution function (ODF), an angular feature of the EAP. 7 Recently, several analytical EAP reconstruction schemes for multiple q-shell acquisitions have been proposed, 8 such as diffusion propagator imaging (DPI) and spherical polar Fourier imaging (). In this study, a new 9 analytical EAP reconstruction method is proposed, called Bessel Fourier Orientation Reconstruction (), 3 whose solution is based on heat equation estimation of the diffusion signal for each shell acquisition, and 3 is validated on both synthetic and real datasets. A significant portion of the paper is dedicated to comparing 3,, and DPI using hybrid, non-cartesian sampling for multiple b-value acquisitions. Ways to mitigate 33 the effects of Gibbs ringing on EAP reconstruction are also explored. In addition to analytical EAP reconstruc- 34 tion, the aforementioned modeling bases can be used to obtain rotationally invariant q-space indices of 35 potential clinical value, an avenue which has not yet been thoroughly explored. Three such measures are 36 computed: zero-displacement probability (Po), mean squared displacement (MSD), and generalized frac- 37 tional anisotropy (GFA). 38 Published by Elsevier Inc. 39 distribution of water molecules during diffusion. Unlike apparent 6 diffusion coefficient (ADC) profiles, the maxima of the ODF are aligned 6 with the fiber directions, making it useful in fiber tractography applica- 63 tions. However, the ODF only retrieves the angular content of the diffu- 64 sion process. 65 The ensemble average propagator (EAP) provides more informa- 66 tion about tissue microstructure than the ODF because it captures 67 both the radial and angular information contained in the diffusion 68 signal. The ODF, mathematically defined as the radial proection of 69 the EAP, is simply an angular feature of the EAP. Unlike the diffusion 7 tensor, the EAP profiles illustrate and recover crossing fibers. The 7 authors in Cohen and Assaf () have suggested that the radial 7 part of the diffusion signal may be sensitive to WM disorders caused 73 by demyelination and could be used to infer the axonal diameter. 74 The estimation of the EAP requires combination of high angular 75 sampling at multiple levels of diffusion weighting. Under the narrow 76 pulse assumption (Steskal and Tanner, 965), the diffusion signal at- 77 tenuation, E(q) inq-space and the EAP, P(p), are Fourier Transform 78 (FT) pairs (Callaghan, 99): Corresponding author at: Department of Medical Physics, University of Wisconsin- Madison, Madison, WI, USA. address: hosseinbor@wisc.edu (A.P. Hosseinbor). PðpÞ ¼ EðqÞe πiq p d 3 q; ðþ 53-89/$ see front matter Published by Elsevier Inc. for hybrid diffusion imaging a..., NeuroImage (),

2 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 8 8 where E(q)=S(q)/S o is the normalized q-space diffusion signal, S(q) 8 is the diffusion signal measured at position q in q-space, and S o is 83 the baseline image acquired without any diffusion gradients (q=). 84 We denote q=q u(θ,ϕ) and p=p r(θ,ϕ ), where u and r are 3D 85 unit vectors. The wave vector q is q=γδg/π, where γ is the nuclear 86 gyromagnetic ratio and G=gu is the applied diffusion gradient direc- 87 tion. The norm of the wave vector, tq, is related to the diffusion 88 weighting level (b-value) via b=4π q (Δ δ/3) (Basser, ), 89 where δ is the duration of the applied diffusion gradients and Δ the 9 time between the two pulses. Eq. () is valid only if the narrow 9 pulse condition is met, which is rarely the case for q-space diffusion 9 MRI performed under experimental conditions. Several studies 93 (Bar-Shir et al., 8; Mair et al., ; Weeden et al., 5) however, 94 have shown that even when these assumptions do not hold, the 95 Fourier relationship in Eq. (3) is still a reasonable approximation of 96 the microstructural features. The diffusion displacements, however, 97 will be consistently underestimated (Weeden et al., 5). 98 Various methods already exist to reconstruct the EAP. Using the 99 diffusion tensor framework, the EAP is simply described by a multi- variate Gaussian function (Basser et al., 994). The authors in Ghosh and Deriche () presented a closed-form approximation of the EAP using higher order tensors, specifically the 4th order diffusion 3 tensor. The diffusion orientation transform (DOT) (Ozarslan et al., 4 6) is a HARDI technique that computes the iso-radius of the EAP. 5 DOT assumes the radial diffusion follows a mono-exponential decay, 6 which allows the radial integration in Eq. () to be solved analytically. 7 The spherical integration is then solved numerically. The application 8 of this technique, however, is limited by its mono-gaussian assump- 9 tion of the radial diffusion decay. In addition, the single shell approach of DOT gives an incomplete picture of the EAP, whose esti- mation requires signal measurements along all of q-space. EAP reconstruction techniques using multiple diffusion weighting 3 acquisitions can be divided into two strategies: Fast Fourier Trans- 4 form (FFT) based and analytical. FFT based methods include diffusion 5 spectrum imaging (DSI) (Canales-Rodriguez et al., ; Weeden 6 et al., 5) and hybrid diffusion imaging (HYDI) (Rathi et al., ; 7 Wu and Alexander, 7). DSI is based on direct sampling of the dif- 8 fusion signal on a Cartesian q-space lattice. The FT in Eq. () is then 9 numerically evaluated via FFT to obtain the EAP. A maor advantage of DSI is that the EAP is estimated without any prior assumptions of behavior of the diffusion signal. However, DSI requires dense sam- pling of the Cartesian lattice, resulting in very long acquisition 3 times. HYDI samples the diffusion signal along concentric spherical 4 shells in q-space, with the measurements then being interpolated 5 and regridded onto a Cartesian lattice so that the EAP can 6 be similarly reconstructed as in DSI. HYDI uses much fewer samples 7 than DSI, making it more clinically feasible. However, the HYDI prop- 8 agator reconstruction may suffer from the ad hoc signal interpolation 9 and regridding. 3 The FFT is impractical for methods employing spherical q-space 3 sampling schemes, such as HYDI, since the FFT requires data to lie 3 on a Cartesian grid. It is also quite computationally expensive. Solving 33 the FT in spherical coordinates (i.e. spherical Fourier Transform) in- 34 stead, obviates the need for FFT and ad hoc processing. Analytic 35 methods, seeking to obtain a closed-form solution of the EAP, pursue 36 such a route. Currently, the two main analytical EAP reconstruction 37 schemes are diffusion propagator imaging (DPI) (Descoteaux et al., 38 ) and spherical polar Fourier imaging () (Assemlal et al., 39 9a; Cheng et al., a, b). 4 DPI assumes that E(q) is a solution to the 3D Laplace's equation 4 E=, which results in the signal basis being composed of the reg- 4 ular and irregular solid harmonics. It is fast, and seems to work well 43 with only a small number of samples. However, the DPI signal basis 44 is an unrealistic model of E(q) because Laplacian modeling of diffu- 45 sion signal entails that () E() does not exist, which arises from 46 the irregular solid harmonic term, and () MSD of water molecules is zero, which will be proved in the Theory section. In addition, the DPI signal basis lacks orthonormality, and hence does not possess the robust numerical stability that would otherwise feature in an orthonormal basis. models the diffusion signal in terms of an orthonormal basis comprising the spherical harmonics (SH) and Gaussian Laguerre polynomials. The signal basis is, in fact, a modified solution of the 3D quantum mechanical simple harmonic oscillator problem. It is robust to noise and low anisotropy, and works well with ust a few number of samples. However, has not been tested at b>3 s/mm. A slightly modified version of the signal basis was proposed ust recently by the authors in Caruyer and Deriche (). This paper, however, will only be concerned with the original basis. A closely related basis to was proposed in Ozarslan et al. (8, 9), which use the Hermite polynomials to estimate the D q-space diffusion signal. In addition to forming a complete orthogonal basis, the Hermite polynomials are also eigenfunctions of the Fourier transform. However, the 3D EAP solution has yet to be derived using this basis. With respect to analytical EAP reconstruction methods, one valuable though overlooked use is in extracting rotationally invariant quantitative measures from them. Recently, the authors in Assemlal et al. () used the signal basis to compute the novel fiber population dispersion (FPD), an index which assess the presence of crossing fibers within a voxel. The FPD, however, is a relatively new measure that has not yet been computed for an actual human brain. More well-established q-space metrics include generalized fractional anisotropy (GFA) (Tuch, 4), mean squared displacement (MSD) (Assaf et al., ; Wu and Alexander, 7), and zero-displacement probability (Po) (Assaf et al., ). All three are simply scalar features of the EAP, the GFA and the MSD can be viewed as high angular resolution analogues of the DTI indices fractional anisotropy (FA) and mean diffusivity (MD) (Basser and Pierpaoli, 996), respectively. An analytical representation of the EAP (and hence diffusion signal) can facilitate either analytic computation of such features or numerical efficiency in estimating them. In this paper, we present Bessel Fourier Orientation Reconstruction () (Hosseinbor et al., ). Rather than assuming the signal satisfies Laplace's equation, we reformulate the problem into a Cauchy problem and assume E(q) satisfies the heat equation. The heat equation is a generalization of Laplace's equation, which the latter approaches at the steady state (i.e. t ). provides an analytical reconstruction of the EAP profile from diffusion signal and models the diffusion signal in terms of an orthonormal basis. In addition, it contains an intrinsic exponential smoothing term that allows one to control the amount of smoothing in the EAP estimation. The last point is significant because, although the Laplacian modeling intrinsically smoothes the diffusion signal, the amount of smoothing cannot be controlled, and hence it may oversmooth the signal. In addition to heat diffusion smoothing, we also look at linear signal extrapolation as a potential means to mitigate the effects of common artifacts afflicting the reconstructed EAP profile, such as Gibbs ringing and signal truncation. Employing a hybrid, non-cartesian encoding scheme in both synthetic and in vivo datasets, we reconstruct the EAP using, SFPI, and DPI and assess their performances. Lastly, we use to compute GFA, Po, and MSD, and compare 's accuracy in estimating such indices to that of DPI and. The paper is organized as follows: in Theory section, we develop, first by describing how to estimate the diffusion signal, and then deriving the analytical solution for the EAP using Eq. (). Scalar features of the EAP are also introduced in this section. The Appendix carefully details the derivations of the signal basis, EAP, and q-space indices. In Materials and methods section, we describe the implementation details of and present the synthetic and in vivo human brain datasets that will be used to validate and for hybrid diffusion imaging a..., NeuroImage (),

3 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 3 3 illustrate and compare it to and DPI in Results section. 4 Lastly, we discuss our results and future applications of analytical 5 EAP methods in Discussion section. 6 Theory signal can be expanded in terms of the spherical orthonormal basis ψ n given in Eq. (6): Eðq; tþ ¼ XN X R n¼ ¼ C n e α nlðþ t l ðþ α nl ðþ q Y ðuþ; ð7þ The heat equation is ubiquitous in the natural sciences, arising in 8 Fick's law of diffusion, Brownian motion, Fourier's law of thermal con- 9 duction, and the price variation over time of stocks. A nice feature of any solution to the heat equation is its temporal dependence, which can be viewed as an inherent smoothing control mechanism. DPI, by solving the heat equation at the steady state, has no smoothing 3 control mechanism, and so can potentially oversmooth the signal. 4 Smoothing can be useful in situations where reconstructions greatly 5 suffer from noise. Although our method is similar in spirit to DPI 6 and, it significantly differs from them due to its inclusion of a 7 smoothing term. 8 Consider the eigenvalue/boundary condition problem I q ψ i ðqþ ¼ λ i ψ i ðqþ; 9 3 which we use to solve the Cauchy problem t gðq; tþ I qgðq; tþ ¼ ; gðq; t ¼ Þ ¼ f ðqþ; ψ i ðq ¼ ; uþ ¼ ðþ 3 3 where f(q) is simply the acquired signal and I is some self-adoint 33 linear operator. We require λ>. Chung et al. (7) derived a unique 34 solution for Eq. (3): gðq; tþ ¼ X i¼ a i e λ i t ψ i ðqþ; where e λ i t is a smoothing term controlled by parameter t and the 37 coefficients are given by a i = f,ψ i. The implication of Eq. (4) is that the 38 solution decreases exponentially as t increases and smoothes out high 39 spatial frequency noise much faster than low-frequency noise. In DPI, 4 however, the steady state assumption permanently removes any tem- 4 poral term, which governs the extent of smoothing, so there is no 4 smoothing control mechanism. Note that t= corresponds to no 43 smoothing being applied. 44 Assuming that I ¼, where is the 3D Laplacian operator in 45 spherical coordinates, Eq. () becomes " q q q þ sinθ # þ ψ q q sin θ θ θ q sin θ ϕ i ðqþ ¼ λ i ψ i ðqþ ð5þ Eq. (5) can be solved via separation of variables to obtain an 49 orthonormal basis, which we show in Appendix A: ψ n ðqþ ¼ l ðþ α nlðþ q Y ðuþ; ð3þ ð4þ ð6þ 5 5 where α nl() is nth root of lth order spherical Bessel function of first 5 kind l and is the radial distance in q-space at which the Bessel func- 53 tion goes to zero. Y are a modified real and symmetric SH basis pro- 54 posed in Descoteaux et al. () to reflect the symmetry and 55 realness of the diffusion signal. The index :=(l,m)=(l +l+)/ 56 +m is defined for l=,, 4,...and m= l,,,, l. Hence, 57 for ={,,3,4,5,6,7, }, l() ={,,,,,,4, }. The eigenvalues are 58 λ nlðþ ¼ α nlðþ. The SH are also used to model the angular profile of 59 the diffusion signal in DPI and. 6 Within the context of our problem, g(q,t) is the diffusion signal. 6 The assumption of a Laplacian operator results in Eq. (3) becoming 6 the heat equation: E q;t Eðq; tþ ¼. From Eq. (4) then, the diffusion t where C n are the expansion coefficients, R ¼ is the number of terms in the modified SH basis of truncation order L, and N is the 67 number of roots for any spherical Bessel function of order l. The 68 total number of coefficients in the expansion is W ¼ NLþ. Note 69 that the actual acquired signal from scanner is given at t=. In 7 DWI, E()=, and so for our basis, we obtain the following identity 7 (derived in Appendix B): 7 Eq¼ ð ; u; t ¼ Þ ¼ p ffiffiffiffiffiffi C n ¼ ; 4π n which holds for any u within the unit sphere S (i.e. u S ) An important property of the diffusion signal is that it asymptoti- 75 cally approaches zero as q. However, the spherical Bessel func- 76 tions infinitely oscillate about zero, as shown in Fig., soafinite 77 upper bound is needed at which the signal model becomes 78 zero. The fact that the radial basis in does not radially decay 79 to zero but becomes zero at some point in q-space is the main limita- 8 tion of the algorithm. 8 In deriving the EAP, the spherical integration of Eq. () is made 8 easier by expressing the Fourier kernel as a plane wave expansion: 83 e πiq p ¼ 4π X ¼ ð i Þ l ðþ l ðþ ðπqpþy ðuþy ðþ r Substituting Eqs. (7) and (9) into Eq. (), we obtain Pðp; tþ ¼ 4π XN l (q) X X R n¼ ¼ ¼ X R ¼ 4π XN C n e α nlðþt α nlðþ q l ðþ Y ðuþ ð iþ l ðþ l ðþ ðπqpþy ðuþy r ðþd3 q n¼ ¼ ð Þ l ðþ= e α C nl ðþt n Y r ðþi nlðþ p ð Þ; Spherical Bessel Function of st Kind q 3 ð8þ ð9þ ðþ Fig.. Plots of spherical Bessel functions of first kind, which form the radial basis of the signal solution, for different orders l. As q approaches infinity, they infinitely oscillate about zero for hybrid diffusion imaging a..., NeuroImage (),

4 4 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx where we use the orthonormal property of SH, i.e. Y ðuþy ðuþd u ¼ 89 δ and define I nlðþ ðpþ ¼ α nlðþ q q l ðþ l ðþ ðπqpþdq α nlðþ q q l ðþ l ðþ ðπqpþdq ðþ The integral in Eq. () is solved in Appendix C, and we can write 93 the EAP as Pðp; t pffiffiffiffiffiffiffiffi Þ ¼ π 3 X N X R n¼ ¼ ð Þ l ðþ C ne α nlðþ t Y ðþ r pffiffiffiffiffiffiffiffiffiffi α nlðþ J l ðþ = α nlðþ 4π p α nlðþ l ðþ ðπpþ ðþ The theoretical solution can be summarized in five steps: 97. ψ i (q)= λ i ψ i (q), ψ i (q=, u)= 98. E(q,t)= i= a i h i (t)ψ i (q) E q;t Eðq; tþ ¼ ; t Eðq; t ¼ Þ ¼ f ðqþ 3 4. h i ðþ¼e t ; a i ¼ hf ðqþ; ψ i ðqþi 3 5. P(p,t)= E(q,t)e πiq p d 3 q=4π = ( i) l() Y (r){ E(q,u,t) l() 3 (πqp)y (u)d 3 q} 33 Rotationally invariant q-space indices 34 In addition to analytical EAP reconstruction, the DPI,, and 35 modeling bases can be used to obtain rotationally invariant q-space 36 indices. Here, we look at three such measures: Po, MSD,andGFA. 37 Po=P(p=) is the probability density that a water molecule 38 returns back to its initial position within the diffusion time [6,5]. In 39 a healthy adult brain, Po is greater in white matter (WM) than gray 3 matter (GM) because WM has more restricting barriers including 3 multi-layer myelin sheaths, axonal membranes, and microtubules. 3 Such restrictivity increases the likelihood of a water molecule re- 33 turning to its original position, whereas it has a very low probability 34 of returning to its original position in areas of unrestricted isotropic 35 diffusion (CSF). Hence, Po can be viewed as a measure of restricted 36 diffusion. Several studies have shown Po to be sensitive to brain pa- 37 thology, and suggesting that changes in myelin are the primary 38 mechanism for differences in Po (Assaf et al., ; Bar-Shir et al., 39 9; Wu et al., ). 3 Po can be evaluated either numerically or analytically. The authors 3 in Wu et al. (8) computed Po by numerically summing the nor- 3 malized diffusion signal E(q) over all diffusion measurements in 33 q-space, and then correcting the sum by the sampling density. Analyt- 34 ical formulations of Po were derived for the and DPI signal bases 36 Q3363 Materials and methods (Cheng et al., b; Descoteaux et al., ). Similarly, an analytical 35 Po expression can be obtained using the basis, which is derived 36 in Appendix D: 37 p Po ¼ ffiffiffi 3 π XN C n n¼ ð α n Þ nþ ð3þ The MSD, which we will denote as p, is simply the second mo- 33 ment of the EAP (Wu and Alexander, 7): p = p P(p)d 3 p.itis 33 related to the MD, which in the case of Gaussian diffusion is given by 33 the well-known Einstein relation p =6(Δ δ/3)md. Thus far, an 333 analytical formulation of MSD exists only within the DTI framework. 334 It is calculated numerically in q-space imaging, either by extracting 335 the full width at half maximum of the EAP (Assaf et al., ) or taking 336 the geometric mean of the diffusion signal over all directions on a 337 HDYI shell (Wu et al., 8)InAppendix E, we will show that, in gen- 338 eral, the MSD is related to the diffusion signal by the relation 339 D p E ¼ 4π EðqÞ q¼ ð4þ Since DPI assumes E=, it predicts the MSD to be zero: 34 ¼. The MSD is (derived in Appendix E) 343 p DPI D E p X N C 8π 5 n α n n¼ ð5þ Tuch (4) introduced the concept of GFA and defined it as 346 std(odf)/rms(odf). Since ODF is only a feature of the EAP, the subse- 347 quent GFA map is derived solely from the angular content of the dif- 348 fusion profile. Incorporating both the angular and radial contents of 349 the diffusion profile into the definition of GFA will result in a radial 35 dial of GFA maps, illustrating how anisotropy varies with diffusion 35 displacement p. Therefore, we define a new GFA: 35 GFAðp ¼ p o Þ ¼ std½pðp ¼ p o ; rþš ð6þ rms½pðp ¼ p o ; rþš Another advantage of Eq. (6) is that it is better suited for multiple 355 diffusion weighted MR experiments, unlike Tuch's definition, which 356 is single-shell HARDI-based. In order to capture the 3D anisotropy of 357 the EAP-defined GFA maps, uniformly distributed vertices on a 358 unit sphere in propagator space (i.e. values of θ and ϕ ) were 359 acquired using the approach described in Wong and Roos (994) In general, we are given k HARDI shell datasets. The number of encoding directions in each shell does not have to be the same. Each HARDI 365 dataset corresponds to a different b-value. Across all k shells, we have a total of exitm diffusion measurements (including the b= measure- 366 ment). Hence, from these multiple shell datasets, we want to reconstruct the EAP, P(p) Numerical implementation of 368 The task is to estimate coefficients C n in Eq. () from the observed signal E(q, u, t=). We achieve this by carrying out a linear least square 369 (LLS) fitting with regularization in the radial and angular parts. We let S ¼ ½Eðq ; t ¼ Þ Eðq M ; t ¼ ÞŠ be the M vector representing the 37 M diffusion signal measurements across all k shells. We also let C represent the W vector of unknown expansion coefficients C n,where Q37 W ¼ NLþ.Defining Z n (q,u)= l() (α nl() q/)y (u), we let Z denote our M exitm design matrix: Z ; ðq ; u Þ Z ; ðq ; u Þ Z N; ðq ; u Þ Z ;R ðq ; u Þ Z ;R ðq ; u Þ Z N;R ðq ; u Þ Z ts A Z ; ðq M ; u M Þ Z ; ðq M ; u M Þ Z N; ðq M ; u M Þ Z ;R ðq M ; u M Þ Z ;R ðq M ; u M Þ Z N;R ðq M ; u M Þ for hybrid diffusion imaging a..., NeuroImage (),

5 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 5 t: Table t: HYDI encoding scheme for synthetic and human datasets. t:3 Shell Ne q (mm ) Δq (mm ) b (s/mm ) t:4 t:5 st t:6 nd t:7 3rd t:8 4th t:9 5th t: Total=6 q max =76 Mean=5. b max = Thus, we have a simple linear model of the form S=ZC. This system of over-determined equations is solved with a regularized LLS solution 375 yielding vector ^C given by ^C ¼ Z T Z T Z þ λ l L reg þ λ n N reg S; ð7þ where L reg is the Laplace Beltrami regularization diagonal matrix with l (l+) entries on the diagonal and N reg is the regularization diagonal 378 matrix for the radial basis, with entries n (n+) on the diagonal. The angular and radial regularization matrices penalize, respectively, high de- 379 grees of the angular and radial parts of Eq. (7) in the estimation under the assumption that they are likely to capture noise (Assemlal et al., 9a). 38 They also serve to reinforce the positivity constraint of the EAP. λ l and λ n are the regularization terms for angular and radial bases, respectively. 38 Visualization of EAP 38 Lastly, from the estimated vector ^C, we can extract the C n coefficients needed to compute the EAP, Po, MSD, and GFA. The spherical function 383 P(p,r,t) is the iso-probability profile at some instant of smoothening t for a given p that is, the probability density that a water molecule, initially 384 at the origin, diffuses a distance p along the direction r. It is computed by generating 8 equidistant points along the equator of a sphere of 385 radius p i.e. the polar angle θ is fixed at π/ and the azimuthal angle ϕ is uniformly varied from to π. The EAP profile P(p,r,t) is then interpo- 386 lated along these 8 points. Thus, the resulting profiles are D with the equator perpendicular to the z-axis. It is important to note that in this 387 paper smoothening was applied only to the EAP, itself, and not on the diffusion signal. 388 Value of parameter ln(s/so) Head Diffusion Signal Decay Curves Bi Exponential Mono Exponential b (s/mm ) Fig.. Plots of diffusion signal attenuation as a function of b-value illustrating mono-exponential and bi-exponential decays. The tail of the bi-exponential can be interpreted as the slow diffusion component, while the head the fast diffusion component. 389 An important point to consider in the implementation is how to determine the parameter in the signal basis. In practice, the diffusion signal 39 is bounded by the maximal q-value q max achievable by the imaging system. The authors in Assaf and Cohen (998) have shown that, depending Tail t: Table t: Fast/Slow diffusion ADCs and component size fractions (from Maier et al., 4). t:3 Region of interest Corpus callosum Internal capsule t:4 ADC f (μm /ms).76. t:5 ADC s (μm /ms) t:6 f f t:7 f s for hybrid diffusion imaging a..., NeuroImage (),

6 6 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 39 on the length of diffusion time, the amount of signal present at b-values near 3, s/mm varies from about half a percent to about 5%, which 39 means that the signal does not approach zero at q max unless the diffusion weighting/diffusion time are very high/long. Thus, we conclude 393 q max. Based on numerical simulations, we find the value of that best reconstructs the EAP to be optimal =q max +Δq, where Δq is the (uni- 394 form) q-space sampling interval. 395 Diffusion MRI data acquisitions for synthetic and human brain data 396 The synthetic and in vivo datasets use a hybrid, non-cartesian sampling scheme (Wu and Alexander, 7), shown in Table. Since EAP re- 397 construction is sensitive to angular resolution, the number of encoding directions is increased with each shell to increase the angular resolution 398 with the level of diffusion weighting. The number of directions in the outer shells was increased to better characterize complex tissue organi- 399 zation. Diffusion tensor elements for measurements in the second shell were calculated using non-linear least squares estimation with the 4 Camino software package (Cook et al., 6), which were then used to obtain the FA and principal eigenvector. 4 Synthetic data 4 The mono-exponential (also referred to as mono-gaussian) mixture model (Tuch et al., ) is frequently used to generate synthetic data to 43 validate a given EAP reconstruction, such as in Assemlal et al. (9a), Cheng et al. (b), where the maximum b-value used was 3 s/mm. 44 However, diffusion MR imaging experiments using high b-values (> s/mm ) have shown that the diffusion signal decay is no longer 45 mono-exponential. Studies in normal human brain, with b-values over an extended range of up to 6 s/mm, have shown that the signal 46 decay is better described with a bi-exponential i.e. bi-gaussian curve (Clark and Le Bihan, ; Mulkern et al., 999). Similar findings were 47 made for rat brain, using multiple b-values of up to, s/mm (Niendorf et al., 996). According to Assaf and Cohen (998), a 48 bi-exponential fit gives very good agreement with the observed water signal attenuation in excised brain tissue from rats for b-values of up 49 to 3 4 s/mm. Thus,,, and DPI were applied to simulations of crossing fiber configurations generated by a bi-gaussian mixture 4 model. Fig. illustrates mono-exponential and bi-exponential decay curves, where the latter has a pronounced tail at high q values, indicating 4 that it takes longer for the signal to decay to zero than under the mono-exponential assumption. The head and tail of the bi-exponential decay 4 curve can be viewed as the fast and slow diffusion components, respectively (Clark and Le Bihan, ; Maier et al., 4). 43 In bi-gaussian mixture, Eq; ð uþ ¼ XN b h i f kf e but D u kf þ f ks e but D ks u ; ð8þ k¼ Signal Fit (a) (b) (c) (d) (e) Signal Fit (f) (g) (h) (i) () DPI Signal Fit (k) (l) (m) (n) (o) Fig. 3. The ground truth diffusion signal (green) and estimated signal (red) using,, and DPI when noise was absent. Two equally weighted WM fibers were simulated crossing at 6. Measurements from all 5 shells were used. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) for hybrid diffusion imaging a..., NeuroImage (),

7 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 7 Fast EAP reconstruction at (a) (b) (c) (d) Fast EAP Reconstruction (e) (f) (g) (h) DPI Fast EAP Reconstruction (i) () (k) (l) Fig. 4. Reconstruction of the fast component EAP (red) using,, and DPI compared with the ground truth (green). Two equally weighted WM fibers were simulated crossing at 6. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) where N b is the total number of simulated fibers, f kf the volume fraction of the fast component of the k th fiber, and f ks the volume fraction of the 46 Nb slow component. The summation of all volume fractions is, i.e., k= [f kf +f ks ]=. D kf and D ks describe the diffusion tensor for the fast and 47 slow components, respectively, of the k th fiber assuming no exchange between the fast- and slow-diffusion compartments. It should be noted Slow EAP Reconstruction at (a) (b) (c) (d) Slow EAP Reconstruction (e) (f) (g) (h) DPI Slow EAP Reconstruction (i) () (k) (l) Fig. 5. Reconstruction of the slow component EAP (red) using,, and DPI compared with the ground truth (green). Two equally weighted WM fibers were simulated crossing at 6. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) for hybrid diffusion imaging a..., NeuroImage (),

8 8 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 48 that there is controversy over the assignment of these components and whether the bi-gaussian model should take into account exchange be- 49 tween compartments (Mulkern et al., 999). The ground truth of EAP is then Pp; ð rþ ¼ XN b k¼ " f kf rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4πþ 3 D kf e p r T D kf r=4 f þ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ks e p r T D ks #; r=4 ð9þ ð4πþ 3 ctd ks 4 4 where =Δ δ/3. For the synthetic data, the diffusion gradient duration is δ=45 ms and diffusion gradient separation Δ=56 ms. 4 In reconstructing the EAP, we look at two equally weighed fibers crossing at 6, and set eigenvalues of each diffusion tensor to be 43 [.6,.4,.4]e 3, which gives an FA value of.77. The values of the fast and slow Gaussian diffusion functions were taken from Maier et al. 44 (4) and are shown in Table. Monte Carlo noise simulations were then performed to investigate the effect of on the estimation of 45 Po, MSD, and GFA for a single voxel for each EAP method. Seven levels ([ ]) for the b= image were simulated, 46 times each, by adding Rician noise in a similar manner as in Descoteaux et al. (7) for four different scenarios: a fast isotropic component 47 (D=.5 mm /s); a slow isotropic component (D=.45 mm /s); fast anisotropic components of a corpus callosum fiber and internal 48 capsule fiber crossing at 6 ; and the slow anisotropic components for the previous scenario. The parameters are {L=4, N=6, =9. mm, 49 λ l = 6, λ n = 6 }, DPI parameters {L=4, λ l = (no noise)/λ l =.6 (with noise)}, and parameters {L=4, N=3, ζ=5, λ l = 8, 43 λ n = 8 }. For each method, model parameters were chosen based on giving the optimal EAP reconstruction when no noise was present. 43 Human brain data 43 HYDI was performed on a healthy, adult human using a 3.T GE-SIGNA scanner with an 8-channel head coil and ASSET parallel imaging. The 433 DW pulse sequence was a single-shot, spin-echo, echo-planar imaging (SS-SE-EPI) with pulse-oximeter gating. The MR parameters were as Fast EAP Reconstruction at with Linear Extrapolation (a) (b) (c) (d) Fast EAP Reconstruction with Linear Extrapolation (e) (f) (g) (h) Slow EAP Reconstruction at with Linear Extrapolation (i) () (k) (l) Slow EAP Reconstruction with Linear Extrapolation (m) (n) (o) (p) Fig. 6. Extrapolated samples were acquired by linearly damping the signal measurements in the outermost shell. Reconstruction of the EAP (red) using and compared with the ground truth (green). Two equally weighted WM fibers were simulated crossing at 6. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) for hybrid diffusion imaging a..., NeuroImage (),

9 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx follows: TE= ms, TR s, FOV=56 mm, matrix=8 8, voxel size= mm, 3 slices with slice thickness=3 mm, and a total 435 scan time of about 3 min. Diffusion parameters were maximum b-value b max =9375 s/mm, diffusion gradient duration δ=45 ms, diffusion 436 gradient separation Δ=56 ms, q-space sampling interval Δq=5. mm, maximum length of the q-space wave vector q max =76 mm, 437 field of view of the diffusion displacement space FOV p =(/Δq)=65 μm, and resolution of the diffusion displacement space Δp=(/q max )= μm (Callaghan, 99). The same, DPI, and modeling parameters utilized for synthetic data were also used for in vivo data Results Results of synthetic data , DPI, and are first applied to the numerical phantom and 44 then on the real dataset. The numerical phantom is used to validate 443, compare its performance to those of DPI and, assess all 444 three methods' robustness in estimating the scalar measures Po, 445 MSD, and GFA, and answer the following questions: () Can these 446 methods properly reconstruct a diffusion signal acquired via hybrid 447 sampling? () How does the slow diffusion component affect the 448 EAP reconstruction and the estimations of the scalar quantities? 449 (3) What can be done to reduce the effects of Gibbs ringing on the 45 EAP reconstructions? It is also important to note that the EAP and 45 quantitative scalar measures were reconstructed using only 5 dif- 45 fusion measurements, while those presented in Descoteaux et al. 453 () used Can these methods be used for diffusion signal estimation? Fig. 3 displays the, DPI, and signal fit for each shell and the corresponding ground truth. The signal basis fits the diffusion signal nearly perfectly for all shells. Both and DPI reasonably fit the diffusion signal for b 5 s/mm, but poorly reconstruct the inner most shell (b=375 s/mm ). As expected, DPI also tends to oversmooth the diffusion signal, especially so at b=3375, 6 s/mm. Results of DPI signal fitting were also reported in Descoteaux et al. (), where uniform sampling along four spherical shells (b=, 4, 6, and 8 s/mm ) was done. Although a hybrid sampling scheme is used here, our results are consistent with those of Descoteaux et al. () for b 5 s/mm. 's and DPI's poor signal fit of the diffusion signal in inner most shell may be due to the fact that (a) Fast Anisotropic Po (b) Slow Anisotropic Po Po Reconstruction: Fast Anisotropic Po Reconstruction: Slow Anisotropic x 5 x 6. 7 Po (c) Fast Isotropic Po Po 4 3 Extrap DPI Extrap Actual Po 7 x Po Reconstruction: Fast Isotropic 5 x Extrap 8 DPI Extrap Actual Po Po Po (d) Slow Isotropic Po Po Reconstruction: Slow Isotropic Extrap DPI Extrap Actual Po Extrap DPI Extrap Actual Po Fig. 7. Monte Carlo simulation investigating the effect of noise on estimation of Po using fast/slow anisotropic components and fast/slow isotropic components. Error bars denote one standard deviation across trials. for hybrid diffusion imaging a..., NeuroImage (),

10 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 468 only six measurements were acquired in this shell (see Table ), which 469 may be inadequate for their respective radial bases. 47 Performances of, DPI, and in reconstructing EAP. Fig shows the EAP reconstruction for the fast diffusion component across 47 propagator space using each method. Modeling the fast component 473 (i.e. head of the bi-exponential in Fig. ) is tantamount to fitting a 474 mono-exponential curve, and so the EAP reconstruction for the fast 475 diffusion component can be viewed as if the diffusion signal decay 476 was mono-exponential. Both and model the fast compo- 477 nent EAP very well, accurately capturing the geometry and orienta- 478 tion of the EAP profile, and the reconstruction is nearly 479 identical to that of. DPI performs reasonably well, but tends to 48 overestimate the EAP. 48 Fig. 5 shows the EAP reconstruction for the slow diffusion compo- 48 nent, which can be viewed as modeling the tail of the bi-exponential 483 curve in Fig.. Note that the and reconstructions are quite 484 alike. At p= μm, all three methods capture the correct geometry of 485 the ground truth EAP profile, but underestimate it, DPI more so. At 486 p=5 μm, all three methods are unsuccessful in capturing the correct 487 geometry, in particular failing to capture the peaks of the ground 488 truth EAP profile. However, they do capture the correct orientation. 489 At p= and especially p=5 μm, the and EAP MSD Extrap Extrap Actual MSD x 4 4 reconstructions of the slow diffusion component begin to suffer from Gibbs ringing, which arise from the truncation of the signal bases at high q. The DPI reconstruction at p=5 μm benefits from the inherent smoothening of Laplacian signal modeling, with no spurious peaks present. Although oversmoothened and overestimated, it does a much better ob than and in resolving the correct fiber orientation at p=5 μm. The difficulty in reconstructing the EAP for the slow diffusion component is due to the slow diffusion component being sensitive to truncation effects. The reality of finite sampling makes it challenging to capture the tail of the bi-exponential curve. How then should one combat the effects of truncation artifacts? Signal extrapolation. Extrapolating the diffusion signal to higher q-values so that q-space is more thoroughly explored could mitigate the truncation effects. Signal extrapolation can increase the spatial resolution of the EAP (Cohen and Assaf, ), and in the case of DSI, significantly reduce the cumbersome q-space sampling (Yeh et al., 8). By linearly damping the signal measurements in the outermost shell (b=9375 shell), we were able to (linearly) extrapolate samples onto three new pseudo-shells. Specifically, the outermost signal measurements were attenuated by a factor of.7,.4, and. to form the three pseudo-shells. The and scaling factors, and ζ, respectively, were changed for the extrapolation to (a) Fast Anisotropic MSD (b) Slow Anisotropic MSD 3.5 x 4 MSD Reconstruction: Fast Anisotropic x 4 MSD Reconstruction: Slow Anisotropic (c) Fast Isotropic MSD MSD 3 MSD Reconstruction: Fast Isotropic Extrap Extrap Actual MSD MSD x (d) Slow Isotropic MSD MSD MSD Reconstruction: Slow Isotropic Extrap Extrap Actual MSD Extrap Extrap Actual MSD Fig. 8. Monte Carlo simulation investigating the effect of noise on estimation of MSD using fast/slow anisotropic components and fast/slow isotropic components. Error bars denote one standard deviation across trials. for hybrid diffusion imaging a..., NeuroImage (),

11 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 5 =36.8 mm and ζ=. Note that the q-space sampling interval 53 Δq was not changed for the extrapolation. 54 Fig. 6 shows that signal extrapolation improves the reconstruction 55 of the slow EAP component for both and. At p= μm, the 56 and slow EAP reconstructions with signal extrapolation, al- 57 though not perfectly capturing the ground truth geometry, better 58 capture the angular features of the ground truth than those without 59 signal extrapolation. The biggest improvement, however, is seen at 5 p=5 μm, where the pronounced Gibbs ringing is greatly reduced 5 by the signal extrapolation. In particular, the and slow 5 EAP reconstructions with signal extrapolation at p=5 μm are not 53 spiky and much closer to the ground truth, although their orienta- 54 tions are slightly off, than those without extrapolation (Fig. 5). Note 55 that both the and slow EAP reconstructions with extrapo- 56 lation are quite alike. The fast EAP reconstruction was not 57 affected by extrapolation, being nearly identical to its counterpart 58 without extrapolation. However, the fast EAP reconstruction 59 with extrapolation was moderately less accurate than that with- 53 out extrapolation. In general, signal extrapolation can significantly 53 improve EAP reconstructions at larger diffusion displacements 53 (i.e. p=5 μm). (a) Fast Anisotropic GFA GFA GFA Reconstruction: Fast Anisotropic Extrap DPI Extrap Actual GFA (c) Isotropic GFA Estimation of q-space indices. Fig. 7 shows the results for the Po measurements. Without signal extrapolation, and asymptotically approach the ground truth fast anisotropic Po, whereas DPI overestimates it. All three methods, without signal extrapolation, severely underestimate slow anisotropic Po, which is due to the truncation of the signal bases at high q. Both and asymptotically approach the ground truth fast/slow isotropic Po, while DPI overestimates both. At low levels of (e.g. and ), which is quite common in diffusion MRI, all three methods (without signal extrap.) have biased estimates of Po, though the variance is fairly small. When signal extrapolation is applied, the estimation of the slow anisotropic Po by and significantly improves, as shown in Fig. 7b. According to Fig. 7a, signal extrapolation does not asymptotically affect 's estimation of fast anisotropic Po, but slightly worsens that of 's. At low levels of, however, signal extrapolation results in more severe overestimation of fast anisotropic Po than without signal extrapolation for both and. Fig. 8 shows the results for the MSD measurements. estimates both the fast and slow anisotropic/isotropic MSD very well across levels. However, at low levels of, the variability (given by the standard deviation) of the estimation of MSD is GFA (b) Slow Anisotropic GFA GFA GFA Reconstruction: Isotropic Component GFA Reconstruction: Slow Anisotropic Extrap DPI Extrap Actual GFA Extrap DPI Extrap Actual GFA Fig. 9. Monte Carlo simulation investigating the effect of noise on estimation of GFA using fast/slow anisotropic components and an isotropic component. GFA was computed at p= μm. Error bars denote one standard deviation across trials. for hybrid diffusion imaging a..., NeuroImage (),

12 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 554 quite large, indicating strong sensitivity to noise. without signal 555 extrapolation severely underestimates anisotropic MSD, giving nega- 556 tive values for slow anisotropic MSD, which indicates that it will give 557 inaccurate measurements of the MSD of WM. also underesti- 558 mates fast isotropic MSD, but asymptotically approaches ground 559 truth slow isotropic MSD. Interestingly, there is less variability in 56 's estimation of MSD than. 56 When signal extrapolation is applied to the MSD measurements, 56 's estimations of slow anisotropic and slow isotropic MSD signif- 563 icantly improve, well-estimating them across levels, as shown 564 in Figs. 8b and d, respectively. Specifically, the slow anisotropic 565 MSD estimation is no longer negative with signal extrapolation. 566 's estimations of slow anisotropic and slow isotropic MSD with 567 extrapolation are nearly identical to those without it. However, the 568 variability of the and MSD measurements is much less 569 than those without extrapolation. A negative consequence of signal 57 extrapolation is that it increases the inaccuracy (i.e. asymptotically 57 worsening) of the / fast anisotropic and fast isotropic MSD 57 estimations, all of which are greatly underestimated. The simulations 573 indicate signal extrapolation may be more beneficial to 's estima- 574 tion of MSD than that of 's. 575 Fig. 9 shows the results for the GFA measurements. Asymptotical- 576 ly, and without signal extrapolation approach the ground 577 truth fast anisotropic GFA, while DPI underestimates it. Across 578 levels, DPI severely underestimates slow anisotropic GFA, while 579 both and greatly overestimate it. For the case of isotropic 58 diffusion, where GFA=, the GFA estimated by each method ap- 58 proaches zero at high. However, at =, both and 58 severely overestimate the isotropic GFA, giving values compara- 583 ble to the GFA of WM. DPI's estimation of the isotropic component is 584 more robust to noise than and. 585 with signal extrapolation still overestimates slow anisotropic 586 GFA, though slightly less so than without it, but the extrapolation in- 587 creases the estimation's variability at the same time. Signal extrapola- 588 tion has negligible effects on 's estimation of slow anisotropic 589 GFA. Both 's and 's estimation of fast anisotropic GFA are (a) FA not asymptotically affected by signal extrapolation, having similar convergences as those without signal extrapolation, but the extrapolation causes both methods to overestimate fast anisotropic GFA to a larger degree at =,, and 3 than without it. Based on Fig. 9c, both and with signal extrapolation overestimate isotropic GFA, across levels, to a larger extent than without extrapolation, implying that extrapolation is quite sensitive to noise in CSF regions. Results of human brain data Resolving single fibers. In Fig., a4 4ROIwasdrawnonthe splenium of corpus callosum. The EAP profiles reconstructed at p= μm by each method have the fundamental peanut shape of a single fiber. Note that the and reconstructions in both cases are very similar. We see that application of smoothing t=55 removes the center peaks of the EAP profiles. Whether these center peaks in the EAP profiles are the result of Gibbs ringing (i.e. artificial peaks) or describe some underlying biological process is an open question. Resolving crossing fibers. In Fig., a 4 9 ROI was drawn in a region of fiber crossing, where the EAP profiles were reconstructed at p= μm. Although not identical, the and reconstructions are quite similar, and they recover and well discriminate crossing fiber configurations in the EAP. DPI, however, tends to oversmooth the EAP profiles of crossing WM fibers, resulting in spherical/oblate shapes that give the impression of isotropic diffusion. Based on Fig. d, the application of smoothing t=6 to removes the center peaks from several voxels, but at the expense of slight angular smoothening of EAP profiles themselves. Fig. shows the reconstructed EAP profiles for the same crossing fiber region, but at p=5 μm. Fiber crossing configurations are recovered and well discriminated in the EAP for each method. Unlike at p= μm, the DPI EAP reconstruction at p=5 μm is sharper and does not suffer from oversmoothening. In fact, as the propagator radius p increases, the angular resolution improves, at the expense of the EAP profiles becoming spiky, as is evident in Figs. a, e, and g. When a smoothing of t=6 is applied to the EAP reconstruction (b) FA modulated by principal eigenvector (c) at (d) at (e) DPI (f) Fig.. Axial slice of FA map of adult human brain at b=5 s/mm (second shell), where 4 4 ROI is drawn on splenium of corpus callosum. Plotted is the EAP profile at p= μm overlaid on FA map in 4 4 ROI using at (c) t=, (d) t=55, and (e) DPI and (f). for hybrid diffusion imaging a..., NeuroImage (),

13 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 3 (a) FA (b) FA modulated by principal eigenvector (c) at (d) at (e) DPI (f) Fig.. Axial slice of FA map of adult human brain at b=5 s/mm (second shell), where a 4 9 ROI is drawn on a region of crossing fibers. Plotted is the EAP profile at p= μm overlaid on FA map in 4 9 ROI using (c) at t=, (d) t=6, and (e) DPI and (f). The splenium is to the left of the ROI. 64 (without signal extrapolation), the subsequent EAP profiles are still 65 spiky. At t=35, the spikiness is smoothed out, but many of the 66 WM voxels have EAP reconstructions significantly differing, with re- 67 spect to orientation and geometry, from those at t=. Figs. d and 68 f show the and EAP profiles reconstructed at p=5 μm 69 with signal extrapolation, respectively, which are much less spiky 63 than the corresponding ones without signal extrapolation, which is 63 consistent with the synthetic results shown in Fig. 6. The signal ex- 63 trapolation also smoothes the reconstructed EAP profiles, but unlike 633 at t=35, none of the WM voxels are oversmoothed to such a 634 degree that their EAP profiles have oblate shapes. Unlike at 635 t=35, the underlying EAP geometry and orientation of the / 636 reconstructions with signal extrapolation are fairly consistent 637 to those without extrapolation (at t=). As observed at p= μm, 638 the and EAP reconstructions at p=5 μm are quite similar, 639 which is consistent with the synthetic data results. 64 Q-space indices. Table 3 shows the mean index value and corre- 64 sponding standard deviation for genu and splenium of corpus 64 callosum (WM) and putamen (GM). Three 4 4 ROIs were drawn 643 on both the genu and splenium and one such ROI on both the left 644 and right putamen, across several slices. The table shows that the 645 MSD (without signal extrapolation) erroneously gives negative 646 values for the MSD of genu and splenium. With extrapolation, the 647 MSD of genu and splenium are positive. 648 Based on the numerical simulations, signal extrapolation was 649 applied to both 's and 's estimation of Po. Fig. 3 displays an axial slice of Po generated by each method. In the first row, we show Po computed from,, DPI, and numerically (Wu et al., 8). The,, and numerical Po maps are quite similar, exhibiting rich GM/WM and tissue/csf contrasts while the DPI Po map has less GM/WM contrast. In particular, based on Table 3, the Po ratio of WM to GM is slightly above for and, while less than for DPI. The MSD maps computed from,, DPI, and numerically (Wu et al., 8) are shown in Fig. 4. Both the and numerical MSD maps exhibit rich tissue/csf contrast, but have little WM/GM contrast, which is similar to the DTI MD. Table 3 shows that the MSD values for WM and GM are quite similar. In the (without signal extrapolation) MSD map, WM regions are completely dark, having negative MSD values. This is consistent with the results of the noise simulations, which showed that severely underestimates the MSD of WM. Signal extrapolation has the effect of enforcing the positivity constraint on the MSD for. However, both the and MSD maps with signal extrapolation have poor tissue/csf contrast because of the noise induced by the signal extrapolation. With regards to, both the synthetic and in vivo data suggest that it's best not to use signal extrapolation in estimating MSD., however, does not generate reliable MSD maps either with or without signal extrapolation. Although DPI predicts the MSD to be zero, an MSD map was computed for it by numerically estimating the second moment of the diffusion propagator. The contrast of the MSD DPI map is completely inverted, with WM appearing bright and CSF dark for hybrid diffusion imaging a..., NeuroImage (),

14 4 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx (a) at (b) at (c) at (d) with Signal (f) with Signal (g) DPI P (e) R O O F Extrap. O R R E C T E D Extrap N Fig. 6 displays axial slices of the GFA computed at p = 5,, and 5 μm for each method, illustrating how the anisotropy of different WM regions, such as the corpus callosum and capsules, varies with diffusion displacement p. According to Table 3, at p = 5 μm, the anisotropy of corpus callosum is lower with respect to levels seen in DTI. At p = μm, the corpus callosum is very anisotropic, as can be seen from Table 3, indicating that p = μm is a diffusion displacement worth reconstructing the EAP at. The GFA at p = 5 μm is more noisy, which is due to truncation of signal basis at high q-values and 5 μm being well beyond the resolvable resolution (of diffusion displacement) limit. The and GFA maps without signal extrapolation are very similar, while WM regions in the DPI computed GFA maps at p = 5 and μm have lower intensity than those of and, which is consistent with the underestimation of GFA() by DPI observed in the Monte Carlo noise simulations (Fig. 9). CSF regions in the GFA() map with signal extrapolation are more noisy than without it, which is consistent with the simulation results shown in Fig. 9c. In the case of, however, the noise U C Fig.. EAP profiles reconstructed at p = 5 μm for same crossing fiber region shown in Fig.. level is very severe in CSF regions in the GFA(5) and GFA() maps 694 with signal extrapolation than those without it. 695 Discussion 696 The three analytical EAP reconstruction schemes examined in this paper possess both certain advantages and disadvantages. Among the three, the DPI reconstruction uses the least number of expansion coefficients. According to both synthetic and tin vivo data, DPI tends to greatly oversmoothen the EAP, especially p = μm, but performs well at p = 5 μm where it did not make use of signal extrapolation. DPI's assumption of Laplacian signal modeling, however, entails that the MSD is zero (refer to Eq. (4)). Fig. 9 indicates that DPI greatly underestimates GFA(), which is reflected in Fig. 6 and Table 3. In addition, the DPI signal basis is only applicable at q >, which is unrealistic since the diffusion signal is defined at q =. The signal basis possesses several advantages in that it radially decays to zero and has no singularity at q =. In addition, the EAP for hybrid diffusion imaging a..., NeuroImage (),

15 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 5 t3: Table 3 t3: Values of indices for various WM and GM structures. t3:3 Index Splenium Genu Putamen t3:4 GFA(5).±.6.88± ±.3 t3:5 GFA(5) Extrap.46±.3.37± ±.96 t3:6 GFA(5).34±.78.9±.79.44±.5 t3:7 GFA(5) Extrap.46±.34.4±.387.3±.34 t3:8 DPI GFA(5).9±.849.±.3.66±.64 t3:9 GFA().998±.3.99±.84.54±.686 t3: GFA() Extrap.999±.7.99±.87.36±.759 t3: GFA().999± ±.56.63±.735 t3: GFA() Extrap.996± ± ±.7 t3:3 DPI GFA().83±.8.766±.533.3±.38 t3:4 GFA(5).97± ± ±.83 t3:5 GFA(5) Extrap.859± ±.4.349±.737 t3:6 GFA(5).957± ± ±.797 t3:7 GFA(5) Extrap.858± ±.4.38±.7 t3:8 DPI GFA(5).95±.53.96± ±.68 t3:9 MSD ( 3 mm ).7±.86.9±.98.±.8 t3: MSD Extrap ( 3 mm ).37±.3.6±.33.58±.6 t3: MSD ( 3 mm ).67±.77.±.95.7±.5 t3: MSD Extrap ( 3 mm ).83±..3±..37±.5 t3:3 DPI MSD ( 3 mm ) 4.6± ± ±.36 t3:4 Po ( 5 mm 3 ) 6.63± ±.63.95±.63 t3:5 Po Extrap ( 5 mm 3 ).8±.7 9.4±.7 4.4±.396 t3:6 Po ( 5 mm 3 ) 7.± ± ±.58 t3:7 Po Extrap ( 5 mm 3 ).± ±. 4.33±.449 t3:8 DPI Po ( 5 mm 3 ) 5.± ± ±.3 t3:9 QIV ( mm 5 ) 4.4± ±.563.±.8 7 is derived via integration over the entire q-space, unlike and 7 DPI, where the q-space integration is up to a certain bound that is 7 related to q max. Interestingly, however, the EAP reconstructions 73 for both synthetic and real datasets are quite similar to those of, 74 suggesting the two methods may be inherently related. According to 75 the synthetic data, signal extrapolation greatly improves the EAP 76 reconstruction at p=5 μm. Although not shown in this paper, heat dif- 77 fusion smoothening can be applied to. 's estimation of Po and 78 GFA, either with or without signal extrapolation, is quite comparable (a) (d) with Signal Extrapolation (b) with Signal Extrapolation (e) DPI to those of 's. However, it poorly estimates the MSD, which recalling Eq. (4), may be due to, from a computational standpoint, 's signal basis not being an eigenfunction of the Laplacian operator. The main limitation of the signal model, as mentioned in the Theory section, is that it infinitely oscillates about zero, which entails a finite integration of the signal over q-space ( being the upperbound) to retrieve the EAP. However, based on Fig. 3, outperforms DPI and in modeling the diffusion signal. Heat diffusion smoothening helps in removing potentially spurious peaks, and signal extrapolation (c) (f) Numerical Fig. 3. Axial slice of Po generated by each method. for hybrid diffusion imaging a..., NeuroImage (),

16 6 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx (a) (b) with Signal Extrapolation (c) (d) with Signal Extrapolation (e) DPI 78 significantly improves the EAP reconstruction at p=5 μm. According 79 to both the synthetic and real data, gives reasonable estimates 73 of all three q-space indices. 73 The slow component of diffusion is the most sensitive to trunca- 73 tion artifacts, which can induce severe Gibbs ringing and adversely 733 affect the orientation of reconstructed EAP. In this paper, signal 734 extrapolation was proposed as a means to mitigate the effects of 735 such artifacts, and was observed to be most effective at higher radii 736 (i.e. p=5 μm), where the effects of signal truncation artifacts are 737 most pronounced. The significant improvement in the / 738 EAP reconstruction at p=5 μm via linear signal extrapolation 739 hence suggests that extrapolation may be a useful preprocessing 74 step for EAP reconstruction at large diffusion displacements. Signal 74 extrapolation also greatly improves the accuracy of the / 74 Po estimation, according to the synthetic data. However, signal 743 extrapolation increases the severity of noise in CSF regions in the 744 GFA and MSD maps for both and, as evidenced by the Fig. 4. Axial slice of MSD generated by each method. (f) Numerical synthetic and real datasets, reducing tissue/csf contrast. Hence, extrapolation may not be desirable for GFA and MSD estimation. Future work includes optimizing the signal extrapolation for a given signal basis. The degree of heat diffusion smoothing desired depends on the propagator radius and whether the fibers are single or crossing. Based on Fig., a smoothing of t=55 was applied to splenium at p= μm to remove the center peaks. However, for a crossing fiber region at p= μm, a smoothing of t=6 was only applied because the EAP profiles of crossing fibers can easily become oversmoothed, resulting in oblate shapes. The fact that the smoothing factor is different for different brain regions poises a problem for whole brain EAP processing. One way to address this issue is to use an optimal bandwidth selection framework from statistics to estimate the optimal t. Specifically, the bandwidth t is selected to minimize a certain cost function. In a spline setting, the cost function will be a generalized cross-validation (GCV) criterion (Katkovnik, 999). In a more simple setting like ours, we can choose the t that minimizes the sum of the squared residuals, where the residual is simply the difference between the actual data and model fit. Although the encoding scheme in this study consisted of equally spaced concentric spherical shells, the framework does not require such a scheme. only requires a minimum of two diffusion weightings and use of a spherical coordinate system. Random sampling along q-space or even the use of unequally spaced concentric shells is perfectly valid. This, however, leads to the important question of what is the best way to sample N diffusion measurements in q-space, which have started to be addressed (Assemlal et al., 9b; Caruyer et al., ; Merlet et al., ). Future work includes optimizing the q-space sampling and applying compressed sensing to. Both the ODF and EAP profiles are not sharp enough to extract the true fiber orientation. Rather, the fiber orientation is given by the fiber orientation distribution function (fodf), which can be computed via spherical deconvolution of some assumed kernel (i.e. response Fig. 5. Axial slice of QIV. Within the CSF regions, some voxels were zeroed out because they blew up upon the division operation in computing QIV for hybrid diffusion imaging a..., NeuroImage (),

17 A.P. Hosseinbor et al. / NeuroImage xxx () xxx xxx 7 (a) (b) (c) with Signal Extrapolation (d) (e) (g) (h) with Signal Extrapolation () (k) DPI (m) (n) (f) (i) (l) (o) Fig. 6. GFA maps computed at p=5,, and 5 μm. for hybrid diffusion imaging a..., NeuroImage (),