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- 1. Introduction
- 2. Theory
- 3. Materials and Methods
- 4. Results
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Neuroimage. Author manuscript; available in PMC 2014 January 1.

Published in final edited form as:

Published online 2012 August 31. doi: 10.1016/j.neuroimage.2012.08.072

PMCID: PMC3508305

NIHMSID: NIHMS404564

A. Pasha Hosseinbor,^{a,}^{b,}^{*} Moo K. Chung,^{b,}^{c} Yu-Chien Wu,^{b,}^{d,}^{e} and Andrew L. Alexander^{a,}^{b,}^{f}

The publisher's final edited version of this article is available at Neuroimage

See other articles in PMC that cite the published article.

The ensemble average propagator (EAP) describes the 3D average diffusion process of water molecules, capturing both its radial and angular contents. The EAP can thus provide richer information about complex tissue microstructure properties than the orientation distribution function (ODF), an angular feature of the EAP. Recently, several analytical EAP reconstruction schemes for multiple **q**-shell acquisitions have been proposed, such as diffusion propagator imaging (DPI) and spherical polar Fourier imaging (SPFI). In this study, a new analytical EAP reconstruction method is proposed, called Bessel Fourier orientation reconstruction (BFOR), whose solution is based on heat equation estimation of the diffusion signal for each shell acquisition, and is validated on both synthetic and real datasets. A significant portion of the paper is dedicated to comparing BFOR, SPFI, and DPI using hybrid, non-Cartesian sampling for multiple *b*-value acquisitions. Ways to mitigate the effects of Gibbs ringing on EAP reconstruction are also explored. In addition to analytical EAP reconstruction, the aforementioned modeling bases can be used to obtain rotationally invariant *q*-space indices of potential clinical value, an avenue which has not yet been thoroughly explored. Three such measures are computed: zero-displacement probability (*Po*), mean squared displacement (MSD), and generalized fractional anisotropy (GFA).

The aim of diffusion-weighted imaging (DWI) is to non-invasively recover information about the diffusion of water molecules in biological tissues. The most common form of DWI is diffusion tensor imaging (DTI) (Basser et al., 1994), which is a good model of diffusion-weighted signal behavior at low levels of diffusion weighting. However, DTI is limited by the Gaussian assumption, which is invalid at higher levels of diffusion weighting (*b* > 2000 s/mm^{2}) and its inability to resolve multiple fiber orientations within a voxel (Wiegell et al., 2000; Alexander et al., 2001; Frank, 2001). In order to recover complex white matter (WM) geometry, high angular resolution diffusion imaging (HARDI) (Tuch et al., 2002), which reduces the diffusion signal sampling to a single sphere (i.e. single level of diffusion weighting) within *q*-space, was proposed. Many HARDI techniques (Tuch, 2004; Hess et al., 2006; Descoteaux et al., 2007; Canales-Rodriguez et al., 2009; Aganj et al., 2010) seek to extract the orientation distribution function (ODF), a probability density function describing the angular distribution of water molecules during diffusion. Unlike apparent diffusion coefficient (ADC) profiles, the maxima of the ODF are aligned with the fiber directions, making it useful in fiber tractography applications. However, the ODF only retrieves the angular content of the diffusion process.

The ensemble average propagator (EAP) provides more information about tissue microstructure than the ODF because it captures both the radial and angular information contained in the diffusion signal. The ODF, mathematically defined as the radial projection of the EAP, is simply an angular feature of the EAP. Unlike the diffusion tensor, the EAP profiles illustrate and recover crossing fibers. The authors in (Cohen and Assaf, 2002) have suggested that the radial part of the diffusion signal may be sensitive to WM disorders caused by demyelination and could be used to infer the axonal diameter.

The estimation of the EAP requires combination of high angular sampling at multiple levels of diffusion weighting. Under the narrow pulse assumption (Stejskal and Tanner, 1965), the diffusion signal attenuation, *E*(**q**) in *q*-space and the EAP, *P*(**p**), are Fourier Transform (FT) pairs (Callaghan, 1991):

$$P(\mathbf{p})=\int E(\mathbf{q}){e}^{-2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{q},$$

(1)

where *E*(**q**) = *S*(**q**)/*S _{o}* is the normalized

Various methods already exist to reconstruct the EAP. Using the diffusion tensor frame-work, the EAP is simply described by a multivariate Gaussian function (Basser et al., 1994). The authors in (Ghosh and Deriche, 2010) presented a closed-form approximation of the EAP using higher order tensors, specifically the 4* ^{th}* order diffusion tensor. The diffusion orientation transform (DOT) (Ozarslan et al., 2006) is a HARDI technique that computes the iso-radius of the EAP. DOT assumes the radial diffusion follows a mono-exponential decay, which allows the radial integration in Eq. (1) to be solved analytically. The spherical integration is then solved numerically. The application of this technique, however, is limited by its mono-Gaussian assumption of the radial diffusion decay. In addition, the single shell approach of DOT gives an incomplete picture of the EAP, whose estimation requires signal measurements along all of

EAP reconstruction techniques using multiple diffusion weighting acquisitions can be divided into two strategies: Fast Fourier Transform (FFT) based and analytical. FFT based methods include diffusion spectrum imaging (DSI) (Weeden et al., 2005; Canales-Rodriguez et al., 2010) and hybrid diffusion imaging (HYDI) (Wu and Alexander, 2007; Rathi et al., 2011). DSI is based on direct sampling of the diffusion signal on a Cartesian *q*-space lattice. The FT in Eq. (1) is then numerically evaluated via FFT to obtain the EAP. A major advantage of DSI is that the EAP is estimated without any prior assumptions of behavior of the diffusion signal. However, DSI requires dense sampling of the Cartesian lattice, resulting in very long acquisition times. HYDI samples the diffusion signal along concentric spherical shells in *q*-space, with the measurements then being interpolated and regridded onto a 9 × 9 × 9 Cartesian lattice so that the EAP can be similarly reconstructed as in DSI. HYDI uses much fewer samples than DSI, making it more clinically feasible. However, the HYDI propagator reconstruction may suffer from the *ad hoc* signal interpolation and regridding.

The FFT is impractical for methods employing spherical *q*-space sampling schemes, such as HYDI, since the FFT requires data to lie on a Cartesian grid. It is also quite computationally expensive. Solving the FT in spherical coordinates (i.e. spherical Fourier Transform) instead, obviates the need for FFT and *ad hoc* processing. Analytic methods, seeking to obtain a closed-form solution of the EAP, pursue such a route. Currently, the two main analytical EAP reconstruction schemes are diffusion propagator imaging (DPI) (Descoteaux et al., 2011) and spherical polar Fourier imaging (SPFI) (Assemlal et al., 2009a; Cheng et al., 2010a,b).

DPI assumes that *E*(**q**) is a solution to the 3D Laplace’s equation ^{2}*E* = 0, which results in the signal basis being composed of the regular and irregular solid harmonics. It is fast, and seems to work well with only a small number of samples. However, the DPI signal basis is an unrealistic model of *E*(**q**) because Laplacian modeling of diffusion signal entails that (1) *E*(0) does not exist, which arises from the irregular solid harmonic term, and (2) 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.

SPFI models the diffusion signal in terms of an orthonormal basis comprising the spherical harmonics (SH) and Gaussian-Laguerre polynomials. The SPFI 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 just a few number of samples. However, SPFI has not been tested at *b* > 3000 s/mm^{2}. A slightly modified version of the SPFI signal basis was proposed just recently by the authors in (Caruyer and Deriche, 2012). This paper, however, will only be concerned with the original SPFI basis.

A closely related basis to SPFI was proposed in (Ozarslan et al., 2008, 2009), which use the Hermite polynomials to estimate the 1D *q*-space diffusion signal. In addition to forming a complete orthogonal basis, the Hermite polynomials are also eigenfuctions 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., 2011) used the SPFI 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, 2004), mean squared displacement (MSD) (Assaf et al., 2000; Wu and Alexander, 2007), and zero-displacement probability (*Po*) (Assaf et al., 2000). All three are simply scalar features of the EAP, and the GFA & MSD can be viewed as high angular resolution analogues of the DTI indices fractional anisotropy (FA) and mean diffusivity (MD) (Basser and Pierpaoli, 1996), 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 (BFOR) (Hosseinbor et al., 2011). 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* → ∞). BFOR 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 can not 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 BFOR, SFPI, and DPI and assess their performances. Lastly, we use BFOR to compute GFA, *Po*, and MSD, and compare BFOR’s accuracy in estimating such indices to that of DPI and SPFI.

The paper is organized as follows: in **Theory Section**, we develop BFOR, first by describing how to estimate the diffusion signal, and then deriving the analytical solution for the EAP using Eq. (1). Scalar features of the EAP are also introduced in this section. The Appendix carefully details the derivations of the BFOR signal basis, EAP, and *q*-space indices. In **Material and Methods Section**, we describe the implementation details of BFOR and present the synthetic and *in vivo* human brain datasets that will be used to validate and illustate BFOR and compare it to SPFI and DPI in **Results Section**. Lastly, we discuss our results and future applications of analytical EAP methods in **Discussion Section**.

The heat equation is ubiquitous in the natural sciences, arising in Fick’s law of diffusion, Brownian motion, Fourier’s law of thermal conduction, 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 control mechanism, and so can potentially oversmooth the signal. Smoothing can be useful in situations where reconstrutions greatly suffer from noise. Although our method is similar in spirit to DPI and SPFI, it significantly differs from them due to its inclusion of a smoothing term.

Consider the eigenvalue/boundary condition problem

$${\mathfrak{J}}_{\mathbf{q}}{\psi}_{i}(\mathbf{q})=-{\lambda}_{i}{\psi}_{i}(\mathbf{q}),\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{\psi}_{i}(q=\tau ,\mathbf{u})=0$$

(2)

which we use to solve the Cauchy problem

$$\frac{\partial}{\partial t}g(\mathbf{q},t)-{\mathfrak{J}}_{\mathbf{q}}g(\mathbf{q},t)=0,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}g(\mathbf{q},t=0)=f(\mathbf{q}),$$

(3)

where *f*(**q**) is simply the acquired signal and is some self-adjoint linear operator. We require *λ* > 0. Chung et al. in (Chung et al., 2007) derived a unique solution for (3):

$$g(\mathbf{q},t)=\sum _{i=0}^{\infty}{a}_{i}{e}^{-{\lambda}_{i}t}{\psi}_{i}(\mathbf{q}),$$

(4)

where *e*^{−λit} is a smoothing term controlled by parameter *t* ≥ 0 and the coefficients are given by *a _{i}* =

Assuming that = ^{2}, where ^{2} is the 3D Laplacian operator in spherical coordinates, Eq. (2) becomes

$$\left[\frac{1}{{q}^{2}}\frac{\partial}{\partial q}({q}^{2}\frac{\partial}{\partial q})+\frac{1}{{q}^{2}sin\theta}\frac{\partial}{\partial \theta}(sin\theta \frac{\partial}{\partial \theta})+\frac{1}{{q}^{2}{sin}^{2}\theta}\frac{{\partial}^{2}}{\partial {\phi}^{2}}\right]\phantom{\rule{0.16667em}{0ex}}{\psi}_{i}(\mathbf{q})=-{\lambda}_{i}{\psi}_{i}(\mathbf{q})$$

(5)

Eq. (5) can be solved via separation of variables to obtain an orthonormal basis, which we show in Appendix A:

$${\psi}_{nj}(\mathbf{q})={j}_{l(j)}(\frac{{\alpha}_{nl(j)}q}{\tau}){Y}_{j}(\mathbf{u}),$$

(6)

where *α _{nl}*

Within the context of our problem, *g*(**q**, *t*) is the diffusion signal. The assumption of a Laplacian operator results in Eq. (3) becoming the heat equation:
${\nabla}^{2}E(\mathbf{q},t)={\scriptstyle \frac{\partial E(\mathbf{q},t)}{\partial t}}$ . From Eq. (4) then, the diffusion signal can be expanded in terms of the spherical orthonormal basis *ψ _{nj}* given in Eq. (6):

$$E(\mathbf{q},t)=\sum _{n=1}^{N}\sum _{j=1}^{R}{C}_{nj}{e}^{\frac{-{\alpha}_{nl(j)}^{2}t}{{\tau}^{2}}}{j}_{l(j)}(\frac{{\alpha}_{nl(j)}q}{\tau}){Y}_{j}(\mathbf{u}),$$

(7)

where *C _{nj}* are the expansion coefficients,
$R={\scriptstyle \frac{(L+1)(L+2)}{2}}$ is the number of terms in the modified SH basis of truncation order

$$E(q=0,\mathbf{u},t=0)=\frac{1}{\sqrt{4\pi}}\sum _{n}{C}_{n1}=1,$$

(8)

which holds for any **u** within the unit sphere *S*^{2} (i.e. **u** *S*^{2}).

An important property of the diffusion signal is that it asymptotically approaches zero as *q* → ∞. However, the spherical Bessel functions infinitely oscillate about zero, as shown in Fig. 1, so a finite upper bound *τ* is needed at which the BFOR signal model becomes zero. The fact that the radial basis in BFOR does not radially decay to zero but becomes zero at some point in *q*-space is the main limitation of the BFOR algorithm.

Plots of spherical Bessel functions of first kind, which form the radial basis of the BFOR signal solution, for different orders *l*. As *q* approaches infinity, they infinitely oscillate about zero.

In deriving the EAP, the spherical integration of Eq. (1) is made easier by expressing the Fourier kernel as a plane wave expansion:

$${e}^{-2\pi i\mathbf{q}\xb7\mathbf{p}}=4\pi \sum _{j=1}^{\infty}{(-i)}^{l(j)}{j}_{l(j)}(2\pi qp){Y}_{j}(\mathbf{u}){Y}_{j}(\mathbf{r})$$

(9)

Substituting Eq. (7) and (9) into Eq. (1), we obtain

$$\begin{array}{l}P(\mathbf{p},t)=4\pi \int \sum _{n=1}^{N}\sum _{j=1}^{R}{C}_{nj}{e}^{\frac{-{\alpha}_{nl(j)}^{2}t}{{\tau}^{2}}}{j}_{l(j)}(\frac{{\alpha}_{nl(j)}q}{\tau}){Y}_{j}(\mathbf{u})\sum _{{j}^{\prime}=1}^{\infty}{(-i)}^{l({j}^{\prime})}{j}_{l({j}^{\prime})}(2\pi qp){Y}_{{j}^{\prime}}(\mathbf{u}){Y}_{{j}^{\prime}}(\mathbf{r})\phantom{\rule{0.16667em}{0ex}}{d}^{3}\mathbf{q}\\ =4\pi \sum _{n=1}^{N}\sum _{j=1}^{R}{(-1)}^{l(j)/2}{C}_{nj}{e}^{\frac{-{\alpha}_{nl(j)}^{2}t}{{\tau}^{2}}}{Y}_{j}(\mathbf{r}){I}_{nl(j)}(p),\end{array}$$

(10)

where we use the orthonormal property of SH, i.e. ∫*Y _{j}*(

$${I}_{nl(j)}(p)={\int}_{0}^{\infty}{q}^{2}{j}_{l(j)}(\frac{{\alpha}_{nl(j)}q}{\tau}){j}_{l(j)}(2\pi qp)dq\approx {\int}_{0}^{\tau}{q}^{2}{j}_{l(j)}(\frac{{\alpha}_{nl(j)}q}{\tau}){j}_{l(j)}(2\pi qp)dq$$

(11)

The integral in (11) is solved in Appendix C, and we can write the EAP as

$$P(\mathbf{p},t)=2\tau \sqrt{2{\pi}^{3}}\sum _{n=1}^{N}\sum _{j=1}^{R}{(-1)}^{{\scriptstyle \frac{l(j)}{2}}}{C}_{nj}{e}^{\frac{-{\alpha}_{nl(j)}^{2}t}{{\tau}^{2}}}{Y}_{j}(\mathbf{r})\frac{\sqrt{{\alpha}_{nl(j)}}{J}_{l(j)-1/2}({\alpha}_{nl(j)}){j}_{l(j)}(2\pi \tau p)}{\left(4{\pi}^{2}{p}^{2}-\frac{{\alpha}_{nl(j)}^{2}}{{\tau}^{2}}\right)}$$

(12)

The BFOR theoretical solution can be summarized in five steps:

^{2}*ψ*(_{i}**q**) = −*λ*(_{i}ψ_{i}**q**),*ψ*(_{i}*q*=*τ*,**u**) = 0- $E(\mathbf{q},t)={\sum}_{i=0}^{\infty}{a}_{i}{h}_{i}(t){\psi}_{i}(\mathbf{q})$
- ${\nabla}^{2}E(\mathbf{q},t)={\scriptstyle \frac{\partial E(\mathbf{q},t)}{\partial t}}$,
*E*(**q**,*t*= 0) =*f*(**q**) *h*(_{i}*t*) =*e*^{−λit};*a*=_{i}*f*(**q**),*ψ*(_{i}**q**)- $P(\mathbf{p},t)=\int E(\mathbf{q},t){e}^{-2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{q}=4\pi {\sum}_{j=1}^{\infty}{(-i)}^{l(j)}{Y}_{j}(\mathbf{r})\phantom{\rule{0.16667em}{0ex}}\{\int E(q,\mathbf{u},t){j}_{l(j)}(2\pi qp){Y}_{j}(\mathbf{u}){d}^{3}\mathbf{q}\}$

In addition to analytical EAP reconstruction, the DPI, BFOR, and SPFI modeling bases can be used to obtain rotationally invariant *q*-space indices. Here, we look at three such measures: *Po*, MSD, and GFA.

*Po* = *P*(**p** = 0) is the probability density that a water molecule returns back to its initial position within the diffusion time (Assaf et al., 2000; Wu and Alexander, 2007). In a healthy adult brain, *Po* is greater in white matter (WM) than gray matter (GM) because WM has more restricting barriers including multi-layer myelin sheaths, axonal membranes, and microtubules. Such restrictivity increases the likelihood of a water molecule returning to its original position, whereas it has a very low probability of returning to its original position in areas of unrestricted isotropic diffusion (CSF). Hence, *Po* can be viewed as a measure of restricted diffusion. Several studies have shown *Po* to be sensitive to brain pathology, and suggesting that changes in myelin are the primary mechanism for differences in *Po* (Assaf et al., 2002; Bar-Shir et al., 2009; Wu et al., 2011).

*Po* can be evaluated either numerically or analytically. The authors in (Wu et al., 2008) computed *Po* by numerically summing the normalized diffusion signal *E*(**q**) over all diffusion measurements in *q*-space, and then correcting the sum by the sampling density. Analytical formulations of *Po* were derived for the SPFI and DPI signal bases (Cheng et al., 2010b; Descoteaux et al., 2011). Similarly, an analytical *Po* expression can be obtained using the BFOR basis, which is derived in Appendix D:

$${Po}_{\mathit{BFOR}}=2\sqrt{\pi}{\tau}^{3}\sum _{n=1}^{N}{C}_{n1}\frac{{(-1)}^{n+1}}{{\alpha}_{n0}^{2}}$$

(13)

The MSD, which we will denote as *p*^{2}, is simply the second moment of the EAP (Wu and Alexander, 2007): *p*^{2} = ∫*p*^{2}*P*(**p**)*d*^{3}**p**. It is related to the MD, which in the case of Gaussian diffusion is given by the well-known Einstein relation *p*^{2} = 6(Δ − *δ*/3)MD. Thus far, an analytical formulation of MSD exists only within the DTI framework. It is calculated numerically in *q*-space imaging, either by extracting the full width at half maximum of the EAP (Assaf et al., 2000) or taking the geometric mean of the diffusion signal over all directions on a HDYI shell (Wu et al., 2008) In Appendix E, we will show that, in general, the MSD is related to the diffusion signal by the relation

$$\langle {p}^{2}\rangle =\frac{-1}{4{\pi}^{2}}{\nabla}^{2}E(\mathbf{q}){\mid}_{\mathbf{q}=0}$$

(14)

Since DPI assumes ^{2}*E* = 0, it predicts the MSD to be zero: *p*^{2}* _{DP I}* = 0. The BFOR MSD is (derived in Appendix E)

$${\langle {p}^{2}\rangle}_{\mathit{BFOR}}=\frac{1}{8{\pi}^{{\scriptstyle \frac{5}{2}}}{\tau}^{2}}\sum _{n=1}^{N}{C}_{n1}{\alpha}_{n0}^{2}$$

(15)

Tuch in (Tuch, 2004) introduced the concept of GFA and defined it as std(ODF)/rms(ODF). Since ODF is only a feature of the EAP, the subsequent GFA map is derived solely from the angular content of the diffusion profile. Incorporating both the angular and radial contents of the diffusion profile into the definition of GFA will result in a radial dial of GFA maps, illustrating how anisotropy varies with diffusion displacement *p*. Therefore, we define a new GFA:

$$\mathit{GFA}(p={p}_{o})=\frac{\mathit{std}\phantom{\rule{0.16667em}{0ex}}[P(p={p}_{o},\mathbf{r})]}{\mathit{rms}\phantom{\rule{0.16667em}{0ex}}[P(p={p}_{o},\mathbf{r})]}$$

(16)

Another advantage of Eq. (16) is that it is better suited for multiple diffusion weighted MR experiments, unlike Tuch’s definition, which is single-shell HARDI-based. In order to capture the 3D anisotropy of the EAP-defined GFA maps, 1000 uniformly distributed vertices on a unit sphere in propagator space (i.e. 1000 values of *θ*′ and ′) were acquired using the approach described in (Wong and Roos, 1994).

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 dataset corresponds to a different *b*-value. Across all *k* shells, we have total of *M* diffusion measurements (including the *b* = 0 measurement). Hence, from these multiple shell datasets, we want to reconstruct the EAP, *P*(**p**).

The task is to estimate coefficients *C _{nj}* in Eq. (12) from the observed signal

$$\mathbf{z}=\left(\begin{array}{ccccccccc}{z}_{1,1}({q}_{1},{\mathbf{u}}_{1})& {z}_{2,1}({q}_{1},{\mathbf{u}}_{1})& \cdots & {z}_{N,1}({q}_{1},{\mathbf{u}}_{1})& \cdots & {z}_{1,R}({q}_{1},{\mathbf{u}}_{1})& {z}_{2,R}({q}_{1},{\mathbf{u}}_{1})& \cdots & {z}_{N,R}({q}_{1},{\mathbf{u}}_{1})\\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ {z}_{1,1}({q}_{M},{\mathbf{u}}_{M})& {z}_{2,1}({q}_{M},{\mathbf{u}}_{M})& \cdots & {z}_{N,1}({q}_{M},{\mathbf{u}}_{M})& \cdots & {z}_{1,R}({q}_{M},{\mathbf{u}}_{M})& {z}_{2,R}({q}_{M},{\mathbf{u}}_{M})& \cdots & {z}_{N,R}({q}_{M},{\mathbf{u}}_{M})\end{array}\right)$$

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 yielding vector **Ĉ** given by

$$\widehat{\mathbf{C}}={({\mathbf{Z}}^{T}\mathbf{Z}+{\lambda}_{l}{\mathbf{L}}_{\mathit{reg}}+{\lambda}_{n}{\mathbf{N}}_{\mathit{reg}})}^{-1}{\mathbf{Z}}^{T}\mathbf{S},$$

(17)

where *L _{reg}* is the Laplace-Beltrami regularization diagonal matrix with

Lastly, from the estimated vector **Ĉ**, we can extract the *C _{nj}* coefficients needed to compute the EAP,

An important point to consider in the implementation is how to determine the parameter *τ* in the signal basis. In practice, the diffusion signal is bounded by the maximal *q*-value *q _{max}* achievable by the imaging system. The authors in (Assaf and Cohen, 1998) have shown that, depending on the length of diffusion time, the amount of signal present at

The synthetic and *in vivo* datasets use a hybrid, non-Cartesian sampling scheme (Wu and Alexander, 2007), shown in Table 1. Since EAP reconstruction is sensitive to angular resolution, the number of encoding directions is increased with each shell to increase the angular resolution with the level of diffusion weighting. The number of directions in the outer shells were increased to better characterize complex tissue organization. Diffusion tensor elements for measurements in the second shell were calculated using non-linear least squares estimation with the *Camino* software package (Cook et al., 2006), which were then used to obtain the FA and principal eigenvector.

The mono-exponential (also referred to as mono-Gaussian) mixture model (Tuch et al., 2002) is frequently used to generate synthetic data to validate a given EAP reconstruction, such as in (Assemlal et al., 2009a; Cheng et al., 2010b), where the maximum *b*-value used was 3000 s/mm^{2}. However, diffusion MR imaging experiments using high *b*-values (> 2000 s/mm^{2}) have shown that the diffusion signal decay is no longer mono-exponential. Studies in normal human brain, with *b*-values over an extended range of up to 6000 s/mm^{2}, have shown that the signal decay is better described with a bi-exponential i.e. bi-Gaussian curve (Mulkern et al., 1999; Clark and Le Bihan, 2000). Similar findings were made for rat brain, using multiple *b*-values of up to 10000 s/mm^{2} (Niendorf et al., 1996). According to (Assaf and Cohen, 1998), a bi-exponential fit gives very good agreement with the observed water signal attenuation in excised brain tissue from rats for *b*-values of up to 2 – 3 × 10^{4} s/mm^{2}. Thus, BFOR, SPFI, and DPI were applied to simulations of crossing fiber configurations generated by a bi-Gaussian mixture model. Fig. 2 illustrates mono-exponential and bi-exponential decay curves, where the latter has a pronounced tail at high *q* values, indicating 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 curve can be viewed as the fast and slow diffusion components, respectively (Clark and Le Bihan, 2000; Maier et al., 2004).

Plots of diffuison signal attenuation as a function of *b*-value illustrating mono-exponential and bi-exponenial decays. The tail of the bi-exponential can be interpreted as the slow diffusion component, while the head the fast diffusion component.

In bi-Gaussian mixture,

$$E(q,\mathbf{u})=\sum _{k=1}^{{N}_{b}}[{f}_{kf}{e}^{-{b\mathbf{u}}^{T}{\mathbf{D}}_{kf}\mathbf{u}}+{f}_{ks}{e}^{-{b\mathbf{u}}^{T}{\mathbf{D}}_{ks}\mathbf{u}}],$$

(18)

where *N _{b}* is the total number of simulated fibers,

$$P(p,\mathbf{r})=\sum _{k=1}^{{N}_{b}}[\frac{{f}_{kf}}{\sqrt{{(4\pi \epsilon )}^{3}\mid {\mathbf{D}}_{kf}\mid}}{e}^{-{p}^{2}{\mathbf{r}}^{T}{\mathbf{D}}_{kf}^{-1}\mathbf{r}/4\epsilon}+\frac{{f}_{ks}}{\sqrt{{(4\pi \epsilon )}^{3}\mid {\mathbf{D}}_{ks}\mid}}{e}^{-{p}^{2}{\mathbf{r}}^{T}{\mathbf{D}}_{ks}^{-1}\mathbf{r}/4\epsilon}],$$

(19)

where *ε* = Δ − *δ*/3. For the synthetic data, the diffusion gradient duration is *δ* = 45 ms and diffusion gradient separation Δ = 56 ms.

In reconstructing the EAP, we look at two equally weighed fibers crossing at 60°, and set eigenvalues of each diffusion tensor to be [1.6,0.4,0.4]e-3, which gives an FA value of 0.7071. The values of the fast and slow Gaussian diffusion functions were taken from (Maier et al., 2004) and are shown in Table 2. Monte Carlo noise simulations were then performed to investiage the effect of SNR on the estimation of *Po*, MSD, and GFA for a single voxel for each EAP method. Seven SNR levels ([10 20 30 40 50 60 100]) for the *b* = 0 image were simulated, 1000 times each, by adding Rician noise in a similar manner as in (Descoteaux et al., 2007) for four different scenerios: a fast isotropic component (*D* = 0.00115 mm^{2}/s); a slow isotropic component (*D* = 0.00045 mm^{2}/s); fast anisotropic components of a corpus callosum fiber and internal capsule fiber crossing at 60°; and the slow anisotropic components for the previous scenario. The BFOR parameters are {*L* = 4, *N* = 6, *τ* = 91.2 mm^{−1}, *λ _{l}* = 10

Fast/Slow Diffusion ADCs & Component Size Fractions (from (Maier et al., 2004))

HYDI was performed on a healthy, adult human using a 3.0 T GE-SIGNA scanner with an 8-channel head coil and ASSET parallel imaging. The DW pulse sequence was a single-shot, spin-echo, echo-planar imaging (SS-SE-EPI) with pulse-oximeter gating. The MR parameters were as follows: TE = 122 ms, TR 12 s, FOV = 256 mm, matrix = 128 × 128, voxel size = 2 × 2 mm^{2}, 30 slices with slice thickness = 3 mm, and a total scan time of about 30 min. Diffusion parameters were maximum *b*-value *b _{max}* = 9375 s/mm

BFOR, DPI, and SPFI are first applied to the numerical phantom and then on the real dataset. The numerical phantom is used to validate BFOR, compare its performance to those of DPI and SPFI, assess all three methods’ robustness in estimating the scalar measures *Po*, MSD, and GFA, and answer the following questions: (1) Can these methods properly reconstruct a diffusion signal acquired via hybrid sampling? (2) How does the slow diffusion component affect the EAP reconstruction and the estimations of the scalar quantities? (3) What can be done to reduce the effects of Gibbs ringing on the EAP reconstructions? It is also important to note that the EAP and quantitative scalar measures were reconstructed using only 125 diffusion measurements, while those presented in (Descoteaux et al., 2011) used 256.

Fig. 3 displays the BFOR, DPI, and SPFI signal fit for each shell and the corresponding ground truth. The BFOR signal basis fits the diffusion signal nearly perfectly for all shells. Both SPFI and DPI reasonably fit the diffusion signal for *b* ≥ 1500 s/mm^{2}, but poorly reconstruct the inner most shell (*b* = 375 s/mm^{2}). As expected, DPI also tends to oversmooth the diffusion signal, especially so at *b* = 3375, 6000 s/mm^{2}. Results of DPI signal fitting were also reported in (Descoteaux et al., 2011), where unform sampling along four spherical shells (*b* = 2000, 4000, 6000, and 8000 s/mm^{2}) was done. Although a hybrid sampling scheme is used here, our results are consistent with those of (Descoteaux et al., 2011) for *b* ≥ 1500 s/mm^{2}. SPFI’s and DPI’s poor signal fit of the diffusion signal in inner most shell may be due to the fact that only six measurements were acquired in this shell (see Table 1), which may be inadequate for their respective radial bases.

Fig. 4 shows the EAP reconstruction for the fast diffusion component across propagator space using each method. Modeling the fast component (i.e. head of the bi-exponential in Fig. 2) is tantamount to fitting a mono-exponential curve, and so the EAP reconstruction for the fast diffusion component can be viewed as if the diffusion signal decay was mono-exponential. Both BFOR and SPFI model the fast component EAP very well, accurately capturing the geometry and orientation of the EAP profile, and the BFOR reconstruction is nearly identical to that of SPFI. DPI performs reasonably well, but tends to overestimate the EAP.

Reconstruction of the fast component EAP (red) using BFOR, SPFI, and DPI compared with the ground truth (green). Two equally weighted WM fibers were simulated crossing at 60°.

Fig. 5 shows the EAP reconstruction for the slow diffusion component, which can be viewed as modeling the tail of the bi-exponential curve in Fig. 2. Note that the BFOR and SPFI reconstructions are quite alike. At *p* = 1 *μ*m, all three methods capture the correct geometry of the ground truth EAP profile, but underestimate it, DPI more so. At *p* = 5 *μ*m, all three methods are unsuccessful in capturing the correct geometry, in particular failing to capture the peaks of the ground truth EAP profile. However, they do capture the correct orientation. At *p* = 10 and especially *p* = 15 *μ*m, the BFOR and SPFI EAP 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* = 15 *μ*m benefits from the inherent smoothening of Laplacian signal modeling, with no spurious peaks present. Although oversmoothened and overestimated, it does a much better job than BFOR and SPFI in resolving the correct fiber orientation at *p* = 15 *μ*m. 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?

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, 2002), and in the case of DSI, significantly reduce the cumbersome *q*-space sampling (Yeh et al., 2008). By linearly damping the signal measurments 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 0.7, 0.4, and 0.1 to form the three ‘pseudo-shells.’ The BFOR and SPFI scaling factors, *τ* and *ζ*, respectively, were changed for the extrapolation to *τ* = 136.8 mm^{−1} and *ζ* = 1100. Note that the *q*-space sampling interval Δ*q* was not changed for the extrapolation.

Fig. 6 shows that signal extrapolation improves the reconstruction of the slow EAP component for both BFOR and SPFI. At *p* = 10 *μ*m, the BFOR and SPFI slow EAP reconstructions with signal extrapolation, although not perfectly capturing the ground truth geometry, better capture the angular features of the ground truth than those without signal extrapolation. The biggest improvement, however, is seen at *p* = 15 *μ*m, where the pronounced Gibbs ringing is greatly reduced by the signal extrapolation. In particular, the BFOR and SPFI slow EAP reconstructions with signal extrapolation at *p* = 15 *μ*m are not spiky and much closer to the ground truth, although their orientations are slightly off, than those without extrapolation (Fig. 5). Note that both the BFOR and SPFI slow EAP reconstructions with extrapolation are quite alike. The BFOR fast EAP reconstruction was not affected by extrapolation, being nearly idential to its counterpart without extrapolation. However, the SPFI fast EAP reconstruction with extrapolation was moderately less acurate than that without extrapolation. In general, signal extrapolation can significantly improve EAP reconstructions at larger diffusion displacements (i.e. *p* = 15 *μ*m).

Fig. 7 shows the results for the *Po* measurements. Without signal extrapolation, BFOR and SPFI 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 BFOR and SPFI asymptotically approach the ground truth fast/slow isotropic *Po*, while DPI overestimates both. At low levels of SNR (e.g. 10 & 20), which is quite common in diffusion MRI, all three methods (without signal extrap.) have biased estimates of *Po*, though the variance is fairly small.

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 1000 trials.

When signal extrapolation is applied, the estimation of the slow anisotropic *Po* by BFOR and SPFI significantly improves, as shown in Fig. 7b. According to Fig. 7a, signal extrapolation does not asymptotically affect SPFI’s estimation of fast anisotropic *Po*, but slightly worsens that of BFOR’s. At low levels of SNR, however, signal extrapolation results in more severe overestimation of fast anisotropic *Po* than without signal extrapolation for both BFOR and SPFI.

Fig. 8 shows the results for the MSD measurements. BFOR estimates both the fast and slow anisotropic/isotropic MSD very well across SNR levels. However, at low levels of SNR, the variability (given by the standard deviation) of the BFOR estimation of MSD is quite large, indicating strong sensitivity to noise. SPFI without signal extrapolation severely underestimates anisotropic MSD, giving negative values for slow anisotropic MSD, which indicates that it will give inaccurate measurements of the MSD of WM. SPFI also underestimates fast isotropic MSD, but asymptotically approaches ground truth slow isotropic MSD. Interestingly, there is less variability in SPFI’s estimation of MSD than BFOR.

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 1000 trials.

When signal extrapolation is applied to the MSD measurements, SPFI’s estimations of slow anisotropic and slow isotropic MSD significantly improve, well-estimating them across SNR levels, as shown in Figs. 8b and 8d, respectively. Specifically, the SPFI slow anisotropic MSD estimation is no longer negative with signal extrapolation. BFOR’s estimations of slow anisotropic and slow isotropic MSD with extrapolation are nearly identical to those without it. However, the variability of the BFOR and SPFI MSD measurements is much less than those without extrapolation. A negative consequence of signal extrapolation is that it increases the inaccuracy (i.e. asymptotically worsening) of the BFOR/SPFI fast anisotropic and fast isotropic MSD estimations, all of which are greatly underestimated. The simulations indicate signal extrapolation may be more beneficial to SPFI’s estimation of MSD than that of BFOR’s.

Fig. 9 shows the results for the GFA measurements. Asymptotically, BFOR and SPFI without signal extrapolation approach the ground truth fast anisotropic GFA, while DPI underestimates it. Across SNR levels, DPI severely underestimates slow anisotropic GFA, while both BFOR and SPFI greatly overestimate it. For the case of isotropic diffusion, where GFA=0, the GFA estimated by each method approaches zero at high SNR. However, at SNR=10, both SPFI and BFOR severely overestimate the isotropic GFA, giving values comparable to the GFA of WM. DPI’s estimation of the isotropic component is more robust to noise than BFOR and SPFI.

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* = 10 *μ*m. Error bars denote one standard deviation across 1000 trials.

SPFI with signal extrapolation still overestimates slow anisotropic GFA, though slightly less so than without it, but the extrapolation increases the estimation’s variability at the same time. Signal extrapolation has negligible effects on BFOR’s estimation of slow anisotropic GFA. Both BFOR’s and SPFI’s estimation of fast anisotropic GFA are 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 SNR=10, 20, & 30 than without it. Based on Fig. 9c, both SPFI and BFOR with signal extrapolation overestimate isotropic GFA, across SNR levels, to a larger extent than without extrapolation, implying that extrapolation is quite sensitive to noise in CSF regions.

In Fig. 10, a 4 × 4 ROI was drawn on the splenium of corpus cal-losum. The EAP profiles reconstructed at *p* = 10 *μ*m by each method have the fundamental peanut shape of a single fiber. Note that the BFOR and SPFI reconstructions in both cases are very similar. We see that application of smoothing *t* = 550 removes the center peaks of the BFOR EAP profiles. Whether these center peaks in the EAP profiles are the result of Gibbs ringing (i.e. artifical peaks) or describe some underlying biological process is an open question.

In Fig. 11, a 4 × 9 ROI was drawn in a region of fiber crossing, where the EAP profiles were reconstructed at *p* = 10 *μ*m. Although not identical, the BFOR and SPFI 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. 11d, the application of smoothing *t* = 60 to BFOR removes the center peaks from several voxels, but at the expense of slight angular smoothening of EAP profiles themselves.

Axial slice of FA map of adult human brain at *b*=1500 s/mm^{2} (second shell), where a 4 × 9 ROI is drawn on a region of crossing fibers. Plotted is the EAP profile at *p* = 10 *μ*m overlaid on FA map in 4 × 9 ROI using (c) BFOR at *t* = **...**

Fig. 12 shows the reconstructed EAP profiles for the same crossing fiber region, but at *p* = 15 *μ*m. Fiber crossing configurations are recovered and well discriminated in the EAP for each method. Unlike at *p* = 10 *μ*m, the DPI EAP reconstruction at *p* = 15 *μ*m is sharper and does not suffer from oversmoothening. In fact, as the propagator radius *p* increases, the angular resolution improves, at the enpense of the EAP profiles becoming spiky, as is evident in Figs. 12a, 12e, and 12g. When a smoothing of *t* = 60 is applied to the BFOR EAP reconstruction (without signal extrapolation), the subsequent EAP profiles are still spiky. At *t* = 350, the spikiness is smoothed out, but many of the WM voxels have EAP reconstructions significantly differing, with respect to orientation and geometry, from those at *t* = 0. Figs. 12d and 12f show the BFOR and SPFI EAP profiles reconstructed at *p* = 15 *μ*m with signal extrapolation, respectively, which are much less spiky than the corresponding ones without signal extrapolation, which is consistent with the synthetic results shown in Fig. 6. The signal extrapolation also smoothes the reconstructed EAP profiles, but unlike BFOR at *t* = 350, none of the WM voxels are oversmoothed to such a degree that their EAP profiles have oblate shapes. Unlike BFOR at *t* = 350, the underlying EAP geometry and orientation of the BFOR/SPFI reconstructions with signal extrapolation are fairly consistent to those without extrapolation (at *t* = 0). As observed at *p* = 10 *μ*m, the BFOR and SPFI EAP reconstructions at *p* = 15 *μ*m are quite similar, which is consistent with the synthetic data results.

Table 3 shows the mean index value and corresponding standard deviation for genu and splenium of corpus callosum (WM) and putamen (GM). Three 4 × 4 ROIs were drawn on both the genu and splenium and one such ROI on both the left and right putamen, across several slices. The table shows that the SPFI MSD (without signal extrapolation) erroneously gives negative values for the MSD of genu and splenium. With extrapolation, the SPFI MSD of genu and splenium are positive.

Based on the numerical simulations, signal extrapolation was applied to both BFOR’s and SPFI’s estimation of *Po*. Fig. 13 displays an axial slice of *Po* generated by each method. In the first row, we show *Po* computed from BFOR, SPFI, DPI, and numerically (Wu et al., 2008). The BFOR, SPFI, 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 2 for BFOR and SPFI, while less than 2 for DPI.

The MSD maps computed from BFOR, SPFI, DPI, and numerically (Wu et al., 2008) are shown in Fig. 14. Both the BFOR 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 BFOR MSD values for WM and GM are quite similar. In the SPFI (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 SPFI severely underestimates the MSD of WM. Signal extrapolation has the effect of enforcing the positivity constraint on the MSD for SPFI. However, both the BFOR and SPFI MSD maps with signal extrapolation have poor tissue/CSF contrast because of the noise induced by the signal extrapolation. With regards to BFOR, both the synthetic and *in vivo* data suggest that it’s best not to use signal extrapolation in estimating MSD. SPFI, 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 constrast of the MSD DPI map is completely inverted, with WM appearing bright and CSF dark.

Fig. 16 displays axial slices of the GFA computed at *p* = 5, 10, and 15 *μ*m 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 *μ*m, the anisotropy of corpus callosum is lower with respect to levels seen in DTI. At *p* = 10 *μ*m, the corpus callosum is very anisotropic, as can be seen from Table 3, indicating that *p* = 10 *μ*m is a diffusion displacement worth reconstructing the EAP at. The GFA at *p* = 15 *μ*m is more noisy, which is due to truncation of signal basis at high *q*-values and 15 *μ*m being well beyond the resolvable resolution (of diffusion displacement) limit. The BFOR and SPFI GFA maps without signal extrapolation are very similar, while WM regions in the DPI computed GFA maps at *p* = 5 and 10 *μ*m have lower intensity than those of BFOR and SPFI, which is consistent with the underestimation of GFA(10) by DPI observed in the Monte Carlo noise simulations (Fig. 9). CSF regions in the BFOR GFA(10) 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 SPFI, however, the noise level is very severe in CSF regions in the GFA(5) & GFA(10) maps with signal extrapolation than those without it.

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 *in vivo* data, DPI tends to greatly oversmoothen the EAP, especially *p* = 10 *μ*m, but performs well at *p* = 15 *μ*m 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. (14)). Fig. 9 indicates that DPI greatly underestimates GFA(10), which is reflected in Fig. 16 and Table 3. In addition, the DPI signal basis is only applicable at *q* > 0, which is unrealistic since the diffusion signal is defined at *q* = 0.

The SPFI signal basis possesses several advantages in that it radially decays to zero and has no singularity at *q* = 0. In addition, the EAP is derived via integration over the entire *q*-space, unlike BFOR and DPI, where the *q*-space integration is up to a certain bound that is related to *q _{max}*. Interestingly, however, the SPFI EAP reconstructions for both synthetic and real datasets are quite similar to those of BFOR, suggesting the two methods may be inherently related. According to the synthetic data, signal extrapolation greatly improves the SPFI EAP reconstruction at

The main limitation of the BFOR 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 (*τ* being the upperbound) to retrieve the EAP. However, based on Fig. 3, BFOR outperforms DPI and SPFI in modeling the diffusion signal. Heat diffusion smoothening helps in removing potentially spurious peaks, and signal extrapolation significantly improves the EAP reconstruction at *p* = 15 *μ*m. According to both the synthetic and real data, BFOR gives reasonable estimates of all three *q*-space indices.

The slow component of diffusion is the most sensitive to truncation artificats, which can induce severe Gibbs ringing and adversely affect the orientation of reconstructed EAP. In this paper, signal extrapolation was proposed as a means to mitigate the effects of such artifacts, and was oberseved to be most effective at higher radii (i.e. *p* = 15 *μ*m), where the effects of signal truncation artifacts are most pronounced. The significant improvement in the BFOR/SPFI EAP reconstruction at *p* = 15 *μ*m via linear signal extrapolation hence suggests that extrapolation may be a useful preprocessing step for EAP reconstruction at large diffusion diplacements. Signal extrapolation also greatly improves the accuracy of the BFOR/SPFI *Po* estimation, according to the synthetic data. However, signal extrapolation increases the severity of noise in CSF regions in the GFA and MSD maps for both BFOR & SPFI, as evidenced by the 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. 10, a smoothing of *t* = 550 was applied to splenium at *p* = 10 *μ*m to remove the center peaks. However, for a crossing fiber region at *p* = 10 *μ*m, a smoothing of *t* = 60 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, 1999). 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 BFOR framework does not require such a scheme. BFOR 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., 2009b; Merlet et al., 2011; Caruyer et al., 2011). Future work includes optimizing the q-space sampling and applying compressed sensing to BFOR.

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 function) from *q*-space diffusion signal (Tournier et al., 2004). Mathematically, the angular convolution is given by

$$E(q,\mathbf{u})={\int}_{{S}^{2}}F(\mathbf{r})K(q,\mathbf{u},\mathbf{r}){d}^{2}\mathbf{r},$$

(20)

where *F*(**r**) is the fODF and *K* the kernel. The derivation of the fODF using the BFOR, SPFI, and DPI signal bases is worth exploring in the future.

In any future clinical study employing HYDI to examine brain pathology, where rotationally invariant indices like GFA and *Po* can be used to assess changes between diseased and normal subjects, voxel-wise analysis is desired. However, spatial normalization of multiple b-value datasets is no easy task. Recently, the authors in (Du et al., 2012) proposed a registration algorithm to align HARDI datasets using the ODFs. Specifically, the algorithm seeks an optimal diffeomorphism of large deformation between two ODF fields across a spatial volume domain and at the same time, locally reorients an ODF in a manner consistent with the underlying anatomical structure. HYDI images could be aligned using the same algorithm, except replacing the ODF with the EAP.

The MSD measure is quite sensitive to noise (Assaf and Cohen, 2000; Wu and Alexander, 2007). The authors in (Wu et al., 2008) proposed an alternative measure to MSD called the *q*-space inverse variance (QIV), which is a pseudo-diffusivity measure. Mathematically, the QIV is defined as

$$\mathit{QIV}={\left[\int {q}^{2}E(\mathbf{q}){d}^{3}\mathbf{q}\right]}^{-1}$$

(21)

The QIV can thus be interpreted as the inverse of the “variance” of *q* (i.e. *QIV* = 1/*q*^{2}). It is not a real variance in the statistical sense because *E*(**q**) does not constitute a probability density function. The QIV is not an arbitrary meausre, but related to the EAP in a manner analogous to which the MSD is related to the diffusion signal - in Appendix F, we will show that
${\mathit{QIV}}^{-1}={\scriptstyle \frac{-{\nabla}^{2}P(\mathbf{p}){\mid}_{\mathbf{p}=0}}{4{\pi}^{2}}}$. The QIV within the BFOR framework is (see Appendix F for derivation)

$${\mathit{QIV}}_{\mathit{BFOR}}=\frac{1}{2\sqrt{\pi}{\tau}^{5}{\sum}_{n=1}^{N}{(-1)}^{n}{C}_{n1}\frac{(6-{\alpha}_{n0}^{2})}{{\alpha}_{n0}^{4}}}$$

(22)

Fig. 15 displays an axial slice of the BFOR QIV, illusrating rich tissue/CSF contrast. The tissue/CSF contrast in the QIV is more enhanced than that of the MSD, and unlike the MSD, the QIV map also exhibits WM/GM contrast (the right and left putamen are visible in Fig. 15 but not in the MSD maps). According to Table 3, the QIV of the corpus callosum is about a third of that of the putamen.

We have introduced a new orthonormal basis to model the *q*-space diffusion signal and from which the EAP can be analytically reconstructed using hybrid, non-Cartesian sampling with multiple *q*-shell measurements. BFOR is a linear and efficient reconstruction based on heat equation estimation of the diffusion signal. Compared to DSI, BFOR employs much fewer diffusion measurements. Rotationally invariant *q*-space indices such as GFA, *Po*, and MSD can then be obtained using the derived EAP.

- Propose a new analytical diffusion propagator scheme called BFOR.
- Derive rotationally invariant
*q*-space indices using BFOR, like*Po*, MSD, & GFA. - Validate BFOR using HYDI-acquired datasets, both synthetic and
*in vivo*. - Compare BFOR to existing analytical diffusion propagator schemes, like DPI and SPFI.
- Explore ways to mitigate the effects of Gibbs ringing on propagator reconstruction.

The authors are thankful to Cheng Guan Koay and Steve Kecskemeti of the University of Wisconsin-Madison for insightful discussions on the estimation of the diffusion propagator.

We want to solve the following boundary value partial differential equation:

$$\left[\frac{1}{{q}^{2}}\frac{\partial}{\partial q}({q}^{2}\frac{\partial}{\partial q})+\frac{1}{{q}^{2}sin\theta}\frac{\partial}{\partial \theta}(sin\theta \frac{\partial}{\partial \theta})+\frac{1}{{q}^{2}{sin}^{2}\theta}\frac{{\partial}^{2}}{\partial {\phi}^{2}}\right]\phantom{\rule{0.16667em}{0ex}}{\psi}_{i}(\mathbf{q})=-{\lambda}_{i}{\psi}_{i}(\mathbf{q}),\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\psi (q=\tau ,\theta ,\phi )=0$$

(Appendix A.1)

where we require *λ* > 0. Substituting the separable solution of the form

$$\frac{{q}^{2}}{f}\frac{{d}^{2}}{{dq}^{2}}f+\frac{2q}{f}\frac{d}{dq}f+{q}^{2}\lambda =-\frac{{\mathrm{\Delta}}_{LB}h}{h}=l(l+1),$$

(Appendix A.3)

where
${\mathrm{\Delta}}_{LB}={\scriptstyle \frac{1}{sin\theta}}{\scriptstyle \frac{\partial}{\partial \theta}}(sin\theta {\scriptstyle \frac{\partial}{\partial \theta}})+{\scriptstyle \frac{1}{{sin}^{2}\theta}}{\scriptstyle \frac{{\partial}^{2}}{\partial {\phi}^{2}}}$ is the Laplace-Bertrami operator and *l* is some real-valued constant.

We first solve for the second equation in (Appendix A.3):

$${\mathrm{\Delta}}_{LB}h+l(l+1)h=0$$

(Appendix A.4)

The solutions to Eq. (Appendix A.4) are the spherical harmonics $(\text{SH})\phantom{\rule{0.16667em}{0ex}}{Y}_{l}^{m}(\theta ,\phi )$.

The second equation in (Appendix A.3) can be written as

$${q}^{2}\frac{{d}^{2}}{{dq}^{2}}f+2q\frac{d}{dq}f+[{q}^{2}\lambda -l(l+1)]f=0$$

(Appendix A.5)

Defining a new variable $f(q)=\sqrt{{\scriptstyle \frac{\pi}{2\sqrt{\lambda}q}}}F(q)$, we can transform Eq. (Appendix A.5) to

$${q}^{2}\frac{{d}^{2}}{{dq}^{2}}F+q\frac{d}{dq}F+[{q}^{2}\lambda -{(l+1/2)}^{2}]F=0,$$

(Appendix A.6)

which is simply a sclaed version of the Bessel differential equation. The only bounded solution at the origin is given in terms of the Bessel funciton of the first kind as
$F(q)={J}_{l+1/2}(\sqrt{\lambda}q)$. The solution to Eq. (Appendix A.5) is then
$f(q)=\sqrt{{\scriptstyle \frac{\pi}{2\sqrt{\lambda}q}}}{J}_{l+1/2}(\sqrt{\lambda}q)={j}_{l}(\sqrt{\lambda}q)$, where *j _{l}* is the spherical Bessel function of the first kind and we invoke the relation
${j}_{l}(x)=\sqrt{{\scriptstyle \frac{\pi}{2x}}}{J}_{l+1/2}(x)$.

Imposing the boundary condition from Eq. (Appendix A.1), we have
${j}_{l}(\sqrt{\lambda}\tau )=0$. Defining *α _{nl}* as the

Multiplying the spherical Bessel functions and the spherical harmonics together, we obtain the eigenfuctions (i.e. our orthonormal basis) to (Appendix A.1): ${Z}_{\mathit{nlm}}(q,\theta ,\phi )={j}_{l}(\sqrt{{\lambda}_{nl}}q){Y}_{l}^{m}(\theta ,\phi )$. Thus, the complete set of solutions to Eq. (Appendix A.1) is

$$\psi (q,\theta ,\phi )\approx \sum _{n=1}^{N}\sum _{l=0}^{L}\sum _{m=-l}^{l}{C}_{\mathit{nlm}}{j}_{l}(\frac{{\alpha}_{nl}q}{\tau}){Y}_{l}^{m}(\theta ,\phi ),$$

(Appendix A.7)

where *N* is the truncation order of the number of roots of spherical Bessel function and *L* the truncation order of the SH.

Now, to derive the diffusion signal, we make two important assumptions. First, we assume the diffusion signal *E*(**q**) is a solution to the heat equation:

$${\nabla}^{2}E(\mathbf{q},t)=\frac{\partial E}{\partial t},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}E(\mathbf{q},t=0)=H(\mathbf{q}),$$

(Appendix A.8)

where *H*(**q**) is simply the acquired signal. Second, we assume that the diffusion signal can be expressed as a linear combination of the orthonormal basis derived in Eq. (Appendix A.7):

$$E(q,\mathbf{u},t)=\sum _{n=1}^{N}\sum _{l=0}^{L}\sum _{m=-l}^{l}{C}_{\mathit{nlm}}{g}_{\mathit{nlm}}(t){j}_{l}(\frac{{\alpha}_{nl}q}{\tau}){Y}_{l}^{m}(\mathbf{u})$$

(Appendix A.9)

Substituting (Appendix A.9) back into (Appendix A.8), we obtain

$$\sum _{n=1}^{N}\sum _{l=0}^{L}\sum _{m=-l}^{l}{C}_{\mathit{nlm}}{j}_{l}(\frac{{\alpha}_{nl}q}{\tau}){Y}_{l}^{m}(\mathbf{u})\phantom{\rule{0.16667em}{0ex}}[-\frac{{\alpha}_{nl}^{2}}{{\tau}^{2}}{g}_{\mathit{nlm}}(t)-\frac{d}{dt}{g}_{\mathit{nlm}}(t)]=0$$

(Appendix A.10)

A unique solution exists if and only if ${g}_{nl}(t)={b}_{nl}{e}^{-\frac{{\alpha}_{nl}^{2}t}{{\tau}^{2}}}$ (Chung et al., 2007), and so

$$E(q,\mathbf{u},t)=\sum _{n=1}^{N}\sum _{l=0}^{L}\sum _{m=-l}^{l}{C}_{\mathit{nlm}}{e}^{-\frac{{\alpha}_{nl}^{2}t}{{\tau}^{2}}}{j}_{l}(\frac{{\alpha}_{nl}q}{\tau}){Y}_{l}^{m}(\mathbf{u})$$

(Appendix A.11)

Note that all constants are absorbed into *C _{nlm}*. In the following sections, we will use the SH basis

In diffusion weighted imaging (DWI), *E*(0) = 1. Thus, for our basis, we obtain the following identity:

$$E(0,t=0)=\sum _{n}\sum _{j}{C}_{nj}{j}_{l(j)}(0){Y}_{j}(\mathbf{u})=\frac{1}{\sqrt{4\pi}}\sum _{n}{C}_{n1}=1,$$

(Appendix B.1)

which holds for any **u** within the unit sphere *S*^{2} (i.e. **u** *S*^{2}). In deriving Eq. (Appendix B.1), we invoked a basic property of the spherical Bessel function that

$${j}_{l}(0)=\{\begin{array}{cc}1,& \text{if}\phantom{\rule{0.16667em}{0ex}}l=0\\ 0,& \text{if}\phantom{\rule{0.16667em}{0ex}}l\ne 0\end{array}$$

and the identity ${Y}_{0}^{0}={\scriptstyle \frac{1}{\sqrt{4\pi}}}$.

In the **Theory** section, we showed that

$$P(p,\mathbf{r},t)=4\pi \sum _{n=1}^{N}\sum _{j=1}^{R}{(-1)}^{l(j)/2}{C}_{nj}{e}^{\frac{-{\alpha}_{nl(j)}^{2}t}{{\tau}^{2}}}{Y}_{j}(\mathbf{r}){I}_{nl(j)}(p),$$

(Appendix C.1)

where
${I}_{nl(j)}(p)={\int}_{0}^{\tau}{q}^{2}{j}_{l(j)}(\frac{{\alpha}_{nl(j)}q}{\tau}){j}_{l(j)}(2\pi qp)dq$. We rewrite *I _{n}*

$${I}_{nl(j)}(p)=\frac{1}{2}\sqrt{\frac{\pi \tau}{2{\alpha}_{nl(j)}p}}{\int}_{0}^{\tau}{qJ}_{l(j)+1/2}(\frac{{\alpha}_{nl(j)}q}{\tau}){J}_{l(j)+1/2}(2\pi qp)dq$$

(Appendix C.2)

Recall the Bessel function of first kind *J _{k}*(

$$\left({x}^{2}\frac{{d}^{2}}{{dx}^{2}}+x\frac{d}{dx}+[{a}^{2}{x}^{2}-{k}^{2}]\right)\phantom{\rule{0.16667em}{0ex}}{J}_{k}(ax)=0$$

(Appendix C.3)

Thus, by definition of the Bessel function,
${J}_{l(j)+1/2}(\frac{{\alpha}_{nl(j)}q}{\tau})$ and *J _{l}*

$$\left(q\frac{{d}^{2}}{{dq}^{2}}+\frac{d}{dq}+[\frac{{\alpha}_{nl(j)}^{2}q}{{\tau}^{2}}-\frac{{(l(j)+1/2)}^{2}}{q}]\right)\phantom{\rule{0.16667em}{0ex}}{J}_{l(j)+1/2}(\frac{{\alpha}_{nl(j)}q}{\tau})=0$$

(Appendix C.4)

$$\left(q\frac{{d}^{2}}{{dq}^{2}}+\frac{d}{dq}+[4{\pi}^{2}{p}^{2}q-\frac{{(l(j)+1/2)}^{2}}{q}]\right)\phantom{\rule{0.16667em}{0ex}}{J}_{l(j)+1/2}(2\pi qp)=0,$$

(Appendix C.5)

respectively. Multiplying Eq. (Appendix C.4) by *J _{l}*

$$\begin{array}{r}{J}_{l(j)+1/2}(2\pi qp)\frac{d}{dq}\left[q\frac{d}{dq}{J}_{l(j)+1/2}(\frac{{\alpha}_{nl(j)}q}{\tau})\right]-{J}_{l(j)+1/2}(\frac{{\alpha}_{nl(j)}q}{\tau})\frac{d}{dq}\left[q\frac{d}{dq}{J}_{l(j)+1/2}(2\pi qp)\right]\\ =q\phantom{\rule{0.16667em}{0ex}}\left(4{\pi}^{2}{p}^{2}-\frac{{\alpha}_{nl(j)}^{2}}{{\tau}^{2}}\right)\phantom{\rule{0.16667em}{0ex}}{J}_{l(j)+1/2}(\frac{{\alpha}_{nl(j)}q}{\tau}){J}_{nl(j)+1/2}(2\pi qp)\end{array}$$

Integrating the above from *q* = 0 to *q* = *τ* via integration by parts and noting that *J _{l}*

$${\int}_{0}^{\tau}{qJ}_{l(j)+1/2}(\frac{{\alpha}_{nl(j)}q}{\tau}){J}_{l(j)+1/2}(2\pi qp)dq=\frac{\tau {J}_{l(j)+1/2}(2\pi \tau p)\phantom{\rule{0.16667em}{0ex}}\left[{\scriptstyle \frac{d}{dq}}{J}_{l(j)+1/2}({\scriptstyle \frac{{\alpha}_{nl(j)}q}{\tau}}){\mid}_{q=\tau}\right]}{4{\pi}^{2}{p}^{2}-{\scriptstyle \frac{{\alpha}_{nl(j)}^{2}}{{\tau}^{2}}}}$$

(Appendix C.6)

The right side of Eq. (Appendix C.6) can be simplified via the Bessel recurrence relations ${\scriptstyle \frac{d}{dx}}{J}_{k}(x)={\scriptstyle \frac{1}{2}}({J}_{k-1}(x)-{J}_{k+1}(x))$:

$${\int}_{0}^{\tau}{qJ}_{l(j)+1/2}(\frac{{\alpha}_{nl(j)}q}{\tau}){J}_{l(j)+1/2}(2\pi qp)dq=\frac{{\alpha}_{nl(j)}{J}_{l(j)+1/2}(2\pi \tau p)}{2\phantom{\rule{0.16667em}{0ex}}\left(4{\pi}^{2}{p}^{2}-\frac{{\alpha}_{nl(j)}^{2}}{{\tau}^{2}}\right)}({J}_{l(j)-1/2}({\alpha}_{nl(j)})-{J}_{l(j)+3/2}({\alpha}_{nl(j)}))$$

(Appendix C.7)

Using the Bessel recurrence relation
${J}_{k+1}(x)={\scriptstyle \frac{2k}{x}}{J}_{k}(x)-{J}_{k-1}(x)$, we obtain *J _{l}*

$${\int}_{0}^{\tau}{qJ}_{l(j)+1/2}(\frac{{\alpha}_{nl(j)}q}{\tau}){J}_{l(j)+1/2}(2\pi qp)dq=\frac{{\alpha}_{nl(j)}{J}_{l(j)-1/2}({\alpha}_{nl(j)}){J}_{l(j)+1/2}(2\pi \tau p)}{4{\pi}^{2}{p}^{2}-\frac{{\alpha}_{nl(j)}^{2}}{{\tau}^{2}}},$$

(Appendix C.8)

Thus, substituting Eq. (Appendix C.8) back into Eq. (Appendix C.2), we obtain

$${I}_{nl(j)}(p)=\frac{1}{2}\sqrt{\frac{\pi \tau}{2p}}\frac{\sqrt{{\alpha}_{nl(j)}}{J}_{l(j)-1/2}({\alpha}_{nl(j)}){J}_{l(j)+1/2}(2\pi \tau p)}{\left(4{\pi}^{2}{p}^{2}-\frac{{\alpha}_{nl(j)}^{2}}{{\tau}^{2}}\right)},$$

and so the diffusion propagator is then

$$P(p,\mathbf{r},t)=2\tau \sqrt{2{\pi}^{3}}\sum _{n=1}^{N}\sum _{j=1}^{R}{(-1)}^{l(j)/2}{C}_{nj}{e}^{\frac{-{\alpha}_{nl(j)}^{2}t}{{\tau}^{2}}}{Y}_{j}(\mathbf{r})\frac{\sqrt{{\alpha}_{nl(j)}}{J}_{l(j)-1/2}({\alpha}_{nl(j)}){j}_{l(j)}(2\pi \tau p)}{\left(4{\pi}^{2}{p}^{2}-\frac{{\alpha}_{nl(j)}^{2}}{{\tau}^{2}}\right)}$$

(Appendix C.9)

We can derive *Po* by evaluating Eq. (Appendix C.9) at *p* = 0:

$$\begin{array}{l}Po=P(p=0,\mathbf{r})=2\tau \sqrt{2{\pi}^{3}}\sum _{n=1}^{N}{C}_{n1}{Y}_{1}(\mathbf{r})\frac{\sqrt{{\alpha}_{n0}}{J}_{-1/2}({\alpha}_{n0})}{\frac{-{\alpha}_{n0}^{2}}{{\tau}^{2}}}=\frac{2\tau \sqrt{2{\pi}^{3}}}{\sqrt{4\pi}}\sum _{n=1}^{N}{C}_{n1}\frac{\sqrt{{\alpha}_{n0}}\sqrt{\frac{2}{\pi {\alpha}_{n0}}}cos({\alpha}_{n0})}{\frac{-{\alpha}_{n0}^{2}}{{\tau}^{2}}}\\ =2\sqrt{\pi}{\tau}^{3}\sum _{n=1}^{N}{C}_{n1}\frac{{(-1)}^{n+1}}{{\alpha}_{n0}^{2}},\end{array}$$

(Appendix D.1)

where we used the relation ${J}_{-1/2}(x)=\sqrt{{\scriptstyle \frac{2}{\pi x}}}cos(x)$.

We define the wave vector **q** as **q** = *q _{x}î* +

Since the diffusion signal and EAP are FT pairs, then the inversion of Eq. (1) gives

$$E(\mathbf{q})=\int P(\mathbf{p}){e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{p}=\int P(\mathbf{p}){e}^{2\pi i({q}_{x}{p}_{x}+{q}_{y}{p}_{y}+{q}_{z}{p}_{z})}{d}^{3}\mathbf{p}$$

(Appendix E.1)

Taking the second derivative of *E*(**q**) with respect to *q _{x}*,

$$\begin{array}{l}\frac{{\partial}^{2}E(\mathbf{q})}{\partial {q}_{x}^{2}}={(2\pi i)}^{2}\int {p}_{x}^{2}P(\mathbf{p}){e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{p}\\ \frac{{\partial}^{2}E(\mathbf{q})}{\partial {q}_{y}^{2}}={(2\pi i)}^{2}\int {p}_{y}^{2}P(\mathbf{p}){e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{p}\\ \frac{{\partial}^{2}E(\mathbf{q})}{\partial {q}_{z}^{2}}={(2\pi i)}^{2}\int {p}_{z}^{2}P(\mathbf{p}){e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{p}\end{array}$$

The sum of the derivatives is simply the Laplacian operator acting on *E*(**q**):

$${\nabla}^{2}E(\mathbf{q})=\frac{{\partial}^{2}E(\mathbf{q})}{\partial {q}_{x}^{2}}+\frac{{\partial}^{2}E(\mathbf{q})}{\partial {q}_{y}^{2}}+\frac{{\partial}^{2}E(\mathbf{q})}{\partial {q}_{z}^{2}}={(2\pi i)}^{2}\int {p}^{2}P(\mathbf{p}){e}^{2\pi i\mathbf{q}.\mathbf{p}}{d}^{3}\mathbf{p}$$

(Appendix E.2)

Note that the Laplacian of *E*(**q**) evaluated at **q** = 0 is related to the second moment of the EAP. Thus, the MSD is

$$\langle {p}^{2}\rangle =\frac{-1}{4{\pi}^{2}}{\nabla}^{2}E(\mathbf{q}){\mid}_{\mathbf{q}=0}$$

(Appendix E.3)

For the case of DTI, where *E*(*q*, **u**) = *e*^{−4π2q2(Δ−δ/3)uTDu}, Eq. (Appendix E.3) simplifies to the Einstein relation

$$\langle {p}^{2}\rangle =6(\mathrm{\Delta}-\delta /3)MD$$

(Appendix E.4)

The BFOR signal basis is an eigenfunction of the Laplacian operator, with eigenvalues $-{\scriptstyle \frac{{\alpha}_{nl(j)}^{2}}{{\tau}^{2}}}$. Hence

$${\nabla}^{2}E(\mathbf{q})=-\sum _{n=1}^{N}\sum _{j=1}^{R}{C}_{nj}\phantom{\rule{0.16667em}{0ex}}{\left(\frac{{\alpha}_{nl(j)}}{\tau}\right)}^{2}{e}^{-{\alpha}_{nl(j)}^{2}t/{\tau}^{2}}{j}_{l(j)}(\frac{{\alpha}_{nl(j)}q}{\tau}){Y}_{j}(\mathbf{u})$$

(Appendix E.5)

Evaluating the Laplacian of *E*(**q**) at *q* = 0 gives

$${\nabla}^{2}E(\mathbf{q}){\mid}_{\mathbf{q}=0}=\frac{-1}{2\sqrt{\pi}{\tau}^{2}}\sum _{n=1}^{N}{C}_{n1}{\alpha}_{n0}^{2}{e}^{-{\alpha}_{n0}^{2}t/{\tau}^{2}}$$

(Appendix E.6)

Substituting Eq. (Appendix E.6) into (Appendix E.3), we obtain

$${\langle {p}^{2}\rangle}_{\mathit{BFOR}}=\frac{1}{8{\pi}^{{\scriptstyle \frac{5}{2}}}{\tau}^{2}}\sum _{n=1}^{N}{C}_{n1}{\alpha}_{n0}^{2},$$

(Appendix E.7)

where we dropped the smoothing term.

Using the definition of QIV, we have

$$\begin{array}{l}{\mathit{QIV}}^{-1}=\int {q}^{2}E(\mathbf{q}){d}^{3}\mathbf{q}=\int {q}^{2}\phantom{\rule{0.16667em}{0ex}}\left[\int P(\mathbf{p}){e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{p}\right]\phantom{\rule{0.16667em}{0ex}}{d}^{3}\mathbf{q}\\ =\int P(\mathbf{p})\phantom{\rule{0.16667em}{0ex}}\left[\int {q}^{2}{e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{q}\right]\phantom{\rule{0.16667em}{0ex}}{d}^{3}\mathbf{p}\end{array}$$

(Appendix F.1)

The Dirac delta function is defined as

$$\delta (\mathbf{p})=\int {e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{q}$$

(Appendix F.2)

Taking the second derivative of *δ*(**p**) with respect to *p _{x}*,

$$\begin{array}{l}\frac{{\partial}^{2}\delta (\mathbf{p})}{\partial {p}_{x}^{2}}={(2\pi i)}^{2}\int {q}_{x}^{2}{e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{q}\\ \frac{{\partial}^{2}\delta (\mathbf{p})}{\partial {p}_{y}^{2}}={(2\pi i)}^{2}\int {q}_{y}^{2}{e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{q}\\ \frac{{\partial}^{2}\delta (\mathbf{p})}{\partial {p}_{z}^{2}}={(2\pi i)}^{2}\int {q}_{z}^{2}{e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{q}\end{array}$$

The sum of the derivatives is simply the Laplacian operator acting on *δ*(**p**):

$${\nabla}^{2}\delta (\mathbf{p})=\frac{{\partial}^{2}\delta (\mathbf{p})}{\partial {p}_{x}^{2}}+\frac{{\partial}^{2}\delta (\mathbf{p})}{\partial {p}_{y}^{2}}+\frac{{\partial}^{2}\delta (\mathbf{p})}{\partial {p}_{z}^{2}}={(2\pi i)}^{2}\int {q}^{2}{e}^{2\pi i\mathbf{q}\xb7\mathbf{p}}{d}^{3}\mathbf{q}$$

(Appendix F.3)

Thus, Eq. (Appendix F.1) becomes

$${\mathit{QIV}}^{-1}=\frac{-1}{4{\pi}^{2}}\int P(\mathbf{p})\phantom{\rule{0.16667em}{0ex}}[{\nabla}^{2}\delta (\mathbf{p})]\phantom{\rule{0.16667em}{0ex}}{d}^{3}\mathbf{p}$$

(Appendix F.4)

Since the Laplacian operator ^{2} is Hermitian and the EAP & *δ*(**p**) are real-valued, Eq. (Appendix F.4) can be equivalently stated as

$${\mathit{QIV}}^{-1}=\frac{-1}{4{\pi}^{2}}\int P(\mathbf{p})\phantom{\rule{0.16667em}{0ex}}[{\nabla}^{2}\delta (\mathbf{p})]\phantom{\rule{0.16667em}{0ex}}{d}^{3}\mathbf{p}=\frac{-1}{4{\pi}^{2}}\int \delta (\mathbf{p})\phantom{\rule{0.16667em}{0ex}}[{\nabla}^{2}P(\mathbf{p})]\phantom{\rule{0.16667em}{0ex}}{d}^{3}\mathbf{p}$$

Exploiting the property of the Dirac delta function that ${\int}_{-\infty}^{\infty}f(x)\delta (x)dx=f(0)$, we have

$${\mathit{QIV}}^{-1}=\frac{-{\nabla}^{2}P(\mathbf{p}){\mid}_{\mathbf{p}=0}}{4{\pi}^{2}},$$

(Appendix F.5)

which is very similar in form to Eq. Appendix E.3. Thus, whereas the MSD directly varies with the Laplacian of the diffusion signal evaulated at the origin, the QIV inversely varies with the Laplacian of the EAP evaluated at the origin.

$$\begin{array}{l}{\mathit{QIV}}^{-1}=\int {q}^{2}E(\mathbf{q}){d}^{3}\mathbf{q}\approx \sum _{n}\sum _{j}{C}_{nj}\int {Y}_{j}(\mathbf{u}){d}^{2}\mathbf{u}{\int}_{0}^{\tau}{q}^{4}{j}_{l(j)}(\frac{{\alpha}_{nl(j)}q}{\tau})dq\\ =\sqrt{4\pi}\sum _{n}{C}_{n1}{\int}_{0}^{\tau}{q}^{4}{j}_{0}(\frac{{\alpha}_{n0}q}{\tau})dq=\sqrt{4\pi}\tau \sum _{n}\frac{{C}_{n1}}{{\alpha}_{n0}}{\int}_{0}^{\tau}{q}^{3}sin(\frac{{\alpha}_{n0}q}{\tau})dq\end{array}$$

The last integral can easily be solved via integration by parts, and so the QIV is

$${\mathit{QIV}}_{\mathit{BFOR}}=\frac{1}{2\sqrt{\pi}{\tau}^{5}{\sum}_{n=1}^{N}{(-1)}^{n}{C}_{n1}\frac{(6-{\alpha}_{n0}^{2})}{{\alpha}_{n0}^{4}}}$$

(Appendix F.6)

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