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Hum Brain Mapp. Author manuscript; available in PMC 2014 February 1.

Published in final edited form as:

Published online 2012 January 16. doi: 10.1002/hbm.21454

PMCID: PMC3538903

NIHMSID: NIHMS320113

Nathan S. White,^{1,}^{*} Trygve B. Leergaard,^{2} Helen D’Arceuil,^{3} Jan G. Bjaalie,^{2} and Anders M. Dale^{1,}^{4}

The publisher's final edited version of this article is available at Hum Brain Mapp

See other articles in PMC that cite the published article.

Diffusion magnetic resonance imaging (dMRI) is a powerful tool for studying biological tissue microarchitectures *in vivo*. Recently, there has been increased effort to develop quantitative dMRI methods to probe both length scale and orientation information in diffusion media. Diffusion spectrum imaging (DSI) is one such approach that aims to resolve such information on the basis of the three-dimensional diffusion propagator at each voxel. However, in practice only the orientation component of the propagator function is preserved when deriving the orientation distribution function. Here, we demonstrate how a straightforward extension of the linear spherical deconvolution (SD) model can be used to probe tissue orientation structures over a range (or “spectrum”) of length scales with minimal assumptions on the underlying microarchitecture. Using high b-value Cartesian *q*-space data on a fixed rat brain sample, we demonstrate how this “restriction spectrum imaging” (RSI) model allows for separating the volume fraction and orientation distribution of hindered and restricted diffusion, which we argue stems primarily from diffusion in the extra- and intra-neurite water compartment, respectively. Moreover, we demonstrate how empirical RSI estimates of the neurite orientation distribution and volume fraction capture important additional structure not afforded by traditional DSI or fixed-scale SD-like reconstructions, particularly in grey matter. We conclude that incorporating length scale information in geometric models of diffusion offers promise for advancing state-of-the-art dMRI methods beyond white matter into grey matter structures while allowing more detailed quantitative characterization of water compartmentalization and histoarchitecture of healthy and diseased tissue.

Owing to its exquisitely sensitive contrast mechanism, diffusion MRI (dMRI) is a powerful technique for studying the microstructural and physiological properties of biological tissue *in vivo* (Le Bihan, et al. 1986). At normal brain temperatures, the diffusion coefficient of water in the CNS is about 1µm^{2}/ms. During typical diffusion times of 20–80 ms water molecules probe lengths scales on the order of 5–20 µm, making dMRI signals uniquely sensitive to a wide range of microstructural information at the cellular and sub-cellular level. Over the past decade, a number of important methodological advances have been made to probe both length scale and orientation information in biological tissue samples (for a review see Yablonskiy and Sukstanskii 2010).

Q-space imaging (QSI) is one such technique that utilizes data collected over multiple *q*-values (defined as *q* = γδG/2π where γ is the gyromagnetic ratio of the hydrogen nucleus, and δ is the diffusion gradient duration) under the so-called narrow pulse-field regime (where δ is infinitesimally small, and δ<<Δ) (Callaghan, et al. 1990; Cory and Garroway 1990) to probe length scale information on the basis of the statistical probability of water displacements along a single (1D) dimension (Assaf, et al. 2000; Cohen and Assaf 2002). One of the major contributions of QSI has been the identification and characterization of separable hindered and restricted diffusion pools in nerve tissue samples, with the restricted pool offering information on the size scale of the compartment (Assaf and Cohen 2000). Length scale information can also be probed on the basis of the apparent diffusion coefficient of water (ADC) using multi-exponential signal models for diffusion data collected over multiple *b*-values (*b*~*q*^{2} Δ, where Δ is the diffusion time) (Mulkern, et al. 1999). These parametric methods assume a Gaussian mixture model for the displacement distribution with different ADCs. Similar to QSI, these multi-exponential signal models routinely show evidence for water compartmentation in biological tissue when measured over an extended *b*-value range. Typically, at least two Gaussian diffusion pools (exponentials) are observed, one with a high ADC, corresponding to coarse scale (“fast”) diffusion, and one with a small ADC, corresponding to fine scale (“slow”) diffusion (Mulkern, et al. 1999). While many have attributed the “fast” and “slow” components to hindered and restricted diffusion in the extra- (ECS) and intracellular (ICS) space, respectively (Mulkern, et al. 2009), this theory is not without controversy, particularly due to paradoxically reversed estimates of their partial volume fractions (Mulkern, et al. 2009).

There is also a large body of literature describing the use of multi-directional dMRI acquisitions at a fixed *b*-value (or *q*-value) for studying the geometric organization of white matter tissue *in vivo* (Basser, et al. 1994b; Behrens, et al. 2003; Jansons and Alexander 2003; Tournier, et al. 2004; Tuch 2004; Alexander 2005b; Anderson 2005; Hess, et al. 2006; Ozarslan, et al. 2006; Kaden, et al. 2007). The most popular of these methods is diffusion tensor imaging (DTI), which uses a set of six or more images with non-collinear diffusion directions, plus one or more images with no diffusion weighting (*b*=0) to estimate the apparent diffusion tensor in three-dimensional (3D) space at each voxel (Basser, et al. 1994a). The eigensystem of the tensor is commonly used to quantify both the degree of anisotropy and the principal directions of the diffusion process (Basser and Pierpaoli 1996), with the later forming the basis for white matter fiber tracking studies (Basser, et al. 2000). However, the well-known limitation of the tensor model in describing diffusion in non-homogenous media (Beaulieu 2002) has led to the development of numerous high angular resolution diffusion imaging (HARDI) techniques to resolve complex (e.g. crossing or bending) fiber orientations within voxels (Alexander, et al. 2002; Frank 2002; Tournier, et al. 2004; Tuch 2004; Alexander 2005b; Anderson 2005; Hess, et al. 2006; Ozarslan, et al. 2006; Jian and Vemuri 2007; Kaden, et al. 2007). In contrast to DTI, these HARDI methods require a large number of diffusion directions (>> 6) to be collected at a fixed diffusion weighting, and include techniques such as spherical harmonic modeling of the ADC (Frank 2002), Q-ball numerical approximation of the 3D water diffusion orientation distribution (dODF) (Tuch 2004), and spherical deconvolution (SD) analysis of the fiber orientation distribution (FOD) (Tournier, et al. 2004; Alexander 2005a; Dell'Acqua, et al. 2007; Jian and Vemuri 2007; Kaden, et al. 2007). However, despite their successful application in resolving complex white matter fiber orientations, these HARDI methods are generally limited to studying only the geometric aspect of the ADC and the underlying fiber architecture, and are generally agnostic to length scale information (e.g. fast/slow hindered/restricted diffusion) due to the adherence of a single diffusion weighting factor with fixed diffusion time.

In recent years there has been increased interest in developing alternative quantitative methods to probe both length scale and orientation information from multi-directional diffusion acquisitions with multiple diffusion weightings. Wedeen *et al* introduced a non-parametric technique called diffusion spectrum imaging (DSI), which generalizes the QSI method to a 3D Cartesian sampling of *q*-space in order to obtain the 3D water displacement probability density function, or diffusion propagator (Kärger and Heink 1983) at each voxel (Wedeen, et al. 2005). However, it is common practice in DSI to integrate the propagator in the radial direction to yield the dODF, which while highlighting the orientation structure of the diffusion function, removes all scale information inherent in the propagator itself. In addition, DSI adopts the classic *q*-space formalism to obtain the propagator through Fourier Transform, but it does so without satisfying the narrow (field) gradient pulse requirement, which challenges the interpretation of the propagator and dODF (Basser 2002). Assaf *et al* proposed a parametric multi-compartmental hindered and restricted model of diffusion (CHARMED) for white matter, which was later extended to measure axon diameter distributions in a technique called AxCaliber (Assaf, et al. 2008; Barazany, et al. 2009). This composite framework makes use of a more efficient concentric shell sampling of *q*-space (sometimes referred to as a multi-shell HARDI acquisition), where each shell measurement is differential sensitivity to diffusion at multiple length scales. Alexander *et al* used a simplified version of the CHARMED model to measure an axon diameter index *in vivo* (Alexander, et al. 2010). Jespersen *et al* introduced a parametric multi-compartmental model of the cytoarchitecture that uses a distribution of cylinders to model fine scale diffusion in axons and dendrites (collectively called neurites), and a tensor model for coarse scale diffusion elsewhere (cell bodies, glia, ECS) (Jespersen, et al. 2007; Jespersen, et al. 2010). In this model, a multi-shell HARDI acquisition is harnessed to disambiguate the intra- versus extra-neurite water fraction yielding estimates of the neurite volume fraction and 3D orientation distribution (Jespersen, et al. 2007; Jespersen, et al. 2010). However, an important limitation of many of these hybrid diffusion methods is that they often require complex and time consuming non-linear optimization of the model parameters.

In this study, we show how a rather straightforward extension of the SD model for HARDI acquisitions can be used to probe the orientation structure of tissue microstructures over a range (or “spectrum”) of length scales with minimal assumptions on the underlying microstructure and while preserving an efficient linear implementation. Using high *b*-value Cartesian *q*-space data collected on a fixed rat brain, we show how this linear analysis approach, which we call “restriction spectrum imaging” (RSI), can be used to separate the volume fraction and orientation structure of fine and coarse scale diffusion processes in rat brain tissue, which we believe stems from restricted and hindered diffusion in the intra- and extra-neurite water compartment, respectively. We support this hypothesis using an abundance of both theoretical and empirical evidence, including a diverse set of histological material from rat brain tissue. We further demonstrate how the resultant neurite orientation distribution provides additional structure beyond that which can be gleaned from traditional DSI and fixed-scale SD-like reconstructions.

The diffusion data and histological materials used in this study were the same as used in a previous report (Leergaard, et al. 2010). Briefly, an adult Sprague Dawley® Rat (Charles River Laboratories International, Inc. Wilmington, MA) was deeply anesthesized (ketamine hydrochloride 50 mg/kg, and sodium pentobarbital 12 mg/kg, i.p.) and transcardially perfused with 4 % paraformaldehyde. The brain was extracted and immersed in contrast enhancing Magnevist® solution (Bayer HealthCare Pharmaceuticals, Inc.) for approximately two weeks (D'Arceuil and de Crespigny 2007; Leergaard, et al. 2010). For image acquisition, the brain was immobilized in a molded plastic holder and placed in a sealable custom-built plastic chamber filled with perfluorocarbon liquid (Fomblin® LC/8, Solvay Solexis, Thorofare, N.J., USA) to fixate the tissue. High resolution diffusion images were collected using a 4.7T Bruker BioSpec Avance scanner (Bruker Instruments, Freemont, CA, USA) featuring a 40cm warm bore diameter and equipped with a 3 cm solenoid receiver coil. Data were acquired using a single-shot pulsed gradient spin echo (PGSE) echo planar imaging sequence with Cartesian *q*-space sampling and the following pulse sequence parameters: TR/TE = 650/49 ms, Δ/δ = 23/12 ms, 515 *q*-space vectors (Wedeen, et al. 2005), |G|_{max} = 380 mTm^{−1}, *b*_{max} = 30452 sec/mm^{2}, matrix = 64×64×128, voxel size = 265 µm isotropic, total imaging time ~ 12 hours. Following tomographic imaging, the brain was coronally sectioned at 50 µm using a freezing microtome (Microm HM450, Microm Gmbh, Waldorf, Germany). Every fourth section was stained for myelin using a standard procedure modified from Woelche (Woelche 1942).

Supplementary histological material was derived from another Sprague Dawley rat (Scanbur, Norway) that was sacrificed as described above. This brain was sectioned sagitally (at 50 µm), and selected sections stained for myelin. High-resolution mosaic images of the histological sections were obtained through UPlanApo 20/0.70 and 40/0.85 dry objectives using a motorized Olympus BX52 microscope running the Neurolucida 7.0 software (Virtual Slice module, MBF Bioscience, Inc, Williston, VT, USA) or a slide scanner (Mirax Scan, Carl Zeiss MicroImaging GmbH, Jena, Germany). Additional histological reference images were downloaded from the BrainMaps.org website (www.brainmaps.org) and the Rodent Brain Workbench (www.rbwb.org). This concerned images showing 40 µm thick coronal sections from an adult Sprague Dawley rat stained for potassium channel interacting protein (KChlP1) using the K55/7 monoclonal antibody (NeuroMab; http://neuromab.ucdavis.edu/), and images showing the distribution of axonal plexuses anterogradely labelled with *Phaseolus vulgaris* leucoagglutinin (www.rbwb.org; Whole Brain Connectivity Atlas; case R606; see also Zakiewicz et al., 2009).

DSI is based on the Fourier relationship between the diffusion signal *S*(**q**, Δ) and the 3D propagator *P*(**R**, Δ) (Kärger and Heink 1983)

$$S(\mathbf{q},\mathrm{\Delta})={\displaystyle \int}P(\mathbf{R},\mathrm{\Delta}){e}^{-2\pi \mathbf{q}\xb7\mathbf{R}}{d}^{3}\mathbf{R}$$

[1]

where **R** is the net displacement of water molecules during the diffusion time Δ, and **q**=γδ**G** is the *q*-space diffusion wave vector. To obtain *P*(**R**, Δ), the signal values were filled into 3D Cartesian coordinate space consisting of a 17 × 17 × 17 voxel grid according to their respective position in *q*-space. Then, the inverse Fourier Transform was applied directly to the gridded data. Prior to the Fourier inversion, a 3D Hanning window *h*(*r*) = 0.5 · (1 + cos(2π*r* / 17)) was used to filter the data at high |**q**| to reduce truncation artifacts (Gibbs ringing) in the reconstructed propagator (Wedeen, et al. 2005). Finally, the dODF(**x**) in the direction of the unit vector **x** was obtained by evaluating the integral

$$\text{dODF}(\mathbf{x})={\displaystyle {\int}_{0}^{R\phantom{\rule{thinmathspace}{0ex}}\text{max}}}P(R\mathbf{x},\mathrm{\Delta})\phantom{\rule{thinmathspace}{0ex}}|R{|}^{2}\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{dR}}.$$

[2]

To evaluate this integral, we used a 3^{rd}-order tessellation of the sphere (642 vertices) to define each point **x** of dODF(**x**), and Sinc interpolation along 20 equally spaced points between 0 and *R*_{max}.

In traditional spherical deconvolution (SD), the diffusion signal magnitude in each voxel *s*(**q**) can be written as

$$s(\mathbf{q})={s}_{0}{\displaystyle \int}R(\mathbf{q};\mathbf{x})f(\mathbf{x}){d}^{3}\mathbf{x}$$

[3]

where **q**=γδ**G** is the *q*-space diffusion wave vector, *f* (·) is the fiber orientation distribution (FOD), *s*_{0} is the signal measured with no diffusion weighting (i.e. *s*_{0} *s*(**q**=0)), and *R*(·, **x**) is the signal attenuation to a single “fiber” with orientation given by the unit vector **x**. To obtain the FOD, a popular approach uses an axially-symmetric Gaussian model for the fiber response function (Anderson 2005)

$$R(\mathbf{q};\mathbf{x})=\text{exp}(-b(({D}_{L}-{D}_{T}){\text{cos}}^{2}\alpha +{D}_{T}\left)\right)$$

[4]

where *b* = | **q** |^{2} (Δ − δ / 3) is the diffusion weighting factor (or b-value), α is the measurement angle relative to the fiber axis (i.e. α = |**r** · **x**|, where **r** = **q**/ | **q** | is a unit vector oriented along **q**), and *D _{L}* and

$$\begin{array}{c}\hfill \mathbf{s}=\mathbf{\text{Rf}},\hfill \\ \hfill [\begin{array}{c}\hfill \mathrm{({\mathbf{q}}_{1})}\hfill & \hfill \hfill & \hfill \mathrm{({\mathbf{q}}_{N})}\hfill \\ ]=[\begin{array}{cc}\hfill R({\mathbf{q}}_{1};{\mathbf{x}}_{1})\hfill & \hfill \hfill R({\mathbf{q}}_{1};{\mathbf{x}}_{M})\hfill \hfill & \hfill \hfill \hfill \hfill & \hfill R({\mathbf{q}}_{N};{\mathbf{x}}_{1})\hfill & \hfill \hfill R({\mathbf{q}}_{N};{\mathbf{x}}_{M})\hfill \hfill \\ \left]\phantom{\rule{thinmathspace}{0ex}}\right[\begin{array}{l}\hfill f({\mathbf{x}}_{1})\hfill \\ \hfill \hfill & \hfill f({\mathbf{x}}_{M})\hfill \\ ]\end{array}\hfill \hfill \end{array}\end{array}\hfill \end{array}$$

[5]

where = *s* / *s*_{0} denotes the normalized signal. However, rather than treating each of the *M* values of *f* (·) as unknown parameters, the FOD is often parameterized using a set of even order spherical harmonics (SH) ${Y}_{l}^{m}(\xb7)$ with order *l* = 0, 2, …*L* and degree *m* = −*l*, …, 0, …, *l*,

$$f(\mathbf{x})={\displaystyle \sum _{k=1}^{P}}{\beta}_{k}{Y}_{k}\phantom{\rule{thinmathspace}{0ex}}(\mathbf{x}),$$

[6]

where *P* = (*L* + 2)(*L* + 1) / 2 is the total number of SH basis functions in the series,

$$\begin{array}{c}{Y}_{k}\phantom{\rule{thinmathspace}{0ex}}(\mathbf{x})=\{\begin{array}{ccc}\hfill \sqrt{2}\phantom{\rule{thinmathspace}{0ex}}\text{Re}\{{Y}_{l}^{m}(\mathbf{x})\},\hfill & \hfill \mathit{\text{if}}\hfill & \hfill -l\le m<0\hfill \\ \hfill {Y}_{l}^{0}(\mathbf{x}),\hfill & \hfill \mathit{\text{if}}\hfill & \hfill m=0\hfill \\ \hfill \sqrt{2}\phantom{\rule{thinmathspace}{0ex}}\text{Im}\{{Y}_{l}^{m}(\mathbf{x})\},\hfill & \hfill \mathit{\text{if}}\hfill & \hfill 0<m\le l\hfill \end{array},\hfill \\ kk(l,m)=({l}^{2}+l+2)/2+m\hfill \end{array}$$

[7]

is the *k*^{th} SH basis function, and {β_{1},β_{2},…,β_{P}} are the unknown real-valued parameters (weights) to be estimated. This particular basis in Eq. [7] ensures that the recovered FOD is both real and symmetric (Descoteaux, et al. 2007). Substituting Eqs. [6] and [7] into [5] yields the parameterized SD signal model

$$\begin{array}{c}\hfill \mathbf{s}={\mathbf{\text{RY}}}_{L}\mathbf{\beta},\hfill \\ \hfill [\begin{array}{c}\hfill \mathrm{({\mathbf{q}}_{1})}\hfill & \hfill \hfill & \hfill \mathrm{({\mathbf{q}}_{N})}\hfill \\ ]=[\begin{array}{cc}\hfill R({\mathbf{q}}_{1};{\mathbf{x}}_{1})\hfill & \hfill \hfill R({\mathbf{q}}_{1};{\mathbf{x}}_{M})\hfill \hfill & \hfill \hfill \hfill \hfill & \hfill R({\mathbf{q}}_{N};{\mathbf{x}}_{1})\hfill & \hfill \hfill R({\mathbf{q}}_{N};{\mathbf{x}}_{M})\hfill \hfill \\ \left]\phantom{\rule{thinmathspace}{0ex}}\right[\begin{array}{cc}\hfill {Y}_{1}({\mathbf{x}}_{1})\hfill & \hfill \hfill {Y}_{P}({\mathbf{x}}_{1})\hfill \hfill & \hfill \hfill \hfill \hfill & \hfill {Y}_{1}({\mathbf{x}}_{M})\hfill & \hfill \hfill {Y}_{P}({\mathbf{x}}_{M})\hfill \hfill \\ \left]\phantom{\rule{thinmathspace}{0ex}}\right[\begin{array}{c}\hfill {\beta}_{1}\hfill \\ \hfill \hfill & \hfill {\beta}_{P}\hfill \\ ]\end{array}\hfill \hfill \\ ,\end{array}\hfill \hfill \end{array}\end{array}\hfill \end{array}$$

[8]

where the subscript *L* is used to make the SH expansion order of **Y** explicit.

The two main assumptions of the SD model in Eq. [8] are that 1) the orientation structure of the FOD lies within the subspace spanned by the spherical harmonic basis vectors in **Y**_{L}, which places an intrinsic upper limit on the angular resolution of the FOD estimate (White and Dale 2009), and 2) the tissue architecture can be described by a linear mixture of (non-exchanging) cylindrical fiber elements with identical diffusion characteristics. In other words, the diffusion length scale is presumed fixed for all fibers within the voxel. To do away with this assumption, we implement a straightforward extension to the deconvolution model that allows for a mixture of cylindrically-symmetric Gaussian kernel functions with different *D _{T}* and fixed (constant)

$$\mathbf{s}=\mathbf{R}\left({D}_{T}^{(1)}\right)\phantom{\rule{thinmathspace}{0ex}}{\mathbf{f}}_{1}+\mathbf{R}\left({D}_{T}^{(2)}\right)\phantom{\rule{thinmathspace}{0ex}}{\mathbf{f}}_{2}+\dots +\mathbf{R}\left({D}_{T}^{(J)}\right)\phantom{\rule{thinmathspace}{0ex}}{\mathbf{f}}_{J}$$

[9]

where ${D}_{T}^{(1)}<{D}_{T}^{(2)}<\dots <{D}_{T}^{(J)}<{D}_{L}$ are the transverse diffusivities for each of the *J* total FODs (Fig. 1). The distribution of FODs, **f**_{1}, **f**_{2}, …, **f**_{J}, which we call the “FOD spectrum”, models the orientation structure of diffusion processes at the length scales given by ${D}_{T}^{(1)},{D}_{T}^{(2)},\dots ,{D}_{T}^{(J)}$. Once again, if the diffusivities are known *a priori*, the multi-scale model retains its efficient linear implementation and can be written

$$\begin{array}{c}\hfill \mathbf{s}=\mathbf{A}\mathbf{\beta}\text{'},\hfill \\ \hfill =[\begin{array}{cc}\hfill \mathbf{R}\left({D}_{T}^{(1)}\right){\mathbf{Y}}_{{L}_{1}}\hfill & \hfill \hfill \mathbf{R}\left({D}_{T}^{(J)}\right){\mathbf{Y}}_{{L}_{J}}\hfill & \hfill {e}^{-{\mathit{\text{bD}}}_{L}}\hfill & \hfill {e}^{-{\mathit{\text{bD}}}_{F}}\hfill \hfill \\ \left]\phantom{\rule{thinmathspace}{0ex}}\right[\begin{array}{c}\hfill {\mathbf{\beta}}^{(1)}\hfill \\ \hfill \hfill & \hfill {\mathbf{\beta}}^{(J)}\hfill \\ \hfill {\beta}_{L}\hfill \\ \hfill {\beta}_{F}\hfill \\ ]\end{array}\end{array}\hfill & .\end{array}$$

[10]

where **A** is now a large $N\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}({\displaystyle \sum _{i=1}^{J}}{P}_{i})+2$ matrix, where *P _{i}* = (

To fit the RSI model in Eq. [10] we first derived an estimate of *D _{L}* by fitting a tensor model to the diffusion data in a small region of the corpus callosum known to have a high density of uniformly oriented white matter fibers. In this region, we estimated

$$\begin{array}{c}\hfill \mathbf{\text{'}={\mathbf{A}}^{\u2020}\mathbf{s}}\hfill & \hfill {\mathbf{A}}^{\u2020}={({\mathbf{A}}^{T}\mathbf{A}+\alpha \mathbf{I})}^{-1}{\mathbf{A}}^{T}.\hfill \end{array}$$

[11]

where α is the Tikhonov regularization factor. To optimize the regularization level, we selected the α which minimized the Bayesian Information Criterion (BIC) (Schwarz 1978) over a large region of our tissue specimen, which included both grey and white matter (not shown). The BIC is defined as $n\phantom{\rule{thinmathspace}{0ex}}\text{ln}({\mathrm{e2}}_{)}^{+}$, where *n* is the number of measurements, ${\mathrm{e2}}_{}^{}$ is the error variance, and *k* = trace(**A**^{†}
**A**) is the *effective* number of model parameters. The BIC can be interpreted as managing the tradeoff between goodness-of-fit on the one hand, by penalizing models with large residual error (i.e. $n\text{ln}({\mathrm{e2}}_{)}^{}$ term), and reducing model complexity on the other, by penalizing models with a large number of free parameters (i.e. *k*ln(*n*) term). At the optimum regularization level (minimum BIC), the effective number of model parameters was reduced to approximately *k* = 20.

The theoretical compartment size diameter for restricted diffusion was estimated using Monte-Carlo simulations consisting of a population of “spins” undergoing random-walk diffusion within a single impermeable cylinder with known diameter. The exact diameter (*d*) varied for each simulation experiment and ranged between 0.1 and 100 µm. Spins were initially randomly distributed within the cylinder and allowed to diffuse with an intrinsic diffusion coefficient of *D _{L}*. Spins that encountered a boundary were reflected off the cylinder wall while preserving their un-reflected diffusion path length (Hall and Alexander 2009). The accumulated spin phase was used to generate synthetic diffusion signals for our Cartesian

The goal of this study was to investigate whether a rather straightforward extension of traditional fixed-scale SD model for HARDI acquisitions could be used to probe the orientation structure of our tissue sample at multiple length scales in a manner that reflects the underlying biology. The results of our RSI analysis is illustrated in Figure 2 for a single horizontal slice taken at the level of the dorsal striatum, hippocampus, and tectum. We found that the majority of the diffusion signal in our fixed tissue sample occurred at the fine (Fig. 2, left hand side; red frame) and coarse scale (Fig. 2, right hand side, blue frame), with little or no signal at intermediate scales. The fine scale diffusion processes were characterized by transverse diffusivities significantly smaller than the longitudinal diffusivity (*D _{T}/D_{L}* <0.1), while the coarse scale processes were characterized by transverse diffusivities approximately 60–90% of the longitudinal diffusivity (0.6<

We were also interested in the physical nature of diffusion at the length scales probed by our RSI model. In biological tissue, there are two general modes of diffusion: *hindered* and *restricted* (Le Bihan 1995). Hindered diffusion relates to the increase in diffusion path length molecules must travel when diffusing around cellular obstructions, and is classically described in terms of the “tortuosity” $\lambda =\sqrt{D/\mathit{\text{ADC}}}$, which relates the ADC to the diffusion coefficient measured in the absence of any obstacles *D* (Sykova and Nicholson 2008). Restricted diffusion, on the other hand, relates to the physical blockage of molecules trapped within cellular compartments. If the diffusion time Δ is long enough, the length scale of diffusion will vary dramatically depending on whether diffusion is hindered or restricted (Le Bihan 1995). To provide insight into this phenomenon in our data, we computed the geometric tortuosity ${\lambda}_{g}=\sqrt{{D}_{L}/{D}_{T}}$ (assuming hindered diffusion) and compartment size diameter (assuming restricted diffusion, estimated using Monte-Carlo simulations, see Section 2.4) for each scale and plot these in Figure 2 (bottom rows). Based on these tortuosity and compartment size calculations, together with the nature of the FOD spectrum, we believe that the fine scale diffusion processes probed by our model has to be restricted and not hindered, and that the physical compartment of restriction is consistent with neurites (axons and dendrites), and possibly even long slender glial cell processes. Moreover, we believe that the coarse scale diffusion processes reflect water hindered within the extra-neurite compartment, including large cell bodies, ECS, and glia. In the remainder of this paper, we provide theoretical and experimental support for this *neurite-specific* hypothesis. In Section 3.2, we begin by explaining the rationale for the neurite hypothesis using prior theoretical and empirical evidence, together with our own calculations for the geometric tortuosity and compartment size diameters. In Section 3.3, we provide further empirical support for this hypothesis in our *q*-space data by comparing the purportedly *restricted* FOD (*r*-FOD) and *hindered* FOD (*h*-FOD) at two representative length scales (*D _{T}/D_{L}* = 0 and 0.82, respectively; cf. Fig. 2) against an abundance of histological material from corresponding anatomical regions.

In this section we explain the rationale for the neurite hypothesis using prior theoretical and empirical evidence along with our own calculations for the tortuosity and compartment size diameters. The main source of water restriction in biological tissues comes from cell membranes, and the vast majority of oriented cylindrical compartments in the brain relate to neuronal extensions, or neurites (axons and dendrites). In the rat brain, the diameter of unmyelinated and myelinated subcortical axons range between 0.02 and 3.0 µm, with a mean of 0.2–0.6 µm (Partadiredja, et al. 2003; Barazany, et al. 2009), which is consistent with our own diameter calculations at the fine scale (see the values for *d* on the bottom of Fig. 2, left). If we assume that the fine scale diffusion is rather hindered and not restricted by neurites, the geometric tortuosity at this scale would have to be greater than 2 (see the values for λ_{g} on the bottom of Fig. 2, left). This is quite certainly not the case. We know from previous reports that tortuosities greater than 2 can only be achieved under extreme pathophysiological states such as severe brain ischemia (Nicholson and Sykova 1998; Chen and Nicholson 2000). In fact, the maximum geometric tortuosity introduced by various packed cellular objects in the brain ECS has previously been estimated to be no greater than 1.225 (Tao, et al. 2005). Even in an ideal simulated environment consisting of a bundle of cylindrical fibers organized in the most compact way, the transverse ADC (*D _{T}*) cannot mathematically exceed (2/π)

In summary, given the aforementioned empirical and theoretical evidence, we surmise that our RSI model has separated the intra-neuritic restricted water from extra-neuritic hindered water in our fixed tissue sample. Note, this *neurite-specific* assignment is fundamentally different from the common (albeit controversial) view that fine (“slow”) and coarse (“fast”) scale diffusion stems from water in the intracellular and extracellular space of tissue (see Mulkern, et al. 2009 for a review). Also note that this neurite assignment fits well with model presented by Jespersen for measuring the neurite density and orientation distribution (Jespersen, et al. 2007). In the next section (Section 3.3), we provide further empirical support for this hypothesis in our *q*-space data, by comparing RSI estimates of the *r*-FOD and *h*-FOD with an abundance of histological material taken from corresponding anatomical regions.

If the neurite hypothesis holds true, then the volume fraction and orientation distribution of neurites should be reflected in the volume fraction and orientation distribution of the *r*-FOD, while the *h*-FOD should reflect the orientation distribution of hindered extra-neuritic water. To investigate the anatomical substrate of the *r*-FOD and *h*-FOD, we compared their orientation distribution and volume fraction against the histoarchitecture in selected brain regions (the striatum, globus pallidus, cerebral cortex, and cerebellum) known to have complex but characteristic tissue architectures.

In both the striatum and globus pallidus, the tissue architecture is characterized by the dense bundles of penetrating corticofugal axons, and by a relatively complex architecture with topographically organized axonal terminal fields (Brown, et al. 1998; Alloway, et al. 1999; Gerfen and Paxinos 2005), neurons and glial cells. The penetrating corticofugal axons appear dark in the myelin stain (Fig. 4B,E, Fig. 5G), while the complex network of dendritic arbors and axonal terminal fields can be visualized using voltage-gated potassium channel stains (KChlP1, Fig. 5F,I) and axonal tracing techniques (Fig. 6D), respectively. In the striatum and globus pallidus, the dissociation between the *h*-FOD and *r*-FOD is pronounced (Fig. 4A,C and D,F; Fig. 5E,H)

For example, in the anterior dorsal striatum, the *r*-FOD features three peaks, of which the anterioposterior component corresponds nicely with the through-plane orientations of penetrating corticofugal axons (compare blue peaks in Figs. 4A with B), while the more complex, crossing orientations appear to reflect the more random organization of corticostriatal terminal plexuses (Brown, et al. 1998; Alloway, et al. 1999; Veinante and Deschenes 2003), dendritic arbors, and cellular processes in the region (compare red and green peaks of the *r*-FOD in Fig. 5E with the histoarchitecture in Fig. 5F,I and Fig. 6D).

In the globus pallidus, a similar pattern of *r*-FOD orientations is observed, with prominent peaks well aligned with the oblique axial fibers penetrating the region (compare blue peaks in Fig. 4D with E), and less prominent, dorsoventrally (green) and mediolaterally (red) oriented peaks. While some of the smaller amplitude green and red peaks may be due to ringing artifacts introduced during reconstruction (see Appendix 5.2 for discussion and simulation of these artifacts), the larger amplitude peaks cannot be explained on the basis of ringing artifacts alone, and are consistent with the orientation of pallidal dendrites and striatopallidal axonal terminations in the region (Gerfen and Paxinos 2005; Sadek, et al. 2007).

By contrast, the *h*-FOD in the globus pallidus and dorsal striatum has a broader disk-like shape, with oblique dorsoventral orientation in the globus pallidus (Fig. 4F), and horizontal orientation in the striatum (Fig. 4C, ,5H).5H). Recall, the *h*-FOD was normalized by subtracting the minimum to highlight their preferred orientation, which turns spherical FODs into disk-like distributions. Interestingly, in none of these regions is the primary orientation of the *h*-FOD aligned consistently with the penetrating myelinated fibers (compare Fig. 4B with C, and E with F). While we lack data to interpret this observation conclusively, we surmise that the primary orientation of the *h*-FOD in this region reflects the combined hindrance of extra-neurite water by both penetrating cortico- and striatofugal axonal projections, and the highly oriented striatal (Brown, et al. 1998; Alloway, et al. 1999; Veinante and Deschenes 2003) and pallidal neurites (Sadek, et al. 2007). This can explain why the *h*-FOD appears to align somewhere in between the primary (blue) and secondary and tertiary (green and red) peaks of the *r*-FOD (Fig. 4B,C,E,F). This also supports the existing notion that in most regions (excluding uniformly oriented dense white matter fiber tracts) the fiber architecture is less well characterized by the fast hindered water fraction compared to the slow restricted fraction (Ronen, et al. 2003; Assaf and Basser 2005).

The cerebral cortex features a characteristic laminar and columnar organization with radially and tangentially oriented neurites (Lorente de No and Fulton 1938; Szentagothai 1975; Mountcastle 1997). Here, the *h*-FOD is aligned radially to the cortical surface (Fig. 5D), consistent with the orientation of pyramidal cell axons and dendrites, as well as most corticoefferent and corticoafferent fibers (Deschenes, et al. 1998) (Fig. 5B, Fig. 6B,C). By contrast, the *r*-FOD not only reflects the same radial orientations, but also displays prominent orientations tangential to the cortical surface (Fig. 5A), with the largest tangential amplitude occurring in the superficial cortical layers I/II. Given their large amplitude relative to the main (radial) peak, is it unlikely that the tangential peaks of the *r*-FOD are caused solely by ringing artifacts (again, see Appendix 5.2 for a detailed discussion and simulation of these artifacts), rather they likely reflect the well-known horizontal orientation of cortical neurites, see e.g. (Kristt 1978; Cowan and Wilson 1994; Veinante and Deschenes 2003). The complex, characteristic radial and tangential organization of cortical neurites is demonstrated in Figure 6 (A–C).

In the external capsule (ec) below the cortical grey matter, both the *h*-FOD and *r*-FOD are elongated parallel with mediolaterally oriented fibers (Fig. 5A–D), while the *r*-FOD displays additional perpendicular orientations (Fig. 5A), which appear to reflect corticoefferent fibers passing through the external capsule towards the underlying striatum (Cowan and Wilson 1994; Veinante and Deschenes 2003) and the fibers peeling off the ec into the cortex (arrows in Fig 5C).

To lend further support for the notion that the fine (restricted) scales probed by the RSI model reflect water restricted within neurites, we qualitatively compared RSI maps of the total restricted volume fraction (summing scales *D _{T}/D_{L}* = 0, 0,08, 0.16, 0.25, Fig. 2, red frame) and against the myelin and KChlP1 stained sections (Fig. 7). We found that the RSI maps were indeed largely consistent with the expected neurite volume fraction within grey and white matter as the model predicts. RSI estimates of the neurite volume fraction in the cerebral cortex varied from approximately 18–31%, with the highest fraction occurring in positions corresponding to layers I/II followed by layer V. This pattern is consistent with the high density of apical dendrites and horizontally oriented neurites in layers I/II (Zhou and Hablitz 1996) (see, also Fig. 6B), as well as numerous dendritic arbors and profuse plexuses of thalamocortical afferents in layer V (Deschenes, et al. 1998) (see Fig. 6C). Histological substrates for this pattern are seen in the KChlP1 stain (Fig. 5B, Fig. 7), and the axonal tracer data (Fig 6B,C). In the underlying white matter (ec, and corpus callosum, cc) RSI estimates of the total restricted volume fraction increased dramatically to approximately 42–84%, consistent with the presence of densely packed white matter fibers in this region, which appears dark red on the myelin stain (Fig. 7). Taken together, these results compare favorably with the expected neurite volume fraction in grey and white matter, and indicate that the restricted volume fraction is a likely surrogate measure of the neurite volume fraction.

We finally explored the anatomical substrates of the *r*-FOD and *h*-FOD in the cerebellum (Fig. 8), which features a well-known stereotypic architecture, see e.g. (Voogd and Paxinos 1995). Our analysis focused on the cerebellar vermis in a horizontal slice at the level of lobules 3, 5, and 8, where the cerebellar folia are mediolaterally oriented, and orientation structure is easier to interpret in relation to the employed slice planes. The cerebellar white matter, granule cell layer, and molecular layers were all readily identified on basis of the *r*-FOD and *h*-FOD volume fraction and orientation (Fig. 8). The narrow Purkinje cell layer could not be distinguished due to partial voluming.

In white matter (wm), the cerebellar afferent and efferent fibers largely follow the foliated structure of the cerebellum, but also form more complex (crossing) orientations in some regions, see e.g. (Wu, et al. 1999). As observed in the earlier examples, the preferred orientation of white matter fibers was clearly reflected in both *r*-FOD and *h*-FOD, while the *r*-FOD also showed additional secondary and tertiary orientations (Fig. 8), which are consistent with crossing afferent and efferent fibers in this region (Wu, et al. 1999), but may also reflect partial voluming with neighboring layers.

The granule cell layer (gcl) is characterized by the presence of numerous small granule cells, and further also smaller numbers of other cell types. Radially (translobularly) oriented mossy- and climbing fibers penetrate the gcl and ascend together with granule cell axons through the Purkinje cell layer into the overlying molecular layer (Voogd and Paxinos 1995). In this layer, the *r*-FOD showed prominent radial (blue and green) peaks (Fig. 8), consistent with the orientation of mossy- and climbing fibers, and less prominent mediolaterally (red) oriented peaks (Fig. 8), which may be associated with the different cellular extensions in the gcl. The *h*-FOD, on the other hand, tended to be more variably elongated in both translobular or parlobular directions (Fig. 8).

The molecular layer (ml) contains several categories of radially (translobularly) oriented neurites (including Purkinje cell dendrites, ascending granule cell axons, dendrites of Golgi- and stellate cells, and glial extensions (Voogd and Paxinos 1995). These orientations are clearly visible as blue and green peaks in the *r*-FOD (Fig. 8). Another characteristic feature of the ml is the presence of numerous, mediolaterally (parlobularly) oriented parallel fibers, which fit well with the prominent mediolateral (red) peaks of the *r*-FOD (Fig. 8). The *h*-FOD, on the other hand, illustrates a single peak pointed in the direction of the parallel fibers (Fig. 8), which makes sense given the high packing density of parallel fibers in this region compared with other neurites in the ml (e.g Purkinje cell dendrites and ascending granule cell axons), leading to increased tortuosity of extra-neurite water in the orthogonal direction.

Taken together, we found that the *r*-FOD demonstrates the stereotypic organization of cerebellar neurites in each layer, while the *h*-FOD tends to be more indicative only of the densely packed white matter and parallel fibers, which provides further support for the notion that the coarse scale hindered fraction of diffusion is less well characteristic of the underlying fiber architecture compared with the fine scale restricted fraction.

While we have demonstrated how the *r*-FOD and *h*-FOD clearly reflect different aspects of the neurite architecture in both white and grey matter, we were also interested in how these multi-scale RSI measures compared with traditional fixed-scale SD and DSI. To address this issue, we computed fixed-scale estimates of the fiber orientation distribution (FS-FOD) using a single deconvolution kernel (*D _{T}/D_{L}* = 0.08), which can be viewed as an extension of the traditional SD for HARDI acquisitions to arbitrary

We found that in the cerebral cortex, the FS-FOD and DSI-dODF reflected mainly the radial orientation of cortical neurites in contrast to the RSI *r*-FOD, which again was able to resolve both the well-known radial and tangential organization of cortical neurites (Fig. 9). It is also interesting to note the qualitative similarity between the *h*-FOD and the DSI-dODF (Fig. 9), which were both normalized by subtracting the minimum to highlight their preferred orientation (the *r*-FOD and FS-FOD were not normalized as their minima was effectively zero). This similarity is not surprising given the DSI-dODF quantifies the statistical likelihood of water diffusion in a given direction, which is heavily weighted by the volume fraction of “fast” hindered diffusion along that direction and less so by the volume fraction of “slow” restricted diffusion.

Figure 10 shows a comparison of RSI estimates of the *r*-FOD and *h*-FOD versus the FS-FOD and DSI-dODF in three representative voxels from different layers of the cerebellum. In white matter (Fig. 10, bottom row), the main (red) peak of the *r*-FOD and *h*-FOD were well aligned with the main peak of the FS-FOD and DSI-dODF, which follow the dominant direction of cerebellar afferent and efferent fibers in this region. Note that in these voxels, the secondary and tertiary peaks of the *r*-FOD and FS-FOD may be due in part to ringing artifacts introduced during reconstruction, as their amplitudes are quite small compared to the main peak (see Appendix 5.2). Small spurious peaks in the *r*-FOD, FS-FOD, and DSI-dODF may also be introduced as a result of the numerical sampling bias associated with non-uniform (Cartesian) sampling of *q*-space.

In the granule cell layer (gcl, Fig. 10 top row), the FS-FOD and DSI-dODF both demonstrated prominent mediolateral (red) and radial (blue) peaks consistent with the orientation of mossy- and climbing fibers in the region. These same peaks were also present in the RSI *r*-FOD and *h*-FOD. Yet additional information can be gleaned from RSI. For example, the strong mediolateral (red) orientation of the *h*-FOD suggests mediolaterally oriented neurites may have higher packing densities compared with radially oriented neurites, producing a higher degree of tortuosity of extra-neurite water.

The ability to separate hindered and restricted diffusion may also increase the sensitivity of RSI to resolve neurite orientations in voxels with a high partial volume fraction of hindered diffusion. For example, in the molecular layer (ml) (Fig. 10 middle row), the FS-FOD, DSI-dODF, and RSI *h*-FOD again demonstrated prominent mediolateral (red) peaks consistent with the orientation of parallel fibers in the ml. However, here the RSI *r*-FOD showed strong additional radially (translobularly) oriented blue and green peaks, which are consistent with the known orientation of Purkinje cell dendrites, ascending granule cell axons, dendrites of Golgi- and stellate cells, and glial extensions in this region (Voogd and Paxinos 1995). These results were generally consistent across other cerebellar voxels (see Fig. 8). Taken together, the RSI *r*-FOD seems to capture additional anatomical structure not afforded by fixed-scale SD-like reconstruction or DSI.

The RSI analysis method presented in this study is similar in spirit to the earlier model of neurite architectures proposed by Jespersen and co-workers (Jespersen et al. 2007; Jespersen et al. 2010). Both use a distribution of non-exchanging cylinders to model the tissue microstructure and spherical harmonics to describe their orientation distribution. Both also harness multiple *b*-value data to separate scale information at the hindered and restricted level. Similar to RSI, Jespersen’s model also showed good agreement with the neurite volume fraction across various grey and white matter regions (Jespersen et al. 2010). The major difference between the two methods is that our RSI model does not explicitly fit for the intra- and extra-cylindrical diffusivities (which requires nonlinear optimization of model parameters), but rather assumes the diffusivities can take on a broad range (or spectrum) of values when fitting the data. This allows RSI to preserve a linear implementation, which greatly simplifies estimation and decreases computation time significantly. The spectrum of independent scales and orientation distributions further allows RSI to model the tissue microstructure with minimal *a priori* assumptions on the number of hindered and restricted water compartments, their respective geometries, and partial volume fractions. This is in contrast to other biophysical models of diffusion in white matter, which often impose some form of geometric dependence between the hindered and restricted compartment to infer on nonlinear parameters describing the axon diameter distribution (Assaf, et al. 2008; Alexander, et al. 2010). While it may be reasonable to equate the geometry of the hindered and restricted compartment in white matter, as axons provide the major source of both diffusion hindrance and restriction in this region, in grey matter, the physical relationship between the two compartments is more complex. For example, in the current study, and in Jespersen’s model, the geometries of the hindered and restricted compartments are modeled independently to account for the fact that there are numerous additional structures in the extra-neurite compartment (cell bodies, glia, ECS) that can alter the tortuosity of the hindered compartment besides neurites alone. Also, the tortuosity is known to fall off rapidly with increasing ECS volume fraction α (Chen and Nicholson, 2000; Kume-Kick et al, 2002) and therefore it’s likely only populations of tightly packed neurites (i.e. small α) will produce measureable increases in tortuosity in grey matter voxels, while sparsely packed neurites will not. This may explain why in the current study not all peaks in the *r*-FOD were demonstrated in the *h*-FOD and helps validate the use of independent geometries for the hindered and restricted compartment. With that said, the disadvantage of using a relatively unconstrained mixture model for the tissue architecture is the risk of over-fitting and blurring of model parameters, as well as reduced inferential power. In the current study, we mitigated the risk of over-fitting using carefully chosen model regularization techniques in conjunction with the Bayesian information criterion (BIC) to optimize the regularization level. We also quantified the extent and nature of model blurring in the optimally regularized model using the resolution matrix (as described in Appendix 5.1). Finally, we gained inferential power in the current study using the combination of simulations and theoretical (tortuosity) models of hindered and restricted diffusion in known geometries to relate our observations to the underlying histoarchitecture in various anatomical regions.

Some care must be taken when extrapolating our findings in a fixed tissue sample to the *in vivo* situation. Formalin fixation is known to reduce the mean diffusivity in grey and white matter by about 64% and 80%, respectively, but has little effect on the fractional anisotropy (FA) (D'Arceuil, et al. 2007). Given the FA is preserved, we would not expect the fixation process to alter the orientation distribution of the *r*-FOD or *h*-FOD, nor the overall pattern of the length scale spectrum per se (i.e. separation of hindered and restricted diffusion), as both are based on measures of the relative diffusivity *D _{T}/D_{L}*. However, as the fixation process has different effects on grey and white matter (D'Arceuil, et al. 2007) and we assumed a constant

In this work we presented an efficient linear reconstruction and modeling framework for multi-dimensional (multi-direction and multi-b-value) diffusion data called restriction spectrum imaging (RSI) that allows for resolving both length scale and orientation information of biological tissue microstructures. Our model is based on a straightforward extension of the linear spherical deconvolution (SD) HARDI reconstruction method, by including a range (or spectrum) of Gaussian deconvolution kernels. We demonstrate in high *b*-value Cartesian *q*-space data how RSI analysis can be used to separate fine and coarse scale diffusion processes, which we argue stems mainly from restricted and hindered diffusion in the intra- and extra-neurite water compartment, respectively. We support this hypothesis both theoretically and empirically, and demonstrate the correspondence of the restricted volume fraction and corresponding orientation distribution (*r*-FOD) with the underlying neurite volume fraction and three-dimensional histoarchitecture of white and grey matter structures. We further demonstrate how RSI reconstructions of the *r*-FOD captures additional anatomical structure beyond that which can be gleaned from traditional fixed-scale SD and DSI analyses, particularly in grey matter regions. Important future work is needed to understand the conditions and experimental protocols required to further probe hindered and restricted diffusion length scales in greater detail, which may include using multi-diffusion time protocols to explore the diameter distribution of various cellular compartments, similar to the AxCaliber method for quantifying axon diameter distributions *in vivo* (Barazany, et al. 2009; Assaf, et al. 2008).

We conclude that incorporating diffusion length scale information in geometric models of biological diffusion will be of upmost importance for advancing state-of-the-art diffusion MRI methods beyond quantifying white matter orientations, to providing a more detailed quantitative characterization of tissue microstructures in health and disease. While future work will be required to test the application of this method on clinical scanners, we anticipate the general multi-scale linear mixture model framework of RSI will find important clinical applications in probing salient microstructural features of normal and pathological tissue *in vivo*.

This project was funded in part by grants from The Research Council of Norway, the NIH (R01-EB00790, U24-RR021382, NS41285, EB00790) and the Athinoula A. Martinos Center for Biomedical Imaging (NIH/NCRR: P41RR14075, 1S10RR016811, NIBIB: EB00790 and the MIND Institute).

The concept of the model resolution matrix in linear estimation problems is a well established and important way to characterize the bias in linear inverse problems (Aster, et al. 2005). The basic idea is to see how well a particular inverse solution matches the original model parameters β through the expression = **A**^{†}
**A** β, where **A**^{ψ} is the regularized inverse matrix, and **A**^{†}
**A** is the model resolution matrix. In practice, the model resolution matrix is commonly used in two different ways. The first is to examine the diagonal elements of **A**^{†}
**A** for their deviation from unity. A value of one would indicate perfect resolution, i.e. where =**I** β, with **I** being the identity matrix. In this way, the trace of **A**^{†}
**A** can be used to indicate the intrinsic dimensionality of the problem, by quantifying the total number of resolvable parameters. The second is to multiply **A**^{†}
**A** by a particular “test model” β^{test} to see how well the true (test) model would be resolved using the regularized inverse matrix. Often the test models are column vectors of all zeros, except for a single element equal to one, which when multiplied on the left hand side by **A**^{†}
**A** equals the corresponding column of **A**^{†}
**A**. Thus, the columns of **A**^{†}
**A**, or “resolution kernels”, describe how well the true parameters are recovered or blurred by the regularized inverse matrix. Similarly the rows of **A**^{†}
**A** can be viewed as quantifying how sensitive a particular parameter *estimate* is to the true model.

In this section, we use the model resolution matrix to quantify the blurring and intrinsic dimensionality of our optimally regularized RSI model. To do this, we first projected the regularized model resolution matrix **A**^{†}
**A** (cf. Eqs 10,11) onto the FOD surface via ${\mathbf{L}}_{{\mathbf{A}}^{\u2020}}$, where _{L} is a large block diagonal matrix of SH vectors. In this way the model resolution can be visualized directly on the FOD surface, rather than in spherical harmonic space, which is more difficult to interpret. The resultant matrix was then row normalized (i.e. the same axis limits were used for each row element) and plotted in Figure 11. For simplicity, and without loss of generality, we plot only the columns of the resolution matrix that corresponds to unit fibers pointed along the horizontal axis (i.e. “test models” were horizontal unit fibers), and ignore the two isotropic terms. Each row of the matrix can be interpreted as the sensitivity of the recovered (estimated) FOD at that scale to FODs at all other scales. Thus, perfect resolution would result in a diagonal matrix of cigar-like FODs.

As expected, we found that FODs at the fine and coarse scale were separable, despite significant blurring across neighboring scales. For example, rows 1–2 have little or no contribution from columns 10–12, and rows 10–12 have little to no contribution from columns 1–2. This separablility fits well with our empirical observations (see Fig. 2). Also, the isotropic FODs in the bottom right hand corner demonstrate how estimates of coarse scale hindered FODs have higher angular uncertainly compared with the restricted FODs in the upper left, which also fits well with our empirical observations. As mentioned above, the resolution matrix in Figure 11 quantitatively demonstrates how information is blurred across neighboring scales using our optimally regularized model and *q*-space acquisition parameters (i.e. **G**, Δ, and δ). The model resolution matrix can also be used to optimize the experimental acquisition parameters offline to achieve finer scale resolutions. For example, it may be possible to employ multi-diffusion time protocols to resolve restricted length scales in more detail, which would result in diagonal terms in the upper left-hand corner of Figure 11. One can also use the trace of the resolution matrix to quantify the *effective* number of resolvable scales, which we calculated in our *q*-space data to be just under 3 (2.67).

An additional source of bias well-known to all deconvolution-based HARDI methods employing spherical harmonic basis functions, are ringing artifacts (spurious sidelobes) introduced during reconstruction. These artifactual sidelobes result from truncation of the spherical harmonic series, leading to a form of Gibbs ringing on the surface. For an L=4^{th} order harmonic expansion there will be exactly L/2-1 = 1 sidelobes that occurs at 90° to the main peak (Hess et al, 2006). Often, these artifactual peaks can be mitigated with appropriate model regularization techniques such as Tikhonov regularization (employed in this study) or Laplace-Beltrami regularization while balancing the tradeoff between angular resolution on the one hand and suppression of ringing artifacts on the other.

To test the extent to which our regularized RSI model was prone to ringing artifacts, we performed an additional Monte Carlo simulation study. Briefly, we generated synthetic diffusion signals using a simplified RSI model consisting of a single fiber system with two diffusion scales, one restricted (DT/DL = 0), and one hindered (DT/DL = 0.82) with various volume fractions. For each Monte Carlo run, we added random Rician noise to the synthetic signal at a given signal-to-noise (SNR) ratio with respect to the signal power at b=0. Finally, we used the fully regularized RSI model (all 12 scales as implemented in this study) to estimate the model parameters. We then repeated the experiment for various SNR levels and hindered and restricted volume fractions.

The results of our Monte Carlo simulations are shown in Figure 12. Note that we only plot RSI estimates of the *r*-FOD, as the *h*-FOD was found to be largely insensitive to these ringing artifacts. Not surprisingly we found that the ringing artifacts increased both as the SNR decreased and as the volume fraction of hindered diffusion increased (Fig. 12). This is consistent with prior simulation studies showing both the SNR dependence of these artifacts (Tournier, et al. 2008) and their sensitivity to isotropic “background” diffusion (Dell'acqua, et al. 2010). However, we also found that the amplitude of these ringing artifacts was generally very small compared to the main peak, even at very low SNR (Fig. 12). Thus, given the SNR of our *q*-space data was in the range of 29.8 and 55.1 for white and grey matter, respectively, we would not expect the amplitude of the ringing artifacts to be greater than about 7% of the main peak, even in regions with a high partial volume fraction of hindered diffusion, such as cortical grey matter (~70–80%). Taken together with the fact that many of our *r*-FOD peaks were not always at 90° to each other (see Fig. 5E), we argue that it is unlikely that many of the secondary and tertiary orientations of the *r*-FOD presented in this study can be explained on the basis of ringing artifacts alone, but rather likely reflects to a large extent the underlying neurite architecture of the region.

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