1 Axon Diameter Mapping in Crossing Fibers with Diffusion MRI Hui Zhang 1,TimB.Dyrby 2, and Daniel C. Alexander 1 1 Microstructure Imaging Group, Depa...

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Alvin Cain

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Microstructure Imaging Group, Department of Computer Science, University College London, London WC1E 6BT, United Kingdom Danish Research Center for Magnetic Resonance, Copenhagen University Hospital, Hvidovre, Denmark

Abstract. This paper proposes a technique for a previously unaddressed problem, namely, mapping axon diameter in crossing ﬁber regions, using diﬀusion MRI. Direct measurement of tissue microstructure of this kind using diﬀusion MRI oﬀers a new class of biomarkers that give more speciﬁc information about tissue than measures derived from diﬀusion tensor imaging. Most existing techniques for axon diameter mapping assume a single axon orientation in the tissue model, which limits their application to only the most coherently oriented brain white matter, such as the corpus callosum, where the single orientation assumption is a reasonable one. However, ﬁber crossings and other complex conﬁgurations are widespread in the brain. In such areas, the existing techniques will fail to provide useful axon diameter indices for any of the individual ﬁber populations. We propose a novel crossing ﬁber tissue model to enable axon diameter mapping in voxels with crossing ﬁbers. We show in simulation that the technique can provide robust axon diameter estimates in a two-ﬁber crossing with the crossing angle as small as 45o . Using ex vivo imaging data, we further demonstrate the feasibility of the technique by establishing reasonable axon diameter indices in the crossing region at the interface of the cingulum and the corpus callosum.

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Introduction

Axon diameter mapping using diﬀusion MRI oﬀers exciting new possibilities for investigating white matter in health and disease beyond diﬀusion-tensor imaging. Information about axon diameter and density informs the role and performance of white matter pathways [1, 2]. Speciﬁc changes in axon diameter have also been linked to diseases such as multiple sclerosis [3] and amyotrophic lateral sclerosis [4]. Direct measurement of such features can therefore shed new light into white matter development and disease mechanisms. Most techniques for axon diameter mapping adopt the model-based strategy in which a geometric model of the tissue predicts the MR signal from water diﬀusing within. Earlier methods [5,6] assume a single and known orientation of axons in the tissue model, which limits their application to nerve tissue samples or small regions of brain with uniform orientation. More recently, estimating G. Fichtinger, A. Martel, and T. Peters (Eds.): MICCAI 2011, Part II, LNCS 6892, pp. 82–89, 2011. c Springer-Verlag Berlin Heidelberg 2011

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axon diameters of unknown orientation on clinical scanners has been shown to be feasible, ﬁrst in simulation [7], and later in live human brains [8]. However, the models in [7, 8] still assume a single albeit unknown ﬁber orientation. Most recently, Zhang et al [9] relax this assumption to some extent by modeling the axonal-orientation distribution as a Watson distribution. The Watson model accommodates the presence of axonal-orientation dispersion and extends axon diameter mapping beyond the most coherent white matter regions like the corpus callosum to a much wider subset of the white matter. Nevertheless, its model of axonal-orientation distribution remains unimodal. Like the earlier models, it is inappropriate in crossings or partial volume between tracts with signiﬁcantly diﬀerent orientations. Fiber crossings occur in many areas of the brain, resulting in the observation of two or more distinct ﬁber populations in a signiﬁcant number of voxels. In such voxels, considerable success has been achieved in resolving the precise underlying crossing ﬁber conﬁgurations. Since the earliest breakthroughs in crossing ﬁber resolution [10,11], major progress has produced very eﬀective modern techniques (See [12] for a review). Despite that, there has been no attempt at direct measurement of additional microstructure features in these locations. A solution to this problem is prerequisite to realizing whole-brain axon diameter mapping. This paper describes a technique that addresses the combined problem of crossing ﬁber resolution and microstructure imaging. We propose a crossing ﬁber tissue model that enables the simultaneous estimation of crossing conﬁgurations and microstructure features. We demonstrate the technique both in simulation and in brain data. The rest of the paper is organized as follows: Section 2 describes the proposed crossing-ﬁber tissue model and the data ﬁtting procedure; Section 3 details the design of the simulation and brain data experiments for validating the proposed technique and reports the ﬁndings; Section 4 summarizes the contribution and discusses future work.

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Crossing-Fiber Tissue Model

The proposed model generalizes the minimal model of white matter diﬀusion (MMWMD) in [7, 8] to accommodate the presence of multiple ﬁber populations in a single voxel. The MMWMD represents white matter as an ensemble of impermeable cylindrical axons, with both a single diameter and a single orientation, embedded in a homogeneous medium. Water molecules diﬀuse with an identical intrinsic diﬀusivity both inside and outside the axons but without exchanges between the compartments. By assuming a single axon diameter rather than a distribution as in [6, 13], the MMWMD enables orientation-invariant estimation of axon diameter for the ﬁrst time, providing a single summary statistic of the axon diameter distribution, called the axon diameter index [8]. The axon diameter index is simpler to estimate than models of the full distribution in [6, 13], but still discriminates naturally occurring axon diameter distributions [8]. To model crossing ﬁbers, we instead represent white matter as a set of N ≥ 1 distinct ﬁber populations embedded in a common homogeneous medium. Each

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ﬁber population is individually modeled as prescribed by the MMWMD. Assuming that no exchange occurs between the ﬁber populations and their common surrounding medium, as well as between the ﬁber populations the themselves, i i normalized (MR) signal from white matter, Awm , is thus νic N i=1 fic Aic + (1 − νic )Aec , where νic ∈ [0, 1] is the intra-cellular volume fraction within the white matter, Aiic the normalized signal from the i-th ﬁber population with a voli ume fraction fic ∈ [0, 1] relative to the whole population of ﬁbers, and Aec the normalized signal from the extra-cellular medium. The MMWMD contains two additional compartments: one models isotropically free Gaussian diﬀusion to capture the partial volume with cerebrospinal ﬂuid (CSF); the other models isotropically restricted diﬀusion to capture observed restrictions parallel to axons in ﬁxed tissue, potentially due to water trapped within glial cells [5]. Here we include only the latter, because the white matter area considered in our brain data experiment is several voxels away from the boundary with the ventricles and thus void of CSF contamination. The full normalized signal model, A, is therefore (1 − νir )Awm + νir Air , where Air is the normalized signal from the isotropically restricted compartment with a volume fraction νir . We set Air = 1, following the stationary water assumption in [8], i.e., the compartment signal remains unattenuated by diﬀusion weighting. The subsequent sections detail the modeling of Aiic and Aec and the ﬁtting procedure. Intra-Cellular Model. Water diﬀusion in this compartment is cylindrically restricted. The intra-cellular signal from the i-th ﬁber population, Aiic , depends on the axon diameter and orientation of the population, denoted by ai and ni respectively, the intrinsic diﬀusivity of water, denoted by d, as well as the imaging protocol. For the pulsed-gradient spin-echo (PGSE) sequence, as used in our experimental evaluation, we compute Aic using the Gaussian phase distribution approximation [14] as in [7, 8] to model the signal from restricted diﬀusion. Extra-Cellular Model. Water diﬀusion in this compartment is hindered. The extra-cellular signal, Aec , is modeled with simple (Gaussian) anisotropic diﬀusion using an (apparent) diﬀusion tensor as in [15]. We model the diﬀusion tensor, N i Dcyl (νic , ni , d), where Dcyl (νic , ni , d) is the diﬀusion tensor repDh , as i=1 fic resenting the hindered diﬀusion in the i-th ﬁber population, deﬁned according to the MMWMD. This follows from the common medium assumption, i.e., water exchanges freely within diﬀerent regions of the extra-cellular space. Model Fitting. We ﬁt the proposed model to data with the three-stage routine described in [8]. It provides robust estimates of the model parameters with the Rician Markov Chain Monte Carlo (MCMC) procedure in [7], after an initial grid search and then gradient descent to determine the maximum likelihood (ML) estimates of the parameters. The full set of model parameters are the N axon diameters (a1 , a2 ,..., aN ) 1 2 and orientations (n1 ,n2 , ..., nN ), the N − 1 relative volume fractions (fic , fic , N −1 ..., fic ), the other two volume fractions νic and νir , and the intrinsic diﬀusivity

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i d. Note that, because N i=1 fic = 1, only N − 1 of the relative volume fractions are independent. Throughout, we ﬁx d to 0.6 × 10−9 m2 s−1 , its expected value in the ex vivo data, as in [8].

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Experiments and Results

Ex Vivo Imaging of Monkey Data Set. Ex vivo diﬀusion weighted imaging (DWI) of a 32-month perfusion-ﬁxed Vervet monkey was acquired on a 4.7T Varian system with maximum gradient strength |G| = 400 mT/m. (See [16] for brain preparation.) A total of 360 images were collected using a PGSE DWI sequence with single-line spin-echo readout (TE = 36ms, TR = 2500ms). Each has isotropic 0.5x0.5x0.5 mm3 voxels and 30 sagittal slices centered on the midsagittal plane of the corpus callosum with in-plane matrix size of 128x256. The protocol, determined using the experimental design optimization [7] adapted for ﬁxed tissue, consists of three HARDI shells with the number of diﬀusion gradients [103, 106, 80], |G| = [300, 210, 300] mT/m, δ = [5.6, 7.0, 10.5] ms, and Δ = [12, 20, 17] ms, corresponding to b-values = [2084, 3084, 9550] s/mm2 . Ethical rules concerning the handling and care of live animals were followed. Synthetic Data Experiment. We synthesize diﬀusion MR signal from a broad set of two-ﬁber crossing substrates using the publicly available Monte-Carlo diffusion simulator in Camino [17] with the ex vivo imaging protocol described above. Synthetic Rician noise with σ = 0.05 is added to match the SNR of the ex vivo data (around 20). For each substrate, the proposed model with N = 2 is then ﬁtted to its synthetic data and the parameter estimates are compared against the known ground-truth settings of the substrate. The synthetic substrates assume reasonable crossing conﬁgurations and microstructure features and are constructed as follows: We consider six representative axon diameter combinations: {a1 , a2 } ∈ {{2, 2}, {2, 4}, {2, 6}, {4, 4}, {4, 6}, 1 {6, 6}} μm. For each combination, we test three relative volume fractions, fic ∈ o o {0.3, 0.5, 0.7}, and a broad range of crossing angles, varying from 30 to 90 in 15o increment. Similar to [7], νic is set to 0.7, its typical value in brain white matter. Since the diﬀusion simulator does not model isotropically restricted compartment, νir is 0. This results in a total of 90 diﬀerent crossing ﬁber substrates. Finally, to avoid possible orientation dependence, 20 diﬀerent instances of each substrate are created by applying random 3-D rotations to the initial conﬁguration. Independent Rician noise described above are added to each instance. For each substrate, we compute the mean and the standard deviation of the parameter estimates for its 20 random instances. This accounts for the eﬀect of noise and the dependence to orientation. We report our ﬁndings from the assessment of all 90 substrates. Due to limit in space, the ﬁndings are illustrated 1 = 0.3 in Fig. 1. with only one axon diameter combination ({2, 6}μm) and fic 1 The relative volume fraction fic can be estimated accurately for all axon diameter combinations and for the crossing angles larger than 45o ; for the crossing angle of 30o , it is consistently over-estimated. The intra-cellular volume fraction

H. Zhang, T.B. Dyrby, and D.C. Alexander

A1 = 2μm, A2 = 6μm Angular Error (degrees)

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Fig. 2. The manually deﬁned crossing ﬁber ROI (in red) overlaid on the FA map in three orthogonal views

νic is consistently under-estimated for all substrates, by about 10%. The volume fraction of the isotropically restricted compartment νir can be consistently estimated for all substrates, with a slight over-estimation. For axon diameters, 2μm and 4μm are diﬃcult to diﬀerentiate from one an1 and crossing angles. However, both can be diﬀerentiated other for all values of fic 1 from 6μm for all values of fic and crossing angles larger than 45o . Orientations can be estimated accurately for all axon diameter combinations, 1 , and the crossing angles larger than 45o . The error in the oriall values of fic entation estimates increases as the crossing angle decreases and as the axon diameter increases. The error in the estimate of the ﬁber population with the lower relative volume fraction is higher than that of the other population. In summary, under practical imaging protocol, the proposed model is able to both resolve the crossing ﬁber conﬁguration and estimate microstructure features for crossing angles larger than 45o . Monkey Data Experiment. This experiment uses the ex vivo monkey data set described earlier to demonstrate the eﬃcacy of the proposed model for mapping axon diameters in brain tissue with crossing ﬁbers. We manually delineate a welldeﬁned two-ﬁber crossing region-of-interest (ROI) between the corpus callosum and the cingulum bundle as shown in Fig. 2. Note the dark band in the center

Axon Diameter Mapping in Crossing Fibers with Diﬀusion MRI

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Fig. 3. The orientation and axon diameter estimates for both the two-ﬁber and single-ﬁber models. The orientation estimates are represented using the standard RGBencoding (Red: medial-lateral, Blue: inferior-superior, Green: anterior-posterior) [18].

of the ROI. The appearance of such voxels ﬂanked by the ones with higher FA is characteristic of crossing ﬁbers due to partial volume between two tracts with substantially diﬀerent orientations. We ﬁt both the proposed two-ﬁber model and the MMWMD (single-ﬁber) to each voxel in this region, and compare their respective estimates. Fig. 3 shows the orientation and axon diameter estimates for both the twoﬁber and single-ﬁber models. For the two-ﬁber model, the ﬁber population 1 corresponds to the population initialized, during model ﬁtting, with the primary eigenvector of the diﬀusion tensor estimate at each voxel. The well-deﬁned trajectory of the cingulum, which traverses from anterior to posterior, allows us to identify the voxels colored in green in Fig. 3(1a) as the ones with the ﬁber population 1 being the cingulum. The dashed line separates these voxels from the ones below them, which are primarily the corpus callosum ﬁbers. The most striking observation is that, despite the orientation estimates of the single-ﬁber model (Fig. 3(1c)) being largely consistent with those of the ﬁber population 1 of the two-ﬁber model (Fig. 3(1a)), their respective axon diameter estimates are distinctly diﬀerent. In particular, among the voxels immediately adjacent to the dashed line, the putative interface between the cingulum and the corpus callosum, the axon diameter estimates of many, from the single-ﬁber model (Fig. 3(2c)), are close to or higher than 10μm, much higher than the values determined from histology (about 2μm) [2]. In contrast, their corresponding estimates for the ﬁber population 1 from the two-ﬁber model (Fig. 3(2a)) are in the same range as the histologically estimated values, highlighting a key beneﬁt of using the two-ﬁber model in such crossing ﬁber voxels.

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The orientation and axon diameter estimates for the ﬁber population 2 from the two-ﬁber model require more care in their interpretation. The most apparent feature of the axon diameter estimates (Fig. 3(2b)) is that a large majority of them are higher than 10μm. Further inspection conﬁrms that the values for almost all these voxels are close or equal to 40 μm, the maximum value allowed in our ﬁtting routine to indicate the negligible presence of the corresponding ﬁber population, in this case, the ﬁber population 2. On the other hand, a number of voxels have axon diameter estimates with values comparable to those from histology. They can be divided into two groups according to their spatial locations and orientations. In the ﬁrst group are the voxels that are immediately adjacent to the putative cingulum and corpus callosum interface and have their orientation estimates for the ﬁber population 2 consistent with being part of the cingulum, i.e., colored in green in Fig. 3(1b). For these voxels, the two-ﬁber model is able to both resolve the crossing conﬁguration and estimate the axon diameters of individual ﬁber populations. In the second group are the voxels that are away from the interface and inside the cingulum region. The ﬁber population 2 of these voxels may correspond to the outward-projecting cingulum ﬁbers.

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Discussion

We have described a technique for joint estimation of crossing ﬁber conﬁguration and microstructure features using a new crossing-ﬁber white matter model that includes individual axon diameter and volume fraction parameters for each ﬁber population. The results from simulation and brain data demonstrate promising possibilities for extending axon diameter mapping to the whole brain. In particular, the brain data ﬁndings suggest that the crossing-ﬁber model can resolve crossing conﬁgurations and estimate axon diameters simultaneously under some circumstances; when it fails to do so, it can still provide more sensible axon diameter estimates for the dominant ﬁber population. Given that it is challenging to estimate axon diameter in voxels with only a single orientation, it is remarkable that we can make progress at crossing. Even improving the estimate of the dominant popualtion is itself useful and a signiﬁcant step forward. Nevertheless, the proposed crossing ﬁber model can be extended in several ways: First, the current model assumes that axons are strictly parallel within each ﬁber population. This assumption can be relaxed to account for axon spreading in each distinct population using the orientation dispersion model in [9]. Second, the current model assumes that the extra-cellular space is a common homegeneous medium. A more general model may require the modeling of the extra-cellular space of each ﬁber population individually. Lastly, we will examine other crossing regions, such as the pons and the crossing regions between the corpus callosum and the corona radiata in centrum semiovale. Acknowledgement. We thank Prof Maurice Ptito, University of Montreal, for providing the monkey brain. The future and emerging technologies program of the EU FP7 framework funds the CONNECT consortium (brain-connect.eu), which supports this work. EPSRC fund DCA under grant EP/E007748.

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