Structural connectivity analysis using Finsler geometry · Structural Connectivity Analysis Using...

$: Structural connectivity analysis using Finsler geometry · Structural Connectivity Analysis Using Finsler Geometry\ast Tom Dela Haije\dagger \ddagger \S , Peter Savadjiev\dagger \P$
Structural connectivity analysis using Finsler geometry

Citation for published version (APA):Dela Haije, T., Savadjiev, P., Fuster, A., Schultz, R. T., Verma, R., Florack, L., & Westin, C-F. (2019). Structuralconnectivity analysis using Finsler geometry. SIAM Journal on Imaging Sciences, 12(1), 551-575.https://doi.org/10.1137/18M1209428

DOI:10.1137/18M1209428

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https://doi.org/10.1137/18M1209428

https://doi.org/10.1137/18M1209428

https://research.tue.nl/en/publications/structural-connectivity-analysis-using-finsler-geometry(46cc5af5-4939-4acc-9967-b2e2eb68fa01).html

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SIAM J. IMAGING SCIENCES c\bigcirc 2019 Society for Industrial and Applied MathematicsVol. 12, No. 1, pp. 551--575

Structural Connectivity Analysis Using Finsler Geometry\ast

Tom Dela Haije\dagger \ddagger \S , Peter Savadjiev\dagger \P , Andrea Fuster\S , Robert T. Schultz\| ,Ragini Verma\#, Luc Florack\S , and Carl-Fredrik Westin\dagger \dagger

Abstract. In this work we demonstrate how Finsler geometry---and specifically the related geodesic tracto-graphy---can be levied to analyze structural connections between different brain regions. We presentnew theoretical developments which support the definition of a novel Finsler metric and associatedconnectivity measures, based on closely related works on the Riemannian framework for diffusionMRI. Using data from the Human Connectome Project, as well as population data from an autismspectrum disorder study, we demonstrate that this new Finsler metric, together with the new connec-tivity measures, results in connectivity maps that are much closer to known tract anatomy comparedto previous geodesic connectivity methods. Our implementation can be used to compute geodesicdistance and connectivity maps for segmented areas and is publicly available.

Key words. diffusion MRI, Finsler geometry, connectivity analysis

AMS subject classification. 92C02

DOI. 10.1137/18M1209428

1. Introduction. Finsler geometry was proposed as a means to analyze diffusion MRIdata in works by Pichon, Westin, and Tannenbaum [41] and Melonakos et al. [36, 37], wherethe authors computed shortest geodesic tracts based on an ad hoc relation with the diffu-sion MRI signal. A different definition of the Finsler function was employed by Astola andFlorack [3], who illustrate some more technical applications of Finsler geometry, while the

\ast Received by the editors September 21, 2018; accepted for publication (in revised form) January 9, 2019; publishedelectronically March 26, 2019.

http://www.siam.org/journals/siims/12-1/M120942.htmlFunding: This work was supported by NIH grants P41EB015902, R01MH074794, and R01MH092862, and by a

NARSAD Young Investigator award (grant number 22591) by the Brain and Behavior Research Foundation to PeterSavadjiev. The authors also gratefully acknowledge NWO (No. 617.001.202) and the Villum Foundation for financialsupport. The work of Andrea Fuster is part of the research programme of the Foundation for Fundamental Researchon Matter (FOM), which is financially supported by the Netherlands Organisation for Scientific Research (NWO).Results presented in this work have been published as part of the main author's Ph.D. thesis ``Finsler Geometry andDiffusion MRI"" (Eindhoven University of Technology, ISBN: 978-90-386-4274-1).

\dagger Joint first authors.\ddagger Department of Computer Science, University of Copenhagen, 2100 Copenhagen, Denmark ([email protected]).\S Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven,

The Netherlands ([email protected], [email protected]).\P Department of Radiology, McGill University Health Center, Montreal, QC H3G 1A4, Canada (petersv@cim.

mcgill.ca).\| Center for Autism Research, Department of Pediatrics and Psychiatry, University of Pennsylvania, Philadelphia,

PA 19104 ([email protected]).\#Section of Biomedical Image Analysis, Department of Radiology, University of Pennsylvania, Philadelphia, PA

19104 ([email protected]).\dagger \dagger Laboratory for Mathematics in Imaging, Department of Radiology, Brigham and Women's Hospital, Harvard

Medical School, Boston, MA 02115 ([email protected]).

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552 DELA HAIJE ET AL.

work of Sepasian et al. [48] extends shortest geodesic tractography to allow multiple geodesicconnections between points. Finally, the first geodesic connectivity analyses were performedby de Boer et al. [16], which also marks the first time Finsler geometry was used to studygroup differences in diffusion MRI data. Though these results demonstrate the potential valueof Finsler geometry in diffusion MRI, one downside is the limited availability of tools that canperform these types of analyses. For this reason, along with the highly technical content ofeven introductory works on Finsler geometry, Finsler-based methods remain inaccessible tothe majority of diffusion MRI researchers.

The goal of this paper is to provide an informal introduction to the core concepts involvedwith Finsler connectivity computations, based on our recently released Finsler connectivitymodule (FCM). This module is an adaptation of the finslertract code by Antonio Trist\'an-Vega (nitrc.org/projects/finslertract), which is used to perform Finsler tractography. To thisend, we describe the ideas behind the application of Finsler geometry to diffusion-weightedMRI data in section 2, and we describe the tractography and connectivity algorithm insection 3. This section introduces a new Finsler metric definition and a generalization of aconnectivity measure used for diffusion tensor imaging (DTI) [6], in order to address theoret-ical limitations of existing geometrical methods. As a practical illustration of the method, wecompute the connectivity for a number of well-known major fiber bundles in a high-resolutionpublic data set. We also discuss how the connectivity algorithm can be used in group studiesand, as a proof of concept, use our method to study group differences in autism spectrumdisorder (ASD) data using network-based analysis techniques. The results of these experi-ments are presented in section 4. Finally we discuss some strengths and shortcomings of theapproach in section 5.

2. Theoretical background.

2.1. Finsler geometry. Finsler geometry is concerned with measuring distances on ab-stract spaces called Finsler manifolds. A Finsler manifold consists of a base space, in thecontext of this paper always the set of positions in \BbbR 3 where diffusion is measured, and a realscalar-valued function F : \BbbR 3 \times \BbbR 3 \rightarrow \BbbR + that captures the additional structure of the space.The distance between two points on a Finsler manifold is defined similar to the standardEuclidean distance, namely, in terms of the length of the shortest curve connecting the twopoints; the difference lies in the way the length of the curve is computed. The length \scrL F (\gamma )of a curve \gamma : [0, L] \rightarrow \BbbR 3 is still a sum over infinitesimal line elements d\gamma , but the associatedlength of each line element is now weighted depending on both its position and its orientation:

\scrL F (\gamma ) :=

\int L

0F

\biggl( \gamma (t),

\.\gamma (t)

\| \.\gamma (t)\|

\biggr) \| \.\gamma (t)\| dt,(2.1)

where \.\gamma (t) = \mathrm{d}\gamma \mathrm{d}t (t). With F = 1 this reduces to the Euclidean length of the curve,

\scrL E(\gamma ) :=

\int L

0\| \.\gamma (t)\| dt.(2.2)

The Finsler function F is (positively) homogeneous of degree one in its second argument,which means that for any point \bfitx and any direction \bfity the following relation holds:

F (\bfitx , \lambda \bfity ) = | \lambda | F (\bfitx ,\bfity ).(2.3)

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STRUCTURAL CONNECTIVITY ANALYSIS 553

We can thus write

\scrL F (\gamma ) =

\int L

0F (\gamma (t), \.\gamma (t)) dt(2.4)

and

F

\biggl( \gamma (t),

d\gamma

dt(t)

\biggr) dt = F (\gamma (t), d\gamma (t)) ,(2.5)

which clarifies the interpretation of F as a function acting locally on a line element d\gamma . At thesame time we note that the homogeneity of F , (2.3), guarantees that the length F (\gamma (t), d\gamma (t))associated to d\gamma is determined solely by its orientation---not by its ``magnitude""---and that\scrL F (\gamma ) is independent of the (proper) parametrization of \gamma . As a technical aside we observethat homogeneity also implies that F is strictly speaking defined only when \| \.\gamma (t)\| \not = 0 forall t, and for this reason we have to assume a parametrization of \gamma that avoids this issue. Tosimplify the following discussion we assume without loss of generality that \| \.\gamma (t)\| = 1, i.e., weassume that \.\gamma (t) is an element of the sphere \BbbS 2. Some additional properties and technicalitiesrelated to F are given in the paper by Florack and Fuster [23], and a thorough exposition onFinsler geometry can be found, for example, in the book by Bao, Chern, and Shen [5].

2.2. The Finslerian framework for diffusion MRI. The Finslerian framework for diffusionMRI models the brain as a Finsler manifold, by deriving a Finsler function F from diffusion-weighted data. The principal idea behind this is the well-known correlation between thelocal amount of diffusion in a certain direction and the large-scale structural orientation ofwhite matter [8]. By defining (or assuming) a correspondence between the Finslerian lengthof a curve and the amount of diffusion along a curve, we can leverage a rich set of Finslergeometrical tools for the analysis of diffusion data.

Bearing this in mind, the Finsler function is generally defined such that some measureof diffusivity at a given point and in a certain direction is inversely related to the associatedlength. In other words, we have that a large diffusivity at a point \bfitx along a vector \bfity corre-sponds to a small value F (\bfitx ,\bfity ). This leads to the useful alternative viewpoint of the Finslerfunction as a kind of cost function. If we consider a displacement in direction \bfity as a parameterthat can be controlled, then F can be interpreted as associating a high cost to movement ina direction with low diffusivity and a low cost to movement in directions of high diffusivity.

One typical measure of the amount of diffusion along a certain orientation is the diffusionorientation distribution function, or dODF, which was originally defined as [53]

\itPsi \mathrm{d}(\bfitx ,\bfity ) =

\int \infty 0

P (\bfitx , r\bfity ) dr,(2.6)

where P is the probability density function describing the diffusion observed at \bfitx , and whichis directly related to the local diffusion-weighted signal through a Fourier transform [12]. Thedefinition given in (2.6) was later revised [1] to ensure a proper normalization of the ODF:

\itPsi \mathrm{s}\mathrm{a}(\bfitx ,\bfity ) =

\int \infty 0

P (\bfitx , r\bfity ) r2 dr.(2.7)

We will make use of these definitions in section 3 when we reconsider the Finsler functionsused in earlier works.

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2.3. Geodesics. A prime example of analysis tools made available by the geometricalframework is geodesics. Geodesics can be regarded as connections along which one encounters,in some sense, optimal diffusivity. More specifically, geodesics connecting two given pointsare those curves for which the length \scrL F is (locally) minimal and can thus be viewed as theFinslerian analogue of ``straight lines."" The existence of a geodesic between any two pointsin the Finsler manifold is guaranteed [5, Chapter 6.6], which means that we can find optimalconnections between any two points or regions of interest. For now we will assume that theshortest geodesic between two points, called the minimal geodesic, is uniquely defined.

In practice we determine minimal geodesics using a fast-sweeping algorithm [36, 37] basedon the principle of optimality [9], which states that given a unique minimal geodesic \gamma :[0, L] \rightarrow \BbbR 3, the geodesic segment between the points \gamma (a) and \gamma (b), a, b \in [0, L], is necessarilyidentical to the geodesic between these two points. If we write \scrL \ast F (\bfitx ) for the shortest geodesicdistance from a point \bfitx \in \BbbR 3 to a seed region \Omega \subset \BbbR 3 relative to F , then

\scrL \ast F (\bfitx ) := inf\gamma \{ \scrL F (\gamma ) | \gamma (0) = \bfitx , \gamma (L) \in \Omega \}

= inf\gamma

\Biggl\{ \scrL \ast F (\gamma (T )) +

\int T

0F (\gamma (t), \.\gamma (t)) dt

\bigm| \bigm| \bigm| \bigm| \bigm| \gamma (0) = \bfitx

\Biggr\} (2.8)

for all T \in (0, L); cf. Figure 1. From the Taylor expansion

\scrL \ast F (\gamma (T )) = \scrL \ast F (\bfitx ) +\nabla \bfity \scrL \ast F (\bfitx )T +\scrO (T 2),(2.9)

where \nabla \bfity denotes the directional derivative along \bfity and where we use the shorthand notation\bfitx = \gamma (0) and \bfity = \.\gamma (0) \in \BbbS 2, we find the Hamilton--Jacobi--Belmann equation [45] in the limitT \rightarrow 0:

inf\bfity \in \BbbS 2

\nabla \bfity \scrL \ast F (\bfitx ) + F (\bfitx ,\bfity ) = 0.(2.10)

Figure 1. The principle of optimality states that segments of the minimal geodesic between two points arethemselves geodesic. The black curve represents the optimal curve \gamma (minimal geodesic) connecting the point \bfitx to the seed region \Omega , i.e., the curve \gamma that minimizes the Finslerian length functional \scrL F . The distance \scrL \ast

F (\bfitx )from \bfitx to \Omega is defined as the length of the optimal curve \gamma that connects the two. If \scrL \ast

F (\gamma (T )) is known for\gamma (T ) near \bfitx , the principle of optimality allows us to compute \scrL \ast

F (\bfitx ) by solving the Hamilton--Jacobi--Bellmanequation (2.10). As \scrL \ast

F (\bfitx ) = 0 for all \bfitx \in \Omega , repeated application of (2.10) allows us to compute \scrL \ast F for all

\bfitx \in \BbbR 3.

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Together with the initial condition

\scrL \ast F (\bfitx ) = 0 for \bfitx \in \Omega ,(2.11)

repeated application of (2.10) allows us to compute the complete \scrL \ast F map for all \bfitx \in \BbbR 3. SeeAppendix A for implementation details.

This method is relatively fast, but as explained before it is limited in that it only findsthe shortest geodesic out of the possibly many geodesics connecting two given points [48].In the following we will make the usual assumption that all relevant information is capturedby the minimal geodesic. More information on fast-sweeping algorithms can be found in thereferences [30, 29, 59].

Returning to the context of diffusion MRI, the optimal diffusivity along geodesics is mademore precise by the inverse relation discussed in subsection 2.1. With the cost functioninterpretation of F , we note that geodesics correspond to curves along which the accruedcost is minimal. Because of the inverse relation between the cost and the local diffusivity,geodesics thus minimize movement in directions of low diffusivity. Additionally, we see thatthe Finslerian length \scrL F of a curve approximately corresponds to the average reciprocaldiffusivity along the curve.

3. Methods. Geodesics have been used as a tool in tractography, based on the hypothesisthat (some small subset of) geodesics between two points coincide with the physical connec-tions between them. This approach has certain practically advantageous features; it relieson the full diffusion information available (unlike most deterministic streamline tractographyalgorithms, for instance) and has essentially no parameters that have to be tuned. It should,however, be noted that since there is no canonical definition of the Finsler function, this choiceof ``metric"" does add an additional degree of freedom to the algorithm. The riemanntract andfinslertract packages available at nitrc.org can be used to perform geodesic tractographywith the different metrics.

Geodesic connectivity analysis is based on the hypothesis that curves of optimal diffusivitybetween two points reflect the likelihood of a structural connection, resulting in high connec-tivity values within anatomical bundles connecting two seed regions. In this section we discussthe options for the Finsler function in the FCM, as well as two available geodesic-based pathand connectivity measures that have been studied in recent works.

3.1. The Finsler function. The choice for the Finsler function F in terms of the diffusionsignal is the primary degree of freedom in the Finslerian framework and determines to a largeextent how geometric features such as geodesics can be interpreted. The most common choicefor F found in the literature was proposed by Melonakos et al. [37] and is given by

F\mathrm{o}\mathrm{l}\mathrm{d}(\bfitx ,\bfity ) :=

\biggl( S(\bfitx ,\bfity )

\itPsi \mathrm{d}(\bfitx ,\bfity )

\biggr) 3

(3.1)

with \bfity \in \BbbS 2 and S the diffusion MRI signal acquired on a fixed b-value shell, and where \itPsi \mathrm{d}

is the dODF (2.6) defined in terms of the Funk--Radon transform (see, e.g., Tuch [53]). Thepower 3 is used as a type of sharpening. The choice F = F\mathrm{o}\mathrm{l}\mathrm{d} has been shown to producereasonable tractography results, e.g., near the cingulum bundle [36], and it has been used in

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the literature by de Boer et al. [16]. However, it lacks a clear relation with the more well-founded existing metrics in the Riemannian framework---we expect the Finsler function incase of Gaussian diffusion to reduce to a Riemannian norm compatible with the canonicalRiemannian metric [39, 32] or one of the metrics derived therefrom [27, 25, 28, 24].

As an alternative we postulate a new choice for F :

F\mathrm{n}\mathrm{e}\mathrm{w}(\bfitx ,\bfity ) :=MD(\bfitx )

\itPsi \mathrm{s}\mathrm{a}(\bfitx ,\bfity ),(3.2)

where MD is a generalization of the mean diffusivity used in DTI [7] defined as the average ap-parent diffusion coefficient [58], and \itPsi \mathrm{s}\mathrm{a} is the solid angle dODF [1]; cf. (2.7). In Appendix Bwe show that for purely Gaussian diffusion, F\mathrm{n}\mathrm{e}\mathrm{w} corresponds to (the cost function of) asharpened version of the Riemannian metric given by the adjugate diffusion tensor [25, 24],which can be considered the preferred choice of metric in the Riemannian setting [46]. Notefurther that we have \itPsi \mathrm{s}\mathrm{a} \propto \itPsi \mathrm{d}

3 in the special case of Gaussian diffusion (Appendix B), whichmeans that in this case both F\mathrm{o}\mathrm{l}\mathrm{d} and F\mathrm{n}\mathrm{e}\mathrm{w} are inversely proportional to \itPsi \mathrm{d}

3. This sug-gests that we are effectively retaining the sharpening strategy deemed optimal by Melonakoset al. [37].

3.2. Path measures. Geodesic-based connectivity analysis combines a variety of curveshape measures with measures derived from the diffusion signal along the curve into a singlepath measure. This shape measure could be as simple as the Euclidean length, while moreadvanced shape measures such as local curvature and torsion are possible but are used lessfrequently. The diffusion signal is encoded in the Finslerian length, representative of thetotal diffusivity, or the Finslerian speed, representative of the local diffusivity, or in a setof statistical measures such as the quantiles, mean, and standard deviation of the Finslerfunction evaluated along the geodesic. The heuristic definition of connectivity in terms ofpath measures is discussed in subsection 3.4 and in Appendix C.

The basic path measure used in the FCM is defined as

\scrC \mathrm{a}\mathrm{v}\mathrm{g}(\gamma ) :=\scrL F (\gamma )

\scrL E(\gamma ),(3.3)

which is the Finslerian generalization of the most commonly used measure in Riemanniangeodesic connectivity analysis [4, 33]. This measure can be interpreted as the average costincurred along the geodesic, which is expected to be low for curves between two well-connectedregions. The normalization with respect to Euclidean length in (3.3) ensures that the pathmeasure does not depend on the length of the path, i.e., it guarantees scale invariance. Furtherdetails regarding the necessity of this invariance can be found in the literature [22, 42, 33].

The second available path measure is defined as the largest local cost along a geodesic,given by

\scrC \mathrm{m}\mathrm{a}\mathrm{x}(\gamma ) := maxt

F

\biggl( \gamma (t),

\.\gamma (t)

\| \.\gamma (t)\|

\biggr) .(3.4)

The \scrC \mathrm{m}\mathrm{a}\mathrm{x} path measure was originally proposed in the Riemannian setting by Pechaud,Descoteaux, and Keriven [40]. Although this measure might be expected to be very sen-sitive to noise, it should be noted that it is based on the same (intrinsically smooth) geodesics

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as the \scrC \mathrm{a}\mathrm{v}\mathrm{g} measure and is thus as stable as \scrC \mathrm{a}\mathrm{v}\mathrm{g}. The \scrC \mathrm{m}\mathrm{a}\mathrm{x} measure highlights geodesicsthat have continuously strong diffusivity along their paths, in contrast to the \scrC \mathrm{a}\mathrm{v}\mathrm{g} measurefor which a locally weak diffusivity might be offset by very strong diffusivities further alongthe geodesic. Again, a low value of the path measure implies a high connectivity.

Details on the implementation of these measures can be found in Appendix A.

3.3. The Finsler connectivity module. The FCM typically returns two scalar maps---adistance map \scrL \ast F and the corresponding path measure map---based on an input diffusion-weighted data set that provides F , an optional mask image, and a label map containinglabeled seed regions. The choice between F\mathrm{o}\mathrm{l}\mathrm{d} and F\mathrm{n}\mathrm{e}\mathrm{w} can be supplied as well, along with achoice of the path measures \scrC \mathrm{a}\mathrm{v}\mathrm{g} and \scrC \mathrm{m}\mathrm{a}\mathrm{x} and further resolution and convergence parameters(e.g., the maximum order used in the spherical harmonic representation of F ). The distancemap provides at each position the shortest Finslerian distance to the seed region in accordancewith (2.8), and the path measure map gives the path measure associated with the shortestgeodesic connecting each point to the seed region. Geodesics can be computed if the internallygenerated tangent vector map is returned as well. The FCM is available at github.com/tomdelahaije/fcm and can be used as a command-line tool.

3.4. Experimental design. We demonstrate the performance of the proposed Finsler-based connectivity analysis in two different settings: (1) a qualitative validation based ondata from the WU-Minn Human Connectome Project (HCP) and (2) a quantitative networkanalysis of connectivity in ASD.

Qualitative validation. The qualitative Finsler connectivity validation experiments pre-sented in subsection 4.1 are based on the preprocessed data of a single healthy subject in theWU-Minn HCP [55], released as part of the HCP 500 Subject Release. The data was acquiredon a modified 3T Siemens scanner with 1.25mm isotropic voxels. The diffusion-weighted im-ages are acquired on three shells (b = \{ 1000, 2000, 3000\} s/mm2) with 90 uniformly distributedgradient directions each, together with 18 baseline images. We express F\mathrm{o}\mathrm{l}\mathrm{d} and F\mathrm{n}\mathrm{e}\mathrm{w} in termsof spherical harmonics based on the b = 3000 s/mm2 shell (maximum order 6 based on pre-liminary investigations for this data set; see also [56, 51]). T1 data with an isotropic voxelsize of 0.7mm was also available. Further details on the acquisition protocol can be found onthe HCP website and in [55, 26, 2].

All maps based on the HCP data are seeded from manually selected voxels within thewhite matter, one voxel per bundle, based on the DTI white matter atlas by Catani andThiebaut de Schotten [14]. Seeds are placed in four well-known major white matter bundles:the cingulum, the arcuate fasciculus, the corticospinal tract, and the splenium of the corpuscallosum.

Network analysis in autism spectrum disorder. In order to provide an illustration of howthe Finsler connectivity framework could be applied to population studies in a clinical setting,we present here a proof of concept network-based analysis of our Finsler connectivity approach,applied to ASD data. To do so, we use a paradigm that is commonly used in network-basedstudies of brain connectivity. Specifically, we build a graph model of the brain, where graphnodes represent gray matter regions as defined (in this case) through a FreeSurfer parcellationbased on the Desikan--Killiany atlas [19]. The edge weights in this model represent the Finsler

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connectivity between pairs of FreeSurfer-defined gray matter regions. Once this network modelis constructed for each subject, we can compute any number of graph-theoretical measuresto quantitatively summarize the network structure. For the purpose of this illustration wefocus on only a single measure---the ``local network efficiency"" [43]---that has been previouslyimplicated in ASD [44, 34]. Details on the connectivity analysis pipeline can be found inAppendix C.

Diffusion and structural MRI data were acquired from 69 typically developing male con-trols (TDC; age range 8.0--14.4 years, mean 10.7, standard deviation 1.8) and 46 age-matchedmale ASD patients (age range 8.1--14.1 years, mean 10.7, standard deviation 2.0). A t-testfor difference in age between the two groups resulted in a p-value of 0.98. All imaging wasperformed using a Siemens 3T Verio scanner with a 32 channel head coil. Structural im-ages were acquired on all subjects using an MP-RAGE imaging sequence [10] (TR = 19 s,TE = 2.54ms, TI = 0.9 s, 0.8mm in-plane resolution, 0.9mm slice thickness). Addition-ally, a HARDI acquisition [54, 18] was performed using a monopolar Stejskal--Tanner [50]diffusion-weighted spin-echo, echo-planar imaging sequence with the parameters TR = 14.8 s,TE = 110ms, 2mm isotropic resolution, and b = 3000 s/mm2, with 64 gradient directions andwith two baseline images. The diffusion-weighted images of each subject were filtered usinga joint linear minimum mean squared error filter to suppress Rician noise [52]. Eddy currentcorrection was performed using registration of each volume to one of the baseline images. Thesame data was used in a study by Caruyer and Verma [13].

4. Results.

4.1. Qualitative validation. All connectivity and path measure maps are shown in radi-ological convention. Path measure maps are shown using a temperature color map, wheredark blue indicates a low connectivity/high path measure, and bright red indicates a highconnectivity/low path measure. The used color maps are shown on the right of each figure.Note that the results for the different path measures \scrC \mathrm{m}\mathrm{a}\mathrm{x} and \scrC \mathrm{a}\mathrm{v}\mathrm{g} cannot be compared quan-titatively and that connectivity is strongest in regions that are connected to a seed point,typically leading to comparable high connectivity values within a \its \itu \itp \ite \itr \its \ite \itt of the bundle inwhich the seed is placed.

Cingulum. The first major bundle we consider is the cingulum, which consists of a set offibers that project from the cingulate gyrus to the entorhinal cortex. In the work of Melonakoset al. [36] geodesic tractography was successfully used to trace this bundle, so we can expectour geodesic-based connectivity analysis to produce reasonable results for seeds placed in thisbundle. The distance map from which geodesics can be computed is shown in Figure 2 forboth F = F\mathrm{o}\mathrm{l}\mathrm{d} and the newly proposed F\mathrm{n}\mathrm{e}\mathrm{w}-based metric. The Finslerian distance increasesas expected from the seed outward, and the boundaries between white matter, gray matter,and cerebral spinal fluid can be identified very roughly. It is, however, difficult to judge therelative merit of the different metrics from these maps alone. Because all the connectivitymeasures under investigation in this section are derived from the same \scrL \ast F distance maps, andbecause the distance maps themselves provide little information, we omit the distance mapshenceforth.

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F\mathrm{n}\mathrm{e}\mathrm{w} F\mathrm{o}\mathrm{l}\mathrm{d}

Figure 2. Axial (top row) and sagittal (bottom row) slices of distance maps \scrL \ast F (2.8) for F = F\mathrm{n}\mathrm{e}\mathrm{w} (left

column, (3.2)) and F = F\mathrm{o}\mathrm{l}\mathrm{d} (right column, (3.1)) seeded at a single voxel in the cingulum (annotated point)of the HCP data set. Greater brightness encodes a greater distance. Other than minor differences in contrast,there are no conspicuous differences between the two choices for F deducible from these maps.

The connectivity maps shown in Figures 3 and 4 provide more information. First, a high-level comparison between Figures 3 and 4 reveals that the \scrC \mathrm{m}\mathrm{a}\mathrm{x} connectivity measure (3.4)provides much more anatomical detail than the \scrC \mathrm{a}\mathrm{v}\mathrm{g} connectivity measure (3.3). In Figure 3we see furthermore that the standard \scrC \mathrm{a}\mathrm{v}\mathrm{g} path measure map seeded in the cingulum bundleleaks into the corpus callosum, and to large sections of the posterior part of the brain. Thiseffect is much less in the \scrC \mathrm{m}\mathrm{a}\mathrm{x} map shown in Figure 4. Overall, the \scrC \mathrm{m}\mathrm{a}\mathrm{x} map is much morespecific to the known anatomy of the cingulum bundle, which allows us to perform a moredetailed comparison between the F = F\mathrm{o}\mathrm{l}\mathrm{d} metric and our newly proposed F = F\mathrm{n}\mathrm{e}\mathrm{w} metric.We see that the former does result in leakage from the cingulum into the posterior parts of thecorpus callosum (the splenium), and from there into large sections of posterior white matter.The choice F = F\mathrm{n}\mathrm{e}\mathrm{w} results in a path measure map that closely follows the known anatomyof the cingulum bundle, without leaking into the corpus callosum.

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Figure 3. Axial (top row) and sagittal (bottom row) slices of maps based on the \scrC \mathrm{a}\mathrm{v}\mathrm{g} path measure (3.3),derived from the data shown in Figure 2 (seeded in the cingulum). The left column shows the results for thenewly proposed F = F\mathrm{n}\mathrm{e}\mathrm{w} metric (3.2), and the right column shows results for the F = F\mathrm{o}\mathrm{l}\mathrm{d} metric (3.1).Bright red voxels are strongly connected to the seed region (white) according to the used path measures, whiledark voxels are weakly connected. The displayed maps provide more anatomical detail than the distance mapsin Figure 2 but appear to wrongly ascribe a high connectivity from the cingulum to the corpus callosum and tothe posterior part of the brain. The choice F = F\mathrm{n}\mathrm{e}\mathrm{w} suffers from artificially high values at the edges, due toerrors in the mask.

Finally we note that all maps suffer to a certain extent from errors in the mask, which cancause erroneous and problematic high path measure values near the boundaries of the mask.These false positives are clearly visible at the edges in Figure 3 (left) and near the brainstemin Figure 4 (bottom). With the \scrC \mathrm{a}\mathrm{v}\mathrm{g} measure these errors can propagate throughout the brain,which is particularly grievous in combination with the new F\mathrm{n}\mathrm{e}\mathrm{w}-based metric. With the \scrC \mathrm{m}\mathrm{a}\mathrm{x}

path measure this propagation is, however, completely suppressed, which means the errorsremain localized.

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Figure 4. Axial (top row) and sagittal (bottom row) slices of maps based on the \scrC \mathrm{m}\mathrm{a}\mathrm{x} path measure (3.4),derived from the data shown in Figure 2 (seeded in the cingulum). The left column shows the results for thenewly proposed metric F = F\mathrm{n}\mathrm{e}\mathrm{w} (3.2), and the right column shows results for the F = F\mathrm{o}\mathrm{l}\mathrm{d} metric (3.1).We observe again that anatomical features are more distinguished compared to Figure 2 but also compared toFigure 3. Note that the F\mathrm{o}\mathrm{l}\mathrm{d} metric leads to much more significant ``leakage"" into the corpus callosum. The\scrC \mathrm{m}\mathrm{a}\mathrm{x} path measure shown in this figure is largely unaffected by masking errors; cf. Figure 3.

Arcuate fasciculus. Maps of the two path measures, \scrC \mathrm{a}\mathrm{v}\mathrm{g} and \scrC \mathrm{m}\mathrm{a}\mathrm{x}, are shown for thearcuate fasciculus in Figures 5 and 6. The arcuate fasciculus is a functionally importantbundle, involved in aspects of language processing. It connects frontal cortical areas with thesuperior temporal gyrus. It also includes connections to the inferior parietal lobe. The mapsfor this bundle very clearly highlight a major issue with the widely used \scrC \mathrm{a}\mathrm{v}\mathrm{g} measure: despitethe seed being placed in the right hemisphere, the left hemisphere shows an overall \its \itt \itr \ito \itn \itg \ite \itr connection to the seed than the voxels in the right hemisphere. This problem is to a largeextent resolved with the introduction of the \scrC \mathrm{m}\mathrm{a}\mathrm{x} measure. For the remaining bundles weshow only results with the \scrC \mathrm{m}\mathrm{a}\mathrm{x} measure, though it should be noted that the \scrC \mathrm{a}\mathrm{v}\mathrm{g} measurecan produce good results \itl \ito \itc \ita \itl \itl \ity as shown in the top-left map in Figure 5.

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Figure 5. Axial (top row) and coronal (bottom row) slices of \scrC \mathrm{a}\mathrm{v}\mathrm{g}-based maps (3.3) for F = F\mathrm{n}\mathrm{e}\mathrm{w} (leftcolumn, (3.2)) and F = F\mathrm{o}\mathrm{l}\mathrm{d} (right column, (3.1)) seeded in the arcuate fasciculus (annotated point) of theHCP data set. Connections over the corpus callosum typically have large average diffusivities, which results inthe undesired high values in the unseeded hemisphere visible for both metrics.

Again, a comparison between the \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based maps with F = F\mathrm{o}\mathrm{l}\mathrm{d} and F = F\mathrm{n}\mathrm{e}\mathrm{w} revealsa superior performance of the new F\mathrm{n}\mathrm{e}\mathrm{w}-based metric (barring the artifacts due to maskingerrors). With the F\mathrm{o}\mathrm{l}\mathrm{d}-based metric, there is again leakage into the corpus callosum andthen into the opposite hemisphere, which is not observed with the F\mathrm{n}\mathrm{e}\mathrm{w}-based metric. As it isdifficult to judge the reconstruction of the arcuate fasciculus from two-dimensional (2D) slices,we show a 3D reconstruction of a thresholded path measure map obtained with F = F\mathrm{n}\mathrm{e}\mathrm{w}

in Figure 7, which shows a fairly complete reconstruction of the regions connected to andincluding the arcuate fasciculus with minimal leakage into other bundles. This reconstructionwas impossible with F = F\mathrm{o}\mathrm{l}\mathrm{d}.

Corticospinal tract. The next tract we examine is the corticospinal tract (CST), whichis a major fiber tract that conducts sensorimotor signals between the cortex and the spinal

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Figure 6. \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based maps (3.4) for F = F\mathrm{n}\mathrm{e}\mathrm{w} (left column, (3.2)) and F = F\mathrm{o}\mathrm{l}\mathrm{d} (right column, (3.1))corresponding to Figure 5 (seeded in the arcuate fasciculus). Unlike the corresponding \scrC \mathrm{a}\mathrm{v}\mathrm{g}-based maps shownin Figure 5, the \scrC \mathrm{m}\mathrm{a}\mathrm{x} path measure does not have a bias for cross-hemispheric connections.

cord. \scrC \mathrm{m}\mathrm{a}\mathrm{x} path measure maps obtained with a single voxel seed in the CST are provided inFigure 8. These maps show some of the advantages and disadvantages of the different metrics.The traditionally used F\mathrm{o}\mathrm{l}\mathrm{d} metric appears to be better equipped to avoid leakage into theopposite hemisphere, at the level of the brain stem, and specifically at the pons, where pontinecrossing fibers may cause the connectivity to cross the midline into the other hemisphere [57].To see this, compare the images in the bottom row of Figure 8. This result is anatomicallyincorrect and is a well-known issue with many tractography algorithms [57]. While the F\mathrm{n}\mathrm{e}\mathrm{w}

metric is more sensitive to this issue, the regions of high connectivity extend much further inthe superior direction toward the cortex, better reproducing the known fiber fanning in thisregion. Furthermore, F\mathrm{n}\mathrm{e}\mathrm{w}-based connectivity of the CST shows only a minor leak into thecorpus callosum, while the F\mathrm{o}\mathrm{l}\mathrm{d} metric produces a much more extensive leak into the posteriorpart of the corpus callosum and into the opposite hemisphere.

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Figure 7. A left-right view of a rendering of the regions connected to the arcuate fasciculus obtainedby thresholding the \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based map (3.4) (with F = F\mathrm{n}\mathrm{e}\mathrm{w}) shown in Figure 6. The rendered segmentationenvelops the arcuate fasciculus, is entirely contained within the right hemisphere, and contains further offshootsof structures connected to the seeded bundle. Attempts to reconstruct a similar segmentation using the \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based map with F = F\mathrm{o}\mathrm{l}\mathrm{d} expectedly included large regions contralateral to the seeded bundle, without capturingthe full structure of the arcuate.

Corpus callosum. The last bundle we consider is the corpus callosum, a massive whitematter highway that connects the two hemispheres. The \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based path measure maps forthe F\mathrm{o}\mathrm{l}\mathrm{d} and F\mathrm{n}\mathrm{e}\mathrm{w} metrics were seeded in the splenium (posterior part) of the corpus callosum,as shown in Figure 9. With the F\mathrm{n}\mathrm{e}\mathrm{w} metric, we observe that the path measure map remainsconcentrated in the posterior parts of the corpus callosum, while with the F\mathrm{o}\mathrm{l}\mathrm{d} metric itprogresses much further in the anterior direction, which should not be happening with a seedlocated posteriorly.

4.2. Network analysis in autism spectrum disorder. We performed the network analysisstudy with the newly proposed F\mathrm{n}\mathrm{e}\mathrm{w}-based Finsler metric, using three different maximumorders for the spherical harmonic representation of the ODF. For comparison we also repeatedthe experiment, for the same three different spherical harmonic orders, using the F\mathrm{o}\mathrm{l}\mathrm{d}-basedmetric proposed by Melonakos et al. [37] (see (3.1)). The resulting p-values are presented inTable 1. All experiments used the \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based connectivity measure.

5. Discussion. In this work we described the basic ideas of Finsler geometry for thepurpose of geodesic connectivity analyses and introduced the open-source Finsler connectivitymodule, or FCM, for Finsler geodesic tractography and connectivity studies. The FCM isbased on the finslertract project of Antonio Trist\'an-Vega (nitrc.org/projects/finslertract)and includes changes that reflect recent advances in Riemannian geodesic tractography [24, 40].The connectivity analysis capacities of the module were primarily evaluated onWU-Minn HCPdata, with results depending heavily on the choice of Finsler function and path/connectivitymeasure.

We have considered two different Finsler metrics, one based on a newly proposed Finslerfunction F\mathrm{n}\mathrm{e}\mathrm{w} derived from work by Fuster et al. [24] in the Riemannian setting, and theF\mathrm{o}\mathrm{l}\mathrm{d}-based metric originally proposed by Melonakos et al. [37], which is the one typically usedin the literature. While a visual assessment of the distance maps obtained with each approach

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nitrc.org/projects/finslertract




Figure 8. Axial (top row) and coronal (bottom row) slices of \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based maps (3.4) for F = F\mathrm{n}\mathrm{e}\mathrm{w} (leftcolumn, (3.2)) and F = F\mathrm{o}\mathrm{l}\mathrm{d} (right column, (3.1)) seeded in the CST (annotated point) of the HCP data set.For F = F\mathrm{n}\mathrm{e}\mathrm{w} connectivity leaks to the opposite hemisphere at the level of the brain stem where pontine fiberscross into the other hemisphere, while the regions of high connectivity fan and extend superiorly to a greaterdegree than for F = F\mathrm{o}\mathrm{l}\mathrm{d}.

is not informative (see, e.g., Figure 2), the corresponding path measure maps highlight someinteresting differences. In particular, the maps obtained using the Finsler function F =F\mathrm{n}\mathrm{e}\mathrm{w} are more faithful to the known anatomy of tracts, as illustrated by the examples insubsection 4.1. Although both the F\mathrm{n}\mathrm{e}\mathrm{w}- and F\mathrm{o}\mathrm{l}\mathrm{d}-based maps suffer some ``leakage"" problems,i.e., high connectivity values spreading to nearby but unrelated tracts, the F\mathrm{n}\mathrm{e}\mathrm{w}-based metricis much more robust to this issue compared to the F\mathrm{o}\mathrm{l}\mathrm{d}-based metric. This is especially clearnear the corpus callosum, as can be seen in the cingulum results shown in Figure 4. Thus,in addition to its more rigorous theoretical foundation, the F\mathrm{n}\mathrm{e}\mathrm{w} Finsler function typicallyresults in connectivity maps which are anatomically more reliable.

In contrast to the F\mathrm{o}\mathrm{l}\mathrm{d}-based Finsler metric, the new metric is designed to correspond toa theoretically well-founded Riemannian metric [24]; cf. Appendix B. However, the fact that

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Figure 9. Axial (top row) and sagittal (bottom row) slices of \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based maps (3.4) for F = F\mathrm{n}\mathrm{e}\mathrm{w} (leftcolumn, (3.2)) and F = F\mathrm{o}\mathrm{l}\mathrm{d} (right column, (3.1)) seeded in the splenium of the corpus callosum (annotatedpoint) of the HCP data set. The F\mathrm{n}\mathrm{e}\mathrm{w}-based metric follows anatomy more closely than the F\mathrm{o}\mathrm{l}\mathrm{d}-based one,exemplified here by the high values of the latter observed in the frontal part of the corpus callosum.

the generalized Finslerian metrics used here are still defined in an ad hoc manner remains asignificant issue with the interpretation of Finsler geometrical analyses. Given the improvedresults obtained with the relatively simple F\mathrm{n}\mathrm{e}\mathrm{w} Finsler function, it will be worthwhile toinvestigate the application of the fundamental results presented in the work of Dela Haije [17]to the current analysis pipeline in future work.

In subsection 3.2 and Appendix A we explain how the \scrC \mathrm{m}\mathrm{a}\mathrm{x} measure used in the Rieman-nian setting by, e.g., P\'echaud, Descoteaux, and Keriven [40] can be applied in the Finsleriansetting. This measure is based on the ``weakest link"" of the geodesic. That is, geodesicsbetween strongly connected points should have a continuously low cost along the entire tract,while even small regions of high cost along the geodesic are taken to significantly decrease thelikelihood of the points being structurally connected. Compared to the more common \scrC \mathrm{a}\mathrm{v}\mathrm{g}

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Table 1p-values for the MANOVA comparison between the local efficiency vectors of the control and ASD groups.

Column heading F\mathrm{n}\mathrm{e}\mathrm{w} indicates connectivity computed using the F = F\mathrm{n}\mathrm{e}\mathrm{w} metric, whereas F\mathrm{o}\mathrm{l}\mathrm{d} denotes themetric proposed by Melonakos et al. [37]. Connectivity matrices were computed using a spherical harmonicrepresentation of the diffusion ODF with three different orders: 2, 4, and 6. All connectivity measures werecomputed with the \scrC \mathrm{m}\mathrm{a}\mathrm{x} path measure. Since a total of six tests were performed, the Bonferroni-correctedthreshold for significance was 0.05/6 \approx 0.0083. p-values below this threshold (shown in bold) indicate rejectionof the null hypothesis that there is no difference between the two groups.

Order F\mathrm{n}\mathrm{e}\mathrm{w} F\mathrm{o}\mathrm{l}\mathrm{d}

2 \bfzero .\bfzero \bfzero \bfthree \bffive \bfzero .\bfzero \bfzero \bfsix \bfone 4 \bfone .\bfseven \times \bfone \bfzero - \bffour 0.00876 0.035 0.030

measure, which considers the average of some diffusivity measure along a geodesic, the \scrC \mathrm{m}\mathrm{a}\mathrm{x}

path measure has a number of important advantages, both theoretical and practical.Primarily, the \scrC \mathrm{m}\mathrm{a}\mathrm{x} measure associates a relatively low connectivity to geodesics taking

shortcuts, a notorious problem of geodesic tractography [48, 28]. In the same vein, provided asufficiently fine spatial resolution, low connectivity is associated with geodesics that jump fromone fiber bundle to another across a small region with high cost. Because of this, in practice,regions of high connectivity tend to concentrate much more on the seeded fiber bundles,which again significantly reduces leakage artifacts. One can appreciate this especially in thedifference between the \scrC \mathrm{a}\mathrm{v}\mathrm{g} and \scrC \mathrm{m}\mathrm{a}\mathrm{x} path measure maps seeded in the arcuate fasciculus(Figures 5 and 6), where the consistently low cost in the corpus callosum results in an above-average \scrC \mathrm{a}\mathrm{v}\mathrm{g}-based connectivity for all geodesics that cross the corpus callosum to the otherhemisphere. This in fact highlights another theoretical advantage; the \scrC \mathrm{m}\mathrm{a}\mathrm{x} measure hasthe very natural property of being monotonic, i.e., it cannot increase in connectivity withdistance along a path. Combined, these properties result in \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based connectivity mapsthat are generally much closer to anatomy than maps produced using the \scrC \mathrm{a}\mathrm{v}\mathrm{g} measure.

However, the \scrC \mathrm{a}\mathrm{v}\mathrm{g} measure can lead to greater contrast than \scrC \mathrm{m}\mathrm{a}\mathrm{x} at a \itl \ito \itc \ita \itl level, as canbe seen in the left columns (F\mathrm{n}\mathrm{e}\mathrm{w}-based metric) of the arcuate fasciculus results, Figures 5and 6. \scrC \mathrm{m}\mathrm{a}\mathrm{x}-based maps typically have low homogeneous connectivity throughout the whitematter, which essentially results in a default situation in which everything is (at least weakly)connected to everything else, while \scrC \mathrm{a}\mathrm{v}\mathrm{g}-based maps can efficiently extract the local, moredirect structural connections.

The large differences between the two path measures highlight an issue with the currentlyemployed definitions. The different types of information captured by these measures makeit clear that other path measures, or more likely combinations of various measures, can andshould be developed to obtain a more complete characterization of the structural connectivitycaptured by the geodesics. Because the validation of connectivity measures is very challenging,a possible next step could be a structured inclusion of a complete set of descriptive measures,e.g., shape measures of increasing complexity, subject to natural constraints like scale and ori-entation invariance. The introduction of anatomical priors, which has become more commonin recent works [49], could also be used to further improve geodesic-based connectivity analysis.

Finally, we have studied group differences in a graph-theoretical analysis of brain networksin ASD, where we found significant differences in the local network efficiency between the ASD

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group and the normally developing controls. These results depend on the spherical harmonicsused to represent the Finsler function (Table 1), which needs to be chosen in a manner thatbalances angular resolution and sensitivity to noise. Differences in local network efficiencyhave been widely reported in previous works, and our results corroborate previous findings ofabnormalities [44, 34]. Note that this analysis was not intended to be exhaustive but ratherto illustrate the application of our new connectivity framework in a standard brain networkanalysis setting. In future work, we will conduct a more thorough analysis of brain networksbased on Finsler connectivity.

The main problem addressed only superficially in this work is validation of the connectivitymeasures. As with available alternatives, the absence of ground truth data makes it very chal-lenging to validate and compare connectivity measures. In the near future this may becomefeasible, with the way being paved by recent efforts on validation of microstructural modelsthrough, e.g., histology [47] and on validation of tractography by evaluating methods on sim-ulated data with a known ground truth [38, 35]. Although outside of the scope of this work, adirect comparison of different connectivity measures in specific applications (such as popula-tion studies) could provide additional insight into the performance of the proposed measure.

In conclusion, we have presented a new tool, the Finsler connectivity module, that imple-ments a connectivity pipeline for diffusion MRI. To our knowledge this is the first non-Gaussiangeodesic connectivity method made available to the neuroimaging community in the form of aready-to-use tool. Our implementation improves the existing framework used by, e.g., de Boeret al. [16], as demonstrated in the presented HCP experiments. These improvements derivefrom a novel Finsler function defined in terms of the solid angle ODF and from the considera-tion of more appropriate connectivity measures. In future work we will evaluate our approachwith respect to the currently employed tractography-based connectivity methods.

Appendix A. Implementation details. In the pseudocode below (Algorithm 1) we presenta modification of Melonakos' fast-sweeping algorithm [37], with the additional steps needed tocompute the proposed path measures ((3.3) and (3.4)). In analogy with the notation for thedistance map \scrL \ast F , which gives the shortest distance between each point and the seed region \Omega ,we define \scrC \ast \mathrm{a}\mathrm{v}\mathrm{g} and \scrC \ast \mathrm{m}\mathrm{a}\mathrm{x} as the path measure maps that give for each point the path measureassociated to the geodesic from that point to the considered seed region. Similarly, we define\scrL \ast E as the shortest Euclidean distance between each point and the seed region, and \bfity \ast asthe tangent to the geodesic at each point. The algorithm is based on the observation thatfor a solution of (2.10), the Finslerian distance between \bfitx and \bfitx + \bfity for \| \bfity \| small is simplygiven by F (\bfitx ,\bfity ). The exact definition of the addition \bfitx + \bfity requires the exponential map onthe Finsler manifold, which goes beyond the scope of this paper---see, e.g., the work by Bao,Chern, and Shen [5] for more information. In the FCM interpolation is done linearly [37], butalternative methods are possible.

Appendix B. Derivation of \bfitF \bfn \bfe \bfw . In the usual Riemannian framework for DTI [39,32] the metric tensor \bfitg is defined by the inverse diffusion tensor, i.e., \bfitg = \bfitD - 1. A minormodification of this relation has recently been proposed on theoretical grounds [25, 24], inwhich a new metric \~\bfitg is defined by the adjugate diffusion tensor:

\~\bfitg := det(\bfitD ) \bfitD - 1,(B.1)

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Algorithm 1. The fast-sweeping algorithm used to compute the maps \scrL \ast F , \scrC \ast \mathrm{a}\mathrm{v}\mathrm{g},\scrC \ast \mathrm{m}\mathrm{a}\mathrm{x}, and \bfity \ast , adapted from Melonakos et al. [37]. The implementation is based onthe finslertract project of Antonio Trist\'an-Vega.

\bfD \bfa \bft \bfa : \mathrm{A} \mathrm{s}\mathrm{e}\mathrm{e}\mathrm{d} \mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n} \Omega \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{a} \mathrm{F}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{l}\mathrm{e}\mathrm{r} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} F ;\bfR \bfe \bfs \bfu \bfl \bft : \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{m}\mathrm{a}\mathrm{p} \scrL \ast F , \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{m}\mathrm{a}\mathrm{p} \bfity \ast , \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h} \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e} \mathrm{m}\mathrm{a}\mathrm{p}\mathrm{s} \scrC \ast \mathrm{a}\mathrm{v}\mathrm{g} \mathrm{a}\mathrm{n}\mathrm{d} \scrC \ast \mathrm{m}\mathrm{a}\mathrm{x};

\mathrm{I}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e} \scrL \ast F (\bfitx /\in \Omega )\leftarrow \infty , \scrL \ast F (\bfitx \in \Omega )\leftarrow 0, \scrL \ast E(\bfitx )\leftarrow 0;\bfr \bfe \bfp \bfe \bfa \bft

\bff \bfo \bfr \bfe \bfa \bfc \bfh \itp \ito \its \iti \itt \iti \ito \itn \bfitx \bfd \bfo (\bfity \ast )\prime (\bfitx )\leftarrow \mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}\bfity \in \BbbS 2 \scrL \ast F (\bfitx + \bfity ) + F (\bfitx ,\bfity );

(\scrL \ast F )\prime (\bfitx )\leftarrow \scrL \ast F (\bfitx + (\bfity \ast )\prime (\bfitx )) + F (\bfitx , (\bfity \ast )\prime (\bfitx ));\bfi \bff (\scrL \ast F )\prime (\bfitx ) < \scrL \ast F (\bfitx ) \bft \bfh \bfe \bfn

\bfity \ast (\bfitx )\leftarrow (\bfity \ast )\prime (\bfitx );\scrL \ast F (\bfitx )\leftarrow (\scrL \ast F )\prime (\bfitx );\scrL \ast E(\bfitx )\leftarrow \scrL \ast E (\bfitx + \bfity \ast (\bfitx )) + 1;\scrC \ast \mathrm{a}\mathrm{v}\mathrm{g}(\bfitx )\leftarrow \scrL \ast F (\bfitx )/\scrL \ast E(\bfitx );

\bfi \bff F (\bfitx ,\bfity \ast (\bfitx )) > \scrC \ast \mathrm{m}\mathrm{a}\mathrm{x} (\bfitx + \bfity \ast (\bfitx )) \bft \bfh \bfe \bfn \scrC \ast \mathrm{m}\mathrm{a}\mathrm{x}(\bfitx )\leftarrow F (\bfitx ,\bfity \ast (\bfitx ));

\bfe \bfl \bfs \bfe \scrC \ast \mathrm{m}\mathrm{a}\mathrm{x}(\bfitx )\leftarrow \scrC \ast \mathrm{m}\mathrm{a}\mathrm{x} (\bfitx + \bfity \ast (\bfitx ));

\bfe \bfn \bfd

\bfe \bfn \bfd

\bfe \bfn \bfd

\bfu \bfn \bft \bfi \bfl \itc \ito \itn \itv \ite \itr \itg \ite \itn \itc \ite \ito \itf \scrL \ast F ;

where det\bfitD is the determinant of the diffusion tensor \bfitD . This relation between F\mathrm{n}\mathrm{e}\mathrm{w} and theadjugate \~\bfitg can be recognized in the exact expression for the normalized (Gaussian) ODF [31]

\itPsi \mathrm{d}(\bfitx ,\bfity ) =MD(\bfitx )\sqrt{}

\bfity T \cdot det\bfitD (\bfitx ) \bfitD - 1(\bfitx ) \cdot \bfity ,(B.2)

where MD is the mean diffusivity and \bfity \in \BbbS 2 is a direction unit vector. We note that thedenominator in this equation is the norm of direction vector \bfity in the Riemannian spaceequipped with the metric \~\bfitg ,

\| \bfity \| \~\bfitg =\sqrt{}

\bfity T \cdot \~\bfitg \cdot \bfity =

\sqrt{} \bfity T \cdot det(\bfitD ) \bfitD - 1 \cdot \bfity ,(B.3)

and combining (B.2) and (B.3) we find

\| \bfity \| \~\bfitg (\bfitx ) =MD(\bfitx )

\itPsi \mathrm{d}(\bfitx ,\bfity ).(B.4)

Replacing \itPsi \mathrm{d} with the sharper [1]

\itPsi \mathrm{s}\mathrm{a}(\bfitx ,\bfity ) =1

4\pi \sqrt{} det\bfitD (\bfitx )

\bigl( \bfity T \cdot \bfitD - 1(\bfitx ) \cdot \bfity

\bigr) 32

(B.5)

then produces the F\mathrm{n}\mathrm{e}\mathrm{w} metric proposed in (3.2).

Appendix C. Connectivity analysis pipeline. In recent years, the view that the functionaland structural systems of the brain can be modeled as complex networks has motivated a

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large amount of research on the application of graph-theoretical concepts to brain networkanalysis [43, 11]. The standard graph network model of the brain consists of a set of nodes,which represent a partitioning of the cortex and other gray matter structures. These nodes areconnected via a set of edges, or links, that represent structural and/or functional connectionsbetween gray matter partition units. Such a graph model of the brain's network organizationcan be constructed from a variety of imaging modalities such as structural MRI, diffusionMRI, functional MRI, or EEG/MEG. In this setting, a characterization of the organization ofthe different computational nodes and the functional interaction between them is achieved viagraph-theoretical analysis [43, 11]. This appendix describes the details of the pipeline used toperform the group analyses reported in this work, covering the cortical surface parcellation,the connectivity measure computation, and the final statistical analysis.

FreeSurfer parcellation. FreeSurfer is a freely available software toolbox that reconstructsmesh-based models of the cortical surface [15, 21] and provides a parcellation of the cortexinto neuro-anatomical areas using both the geometrical model of the cortical surface andneuro-anatomical convention [20].

For the present set of experiments, we computed a FreeSurfer parcellation for each subjectbased on the Desikan--Killiany atlas [19], which resulted in a total of 86 cortical and sub-cortical gray matter regions. FreeSurfer parcellations are initially defined in each subject'sT1 standard space. In order to allow for these gray matter regions of interest to be used asseed regions for Finsler connectivity analysis, we registered the FreeSurfer parcellation to thediffusion MRI space using nonlinear registration.

Connectivity matrix construction. The path measures discussed in subsection 3.2 arelength-based, i.e., low values indicate a short geodesic distance, which implies a high con-nection strength. From these path measures we can heuristically derive a new measure thatsignifies a general notion of connectivity as follows.

For each subject in our study, we constructed an 86\times 86 connectivity matrix \bfitC , whose ijthentry Cij represents the connectivity between the ith and jth FreeSurfer regions. Each entryCij is computed as follows. First, seeding in the ith region, we run our Finsler connectivityalgorithm to produce a path measure map for the entire brain. Then, the values for the pathmeasure in the spatial extent of the jth region are averaged for each j \not = i, which results inthe average path measure values mi\rightarrow j . This process is repeated for all seed regions i. Finally,we construct the connectivity matrix \bfitC :

Cij =

\Biggl\{ exp

\Bigl( - \nu

mi\rightarrow j+mj\rightarrow i

2

\Bigr) if i \not = j,

1 if i = j.(C.1)

Note that \bfitC is a symmetric matrix by construction, so each pair of regions has a uniquelydefined connectivity value. The scaling parameter \nu is set to 0.1 for the presented experiments,producing a reasonable spread of connectivity values over the range [0, 1] with small valuesindicating a low connection strength.

As an example, Figure 10 shows a coronal slice through a path measure map computed onone of the TDC subjects. This map was seeded in the left caudal middle frontal gyrus region,as defined by a FreeSurfer segmentation using the Desikan--Killiany atlas as described above.

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Figure 10. A comparison of \scrC \mathrm{m}\mathrm{a}\mathrm{x} path measure maps, using the F\mathrm{n}\mathrm{e}\mathrm{w}- and F\mathrm{o}\mathrm{l}\mathrm{d}-based metrics on data fromone of the TDC subjects. These maps are seeded in the left caudal middle frontal gyrus region, as defined byFreeSurfer. The seed voxels that intersect this particular coronal slice are shown in white. Bright red voxelsare strongly connected to the seed region according to the used path measures, while dark voxels are weaklyconnected.

The seed voxels that intersect this particular coronal slice are shown in white with a blackoutline. The path measure map itself is shown in a modified temperature color map, suchthat bright red colors indicate a low path measure and a dark blue color indicates a high pathmeasure value. Thus, in order to compute mi\rightarrow j , where i indicates the caudal middle frontalregion, and j indicates any of the other FreeSurfer-defined regions, we average the values ofthis path measure map over the voxels that comprise FreeSurfer region j. Intuitively, regionsthat are well-connected to region i will result in low values for mi\rightarrow j , which in turn will leadto higher connectivity values Cij according to (C.1).

In addition to illustrating the concept of a path measure map and the correspondingcomputation of mi\rightarrow j , Figure 10 also provides an initial comparison between the F\mathrm{n}\mathrm{e}\mathrm{w}- andthe F\mathrm{o}\mathrm{l}\mathrm{d}-based metrics. With the caudal middle frontal gyrus as a seed region, the newlyproposed F\mathrm{n}\mathrm{e}\mathrm{w}-based metric results in the recovery of the known transcallosal connectivity tothe opposite hemisphere (Figure 10, left). In contrast, the F\mathrm{o}\mathrm{l}\mathrm{d}-based metric produces a map(Figure 10, right) that has drawbacks similar to known artifacts observed with tractography onDTI data. In particular, the connectivity does not reach the contralateral cortical regions, as itappears to stop in the well-known region of three-way crossings between the corpus callosum,the CST, and the superior longitudinal fasciculus. On the other hand, connectivity appearsto ``leak"" into the CST of both hemispheres, as well as into other white matter tracts that donot have direct anatomical connectivity with the seed region. Furthermore, the F\mathrm{n}\mathrm{e}\mathrm{w}-basedmetric correctly identifies CSF areas, such as the ventricles, and assigns these a high pathmeasure (low connectivity). In contrast, the F\mathrm{o}\mathrm{l}\mathrm{d}-based metric does not detect CSF and as aresult propagates connectivity through CSF regions, which is clearly anatomically incorrect.The differences between the two metrics is addressed in more detail in subsection 4.1.

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Computation of local efficiency and statistical analysis. Once the connectivity matrix\bfitC is computed for each subject, we compute its local efficiency measures using the BrainConnectivity Toolbox [43]. The local efficiency of a network is a quantity computed at eachnode of the network such that it quantifies the network's resistance to failure at the localscale. In other words, it quantifies the importance of a graph node by measuring how wellinformation is exchanged by the immediate neighbors of the node when it is removed. Thus,in the present experiment, this computation results in a 1\times 86 vector, such that its ith elementcorresponds to the local efficiency measure of the ith FreeSurfer region.

In the present experiment, we are interested in performing a statistical test for a groupdifference between the TDC and ASD groups of subjects based on their local efficiency vectors.To reduce the number of multiple comparisons, we do not test each region individually butperform a one-way multivariate analysis of variance to compare the mean vectors for the twogroups. This is a statistical test for the null hypothesis that the mean local efficiency vectorsof the TDC and ASD groups are the same. If we can reject this null hypothesis, we concludethat the two groups differ in terms of their local efficiency measure, although the test doesnot identify specific regions that may be responsible for this difference.

Acknowledgments. Data were provided in part by the Human Connectome Project,WU-Minn Consortium (principal investigators: David Van Essen and Kamil Ugurbil;1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blue-print for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience atWashington University. Andrea Fuster and Tom Dela Haije would like to thank their hostsCarl-Fredrik Westin and Peter Savadjiev at the Laboratory for Mathematics in Imaging atthe Brigham and Women's hospital, Harvard Medical School, for their hospitality.

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Structural connectivity analysis using Finsler geometry · Structural Connectivity Analysis Using...

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