O R I G I N A L R E S E A R C H A R T I C L E

Quantitative streamlines tractography: methods and

inter-subject normalisation

Robert E. Smitha,b, David Raffelta, J-Donald Tournierc, Alan Connellya,b,d

a The Florey Institute of Neuroscience and Mental Health, Heidelberg, Victoria, Australia

b Florey Department of Neuroscience and Mental Health, University of Melbourne, Melbourne, Victoria, Australia

c Centre for the Developing Brain, School of Biomedical Engineering & Imaging Sciences, King’s College London, London, UK

d Department of Medicine, Austin Health and Northern Health, University of Melbourne, Melbourne, Victoria, Australia

This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (CC BY 4.0), which permits authors to copy and redistribute

the material in any medium or format, remix, transform and build upon material, for any purpose, even commercially.

: 2022, Volume 2 - 1 - CC By 4.0: © Smith etal.

ABSTRACT

Recent developments in semi-global tractogram optimisation algorithms have opened the eld of diffusion magnetic resonance

imaging (MRI) to the possibility of performing quantitative assessment of structural bre ‘connectivity’. The proper application of

these methods in neuroscience research has, however, been limited by a lack of awareness, understanding, or appreciation for the

consequences of these methods; furthermore, particular steps necessary to use these tools in an appropriate manner to fully exploit

their quantitative properties have not yet been described. This article therefore serves three purposes: to increase awareness of

the fact that there are existing tools that attempt to address the well-known non-quantitative nature of streamlines counts; to illus-

trate why these algorithms work the way they do to yield quantitative estimates of white matter ‘connectivity’ (in the form of total

intra-axonal cross-sectional area: ‘bre bundle capacity (FBC)’); and to explain how to properly utilise these results for quantitative

tractography analysis across subjects.

Keywords: Magnetic resonance imaging, diffusion, white matter, streamlines tractography, quantication

Corresponding author: Robert E Smith, Florey Institute of Neuroscience and Mental Health, Melbourne Brain Centre, 245 Burgundy Street, Heidelberg, Victoria 3084, Australia,

Phone: (+61 3) 9035 7128, Fax: (+61 3) 9035 7301, Email: robert.smith@orey.edu.au

Received: 02.10.2020

Accepted: 24.01.2022

DOI: 10.52294/ApertureNeuro.2022.2.NEOD9565

ABBREVIATIONS

AFD: apparent bre density;

COMMIT: convex optimisation modelling for microstructure-informed tractography;

DWI: diffusion-weighted imaging (/image)

FBA: xel-based analysis;

FBC: bre bundle capacity (an estimate of the bandwidth of a white matter pathway);

FC: bre cross-section (NB: macroscopic change in);

FD: bre density (microscopic);

FDC: bre density and cross-section (combined measure of FD and FC);

‘xel’: specic ﬁbre population within a voxel;

FOD: bre orientation distribution;

LiFE: linear fascicle evaluation;

SIFT: spherical-deconvolution informed ltering of tractograms.

INTRODUCTION

Since the introduction of tractography to the eld of dif-

fusion magnetic resonance imaging (MRI), there has been

extensive interest in the use of this technology to assess

bre ‘connectivity’ in the brain for various neuroscientic

applications (1–4). The vast majority of tractography al-

gorithms operate on the same fundamental mechanism:

the ‘streamlines’ algorithm, where plausible white mat-

ter bre pathways are constructed by iteratively propa-

gating along the local estimated bre orientation (5–9).

Unfortunately, this mechanism of reconstruction does not

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directly facilitate one of the most fundamental parameters

of interest: the density of ‘connectivity’ between two brain

regions (10). A major contributing factor to this limitation

is that while the streamlines algorithm enforces that the re-

constructed trajectories obey the estimated orientations

of the underlying bre bundles, it provides no meaningful

control over the reconstructed densities of those bundles.

The class of ‘global tractography’ methods (11–15) has

for many years shown promise to circumvent this prob-

lem. While in the ‘streamlines’ algorithm individual white

matter trajectories are propagated independently and

using only local bre orientation information, these ‘glob-

al’ methods simultaneously solve for all connections at

once, in a manner that enforces the entire tractogram re-

construction to be consistent with the raw diffusion image

data. Even the most modern of these methods, however,

incur considerable computational expense (particularly

as reconstructions with greater numbers of connections

are sought), and typically do not provide any guarantees

regarding the construction of connections with biologi-

cally meaningful terminations, for instance, resulting in

terminations in the white matter or cerebrospinal uid

(CSF) that are otherwise considered erroneous (16,17).

A new class of ‘semi-global’ tractogram optimisation

algorithms offers a potential compromise (18–22); these

have additionally been referred to as ‘tractogram lter-

ing’, ‘microstructure-informed tractography’, and ‘top-

down’ algorithms in various contexts. These approaches

take as input a whole-brain tractogram generated using

one or more streamlines tractography algorithms and

modify the reconstruction in some way such that the local

streamlines densities become consistent with the density

of underlying bres evidenced by the image data. These

methods therefore enable quantitative assessment of

bre ‘connectivity’ (within the myriad other associated

limitations of diffusion MRI and streamlines tractogra-

phy), with whole-brain reconstructions that are sufciently

dense to enable higher-level analyses (e.g. connectomics

(23,24)) within reasonable computational requirements.

Despite the potential inuence of these methods on

the neuroimaging eld, they have had only limited up-

take. This may be due to a lack of awareness of the public

availability of such methods, or a lack of understanding

that these methods address some of the origins of the

limitations of raw streamline count as a metric of ‘con-

nectivity’. Furthermore, although these methods seek to

modulate the relative connection densities of different

white matter pathways within a single brain, the appro-

priate mechanism by which these quantities should be

compared across subjects has not yet been comprehen-

sively explained in the literature. This article therefore

serves three purposes, with the aim of increasing the util-

ity of these tools in the eld:

• Alert a wider audience to the fact that a primary

contributing factor to the non-quantitative nature

of streamlines counts can be addressed using freely

available methods;

• Carefully explain and demonstrate why the design of

these methods is appropriate to provide estimates of

white matter connection density, including in the con-

text of structural connectome construction;

• Explain how these estimates of connection density

should be handled when performing direct compari-

sons between subjects.

BACKGROUND

Before addressing the major points of this article, we rst

clarify the specic position and role of these ‘semi-glob-

al’ tractography optimisation algorithms, the ‘connectiv-

ity’ metric of interest to be derived from them, and the

limitations within which they operate.

Requisite knowledge

The specic ‘semi-global’ methods under discussion

here are intrinsically dependent on both voxel-level

modelling of diffusion MRI data and streamlines trac-

tography. As such, an adequate understanding of those

concepts will be necessary for readers to follow the logic

presented here; these topics are covered extensively by

prior publications (2,5,9,10,25–33).

Context and role of semi-global algorithms

Figure 1 presents the role of these methods within a trac-

tography-based reconstruction pipeline.

• Some biological white matter bundle of interest (Figure

1a; the connection between homologous motor areas

in this example) is interrogated using diffusion-weight-

ed imaging (Figure 1b). Due to the sizes of the under-

lying axons within the white matter relative to the im-

aging resolution, there will typically be of the order of

a million axons traversing any given image voxel.

The notion of a single scalar quantity of ‘connectivity’

of a white matter pathway is intrinsically ambiguous. If

quantifying such a property of the underlying biological

bundle, a reasonable interpretation would be the num-

ber of axons constituting the connection, as the informa-

tion-carrying capacity of the bundle could be reasonably

expected to scale in direct proportion to such. However,

precisely estimating this parameter is prohibited by the

limitations of diffusion-weighted imaging (DWI). The logic

behind the proposed total intra-axonal cross-sectional

area metric mentioned here in Figure 1 is discussed fur-

ther in the ‘Metric of “connectivity”’ section.

• A diffusion model estimates from these data, within

each image voxel, the orientations and densities of the

bre bundles within that voxel (Figure 1c–d).

• These orientation estimates are used by a streamlines

tractography algorithm to attempt to reconstruct in

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construction and subsequent interrogation of the brain

‘structural connectome’ (23,24). Within this framework, a

parcellation of the grey matter is dened, and for every

possible unique pair of grey matter regions, a scalar

measure of ‘connectivity’ is quantied, with these values

together forming a connectivity matrix that encodes the

value of this connectivity metric between pairs of regions

in their corresponding rows/columns (29,30,40). Such

connectome construction can therefore be thought of as

simply repeating this quantiﬁcation process many times,

where each ‘bundle of interest’ is dened based on the

streamline endpoints being ascribed to a specic pair of

grey matter regions. So, in the context of connectomics,

the techniques described here for characterising such

‘connectivity’ are intended to:

• Supersede the use of streamline count, which contin-

ues to be used in neuroscientic applications despite

being known to be biased by many reconstruction-re-

lated parameters (10,19,41);

• Provide a measure of ‘connectivity’ for which, when

applying higher-order analyses that implicitly interpret

such data as the bandwidth of information ow around

a network (42,43), such an interpretation is more direct

and intuitive than alternative measures such as aggre-

gate microstructural quantities.

Metric of ‘connectivity’

The notion of ‘connectivity’ in the context of diffusion

MRI tractography remains ambiguous without a very ex-

plicit description of exactly what metric is derived from

a piecewise fashion the bres within the pathway of

interest (Figure 1e). Unlike biological axons, recon-

structed streamlines have no associated volume and

are therefore shown as innitesimally thin in Figure 1e.

The number of streamlines traversing any given image

voxel may be of the order of 1,000, but varies wildly

depending on reconstruction parameters.

• The role of such a ‘semi-global’ tractogram optimis-

ation algorithm is to combine the reconstructed trac-

togram with bre density (FD) information from the

diffusion model (or alternatively the diffusion-weight-

ed image data themselves; see the ‘Comparing tracto-

grams and image data’ section), relying on the quan-

titative nature of these FD estimates to overcome the

non-quantitative nature of streamlines tractography.

• The outcome of such a process is a derived measure of

‘connectivity’ of the pathway of interest (here named

‘bre bundle capacity (FBC)’; more in the ‘Metric of

“connectivity”’ section). If calculated appropriately,

this measure should be a reasonable proxy for the in-

formation-carrying capacity of the biological pathway.

Unlike other analysis techniques that interrogate the

values of quantitative properties as they vary along the

length of a white matter bundle of interest (34–39), here

the derived experimental output is a single scalar mea-

sure reecting the total ‘connectivity’ of that pathway be-

tween its two endpoints.

While this explanation (and subsequent demon-

strations in this article) focuses on performing such a

quantication for only a single pathway of interest, all

of the content of this article is directly relevant to the

Fig. 1. Contextualisation of semi-global tractogram optimisation algorithms. Given the existence of a biological white matter pathway of interest (a), diffu-

sion-weighted imaging is performed (b). A diffusion model is tted to these data to yield bre orientation (c) and density (d) estimates. Fibre orientations are utilised

by a tractography algorithm to produce streamlines (e). An optimisation algorithm operates on both the tractogram reconstruction and FD information to yield an

estimate of bundle connectivity, here named ‘bre bundle capacity (FBC)’, which should ideally be proportional to the connectivity of the biological bundle.

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O R I G I N A L R E S E A R C H A R T I C L E

the pathway being reconstructed. While for simplici-

ty this metric can be thought of as a proportional esti-

mate of axon count, the precise attributes of this metric

are discussed later in the ‘Qualifying the “Fibre Bundle

Capacity (FBC)” metric’ section.

The important distinction between FD estimates quan-

tied at the voxel level, and bre connection density es-

timates (i.e. FBC) quantied at the level of pathways of

interest, is demonstrated in Figure 3. There are various

diffusion models that include some parameters related

to bre volume for each image voxel (Figure 3a–d). In

the context of this article, however, we seek to quantify

the total bre cross-sectional area associated with some

specic pathway of interest (Figure 3e–f).

Limitations of semi-global optimisation

algorithms

• We do not consider the sub-voxel spatial congu-

rations of bre bundles in either the image (45) or

tractogram (46) domains; we consider only that each

voxel is the sum of its constituent parts, irrespective of

sub-voxel position;

the image data/tractogram reconstruction. In the diffu-

sion MRI tractography literature, myriad metrics have

been utilised, all of which have been referred to at some

point as simply ‘connectivity’.

In the context of the methods discussed here, our

target scalar metric of interest when quantifying ‘white

matter connectivity’ is the total ﬁbre cross-sectional area

of a bre bundle (ideally, the intra-axonal cross-sectional

area). The nature of this metric is presented visually in

Figure 2, where a bundle of interest is dened based on

those bres connecting two endpoints of interest, and

the intra-axonal cross-sectional areas of only those ﬁbres

attributed to the bundle of interest are summed to de-

rive this estimate. This metric has previously been shown

to converge white matter connection density estimates

towards gross axon count estimates from post mortem

dissection (44). To facilitate discussion of higher-level

concepts in the context of this metric, we henceforth

refer to this metric as the ‘FBC’. This term communicates

that the intent of this measure is the capacity of a white

matter bre bundle to transmit information between its

endpoints. Ideally, derivation of this measure should be

as sensitive and specic as possible to the intra-axonal

cross-sectional area of the biological bres constituting

Fig. 2. Visual depiction of the bre bundle capacity (FBC) metric. A white matter bundle of interest is dened based on its endpoints, shown as yellow cuboids.

Only those bres that are attributed to both endpoints are constituent members of that bundle (green cylinders). The FBC is dened as the sum of the intra-axonal

cross-sectional areas of these bres.

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which to compare the tractogram to those image data.

There are two principal mechanisms by which this may

be done (demonstrated in Figure 4):

1. Each streamline contributes some intensity to the

reconstructed diffusion signal based on a forward

model. Typically, a diffusion tensor with xed diffusivi-

ties is chosen. Signal intensity from sources other than

white matter bres may additionally be modelled as

isotropic or anisotropic sources contributing to the

diffusion signal and included in the optimisation. The

reconstructed image data from the tractogram (and

possibly other tissue sources) is compared directly to

the empirical diffusion data.

2. The white matter bre density within each xel is rst

estimated based on an inverse model, potentially

with estimation and separation of other signal sources

(e.g. other tissues or uid) (50–58). Here, we focus on

the spherical deconvolution model (31,59,60), though

other approaches can certainly be used. The recon-

structed streamlines density from the tractogram as-

cribed to each xel is compared directly to the corre-

sponding white matter bre densities estimated from

the diffusion model.

• We assume that the diffusion signal measured in a

voxel is the sum of signal contributions from the mat-

ter bre bundles and other tissues within that volume

(i.e. the ‘slow exchange’ regime);

• We do not consider inuencing streamlines trajectories

based on microstructural information, as discussed (47)

and proposed (48) recently; we consider only the use of

microstructural/image information to modulate the re-

constructed densities of different white matter pathways.

Comparing tractograms and image data

Throughout this discussion, we use the term ‘ﬁxel’ (49)

to refer to a specic ﬁbre population within a particu-

lar voxel. Each voxel in the diffusion image may contain

multiple xels (‘crossing bres’), and the number of xels

may vary between different voxels. Use of such terminol-

ogy assists in disambiguating this concept from a mac-

roscopic white matter fascicle that connects two areas of

grey matter, each of which will be associated with many

xels along its length and breadth.

Using diffusion image data to provide a tractogram

with quantitative attributes requires a mechanism by

Fig. 3. Relationship between quantication of intra-axonal volume within individual voxels (FD), and quantication of intra-axonal cross-sectional area of a white

matter bundle of interest (bre bundle capacity (FBC)). A bundle of interest is dened based on selection of two parcels (yellow surfaces) within a grey matter sur-

face segmentation (planes at far left and far right of the gure). (a) A diffusion model may provide, within each individual image voxel, an estimate of intra-axonal

volume (FD). Any bundle of interest will likely traverse a large number of image voxels along its length and breadth. (b) The parameter of interest for quantifying

the ‘connectivity’ of this white matter pathway is the total intra-axonal cross-sectional area of the axons attributed to the bundle of interest (FBC). Note that this is

irrespective of the length of the pathway or the total dimensions of the plane necessary to encapsulate all axons within that pathway.

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intersection, which is then attributed to the appropriate

xel within that voxel. As such, each streamline has asso-

ciated with it a set of xels traversed, and each xel has

associated with it both an FD as estimated by the diffu-

sion model and a total streamlines density based on the

set of streamlines traversing it. The algorithms described

below involve direct utilisation of this per-xel informa-

tion (to a greater or lesser extent). While mechanism (1) is

potentially sensitive to bre/streamlines orientation dis-

tribution information that is more complex than what can

be represented using a small nite number of discrete

xels in each voxel, mechanism (2) essentially utilises

the xels provided by the diffusion model as a sparsi-

fying transform, reducing the size of the computational

problem; further, the fact that it is possible to perform

for each xel a direct scalar comparison between FD and

total streamlines density makes this mechanism more

amenable to promoting an understanding of the logic

underlying the methods described in this article.

METHODS

The algorithmic basis of quantitative streamlines

tractography

In order to demonstrate the fundamental operation of

the algorithms under discussion (and hence the quanti-

tative properties they provide), we begin with a simple

Although the former approach is more ‘conventional’,

and additionally has a long history of use in the context

of global tractography methods, for demonstration pur-

poses we use the latter model, as it provides a more intu-

itive course of reasoning in the following sections.

Note that these two approaches are directly related via

the invertibility of the spherical convolution transform:

the a priori denition of the forward models to be used

for each tissue component/compartment in approach

(1) serves the same purpose as the a priori denition of

the tissue ‘response functions’ for spherical deconvolu-

tion in approach (2) (Figure 4). Note also that in Figure

1, the ‘semi-global algorithm’ is shown to be utilising

information from estimated bre densities rather than

the diffusion-weighted images, corresponding to case

(2) described here.

There is an important difference between these two

cases that is requisite for proper understanding of sub-

sequent sections of this article. In mechanism (1), the

model considers not only the density of streamlines

within any particular image voxel, but also the precise

orientation distribution of those streamlines: the con-

tribution of each streamline towards the reconstructed

diffusion-weighted signal is based on the tangent of

the streamline at each location along its trajectory. In

mechanism (2), the diffusion model denes a small nite

number of discrete xels in each image voxel; the algo-

rithm that maps each streamline to the DWI voxel grid

(19) determines the orientation of each streamline-voxel

Fig. 4. Relationship between two different modelling approaches used in ‘semi-global’ tractography optimisation algorithms. (a) Based on estimates of local densi-

ties of different types of tissue (including orientation information in the case of white matter streamlines), and functions describing how each tissue contributes to the

diffusion signal at different b-values (often derived from e.g. the diffusion tensor model), a spherical convolution is performed to estimate the diffusion signal from

the current tractogram reconstruction. This is compared to the empirical diffusion signal intensity data, and the tissue density estimates within the reconstruction

are revised accordingly. (b) Based on tissue response functions describing the appearance of each type of tissue in the diffusion data (determined either from some

model or from the image data directly), a spherical deconvolution is performed to obtain estimates of tissue densities (including orientation information in the case

of white matter bres). The densities of a reconstructed tractogram are compared to the white matter bre density estimates, and the parameters of the tractogram

are revised accordingly.

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that a pathway volume can be derived by computing a

mask of xels that are traversed by the streamlines be-

longing to that pathway, and this can be converted to a

pathway cross-sectional area based on the length of the

pathway (Figure 6; see also pseudocode in Appendix):

Step 1. Identify all xels that are traversed by at least

one streamline belonging to the pathway.

Step 2. Sum the bre volumes of all selected xels to

calculate the total bre volume of the pathway.

Step 3. Divide this value by the mean streamline length

to estimate FBC for the pathway.

Figure 7a shows the assignment of FD from the underly-

ing diffusion model eld to the quantication of volume

(and hence cross-sectional area) of this specic pathway.

There are a couple of weaknesses in this algorithm ob-

served in Figures 6 and 7a:

• The effect of ‘outlier streamlines’: those streamlines

that are attributed to the pathway of interest but fol-

low a trajectory drastically different from the rest of

the streamlines assigned to the pathway. When this

occurs, all of the xels traversed by that streamline

are added to the xel mask, and all of the FD within

each of those xels contributes to the total bre vol-

ume of the pathway. One or a small number of stray

streamlines may therefore drastically increase the nal

quantication of FBC (e.g. the streamline in Figure 5

that travels partway down the corticospinal tracts as it

traverses between the two regions of interest).

Utilisation of some more stringent criteria for inclusion

of xels in the mask (e.g. an increased streamline count

threshold) could theoretically mitigate this effect,

though what form such criteria should take is subjec-

tive. Alternatively, such errors may be addressed using

additional tractography regions-of-interest or manual

quality control procedures; but such mitigation does

denition of the fundamental data and research question

that may be applicable to an example analysis involving

diffusion MRI tractography.

• What we want:

{ An estimate of the FBC metric for some pathway, as

dened in the Background section and demonstrat-

ed in Figures 2 and 3.

• What we have:

{ A measure of bre volume for each xel as estimat-

ed via a diffusion model;

{ A set of streamlines delineating the trajectory of

the pathway of interest, typically based on a prio-

ri regions of interest or other criteria to isolate the

pathway;

{ A whole-brain tractogram, of which the set of

streamlines ascribed to the pathway of interest is a

subset (while this is not required for Algorithm 1, its

necessity will be demonstrated in later algorithms).

The example to be used for demonstration in this article

is the connection between left and right precentral sulci, as

derived from the ‘Desikan-Killiany’ parcellation (61) provid-

ed by the FreeSurfer software (62); this is shown in Figure 5.

We now demonstrate in this section several plausi-

ble algorithmic approaches by which our goal may be

achieved. We start by proposing a relatively simple and

naïve algorithm, observing its benets and shortcomings,

and then use these observations to derive increasingly

advanced approaches, eventually presenting a total of

four algorithms.

Algorithm 1: ‘Fixel mask’

Algorithm 1 is the simplest possible approach for incorpo-

rating the FD information from a diffusion model into esti-

mating FBC for a pathway of interest reconstructed using

streamlines tractography. It is based on the observation

Fig. 5. Visualisation of the pathway of interest to be used in a demonstration of the four bre bundle capacity (FBC) quantication algorithms presented in the

‘Methods’ section: three orthogonal views. The left and right precentral sulci are highlighted in orange; streamlines are coloured according to their local tangent

orientation (red = left-right; green = anterior-posterior; blue = inferior-superior).

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Fig. 6. Visual demonstration of the operation of Algorithm 1 (‘Fixel mask’). For a particular pathway of interest (left panel: grey matter regions labelled orange,

leading to selection of streamlines shown), those xels traversed by the streamlines corresponding to that pathway are selected (right panel: red xels within yellow

voxels). The sum of the microscopic bre densities of these selected xels (equation numerator; encoded visually as xel lengths) is divided by the mean streamline

length (equation denominator: sum of streamline lengths divided by the number of streamlines) to yield the bre bundle capacity (FBC) measure.

Fig. 7. (Left) Coronal projection of brain grey matter, with regions of interest used in the reconstruction of the pathway of interest highlighted; (a–d) Maximum

intensity projection (MIP) spatial distributions of the density of white matter bres attributed to the pathway of interest resulting from quantication using the four

algorithms described in the ‘Methods’ section. The reconstructed bundle is that shown in Figure 5.

Fig. 8. The effects of partial volume on Algorithm 1. Left: A white matter bre pathway connecting between two grey matter regions, shown as both bre trajec-

tories and per-voxel bre orientation/density. Right: The pathway is split into two bundles of interest based on parcellation of the voxels at the endpoints of the

pathway; a subset of voxels (highlighted red) is intersected by both bundles.

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It is based on the observation that when a xel is tra-

versed by streamlines belonging to the pathway of inter-

est in addition to other streamlines not belonging to that

pathway, then rather than the entire ﬁbre volume of that

xel contributing to the pathway, ideally only the fraction

of that ﬁxel attributed to the pathway of interest should

be included. This is achieved as follows (Figure 9; see

also pseudocode in Appendix):

Step 1. Generate a whole-brain tractogram; dene the

pathway of interest as a subset of those whole-

brain streamlines.

Step 2. For every xel, calculate the fraction of the total

streamlines density in that xel that belongs to

the pathway of interest.

Step 3. The contribution of the bre volume of each xel

to the bre volume of the pathway of interest is

modulated by the fraction of that xel ascribed

to the pathway of interest in step 2.

Step 4. As in Algorithm 1, divide the total volume of the

pathway by the mean streamline length to esti-

mate FBC.

The primary advantage of this approach over Algorithm

1 is that if a xel is only traversed by a small number of

streamlines within the pathway of interest, that xel only

contributes a small amount of its bre volume to the

FBC result. The effect of this change from Algorithm 1

is particularly evident in the inferior half of the brain in

Figure 7b, where individual erroneous streamlines trajec-

tories contribute far less to the calculated bre pathway

volume.

While various streamline reconstruction biases mean

that the xel bre volume fractions ascribed to the

streamlines within the pathway of interest may not be

precisely equivalent to the total fraction of the underly-

ing ﬁbres within that xel that belong to the biological

pathway of interest, this algorithm certainly provides a

not trivially extend to studies where many different

pathways are assessed (e.g. when building the struc-

tural connectome over the whole brain).

• The local FD per voxel attributed to the pathway is rel-

atively consistent throughout the entire pathway, from

the corpus callosum through the centrum semiovale

and to the interface between grey and white matter.

This is, however, contrary to how such tracts are con-

structed physically: as white matter bres fan out from

the narrow cross-section of the corpus callosum to a

long strip of grey matter, the local voxel-wise density

of the bres within this specic pathway would be ex-

pected to decrease.

The way in which these effects can manifest, as well as

the source of the limitation, is demonstrated in Figure 8.

Here the selection of two bundles of interest from a

larger white matter pathway is shown, both in the corre-

sponding streamlines and in the xels to which they are

ascribed. What is highlighted in red is the fact that if each

of the two bundles is independently mapped to the cor-

responding voxels traversed, then for the set of voxels

intersected by both bundles, all of the FD within those

voxels will be attributed to both bundles. This has two

effects: rstly, the spatial distribution of FD attributed to

each bundle individually does not vary smoothly, failing

to represent partial volume at the outer edge of each

bundle similarly to that observed in Figure 7a; secondly,

the sum of the calculated bre connectivity of the two

bundles would be greater than that of the actual white

matter structure — without even necessitating tractog-

raphy reconstruction errors — as the FD in those voxels

would contribute to the quantication of both bundles.

Algorithm 2: ‘Weighted xel mask’

Algorithm 2 directly addresses the major imperfections

of Algorithm 1 that arise due to partial volume effects.

Fig. 9. Visual demonstration of the operation of Algorithm 2 (‘Weighted xel mask’). The streamlines corresponding to the pathway of interest (left panel: solid

lines) are a subset of a whole-brain tractogram (left panel: dashed lines). For each xel in the image (right panel), the fraction of the streamlines density in that xel

corresponding to the pathway of interest can be quantied (right panel: red intensity). The contribution of the bre volume within each xel (encoded visually as

xel lengths) to the pathway of interest is modulated by the fraction of the streamlines density in that xel attributed to the pathway of interest (multiplication in

equation numerator); as with Algorithm 1, this volume is then divided by the mean streamline length (equation denominator: sum of streamline lengths divided by

the number of streamlines) in the calculation of bre bundle capacity (FBC).

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perceptible improvement over Algorithm 1, with the spa-

tial distribution of bre volume within the pathway hav-

ing a much more biologically plausible appearance.

An inherent problematic issue in the design of

Algorithm 2, however, is that it fails to enforce a consis-

tent intra-axonal cross-sectional area within the pathway.

Along the length of the bre bundle shown in Figure

7b, there are local ‘hot-spots’ of supposedly increased

bre volume, both within the supercial white matter

and where the bundle intersects the grey matter targets.

While individual axons may have some modulation in

their diameter along their length, such gross modula-

tion of macroscopic intra-axonal cross-sectional area is

physically unrealistic for white matter pathways at the

macro-scale. Observation of such in diffusion MRI data is

therefore far more likely to be an artefact of image anal-

ysis and reconstruction. Furthermore, at the endpoints of

the pathway, voxels containing partial volume between

grey and white matter are likely to contain a smaller

number of streamlines than those voxels entirely with-

in the white matter, which results in the fraction of xel

FD assigned to the pathway of interest being prone to

quantisation effects; this contributes to the ‘speckly’ ap-

pearance of the density map in Figure 7b near the grey

matter.

Algorithm 3: ‘Volume-averaged streamline weights’

In order to overcome the fundamental limitation of

Algorithm 2, a solution is sought for deriving FBC that en-

forces a constant intra-axonal cross-sectional area along

the length of the pathway. We clarify here that this con-

straint does not apply to the macroscopic span of space

traversed by bres of that bundle, but applies specical-

ly to the intra-axonal portion of the bundle. For exam-

ple: when bres within a tightly packed bundle diverge

Fig. 10. Visual demonstration of the operation of Algorithm 3 (‘Volume-averaged streamline weights). For each individual streamline, the total bre volume at-

tributed to that streamline (numerator, rst equation) is based on the products of the xel bre volumes (encoded visually as xel lengths) and the fraction of the

streamlines density in each of those xels attributed to the streamline of interest (left panel: encoded as red intensity); this bre volume is then divided by the length

of that individual streamline (denominator, rst equation) to ascribe a ‘weight’ ws to each individual streamline s (right panel: streamline colours). The bre bundle

capacity (FBC) of the pathway of interest is then the sum of the weights ascribed to the streamlines attributed to the pathway of interest (right panel: solid lines).

from one another (commonly referred to as ‘fanning’),

the surface area of the subset of a plane encapsulating

all bres in the bundle may increase (the ‘macroscopic

cross-section’), but the sum of intra-axonal cross-sectional

areas should remain unchanged if the axon diameters are

consistent along their length (e.g. Figure 2; Figure 3e–f);

here it is the latter denition that we advocate should be

constrained.

Algorithm 3 is an initial realisation of this concept. It

enforces constant intra-axonal cross-sectional area of the

pathway, by requiring that each streamline in the path-

way contributes a constant ﬁbre cross-sectional area

along its entire length. Hence, unlike Algorithms 1 and 2,

here contributions towards FBC are made not per ﬁxel,

but per streamline (in the context of semi-global trac-

tography algorithms, these parameters are sometimes

referred to as ‘weights’).

This algorithm operates as follows (Figure 10; see also

pseudocode in Appendix):

Step 1. Using the whole-brain tractogram, calculate the

total streamlines density in each xel.

Step 2. For each streamline, calculate the ﬁbre volume

to be attributed to that streamline. Every xel

traversed by the streamline contributes a fraction

of its bre volume to the sum for that streamline,

based on the fraction of the total streamlines

density in that xel that was contributed by that

particular streamline.

Step 3. Convert the ﬁbre volume of each streamline to

a ﬁbre cross-sectional area, by dividing by the

length of that streamline.

Step 4. Sum the cross-sectional areas of the streamlines

belonging to the pathway of interest to derive

FBC.

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FD derived from spherical deconvolution (Figure 11a)

than the original tractogram where every streamline con-

tributes equally (Figure 11b). However, it still does not

provide an entirely faithful representation of the under-

lying white matter FD eld. This is perhaps expected

given the nature of the algorithm itself. From a physi-

cal perspective, the operation of this algorithm can be

thought of as taking the sum of bre volumes attribut-

ed to the streamline by each xel (which may be great-

er or lesser at different points along the streamline) and

spreading this bre volume evenly along the length of

the streamline in order to ascribe to it a constant bre

cross-sectional area. While this process does go some

way to incorporating bre volume information from the

diffusion model into the tractogram, it fails to directly en-

force consistency between the estimated FD and recon-

structed streamlines density for each xel of the image

(the residual discrepancy visible between Figure 11a and

11c). In the case of the specic pathway of interest used

in this demonstration, the FD within the corpus callosum

projected by the tractogram is clearly greater than that

indicated by the image data; this means that the calcu-

lated FBC for this connection relative to other pathways

when using Algorithm 3 would likely be an over-estimate.

Algorithm 4: ‘Optimised streamline weights’

Addressing the remaining problem with the approach

described in Algorithm 3 — the fact that the bre vol-

umes estimated from the tractogram are not a sufciently

accurate reconstruction of the bre volumes estimated

from the diffusion model, as shown in Figure 11 — is

a fundamental requirement if we are to consider the

streamline weights truly quantitative. If the streamlines

trajectories and ascribed weights are reective of the

underlying biological connectivity, then the spatial distri-

bution of FD throughout the white matter represented

within this connectivity-based reconstruction should ac-

curately match estimates of this measure that are derived

from the image data directly.

While this algorithm produces a measure of bre

cross-sectional area per streamline rather than bre vol-

ume per xel, it is still possible to reconstruct the latter;

this allows us to generate a spatial map of FD attribut-

ed to the pathway of interest that can be compared to

Algorithms 1 and 2. The product of a cross-sectional area

with a length yields a measure of volume; hence, each

streamline in the tractogram contributes a bre volume

to every voxel it traverses, based on the product of its

weight and the length of the streamline segment that in-

tersects that voxel. The result of this process is shown in

Figure 7c. Compared to the previous two algorithms, this

approach produces an FD map for the resulting pathway

that appears quite biologically reasonable, with a max-

imal microscopic FD within the narrow connes of the

corpus callosum that decreases as those bres fan out

towards the cortex.

There does, however, remain one slight inadequacy

with this algorithm. Consider an experiment where, in-

stead of deriving a spatial map of reconstructed bre

volume for a particular pathway of interest only (as we

have been doing here), we instead map the spatial dis-

tribution of reconstructed bre volume of the entire trac-

togram. If the streamlines weights are faithful to the in-

tra-axonal cross-sectional areas of the biological bres

following the trajectories reconstructed by those stream-

lines, then one would expect an accurate reproduction

of the bre volumes that were estimated from the vox-

el-wise diffusion model throughout the white matter (or

equivalently: applying the forward model to the whole-

brain tractogram should yield the empirical diffusion sig-

nal, as demonstrated in the ‘Comparing tractograms and

image data’ section and Figure 3). Note this process is

very similar to track density imaging (TDI) at native DWI

resolution,63–65 incorporating the ability for streamlines to

contribute differentially towards the image.

This experiment is shown in Figure 11. The recon-

structed FD from the outcome of Algorithm 3 (Figure

11c) is closer to the voxel-wise estimate of white matter

Fig. 11. Comparison of spatial distributions of track densities from whole-brain tractogram data with the density of white matter bres as estimated through spherical

deconvolution. (a) The orientationally averaged mean of the white matter orientation distribution functions (the l=0 term of the spherical harmonic expansion) as a mea-

sure of total bre density within each voxel. The distribution of TD within a whole-brain tractogram should ideally match these data. (b–d) The density of streamlines

in the whole-brain tractogram when the contribution of each streamline to the map is modulated as follows: (b) no modulation (all streamlines contribute equally); (c)

modulated by the weights ascribed to the streamlines by Algorithm 3 (‘Volume-averaged streamline weights’); (d) modulated by the weights ascribed to the streamlines

by Algorithm 4 (‘Optimised streamline weights’).

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distribution of FD of the entire tractogram is shown in

Figure 11d. Crucially, the latter demonstrates a highly ac-

curate reproduction of the underlying FD eld estimated

from the diffusion model (Figure 11a), highlighting how

this approach provides a tractogram-based connectivi-

ty model that obeys the fundamental spatial constraints

imposed by the physical nature of the underlying biolog-

ical bre structure. It is this observation that permits the

streamline weights estimated by such algorithms to be

used in the quantication of FBC (within the constraints

imposed by other limitations associated with diffusion

MRI streamlines tractography (66)).

Inter-subject connection density normalisation

Whenever quantitative data are to be compared across

subjects, an important distinction must be made be-

tween absolute and relative quantitative measures.

For instance, when assessing the fractional anisotropy

(FA) measure (67) from the diffusion tensor model (68),

there is no need to modulate these values differentially

between subjects, as it is an absolute measure (for any

given DWI acquisition scheme) and therefore any differ-

ences in this metric between individuals can be inter-

preted according to the properties encapsulated within

that metric. However, if a quantitative measure is relative

to other latent parameters that vary across individuals,

care must be taken to appropriately handle the effects of

those latent parameters, in order to be able to interpret

This limitation is addressed by designing an algorithm

that explicitly seeks to derive a set of streamline weights

that result in an accurate reconstruction of the underly-

ing bre volumes estimated from the diffusion model (or

equivalently, an accurate reconstruction of the empirical

diffusion signal using a forward model). This basic con-

cept is shown diagrammatically in Figure 12 and might

proceed, for example, as follows (see also pseudocode

in Appendix):

Step 1. Initially assign a unity weight to each streamline.

Step 2. Using the whole-brain tractogram, based on the

current streamline weights, calculate the total

streamlines density in each xel.

Step 3. For each xel, calculate the difference between

the FD estimated from the diffusion model and

the total attributed streamlines density.

Step 4. For each streamline, increase or decrease the

weight in order to minimise the error quantied

in step 3.

Step 5. Loop back to Step 2 until some termination crite-

rion is met.

Step 6. Sum the weights of those streamlines belonging

to the pathway of interest to derive FBC.

This is the mechanism by which the ‘spherical-

deconvolution informed ltering of tractograms 2

(SIFT2)’ method (22) operates. The spatial distribution of

FD within the pathway of interest after having applied

the SIFT2 algorithm is shown in Figure 7d, and the spatial

Fig. 12. Visual demonstration of the operation of Algorithm 4 (‘Optimised streamline weights’). For a whole-brain tractogram (top left panel), the total streamlines

density traversing each xel (bottom left panel; encoded visually as xel lengths) may not match the bre volumes estimated from the diffusion model (bottom

panel; xel lengths). This algorithm modulates the weight ws ascribed to each streamline s (top right panel: streamline colours) in order to achieve correspondence

between the total streamlines density traversing each xel (bottom right; xel lengths) and the diffusion model bre density estimate (bottom panel). The bre

bundle capacity (FBC) measure for the pathway of interest is the sum of the weights ascribed to those streamlines attributed to the pathway of interest (top right

panel; solid lines only; streamline colours).

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incontrovertible, it is in fact an imperfect solution to con-

nection density normalisation. By using a xed number of

streamlines per subject, each streamline effectively rep-

resents a ﬁxed, subject-speciﬁc fraction of the total white

matter ﬁbre connectivity. While comparing such quanti-

tative measures across subjects is acceptable as long as

they are properly interpreted as such, this metric fails to

take into account a number of factors that may differ be-

tween subjects that may correspondingly introduce biases

or unwanted variance into such an analysis; this includes

both biological differences (e.g. widespread reductions in

FD) and features of the tractogram reconstructions (e.g.

differences in streamline lengths). For a genuine quanti-

tative comparison of absolute bre connection densities

between subjects, greater care must therefore be taken.

Here, we demonstrate our recommendation for how

this normalisation should be achieved when using specif-

ically the model underlying the ‘SIFT’ (19) and ‘SIFT2’ (22)

methods, which is itself directly dependent on the AFD

measure. Use of alternative reconstruction techniques (in

terms of either the underlying diffusion model or alter-

native semi-global tractography methods) would require

that appropriate comparable steps be taken.

The SIFT model denes the proportionality coefﬁcient

μi for subject i, which relates the global sum of track den-

sity (TD) to the global sum of estimated FD in the sin-

gle-subject reconstruction, computed across all xels f

in that subject:

∑

=FD

(2)

(For simplicity, the inuence of the processing mask with-

in the SIFT model (19) is omitted here.)

In the original SIFT method, this parameter permits

direct comparison between the streamlines density

and bre volume within each xel, in order to drive the

streamlines ltering process. In SIFT2, it approximately

centres the distribution of streamlines weights about

unity. Note that all parameters within this expression are

subject-speciﬁc.

As FD is a measure of volume (dimensions L3) and

TD is a sum of streamline lengths (dimensions L), µ is

a measure of cross-sectional area, with dimensions L2.

For every streamline, this parameter (multiplied by the

weight assigned to that streamline in the case of SIFT2)

is a measure of the intra-axonal cross-sectional area rep-

resented by that streamline. For each voxel traversed by

the streamline, the product of this cross-sectional area

with the length of the streamline intersection within that

voxel produces the bre volume contributed to that xel

by that particular streamline within the model.

In the context of FBC quantication, we are interested

not in these xel-wise bre volumes, but the connection

densities of specic macroscopic pathways of interest.

Any such pathway is represented as a subset of stream-

lines in the whole-brain tractogram. For an example

any differences in that metric as being speciﬁc to that

metric rather than some nuisance confound.

In the xel-based analysis (FBA) framework (69), which

enables statistical analysis of white matter quantitative

measures in the presence of crossing bres, FD estimates

must be comparable across subjects throughout some

common template space. In the context of apparent

bre density (AFD) quantication (70) using the spherical

deconvolution model (59), the bre orientation distribu-

tions (FODs) (i.e. the representation of estimated bre di-

rections and densities in each voxel) are deliberately not

normalised either to a unit integral in each voxel or to the

intensity of the b=0 image (i.e. volume acquired with no

diffusion sensitisation) in each voxel. This makes the size

of the FOD directly proportional to the magnitude of the

DWI signal (which is itself proportional to intra-cellular

volume at high b-values (70)); the size of the FOD is also

inversely proportional to the magnitude of the response

function used for deconvolution (70):

∗∗

−

FODRF=DWIFOD =DWI RF (1)

(‘RF’: response function; ‘*’: convolution operation;

‘*−1’: deconvolution operation)

Enabling direct comparison of this measure across

subjects therefore necessitates global inter-subject in-

tensity normalisation, in order for AFD to be minimally

inuenced by nuisance variables. This typically includes

B1 bias eld correction, scaling of DWI intensities to a

common intensity value according to some representa-

tive image statistic (e.g. mean b=0 magnitude in white

matter), and use of a group average response function

for deconvolution (70): these together ensure that ‘one

unit of AFD’ is comparable across subjects, rather than

being dened relative to subject-specic parameters

(e.g. coil loading, scanner receiver gain, subject-specic

response function magnitudes).

In a similar manner, quantitative analysis of FBC re-

quires inter-subject connection density normalisation:

that is, if we quantify the intra-axonal cross-sectional

area of a particular pathway (e.g. edge of a connectome)

across multiple subjects, we want these quantities to be

directly comparable across subjects, without being bi-

ased by confounding factors that destroy the physical

interpretation of this measure or introduce substantial

correlations with nuisance reconstruction parameters.

In the brief history of diffusion MRI tractography con-

nectomics, this normalisation has most commonly been

achieved by simply generating the same number of

streamlines for each subject. Or, expressed in an alterna-

tive way: if half as many streamlines were generated for

one subject as there were for all other subjects, it would

seem intuitively logical that the streamline counts in each

edge for that subject should be doubled in order for the

raw values stored in the connectome matrices to be com-

parable to other subjects. While generation of an iden-

tical number of streamlines across subjects may seem

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The global intensity normalisation and group average re-

sponse function components of the recommended

pre-processing pipeline for AFD analysis are tailored to

make equivalent across subjects the values of DWIref and

AFDref, respectively. As such, if this pipeline is followed,

the term

AAFFDD

DDWWII

rreeff

is identical across subjects by construc-

tion, and simply multiplying the sum of streamline weights

within a pathway of interest (e.g. a connectome edge) by

μi permits direct quantitative comparison of FBC between

subjects, in a manner that appropriately accounts for

many variables that would otherwise confound the inter-

pretation of streamline counts as ‘connection density’.

The consequences of this connection density normal-

isation are demonstrated in Figure 13. This demonstra-

tion consists of 16 individual synthetic subjects, each of

whom possesses a single white matter bre bundle. The

fundamental properties of this bundle — length, width,

and microscopic FD per voxel — vary among the sub-

jects; additionally, the number of streamlines seeded in

each white matter voxel is also varied. The lower part

of Figure 13 then demonstrates visually how the quanti-

ed ‘connectivity’ of this bundle across the 16 subjects

changes in magnitude across the different subjects, de-

pending on the exact measure of ‘connectivity’ that is

utilised. The connectivity measures demonstrated are as

follows:

• The macroscopic bundle volume V;

• The streamline count N;

• The number of streamlines divided by the length of

the streamlines41,71;

• The mean FD sampled along streamlines within the

bundle;

• FBC, incorporating the proposed connection density

normalisation.

We assert that the proposed connection density normali-

sation (highlighted in green in Figure 13) is most appropri-

ate for quantitative comparison of endpoint-to-endpoint

connectivity across subjects; it matches the theoretical

properties of the FBC metric:

• Scales with the cross-sectional area of the bundle;

• Scales with the underlying FD in each voxel;

• Does not scale with the length of the bundle;

• Does not scale with the number of streamlines

generated.

DISCUSSION

As stated in the ‘Introduction’ section, the information

presented in the ‘Methods’ section is intended to serve

two principal purposes:

1. To properly contextualise a class of methods already

present in the literature that perform tractogram

pathway p (which is reconstructed by a subset of stream-

lines s), it is the sum of intra-axonal cross-sectional areas

of the streamlines within that pathway that gives a mea-

sure of the intra-axonal cross-sectional area of the path-

way FBCp,i:

µ∑ε

FBC=

p,ii

Sp s

(3)

For subject i, the connection density FBCp,i of pathway

p is the product of the subject-specic proportionality

coefcient µi and the sum of streamline weights wi of

those streamlines si belonging to pathway p.

(Note that in the original SIFT method, ws = 1 for all

retained streamlines after ltering, but µi is modulated

during the ltering process.)

This equation suggests that if one wants to compare

FBC across subjects (whether for an individual bun-

dle of interest or an entire connectome matrix), simply

multiplying the sum of bundle streamlines weights by μi

is sufcient to produce a measure of FBC that can be

compared across subjects. However, parameter μi only

considers the bre densities and track densities within

a single subject; in order to compare these quantities

between subjects, we must ensure that parameter μi is

adjusted appropriately to account for between-subject

variation, by ensuring that the fundamental scaling un-

derlying this parameter is equivalent between subjects.

We can extend Equation (2) as follows:

∑

=AFD

DWI.x yz.FD

i, adj

ref

(4)

• μi,adj is the proportionality coefcient of subject i ‘ad-

justed’ for facilitation of inter-subject comparison;

• The rst term,

AAFFDD

DDWWII

rreeff

, is specic to the spherical de-

convolution model if processing were to be performed

independently for each subject. It relates to the global

scaling of AFD magnitudes within that subject, which

is dependent on the magnitude of the diffu-

sion-weighted signal (DWIref) and the size of the re-

sponse function for deconvolution that forms the ref-

erence unit of AFD (AFDref).

• The second term,

xyz

, is the volume of each voxel in

the image. This multiplier converts AFD (or bre volume

fractions from a partial volume-based diffusion model)

into estimated intra-axonal volumes in mm3, thereby ap-

propriately scaling connectivity estimates in cases where

the DWI voxel size differs across subjects (this also coinci-

dentally gives μi,adj and hence FBC, units of mm2).

• In the specic case of the SIFT model, no term relat-

ing to the inter-subject scaling of TD appears in this

expression: this is calculated in xed units of mm re-

gardless of DWI spatial resolution, and therefore can-

not vary across subjects (this may, however, not be the

case for alternative models or methods).

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within a single individual to estimation of relative

connection strengths of a bundle across individuals

(Figure 13).

Relationship to existing methods

In ‘Algorithm 4: “Optimizsed streamline weights”’, the

fact that the existing SIFT2 algorithm operates on the

same mechanism as that arrived at through the course

of logic presented is not coincidental. Here, we note that

there are a number of other existing methods that also

operate similarly:

• The ‘linear fascicle evaluation (LiFE)’ (21) and ‘convex op-

timisation modelling for microstructure-informed trac-

tography (COMMIT)’ (20) methods both operate on an

identical premise — modulating the weight of contribu-

tions from individual streamlines within the tractogram,

manipulation for the purposes of quantitative tractog-

raphy by elucidating:

a. Why these methods are designed the way they are

(see also ‘Relationship to existing methods’ section);

b. That many ‘alternative’ methods for white matter

connectivity analysis frequently suggested infor-

mally by community members have already been

considered, but have specic weaknesses com-

pared to established methods;

c. That the goal of ‘quantifying bundle connectivity’

is in fact accessible using these existing methods

(while of course acknowledging all of the limita-

tions of such methods and other components of

the analysis pipeline).

• To demonstrate how to appropriately extend the ap-

plication of these methods from the correct estimation

of relative connection strengths of different bundles

Fig. 13. Demonstration of the efcacy of the proposed inter-subject connection density normalisation. Each of the 16 panels represents a synthetic subject, contain-

ing one white matter bundle reconstructed by streamlines. For each, the number of streamlines generated N, and the ‘proportionality coefcient’ µ within the SIFT

model (derived from Equation 2), are provided. The matrix representations at the bottom show visually the relative connection densities quantied for the different

bundles, for various ‘connectivity’ metrics of interest: volume of bundle V; number of streamlines N; number of streamlines divided by the streamline length; the

mean bre density (FD) sampled along streamlines within the bundle; the proposed bre bundle capacity (FBC) measure incorporating connection density normali-

sation (highlighted in green).

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the capacity for information transmission, this metric

exhibits somewhat desirable behaviour in the pres-

ence of such heterogeneity, at least within the limita-

tions of expressing such as a single scalar quantity. It

should also be noted that conventionally acquired dif-

fusion-weighted image data are unable to resolve dif-

ferences in axon diameter (89,90) — indeed even state-

of-the-art acquisitions and hardware struggle to resolve

such in the presence of crossing bres (91–93) — so

overcoming this limitation in specicity is non-trivial.

• While the presence of myelin does inuence the con-

duction of action potentials, conventionally acquired

diffusion MRI data are relatively insensitive to the pres-

ence or absence of such (94).

• This metric is intentionally not proportional to length,

unlike bundle volume. Although bundle length may

alter axon conduction delays (95), we consider such

to be an independent property of white matter con-

nections that does not directly inuence the notion of

‘bandwidth’.

• Unlike macroscopic bundle volume (or indeed

cross-sectional area), this metric additionally considers

the density of axonal packing within the white matter

traversed.

The prevalence of studies utilising quantitative metrics of

white matter bundles such as streamline count or bundle

volume highlight the demand for an appropriate quan-

tication of ‘connectivity’ of white matter bundles. We

posit that, compared to other univariate metrics already

in use in the community, the FBC metric is in fact more

faithful to a subjective notion of ‘connectivity’, based on

both the logic presented above and the evidence shown

in Figure 13.

Whole-brain tractography is compulsory

With the exception of Algorithm 1, all other algorithms

described here necessitate the use of a whole-brain trac-

togram reconstruction, even in the scenario where it is

only the connectivity of one specic white matter bundle

that is of interest. This includes the SIFT2 algorithm shown

as Algorithm 4, and similar methods described above in

the ‘Relationship to existing methods’ section. The rea-

son for this is as demonstrated in Figure 8 in relation to

Algorithm 1. If a white matter bundle (as dened as a

set of streamlines that is not a whole-brain tractogram)

is interpreted in isolation, then anywhere there exists a

voxel that contains collinear bres not belonging to that

bundle, such a quantication will erroneously attribute

the entire FD in that orientation to that bundle. Given the

contrast between the complexity of white matter bundle

trajectory/shape and the xed lattice of an image voxel

grid, this will always occur. Note that this is an issue re-

gardless of whether or not any adjacent bundle sharing

that voxel is or is not also ‘of interest’ experimentally.

in such a way that the streamlines densities are faith-

ful to the underlying image information — except

that they operate directly on the diffusion image data,

as explained in the ‘Comparing tractograms and

image data’ section.

• The earlier ‘BlueMatter’ (18), ‘MicroTrack’ (72), and

‘SIFT’ (19) methods also lie within this classication:

while these algorithms instead select a subset of

streamlines that together produce a faithful repre-

sentation of the image data, this is mathematically

equivalent to setting the contribution weights of those

streamlines omitted from the tractogram to zero.

• Another method entitled ‘global tractography with

embedded anatomical priors’ (73) optimises stream-

line weights based on a tractogram initially construct-

ed using streamlines tractography, but also optimises

other features of the reconstruction (such as stream-

lines trajectories) based on the diffusion image data in

a manner more similar to genuinely global tractogra-

phy algorithms.

It has been shown that utilisation of such methods yields

white matter connectivity estimates with properties that

are more faithful to biological reality (44) and that their

inuence on network connectivity analyses is non-neg-

ligible (41). Such methods are publicly accessible and

have already been adopted in some studies in the neu-

roscience research literature (74–87).

Qualifying the ‘FBC’ metric

The origins and biological signicance of the FBC metric

require explicit communication, as it is a frequent source

of confusion.

• It should be noted that the quantication of speci-

cally total intra-axonal cross-sectional area is in part

a direct consequence of the proportionality of the

diffusion MRI signal (under certain conditions) to the

local intra-axonal volume of bre bundles (70), in con-

junction with the spatial/orientation information of the

tractogram reconstruction. It is not a metric that was

devised in isolation, with methods then developed to

quantify such, but a natural consequence of what can

be quantiﬁed given the data available.

• Where there is heterogeneity in axon diameters, the

FBC metric will not be a proportional estimate of axon

count, as was proposed for simplicity in the ‘Metric of

“connectivity”’ section. The inuence of heterogeneity

in axon diameters can be considered as follows. For a

given xed value of FBC, there could be many axons of

small diameter or few axons of larger diameter. In the

latter case, while there are fewer connections, conduc-

tion velocity will be greater, as will be the potential r-

ing rate (88). As such, when seeking a univariate quan-

tication of ‘connectivity’, which specically relates to

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O R I G I N A L R E S E A R C H A R T I C L E

constrained tractography (ACT) (16), though technical

improvements can reduce this variance (97,98). While it

is tempting to normalise the connectome edge values by

scaling for the number of streamlines in the connectome

(rather than using the number of streamlines in the tracto-

gram as reference) to ‘compensate’ for this inter-subject

discrepancy, it is also entirely possible for differences in this

parameter to be reective of a genuine effect of interest.

For instance, consider the case where a large tumour

within the white matter in one hemisphere results in a

substantial number of streamlines terminating within

that tumour, rather than reaching some alternative grey

matter target. If the subject-specic tumour does not ap-

pear as a node within the connectome parcellation, and

the total number of streamlines generated across sub-

jects is equivalent, then the total number of streamlines

in the connectome for this particular subject would be

decreased relative to other subjects (as the total num-

ber of streamlines is the same, but the fraction of those

assigned to the connectome is reduced). A useful inter-

pretation here is to treat the tumour as a ‘latent connec-

tome node’. Consider the situation if the tumour were

to be segmented and included in the connectome par-

cellation, with streamlines terminating within that node

and being assigned as such; but following connectome

construction, that node would then be erased from the

connectome matrix. We now consider two options for

normalisation:

1. If a xed number of streamlines in the tractogram per

subject were to be used, then the subject with the tu-

mour would have a reduced total connection density

within the connectome, particularly within bundles

affected by the tumour, which is likely to be at least

somewhat faithful to biological reality.

2. If instead the connectomes were scaled based on the

number of streamlines in the connectome in each

subject (bearing in mind that this scaling would by

necessity occur after removal of the tumour node if

the total connection density is to be equivalent across

subjects), the connection densities of all pathways in

that subject would be increased as a consequence of

that process. This would be misleading, as it would

suggest that all white matter bundles not affected by

the tumour in that subject have increased connectivity

in that subject relative to healthy controls.

The important observation here is that while the number

of streamlines in the tractogram reconstruction may not

be equal to the number of streamlines in the connec-

tome (and this ratio may vary across participants) and this

effect can be inuenced by inadequacies in data process-

ing and reconstruction (96), this is not the only source of

such mismatch and should therefore be interpreted with

caution. For instance, consider the inuence of the re-

constructed corticospinal tract, where streamlines exit

the inferior edge of the image data via the spinal column.

The functionality afforded by the use of a whole-brain

tractogram — whether very directly and explicitly in the

case of Algorithm 2, or more indirectly/implicitly in other

algorithms — is to determine the fraction of the bre vol-

ume within each xel that should be attributed to the

bundle of interest, in order to prevent the over-attribution

of FD that occurs in Algorithm 1. As such, we here reca-

pitulate a message crucial to prevent further erroneous

use of these methods in the community:

Semi-global methods must be applied to a whole-brain

tractogram, with interrogation of bundles of interest

performed after the fact (as shown in Algorithm 4).

If this condition is not satised, then when interrogating

discrepancies between streamlines density and FD, such

algorithms are unable to distinguish between differences

that arise due to tractogram reconstruction biases and

differences that arise due to the presence of biological

bre pathways that contribute to the diffusion-weighted

signal but are absent from the streamlines reconstruc-

tion due to erroneous prior removal (see Figure 8). Note

that this description deliberately does not exclude the

prospect of tractogram manipulation in between whole-

brain bre-tracking and semi-global algorithm appli-

cation — for instance, one could envisage that some

approach for data-driven classication and removal of

false-positive streamlines could be applied in between

these steps — but it is vital that the tractogram provided

to a semi-global optimisation process comprehensively

cover the domain of plausible bre trajectories within the

imaged region, as such a comprehensive set is necessary

for explaining the diffusion-weighted signal.

Alternatives for connection density normalisation

A common idea in applications of white matter tractog-

raphy, particularly in the construction of the structural

connectome, is that there is a range of other parameters

that should be applied as multiplicative factors to con-

nectivity measures, in order to ‘compensate’ for varia-

tions in those parameters that may indirectly inuence

the results of streamlines tractography or connectome

construction. Here, we take the opportunity to clarify a

few concepts that have arisen in our own communica-

tions on the topic.

Number of streamlines in the connectome

Although it is trivial to use a xed number of streamlines

for tractogram construction across subjects for the implicit

purpose of connection density normalisation, typically the

proportion of those streamlines successfully assigned to

a pair of parcels (and that hence contribute to the con-

nectome) will vary between subjects (96), due to a range

of factors; note that this still occurs even with the con-

trolled termination of streamlines such as in anatomically

: 2022, Volume 2 - 18 - CC By 4.0: © Smith et al.

O R I G I N A L R E S E A R C H A R T I C L E

We instead suggest that intracranial/brain volume (and

other such regressors) may be better handled as nui-

sance parameters when performing statistical testing. In

this way, the hypothesis being tested would better re-

ect the intention of the experiment, for example, ‘the

connection density of this pathway is not equivalent be-

tween the two groups, the magnitude of which cannot

be attributed to differences in brain volume alone’.

The effect of inter-node distance in streamlines

tractography

The relationship between white matter pathway length

and estimated connection density has attracted consid-

erable interest in this eld (19,41,71). There are, howev-

er, multiple mechanisms by which pathway length may

inuence streamlines-based connectivity; and in our ex-

perience these are regularly conated or confused. We

therefore take this opportunity to disambiguate the ef-

fects of which we are aware.

Most frequently, discussion regarding bundle length

biases are in reference to the effect arising from homoge-

neous seeding throughout the white matter: because lon-

ger pathways present a greater volume in which streamline

seeds may be drawn, they will typically be reconstructed

by a greater number of streamlines than shorter pathways.

A naïve direct correction of this seeding density effect is to

make the contribution of each streamline to the connec-

tome the reciprocal of its length (71); this has been shown

to be incomplete, as the graph theory metrics derived

from connectomes calculated in such a manner differ sig-

nicantly from those produced using more comprehensive

data-driven correction of bre tracking biases (41).

A distinctly different effect is attributed to probabi-

listic streamlines algorithms. Due to the spatial disper-

sion of streamlines when using a probabilistic tracking

algorithm, biologically connected nodes that are dis-

tant from one another are likely to have a reduced re-

constructed connection density: streamlines emanating

from one parcel increasingly disperse from one another

as a function of distance from that parcel, such that the

fraction of those streamlines reaching the intended tar-

get decreases as a function of distance (1). This effect is,

however, not a bias that can be corrected naïvely. For in-

stance, consider two distant nodes that are not connect-

ed biologically, yet their immediate spatial neighbouring

parcels are connected biologically, and therefore there

is a plausible white matter pathway between them. The

connectivity estimated between these two nodes using a

probabilistic streamlines algorithm will be increased by

this probabilistic dispersion effect relative to if the white

matter pathway were short. Data regarding inter-node

1 We note that deterministic streamlines algorithms do not solve the issue de-

scribed here. With such methods, instead of the fraction of streamlines reaching

the intended target decreasing smoothly as a function of distance, the likeli-

hood of a dichotomous switch from all true connections to all false connections

increases as a function of distance.

If no connectome parcel is explicitly dened at this lo-

cation, then these streamlines will not contribute to the

connectome, despite the known anatomical validity of

this bundle and its non-negligible contribution to the

diffusion-weighted signal. Variance in the density of this

bundle across subjects could therefore manifest as dif-

ferences in streamline count within connectomes across

subjects; the latter would not ideally be interpreted as

either indicative of a difference in reconstruction efcacy,

or a ‘nuisance’ effect between individuals,

Intracranial/brain/white matter volume

Another concept commonly raised in the discussion on

this topic is that: if subject brains vary considerably in

physical size, but the same number of streamlines is gen-

erated for each, then a comparison of streamlines den-

sity between them must be biased, as each individual

streamline reconstructed in a physically larger brain likely

represents a larger density of biological bres than does

each individual streamline reconstructed in a physically

smaller brain. Users of such methods correspondingly

often propose dividing all estimates of connectivity by

the estimated intracranial/brain/white matter volume of

that subject, as a ‘correction’ for this effect. There are a

number of comments to be made on this concept:

• It pre-supposes that if the cross-sectional area of a

bundle scales directly in proportion to brain size, then

that bundle should be reported as possessing equiv-

alent ‘connectivity’ between brains of different sizes;

this is the intent of such scaling and so should be ap-

preciated as such.

• There is an implicit assumption in this proposal that,

even between the largest and smallest of brains, the

voxel-wise FD within the white matter is equivalent;

this is, however, not guaranteed to be the case.

• Performing scaling as proposed in the ‘Inter-subject

connection density normalizsation’ section intrinsically

handles this confound in an appropriate fashion. A larg-

er brain will likely have more white matter voxels and

therefore a larger FD sum, but it will also have a great-

er sum of streamline lengths. Consequently, while FBC

may be greater in a large brain than in a small brain if the

bundle size scales in direct proportion to brain size, this

would not be an unwanted confound of brain size, but a

realistic measurement of a greater information-carrying

capacity of that bre bundle in the larger brain.

• The interpretation of experimental outcomes also

changes by necessity through such scaling. For in-

stance, if one were to compare the connection density

of a specic bundle of interest between two groups,

where this brain volume scaling factor were applied

to the connectivity estimates prior to the comparison,

then the actual hypothesis being tested would be ‘the

connection-density-divided-by-white-matter-volume

of this pathway is not equivalent between two groups’.

O R I G I N A L R E S E A R C H A R T I C L E

There is an additional clarication required regarding

terminology between these two cases:

• In FBA, the ‘bre bundle cross-section (FC)’ metric is in

fact a change in the cross-sectional area; in the calcu-

lation of the FDC metric, this is used to modulate the

local quantitative measure;

• In tractogram-based connectivity quantication, FBC

of any particular pathway is typically directly propor-

tional to the absolute bundle cross-sectional area.

Another concept frequently raised in communications

in this context is the prospect of an alternative quantita-

tive metric for connectome construction, which exploits

the quantitative nature of these per-xel metrics. That

is, instead of summing streamlines weights within a

pathway (as an estimate of intra-axonal cross-sectional

area), quantitative values from some metric of interest

are instead sampled along the corresponding stream-

lines trajectories. This could conceivably be done in

one of two ways, shown as Algorithms 5a and 5b in

Figure 14:

a. From the set of streamlines constituting a pathway of

interest/connectome edge, derive a mask correspond-

ing to the areas (either voxels or xels) in which the

values of the quantitative metric should be sampled;

some statistic from these elements (e.g. the mean) is

then calculated to produce a single scalar value per

connectome edge.

b. For each streamline, measure the value of the metric at

every point along the streamline trajectory; calculate

some statistic from the samples along each stream-

line (e.g. the mean) in order to produce a single sca-

lar value per streamline; calculate some statistic from

these per-streamline values (e.g. the mean) to produce

a single scalar value per connectome edge.

The intent behind such suggestions is that these

quantities would exhibit a reduced inuence from the

errors and biases associated with streamlines tractog-

raphy compared to the FBC metric, would incorporate

the quantitative nature of those underlying metrics,

and would inherit the xel specicity of the FBA metrics

(there are already many applications that have utilised

such sampling along streamline trajectories but in con-

junction with voxel-wise imaging metrics). Such quanti-

cation should, however, be interpreted in accordance

with the relevant calculations. For instance, calculating

the mean of the FD metric along a pathway using one

of the two approaches discussed previously provides a

measure that could be interpreted as ‘mean intra-axo-

nal volume fraction within the bundle’; which, while po-

tentially informative, would not be an absolute measure

of bundle connectivity, as it neither scales with bundle

width (Figure 13), nor does it consider partial volume

with other bundles (as demonstrated in Algorithm 1,

Figures 7 and 8).

distances alone are therefore not sufcient to ‘correct’

for this effect.

We propose that this particular effect is better under-

stood as a distance-dependent connectome blurring:

biologically strong connections are ‘spread out’ in the

reconstruction to edges corresponding to spatially ad-

jacent nodes, with the extent of that blurring being a

function of the pathway length. The way in which bre

orientation uncertainty/dispersion is modelled and uti-

lised in the tractography algorithm is likely to inuence

the magnitude of this effect. While there exists a tailored

correction mechanism for addressing this specic issue

in the context of targeted tracking when quantifying a

probability of connectivity (99), to our knowledge there

has been no such mechanism proposed for addressing

this issue when quantifying the density of white matter

connections.

We further clarify that there is another streamlines

tractography effect that bears similarity to that de-

scribed above, but behaves slightly differently and

applies to both deterministic and probabilistic stream-

lines algorithms. Opportunities for the streamlines al-

gorithm to sample from an inappropriate bre orien-

tation (particularly in the presence of crossing bres),

and therefore construct a wholly erroneous trajectory,

increase as a function of bundle length. This effect has

also to our knowledge not been investigated compre-

hensively but should be considered as distinct from

both other ‘inuences of bundle length on streamline

count’ described above.

Relationship to xel-based analysis metrics

We have recently published on the disentanglement of

statistical effects in microscopic FD and macroscopic

changes in cross-sectional area, made possible within the

FBA framework (69). Due to the subsequent interest we

received in the potential incorporation of such xel-wise

measures into tractogram and/or structural connectome

quantication, here we clarify the relationship between

these xel-wise measures and FBC.

For any particular white matter pathway in the brain,

FBC quantied using a global or semi-global tractog-

raphy approach will scale directly proportionally with

both the microscopic FD (resulting in a greater number

of streamlines or increased streamlines weights travers-

ing any particular xel) and the macroscopic pathway

cross-sectional area (resulting in a greater breadth of x-

els traversed by the streamlines within that pathway, and

hence likely also more streamlines being assigned to that

pathway), as shown in Figure 13. FBC quantied in this

manner therefore behaves most comparably to the ‘bre

density and bundle cross-section (FDC)’ measure (69);

but crucially, FBC is quantied as an endpoint-to-end-

point connectivity measure, whereas FDC is a local x-

el-wise quantitative measure (as shown in Figure 3).

O R I G I N A L R E S E A R C H A R T I C L E

global image information into otherwise locally greedy,

streamlines-based tractography data. When used ap-

propriately, these methods address one of the major

fundamental technical limitations in the eld that oth-

erwise precludes the direct comparison of quantitative

estimates of white matter connection density between

subjects. We hope that the explanations and clarica-

tions contained herein assist those readers for whom

the purpose (or indeed existence) of these approaches

was unclear.

ACKNOWLEDGEMENTS

We are grateful to the National Health and Medical

Research Council (NHMRC) of Australia and the Victorian

Government’s Operational Infrastructure Support

Program for their support.

RS is supported by fellowship funding from the

National Imaging Facility (NIF), an Australian Government

National Collaborative Research Infrastructure Strategy

(NCRIS) capability.

Software implementations

The algorithms described in the section ‘The algorithmic

basis of quantitative streamlines tractography’ have been

made available as part of the MRtrix3 software package

(100) (www.mrtrix.org; command and option names as at

version 3.0.0):

• Algorithm 1, ‘Fixel mask’: afdconnectivity;

• Algorithm 2, ‘Weighted xel mask’: afdconnectivity

using the -wbft option;

• Algorithm 3, ‘Volume-averaged streamline weights’:

tcksift2 using the -linear option;

• Algorithm 4, ‘Optimised streamline weights’: tcksift2.

The SIFT method (19) mentioned in the ‘Relationship

to existing methods’ section is additionally available as

command tcksift.

CONCLUSION

We have shown how the algorithmic design of a

class of ‘semi-global’ tractogram optimisation algo-

rithms is the inevitable result of trying to incorporate

Fig. 14. Visual demonstration of the operation of hypothetical alternative Algorithms 5a and 5b. (a) For the pathway of interest (left panel; streamlines), derive a

mask of xels traversed (top panel; red xels within yellow voxels). Within this mask, compute from the bre densities of those xels (encoded visually as xel lengths)

the mean (equation at right shows the sum of xel bre densities divided by the number of traversed xels), as a measure of ‘connectivity’ C of the pathway. (b) For

each individual streamline within the pathway of interest (left panel), derive a mask of xels traversed by that streamline (bottom panel; red xels). Across those xels,

compute the mean bre density (rst equation at right shows the sum of traversed xel bre densities divided by the number of xels traversed by that streamline).

Finally, take the mean of these values across the streamlines corresponding to the pathway of interest (second equation at right shows sum of streamline weights

divided by the number of streamlines) as a measure of ‘connectivity’ C of the pathway.

O R I G I N A L R E S E A R C H A R T I C L E

20. Daducci A, Dal Palú A, Lemkaddem A, Thiran J. COMMIT: Convex optimiza-

tion modeling for micro-structure informed tractography. IEEE Transactions

on Medical Imaging. 2014;34:246–257. doi:10.1109/TMI.2014.2352414

21. Pestilli F, Yeatman JD, Rokem A, Kay KN, Wandell BA. Evaluation

and statistical inference for human connectomes. Nature Methods.

2014;11(10):1058–1063.

22. Smith RE, Tournier J-D, Calamante F, Connelly A. SIFT2: Enabling dense

quantitative assessment of brain white matter connectivity using stream-

lines tractography. NeuroImage. 2015;119:338–351.

23. Sporns O, Tononi G, Kötter R. The human connectome: A structural de-

scription of the human brain. PLoS Computational Biology. 2005;1(4):e42.

24. Hagmann P. From diffusion MRI to brain connectomics. 2005. doi:10.5075/

ep-thesis-3230

25. Christiaens D, Tournier JD. Chapter 20 – Modeling ber orientations using

diffusion MRI. In: Seiberlich N, Gulani V, Calamante F, et al., eds. Advances

in Magnetic Resonance Technology and Applications. Vol 1. Quantitative

Magnetic Resonance Imaging. Academic Press; 2020:509–532. doi:10.1016/

B978-0-12-817057-1.00022-6

26. Smith RE, Connelly A, Calamante F. Chapter 21 – Diffusion MRI ber trac-

tography. In: Seiberlich N, Gulani V, Calamante F, et al., eds. Advances in

Magnetic Resonance Technology and Applications. Vol 1. Quantitative

Magnetic Resonance Imaging. Academic Press; 2020:533–569. doi:10.1016/

B978-0-12-817057-1.00023-8

27. Calamante F. The seven deadly sins of measuring brain structural con-

nectivity using diffusion MRI streamlines bre-tracking. Diagnostics.

2019;9(3):115. doi:10.3390/diagnostics9030115

28. Le Bihan D, Johansen-Berg H. Diffusion MRI at 25: Exploring brain tis-

sue structure and function. Neuroimaging: Then, Now and the Future.

2012;61(2):324–341.

29. Sotiropoulos SN, Zalesky A. Building connectomes using diffusion MRI:

Why, how and but. NMR in Biomedicine. 2019;32(4):e3752. doi:10.1002/

nbm.3752

30. Yeh C-H, Jones DK, Liang X, Descoteaux M, Connelly A. Mapping structur-

al connectivity using diffusion MRI: Challenges and opportunities. Journal

of Magnetic Resonance Imaging. doi:10.1002/jmri.27188

31. Tournier J-D, Calamante F, Connelly A. Robust determination of the bre

orientation distribution in diffusion MRI: Non-negativity constrained su-

per-resolved spherical deconvolution. NeuroImage. 2007;35(4):1459–1472.

32. Derek K. Jones PD. Diffusion MRI: Theory, Methods, and Applications.

Oxford University Press; 2011. http://books.google.com.au/

books?id=dbZCMePD52AC

33. Johansen-Berg H, Behrens TEJ. Diffusion MRI: From Quantitative

Measurement to In-Vivo Neuroanatomy. Academic Press; 2009. http://

books.google.com.au/books?id=N20nnxByjVAC

34. Chamberland M, Raven EP, Genc S, et al. Dimensionality reduction of

diffusion MRI measures for improved tractometry of the human brain.

NeuroImage. 2019;200:89–100. doi:10.1016/j.neuroimage.2019.06.020

35. Jones D, Travis A, Eden G, Pierpaoli C, Basser P. PASTA: Pointwise as-

sessment of streamline tractography attributes. Magnetic Resonance in

Medicine. 2005;53(6):1462–1467.

36. Yeatman JD, Dougherty RF, Myall NJ, Wandell BA, Feldman HM. Tract pro-

les of white matter properties: Automating ber-tract quantication. PLoS

ONE. 2012;7(11):e49790.

37. Corouge I, Fletcher PT, Joshi S, Gouttard S, Gerig G. Fiber tract-orient-

ed statistics for quantitative diffusion tensor MRI analysis. Medical Image

Analysis. 2006;10(5):786–798. doi:10.1016/j.media.2006.07.003

38. Colby JB, Soderberg L, Lebel C, Dinov ID, Thompson PM, Sowell ER. Along-

tract statistics allow for enhanced tractography analysis. NeuroImage.

2012;59(4):3227–3242.

39. Dayan M, Monohan E, Pandya S, et al. Prolometry: A new statistical

framework for the characterization of white matter pathways, with appli-

cation to multiple sclerosis. Human Brain Mapping. 2016;37(3):989–1004.

doi:10.1002/hbm.23082

40. Fornito A, Zalesky A, Breakspear M. Graph analysis of the human con-

nectome: Promise, progress, and pitfalls. Mapping the Connectome.

2013;80(0):426–444.

41. Yeh C-H, Smith RE, Liang X, Calamante F, Connelly A. Correction for diffu-

sion MRI bre tracking biases: The consequences for structural connecto-

mic metrics. NeuroImage. 2016;142:150–162.

42. Rubinov M, Sporns O. Complex network measures of brain connec-

tivity: Uses and interpretations. Computational Models of the Brain.

2010;52(3):1059–1069.

43. Sporns O. Network attributes for segregation and integration in the human

brain. Macrocircuits. 2013;23(2):162–171.

JDT was supported with funding from the European

Research Council under the European Union’s Seventh

Framework Programme [FP7/20072013], ERC grant

agreement no. [319456] (developing Human Connectome

Project) and MRC strategic funds [MR/K006355/1]. JDT

was also supported by the Wellcome/EPSRC Centre

for Medical Engineering at King’s College London

[WT 203148/Z/16/Z] and by the National Institute for

Health Research (NIHR) Biomedical Research Centre at

Guy’s and St Thomas’ NHS Foundation Trust and King’s

College London. The views expressed are those of the

authors and not necessarily those of the NHS, the NIHR,

or the Department of Health.

REFERENCES

1. Mori S, van Zijl PCM. Fiber tracking: Principles and strategies – a technical

review. NMR in Biomedicine. 2002;15(7–8):468–480.

2. Dell’acqua F, Catani M. Structural human brain networks: Hot topics in dif-

fusion tractography. Current Opinions in Neurology. 2012;25(4):375–383.

3. Johansen-Berg H, Behrens TEJ. Just pretty pictures? What diffusion trac-

tography can add in clinical neuroscience. Current Opinion in Neurology.

2006;19(4):379–385.

4. Griffa A, Baumann PS, Thiran J-P, Hagmann P. Structural connectomics in

brain diseases. Mapping the Connectome. 2013;80(0):515–526.

5. Tournier J-D, Mori S, Leemans A. Diffusion tensor imaging and beyond.

Magnetic Resonance in Medicine. 2011;65(6):1532–1556.

6. Mori S, Crain BJ, Chacko VP, van Zijl PCM. Three-dimensional track-

ing of axonal projections in the brain by magnetic resonance im-

aging. Annals of Neurology. 1999;45(2):265–269. doi:10.1002/1531

-8249(199902)45:2%3C265::aid-ana21%3E3.0.co;2-3

7. Conturo TE, Lori NF, Cull TS, et al. Tracking neuronal ber pathways in the

living human brain. Proceedings of the National Academy of Sciences of

the United States of America. 1999;96(18):10422–10427.

8. Basser PJ, Pajevic S, Pierpaoli C, Duda J, Aldroubi A. In vivo ber trac-

tography using DT-MRI data. Magnetic Resonance in Medicine.

2000;44(4):625–632.

9. Jeurissen B, Descoteaux M, Mori S, Leemans A. Diffusion MRI ber tractog-

raphy of the brain. NMR in Biomedicine. 2017;32:e3785.

10. Jones DK, Knösche TR, Turner R. White matter integrity, ber count,

and other fallacies: The do’s and don’ts of diffusion MRI. NeuroImage.

2013;73(0):239–254.

11. Mangin J-F, Poupon C, Cointepas Y, et al. A framework based on spin glass

models for the inference of anatomical connectivity from diffusion-weighted

MR data – a technical review. NMR in Biomedicine. 2002;15(7–8):481–492.

12. Kreher BW, Mader I, Kiselev VG. Gibbs tracking: A novel approach for the

reconstruction of neuronal pathways. Magnetic Resonance in Medicine.

2008;60(4):953–963.

13. Reisert M, Mader I, Anastasopoulos C, Weigel M, Schnell S, Kiselev V. Global

ber reconstruction becomes practical. NeuroImage. 2011;54(2):955–962.

14. Mangin J-F, Fillard P, Cointepas Y, Le Bihan D, Frouin V, Poupon C. Toward

global tractography. Mapping the Connectome. 2013;80(0):290–296.

15. Christiaens D, Reisert M, Dhollander T, Sunaert S, Suetens P, Maes F.

Global tractography of multi-shell diffusion-weighted imaging data using a

multi-tissue model. NeuroImage. 2015;123:89–101.

16. Smith RE, Tournier J-D, Calamante F, Connelly A. Anatomically-constrained

tractography: Improved diffusion MRI streamlines tractography through ef-

fective use of anatomical information. NeuroImage. 2012;62(3):1924–1938.

17. Girard G, Descoteaux M. Anatomical tissue probability priors for trac-

tography. CDMRI. 2012:174–185. https://www.researchgate.net/publica-

tion/237170488_Anatomical_tissue_probability_priors_for_tractography

1 8. Sherbondy AJ, Dougherty RF, Ananthanarayanan R, Modha DS, Wandell BA.

Think Global, Act Local; Projectome estimation with BlueMatter. Medical

Image Computing and Computer-Assisted Intervention. 2009;12:861–868.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3076280/

19. Smith RE, Tournier J-D, Calamante F, Connelly A. SIFT: Spherical-

deconvolution informed ltering of tractograms. NeuroImage.

2013;67(0):298–312.

O R I G I N A L R E S E A R C H A R T I C L E

70. Raffelt D, Tournier J-D, Rose S, et al. Apparent bre density: A novel mea-

sure for the analysis of diffusion-weighted magnetic resonance images.

NeuroImage. 2012;59(4):3976–3994.

71. Hagmann P, Cammoun L, Gigandet X, et al. Mapping the structural core of

human cerebral cortex. PLoS Biology. 2008;6(7):e159.

72. Sherbondy A, Rowe M, Alexander D. MicroTrack: An algorithm for concur-

rent projectome and microstructure estimation. In: Jiang T, Navab N, Pluim

J, Viergever M, eds. Medical Image Computing and Computer-Assisted

Intervention. Vol 6361. Springer Berlin / Heidelberg; 2010:183–190. https://

link.springer.com/chapter/10.1007/978-3-642-15705-9_23

73. Lemkaddem A, Skiöldebrand D, Dal Palú A, Thiran J-P, Daducci A. Global

tractography with embedded anatomical priors for quantitative connectiv-

ity analysis. Frontiers in Neurology. 2014;5:232.

74. Batalle D, Hughes EJ, Zhang H, et al. Early development of structural net-

works and the impact of prematurity on brain connectivity. NeuroImage.

2017;149:379–392. doi:10.1016/j.neuroimage.2017.01.065

75. McColgan P, Seunarine KK, Razi A, et al. Selective vulnerability of Rich Club

brain regions is an organizational principle of structural connectivity loss in

Huntington’s disease. Brain. 2015;138(11):3327–3344. doi:10.1093/brain/

awv259

76. Proix T, Spiegler A, Schirner M, Rothmeier S, Ritter P, Jirsa VK. How do par-

cellation size and short-range connectivity affect dynamics in large-scale

brain network models? NeuroImage. 2016;142:135–149. doi:10.1016/j.

neuroimage.2016.06.016

77. Mitra J, Shen K, Ghose S, et al. Statistical machine learning to identi-

fy traumatic brain injury (TBI) from structural disconnections of white

matter networks. NeuroImage. 2016;129:247–259. doi:10.1016/j.

neuroimage.2016.01.056

78. McColgan P, Gregory S, Seunarine KK, et al. Brain regions showing white

matter loss in Huntington’s disease are enriched for synaptic and meta-

bolic genes. Biological Psychiatry. 2018;83(5):456–465. doi:10.1016/j.

biopsych.2017.10.019

79. Amico E, Goñi J. Mapping hybrid functional-structural connectivity traits

in the human connectome. Network Neuroscience. 2018;2(3):306–322.

doi:10.1162/netn_a_00049

80. Silk TJ, Genc S, Anderson V, et al. Developmental brain trajectories in chil-

dren with ADHD and controls: A longitudinal neuroimaging study. BMC

Psychiatry. 2016;16(1):59. doi:10.1186/s12888-016-0770-4

81. Blesa M, Sullivan G, Anblagan D, et al. Early breast milk exposure modi-

es brain connectivity in preterm infants. NeuroImage. 2019;184:431–439.

doi:10.1016/j.neuroimage.2018.09.045

82. Takemura H, Pestilli F, Weiner KS, et al. Occipital white matter tracts in

human and macaque. Cerebral Cortex. 2017;27(6):3346–3359. doi:10.1093/

cercor/bhx070

83. Weiner KS, Jonas J, Gomez J, et al. The face-processing network is resil-

ient to focal resection of human visual cortex. Journal of Neuroscience.

2016;36(32):8425–8440. doi:10.1523/JNEUROSCI.4509-15.2016

84. Yeatman JD, Weiner KS, Pestilli F, Rokem A, Mezer A, Wandell BA. The

vertical occipital fasciculus: A century of controversy resolved by in vivo

measurements. Proceedings of the National Academy of Sciences of

the United States of America. 2014;111(48):E5214–E5223. doi:10.1073/

pnas.1418503111

85. Gomez J, Pestilli F, Witthoft N, et al. Functionally dened white matter

reveals segregated pathways in human ventral temporal cortex asso-

ciated with category-specic processing. Neuron. 2015;85(1):216–227.

doi:10.1016/j.neuron.2014.12.027

86. Takemura H, Rokem A, Winawer J, Yeatman JD, Wandell BA, Pestilli F. A

major human white matter pathway between dorsal and ventral visual cor-

tex. Cerebral Cortex. 2016;26(5):2205–2214. doi:10.1093/cercor/bhv064

87. Ajina S, Pestilli F, Rokem A, Kennard C, Bridge H. Human blindsight is me-

diated by an intact geniculo-extrastriate pathway. eLife. 2015;4:e08935.

doi:10.7554/eLife.08935

88. Perge JA, Niven JE, Mugnaini E, Balasubramanian V, Sterling P. Why do

axons differ in caliber? The Journal of Neuroscience: The Ofﬁcial Journal

of the Society for Neuroscience. 2012;32(2):626–638.

89. Drobnjak I, Zhang H, Ianus¸ A, Kaden E, Alexander DC. PGSE, OGSE, and

sensitivity to axon diameter in diffusion MRI: Insight from a simulation

study. Magnetic Resonance in Medicine. 2016;75(2):688–700. doi:10.1002/

mrm.25631

90. Whittall KP, Mackay AL, Graeb DA, Nugent RA, Li DKB, Paty DW. In vivo

measurement of T2 distributions and water contents in normal human

brain. Magnetic Resonance in Medicine. 1997;37(1):34–43. doi:10.1002/

mrm.1910370107

91. Fan Q, Nummenmaa A, Witzel T, et al. Axon diameter index estimation inde-

pendent of ber orientation distribution using high-gradient diffusion MRI.

NeuroImage. 2020;222:117197. doi:10.1016/j.neuroimage.2020.117197

44. Smith RE, Tournier J-D, Calamante F, Connelly A. The effects of SIFT on

the reproducibility and biological accuracy of the structural connectome.

NeuroImage. 2015;104(0):253–265.

45. Savadjiev P, Campbell JSW, Descoteaux M, Deriche R, Pike GB, Siddiqi K.

Labeling of ambiguous subvoxel bre bundle congurations in high angu-

lar resolution diffusion MRI. NeuroImage. 2008;41(1):58–68.

46. Close TG, Tournier J-D, Johnston LA, Calamante F, Mareels I, Connelly A.

Fourier Tract Sampling (FouTS): A framework for improved inference of

white matter tracts from diffusion MRI by explicitly modelling tract volume.

NeuroImage. 2015;120(0):412–427.

47. Daducci A, Dal Palú A, Descoteaux M, Thiran J-P. Microstructure Informed

Tractography: Pitfalls and open challenges. Frontiers in Neuroscience.

2016;10:247.

48. Girard G, Daducci A, Petit L, et al. AxTract: Toward microstructure informed

tractography. Human Brain Mapping. 2017;38(11):5485–5500.

49. Raffelt DA, Smith RE, Ridgway GR, et al. Connectivity-based xel enhance-

ment: Whole-brain statistical analysis of diffusion MRI measures in the

presence of crossing bres. NeuroImage. 2015;117:40–55.

50. Behrens TEJ, Woolrich MW, Jenkinson M, et al. Characterization and

propagation of uncertainty in diffusion-weighted MR imaging. Magnetic

Resonance in Medicine. 2003;50(5):1077–1088.

51. Dell’Acqua F, Rizzo G, Scifo P, Clarke RA, Scotti G, Fazio F. A model-based

deconvolution approach to solve ber crossing in diffusion-weighted MR

imaging. IEEE Transactions on Biomedical Engineering. 2007;54(3):462–

472. doi:10.1109/TBME.2006.888830

52. Alexander DC, Hubbard PL, Hall MG, et al. Orientationally invariant in-

dices of axon diameter and density from diffusion MRI. NeuroImage.

2010;52(4):1374–1389.

53. Assaf Y, Basser PJ. Composite hindered and restricted model of dif-

fusion (CHARMED) MR imaging of the human brain. NeuroImage.

2005;27(1):48–58.

54. Assaf Y, Blumenfeld-Katzir T, Yovel Y, Basser PJ. AxCaliber: A method

for measuring axon diameter distribution from diffusion MRI. Magnetic

Resonance in Medicine. 2008;59(6):1347–1354.

55. Behrens TEJ, Johansen-Berg H, Jbabdi S, Rushworth MFS, Woolrich MW.

Probabilistic diffusion tractography with multiple bre orientations: What

can we gain? NeuroImage. 2007;34(1):144–155.

56. Kaden E, Kelm ND, Carson RP, Does MD, Alexander DC. Multi-compartment

microscopic diffusion imaging. NeuroImage. 2016;139:346–359.

57. Panagiotaki E, Schneider T, Siow B, Hall MG, Lythgoe MF, Alexander DC.

Compartment models of the diffusion MR signal in brain white matter: A

taxonomy and comparison. NeuroImage. 2012;59(3):2241–2254.

58. Zhang H, Schneider T, Wheeler-Kingshott CA, Alexander DC. NODDI:

Practical – in vivo neurite orientation dispersion and density imaging of the

human brain. NeuroImage. 2012;61(4):1000–1016.

59. Tournier J-D, Calamante F, Gadian DG, Connelly A. Direct estimation of

the ber orientation density function from diffusion-weighted MRI data

using spherical deconvolution. NeuroImage. 2004;23(3):1176–1185.

60. Jeurissen B, Tournier J-D, Dhollander T, Connelly A, Sijbers J. Multi-tissue

constrained spherical deconvolution for improved analysis of multi-shell

diffusion MRI data. NeuroImage. 2014;103(0):411–426.

61. Desikan RS, Ségonne F, Fischl B, et al. An automated labeling system for

subdividing the human cerebral cortex on MRI scans into gyral based re-

gions of interest. NeuroImage. 2006;31(3):968–980.

62. Fischl B. FreeSurfer. Neuroimage. 2012;62(2):774–781.

63. Calamante F, Tournier J-D, Jackson GD, Connelly A. Track-density imaging

(TDI): Super-resolution white matter imaging using whole-brain track-den-

sity mapping. NeuroImage. 2010;53(4):1233–1243.

64. Stadlbauer A, Buchfelder M, Salomonowitz E, Ganslandt O. Fiber den-

sity mapping of gliomas: Histopathologic evaluation of a diffusion-ten-

sor imaging data processing method. Radiology. 2010;257(3):846–853.

doi:10.1148/radiol.10100343

65. Bozzali M, Parker GJM, Serra L, et al. Anatomical connectivity mapping: A

new tool to assess brain disconnection in Alzheimer’s disease. NeuroImage.

2011;54(3):2045–2051.

66. Maier-Hein KH, Neher PF, Houde J-C, et al. The challenge of map-

ping the human connectome based on diffusion tractography. Nature

Communications. 2017;8(1):1349.

67. Basser PJ, Pierpaoli C. Microstructural and physiological features of tis-

sues elucidated by quantitative-diffusion-tensor MRI. Journal of Magnetic

Resonance, Series B. 1996;111(3):209–219. doi:10.1006/jmrb.1996.0086

68. Basser PJ, Mattiello J, Lebihan D. MR diffusion tensor spectroscopy and

imaging. Biophysical Journal. 1994;66:259–267.

69. Raffelt DA, Tournier J-D, Smith RE, et al. Investigating white matter

bre density and morphology using xel-based analysis. NeuroImage.

2017;144:58–73.

O R I G I N A L R E S E A R C H A R T I C L E

connectome construction. Proceedings of the ISMRM. 2017:0058. https://

www.researchgate.net/publication/315836374_Mesh-based_anatomical-

ly-constrained_tractography_for_effective_tracking_termination_and_

structural_connectome_construction

98. Smith R, Skoch A, Bajada C, Caspers S, Connelly A. Hybrid surface-volume

segmentation for improved anatomically-constrained tractography. In:

Organisation for Human Brain Mapping. 2020:1034. https://www.research-

gate.net/publication/342800028_Hybrid_Surface-Volume_Segmentation_

for_improved_Anatomically-Constrained_Tractography

99. Morris DM, Embleton KV, Parker GJM. Probabilistic bre tracking:

Differentiation of connections from chance events. NeuroImage.

2008;42(4):1329–1339.

100. Tournier J-D, Smith R, Raffelt D, et al. MRtrix3: A fast, exible and open

software framework for medical image processing and visualisation.

NeuroImage. 2019;202:116137. doi:10.1016/j.neuroimage.2019.116137

92. Zhang H, Hubbard PL, Parker GJM, Alexander DC. Axon diameter mapping

in the presence of orientation dispersion with diffusion MRI. NeuroImage.

2011;56(3):1301–1315.

93. Veraart J, Nunes D, Rudrapatna U, et al. Nonivasive quantication of axon

radii using diffusion MRI. eLife. 2020;9:e49855. doi:10.7554/eLife.49855

94. Beaulieu C. The basis of anisotropic water diffusion in the nervous system –

a technical review. NMR in Biomedicine. 2002;15(7–8):435–455.

95. Swadlow H, Waxman S. Axonal conduction delays. Scholarpedia.

2012;7:1451.

96. Yeh C-H, Smith RE, Dhollander T, Calamante F, Connelly A. Connectomes

from streamlines tractography: Assigning streamlines to brain parcellations

is not trivial but highly consequential. NeuroImage. 2019;199:160–171.

doi:10.1016/j.neuroimage.2019.05.005

97. Yeh C-H, Smith RE, Dhollander T, Connelly A. Mesh-based anatomical-

ly-constrained tractography for effective tracking termination and structural

O R I G I N A L R E S E A R C H A R T I C L E

Appendix

Algorithm 1: ‘Fixel mask’

dene pathway.xels = empty_set

dene pathway.volume = 0.0

for s in pathway.streamlines:

for f in s.xels_traversed:

if f not in pathway.xels:

pathway.xels += f

pathway.volume += f.volume

end if

end for

dene pathway.sum_lengths = 0.0

for s in pathway.streamlines:

pathway.sum_lengths += s.length

end for

dene pathway.mean_length = pathway.sum_lengths / pathway.number_of_streamlines

dene pathway.connectivity = pathway.volume / pathway.mean_length

Algorithm 2: ‘Weighted xel mask’

for s in tractogram.streamlines:

for f in s.xels_traversed:

f.tractogram_density += s.length_within_xel[f]

if s in pathway.streamlines:

f.pathway_density += s.length_within_xel[f]

end for

dene pathway.volume = 0.0

for f in xels:

pathway.volume += (f.volume * f.pathway_density / f.tractogram_density)

end for

dene pathway.sum_lengths = 0.0

for s in pathway.streamlines:

pathway.sum_lengths += s.length

end for

dene pathway.mean_length = pathway.sum_lengths / pathway.number_of_streamlines

dene pathway.connectivity = pathway.volume / pathway.mean_length

Algorithm 3: ‘Volume-averaged streamlines weights’

for s in tractogram.streamlines:

for f in xels_traversed_by_s:

f.tractogram_density += s.length_within_xel[f]

end for

dene pathway.connectivity = 0.0

for s in pathway.streamlines:

dene s.volume = 0.0

for f in xels_traversed_by_s:

s.volume += (f.volume * s.length_within_xel[f] / f.tractogram_density)

end for

dene s.crosssection = s.volume / s.length

pathway.connectivity += s.crosssection

end for

O R I G I N A L R E S E A R C H A R T I C L E

Algorithm 4: ‘Optimized streamlines weights’

dene error = sum (f.density - f.tractogram_density)^2 for all xels “f”

for i in iterations:

for s in tractogram.streamlines:

optimize s.crosssection to minimise error

end for

dene pathway.connectivity = 0.0

for s in pathway.streamlines:

add s.crosssection to pathway.connectivity

end for