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.
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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, quantication
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’: specic 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 neuroscientic
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 sufciently
dense to enable higher-level analyses (e.g. connectomics
(23,24)) within reasonable computational requirements.
Despite the potential inuence 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 specic 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 specic ‘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 dened, and for every
possible unique pair of grey matter regions, a scalar
measure of ‘connectivity’ is quantied, 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 dened based on the
streamline endpoints being ascribed to a specic 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 neuroscientic 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 innitesimally 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 reecting 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
quantication 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-
tied at the voxel level, and bre connection density es-
timates (i.e. FBC) quantied 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
specic pathway of interest (Figure 3e–f).
Limitations of semi-global optimisation
algorithms
• We do not consider the sub-voxel spatial congu-
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 dened 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 specic 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 dened 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 dened 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 inuencing 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 specic 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 quantication of intra-axonal volume within individual voxels (FD), and quantication of intra-axonal cross-sectional area of a white
matter bundle of interest (bre bundle capacity (FBC)). A bundle of interest is dened 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 denition of the forward models to be used
for each tissue component/compartment in approach
(1) serves the same purpose as the a priori denition 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 denes 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 quantication of volume
(and hence cross-sectional area) of this specic 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
quantication 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
denition 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
dened 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 benets 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) quantication 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 quantication 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; dene 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 specic 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 quantication 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 quantied (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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O R I G I N A L R E S E A R C H A R T I C L E
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 supercial 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 specical-
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 denition 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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O R I G I N A L R E S E A R C H A R T I C L E
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 specic 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 sufciently
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 reective 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 connes 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’).