# Assessing Streamline Plausibility Through Randomized Iterative Spherical-
Deconvolution Informed Tractogram Filtering

Antonia Hain<EMAIL_ADDRESS>Daniel Jörgens<EMAIL_ADDRESS>Rodrigo Moreno<EMAIL_ADDRESS>
###### Abstract

Tractography has become an indispensable part of brain connectivity studies.
However, it is currently facing problems with reliability. In particular, a
substantial amount of nerve fiber reconstructions (streamlines) in tractograms
produced by state-of-the-art tractography methods are anatomically
implausible. To address this problem, tractogram filtering methods have been
developed to remove faulty connections in a postprocessing step. This study
takes a closer look at one such method, Spherical-deconvolution Informed
Filtering of Tractograms (SIFT), which uses a global optimization approach to
improve the agreement between the remaining streamlines after filtering and
the underlying diffusion magnetic resonance imaging data. SIFT is not suitable
to judge the plausibility of individual streamlines since its results depend
on the size and composition of the surrounding tractogram. To tackle this
problem, we propose applying SIFT to randomly selected tractogram subsets in
order to retrieve multiple assessments for each streamline. This approach
makes it possible to identify streamlines with very consistent filtering
results, which were used as pseudo ground truths for training classifiers. The
trained classifier is able to distinguish the obtained groups of plausible and
implausible streamlines with accuracy above 80%.

###### keywords:

Diffusion MRI , Tractography , Tractogram filtering , Machine Learning

††journal: arXiv

[inst1]organization=Saarland University, Faculty of Mathematics and Computer
Science, addressline=Campus E1.7, city=Saarbruecken, postcode=66041,
state=Saarland, country=Germany

[inst2]organization=Division of Brain, Imaging, and Behaviour, Krembil
Research Institute, Toronto Western Hospital, University Health
Network,city=Toronto,state=Ontario,country=Canada

[inst3]organization=KTH Royal Institute of Technology, Department of
Biomedical Engineering and Health Systems, addressline=Halsovagen 11C,
city=Huddinge, postcode=14157, state=Stockholm, country=Sweden

##  1 Introduction

Tractography uses data acquired with diffusion-weighted magnetic resonance
imaging (DW-MRI) to trace nerve fiber tracts in the brain, producing a model
called a tractogram. A tractogram consists of a set of streamlines, each of
which represents an assumed nerve fiber through an ordered set of
3-dimensional coordinates [1]. The main goal of tractography can be described
as creating a set of streamlines that pose a maximally accurate, digital
representation of the structural connectome [2], which refers to the actual
set of nerve fibers in the brain [3]. Applications for tractography include
neurosurgery planning [4], the study of neurological diseases [5], or the
scientific study of the brain to understand its function and its links to
human behavior [6].

By generating many streamlines (usually millions), current tractography
methods have been shown to be able to recover all relevant bundles [7], at
least if they are run in an appropriate way [8, 9]. However, they are also
prone to generating erroneous and implausible streamlines. It has been
estimated that an average of four false-positive streamlines are generated for
each valid streamline in the tractogram [10]. This can hardly be fully avoided
because tractography methods with high sensitivity tend to show low
specificity and vice versa [11]. In order to take steps towards a reliable
model of the connectome, removing implausible connections is crucial.

A whole field of research, commonly called tractogram filtering, has been
established as a way to deal with the excess of false-positive streamlines.
Tractogram filtering approaches remove streamlines from tractograms by
evaluating their plausibility, for example, by considering the geometrical
properties of streamlines (e.g., [12]), anatomical constraints (e.g., [13]),
through clustering approaches (e.g., [14]) or by correspondence to the
underlying DW-MRI data (e.g., [15]). A review of methods is presented in [16].

Identifying implausible streamlines can be posed as a binary classification
problem: Each streamline is assigned a “positive” (P) or “negative” (N) label
if it appears to be plausible or implausible, respectively. Streamlines with a
“negative” label are subsequently removed from the tractogram. The terms “true
positive” (TP) and “false positive” (FP) frequently appear in tractography
literature as well, even though they express notions that are slightly
different from the meaning used in statistics [17]: Here, a “true positive”
streamline is regarded to be a fully correct reconstruction of an actual,
plausible nerve fiber bundle, while a “false positive” streamline represents a
faulty/noisy reconstruction that does not accurately represent any existing
structure in the brain. Therefore, in this work, we use the terms “plausible”
or “implausible” interchangeably with “positive” and “negative” or “true
positive” and “false positive”, even though the concepts are not entirely
equal in the classical sense.

One popular tractogram filtering method is Spherical-deconvolution Informed
Filtering of Tractograms (SIFT) [15]. This approach belongs to the family of
methods that assess tractogram quality by comparing the acquired DW-MRI data
to the expected one from the tractogram [16]. In particular, SIFT removes
streamlines to increase the consistency of the tractogram with respect to the
acquired data based on a global optimization approach. Other examples of such
methods include LiFE [18], COMMIT [19], SIFT2 [20], and COMMIT2 [21, 22, 23].
SIFT does not operate directly on raw data, but instead uses the fiber
orientation distribution (FOD) [24]. A tractogram can be filtered by
investigating how the effect of removing an individual streamline manifests in
the streamline density in each voxel compared to the corresponding local FOD
along the streamline. Streamlines that increase the mismatch between
streamline density and local FODs are prioritized for filtering. Given the
complexity of the computation and the number of streamlines, trying to find a
globally optimal solution is infeasible. Therefore, a gradient-descent
approach is used to reduce a cost function.

Several criteria for termination of the optimization process are provided by
SIFT. By default, streamlines are removed until the cost function gradient of
the candidate streamlines becomes sufficiently small. This option is referred
to as filtering “to convergence”. In the absence of other termination criteria
which specify a maximum number of streamlines to be removed, filtering to
convergence will naturally lead to the most reliable result set of
streamlines, but with the streamline density being decreased the most. In
fact, it has been argued that the remaining number of streamlines is not
always sufficient for quantitative analyses, and for that reason, the authors
of SIFT recommend applying the method to tractograms with a high number of
streamlines [25].

Due to the design of SIFT’s cost function, streamlines will be removed if the
track density in their path is too high to match (parts of) the FODs. Such a
mismatch could be due to different reasons, including at least the following
cases:

  1. 1.

The FOD representation in a voxel is not accurately describing the underlying
anatomy. This can happen, for example, when the raw data is not able to model
and resolve different fiber populations.

  2. 2.

The streamlines may be (partly) faulty/noisy. This is the target case for
filtering, as the streamline is a false positive and must be removed.

  3. 3.

The streamlines are plausible, but the streamline density in the corresponding
voxel fractions is simply exaggerated. It is common for tractography methods
to create multiple similar streamlines. We will refer to these streamlines as
redundants.

Thus, a streamline being rejected by SIFT is therefore not a sufficient
indicator of it being implausible. Since SIFT removes streamlines from these
three categories, SIFT as such is not completely suitable for assessing the
correspondence of an individual streamline to the DW-MRI data. The focus of
this paper is, therefore, to distinguish cases two and three.

Machine learning has been used for training classifiers to speed up the
processing of expensive tractogram filtering methods [26, 27]. Although it is
appealing to train a binary classifier to distinguish false-positive from
true-positive streamlines based on the raw output of SIFT, this is
unfortunately not possible for the abovementioned reasons. Indeed, redundant
and true positive streamlines share similar features, but SIFT rejects the
former and accepts the latter. Thus, for SIFT to be used in classifiers of
streamlines, it is necessary to have a method to distinguish between false
positives and redundants.

We have found that specific streamlines may be classified differently by SIFT
depending on the composition of the tractogram. In this paper, we take
advantage of this property with the goal of disentangling false positives from
redundant streamlines. More specifically, we apply SIFT on randomized subsets
of tractograms to identify both of those streamline groups. Since SIFT seems
to yield inconsistent results for certain streamlines when found in different
subsets, we spotlight those streamlines and explore how they might be
appropriately labeled. We refer to the proposed approach as randomized SIFT
(rSIFT).

##  2 Materials and Methods

###  2.1 Datasets

We carried out the experiments on pre-processed data [28] of six different
subjects from the young adult data set of the Human Connectome Project [29].
The whole-brain tractograms derived from this data were provided by the
authors of [30]111https://doi.org/10.5281/zenodo.1088277 and computed with the
iFOD2 [31] algorithm. Ten million streamlines were created for each subject,
restricted to be between 40mm and 250mm in length, with a step size of
0.625mm. The tracking was further constrained by anatomical priors based on
the segmentation of different tissue types in the brain [32]. In order to make
computation more feasible, we used an additional post-processing step in which
the streamlines were compressed to smaller sets of coordinates with the method
in [33] using a tolerance error of 0.35mm.

The ten million streamlines in each HCP tractogram covered the entire white
matter volume. An exemplary depiction of the distribution of streamline
length, the number of sampling points, and their correlation for one subject
can be seen in Fig. 1. This figure also shows that the streamline length and
the number of sampling points are still highly correlated after compression.
The tractograms were generated with MRtrix3 [34] with the exception of the
compression step, for which the Python library Dipy [35] was used.

We further studied the phantom data from the DiSCo Challenge [36], which was
used as ground truth to create datasets with known true positive, redundant,
and false-positive streamlines. The ground truth of this dataset consists of
12 196 streamlines. In our first experiment on this data, we used half of the
streamlines as true positives, while the other half was distorted to produce
realistic implausible streamlines by applying random rotation in 3D. The
rotations were done by randomly selecting Euler angles in the range of 45–315
degrees to get streamlines not too close to the ground truth. We filled the
tractogram with false positives to a total of 89 570 streamlines in order to
achieve a similar streamline density compared to the 10 million used in the
HCP case. In a second experiment, half of the ground-truth streamlines were
used as true positives, while the second half was systematically copied to
produce redundant streamlines. Notice that all streamlines are plausible in
this experiment. The second half of the data was split evenly into five
further groups: the first fifth of the streamlines was copied once, such that
each of them would appear twice in the tractogram. The other groups were
copied twice, four, nine, and 48 times (filler data), respectively. This
resulted in a tractogram with around 89 000 streamlines, including groups of
streamlines appearing with very different frequencies.

Figure 1: Distribution of the number of points and length of streamlines in an
exemplary tractogram of ten million streamlines. Top: histogram of the number
of points per streamline. Middle: histogram of streamline length in mm.
Bottom: correlation plot between streamline length and the mean number of
sampling points with standard deviation (light blue).

###  2.2 Randomized SIFT (rSIFT)

Fig. 2 summarizes the method, and Algorithm 1 shows the pseudocode of rSIFT.
The following subsections detail the different components of the method.
Instead of running SIFT once on the complete tractogram, we applied it several
times to random subsets of streamlines. The number of times a streamline was
filtered out or kept over the different runs of SIFT was used to define its
acceptance rate (AR) per subset size. We refer to each evaluation of a
streamline through SIFT as a vote. That means a streamline that was kept in a
tractogram after filtering receives a positive vote and a streamline that was
removed receives a negative vote.

Figure 2: Pipeline of rSIFT. Top: SIFT is run on different random subsets of
the original tractogram. The numbers of positive and negative votes are used
to estimate the acceptance rate (AR) of every streamline. Bottom: streamlines
with AR=100% and 0% are used to compute a pseudo ground truth of streamlines.
The remaining streamlines are inconclusive.

Input: $T=\\{t_{1},t_{2},\ldots,t_{M}\\}$ : tractogram,
---
$SS=\\{s_{1},s_{2},\ldots,s_{N}\\}$ : subset sizes

Output: $P=\\{p_{1},p_{2},\ldots,p_{M}\\}$: plausibility votes,
---
$N=\\{n_{1},n_{2},\ldots,n_{M}\\}$: implausibility votes

Initialize $P$ and $N$ with zeros

forall  _$n\in SS$_ do

Initialize $P_{n}$ and $N_{n}$ with zeros

$k\leftarrow\tau M/n$

for  _$i\in\\{1,\ldots,k\\}$_ do

$subset_{i}\leftarrow\\{r_{1},\ldots,r_{n}\\}\subseteq T$, where
$r_{1,\ldots,n}$ are randomly selected from $T$

$[subset_{i}^{P},subset_{i}^{N}]\leftarrow SIFT(subset_{i})$

$P_{n}(subset_{i}^{P})\leftarrow P_{n}(subset_{i}^{P})+1$

$N_{n}(subset_{i}^{N})\leftarrow N_{n}(subset_{i}^{N})+1$

end for

$P=P+P_{n}$

$N=N+N_{n}$

end forall

Algorithm 1 Pseudocode outlining the procedure of rSIFT. For each chosen
subset size $n$, $k$ random subsets of the tractogram are extracted and
filtered with SIFT to receive the indices of the accepted and rejected
streamlines $subset_{i}^{P}$ and $subset_{i}^{N}$. These are used to update
the number of votes.

In order to examine the influence of not only the composition, but also the
subset size on the results of the SIFT algorithm, we employ different subset
sizes in the rSIFT procedure. The streamline subset sizes were selected to
cover a wide range but simultaneously be able to produce meaningful results
within reasonable computation time.

In pretests, out of the ten million streamlines per HCP tractogram, only
around 2–2.5% remained after filtering. The fact that the algorithm terminated
through convergence indicates that the remaining set of streamlines was too
small to guarantee a stable model and valid removal of streamlines. Any
results for streamline sets smaller than that might not be meaningful. Thus,
the smallest subset size was chosen to be 2.5% of the original tractogram
size, i.e., 250 000 streamlines for the HCP data. We applied a similar
procedure to determine a minimum subset size for the DisCo data, which turned
out to be around 16 000 in the tractogram with ground truth and false-positive
data and around 13 000 in the tractogram with ground truth and redundant data.

Between the largest (the complete tractogram) and smallest subsets, subset
sizes were systematically chosen, usually by halving the subset size. Thus,
the probed sizes were $SS=\\{1\times 10^{7},5\times 10^{6},2.5\times
10^{6},1.25\times 10^{6},6.25\times 10^{5},5\times 10^{5},2.5\times 10^{5}\\}$
for the HCP data.

For the first DiSCo data experiment (with implausible streamlines), we used
$SS=\\{$89 570, 44 785, 22 392, 16 000$\\}$ and for the second experiment
(with redundant streamlines), $SS=\\{$88 996, 44 498, 22249, 13 000$\\}$.

Subsequently, rSIFT was run on all of the pre-defined subset sizes $n\in SS$,
with a number of repetitions $k$ defined for each $n$. $k$, was adapted to
$n$, such that $n\times k$ remained constant. In particular, $k=\tau M/n$,
with $M$ being the total number of streamlines in the tractogram and $\tau$
being a parameter that we set to five in the experiments. This way, we aim to
obtain enough votes for each streamline to compute robust statistics (e.g.,
$k=5$ and $k=20$ for $n=1\times 10^{7}$ and $n=2.5\times 10^{6}$,
respectively, in the experiments with HCP data).

Notice that, since each streamline subset is defined from the complete
tractogram, the number of occurrences across all subsets varies for all
streamlines. That means that the number of received votes is expected to
differ between streamlines and could, for some, even be zero.

After completing the filtering procedure, the total numbers of positive and
negative votes of a streamline $s$ for subset size $n$, denoted as $P_{n}(s)$
and $N_{n}(s)$, respectively, were used to compute its acceptance rate
$AR_{n}$ as:

$AR_{n}(s)=\frac{P_{n}(s)}{P_{n}(s)+N_{n}(s)}.$ (1)

In addition to this, $AR(s)$ (without $n$) refers to the acceptance rate of a
streamline compiling the votes received over all subset sizes. For the
analysis of the method, we analyzed the distribution of $AR_{n}$ over
different choices of $n$.

As mentioned, finding an anatomically reliable ground truth is an ongoing
challenge in tractogram filtering. Having this in mind, our focus here is
limited to extending the notion of streamline acceptance that is possible with
SIFT. Streamlines with AR = 100% and AR = 0% are likely to be plausible and
implausible, respectively, since all runs of SIFT are consistent for these
streamlines regardless of the tractogram configuration they are in. That means
that these streamlines can be used to generate a pseudo ground truth of a
plausible and implausible class of streamlines in the sense of SIFT. The
plausibility of the remaining streamlines with less consistent results is
harder to assess solely based on their AR. Thus, we group them with the label
inconclusive. Notice that inconclusive streamlines may consist of a mix of
plausible and implausible streamlines that SIFT cannot consistently detect or
that SIFT has not evaluated in all subset sizes. In summary, we separate the
streamlines in a tractogram into three classes:

  1. 1.

plausible where $AR=100\%$

  2. 2.

implausible where $AR=0\%$

  3. 3.

inconclusive where $0\%<AR<100\%$.

We use the distinction of plausible and implausible streamlines as pseudo
ground truth for training the classifiers described in the next section.

###  2.3 Neural network-based streamline classifier

We designed rSIFT with the goal of providing useful information about the
plausibility of individual streamlines. In order to gain insight into the
properties of the streamline grouping based on plausible, implausible, and
inconclusive labels, we analyzed the performance of a neural network
classifier trained on these labels in different scenarios. In doing so, we
simultaneously show the feasibility of training a neural network model that
mimics the characteristics of rSIFT, and thus provide a computationally
efficient alternative implementation to the originally proposed method.

In our experiments, we investigated the composition of the three rSIFT labels
by means of their separation based on neural network classifiers. First, we
trained models for the pair-wise separation of labels, i.e., plausible
(positive) vs. implausible (negative), plausible vs. inconclusive, and
implausible vs. inconclusive streamlines. Secondly, we trained a multi-class
model to distinguish between all three rSIFT labels simultaneously.

The training of all classifiers was performed using a 5-fold cross-validation
(CV) approach on the pooled set of streamlines from two HCP subjects. We
report the average performance of all CV models on the respective validation
set. For testing, we chose the best performing model in terms of balanced
sensitivity and specificity in order to minimize bias. This model was used to
evaluate streamlines from four unseen HCP subjects, which we refer to as test
data.

In order to make the classifier applicable to data from multiple subjects and
to provide a basic image registration, streamline coordinates were normalized
to a range of -1 and 1 in each dimension using the minimum and maximum
streamline coordinates per subject. Since the input streamlines comprised a
varying number of points, they were resampled to the same number of points in
order to be used as input for the network. To minimize the impact of this
resampling on as many streamlines as possible, the median number of streamline
points across training subjects, which was determined to be 22 points, was
chosen as the resampling target. Linear interpolation was applied to
approximate the original streamline geometry.

Each classifier was composed of two 1-D convolution layers (kernel sizes 5 and
3), with ReLU activation and max-pooling (pool size 2) applied after each of
them. The convolutional layers were followed by a dense layer with a dropout
chance of 0.5 and connected to either one or multiple output neurons
(depending on the classifier type being binary or multi-class). For the binary
type, the last layer used a sigmoid activation function, while the multi-class
classifier used softmax. Therefore, both classifier architectures were
identical except for the number of output neurons (one vs. multiple) and the
corresponding activation functions. An illustration of the structure of the
classifiers is shown in Fig. 3. The input to the network consisted of a one-
dimensional array of the coordinates of one streamline reordered in an
interleaved manner, such that the x-, y-, and z-coordinates of the same point
were subsequently following each other (i.e.,
$x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},...$).

|
---|---
Figure 3: Architecture of the classifiers. Left: binary classifier. Right:
multi-class classifier. The only difference is in the last layer. The binary
classifier uses one output neuron with sigmoid activation, while the multi-
class classifier uses multiple output neurons with softmax activation.

Since the data labels were determined based on a filtering method that rejects
the vast majority of streamlines, a large disproportion in the number of
training samples available for each class was expected. Thus, a balanced data
generator was used to oversample data of plausible streamlines for network
training and testing. The epoch size was defined by the larger of the classes
such that each training sample was seen at least once per epoch. The smaller
set of samples was shuffled and re-used whenever it was exhausted during an
epoch. Each classifier was trained towards maximizing accuracy using the Adam
optimizer [37] with default settings in TensorFlow 2.8.0 [37]. The training
was done in batches of 50 samples each (except for the categorical classifier
with three classes, which used a batch size of 60 such that each batch could
contain an equal number of samples from each class). Five training epochs were
determined to be sufficient since accuracy and loss saturated quickly and
showed almost no improvement after two epochs.

##  3 Results

###  3.1 Randomized SIFT on HCP data

Fig. 4 and 6 show the distribution of streamlines for the different subset
sizes and AR ranges on the six HCP subjects. As shown, most of the streamlines
received either $AR_{n}=0\%$ or $AR_{n}=100\%$ after rSIFT for all $n$. Notice
that the fractions of those streamlines with $0\%<AR_{n}<100\%$ increased when
the subsets became smaller. Moreover, using smaller subset sizes led to a
substantial increase in streamlines with $AR_{n}=100\%$ even after repeated
evaluation: The number of streamlines with exclusively positive votes rose
from an average of 2.63% to 14.86%, comparing the largest and the smallest
subset sizes. Conversely, the streamlines with $AR_{n}=0\%$ were reduced from
96.64% to 53.67%. As shown in the last column of Fig. 4, the number of
plausible and implausible streamlines for the whole dataset are around 1.7%
and 53.7%, which means that approximately 44.6% of the streamlines can be
considered as inconclusive. Interestingly, SIFT yielded inconclusive results
for on average 0.7% of the streamlines when the complete tractogram was used
(i.e., subset size of ten million), in four of the HCP subjects.

As mentioned, the rSIFT procedure does not guarantee having the same amount of
votes for each streamline. Thus, we assessed the distribution of ARs for
streamlines with exactly five votes, as shown in Fig. 5. Although the values
are slightly different in this case, they follow a very similar trend. Due to
the entirely random choice of streamlines for each round of the experiment, we
found that the number of streamlines that were not included in any of the
subsets in each subset size was negligible. They were not included in Fig. 4.
Every streamline received on average 35 votes over all experiments, with an
average of five votes per subset size.

Figure 4: Distribution of streamlines per acceptance rates $AR_{n}$ and
different subset sizes $n$. Each cell shows the mean ratio of streamlines
across HCP subjects, with standard deviations in parentheses. In the last
column, percentages are computed over all experiment instances and subset
sizes, thus denoting $AR$ instead of $AR_{n}$. Note that the AR intervals do
not include the particular lower bound and include the upper bound, except for
80–100%, where 100% is also excluded. Figure 5: Distribution of streamlines
per vote combination and subset size. Only combinations of exactly 5 votes are
considered. “P: 0, N: 5” refers to “0 positive, 5 negative votes” with the
other rows labeled analogously. The cell values show the mean ratio of
streamlines across HCP subjects, with standard deviations in parentheses.
$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$20$$40$$60$$80$$100$53.67%1.65%Subset
size ($\times 10^{6}$)Percentage of streamlinesrSIFT streamline acceptance
rates ($AR_{n}$) by subset size $n$$AR_{n}$= 0%$AR_{n}>$ 0%$AR_{n}>$
20%$AR_{n}>$ 40%$AR_{n}>$ 60%$AR_{n}>$ 80%$AR_{n}=$ 100% Figure 6: Evolution
of acceptance rates ($AR_{n}$) from rSIFT with the subset size $n$. Each curve
shows the percentage of streamlines with the respective AR, averaged over all
six HCP subjects. Standard deviations are shown in transparent (zoom in for
details). Dashed lines show the respective ARs considering all subset sizes at
once for AR=0% (in grey) and AR=100% (in orange). As shown, streamlines tend
to get higher AR in smaller subset sizes.

We were further interested in assessing the influence of the streamline length
on the filtering. As shown in Fig. 7, SIFT strongly influenced the
distribution of streamline lengths. In fact, streamlines deemed plausible were
among the shortest and barely exceeded the length of 125mm. The same pattern
is shown for rSIFT in Fig. 8. Notice that the new class of inconclusive
streamlines covers the whole range of lengths.

Figure 7: Distribution of streamlines accepted and rejected by SIFT with
respect to their length for one exemplary subject with ten million
streamlines. Figure 8: Distribution of streamlines labeled plausible
(accepted), implausible (rejected), and inconclusive by rSIFT with respect to
their length for one exemplary subject with ten million streamlines.

###  3.2 Randomized SIFT on the DiSCo data

Fig. 9 summarizes the ARs of rSIFT on the DiSCo data with false positives,
considering all subset sizes. Although SIFT filters out false positives more
often, it also filters out true positives. As shown, selecting a single
threshold of AR for distinguishing between true and false positives is
challenging since it involves a compromise between filtering out as few
plausible streamlines as possible at the cost of accepting more implausible
ones or filtering out more plausible ones to minimize the false positives. For
example, using a threshold of 20% will lead to a rejection of 64.8% of false
positives, at the cost of losing 11.4% of true positives in this experiment.
In turn, a threshold of 80% will reject 87.7% false positives, but, at the
same time, it will reject almost half of the true positives (48.5%). As in the
case of the HCP data, analyzing the streamlines independently where SIFT is
not consistent might be beneficial.

Figure 9: Distribution of acceptance rate (AR) for true positive (TP) and
false positive (FP) streamlines in the DiSCo dataset after running rSIFT with
$\tau=5$ and four different subset sizes. The range 80–100% excludes
$AR=100\%$. Note that the values in the columns TP and FP sum to 100%,
respectively.

Fig. 10 shows the results of the experiment with DiSCo data with redundant
streamlines. Redundant streamlines are filtered out more often when they
appear more frequently in the tractogram. For example, the amount of
streamlines with $AR=0\%$ increases from 13.5% to 27.5% when their
multiplicity grows from 2 to 49, respectively. However, this increment in the
amount of rejected streamlines is relatively slow. For example, for $AR=0\%$,
the amount of streamlines only increases by 3.4% (13.5% vs. 16.9%) when
increasing the number of redundant copies from 2 to 10. Redundancy has a
larger effect at the other end: the number of streamlines with $AR=100\%$
decreases rapidly with the number of redundant copies. Comparing Fig. 9 and
10, it can be seen that SIFT rejects more often true positives when there are
no false positives. As a comparison, around 53% of streamlines (6 483 of 12
196) remain in the tractogram when SIFT is used only once on the DiSCo ground
truth streamline set.

Figure 10: Distribution of acceptance rate (AR) for true positive (TP) and
redundant (R) streamlines in the DiSCo dataset after running rSIFT with
$\tau=5$ and four different subset sizes. The number in parentheses indicates
the number of replications per streamline, i.e., how often they were found in
the tractogram. The range 80–100% excludes $AR=100\%$. Each column sums up to
100%.

###  3.3 Classification performance

An overview of the performance of the three binary classifiers on HCP data can
be found in Fig. 11.

Figure 11: Classifier performance for binary classification. “N” denotes
“negative” (implausible), “P” “positive” (plausible), and “I” inconclusive.
The metrics are specified through the mean values as determined by 5-fold
cross-validation (CV) in training or tests with unseen subjects during
training, with standard deviations in parentheses.

As shown, the binary classifier for plausible and implausible streamlines was
around 80% accurate on the validation folds, with similar accuracy for both
classes. This shows that the procedure for balancing the classes was
appropriate for this task. In turn, the performance of the binary classifiers
trained to distinguish inconclusive streamlines from plausible or implausible
was much lower and very similar (e.g., accuracy was 67.80% and 66.09%,
respectively).

As mentioned, we also evaluated the best-performing CV model for
distinguishing negative and positive streamlines on the four unseen subjects.
As shown in Fig. 11, the results of this test were equal to the five-fold CV
experiment, with a mean accuracy of 80.2%.

Fig. 12 and 13 show the performance of the multi-class classifier in terms of
the confusion matrix and accuracy-based metrics, respectively. This network
was trained to recognize the three classes ’P’, ’N’, and ’I’. While it showed
some success on the sets of plausible and implausible streamlines (true
positive rate (TPR) and true negative rate (TNR) around 70% in Fig. 12, column
maximum for ’P’ and ’N’ on the diagonal of confusion matrix in Fig. 13), its
performance remained at chance level for the inconclusive class (true
inconclusive rate (TIR) less than $1/3$ for three classes in Fig. 12, similar
values in all entries of column ’I’ in confusion matrix in Fig. 13).
Therefore, it seems that this classifier is unable to tell the inconclusive
streamlines apart from plausible or implausible ones in our pseudo ground
truth.

Figure 12: Performance metrics of the multi-class classifier. “N” denotes
“negative” (implausible), “P” “positive” (plausible) and “I” inconclusive. TPR
= true positive rate, TNR = true negative rate, TIR = true inconclusive rate.
Figure 13: Confusion matrix for the multi-class task involving samples from
all three sets from rSIFT. “N” denotes “negative” (implausible), “P”
“positive” (plausible), and “I” inconclusive. The cells show the mean
percentage of streamlines belonging to this case in cross-validation, with the
standard deviation in parentheses. The columns show the original labels, and
the rows show the label given by the classifier. Each column sums up to 100%.

For further analyses, we chose the binary model between plausible and
implausible streamlines from the cross-validation fold that yielded the most
balanced sensitivity/specificity values (81.26% and 83.15%). Raw classifier
scores for test samples from the positive class showed a mean of around 0.71
and a median of around 0.76, while the negative test samples received a mean
score of around 0.28 and a median of around 0.21. These results suggest that
the classifier was relatively confident about the results. However, the
mismatch between mean and median hints at the presence of outlier streamlines.
Accordingly, classification scores had both a mean and a median of around 0.48
when the classifier was applied to the inconclusive set, which was not part of
the training data.

We further split this set into two groups using a threshold of 0.5 on the
classification score to get a set of leaning positive and leaning negative
streamlines. The mean scores of these two groups were 0.72 and 0.21, which
shows that they were slightly more off-center for the negative class.

As mentioned before, SIFT tends to filter out long streamlines. Fig. 14 and 15
show the performance of the binary classifier (P vs. N) with respect to the
length of the streamlines. It is apparent that longer streamlines of the
positive class were misclassified more frequently (cf. Fig. 14), and the same
applies to the shorter streamlines of the negative class (cf. Fig. 15).
Applying the P vs. N classifier to the inconclusive set, we found that, also
here, shorter streamlines tended more to be classified as plausible and longer
ones to be classified as implausible (cf. Fig. 16).

Figure 14: Histogram of streamline lengths of positive (P) streamlines,
grouped by the streamline labels obtained from the binary positive vs.
negative (plausible vs. implausible) classifier. The streamlines were taken
from one test subject unseen in training. Longer streamlines are more
frequently misclassified (i.e., labeled differently by the classifier than by
rSIFT). Negative classifications outnumber positive classifications above a
streamline length of around 75mm. Figure 15: Histogram of streamline lengths
of negative (N) streamlines, grouped by the streamline labels obtained from
the binary positive vs. negative (plausible vs. implausible) classifier. The
streamlines were taken from one test subject unseen in training. Shorter
streamlines are more frequently misclassified (i.e., labeled differently by
the classifier than by rSIFT). Positive classifications outnumber negative
classifications below a streamline length of around 60mm. Figure 16: Histogram
of streamline lengths of inconclusive (I) streamlines, grouped by the
streamline labels obtained from the binary positive vs. negative (plausible
vs. implausible) classifier. The streamlines were taken from one test subject
unseen in training. Shorter streamlines are more frequently classified as
positive, and longer streamlines are more frequently as negative, with an
equal amount of positive and negative predictions around a streamline length
of around 65mm.

####  3.3.1 Comparison between classifier scores and rSIFT

The correlation coefficient for the relation between classifier scores and the
ARs of inconclusive streamlines in rSIFT was moderately positive with a value
of 0.28. Fig. 17 depicts scores given by the classifier to each class and how
they compare to the vote distribution from rSIFT.

Figure 17: Scatter plot relating classifier scores of streamlines and their
acceptance rate from rSIFT. The data points (i.e., streamlines) in the plot
are sampled in a balanced manner such that all classes (i.e., plausible,
implausible, and inconclusive) appear with the same frequency, where blue
represents the positive, orange the negative, and gray the inconclusive class.
As seen, scores for the positive and negative classes accumulate around the
ends of the spectrum. For the inconclusive samples, there is a weak
correlation of 0.28 (depicted with the trend line) between scores and AR from
rSIFT.

In order to visualize clustering patterns of the three different sets, we used
a t-stochastic neighbor embedding (t-SNE) [38], which enables high-dimensional
input vectors to be projected into a 2-dimensional plane. This way, hidden
structures in the data as well as the closeness of data points in high-
dimensional space can be made visible. We applied t-SNE to the streamlines’
feature vectors taken from the 4-th (pre-dense) network layer to examine if
the streamline groups were clustered together. For the sets of streamlines, we
used the pseudo ground truth and inconclusive streamlines with $0\%<=AR<20\%$,
or $80\%<AR<=100\%$, respectively. These inconclusive streamlines are expected
to be closer to the pseudo ground truth data.

Plots of t-SNE were fine-tuned to a learning rate and perplexity of 100 each
and 800 iterations. The comparison of t-SNE results with and without
inconclusive streamlines using balanced sampling can be seen in Fig. 18.

---
a) P/N
b) P/N/I($AR\leq 20\%$)
c) P/N/I($AR\geq 80\%$)
Figure 18: t-SNE results. Orange dots represent streamlines from the
implausible pseudo ground truth, blue dots represent streamlines from the
plausible pseudo ground truth, and black dots show inconclusive streamlines.
The upper row does not include inconclusive samples. In the middle row,
inconclusive samples with an acceptance rate AR $\leq$ 20% in rSIFT were used.
The bottom row shows the results with inconclusive streamlines with AR $\geq$
80%.

As shown, the positive and negative streamlines formed multiple small clusters
which were positioned close to each other (cf. Fig. 18a). When additionally
presented with samples from the inconclusive class, small differences were
noticeable depending on if samples with low or high AR were used. Especially
the latter seemed to be more close to the clusters of the plausible
streamlines (cf. blue and black dots in Fig. 18c).

##  4 Discussion

Tractogram filtering based on methods like SIFT offers the opportunity to
improve the match of a given tractogram with the measured diffusion MRI data.
However, the ability of SIFT to assign binary labels of streamline
plausibility is inherently limited. In this paper, we propose a randomized
iterative adaptation, called rSIFT, as a way to address this limitation of
SIFT. We analyzed rSIFT in experiments on human and phantom data and employed
neural network-based streamline classifiers further to characterize the
properties of the output of our method. Additionally, the trained classifiers
underline the potential for tractogram filtering being performed by neural
networks at a reduced computational cost. In the following subsections, we
discuss different aspects of our findings in further detail.

###  4.1 SIFT

Our experiments highlight how the filtering performed by SIFT depends not only
on the characteristics of a streamline but also on the size and composition of
the streamline set used as the input. As shown in the presented results, the
same streamline can be accepted and rejected in different SIFT runs depending
on the streamlines contained in the analyzed tractogram. This ambiguity,
reflected in the inconclusive class from rSIFT, is more prominent with smaller
tractograms. Through the randomized and iterative approach, rSIFT is designed
to exploit this dependency of the SIFT output on the composition of the input
tractogram.

This dependency on the input tractogram is rooted in the global optimization
approach for improving the consistency between tractogram and diffusion data.
It implies that the focus of SIFT lies in fitting all streamlines
simultaneously to the diffusion data instead of assessing an individual
streamline alone. As shown in the experiments, filtering a tractogram
containing fewer streamlines with SIFT will directly lead to an increase in
accepted streamlines (2.6% for the whole tractogram vs. 14.9% for the smallest
subset size), a decrease in rejected ones (96.6% vs. 56.8%, respectively), and
an increase in streamlines for which SIFT gives mixed results (0.7% vs.
28.4%). One possible explanation is that smaller tractograms might contain
fewer redundant streamlines making it more likely for streamlines to be
accepted.

As a consequence, we argue that SIFT in its original form with only a single
run is not perfectly suited for the assessment of individual streamline
plausibility with respect to the diffusion data. In fact, for some of the HCP
tractograms, the results differed even when the set of streamlines did not: In
the experiment instance where the complete tractogram was repeatedly filtered
(i.e. subset size 10 million), around 0.7% of the streamlines received mixed
votes even though the overall tractogram remained the same, but was arranged
in shuffled order due to our random sampling of streamlines.

A general limitation of SIFT is that the method is indifferent to why
streamlines do not match the data. As mentioned in the introduction,
anatomical implausibility is just one possible reason, but another possible
explanation can be that the streamline in question contributes to a white
matter fiber bundle that has been reconstructed with exaggerated streamline
density compared to other bundles, relative to the measured data. This is
supported by our findings on the DiSCo data: As shown in Fig. 10, redundant
streamlines are rejected more often when they appear with a higher frequency
in the dataset. At the same time, SIFT rejects true positives more often when
there are no false positives present in the tractogram. Together, this shows
that SIFT rejects not only implausible, but also both plausible and redundant
streamlines. However, for purposes such as finding training samples for a
machine learning classifier or for the purpose of combining several filtering
methodologies in an ensemble fashion, as showcased in [27], there is a need
for methods performing the distinction between plausible and implausible
streamlines more reliably than what is achievable with a single run of SIFT.

###  4.2 Randomized SIFT

The proposed method, rSIFT, is a randomized, iterative adaptation of SIFT with
the aim to separate plausible (including redundant) streamlines from
implausible ones. Through the repeated application of SIFT to random
tractogram subsets, we defined distilled sets of streamlines that are
consistently categorized as either fitting the diffusion data well or not,
thus deemed plausible or implausible in the sense of SIFT. Despite a large
amount of votes per streamline (35 on average), the repeated SIFT assessments
were consistent for around half of the streamlines in the HCP tractograms.
Based on this, these streamlines might also possess distinct characteristics
that a machine learning classifier may be able to learn. Thus, they are
suitable for creating a pseudo ground truth of plausible and implausible
streamlines with consistent characteristics within each of the two label
groups, as we did in this study.

On the other hand, around half of the streamlines were inconclusive, with ARs
between 0% and 100%. Some of these streamlines may be outliers that received
mostly consistent votes. Apart from those, we suspected that the group of
inconclusives contains a large number of redundant streamlines as well. By
choosing random subsets of the tractogram and thus omitting potentially
similar streamlines that would “rival” a candidate redundant streamline, our
filtering procedure allows such streamlines to receive more positive votes
than in the context of the full tractogram. Indeed, our experiments show that
this group of inconclusive streamlines likely contains a mixture of both
redundant, i.e., actually plausible, as well as implausible streamlines.
Naturally, alternative choices of AR thresholds for the separation of
plausible, inconclusive, and implausible streamlines are possible. Actually,
our results show that streamlines in the inconclusive group with higher and
lower AR (e.g., higher than 80% and lower than 20%) share similar
characteristics with plausible and implausible streamlines, respectively. This
is a promising avenue for future research to better understand the composition
of the group of inconclusive streamlines.

In general, it needs to be noted that the labeling of a streamline as
“plausible” through SIFT represents more of an absence of the label
“implausible”. One could even argue that those plausible streamlines had
remained in the tractogram because there were other streamlines that
demonstrated to be even less plausible before filtering was terminated.
However, we think this to be unlikely as, from our experience, SIFT filters
streamlines quite strictly compared to other tractogram filtering methods,
leading to the removal of a significant amount of streamlines. For example, it
is interesting that SIFT only kept 53% of the streamlines in the original
DiSCo data. This value was even lower for AR=100% after rSIFT (35.0% and 33.2%
for the false positives and redundant experiments). This means that SIFT may,
in general, be too restrictive, and a threshold on AR can be set to a lower
value (e.g., 80%) to distinguish between plausible and implausible
streamlines. Of course, such a threshold involves a trade-off between rejected
false positives and removed true positives.

It is important to emphasize that, as discussed in Smith et al. [39], SIFT
results tend to correlate with streamline length. Since rSIFT builds on SIFT,
this characteristic is inherited and also rSIFT, and the trained classifiers
show this tendency to keep shorter streamlines. It would be interesting to
assess if this correlation is a bias of SIFT or if the longer streamlines are
more likely to be implausible.

Finally, notice that while rSIFT is based on SIFT, they have different goals
and applications. While SIFT assesses the consistency of the whole tractogram
with respect to the measured data, rSIFT augments this goal with the
assessment of the stability of these results and thus becomes more specific
for individual streamlines.

###  4.3 rSIFT neural network classifier

We employed different neural network classifiers to investigate the
inconclusive streamlines in rSIFT as well as to showcase the potential of such
machine learning-based approaches for the purpose of reducing the
computational burden of our method.

The classifier which was trained to tell apart the plausible and implausible
samples showed good accuracy. This indicates that some distinct structural
information that may separate the two sets of streamlines was recognized, even
in data samples unseen during training. Since rSIFT is time-consuming, it is
relevant to assess whether the obtained classification scores could be used as
an alternative to the SIFT-based method. The distributions of classification
scores show that it is unlikely for a plausible streamline to receive a
classifier score that is lower than the average (0.28) or median (0.21) scores
for implausible streamlines. Considering the difference in the size of the
groups of plausible and implausible streamlines, we would argue that
streamlines with scores lower than approximately 0.2 could be safely removed.
Such a strategy could significantly reduce the number of false-positive
streamlines in a tractogram and would require little computational effort once
the classification neural network has been trained.

Since the performance of the binary classifiers distinguishing plausible from
implausible streamlines carried over to unseen data (of plausible/implausible
samples), it may also be suited to identify such information in the
streamlines assigned to the inconclusive group. However, employing a similar
network architecture, the three-class network was unable to recognize the
inconclusive samples (cf. TPR and TNR in Fig. 12). The same tendency was found
for the binary approaches to distinguish inconclusive from plausible or
implausible streamlines, respectively (in both cases, accuracy, sensitivity,
and specificity decreased significantly compared to the N vs. P case, as seen
in Fig. 11). This suggests that there is no inherent structural difference
between inconclusive and likely plausible or implausible samples obtained
through rSIFT. The findings of the t-SNE experiment further show that
inconclusive streamlines are clustered together with the likely plausible and
implausible samples. Additionally, their acceptance rate seems to be related
to the observation of being closer to one group or the other. Taking this into
account, as well as the performance of the classifiers, it can be inferred
that the inconclusive samples are a mixture of plausible (containing
redundant) and implausible streamlines.

Notice that the binary classifier (plausible vs. implausible) yielded good
results in both cross-validation as well as classification of unseen data
despite the fact that its only input is the streamline coordinates. In
contrast, SIFT considers the FOD data. This observation is consistent with the
results in [27] that showed that the coordinates of the streamlines are the
most important features for the prediction of different tractogram filtering
methods, followed by diffusion data.

Further, the presented classification results were achieved with a relatively
simple neural network architecture. In preliminary experiments, we explored
the architecture of the classifier (e.g., adding more layers or batch
normalization) without a large improvement in performance. However, ways of
building more sophisticated classifiers achieving even better accuracy are a
direction of research that may be explored more in the future. For datasets
that suffer from severe data imbalance (such as our training data), specific
care may be put on recognizing the false positive samples in order to help
decrease their number in comparison to the true positives.

###  4.4 Future work

There are many avenues for extending the current study. While we focused our
experiments on SIFT, the same methodology can be applied to similar tractogram
filtering methods such as LiFE [18], COMMIT [19], SIFT2 [20], or COMMIT2 [21,
22, 23]. It may be particularly interesting to investigate if the presented
findings would generalize to these methods as well. Also, further performance
evaluation with other tractography methods and images acquired with clinical
settings is relevant.

Regarding the classifier, one possibility is to add different input features
as done, e.g., in [40, 27], which described the streamline’s structure in a
more sophisticated manner or add diffusion data [41, 27].

As discussed previously, rSIFT can be used to assess streamline plausibility
or to use it for training machine learning-based classifiers, as we did in
this paper. An additional interesting application would be to combine
filtering and tractography in order to improve the quality of the tractogram
from its generation.

##  5 Conclusion

In this paper, we proposed rSIFT, a randomized and iterative adaptation of
SIFT that allows assessing the plausibility of individual streamlines with
improved specificity. rSIFT was used to generate pseudo ground truths for the
training of machine learning-based classifiers. These classifiers were used to
study the characteristics of different types of streamlines (plausible,
implausible, and inconclusive) and to speed up the computations. We show how
to use AR from rSIFT or the classification scores for distinguishing plausible
and implausible streamlines. Streamlines with inconclusive results from rSIFT
are likely to be a mixture of redundant and implausible streamlines.

## Acknowledgments

Data for this study were provided in part by the Human Connectome Project, WU-
Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil;
1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH
Blueprint for Neuroscience Research; and by the McDonnell Center for Systems
Neuroscience at Washington University.

We thank Jakob Wasserthal for computing and providing the tractograms we used
in this paper.

Antonia Hain was affiliated with Saarland University at the time the presented
research was conducted and is now with the Department Quality Assurance,
Characterization and Simulation at Fraunhofer Institute for Solar Energy
Systems ISE, Heidenhofstr. 2, 79110 Freiburg, Baden-Württemberg, Germany.

## Funding sources

This work was partially supported by Digital Futures, project dBrain. The
funding sources had no involvement in the research and preparation of this
article.

## Author contributions

AH: conceptualization; formal analysis; investigation; methodology; software;
validation; visualization; writing - original draft; writing - review &
editing. DJ: conceptualization; formal analysis; investigation; methodology;
software; validation; visualization; writing - original draft; writing -
review & editing. RM: conceptualization; funding acquisition; methodology;
visualization; resources; project administration; supervision; writing -
review & editing.

## Conflicts of interest/Competing interests

The authors declare that they have no conflict of interest.

## Data and code availability statement

We used data from the Human Connectome Project and the DiSCo challenge. The
software code used in the paper and pretrained weights of the classifier are
distributed freely via the Github repository
https://github.com/djoerch/randomised_filtering.

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