aaaffiliationtext: Department of Computer Science, Vanderbilt University,
Nashville, TN, USAbbaffiliationtext: Department of Computer Science,
Université de Sherbrooke, Sherbrooke, Québec, Canadaccaffiliationtext:
Department of Biomedical Engineering, Vanderbilt University, Nashville, TN,
USAddaffiliationtext: NVIDIA Corporation, Bethesda, MD, USAeeaffiliationtext:
Laboratory of Behavioral Neuroscience, National Institute on Aging, Baltimore,
MD, USAffaffiliationtext: Department of Electrical and Computer Engineering,
Vanderbilt University, Nashville, TN, 37235, USA

figure table

# Robust Fiber ODF Estimation Using Deep Constrained Spherical Deconvolution
for Diffusion MRI

Tianyuan Yao  Francois Rheault  Leon Y Cai  Vishwesh Nath  Zuhayr Asad  Nancy
Newlin  Can Cui  Ruining Deng  Karthik Ramadass  Andrea Shafer  Susan Resnick
Kurt Schilling  Bennett A. Landman  Yuankai Huo

###### Abstract

Diffusion-weighted magnetic resonance imaging (DW-MRI) is a critical imaging
method for capturing and modeling tissue microarchitecture at a millimeter
scale. A common practice to model the measured DW-MRI signal is via fiber
orientation distribution function (fODF). This function is the essential first
step for the downstream tractography and connectivity analyses. With recent
advantages in data sharing, large-scale multi-site DW-MRI datasets are being
made available for multi-site studies. However, measurement variabilities
(e.g., inter- and intra-site variability, hardware performance, and sequence
design) are inevitable during the acquisition of DW-MRI. Most existing model-
based methods (e.g., constrained spherical deconvolution (CSD)) and learning
based methods (e.g., deep learning (DL)) do not explicitly consider such
variabilities in fODF modeling, which consequently leads to inferior
performance on multi-site and/or longitudinal diffusion studies. In this
paper, we propose a novel data-driven deep constrained spherical deconvolution
method to explicitly constrain the scan-rescan variabilities for a more
reproducible and robust estimation of brain microstructure from repeated DW-
MRI scans. Specifically, the proposed method introduces a new 3D volumetric
scanner-invariant regularization scheme during the fODF estimation. We study
the Human Connectome Project (HCP) young adults test-retest group as well as
the MASiVar dataset (with inter- and intra-site scan/rescan data). The
Baltimore Longitudinal Study of Aging (BLSA) dataset is employed for external
validation. From the experimental results, the proposed data-driven framework
outperforms the existing benchmarks in repeated fODF estimation. The proposed
method is assessing the downstream connectivity analysis and shows increased
performance in distinguishing subjects with different biomarkers. The plug-
and-play design of the proposed approach is potentially applicable to a wider
range of data harmonization problems in neuroimaging.

###### keywords:

diffusion MRI, harmonization, modeling, deep learning

*Corresponding Author: Yuankai Huo<EMAIL_ADDRESS>
##  1 Introduction

Diffusion-weighted MRI (DW-MRI) provides a non-invasive approach to estimate
the intra-voxel tissue microarchitectures as well as the reconstruction of in-
vivo neural pathways of the human brain [1]. Reproducible fiber orientation
distribution function (fODF) estimation is essential for downstream
tractography and connectivity analyses [2]. Recent advances in imaging
technologies, such as high angular resolution diffusion imaging (HARDI) [3],
provide us with a higher angular resolution for modeling intra-voxel
orientation uncertainty. On the other hand, diffusion tensor imaging (DTI)
[4], while useful, does not inherently provide this higher resolution due to
its limitations in representing multiple fiber orientations within a single
voxel. These new capabilities in imaging, particularly with HARDI, result in
more precise depictions of white matter microstructure. However, they also
necessitate more sophisticated processing methods due to the increased
complexity of the data.

The first family of the ODF esitimation methods is typically called “model-
based”, which links underlying tissue microstructures with observed signals
via sophisticated mathematical modeling such as constrained spherical
deconvolution [5, 6] (CSD), Q-ball [7], and persistent angular structure [8]
(PAS MRI). Among such approaches, CSD is one of the most broadly accepted for
modeling HARDI signals [9]. However, CSD is plagued by limited reproducibility
(Fig. 1). Several studies have highlighted the biases, inaccuracies, as well
as other limitations of HARDI methods in characterizing tissue microstructure
(e.g., parameter selection, noise sensitivity, and assumptions) [9]. Moreover,
such methods exhibit high computational complexity and often require a high
number of acquisition points, which might not be available in clinical
settings [10].

To address such challenges, the second family of approaches – “data-driven”
methods – is attracting increasingly more interest. For example, machine
learning (ML) and deep learning (DL) techniques have demonstrated their
remarkable abilities in neuroimaging [11, 10]. Such approaches have been
applied to the task of microstructure estimation, aiming to directly learn the
mapping between input DW-MRI scans and output fiber tractography. To maintain
the necessary characteristics and reproducibility for clinical translation,
robust training is carried out on a diverse and representative dataset in a
data-driven method, ensuring the model can handle a variety of patient
demographics and scanner variability. The trained model is subsequently
validated and tested on independent datasets to evaluate performance and
confirm its generalizability. The replicability of the model is assessed, with
a thorough evaluation of its consistency across different test sets and
scanners. By not assuming a specific diffusion model, data-driven algorithms
can reduce the dependence on data acquisition schemes and additionally require
less user intervention.

However, measurement variabilities (e.g., inter- and intra-site variability,
hardware performance, and sequence design) are inevitable during the imaging
process of DW-MRI (Fig. 1). Moreover, most existing model-based and learning-
based methods do not explicitly consider such variabilities in modeling, which
consequently leads to inferior performance on multi-site and/or longitudinal
diffusion studies [12]. To alleviate such issues, Nath et al. [13] presented a
multi-layer perceptron (MLP) based deep learning method for estimating
discrete fODF from voxel-wise DW-MRI signals. In this work, Nath incorporates
identical dual networks to minimize the influence of scanner effects via scan-
rescan data while learning the mapping between input DW-MRI signals and fiber
orientation distribution functions. The novelty in this approach is that
paired data is used to drive the training of the network where it learns to
ignore features of scanner noise and inter-scanner bias which would otherwise
lead the network to differentiate between the data.

Assessing scan-rescan consistency in DW-MRI studies is a common metric for
validating the reproducibility of a proposed method because it directly tests
the stability of the method under realistic conditions. Variability between
multiple scans of the same subject could arise from a number of sources such
as slight differences in patient positioning, physiological changes in the
patient, or even minor fluctuations in scanner performance. A method that
produces consistent results across repeated scans is likely to be more
reliable and robust, enhancing its potential for clinical application. We
followed this approach to ensure the modeling reproducibility in our study.

However, the 3D context information was largely ignored in the “voxel-based”
MLP model (i.e., model each voxel independently without considering the nearby
voxels). It might lead to inferior performance for the 3D DW-MRI signals. To
the best of our knowledge, this is the first data-driven approach to
explicitly model the 3D patch-wise scan-rescan reproducibility for fODF
estimation.

In this paper, we propose a novel deep constrained spherical deconvolution
method to explicitly reduce the scan-rescan variabilities, so as to model a
more reproducible and robust brain microstructure from repeated DW-MRI scans.
Different from the voxel-wise learning in Nath et al. [13], we introduce a new
volumetric patch-based modeling method for 3D DW-MRI signals. Briefly, the
3×3×3 3D patches from spherical harmonics (SH)-represented DW-MRI signals are
employed for a single shell microstructure estimation. The deep convolutional
neural network (CNN) is deployed as the computational model in our approach to
derive the coefficients.

Another innovation is that we add intra-subject data augmentation in order to
alleviate the impacts of a smaller number of diffusion directions on both
reproducibility and the accuracy of metrics derived from CSD. The scan/rescan
data are employed to facilitate our new loss function in reducing the intra-
subject variability. The method has been trained, validated, and tested on
both the HCP young adults (HCP-ya) test-retest group [14] and the MASiVar
dataset [15]. To assess the model’s generalizability, we applied our model to
the Baltimore Longitudinal Study of Aging (BLSA) dataset as the external
validation [16]. With both direct deployment and further finetuning, we
witnessed an increasing consistency between intra-subject scans. Additionally,
the brain structural connectomes are computed from the deep CSD as the
downstream task for further model evaluation.

Our contribution is four-fold:

  * •

We propose a novel deep constrained spherical deconvolution method to
explicitly reduce the scan-rescan variabilities, so as to model a more
reproducible and robust brain microstructure from repeated DW-MRI scans. To
the best of our knowledge, this is the first data-driven approach to
explicitly model the 3D patch wise scan-rescan reproducibility for fODF
estimation.

  * •

Different from the previous voxel-wise learning studies, we introduce a new 3D
volumetric representation of DW-MRI signals for a single-shell microstructure
estimation.

  * •

We propose a new intra-subject augmentation strategy that increases model
robustness under the “fewer diffusion directions” scenarios.

  * •

The proposed method is a “plug-and-play” design as a simple multi-layer
regression network, which can be easily aggregated with downstream
connectivity analysis.

Figure 1: Overview of the proposed method. The constrained spherical
deconvolution (CSD) is affected by measurement factors of DW-MRI signals
(e.g., hardware, reconstruction algorithms, acquisition parameters). The right
panel shows the inter-site variability during modeling, even for the data that
are collected from the same patient.

##  2 Related work

Ensuring reproducibility has been an important research topic in MR imaging,
specifically in diffusion imaging [17, 18]. To control the site effect in
multi-site imaging data, several strategies have been proposed. Such
approaches can be summarized into two major categories: conventional
statistics-based method methods [19, 20, 17, 21, 22, 23] and more recent
machine learning based methods [24, 25, 26].

###  2.1 Statistics-based Methods

Conventional statistical methods are usually applied in a linear regression
manner on univariate metrics with sites indexed as a categorical covariate,
such as the least squares-based general linear model [19] and Bayesian
estimation-based ComBat [20, 17]. These methods have been utilized in
multisite imaging studies and have shown a powerful capacity for removing
linear site effects in brain metrics [21, 22, 23]. However, noticeable
limitations have been observed for this type of method. First, the site effect
is mathematically assumed to be linear, while the actual effect can be
fundamentally more complicated. Second, brain characteristics are considered
independently in these models, largely neglecting the spatial and topological
relationships among brain regions.

####  2.1.1 Machine Learning based Approaches

Recently proposed DL-based harmonization methods, including U-Net [24], cycle
generative adversarial network [27], or three-dimensional convolutional neural
network [28], allow for mapping the complex abstract representations of the
nonlinear spatial pattern of the site effects. These models have been
primarily applied to ensure reproducibility of diffusion images [29],
structural images [25], and morphological measurements [26], successfully
eliminating the site effect with complex spatial or topological information.
The model training strategy of site pairing is a common approach for DL-based
methods. Briefly, the fusion of data from multiple sites from a single model
greatly increases the generalizability of deep learning models. Most methods
are ensuring test-retest reliability on the signal level.

As for tractography, most DL-based methods are dealing with the modeling
process [30, 31, 32]. By regarding different data as the gold standard, these
approaches are chasing perfect fitting with different ground truth (GT) data,
but the feasibility/confidence of the GT remains risky. For instance, Nath et
al [13] used monkey histology data as GT for fiber orientation distributions.
Such data is hard to obtain for a large human cohort, risking the
generalizability of the trained model. Sedlar et al. [33] regarded the human
data from HCP as GT, which has over three shells with b-values of 1000, 2000,
and 3000 $s/mm^{2}$ (each with 90 gradient directions). However, such
acquisition settings are not common in clinical settings. Additionally, even
the same subject collected at two different settings may have some degrees
apart. As a result, they might have very similar diffusion signals but
fundamentally different fODFs. In our study, we proposed a robust fitting with
GT while constraining the scan-rescan variabilities through modeling.

##  3 Method

###  3.1 Data Representation

Spherical Harmonics (SH) are functions defined on the sphere. A collection of
SH can be used as a basis function to represent and reconstruct any function
on the surface of a unit sphere [34]. All diffusion signals are transformed to
SH basis signal ODF as a unified input for deep learning models, using
spherical harmonics with the ’tournier07’ basis [35]. For the spherical
harmonics coefficients $c^{m}_{k}$, $k$ is the order, $m$ is the degree. For a
given value of $k$, there are $2m+1$ independent solutions of this form, one
for each integer $m$ with $-k\leq m\leq k$. In practice, a maximum order $L$
is used to truncate the SH series. By only taking into account even-order SH
functions, the above bases can be used to reconstruct symmetric spherical
functions.

###  3.2 Architecture

Inspired by Nath et al. [13, 32], we employ a 3D CNN with a residual block and
utilize $3\times 3\times 3$ cubic patches as inputs. The rationale is that 3D
patches might provide more complete spatial information for deep learning
networks (Fig. 2) compared with modeling each individual voxel independently.
$8^{th}$ order SH is used as data representation in our study. Briefly, the
input size of the network is $3\times 3\times 3$ with 45 channels, while the
outputs are 45 $8^{th}$ order spherical harmonics coefficients. During
training, the architecture takes 3 patches as input, where the first comes
from one subject, the network learns the direct mapping between DW-MRI signals
and fiber orientation distribution functions of the center voxel. The other
two are paired patches extracted from scan/rescan DW-MRI of another subject
and the network learns to minimize the difference of the fiber orientation
distribution functions of the center voxel. For validation of the proposed
method, it is in terms of accuracy relative to withheld voxels and
reproducibility with paired voxels in scan-rescan imaging.

In this network, three subsequent convolutional layers serve as the critical
components of the CNN with 3D convolutional filters. One residual block is
included to allow for the direct transmission of information from input and
the third convolutional layer, thereby enabling the effective training of deep
learning networks. The convolutional filters are then flattened and connected
to two dense layers for predicting fODF at the center voxel locations. All
layers have ReLU as the activation function. To enhance model training
efficiency and stability, we employ batch normalization (BN) in our deep
learning architecture to standardize the inputs to each layer of the network,
reducing internal covariate shifts.

Figure 2: Deep learning architecture. A 3D patch-wise convolutional neural
network is proposed to fit the fiber orientation distribution function (fODF)
from the spherical harmonics (SH) using $3\times 3\times 3$ cubic DW-MRI
signals. A contrastive loss is introduced to reduce the intra-subject
variability.

###  3.3 Loss Function

We introduce a customized loss function, as shown in Eqs. 1, 2 and 3. The
first term is the MSE loss between the network’s FOD prediction and the ground
truth with the hyper-parameter ‘$\alpha$’. N is the number of samples, $m$ is
the order of the spherical harmonic (SH) basis and $c^{m}_{k}$ is the SH
coefficients. The second term is the MSE loss between a corresponding/paired
of voxels ($u$ and $v$). The second term has an expectation to be 0 and the
hyper-parameter is ‘$\beta$’. Specifically, if no scan/rescan data participate
during training, ‘$\beta$’ is set to 0.

$\begin{split}loss1=\frac{1}{N}\sum_{i=1}^{N}\sum_{k=0}^{L}\sum_{m=-k}^{k}((c^{m}_{k})_{true,i}-(c^{m}_{k})_{pred,i})^{2}\\\
\end{split}$ (1)

$\begin{split}loss2=\frac{1}{N}\sum_{i=1}^{N}\sum_{k=0}^{L}\sum_{m=-k}^{k}((c^{m}_{k})_{u,i}-(c^{m}_{k})_{v,i})^{2}\\\
\end{split}$ (2)

$\begin{split}Loss=\alpha*loss1+\beta*loss2\\\ \end{split}$ (3)

###  3.4 Intra-subject data augmentation

In order to provide a robust microstructure estimation, we introduce intra-
subject data augmentation during our network training. By performing random
diffusion directions dropout and feeding the model with augmented data during
training, the model learns to handle situations like missing or corrupt
diffusion directions due to factors like patient movement or hardware
malfunction and make accurate predictions even with incomplete data.
Furthermore, this augmentation strategy can improve the model’s
generalizability by exposing it to a wider range of data scenarios, thus
enabling it to better handle the variability and complexity inherent in real-
world diffusion data. An additional b-vector check is performed to ensure the
rest directions are still well distributed on a sphere. Thus, we have
reconstruction results from different total numbers of diffusion directions
from the same DW-MRI signal. The CSD methods are sensitive to the number of
diffusion directions and therefore the generated fODF are augmented. By
applying this augmentation, the diffusion signal ODF generated from fewer
diffusion directions is labeled by the CSD results with the full numbers of
gradient directions during the training process.

##  4 Experiment

The experiments can be summarized as with/without scan/rescan data, and
with/without intra-subject augmentation on two deep learning models (voxel-
wise MLP presented by Nath et al. [13] and ours in Fig. 2). We assessed the
models, as well as benchmarks, using the overall mean ACC on white matter
voxels between the prediction and the ground truth on the HCP and MASiVar
dataset. The BLSA dataset has been introduced to further test model
reproducibility.

We conducted studies to examine the robustness of the model with ‘fewer
gradient directions.’ The results are quantitatively evaluated with single
shell single tissue CSD (ssst-CSD). Full direction ssst-CSD is regarded as the
silver standard.

Eventually, complex network measures of brain structural connectomes
(modularity, average betweenness centrality, characteristic path length, and
global efficiency) are computed as an example of the downstream task
evaluation [36].

###  4.1 Data & Data process

For the HCP-ya dataset [14], 45 subjects with the scan-rescan acquisition were
used (a total of 90 images). A T1 volume of the same subject was used for WM
segmentation using SLANT [37]. HCP was distortion corrected with topup and
eddy [38, 39]. The acquisitions at b-value of 1000 and 2000 $s/mm^{2}$ (each
shell with 90 diffusion directions) were extracted for the study. 30 subjects
were used as training data, 10 were used for validation, and 5 were used for
testing.

For the MASiVar dataset [15], five subjects were acquired on three different
sites referred to as ‘A’, ‘B’ and ’C’. Structural T1 was acquired for all
subjects at all sites. All in-vivo acquisitions were pre-processed with the
PreQual pipeline [40] and then registered pairwise per subject. The
acquisitions at b-value of 1000 and 2000 $s/mm^{2}$ (each shell with 96
diffusion directions) were extracted for the study. Two subjects from sets ’A’
and ‘B’ were used as paired training data. One subject was used for
validation, while two subjects were used for testing. Scans from site ‘C’ were
only used for evaluation.

Twenty-four subjects with the scan-rescan acquisition from The Baltimore
Longitudinal Study of Aging (BLSA) [16] were used as an additional model
evaluation cohort. The BLSA dataset was acquired at a b-value of 700
$s/mm^{2}$ using a Philips 3T scanner. The data were pre-processed with Pre-
Qual. 18 subjects were used for training, 2 subjects were for validation, and
4 subjects were for testing. Additionally, another cohort of 198 subjects with
biological information (Age, Sex, and $\epsilon 4$ allele of apolipoprotein E
(APOE) states) was used for analyzing connectomics. Between them, 46 subjects
were APOE positive while 152 subjects were APOE negative. They were pre-
processed with the same protocol.

###  4.2 Model-based methods

We processed all the data to generate fODF with single shell single tissue CSD
(ssst-CSD) using the DIPY library (version 1.50) with its default setting
[34]. Reconstructions from full diffusion direction DMRI were regarded as
ground truth. Voxel-wise agreement (metrics see 4.6) between paired data was
calculated as silver standard.

###  4.3 Machine learning-based methods

HCP and MASiVar datasets were employed to evaluate the learning-based method.
The image data at a b-value of 2000 $s/mm^{2}$ were used. We use a voxel-wise
neural network as our deep learning baseline. The network consists of four
fully connected layers. The number of neurons per layer is 400, 45, 200, and
45. The input is the $1\times 45$ vector of the SH basis signal ODF, and the
output is the $1\times 45$ vector of the SH basis Fiber ODF. The architecture
in Fig. 2 was used to evaluate the proposed patch-wise deep learning
experiment.

Furthermore, the generalizability of different approaches was assessed by a
rigorous external validation using BLSA dataset. The BLSA dataset was acquired
at a b-value of 700 $s/mm^{2}$. We used the image data at b-value of 1000
$s/mm^{2}$ from both HCP and MASiVar to train the deep learning model using
the same approaches in the previous experiments. Briefly, we trained the deep
learning model with only BLSA data as baseline performance. To further test
our model capacity, we further assess the scenarios of finetuning the last two
linear layers, beyond applying our trained model on BLSA. Thus, we have three
general approaches to compare the performance. Moreover, with/without
scan/rescan data, and with/without intra-subject augmentation on two deep
learning models (voxel-wise MLP presented by Nath et al.[13] and our model in
Fig. 2) are included as additional evaluation.

###  4.4 Ablation study

In the ablation study, we evaluated the intra-subject consistency on all white
matter voxels with a different number of diffusion directions. The deep
learning model which had the best performance on the validation set is chosen
for comparison. Different reconstruction results from a different number of
diffusion directions were visualized as the qualitative results, while their
agreements with the silver standard were assessed as the quantitative results.

Additionally, we evaluated the effectiveness of the random dropout of
diffusion directions using the MASiVar dataset. A subset of 45 diffusion
directions from the same shell was first determined. A number of 45 was the
basic requirement for $8^{t}h$ order spherical harmonics and then visualized.
Then the dropout (drops from 96 to the subset) was performed randomly 10
times. With both results of CSD reconstruction and DL prediction on the DW-MRI
signal, the ACC and the MD (0th order SH of the fODF) were computed. The
spheres in both single fiber and crossing fiber areas were presented to assess
the effectiveness of the intra-subject augmentation by examining shapeshift.

###  4.5 Downstream task evaluation

To evaluate the performance on downstream tasks, we predicted APOE status
using different brain structure connectome maps via different fODF modeling
methods. Complex network measures of brain structural connectomes (modularity,
average betweenness centrality, characteristic path length, and global
efficiency) were computed from the results as the downstream task [36]. During
the process, fODFs were computed from both CSD and our DL networks. We used
the MRTrix (version-3.0.3) default probabilistic tracking algorithm of second-
order integration over fiber orientation distributions (FODs) for tractography
[41]. We generated 10 million streamlines to build each tractogram, seeding
and termination using the five-tissue-type mask. We allowed backtracking.
After, we converted the tractography to a connectome with the Desikan-Killany
atlas [42] with 84 cortical parcellations from Freesurfer [43]. Graph theory
measures were computed with the Brain Connectivity Toolbox (BCT,
version-2019-03-03) [36].

Modularity was the degree to which the network may be subdivided into clearly
delineated and nonoverlapping groups [36]. Betweenness centrality was the
fraction of the shortest paths in the network that contained a given node
[36]. Average betweenness centrality was the average fraction of shortest
paths that nodes in a network participate in [36]. The characteristic path
length was the average shortest path between nodes in millimeters. Global
efficiency was the average inverse shortest path length [36]. Using the graph
measures and biological information (age, sex) as input, we used a three-layer
MLP network to perform classification across different APOE groups between CSD
and the deep learning approaches.

We performed leave-one-out cross-validation as the evaluation strategy.
Weighted cross-entropy was applied as we had unbalanced groups. The cross-
validation had been performed 20 times, and the 95

###  4.6 Evaluation metrics

To compare the predictions of the proposed deep learning methods, we used
angular correlation coefficient (ACC, Eq. 4) to evaluate the similarity of the
prediction when compared with the ground truth estimate of CSD. ACC was a
generalized measure for all fiber population scenarios. It assessed the
correlation of all directions over a spherical harmonic expansion. It is
calculated between fODF of two voxels ($u$ and $v$), where $u_{km}$ and
$v_{km}$ are the SH coefficients. In brief, it provided an estimate of how
closely a pair of fODF’s were related on a scale of -1 to 1, where 1 was the
best measure. Here ‘u’ and ‘v’ represented sets of SH coefficients.

$\begin{split}ACC=\frac{\sum_{k=1}^{L}\sum_{m=-k}^{k}(u_{km})(v^{*}_{km})}{[\sum_{k=1}^{L}\sum_{m=-k}^{k}|u_{km}|^{2}]^{0.5}\cdot[\sum_{k=1}^{L}\sum_{m=-k}^{k}|v_{km}|^{2}]^{0.5}}\end{split}$
(4)

For downstream task evaluation, we used the classical classification metrics
(Accuracy, precision, recall, and F1) to evaluate the biomarker prediction.
Macro precision, recall, and F1 were used in our study as we had unbalanced
positive/negative APOE groups.

##  5 Results

###  5.1 Deep learning results

The results of the fODF estimation are presented in Table 1. The mean ACC over
white matter regions are shown in the table to evaluate the similarity of the
prediction when compared with the truth estimate of CSD. This serves as the
accuracy of prediction. The mean ACC over white matter regions are calculated
in the test cohort of scan/rescan imaging, shown as scan/rescan consistency in
the table. This serves as validation of reproducibility.

The implementation of the CNN network for 3D-patch inputs led to a superior
spherical harmonics coefficients estimation by incorporating more information
from neighboring voxels. Meanwhile, by introducing the identity loss with
scan/rescan data, the proposed method achieved a higher consistency while
maintaining higher angular correlation coefficients with CSD.

Table 1: Performance of fODF prediction on HCP&MASiVar

model & Method | scan/rescan |  intra-subject augmentation |  ACC, compare with full direction CSD |  scan/rescan consistency
---|---|---|---|---
CSD (silver standard) | N/A | N/A | 1 | 0.826
voxel-wise |  |  | 0.942 | 0.830 ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
✓ |  | 0.938 | 0.878 ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
✓ | ✓ | 0.939 | 0.882 ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
patch-wise |  |  | 0.949 | 0.834 ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
✓ |  | 0.954 | 0.886 ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
✓ | ✓ | 0.953 |  0.891 ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$

*Mean ACC are calculated over white matter voxels.

Table 1: Mean ACC over white matter region voxels between 1) prediction and
ground truth signals, 2) prediction of paired scan/rescan DWI signals are
calculated. The Wilcoxon signed-rank test is applied to voxel-wise ACC as a
statistical assessment between the derived results with silver standard CSD.
The statistical difference ($p<0.001$) compared with scan/rescan consistency
from CSD is marked as
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$.

###  5.2 Model evaluation on unseen dataset

By applying our method on BLSA dataset, as shown in Table 5.2. It shows a
great improvement in scan/rescan consistency while applying our approaches and
using BLSA as finetuning data as compared with both the silver standard and
the model trained directly on the BLSA dataset (0.838 vs. 0.834 vs. 0.635).
More importantly, by directly applying the model to unseen data, we still show
significant intra-subject consistency and maintain high agreement (0.836 vs.
0.872) with the full direction CSD.

Table 2: Performance on BLSA dataset

ccccccc model & method training data test data scan/rescan  intra-subject
augmentation ACC, compare with full direction CSD scan/rescan consistency
CSD (silver standard) N/A BLSA N/A N/A 1 (upper bound) 0.635
voxel-wise DL (Nath et al. 2019) HCP, MASiVar  BLSA  0.845 0.747
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
HCP, MASiVar BLSA ✓ 0.832 0.813
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
Patch-wise DL HCP, MASiVar  BLSA  0.829  0.763
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
HCP, MASiVar BLSA ✓ 0.834 0.812
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
HCP, MASiVar BLSA  ✓ ✓0.836 0.824
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
BLSA BLSA ✓ 0.872 0.834
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
HCP, MASiVar, BLSA (finetune) BLSA ✓ 0.849 0.828
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
HCP, MASiVar, BLSA (finetune) BLSA ✓ ✓0.842 0.838
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$

Table 2: The performance of different methods on the BLSA dataset. Wilcoxon
signed-rank test is applied to compare the derived results with silver
standard CSD. The statistical difference ($p<0.001$) compared with scan/rescan
consistency from CSD is marked as
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$.

###  5.3 Ablation study

In the ablation study (Table 3), we evaluate the intra-subject augmentation by
comparing the intra-subject consistency on all white matter voxels with
different number of diffusion directions. The deep learning model which has
the best performance on the validation set is chosen for comparison. In Fig.
3, the right side shows a qualitative result of the visualization of the
estimated SH coefficients and the left side shows the comparison with full-
direction CSD. By performing CSD, the test subjects with a mean of 72
diffusion directions can only maintain a mean ACC of 0.848 as compared with
their same acquisition with 96 directions. By adding the intra-subject
augmentation during the training process, both voxel-wise and patch-wise
models have significant improvement, which shows that deep learning reveals
untapped information during the ODF estimation. Fig. 4 shows the result of 1)
estimation on a signal with fewer diffusion directions using a patch-wise DL
model with scan/rescan data and intra-subject augmentation participated during
training and 2) CSD reconstruction result.

We’re evaluating our deep learning model by testing its performance with
degraded signals, which involve artificially reducing input data quality
(diffusion direction dropout). This assessment allows us to understand the
model’s robustness under less ideal conditions by comparing its results with
CSD. Fig. 5 shows how different methods maintain consistency in their result
compared with silver standard and methods’ self-full direction-estimation when
faced with degraded input data.

Fig. 6 and Fig. 7 provide an additional qualitative visualization of the
robust deep learning fitting during diffusion direction dropout. In Fig. 6 we
plot the mean diffusivity and the ACC spatial map where MD indicates the
fitting of the $0^{th}$ order coefficients (No.1 in 45) and the ACC focus on
the rest of the SH coefficients (No.2-45 in 45). In Fig. 7, we focus on two
unique voxels belonging to the single fiber population and crossing fiber
population respectively, the result shows that CSD reconstruction has an
obvious shapeshift in the low-resolution scheme while deep learning with the
data augmentation strategies remains higher consistency, especially in the
visualization of the crossing-fiber voxel.

Table 3: Performance in fewer diffusion direction situation model & method | scan/rescan |  intra-subject augmentation |  intra-subject consistency
---|---|---|---
CSD | N/A | N/A | 0.848 ± 0.189
voxel-wise |  |  | 0.838 ± 0.195${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
✓ |  | 0.849 ± 0.175${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
✓ | ✓ | 0.879 ± 0.138${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
Patch-wise |  |  | 0.842 ± 0.185${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
✓ |  | 0.856 ± 0.173${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$
✓ | ✓ | 0.902 ± 0.128${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$

Table 3: Model/Method degradation is evaluated by assessing the prediction
consistency with fewer diffusion direction signals. Wilcoxon signed-rank test
is applied as a statistical assessment. The statistical difference ($p<0.001$)
compared with full direction reconstruction from CSD is marked as
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bm{\star}}$.
Figure 3: Qualitative results of fODF modeling. Visualizations of fODF of the
proposed deep learning (DL) method and the results from CSD modeling on two
testing subjects in MASiVar. We took the same patch(matched with same color of
the border) from the results in the crossing fiber area. The same voxel is
matched with the same color of the circle. Figure 4: ACC histogram and ACC
spatial map. This figure depicts the histogram of ACC between full diffusion
directions’ reconstruction and fewer directions’ reconstruction while using
the proposed deep learning (DL) method and the results from CSD modeling on
the 2 testing subjects in MASiVar. The ACC spatial maps are the comparison
between (1) the fODFs of reconstruction from CSD with full diffusion
directions and (2) fewer diffusion directions’ CSD and DL estimator on two
testing subjects. Figure 5: Quantitative result of performances of
methods/modeling with diffusion direction dropout. ACC is calculated at
specific intervals - every 5 diffusion directions - such as at 45, 50, 55, and
so on, enabling us to evaluate how the consistency of the model’s/method’s
output was preserved despite the reduction in diffusion gradient directions.
The dropout (drops from 96 to the subset of 45 directions) was performed
randomly 10 times. The mean ACC and the std are calculated and shown in the
line chart. The left panel shows that the deep learning-based method maintains
high consistency than the CSD reconstruction when both are compared with the
silver standard (full-direction CSD) during the diffusion direction dropout.
The right panel presents a similar assessment, except comparing itself using
the full-direction modeling. Figure 6: Qualitative visualization of the
predicted coefficients in fewer diffusion directions scenario. The mean
diffusivity map(MD, $0^{th}$ order of the SH coefficient) and ACC agreement
spatial map (even orders of the SH coefficient without $0^{th}$ order,
compared with the silver standard–full direction CSD) of deep learning results
of the two MASiVar test subjects. The dropout was performed from 96 to the
subset of 45 directions while the visualization was shown at intervals of
every 5 diffusion directions. Figure 7: Qualitative results of synthetic
spheres. Visualization of reconstruction from different numbers of diffusion
directions of voxel from single/crossing fiber population. The dropout was
performed from 96 to the subset of 45 directions while the visualization was
shown at intervals of every 5 diffusion directions. CSD reconstruction of a
voxel in crossing fiber population has an obvious shapeshift in the low-
resolution scheme while deep learning with the data augmentation strategies
remains a robust estimation.

###  5.4 Downstream task evaluation

The APOE states are predicted using different brain structure connectome maps.
This task is employed as a downstream task to evaluate the performance of
different fODF modeling methods. Briefly, we calculated the accuracy,
precision, recall, and F1 for each method via 20 bootstraps. Then, the 95%
confidence intervals are calculated for all metrics to demonstrate the
variability. As shown in Table 4, all lower bounds of deep learning metrics
are higher than the upper bounds of the CSD method.

Table 4: Confidence interval on metrics of biomarker predictions | DL | 95% CI | CSD | 95% CI | $p$-value
---|---|---|---|---|---
Metrics |  Lower bound |  Upper bound |  Lower bound |  Upper bound
Accuracy | 0.568 | [0.560 | 0.577] | 0.526 | [0.517 | 0.535] | $p<0.001$
Precision | 0.520 | [0.513 | 0.528] | 0.463 | [0.456 | 0.472] | $p<0.001$
Recall | 0.525 | [0.515 | 0.534] | 0.450 | [0.438 | 0.461] | $p<0.001$
F1 | 0.511 | [0.502 | 0.520] | 0.441 | [0.432 | 0.480] | $p<0.001$
Table 4: The Mean and the range of the 95% confidence interval for the deep
learning metrics are reported. The $p$-value of the metrics from both methods
are statistically significant.

##  6 Discussion

Ensuring scan-rescan variability is of paramount importance in diffusion
signal modeling, particularly when employing a data-driven approach. Data-
driven methods, which learn directly from the data, inherently rely on the
consistency and reliability of the input data to produce accurate and robust
models. As such, the issue of scan-rescan variability comes into sharp focus.
In the context of diffusion signal modeling, variations between repeated scans
of the same subject can introduce inconsistencies that could significantly
affect the performance of the data-driven models. This is because these models
are sensitive to the statistical properties of the training data. If the input
data are inconsistent due to scan-rescan variability, the learned model may
not generalize well, leading to less accurate predictions. Thus, in our study,
addressing scan-rescan variability is not just a quality control issue, but a
crucial factor that directly impacts the reliability and clinical
applicability of data-driven diffusion signal models. A consistent focus on
minimizing this variability can result in models that provide more accurate,
reliable, and clinically meaningful results.

In our study, we developed a data-driven fODF modeling algorithm to provide
robust microstructure estimation for modeling tractography. The proposed
method (1) learns the mapping from SH basis 3D DW-MRI signal to a fiber ODF,
(2) improves consistency and alleviates the effects that occur between
different scanners with a new loss function, (3) increases model robustness in
the ’fewer diffusion directions’ scenarios, and (4) empowers better a
predicative power in downstream tasks (e.g., APOE states estimation from
connectomes).

The improvement from voxel-vise input to patch-vise input is based on the
assumption that the model is trained on a diverse patch selection, it sees a
wide range of scenarios. And meanwhile, a deep learning model can capture the
complex relationships as a deeper network is able to account for the 3D
context without being unduly influenced by neighboring voxels in some
scenarios. This is the advantage of data-driven method, we do not need to
explicitly ensure that neighboring voxels have similar distribution functions
to central voxel.

However, there are still several limitations in our approaches. First, one key
limitation of our approaches is the cost of computing resources by performing
patch-to-center predictions. We need to generate 27 times storage for the
3x3x3 patches for one single DMRI. Second, to provide more precise
microstructure estimation, we need to target our model to multi-tissue multi-
shell CSD (MSMT-CSD) which provides different fiber response functions to
different tissues. It also leads to a potential future question on how to
encode the b-value information into SH representation during the transform of
multi-shell DWI and signal ODFS.

In general, our study is a step towards the direct harmonization of the
estimated microstructure (FOD) using deep learning and a data-driven scheme
when scan-rescan data are available for training. The proposed method is
potentially applicable to a wider range of data harmonization problems in
neuroimaging.

##  7 Conclusion

In this paper, we propose a novel deep constrained spherical deconvolution
method to explicitly reduce the scan-rescan variabilities, so as to model a
more reproducible and robust brain microstructure of repeated DW-MRI scans
that are acquired from the same patient. From the experimental results, the
proposed data-driven framework outperforms the existing benchmarks in fODF
estimation. In general, our study is a step towards the direct harmonization
of the estimated microstructure (e.g., FOD) using deep learning and a data-
driven scheme when scan-rescan data are available for training. The proposed
method is potentially applicable to a wider range of data harmonization
problems in neuroimaging.

## Disclosures

The authors of the paper have no conflicts of interest to report.

## Acknowledgments

This work was supported by the National Institutes of Health under award
numbers R01EB017230, T32EB001628, and 5T32GM007347, and in part by the
National Center for Research Resources and Grant UL1 RR024975-01. This study
was also supported by National Science Foundation (1452485, 1660816, and
1750213). The content is solely the responsibility of the authors and does not
necessarily represent the official views of the NIH or NSF.

## References

  * [1] P. W. Schaefer, P. E. Grant, and R. G. Gonzalez, “Diffusion-weighted mr imaging of the brain,” Radiology(2), 331–345 (2000).
  * [2] P. Hagmann, J.-P. Thiran, L. Jonasson, et al., “Dti mapping of human brain connectivity: statistical fibre tracking and virtual dissection,” Neuroimage(3), 545–554 (2003).
  * [3] D. S. Tuch, T. G. Reese, M. R. Wiegell, et al., “High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity,” Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine(4), 577–582 (2002).
  * [4] D. Le Bihan, J.-F. Mangin, C. Poupon, et al., “Diffusion tensor imaging: concepts and applications,” Journal of Magnetic Resonance Imaging: An Official Journal of the International Society for Magnetic Resonance in Medicine(4), 534–546 (2001).
  * [5] J.-D. Tournier, C.-H. Yeh, F. Calamante, et al., “Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data,” Neuroimage(2), 617–625 (2008).
  * [6] B. Jeurissen, J.-D. Tournier, T. Dhollander, et al., “Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion mri data,” NeuroImage, 411–426 (2014).
  * [7] I. Aganj, C. Lenglet, G. Sapiro, et al., “Reconstruction of the orientation distribution function in single-and multiple-shell q-ball imaging within constant solid angle,” Magnetic resonance in medicine(2), 554–566 (2010).
  * [8] K. M. Jansons and D. C. Alexander, “Persistent angular structure: new insights from diffusion magnetic resonance imaging data,” Inverse problems(5), 1031 (2003).
  * [9] K. Schilling, V. Janve, Y. Gao, et al., “Comparison of 3d orientation distribution functions measured with confocal microscopy and diffusion mri,” Neuroimage, 185–197 (2016).
  * [10] I. Benou and T. Riklin Raviv, “Deeptract: A probabilistic deep learning framework for white matter fiber tractography,” in International conference on medical image computing and computer-assisted intervention, 626–635, Springer (2019).
  * [11] P. Poulin, D. Jörgens, P.-M. Jodoin, et al., “Tractography and machine learning: Current state and open challenges,” Magnetic resonance imaging, 37–48 (2019).
  * [12] M. Afzali, T. Pieciak, S. Newman, et al., “The sensitivity of diffusion mri to microstructural properties and experimental factors,” Journal of Neuroscience Methods, 108951 (2021).
  * [13] V. Nath, P. Parvathaneni, C. B. Hansen, et al., “Inter-scanner harmonization of high angular resolution dw-mri using null space deep learning,” in International Conference on Medical Image Computing and Computer-Assisted Intervention, 193–201, Springer (2019).
  * [14] D. C. Van Essen, S. M. Smith, D. M. Barch, et al., “The wu-minn human connectome project: an overview,” Neuroimage, 62–79 (2013).
  * [15] L. Y. Cai, Q. Yang, P. Kanakaraj, et al., “Masivar: Multisite, multiscanner, and multisubject acquisitions for studying variability in diffusion weighted mri,” Magnetic resonance in medicine(6), 3304–3320 (2021).
  * [16] L. Ferrucci, “The baltimore longitudinal study of aging (blsa): a 50-year-long journey and plans for the future,” (2008).
  * [17] J.-P. Fortin, D. Parker, B. Tunç, et al., “Harmonization of multi-site diffusion tensor imaging data,” Neuroimage, 149–170 (2017).
  * [18] M. S. Pinto, R. Paolella, T. Billiet, et al., “Harmonization of brain diffusion mri: Concepts and methods,” Frontiers in Neuroscience, 396 (2020).
  * [19] A. Rao, J. M. Monteiro, J. Mourao-Miranda, et al., “Predictive modelling using neuroimaging data in the presence of confounds,” Neuroimage, 23–49 (2017).
  * [20] J.-P. Fortin, N. Cullen, Y. I. Sheline, et al., “Harmonization of cortical thickness measurements across scanners and sites,” Neuroimage, 104–120 (2018).
  * [21] R. Pomponio, G. Erus, M. Habes, et al., “Harmonization of large mri datasets for the analysis of brain imaging patterns throughout the lifespan,” NeuroImage, 116450 (2020).
  * [22] M. Xia, T. Si, X. Sun, et al., “Reproducibility of functional brain alterations in major depressive disorder: Evidence from a multisite resting-state functional mri study with 1,434 individuals,” Neuroimage, 700–714 (2019).
  * [23] M. Yu, K. A. Linn, P. A. Cook, et al., “Statistical harmonization corrects site effects in functional connectivity measurements from multi-site fmri data,” Human brain mapping(11), 4213–4227 (2018).
  * [24] B. E. Dewey, C. Zhao, J. C. Reinhold, et al., “Deepharmony: A deep learning approach to contrast harmonization across scanner changes,” Magnetic resonance imaging, 160–170 (2019).
  * [25] L. Zuo, B. E. Dewey, Y. Liu, et al., “Unsupervised mr harmonization by learning disentangled representations using information bottleneck theory,” NeuroImage, 118569 (2021).
  * [26] F. Zhao, Z. Wu, L. Wang, et al., “Harmonization of infant cortical thickness using surface-to-surface cycle-consistent adversarial networks,” in International Conference on Medical Image Computing and Computer-Assisted Intervention, 475–483, Springer (2019).
  * [27] G. Modanwal, A. Vellal, M. Buda, et al., “Mri image harmonization using cycle-consistent generative adversarial network,” in Medical Imaging 2020: Computer-Aided Diagnosis, 11314, 259–264, SPIE (2020).
  * [28] Q. Tong, T. Gong, H. He, et al., “A deep learning–based method for improving reliability of multicenter diffusion kurtosis imaging with varied acquisition protocols,” Magnetic Resonance Imaging, 31–44 (2020).
  * [29] D. Moyer, G. Ver Steeg, C. M. Tax, et al., “Scanner invariant representations for diffusion mri harmonization,” Magnetic resonance in medicine(4), 2174–2189 (2020).
  * [30] D. Karimi, L. Vasung, C. Jaimes, et al., “Learning to estimate the fiber orientation distribution function from diffusion-weighted mri,” NeuroImage, 118316 (2021).
  * [31] Z. Lin, T. Gong, K. Wang, et al., “Fast learning of fiber orientation distribution function for mr tractography using convolutional neural network,” Medical physics(7), 3101–3116 (2019).
  * [32] V. Nath, S. K. Pathak, K. G. Schilling, et al., “Deep learning estimation of multi-tissue constrained spherical deconvolution with limited single shell dw-mri,” in Medical Imaging 2020: Image Processing, 11313, 162–171, SPIE (2020).
  * [33] S. Sedlar, T. Papadopoulo, R. Deriche, et al., “Diffusion mri fiber orientation distribution function estimation using voxel-wise spherical u-net,” in Computational Diffusion MRI, 95–106, Springer (2021).
  * [34] E. Garyfallidis, M. Brett, B. Amirbekian, et al., “Dipy, a library for the analysis of diffusion mri data,” Frontiers in neuroinformatics, 8 (2014).
  * [35] J.-D. Tournier, R. Smith, D. Raffelt, et al., “Mrtrix3: A fast, flexible and open software framework for medical image processing and visualisation,” Neuroimage, 116137 (2019).
  * [36] M. Rubinov and O. Sporns, “Complex network measures of brain connectivity: uses and interpretations,” Neuroimage(3), 1059–1069 (2010).
  * [37] Y. Huo, Z. Xu, Y. Xiong, et al., “3d whole brain segmentation using spatially localized atlas network tiles,” NeuroImage, 105–119 (2019).
  * [38] J. L. Andersson, S. Skare, and J. Ashburner, “How to correct susceptibility distortions in spin-echo echo-planar images: application to diffusion tensor imaging,” Neuroimage(2), 870–888 (2003).
  * [39] J. L. Andersson and S. N. Sotiropoulos, “An integrated approach to correction for off-resonance effects and subject movement in diffusion mr imaging,” Neuroimage, 1063–1078 (2016).
  * [40] L. Y. Cai, Q. Yang, C. B. Hansen, et al., “Prequal: An automated pipeline for integrated preprocessing and quality assurance of diffusion weighted mri images,” Magnetic resonance in medicine(1), 456–470 (2021).
  * [41] J. D. Tournier, F. Calamante, A. Connelly, et al., “Improved probabilistic streamlines tractography by 2nd order integration over fibre orientation distributions,” in Proceedings of the international society for magnetic resonance in medicine, 1670, John Wiley & Sons, Inc. New Jersey, USA (2010).
  * [42] A. Klein and J. Tourville, “101 labeled brain images and a consistent human cortical labeling protocol,” Frontiers in neuroscience, 171 (2012).
  * [43] B. Fischl, “Freesurfer,” Neuroimage(2), 774–781 (2012).

Tianyuan Yao is currently a PhD student in Computer Science at Vanderbilt
University. He is supervised by Prof. Yuankai Huo at HRLB Lab. He received his
bachelor’s degree from Shandong University in 2019, with major in
Communication Engineering, and MS degree of Computer Science from Vanderbilt
University in 2021. His main research interests are medical image analysis,
deep learning, and computer vision. He is passionate about their applications
in pathology and radiology imaging.

Biographies and photographs of the other authors are not available.

###### List of Figures

  1. 1 Overview of the proposed method. The constrained spherical deconvolution (CSD) is affected by measurement factors of DW-MRI signals (e.g., hardware, reconstruction algorithms, acquisition parameters). The right panel shows the inter-site variability during modeling, even for the data that are collected from the same patient.
  2. 2 Deep learning architecture. A 3D patch-wise convolutional neural network is proposed to fit the fiber orientation distribution function (fODF) from the spherical harmonics (SH) using $3\times 3\times 3$ cubic DW-MRI signals. A contrastive loss is introduced to reduce the intra-subject variability.
  3. 3 Qualitative results of fODF modeling. Visualizations of fODF of the proposed deep learning (DL) method and the results from CSD modeling on two testing subjects in MASiVar. We took the same patch(matched with same color of the border) from the results in the crossing fiber area. The same voxel is matched with the same color of the circle.
  4. 4 ACC histogram and ACC spatial map. This figure depicts the histogram of ACC between full diffusion directions’ reconstruction and fewer directions’ reconstruction while using the proposed deep learning (DL) method and the results from CSD modeling on the 2 testing subjects in MASiVar. The ACC spatial maps are the comparison between (1) the fODFs of reconstruction from CSD with full diffusion directions and (2) fewer diffusion directions’ CSD and DL estimator on two testing subjects.
  5. 5 Quantitative result of performances of methods/modeling with diffusion direction dropout. ACC is calculated at specific intervals - every 5 diffusion directions - such as at 45, 50, 55, and so on, enabling us to evaluate how the consistency of the model’s/method’s output was preserved despite the reduction in diffusion gradient directions. The dropout (drops from 96 to the subset of 45 directions) was performed randomly 10 times. The mean ACC and the std are calculated and shown in the line chart. The left panel shows that the deep learning-based method maintains high consistency than the CSD reconstruction when both are compared with the silver standard (full-direction CSD) during the diffusion direction dropout. The right panel presents a similar assessment, except comparing itself using the full-direction modeling.
  6. 6 Qualitative visualization of the predicted coefficients in fewer diffusion directions scenario. The mean diffusivity map(MD, $0^{th}$ order of the SH coefficient) and ACC agreement spatial map (even orders of the SH coefficient without $0^{th}$ order, compared with the silver standard–full direction CSD) of deep learning results of the two MASiVar test subjects. The dropout was performed from 96 to the subset of 45 directions while the visualization was shown at intervals of every 5 diffusion directions.
  7. 7 Qualitative results of synthetic spheres. Visualization of reconstruction from different numbers of diffusion directions of voxel from single/crossing fiber population. The dropout was performed from 96 to the subset of 45 directions while the visualization was shown at intervals of every 5 diffusion directions. CSD reconstruction of a voxel in crossing fiber population has an obvious shapeshift in the low-resolution scheme while deep learning with the data augmentation strategies remains a robust estimation.

###### List of Tables

  1. 1 Performance of fODF prediction on HCP&MASiVar
  2. 3 Performance in fewer diffusion direction situation
  3. 4 Confidence interval on metrics of biomarker predictions