Registration, atlas generation, and statistical analysis of high angular resolution diffusion imaging based on riemannian structure of orientation distribution functions 6

6
Conclusion and Future Work
Recent advances in diffusion-weighted imaging (DWI), such as High Angular Reso-
lution Diffusion Imaging (HARDI), allows us to model the water diffusion at a voxel
with an orientation distribution function (ODF) that can capture multiple orientation at
a voxel. Major research questions are: how would one analyze the HADRI data and
make the correct inferences from the rich information provided? Whether new insights
into the human brain, in particular white matter, would surface? Before HARDI can be
useful in both diagnosis and clinical applications, an ODF-based computational frame-
work, including registration, atlas generation and regression analysis, etc, is needed for
HARDI-based analysis across populations.
Firstly, we proposed a novel large deformation diffeomorphic registration algorithm
to align HARDI data characterized by ODFs. The proposed algorithm seeks an optimal
diffeomorphism of large deformation between two ODF fields in a spatial volume
domain and at the same time, locally reorients an ODF in a manner such that it remains
consistent with the surrounding anatomical structure. To this end, we first reviewed
the Riemannian manifold of ODFs. We then defined the reorientation of an ODF when
111
6. CONCLUSION AND FUTURE WORK
an affine transformation is applied and subsequently, defined the diffeomorphic group
action to be applied to the ODFs based on this reorientation. We incorporated the
Riemannian metric of ODFs for quantifying the similarity of two HARDI images into a
variational problem defined under the large deformation diffeomorphic metric mapping
(LDDMM) framework. We finally derived the gradient of the cost function in both
Riemannian spaces of diffeomorphisms and the ODFs, and presented its numerical
implementation. Both synthetic and real brain HARDI data were used to illustrate the
performance of our registration algorithm.
We also presented a Bayesian probabilistic model to estimate the HARDI atlas of
the brain white matter. First of all, we assumed that the HARDI atlas is generated
from a known hyperatlas through a flow of diffeomorphisms. A shape prior of the
HARDI atlas can thus be constructed based on LDDMM. LDDMM characterizes a

nonlinear diffeomorphic shape space in a linear space of initial momentum that uniquely
determines diffeomorphic geodesic flows from the hyperatlas. Therefore, the shape
prior of the HARDI atlas can be modeled using a centered Gaussian random field
(GRF) model of the initial momentum. In order to construct the likelihood of observed
HARDI datasets, it is necessary to study the diffeomorphic transformation of individual
observations relative to the atlas and the probabilistic distribution of ODFs. To this end,
we constructed the likelihood related to the transformation using the same construction
as discussed for the shape prior of the atlas. The probabilistic distribution of ODFs
was then constructed based on the ODF Riemannian manifold. We assumed that the
observed ODFs are generated by an exponential map of random tangent vectors at
the deformed atlas ODFs. Hence, the likelihood of the ODFs can be modeled using a
GRF of their tangent vectors in the Riemannian manifold of ODFs. We solved for the
maximum a posteriori using the Expectation-Maximization algorithm and derive the
112
corresponding update equations. Finally, we illustrated the HARDI atlas constructed
based on a Chinese aging cohort of
94
adults and compared it with that generated by
averaging the coefficients of spherical harmonics of the ODFs across subjects.
We further proposed a geodesic regression algorithm on the Riemannian manifold
of ODFs. We derived the algorithm for the geodesic regression of ODFs and conducted
the simulation experiment to evaluate its performance. We then examined the effects
of aging via geodesic regression of ODFs in a large group of healthy men and women,
spanning the adult age range. Results show that the proposed method is able to capture
more regions with age effect on white matter changes as compared to the conventional
regression based on DTI. The evolution of ODF fields along the geodesic regression
line depicts in great detail the changes of white matter including the breakdown of the
myelin sheath with aging and the anterior-posterior gradient of corpus callosum. In
the investigation of the regional aging effects where corpus callosum and corticospinal
tracts come across, the result suggests that the diffusivity in corpus callosum declines

more than in corticospinal tracts in the selected region. To sum up, experiments have
shown that the HARDI-based computational framework offers valuable clues about
the changes of white matter in population studies that were previously undetected with
existing methods.
Future work will aim to find new biomarkers sensitive to white-matter pathologies
related to neuropsychiatric disorders such as Alzheimer’s disease (AD). Together with
the HARDI-based tractography e.g., [
32
], we can quantify in detail the strength of
connections along a specific pathway, which can be useful in the diagnosis and prognosis
of AD. In addition, there were a few methods proposed for the reconstruction of
ensemble average propagator (EAP) from original DWI images recently [e.g.
114
,
115
,
116
]. In order to accurately reconstruct the diffusion signal and EAP, a thorough
113
6. CONCLUSION AND FUTURE WORK
exploration of q-space is needed, which requires multiple b-value diffusion weighted
imaging (mDWI). MDWI can characterize more complex neural fiber geometries when
compared to single b-value techniques like diffusion tensor imaging (DTI) or high
angular resolution diffusion imaging (HARDI). Hybrid diffusion imaging (HYDI) [
117
]
is a mDWI technique that samples the diffusion signal along concentric spherical shells
in q-space, with the number of encoding directions increased with each shell to increase
the angular resolution with the level of diffusion weighting. MDWI techniques like
HYDI, however, have not been widely used by clinicians and neuroscientists partially

due to their relatively long acquisition times. In addition, there is a lack of fundamental
image analysis tools, such as registration, that can fully utilize their information. Our
future work will also include the extension of our framework to HDYI dataset.
114
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