Conference PaperPDF Available

Prediction of the Progression of Subcortical Brain Structures in Alzheimer’s Disease from Baseline


Abstract and Figures

We propose a method to predict the subject-specific longitudinal progression of brain structures extracted from baseline MRI, and evaluate its performance on Alzheimer's disease data. The disease progression is modeled as a trajectory on a group of diffeomorphisms in the context of large deformation diffeomorphic metric mapping (LDDMM). We first exhibit the limited predictive abilities of geodesic regression extrapolation on this group. Building on the recent concept of parallel curves in shape manifolds, we then introduce a second predictive protocol which personalizes previously learned trajectories to new subjects, and investigate the relative performances of two parallel shifting paradigms. This design only requires the baseline imaging data. Finally, coefficients encoding the disease dynamics are obtained from longitudinal cognitive measurements for each subject, and exploited to refine our methodology which is demonstrated to successfully predict the follow-up visits.
Content may be subject to copyright.
Prediction of the progression of subcortical brain
structures in Alzheimer’s disease from baseline
Alexandre one12 , Maxime Louis12 , Alexandre Routier12 , Jorge Samper12,
Michael Bacci12, Benjamin Charlier2 3, Olivier Colliot1 2, Stanley Durrleman1 2,
and the Alzheimer’s Disease Neuroimaging Initiative
1Sorbonne Universit´es, UPMC Universit´e Paris 06, Inserm, CNRS, Institut du
Cerveau et de la Moelle (ICM) opital Piti´e-Salpˆetri`ere, 75013 Paris, France,
2Inria Paris, Aramis project-team, 75013 Paris, France,
3Universit´e de Montpellier, France
Abstract. We propose a method to predict the subject-specific longitu-
dinal progression of brain structures extracted from baseline MRI, and
evaluate its performance on Alzheimer’s disease data. The disease pro-
gression is modeled as a trajectory on a group of diffeomorphisms in the
context of large deformation diffeomorphic metric mapping (LDDMM).
We first exhibit the limited predictive abilities of geodesic regression
extrapolation on this group. Building on the recent concept of parallel
curves in shape manifolds, we then introduce a second predictive protocol
which personalizes previously learned trajectories to new subjects, and
investigate the relative performances of two parallel shifting paradigms.
This design only requires the baseline imaging data. Finally, coefficients
encoding the disease dynamics are obtained from longitudinal cognitive
measurements for each subject, and exploited to refine our methodology
which is demonstrated to successfully predict the follow-up visits.
1 Introduction
The primary pathological developments of a neurodegenerative disease such as
Alzheimer’s are believed to spring long before the first symptoms of cognitive
decline. Subtle gradual structural alterations of the brain arise and develop along
the disease course, in particular in the hippocampi regions, whose volumes are
classical biomarkers in clinical trials. Among other factors, those transformations
ultimately result in the decline of cognitive functions, which can be assessed
through standardized tests. Being able to track and predict future structural
changes in the brain is therefore key to estimate the individual stage of disease
progression, to select patients and provide endpoints in clinical trials.
To this end, our work settles down to predict the future shape of brain
structures segmented from MRIs. We propose a methodology based on three
building blocks : extrapolate from the past of a subject ; transfer the progression
of a reference subject observed over a longer time period to new subjects ;
Equal contributions.
and refine this transfer with information about the relative disease dynamics
extracted from cognitive evaluations. Instead of limiting ourselves to specific
features such as volumes, we propose to see each observation of a patient at a
given time-point as an image or a segmented surface mesh in a shape space.
In computational anatomy, shape spaces are usually defined via the action of
a group of diffeomorphisms [1,16, 17]. In this framework, one may estimate a flow
of diffeomorphisms such that a shape continuously deformed by this flow best fits
repeated observations of the same subject over time, thus leading to a subject-
specific spatiotemporal trajectory of shape changes [8,12]. If the flow is geodesic
in the sense of a shortest path in the group of diffeomorphisms, this problem is
called geodesic regression [4, 5, 8, 12] and may be thought of as the extension to
Riemannian manifolds of the linear regression concept. It is tempting then to
use such regression to infer the future evolution of the shape given several past
observations. To the best of our knowledge, the predictive power of such a method
has not yet been extensively assessed. We will demonstrate that satisfying results
can only be obtained when large numbers of data points over extensive periods of
time are available, and that poor ones should be expected in the more interesting
use-case scenario of a couple of observations.
In such situations, an appealing workaround would be to transfer previ-
ously acquired knowledge from another patient observed over a longer period of
time. This idea requires the definition of a spatiotemporal matching method to
transport the trajectory of shape changes into a different subject space. Several
techniques have been proposed to register image time series of different sub-
jects [11,18]. They often require time series to have the same number of images,
or to have correspondences between images across time series, and are therefore
unfit for prognosis purposes. Parallel transport in groups of diffeomorphisms has
been recently introduced to infer deformation of follow-up images from baseline
matching [10,15]. Such paradigms have been used mostly to transport spatiotem-
poral trajectories to the same anatomical space for hypothesis testing [6,13]. Two
main methodologies have emerged: either by parallel-transporting the time se-
ries along the baseline matching as in [5], or by parallel-transporting the baseline
matching along the time series as in [14]. We evaluate both in this paper.
In any case, these approaches require to match the baseline shape with one
in the reference time series. Ideally, we should match observations corresponding
to the same disease stage, which is unknown. We propose to complement such
approaches with estimates of the patient stage and pace of progression using
repeated neuropsychological assessments in the spirit of [14]. These estimates
are used to adjust the dynamics of shape changes of the reference subject to the
test one, according to the dynamical differences observed in the cognitive tests.
Among the main contributions of this papers are : the first quantitative
study of the predictive power of geodesic regression ; a new methodology for the
prediction of shape progression from baseline ; the evaluation of its accuracy for
two different parallel shifting protocols ; new evidence of the utter importance
of capturing the individual dynamics in Alzheimer’s disease models.
Section 2 sets the theoretical background and incrementally describes our
methodology. Section 3 presents and discusses the resulting performances.
2 Method
Let (yj)j=1,..,nibe a time series of segmented surface meshes for a given subject
i {1, ..., N }, obtained at the ages (tj)j=1,..,ni. We build a group of diffeomor-
phisms of the ambient space which act on the segmented meshes, following the
procedure described in [3]. Flows of diffeomorphisms of R3are generated by in-
tegrating time-varying vector fields of the form v(t, x) = Pncp
k=1 K[x, ck(t)]βk(t)
where Kis a Gaussian kernel, c(t) = [ck(t)]k=1,..,ncp and β(t) = [βk(t)]k=1,..,ncp
are respectively the control points and the momenta of the deformation.
We endow the space of diffeomorphisms with a norm which measures the cost
of the deformation. In the following, we only consider geodesic flows of diffeomor-
phisms i.e. flows of minimal norm connecting the identity to a given diffeomor-
phism. Such flows are uniquely parametrized by their initial control points and
momenta c0=c(0), β0=β(0). Under the action of the flow of diffeomorphisms,
an initial template shape Tis continuously deformed and describes a trajectory
in the shape space, which we will note tγ(c00)(T, t). Simultaneously, we
endow the surface meshes with a varifold norm k · k which allows to measure a
data attachment term between meshes without point correspondence [3].
2.1 Geodesic regression
In the spirit of linear regression, one can perform geodesic regression in the shape
space by estimating the intercept Tand the slope (c0, β0) such that γ(c00)(T , ·)
minimizes the following functional :
kγ(c00)(T , tj)yjk2+R(c0, β0) (1)
where Ris a regularization term which penalizes the kinetic energy of the defor-
mation. We estimate a solution of equation (1) with a Nesterov gradient descent
as implemented in the software Deformetrica (, where
the gradient with respect to the control points, the momenta and the template
is computed with a backward integration of the data attachement term along
the geodesic [2].
Once an optimum is found, we obtain a description of the progression of
the brain structures which lies in the tangent space at the identity of the group
of diffeomorphisms. It is natural to attempt to extrapolate from the obtained
geodesic to obtain a prediction of the progression of the structures.
2.2 Two methods to transport spatiotemporal tra jectories of shapes
As it will be demonstrated in section 3, geodesic regression extrapolation pro-
duces an accurate prediction only if data over a long time span is available for
the subject, which is not compatible with the goal of early prognosis.
As proposed in [10,19], given a reference geodesic, we use the Riemannian par-
allel transport to generate a new trajectory. We first perform a baseline matching
between the reference subject and the new subject, which can be described as
a vector in the tangent space of the group of diffeomorphisms. Two paradigms
are available to obtain a parallel trajectory. [15] advises to transport the ref-
erence regression along the matching and then shoot. In the shape space, this
generates a geodesic starting at the baseline shape ; for this reason, we call
this solution geodesic parallelization, and is illustrated on Figure (A1). On the
other hand, [14] advocates to transport the matching vector along the reference
geodesic and then build a trajectory with this transported vector from every
point of the reference geodesic, as described on Figure (B1). We will call this
procedure exp-parallelization.
In such a high-dimensional setting, the computation of parallel transport
classically relies on the Schild’s ladder scheme [9]. However, in our case the com-
putation of the Riemannian logarithm may only be computed by solving a shape
matching problem, resulting not only in an computationally expensive algorithm
but also in an uncontrolled approximation of the scheme. To implement these
parallel shifting methods, we use the algorithm suggested in [19], which relies
on an approximation of the transport to nearby points by a well-chosen Jacobi
field, with a sharp control on the computational complexity. The same rate of
convergence as Schild’s ladder is obtained at a reduced cost.
2.3 Cognitive scores dynamics
The protocol described in the previous section has two main drawbacks. First,
the choice of the matching time in the reference trajectory is arbitrary : the base-
line is purely a convenience choice and ideally the matching should be performed
at similar stages of the disease. Second, it does not take into account the pace of
progression of the subject. In [14], the authors propose a statistical model allow-
ing to learn, in an unsupervised manner, dynamical parameters of the subjects
from ADAS-cog test results, a standardized cognitive test designed for disease
progression tracking. More specifically, they suppose that each patient follows a
parallel to a mean trajectory, with a time reparametrization :
ψ(t) = α(tt0τ) + t0(2)
which maps the subject time to a normalized time frame, where α > 0 and τ
are scalar parameters. A high (resp. low) αhence corresponds to a fast (resp.
slow) progression of the scores, when a negative (resp. positive) τcorresponds
to an early decay (resp. late decay) of those scores. In the dataset introduced
below, the acceleration factors (αi)irange from 0.15 to 6.01 and the time-shifts
(τi)ifrom 20.6 to 22.8, thus showing a tremendous variability in the individual
dynamics of the disease, which must be taken into account.
With these dynamic parameters, the shape evolution can be adjusted by
reparametrizing the parallel trajectory with the same formula (2), as illustrated
on Figures (A2) and (B2).
(A1) Geodesic parallelization. Blue ar-
row: baseline matching. Red arrows:
transported regression. Black dotted
line : exponentiation of the transported
(A2) Reparametrized geodesic paral-
lelization. Matching time and exp-
parallel trajectory are reparametrized.
(B1) Exp-parallelization. Red arrow:
geodesic regression. Blue arrows: trans-
ported baseline matching. Black dotted
line : exp-parallelization of the reference
geodesic for the given subject.
(B2) Reparametrized exp-
parallelization. Matching time and exp-
parallel trajectory are reparametrized.
3 Results
3.1 Data, preprocessing, parameters and performance metric
MRIs are extracted from the ADNI database, where only MCI converters with 7
visits or more are kept, for a total of N= 74 subjects and 634 visits. Subjects are
observed for a period of time ranging from 4 to 9 years (5.9 on average), with 12
visits at most. The 634 MRIs are segmented using the FreeSurfer software. The
extracted brain masks are then affinely registered towards the Colin 27 Average
Brain using the FSL software. The estimated transformations are finally applied
to the pairs of caudates, hippocampi and putamina subcortical structures.
All diffeomorphic operations i.e. matching, geodesic regression estimation,
shooting, exp-parallelization and geodesic parallelization are performed thanks
to the Deformetrica software previously mentioned. A varifold distance with
Gaussian kernel width of 3 mm for each structure and a deformation kernel
width of 5 mm are chosen. The time discretization resolution is set to 2 months.
The chosen performance metric between two sets of meshes is the Dice coef-
ficient, that is the sum of the volumes of the intersections of the corresponding
meshes, divided by the total sum of the volumes. We only measure the vol-
ume of the intersection between corresponding structures. The Dice coefficient
is comprised between 0 and 1 : it equals 1 for a perfect match, and 0 for disjoint
3.2 Geodesic regression extrapolation
The acceleration factor αin equation (2) encodes the rate of progression of each
patient. Multiplying this coefficient with the actual observation window gives a
notion of the absolute observation window length, in the disease time referential.
Only the 22 first subjects according to this measure have been considered for
this section : they are indeed expected to feature large structural alterations,
making the geodesic regression procedure more accurate. The geodesic regression
predictive performance is compared to a naive one consisting in leaving the last
observed brain structures in the learning dataset unchanged.
Table 1 presents the results obtained for varying learning dataset and extrap-
olation extents. We perform a Mann-Whitney test with the null hypothesis that
the observed Dice coefficients distributions are the same to obtain the statistical
significance levels. The extrapolated meshes are satisfying only in the case where
all but one data points are used to perform the geodesic regression, achieving
a high Dice index and outperforming the naive one, by a small margin though
and failing to reach the significance level (p= 0.25). When the window of obser-
vation becomes narrower, the prediction accuracy decreases and becomes worse
than the naive one. Indeed, the lack of robustness of the although standard
segmentation pipeline imposes a high noise level, which seems to translate into
a too low signal-to-noise ratio after extrapolation from only a few observations.
Predicted follow-up visit
Method M12 M24 M36 M48 M72 M96
N=22 N=21 N=19 N=18 N=16 N=5
6[reg] .878
12 [reg] - .839
[naive] -
18 [reg] - .885
[naive] -
24 [reg] - - .864
[naive] - -
max - 1 [reg] .807
Prediction at the most remote possible time
60 months [naive] point (76 months) for all subjects (N=22).
Table 1: Averaged Dice performance measures between predictions and obser-
vations for varying extents of learning datasets and extrapolation. The [reg] tag
indicates the regression-based prediction, and [naive] the naive one. Each row
corresponds to an increasingly large learning dataset, patients being observed
for widening periods of time. Each column corresponds to an increasingly re-
mote predicted visit from baseline. Significance levels [.05, .01, .001, .0001] for
the Mann-Whitney test.
Fig. 1: Extrapolated geodesic regression for the subject s0671. Are only repre-
sented the right hippocampus, caudate and putamen brain structures in each
subfigure. The three first rows present the interpolated brain structures, cor-
responding to ages 61.2, 64.2 and 67.2 (years). The last row presents the ex-
trapolation result at age 70.2. On the right column are added the target brain
structures (red wireframes), segmented from the original images.
Figure 1 displays an extrapolated geodesic regression for a specific subject,
with a large learning period of 72 months, and a prediction at 108 months from
the baseline (Dice performance of 0.74 versus 0.65 with the naive approach).
3.3 Non reparametrized transport
Among the 22 subjects whose regression-based predictive power has been evalu-
ated in the previous section, the two which performed best are chosen as refer-
ences for the rest of this paper. Their progressions are transported onto the 73
other subjects with the two different parallel shifting methods.
In more details, for each pair of reference and target subjects, the baseline
target shape is first registered to the reference baseline. The reference geodesic
regression is then either geodesically or exp-parallelized. Prediction performance
is finally assessed : the Dice index between the prediction and the actual ob-
servation, for the two modes of transport, are computed and compared to the
Dice index between the baseline meshes and the actual observation the only
available information in the absence of a predictive paradigm.
The upper part of Table 2 presents the results. In most cases, the obtained
meshes by the proposed protocol are of lesser quality than the reference ones,
according to the Dice performance metric. The two methods of transport are
essentially similarly predictive, although geodesic parallelization slightly outper-
forms the exp-parallelization for the M12 prediction.
Time Method Predicted follow-up visit
reparam. M12 M24 M36 M48 M72 M96
N=144 N=138 N=130 N=129 N=76 N=11
Without [exp] .878
reparam. [geod]
N=140 N=134 N=123 N=113 N=62 N=17
With [exp] .882
reparam. [geod]
Table 2: Averaged Dice performance measures between predictions and oberva-
tions for two modes of transport, with or without refinement by the cognitive
scores. In each cell, the first line corresponds to the exp-parallelization-based
prediction [exp], the middle line to the geodesic parallelization-based one [geod],
and the last line to the naive approach [naive]. Each column corresponds to
an increasingly remote predicted visit from baseline. Significance levels for the
Mann-Whitney test [.05, .01, .001, .0001].
Fig. 2: Exp-parallelization of the reference subject s0906 (first column) towards
the subject s1080 (second column), giving predictions for ages 81.6, 82.6, 83.6,
84.6 and 85.6 (years). On the third column are added the target brain structures
(red wireframes), segmented from the original images.
3.4 Refining with cognitive dynamical parameters
The two reference progressions are transported through geodesic and exp-paral-
lelization onto all remaining subjects. After time-reparametrization, the obtained
parallel trajectories then deliver predictions for the brain structures.
Figure 3 displays a reference geodesic and an exp-parallelized curve. The
predicted progression graphically matches the datapoints, and it can be noticed
that the final prediction at age 85.6 (Dice 0.73) outperforms the corresponding
one on Figure 2, obtained without time-reparametrization (Dice 0.69).
Quantitative results are presented in the lower part of Table 2. At the excep-
tion of the M12 prediction, both protocols outperform the naive one. The M36,
M48, M72 and M96 predictions are the most impressive ones, with p-values al-
ways lesser than 1%. This shows that the pace of cognitive score evolution is
well correlated with the pace of structural brain changes, and therefore allows
an enhanced prediction of follow-up shapes.
No conclusion can be drawn concerning the two parallel shifting method-
ologies, a single weak significance result being obtained only for the M12 pre-
diction where the geodesic parallelization method slightly outperforms the exp-
parallelization one with a Dice score of 0.888 versus 0.882.
4 Conclusion
We conducted a quantitative study of geodesic regression extrapolation, exhibit-
ing its limited predictive abilities. We then proposed a method to transport a
spatiotemporal trajectory into a different subject space with cognitive decline-
derived time reparametrization, and demonstrated its potential for prognosis.
The results show how crucial the dynamics are in disease modeling, and how
cross-modality data can be exploited to improve a learning algorithm. The two
main paradigms that have emerged for the transport of parallel trajectories
were shown to perform equally well in this prediction task. Nonetheless, the
exp-parallelization offers a methodological advantage in that the generated tra-
jectories do not depend on a particular choice of point on the reference geodesic,
in contrast with the trajectories obtained by geodesic parallelization. It takes
full advantage of the isometric property of the parallel transport, and eases the
combination with time-warp functions based on the individual disease dynamics.
In future work, more complex time reparametrization could be considered as
in [7]. Finally, the robustness of the proposed protocol to the choice of reference
subject has not been assessed. Such a choice could be avoided by constructing
an average disease model as in [15], or by translating for shapes the method
of [14]. We may also use this framework to estimate a joint image and cognitive
model to better estimate individual dynamical parameters of disease progression.
Acknowledgments. This work has been partly funded by the European Research
Council (ERC) under grant agreement No 678304, European Union’s Horizon 2020
research and innovation program under grant agreement No 666992, and the program
Investissements d’avenir ANR-10-IAIHU-06.
Fig. 3: Time-reparametrized exp-parallelization of the reference subject s0906
(first column) towards the subject s1080 (second column), giving predictions for
ages 81.6, 82.6, 83.6, 84.6 and 85.6 (years). On the third column are added the
target brain structures (red wireframes), segmented from the original images.
1. Beg, M., Miller, M., Trouv´e, A., Younes, L.: Computing large deformation metric
mappings via geodesic flows of diffeomorphisms. IJCV 61(2), 139–157 (2005)
2. Durrleman, S., Allassonni`ere, S., Joshi, S.: Sparse adaptive parameterization of
variability in image ensembles. IJCV 101(1), 161–183 (2013)
3. Durrleman, S., Prastawa, M., Charon, N., Korenberg, J.R., Joshi, S., Gerig, G.,
Trouv´e, A.: Morphometry of anatomical shape complexes with dense deformations
and sparse parameters. NeuroImage (2014)
4. Fishbaugh, J., Prastawa, M., Gerig, G., Durrleman, S.: Geodesic regression of
image and shape data for improved modeling of 4D trajectories
5. Fletcher, T.: Geodesic regression and the theory of least squares on riemannian
manifolds. IJCV 105(2), 171–185 (2013)
6. Hadj-Hamou, M., Lorenzi, M., Ayache, N., Pennec, X.: Longitudinal analysis of
image time series with diffeomorphic deformations: A computational framework
based on stationary velocity fields. Frontiers in Neuroscience 10, 236 (2016)
7. Hong, Y., Singh, N., Kwitt, R., Niethammer, M.: Time-warped geodesic regression.
In: MICCAI. vol. 17, p. 105 (2014)
8. Lorenzi, M., Ayache, N., Frisoni, G., Pennec, X.: 4D registration of serial brains
MR images: a robust measure of changes applied to Alzheimer’s disease. Spatio
Temporal Image Analysis Workshop (STIA), MICCAI (2010)
9. Lorenzi, M., Ayache, N., Pennec, X.: Schild’s ladder for the parallel transport of
deformations in time series of images. pp. 463–474. Springer (2011)
10. Lorenzi, M., Pennec, X.: Geodesics, parallel transport & one-parameter subgroups
for diffeomorphic image registration. IJCV 105(2), 111–127 (Nov 2013)
11. Metz, C., Klein, S., Schaap, M., van Walsum, T., Niessen, W.: Nonrigid registra-
tion of dynamic medical imaging data using nd + t b-splines and a groupwise
optimization approach. Medical Image Analysis 15(2), 238 249 (2011)
12. Peyrat, J., Delingette, H., Sermesant, M., Pennec, X., Xu, C., Ayache, N.: Registra-
tion of 4D time-series of cardiac images with multichannel diffeomorphic Demons.
MICCAI (2008)
13. Qiu, A., Younes, L., Miller, M.I., Csernansky, J.G.: Parallel transport in diffeomor-
phisms distinguishes the time-dependent pattern of hippocampal surface deforma-
tion due to healthy aging and the dementia of the Alzheimer’s type. NeuroImage
40(1), 68–76 (2008)
14. Schiratti, J.B., Allassonni`ere, S., Colliot, O., Durrleman, S.: Learning spatiotem-
poral trajectories from manifold-valued longitudinal data. In: NIPS 28
15. Singh, N., Hinkle, J., Joshi, S., Fletcher, P.T.: Hierarchical geodesic models in
diffeomorphisms. IJCV 117(1), 70–92 (2016)
16. Vercauteren, T., Pennec, X., Perchant, A., Ayache, N.: Non-parametric diffeomor-
phic image registration with the Demons algorithm. In: MICCAI
17. Wang, L., Beg, F., Ratnanather, T., Ceritoglu, C., Younes, L., Morris, J.C., Cser-
nansky, J.G., Miller, M.I.: Large deformation diffeomorphism and momentum
based hippocampal shape discrimination in dementia of the Alzheimer type. IEEE
Transactions on Medical Imaging 26(4), 462–470 (2007)
18. Wu, G., Wang, Q., Lian, J., Shen, D.: Estimating the 4D respiratory lung motion
by spatiotemporal registration and building super-resolution image. In: MICCAI.
pp. 532–539 (2011)
19. Younes, L.: Jacobi fields in groups of diffeomorphisms and applications. Quarterly
of Applied Mathematics 65(1), 113–134 (2007)
... This allows capturing subtle variations between or within subjects, which would be lost just by looking at flat imaging markers such as volume or voxel intensities. Current research shows longitudinal changes in shape in key structures of the brain (such as lateral ventricles or hippocampus) that are strongly related to cognitive degeneration [73,74] and can reveal differences between groups of patients [75][76][77] . Mixed effect models are often used for modelling shape changes [73][74][75][76] , with [46] proposing a novel vertex clustering method to model shape changes over time, using a similar mixed effect model already proposed in other reviewed works [71,82,114] . ...
... Current research shows longitudinal changes in shape in key structures of the brain (such as lateral ventricles or hippocampus) that are strongly related to cognitive degeneration [73,74] and can reveal differences between groups of patients [75][76][77] . Mixed effect models are often used for modelling shape changes [73][74][75][76] , with [46] proposing a novel vertex clustering method to model shape changes over time, using a similar mixed effect model already proposed in other reviewed works [71,82,114] . ...
Background and objectives: Recently, longitudinal studies of Alzheimer's disease have gathered a substantial amount of neuroimaging data. New methods are needed to successfully leverage and distill meaningful information on the progression of the disease from the deluge of available data. Machine learning has been used successfully for many different tasks, including neuroimaging related problems. In this paper, we review recent statistical and machine learning applications in Alzheimer's disease using longitudinal neuroimaging. Methods: We search for papers using longitudinal imaging data, focused on Alzheimer's Disease and published between 2007 and 2019 on four different search engines. Results: After the search, we obtain 104 relevant papers. We analyze their approach to typical challenges in longitudinal data analysis, such as missing data and variability in the number and extent of acquisitions. Conclusions: Reviewed works show that machine learning methods using longitudinal data have potential for disease progression modelling and computer-aided diagnosis. We compare results and models, and propose future research directions in the field.
... The main approaches are based off of the geodesic regression framework [3,25] and allow learning a deformation map that models the effect of time on the images for a given subject. While providing high resolution predictions, these methods show limited predictive abilities further in time when compared to mixed-effects models, that aggregate information from all the subjects at different stages of the disease [6]. ...
Full-text available
Disease progression models are crucial to understanding degenerative diseases. Mixed-effects models have been consistently used to model clinical assessments or biomarkers extracted from medical images, allowing missing data imputation and prediction at any timepoint. However, such progression models have seldom been used for entire medical images. In this work, a Variational Auto Encoder is coupled with a temporal linear mixed-effect model to learn a latent representation of the data such that individual trajectories follow straight lines over time and are characterised by a few interpretable parameters. A Monte Carlo estimator is devised to iteratively optimize the networks and the statistical model. We apply this method on a synthetic data set to illustrate the disentanglement between time dependant changes and inter-subjects variability, as well as the predictive capabilities of the method. We then apply it to 3D MRI and FDG-PET data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) to recover well documented patterns of structural and metabolic alterations of the brain.KeywordsVariational auto encodersMixed-effects modelsDisease progression modelsAlzheimer’s Disease
... An interesting example occurs when q α , µ α describes a known progression, for example a geodesic regression learned from repeated observation of a reference subject and when q β , µ β describes a registration between an observation of the reference subject and a new subject. In that case, the flow of the paralleltransported deformation can be used to obtain a prediction of the future state of the subject [3]. It is in some sense a transfer learning operation. ...
Conference Paper
Full-text available
Deformetrica is an open-source software for the statistical analysis of images and meshes. It relies on a specific instance of the large deformation diffeomorphic metric mapping (LDDMM) framework, based on control points: local momenta vectors offer a low-dimensional and interpretable parametrization of global diffeomorphims of the 2/3D ambient space, which in turn can warp any single or collection of shapes embedded in this physical space. Deformetrica has very few requirements about the data of interest: in the particular case of meshes, the absence of point correspondence can be handled thanks to the current or var-ifold representations. In addition to standard computational anatomy functionalities such as shape registration or atlas estimation, a bayesian version of atlas model as well as temporal methods (geodesic regression and parallel transport) are readily available. Installation instructions, tutorials and examples can be found at
Predicting the future development of an anatomical shape from a single baseline observation is a challenging task. But it can be essential for clinical decision-making. Research has shown that it should be tackled in curved shape spaces, as (e.g., disease-related) shape changes frequently expose nonlinear characteristics. We thus propose a novel prediction method that encodes the whole shape in a Riemannian shape space. It then learns a simple prediction technique founded on hierarchical statistical modeling of longitudinal training data. When applied to predict the future development of the shape of the right hippocampus under Alzheimer’s disease and to human body motion, it outperforms deep learning-supported variants as well as state-of-the-art.
This PhD proposes new Riemannian geometry tools for the analysis of longitudinal observations of neuro-degenerative subjects. First, we propose a numerical scheme to compute the parallel transport along geodesics. This scheme is efficient as long as the co-metric can be computed efficiently. Then, we tackle the issue of Riemannian manifold learning. We provide some minimal theoretical sanity checks to illustrate that the procedure of Riemannian metric estimation can be relevant. Then, we propose to learn a Riemannian manifold so as to model subject's progressions as geodesics on this manifold. This allows fast inference, extrapolation and classification of the subjects.
Predicting the age progression of individual brain images from longitudinal data has been a challenging problem, while its solution is considered key to improve dementia prognosis. Often, approaches are limited to group-level predictions, lack the ability to extrapolate, can not scale to many samples, or do not operate directly on image inputs. We address these issues with the first approach to artificial aging of brain images based on Wasserstein Generative Adversarial Networks. We develop a novel recursive generator model for brain image time series, and train it on large-scale longitudinal data sets (ADNI/AIBL). In addition to thorough analysis of results on healthy and demented subjects, we demonstrate the predictive value of our brain aging model in the context of conversion prognosis from mild cognitive impairment to Alzheimer’s disease. Conversion prognosis for a baseline image is achieved in two steps. First, we estimate the future brain image with the Generative Adversarial Network. This follow-up image is passed to a CNN classifier, pre-trained to discriminate between mild cognitive impairment and Alzheimer’s disease. It estimates the Alzheimer probability for the follow-up image, which represents an effective measure for future disease risk.
Full-text available
We propose and detail a deformation-based morphometry computational framework, called Longitudinal Log-Demons Framework (LLDF), to estimate the longitudinal brain deformations from image data series, transport them in a common space and perform statistical group-wise analyses. It is based on freely available software and tools, and consists of three main steps: (i) Pre-processing, (ii) Position correction, and (iii) Non-linear deformation analysis. It is based on the LCC log-Demons non-linear symmetric diffeomorphic registration algorithm with an additional modulation of the similarity term using a confidence mask to increase the robustness with respect to brain boundary intensity artifacts. The pipeline is exemplified on the longitudinal Open Access Series of Imaging Studies (OASIS) database and all the parameters values are given so that the study can be reproduced. We investigate the group-wise differences between the patients with Alzheimer's disease and the healthy control group, and show that the proposed pipeline increases the sensitivity with no decrease in the specificity of the statistical study done on the longitudinal deformations.
Full-text available
Hierarchical linear models (HLMs) are a standard approach for analyzing data where individuals are measured repeatedly over time. However, such models are only applicable to longitudinal studies of Euclidean data. This paper develops the theory of hierarchical geodesic models (HGMs), which generalize HLMs to the manifold setting. Our proposed model quantifies longitudinal trends in shapes as a hierarchy of geodesics in the group of diffeomorphisms. First, individual-level geodesics represent the trajectory of shape changes within individuals. Second, a group-level geodesic represents the average trajectory of shape changes for the population. Our proposed HGM is applicable to longitudinal data from unbalanced designs, i.e., varying numbers of timepoints for individuals, which is typical in medical studies. We derive the solution of HGMs on diffeomorphisms to estimate individual-level geodesics, the group geodesic, and the residual diffeomorphisms. We also propose an efficient parallel algorithm that easily scales to solve HGMs on a large collection of 3D images of several individuals. Finally, we present an effective model selection procedure based on cross validation. We demonstrate the effectiveness of HGMs for longitudinal analysis of synthetically generated shapes and 3D MRI brain scans.
Conference Paper
Full-text available
We propose a Bayesian mixed-effects model to learn typical scenarios of changes from longitudinal manifold-valued data , namely repeated measurements of the same objects or individuals at several points in time. The model allows to estimate a group-average trajectory in the space of measurements. Random variations of this trajectory result from spatiotemporal transformations , which allow changes in the direction of the trajectory and in the pace at which trajectories are followed. The use of the tools of Riemannian geometry allows to derive a generic algorithm for any kind of data with smooth constraints , which lie therefore on a Riemannian manifold. Stochastic approximations of the Expectation-Maximization algorithm is used to estimate the model parameters in this highly non-linear setting. The method is used to estimate a data-driven model of the progressive impairments of cognitive functions during the onset of Alzheimer ' s disease. Experimental results show that the model correctly put into correspondence the age at which each individual was diagnosed with the disease , thus validating the fact that it effectively estimated a normative scenario of disease progression. Random effects provide unique insights into the variations in the ordering and timing of the succession of cognitive impairments across different individuals .
Full-text available
A variety of regression schemes have been proposed on images or shapes, although available methods do not handle them jointly. In this paper, we present a framework for joint image and shape regression which incorporates images as well as anatomical shape information in a consistent manner. Evolution is described by a generative model that is the analog of linear regression, which is fully characterized by baseline images and shapes (intercept) and initial momenta vectors (slope). Further, our framework adopts a control point parameterization of deformations, where the dimensionality of the deformation is determined by the complexity of anatomical changes in time rather than the sampling of the image and/or the geometric data. We derive a gradient descent algorithm which simultaneously estimates baseline images and shapes, location of control points, and momenta. Experiments on real medical data demonstrate that our framework effectively combines image and shape information, resulting in improved modeling of 4D (3D space + time) trajectories.
Full-text available
This paper introduces a new parameterization of diffeomorphic deformations for the characterization of the variability in image ensembles. Dense diffeomorphic deformations are built by interpolating the motion of a finite set of control points that forms a Hamiltonian flow of self-interacting particles. The proposed approach estimates a template image representative of a given image set, an optimal set of control points that focuses on the most variable parts of the image, and template-to-image registrations that quantify the variability within the image set. The method automatically selects the most relevant control points for the characterization of the image variability and estimates their optimal positions in the template domain. The optimization in position is done during the estimation of the deformations without adding any computational cost at each step of the gradient descent. The selection of the control points is done by adding a L 1 prior to the objective function, which is optimized using the FISTA algorithm.
Conference Paper
We consider geodesic regression with parametric time-warps. This allows for example, to capture saturation effects as typically observed during brain development or degeneration. While highly-flexible models to analyze time-varying image and shape data based on generalizations of splines and polynomials have been proposed recently, they come at the cost of substantially more complex inference. Our focus in this paper is therefore to keep the model and its inference as simple as possible while allowing to capture expected biological variation. We demonstrate that by augmenting geodesic regression with parametric time-warp functions, we can achieve comparable flexibility to more complex models while retaining model simplicity. In addition, the time-warp parameters provide useful information of underlying anatomical changes as demonstrated for the analysis of corpora callosa and rat calvariae. We exemplify our strategy for shape regression on the Grassmann manifold, but note that the method is generally applicable for time-warped geodesic regression.
We propose a generic method for the statistical analysis of collections of anatomical shape complexes, namely sets of surfaces that were previously segmented and labeled in a group of subjects. The method estimates an anatomical model, the template complex, that is representative of the population under study. Its shape reflects anatomical invariants within the dataset. In addition, the method automatically places control points near the most variable parts of the template complex. Vectors attached to these points are parameters of deformations of the ambient 3D space. These deformations warp the template to each subject's complex in a way that preserves the organization of the anatomical structures. Multivariate statistical analysis is applied to these deformation parameters to test for group differences. Results of the statistical analysis are then expressed in terms of deformation patterns of the template complex, and can be visualized and interpreted. The user needs only to specify the topology of the template complex and the number of control points. The method then automatically estimates the shape of the template complex, the optimal position of control points and deformation parameters. The proposed approach is completely generic with respect to any type of application and well adapted to efficient use in clinical studies, in that it does not require point correspondence across surfaces and is robust to mesh imperfections such as holes, spikes, inconsistent orientation or irregular meshing.
This paper develops the theory of geodesic regression and least-squares estimation on Riemannian manifolds. Geodesic regression is a method for finding the relationship between a real-valued independent variable and a manifold-valued dependent random variable, where this relationship is modeled as a geodesic curve on the manifold. Least-squares estimation is formulated intrinsically as a minimization of the sum-of-squared geodesic distances of the data to the estimated model. Geodesic regression is a direct generalization of linear regression to the manifold setting, and it provides a simple parameterization of the estimated relationship as an initial point and velocity, analogous to the intercept and slope. A nonparametric permutation test for determining the significance of the trend is also given. For the case of symmetric spaces, two main theoretical results are established. First, conditions for existence and uniqueness of the least-squares problem are provided. Second, a maximum likelihood criteria is developed for a suitable definition of Gaussian errors on the manifold. While the method can be generally applied to data on any manifold, specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age.
The aim of computational anatomy is to develop models for understanding the physiology of organs and tissues. The diffeomorphic non-rigid registration is a validated instrument for the detection of anatomical changes on medical images and is based on a rich mathematical background. For instance, the “large deformation diffeomorphic metric mapping ” framework defines a Riemannian setting by providing an opportune right invariant metric on the tangent space, and solves the registration problem by computing geodesics parametrized by time-varying velocity fields. In alternative, stationary velocity fields have been proposed for the diffeomorphic registration based on the oneparameter subgroups from Lie groups theory. In spite of the higher computational efficiency, the geometric setting of the latter method is more vague, especially regarding the relationship between one-parameter subgroups and geodesics. In this study, we present the relevant properties of the Lie groups for the definition of geometrical properties within the one-parameter subgroups parametrization, and we define the geometric structure for computing geodesics and for parallel transporting. The theoretical results are applied to the image registration context, and discussed in light of the practical computational problems. 1