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

Abstract

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.

Full text

Prediction of the progression of subcortical brain
structures in Alzheimer’s disease from baseline
Alexandre Bˆone†12 , Maxime Louis†12 , 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) – Hˆopital 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 t→γ(c0,β0)(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(c0,β0)(T , ·)
minimizes the following functional :
inf
c0,β0,T
ni
X
j=1
kγ(c0,β0)(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 (www.deformetrica.org), 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(t−t0−τ) + 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
regression.
(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
structures.

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.
Learning
period
(months)
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
.888
.800
.850 ∗
∗
∗
.737
.803 ∗
∗
∗
.624
.708 ∗
∗
.509
.626 ∗
∗
.483
.602
[naive]
12 [reg] - .839
.875 ∗
∗
.769
.832 ∗
∗
∗
∗
.658
.735 ∗
∗
.523
.644 ∗
∗
.465
.608 ∗
[naive] -
18 [reg] - .885
.890
.823
.851 ∗
.738
.764
.611
.661
.579
.627
[naive] -
24 [reg] - - .864
.869
.778
.779
.681
.689
.657
.653
[naive] - -
max - 1 [reg] .807
.797
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
.883
.882
.841
.847
.850



∗
.799
.806
.806
.744
.753
.754
.650
.664
.682
.647
.661
.611
reparam. [geod]
[naive]
N=140 N=134 N=123 N=113 N=62 N=17
With [exp] ∗.882
.888
.884 ∗
.852
.858
.852
∗
∗
.825
.831
.809



∗
∗∗
∗
∗
∗
.796
.802
.764



∗
∗
∗∗
∗
∗
.756
.762
.706



∗
∗
∗∗
∗
.730
.732
.636



∗
∗
reparam. [geod]
[naive]
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.

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