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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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