Matthieu Fradelizi

Grigoris Paouris
email: grigorispaouris@yahoo.co.uk

Carsten Schütt
email: schuett@math.uni-kiel.de

Simplices in the Euclidean ball

HAL is

Introduction

The starting point of this paper is the article [START_REF] Böröczky | Convex bodies of minimal volume, surface area and mean width with respect to thin shells[END_REF], where it was shown that if all the extreme points of a convex body K in R n have Euclidean norm greater than r > 0, then 1

|K| K |x| 2 2 dx r 2 9n ( 1
)
where |x| 2 stands for the Euclidean norm of x and |K| for the volume of K.

We improve here this inequality showing that the optimal constant is r 2 n + 2 , with equality for the regular simplex, with vertices on the Euclidean sphere of radius r. We also prove the same inequality under the different condition that K is in Löwner position. More generally, we investigate upper and lower bounds on the quantity

C 2 (K) := 1 |K| K |x| 2 2 dx , (2)
under various assumptions on the position of K. Some hypotheses on K are necessary because C 2 (K) is not homogeneous, one has C 2 (λK) = λ 2 C 2 (K). Let n 2. We denote by K n the set of all convex bodies in R n , i.e. the set of compact convex sets with non empty interior and by ∆ n the regular simplex in R n with vertices in S n-1 , the Euclidean unit sphere. For K ∈ K n , we denote by g K , its centroid,

g K = 1 |K| K xdx.
Under these notations we prove the following theorem.

Theorem 1.1. Let r > 0, K ∈ K n such that all its extreme points have Euclidean norm greater than r. Then

C 2 (K) = 1 |K| K |x| 2 2 dx C 2 (r∆ n ) + n + 1 n + 2 |g K | 2 2 = r 2 + (n + 1)|g K | 2 2 n + 2 .
Moreover, if K is a polytope there is equality if and only if K is a simplex with its vertices on the Euclidean sphere of radius r.

In Theorem 1.1, for a general K, we don't have a characterization of the equality case because we deduce it by approximation from the case of polytopes. We conjecture that the equality case is still the same.

Notice that the condition imposed on K that all its extreme points have Euclidean norm greater than r is unusual. For example, if K has positive curvature, it is equivalent to either K ⊃ rB n 2 or K ∩ rB n 2 = ∅. Moreover, this hypothesis is not continuous with respect to the Hausdorff distance. Indeed, if we define P = conv(∆ n , x), where x / ∈ ∆ n is a point very close to the centroid of a facet of ∆ n then the distance of ∆ n and P is very small but the point x will be an extreme point of P of Euclidean norm close to 1/n, i.e. much smaller than 1, the Euclidean norm of the vertices of ∆ n .

Other conditions on the position of K may be imposed. To state it, let us first recall the classical definitions of John and Löwner position. Let K ∈ K n . We say that K is in John position if the ellipsoid of maximal volume contained in K is B n 2 . We say that K is in Löwner position if the ellipsoid of minimal volume that contains K is B n 2 . It was proved by Guédon in [START_REF] Guédon | Sections euclidiennes des corps convexes et inégalités de concentration volumique[END_REF] (see also [START_REF] Guédon | On the symmetric average of a convex body[END_REF]) that if K ∈ K n satisfies g K = 0 and if K ∩(-K) is in Löwner position (which is equivalent to say that B n 2 is the ellipsoid of minimal volume containing K and centered at the origin) then

C 2 (K) ≥ C 2 (∆ n ).
Using the same ideas, we prove the following theorem.

Theorem 1.2. Let K be a convex body in Löwner position. Then

n n + 2 = C 2 (B n 2 ) ≥ C 2 (K) ≥ C 2 (∆ n ) + (n + 1) 2 n(n + 2) |g K | 2 2 = n + (n + 1) 2 |g K | 2 2 n(n + 2) . Moreover, if K is symmetric, then n n + 2 = C 2 (B n 2 ) ≥ C 2 (K) ≥ C 2 (B n 1 ) = 2n (n + 1)(n + 2)
.

Let K be a convex body in John position. Then

n n + 2 = C 2 (B n 2 ) ≤ C 2 (K) ≤ C 2 (n∆ n ) + 2 n + 1 n + 2 |g K | 2 2 = n 2 + 2(n + 1)|g K | 2 2 n + 2 .
Moreover, if K is symmetric, then

n n + 2 = C 2 (B n 2 ) ≤ C 2 (K) ≤ C 2 (B n ∞ ) = n 3 .
The inequalities involving the Euclidean ball in Theorem 1.2 are deduced from the following proposition.

Proposition 1.3. Let K be a convex body.

1. If K ⊂ B n 2 and 0 ∈ K then C 2 (K) ≤ C 2 (B n 2 ) = n n+2 with equality if and only if K = {tx; 0 ≤ t ≤ 1, x ∈ S}, where S ⊂ S n-1 . 2. If K ⊃ B n 2 then C 2 (K) ≥ C 2 (B n 2 ) = n n+2 , with equality if and only K = B n 2 .
In view of Proposition 1.3, it could be conjectured that for every centrally symmetric convex bodies K, L such that K ⊂ L one has C 2 (K) ≤ C 2 (L). But this is not the case. It can be seen already in dimension 2, by taking L = conv((a, 0), (-a, 0), (0, 1), (0, -1)) K = {(x, y) ∈ L; |y| ≤ 1/2} with a large enough. Indeed, C 2 (L) = 1+a and C 2 (K) = 5+15a 2 72 . The paper is organized as follows. In §2, we gather some background material needed in the rest of the paper. We prove Theorem 1.1 in §3, Theorem 1.2 in §4 and Proposition 1.3 in §5.

Acknowledgment. We would like to thank B. Maurey and O. Guédon for discussions.

Preliminaries

As mentioned before the quantity C 2 (K) is not affine invariant. Let us investigate the behaviour of C 2 (K) under affine transform. We start with translations. For a ∈ R n , and K ∈ K n , one has

C 2 (K -a) = 1 |K| K |x -a| 2 2 dx = C 2 (K) -2 g K , a + |a| 2 2 . Hence C 2 (K -g K ) = C 2 (K) -|g K | 2 2 (3) 3 minimizes C 2 (K -a) among translation a ∈ R n .
Let T be a non-singular linear transform, then

C 2 (T K) = 1 |T K| T K |x| 2 2 dx = 1 |K| K |T x| 2 2 dx.
The preceding quantity may be computed in terms of

C 2 (K) if K is in isotropic position (see below).

Decomposition of identity

Let u 1 , . . . , u N be N points in the unit sphere S n-1 . We say that they form a representation of the identity if there exist c 1 , . . . c N positive integers such that

I = N i=1 c i u i ⊗ u i and N i=1 c i u i = 0.
Notice that in this case, one has, for

x ∈ R n x = N i=1 c i x, u i u i , |x| 2 2 = N i=1 c i x, u i 2 and N i=1 c i = n. (4)
Moreover, for any linear map T on R n , its Hilbert-Schmidt norm is given by

T 2 HS := tr(T ⋆ T ) = N i=1 c i u i , T ⋆ T u i = N i=1 c i |T u i | 2 2 .
If A is an affine transformation and T its linear part, i.e. A(x) = T (x) + A(0), then

N i=1 c i |Au i | 2 2 = T 2 HS + n|A(0)| 2 2 . (5) Indeed,
N i=1 c i |Au i | 2 2 = N i=1 c i |T u i + A(0)| 2 2 = T 2 HS + 2 N i=1 c i T u i , A(0) + N i=1 c i |A(0)| 2 2 = T 2 HS + n|A(0)| 2 2 .

John, Löwner and isotropic positions

Let K ∈ K n . Recall that K is in John position if the ellipsoid of maximal volume contained in K is B n 2 and that K is in Löwner position if the ellipsoid of minimal volume that contains K is B n 2 .
The following theorem ( [START_REF] John | Extremum problems with inequalities as subsidiary conditions[END_REF], see also [START_REF] Ball | Ellipsoids of maximal volume in convex bodies[END_REF]) characterizes these positions.

Theorem 2.1. Let K ∈ K n . Then K is in John position if and only if B n 2 ⊆ K and there exist u 1 , . . . , u N ∈ ∂K ∩ S n-1 that form a representation of identity. Also K is in Löwner position if and only if B n 2 ⊇ K and there exist u 1 , . . . , u N ∈ ∂K ∩ S n-1 that form a representation of identity.

Let K ∈ K n . We say that K is in isotropic position if |K| = 1, g K = 0 and K x, θ 2 dx = L 2 K , ∀ θ ∈ S n-1 . ( 6
) If K is in isotropic position then C 2 (K) = nL 2 K .
Note that the isotropic position is unique up to orthogonal transformations and that for any convex body K ∈ K n there exist an affine transformation A such that AK is in isotropic position (see [START_REF] Milman | Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space[END_REF] or [START_REF] Giannopoulos | Notes on isotropic convex bodies[END_REF]). The quantity L K is called the isotropic constant of K. For any non singular linear transform

T on R n , K x, T x dx = L 2 K trT .
In particular if K is isotropic and T ∈ GL n then

C 2 (T K) = 1 |K| K |T x| 2 2 dx = L 2 K trT * T = T 2 HS n C 2 (K) . (7)
From the arithmetic geometric inequality, it implies that 3), it gives, as it is well known, that the isotropic position minimizes C 2 (AK) among affine transforms A that preserve volume.

C 2 (T K) ≥ | det(T )| 2 n C 2 (K). With (

Proof of Theorem 1.1

We start with the following Lemma 3.1. For every n 1

C 2 (∆ n ) = 1 n + 2
.

Proof: The volume of the regular simplex ∆ n with vertices u 1 , . . . , u n+1 in S n-1 is

|∆ n | = √ n + 1 n! n + 1 n n 2 . Let f (t) = |{x ∈ ∆ n : x, u 1 = t}|. One has f (t) = n -1 n + 1 n-1 2 |∆ n-1 |(1-t) n-1 1 [-1 n ,1] (t) = |∆ n | n n + 1 n n(1-t) n-1 1 [-1 n ,1] (t) .
Hence, by Fubini

1 |∆ n | ∆ n x, u 1 2 dx = 1 |∆ n | 1 -1 n t 2 f (t)dt = 1 n(n + 2)
.

Since λ∆ n is in isotropic position for some λ > 0 we conclude that

C 2 (∆ n ) = 1 |∆ n | ∆ n |x| 2 2 dx = n |∆ n | ∆ n x, u 1 2 dx = 1 n + 2 . 2 Lemma 3.2. Let S = conv(x 1 , . . . , x n+1 ) ⊂ R n be a non degenerate simplex. Then C 2 (S) = 1 n + 1 n+1 i=1 |x i | 2 2 C 2 (∆ n ) + |g S | 2 2 (1 -C 2 (∆ n )).
In particular, if

1 n + 1 n+1 i=1 |x i | 2 2 ≥ r 2 then C 2 (S) ≥ r 2 C 2 (∆ n ) + n + 1 n + 2 |g S | 2 2 .
Remark: This implies that a non degenerate simplex S = conv(x 1 , . . . , x n+1 ) such

that |x i | 2 ≥ r for all i satisfies C 2 (S) ≥ C 2 (r∆ n ) = r 2 n+2
, with equality if and only if |x i | 2 = r for all i and g S = 0. In dimension 2, these conditions for equality imply that S is regular but in dimension n ≥ 3, it is not the case anymore. For example in dimension 3, if one takes the regular simplex and that one moves symmetrically two vertices along the geodesic between them to make them closer and if one does the same to the two other opposite vertices then the centroid stays at 0 and the vertices stay on the sphere. In any dimension n ≥ 4, one chooses the north pole as the first vertex of our simplex and the n other vertices are the vertices of a simplex in dimension n -1, which is not regular and satisfies the equality case, we put this simplex in an horizontal hyperplane in such a way that the centroid is at 0. Proof. Let A be an affine map such that S = A∆ n and denote by T its linear part. One has g S = Ag ∆ n = A(0), hence A = T + g S and S = g S + T ∆ n . Denote by u i ∈ S n-1 the vertices of ∆ n , so that x i = g S + T u i , for 1 ≤ i ≤ n + 1. Hence, by (3) and ( 7)

C 2 (S) = |g S | 2 2 + C 2 (T ∆ n ) = |g S | 2 2 + T 2 HS n C 2 (∆ n )
Since u 1 , . . . , u n+1 form a decomposition of identity

I = n n + 1 n+1 i=1 u i ⊗ u i , one has T 2 HS = n n + 1 n+1 i=1 |T u i | 2 2 = n n + 1 n+1 i=1 |Au i | 2 2 -n|g S | 2 2 .
Therefore, we get the equality. The inequality is obvious.

Proof of Theorem 1.1 We first consider the case where K is a polytope. Let us prove by induction on the dimension that there exists non degenerate simplices S 1 , . . . , S m with the following properties: K = m i=1 S i , the interiors of S i and S j are mutually disjoint for i = j and the vertices of the S i 's are among the vertices of K.

In dimension two, it is enough to fix a vertex x of K, to consider the edges ∆ 1 , . . . , ∆ m which don't contain x and to choose S i = conv(x, ∆ i ), for i = 1, . . . , m.

Assuming that the property has been proved in dimension n -1, let us prove it in dimension n. We proceed in the same way as in dimension two. Let x be a fixed vertex of K. Let F 1 , . . . , F N be the facets of K which don't contain x. From the induction hypothesis, we can split each of them into simplices with the required properties. Let ∆ 1 , . . . , ∆ m be the collection of these simplices. Then it is not difficult to check that the simplices S i = conv(x, ∆ i ), for i = 1, . . . , m have the required properties. Now we may apply Lemma 3.2 to the S i 's and we get

C 2 (K) = 1 |K| m i=1 |S i |C 2 (S i ) ≥ 1 |K| m i=1 |S i | r 2 C 2 (∆ n ) + n + 1 n + 2 |g Si | 2 2 = r 2 C 2 (∆ n ) + n + 1 n + 2 m i=1 |S i | |K| |g Si | 2 2
Then we use the convexity of the function x → |x| 2 2 to deduce that

C 2 (K) ≥ r 2 C 2 (∆ n ) + n + 1 n + 2 m i=1 |S i | |K| g Si 2 2 = r 2 C 2 (∆ n ) + n + 1 n + 2 |g K | 2 2 .
This proves the inequality. If there is equality then from the strict convexity of the function x → |x| 2 2 one deduces that g Si = g K , for every i. Since the S i 's have disjoint interiors, this implies that there is only one of them. Hence K is a simplex and its vertices have Euclidean norm r. Let us prove the general case. Let K be a convex body such that such that all its extreme points have Euclidean norm greater than r. Then there is a sequence of polytopes (P n ) n converging to K in the Hausdorff metric in K n such that for every n ∈ N , the extreme points of P n have Euclidean norm greater than r. Since C 2 is continuous with respect to the Hausdorff distance we get the inequality for K. 2 4 Proof of Theorem 1.2

As in [START_REF] Guédon | Sections euclidiennes des corps convexes et inégalités de concentration volumique[END_REF], our main tools are the following inequalities proved by Milman-Pajor [START_REF] Milman | Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space[END_REF] in the symmetric case and by Kannan-Lovász and Simonovits [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF] in the non-symmetric case. Recall that if K is a convex body and u ∈ R n , the support function of K is defined by

h K (u) = sup x∈K x, u . Lemma 4.1. Let K be a convex body and u ∈ S n-1 . 1) If K is symmetric then 2h K (u) 2 (n + 1)(n + 2) ≤ 1 |K| K x, u 2 dx ≤ h K (u) 2 3 ,
with equality on the left hand side if and only if K is a double-cone in direction u, which means that there exists x in R n and a symmetric convex body L in u ⊥ such that x, u = 0 and K = conv(L, x, -x) and equality in the right hand side if and only if K is cylinder in direction u, which means that there exists x in R n and a symmetric convex body L in u ⊥ such that x, u = 0 and

K = L + [-x, x]. 2) If g K = 0 then h K (u) 2 n(n + 2) ≤ 1 |K| K x, u 2 dx ≤ nh K (u) 2 n + 2 ,
with equality on the left hand side if and only if K is a cone in direction u, which means that there exists x in R n and a convex body L in u ⊥ such that x, u > 0, g L = 0 and K = conv(L, x) -x n and equality in the right hand side if and only if K is cone in direction -u.

Notice that the proof of 1) given in [START_REF] Milman | Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space[END_REF] is beautiful and elementary but the proof of 2) given in [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF] is not as simple, it used the much more elaborated tool called the localization lemma. A simple proof of 2) was given in [START_REF] Fradelizi | Inégalités fonctionnelles et volume des sections des corps convexes[END_REF] but since it is not easily available, we reproduce it partially here for completeness.

Proof of of 2):

With a change of variable, we may assume that |K| = 1 and

h K (u) = 1. Let f (t) = |{x ∈ K; x, u = t}|. The support of f is [-h K (-u), h K (u)]
and from Brunn's theorem, f 1/(n-1) is concave on its support. Moreover, from Fubini, for any continuous function

φ on R K φ( x, u )dx = ∞ -∞ φ(t)f (t)dt.
Since |K| = 1 we have f = 1 and since g K = 0, tf (t)dt = 0. Define

g 1 (t) = n n+1 (n + 1) n (1 -t) n-1 1 [-1 n ,1] , g 2 (t) = n (n + 1) n (t + n) n-1 1 [-n,1] .
Then the functions g 1/(n-1) i are affine on their support and satisfy g i = 1 and tg i (t)dt = 0. Let h 1 = fg 1 and h 2 = g 2f . Assume that h i = 0. Since h i = th i = 0, the function h i changes sign at least twice at s i < t i for i = 1, 2. Moreover because of the concavity of f 1/(n-1) , the function h i is negative in (s i , t i ) and positive outside. By looking at its variations, one sees that the function

W i (t) := t -∞ s -∞ h i (x)dxds
is non-negative on its support. Integrating by part twice and assuming that φ is twice differentiable and convex one has

φ(t)h i (t)dt = φ ′′ (t)W i (t)dt ≥ 0.
Therefore φ(t)g 1 (t)dt ≤ φ(t)f (t)dt ≤ φ(t)g 2 (t)dt. For φ(t) = t 2 , we get the inequality. If there is equality for example in the left hand side then W 1 = 0, thus h 1 = 0, hence f = g 1 , from the equality case in Brunn's theorem we deduce that all the sets {x ∈ K; x, u = t} are homothetic.

2

Using Lemma 4.1, we prove the following proposition.

Proposition 4.2. Let K be a convex body such that there exist vectors u

1 , . . . , u m ∈ S n-1 , with h K (u i ) = 1 which form a representation of identity I = m i=1 c i u i ⊗ u i , with m i=1 c i u i = 0. 1) If K is symmetric then 2n (n + 1)(n + 2) = C 2 (B n 1 ) ≤ C 2 (K) ≤ C 2 (B n ∞ ) = n 3 .
2) In general one has

1 n + 2 + |g K | 2 2 n(n + 2) ≤ C 2 (K) -|g K | 2 2 ≤ n 2 n + 2 + n n + 2 |g K | 2 2 .
Proof: 1) Using the decomposition of identity, we deduce that

C 2 (K) = 1 |K| K |x| 2 2 dx = m i=1 c i 1 |K| K x, u i 2 dx.
We apply the preceding lemma to the vectors u i and use that h

K (u i ) = 1 to get 2n (n + 1)(n + 2) ≤ C 2 (K) ≤ n 3 . For K = B n 1 or K = B n
∞ , the calculation is trivial but we could also check that we are in the case of equality of Lemma 4.1.

2) One has

C 2 (K) -|g K | 2 2 = 1 |K| K-g K |x| 2 2 dx = m i=1 c i 1 |K| K-g K x, u i 2 dx.
We apply the preceding lemma to the vectors u i

C 2 (K) -|g K | 2 2 ≤ m i=1 c i n n + 2 h K-g K (u i ) 2 = n n + 2 m i=1 c i (h K (u i ) -g K , u i ) 2 . Since h K (u i ) = 1 and c i u i = 0 C 2 (K) -|g K | 2 2 ≤ n n + 2 (n + |g K | 2 2 ).
The lower estimate follows in the same way.

Theorem 1.2 follows from Proposition 1.3 and Proposition 4.2.

Proof of Proposition 1.3

We first state and prove a standard lemma that we shall use in the proof. Two changes of variables finish the proof. 2

Proof of Proposition 1.3:

1) Let • K be the gauge function of K, i.e. x K = inf{t > 0; x ∈ tK}, for every x ∈ R n . Since K ⊂ B n 2 one has |x| 2 ≤ x K , for every x ∈ R n . Hence

K |x| 2 2 dx ≤ K x 2 K dx = K x K 0 2tdtdx = 1 0 2t|{x ∈ K; x K ≥ t}|dt = |K| 1 0 2t(1 -t n )dt = |K| n n + 2 .
This gives the inequality. If there is equality then x K = |x| 2 for every x ∈ K hence {x; x K = 1} = K ∩ S n-1 := S therefore K = {tx; 0 ≤ t ≤ 1, x ∈ S}, with S ⊂ S n-1 .

2) Applying Lemma 5.1 to ϕ(t) = t 2 , we deduce that

C 2 (K) = 1 |K| K |x| 2 2 dx ≥ λ 2 C 2 (B n 2 ) = |K| |B n 2 | 2 n C 2 (B n 2 ).
If we assume that K ⊃ B n 2 then it follows that C 2 (K) ≥ C 2 (B n 2 ), with equality if and only if K = B n 2 , which is the content of the second part of Proposition 1.3. 2

2

 2

6

 6

Lemma 5 . 1 . 2 | B n 2 ϕ(λ|x| 2 )dx = n 1 0ϕn 2 ϕ

 5122212 Let K be a Borel set such that 0 < |K| < +∞, ϕ : R + → R + be a non-decreasing function and λ= (|K|/|B n 2 |) 1/n . Then 1 |K| K ϕ(|x| 2 )dx ≥ 1 |B n (λr)r n-1 dr. Proof: One has |λB n 2 | = |K|, hence |K \ (λB n 2 )| = |(λB n 2 ) \ K| Since ϕ is nondecreasing, we deduce that K\(λB n 2 ) ϕ(|x| 2 )dx ≥ ϕ(λ)|K \ (λB n 2 )| = ϕ(λ)|(λB n 2 ) \ K| ≥ (|x| 2 )dx.


Renaud La Joie

Audrey Perrotin

Louisa Barre

Caroline Hommet

Florence Mézenge

Méziane Ibazizene

Vincent Camus

Brigitte Landeau

Denis Guilloteau

Vincent De

La Sayette

Ahmed Abbas

Francis Eustache

Béatrice Desgranges

Dr Gaël Chételat
email: chetelat@cyceron.fr

Region-specific hierarchy between atrophy, hypometabolism and β-amyloid load in Alzheimer"s disease dementia

come

Figures width:

Figures

INTRODUCTION

Alzheimer"s disease is characterised by major brain changes including gray matter atrophy, hypometabolism and β-amyloid (Aβ) deposition. These changes have been extensively described using neuroimaging techniques, i.e. magnetic resonance imaging (MRI) and positron emission tomography (PET) combined with specific radiotracers: 18 Ffluorodeoxyglucose (FDG) for glucose metabolism; 11 C-Pittsburgh compound B, 18 F-Florbetapir or others for Aβ deposition [START_REF] Herholz | Clinical amyloid imaging in Alzheimer"s disease[END_REF]. These techniques have proved to be useful for early diagnosis of Alzheimer"s disease, but also for further understanding the pathological mechanism(s) underlying the disease [START_REF] Rabinovici | Beyond diagnosis: what biomarkers are teaching us about the "bio"logy of Alzheimer disease[END_REF].

Dynamic models of changes in these neuroimaging biomarkers over the course of the disease have recently been proposed [START_REF] Perrin | Multimodal techniques for diagnosis and prognosis of Alzheimer"s disease[END_REF][START_REF] Jack | Hypothetical model of dynamic biomarkers of the Alzheimer"s pathological cascade[END_REF][START_REF] Ewers | Neuroimaging markers for the prediction and early diagnosis of Alzheimer"s disease dementia[END_REF].

According to the amyloid cascade hypothesis [START_REF] Hardy | The amyloid hypothesis of Alzheimer"s disease: progress and problems on the road to therapeutics[END_REF], the sequence of events would start with Aβ deposition in a very early, presymptomatic stage. Then, Aβ is thought to trigger functional and structural changes that would appear latter, closer to clinical symptom onset. Yet, in spite of experimental studies showing that Aβ impairs synaptic functioning [START_REF] Palop | Amyloid-β-induced neuronal dysfunction in Alzheimer"s disease: from synapses toward neural networks[END_REF][START_REF] Parihar | Amyloid-$\beta$ as a Modulator of Synaptic Plasticity[END_REF], results from the human neuroimaging literature are ambiguous. Correlations between increased Aβ and decreased metabolism in Alzheimer"s disease patients have been found in some studies [START_REF] Engler | Two-year follow-up of amyloid deposition in patients with Alzheimer"s disease[END_REF][START_REF] Edison | Amyloid, hypometabolism, and cognition in Alzheimer disease: an [11C]PIB and [18F]FDG PET study[END_REF][START_REF] Cohen | Basal cerebral metabolism may modulate the cognitive effects of Abeta in mild cognitive impairment: an example of brain reserve[END_REF] but not in others [START_REF] Li | Regional analysis of FDG and PIB-PET images in normal aging, mild cognitive impairment, and Alzheimer"s disease[END_REF]Rabinovici et al., 2010;[START_REF] Furst | Cognition, glucose metabolism and amyloid burden in Alzheimer"s disease[END_REF]. Previous works showed that the relationship between Aβ load and hypometabolism [START_REF] Cohen | Basal cerebral metabolism may modulate the cognitive effects of Abeta in mild cognitive impairment: an example of brain reserve[END_REF], or atrophy [START_REF] Chételat | Relationship between atrophy and beta-amyloid deposition in Alzheimer disease[END_REF] varies throughout the evolution of the disease, suggesting that neurodegeneration could be independent of Aβ pathology in later stages [START_REF] Hyman | Amyloid-Dependent and Amyloid-Independent Stages of Alzheimer Disease[END_REF].

Moreover, there may also be differences in the sequence of events according to brain regions, as suggested by discrepancies in the regional pattern of atrophy, hypometabolism and Aβ deposition. More specifically, while all three alterations are found in some regions such as the posterior cingulate, precuneus and temporo-parietal areas [START_REF] Buckner | Molecular, structural, and functional characterization of Alzheimer"s disease: evidence for a relationship between default activity, amyloid, and memory[END_REF][START_REF] Jack | 11C PiB and structural MRI provide complementary information in imaging of Alzheimer"s disease and amnestic mild cognitive impairment[END_REF], regional discrepancies have been highlighted when comparing modalities two by two.

Thus, differential degrees of atrophy and hypometabolism are found in the posterior cingulate cortex versus the hippocampus for example [START_REF] Alsop | Hippocampal hyperperfusion in Alzheimer"s disease[END_REF][START_REF] Chételat | Direct voxel-based comparison between grey matter hypometabolism and atrophy in Alzheimer"s disease[END_REF], and both atrophy and hypometabolism can be found in regions with low Aβ load, or can be absent in regions with high Aβ deposition [START_REF] Edison | Amyloid, hypometabolism, and cognition in Alzheimer disease: an [11C]PIB and [18F]FDG PET study[END_REF][START_REF] Jack | 11C PiB and structural MRI provide complementary information in imaging of Alzheimer"s disease and amnestic mild cognitive impairment[END_REF][START_REF] Li | Regional analysis of FDG and PIB-PET images in normal aging, mild cognitive impairment, and Alzheimer"s disease[END_REF].

The main objective of this study was therefore to characterise and compare the regional degrees of gray matter atrophy, hypometabolism and Aβ deposition as measured in the same patients with Alzheimer"s disease. In addition we aimed at assessing for the first time the local correlations between these three alterations using a voxel-based method. For these purposes, we thoroughly selected patients as having a high probability of Alzheimer"s disease etiology according to recent recommendations [START_REF] Mckhann | The diagnosis of dementia due to Alzheimer"s disease: recommendations from the National Institute on Aging-Alzheimer"s Association workgroups on diagnostic guidelines for Alzheimer"s disease[END_REF] and used a methodology especially designed to compare data from different imaging modalities [START_REF] Chételat | Direct voxel-based comparison between grey matter hypometabolism and atrophy in Alzheimer"s disease[END_REF].

MATERIALS AND METHODS

Participants

Twenty two patients diagnosed with probable Alzheimer"s disease were first selected according to NINCDS-ADRDA clinical criteria [START_REF] Mckhann | Clinical diagnosis of Alzheimer"s disease: report of the NINCDS-ADRDA Work Group under the auspices of Department of Health and Human Services Task Force on Alzheimer"s Disease[END_REF]. Following the recent recommendations from the National Institute on Aging and Alzheimer"s Association workgroup [START_REF] Mckhann | The diagnosis of dementia due to Alzheimer"s disease: recommendations from the National Institute on Aging-Alzheimer"s Association workgroups on diagnostic guidelines for Alzheimer"s disease[END_REF] for research studies, only those 20 patients with a high probability of Alzheimer"s disease etiology (i.e. who have positive neuroimaging biomarkers for both Aβ deposition and neurodegeneration) were selected for further analyses (see below).

Thirty-seven healthy controls were also enrolled in this study following clinical and neuropsychological examination. They had no history or clinical evidence of major neurological or psychiatric disorder and performed in the normal range in all neuropsychological tests (including tests of episodic memory, working memory, language skills, executive functions and visuospatial abilities).

The study was approved by the local ethics committee and all participants gave written consent for participation prior to the scans.

Neuroimaging data acquisition

All participants were scanned on the same MRI and PET cameras at the CYCERON center (Caen, France). The median time lapse between the first and last examination was 16.5 days (min= 1; max =60) for patients and 28 days (min= 8; max =215) for controls.

MRI data.

For each participant, a high-resolution T1-weighted anatomical image was acquired on a Philips (Eindhoven, The Netherlands) Achieva 3T scanner using a 3D fast field echo sequence (3D-T1-FFE sagittal; repetition time = 20 ms; echo time = 4.6 ms; flip angle = 20°; 170 slices; slice thickness = 1 mm; field of view = 256 x 256 mm 2 ; matrix = 256 x 256).

PET data.

Both FDG and Florbetapir PET scans were acquired on a Discovery RX VCT 64 PET-CT device (General Electric Healthcare) with a resolution of 3.76 x 3.76 x 4.9 mm (field of view = 157 mm). Forty-seven planes were obtained with a voxel size of 2.7 x 2.7 x 3.27 mm. A transmission scan was performed for attenuation correction before the PET acquisition. FDG-PET. Participants were fasted for at least 6h before scanning. After a 30-min resting period in a quiet and dark environment, ≈ 180 MBq of FDG were intravenously injected as a bolus. A 10-min PET acquisition scan began 50 min post-injection.

Florbetapir-PET. Each participant underwent a 20-min PET scan, beginning 50 min after the intravenous injection of ≈ 4MBq/kg of Florbetapir.

Neuroimaging Data Handling and Transformation

Preprocessing MRI data were segmented, normalized and modulated using the VBM5.1 toolbox (http://dbm.neuro.uni-jena.de), implemented in the Statistical Parametric Mapping 5 (SPM) software (Wellcome Trust Centre for Neuroimaging, London, UK) to obtain maps of local gray matter volume corrected for brain size. PET data (both FDG and Florbetapir) were corrected for partial volume effects (PMOD Technologies Ltd., Adliswil, Switzerland), coregistered onto their corresponding MRI and normalized using the deformation parameters defined from the MRI procedure. Resultant images underwent quantitative scaling using the cerebellar gray matter as a reference to obtain standardized uptake value ratio (SUVr) images.

Since MRI and PET data have different original spatial resolutions, a differential smoothing was applied to equalize the effective smoothness [START_REF] Richardson | Cortical grey matter and benzodiazepine receptors in malformations of cortical development. A voxel-based comparison of structural and functional imaging data[END_REF][START_REF] Chételat | Direct voxel-based comparison between grey matter hypometabolism and atrophy in Alzheimer"s disease[END_REF]Villain et al., 2008): a Gaussian kernel of 10 x 10 x 10 (x, y, z) mm was used for the MRI and 9.3 x 9.3 x 8.8 mm for the PET data. Finally, images were masked to exclude nongray matter voxels from the analyses.

Creation of W-score maps

In order to obtain measurements of atrophy, hypometabolism and amyloid load expressed in the same unit, therefore enabling a direct comparison of different imaging modalities, Wscore maps were computed for each patient and each imaging modality using the healthy control group as a reference (see Figure 1). W-scores are analogous to Z-scores but they are ajusted for specific covariate(s) [START_REF] Jack Cr | Medial temporal atrophy on MRI in normal aging and very mild Alzheimer"s disease[END_REF][START_REF] Boccardi | The MRI pattern of frontal and temporal brain atrophy in frontotemporal dementia[END_REF][START_REF] Jack | 11C PiB and structural MRI provide complementary information in imaging of Alzheimer"s disease and amnestic mild cognitive impairment[END_REF], age in the present case. Like Z-scores, W-scores have a mean value of 0 and a standard deviation of 1 in the control group, and values of +1.65 and -1.65 correspond to the 95th and 5 th percentiles, respectively. To create W-score maps, voxel-wise regressions were first performed in the control group between age and each imaging data using the SPM software.

Then, W-score maps was computed using the following formula: W-score = [ (patient"s raw value) -(value expected in the control group for the patient"s age) ] / standard deviation of the residuals in controls.

For MRI and FDG-PET, W-scores were reversed so that positive W-scores indicate pathology for all modalities (i.e. less gray matter volume, less glucose metabolism, more Aβ deposition).

For each modality, individual W-score maps were averaged across the patients to provide whole-brain profiles of atrophy, hypometabolism and Aβ deposition, expressed as mean Wscores.

Regions of interest

Finally, to obtain quantitative and statistical information in specific brain areas, mean values of gray matter volume, FDG SUVr and Florbetapir SUVr were extracted from the corresponding images before the smoothing step in 11 regions of interest defined using the automated anatomical labelling atlas [START_REF] Tzourio-Mazoyer | Automated anatomical labeling of activations in SPM using a[END_REF]: hippocampus, amygdala, parahippocampus, temporal pole, angular gyrus, precuneus/posterior cingulate, lateral temporal, anterior cingulate, orbitofrontal, dorsomedial prefrontal, and dorsolateral prefrontal cortices. The neocortical values of Florbetapir retention were also extracted from the Florbetapir SUVr images for the sake of classification of patients as amyloid-positive versus negative (see below).

Selection of patients and subjects using neuroimaging biomarkers.

Patients

As mentioned above, neuroimaging biomarkers were used to further select those Alzheimer"s disease patients with a high likelihood of Alzheimer"s disease etiology, i.e. showing positive biomarkers of both Aβ deposition (from Florbetapir PET) and neurodegeneration (from MRI or FDG-PET), as recommended [START_REF] Mckhann | The diagnosis of dementia due to Alzheimer"s disease: recommendations from the National Institute on Aging-Alzheimer"s Association workgroups on diagnostic guidelines for Alzheimer"s disease[END_REF]. First, values of Florbetapir neocortical SUVr were used to separate amyloid-positive from amyloid-negative patients. For that purpose, a Florbetapir SUVr cutoff value was determined using a cluster analysis performed on the whole sample [START_REF] Bourgeat | Beta-amyloid burden in the temporal neocortex is related to hippocampal atrophy in elderly subjects without dementia[END_REF][START_REF] Rowe | Amyloid imaging results from the Australian Imaging, Biomarkers and Lifestyle (AIBL) study of aging[END_REF], yielding to a value of 1.1. This value is consistent with previous studies using Florbetapir [START_REF] Fleisher | Using positron emission tomography and florbetapir f 18 to image cortical amyloid in patients with mild cognitive impairment or dementia due to Alzheimer disease[END_REF][START_REF] Camus | Using PET with (18)F-AV-45 (florbetapir) to quantify brain amyloid load in a clinical environment[END_REF].

Then, neurodegeneration was assessed using two specific indexes. For atrophy, the hippocampus, amygdala and lateral temporal cortex were combined and a global W-score was computed. Hypometabolism W-score included the angular gyrus, precuneus and posterior cingulate. These areas where chosen as the prototypical neural substrates of Alzheimer"s disease, according to a recent meta-analyses of MRI and FDG-PET studies [START_REF] Schroeter | Neural correlates of Alzheimer"s disease and mild cognitive impairment: a systematic and quantitative meta-analysis involving 1351 patients[END_REF].

Patients were included if they had a positive Florbetapir-PET scan and at least one positive biomarker of neurodegeneration (W-score ≥ 1.65 for atrophy and / or hypometabolism).

Individual values for the three biomarkers are shown in Table 1. Out of the 22 clinicallydiagnosed patients, 2 showed a negative Florbetapir PET-scan. For the 20 remaining patients, at least one of the two biomarkers of neuronal injury was positive. Consequently, these 20

Alzheimer"s disease patients were all included in the following analyses. Only three out of these 20 patients were treated with cholinesterase inhibitors when included while the others were de novo (untreated) patients. All analyses were repeated without these three patients and the conclusions remained unchanged (data not shown) so that all results will be presented on the complete sample of 20 patients.

Healthy controls

In addition, to avoid the presence of healthy subjects at a preclinical stage of Alzheimer"s disease in the control group that could bias our analyses, amyloid-positive subjects were excluded from further analysis. Indeed, the presence of cerebral amyloidosis is believed to indicate the first stage of preclinical Alzheimer"s disease [START_REF] Sperling | Toward defining the preclinical stages of Alzheimer"s disease: Recommendations from the National Institute on Aging and the Alzheimer"s Association workgroup. Alzheimer"s and Dementia[END_REF]. Therefore, three subjects with a Florbetapir SUVr value over 1.1 were withdrawn from the control group.

Demographic data and mini-mental state examinations scores for the 34 controls (14 males, 20 females) and 20 patients with highly probable Alzheimer"s disease (10 males, 10 females) are displayed in Table 2.

Statistical analyses

Comparing local atrophy, hypometabolism and Aβ burden

Voxel-wise analyses. Individual W-score maps of atrophy, hypometabolism and Aβ deposition were compared in a voxel-wise factorial analysis using the SPM software, with MRI, FDG-PET and Florbetapir-PET images as within-subject measurements. Results were considered as significant at a p (family wise error (FWE)-corrected for multiple comparisons) < 0.05 threshold and cluster size k ≥ 20 (160 mm 3 ).

Regions of interest analyses.

Using the values extracted from the 11 regions of interest, regional W-scores were calculated and compared using non-parametric statistical tests. To compare the local degrees of atrophy, hypometabolism and Aβ deposition, a Friedman ANOVA was performed within each region of interest. If significant (i.e. p < 0.05), post hoc two-by-two comparisons were conducted using Wilcoxon signed-rank test. So as to provide a more global insight on the different gradients of alteration across brain regions, a Ward"s hierarchical clustering method was used to group brain regions with closely similar patterns.

Correlation between local alterations

To assess the link between local atrophy, hypometabolism and Aβ deposition, W-score maps were entered two-by-two in voxel-wise correlation analyses using the Biological Parametric Mapping toolbox implemented in SPM and especially designed to analyze brain images from different modalities [START_REF] Casanova | Biological parametric mapping: A statistical toolbox for multimodality brain image analysis[END_REF]. Three independent correlation analyses were performed: W β-amyloid vs W hypometabolism , W β-amyloid vs W atrophy and W atrophy vs W hypometabolism .

Results were considered as significant at a p FWE < 0.05 threshold and cluster size k ≥ 20 (160 mm 3 ).

RESULTS

Patterns of gray matter atrophy, hypometabolism and Aβ deposition in Alzheimer's disease

Averaged W-score maps for each modality are shown in Figure 2. Briefly, gray matter atrophy was found in the medial and lateral temporal, inferior parietal cortex and precuneus.

Hypometabolism concerned the precuneus, posterior cingulate, lateral temporal and parietal cortices, and to a lesser degree the medial temporal lobe. Aβ deposition was found in most brain areas, with highest W-scores in the medial and orbital prefrontal cortex, precuneus and posterior cingulate (with W-score ≥ 9), while the primary sensori-motor cortex, occipital cortex, thalamus and medial temporal lobe were relatively spared.

Voxel-wise comparisons between alterations

The results of the direct between-modality comparison analyses are displayed in Figure 3.

First, the comparison between atrophy and hypometabolism revealed significant differences in both directions. Hypometabolism was greater than atrophy in the precuneus, posterior cingulate, parietal and middle / inferior temporal gyrus while atrophy significantly exceeded hypometabolism in the anterior part of medial temporal lobe. Second, Aβ deposition significantly exceeded atrophy in most brain areas, with greatest differences in frontal areas, precuneus and posterior cingulate, and less but still significant differences in lateral temporal and parietal regions. Conversely, atrophy was superior to Aβ deposition in the anterior part of the medial temporal lobe and in the posterior hippocampus. Third, Aβ deposition considerably exceeded hypometabolism in prefrontal cortex, precuneus and posterior cingulate while the difference was less marked, but still significant, in the lateral temporal and parietal cortices.

Finally, hypometabolism significantly exceeded Aβ deposition in the thalamus and posterior hippocampus.

Voxel-wise correlations between alterations

The two-by-two inter-modality correlation analyses revealed significant results for the atrophy versus hypometabolism analysis only (Figure 4). Specifically, hypometabolism was correlated to local atrophy in the hippocampus, temporoparietal cortex, cuneus / precuneus junction as well as the dorsolateral prefrontal cortex. In contrast, correlations between Aβ deposition and atrophy or hypometabolism did not reach significance at the p FWE < 0.05 threshold. Using a more permissive statistical threshold of p uncorrected < 0.001, a significant positive correlation was found between Aβ load and hypometabolism in a small cluster within the retrosplenial cortex / posterior cingulate (MNI coordinates at the peak: -4, -52, 16; t-value = 4.53; cluster size = 14 voxels).

Regions of interest analyses

The results of the ANOVA comparing the regional W-scores of the three alterations within the regions of interest are displayed in Figure 5. Significant differences were found between the degrees of atrophy, hypometabolism and Aβ deposition in all investigated areas, except for the amygdala (Friedman ANOVA did not reach significance, p = 0.55). Post-hoc comparisons performed with Wilcoxon signed-rank test showed clearly different gradients according to regions (Figure 5), with areas of predominant atrophy (such as the hippocampus) and areas of excessive Aβ deposition as compared to atrophy and hypometabolism (such as the prefrontal cortex).

Using the three mean W-scores of each region, Ward"s hierarchical clustering method led to the distinction between 4 subsets of brain areas (Figure 6, left). A repeated measures ANOVA was conducted on the mean W-score values from the 4 sets of regions, with two factors: brain region (with 4 levels) and imaging modality (with 3 levels). It revealed a highly significant region x modality interaction (F(6; 152) = 35.0; p < 10 -5 ), which confirmed the differential hierarchy between the levels of the three markers in these 4 sets of brain regions (Figure 6, right). In the hippocampo-amygdala complex, atrophy was marked (median W = 2.16), and significantly exceeded hypometabolism (W = 1.99), and Aβ load (W = 0.50). In the parahippocampus and temporal pole, Aβ load was relatively high (W = 4.5) and exceeded local degrees of atrophy (W = 1.45) and hypometabolism (W = 0.78). In posterior association areas (lateral temporal cortex, angular gyrus and precuneus/posterior cingulate), all modalities showed high to very high W scores, with a predominance of Aβ deposition (W = 7.36), exceeding local hypometabolism (W = 3.69), itself exceeding atrophy (W = 2.31). Lastly, in frontal areas (dorso-lateral prefrontal cortex, dorso-medial prefrontal cortex, orbito-frontal cortex and anterior cingulate), extreme Aβ deposition (W = 7.99) contrasted to the weakness of structural and metabolic defects (respectively 0.61 and 0.85).

DISCUSSION

In this study, three major brain alterations in Alzheimer's disease, namely gray matter atrophy, hypometabolism and Aβ deposition, were measured concomitantly in the same patients with Alzheimer's disease dementia. Their relative expression and relationships were assessed both regionally and voxel-wise. We showed marked regional variability in the hierarchy between these different brain alterations with three main profiles: i) extreme Aβ deposition with low hypometabolism and atrophy (frontal areas); ii) predominance of Aβ deposition, together with high hypometabolism and lower but still significant atrophy (posterior association areas); and iii) predominance of atrophy and hypometabolism faced to low Aβ burden (in medial temporal areas). In addition, local correlations were found between atrophy and hypometabolism within a large parieto-temporal network, while Aβ deposition did not correlate to either atrophy or hypometabolism.

One of the most striking results of this study is the considerable discrepancy between Aβ deposition on the one hand, and atrophy and hypometabolism on the other hand in patients with Alzheimer"s disease dementia. First, the brain distribution of Aβ deposition clearly differed from that of the two other alterations (Figure 3, middle and right). Second, Aβ deposition did not correlate to the other alterations. Previous studies assessing the correlations between Aβ deposition and metabolism in patients with Alzheimer's disease reported conflicting results, and, when significant, correlations were only found in restricted areas, mostly in temporal and parietal cortices [START_REF] Engler | Two-year follow-up of amyloid deposition in patients with Alzheimer"s disease[END_REF][START_REF] Edison | Amyloid, hypometabolism, and cognition in Alzheimer disease: an [11C]PIB and [18F]FDG PET study[END_REF][START_REF] Cohen | Basal cerebral metabolism may modulate the cognitive effects of Abeta in mild cognitive impairment: an example of brain reserve[END_REF][START_REF] Forsberg | High PIB retention in Alzheimer"s disease is an early event with complex relationship with CSF biomarkers and functional parameters[END_REF]. This is consistent with our finding of a restricted correlation in the posterior cingulate cortex when using a more permissive threshold. As for the link between atrophy and Aβ deposition, while relationships have been reported in preclinical stages, most studies reported no significant correlations in demented patients [START_REF] Fagan | Decreased cerebrospinal fluid Abeta(42) correlates with brain atrophy in cognitively normal elderly[END_REF][START_REF] Chételat | Relationship between atrophy and beta-amyloid deposition in Alzheimer disease[END_REF].

These findings are in line with the idea that, at symptomatic stages of the disease, Aβ does not play a prominent role in on-going neurodegeneration [START_REF] Hyman | Amyloid-Dependent and Amyloid-Independent Stages of Alzheimer Disease[END_REF]. Instead, Tau pathology, disturbance in axonal transport, or inflammation processes, could contribute to brain atrophy and metabolic disruption (see below). They could also be interpreted in the light of current models suggesting a time decay between the appearance of Aβ plaques, more than one decade before the first symptoms, and hypometabolism and atrophy thought to appear years later, closer to cognitive decline [START_REF] Jack | Hypothetical model of dynamic biomarkers of the Alzheimer"s pathological cascade[END_REF]. Another interesting hypothesis is the possibility of distant, rather than local, effects of Aβ. Notably, [START_REF] Cohen | Basal cerebral metabolism may modulate the cognitive effects of Abeta in mild cognitive impairment: an example of brain reserve[END_REF] observed that metabolism in the precuneus of amyloid-positive patients with Alzheimer"s disease was inversely correlated to Aβ deposition in frontal areas. [START_REF] Bourgeat | Beta-amyloid burden in the temporal neocortex is related to hippocampal atrophy in elderly subjects without dementia[END_REF] also reported a distant relationship (between temporal neocortical Aβ deposition and hippocampal atrophy); yet the correlation was observed in the cognitively normal group and not in demented patients.

In contrast, atrophy and hypometabolism were found in the same brain areas and were highly correlated (Figure 4), suggesting that both alterations share at least partly common underlying mechanisms. The direct comparison between both processes also revealed regional differences in their relative degrees (Figure 3) indicating that additional biochemical mechanisms may amplify or attenuate hypometabolism or atrophy in specific brain regions.

The other notable finding is the regional variability in the hierarchy between the relative degrees of atrophy, hypometabolism and Aβ deposition, which suggests that these alterations are subtended by multiple region-specific processes. Thus, by contrast to current global models that propose a general chronology in the appearance of the different alterations (independently of brain regions), the present study rather highlights the regional discrepancy in this sequence, suggesting that different subtending processes may be involved in the different brain regions. While the absolute number of sets of brain regions is arbitrary as it depends on the subjective dendograms cutoff, our findings however revealed clearly distinguishable patterns with three poles: the hippocampus, the posterior association areas, and the frontal cortex.

In the hippocampus, the presence of abnormally phosphorylated tau proteins that aggregate to form neurofibrillary tangles (NFT), may be one of the processes underlying severe atrophy and moderate hypometabolism in spite of low Aβ burden. Indeed, neurodegeneration seems to be closely related to tau pathology: progression of both gray matter atrophy (Whitwell et al., 2008) and hypoperfusion [START_REF] Bradley | Cerebral perfusion SPET correlated with Braak pathological stage in Alzheimer"s disease[END_REF] through the brain seems to follow NFT spreading as evaluated with Braak staging. In addition, neuronal loss has been shown to correlate with local counts of NFT in different brain areas such as the entorhinal cortex [START_REF] Gómez-Isla | Profound Loss of Layer II Entorhinal Cortex Neurons Occurs in Very Mild Alzheimer"s Disease[END_REF] and superior temporal sulcus [START_REF] Gómez-Isla | Neuronal loss correlates with but exceeds neurofibrillary tangles in Alzheimer"s disease[END_REF].

Consequently, as NFT first appear and remain predominant in medial temporal structures as the disease pathology progresses [START_REF] Braak | Neuropathological stageing of Alzheimer-related changes[END_REF][START_REF] Delacourte | The biochemical pathway of neurofibrillary degeneration in aging and Alzheimer"s disease[END_REF], it is likely that NFT, rather than Aβ deposition, are responsible for hippocampal atrophy and hypometabolism. However, the possibility of a distant effect of Aβ on the hippocampus cannot be ruled out, especially because the brain structures to which the hippocampus projects the most, i.e. the retrosplenial cortex and medial and orbital prefrontal cortex [START_REF] Aggleton | Multiple anatomical systems embedded within the primate medial temporal lobe: Implications for hippocampal function[END_REF] are also those showing the highest Aβ load (Figure 2, right).

In posterior association areas, several factors have been proposed to induce neuronal dysfunction over and above local Aβ pathology. Notably, recent studies suggested that hypometabolism observed in patients with mild cognitive impairment and Alzheimer's disease dementia (especially in the posterior cingulate cortex) at least partly results from hippocampal atrophy through cingulate bundle disruption (Villain et al., 2008;[START_REF] Choo | Posterior cingulate cortex atrophy and regional cingulum disruption in mild cognitive impairment and Alzheimer"s disease[END_REF]Villain et al., 2010). Besides, as mentioned earlier, relationships between hypometabolism in the precuneus and prefrontal Aβ load have been observed [START_REF] Cohen | Basal cerebral metabolism may modulate the cognitive effects of Abeta in mild cognitive impairment: an example of brain reserve[END_REF], possibly revealing distant Aβ effects. Actually, the posterior cingulate, precuneus, angular gyrus and lateral temporal cortex are "cortical hubs" [START_REF] Buckner | Cortical Hubs Revealed by Intrinsic Functional Connectivity: Mapping, Assessment of Stability, and Relation to Alzheimer"s Disease[END_REF], i.e. nodes in brain networks that are tightly connected to numerous brain structures. This may explain their vulnerability to Alzheimer"s disease-related pathological processes in widely distributed brain regions.

Lastly, in frontal areas, Aβ load reached its maxima while atrophy and hypometabolism were minimal. Although already noticed by previous authors [START_REF] Edison | Amyloid, hypometabolism, and cognition in Alzheimer disease: an [11C]PIB and [18F]FDG PET study[END_REF][START_REF] Jack | 11C PiB and structural MRI provide complementary information in imaging of Alzheimer"s disease and amnestic mild cognitive impairment[END_REF][START_REF] Li | Regional analysis of FDG and PIB-PET images in normal aging, mild cognitive impairment, and Alzheimer"s disease[END_REF], the present study provides a quantitative evidence for this discrepancy. This may appear surprising considering studies in animal models showing that severe neurite abnormalities especially develop in the vicinity of Aβ deposits [START_REF] Tsai | Fibrillar amyloid deposition leads to local synaptic abnormalities and breakage of neuronal branches[END_REF] because they are surrounded by a halo of soluble oligomeric forms of Aβ [START_REF] Koffie | Oligomeric amyloid beta associates with postsynaptic densities and correlates with excitatory synapse loss near senile plaques[END_REF], themselves shown to be toxic to synapses [START_REF] Lacor | Abeta oligomer-induced aberrations in synapse composition, shape, and density provide a molecular basis for loss of connectivity in Alzheimer"s disease[END_REF]. Our findings may reflect i) a lack of deleterious effect of local Aβ deposits in humans; ii) a difference in the timing of the different biomarkers (see above); or iii) the presence of compensation processes. This later hypothesis is supported by previous functional MRI studies showing greater activations in patients with Alzheimer"s disease relative to healthy controls (notably in the frontal cortex), positively associated with cognitive performances and interpreted as the reflect of compensation processes (see [START_REF] Schwindt | Functional imaging studies of episodic memory in Alzheimer"s disease: a quantitative meta-analysis[END_REF] for review). It is thus also possible that, faced to Aβ deposition, neuronal and synaptic plasticity occurs, allowing to maintain neuronal integrity, glucose consumption and brain volume.

Note that the patient sample size is relatively limited in the present study as compared to large multicentre studies. However, i) our patients have been thoroughly selected as having a high probability of Alzheimer's disease etiology based on neuroimaging biomarkers as recently recommended; ii) the use of multimodal data obtained in a single laboratory has clear advantage upon multicentre data for complex and precise voxel-wise analyses as performed here; iii) our findings were confirmed using non-parametric tests; iv) our results are entirely consistent with previous studies when applicable. Thus, whole brain patterns of atrophy, hypometabolism and Aβ deposition (Figure 2) are highly similar to those reported in studies from different samples [START_REF] Buckner | Molecular, structural, and functional characterization of Alzheimer"s disease: evidence for a relationship between default activity, amyloid, and memory[END_REF][START_REF] Edison | Amyloid, hypometabolism, and cognition in Alzheimer disease: an [11C]PIB and [18F]FDG PET study[END_REF][START_REF] Alsop | Hippocampal hyperperfusion in Alzheimer"s disease[END_REF][START_REF] Chételat | Direct voxel-based comparison between grey matter hypometabolism and atrophy in Alzheimer"s disease[END_REF][START_REF] Jack | 11C PiB and structural MRI provide complementary information in imaging of Alzheimer"s disease and amnestic mild cognitive impairment[END_REF][START_REF] Caroli | Metabolic Compensation and Depression in Alzheimer"s Disease[END_REF][START_REF] Fleisher | Using positron emission tomography and florbetapir f 18 to image cortical amyloid in patients with mild cognitive impairment or dementia due to Alzheimer disease[END_REF], and the map of local mismatch between atrophy and hypometabolism (Figure 3, left) is almost identical to previous studies by [START_REF] Alsop | Hippocampal hyperperfusion in Alzheimer"s disease[END_REF] and [START_REF] Caroli | Metabolic Compensation and Depression in Alzheimer"s Disease[END_REF] despite the use of different methodologies and samples. Yet, we cannot exclude the possibility of a lack of statistical power in the analyses, for instance to detect subtle relationships between Aβ deposition and neurodegeneration.

As a whole, our findings highlight the complex local relationships between atrophy, hypometabolism and Aβ load and the regional heterogeneity in their hierarchy, suggesting the involvement of different underlying processes. It would be of particular interest to assess these relationships in earlier stages of the disease, and to consider not only local but also distant phenomena to get a more comprehensive overview of these different brain alterations and their interactions. Finally, longitudinal studies are warranted to take into account the potential lapse between the different pathological processes while assessing their relationships and eventually propose a more complete, region-specific, model of Alzheimer"s disease pathology evolution.

FIGURE LEGENDS Figure 1 (width: 1.5 columns)

Illustration of the different steps required to create W-score maps using the SPM software.

This figure shows the procedure for the atrophy W-score map of a 65-year-old patient with Alzheimer"s disease as compared to 6 healthy controls (HC) for the sake of illustration.

First, a simple linear regression was performed in the HC group to estimate age-related changes (1a), resulting in several files: the β1 map containing voxel-wise age-related regression coefficients, the β2 map containing intercept values, and the individual maps of residuals. The standard deviation of residuals was computed voxel-wise (1b). A W-score map was then created using the corresponding formula and previously-computed maps using SPM"s "ImCalc" function (2a). Lastly, for MRI and FDG-PET data, W-values were reversed so that positive numbers represent pathological features in all three imaging modalities (2b).

Figure 2 (width: 2 columns)

Brain patterns of alteration in the 20 patients with Alzheimer"s disease dementia. For each imaging modality, local degrees of alteration are expressed as mean W-score as compared to the control group (n = 34) in each gray matter voxel. Note that for all imaging modalities, a positive W-score indicates a pathological feature. Colors have been scaled to the range of each modality to fit to the regional distribution of each process. For clarity, only the left hemisphere is represented here, as results were mainly symmetrical.

Figure 3 (width: 2 columns)

Voxel-wise comparisons between the local degrees of atrophy, hypometabolism and βamyloid deposition in the 20 patients with Alzheimer"s disease dementia.

The T-value of 5.2 used as a threshold in this figure corresponds to the P (family wise error corrected) < 0.05 threshold described in the text.

  Figures 1 and 6: 1.5 columns

Figure 4 (

 4 Figure 4 (width: 1 column)

Figure 5 (

 5 Figure 5 (width: 2 columns)

Figure

  Figure 6 (width: 1.5 columns)

  Figure 1

Table 1

 1

	type of								Patients with a clinical diagnosis of probable Alzheimer's disease
		index used
	biomarker
			1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22
	Aβ deposition	Florbetapir neocortical	1.46 1.54 1.86 1.20 1.48 1.81 0.87 1.80 1.92 1.71 1.41 1.77 1.56 1.28 1.29 1.83 1.72 0.89 1.59 1.68 1.45 1.76
		SUVr
		Atrophy W score 3.36 1.82 5.71 1.33 3.52 4.77 6.11 3.24 2.37 2.07 1.36 2.51 0.13 1.08 2.84 1.91 4.25 2.01 1.72 2.85 3.80 1.38
	neuronal
	degeneration
		Hypometabolism	1.39 4.44 4.88 1.75 5.51 5.83 1.96 3.26 3.26 4.33 2.97 4.42 1.73 1.89 3.05 5.52 4.83 2.30 2.40 4.84 1.62 4.30
		W score
	Presence of both biomarkers ?	yes yes yes yes yes yes	no	yes yes yes yes yes yes yes yes yes yes	no	yes yes yes yes

Acknowledgments

This study was funded by Agence Nationale de la Recherche (ANR LONGVIE 2007), Programme Hospitalier de Recherche Clinique (PHRC National 2008), Région Basse Normandie, and Institut National de la Santé et de la Recherche Médicale (Inserm), including the Inserm Liliane Bettencourt School (MD-PhD Program). The authors are grateful to M. Fouquet and N. Villain for their precious comments and support. The authors would like to thank K. Mevel, H. Mirabel, M. Gaubert, S. Egret, A. Pélerin, C. Schupp, J. Dayan, A. Quillard, A. Manrique, C. Lebouleux, M.H. Noel & M.C. Onfroy for their help in the data acquisition and the volunteers who participated in this study.

Selection of patients with a high probability of Alzheimer"s disease etiology based on neuroimaging biomarkers of both amyloid deposition and neuronal degeneration [START_REF] Mckhann | The diagnosis of dementia due to Alzheimer"s disease: recommendations from the National Institute on Aging-Alzheimer"s Association workgroups on diagnostic guidelines for Alzheimer"s disease[END_REF]. Amyloid load was assessed through neocortical Florbetapir SUVr and considered as positive over 1.1 (see Methods for more detail). Alzheimer"s disease type neuronal degeneration was considered using two indexes: i) atrophy in the hippocampus, amygdala and lateral temporal lobe and ii) hypometabolism in the angular gyrus, precuneus and posterior cingulate. Both indexes are expressed in W-scores (i.e. age-adjusted Z-scores) as compared to the control group. Patients were selected for further analysis if they had a Florbetapir-positive PET scan together with at least one neurodegeneration degeneration W-score above 1.65.

Indexes considered as positive are bolded. This condition was fulfilled for all patients, except for patients 7 and 18 that were consequently excluded from further analyses.

Table 2

Demography and mini mental state examination (MMSE) scores in the groups used for data analyses. Statistical analyses were performed using Mann-Whitney U test (for age, education and MMSE) and Yates's chi-squared test (sex ratio). AD: Alzheimer"s disease patients; HC: healthy controls.