Yann Bugeaud
email: bugeaud@math.unistra.fr

AND Lingmin Liao
email: lingmin.liao@u-pec.fr

Micha L Rams
email: m.rams@impan.pl

METRICAL RESULTS ON THE DISTRIBUTION OF FRACTIONAL PARTS OF POWERS OF REAL NUMBERS

Keywords: 2010 Mathematics Subject Classification: Primary 11K36 Secondary 11J71, 28A80 fractional parts of powers, Diophantine approximation, Hausdorff dimension

Denote by {•} the fractional part. We establish several new metrical results on the distribution properties of the sequence ({x n }) n≥1 . Many of them are presented in a more general framework, in which the sequence of functions (x → x n ) n≥1 is replaced by a sequence (fn) n≥1 , under some growth and regularity conditions on the functions fn.

Introduction

Let {•} denote the fractional part and • the distance to the nearest integer. For a given real number x > 1, only few results are known on the distribution of the sequence ({x n }) n≥1 . For example, we still do not know whether 0 is a limit point of ({e n }) n≥1 , nor of ({( 32 ) n }) n≥1 ; see [START_REF] Bugeaud | Distribution modulo one and Diophantine approximation[END_REF] for a survey of related results.

However, several metric statements have been established. The first one was obtained in 1935 by Koksma [START_REF] Koksma | Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins[END_REF], who proved that for almost every x > 1 the sequence ({x n }) n≥1 is uniformly distributed on the unit interval [0, 1]. Here and below, almost every always refers to the Lebesgue measure. In 1967, Mahler and Szekeres [START_REF] Mahler | On the approximation of real numbers by roots of integers[END_REF] studied the quantity P (x) := lim inf x n 1/n (x > 1).

They proved that if P (x) = 0 then x is transcendental, and P (x) = 1 for almost all x > 1. The function x → P (x) was subsequently studied in 2008 by Bugeaud and Dubickas [START_REF] Bugeaud | On a problem of Mahler and Szekeres on approximation by roots of integers[END_REF]. Among other results, it was shown in [START_REF] Bugeaud | On a problem of Mahler and Szekeres on approximation by roots of integers[END_REF] that, for all v > u > 1 and b > 1, we have dim H {x ∈ (u, v) : P (x) ≤ 1/b} = log v log (bv) ,

where dim H denotes the Hausdorff dimension. In a different direction, Pollington [START_REF] Pollington | The Hausdorff dimension of certain sets related to sequences which are not dense mod 1[END_REF] showed in 1980 that there are many real numbers x > 1 such that ({x n }) n≥1 is very far from being well distributed, namely he established that, for any ε > 0, we have dim H x > 1 : {x n } < ε for all n = 1. This result has been subsequently extended by Bugeaud and Moshchevitin [START_REF] Bugeaud | On fractional parts of powers of real numbers close to 1[END_REF] and, independently, by Kahane [START_REF] Kahane | Sur la répartition des puissances modulo 1[END_REF], who proved that for any ε > 0, for any sequence of real numbers (y n ) n≥1 , we have dim H x > 1 : x n -y n < ε for all n = 1.

In the present paper, we further investigate, from a metric point of view, the Diophantine approximation properties of the sequence ({x n }) n≥1 , where x > 1, and extend several known results to more general families of sequences ({f n (x)}) n≥1 , under some conditions on the sequence of functions (f n ) n≥1 .

As a consequence of our main theorem, we obtain an inhomogeneous version of the result of Bugeaud and Dubickas [START_REF] Bugeaud | On a problem of Mahler and Szekeres on approximation by roots of integers[END_REF] mentioned above.

Theorem 1. Let b > 1 be a real number and y = (y n ) n≥1 an arbitrary sequence of real numbers in [0, [START_REF] Baker | On the distribution of powers of real numbers modulo 1[END_REF]. Set E(b, y) := {x > 1 : x n -y n < b -n for infinitely many n}.

For every v > 1, we have

lim ε→0 dim H ([v -ε, v + ε] ∩ E(b, y)) = log v log(bv)
.

In the homogeneous case (that is, the case where y n = 0 for n ≥ 1), Theorem 1 was proved in [START_REF] Bugeaud | On a problem of Mahler and Szekeres on approximation by roots of integers[END_REF] by using a classical result of Koksma [START_REF] Koksma | Sur la théorie métrique des approximations diophantiques[END_REF] and the mass transference principle developed by Beresnevich and Velani [START_REF] Beresnevich | A Mass Transference Principle and the Duffin-Schaeffer conjecture for Hausdorff measures[END_REF]. The method of [START_REF] Bugeaud | On a problem of Mahler and Szekeres on approximation by roots of integers[END_REF] still works when y is a constant sequence, but one then needs to apply the inhomogeneous version of Koksma's theorem in [START_REF] Koksma | Sur la théorie métrique des approximations diophantiques[END_REF].

Here, for an arbitrary sequence (y n ) n≥1 , we use a direct construction.

Letting v tend to infinity in Theorem 1, we obtain the following immediate corollary.

Corollary 2. For an arbitrary sequence y of real numbers in [0, 1] and any real number b > 1, the set E(b, y) has full Hausdorff dimension.

Theorem 1 gives, for every v > 1, the value of the localized Hausdorff dimension of E(b, y) at the point v. We stress that, in the present context, the localized Hausdorff dimension varies with v, while this is not at all the case for many classical results, including the Jarník-Besicovitch Theorem and its extensions. Taking this point of view allows us also to place Theorem 1 in a more general context, where the family of functions x → x n is replaced by an arbitrary family of functions f n satisfying some regularity and growth conditions.

We consider a family of strictly positive increasing

C 1 functions f = (f n ) n≥1 defined on an open interval I ⊂ R and such that f n (x), f n (x) > 1 for all x ∈ I. For τ > 1, define E(f, y, τ ) := {x ∈ I : f n (x) -y n < f n (x) -τ for infinitely many n}. For v ∈ I, put u(v) := lim sup n→∞ log f n (v) log f n (v) , (v) := lim inf n→∞ log f n (v) log f n (v) .
We will assume the regularity condition

(1.1) lim r→0 lim sup n→∞ sup |x-y|<r log f n (x) log f n (y) = 1,
which guarantees the continuity of the functions u and .

For non-linear functions f n , i.e., when f n is not of the form f n (x) = a n • x + b n , we also need the following condition:

(1.2) M := sup n≥1 log f n+1 (v) log f n (v) < ∞ for all v ∈ I.
Theorem 1 is a particular case of the following general statement.

Theorem 3. Consider a family of strictly positive increasing

C 1 functions f = (f n ) n≥1 defined on an open interval I ⊂ R and such that f n (x), f n (x) > 1 for all x ∈ I. Assume (1.1) and (1.2). If for all x ∈ I, ∀ε > 0, ∞ n=1 f n (x) -ε < ∞, (1.3)
then, for any v ∈ I and any τ > 1, we have

1 1 + τ u(v) ≤ lim ε→0 dim H ([v -ε, v + ε] ∩ E(f, y, τ )) ≤ 1 1 + τ (v)
.

If the functions f n are linear then we do not need to assume (1.2), and the assertion gets strengthened to

lim ε→0 dim H ([v -ε, v + ε] ∩ E(f, y, τ )) = 1 1 + τ (v)
.

We remark that the condition (1.3) is satisfied if

∀x ∈ I, lim n→∞ log f n (x) log n = ∞. (1.4)
We also observe that the condition (1.1) implies that (v) ≥ 1 for v in I. In many cases (in particular, for f n (x) = x n ), we have u(v) = (v) = 1 for v in I.

It follows from the formulation of Theorem 3 that the real number τ can be replaced by a continuous function τ : I → (0, ∞), in which case the set E(f, y, τ ) is defined by E(f, y, τ ) := {x ∈ I : f n (x) -y n < f n (x) -τ (x) for infinitely many n}.

We get at once the following localized version of Theorem 3. For the classical Jarník-Besicovitch Theorem, such a localized theorem was obtained by Barral and Seuret [START_REF] Barral | A localized Jarnik-Besicovich theorem[END_REF], who were the first to consider localized Diophantine approximation.

Corollary 4. With the above notation and under the hypotheses of Theorem 3, we have

1 1 + τ (v)u(v) ≤ lim ε→0 dim H ([v -ε, v + ε] ∩ E(f, y, τ )) ≤ 1 1 + τ (v) (v)
.

We illustrate Theorem 3 and Corollary 4 by some examples. If the family of functions f = (f n ) n≥1 in Theorem 3 is such that, for every x in I, the sequence (f n (x)) n≥1 increases sufficiently rapidly, then

lim ε→0 dim H ([v -ε, v + ε] ∩ E(f, y, τ )) = 1 1 + τ ,
independently of the family f . This applies, for example, to the families of functions x n 2 , x n , 2 n x and x √ n . The case f n (x) = a n x, where (a n ) n≥1 is an increasing sequence of positive integers, has been studied by Borosh and Fraenkel [START_REF] Borosh | A generalization of Jarník's theorem on Diophantine approximations[END_REF] (but only in the special case of a constant sequence y equal to 0). Let I be an open, non-empty, real interval. They proved that dim H {x ∈ I :

a n x < a -τ n } = 1 + s 1 + τ ,
where s (usually called the convergence exponent of the sequence (a n ) n≥1 ) is the largest real number in [0, 1] such that n≥1 a -s-ε n converges for any ε > 0.

The case s = 0 of their result, which corresponds to rapidly growing sequences (a n ) n≥1 , follows from Theorem 3. The case a n = n for n ≥ 1 corresponds to the Jarník-Besicovitch Theorem. We stress that the assumption (1.3) is satisfied only if (a n ) n≥1 increases sufficiently rapidly.

Questions of uniform Diophantine approximation were recently studied by Bugeaud and Liao [START_REF] Bugeaud | Uniform Diophantine approximation related to b-ary and β-expansions[END_REF] for the b-ary and β-expansions and by Kim and Liao [START_REF] Kim | Dirichlet uniformly well-approximated numbers[END_REF] for the irrational rotations. In this paper, we consider the uniform Diophantine approximation of the sequence ({x n }) n≥1 with x > 1.

For a real number B > 1 and a sequence of real numbers y = (y n ) n≥1 in [0, 1], set

F (B, y) := {x > 1 : for all large integer N, x n -y n < B -N has a solution 1 ≤ n ≤ N }.
Our next theorem gives a lower bound for the Hausdorff dimension of F (B, y) intersected with a small interval.

Theorem 5. Let B > 1 be a real number and y an arbitrary sequence of real numbers in [0, 1]. For any v > 1, we have

lim ε→0 dim H ([v -ε, v + ε] ∩ F (B, y)) ≥ log v -log B log v + log B 2 .
Unfortunately, we are unable to decide whether the inequality in Theorem 5 is an equality. Observe that the lower bound we obtain is the same as the one established in [START_REF] Bugeaud | Uniform Diophantine approximation related to b-ary and β-expansions[END_REF] for a question of uniform Diophantine approximation related to b-ary and β-expansions.

Letting v tend to infinity, we have the following corollary.

Corollary 6. For an arbitrary sequence y of real numbers in [0, 1] and any real number B > 1, the set F (B, y) has full Hausdorff dimension.

We end this paper with results on sequences ({x n }) n≥1 , with x > 1, which are badly distributed, in the sense that all of their points lie in a small interval. As above, we take a more general point of view. Consider a family of C 1 strictly positive increasing functions f = (f n ) n≥1 defined on an open interval I ⊂ R and such that f n (x), f n (x) > 1 for all x ∈ I and for all n ≥ 1. Let δ = (δ n ) n≥1 be a sequence of positive real numbers such that

δ n < 1/4 for n ≥ 1. Set G(f, y, δ) := {x ∈ I : f n (x) -y n ≤ δ n , ∀n ≥ 1}.
We need the following hypotheses:

∀ε > 0, ∀n ≥ 1, inf x∈(v-ε,v+ε) f n+1 (x) sup x∈(v-ε,v+ε) f n (x) • δ n ≥ 2, (1.5) ∀x ∈ I, lim n→∞ log f n+1 (x) log f n (x) = ∞. (1.6)
Our last main theorem is as follows.

Theorem 7. Keep the above notation. Under the hypotheses (1.1), (1.5), and (1.6), for all v ∈ I, we have

(1.7) lim ε→0 dim H ([v -ε, v + ε] ∩ G(f, y, δ)) = lim inf n→∞ log f n (v) + n-1 j=1 log δ j log f n (v) -log δ n .
We remark that our result extends a recent result of Baker [START_REF] Baker | On the distribution of powers of real numbers modulo 1[END_REF]. In fact, in [START_REF] Baker | On the distribution of powers of real numbers modulo 1[END_REF], the author studied the special case f n (x) = x qn with (q n ) n≥1 being a strictly increasing sequence of real numbers such that lim n→∞ (q n+1 -q n ) = +∞.

Our result also gives the following corollary. Then, for any sequence (y n ) n≥1 of real numbers, we have

dim H {x ∈ R : lim n→+∞ a n x -y n = 0} = 1.

Basic tools

We present two lemmas which serve as important tools for estimating the Hausdorff dimension of the sets studied in this paper.

Let [0, 1] = E 0 ⊃ E 1 ⊃ E 2 ⊃ • • • be a decreasing sequence of sets, with each E k a finite union of disjoint closed intervals. The components of E k are called k-th level basic intervals. Set F = ∩ ∞ k=0 E k .
We do not assume that each basic interval in E k-1 contains the same number of next level basic intervals, nor that they are of the same length, nor that the gaps between two consecutive basic intervals are equal. Instead, for x ∈ E k-1 , we denote by m k (x) the number of k-th level basic intervals contained in the (k -1)-th level basic interval containing x, and by εk (x) the minimal distance between two of them. Set

ε k (x) = min i≤k εi (x).
In the following, we generalize a lemma in Falconer's book [START_REF] Falconer | Fractal Geometry, Mathematical Foundations and Application[END_REF]Example 4.6].

Lemma 9.

For any open interval I ⊂ [0, 1] intersecting F , we have

dim H (I ∩ F ) ≥ inf x∈I∩F lim inf k→∞ log(m 1 (x) • • • m k-1 (x)) -log(m k (x)ε k (x)) .
Proof. The proof is similar to that in the book of Falconer. We define a probability measure µ on F by assigning the mass evenly. Precisely, for k ≥ 1, let I k (x) be the k-th level interval containing x. For x ∈ F and k ≥ 1, we put a mass (m

1 (x) • • • m k (x)) -1
to the interval I k (x). Note that any two k-th basic intervals contained in the same (k -1)-th interval have the same measure. One can check that the measure µ is well defined. Now let us calculate the local dimension at the point x. Let B(x, r) be the ball of radius r centered at x. Suppose that

ε k (x) ≤ 2r < ε k-1 (x). The number of k-th level intervals intersecting B(x, r) is at most min m k (x), 2r ε k (x) + 1 ≤ min m k (x), 4r ε k (x) ≤ m k (x) 1-s 4r ε k (x) s , for any s ∈ [0, 1]. Thus µ(B(x, r)) ≤ m k (x) 1-s 4r ε k (x) s • (m 1 (x) • • • m k (x)) -1 . Hence log µ(B(x, r)) log r ≥ s log m k (x)ε k (x) -s log(4r) + log(m 1 (x) • • • m k-1 (x)) -log r .
Let s be in (0, 1) such that

s < inf z∈I∩F lim inf k→∞ log(m 1 (z) • • • m k-1 (z)) -log m k (z)ε k (z) ≤ lim inf k→∞ log(m 1 (x) • • • m k-1 (x)) -log m k (x)ε k (x) . Then s log m k (x)ε k (x) -s log 4 + log(m 1 (x) • • • m k-1 (x)) ≥ 0, for k large enough. Therefore lim inf r→0 log µ(B(x, r)) log r ≥ s.
The proof is completed by applying the mass distribution principle (see [START_REF] Falconer | Techniques in Fractal Geometry[END_REF], Proposition 2.3).

We also have an upper bound for the dimension of the set I ∩ F . Denote by |I k (x)| the length of the k-th basic interval I k (x) containing x.

Lemma 10. For any open interval

I ⊂ [0, 1] intersecting F , we have dim H (I ∩ F ) ≤ sup x∈I∩F lim inf k→∞ log(m 1 (x) • • • m k (x)) -log |I k (x)| .
Proof. We define the same probability measure µ as in Lemma 9, i.e., the interval

I k (x) has measure (m 1 (x) • • • m k (x)) -1 . Then lim inf r→0 log µ(B(x, r)) log r ≤ lim inf k→∞ log µ(I k (x)) -log |I k (x)| = lim inf k→∞ log(m 1 (x) • • • m k (x)) -log |I k (x)| .
We finish the proof by applying again the mass distribution principle (see [START_REF] Falconer | Techniques in Fractal Geometry[END_REF], Proposition 2.3).

Asymptotic approximation

In this section, we prove Theorem 3. To see that Theorem 1 is a special case of it, take the family of functions f defined by

f n (x) = x n , ∀n ≥ 1, we have u(v) = (v) = 1 and [v -ε, v + ε] ∩ E f, y, log b log(v + ε) ⊂ [v -ε, v + ε] ∩ E(b, y) ⊂ [v -ε, v + ε] ∩ E f, y, log b log(v -ε)
.

Then, Theorem 1 follows directly from Theorem 3. Now we prove Theorem 3.

Proof of Theorem 3. Lower bound: We can assume that u(v) is finite, since otherwise there is nothing to prove. Let us start by the simple observation about the condition (1.1). Given an integer n ≥ 1, set (3.1)

η(n) = sup log f n (w) log f n (z) -1; w, z ∈ [v -ε, v + ε], |f n (w) -f n (z)| ≤ 1 .
Lemma 11. If (1.1) and (1.3) hold, then

lim n→∞ η(n) = 0.
Proof. Assume this is not true. Then there exists a sequence of integers (n i ) and a sequence of pairs of points (w i , z i ) such that

|f n i (w i ) -f n i (z i )| ≤ 1 and log f n i (w i ) log f n i (z i ) > Z > 1.
By compactness of [v -ε, v + ε], taking a subsequence if necessary, we can assume that (w i ) i≥1 converges to some point w 0 . By (1.3), f n (v) → ∞. Hence, (1.1) gives us

lim n→∞ inf x∈[v-ε,v+ε] f n (x) = ∞.
This implies that

|w i -z i | ≤ 1 inf x∈[v-ε,v+ε] f n i (x)
→ 0 as i → ∞, and hence any neighborhood of w 0 contains all except finitely many points w i , z i . Thus, in any neighbourhood U of w 0 we have lim sup

n→∞ sup w,z∈U log f n (w) log f n (z) > Z,
which is a contradiction with (1.1). Now we construct a nested Cantor set which is the intersection of unions of subintervals at level n i , where (n i ) i≥1 is an increasing sequence of positive integers which will be defined precisely later. Suppose we have already well chosen this subsequence. Let us describe the nested family of subintervals. For each level i, we need to consider the set of points x such that

f n i (x) -y n i ≤ f n i (x) -τ .
By the property f n 1 (x) -y n 1 ≤ f n 1 (x) -τ , we take the intervals at level 1 as

I 1 (k, v, f, y, τ ) := [f -1 n 1 (k + y n 1 -f n 1 (v + ε) -τ ), f -1 n 1 (k + y n 1 + f n 1 (v + ε) -τ )], with k being an integer in [f n 1 (v -ε) + 1, f n 1 (v + ε) -1].
Suppose we have constructed the intervals at level i -

1. Let [c i-1 , d i-1 ] be an interval at such level. A subinterval of [c i-1 , d i-1 ] at level i is such that [f -1 n i (k + y n i -f n i (d i-1 ) -τ ), f -1 n i (k + y n i + f n i (d i-1 ) -τ )], with k being an integer in [f n i (c i-1 ) + 1, f n i (d i-1 ) -1]
. By continuing this construction, we obtain intervals I i (•) for all levels.

Finally, the intersection F of these nested intervals is obviously a subset

of [v -ε, v + ε] ∩ E(f, y, τ ).
Let z ∈ F and [c i (z), d i (z)] be the i-th level interval containing z. Then we have

(3.2) m i+1 (z) ≥ f n i+1 (w i ) • (d i -c i ) -2 ≥ f n i+1 (w i ) • 2f n i (d i ) -τ f n i (z i ) -2, where w i , z i ∈ [c i (z), d i (z)]. Furthermore, (3.3) ε i+1 (z) ≥ 1 -2f n i+1 (c i (z)) -τ f n i+1 (u i ) ≥ 1 2f n i+1 (u i )
,

where u i ∈ [c i (z), d i (z)
]. Now we are going to define the subsequence (n i ) i≥1 .

Lemma 12. Assume (1.1) and (1.2). For any γ > 0, we can find a subsequence (n i ) i≥1 such that

(3.4) f n i+1 (w) f n i+1 (u) ≤ f n i (z) γ ∀w, u ∈ [c i (z), d i (z)],
and for any small ε > 0, we have

∀x ∈ (v -ε, v + ε), lim i→∞ log f n i (x) log f n i-1 (x) = lim i→∞ log f n i (x) log f n i-1 (x) = ∞, (3.5) and inf x∈(v-ε,v+ε) f n i+1 (x) sup x∈(v-ε,v+ε) f n i (x) • f n i (x) τ ≥ 2. (3.6)
If f n are linear then we do not need to assume (1.2), moreover we can choose (n i ) in such a way that we have (in addition to the other parts of the assertion)

(3.7) lim i→∞ log f n i (v) log f n i (v) = (v).
Proof. In the linear case (3.4) is automatically true, and to have (3.5) and (3.6) we just need that (n i ) i≥1 increases sufficiently fast (as will be clear from the proof for the general case). Hence, we will be free to choose (n i ) satisfying in addition (3.7).

Let us proceed with the general case. For any γ > 0, by Lemma 11, there exists n 0 ∈ N such that

∀n ≥ n 0 , η(n) < γ 2M ,
where M is the constant in assumption (1.2).

Starting with this n 0 , by the assumption (1.2), we can then construct a subsequence (n i ) i≥1 satisfying

(3.8) γ 2η(n i ) • M ≤ log f n i+1 (v) log f n i (v) ≤ γ 2η(n i ) .
Observe that, as η(n i ) → 0 by Lemma 11, the lefthand side of (3.8) implies the first part of (3.5). As u < ∞, the second part of (3.5) follows. The condition (3.6) will also follow, provided that n 0 was selected large enough.

We need now to prove (3.4). By (3.1), for any w, u in the interval [c i (z), d i (z)], (3.9) f n i+1 (w)

f n i+1 (u) ≤ f n i+1 (z) 1+η(n i ) f n i+1 (z) 1-η(n i ) = f n i+1 (z) 2η(n i ) .
Combining (3.8) and (3.9), we get (3.4).

We continue the proof of the lower bound of Theorem 3. By (3.2) and (3.6),

m i+1 (z) ≥ f n i+1 (w i ) • 2f n i (d i ) -τ f n i (z i ) -2 ≥ 2,
which then implies that F is non-empty. Further, by (3.4), for any γ > 0,

m i+1 (z) ≥ f n i+1 (z) • f n i (z) -γ • f n i (d i ) -τ f n i (z i ) . ( 3
m i+1 (z)ε i+1 (z) ≥ f n i (z) -γ • f n i (d i ) -τ 2f n i (z i ) . ( 3
lim inf i→∞ log(m 2 (z) • • • m i (z)) -log m i+1 (z)ε i+1 (z) ≥ lim inf i→∞ i j=2 (log f n j (z) -γ log f n j-1 (z) -τ log f n j-1 (d j ) -log f n j-1 (z j )) log 2 + log f n i (z i ) + γ log f n i (z) + τ log f n i (d i ) = lim inf i→∞ log f n i (z) log f n i (z i ) + γ log f n i (z) + τ log f n i (d i )
.

Hence, by the definition of η(n i ), we have

lim inf i→∞ log(m 2 (z) • • • m i (z)) -log m i+1 (z)ε i+1 (z) ≥ 1 lim sup i→∞ 1 + η(n i ) + γ + τ (1 + η(n i )) • log fn i (d i ) log f n i (d i )
.

In the linear case, log

f n i (d i )/ log f n i (d i ) converges to ( lim i→∞ d i ).
In the general situation, we have lim sup

i→∞ log f n i (d i ) log f n i (d i ) ≤ u( lim i→∞ d i ).
As γ can be chosen arbitrarily small, η(n i ) → 0 by Lemma 11, and

lim i→∞ d i ∈ [v -ε, v + ε],
the lower bound is obtained by applying Lemma 9.

Upper bound: Since for all x ∈ [v -ε, v + ε] ∩ E(f, y, τ ), we have f n (x) -y n < f n (x) -τ for infinitely many n ≥ 1. Then the set [v -ε, v + ε] ∩ E(f, y, τ
) is covered by the union of the family of intervals

I n (k) := [f -1 n (k + y n -f n (v -ε) -τ ), f -1 n (k + y n + f n (v -ε) -τ )], where k ∈ [f n (v -ε), f n (v + ε)
] is an integer. Note that the length of the interval I n (k) satisfies

|I n (k)| ≤ 2f n (v -ε) -τ f n (z) for some z ∈ (v -ε, v + ε).
The number of the intervals at level n is less than

f n (v + ε) -f n (v -ε) ≤ 2εf n (w) for some w ∈ (v -ε, v + ε). Thus for s > 0 ∞ n=1 k∈[fn(v-ε),fn(v+ε)] |I n (k)| s ≤ ∞ n=1 2εf n (w) • 2f n (v -ε) -τ f n (z) s . (3.12)
By the definition of (v), for any η > 0, there exists

n 0 = n 0 (η) ∈ N such that for any n ≥ n 0 f n (v -ε) > f n (v -ε) (v-ε)-η .
Thus by ignoring the first n 0 terms, we have (3.12) is bounded by

2 1+s ε ∞ n=n 0 f n (w) • f n (z) -s • f n (v -ε) -τ s( (v-ε)-η) . (3.13)
Hence by the assumption (1.3) if

s > lim sup n→∞ log f n (w) log f n (z) + τ ( (v -ε) -η) log f n (v -ε)
the sum in (3.12) converges. By (1.1),

lim n→∞ log f n (w) log f n (z) = 1, lim n→∞ log f n (w) log f n (v -ε) = 1. Therefore lim ε→0 dim H [v -ε, v + ε] ∩ E(f, y, τ ) ≤ 1 1 + τ (v)
.

Uniform Diophantine approximation

In this section, we study the uniform Diophantine approximation of the sequence ({x n }) n≥1 with x > 1.

Recall that for any sequence of real numbers y = (y n ) n≥1 in [0, 1], we are interested in the set

F (B, y) := {x > 1 : for all large integer N, x n -y n < B -N has a solution 1 ≤ n ≤ N }.
For any v ∈ F (B, y), for any ε > 0, we will give a lower bound for the Hausdorff dimension of [v -ε, v +ε]∩F (B, y). To this end, we investigate the uniform Diophantine approximation and asymptotic Diophantine approximation together. We consider the following subset of Proof of Theorem 5. We first construct a subset F ⊂ F (v, ε, b, B, y). Suppose that b = B θ with θ > 1. Let n k = θ k . Consider the points z such that

[v -ε, v + ε] ∩ F (B, y) F (v,
z n k -y n k < b -n k . Then one can check that z ∈ F (v, ε, b, B, y) = F (v, ε, B θ , B, y) = F (v, ε, b, b 1 θ , y).
We do the same construction as in Section 3. We will obtain a Cantor set F ⊂ F (v, ε, b, b 1 θ , y), which is the intersection of a nested family of intervals with

m k (z) = 2n k+1 c k (z) n k+1 -1 n k b n k d k (z) n k -1 and ε k (z) = 1 - 2 b n k+1 1 n k+1 d k (z) n k+1 -1 , where [c k (z), d k (z)] is the k-th level interval containing z.
By the choice of n k , we will have the following estimations: 1) .

m k (z) ≥ 2(θ k+1 -1)c k (z) θ k+1 -1 θ k b θ k d k (z) θ k -1 ≥ θ • b -θ k • c k (z) d k (z) θ k • c k (z) θ k (θ-
and ε k (z) ≥ 1 2θ k+1 • d k (z) -θ k+1 . Since d k (z) -c k (z) ≤ b -n k n k c k (z) n k -1 ≤ b -θ k is much more smaller than 1/θ k , c k (z) d k (z) θ k = 1 - d k (z) -c k (z) d k (z) θ k ≥ 1 2 1 bz θ k .
Thus by Lemma 9, for any z ∈ F , we have

lim inf k→∞ log(m 1 (z) • • • m k-1 (z)) -log m k (z)ε k (z) ≥ lim inf k→∞ (θ -1) log z -log b k-1 j=1 θ j θ k log bz = lim inf k→∞ (θ -1) log z -log b (θ -1) log bz • θ k-1 -1 θ k-1 = (θ -1) log z -log b (θ -1) log bz .
Hence, by the relation b = B θ , we deduce that the Hausdorff dimension of the set

F (v, ε, b, B, y) = F (v, ε, B θ , B, y) is at least equal to (θ -1) log(v -ε) -log b (θ -1) log(b(v -ε)) = log(v -ε) -θ θ-1 log B log(v -ε) + θ log B .
Taking θ → ∞ in the left side of the equality, we get the lower bound log(v -ε)/ log(b(v -ε)) for the Hausdorff dimension of the set considered in Theorem 1:

[v -ε, v + ε] ∩ E(b, y) ={v -ε ≤ x ≤ v + ε : x n -y n < b -n for infinitely many n}.
By maximizing the right side of the equality with respect to θ > 1, we obtain the lower bound

log(v -ε) -log B log(v -ε) + log B 2 for the Hausdorff dimension of the set [v -ε, v + ε] ∩ F (B, y) ={v -ε ≤ x ≤ v + ε : ∀N 1, ||x n -y|| < B -N has a solution 1 ≤ n ≤ N }.
By letting ε tend to 0, this completes the proof of Theorem 5.

Bad approximation

In this section, we study the bad approximation properties of the sequence ({x n }) n≥1 , where x > 1.

Let q = (q n ) n≥1 be a sequence of positive real numbers and y = (y n ) n≥1 be an arbitrary sequence of real numbers in [0, 1]. Define G(q, y) = {x > 1 : lim n→∞ x qn -y n = 0}, and, for v > 1, define

G(v, q, y) = {1 < x < v : lim n→∞ x qn -y n = 0}.
Recently Baker [START_REF] Baker | On the distribution of powers of real numbers modulo 1[END_REF] showed that if q = (q n ) n≥1 is strictly increasing and lim n→∞ (q n+1 -q n ) = ∞, then the set G(q, y) has Hausdorff dimension 1.

We want to generalize Baker's result. Consider a family of C 1 functions f = (f n ) n≥1 from an interval I ⊂ R to R such that f n (x) ≥ 1 for all x ∈ I and for all n ≥ 1. Let δ = (δ n ) n≥1 be a sequence of positive real numbers tending to 0. For ε > 0, set

G(ε, v, f, y, δ) := {v -ε < x < v + ε : f n (x) -y n ≤ δ n , ∀n ≥ 1}.
To prove Theorem 7, we need to estimate dim H G(ε, v, f, y, δ). Sketch proof of Theorem 7. Lower bound: We do the same construction as in the proof of the lower bound in Theorem 3. If the right-hand side inequality in (3.8) is satisfied, that is, if

(5.1) log f n+1 (v) log f n (v) ≤ γ 2η(n) ,
for some γ > 0, for large enough n, and for η defined in (3.1), then the distortion estimation (3.4) holds and we estimate the dimension in exactly the same way as in Theorem 3. If, however, (5.1) is not satisfied, that is, at some place f n is too sparse, with log f n+1 (v) log f n (v) then we can apply the idea of Baker ([1], page 69): we add some new functions fm between f n and f n+1 , in such a way that the resulting, expanded, sequence of their logarithms of derivatives is not too sparse anymore. We also add some δm = 1 for each added fm . Observe that the right-hand side of (1.7) does not change. Naturally, the resulting set G(ε, v, f , y, δ) is exactly the same as G(ε, v, f, y, δ). So, for the lower bound, we need only to estimate the lower bound of dim H G(ε, v, f , y, δ).

This means that we can freely assume that (5.1) holds. We will construct a subset of G(ε, v, f, y, δ) which is the intersection of a nested family of subintervals I n (•).

For n = 1, by the property f 1 (x) -y 1 ≤ δ 1 , we take the intervals at level 1 as

I 1 (k, v, f
m n+1 (z) ≥ f n+1 (w n ) • (d n -c n ) -2 ≥ f n+1 (w n ) • 2δ n f n (z n ) -2 ≥ 2,
and

ε n+1 (z) ≥ 1 -2δ n+1 f n+1 (u n ) ≥ 1 2f n+1 (u n ) .
with w n , z n , u n ∈ [c n (z), d n (z)]. As we are assuming (5.1), we have (3.4) and then for any γ > 0

m n+1 (z) ≥ f n+1 (z) • f n (z) -γ • δ n f n (z n ) .
and Hence the lower bound of Theorem 7 is obtained by Lemma 9.

m n+1 (z)ε n+1 (z) ≥ f n (z) -γ • δ n 2f n (z n ) Thus,
Upper bound: We will apply Lemma 10. For each basic interval I n (z), by (1.1), we have for any γ, for n large enough

δ n f n (z)f n-1 (z) γ ≤ |I n (z)| ≤ δ n f n-1 (z) γ f n (z) .
Thus,

m n (z) ≤ |I n-1 (z)| • f n (z)f n-1 (z) γ ≤ δ n-1 f n-2 (z) γ f n-1 (z) f n (z)f n-1 (z) γ .
Hence, By Lemma 10, we conclude the proof.

Corollary 8 .

 8 Let (a n ) n≥1 be a sequence of positive real numbers such that lim n→∞ a n+1 a n = +∞.

. 10 )

 10 By (3.2), (3.3) and (3.4),

. 11 )

 11 Thus, (1.3) and (3.5) imply that -log m i+1 (z)ε i+1 (z) is unbounded. So by (3.10),(3.11) and(3.5)

  ε, b, B, y) := {z ∈ [v -ε, v + ε] : z n -y n < b -n for infinitely many n and ∀N 1, z n -y n < B -N has a solution 1 ≤ n ≤ N }. The proof of Theorem 5 will be completed by maximizing the lower bounds of F (v, ε, b, B, y) with respect to b > B.

  , y, δ) := [f -1 1 (k + y 1 -δ 1 ), f -1 1 (k + y 1 + δ 1 )], with k being an integer in [f 1 (v -ε) + 1, f 1 (v + ε) -1].Suppose we have constructed the intervals at leveln -1. Let [c n-1 , d n-1 ] be an interval at this level. A subinterval of [c n-1 , d n-1 ] at level n is [f -1 n (k + y n -δ n ), f -1 n (k + y n + δ n )], with k being an integer in [f n (c n-1 ) + 1, f n (d n-1 ) -1]. By continuing this construction, we obtain intervals I n (•) for all levels. Finally, the intersection F of these nested intervals is obviously a subset of G(ε, v, f, y, δ).Let z ∈ F and [c n (z), d n (z)] be the n-th level interval containing z. Then by(1.5)

log(m 2 1 j=1

 21 (z) • • • m n (z)) -log m n+1 (z)ε n+1 (z) ≥ log f n (z) -log f 1 (z) + z) -γ f j (z j ) log 2 + log f n (z n ) + γ log f n (z) -log δ n .By (1.6), we havelim inf n→∞ log(m 2 (z) • • • m n (z)) -log m n+1 (z)ε n+1 (z) log f n (z n ) + γ log f n (z) -log δ n .Since γ can be chosen arbitrary small and z n tends to z, by (1.1) we havelim inf n→∞ log(m 2 (z) • • • m n (z)) -log m n+1 (z)ε n+1 (z) ≥ lim inf n→∞ log f n (z) + nlog δ j log f n (z) -log δ n .

  1 (z) • • • m n (z)) -log |I n (z)| ≤ lim inf n→∞ log f n (z) + n-1 j=1 log δ j log f n (z) -log δ n .

M. R. was supported by National Science Centre grant 2014/13/B/ST1/01033 (Poland).


Pierre Daudé
email: pierre-daude@hotmail.fr

Patricia Ancel
email: patricia.ancel@univ-amu.fr

Sylviane Confort Gouny

Alexis Jacquier
email: alexis.jacquier@ap-hm.fr

Frank Kober
email: frank.kober@univ-amu.fr

Anne Dutour

Monique Bernard
email: monique.bernard@univ-amu.fr

Bénédicte Gaborit
email: dicte.gaborit@ap-hm.fr

Stanislas Rapacchi
email: stanislas.rapacchi@gmail.com

Deep-Learning Segmentation of Epicardial Adipose Tissue Using Four-Chamber Cardiac Magnetic Resonance Imaging

Keywords: epicardial adipose tissue quantification, automatic segmentation, cine four-chamber, fully convolutional networks, machine learning

In magnetic resonance imaging (MRI), epicardial adipose tissue (EAT) overload remains often overlooked due to tedious manual contouring in images. Automated four-chamber EAT area quantification was proposed, leveraging deep-learning segmentation using multi-frame fully convolutional networks (FCN). The investigation involved 100 subjects-comprising healthy, obese, and diabetic patients-who underwent 3T cardiac cine MRI, optimized U-Net and FCN (noted FCNB) were trained on three consecutive cine frames for segmentation of central frame using dice loss. Networks were trained using 4-fold cross-validation (n = 80) and evaluated on an independent dataset (n = 20). Segmentation performances were compared to inter-intra observer bias with dice (DSC) and relative surface error (RSE). Both systole and diastole four-chamber area were correlated with total EAT volume (r = 0.77 and 0.74 respectively). Networks' performances were equivalent to inter-observers' bias (EAT: DSCInter = 0.76, DSCU-Net = 0.77, DSCFCNB = 0.76). U-net outperformed (p < 0.0001) FCNB on all metrics. Eventually, proposed multi-frame U-Net provided automated EAT area quantification with a 14.2% precision for the clinically relevant upper three quarters of EAT area range, scaling patients' risk of EAT overload with 70% accuracy. Exploiting multi-frame U-Net in standard cine provided automated EAT quantification over a wide range of EAT quantities. The method is made available to the community through a FSLeyes plugin.

Introduction

Epicardial adipose tissue (EAT) is a visceral fat depot surrounding the heart between the myocardium and the pericardium [START_REF] Gaborit | Role of Epicardial Adipose Tissue in Health and Disease: A Matter of Fat? In Comprehensive Physiology[END_REF]. Its volume quantification holds potential as a novel biomarker for risks of coronary heart disease [START_REF] Mahabadi | Association of Epicardial Adipose Tissue With Progression of Coronary Artery Calcification Is More Pronounced in the Early Phase of Atherosclerosis[END_REF]. Pericardial fat, merging EAT and paracardial (PAT) fat, has been studied in the past in association with atherosclerotic disease [START_REF] Ding | The Association of Pericardial Fat with Incident Coronary Heart Disease: The Multi-Ethnic Study of Atherosclerosis (MESA)[END_REF] but these results have since been heavily criticized [START_REF] Iacobellis | Epicardial and Pericardial Fat: Close, but Very Different[END_REF]. The inclusion of two fat depots as one single entity may not reflect the separate functions and clinical implications of each adipose tissue. Indeed, recent studies focusing on separating EAT and PAT concluded that EAT alone was involved in the corresponding disease [START_REF] Nakanishi | Persistent Epicardial Adipose Tissue Accumulation Is Associated with Coronary Plaque Vulnerability and Future Acute Coronary Syndrome in Non-Obese Subjects with Coronary Artery Disease[END_REF][START_REF] Rosito | Pericardial Fat, Visceral Abdominal Fat, Cardiovascular Disease Risk Factors, and Vascular Calcification in a Community-Based Sample: The Framingham Heart Study[END_REF]. Indeed, EAT is a metabolically active adipose tissue [START_REF] Gaborit | Role of Epicardial Adipose Tissue in Health and Disease: A Matter of Fat? In Comprehensive Physiology[END_REF] compared to PAT. Its accumulation and subsequent inflammation add to cardiovascular risks, potentially impacting left ventricle (LV) diastolic dysfunction [START_REF] Dabbah | Epicardial Fat, Rather than Pericardial Fat, Is Independently Associated with Diastolic Filling in Subjects without Apparent Heart Disease[END_REF][START_REF] Antonopoulos | The Role of Epicardial Adipose Tissue in Cardiac Biology: Classic Concepts and Emerging Roles[END_REF]. Even more recently, EAT overload has raised concern as a risk factor in generalized inflammation from COVID-19 [START_REF] Iacobellis | Epicardial Fat Inflammation in Severe COVID-19[END_REF][START_REF] Grodecki | Epicardial Adipose Tissue Is Associated with Extent of Pneumonia and Adverse Outcomes in Patients with COVID-19[END_REF]. It is now recognized that the amount of EAT is prospectively and independently associated with the number of coronary events in at-risk populations [START_REF] Mahabadi | Thoracic Adipose Tissue Density as a Novel Marker of Increased Cardiovascular Risk[END_REF]. Consequently, reproducibly quantifying EAT is a major public health objective aiming at a better identification of patients at high CV risk. EAT can indeed be visualized from standard cardiac magnetic resonance imaging (MRI) images, but its analysis is currently not performed in clinical routine, because the necessary manual image segmentation is extremely time-consuming, and its measurement is not sufficiently standardized.

Different imaging modalities have been proposed to quantify EAT burden. Transthoracic echocardiography, a forefront modality in cardiology, was used to measured EAT thickness on the free wall of the right ventricle [START_REF] Iacobellis | Epicardial Fat from Echocardiography: A New Method for Visceral Adipose Tissue Prediction[END_REF]. However, only a single distance measurement was used to estimate EAT volume, strongly limiting the precision of this method because EAT is irregularly distributed around the heart. Cardiac computed tomography (CCT) imaging has become the gold standard for the quantification of EAT volume [START_REF] Dey | Automated Quantitation of Pericardiac Fat From Noncontrast CT[END_REF]. In more recent studies, semi-automatic and deep learning methods have been implemented to achieve the EAT segmentation [START_REF] Nichols | Volumetric Measurement of Pericardial Adipose Tissue from Contrast-Enhanced Coronary Computed Tomography Angiography: A Reproducibility Study[END_REF][START_REF] Commandeur | Fully Automated CT Quantification of Epicardial Adipose Tissue by Deep Learning: A Multicenter Study[END_REF]. However, the requirement of high spatial resolution CCT led to the use of elevated ionizing radiation doses, which could be a risk for patients' follow-up.

Cardiac MRI is a versatile tool than can measure cardiac function, morphology, perfusion and characterizes myocardial tissue in a single exam [START_REF] Kramer | Standardized Cardiovascular Magnetic Resonance Imaging (CMR) Protocols: 2020 Update[END_REF]. Cardiac MRI is also highly sensitive to fat, which has long been considered as an obstacle for myocardial visualization. Cardiac fat remains under-appreciated as a diagnostic feature of cardiac MRI. To specifically probe fat around the beating heart with MRI, one can use a dedicated acquisition technique such as watersuppressed MRI [START_REF] Flüchter | Volumetric Assessment of Epicardial Adipose Tissue With Cardiovascular Magnetic Resonance Imaging*[END_REF] or Dixon MRI (3D) [START_REF] Homsi | 3D-Dixon MRI Based Volumetry of Peri-and Epicardial Fat[END_REF][START_REF] Henningsson | Quantification of Epicardial Fat Using 3D Cine Dixon MRI[END_REF]. Alternatively, EAT volume may also be measured from a routine stack of short-axis cine images [START_REF] Gaborit | Assessment of Epicardial Fat Volume and Myocardial Triglyceride Content in Severely Obese Subjects: Relationship to Metabolic Cardiac Function and Visceral Fat[END_REF]. EAT quantification is usually performed manually, which is a time-consuming and tedious task subjected to inter-observer variability. To help observers, first the cine temporal information could ease distinguishing EAT from paracardial fat. Indeed, EAT is attached to the myocardium and moves at pace with cardiac contraction and torsion, whereas PAT is only moderately pulled by the cardiac contraction and expansion. Second, while the pericardial fascia is not clearly visible on short axis views, which was often reduced to a thin line that may be blurred by partial volume effects, on four-chamber (4Ch) views, the pericardium is generally less affected by partial volume effects resulting in better visualization. As such, the four-chamber view is recommended for evaluating pericarditis [START_REF] Chetrit | Imaging-Guided Therapies for Pericardial Diseases[END_REF] and is a frequent choice of orientation to quantify EAT, PAT, and pericardial fat [START_REF] Rado | MRI-Based Assessment and Characterization of Epicardial and Paracardial Fat Depots in the Context of Impaired Glucose Metabolism and Subclinical Left-Ventricular Alterations[END_REF][START_REF] Bard | Automated Quality-Controlled Cardiovascular Magnetic Resonance Pericardial Fat Quantification Using a Convolutional Neural Network in the UK Biobank[END_REF][START_REF] Nyman | Cardiac Steatosis and Left Ventricular Function in Men with Metabolic Syndrome[END_REF][START_REF] Granér | Cardiac Steatosis Associates With Visceral Obesity in Nondiabetic Obese Men[END_REF][START_REF] Nakanishi | Relationships Between Periventricular Epicardial Adipose Tissue Accumulation, Coronary Microcirculation, and Left Ventricular Diastolic Dysfunction[END_REF][START_REF] Perseghin | Increased Mediastinal Fat and Impaired Left Ventricular Energy Metabolism in Young Men with Newly Found Fatty Liver[END_REF]. Consequently, the EAT analysis in this study were based on quantification of its 2D area representation in 4Ch long-axis cine MRI views. To address specifically this kind of segmentation challenges, deep learning approaches have recently bloomed. Indeed, fully automated methods applied on routine images, such as cine MRI, could be rapidly translated to the clinics. Bard et al. [START_REF] Bard | Automated Quality-Controlled Cardiovascular Magnetic Resonance Pericardial Fat Quantification Using a Convolutional Neural Network in the UK Biobank[END_REF] developed a deep learning method to quantify pericardial fat in 4Ch long-axis cine MRI and evaluated it on the UK BioBank dataset. However, the segmentation of pericardial fat (EAT + PAT) limits the evaluation of the distinct roles and clinical implications of epicardial fat compared to paracardial fat.

Thus, we propose here to segment the thin EAT area on 4Ch cine MRI multi-frame images using state-of-the-art fully convolutional networks (FCNs) for cardiac image segmentation, that were adapted to segment EAT, PAT, and cardiac ventricles. A specific database of 4Ch cine MRI spanning diabetic, obese, and healthy subjects was leveraged to train, validate, and evaluate proposed FCN networks.

Materials and Methods

Study Population

A retrospective mono-centric database was defined totaling 153 subjects, out of which 100 exams could be exploited. The 100 enrolled subjects including healthy controls, type-2 diabetic patients, and non-diabetic obese patients were selected based on 4Ch orientation and the absence of severe artifacts as shown in Figure 1. Patients were defined as having type 2 diabetes mellitus if they fulfilled any of the WHO criteria: HbA1c ≥ 6.5%, FBG level ≥ 7.0 mmol/L, oral glucose tolerance test result ≥ 11.1 mmol/L, or current treatment with antidiabetic agents. Obese non-diabetic patients were defined as the absence of any WHO criteria and a BMI ≥ 30 kg/m 2 . All enrolled subjects had normal left ventricular function, no history of heart failure or coronary heart disease.

MRI Acquisition

All subjects underwent cardiac MRI including the acquisition of a full stack of short-axis slices and a single slice four-chamber cine on a 3-T MRI system (Magnetom Verio, Siemens Healthineers, Erlangen, Germany) with a dedicated cardiac 32-channel coil array (Invivo, Gainesville, FL, USA). The cine series were acquired with a retrospectively ECGgated balanced steady-state free precession (bSSFP) sequence with in-plane image resolution varying from 1.3 × 1.3 mm 2 to 1.8 × 1.8 mm 2 (depending on subjects), slice thickness of 6 mm, TE/TR = 1.2/3.2 ms, GRAPPA 2 (24 auto-calibration signal lines), temporal resolution of 28-35 ms, with 25 frames reconstructed. Further details of the cardiac MRI protocols were previously described [START_REF] Gaborit | Assessment of Epicardial Fat Volume and Myocardial Triglyceride Content in Severely Obese Subjects: Relationship to Metabolic Cardiac Function and Visceral Fat[END_REF][START_REF] Gaborit | Epicardial Fat Volume Is Associated With Coronary Microvascular Response in Healthy Subjects: A Pilot Study[END_REF][START_REF] Dutour | Exenatide Decreases Liver Fat Content and Epicardial Adipose Tissue in Patients with Obesity and Type 2 Diabetes: A Prospective Randomized Clinical Trial Using Magnetic Resonance Imaging and Spectroscopy[END_REF][START_REF] Gaborit | Effects of Bariatric Surgery on Cardiac Ectopic Fat[END_REF][START_REF] Gaborit | Effect of Empagliflozin on Ectopic Fat Stores and Myocardial Energetics in Type 2 Diabetes: The EMPACEF Study[END_REF]. N4 bias field correction [START_REF] Tustison | N4ITK: Improved N3 Bias Correction[END_REF] was applied to all image series before further processing.

EAT Segmentation

For reference, EAT volume was segmented by expert readers provided with full stack short-axis series using Argus viewer (Siemens Medical Solutions, Erlangen, Germany). In an independent session, two expert readers were provided with full 4Ch series and performed blinded segmentation of three labels using the FSLeyes viewer [33] (version 0.31, Paul McCarthy, University of Oxford, UK): heart ventricles (HV) (including both ventricle muscles and blood pools), epicardial (EAT), and paracardial (PAT) adipose tissues. EAT was defined as hyperintense signal within the pericardium around the ventricles. Peri-atrial fat was not included as it has been shown that peri-ventricle EAT alone had a stronger correlation with coronary diseases than total EAT [START_REF] Nakanishi | Relationships Between Periventricular Epicardial Adipose Tissue Accumulation, Coronary Microcirculation, and Left Ventricular Diastolic Dysfunction[END_REF]. All isles of periventricular fat were included to form EAT area. PAT was defined as fat adjacent but outside the pericardium. Segmentations were performed on three cardiac phases determined by readers having the entire series at their disposal: first phase, peak systole, and late diastole. The three segmented masks were propagated to the remaining frames using an automatic label propagation algorithm based on non-linear registrations, as previously described [START_REF] Ogier | Individual Muscle Segmentation in MR Images: A 3D Propagation through 2D Non-Linear Registration Approaches[END_REF] resulting in 25 images segmented per subjects. Series in the test dataset were segmented by both readers, and reader 1 repeated blinded segmentations 6 weeks later.

Network Architecture

Two different fully convolutional networks (FCNs) were investigated: U-Net [START_REF] Ronneberger | Convolutional Networks for Biomedical Image Segmentation[END_REF] with 48 filters for the first layer and FCN developed by Bai et al. [START_REF] Bai | Automated Cardiovascular Magnetic Resonance Image Analysis with Fully Convolutional Networks[END_REF] with 48 filters for the first layer, later referenced as FCNB. These networks are based on an encoder-decoder structure but differ in their decoder structure. The encoder part processes an image of arbitrary size as input and applies convolutional layers for extracting image features while the decoder upsamples and combines low-resolution featured map to the original input resolution. The absence of a dense layer allows these networks to process images of various sizes.

The U-Net [START_REF] Ronneberger | Convolutional Networks for Biomedical Image Segmentation[END_REF] has been the most popular 2D segmentation network for biomedical images and a fundamental component of many state-of-the-art cardiac image segmentation approaches [START_REF] Zheng | 3-D Consistent and Robust Segmentation of Cardiac Images by Deep Learning With Spatial Propagation[END_REF][START_REF] Baumgartner | An Exploration of 2D and 3D Deep Learning Techniques for Cardiac MR Image Segmentation[END_REF][START_REF] Isensee | Automatic Cardiac Disease Assessment on Cine-MRI via Time-Series Segmentation and Domain Specific Features[END_REF]. The specificity of the U-Net is to employ skip connections between encoder and decoder to recover spatial information lost in downsampling layers as shown in Figure 2. The second network investigated is the FCN developed by Bai et al. [START_REF] Bai | Automated Cardiovascular Magnetic Resonance Image Analysis with Fully Convolutional Networks[END_REF], later referred to as FCNB. FNCB has demonstrated excellent segmentation performances on the largest available cardiac MR dataset (UK-Biobank [START_REF] Sudlow | UK Biobank: An Open Access Resource for Identifying the Causes of a Wide Range of Complex Diseases of Middle and Old Age[END_REF]Its specificity is based on the decoder that only consists of the concatenation of all featured maps, upsampled to the original resolution, as shown in Figure 2.

In their original papers, the cross-entropy loss was used to train those networks. However, this loss has shown limits to address class imbalance. In our study, regions of interest (ROI) were sparsely represented compared to the background and cross-entropy loss is inadequate to handle it. Thus, the loss function was defined as the mean dice between the probabilistic label map without background and the manually annotated label map.

Training

Specifically, optimized FCNB and U-Net were trained on three consecutive cine frames for segmentation of the central frame, providing a crucial temporal information often necessary for the experts to segment EAT. Input images were normalized to the range of [0,1] with fixed size (256 × 192 × 3), mask zero-padding or cropping was applied when needed.

For each batch (N = 30), on-the-fly data augmentation was performed using rotational transformation and/or image scaling before feeding them to the network. Both data augmentation were set using a random clipped normal distribution spanning from -30°/0.4 up to 30°/1.6 for rotational transformation and image scaling respectively. The Adam optimization [START_REF] Kingma | A Method for Stochastic Optimization[END_REF] was used for minimizing the dice loss function with a constant learning rate of 1e-3. It took approximatively 35 min to train either the U-Net or FCNB on a Graphics Processing Unit (GPU) (NVidia Tesla K80).

The networks investigated were implemented using Python within the TensorFlow 2 framework. The FCNB model was adapted from the original implementation [START_REF] Bai | Ukbb_cardiac Available online[END_REF], whereas U-Net was custom-designed. To adapt to the proposed multi-frame approach, both 2D networks were modified to accept 2D+t inputs, considering the cardiac time dimension as a third dimension with limited horizon. Thus, the first convolution layer of each network was replaced with a 3D convolution layer with valid padding. The following layers were kept identical, processing extracted features independently of the input dimensions.

To perform a robust evaluation, networks were trained using cross-validation and evaluated on an independent dataset: the database was split in five subsets (500 images/20 subjects each reflecting our database populations distribution: 4 healthy controls, 13 type 2-diabetics, 3 nondiabetic obese patients). One subset (500 images) was used as a test set whereas the 4 other subsets were used for stratified cross-validation training, resulting in a 4-fold cross-validation. Thus, a single subset is retained as validation (500 images) whereas the 3 others (1500 images) are used for training, ensuring that validation and training dataset reflects the database population distribution.

Evaluation Metrics

Segmentation performances were evaluated for accuracy, propinquity, and surface estimation error. Dice similarity coefficient (DSC) measured segmentation accuracy from the overlap between the manual and automatic segmented surfaces (SM and SA), defined as

DSC = 2 S M ∩ S A S M + S A (1)
The mean surface distance (MSD) calculated the propinquity between segmentations as is the mean distance (in mm) between segmented contours, defined as

MSD = 1 n M + n A ((∑ d(k, S A ) n M k=1 )) + (∑ d(k, S M ) n A k=1 )) (2)
To evaluate the clinical final purpose, which is the quantitative measurement of EAT area, absolute relative surface error (RSE) was utilized, defined as

RSE = |S M -S A | S M (3)
To further assess accuracy, positive predicted value (PPV) which is an indicator of over-segmentation (PPV << 1) was calculated on the entire database, defined as

PPV = S M ∩ S A S A (4)

Statistical Analysis

Statistical analysis was conducted using R (version 3.6.3) [START_REF] Team | A Language and Environment for Statistical Computing[END_REF]. Analysis of linear regression was used to study the correlation between manually evaluated EAT volume and 4Ch area. The metrics' distribution normality was assessed using the Shapiro-Wilk test. Wilcoxon signed rank and Wilcoxon rank sum tests were used to investigate significant differences for each metrics between intra-inter observers and FCNs. To account for segmentation difficulty and clinical relevance [START_REF] Mahabadi | Association of Epicardial Fat With Cardiovascular Risk Factors and Incident Myocardial Infarction in the General Population[END_REF] that scale with the quantity of EAT, networks' performances were assessed per quartile of manually segmented EAT area (Q1 < 8.22 cm 2 ≤ Q2 < 12.70 cm 2 ≤ Q3 < 15.55 cm 2 ≤ Q4).

Results

The selected 100 subjects were divided into three groups (21 healthy controls, 67 type-2 diabetic patients and 12 non-diabetic obese patients) as detailed in Table 1. In studied database, EAT volumes spanned a wide range from 29 to 376 cm 3 , defining quartiles by: Q1 < 77.8 cm 3 ≤ Q2 < 94.6 cm 3 ≤ Q3 < 114.3 cm 3 ≤ Q4.

Corresponding EAT areas as measured on 4Ch views correlated well with total EAT volume measured from the stack of short-axis cine (Figure 3) with a slightly higher correlation in systole (Pearson r = 0.77) than in diastole (Pearson r = 0.74). Thus, a wide range of EAT 4Ch areas was available from 1.2 cm 2 to 37.2 cm 2 , with a lower range for healthy subjects from 2.5 to 13.7 cm 2 , from 1.2 cm 2 to 23.2 cm 2 for non-diabetic obese subjects and from 5.3 cm 2 to 37.2 cm 2 for type 2 diabetic patients. As shown in Table 2, intra and inter-observer DSC confirmed excellent reproducibility for HV segmentation (DSCIntra = 0.98 and DSCInter = 0.96 resp.). EAT and PAT differed between the two observers (DSCInter = 0.76 and 0.78 for EAT and PAT resp.), although segmentations performed twice by the same observer proved to be more reproducible (DSCIntra = 0.83 and 0.85 for EAT and PAT resp.). Intra-observer DSC and MSD were significatively lower (p < 0.05) concerning EAT segmentation in the diastolic frame compared to the segmentation in the systolic frame. For inter-observer bias, differences in DSC, MSD, or RSE metrics were not statistically significant between diastolic and systolic frames.

FCNB and U-Net segmentations performance measured by DSC, were significantly lower (p < 0.05) than intraobserver bias for all labels (for EAT: DSCIntra = 0.83, DSCU-Net = 0.77, DSCFCNB = 0.76). Both networks provided equivalent DSC, MSD, and RSE performance than inter-observer bias for all labels (for instance PAT: DSCInter = 0.78, DSCU-Net = 0.80, DSCFCNB = 0.78). Across the four quartiles of data defined by equally populated ranges of EAT areas, both networks provided reliable segmentation of the heart ventricles (HV, FCNB: DSCQ1-Q4 = 0.97-0.96, U-Net: DSCQ1-Q4 = 0.97) as shown in Table 3. Interestingly, the network performances to segment EAT strongly depended on the population quartile. Indeed, U-Net DSC was significantly higher (p < 0.001) for upper quartiles as observed using U-Net: DSCQ4 = 0.83 > DSCQ3 = 0.80 > DSCQ2 = 0.76 > DSCQ1 = 0.69 as illustrated in Figure 4. DSC and RSE metrics demonstrated a gap of segmentation quality between the lower two quartiles and the upper two quartiles for both PAT and EAT segmentation (for EAT FCN: RSEQ4 = 15.60, RSEQ3 = 15.87 < RSEQ2 = 21.91 < RSEQ1 = 27.98). Across all quartiles, both networks had more difficulty separating PAT from EAT than identifying total pericardial fat (EAT+ PAT) in the image (with U-Net, RSEEAT + PAT << RSEEAT or RSEPAT for all quartiles). Over the database and for all labels, U-net outperformed (p < 0.0001) FCNB for segmenting accurately (DSC), nearer to the ground truth (MSD), thus providing a more reliable (i.e., accurate) measurement (RSE).

FCNB and U-net performed significantly better (p < 0.05) for segmenting EAT area on the systolic frame compared to the diastolic frame (DSCUNet-diastole = 0.76 DSCUNet-systole = 0.80). These differences were not significant in PAT (see Appendix A Figure A1).

Classification of our database split by quartile of EAT burden was observed by confusion matrices. From Figure 5, the confusion matrices diagonal (in green) gave a measure of correct classification (66% for FCNB and 71% for U-Net), whereas the subdiagonal and the superdiagonal (in yellow) allowed evaluating a misclassification by one quartile (32% for FCNB and 27% U-Net) and the second subdiagonal and superdiagonal (in red) gave an estimate of a misclassification by two quartiles (2% for FCNB 2% for U-Net). As shown by subdiagonal confusion matrices and confirmed by PPV, FCNB significantly over-estimated EAT area compared to U-Net (PPVFCNB = 0.73 < PPVU-Net = 0.75, p < 0.0001).

Discussion

This study aimed at providing a rapid and fully integrable evaluation of epicardial fat burden. To achieve this evaluation, automated segmentation of the EAT layer was performed on four-chamber cine MRI series using Deep Learning approaches.

Four-Chamber-View Intrapericardial Fat Area Is a Relevant Measure of EAT

Confirming previous literature [START_REF] Nyman | Cardiac Steatosis and Left Ventricular Function in Men with Metabolic Syndrome[END_REF][START_REF] Granér | Cardiac Steatosis Associates With Visceral Obesity in Nondiabetic Obese Men[END_REF], the correlation found in this work between EAT area and volume across a wide range of EAT volumes (from 29 to 376 cm 3 ) comforted the relevant use of four-chamber EAT area as a rapid but realistic measure of EAT burden. Already in past studies, the 2D EAT area has been linked to left ventricular diastolic dysfunction [START_REF] Rado | MRI-Based Assessment and Characterization of Epicardial and Paracardial Fat Depots in the Context of Impaired Glucose Metabolism and Subclinical Left-Ventricular Alterations[END_REF][START_REF] Nakanishi | Relationships Between Periventricular Epicardial Adipose Tissue Accumulation, Coronary Microcirculation, and Left Ventricular Diastolic Dysfunction[END_REF], hypertension and severity of insulin resistance [START_REF] Granér | Cardiac Steatosis Associates With Visceral Obesity in Nondiabetic Obese Men[END_REF], and non-alcoholic fatty liver disease patients [START_REF] Perseghin | Increased Mediastinal Fat and Impaired Left Ventricular Energy Metabolism in Young Men with Newly Found Fatty Liver[END_REF]. Thus, four-chamber view holds potential as a surrogate to quantify EAT in routine clinical practice. Moreover, in four-chamber view, the pericardium beyond the apex of the heart could be visualized with more reliability. However, our database gathered retrospective studies in which EAT volume segmentation had been measured in short-axis views by different investigators over the years, which could lead to unaccounted volume imprecision. Ideally, the gold standard CCT EAT volume quantification would have been preferred but this examination is not commonly indicated for metabolic patients.

A Specific Database with Possible Extensions

This work leverages a unique database that combines a population spanning a large range of EAT quantity and manual segmentation of EAT on cine series. The strength of our dedicated database stands in its diversity in BMI, sex, age, health condition across many subjects (n = 100) (Table 1). Despite a large diversity of subjects, a disparity of age remains between younger healthy subjects and diabetic and/or obese patients. The addition of data from older healthy subjects, as well as elderly subjects (>65 years) would benefit the current database to reinforce our network training as elderly have been shown to be significantly more EAT burdened than younger individuals [START_REF] Foldyna | Epicardial Adipose Tissue in Patients With Stable Chest Pain[END_REF]. Our database could also be extended by including image sets from different MRI scanner types. Currently, this is a monocentric study and database. As a result, the trained models might not adapt well on datasets from scanners of different vendors and field strengths. Nevertheless, the database was made up of multiple protocols acquired over a decade, which already featured a variety of acquisition parameters and image quality levels. To further leverage the number of annotated data (2500 ground-truth, 25 images segmented per subject), generative adversarial network could be explored to extend beyond proposed data augmentation [START_REF] Chaitanya | Semi-Supervised and Task-Driven Data Augmentation[END_REF]. Another challenge are recurrent artifacts (aliasing, dark bands, flux artifacts) commonly observed in 3T bSSFP cine-MRI images, particularly pronounced in obese patients. This might preclude EAT segmentation and disturb networks accuracy. Training networks on artifacted images is another important addition to strengthen models for them to be ready for the clinic.

Challenge of EAT Segmentation

Experts and networks provided excellent results on large structures such as heart ventricles (DSC ≥ 0.96) and pericardial fat (DSC ≥ 0.88). However, one major challenge for the segmentation of EAT on cine MRI is to distinguish between burdening EAT and its extra-pericardial neighbor PAT. The pericardial fascia that separates those two fat compartments is about 2 mm thick [START_REF] Bull | CT Dimensions of the Normal Pericardium[END_REF][START_REF] Bogaert | Cardiovascular Magnetic Resonance in Pericardial Diseases[END_REF] which is of the same order of magnitude as the image resolution (1.3-1.8 mm). This explains why both networks were able to segment combined EAT + PAT pericardial fat with appreciable precision, but the identification of individual fat was less satisfying. Nevertheless, FCN networks provided segmentation results on par with experts' precision. Additionally, since cardiac contraction pulls onto the pericardium, its visualization improves in peak-systole [START_REF] Rado | MRI-Based Assessment and Characterization of Epicardial and Paracardial Fat Depots in the Context of Impaired Glucose Metabolism and Subclinical Left-Ventricular Alterations[END_REF], making this frame more suitable for the measurement of EAT when compared to diastole (pintra(DSCdia/DSCsys) = 0.0282).

One novelty has been to input multiple cardiac frames from the cardiac cycle to networks using a 3D first convolutional layer. It could be interesting in future work to enhance temporal information which is essential to detect the pericardial fascia. A map of cardiac deformations could enhance input images to be supplied to the network. It would be also interesting to investigate other network architectures, such as recurrent neural network, that could memorize information from adjacent slices to improve inter-slices coherence [START_REF] Poudel | Recurrent Fully Convolutional Neural Networks for Multi-Slice MRI Cardiac Segmentation[END_REF], but these extensions fall outside the scope of this work.

Comparing FCNs Performances

Specific complementary metrics (DSC, MSD, and RSE) have been chosen to evaluate EAT area segmentation and quantification. Alternatively, the Hausdorff distance metric is a common choice to evaluate segmentation performance [START_REF] Huttenlocher | Comparing Images Using the Hausdorff Distance[END_REF], measuring the maximal pixel distance error between segmentations. However, EAT region is sparsely distributed around the heart, thus the Hausdorff distance was not considered in this work since it might range rapidly high, even when comparing two segmentations with similar areas.

From chosen metrics, U-Net outperformed FCNB for all labels, thus appearing preferrable to quantify EAT 4Ch area. Alternative semi-and fully automatic methods have been proposed for the EAT quantification on MRI-cine. Cristobal-Huerta et al. [START_REF] Cristobal-Huerta | Automated Quantification of Epicardial Adipose Tissue in Cardiac Magnetic Resonance Imaging[END_REF] developed an automatic pipeline composed of Law texture filters, snakes and K-cosine curvature analysis to partially quantify EAT volume, albeit on 10 subjects only. In a semi-automatic processing, Fulton et al. [START_REF] Fulton | Segmentation of Epicardial Adipose Tissue in Cardiac MRI Using Deep Learning[END_REF] applied landmarks on short-axis images from 12 subjects to unroll images into polar coordinates before employing a neural network for detection of epicardial fat contours. We were unable to compare our results with those previous works as segmentation metrics (e.g., DSC metric or Jaccard similarity index) were not provided. Recently, automatic total pericardial fat quantification has been developed in 4Ch cine MRI. Bard, Raisi-Estabragh et al. [START_REF] Bard | Automated Quality-Controlled Cardiovascular Magnetic Resonance Pericardial Fat Quantification Using a Convolutional Neural Network in the UK Biobank[END_REF] obtained segmentation performances (DSC EAT+PAT = 0.8) very similar to ours (DSC EAT+PAT = 0.88) on their respective test-set. In their study, only the end-diastolic frame had been segmented while we segmented the full 4Ch cine MRI and trained on three consecutive cine frames to leverage cine temporal information. Finally, the optimized multi-frame U-Net was integrated in a FSLeyes plugin made available to the community [START_REF] Daudé | Fsleyes-Plugin-Epicseg Available online[END_REF] allowing comparison with further work and providing clinicians with a rapid EAT area segmentation (see Appendix A Figure A2).

Performances across Quartiles

Splitting the database in quartiles of EAT enabled to differentiate segmentation performances depending on EAT area. Indeed, segmentations quality from FCNs proved to be degraded in group Q1, in which EAT (as well as PAT) was thin and sparse as illustrated in Figure 4. However, EAT segmentations were on a par with inter-observers' manual segmentation for the three upper quartiles and remained relevant for identifying patient at risk (Q2, Q3, Q4 ≥ 8.22 cm 2 ) by measuring their EAT burden within 14% and 18% precision for U-Net and FCNB respectively.

Conclusions

This study provides a methodology for fully automated segmentation of epicardial fat on multi-frame cardiac cine MRI, demonstrated across 100 subjects exhibiting low to high EAT quantities. EAT is often overseen in diagnosis but has received increasing attention as a relevant biomarker of cardiac risk. Automatic EAT evaluation could help to identify patients at risk, especially for diabetic patients. The comparison with EAT volume supports the potential of fourchamber cine EAT area as a surrogate for clinical evaluation, with higher segmentation robustness in systolic frame.

Between the two FCNs investigated, the optimized U-Net was better suited to provide EAT area estimation with a 14.2% precision for the clinically relevant upper three quarters of targeted EAT range. EAT evaluation on cine, leveraging multi-frame information, could be further integrated to explore both retrospective and prospective cardiac studies without the need for a specific acquisition thanks to publicly provided automatic EAT area segmentation.

Figure 1 .

 1 Figure 1. Overview of the study design.

Figure 2 .

 2 Figure 2. Networks' optimized architecture. The two networks evaluated in this study: U-Net and fully-convolutional network (FCNB) architectures included a first 3D convolution layer to allow multiple cardiac frames as input. Following 2D convolution layers encoded images from 48 features up to 768 features. Eventually, the decoder targeted three labels for segmentation in the central input frame: epicardial adipose tissue (EAT), paracardial adipose tissue (PAT), and heart ventricles (HV).

Figure 3 .

 3 Figure 3. Comparison of reference total epicardial fat volume andproposed EAT area measured on four-chamber cine. EAT area was measured in end-systolic or end-diastolic frame across the 100 subjects' database. The three cohorts merged for the database were identified by markers color.

Figure 4 .

 4 Figure 4. Representative segmentation results for each population defined by quartile of EAT area. Images were cropped around the heart for visualization. White arrows point out discrepancies between manual and automatic segmentations. As detailed in the methods, only periventricular EAT was segmented.

Figure 5 .

 5 Figure 5. Quartile classification results from EAT area estimated from networks segmentation against classification from manual EAT area. Only segmentations from preferred systolic frames were shown here. Markers colors were defined by manual EAT quartiles. Red squares delineate manual EAT quartiles.

Table 1 .

 1 Study population clinical characteristics

		Healthy	Non-Diabetic Obese Type-2-Diabetic
	Clinical characteristics
	Number of participants	21	12	67
	Age, years	25 ± 10	41 ± 13	53 ± 10
	Gender: female, n (%)	11 (52)	10 (83)	41 (61)
	BMI, kg/m²	21.9 ± 2.6	40.8 ± 5.9	35.6 ± 6.8
	T2D
	Duration of diabetes, years			8 ± 6
	Cardiovascular risk factors, n (%)

Table 2 .

 2 Mean values and standard deviations (in parenthesis) of segmentation results on the test set Metrics are reported as mean values (standard deviation). Systole and diastole segmentations were not differentiated in these metrics. DSC, dice similarity coefficient; MSD, mean surface distance; RSE, absolute relative surface error. Epicardial Fat values are highlighted in bold.

			DSC			MSD (mm)			RSE (%)
		Intra Inter U-Net FCNB Intra Inter U-Net FCNB Intra Inter U-Net FCNB
	Paracardial Fat	0.85	0.78	0.80	0.78	1.15	2.08	2.38	2.29	11.78	20.43	14.29	17.43
	(PAT)	(0.06)	(0.09)	(0.08)	(0.10)	(0.63)	(1.49)	(1.78)	(1.47)	(8.09)	(18.77)	(10.44)	(17.50)
	Epicardial Fat	0.83	0.76	0.77	0.76	1.53	2.65	1.71	2.06	13.02	17.67	20.33	20.97
	(EAT)	(0.07)	(0.10)	(0.07)	(0.07)	(1.32)	(2.98)	(1.06)	(1.96)	(14.59)	(15.07)	(15.70)	(15.66)
	Pericardial Fat (EAT	0.90	0.88	0.88	0.88	1.12	1.55	1.36	1.60	6.92	9.20	7.36	8.92
	+ PAT)	(0.04)	(0.05)	(0.06)	(0.06)	(0.66)	(0.07)	(0.90)	(1.28)	(7.16)	(6.80)	(9.40)	(12.97)
	Heart ventricles	0.98	0.96	0.97	0.96	0.96	1.88	1.33	1.42	2.33	3.69	3.88	4.22
	(HV)	(0.01)	(0.02)	(0.02)	(0.03)	(0.5)	(2.24)	(0.79)	(0.89)	(2.20)	(3.18)	(4.46)	(5.80)

Table 3 .

 3 DSC, MSD, RSE metrics evaluated per quartile (Q1-Q4) of EAT area for U-Net and FCNB

	Q1	DSC		MSD (mm)	RSE (%)
		U-Net	FCNB	U-Net	FCNB	U-Net	FCNB
	Paracardial Fat (PAT)	0.55	0.53	5.82	5.69	36.21	38.54
	Epicardial Fat (EAT)	0.69	0.67	2.14	2.21	22.15	27.98
	Pericardial Fat (EAT + PAT)	0.78	0.77	1.60	1.78	2.08	2.65
	Heart ventricles (HV)	0.97	0.97	1.12	1.35	12.59	16.19
	Q2	DSC		MSD (mm)	RSE (%)
		U-Net	FCNB	U-Net	FCNB	U-Net	FCNB
	Paracardial Fat (PAT)	0.76	0.75	2.68	2.82	17.29	20.83
	Epicardial Fat (EAT)	0.76	0.74	1.22	1.53	17.85	21.91
	Pericardial Fat (EAT + PAT)	0.87	0.87	1.16	1.35	7.55	8.60
	Heart ventricles (HV)	0.97	0.97	1.11	1.65	2.57	3.04
	Q3	DSC		MSD (mm)	RSE (%)
		U-Net	FCNB	U-Net	FCNB	U-Net	FCNB
	Paracardial Fat (PAT)	0.82	0.82	2.26	1.99	12.72	12.14
	Epicardial Fat (EAT)	0.80	0.79	1.30	1.47	13.49	15.87
	Pericardial Fat (EAT + PAT)	0.90	0.90	1.37	1.43	5.86	5.28
	Heart ventricles (HV)	0.97	0.97	1.08	1.50	2.54	3.07
	Q4	DSC		MSD (mm)	RSE (%)
		U-Net	FCNB	U-Net	FCNB	U-Net	FCNB
	Paracardial Fat (PAT)	0.80	0.78	2.46	3.12	13.65	16.72
	Epicardial Fat (EAT)	0.83	0.79	1.40	2.06	11.72	15.60
	Pericardial Fat (EAT + PAT)	0.91	0.90	1.40	1.84	5.64	6.43
	Heart ventricles (HV)	0.97	0.96	1.31	2.60	3.20	4.52
	Epicardial Fat values are highlighted in bold.

Acknowledgments:

The authors thank Danone Institute International for supporting this work.

Conflicts of Interest:

The authors declare no conflict of interest.

Funding: This project has received financial support from the CNRS through a MITI program and was performed within a laboratory member of France Life Imaging network, grant no. ANR-11-INBS-0006. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Institutional Review Board

Statement: The study was conducted according to the guidelines of the Declaration of Helsinki, and approved by the Ethics Committees of CPP Assistance Publique des Hôpitaux de Marseille (NCT00923039 registered the 18 June 2009, NCT01284816 registered 26 January 2011, NCT02042664 registered 20 January 2014, NCT03118336 registered 18 April 2017).

Informed Consent Statement: Written informed consent, including consent for publication, was obtained from the patient.

Data Availability Statement:

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.