Bruno Massé

Dominique Schneider

The mantissa distribution of the primorial numbers by

Keywords: 2010 Mathematics Subject Classification: Primary 11J71; Secondary 60B10, 11A41 uniform distribution, Benford's law, mantissa, primorial number

will appear on the journal's web site only)

We show that the sequence of mantissas of the primorial numbers P n , defined as the product of the first n prime numbers, is distributed following Benford's law. This is done by proving that the values of the first Chebyshev function at prime numbers are uniformly distributed modulo 1. We provide a convergence rate estimate. We also briefly treat some other sequences defined in the same way as P n .

1. Introduction. Fix a numeration base b and define the mantissa of the positive real number x as the unique number M b (x) in [1, b[ such that x = M b (x)b k for some integer k. We will use the following notations: log is the natural logarithm, log b is the logarithm in base b, p n is the nth prime number and P n is the product of the first n prime numbers (P n is sometimes denoted p n #). We are interested in the distribution of the sequence (M b (P n )) n and, secondarily, of some sequences defined in a like manner. When u n is the nth Fibonacci number or u n = n n , u n = n!, u n = a n and u n = a pn (with log b a irrational), the sequence (u n ) n is known to have a mantissa distributed following Benford's law in the sense of the natural density [START_REF] Berger | One-dimensional dynamical systems and Benford's law[END_REF][START_REF] Fuchs | Le problème du premier chiffre décimal pour les nombres premiers[END_REF][START_REF] Pollicott | A note on uniform distribution for primes and closed orbits[END_REF][START_REF] Posch | A survey of sequences and distribution functions satisfying the firstdigit-law[END_REF]. That is, (1.1) lim

N →∞ 1 N N n=1 1 [1,t[ (M b (u n )) = log b t (1 ≤ t < b).
The sequences (u n ) n satisfying (1.1) will be called natural-Benford. In particular, about 30.1 percent of the terms of a natural-Benford sequence have first digit 1, in the sense of (1.1), when b = 10. This kind of property is known as the first digit phenomenon and holds, more or less for many real-life data sets.

The sequences (n) n and (p n ) n are considered in particular in [START_REF] Cohen | Prime numbers and the first digit phenomenon[END_REF], [START_REF] Eliahou | On the mantissa distribution of powers of natural and prime numbers[END_REF], [START_REF] Fuchs | Le problème du premier chiffre décimal pour les nombres premiers[END_REF], [START_REF] Hlawka | Gleichverteilung und das Konvergenzverhalten von Potenzreihen am Rande des Konvergenzkreises[END_REF], [START_REF] Tichy | Logarithmic uniform distribution of (αn + β log n)[END_REF] and [START_REF] Whitney | Initial digits for the sequence of primes[END_REF]. They are not natural-Benford and their mantissa does not admit any distribution in the sense of the natural density. However, they exhibit Benford behavior in a weaker sense: if u n = n or u n = p n , then lim

N →∞ 1 log N N n=1 1 n 1 [1,t[ (M b (u n )) = log b t (1 ≤ t < b).
Such sequences (u n ) n will be called log-Benford.

The sequence (log n) n is neither natural-Benford nor log-Benford [START_REF] Massé | On weighted densities and their connection with the first digit phenomenon[END_REF]. But it can be called loglog-Benford because lim

N →∞ 1 log log N N n=2 1 n log n 1 [1,t[ (M b (log n)) = log b t (1 ≤ t < b).
All the natural-Benford sequences are log-Benford and all the log-Benford sequences are loglog-Benford. The converses are false [START_REF] Massé | On weighted densities and their connection with the first digit phenomenon[END_REF].

More generally, let (w n ) n be a sequence of positive real numbers summing to infinity and, for each

N ≥ 1, let W N = w 1 + • • • + w N . The w n -density of a set ∆ of positive integers is the number lim N →∞ 1 W N N n=1 w n 1 ∆ (n)
provided that it exists. This is the limit of the weighted frequency of the elements of ∆ among the positive integers. We shall say that the sequence (u n ) n is Benford in the sense of the w n -density when, for all t ∈ [1, b[, the set ∆ = {n : M b (u n ) < t} has w n -density log b t. See [START_REF] Massé | On weighted densities and their connection with the first digit phenomenon[END_REF] for a general treatment of this kind of densities and their connection with Benford sequences.

As pointed out by an anonymous referee, many references on Benford's law and special sequences are available in [START_REF] Drmota | Sequences, Discrepancies and Applications[END_REF] (Schatte's contributions should be mentioned), and [START_REF] Hlawka | Gleichverteilung und das Konvergenzverhalten von Potenzreihen am Rande des Konvergenzkreises[END_REF], [START_REF] Tichy | Logarithmic uniform distribution of (αn + β log n)[END_REF], [START_REF] Tichy | Diskrepanz bezüglich gewichteter Mittel und das Konvergenzverhalten von Potenzreihen[END_REF] and [START_REF] Niederreiter | Beiträge zur Diskrepanz bezüglich gewichteter Mittel[END_REF] contain quantitative theorems, with respect to weighted means, closely related to those featuring in the present paper.

Content.

In Section 2, we present some useful properties of the theory of uniform distribution. In particular, we pay a lot of attention to van der Corput's Difference Theorem and to some of its generalizations.

In Section 3 we show that the sequence (P n ) n of primorial numbers is natural-Benford (while (p n ) n is not). We provide an estimate of the convergence rate in (1.1) depending only on the numeration base b. This is done by proving the uniform distribution of (aϑ(p n )) n , where ϑ denotes the first Chebyshev function and a any nonzero real, with convergence rate depending only on a.

In order to illustrate by a second example the utility of van der Corput's methods, we provide in Section 4 a brief discussion of the sequence (log 2 × • • • × log n) n . We show that it is log-Benford (while (log n) n is not).

Preliminaries.

We collect here the main tools we use. They are all connected with the theory of uniform distribution modulo 1. Most of them are known results.

The fractional part of a real number x will be denoted by {x}. For every real λ, we set e λ (x) = exp(2iπλx) with i 2 = -1. All along this section, (w n ) n is a sequence of positive numbers summing to infinity. Recall that

W N = w 1 + • • • + w N .
2.1. Basic properties. A sequence (x n ) n of real numbers is said to be uniformly distributed modulo 1 (u.d. mod 1) in the sense of the w n -density when, for all t ∈ [0, 1[, lim

N →∞ 1 W N N n=1 w n 1 [0,t[ ({x n }) = t.
Due to Dini's Theorem, this is equivalent to

lim N →∞ 1 W N sup t∈[0,1[ N n=1 w n 1 [0,t[ ({x n }) -t = 0.
The simple fact that log b x and log b (M b (x)) are equal modulo 1 yields the following.

Lemma 2.1. A sequence (u n ) n of positive numbers is Benford in the sense of the w n -density if and only if (log b u n ) n is u.d. mod 1 in the sense of the same w n -density.

The next statement is known as the Weyl Criterion. A simple proof in the case w n = 1 is available in [7, p. 7]. The proof in the general case proceeds along the same lines. Lemma 2.2. A sequence (x n ) n of real numbers is u.d. mod 1 in the sense of the w n -density if and only if for all integers k = 0, lim

N →∞ 1 W N N n=1 w n e k (x n ) = 0.
As a direct consequence of this lemma, if a sequence (u n ) n of positive numbers is Benford in the sense of the w n -density, then so is (λu m n ) n for all integer m = 0 and positive λ.

Lemma 2.3 is elementary but crucial and we did not find any reference for it. Lemma 2.3. Let (a n ) n and (b n ) n be two bounded equivalent sequences of complex numbers, and let (w n ) n be a sequence of positive numbers summing to infinity and equivalent to

(w n ) n . Set W N = w 1 + • • • + w N (N ≥ 1). Then lim N →∞ 1 W N N n=1 w n a n - 1 W N N n=1 w n b n = 0. Proof. Set b n = a n +a n ε n and w n = w n +w n ε n with lim n ε n = lim n ε n = 0. Then, for all N ≥ 1, 1 W N N n=1 w n a n - 1 W N N n=1 w n b n ≤ W N -W N W N 1 W N N n=1 w n a n + 1 W N N n=1 w n a n (ε n + ε n + ε n ε n ) .
By the Stolz-Cesàro Theorem, the sequences (W N ) N and (W N ) N are equivalent. So the first term in the above sum tends to 0 as N tends to infinity because the sequence ((1/W N ) N n=1 w n a n ) N is bounded. Moreover, by the classical generalization of the Cesàro Theorem, the second term tends to 0 too, because

(W N ) N and (W N ) N are equivalent and lim n a n (ε n + ε n + ε n ε n ) = 0.
If two sequences of positive numbers (u n ) n and (v n ) n are equivalent, then for every integer k the sequences of complex numbers (e k (log b (u n ))) n and (e k (log b (v n ))) n are equivalent too. This and Lemmas 2.1, 2.2 and 2.3 prove Lemma 2.4 below.

Lemma 2.4. Two equivalent sequences of positive numbers are simultaneously natural-Benford (respectively log-Benford, loglog-Benford).

Van der

Corput's Difference Theorem. Van der Corput's Difference Theorem [7, p. 26] says that, in the context of the natural density, a sequence (x n ) n is u.d. mod 1 when, for every positive integer h, the sequence (x n+hx n ) n is u.d. mod 1. It derives from Lemma 2.5 below. We present here three generalizations of this theorem. The first one is due to Tsuji [START_REF] Tsuji | On the uniform distribution of numbers mod. 1[END_REF] and extends van der Corput's Difference Theorem to the general context of weighted densities. The second and the third say that, again in the context of weighted densities, (x n ) n is u.d. mod 1 if (x n+hx n ) n tends to be u.d. mod 1 as h tends to infinity in a sense made precise in the statements of Lemmas 2.9 and 2.10. To the best of our knowledge, Lemmas 2.8-2.10 are new results.

Lemma 2.5 is a direct consequence of van der Corput's Fundamental Inequality [7, p. 25] and is crucial in the proof of Theorem 3.4 below. Lemma 2.5. Let N be a positive integer greater than 1, and a 1 , . . . , a N be N complex numbers of modulus 1. Then there exists an absolute constant C such that for all positive integers H < N,

1 N N n=1 a n 2 ≤ C H + C H H h=1 1 N -h N -h n=1 a n a n+h .
Lemma 2.6 is Tsuji's extension of the above lemma to the case of general weighted densities. The real part of a complex number z is denoted by R(z).

Lemma 2.6. Let N be a positive integer greater than 1, and a 1 , . . . , a N be N complex numbers. Then for all positive integers H < N,

| N n=1 w n a n | 2 W N +H-1 ≤ 1 H 2 N n=1 w 2 n |a n | 2 H-1 j=0 1 w n+j + 2R 1
H 2 H-1 h=1 N -h n=1 w n w n+h a n a n+h H-1 j=h 1 w n+j .
Tsuji [START_REF] Tsuji | On the uniform distribution of numbers mod. 1[END_REF]Theorem 10] used this lemma to prove the following generalization of van der Corput's Difference Theorem.

Lemma 2.7. Let (x n ) n be a sequence of real numbers. Suppose that, for all positive integers h, (w n /w n+h ) n is decreasing and (x n+hx n ) n is u.d. mod 1 in the sense of the w n -density. Then (x n ) n is u.d. mod 1 in the sense of the w n -density.

We now present the second generalization. The next lemma uses the bound in Lemma 2.6 more precisely than Lemma 2.7.

Lemma 2.8. Let (a n ) n be a sequence of complex numbers bounded in modulus by 1. Assume that (w n ) n and (w n+1 ) n are equivalent. For h ≥ 1, set

l h = lim sup N →∞ 1 W N N n=1 w n a n a n+h . Then lim H→∞ 1 H H-1 h=1 l h = 0 ⇒ lim N →∞ 1 W N N n=1 w n a n = 0.
Proof. Note that the hypotheses on (w n ) n and the Stolz-Cesàro Theorem imply that for all j ≥ 1 the sequences (W N ) N and (W N +j ) N are equivalent.

Since

W N ≥ W N -h and H ≥ H -h, Lemma 2.6 leads to | N n=1 w n a n | 2 W 2 N W N W N +H-1 ≤ 1 HW N N n=1 w n H H-1 j=0 w n w n+j + 2 H H-1 h=1 1 W N -h N -h n=1 w n H -h a n a n+h H-
N →∞ 1 W N N n=1 w n a n 2 ≤ 1 H + 2 H H-1 h=1 l h , which implies lim sup N →∞ 1 W N N n=1 w n a n 2 ≤ 2 lim sup H→∞ 1 H H-1 h=1 l h .
Lemmas 2.8 and 2.2 prove the following generalization of Tsuji's Theorem. The condition "(x n+hx n ) n is u.d. mod 1 for all h" is replaced by a weaker one which may be interpreted as "(x n+hx n ) n tends to be u.d. mod 1 as h → ∞". Lemma 2.9. Let (x n ) n be a sequence of real numbers. For integers h ≥ 1 and k = 0, set

l h,k = lim sup N →∞ 1 W N N n=1
w n e k (x n+hx n ) and suppose that, for all k, (l h,k ) h converges to 0 as h → ∞ or, more generally, that

lim H→∞ 1 H H-1 h=1 l h,k = 0.
Then (x n ) n is u.d. mod 1 in the sense of the w n -density.

Lemma 2.10 is a kind of Paul Lévy's Theorem for arrays of distribution functions and is essential in the proof of Theorem 4.1.

Lemma 2.10. Let (y h,n ) h≥1,n≥1 be an array of numbers in [0, 1[. For positive integers N and h, for integers k different from zero and t ∈ [0, 1], set

α h (k) = lim sup N →∞ 1 W N N n=1 w n e k (y h,n ) , F N,h (t) = 1 W N N n=1 w n 1 [0,t[ (y h,n ), β h (t) = lim sup N →∞ |F N,h (t) -t|. Then ∀t lim h→∞ β h (t) = 0 ⇒ ∀k lim h→∞ α h (k) = 0 .
Proof. We first apply classical methods used in the proof of Dini's Theorem. For a positive integer h and t ∈ [0, 1], set γ h = lim sup N sup t |F N,h (t)-t|, and let M be a positive integer. Fix t ∈ [0, 1] and m ∈ {0, . . . , M -1} such that m/M ≤ t < (m + 1)/M , and positive integers N and h. Then

F N,h (t) -t ≤ F N,h m + 1 M - m + 1 M + 1 M , F N,h (t) -t ≥ F N,h m M - m M - 1
M , because all the functions considered here are nondecreasing. Hence

sup t∈[0,1] |F N,h (t) -t| ≤ max m=0,...,M F N,h m M - m M + 1 M .
This implies, for all h and M , Fix now k ∈ Z * . For each h ∈ N * , let N h be an integer such that (2.1)

γ h ≤ lim sup N →∞ max m=0,...,M F N,h m M - m M + 1 M ≤ max
1 W N h N h n=1 w n e k (y h,n ) -α h (k) ≤ 1 h
and large enough to ensure that

sup t∈[0,1[ |F N h ,h (t) -t| ≤ γ h + 1/h.
W N h N h n=1 w n e k (y h,n ) = 0.
The proof is completed by using (2.1).

Remark 2.11. The converse of Lemma 2.10 is true and is a consequence of the Erdős-Turán inequality (see Lemma 3.1 for the version of this inequality involving the natural density and [START_REF] Drmota | Sequences, Discrepancies and Applications[END_REF] for the versions involving weighted densities). Applying Lemma 2.9 and Lemma 2.10 with y h,n = x n+hx n leads to a new criterion, more telling than Lemma 2.9: if for all t ∈ [0, 1[,

lim h→∞ lim sup N →∞ 1 W N N n=1 w n 1 [0,t[ (x n+h -x n ) -t = 0,
then (x n ) n is u.d. mod 1 in the sense of the w n -density.

3. The primorial numbers. Throughout this section we set, for n ≥ 2,

P n = p 1 × • • • × p n
, where p n denotes the nth prime number, and

Q n = 2 log 2 × • • • × n log n.
Recall that the sequence (p n ) n is log-Benford and is not natural-Benford. We shall now prove that (P n ) n is natural-Benford and provide a convergence rate estimate. We shall make use of Lemma 2.5 above and of Lemmas 3.1 and 3.3 below. Lemma 3.1 is available, among many other references, in [START_REF] Rivat | Constantes d'Erdős-Turán[END_REF] and known as the Erdős-Turán inequality. Let (x n ) n be a real sequence in [0, 1[ and N be a positive integer. The number D N (x n ), defined by

D N (x n ) = sup 0<c<d<1 1 N N n=1 1 [c,d[ (x n ) -(d -c) ,
is called the discrepancy of (x n ) N n=1 (see [START_REF] Drmota | Sequences, Discrepancies and Applications[END_REF] for more information on this subject). Lemma 3.1. For every positive integer K,

D N (x n ) ≤ 1 K + 1 + K k=1 1 k 1 N N n=1 e k (x n ) .
We point out that, in the above inequality, the choice of the integer K ≥ 1 is free. Lemma 3.2 is known as van der Corput's Theorem [7, p. 17 Lemma 3.3. There exists an absolute constant C such that for all θ > 0 and all positive integers L, M and h with L < M and h ≤ M ,

M n=L e θ (log(Q n+h /Q n )) ≤ C M √ hθ L + M √ hθ + hθ L + 1
.

Proof. Let L, M , θ and h satisfy the hypothesis. If

f (x) = θ h j=1 log (x + j) log(x + j) , then for all x ∈ [L, M ], 0 < f (x) ≤ f (L) ≤ 2hθ L and -f (x) ≥ -f (M ) ≥ hθ 4M 2 . Thus, by Lemma 3.2, M n=L e θ (log(Q n+h /Q n )) ≤ 2hθ L + 2 8M √ hθ + 3
.

The next theorem is stated in terms of mantissa distribution of (P n ) n . It could have been equivalently stated in terms of distribution modulo 1 of (aϑ(p n )) n where ϑ denotes the first Chebyshev function and a any nonzero real.

Theorem 3.4. There exists a positive constant C b , depending only on the numeration base b, such that for every positive integer N ,

sup t∈[1,b[ 1 N N n=1 1 [1,t[ (M b (P n )) -log b t ≤ C b (log log N ) 1/2 (log N ) 1/9 .
In particular the sequence (P n ) n is natural-Benford.

Proof. In what follows, C denotes an absolute positive constant which may vary from line to line. It is written C b when it depends on b. Recall that for all n ≥ 1, n log n ≤ p n ≤ n log n + Cn log log n [START_REF] Ribenboim | The Little Book of Bigger Primes[END_REF].

Fix N > 1. By Lemma 3.1, for every positive integer K,

(3.1) sup t∈[1,b[ 1 N N n=1 1 [1,t[ (M b (P n )) -log b t ≤ D N ({log b P n }) ≤ 1 K + 1 + K k=1 1 k 1 N N n=1 e k (log b P n )
and by Lemma 2.5, for all positive integers h, k and H < N,

1

N N n=1 e k (log b P n ) 2 ≤ C H + C H H h=1 1 N -h N -h n=1 e k (log b (P n+h /P n ))
.

By the Triangular Inequality and the Mean Value Theorem, for all h, k and M ,

M n=2 e k (log b (P n+h /P n )) ≤ M n=2 2kπ log b P n+h /P n Q n+h /Q n (3.3) + M n=2 e k (log b (Q n+h /Q n ))
.

We are going to calculate a bound for each term of (3.3) and then choose the best possible H in (3.2) and the best possible K in (3.1). Fix M ≥ 3 and h ≥ 1. For n ≥ 3,

1 ≤ P n+h /P n Q n+h /Q n ≤ h j=1 (n + j) log(n + j) + C(n + j) log log(n + j) (n + j) log(n + j) ≤ 1 + C log log n log n h . If we set θ = k/log b, this leads to M n=3 2kπ log b P n+h /P n Q n+h /Q n ≤ Chθ M n=3 log log n log n (3.
(Q n+h /Q n )) ≤ √ M + M n= √ M e θ (log(Q n+h /Q n )) ≤ √ M + C √ M √ hθ + M √ hθ + hθ √ M + 1
.

Combining (3.3)-(3.5) gives M n=1 e k (log b (P n+h /P n )) ≤ C b hk M log log M log M + √ hk √ M + M √ hk + hk √ M .
Since N -(log N ) 2/3 and N are equivalent, since log(N -(log N ) 2/3 ) and log N are equivalent, and so on, we get, for all h ≤ (log N ) 2/3 ,

1 N -h N -h n=1 e k (log b (P n+h /P n )) ≤ C b hk log log N log N + √ hk √ N + 1 √ hk + hk N 3/2 . Hence, if H = (log N ) 2/3 , then 1
H H h=1 1 N -h N -h n=1 e k (log b (P n+h /P n )) ≤ C b k log log N (log N ) 1/3 ,
and so, by (3.2),

1 N N n=1 e k (log b P n ) ≤ C b 1 (log N ) 2/3 + k log log N (log N ) 1/3 1/2 ≤ C b √ k (log log N ) 1/2
(log N ) 1/6 . Thus, for every positive integer K,

K k=1 1 k 1 N N n=1 e k (log b P n ) ≤ C b √ K (log log N ) 1/2 (log N ) 1/6 .
The proof is completed by taking K = (log N ) 1/9 in (3.1).

4. Product of the first n logarithms. Recall that the sequence (log n) n is loglog-Benford and is not log-Benford [START_REF] Massé | On weighted densities and their connection with the first digit phenomenon[END_REF]. We shall now prove that the sequence

(log 2 × • • • × log n) n is log-Benford. Lemma 3.2 is ineffi- cient in
F N,h (s) = 1 log N N n=1 1 n 1 [1,s[ M b ((log n) h ) .
For all h, u n+h /u n is equivalent to (log n) h as n → ∞, and this implies that, for all integer k, e k (log b (u n+h /u n )) is equivalent to e k (log b ((log n) h )). So, using Lemma 2.3 with w n = w n = 1/n, for all k and all h we obtain

N →∞ 1 log N N n=1 1 n 1 [0,log b s[ ({log b ((log n) h )}) -log b s = 0.
Then Lemma 2.10 will prove that, for all k,

lim h→∞ lim sup N →∞ 1 log N N n=1 1 n e k (log b ((log n) h )) = 0,
and (4.1), Lemma 2.9 with w n = 1/n, and Lemmas 2.2 and 2.1 will conclude the proof. But direct calculations, using Fuchs' and Letta's methods (see [START_REF] Fuchs | Le problème du premier chiffre décimal pour les nombres premiers[END_REF] and [9, p. 11]), show that lim inf

N →∞ F N,h (s) = s 1/h -1 b 1/h -1 and lim sup N →∞ F N,h (s) = b 1/h (s 1/h -1) s 1/h (b 1/h -1)
.

These two limits tend to log b s as h → ∞. This proves (4.2).

Concluding remarks.

Van der Corput's methods, described in Section 2.2, can also be used to prove that the product sequence

(U n ) n = (u 1 × • • • × u n ) n and the corresponding iterated product sequence (U 1 × • • • × U n ) n are log-Benford or natural-Benford when u n = log 2 × • • • × log n, u n = n!,
u n = n n and u n = P n . When u n = n! (respectively u n = n n ), the numbers U n are called superfactorials (respectively hyperfactorials). A detailed discussion of this subject is in preparation.

It is known that P n = e (1+εn)n log n with lim n ε n = 0. But this does not provide an equivalent of P n . In contrast, the hyperfactorial sequence is equivalent to

An n 2 /2+n/2+1/12 × e -n 2 /4
where A is a constant, and this can be used together with Lemma 2.4 to prove that the hyperfactorial sequence is natural-Benford.

We do not know if the sequence (n#) n , defined by n# = pr≤n p r , is Benford or not because of the irregularity of the prime gaps. So we do not know if the values ϑ(n) of the first Chebyshev function at integers are u.d. mod 1 or not. Of course, since they are respectively equivalent to n and n log n, the sequences (ϑ(n)) n and (ϑ(p n )) n are log-Benford and are not natural-Benford. But this is off topic.

The fact that the sequence (P n ) n is natural-Benford can be proved quickly by combining Lemmas 3.3, 2.3 and 2.9. Indeed, Lemma 3.3 shows that lim h→∞ lim sup

N →∞ 1 N N n=1 e k (Q n+h /Q n ) = 0
where Q n = 2 log 2ו • •×n log n. Since Q n+h /Q n and P n+h /P n are equivalent as n → ∞, Lemma 2.3 shows that this is also true when we replace Q n+h /Q n by P n+h /P n . Lemma 2.9 with w n = 1 concludes the proof. The main term in the calculations in the proof of Theorem 3.4 comes from the term Cn log log n in the formula p n ≤ n log n + Cn log log n. If we replace p n by n log n in Theorem 3.4, we obtain a far better convergence rate:

sup t∈[1,b[ 1 N N n=2 1 [1,t[ (M b (Q n )) -log b t ≤ C b N -1/10
where C b is a positive constant depending only on b. The proof proceeds along the same lines as the proof of Theorem 3.4: it suffices to replace P n by Q n in (3.1) and (3.2), to use (3.5) and to choose H = √ N and K = N 1/10 . The numbers P n -1 and P n + 1 have few divisors and many of them are prime numbers (in this case they are called primorial primes). Yet (P n -1) n and (P n + 1) n are natural-Benford since they are equivalent to (P n ) n .

We cannot prove or disprove that (log 2ו • •×log n) n is actually natural-Benford with our methods. Indeed, if we replace 1/n by 1 in the proof of Theorem 4.1, we get lim inf N F N,h (s) = 0 and lim sup N F N,h (s) = 1 for all s and all h. So (x nx n+h ) n does not tend to be u.d. mod 1 as h → ∞ when

x n = log 2 × • • • × log n (see the remark at the end of Section 2).

Fuchs and Letta's methods (see [START_REF] Fuchs | Le problème du premier chiffre décimal pour les nombres premiers[END_REF] and [9, p. 11]) and direct calculations prove that the sequence (log log n) n is not loglog-Benford. The arguments used in the proof of Theorem 4.1 show that the sequence (u 1 × • • • × u n ) n is loglog-Benford when u n = log log n. Of course we can also consider (log log log n) n with the weights 1/(n log n log log n) and so on.

1. 1 .

 1 Benford sequences. Benford's law in base b is the probability measure µ b on [1, b[ defined by µ b ([1, t[) = log b t (1 ≤ t < b).

  {0, . . . , M} is finite. It follows that ∀t lim h→∞ β h (t) = 0 ⇒ lim h→∞ γ h = 0 .

]. Lemma 3 . 2 .√ ρ + 3 .

 323 Let a and b be two integers with a < b, let ρ > 0, and let f be twice differentiable in [a, b] and such that f (x) ≥ ρ > 0 or -f (x) ≥ ρ > 0 for all x ∈ [a, b] and some positive real ρ. Then b n=a exp (2iπf (n)) ≤ (|f (a)f (b)| + 2) 4 Lemma 3.3 is a direct consequence of Lemma 3.2.

4 )≤

 4 Chθ(log log M ) M log M by the classical properties of the logarithmic integral function. On the other hand, by Lemma 3.3, if h ≤ M , (3.5) M n=3 e k (log b

4 . 1 .

 41 dealing with log 2 × • • • × log n because of the value of the second derivative of x → log log x. Fortunately, we can use direct calculations which are themselves fruitless for sequences like (P n ) n . Theorem The sequence (log 2 × • • • × log n) n is log-Benford. Proof. Set u n = log 2 × • • • × log n and, for N ≥ 2 and s ∈ [1, b],

F

  log b ((log n) h )) = 0. If we prove that for all s ∈ [1, b], N,h (s) = log b s, this will imply lim h→∞ lim sup

  Fix h and H. Then W N and W N +H-1 are equivalent as N → ∞. Moreover n+j converges to Hh as n → ∞, since (w n ) n and (w n+1 ) n are supposed to be equivalent.Hence, by Lemma 2.3, for all H,

	H-1 j=0	wn w n+j converges to H and	j=h H-1	w n+h
		lim sup
				1
					w n+h	.
				j=h	w n+j

w

  If we suppose lim h γ h = 0 and denote the Dirac measure at y h,n by δ y h,n , then the sequence of probability measures 1

				N h
		W N h	n=1	w n δ y h,n	h
	converges weakly to the uniform distribution on [0, 1[. So Paul Lévy's Theo-
	rem, characterizing weak convergence by means of Fourier transform, implies
	lim	1
	h→∞
	The integer N h exists because at least one subsequence of
		1 W N	N n=1	w n e k (y h,n )

N

converges to α h (k) and every subsequence of sup

t∈[0,1[

|F N,h (t) -t| N has upper limit lower than or equal to γ h .


Ghoshana Bista
email: ghoshana.bista@orange.com

Eddy Caron
email: eddy.caron@ens-lyon.fr

Anne-Lucie Vion

Total Cost Modeling for VNF Based on Licenses and Resources

Keywords: License Cost, Simultaneous Active Users, Software licensing, VNF, Software Cost, Cloudification, Softwarization

Moving to NFV (Network Function Virtualization) and SDN (Software Defined Network), Telco cloud architectures face four key challenges: interoperability, automation, reliability, and adaptability. All these challenges involve the optimization of resources; whether it is to increase the utilization of hardware resources (virtualization) or to deliver shared computing resources and functions in real-time (cloudification). Softwarization of networks is a consequence of telecom cloudification. Virtual Network Function (VNF) is protected by IPR (Intellectual Property Right) like any software, ensured by a license describing usage rights and restrictions at a given cost. Until now limited studies have happened in the economic dimension linked to softwarisation. Currently, the telco industry struggles to converge and standardize licensing and cost models. At risk: the network cloudification benefits could be swept away by poor management of resources (Hardware and Software). This article presents a preliminary model for optimizing the total cost of a VNF, based on the Resource Cost (RC) and License Cost (LC). This analysis is inspired by measurement and licensing practices commonly observed in the Telcos industries,i.e consumption and capacity.

INTRODUCTION

NFV aims to increase automation and network reliability for better and quicker service delivery. From the research and markets, the global NFV market is projected to grow from 12.9bn dollars in 2019 to 36.3bn dollars by 2024 1 . Also, 60 percent of service providers will adopt NFV in the next two year 1 . Service providers, more specifically telco companies, need to adapt to this shift quickly and efficiently. More than only technological challenges, historical telcos must face the arrival of large hyper-scalers, partners, and also aggressive competitors.

Service Providers will benefit from NFV if they can enable new services with a faster time-to-market, rapidly scaling resources up and down, lowering the costs. The key challenges facing NFV are thus linked with resource optimization. Success relies on the ability to monitor and use standards and interoperable resources: in other words, to mix and match various software components on standard COTS (Commercial -Off-The-Shelf) hardware. a https://orcid.org/0000-0001-6626-3071 1 https://www.f5.com/ f r f r/company/blog/why-nfv-ismore-relevant-than-ever As the network becomes software, failure in controlling software spending destroys the promises of NFV efficiency. The paradigm shift from equipment property toward SW (Software) Right To Use (SW RTU) is adding complexity in resource management. As SW is protected by IPR over a license, it becomes essential to ensure the compliance of SW deployments regarding acquired rights. As well, it becomes essential to optimize license costs. For this, one of the common practices in IT, extending to Network, is to practice Software Asset Management (SAM). Implementing end-to-end SAM guarantees that users buy all the licenses user need, only the license need: to avoid counterfeiting and waste.

SW license frames the rights and obligations of the CSP (Communication Service Providers) to use SW. License is associated with a cost (LC) which depends on the volume of rights granted. The volume granted and associated conditions of uses are contractually defined by one or several metrics. There are currently, no standards on metrics and their definition, they depend on the creativity of software providers. Which could facilitate every supplier of VNF to propose their metrics, model, and tools. And, this is being an intriguing and complex task for VNF services providers. Based on our observation on the telcos industries, metrics can be linked, among with usage, resource allocation, and resource consumption. Thus, it is not wiser to consider LC and RC independently, but we need to consider that LC can be dependent on the resources and vice versa.

In this paper, we propose a model to evaluate and optimize TCO (Total Cost of Ownership) of VNF deployment based on LC and RC. We base our model on two metrics; Simultaneous Active User (SAU) and Bandwith (BW) as we observed that they are well known in the telcos industries and can fit both resource consumption, allocation, and capacity textitOur contribution:

• We formulate the software cost model for VNF based on LC, RC, and TC

• We introduce the concept of license reference, which is crucial to anticipate the LC, RC, and TC.

• We provide different methods for licensing and estimating its associated cost as LC, RC, and TC.

The rest of the paper is organized as follows: Section 2 provides our related work and background. Section 3 present our models and simulation and Section 4 concludes this research.

BACKGROUND AND RELATED WORK

Business Model

Our assumption of the business model is similar to [START_REF] Ahvar | Ccvp: Costefficient centrality-based vnf placement and chaining algorithm for network service provisioning[END_REF], which is defined as (i)

Related Work

Some researches were focused on optimizing the network cost and the network path most of them include LC as a constant entity. In [START_REF] Mouaci | Virtual network functions placement and routing problem: Path formulation[END_REF] they have dealt with finding the best place of VNF for a better routing path for each demand this article helps our research for finding the placement of SFC (Service Function Chain). [START_REF] Ahvar | Ccvp: Costefficient centrality-based vnf placement and chaining algorithm for network service provisioning[END_REF]] had a significant impact on licensing but still, the authors did not focus on the licensing cost or providing better options for clients. In [START_REF] Kiran | Vnf placement and resource allocation in sdn/nfv-enabled mec networks[END_REF] they developed VNFPRA problem which finds the optimal placement of VNFs in SDN/NFV-enabled MEC (Mobile Edge Computing) nodes to reduce the deployment and resource cost using genetic and mixed-integer problems. From this article, we get an idea about resource allocation. In [START_REF] Liu | Cost-efficient virtual network function placement and traffic steering[END_REF] they focused on the placement of the VNF and traffic steering using network cost, node cost, and VNF placement cost. However, this article did not discussed the total cost and optimizing LC but has illustrated a good insight into in-network cost. In [START_REF] Pham | Traffic-aware and energy-efficient vnf placement for service chaining: Joint sampling and matching approach[END_REF], Chuan Pham et al. formulated problems for joint optimization and traffic cost optimization using the Markov Approximation (MA) in which they added their matching approach called SAMA. This research helps us to get a proper idea of VNF instances, Although this work was huge and concrete, they have not considered license cost or overall total cost. In [START_REF] Shi | Mdp and machine learning-based cost-optimization of dynamic resource allocation for network function virtualization[END_REF] authors focused on resources allocation of NFV components using Markov Decision Process and Bayesian learning which helped to dynamically allocate NFV components.

FROM SYSTEM MODEL TO SIMULATION

System Model

In this section, we provided cost models based on VNF and traditional ways.

Traditional ways

The popular traditional model is a perpetual license and pay as you grow.

• Perpetual license: In this system users have to pay upfront then only users have the right to use the software. Depending upon the license entitlement user can upgrade and update their software. Since it is one-time pay generally it is highly expensive.

The dimension parameters used for a perpetual license are:

-LC prl : One time cost, upfront payment for license. -RC prl : One time cost , upfront payment for resources.

-TC prl : Total cost for perpetual license.

TC prl = LC prl + RC prl (1)
• Pay as you grow (PAYG): It is the model in which end-user have to pay according to their capacity increment, it can be usages based on resources, services, or others. There are lots of pay-as-you methods such as pay as you use, pay as you eat, pay as you go, but for us, it is paid as you grow, and grow here is in terms of SAU/BW. The dimension parameters used for the PAYG license model are:

no SAU = Number of SAU at a time.

-CS PAY G : Unit cost per SAU for license.

-CSr PAY G : Unit cost per SAU for resources.

-LCs PAY G : License cost for SAU.

-RCs PAY G : Resource cost for SAU.

-TCs PAY G : Total cost for SAU using PAYG.

LCs PAY G = Cs PAY G × no SAU RCs PAY GS = CSr PAY G × no SAU (2)
TCs P AY G = LCs PAY G + RCs PAY G (3)
Now for the BW, dimension parameters are;

no BW : number of SAU at a time.

-CB PAY G : Unit cost per BW for license.

-CBr PAY G : Unit cost per BW for resources.

-LCB PAY G : License cost for BW.

-RCB PAY G : Resource cost for BW.

-TCB PAY G : Total cost for BW using PAYG.

LCB PAY G = CB PAY G × no SAU RCB PAY GS = CSr PAY G × no SAU (4) TCB PAY G = LCB PAY G + RCB PAY G (5)
One of the important points here not to forget is that these are actually business models, not actual licensing models, they are only used as license models due to lack of the standard license models and metrics. To fill this gap we proposed license models which ultimately help to construct optimized business models in a virtual environment, VNF.

Virtual Network Function

Our crucial problem was to find reliable and authentic methods for licensing which would ultimately help to model optimize the total cost in VNF. To address this task we used the two important license metrics that are SAU and BW (Bandwidth).

• SAU: SAU generally means simultaneous active users connected with VNF who are consuming some resources, and using services provided by VNF.

• BW: It is related to the amount of bandwidthconsuming/consumed by SAU.

• License reference: it is considered to be the estimated number of licenses required for the VNF system for a certain period.

We choose these metrics among the other existing metrics such as Transmission Per Second (TPS), Request Per Second (RPS), etc. because these metrics are convenient to measure the usages and scalable parameters. Using these two metrics we created two license reference (LR) models LR SAU (Equation ( 6)) and LR BW (Equation ( 7)) respectively for SAU and bandwidth, they can be formulated as follows;

LR SAU = max ∀ j∈D ∑ ∀i∈V average(SAU i , j) (6)
Similarly using BW,

LR BW = max ∀ j∈D ∑ ∀i∈V average(BW i , j) (7)
where, V = (1, 2,.....v) is a set of all concerned VNF (can be same or different type) in node. We define H as a set of hours (1, 2, 3, ..., h), D as a set of days (1, 2, 3, ..., d) and R as a set of License Reference (LR). LR corresponds to LR SAU or LR BW which we get from the Equations ( 6) and ( 7). Table 1 delineates the parameters use in our formulation. Our assumption for this research was that all VNFs were deployed properly in their respective places and they were functioning accurately in full capacity. These license references helped to generate an optimized total cost model which includes license and resource cost calculated as;

TC = RC j + LC j (8)
Now, LC and RC can be calculated for capacity model for a day, j ∈ D and r ∈ R, as:

LC ca j = φ r + σ r α r τ r , (9)
RC ca j = θ r + δ r τ r β r (10)
LC and RC can be calculated for consumption model for a day, j ∈ D, as: Now, implementing these equations ( 6), ( 7) and ( 8) in different scenario, such as i) VNF instances scenario ii) Users dependent iii) Using flavour in nodes. These scenarios are depends on users, usages, and nodes.

LC cp j = γ j × α r , (11)
RC cp j = γ j × β r , (12)

Simulations

In this section, we deal with two types of use case scenarios and a real scenario.

Scenario 1: VNF instances

In this scenario, we presented different available clients' usages like; Web, VoIP, and Online Game. We modified the table of [START_REF] Pham | Traffic-aware and energy-efficient vnf placement for service chaining: Joint sampling and matching approach[END_REF] to adjust it to our model. For this scenario, users need to be aware of their requirements based on SAU or BW. Using SAU and BW the flavour base table is proposed on Table 2 and Table 3 which is supposed to meet the user's requirement. Importantly, license cost and resource cost mentioned in our tables are LC ca and RC ca i.e. license capacity and resources capacity cost.

So, if the traffic requirement is BW (65Mbps) for web services from Table 4 then for this services simulator proposed flavour E and for VoIP simulator proposed flavour D. Whenever exact traffic range requirement is not available simulator proposed higher value flavour, and for the last online game since users requirement is higher than the available option so users either can be satisfied with flavour D or the best option would be to customize their requirements with services provider, i.e. flavour E. So, in these kinds of situation simulator directly proposed flavour E. A similar phenomenon goes for SAU as well. For web services usage, flavour C was proposed from Table 3. Similarly, for VoIP, flavour C matched the requirement too and for the online game, flavour E covered the user's requirement. So using these flavours based tables the end-user can estimate the optimized total cost.

Scenario 2: Users

The second scenario depends on the user. In this research users were categorized in two types, Resources Know Users and Resources Unknown Users.

Resources Known Users (RKU): These are the users who know the estimated amount of resources in terms of SAU/BW required for their system. In this scenario, flavour table was created based on SAU, BW like in previous Table 2 andTable 3, also in these tables total cost was introduced so that depend-ing upon the user's budget they can choose flavours too.

For this evaluation, a simulator had been created named flavour selector where users can input the range of SAU, BW, and total cost depending upon their requirement our simulator will propose the approximate flavor. For example, if the SAU range given by the client is 100-120 our selector suggests flavor A from Table 5. Another case is a range between 130-160 then our selector suggests flavour B also whenever the user inputs range value then it will suggest customizing flavor, i.e. flavour E. Similar to BW, if the client provides the range of bandwidth between 18-20 Mbps the selector will propose flavour B from Table 6. Another interesting case is using total cost. For the total cost, we added the value of Resources Cost (RC ca ) and License Cost (LC ca ) as shown in column (9 th ) in Table 5 andTable 6. So, depending upon the customer's budget simulator proposed flavour between SAU and BW. For example: if the client's budget range is from 800-900k€, the simulator provides flavour D from Table 5. Furthermore, if the range is from 600-700k€ then it could be from flavour C from SAU or BW table. So, to avoid the confusion of choosing between SAU and BW the simulator asks the client preference between SAU and BW. Thus, depending upon the client's needs simulator provides the result either from BW, SAU or cost.

Resources Unknown Users (RUU): These are the users who don't have the estimated knowledge about resources requirements (SAU, BW) for their system. So for these types of users, a simulator was created to provide the users with several choices. At first, users need to provide their range after which the simulator will propose the least SAU value from the Table 5. If the user is not satisfied with that proposed then they can processed further simulator will propose from BW Table 6, least range from BW. If this range is also not satisfactory to the client requirement then the simulator proposed the mean value from the SAU flavour table, if this also fails to meet the user's needs the simulator proposed the mean value from the BW flavour table. After this, the simulator proposed the highest value of SAU and BW from SAU and BW flavour table respectively. So the simulator proposed from least to maximum flavour value from tables based on SAU, BW. Thus, our aim here is to provide as many options as possible to the user. Additionally, the offer can be made concerning the total cost as performed in RKU.

Scenario 3: Node analysis in Real scenario

For this analysis, we used the two techno-economic friendly models to estimate the LC, RC. They are capacity and consumption. To adapt these models from a business point of view we have considered some thresholds, constraints related to license and resources like license threshold, resources threshold, license factors, etc.

• Capacity: The capacity analysis is similar to prepaid service where a certain amount of cost is paid upfront to a certain capacity (license reference for our research) of VNF. When it surpasses the threshold extra costs will be incurred. The threshold can be a license or resource or both.

In this research both LC and RC were estimated using unit cost and license reference using Equations ( 9) and ( 11). In this mode, once the capacity is increased it cannot be reversed even if the consumption (SAU/BW) is lower than the threshold.

• Consumption: Clients will pay for the resources they had consumed or will consume during a certain time. As a consequence, there is no contractual threshold limiting the user's ability to consume resources (SAU, BW). It was calculated using Equations ( 11) and ( 12).

• LC threshold: This is the threshold for calculating license cost. LC threshold was implemented for both LC ca and LC cp . In this research, wherever license is being used it includes both (LC ca , LC cp ).

It is defined at the time of negotiation of the contract based on estimated needs. If the uses exceed the threshold, the cost of a license is increased by 1.5, 2, 3, . . . , n, which is known as license factor σ. The license threshold is based on LR. • RC threshold: It is the threshold in the resources.

Whenever a threshold is exceeded, it requires a careful evaluation to understand whether the exceeded threshold can be covered by a single resource or more. If it can be covered with one resource then our research will use resources factor (δ)=1, and if requires more than one it will be from 1,5, 2, 3, 4, 5, . . . , n. This threshold is also dependent upon the LR. Alike the LC threshold, it can be negotiable between the VNF provider and the SP. • LC factor (σ): It is a multiplicative factor after exceeding threshold, σ= 1.5, 2, . . . , n. License factor and resources factor were introduced here to create a proper business model because whenever the threshold is exceeded in the capacity model, the service provider will charge some extra amount. • RC factor (δ): It is similar to the LC factor but it is in the resources aspect. δ= 1.5, 2, 3, . . . , n. So, now using all these metrics and equations ( 12) and ( 11) the total cost in consumption model becomes: max

∀ j∈D ∑ (α r γ j + β r γ j ) (13)
Similarly, from equations ( 10) and ( 9) total cost for capacity model became: max ∀ j∈D ∑ (φ r + σ r τ r α r + θ r + δ r τ r β r ) (14)

A presumption was made that it will meet the QoS threshold, T H f , i.e min TC ≤ T H f . T H f is not a numerical value but a condition. Now, for this situation, we considered a similar scenario as in [START_REF] Malandrino | Reducing service deployment cost through vnf sharing[END_REF] i.e ICA (Intersection Collision Avoidance). ICA issues the alert signal if any pair of them are about to collision. All the parameters which were adapted to replicate the business models are given in Table 7 andTable 8. This simulation is executed on the Intel(R) Core(TM) i7-6600U CPU @ 2.60GHz 2.81 GHz, 16GB RAM, Windows 10. For the sake of simplicity, we assume that resources can be scaled easily. We generated SAU and BW randomly on each virtual node, also known as Virtual Evolve Packets (vEPC) such as MME, SGW, PGW, etc. After the SAU and BW were generated in each vEPC we implemented our license reference model as in Equations ( 6) and ( 7) and obtain the Figures 2 and6 for SAU and BW. After the estimation of license reference, we estimated the license cost using the Equations ( 9) and ( 11) and its result is shown in the Figures 3 and7. After successfully evaluating the LC we analyze the RC using the Equations ( 10) and ( 12) with the help of the license reference as shown in Figures 4 and8. Further, we estimated the total cost using Equations ( 13) and ( 14) which were shown in Figures 5 and9. The experiment is carried out three times with three different random values for thirty days and its cumulative average results are presented in all figures.

Evaluation

Figures 2 and6 show the estimated license reference based on randomly generated SAU/BW. Randomly generated SAU/BW is also shown in the figures with help of which pay as you grow and perpetual license and its related cost were estimated. We can see on both figures that daily SAU/BW is higher than the reference. It is due to licensing reference being based on hourly average, maximum over a day from all concerned VNF from Equations ( 6) and ( 7). Not to be confused that daily SAU and BW shown in figures are 7 which are the cumulative license cost for 30 days we can see that license cost using perpetual is lower and license cost using pay as you grow is higher. An interesting case here is license cost using consumption and capacity methods these are lower than pay as you grow, among consumption and capacity, consumption has a lower cost than capacity. One can argue that since the perpetual cost is lower why not choose it but it is not beneficiary for the VNF services provider.

Because the perpetual model did not consider the usages or resources consumption it is not fair for the VNF services provider. Now, coming back to our fig- ures in contrast with LC of SAU, BW LC for consumption is higher and capacity is lower. Figures 4 and8 show that the consumption model estimated the lower resources cost than the capacity in both SAU and BW. Figures 5 and9 show the estimated total cost. Figure 5 is the total cost for the SAU here we can see that the consumption model estimated the lower cost than capacity. While the BW capacity model estimated the lower cost than consumption as shown in Figure 9. One of the interesting points we can depict from these figures is that when we use SAU as metrics with our models' consumption model estimated lower cost but in contrast of this when we use BW as metrics with our models capacity model estimated lower cost. This is due to the consumption of BW and the number of SAU can not be compared, these are two different metrics. SAU is a user connected with the VNF and BW is the consumption of bandwidth based on the SAU or services it consumes. This leads us to the point that licensing could be done in different ways depending upon the end-user requirement, scenario, and QoS parameters (throughput, bit-rate, etc). Also, different licensing metrics can be considered and depending upon the licensing metrics LC, RC, and TC cost can be optimized. Thus, in simulation for ICA when we use SAU as a metric consumption model generated an optimized LC, RC and when BW as a metric capacity model provided optimized LC and RC.

CONCLUSIONS

The study has tried to suggest several models that are relevant to the various scenarios. We compare the traditional ways of estimating total cost and our models (capacity and consumption). Results confirm that our model is far better than the traditional one. We also present different kinds of possible scenarios such as VNF instances and Users which have a huge range of requirements to be fulfilled. To meet the scenario requirement, we proposed the flavours methods (SAU, BW). For users, we presented a solution based on total cost, SAU, and BW. Secondly, we implemented our models in real VNF scenario ICA which clearly shows that our models outperform other traditional models it is because we introduced the concept of license reference based on SAU and BW. So, from all these, we can conclude that licensing is a complex task that depends not only on one factor or metrics but also on several metrics, users, services, and many others. We tried to include potential metrics and constructed a novel model. Thus, we assume that our models are not just limited to one scenario but could be implemented on different circumstances and topologies and will be helpful to estimate the optimized total cost. Our future work will be to enhance this model using DF, green energy, and implement it on the more complex VNF SFCs.

Figure 1 :

 1 Figure 1: VNF graph of the ICA service.Table 7: Simulation parameter for SAU. Parameters Value License threshold for capacity 4725

Figure 2 :

 2 Figure 2: License Reference for SAU

Figure 5 :

 5 Figure 5: Cumulative Total cost using SAU

Figure 8 :

 8 Figure 8: Cumulative Resource cost using BW

Table 1

 1

		: Parameters use in problem formation
	r	A License Reference. r ∈ R,
	φ r	Pre-paid amount for License Refer-
		ence (C),
	τ r	Surpass or exceed License Refer-
		ence r,
	α r	Unit cost of license for License Ref-
		erence r (C/SAU or Mbps),
	σ r	License factor for License Refer-
		ence r,
	θ	Prepaid amount for resources (C),
	δ r	Resources factor for License Refer-
		ence r,
	β r	Unit cost of resources for License
		Reference r,
	γ j	License Reference for a day, j ∈ D,
	LC ca	License cost for capacity,
	RC ca	Resources cost for capacity,
	LC cp	License cost for consumption,
	RC cp	Resources cost for consumption.

D

As a set of days

(1, 2, 3, ..., d)

Table 2 :

 2 BW Flavour Table for Scenario 1 (VNF instances)

	Flavour BW	LC	vStorage	vRAM vCPU Redundancy Resources
		(Mbps)	cost	(TB)			cost (K€)
			(K€)
	A	15	200	2	2 GB 2	1	170
	B	20	250	3	2 GB 3	2	200
	C	30	355	4	3 GB 4	2	285
	D	50	435	4	3 GB 4	2	338
	E	65	549	5	4 GB 4	2	420
	F	Customize your needs

Table 3 :

 3 SAU Flavour Table for Scenario 1 (VNF instances)

	Flavour SAU LC	vStorage	vRAM vCPU Redundancy Resources
		cost	(TB)			cost (K€)
		(K€)
	A	100 250	4	2 GB 2	1	100
	B	150 350	4	3 GB 3	2	150
	C	200 450	6	4 GB 4	2	215
	D	250 550	7	5 GB 5	2	275
	E	350 650	8	5 GB 5	2	325
	F	Customize your needs

Table 4 :

 4 Service chain of different client usages with BW

	Client usage	Service Chain Minimum		Minimum
			Traffic	Re-	Traffic	Re-
			quired (BW)	quired (SAU)
	Web Services NAT-FW-	65Mbps		165
		WOC-IDPS
	VoIP	NAT-FW-	35Mbps		200
		TM-FW-NAT
	Online Game	NAT-FW-	150Mbps		300
		VOC-WOC-
		IDPS

NAT: Network Address Translator, FW: Firewall, TM: Traffic Monitor, WOC: WAN Optimization Controller, IDPS: Intrusion Detection Prevention System, VOC: Video Optimization Controller

Table 5 :

 5 SAU Flavour Table for Scenario 2

	Flavour SAU LC	vStorage	vRAM vCPU	Redundancy Resources	Total
		(K€)	(TB)				cost	cost
							(K€)	(K€)
	A	100 250 4	2 GB	2	1	100	350
	B	150 350 4	3 GB	3	2	150	500
	C	200 450 6	4 GB	4	2	215	665
	D	250 550 7	5 GB	5	2	275	825
	E	350 650 8	5 GB	5	2	325	975
	F	Customize your needs

Table 6 :

 6 BW Flavour Tablefor Scenario 2

	Flavour BW	LC	vStorage	vRAM vCPU	Redundancy Resources	Total
		(Mbps)	(K€)	(TB)				cost (K	cost
	A	15	200	2	2 GB	2	1	€) 170	(K€) 370
	B	20	250	3	2 GB	3	2	200	450
	C	30	355	4	3 GB	4	2	285	640
	D	50	435	4	3 GB	4	2	338	773
	E	65	549	5	4 GB	4	2	420	969
	F	Customize your needs

Table 8 :

 8 Simulation parameter for BW.

	Parameters	Value
	License threshold for ca-	65
	pacity
	Resource threshold for	70
	capacity
	σ for both capacity for	1.5 (for LR less then 65, 2
	BW	for LR >65)
	δ	1.5 for SAU < 70 and 2 for
		LR > 70
	θ	700 (C)
	φ	600 (C)
	BW	random(5,10),random(10,20)
		and random(30,40)
	α for BW	10 C per LR
	β for BW	6 C per Resource
	CB PAY G	0.01 (C) per BW
	CBr PAY G	0.04 (C) per BW
	LC prl	500 (C)
	RC prl	800 (C)


Natacha Linas

Marie-Agnès Peyron

Caroline Eschevins

Martine Hennequin

Emmanuel Nicolas

Dr Valérie Collado
email: valerie.collado@uca.fr

Natural food mastication capability in preschool children according to their oral condition: a preliminary study

Keywords: Preschool Children, Dental Caries, Mastication, Particle Size, Food Oral Processing, Swallowing

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Mature mastication and its influencing factors have been widely studied in adults [START_REF] El Osta | The pertinence of oral health indicators in nutritional studies in the elderly[END_REF][START_REF] Tada | Association of mastication and factors affecting masticatory function with obesity in adults : A systematic review[END_REF][START_REF] Woda | The regulation of masticatory function and food bolus formation[END_REF]. However, the masticatory function, and especially the capability to form a food bolus, has been little explored in early childhood under -real‖ conditions (with natural foods in particular), while it is a crucial period for mastication skills acquisition.

The introduction of diverse and increasingly hard food textures [START_REF] Ferrazzano | Assessing changes in oral health-related quality of life and body growth in 3-5 years old children following dental treatment under general anaesthesia due to severe dental caries[END_REF][START_REF] Linas | Masticatory behavior for different solid foods in preschool children according to their oral state[END_REF] is considered to have a substantial impact on mastication maturation (Wilson, Green, & Weismer, 2012b) and to influence the future food preferences and eating habits of the individual. Indeed, the ability to adjust the shape of the masticatory mandibular movements and the force of masticatory muscular contraction to food textural properties gradually improves during the first years of life, especially through stimulation of proprioceptive periodontal and pulpal receptors of the positioning deciduous dentition, and subsequent cortical memorization [START_REF] Almotairy | Development of the jaw sensorimotor control and chewing-A systematic review[END_REF][START_REF] Lund | Mastication and its control by the brain stem[END_REF][START_REF] Lund | Generation of the central masticatory pattern and its modification by sensory feedback[END_REF]. Proper physiological development of oral function is essential to achieve mature mastication characterised by fine and specific adaptation of the motor program to the physical properties of the food.

Little is known about mastication in children with altered dentition, but it may be hypothesised that any alteration of sensory information from teeth could modify the masticatory capability, defined as the capacity to produce a bolus suitable for safe swallowing and further digestion [START_REF] Laguna | The eating capability : Constituents and assessments[END_REF]). To address this issue, a study model with defined oral health alterations would be useful. Recent works have focused on children with early childhood caries (ECC), which has become a significant public health concern worldwide [START_REF] Chen | Prevalence of early childhood caries among 5-year-old children : A systematic review[END_REF], especially affecting socioeconomically disadvantaged families. Early caries impact children's wellbeing, nutrition, growth and development, and families' quality of life [START_REF] Feitosa | The psychosocial effects of severe caries in 4-year-old children in Recife, Pernambuco, Brazil. Cadernos De Saúde Pública / Ministério[END_REF][START_REF] Schroth | Vitamin D status of children with severe early childhood caries : A case-control study[END_REF]Schroth, Levi, Kliewer, Friel, & Moffatt, 2013;[START_REF] Tinanoff | Early childhood caries epidemiology, aetiology, risk assessment, societal burden, management, education, and policy : Global perspective[END_REF][START_REF] Vania | Early childhood caries underweight or overweight, that is the question[END_REF]. In particular, growth chart alteration or increased risk of iron deficiency anaemia or vitamin deficiencies [START_REF] Schroth | Vitamin D status of children with severe early childhood caries : A case-control study[END_REF]Schroth, Levi, Kliewer, et al., 2013;[START_REF] Vania | Early childhood caries underweight or overweight, that is the question[END_REF] are frequently observed in children with ECC. Such deficiencies might be due to poor diet, which could be associated with sociocultural and environmental determinants and/or with food selection induced by mastication difficulties or pain. They might also result from insufficient oral fragmentation of foods during deficient mastication, leading to altered nutrient absorption. Some previous studies showed a decrease in masticatory performance in school-aged and preschool children with caries by assessing their ability to break down standardized pieces of silicone chewed for a fixed number of cycles [START_REF] Barbosa | The relationship between oral conditions, masticatory performance and oral health-related quality of life in children[END_REF][START_REF] Barrera | Mixed longitudinal evaluation of masticatory performance in children 6 to 17 years of age[END_REF][START_REF] Soares | Factors associated with masticatory performance among preschool children[END_REF][START_REF] De Morais Tureli | Associations of masticatory performance with body and dental variables in children[END_REF]. However, because silicone material is not suitable for swallowing and further digestion, the masticatory performance obtained under such conditions cannot be related to the physiological eating behaviour when consuming real foods [START_REF] Van Der Bilt | Assessment of mastication with implications for oral rehabilitation : A review[END_REF][START_REF] Woda | The regulation of masticatory function and food bolus formation[END_REF]. To date, the state and characteristics of ready-to-swallow food boluses have never been explored in children, while it is known that adults with multiple caries swallowed raw carrot boluses containing a greater proportion of large particles [START_REF] Decerle | Chewing Deficiencies in Adults with Multiple Untreated Carious Lesions[END_REF][START_REF] Woda | The regulation of masticatory function and food bolus formation[END_REF]. Recently, alterations of masticatory kinematic behaviour have been observed in children with severe early childhood caries (ECC) in comparison with children with a healthy oral condition [START_REF] Collado | Impact of early childhood caries and its treatment under general anesthesia on orofacial function and quality of life : A prospective comparative study[END_REF][START_REF] Linas | Masticatory behavior for different solid foods in preschool children according to their oral state[END_REF], but it remains unknown if such modifications are sufficient for the production of a food bolus suitable for swallowing and further transformation for nutrient absorption [START_REF] Van Der Bilt | Assessment of mastication with implications for oral rehabilitation : A review[END_REF][START_REF] Woda | The regulation of masticatory function and food bolus formation[END_REF]. In particular, these young children decreased their masticatory frequencies (number of cycles per second) for three different calibrated solid foods (raw carrot, cheese and breakfast cereals) [START_REF] Linas | Masticatory behavior for different solid foods in preschool children according to their oral state[END_REF]. Nonetheless, the number of cycles performed before swallowing was repeatable for a given food for each child, making it possible to predetermine the individual swallowing threshold and collect ready-toswallow food boluses for granulometry analysis.

The present study aimed to measure, for the first time in children, the impact of oral health on their masticatory capability using natural foods. After recording the kinematic parameters, the granulometry of food boluses collected at the time of swallowing after mastication of three different natural foods was compared between children with severe ECC and children with a healthy oral condition within the same age range. Relationships between masticatory and oral health-related parameters were explored.

MATERIAL AND METHODS

This cross-sectional observational study was designed in accordance with Good Clinical Practice and with the ethical standards (ICH expert working group, 1996). All procedures performed in this study were in accordance with the local Ethical Committee (CECIC, 2010/06; IRB Number 5044). Children and parents, or legal representatives, received oral and written information regarding the design and goals of the study, the potential benefits and constraints related to their participation, and information about data confidentiality. They all signed a consent form.

Participants

Children from 36 to 71 months, without developmental disorders and with strict deciduous dentition, who were visiting the special dental care unit of the University Hospital of Clermont-Ferrand (France) between May 2016 and May 2017 for treatment of Early Childhood Caries (ECC) were included (-ECC group‖).

Children of the same age range who were visiting the dental unit for their routine dental examination and without any oral problems were included in the -control group‖. 13 children were included in each group.

Explanations about the calculation of the required number of subjects is detailed in the -Statistical analysis‖ section (2.6).

Experimental procedures and data collection

The descriptive variables and study criteria were collected by the two investigators (dental practitioners).

At the first appointment, general health-related data (age, height and weight) and oral health criteria, detailed in the following section, were gathered during children's clinical examination and an interview with the caregivers.

A second session, within the following month, was organized according to the participants' schedule for the mastication evaluation tests. It was planned between meals, in the mid-morning or mid-afternoon.

Children were invited to masticate samples of three solid foods with different textures that are commonly used in the research laboratory: cheese (CHS), breakfast cereals (CER), and raw carrot (CAR). For each child, 12 calibrated food samples were prepared: four samples of the breakfast cereal Fitness® Nature This article is protected by copyright. All rights reserved.

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(Nestlé) (1 g ± 10%), four of the raw carrot and four of the cheese Emmental Meule d'or® (Entremont).

Cylindrical samples of carrots and cheese were obtained using a cutter (diameter 2 cm/thickness 1 cm, 3 g ± 10%) [START_REF] Linas | Masticatory behavior for different solid foods in preschool children according to their oral state[END_REF]. During the mastication tests, children sat at a small table with their feet on the floor and one video camera was placed about one meter in front of them, in order to frame the children's face and shoulders. The children chose the food sample order in which they wanted to chew them. Then, they chewed all samples of a given food in a row. The children were encouraged to express any difficulty upon refusal or uneasiness with the food samples, and tests were stopped at will.

As young individuals are supposed not to be mature enough to stop themselves and spit out the food bolus right before swallowing [START_REF] Gisel | Effect of food texture on the development of chewing of children between six months and two years of age[END_REF], the masticatory kinematic parameters had to be recorded for each child in order to determine the individual swallowing threshold. This threshold allows the collection of a ready-to-swallow food bolus for further granulometric analysis. Consequently, four mastication sequences were video recorded for each tested food. The first two masticatory sequences allowed live recording of the natural number of cycles needed before swallowing (Nc) [START_REF] Linas | Masticatory behavior for different solid foods in preschool children according to their oral state[END_REF]. This number was later verified by individual viewing of the video sequences (Imovie® software) in random order by two dental practitioners also involved in a previous study [START_REF] Linas | Masticatory behavior for different solid foods in preschool children according to their oral state[END_REF]. Both investigators were trained in recording masticatory kinematic parameters according to the routinely used procedures in mastication studies in the research laboratory [START_REF] Nicolas | Validation of video versus electromyography for chewing evaluation of the elderly wearing a complete denture[END_REF]. Agreement between the values measured by the two investigators was ensured for each mastication sequence (interindividual variation of less than 10%). If the results varied between them by more than 10%, consultation with a third reader was organized and the sample recording was registered as uninterpretable if any doubt remained. During the next two masticatory sequences, each food bolus was collected for further granulometric analysis, by stopping the child at the predetermined number of cycles (Nc) needed before swallowing. Each food bolus was collected in a receiving cup, and the child was asked to take a small amount of water and spit it out with the remaining particles. If a food bolus seemed incomplete, the child was asked to chew another sample. For one child, food boluses were all incomplete and were not analysed.

The collected food boluses were then stored for at least 24 h at -18°C.

The particle size distribution of each food bolus was determined by manual sieving. For CAR and CHS boluses, a watery sieving was performed. Each bolus was defrosted and particles were separated by rinsing This article is protected by copyright. All rights reserved.

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them onto a stack of nine sieves with apertures of 7.1, 6.3, 4, 2.5, 2, 1.4, 1, 0.8 and 0.4 mm (Saulas, France), using a gentle hand shower. Each sieve was then dried with a paper towel and put into a ventilated oven at 37°C for 6 minutes. For CER food boluses, particle size distribution was assessed using dry sieving. Each bolus was poured onto a 0.3 mm nylon cloth, washed with running water for defrosting, saliva elimination and particle spreading and then left in a ventilated oven for 1 h at 37°C. The dried bolus was poured onto the stack of sieves described above and manually sieved using a paintbrush. For both sieving procedures, the particles retained by each sieve were weighed.

Oral health criteria

Oral state

The dmft index (decayed, missing, filled teeth index) was used to evaluate children's carious status (World Health Organisation, 2013).

International Caries Detection and Assessment System scores (ICDAS) specified the extent of decay, and the clinical consequences of caries were evaluated with the pufa index (visible pulp, ulceration of the oral mucosa due to root fragments, fistula or abscess) [START_REF] Monse | PUFA--an index of clinical consequences of untreated dental caries[END_REF][START_REF] Shoaib | Validity and reproducibility of ICDAS II in primary teeth[END_REF]. The severity of ECC was defined according to the criteria specified by the American Academy of Pediatric Dentistry (American Academy of Paediatric Dentistry, 2016).

The presence of orofacial dysmorphology was controlled in both groups, as it could modify dental occlusion and thus alter mastication [START_REF] Marquezin | Assessment of masticatory performance, bite force, orthodontic treatment need and orofacial dysfunction in children and adolescents[END_REF]. It was checked using the different occlusal deviations of the -dental health‖ component of the Index of Orthodontic Treatment Need (IOTN) [START_REF] Brook | The development of an index of orthodontic treatment priority[END_REF]. In this study, they were clustered in dentomaxillary disharmony, eruption or number anomalies, and sagittal, vertical and transversal dimension abnormalities.

Occlusal parameters

The number of posterior (post-canine) and anterior (incisor and canines) functional units (FUs) was obtained by counting the number of pairs of antagonist teeth with at least one occlusal contact visible after chewing on a 200 µm articulating paper for 2-3 cycles [START_REF] Owens | Masticatory performance and areas of occlusal contact and near contact in subjects with normal occlusion and malocclusion[END_REF].

Orofacial dysfunction criteria

The frequency of orofacial dysfunction was assessed using the French version of the NOT-S questionnaire (Nordic Orofacial Test-Screening) [START_REF] Bakke | Development and evaluation of a comprehensive screening for orofacial dysfunction[END_REF]. This two-part questionnaire consists of a structured interview exploring -sensory function‖ (I), -breathing‖ (II), -habits‖ (III), -chewing and swallowing‖ (IV), -drooling‖ (V), -dryness of the mouth's‖ (VI) functional domains, and a clinical examination evaluating -face at rest‖ (1), -nose breathing‖ (2), -facial expression‖ (3), -masticatory muscle and jaw function‖ (4), -oral motor function‖ (5) and -speech‖ (6). The -chewing and swallowing‖ domain contains the 5 following dysfunction items: -Does not eat with the mouth (nasogastric tube, gastrostomy or other)‖, -Do you find it difficult to eat foods with certain consistencies‖, -Does it take you 30 minutes or more to eat a main meal‖, -Do you swallow large bites without chewing‖ and -Do you often cough during meals‖. Each domain is scored 1 if at least one dysfunction (one item within each domain) is noted. The total score ranges from 0 to 12.

Mastication evaluation criteria

Data collection of mastication evaluation criteria was possible for the food samples accepted and masticated by children. Food refusals were analysed separately.

Mastication capability: Bolus granulometry at the time of swallowing

The food bolus granulometry criterion was the median particle size value (D50 value) calculated for each food bolus collected at the moment of swallowing (Nc). For each sample, a cumulative curve was drawn from the particle mass falling through each sieve. The D50 value, defined as the median particle size, was extracted from this curve by pinpointing the theoretical sieve size that would let 50% of the particle mass pass through. Hence, the higher the D50 value is, the greater the proportion of large particles in the food bolus.

Masticatory behaviour: food refusals, kinematic parameters, and quality of muscular function

For each food, the number of food refusals was noted and reasons were classified in -Don't like‖, -Don't know/Never tried‖, -Painful‖ and -Too difficult to eat‖.

Masticatory behaviour was evaluated by three kinematic parameters. The chewing time (Ti, in seconds) was measured from the moment the food was placed into the mouth until the last food bolus manipulation just before complete swallowing. The number of masticatory cycles (Nc) was the number of chewing actions, with and without lip closure, corresponding to biting movements (tongue and perioral muscles manipulation movements were not counted). The chewing frequency (Fq) was calculated by the ratio Nc/Ti.

The -quality of muscular function‖ was assessed on the video recordings, using a clinical tool developed by speech therapists [START_REF] Pires | Influence of the duration of breastfeeding on quality of muscle function during mastication in preschoolers : A cohort study[END_REF]. The two investigators, trained in orofacial functional evaluation as part as their specialty skills requirement, reached an agreement for each video recording. This scale contains 5 items: incision, lip competence, masticatory pattern (unilateral or bilateral), masticatory movements and use of perioral muscles. The predominant use of tongue to manage the food was added to these items for the present study. Every parameter was scored 1 if performed correctly and 0 otherwise.

For these parameters as well as for kinematic ones, the muscular activity in the anterior region of the oral sphere was closely observed to distinguish biting movements from food manipulation by soft tissues. In particular, when the food is managed mostly by soft tissues, obvious contractions of the lips and cleaning movements of the tongue in the vestibules are noticeable through lips and cheeks. Tongue protrusion is also visible in case of predominant use of the tongue. Conversely, strong contractions of the mandibular elevator muscles are observed when the food is crushed between the teeth. When the pattern is mostly unilateral, a stronger contraction is recognizable on the predominant side of mastication.

Statistical analysis

The required sample size was estimated from a preliminary pilot study that measured the mean D50 value for raw carrot (D50 CAR) between children with healthy teeth and children with ECC (eight first included children of each group). The mean D50 CAR value was higher in the ECC group than in the control group (4.55 µm versus 3.08 µm respectively) (SD = 0.98). Calculations were based on this difference for a continuous criterion with independent values and indicated the need for at least 10 subjects for each group (α = 5%, β = 10%, epiR package 0.9-30). As children could refuse certain food samples, leading to missing data, thirteen children were finally included in each study group.

A statistical analysis was conducted using SPSS® (IBM, v20). The significance threshold value was set at p<0.05. The number of FUs and the NOT-S mean scores were compared between both groups with a series of Mann Whitney U tests. For each food, the individual repeatability of masticatory parameters (Ti, Nc, Fq and D50) was verified (Wilcoxon-Sign-Rank test). Food sample refusal rates (Fisher's exact test) and masticatory parameters (Mann Whitney U test) were compared between groups for each food. In addition, effect size was measured by (η 2 ) (partial eta squared). For the masticatory criteria, all recorded data of all subjects included in the study were analysed. Missing data corresponded to food sample refusals. Potential correlations were sought for all children between dmft scores, number of FUs, NOT-S scores and masticatory parameters (Spearman's test).

RESULTS

Descriptive data

Thirteen children suffering from ECC (9 girls and 4 boys), and thirteen children with healthy oral state (5 girls and 8 boys) were included in the study. Children with ECC all exhibited the severe form of the disease. Data regarding general and oral health are gathered in Table 1. Both groups were similar in age, weight and height (U tests of Mann Whitney, NS), and sex (Fisher's exact test, NS). Both groups were comparable concerning the presence of dysmorphology (Fisher's exact test, NS) (Table 1).

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Oral state and occlusal parameters

For the ECC group, the mean dmft score was 11.54 ± 4.41 (Table 1).

Children from the ECC group had a significantly lower number of posterior FUs than the children from the control group (3.31 ± 1.03 vs. 4.00 ± 0 respectively, U test of Mann Whitney, p≤0.05) (Table 1). The difference was not significant for anterior FUs (U test of Mann Whitney, NS).

Orofacial dysfunction

The description of domain-by-domain dysfunction frequencies (number of children affected by each dysfunction) is detailed in Figure 1. The NOT-S mean global score was significantly higher for children with ECC than for children from the control group (2.62 ± 1.37 vs. 1 ± 0.91, U test of Mann Whitney, p≤0.01). The only specific domain to be significantly altered in children with ECC (higher mean scores) compared to children from the control group was the -Mastication and swallowing‖ domain (0.69 ± 0.48 vs. 0.15 ± 0.38 respectively, U test of Mann Whitney, p≤0.01). When detailing this domain, only 2 children from the control group took longer than 30 minutes to eat their meal and no other dysfunction item was scored for this group. In the ECC group, 9 children (out of 13) showed an altered -chewing and swallowing‖ domain (NOT-S IV): Item-by-item detail for these children is given in Table 2. The other four children did not have any dysfunction in this domain.

Mastication evaluation criteria

During the mastication sequence, children did refuse some food samples. Mastication results presented in the following sections correspond to the number of children that accepted the food samples and were able to spit out interpretable food boluses for granulometric analysis. One child had difficulty to spit out complete food boluses that were all not interpretable.

The food bolus granulometry (D50) and the masticatory kinematic parameters (Ti, Nc and Fq) and) did not differ significantly between the two corresponding replicates for each food (Wilcoxon-Sign-Rank test, NS), regardless of the group.

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Accepted Article

Mastication capability: bolus granulometry at the time of swallowing

For the three foods, the food boluses collected in children with ECC consisted of a greater proportion of large particles, as indicated by significantly higher D50 mean values, compared to the food bolus collected after mastication in children from the control group (U test of Mann Whitney, p≤0.001). For example, the D50 value for raw carrot was 4384µm ± 929, compared to 2960µm ± 627 in children from the control group (U test of Mann Whitney, p≤0.001). The detail of D50 mean values obtained from the food samples accepted and masticated by both groups of children is given in Table 3. Figure 2 gives a box-plot representation of D50 values in both groups for the three foods.

Masticatory behaviour: food refusals, kinematic parameters, and quality of muscular function

The frequency and distribution of food refusals were comparable for both groups (Fisher's exact test, NS). Details of reasons for refusals according to the food are detailed in Table 4.

Mean chewing times for CER and CHS were significantly higher in the ECC group (U test of Mann Whitney, p≤0.001 and p≤0.05, respectively) (Table 5). Mastication in children with ECC was characterized by a lower chewing frequency for all tested foods (U test of Mann Whitney, p≤0.001).

Regarding the quality of muscular function while masticating, -Lip incompetence while masticating‖ was the only altered muscular function observed in children from the control group (4 children/13). Among 13 children with ECC, nine presented one or more alterations in their -quality of muscular function‖. Six children presented impaired incision, four presented lip incompetence, seven presented unilateral mastication, one presented predominant vertical masticatory movements and four presented a predominant use of their tongue to crush the food.

Correlations between masticatory parameters and oral health criteria

For the three foods, the dmft scores were negatively related to chewing frequencies (Fq CER: ρ = -0.63, p≤0.001; Fq CHS: ρ = -0.57, p≤0.001; Fq CAR: ρ = -0.56; p≤0.001) and positively correlated with D50 values (D50 CER: ρ = 0.42, p≤0.01; D50 CHS: ρ = 0.69, p≤0.001, D50 CAR: ρ = 0.63, p≤0.001).

Moreover, the number of posterior FUs was positively correlated with the chewing frequency measured for carrot (ρ = 0.36, p≤0.05) and cereals (ρ = 0.34, p≤0.05). It was negatively related to D50 CHS (ρ = -0.42, p≤0.01) and D50 CAR (ρ = -0.37, p≤0.05).

Finally, the global NOT-S score was negatively correlated to the chewing frequency for the three test food (Fq CER: ρ = -0.57; p≤0.001, Fq CHS: ρ = -0.64; p≤0.001, Fq CAR: ρ = -0.56; p≤0.001), and positively correlated with the cheese D50 (ρ = 0.51; p≤0.001) and the carrot D50 (ρ = 0.51; p≤0.001).

DISCUSSION

This preliminary study is the first to assess the granulometry of food boluses collected at the time of swallowing in preschool children, after masticating natural foods with different textures. Similarly to adults [START_REF] Decerle | Chewing Deficiencies in Adults with Multiple Untreated Carious Lesions[END_REF], young children's ability to reduce different foods was altered in the presence of caries when compared to children with healthy teeth. Both chewing frequency and granulometric alterations of the food bolus were related to the extent of caries, and particularly with the loss of posterior functional dental units responsible for a reduced sensory input from the teeth receptors. Masticatory behavioural adaptations were not sufficient to compensate for this deficiency.

In particular, the present study showed that increasing chewing time is not sufficient for children with ECC to produce a state of food fragmentation comparable to that of children with healthy teeth. Indeed, the food particle size analysis showed a significant increase of the mean D50 values in the presence of ECC for all three of the tested foods. However, the difference is more important for carrot and cheese compared to cereals, suggesting the impact of the type of food tested on masticatory capability should further investigated. The D50 -normality threshold‖ (maximum value of D50 to define a -normal‖ capacity for fragmenting foods) is still unknown in children, while in adults this value, the masticatory normative indicator (MNI), was estimated at 4000 µm for raw carrot [START_REF] Woda | The masticatory normative indicator[END_REF]. In the present study, the mean D50 value obtained for carrot boluses in children with ECC exceeded the MNI determined in adults, whereas children with healthy teeth produced boluses with a mean D50 value around 3000 µm (4384µm ± 929 in children with ECC vs. 2960 µm ± 627 in children from the control group). Acquiring more data for children according to their age and stage of dental development would make it possible to determine the mean progression of D50 values throughout childhood according to each food. Moreover, there is little data available regarding the quantitative and qualitative changes in saliva secretion throughout infancy and childhood, although it is crucial for food bolus formation [START_REF] Dezan | Flow rate, amylase activity, and protein and sialic acid concentrations of saliva from children aged 18, 30 and 42 months attending a baby clinic[END_REF][START_REF] Negoro | Oral glucose retention, saliva viscosity and flow rate in 5-year-old children[END_REF]. Existing studies on children with ECC have only focused on the bacteriological and immunological aspects of saliva [START_REF] Bachtiar | Mutans Streptococci counts from saliva and its protein profile in early childhood caries[END_REF][START_REF] Letieri | Longitudinal Evaluation of Salivary Iga-S in Children with Early Childhood Caries Before and After Restorative Treatment[END_REF] or on the link between saliva properties and glucose retention risk [START_REF] Negoro | Oral glucose retention, saliva viscosity and flow rate in 5-year-old children[END_REF]. The role of saliva parameters on mastication in young children would require further investigation. Impaired masticatory capability can also cause swallowing difficulties, mainly due to inadequate food fragmentation and production of insufficiently lubricated particles that can then escape to the upper respiratory airways. Clinical experience suggests that unpleasant or even traumatic experiences of food aspiration often lead children and/or parents to exclude that food from their diet. In this study, swallowing difficulties were observed in at least five children with ECC out of 13, as -cough occurring at almost every meal‖ (item NOT-S IV E) or -swallowing of large bites without chewing‖ (item NOT-S IV D) were reported by their parents on the orofacial dysfunction test screening (NOT-S). Given the strong correlation between the NOT-S scores and the physiological measurements (D50 and chewing frequency) observed in this study, it can be emphasised that this clinical test could be used as a reliable tool to screen for masticatory deficiencies [START_REF] Collado | Impact of early childhood caries and its treatment under general anesthesia on orofacial function and quality of life : A prospective comparative study[END_REF][START_REF] Linas | Masticatory behavior for different solid foods in preschool children according to their oral state[END_REF]. This accessible tool could be particularly useful to detect masticatory deficiencies in young children or in patients with special needs, in order to adapt their diet and guide their functional rehabilitation.

In addition to masticatory kinematic parameter alterations, various masticatory behaviour modifications depending on the food physical characteristics were observed during the study. As previously shown, some children with ECC exhibited a predominant use of their tongue and lips to crush cheese against the palate, and let breakfast cereals soften in saliva [START_REF] Linas | Masticatory behavior for different solid foods in preschool children according to their oral state[END_REF]. Half of the children with ECC included in this [START_REF] Mc Donnell | Chewing side preferences in children[END_REF]. During the mastication tests, some food refusals were due to mastication difficulties (declared as -painful‖ or -too difficult to eat‖), suggesting that food selection may be another adaptation strategy. In order to protect their child from food aspiration, parents also reported excluding some foods at home, such as meat, from the child's diet or modifying food textures by mixing or mincing food, for example. The occurrence of all these behavioural adaptations and diet modifications throughout children's development may lead to maxillary growth alteration, an aggravating factor of dysfunction [START_REF] Limme | The need of efficient chewing function in young children as prevention of dental malposition and malocclusion[END_REF].

Whether children exclude certain foods from their diet or swallow a food bolus containing large particles, this could potentially modify the contact area with digestive enzymes and nutritional consequences may be expected [START_REF] Schroth | Vitamin D status of children with severe early childhood caries : A case-control study[END_REF]Schroth, Levi, Kliewer, et al., 2013;[START_REF] Vania | Early childhood caries underweight or overweight, that is the question[END_REF]. Early childhood caries is currently considered a chronic disease due to persistent psychosocial feeding risk factors and associated risk of caries relapse and related oral infection [START_REF] Gussy | Early childhood caries : Current evidence for aetiology and prevention[END_REF]. The functional impact of oral pathologies, such as ECC, may also expose children to long-lasting nutritional risks due to persistent mastication impairment and/or subsequent diet adjustments. Previous data has suggested that mastication difficulties persisted in this population three months after extensive dental treatment under general anaesthesia [START_REF] Collado | Impact of early childhood caries and its treatment under general anesthesia on orofacial function and quality of life : A prospective comparative study[END_REF]. Some authors have addressed the evolution of mastication after dental rehabilitation in children [START_REF] Serra | A 1-year intraindividual evaluation of maximum bite force in children wearing a removable partial dental prosthesis[END_REF] but none in children with ECC. In particular, it is still unknown if and how children's capacity to produce a food bolus suitable for swallowing and digestion could recover post-operatively, regardless of the type of treatment provided (prosthesis or conservative treatment). A one-year follow-up mastication study is ongoing in children with ECC, while keeping in mind that these evolutions remain dependent on psycho-socio-cultural determinants.

CONCLUSIONS

For the first time, masticatory capability while masticating natural foods was compared between preschool children with and without ECC. This study suggested that oral condition, and in particular sensory information from teeth, was an important determinant of mastication behaviour and capability. In the presence of dental caries, children's ability to reduce different foods was altered. Children also modified their masticatory behaviour such as increasing their chewing time, resorting to unilateral mastication or even refusing certain foods. Nutritional consequences can be expected, whether children exclude certain foods from their diet or swallow a food bolus containing large particles. Comprehensive dental treatment may include nutritional guidance and mastication rehabilitation programs. Informed consent: Parents or legal guardians, and children, received oral and written information explaining the study's design and the possible benefits and constraints related to their participation, and were asked to sign a consent form. Moreover, the document specified that all data collected during the children's usual follow-up would be used anonymously for research purposes. NOT-S IV items ("chewing and swallowing" domain)

AUTHORS' CONTRIBUTIONS

Natacha

REFERENCES TABLES

NOT-S IV A "Does not eat with the mouth" NOT-S IV B "Have difficulties with certain food consistencies" x x x x x x x NOT-S IV C "Take 30 minutes or more to eat a main meal"

  Linas performed most of the masticatory tests and data acquisition, contributed to data analysis and interpretation and to the drafting of the manuscript. Marie-Agnès Peyron contributed to the design of the study and critically revised the manuscript. Caroline Eschevins contributed to data acquisition and critically revised the manuscript. Martine Hennequin contributed to the conception of the study and critically revised the manuscript. Emmanuel Nicolas contributed to the design of the study, contributed to data analysis and interpretation and critically revised the manuscript. Valérie Collado conceived and design the study, performed part of the masticatory tests and contributed to data acquisition, analysis and interpretation and to the drafting of the manuscript. All authors approved the final manuscript. ETHICAL STATEMENT Declarations of interest: The authors declare that they do not have any conflict of interest Ethical Review: This prospective observational comparative study was designed in accordance with Good Clinical Practice (International Clinical Harmonization. Guideline for good clinical practice 1996, s. d.) and was approved by the local ethical committee (CECIC, 2010/06; IRB Number 5044).

  Childhood Caries, NOT-S: Nordic Orofacial Test-Screening

Fig. 1 Fig. 2

 12 Fig. 1 Domain-by-domain orofacial dysfunction frequency in both ECC and control groups (n=13 per

Table 1 :

 1 General and oral health-related data for the ECC and control groups.

	Criteria	ECC group	Control group	Statistical
				differences
		(n=13 children)	(n=13 children)
	Girls/Boys ratio (n/n)	9/4	5/8	Fisher's
				exact test,
				NS
		mean ± SD (min-max) mean ± SD (min-max)	Mann-
				Whitney U
				test
	Age (months)	57.5 ± 6.5 (46-67)	56.9 ± 10.3 (36-71)	NS
	Weight (kg)	17.2 ± 2.2 (14.6-23.3)	18.8 ± 2.7 (13.7-23)	NS
	Height (cm)	107.7 ± 4.1 (102-116)	109.5 ± 7.5 (93-120)	NS
	dmft index	11.54 ± 4.41 (3-18)	0	p≤0.001
	decayed	10.85 ± 3.89 (3-16)	0	p≤0.001
	missing	0.38 ± 0.77 (0-2)	0	NS
	filled	0.31 ± 1.11 (0-4)	0	NS
	ICDAS scores
	ICDAS 0 (Sound)	9.15 ± 4.16 (3-17)	20 ± 0 (20-20)	p≤0.001
	ICDAS 1 (1 st enamel visual changes)	0	0	NS
	ICDAS 2 (Clear enamel visual changes)	0	0	NS
	ICDAS 3 (Localized enamel disruption)	0	0	NS
	ICDAS 4 (Dark area on underlying dentine)	4.77 ± 2.42 (2-10)	0	p≤0.001
	ICDAS 5 (Distinct cavity with exposed	0.23 ± 0.83 (0-3)	0	NS
	dentine)
	ICDAS 6 (Large cavity with exposed dentine)	5.46 ± 3.62 (0-12)	0	p≤0.001
	PUFA score
	Visible pulp (p)	5.38 ± 3.55 (1-14)	0	p≤0.001
	Ulceration (u)	0	0	NS
	Fistula (f)	0,08 ± 0,28 (0-1)	0	NS
	Abscess (a)	0	0	NS
	Occlusal parameters
	Anterior functional units	3.23 ± 2.42 (0-6)	4.92 ± 1.75 (2-6)	NS
	Posterior functional units	3.31 ± 1.03 (1-4)	4.00 ± 0 (4-4)	p≤0.05
	Dysmorphologies	n	n	Fisher's
				exact test,
				NS
	Dento-maxillary dysharmony	0	0
	Eruption or number anomalies	0	0
	Sagittal dimension abnormality	0	2
	Transversal dimension abnormality	2	1
	Vertical dimension abnormality	2	4
	ECC: Early Childhood Caries; dmft: decayed, missing, filled, teeth index; ICDAS: International Caries Detection and
	Assessment System; PUFA: visible pulp, ulceration of the oral mucosa due to root fragments, fistula or abscess; NS:
	no statistical difference (p>0.05).

Table 2 :

 2 Detail of the impaired -chewing and swallowing‖ NOT-S IV domain (5 items) for the nine children (out of 13) of the ECC group.

		Children from the ECC group with an impaired NOT-S
					IV domain
	Child 1	Child 2	Child 3	Child 4	Child 5	Child 6	Child 7	Child 8	Child 9

Table 3 :

 3 Comparison of the food related D50 value between the ECC group and the control group.

			ECC group	Control group	Statistical	Effect size
			(n=13)	(n=13)	differences
					between groups
	Tested		D50 (µm)	D50 (µm)	Mann-Whitney	η 2 (partial eta
	food				U test	squared)
			n=12 (1 R)	n=12 (1 NI)
	CER	mean ± SD	1120 ± 222	929 ± 186	p≤0.001	0.186
		CI	1027 -1214	848 -1009
			n=11 (2 R)	n=10 (2 R + 1 NI)
	CHS	mean ± SD	4596 ± 1617	2237 ± 756	p≤0.001	0.470
		CI	3879 -5313	1883 -2591
			n=9 (4 R)	n=10 (2 R + 1 NI)
	CAR	mean ± SD	4384 ± 929	2960 ± 627	p≤0.001	0.465
		CI	3922 -4847	2666 -3253

CAR: raw carrot samples; CER: cereals samples, CHS: cheese samples; ECC: Early Childhood Caries; CI: Confidence interval; n correspond to the number of children with no food refusal. R: number of refusals, NI: number of not interpretable food boluses.

Table 5 :

 5 Comparison between groups of the kinematic parameters values for each food.

						Statistical
			ECC group (n=13)	Control group (n=13)	differences between	Effect size
						groups
	Tested food	Parameters	mean ± SD (CI)	mean ± SD (CI)	Mann-Whitney U test	η 2 (partial eta squared)
			n=12 (1 refusal)	n=13 (No refusal)
		Ti (s)	27.46 ± 7.87 (24.13 -30.78)	20.00 ± 4.51 (18.18 -21.82)	p≤0.001	0.264
	CER	Nc (n)	25.79 ± 6.96 (22.85 -28.73)	27.38 ± 5.93 (24.99 -29.78)	NS	0.016
		Fq (n/s)	0.97 ± 0.23 (0.87 -1.07)	1.37 ± 0.22 (1.28 -1.46)	p≤0.001	0.456
			n=11 (2 refusals)	n=11 (2 refusals)
		Ti (s)	24.59 ± 10.59 (19.90 -29.28)	17.95 ± 5.21 (15.64 -20.27)	p≤0.05	0.142
	CHS	Nc (n)	25.32 ± 12.87 (19.61 -31.03)	24.00 ± 7.77 (20.55 -27.45)	NS	0.169
		Fq (n/s)	1.05 ± 0.25 (0.94 -1.16)	1.38 ± 0.18 (1.31 -1.46)	p≤0.001	0.389
			n=9 (4 refusals)	n=11 (2 refusals)
		Ti (s)	44.17 ± 29.33 (29.58 -58.75)	34.82 ± 15.33 (28.02 -41.61)	NS	0.042
	CAR	Nc (n)	45.06 ± 20.71 (34.76 -55.36)	48.82 ± 17.32 (41.14 -56.50)	NS	0.010
		Fq (n/s)	1.08 ± 0.22 -1.19)	(0.97	1.48 ± 0.25 (1.36 -1.59)	p≤0.001	0.403

CAR: carrot sample; CER: cereals sample, CHS: cheese sample; ECC: Early Childhood Carie. CI: Confidence Interval; n correspond to the number of children with no food refusal; NS: no statistical difference (p>0.05)

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