Thomas Bonald

A Recursive Formula for Estimating the Packet Loss Rate in IP Networks

We present a recursive formula for estimating the packet loss rate in IP networks. Specifically, we consider a single link whose capacity is shared dynamically by elastic data flows, each characterized by some peak rate, and derive the formula from the steady-state distribution of the number and peak rates of ongoing flows. The result is particularly useful for dimensioning ISP backhaul networks that aggregate the traffic of various types of users.

Introduction

The teletraffic theory derived from the Erlang formula is not applicable to IP networks whose resources are shared dynamically by elastic data flows. New results are required to predict key performance metrics (throughput, delay, loss) with respect to network capacity and traffic characteristics (intensity, peak rates). A practically interesting case is that of an ISP that aggregates the traffic of a high number of DSL and FTTH users, with peak rates ranging from 500 kbit/s to 100 Mbit/s, and needs to predict the maximum load sustainable by its backhaul network for some target packet loss rate.

In this paper, we focus on a single link whose capacity is shared by various types of elastic data flows. Specifically, we characterize each flow by some peak rate that is typically equal to the speed of the user's access line. Assuming that link capacity is shared according to balanced fairness [START_REF] Bonald | Insensitive bandwidth sharing in data networks[END_REF], a recursive formula has been derived in [START_REF] Bonald | A recursive formula for multirate systems with elastic traffic[END_REF] for evaluating the throughput of each flow. This formula is the analogue of the Kaufman-Roberts formula that has proven very useful for dimensioning circuit-switched networks [START_REF] Kaufman | Blocking in a shared resource environment[END_REF]4]. We here adapt the formula to derive some simple, conservative estimation of the packet loss rate.

Model

Consider a link of capacity C shared by elastic data flows. Each flow is characterized by some peak rate. Specifically, we consider N types of flows, referred to as classes. Each class-i flow has peak rate c i . Class-i flows arrive according to a Poisson process and leave the system once some random volume of data has been transferred. We denote by α i the associate traffic intensity, defined as the product of the flow arrival rate by the mean flow size, and by ρ i = α i /C the corresponding load. The overall link load is given by:

ρ = N i=1 ρ i .
Let x i be the number of class-i flows. The evolution of the system state x = (x 1 , . . . , x N ) depends on the way link capacity is shared between ongoing flows. We assume that this sharing realizes balanced fairness [START_REF] Bonald | Insensitive bandwidth sharing in data networks[END_REF]. Under the stability condition ρ < 1, the steady-state distribution of x is then insensitive to the flow size distribution beyond the mean and given by:

π(x) = π(0)Φ(x)α x , (1) 1
where Φ is the so-called balance function, defined by

Φ(x) = 1 x!c x if x.c ≤ C, and Φ(x) = 1 C N i=1 Φ(x -e i ) otherwise.
In the above expressions, the vectorial notation is that introduced in [START_REF] Bonald | A recursive formula for multirate systems with elastic traffic[END_REF].

The packet loss rate is hard to estimate in practice since it depends on the complex packet-level dynamics induced by TCP. At the considered aggregation link, however, packet losses are most likely due to the flowlevel dynamics, that is to the presence of a too high number of simultaneous flows. A simple yet reasonable appproximation then consists in considering the worst case where each ongoing flow is active at its peak rate. Thus in state x, the intensity of lost traffic is equal to x.c -C if x.c > C and to 0 otherwise; we deduce the following estimate for the packet loss rate:

L = x:x.c>C (x.c -C)π(x) x (x.c)π(x) . (2)
Although this expression can in principle be evaluated directly from (1), the computation is exponential in the number of classes N . The following recursive formula makes the computation linear in N .

Recursive formula

We assume as for the Kaufman-Roberts formula that the link capacity C and the rate limits c 1 , . . . , c N are integers. In the rest of the paper, we denote by π the measure (1) obtained with π(0) = 1. For any integer n, let: p(n) =

x:x.c=n π(x).

We define: p = n>C p(n) and q = n>C np(n)

In view of (2), we have:

L = q -Cp C n=1 np(n) + q For n = 1, . . . , C, we have p(n) = N i=1 α i n p(n -c i ), (3)
with p(0) = 1 and p(n) = 0 for all n < 0. This is the analogue of the Kaufman-Roberts formula. It remains to calculate p and q, that can be derived from the values of p(n) for n = 1, . . . , C:

Proposition 1 We have:

p = N i=1 ρ i p i 1 -ρ , q = N i=1 ρ i (q i + c i (p + p i )) 1 -ρ , (4)
where for all i = 1, . . . , N ,

p i = C-ci<n≤C p(n), q i = C-ci<n≤C np(n).

Proof. Using the fact that, for all x such that x.c > C, π(x) = N i=1 ρ i π(x -e i ), we obtain:

and:

from which (4) follows. [4] J. W. Roberts, A service system with heterogeneous user requirement, in: Performance of Data Communications Systems and Their Applications, G. Pujolle, Ed. Amsterdam, The Netherlands: North-Holland, 1981, pp. 423-431.


Matthias Hieber
email: hieber@mathematik.tu-darmstadt.de

Sylvie Monniaux
email: sylvie.monniaux@univ-amu.fr

WELL-POSEDNESS RESULTS FOR THE NAVIER-STOKES EQUATIONS IN THE ROTATIONAL FRAMEWORK

Consider the Navier-Stokes equations in the rotational framework either on R 3 or on open sets Ω ⊂ R 3 subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.

Introduction

Well-posedness results for linear and non-linear differential equations are important aspects in many scientific articles by Jerry Goldstein. He contributed in many ways to this concept, either by investigating abstract Cauchy problems or by considering concrete partial differential equations in his many papers and in particular in his fundamental book on Semigroups of Linear Operators and Applications, see [START_REF] Goldstein | Semigroups of Operators and Applications[END_REF], which was published in 1985 by the Oxford University Press. Of fundamental importance in this approach are generators of semigroups and the variation of constant formula for inhomogeneous Cauchy problems in Banach spaces. Mapping properties of the semigroup combined with fixed point arguments are often the central tools for proving local or global existence results for non-linear evolution equations.

In this paper, we follow this approach and consider the Navier-Stokes equations with Coriolis force on Ω = R 3 or on domains Ω ⊂ R 3 subject to Dirichlet boundary conditions, i.e. we study the equation (1.1)

           ∂ t u -∆u + ∇π + ω e × u + (u • ∇)u = 0 in (0, ∞) × Ω, div u = 0 in (0, ∞) × Ω, u = 0 on (0, ∞) × ∂Ω, u(0) = u 0 in Ω,
Here ω denotes the speed of rotation and e 3 is the unit vector in x 3 -direction. If ω = 0, the system (1.1) reduces to the classical Navier-Stokes system. This equation recently gained quite some attention due to its importance in applications to geophysical flows; in particular, large-scale atmospheric and oceanic flows are dominated by rotational effects, see e.g. [START_REF] Majda | Introduction to PDEs and Waves for the Atmosphere and Ocean[END_REF] and [START_REF] Chemin | Mathematical Geophysics[END_REF].

Our main technique to study this equation follows the approach sketched above. In fact, considering first the linearization of equation (1.1), we show first that the so called Stokes-Coriolis operator generates a C 0 -semigroup T on certain function spaces. Then, by the method which is nowadays called the Fujita-Kato method, well-posedness results for equation (1.1) in the L 2 -seting will be obtained by considering the integral equation

u(t) = T (t)u 0 + t 0 T (t -s)Pdiv (u ⊗ u)(s)ds.

1

Here P denotes the Helmholtz projection onto the solinoidal vector fields of L 2 (R 3 ). For the classical Navier-Stokes equations, this approach was generalized and pushed further to various scaling invariant function spaces. For details, we refer e.g. to the work of Kato [START_REF] Kato | Strong L p -solutions of Navier-Stokes equations in R n with applications to weak solutions[END_REF], Koch and Tataru [START_REF] Koch | Wellposedness for the Navier-Stokes equations[END_REF] and Cannone [START_REF] Cannone | Harmonic analysis tools for solving the incompressible Navier-Stokes equations[END_REF].

The first ill-posedness result for the classical Navier-Stokes equations is due to Bourgain and Pavlović [START_REF] Bourgain | Ill-posedness of the Navier-Stokes equations in a critical space in 3D[END_REF] for initial data in the Besov space Ḃ-1 ∞,∞ (R 3 ). It means that the solution map sending an initial data to the solution given by the above formula, where T now denotes the classical Stokes semigroup, is now longer continuous with respect to this Besov-norm.

The above equation (1.1) was mainly studied so far in the case of Ω = R 3 . It is a very remarkable fact that in this case the equation (1.1) allows a global, mild solution for arbitrary large data in the L 2 -setting provided the speed Ω of rotation is fast enough, see [START_REF] Babin | Regularity and integrability for the 3D Euler and Navier-Stokes equations for uniformly rotating fluids[END_REF], [START_REF] Babin | 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity[END_REF], [START_REF] Chemin | Mathematical Geophysics[END_REF] and [START_REF] Cao | Global wellposedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics[END_REF]. More precisely, it was proved by Chemin, Desjardins, Gallagher and Grenier in [START_REF] Chemin | Mathematical Geophysics[END_REF] that for initial data u 0 ∈ L 2 (R 2 ) 3 + H 1/2 (R 3 ) 3 satisfying div u 0 = 0, there exists a constant ω 0 > 0 such that for every ω ≥ ω 0 the equation (1.1) admits a unique, global mild solution. The case of periodic intial data was considered before by Babin, Mahalov and Nicolaenko in the papers [START_REF] Babin | Regularity and integrability for the 3D Euler and Navier-Stokes equations for uniformly rotating fluids[END_REF] and [START_REF] Babin | 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity[END_REF].

It is a natural question to ask whether, for given and fixed ω > 0, there exists a unique, global mild solution to (1.1) provided the norm of the initial data is sufficiently small with respect to certain norms. In this context it is natural to extend the classical Fujita-Kato approach for the Navier-Stokes equations to the rotational setting. This was carried out first by Hieber and Shibata in [START_REF] Hieber | The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework[END_REF] for the case of initial data belonging to H 1/2 (R 3 ). Generalizations of this result to the case of Fourier-Besov spaces are due to Konieczny and Yoneda [START_REF] Konieczny | On the dispersive effect of the Coriolis force for stationary Navier-Stokes equations[END_REF] and Iwabuchi and Takada [START_REF] Iwabuchi | Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type[END_REF]. These results will be discussed in some detail for the case Ω = R 3 in the following Section 2. We will further address the question of ill-posedness of (1.1). Starting point here is the pioniering paper by Bourgain and Pavlović, [START_REF] Bourgain | Ill-posedness of the Navier-Stokes equations in a critical space in 3D[END_REF], who showed ill-posedness for the classical Navier-Stokes equation, i.e. for the case ω = 0, in the Besov space Ḃ-1 ∞,∞ (R 3 ). It was recently shown by Iwabuchi and Takada [START_REF] Iwabuchi | Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type[END_REF] that equation (1.1) is also ill posed in certain Fourier-Besov spaces. All of these results rely on a good description of the Stokes-Coriolis semigroup.

In Section 4 we consider equation (1.1) on arbitrary domains Ω ⊂ R 3 . It was shown in [HM12] that the Stokes-Coriolis operator, defined via form methods, generates a contraction semigroup on a certain subspace H of L 2 (Ω). Note that H coincides with L 2 σ (Ω) in the case of domains with smooth boundaries.

The above equation was also studied in the setting of nondecaying initial data in a series of papers by Giga et al, see [START_REF] Giga | Navier-Stokes equations in a rotating frame in R 3 with initial data nondecreasing at infinity[END_REF], [START_REF] Giga | Rotating NS-equations in R 3 + with initial data nondecreasing at infinity: the Ekman boundary layer problem[END_REF]. More precisely, these authors prove local existence of mild solutions to the problem (1.1) for initial data u 0 belonging to L ∞ σ,a (R 3 ), a suitable subspace of L ∞ σ (R 3 ). Global existence results for initial data which may not decay at infinity were obtained by Giga, Inui, Mahalov and Saal in [START_REF] Giga | Uniform global solvability of the Navier-Stokes equations for nondecaying data[END_REF]. For more details, see Section 4. For extensions of this result we refer to [START_REF] Yoneda | Long-time solvability of the Navier-Stokes equations in a rotating frame with spatially almost periodic large data[END_REF].

Linear Theory for Ω = R 3

We start this section by considering the linear equation on R 3 , i.e.

u t + ∆u + ωe 3 × u + ∇p = 0, x ∈ R 3 , t > 0 div u = 0, x ∈ R 3 , t > 0, (2.1) u(0, x) = u 0 (x), x ∈ R 3 ,
and the corresponding resolvent equation in classical L p spaces. To this end, let λ ∈ Σ φ for some φ ∈ [0, π/2), where Σ φ = {z ∈ C\{0}, | arg z| < φ} and let f ∈ L p σ (R 3 ) 3 . Taking Fourier transforms with respect to x in the resolvent equation

λu -ν∆u + ωe 3 × u + ∇p = f, x ∈ R 3 , (2.2) div u = 0, x ∈ R 3 , yields (λ + |ξ| 2 ) u 1 -ω u 2 + iξ 1 p = f 1 (λ + |ξ| 2 ) u 2 + ω u 1 + iξ 2 p = f 2 (2.3) (λ + |ξ| 2 ) u 3 + iξ 3 p = f 3 . It follows that (2.4) p(ξ) = ω |ξ| 2 [iξ 2 u 1 (ξ) -iξ 1 u 2 (ξ)].
Inserting this expression for the pressure p in the above resolvent equation (2.3) yields that its solution is given by p defined as in (2.4) and by u defined by

u = ω 2 det I f + ω det ξ 3 |ξ| R f ,
where I is the identity matrix and

R = R(ξ) =    0 ξ3 |ξ| -ξ2 |ξ| -ξ3 |ξ| 0 ξ1 |ξ| ξ2 |ξ| -ξ1 |ξ| 0    and det = det(ξ) = ω 4 + ω 2 ξ 2 3 |ξ| 2 .
It follows that the solution u of the time dependent linear problem (2.1) in Fourier variables, i.e. of the problem

u t (ξ) + ν|ξ| 2 I u(ξ) + ω e 3 × u(ξ) + iξ pξ = 0, ξ ∈ R 3 , t > 0 iξ • u(ξ) = 0, ξ ∈ R 3 , t > 0, (2.5) u(0, ξ) = u 0 (ξ), ξ ∈ R 3 . is given by u(t, ξ) = cos(ω ξ 3 |ξ| t)e -|ξ| 2 t I u 0 (ξ) + sin(ω ξ 3 |ξ| t)e -|ξ| 2 t R(ξ) u 0 (ξ), t ≥ 0, ξ ∈ R 3 .
This representation combined with Plancherel's theorem implies the following result.

Proposition 2.1 ([HS10], Prop. 2.1). The unique solution of equation (2.1) in L 2 σ (R 3 ) is given by the bounded C 0 -semigroup T 2 , which is explicitly given by

T 2 (t)f := F -1 [cos(ω ξ 3 |ξ| t)e -|ξ| 2 t I f (ξ) + sin(ω ξ 3 |ξ| t)e -|ξ| 2 t R(ξ) f (ξ)], t ≥ 0, f ∈ L 2 σ (R 3 ) 3 .
The above semigroup is called the Stokes-Coriolis semigroup.

Writing (2.6) T p (t)f := F -1 [cos(ω ξ 3 |ξ| t)e -|ξ| 2 t I f (ξ) + sin(ω ξ 3 |ξ| t)e -|ξ| 2 t R(ξ) f (ξ)], t ≥ 0, f ∈ L p σ (R 3 ) 3 ,
for 1 < p < ∞, we may extend the semigroup T by Mikhlin's theorem to a C 0 -semigroup on L p (R 3 ) 3 . Indeed, set R 3 f (ξ) := ξ3 |ξ| f ξ for ξ = 0. Then Mikhlin's theorem implies the following result.

Proposition 2.2. Let 1 < p < ∞ and let T p be defined as in (2.6). Then

T p is a C 0 -semigroup on L p σ (R 3 ) 3 satisfying T p (t)f p ≤ M p ω 2 t 2 f p , t ≥ 1, f ∈ L p σ (R 3
) 3 for some constant M p . Moreover, T p may be represented as

T p (t) = [cos(ωR 3 t)I + sin(ωR 3 t)R]e t∆ f, t ≥ 0, f ∈ L p σ (R 3 ) 3 ].
Aiming for global existence results for equation (1.1), it is interesting to look for function spaces on which the Stokes-Coriolis semigroup defines a bounded semigroup. To this end, let Φ ∈ S(R 3 ) such that 0

≤ Φ(ξ) ≤ 1, supp Φ ⊂ {ξ ∈ R 3 : 2 -1 ≤ |ξ| ≤ 2} and j∈Z Φ j (ξ) = 1, ξ ∈ R 3 \{0},
where Φ j (x) := 2 jn Φ(2 j x). Then the Fourier-Besov space F Ḃs p,q (R 3 ) is defined as follows: let s ∈ R and 1 ≤ p, q ≤ ∞. Then the space F Ḃs p,q (R 3 ) is defined by

F Ḃs p,q (R 3 ) := {f ∈ S(R 3 ) : f ∈ L 1 loc (R 3 ) and f F Ḃs p,q (R 3 ) < ∞}, where f F Ḃs p,q (R 3 ) := {2 sj Φ j f p } j∈Z l q .
Given the representation of Proposition 2.1 it in now not difficult to verify the following assertion.

Proposition 2.3 ([IT11a],Lemma 2.1).

There exists a constant C > 0 such that

T (t)f F Ḃ-1 1,2 (R 3 ) ≤ C f F Ḃ-1 1,2 (R 3 )
, ω ≥ 0. The following L p -L q as well as H 1 2 -L q smoothing properties for the semigroup T are established by [START_REF] Hieber | The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework[END_REF] and are essential in their approach for the nonlinear problem (1.1).

Proposition 2.4. Let 1 ≤ p ≤ 2 ≤ q ≤ ∞. Then for m ∈ N 0 there exists a constant C > 0 such that

∇ m T (t)f q ≤ Ct -m 2 -3 2 (1/p-1/q) f p , t > 0, f ∈ L p (R 3 ), (2.7) ∆ 1 4 T (t)f 2 ≤ Ct -1 4 -3 2 (1/p-1/2) f p , t > 0, f ∈ L p (R 3 ). (2.8)
Moreover, there exists a constant C > 0 such that

T (t)f 1 2 ≤ C f 1 2 , t > 0, f ∈ H 1 2 (R 3 ), (2.9) ∇T (t)f 2 ≤ Ct -1 4 f 1 2 , t > 0, f ∈ H 1 2 (R 3 ), (2.10)
and for q > 3 there exists C > 0 such that

(2.11) T (t)f q ≤ Ct -1 2 + 3 2q f 1 2 , t > 0, f ∈ H 1 2 (R 3 ).

Linear Theory for Domains

In this section we define the Stokes and the Stokes-Coriolis operator by the theory of forms. To this end, let Ω ⊂ R 3 be an open set and let H = L 2 (Ω; R 3 ) by defined by

H = u = (u 1 , u 2 , u 3 ); u i ∈ L 2 (Ω; R 3
), for all i = 1, 2, 3 , endowed with the usual scalar product •, • . Observe that the set G defined by

G := ∇p; p ∈ L 2 loc (Ω; R) with ∇p ∈ L 2 (Ω; R 3 ) ; is a closed subspace of H. Set H := G ⊥ = u ∈ L 2 (Ω; R 3 ); u, g = 0 for all g ∈ G .
Denote by J : H ֒→ H the canonical injection from H onto H and define a scalar product on H by (u, v) → Ju, Jv .

Endowed with this scalar product, H is a Hilbert space and we have

H = H ⊥ ⊕ G.
Finally, denote by P the orthogonal projection from H onto H. Then P is equal to the adjoint J ′ of J and PJ = Id H .

We are now in the position to apply the theory of forms as follows. Define V by V := H ∩ V , where V = H 1 0 (Ω; R 3 ). Then V is a closed subspace of V a hence a Hilbert space. It follows from De Rham's theorem that V is dense in H.

Next, denote by V ′ the dual space of V , i.e. V ′ = H -1 (Ω; R 3 ) and let V ′ be the dual space of V. Let J be the canonical injection V ֒→ V . It is a restriction of the canonical injection J : H ֒→ H, so that its adjoint P = J′ : V ′ → V ′ is an extension of the Helmholtz projection P : H → H.

Given ω ≥ 0, we define sesquilinear forms a and b on V × V by a(u, v) := The Stokes operator A in H is then defined to be the operator associated with the form a, i.e. A is given by

D(A) := u ∈ V; P(-∆ Ω D ) Ju ∈ H Au := P(-∆ Ω D ) Ju.
Here, ∆ Ω D denotes the Dirichlet-Laplacian on V .

Remark 3.1. The Stokes operator A has the following properties:

a) A is a self-adjoint operator in H. b) D(A 1 2 ) = V . c) -A generates an analytic semigroup of contractions on H of angle π 2 . d) D(A) = u ∈ V ; ∃ p ∈ L 2 loc (Ω; R) : -∆ Ju + ∇p ∈ H .

Define the Stokes-Coriolis operator A

C in H to be the operator which is associated to the form a + b. We then have the following result.

Proposition 3.2 ([HM12]

). The operator -A C generates a semigroup of contractions (T C (t)) t≥0 on H satisfying the properties

a) t → T C (t) ∈ L ∞ (0, ∞; L (H)) and t → A 1 2 T C (t) ∈ L 2 (0, ∞; L (H)) with norms less than or equal to 1; b) t → A 1 4 T C (t)A 1 4 ∈ L 2 (0, ∞; L (H)) and t → A 1 2 T C (t)A 1 4 ∈ L 4 3 (0, ∞; L (H))
with norms less than or equal to 1.

Global Existence Results for the non-linear equation in

Ω = R 3
We start this section with a global existence result for data being small in the H 1/2 -norm. More precisely, we have the following result.

Theorem 4.1 ([HS10], Theorem 3.1). There exists ε > 0, independent of ω, such that for

any u 0 ∈ H 1 2 σ (R 3 ) with u 0 1 2 ≤ ε, the equation (1.1) admits a unique, mild solution u ∈ C 0 ([0, ∞), H 1 2 σ (R 3 )) 3 satisfying lim t→0+ u(•, t) -u 0 1/2 = 0.
The proof of the above theorem is based on the Fujita-Kato argument applied to the integral equation

(4.1) u(t) = T (t)u 0 - t 0 T (t -s)P [(u(s) • ∇)u(s)] ds,
where P denotes the Helmholtz projection. In order to do so, one has to estimate the bilinear form associated with the second term on the right hand side above in certain function spaces. For details, see [START_REF] Hieber | The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework[END_REF].

The above result was recently generalized to the setting to Fourier-Besov spaces by Konieczny and Yoneda [START_REF] Konieczny | On the dispersive effect of the Coriolis force for stationary Navier-Stokes equations[END_REF] and Iwabuchi and Takada [START_REF] Iwabuchi | Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type[END_REF]. In fact, their results read as follows.

Theorem 4.2 ([KY11], Thorem 2.2). Let 3 < p ≤ ∞. Then there exists a constant δ > 0, independent of ω, such that for all u 0 ∈ F Ḃ2-3/p p,∞ (R 3 ) with div u 0 = 0 and u 0 F Ḃ2-3/p p,∞ (R 3 ) ≤ δ, the equation (1.1) admits a unique, global mild solution

u ∈ C w ([0, ∞); F Ḃ2-3/p p,∞ ) ∩ L ∞ (0, ∞; F Ḃ2-3/p p,∞ ). Moreover, if u 0 is small in X 0 = F ḂB -1
1,1 (R 3 )∩F Ḃ0 1,1 (R 3 ), then there exists a unique global solution u to (1.1) such that u ∈ C([0, ∞); F Ḃ-1 1,1 ) ∩ L 2 (0, ∞; X 0 ). Theorem 4.3 ([IT11a], Thorem 1.2). For all α ∈ (0, 1), there exist constants C, δ > 0, independent of ω, such that for all u 0 ∈ F Ḃ-1 1,2 (R 3 ) with div u 0 = 0 and u 0 F Ḃ-1 1,2 (R 3 ) ≤ δ, the equation (1.1) admits a unique, global mild solution u ∈ X α , where

X α = {u ∈ C([0, ∞); F Ḃ-1 1,2 (R 3 )) : u X α ≤ 2Cδ, div u = 0} and u X α is defined as u X α := sup t>0 u(t) F Ḃ-1 1,2 + u Z α + u Z -α , where u Z ±α = { j∈Z (2 αj Φ j u L 2 1+α (0,∞;L 1 (R 3 )) ) 2 } 1/2 .
Remark 4.4. Note that due to the embedding Ḣ1/2 (R 3 ) ֒→ F Ḃ-1 1,2 (R 3 ) the above Theorem 4.3 generalizes in particular the corresponding Theorem 4.1.

Given Theorem 4.3, it is natural to ask whether equation (1.1) is well posed in larger function spaces as F Ḃ-1 1,2 (R 3 ). The following result is hence very interesting.

Theorem 4.5 ([IT11a], Theorem 1.5). For q ∈ (2, ∞], the equation (1.1) is ill posed in F Ḃ-1 1,q (R 3 ) in the sense that the solution map from the initial data to the solution is not continuous.

The proof of Theorem 4.5 is based on an ill-posedness approach for nonlinear Schrödinger equations, due to Bejenaru and Tao; see [START_REF] Bejenaru | Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation[END_REF].

Observe that the above result is true even also in the case ω = 0. It is interesting to compare Theorem 4.5 with the ill posedness result for the Navier-Stokes equations in the Besov space Ḃ-1 ∞,∞ (R 3 ), due to Bourgain and Pavlović [START_REF] Bourgain | Ill-posedness of the Navier-Stokes equations in a critical space in 3D[END_REF]. In fact, since F Ḃ-1 1,q (R 3 ) ֒→ Ḃ-1 ∞,q (R 3 ) for all q ∈ [1, ∞], the above Theorem 4.5 generalizes the Bourgain-Pavlović result to the case Ḃ-1 ∞,q (R 3 ) for q ∈ (2, ∞].

In order to describe the situation for nondecaying initial data u 0 denote first by M the space of finite C 3 -valued Radon measures on R 3 and by F M its Fourier transform. Equipped with the norm f F M := F -1 f M , the space F M becomes a Banach space, where F -1 denotes the inverse Fourier transform. Furthermore, denote by F 3 the class of all sum-closed frequency sets in R 3 ; for details see [START_REF] Giga | Uniform global solvability of the Navier-Stokes equations for nondecaying data[END_REF]. Moreover, sum-closed frequency set with distance δ > 0 from zero are denoted by F δ . For F δ ∈ F 3 define the space F M σ,δ := {f ∈ F M : div f = 0, supp f ⊂ F δ }.

The following result is due to Giga, Inui, Maholov and Saal.

Theorem 4.6 ([GIMS08],Thm.1.2). Let δ > 0, ω ∈ R, F δ ∈ F 3 and u 0 ∈ F M σ,δ . If u 0 F M < δ K for a certain K > 0, then there exists a global, mild solution u ∈ BC([0, ∞), F M σ,δ to equation (1.1) satisfying u(t) -u 0 F M → 0 for t → 0 and u(t)| F M ≤ 2e -δ 2 t u 0 F M for t ≥ 0.

Global Existence Results for the non-linear equation in domains

We define the space E by (5.1) E = u ∈ L 4 (0, ∞; Ḣ1 0 (Ω; R 3 )); div u = 0 in (0, ∞) × Ω , endowed with the norm u E = ∇u L 4 (0,∞;L 2 (Ω;R 3 )) .

We reduce the problem of finding mild solutions for (1.1) by solving the equation (5.2) u ′ (t) + Au(t) + Bu(t) = -P ( Ju • ∇) Ju

u(0) = u 0 , u ∈ E ,
for which a mild solution u is given by u = α + φ(u, u), where, for t > 0, α(t) = T C (t)u 0 , and φ(u, v)(t) = The strategy for finding u ∈ E satisfying u = α + φ(u, u) is to apply the contration principle in a suitable "good" space E , for which α ∈ E and φ(u, u) ∈ E .

Proposition 5.1. The application φ : E × E → E is bilinear, continuous and symmetric.

∂

  j Ju, ∂ j Jv u, v ∈ V, and b(u, v) := ω e × Ju, Jv .

t 0 T

 0 C (t -s) -1 2 P ( J u(s) • ∇) Jv(s) + ( J v(s) • ∇) J u(s) ds.(5.3)

Denote by M the norm of the above mapping φ, i.e.

Observe that M is independent on ω.

We define now the space X where we will consider the initial values u 0 for (1.1) by

(5.4)

Then the following result holds.

Theorem 5.2 ([HM12]

). Let Ω ⊂ R 3 be an open set. Then, for all u 0 ∈ X with u 0 X < 1 4M , there exists a unique u ∈ E satisfying u E < 1 2M and such that u is the solution of u = α + φ(u, u), where α and φ are defined as in (5.3). Moreover, in this case, t → A

Remark 5.3. We remark that Theorem 5.2 generalizes in particular Theorem 4.1, since in the case Ω = R 3 , it is not difficult to verify that


Gabriel Courrioux

Simon Lopez

Cécile Allanic

Sunsearé Gabalda

come

From 2D maps to 3D modelling: much more than "just one more dimension" Courrioux, G. 1 , Lopez, S. 1 , Allanic, C. 1 Gabalda, S. 1 and Bourgine, B. 1 1 BRGM, Bureau de Recherches Géologiques et Minières Orléans France, g.courrioux@brgm.fr ________________________________________________________________________ By analogy with geological mapping, a possible definition of 3D cartography is the ability to provide a continuous description of geology at depth from scattered and heterogeneous data. Whatever the dimension is, both activities involve building a geological model which, as in any discipline calls for the use of general concepts and knowledge to produce a synthetic and abstract representation of the subsurface. This representation is inherently 3D. While in traditional mapping an important part of the geologist's cognitive effort is devoted to mentally project into 2D space the geometries of geological structures as imagined in 3D, the advent of computers and numerical geology considerably eased and automated this task. Consequently, 3D modellers (who should also be geologists) can focus much more on the description of geological bodies, their mutual relations and modelling choices (data selection and interpretation) and rely on algorithms to produce maps and random cross-sections or 3D volume explorations which all represent a consistent set of views of the subsurface. As enjoyable as this may seem, things didn't get easier in adding one more dimension.

The price to pay for consistency is a substantial work to integrate all available data and specify exhaustive constraints on geometries and relations without "dashed lines". Then, the final product, the 3D geological model, realizes a coherent synthesis and structure of knowledge available on a specific site and at a specific modelling scale. In itself, it is already a valuable tool for knowledge dissemination. Moreover, its digital nature makes it a natural candidate for quantitative analysis, and as it is often poorly constrained at depth, it immediately calls for uncertainty quantification which is far from a trivial task. Thanks to the spatial discretization of geological models (meshing) it is also possible to perform various dynamic simulations of physical phenomena: seismic wave propagation, mass and energy transfer, resource assessment, water quality… This wide range of applications combined with the increasing computing capabilities already boost the demand for ever more accurate geological models and it is likely that this trend will continue with geological modelling being at the core of both scientific understanding of the subsurface and engineering needs.

Consequently, most of current National Cartographic Programs have set 3D data management and 3D geological modelling as strategic objectives. This implies continuous methodological research both with regard to the complexity of geological objects being modelled and the efficiency of the tools used which should be as geologist friendly as possible. Based on our experience in different application fields such as 3D mapping programs, tunnelling, seismic risk appraisal, geothermal resource assessment… we point out several issues in going from the routine production of 2D maps to the routine production of 3D geological models. We will illustrate the great benefits from 3D modelling with examples from case studies and will focus on the current bottlenecks and development needs.


C Jeunet

B N'kaoua

R N'kambou

F Lotte

Why and How to Use Intelligent Tutoring Systems to Adapt MI-BCI Training to Each User

Introduction: While Mental Imagery based BCIs (MI-BCIs) are promising for many applications, their usability "out-of-the-lab" has been questioned due to their lack of reliability: literature reports that 15% to 30% of users cannot control such a technology, while most of the remaining users obtain only modest performances [1]. Standard MI-BCI training protocols have been suggested to be partly responsible for these modest performances as they do not comply with general human learning principles [2]. The modest performances as well as the flaws in the protocols led to the investigation of solutions to improve MI-BCI training by adapting it to each user. Such an approach is possible using Intelligent Tutoring Systems (ITS), i.e., computerised systems aiming at supporting learning [3]. Hence, we show why ITS are relevant for MI-BCI training and how this technology could be used.

Why? -MI-BCI training resembles distance learning (DL) as it is performed autonomously, with neither teacher nor classmates. Consistently with DL literature, highly anxious and poorly autonomous learners have been shown to struggle with MI-BCI training [START_REF] Jeunet | [END_REF]. Since ITS have been proven efficient for improving DL [3], MI-BCI training may also benefit from ITS. The strength of ITS lies in (1) a personalised support provided by a learning companion [3] and (2) an adaptation of the training process according to the learner's profile and skill evolution.

How? -We are proposing the conceptual framework for an ITS which would support MI-BCI user-training. ITS comprise 4 modules. First, the Student Model is the core component containing information about the user's personality and cognitive profile and state. Second, the Expert module contains the concepts, rules and strategies of the field to be learned. Third, the Tutoring module uses input from the two previous modules to select a tutoring strategy, and finally the Interface provides the user with access to the learning environment. Each module will be described in an MI-BCI training context (see Fig. 1). The Student Model contains 2 kinds of information: 1) the user-profile, as assessed by questionnaires, and more specifically spatial abilities and personality traits (e.g., abstractness, tension or autonomy), which have been shown to be related to MI-BCI performance [START_REF] Moore | Self-Directed Learning & Distance Education[END_REF]; and 2), the user's cognitive state, e.g., fatigue and workload levels and MI-BCI skill development, provided by the BCI system through classification-accuracy measures. The Expert module contains a cognitive model of the skills to be learned, e.g., the ability to generate stable and distinct brain-activity patterns while performing the MI-tasks. It also includes a bank of exercises with different levels of difficulty [6], which would help the user to acquire these skills. Based on the Student Model and on the Expert module, and using specialised algorithms [3], the Tutor selects the appropriate exercises and provides the users with a suitable support, i.e., adapted to their performance and profile. This support will be provided using a physical learning companion [3], which has been proven to increase motivation and learning [3]. In particular, this companion will provide any users who have high tension and low autonomy levels [START_REF] Moore | Self-Directed Learning & Distance Education[END_REF] with a social presence and an emotional support (e.g., empathy). We are currently designing and evaluating the content of these different modules. Significance: Such an ITS represents a promising pluridisciplinary approach for improving MI-BCI performance as it would enable to gather different levers and articulate them in order to optimise the user-training process.

Figure 1 .

 1 Figure 1. Diagram representing the conceptual architecture of an ITS supporting MI-BCI training. Discussion: ITS may be very useful for MI-BCI user training, especially if the Student Model and Expert module are reinforced. The former could include more detail on the user's profile and cognitive state, while the latter could be improved by a better fundamental understanding of MI-BCI related skills and how they are acquired.


C Jeunet

F Lotte

M Hachet

S Subramanian

& B N'kaoua

Spatial Abilities Play a Major Role in BCI Performance

Introduction: Despite their promising potential impact for many applications, Mental-Imagery based BCIs (MI-BCIs) remain barely used outside laboratories. One reason is that 15% to 30% of naïve users seem unable to control them [START_REF] Allison | Could Anyone Use a BCI?[END_REF] and only a few reach high control abilities. Although different predictors of BCI performance (i.e., command classification accuracy) have been investigated to explain this huge inter-user variability [START_REF] Kleih | Psychological Factors Influencing Brain-Computer Interface (BCI) Performance[END_REF][START_REF] Ahn | Performance Variation in Motor Imagery Brain-Computer Interface: A Brief Review[END_REF], no strong predictive model has yet been determined. This could be due to (a) the often small samples used (N=5 or 6) and (b) the fact that these predictors have been mostly determined based on one-session experiments. Yet there is no evidence that performance obtained at the first session is predictive of users' MI-BCI control ability.

Material, Methods and Results: In [START_REF] Jeunet | Predicting Mental-Imagery-Based BCI Performance from Personality, Cognitive Profile and Neurophysiological Patterns[END_REF], we investigated the impact of the user's personality and cognitive profile on MI-BCI performance based on a 6-session experiment. Averaging performances over these sessions reduced the intra-subject variability (e.g., due to fatigue or external factors), and thus led to a better estimation of participants' MI-BCI control ability. Each session comprised 5 runs during which the participants (N=18) had to learn to perform 3 MI tasks: left-hand motor imagery, mental rotation and mental calculation. The results stressed the impact of mental rotation scores (measured using questionnaires), and which reflect Spatial Abilities (SA), on mean MI-BCI performance [r=0.696, p<0.05] (see Fig. 1[A]). SA are the mental capacities which enable the construction, transformation and interpretation of mental images. In a more recent study (to be published), we trained 20 participants to control a 2-class MI-BCI by performing motor-imagery of their left-and right-hands, within 1 session of 5 runs. Results confirmed the role of SA: mental rotation scores were correlated with peak MI-BCI performance [r=0.464, p<0.05]. This suggests that SA are a generic predictor of MI-BCI performances. ) including different exercise types and difficulties. In the coming weeks, we will test this protocol efficiency in terms of MI-BCI performance improvement by comparing it to a standard MI-BCI training approach. We will also investigate the neurophysiological correlates of the SA training (notably the implication of the motor cortex) to improve the understanding of the relationship between SA and MI-BCI performance.

Perspectives for Stroke Rehabilitation: If a causal link between SA and MI-BCI performance is confirmed, this would be a promising way to improve MI-BCI performance and thus MI-BCI-based applications such as stroke rehabilitation [START_REF] Ang | Brain-Computer Interface for Neurorehabilitation of Upper Limb After Stroke[END_REF]. Also, current MI-BCI based stroke rehabilitation procedures [START_REF] Ang | Brain-Computer Interface for Neurorehabilitation of Upper Limb After Stroke[END_REF] require the execution of MI tasks which can induce (or increase) a depressed state in patients by reminding them of the loss of movement in their limb. Since SA training and mental rotation tasks activate the motor cortex [START_REF] Windischberger | Human Motor cortex activity during mental rotation[END_REF], they might also be used as a more transparent way to indirectly induce synaptic plasticity in the motor cortex during rehabilitation.

Significance: Through SA, we propose a new approach for MI-BCI training that could offer promising perspectives for MI-BCI and stroke rehabilitation. We are currently evaluating and validating this approach.

Figure 1 .

 1 Figure 1. [A] Diagram representing the mean classification accuracy for the different subjects as a function of their mental rotation score; [B] One item per exercise included in the Spatial Ability training:the shape on top is the target, and the participant must identify the two shapes that are identical to the target among the four below. Spatial Ability Training: The strong correlation between SA and MI-BCI performance raised a new research question: Is there a causal effect between SA and MI-BCI performance? In other words: Would an improvement of users' SA result in an increase of their MI-BCI control abilities? We implemented an SA training protocol (see Fig. 1[B]) including different exercise types and difficulties. In the coming weeks, we will test this protocol efficiency in terms of MI-BCI performance improvement by comparing it to a standard MI-BCI training approach. We will also investigate the neurophysiological correlates of the SA training (notably the implication of the motor cortex) to improve the understanding of the relationship between SA and MI-BCI performance.


MD † §# Joe-Elie Salem
email: joe-elie.salem@aphp.fr

MD ‡ Maria El-Aissaoui

MD §# Margaux Alazard

MD-PhD † § Jean-Sébastien Hulot

MD-PhD Nadia Aissaoui

MD-PhD Jean-Yves Le-Heuzey

MD-PhD Christian Funck-Brentano

Christophe Faisy

PhD ‖ Saik Urien

Modeling of amiodarone effect on heart rate control in critically ill patients with atrial tachyarrhythmias Running title: Amiodarone in critically ill patients with AT

Keywords: amiodarone, atrial tachyarrhythmia, pharmacology, shock, heart rate

Aims: Amiodarone is the gold standard medication to control heart rate in critically ill patients with atrial tachyarrhythmias (AT). However, effective doses and covariates influencing its efficacy remain unknown. We therefore performed pharmacodynamic modeling of heart rate reduction induced by amiodarone in these patients.

Methods

-Results: This observational study included 80 consecutive severely ill patients receiving amiodarone to treat atrial tachyarrhythmias. A total of 1348 timeheart rate observations with 361 amiodarone dose administrations were analyzed during up to 6 days after hospital treatment initiation using a nonlinear mixed effect model. Pretreatment with amiodarone before intensive care administration, paroxysmal versus persistent AT, catecholamine infusion, fluid and magnesium loading were among covariates assessed in the model.

In case of paroxysmal AT in a patient not pretreated by amiodarone, a 300 mg intravenous loading dose combined with a 800 mg oral dose on the first day followed by 800 mg/day p.o. for 4 days was effective to achieve a heart rate between 80 and 115 bpm within the first day and to maintain it during the next 4 days. Corresponding doses were twice as high in patients with persistent AT. Use of intravenous magnesium (p<0.02) and fluid loading (p<0.02) was associated with an earlier and greater heart rate decrease. Dobutamine use had an opposite influence (p<0.05).

Conclusions:

In critically ill patients with AT, dose of amiodarone required to control heart rate is influenced by the type of AT and by other easily measurable conditions which may allow better individualization of amiodarone dosing.

1/ Amiodarone pre-treatment, paroxysmal versus persistent atrial tachyarrhythmias (AT), dobutamine infusion, intravenous magnesium and fluid loading have significant influence on amiodarone effect on heart rate reduction in critically ill patients with AT.

2/ An optimized dosing regimen of amiodarone refining current guidelines indications is proposed to better control heart rate in patients with AT depending on their clinical presentation: paroxysmal AT amiodarone naive, persistent AT amiodarone naive and paroxysmal AT amiodarone pre-treated.

3/ Co-administration of intravenous magnesium with amiodarone is associated with a greater and earlier heart rate control in patients with AT. This suggests that systematic use of IV magnesium as an add-on therapy might be valuable

INTRODUCTION

Atrial tachyarrhythmias (AT), such as atrial fibrillation are highly prevalent and are associated with a poor outcome in critically ill patients admitted in intensive care units (ICU) [START_REF] Artucio | Cardiac arrhythmias in critically ill patients: epidemiologic study[END_REF] . AT is often characterized by a persistent elevation of ventricular heart rate (HR) above 120-130 beats per minute (bpm) leading to cardiac systolic dysfunction, (i.e ventricular tachycardiomyopathy), responsible for hemodynamic instability, increased morbidity and mortality [START_REF] Packer | Tachycardia-induced cardiomyopathy: a reversible form of left ventricular dysfunction[END_REF][START_REF] Nerheim | Heart failure and sudden death in patients with tachycardia-induced cardiomyopathy and recurrent tachycardia[END_REF][START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] . A rate or rhythm control strategy with pharmacological therapy is recommended in patients with AT for the treatment or prevention of ventricular tachycardiomyopathy [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] . Before initiating antiarrhythmic treatment, factors promoting AT in the critically ill should be sought and corrected (anemia, hypoxia, hypokalemia…). Anticoagulant treatments are also usually administered. Attempts to restore sinus rhythm are frequently unsuccessful in the setting of critically ill patients [START_REF] Kumar | Intravenous amiodarone for therapy of atrial fibrillation and flutter in critically ill patients with severely depressed left ventricular function[END_REF][START_REF] Clemo | Intravenous amiodarone for acute heart rate control in the critically ill patient with atrial tachyarrhythmias[END_REF][START_REF] Mayr | Atrial tachyarrhythmia after cardiac surgery[END_REF] , particularly when using electrical cardioversion without antiarrhythmic pretreatment. HR control is thus the main preferred strategy in this situation [START_REF] Galve | Intravenous amiodarone in treatment of recentonset atrial fibrillation: results of a randomized, controlled study[END_REF][START_REF] Delle Karth | Amiodarone versus diltiazem for rate control in critically ill patients with atrial tachyarrhythmias[END_REF][START_REF] Carlsson | Randomized trial of ratecontrol versus rhythm-control in persistent atrial fibrillation: the Strategies of Treatment of Atrial Fibrillation (STAF) study[END_REF] . Guidelines in stable patients recommend strict control aiming at a resting HR between 60-80 bpm and 90-115 bpm during stress situations, based on the results of the AFFIRM trial [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] Van Gelder | Rate Control versus Electrical Cardioversion for Persistent Atrial Fibrillation Study Group. A comparison of rate control and rhythm control in patients with recurrent persistent atrial fibrillation[END_REF] . There are no specific guidelines for patients in severely ill condition but it is commonly admitted that reaching a HR below 115 bpm is advisable [START_REF] Packer | Tachycardia-induced cardiomyopathy: a reversible form of left ventricular dysfunction[END_REF][START_REF] Nerheim | Heart failure and sudden death in patients with tachycardia-induced cardiomyopathy and recurrent tachycardia[END_REF][START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] .

Amiodarone is the recommended agent to control HR in critically ill patients presenting with AT [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] . However, it can be associated with hypotension and potentially severe liver and pulmonary toxicity, particularly after IV administration [4,13- 15] . This toxic risk justifies the administration of the lowest possible dose to control heart rate. In addition, patients admitted in ICU present with a diversity of adverse conditions that can affect antiarrhythmic drug pharmacokinetics and dynamics, including instable hemodynamics, pre-existent or acquired left ventricular dysfunction, hypoxemia, hyperthermia, hypovolemia or acidosis [START_REF] Lafont | Modeling for critically ill patients: An introduction for beginners[END_REF] .

The choice of amiodarone doses in ICU patients is thus challenging due to interpatient heterogeneity and limited knowledge of optimal dosing in this setting [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] . In general, an IV loading dose of 300 mg is administered over 1 hour followed by an additional dose given either intravenously (IV) over the first day (10 -50mg/h) [START_REF] Delle Karth | Amiodarone versus diltiazem for rate control in critically ill patients with atrial tachyarrhythmias[END_REF] or orally over the next few days (400 mg/day up to 1200 mg /day) [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF][START_REF] Tonet | Electrocardiographic effects and antiarrhythmic action of 1200 mg of oral amiodarone per day[END_REF][START_REF] Kochiadakis | Efficacy of amiodarone for the termination of persistent atrial fibrillation[END_REF] . In daily practice, amiodarone dosing regimens used in these patients vary widely. Factors that influence amiodarone efficacy in critically ill patients have not been reported.

Therefore, a randomized study comparing the efficacy of different amiodarone dosing regimens would be inappropriate without previous knowledge of the sources of variability of drug response in ICU patients. Modelling of amiodarone efficacy in critically ill patients has not been previously reported and is the best initial approach to better characterize the complex interactions of multiple covariates on amiodarone pharmacodynamics.

The objective of this study was to determine optimal doses of amiodarone to achieve a heart rate between 80 and 115 bpm when treating AT after identification and integration of clinical and therapeutic covariates influencing amiodarone efficacy in critically ill patients.

MATERIAL AND METHODS

Patients and Study Design

This observational, cohort study was conducted from January 2007 to April 2012 in

Variables

At the time of first amiodarone administration in ICU the following time-independent covariates were collected: gender, age, body weight, height, etiology of the disease leading to ICU admission, Simplified Acute Physiology Score II (SAPS II score), Acute Physiology And Chronic Health Evaluation II (APACHE II score), diabetes, hypertension, past medical history of AT, withdrawal of a chronic antiarrhythmic drug since less than three times its half-life, total bilirubinemia, previous amiodarone treatment before first dose in ICU (yes, no), type of cardiac rhythm (sinus versus persistent / paroxysmal AT). Paroxysmal and persistent AT were defined as an arrhythmia which started less or more than one week before the first amiodarone administration in ICU, respectively.

Pharmacodynamic Modeling

General model

Amiodarone pharmacokinetics was ascribed to a virtual compartment model including zero or first order input rates. This virtual compartment, A(t), represents the biophase in which the amount or concentration is in equilibrium with the observed effect. It is used to extract the kinetic component, KDE or "pharmacodynamic halflife", from the pharmacodynamic data alone describing the equilibrium between the administration rate and the observed effect. G(t) represents the amiodarone amount in the gut at a given time and ka, the first-order absorption rate (fixed to 8) [19-21].

Amiodarone bioavailability (F) was fixed to 0.33 [19-21]. The effect of amiodarone on heart rate, HR(t) was modeled as follows:

dA(t)/dt = input -KDE × A(t) (1) input = ka × G(t) × F if administered orally (2) IR = KDE × A(t) and Emax = 1 (3) dHR(t)/dt = kin [1 -Emax × IR/(IR + IR50)] -kout HR(t) (4)
In the effect model, equation ( 4), the production rate constant of response HR, kin, is inhibited by the amiodarone virtual infusion rate, IR (drug amount per time unit) via an hyperbolic function. Finally, IR50 and kout are the IR that induces 50% of the maximal response and the first-order dissipation rate constant for the HR response. It was shown that the use of the IR function usually performs well and can be used for predictions of drug effects [START_REF] Jacqmin | Modelling response time profiles in the absence of drug concentrations: definition and performance evaluation of the K-PD model[END_REF] , provided the investigated doses extend well beyond the IR50, which is the case in this study (Table 1).

The initial conditions of these differential equations were HR(0) = heart rate at time zero, A (0) = Dose for IV administration and G(0) = F x Dose for oral route.

Influence of covariates

Several time-dependent variables were tested, and then included in the response model when they significantly (percent relative standard error (%rse) <50%) modified either kin (ModKin) or kout (ModKout) as follows :

dHR(t)/dt = M od K in  k in [1 -IR Amio /(IR Amio + IR Amio 50)] -M od K out  k out HR(t) (5) with M od K in = (1 -t MG MG) (6)
M od K out = (1 -tDOBU DOBU) (1 + t FILL FILL) (1 + t ECV ECV) (Retro HR ) (7)
where MG, DOBU, FILL and ECV denote magnesium loading, beta-agonist administration, fluid therapy and electrical cardio-version (only when effectively restoring sinus rhythm), respectively, with their corresponding t SUBSCRIPT influential parameters.

where Retro HR = (HR/HR REF ) FRetroHR ( 8) HR (0) = 154 x (Age/70) -0.23 x [0.85 (if persistent AT) or 0.67 (if sinus rhythm) ] x 0.86 (if Amio pretreatment) ( 9)

FRetro
k out = 0.9 x 0.37 (if persistent AT or sinus rhythm) ( 10)

HR REF = 78.6 x 1.73 (if persistent AT) (11)
Covariates effects on structural parameters were only tested and included if they were physiologically plausible. For example HR (0) is known to be faster in patients with AT compared to patients in sinus rhythm, so this covariate effect can be accepted if it is significant.

The inclusion of patients in sinus rhythm was useful for they received the lowest amiodarone doses and thus served to validate the model over a large dose range.

These subjects specificities were very well identified during the modeling process, since the "Sinus rhythm" covariate had a significant effect on the three major elements of our model : HR (0) (i,e baseline heart rate at the time of amiodarone administration), Kout and ReferenceHR. Simulation graphics and statistics were derived using the R software program.

Data analysis details

Data

RESULTS

Patients and Observations

Among the 856 consecutive screened patients admitted to ICU, 97 (11.3%) had received at least one amiodarone dose for AT. Seventeen patients had paced rhythm or incomplete chart and were therefore excluded, leaving 80 patients eligible for analysis [Figure 1]. In these Other treatments received by the patients that could interfere with heart rate control are summarized in Table2.

Impact of amiodarone pretreatment and type of atrial tachyarrhythmias on amiodarone doses needed to control heart rate.

Baseline heart rate before the first administration of amiodarone in ICU (defined as HR (0) ) was higher (Table3) in amiodarone-naive patients with paroxysmal AT than in amiodarone-pretreated patients with paroxysmal AT or amiodarone-naive patients with persistent AT (154 ± 4 vs. 132 ± 35 and 131 ± 6 bpm, respectively, p <0.0001).

We modeled the effect of different amiodarone dosing regimen on heart rate reductions in these three different clinical scenarios and took the optimal dosing regimen to achieve a HR of 90 bpm as an example (Figure3). The amiodarone dosages were then extrapolated from these simulations results.

Among patients with paroxysmal AT, slightly higher doses of amiodarone were needed to achieve a heart rate below 115 bpm within the first day and to maintain it around 90 bpm for the 4 subsequent days in amiodarone-naive compared to pretreated patients. This was due to the difference in baseline HR (Figure3A-3B). In silico simulations indicated that the amiodarone dosing regimen to achieve such heart rate control in amiodarone-naive patients with paroxysmal AT should be 300 mg IV loading dose on day 1 plus 800 mg oral dose from day 1 to day 5 (Figure3A). Corresponding doses indicated by the model were slightly lower in patients with paroxysmal AT pretreated by amiodarone: 300 mg IV loading dose on day 1 plus 400 mg oral dose from day 1 to day 5 (Figure3B).

Interestingly, despite lower baseline HR in amiodarone-naive patients with persistent AT, corresponding doses were almost twice those of amiodarone-naive patients with paroxysmal AT (600 mg IV loading dose on day 1 and 1400 mg oral doses from day 1 to day 5, Figure 3C). This relative resistance of persistent AT was associated to a decrease in the amiodarone effect (k out ) in patients with persistent AT compared to paroxysmal AT (Table3).

Impact of dobutamine infusion, electrical cardioversion, magnesium and fluid

loading on amiodarone-effect.

We then assessed the effect of covariates on amiodarone effect. Among the covariates analyzed, electrical cardioversion (p<0.05), absence of dobutamine infusion (p<0.05), intravenous magnesium (p<0.02) and fluid loading (p<0.02) had a significant positive impact on amiodarone pharmacodynamics in ICU patients with AT (Appendix1). Use of furosemide, extra-renal dialysis, epinephrine infusion, curare and sedative drugs had no significant impact. Underlying pathologies, hemodynamic conditions, kidney function and baseline clinico-biological values (Table2) had neither significant impact. Figures 3D,E andF show that the use of magnesium and/or fluid loading on the first day were associated with a greater and earlier HR decrease than suggested by amiodarone pharmacodynamic half-life (3.33±0.9 days, Table3). In contrast, efficacy of amiodarone on HR control decreased if dobutamine was used (Figures 3D, E andF).

DISCUSSION

The present study indicates that amiodarone doses needed to control HR should be almost twice as high in persistent AT compared to paroxysmal AT [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] Anastasiou-Nana | Pharmacokinetics of amiodarone after intravenous and oral administration[END_REF][START_REF] Vardas | Amiodarone as a first-choice drug for restoring sinus rhythm in patients with atrial fibrillation: a randomized, controlled study[END_REF] . Patients who had received amiodarone before first ICU administration required lower doses.

Our pharmacodynamic model also showed that the main interventional covariates influencing amiodarone efficacy were dobutamine use, electrical cardioversion, magnesium and fluid loading.

A mixed oral and IV loading dose over a few days

Due to incomplete bioavailability and delayed oral absorption [START_REF] Anastasiou-Nana | Pharmacokinetics of amiodarone after intravenous and oral administration[END_REF][START_REF] Pourbaix | Absolute bioavailability of amiodarone in normal subjects[END_REF][21] , a 300 mg IV loading dose is recommended [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] to rapidly achieve the aimed HR. Thus, the total daily dose that may be given safely is expected to be higher with the oral route, avoiding hypotension, hemodynamic impairment and phlebitis which can specifically occur with IV amiodarone infusion [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] Lotto | Hemodynamic effects of amiodarone[END_REF][START_REF] Schwartz | Hemodynamic effects of intravenous amiodarone in patients with depressed left ventricular function and recurrent ventricular tachycardia[END_REF][START_REF] Remme | Hemodynamic effects and tolerability of intravenous amiodarone in patients with impaired left ventricular function[END_REF][START_REF] Jacqmin | Modelling response time profiles in the absence of drug concentrations: definition and performance evaluation of the K-PD model[END_REF] . Oral loading dosing regimen up to 30 to 50 mg/kg over 5 days [START_REF] Evans | High dose oral amiodarone loading: electrophysiologic effects and clinical tolerance[END_REF] were proved safe and effective. In our model, we therefore chose to test a rapid and safe therapeutic strategy combining IV and oral dosing regimen over 5 days. According to our model, the different proposed dose scheme (300 to 600 mg IV loading dose on day 1 plus 400 to 1400 mg oral dose from day 1 to day 5) in patients with paroxysmal or persistent AT, amiodarone-pretreated or not, agree with the recommended doses [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF][START_REF] Kochiadakis | Efficacy of amiodarone for the termination of persistent atrial fibrillation[END_REF][START_REF] Evans | High dose oral amiodarone loading: electrophysiologic effects and clinical tolerance[END_REF] but provide a more specific approach in each group.

Pharmacologic cardioversion with amiodarone is not effective in the short and medium term. At 24 hours, amiodarone has inconsistently demonstrated increased restoration of sinus rhythm when compared to placebo in some but not all randomized studies [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] . We therefore, focused on heart rate reduction which is a clinically relevant therapeutic endpoint in critically ill patients [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF][START_REF] Delle Karth | Amiodarone versus diltiazem for rate control in critically ill patients with atrial tachyarrhythmias[END_REF] .

Amiodarone pharmacodynamics in the ICU

We identified four interventions that influenced amiodarone efficacy in critically ill patients with AT. Dobutamine use was associated with decreased amiodarone efficacy while electrical cardioversion, fluid and magnesium loading were associated with earlier and better HR control.

Fluid therapy, defined as infusion of more than 0.5 liter of saline administered over 30 minutes, was administered to patients with apparent or real hypovolemia and was significantly associated with better HR control. By correcting hypovolemia, reflex tachycardia may have decreased [START_REF] Hainsworth | Reflexes from the heart[END_REF] . Dobutamine is a beta adrenergic agonist leading to tachycardia and supra-ventricular arrhythmias [START_REF] Fellahi | Perioperative use of dobutamine in cardiac surgery and adverse cardiac outcome: propensity-adjusted analyses[END_REF] to a greater extent than other catecholamines, such as epinephrine [START_REF] Butterworth | Dobutamine: too dangerous for « routine » administration?[END_REF] . In our model, dobutamine significantly influenced amiodarone pharmacodynamics while this interaction was not found with epinephrine. Interestingly, IV magnesium, which is recommended to prevent postoperative AT [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] Miller | Effects of magnesium on atrial fibrillation after cardiac surgery: a meta-analysis[END_REF] was found to improve HR control of patients with AT treated with amiodarone. Further investigations are required to confirm the present findings before extrapolating our results to other patient populations and procedures for the treatment of AT. Electrical cardioversion (ECV) was rarely used or efficient to restore sinus rhythm in our patients. Ineffective or immediate recurrence of AT after ECV is an important concern in critically ill patients, and even more when AT is persistent [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF] .

ECV success rates as low as 30% are reported in these patients [START_REF] Mayr | Effectiveness of direct-current cardioversion for treatment of supraventricular tachyarrhythmias, in particular atrial fibrillation, in surgical intensive care patients[END_REF][START_REF] Kanji | Epidemiology and management of atrial fibrillation in medical and noncardiac surgical adult intensive care unit patients[END_REF] . Impregnation with antiarrhythmic drugs such as amiodarone increases the probability to restore and maintain sinus rhythm [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] Singh | Sotalol Amiodarone Atrial Fibrillation Efficacy Trial (SAFE-T) Investigators. Amiodarone versus sotalol for atrial fibrillation[END_REF][START_REF] Sticherling | Comparison of acute and long-term effects of single-dose amiodarone and verapamil for the treatment of immediate recurrences of atrial fibrillation after transthoracic cardioversion[END_REF] .

There was no significant influence of etiology of shock, sedative and curare drugs on amiodarone efficacy. Neither furosemide administration nor ultrafiltration were identified as significant covariates. We hypothesize that patients treated with these depletive strategies had high pre-therapeutic filling pressures and did not experience hypovolemia and reactional HR increase [START_REF] Hainsworth | Reflexes from the heart[END_REF] . Type of atrial tachyarrhythmia was not a significant covariate but caution is needed because patients included in this modelling were mostly patients with atrial fibrillation and few data were available for other types of arrhythmia such as atrial flutter. Amiodarone and its metabolite N-desethylamiodarone are eliminated by hepatic route and marginally by the kidney.

In patients on hemodialysis or intermittent peritoneal dialysis, amiodarone and its metabolite were not found in the dialysate [START_REF] Harris | Renal elimination of amiodarone and its desethyl metabolite[END_REF] . In the present study, we did not find any influence of creatinine clearance or extra-renal dialysis on the model parameters.

Beta-blockers and verapamil were used exceptionally in this study because of their negative inotropic effects. Digoxin was also very rarely used because of its poor efficacy to rapidly slow HR and of significant interaction with amiodarone that may lead to adverse events [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] .

Limitations

In this observational study, we may have not taken into consideration other unavailable covariates of interest which may influence drug response. However, this study provides for the first time the basis for designing a future prospective trial of amiodarone dosing in ICU patients with AT. Also, the therapeutic strategy based on our model should now be prospectively tested to compare if a tailored amiodarone dosing strategy according to duration of AT (paroxysmal versus persistent) and other significant covariates (amiodarone pretreatment or not, coadministration of magnesium or not, on/off dobutamine) is more efficient and more convenient (less bradycardia and hypotension issues) to achieve an optimal heart rate control than a lenient strategy relying on physician's decisions without any standardized protocol.

Left atrial volume is inversely correlated to amiodarone dose needed to restore or maintain sinus rhythm after AT [START_REF] Vietti-Ramus | Efficacy and safety of short intravenous amiodarone in supraventricular tachyarrhythmias[END_REF] . To date, influence of left atrial volume on the ability of amiodarone to control heart rate is still unclear [START_REF] Kochiadakis | Efficacy of amiodarone for the termination of persistent atrial fibrillation[END_REF][START_REF] Vardas | Amiodarone as a first-choice drug for restoring sinus rhythm in patients with atrial fibrillation: a randomized, controlled study[END_REF][START_REF] Brodsky | Amiodarone for maintenance of sinus rhythm after conversion of atrial fibrillation in the setting of a dilated left atrium[END_REF][START_REF] Kochiadakis | Amiodarone versus propafenone for conversion of chronic atrial fibrillation: results of a randomized, controlled study[END_REF] . Impact of left atrial size could not be tested in our model due to incomplete data in too many patients.

Plasma levels of amiodarone and its metabolite N-desethylamiodarone, were not tested in our model because amiodarone therapeutic drug monitoring is not recommended [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF]AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] in clinical practice. It is theoretically conceivable that including plasma concentrations of amiodarone and N-desethylamiodarone could have improved our predictions.

It is however unlikely since individual plasma concentrations are poorly predictive of pharmacodynamics response [START_REF] Greenberg | Relation between amiodarone and desethylamiodarone plasma concentrations and electrophysiologic effects, efficacy and toxicity[END_REF][START_REF] Deharo | Clinical electrophysiologic effects of a single high oral dose of amiodarone[END_REF] .

Our patients with paroxysmal AT pretreated with amiodarone had very heterogeneous doses before first administration in ICU. Due to our observational design, these pre-admission doses could not be precisely identified. In our model, we found that this covariate significantly influenced baseline HR. Baseline HR in this subgroup of patients were quite scattered (132 ± 35 bpm) compared to other subgroups and lower than in amiodarone naïve patients with paroxysmal AT (154 ± 3 bpm). This difference in baseline HR between those latter groups could be explained by the fact that patients pretreated by amiodarone were already under its slowing effect on heart rate. Thus, our recommended doses of amiodarone for this subgroup of amiodarone-pretreated patients with paroxysmal AT was the lowest proposed and will be applicable only if baseline HR are in the 130-135 bpm range. Even lower doses would be needed if this baseline HR is lower than 120 bpm and higher doses would be needed in patients with baseline HR above 140 bpm.

Perspectives

This work, modeling amiodarone efficacy to control HR in critically ill patients with AT, identified several clinically relevant covariates that influenced its pharmacodynamics. Taking into account these covariates might achieve better individualization of amiodarone dosing for poorly tolerated AT in critically ill patients.

Compared to paroxysmal AT, double doses are needed in persistent AT. Amiodarone pretreated patients before ICU admission needed lower doses than amiodaronenaive patients. Correction of hypovolemia, magnesium administration and use of lower dobutamine doses also contributed to a better control of HR.

This study suggests that co-administration of magnesium with amiodarone is associated with a greater and earlier HR control in critically ill patients with AT. Lower amiodarone doses might be needed to slow AT in case of co-administration of magnesium, thereby potentially lowering the incidence of amiodarone adverse events. Administration of high doses of magnesium versus placebo as an adjunct to ibutilide to treat AT, a class III agent, was associated with a higher conversion rate to sinus rhythm and less ventricular arrhythmia [START_REF] Patsilinakos | Effect of high doses of magnesium on converting ibutilide to a safe and more effective agent[END_REF] . To date, prophylactic use of IV magnesium has been shown to reduce the probability of post-operative AT [START_REF] Miller | Effects of magnesium on atrial fibrillation after cardiac surgery: a meta-analysis[END_REF] but data regarding its efficacy in acute constituted AT remain scarce, particularly when combined with amiodarone. We are planning to further investigate this issue by conducting a prospective study evaluating IV magnesium as add-on therapy to amiodarone to treat AT in critically ill patients.

CONCLUSION

In critically ill patients with AT, our results indicate that doses of amiodarone needed to control heart rate are higher in persistent than in paroxysmal AT. In contrast, patients who had received amiodarone before first ICU administration required lower doses. In critically ill patients, optimization of fluid therapy, decreasing dobutamine infusion as soon as possible and magnesium supplementation may also help control heart rate in AT.

FUNDING SOURCES

No external funding was used in the preparation of this manuscript. 154 (3) -0.23 (41) 0.85 (27) 0.67 (18) 0.86 (29) 154 bpm (3) . 131 bpm (4) 103 bpm (20) 132 bpm (26) 0.14 (11) k out (

DISCLOSURES

  ABBREVIATION

the 18 -

 18 bed medical ICU of a tertiary teaching hospital. Due to the observational design, and in accordance with French bioethics laws, Institutional Review Board authorization was not necessary. The study was approved by the Commission Nationale de l'Informatique et des Libertés to use computerized medical data with protection of patient confidentiality. Data were extracted from the files of 80 consecutive critically ill patients who had received at least one dose of amiodarone to treat or prevent AT during their hospitalization in ICU. Three investigators (JES, MEA, MA) were each responsible of the data extraction during a specified time period resulting in patients inclusion over three distinct time periods. [Figure1]. Patients with paced rhythm or incomplete chart were excluded. HR data were collected at the moment of the first amiodarone administration in ICU and then 4 to 6 times daily after each dosing over 6 days or until death or ICU discharge. Amiodarone could be administered either IV over 20 to 60 minutes or orally. Route, frequency of administration and doses were prescribed at the discretion of the treating physician without any predefined protocol.

  HR denotes the parameter modulating the importance of spontaneous retrocontrol to a reference value, HR REF , depending on the type of cardiac rhythm at the time of first amiodarone administration in ICU. Several time-independent factors were tested and then included in the response model when they significantly (%rse<50%) modified either baseline HR before first amiodarone dose in ICU (HR (0) ) or k out or HR REF as follows:

  was analyzed using the nonlinear mixed effect modeling software program Monolix version 4.1.3 (www.lixoft.com). Parameters were estimated by computing the maximum likelihood estimator of the parameters without any approximation of the model using the stochastic approximation expectation maximization algorithm combined to a Markov Chain Monte Carlo procedure. A combined, proportional plus constant, error model was used to describe residual variability and between-subject variability (BSV) was ascribed to an exponential error model. The Bayesian information criterion (BIC) was used to test different hypotheses regarding the final model, i.e., the covariate effects on parameters, residual variability model (proportional versus proportional plus additive error model), and structure of the variance-covariance matrix for the BSV parameters. Residuals were presented as normalized prediction distribution errors, based on the estimates of unbiased means and variances of the predictions by using 500 Monte Carlo simulations of the final model (the calculation includes a de-correlation step of the prediction errors). The mean of these normalized residues must not be different from 0. Goodness-of-fit plots for the final model of amiodarone pharmacodynamics are represented in Figure 2. Results are expressed as numbers (%), means ± standard deviation, or medians (interquartile ranges) as appropriate. A p value < 0.05 was considered significant.

Figure 1 .Figure 2 .

 12 FIGURES AND LEGENDS

Figure 3 .

 3 Figure 3. Model-predicted effect of amiodarone (Amio) on heart rate (HR) in beats per

Abbreviations:

  Abbreviations: AT = Atrial fibrillation; Amio 50 = Amiodarone dosage that induces 50% of

  on therapy on top of amiodarone to control heart rate and only two patients received digoxin. The main clinical, demographic and biological patients' characteristics are summarized in Table2. Patients were mainly admitted for cardiogenic shock or sepsis. No stroke was identified during patients stay in ICU.

	80 patients, 1348 time-heart rate observations with 361
	amiodarone dose administrations were available during a maximum of 6 days after
	the first dosing in ICU. In our dataset during the observed period, patients with
	paroxysmal AT (n=47) had heart rate above 115 bpm less frequently than patients
	with persistent AT (n=25): respectively 37% (261/706) versus 47.7% (224/470) of HR
	ill, had cardiovascular comorbidities and received amiodarone during their ICU stay
	mainly for poorly tolerated paroxysmal atrial fibrillation. Dobutamine (10%) or
	epinephrine (56%) were the only IV catecholamine used. Before first amiodarone
	administration in ICU, 52 patients were naive of amiodarone pre-treatment (74%

collected (p=0.02). Among patients initially diagnosed with AT in our dataset (n=72), 33 (46%) never returned to sinus rhythm while 39 had at least one ECG in sinus rhythm. Among those latter 39 patients, AT relapsed in 12 patients. Overall, sinus rhythm was restored in only 14 patients (19%) within 24 hours of first amiodarone administration in ICU without recurrence of AT. Electrical cardioversion was delivered to 9 patients and successful in only 4, of whom 2 after several attempts at distinct times of care. Those data confirm the difficulty to achieve and maintain heart rate control or sinus rhythm in clinical practice.More details about heart rate data collection and amiodarone doses are summarized in Table

1

. Patients were severely patients with paroxysmal AT and 26% with persistent AT) and 28 patients had previously received amiodarone. Thirty patients (37.5%) received magnesium loading as add-

Table 1 . Heart rhythm characteristics and amiodarone dosing of the 80 patients Characteristic Value

 1 ). Use of magnesium or/and fluid loading the first day, as add-on therapy on Amio were associated with a faster and greater HR decrease. Efficacy of Amio on HR control decreased in case of persistent AT or if dobutamine was used.

	Heart rate
	Total number of HR collected

TABLES

Statistics: Values are given as the mean  standard deviation, median (interquartile range

25-75)

or number (%).

Table 2 . Main Characteristics of the 80 patients studied Characteristic Value Demographic status and comorbidities

 2

	Age, year

clinical values at first administration of amiodarone in ICU

	Mean arterial pressure <65mmHg	29(36%)
	Body Temperature, (°Celsius, eardrum)	37.1(36.4-38%)
	Non-invasive/invasive mechanical ventilation	9(11%) / 43(54%)
	Cardiac systolic dysfunction (Ejection fraction<45%)	31(39%)
	Sepsis	35(44%)
	Paroxysmal/persistent atrial tachyarrhythmia	47(59%)/25(31%)
	Sinus rhythm	8(10%)
	pH	7.4(7.26-7.43)
	P a O 2 (mmHg)	85(72-100)
	Na + (mmol/l)	137.5(135-141)
	K + (mmol/l)	3.7(3.2-4.6)
	Hemoglobin	10.9(9.1-12)

Table 3 . Parameter estimates of the final model of amiodarone effect on heart rate in the 80 critically ill patients studied

 3 , (if Parox AT, amiodarone naive), bpm  by Age effect, [(Age/70) -0.23 ]  if Persistent AT [vs parox AT]  if Sinus rhythm [vs parox AT]   if pretreated Amio [vs Amio naive]

	Parameter

  if Parox AT),day 1  if Persistent AT or sinus rhythm [vs parox AT] Reference HR, (if Parox AT or sinus rhythm), bpm  if Persistent AT [vs parox AT]

		0.9 (28)	0.9 day 1 (28)	1.42 (13)
		0.37 (48)	0.33 day 1 (NA)
		78.6 (20)	78.6 bpm (20)	0.8 (11)
		1.73 (47)	136 bpm (29)
	FRetro HR	0.46 (21)		NA
	Amio 50 , mg/day	291 (29)		NA
	t MG	0.4 (32)
	t FILL	0.31 (31)
	t ECV	0.37 (31)
	t DOBU	0.26 (46)
	Residual variability, bpm	16 (2)

* TSH was available in 35 subjects within 10 days before first administration of amiodarone in ICU.

Among them, none had a TSH below lower normal limit [<0.2 mUi/L] and three had a TSH above upper normal limit [>5.2mUi/L].