Nicolae Cîndea
email: nicolae.cindea@inria.fr

Sorin Micu
email: sdmicu@yahoo.com.

Ademir F Pazoto
email: ademir@acd.ufrj.br

Periodic solutions for a weakly dissipated hybrid system

We consider the motion of a stretched string coupled with a rigid body at one end and we study the existence of periodic solution when a periodic force f acts on the body. The main difficulty of the study is related to the weak dissipation that characterizes this hybrid system, which does not ensure a uniform decay rate of the energy. Under additional regularity conditions on f , we use a perturbation argument in order to prove the existence of a periodic solution. In the last part of the paper we present some numerical simulations based on the theoretical results.

Introduction

In this paper we consider the following hybrid system            u tt (x, t) -u xx (x, t) = 0 in (0, 1) × (0, ∞) u(0, t) = 0 on (0, ∞) u tt (1, t) + u x (1, t) + αu t (1, t) = f (t) on (0, ∞) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) on (0, 1), (1.1) where α is a positive constant. System (1.1) models the coupling between a vibrating string located on the interval (0, 1) and a rigid body attached at the right extremity x = 1. Here u(x, t) describes the position of the string at each moment t ≥ 0 and point x ∈ (0, 1) and verifies the linear wave equation. The left extremity x = 0 of the string is supposed to be fixed. In the third equation of (1.1), u (1, t) gives the movement of the body and satisfies the Newton law. When α > 0, the term αu t (1, t) represents a friction proportional with the velocity of the body. Note that the dissipation of system (1.1) acts only through the ordinary differential equation representing the movement of the rigid body. For simplicity, we have chosen the length of the string, the wave velocity and the mass of the solid to be equal to one.

Model (1.1) has been introduced and analyzed in [START_REF] Lee | Stabilization of a hybrid (string/point mass) system[END_REF] (see also [START_REF] Lee | Stabilization of a vibrating string system linked by point masses, Control of boundaries and stabilization[END_REF][START_REF] Littman | Some recent results on control and stabilization of flexible structures[END_REF][START_REF] Morgul | On the stabilization of a cable with a tip mass[END_REF]) and represents the simplest example of hybrid system which couples two different types of differential equations. In [START_REF] Andra-Novel | Feedback stabilization of a hybrid PDE-ODE system: application to an overhead crane[END_REF][START_REF] Littman | Lawrence Exact boundary controllability of a hybrid system of elasticity[END_REF][START_REF] Rao | Uniform stabilization of a hybrid system of elasticity[END_REF][START_REF] Rao | Decay estimates of solutions for a hybrid system of flexible structures[END_REF] similar models in which the string is replaced by a beam with different boundary conditions are considered. Most of the results obtained in this paper can be extended to these models with similar statements. Many more complex one or multi dimensional hybrid systems have been introduced and analyzed (see, for instance, [START_REF] Avalos | Uniform decay rates for solutions to a structural acoustics model with nonlinear dissipation[END_REF][START_REF] Banks | Feedback control of noise in a 2-D nonlinear structural acoustics model[END_REF][START_REF] Castro | Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass[END_REF][START_REF] Cot | Exact controllability of an aeroacoustic model with a Neumann and a Dirichlet boundary control[END_REF][START_REF] Hansen | Controllability and stabilization of strings with point masses[END_REF][START_REF] Micu | Periodic solutions for a bidimensional hybrid system arising in the control of noise[END_REF][START_REF] Micu | Asymptotics for the spectrum of a fluid/structure hybrid system arising in the control of noise[END_REF][START_REF] Raymond | Exact controllability in fluid-solid structure: the Helmholtz model[END_REF]). Some of these models are characterized by a weak dissipation acting on a component of the system only. This does not ensure an uniform decay rate of the corresponding energy and makes more difficult the study of such properties as asymptotic behavior, compactness of trajectories, existence of periodic or almost periodic functions, etc. To establish these properties may be as a delicate problem as the corresponding one for some nonlinear models.

The non homogeneous term f in (1.1) represents an external force acting on the body and it is supposed to be not zero and periodic in time with period T f (t + T ) = f (t) ∀t ≥ 0.

(1.

2)

The aim of this article is to study if there exists or not a periodic solutions of (1.1) under the hypothesis (1.2) for f . In the positive case, any other solution of (1.1) is bounded and converges to the periodic one as t goes to infinity. On the other hand, the non-existence of a periodic solution implies that all the solutions are unbounded in the finite energy space when t tends to infinity. In this case, we are dealing with a resonance phenomenon in which a bounded perturbation f of the system leads to unbounded solutions.

In this paper we show that, if the non homogeneous term f is such that f |(0,T ) ∈ H 1 (0, T ), then there exists a unique periodic solution of (1.1). To prove this result we introduce a perturbed system, depending on a small parameter ε, devoted to go to zero, for which we have an exponential decay rate of the corresponding energy. This property allows us to apply a fixed point argument and to show that the perturbed system has a unique periodic solution for each ε > 0. If f |(0,T ) ∈ H 1 (0, T ), these periodic solutions are uniformly bounded. Hence, we can pass to the limit to obtain a periodic solution for the initial system (1.1).

On the other hand, note that finite energy solutions for (1.1) exist even if f |(0,T ) belongs to L 2 (0, T ). In this case we use Fourier decomposition method in order to characterize the periods T for which there exists a periodic solution. Due to the complexity of this characterization, it is not easy to say if this property holds for any T or not. Thus, in the case f |(0,T ) ∈ L 2 (0, T ), we cannot guarantee the absence of the resonance phenomenon but we mathematically show that its possible occurrence is precisely related to the weak dissipation of the system. In [START_REF] Micu | Periodic solutions for a bidimensional hybrid system arising in the control of noise[END_REF] a more complex hybrid system, coupling two partial differential equations, is considered and the existence of periodic solutions is studied. Due to simpler nature of the model, the results we obtain in the present paper are much sharper.

In the last part of the article we use the perturbation argument and the fixed point method mentioned above in order to give some numerical approximation for the periodic solution. From these computations we can see that the extra dissipation introduced in the perturbed system ensures, in many cases, a better numerical behavior of the approximation scheme. Also, we provide some criteria for the optimal choice of the perturbed parameter ε with respect to the discretization step.

The remaining part of the paper is organized as follows. In Section 2 we present some basic properties of system (1.1). Section 3 is devoted to the analysis of the perturbed system and the proof of existence of periodic solutions for it. Their uniform boundedness is proved in Section 4 supposing that f |(0,T ) ∈ H 1 (0, T ) and, as a consequence, the existence of a periodic solution for the initial system (1.1) follows. In Section 5 we analyze the case f |(0,T ) ∈ L 2 (0, T ) and we give characterization of the periods T for which there exists a periodic solution. Finally, in the last section we present the numerical simulations which are based on and confirm the theoretical results.

Preliminaries

In this section we present some basic properties of system (1.1). Most of these results are well-known (see, for instance, [START_REF] Andra-Novel | Feedback stabilization of a hybrid PDE-ODE system: application to an overhead crane[END_REF][START_REF] Lee | Stabilization of a vibrating string system linked by point masses, Control of boundaries and stabilization[END_REF][START_REF] Lee | Stabilization of a hybrid (string/point mass) system[END_REF][START_REF] Morgul | On the stabilization of a cable with a tip mass[END_REF]) but we prefer to present them in order to make our paper self contained and easier to read. Firstly, we need to introduce some notation and to rewrite (1.1) in an abstract form. We define the Hilbert spaces

V = {u ∈ H 1 (0, 1) : u(0) = 0}, H = V × L 2 (0, 1) × R
endowed with their natural inner product. We shall denote by (

U (0) = U 0 (2.1)
where

U =   u w z   , AU =   w u xx -u x (1) -αw(1)   , F =   0 0 f   and U 0 =   u 0 w 0 z 0   .
The operator A is an unbounded operator in H, with domain

D(A) = {(u, w, z) ∈ H 2 (0, 1) ∩ V × V × R : w(1) = z}.
With this notation, we have the following result. The total energy associated to (1.1) is given by

E(t) = 1 2 1 0 (u 2 t (x, t) + u 2 x (x, t))dx + 1 2 u 2 t (1, t) (2.3)
and under the above boundary conditions we can, formally, deduce that

dE(t) dt = -αu 2 t (1, t) + f (t)u t (1, t). ( 2.4)
When f = 0, (2.4) indicates that the energy is decreasing which has been shown sufficient for the strong stability, but as we have said before, does not ensure the uniform stability. The next theorem describes this asymptotic behavior of solutions of the homogeneous system (2.1). Theorem 2.2. For any U 0 ∈ H we have that lim t→∞ S(t)U 0 = 0.

(2.5)

Moreover, there are no positive constants M and ω such that

S(t)U 0 ≤ M exp(-ωt) U 0 ∀U 0 ∈ H, ∀t ≥ 0. (2.6)
Remark 2.3. The asymptotic properties described by Theorem 2.2 show that the dissipation αu t (1, t) of (1.1), which is concentrated on the rigid body, cannot ensure the exponential decay of the energy (2.4). As it is shown in [START_REF] Littman | Some recent results on control and stabilization of flexible structures[END_REF] this is equivalent to the existence of solutions with arbitrarily slow decay rate.

A spectral analysis of the operator A confirms the asymptotic behavior described in the previous theorem. More precisely, we have the following result.

Theorem 2.4. The operator (D(A), A) has a sequence of eigenvalues (ν n ) n≥1 with the following behavior

ν n = nπi + i nπ - α nπ + o 1 n , as n → ∞. ( 2
W (t) = S(t)(U (0) -V (0)), ∀t ≥ 0. Now, (2.9) follows from (2.5) in Theorem 2.2. If V is a T -periodic solution we deduce that W = U -V is T -periodic too.
From (2.9) we obtain that U ≡ V and the proof ends.

Existence of periodic solution for a perturbed system

As we have said before, the main difficulty in the study of existence of periodic solutions for (1.1) comes from the non uniform decay rate of the corresponding energy (2.3). To overcome this difficulty, we introduce an extra dissipative term, depending on a small parameter ε, which ensures the uniform decay of the energy. More precisely, for any ε > 0, we consider the system

           u tt (x, t) -u xx (x, t) + εu t = 0 in (0, 1) × (0, ∞) u(0, t) = 0 on (0, ∞) u tt (1, t) + u x (1, t) + αu t (1, t) = f (t) on (0, ∞) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) on (0, 1) (3.1)
where f is a T -periodic function verifying (1.2). The aim of this section is to prove that, for each ε > 0, there exists a unique T -periodic solution of (3.1). As shown in [START_REF] Haraux | Semi-linear hyperbolic problems in bounded domains[END_REF], if the corresponding linear system is uniformly stable, the existence of a periodic solution may be obtained from a fixed point argument.

The total energy associated to (3.1) is given by

E ε (t) = 1 2 1 0 (u 2 t (x, t) + u 2 x (x, t))dx + 1 2 u 2 t (1, t). (3.2)
It is easy to see that, at least formally,

dE ε (t) dt = -αu 2 t (1, t) -ε 1 0 u 2 t (x, t)dx + f (t)u t (1, t). (3.3)
By comparing (2.4) and (3.3) we see that the term εu t introduced in (3.1) reinforces the dissipation of the system. In fact, as we shall prove latter on, this term ensures the uniform stability of system (3.1) and allows to prove the existence of a periodic solution for (3.1) as in [START_REF] Haraux | Semi-linear hyperbolic problems in bounded domains[END_REF].

As in the previous section, introducing the new variables w = u t and z = u t (1, t), we may write (3.1) as

   dU (t) dt = A ε U (t) + F (t) U (0) = U 0 (3.4)
where

U =   u w z   , A ε U =   w u xx -εw -u x (1) -αw(1)   , F =   0 0 f   and U 0 =   u 0 w 0 z 0   .
The operator A ε is an unbounded operator in H, with domain

D(A ε ) = {(u, w, z) ∈ H 2 (0, 1) ∩ V × V × R : w(1) = z} which is independent of ε. In fact, D(A ε ) = D(A) for each ε > 0.
With this notation, we have the following result.

Theorem 3.1. If U 0 ∈ H and f ∈ C[0, ∞), there exists a unique weak solution U ∈ C([0, ∞); H) of (3.4). Moreover, if U 0 ∈ D(A ε ) and f |(0,a) ∈ H 1 (0, a)
, for all a > 0, there exists a unique weak solution

U ∈ C 1 ([0, ∞); H) ∩ C([0, ∞); D(A ε )) of (3.4).
Proof. Performing as in [START_REF] Morgul | On the stabilization of a cable with a tip mass[END_REF] we can prove that (D(A ε ), A ε ) is the infinitesimal generator of a semigroup (S ε (t)) t≥0 of contractions in H. Now, the result follows from the classical theory of linear equations of evolution (see, for instance, [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF]).

The following theorem shows that system (3.4) is exponentially stable if f ≡ 0.

Theorem 3.2. Let (S ε (t)) t≥0 be the semigroup of contractions generated by the operator (D(A ε ), A ε ). There exist two positive constants M > 0 and ω = ω(ε) > 0 such that, for any

U 0 ∈ H, S ε (t)U 0 ≤ M e -ωt U 0 t > 0. (3.5)
Moreover, for any

U 0 ∈ D(A ε ), S ε (t)U 0 D(Aε) ≤ M e -ωt U 0 D(Aε) t > 0. (3.6)
Proof. Multiplying the first equation in (3.1) by u it is easy to see that the solution u satisfies

d dt 1 0 uu t dx + ε 2 1 0 u 2 dx + u(1, t)u t (1, t) + α 2 u 2 (1, t) = - 1 0 u 2 x dx + 1 0 u 2 t dx + u 2 t (1, t). (3.7)
For any δ > 0, from (3.3) and (3.7) we deduce that

dL(t) dt = -αu 2 t (1, t) -ε 1 0 u 2 t dx -δ 1 0 u 2 x dx + δ 1 0 u 2 t dx + δu 2 t (1, t)
where L = L(t) is a suitable perturbation of the energy given by

L(t) = E ε (t) + δ 1 0 uu t dx + ε 2 1 0 u 2 dx + u(1, t)u t (1, t) + α 2 u 2 (1, t) . (3.8)
Moreover, for 0 < δ < ε, sufficiently small, we have

dL(t) dt ≤ -c E ε (t), (3.9)
where c = 1 2 min{α-δ, ε-δ, δ}. On the other hand, (3.8) and the Sobolev embedding theorem allow us to conclude that

|L(t) -E ε (t)| ≤ c 1 δE ε (t),
for some c 1 independent of ε as well. Then, choosing δ sufficiently small the above inequality allow to deduce that

1 2 E ε (t) ≤ L(t) ≤ 3 2 E ε (t).
Since δ > 0 is small enough, the above inequality combined with (3.9) leads to the differential inequality

dE ε (t) dt ≤ -c 2 E ε (t),
for some c 2 > 0, which gives us (3.5).

In order to prove (3.6), note that

S ε (t)U 0 D(A ε ) = S ε (t)A ε U 0 + S ε (t
lim ε→0 ω(ε) = 0.
Now we have all the ingredients needed to prove the existence of a periodic solution for (3.4).

Theorem 3.4. If f ∈ C[0, ∞) is a T -periodic function, for any ε > 0, there exists a unique weak solution U ε ∈ C([0, ∞); H) of (3.4) which is T -periodic. Moreover, if f | (0,T ) ∈ H 1 (0, T ) then there exists a unique classical T -periodic solution U ε ∈ C([0, ∞); D(A ε )) of (3.4).
Proof. The proof uses the ideas from [START_REF] Haraux | Semi-linear hyperbolic problems in bounded domains[END_REF] and it is done by using a fixed point argument. We introduce the map Λ :

H -→ H given by ΛU 0 = S ε (T )U 0 + T 0 S ε (T -s)F (s)ds
where (S ε (t)) t≥0 is the semigroup generated by A ε (see Theorem 3.1). Then, from Theorem 3.2 we get

Λ n U 0 -Λ n U 1 ≤ S ε (nT )(U 0 -U 1 ) ≤ M e -nT ω U 0 -U 1 .
Then, for n sufficiently large Λ n is a contraction and, therefore, there exists a unique

U 0 ∈ H such that Λ n U 0 = U 0 . Consequently, Λ n (ΛU 0 ) = Λ(Λ n U 0 ) = ΛU 0 .
From the uniqueness of the fixed point of Λ n it follows that ΛU 0 = U 0 which says that Λ has a fixed point. The solution U of (3.4) with initial data U 0 belongs to C([0, ∞); H) and verifies U (0) = U (T ). This gives us a T -periodic weak solution of (3.4).

To treat the case f | (0,T ) ∈ H 1 (0, T ), we use Theorem 3.1 and we repeat the above fixed point argument in D(A ε ) instead of H. From estimate (3.6), we deduce that there exists a unique fixed point U 0 of Λ in D(A ε ). This gives us a periodic classical solution

U ε ∈ C([0, ∞); D(A ε )) of (3.4).
4 Existence of periodic solution when f ∈ H 1 (0, T )

In this section we prove the existence of a periodic solution for our initial system (1.1) under the additional condition f |(0,T ) ∈ H 1 (0, T ). Theorem 3.4 gives us a family (U ε ) ε>0 of T -periodic functions. Any limit point of this family is a periodic solution for (1.1). However, in order to prove that this family has a limit, we need some uniform estimates with respect to ε. These estimates require additional regularity conditions for f mentioned above. (3.4), then there exists c > 0, independent of ε, with

Theorem 4.1. Let f ∈ C[0, ∞) be a T -periodic function such that f | (0,T ) ∈ H 1 (0, T ) and ε < 1. If U ε is the periodic solution of
U ε C([0,∞);H) ≤ c. (4.1)
Proof. Multiplying the first in (3.1) equation by xu x and integrating by parts over (0, 1) × (0, T ), we deduce that

1 2 T 0 1 0 (u 2 t + u 2 x )dxdt = 1 2 T 0 u 2 t (1, t)dt + 1 2 T 0 u 2 x (1, t)dt -ε T 0 1 0 u t xu x dxdt
since, from the periodicity,

1 0 u t xu x | T 0 dx = 0. Consequently, (1 -ε) 1 2 T 0 1 0 (u 2 t + u 2 x )dxdt ≤ 1 2 T 0 u 2 t (1, t)dt + 1 2 T 0 u 2 x (1, t)dt (4.2)
In order to bound the terms T 0 u 2 t (1, t)dt and T 0 u 2 x (1, t)dt we proceed as follows. Integrating (3.3) in (0, T ) and taking into account the periodicity of the solutions U ε , the following holds

0 = ε T 0 1 0 u 2 t dxdt + T 0 (αu 2 t (1, t) -f (t)u t (1, t))dt which allows to conclude that α 2 T 0 u 2 t (1, t)dt ≤ 1 2α T 0 f 2 (t)dt.
Now, differentiating the equation (3.1) with respect to t and proceeding in the same way we can bound

T 0 u 2 tt (1, t)dt. Indeed, letting v = u t we get            v tt (x, t) -v xx (x, t) + εv t = 0 in (0, 1) × (0, ∞) v(0, t) = 0 on (0, ∞) v tt (1, t) + v x (1, t) + αv t (1, t) = f (t) on (0, ∞) v(x, 0) = u 1 (x), v t (x, 0) = v 1 (x) on (0, 1), (4.3)
where v 1 (x) = u tt (x, 0) = u 0,xx -εu 1 (x). Then, since f ∈ H 1 (0, T ) using the previous estimates we obtain that

α 2 T 0 u 2 tt (1, t)dt ≤ 1 2α T 0 (f ) 2 (t)dt.
By using the third equation from (3.1), we deduce that

T 0 u 2 x (1, t)dt ≤ 1 α 2 T 0 (f ) 2 (t)dt + 1 + 1 α T 0 f 2 (t)dt.
Now, we can bound the term on the right hand side of (4.2) to obtain

T 0 E ε (s)ds ≤ C
for some positive constant C independent of ε. Finally, from (3.3) we have

E ε (0) = E ε (t) + ε t 0 1 0 u 2 t dxdt + α t 0 u 2 t (1, t)dt - t 0 f (t)u t (1, t)dt.
Now, integrating on (0, T ) and using the previous estimates we deduce that

T E ε (0) ≤ (1 -ε2T ) T 0 E ε (t)dt + (T α + T 2 ) T 0 u 2 t (1, t)dt + T 2 T 0 f 2 (t)dt (4.4)
and the result follows.

The uniform estimate from Theorem 4.1 allows us to deduce the existence of a periodic solution for our initial problem (1.1).

Theorem 4.2. Let f ∈ C[0, ∞) be a T -periodic function such that f | (0,T ) ∈ H 1 (0, T ). Then (2.

1) has a unique T -periodic solution U ∈ C([0, ∞); H).

Proof. Following [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF], Proposition 2.3.1, we introduce the completion H of the space H with respect to the norm |||Φ||| = ||(I -A) -1 Φ|| and the unbounded operator in H given by (H, Ã) which extends (D(A), A). Also, let ( S(t)) t≥0 be the semigroup generated by (H, Ã) in H.

According to Theorem 4.1, the family of periodic functions (U ε ) ε∈(0,1) is uniformly bounded in C([0, T ]; H) and verifies

U ε (t) = S(t)U ε (0) + t 0 S(t -s)(F (s) + εP 2 (s))ds t ∈ [0, T ] (4.5)
where P 2 (s) = (0, -u t (s), 0) * . Since the inclusion H ⊂ H is compact, it follows that (U ε ) ε∈(0,1) is relatively compact in C([0, T ]; H). Thus, there exists Ũ ∈ C([0, T ]; H) and a subsequence that we denote in the same way, (U ε ) ε∈(0,1) , which converges to Ũ in C([0, T ]; H). Let us denote Ũ0 = U (0) ∈ H. It is easy to see that

Ũ (t) = S(t) Ũ0 + t 0 S(t -s)F (s)ds t ∈ [0, T ]. (4.6)
On the other hand, since (U ε (0)) ε∈(0,1) is a bounded sequence in H, it follows that there exists U 0 ∈ H and a subsequence that we denote in the same way, (U ε (0)) ε∈(0,1) , which converges weakly to U 0 in H. From the uniqueness of the limit in H, we deduce that Ũ (0) = Ũ0 = U 0 ∈ H.

Since S(t)Φ = S(t)Φ for any Φ ∈ H, it follows from (4.6) that

Ũ (t) = S(t)U 0 + t 0 S(t -s)F (s)ds t ∈ [0, T ]. (4.7)
Consequently, Ũ is a weak solution in C([0, T ]; H) of (2.1) which may be extended to a T -periodic solution of the same equation.

5 Existence of periodic solution when f ∈ L 2 (0, T ) Theorem 4.2 shows that f | (0,T ) ∈ H 1 (0, T ) is a sufficient condition for the existence of a periodic solution of (1.1). If this condition is necessary remains an open question. In this section we use Fourier decomposition to give a characterization of the periods T for which periodic solutions exists even when f ∈ L 2 (0, T ). Firstly, we consider the Fourier expansion of f , f (t) = n∈Z a n e iλ n t (5.1)

where λ n = 2nπ T and a n ∈ C. Note that f given by (5.1) belongs to L 2 (0, T ) if and only if

n∈Z |a n | 2 < ∞ and belongs to H 1 (0, T ) if and only if n∈Z n 2 |a n | 2 < ∞.
It is easy to check that, at least formally, the corresponding periodic solution of (1.1) is given by u

(x, t) = n∈Z a n e iλnt u n (x) (5.2)
where, for each n ∈ Z, the function

u n = u n (x) satisfy the system      -λ 2 n u n (x) -u n,xx (x) = 0 u n (0) = 0 -λ 2 n u n (1) + u n,x (1) + iαλ n u n (1) = 1.
(5.3) From (5.3) it follows that u 0 (x) = x and, for any n ∈ Z * ,

u n (x, t) = sin(λ n x) γ n (5.4)
where γ n = -λ 2 n sin λ n + λ n cos λ n + i αλ n sin λ n . Now, from (5.2), we deduce that

u 2 L 2 (0,T ;V ) = T 2 n∈Z * |λ n | 2 |γ n | 2 1 - sin(2λ n ) 2λ n |a n | 2 + T |a 0 | 2
and consequently u(x, t) gives a finite energy periodic solution if and only if ≤ cn 2 for some positive constant c > 0. Indeed, we have the following cases:

n∈Z * |λ n | 2 |γ n | 2 |a n | 2 < ∞. (5.5) Remark 5.1. When f ∈ H 1 (0, T ), which is equivalent to (a n ) n∈Z * ∈ h 1 ,
• If |λ n sin λ n | > 1 2 , we have | sin λ n | > 1 2|λ n | = T 4nπ
and, therefore,

λ n γ n 2 = 1 α 2 sin 2 λ n + (λ n sin λ n -cos λ n ) 2 ≤ 1 α 2 sin 2 λ n < n 2 4π αT 2 . • |λ n sin λ n | ≤ 1 2 , we have | cos λ n | 2 ≥ 1 - 1 4λ 2 n
. Then there exists c > 0 such that

λ n γ n 2 ≤ 1 (λ n sin λ n -cos λ n ) 2 ≤ 1 (| cos λ n | -|λ n sin λ n |) 2 ≤ 1 1 -1 4λ 2 n -1 2 2 ≤ c.
This is precisely the case we have studied in the previous section, when we have assumed that f | (0,T ) ∈ H 1 (0, T ).

If (5.5) does not hold, no periodic solution for equation (1.1) exists. As we have mentioned in the introduction, the non-existence of periodic solution is equivalent to the presence of the resonance phenomenon. Note that (5.5) does not hold if and only if λ n = 2nπ T verifies sin λ k n → 0 as n → ∞, (5.6)

λ k n sin λ k n -cos λ k n → 0 as n → ∞, ( 5.7)
for some (strictly) increasing sequence of natural numbers

(k n ) n≥1 ⊂ N. Let us recall that ν n = nπi + i nπ -α nπ 2 + o( 1 n
) are the eigenvalues of (2.1) while the eigenvalues of the corresponding conservative system with α = 0 are of the form iz n where

z n = nπ + 1 nπ + o 1 n , as n → ∞ (5.8)
and verifies, for each n ≥ 1, the equation t sin t -cos t = 0.

(5.9)

Remark 5.2. As it follows form (5.5), the existence of a periodic solution in the case f ∈ L 2 (0, T ) is equivalent to the boundedness of the sequence λ n γ n n≥1

. Thus, the resonance phenomenon occurs if, and only if, a subsequence of (λ n ) n≥1 is close to the roots of the equation -z sin z + cos z + i α sin z = 0.

(5.10)

Note that -iν n are exactly the roots of (5.10). Since, as follows from Remark 2.5, the real parts of (ν n ) n≥1 are smaller and smaller as n tends to infinity, some of the real numbers (λ n ) n≥1 may be close to (-iν n ) n≥1 . Consequently, the possible occurrence of the resonance phenomenon is precisely related to the weak dissipation of system (1.1).

With the notation above, we have the following properties, which characterize the periods T for which the resonance phenomenon may occur.

Property 5.3. The resonance phenomenon occurs for the T -periodic function f given by (5.1) if, and only if, there exist two sequences of (strictly) increasing natural numbers

(k i n ) n≥1 , i = 1, 2, such that |λ k 1 n -z k 2 n | |z k 2 n | → 0 as n → ∞.
(5.11)

Proof. First, we suppose that (5.11) holds and we prove (5.6)-(5.7). From (5.11) we deduce that lim n→∞ |λ k 1 n -z k 2 n | = 0, which together with (5.8) gives (5.6). In order to show (5.7), let g(z) = z sin z -cos z. We have that there exists ξ n between λ k 1 n and

z k 2 n such that |g(λ k 1 n )| = |g(λ k 1 n ) -g(z k 2 n )| ≤ |g (ξ n ) λ k 1 n -z k 2 n | ≤ 2|ξ n | |λ k 1 n -z k 2 n
| from which we immediately deduce (5.7). Now suppose that (5.6)-(5.7) holds, which implies that g(λ k 1 n ) → 0 as n → ∞. Let ε > 0 small and k sufficiently large. In each interval

I k = ((2k -1) π 2 + ε, (2k + 1) π 2 -ε) the function g is invertible since g (z) = z cos z + 2 sin(z) = 0.
Then, in each interval I k there is exactly one number z k for which g(z k ) = 0. Let k 2 n the index of the interval I k for which λ k 1 n ∈ I k 2 n . We have that there exists ζ n between g(λ k 1 n ) and g(z (5.11) holds. The following characterization is a direct consequence of Proposition 5.3.

k 2 n ) such that |λ k 1 n -z k 2 n | = |g -1 (g(λ k 1 n )) -g -1 (g(z k 2 n ))| ≤ |(g -1 ) (ζ n )| |g(λ k 1 n ) -g(z k 2 n )| = |(g -1 ) (ζ n )| |g(λ k 1 n )| = 1 g (t n ) |g(λ k 1 n )| = |g(λ k 1 n )| |t n cos t n + 2 sin t n | ≤ |g(λ k 1 n )| c |t n | for some constant c > 0, where t n = g -1 (ζ n ) belongs to I k 2 n and tends to z k 2 n as n goes to infinity. Consequently, |λ k 1 n -z k 2 n z k 2 n | ≤ c |g(λ k 1 n )| and

Property 5.4. The resonance phenomenon occurs for the T -periodic function f given by (5.1) if, and only if, there exist two (strictly) increasing sequences of natural numbers

(k i n ) n≥1 , i = 1, 2 such that k 1 n k 2 n - T 2 (k 2 n ) 2 → T 2π 2
as n → ∞.

(5.12)

Proof. Firstly, observe that |λ k 1 n -z k 2 n | |z k 2 n | → 0 as n tends to infinity if and only if 2k 1 n π T -k 2 n π - 1 k 2 n π + o( 1 k 2 n ) k 2 n π + 1 k 2 n π + o( 1 k 2 n ) → 0 as n → ∞.
It follows that (5.11) is equivalent to

k 1 n k 2 n - T 2 - T 2(k 2 n ) 2 π 2 |k 2 n | 2 → 0 as n → ∞
and the proof ends.

Remark 5.5. Proposition 5. [START_REF] Castro | Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass[END_REF] shows the conditions that the period T has to fulfill in order to have or not a periodic solution. It is not easy to say if there are or not periods T which verifies (5.12). For instance, if T is a rational number (5.12) cannot hold. Indeed, if we suppose that T = p q , with p, q ∈ N * , from (5.12) we deduce that

lim n→∞ 2qk 1 n -pk 2 n = 0 and, consequently, T = 2k 1 n k 2 n
for any n sufficiently large. However, this contradicts (5.12). The existence of irrational numbers T that verifies (5.12) and produces the resonance phenomenon remains an open question.

Numerical approximation

In this section we approximate numerically the periodic solutions of the hybrid system (1.1), using the perturbation argument presented in the theoretical part. More precisely, for a given small value of ε > 0, we use an iterative algorithm to approximate the fixed point of the contractive map Λ introduced in Theorem 3.4. Once we have obtained this fixed point with a given precision, we can compute the periodic solution of the perturbed system (3.1). The proof of Theorem 4.2 shows that taking smaller and smaller values of ε > 0 the periodic solutions of the perturbed system converge weakly to the periodic solution of the initial hybrid system, if f |(0,T ) ∈ H 1 (0, T ).

In order to preserve the precision of the numerical scheme for (1.1), the parameter ε should be small enough. On the other hand, the contractive property of the map Λ gets better when ε increases. This analysis allows us to conclude that the optimal value of ε is of order of the precision of the numerical scheme (see the conclusions from Section 6.2).

First of all, we need to approximate the solution of system (3.1). We choose to semidiscretize in space using P 1 finite elements, paying attention to the special boundary condition in x = 1. More precisely, we denote by N ∈ N * the number of discretization points in (0, 1) and by h = 1 N +1 the discretization step. The vector u h (t) = u 1 (t), • • • , u N +1 (t) * will be the solution of the following differential system

M u h,tt (t) + Ku h (t) + Lu h,t (t) = F h (t) t > 0 u h (0) = u 0h , u h,t (0) = u 1h , ( 6.1)
where

M = h 6        4 1 0 • • • 0 1 4 1 • • • 0 . . . . . . . . . . . . . . . 0 • • • 1 4 1 0 0 • • • 0 6 h        , K = 1 h        2 -1 0 • • • 0 -1 2 -1 • • • 0 . . . . . . . . . . . . . . . 0 • • • -1 2 -1 0 0 • • • -1 1        , L = hε 6        4 1 0 • • • 0 1 4 1 • • • 0 . . . . . . . . . . . . . . . 0 • • • 1 4 1 0 0 • • • 0 6α hε        , F h (t) =        0 0 . . . 0 f (t)        .
In the sequel U * denotes the transposed vector.

Denoting U h = u h u h,t
, we write (6.1) as a first order system

U h,t (t) = A h U h (t) + F h (t), t > 0 U h (0) = U 0h , ( 6.2)
where

A = 0 I -M -1 K -M -1 L , F h (t) = 0 F h (t) , U 0h = u 0h u 1h .
To discretize in time (6.2) we use the following implicit mid-point scheme

   U k+1 h -U k h ∆t = A h U k h + U k+1 h 2 + F k h + F k+1 h 2 , k ∈ N * U 0 h = U 0h , (6.3)
where ∆t > 0 is the time discretization step and U k h and F k h approximate U h (k∆t) and F h (k∆t), respectively. We define the energy associated to (6.3) by the discrete equivalent of (2.3)

E h (U k h ) = 1 2 (U k h ) * K e 0 0 M e U k h + U k h, 2N +2 2 , ( 6.4)
where M e , K e ∈ M N +1 are defined by

M e = h 6        4 1 0 • • • 0 1 4 1 • • • 0 . . . . . . . . . . . . . . . 0 • • • 1 4 1 0 0 • • • 1 4        , K e = 1 h        2 -1 0 • • • 0 -1 2 -1 • • • 0 . . . . . . . . . . . . . . . 0 • • • -1 2 -1 0 0 • • • -1 2        .
Once we have a solver for (3.4), the second step is to compute the fixed point of the application Λ introduced in Theorem 3.4, for a given T -periodic function f . Remark that Λ is nothing else than the application which associates U (T ) to U 0 , where U is the solution of (3.4). Assuming that there exists l ∈ N * such that T = l∆t we define Λ h the application which associates U l h to U 0h , where U k h verifies (6.3). To compute then an approximation of the fixed point of Λ we use the following algorithm 1. Let U 0h be arbitrarily chosen.

2. Let e > 0 be a given tolerance and n = 1.

Repeat

• V n h = Λ n h U 0h . • n = n + 1. until E h (V n h -V n-1 h ) < e.
4. The approximation of the fixed point is V n h .

Numerical experiments

We shall present three different choices of periodic functions f and the corresponding numerical results. In all the computations the parameter α is set to 1 and the tolerance e in the fixed point algorithm is 10 -5 .

Case of an explicit periodic solution

It is easy to verify that considering the 2-periodic function

f (t) = -π cos(πt) for t ∈ [0, 10], (6.5)
the corresponding periodic solution of (1.1) is given by u(x, t) = cos(πt) sin(πx). We take the number of discretization points N ∈ {50, 100, 200} and ∆t = h = 1 N +1 . Table 1 displays the values of max{E h (u h,ε (t) -u(t)) : t ∈ [0, 10]} for different choices of h and ε. In Figure 1 we show the decay of the error in the fixed point algorithm for different values of ε and N = 100.

Value of h ε

= 1 ε = 0.1 ε = 0.01 ε = 0.001 ε = 0.0001 ε = 0 0.02 3
f (t) = 1 - 2 T t - T 2 , for t ∈ [0, T ]. (6.6)
By periodicity, we extend f to the interval [0, 10]. We take N = 100 and ∆t = h = 1 N +1 . The corresponding periodic solution for ε = 0 is displayed in Figure 2. In Figure 3 we show the norm of the error in the fixed point algorithm (a) and the energy of the solution of (6.3) with the initial data being the fixed point of Λ h , (b) for different values of ε.

f (t) = 1, if x ∈ (T /3, 2T /3) 0, if x ∈ (0, T /3) ∪ (2T /3, T ). (6.7)
By periodicity, we extend f to the interval [0, 10]. As in the case f ∈ H 1 (0, T ) we take N = 100 and ∆t = h = 1 N +1 . A very similar behavior of the solutions as in the case f ∈ H 1 can be observed in Figure 4 and Figure 5. In Table 2 we summarize the values of the energy of the computed periodic solution of (6.2) at the time t = 1 for different periodic functions. We remark that the values of the energy for ε = 1 and ε = 0.1 are clearly different from the values obtained for smaller ε.

Conclusion and comments

First of all, we should remark from Table 1 that the solutions of the perturbed systems converge to the exact solution when ε → 0. The results displayed in Table 2 confirm this fact, the difference between the solutions decreasing as ε tends to zero.

Analyzing Figure 1, Figure 3(a) and Figure 5(a) we remark that the number of iterations necessary for the fixed point algorithm increases when ε decreases with different velocities depending on f . This is a consequence of the fact that the application Λ h has better contractive properties when ε is larger. At the numerical level, this is related to Remark 3.3 for the continuous problem. Note that taking ∆t = h, the order of convergence of the approximation (b) Energy of the solution of (6.3) with the initial data being the fixed point of Λ h . scheme using P 1 finite elements for the wave equation is O(h) (see, for instance, [19, p. 213] and [16, p. 96-97]). Coupling the wave equation with the rigid body at one end, and still using a P 1 finite elements approximation of this hybrid system, we maintain the same order of convergence as for the wave equation with homogeneous Dirichlet boundary condition. Therefore, the perturbation εu t added in system (3.1) does not affect the precision of the numerical approximation if ε is at most of the same order as h. Since large values of ε ensure better convergence properties of the fixed point algorithm, we deduce that the optimal choice is ε of order h.

In conclusion, the numerical simulations above are coherent with our theoretical results concerning the existence of periodic solutions for the hybrid system (1.1) if the T -periodic source term f is in H 1 (0, T ). For all T -periodic function f |(0,T ) ∈ L 2 (0, T ) that we have tested, a periodic solution has been numerically found.

Figure 1 :

 1 Figure 1: Decay of the error in the fixed point algorithm for different values of ε, N = 100 and f given by(6.5)

Figure 2 :

 2 Figure 2: Periodic solution of (6.3) for ε = 0 and f given by (6.6).

Figure 3 :

 3 Figure 3: The function f is given by (6.6). (a) Norm of the error in the fixed point algorithm. (b) Energy of the solution of (6.3) with the initial data being the fixed point of Λ h .

Figure 4 :

 4 Figure 4: Periodic solution of (6.3) for ε = 0 and f given by (6.7).

Function ε = 1 εTable 2 :

 12 = 0.1 ε = 0.01 ε = 0.001 ε = 0 f given by (6.6) 10.1315 10.0268 10.0679 10.0671 10.0674 f given by (6.7) 4Values of the energy of the periodic solution at the time t = 1 and different values of ε.

Figure 5 :

 5 Figure 5: Function f is given by (6.7). (a) Norm of the error in the fixed point algorithm. (b) Energy of the solution of (6.3) with the initial data being the fixed point of Λ h .

  )U 0 and use two times(3.5).

	Remark 3.3. Since Theorem 2.2 shows that (3.6) does not hold, we have that

Table 1 :

 1 Values of max{E

		.2384 0.1043	0.0012	0.0021	0.0023	0.0023
	0.01	4.7285 0.1686	0.0015	0.0005	0.0006	0.0006
	0.005	7.6631 0.2859	0.0033	0.0004	0.0005	0.0004

h (u h,ε (t) -u(t)) : t ∈ [0, 10]} for different choices of h and ε.

Acknowledgments: The second author was partially supported by Grants MTM2008-03541 founded by MICINN (Spain) and 420/2008 of CNCSIS (Romania). The third author was partially supported by CNPq (Brazil). Part of this work was done when S. Micu was visiting Universidade Federal do Rio de Janeiro (UFRJ) with the support of this institution and PRONEX E-26/110560/2010-APQ1. He would like to thanks the members of the department for their kindness and warm hospitality.


Q Beaufils

G Tackmann

X Wang

B Pelle

S Pelisson

P Wolf

F Pereira

Dos Santos

Laser controlled tunneling in a vertical optical lattice

Keywords: numbers: 32.80.Qk, 37.10.Jk, 05.60.Gg, 37.25.+k

Raman laser pulses are used to induce coherent tunnelling between neighbouring sites of a vertical 1D optical lattice. Such tunneling occurs when the detuning of a probe laser from the atomic transition frequency matches multiples of the Bloch frequency, allowing for a spectroscopic control of the coupling between Wannier Stark (WS) states. In particular, we prepare coherent superpositions of WS states of adjacent sites, and investigate the coherence time of these superpositions by realizing a spatial interferometer. This scheme provides a powerful tool for coherent manipulation of external degrees of freedom of cold atoms, which is a key issue for quantum information processing.

Trapping and manipulating cold neutral atoms in an optical lattice offers high controllability and robust quantum coherence properties, which makes it an attractive system for many applications such as quantum simulation of solid state systems [1], metrology [2,3], and quantum information processing (QIP) [4]. One key issue in this context is the possibility to coherently control the atoms internal and external degrees of freedom. Combined with the possibility to address single sites [5], this allows for the realization of quantum logic operations [6].

Atom transport control in an optical lattice has been previously reported using microwave fields [7], frequency, phase and amplitude modulation techniques [8][9][10], or an adiabatic change of the trapping potential [11,12]. In this work, we demonstrate coherent laser induced tunneling of cold atoms between neighboring sites of an optical lattice. In contrast with most previous approaches, our technique doesn't require any modification of the trapping potential. It allows an unprecedent control of the atom's external degrees of freedom (displacing the atoms by 1 to 9 lattice periods in this work) in a system showing good coherence properties (up to 1 s).

Our system consists in laser-cooled 87 Rb atoms in the first band of a vertical one-dimensional optical lattice. Due to earth gravity, the ground energy levels of the lattice are shifted out of resonance. For a sufficiently large lattice depth U l , tunneling is highly reduced, leading to a ladder of localized Wannier-Stark (WS) eigenstates separated by the Bloch frequency ν B = m a gλ l /2h. Here, m a is the atomic mass, g is the gravity acceleration, λ l /2 is the distance between two adjacent lattice sites, and h is the Planck constant. The WS states |W m are indexed by the discrete quantum number m characterizing the well containing the center of the wave function x |W m .

We use counterpropagating Raman beams to drive coherent transitions between the ground and excited hyperfine levels |g = 5 2 S 1/2 , F = 1, m F = 0 and |e = 5 2 S 1/2 , F = 2, m F = 0 . Such a transition implies a momentum transfer of k ef f = k 1 + k 2 ≈ 4π/(780 nm) that couples the WS states either in the same well or in neighboring wells, with a coupling strength proportional to W m | e ik ef f x |W m±∆m . Fourier-limited widths of the resonances over excitation times larger than the Bloch period allows resolved intersite transitions |g, m → |e, m ± ∆m , at Raman frequencies :

ν R = ν HF S ± ∆m × ν B
where ν HF S is the hyperfine splitting and ∆m is the number of lattice wells separating the two coupled WS states. The energy spectrum of our system is schematically illustrated in Fig. 1.

Coupling between neighboring wells can be efficiently tuned using the lattice depth when k l is close to k ef f , where k l is the optical lattice wave vector [START_REF] Pereira | Proceedings of the 7 th symposium Frequency Standards and Metrology[END_REF]. We therefore use a mixed trap configuration with a blue detuned lattice generated by a single mode frequency doubled Nd:YVO 4 laser (λ l = 532 nm, beam waist 600 µm) that provides only vertical longitudinal confinement, superposed with a red detuned (λ = 1064 nm, beam waist 200 µm ) Yb fiber laser providing transverse confinement (see Fig. 2). To load this dipole trap, we first accumulate up to 10 7 atoms in a 3D-Magneto-Optical trap (MOT) fed by a 2D-MOT. The cloud is then cooled down to 2 µK by a far detuned molasses, at the end of which we switch off the lasers to let the untrapped atoms fall. At our low lattice depth (U l ≃ 4E R (where E R = (hk l ) 2 /(2m a ) is the lattice recoil energy), only the first band has a non-negligible lifetime and is populated with about 10 5 atoms vertically distributed along 10 4 sites (the second band is centered at 5E R already above the lattice depth). The atoms accumulated in all the Zeeman sublevels of 5 2 S 1/2 , F = 2 are depumped to 5 2 S 1/2 , F = 1 and then optically pumped (95% efficiency) on the 5 2 S 1/2 , F = 1 → 5 2 P 3/2 , F = 0 sition to the 5 2 S 1/2 , F = 1, m F = 0 Zeeman sublevel, which is sensitive to stray magnetic fields only to second order. The remaining 5% unpolarized atoms can easily be removed from the trap with a pushing beam. Our fluorescence detection scheme, based on a time of flight measurement similar to the one used in atomic clocks and inertial sensors, allows us to measure the atomic populations in the two hyperfine states after releasing the atoms from the trap [START_REF] Le Gouët | [END_REF]. The Raman transitions are driven by two counterpropagating beams at 780 nm circularly polarized, detuned from the atomic transition by about 3 GHz, and aligned along the direction of the optical trap beams. The beams are collimated with a 1/e 2 radius of 1 cm, ensuring a good intensity homogeneity along the transverse size of the trap (about 200 µm radius). Fig. 3 shows two typical Raman spectra of the transition probability as a function of the Raman frequency ν R , taken for two different lattice depths. Transitions between the two hyperfine levels at Raman frequencies equal to the hyperfine splitting plus or minus an integer number ∆m of Bloch frequencies (ν B ≈ 569 Hz in our system) are the signature that the atoms actually tunneled across ∆m lattice sites. For those scans, the intensities in the Raman laser beams were 0.25 and 0.54 mW/cm 2 . The resulting Rabi frequencies Ω ∆m , different for each transition, are always smaller than the Bloch frequency, so that each peak is well resolved. The ratio between the Raman intensities was chosen to cancel the differential light shift of the hyperfine transition induced by them [15]. The Rabi frequency for each transition ∆m is written [16]:

Ω ∆m = Ω U l =0 W m | e -ik ef f x |W m±∆m (1)
where Ω U l =0 is the Rabi frequency in free space. Due to the translational symmetry of the WS states, Ω ∆m doesn't depend on the initial well index m but only on the absolute value of ∆m [16]. It also depends on the lattice wavelength λ l and depth U l , which is an important feature of this experiment, as it induces a spatial inhomogeneity on the Rabi frequency seen by the trapped atoms via the transverse inhomogeneity of the lattice depth in the trap. The damping induced on the Rabi oscillations by this inhomogeneity is the main limitation on the trans- fer efficiency of the Raman transitions. We calculated Ω ∆m for the parameters of our system, as a function of U l and for various values of ∆m. The result is shown in Fig. 4 for ∆m = ±1, ∆m = ±2 and ∆m = ±3. To limit the lattice depth inhomogeneity due to the transverse extension of the atomic cloud, the 1/e 2 waist of the beam providing transverse confinement is smaller than the one of the lattice beam. Moreover, we can choose to tune the lattice depth at a value where, for a transition of interest, the variation of the coupling with lattice depth ∆Ω(U l )/∆U l is small, as illustrated by the inset of Fig. 4. On this graph, we compare the shape of Rabi oscillations at resonance for the ∆m = -3 transition and for two different lattice depths. Besides a difference in the period of the Rabi oscillations, we observe that the best contrast is obtained for U l ≃ 4E R , where the coupling inhomogeneity is lower, allowing us to reach a transfer efficiency of about 80 %. The lattice depth is estimated by measuring the Rabi frequencies for different transitions and comparing them to the calculation.

We investigated the question of the coherence time of the trapped states. As a diagnosis tool, we performed Ramsey spectroscopy on the |g, m → |e, m + 3 transition. As there is no initial atomic coherence from one site to it's neighbors, the atoms distributed in many lattice sites can be treated as independent interferometers. The phase is read out by the measurement of the internal atomic state population. Our interferometer consists in two Raman π/2 pulses of frequency ν R scanned close to ν HF S +3ν B , separated by a time interval T . The intensity of the Raman lasers was chosen for the Rabi frequency Ω ∆m=3 /2π to be much smaller than ν B , in order to ensure a good separation with neighboring transitions. The duration of the π/2 pulses is then τ π/2 = 5.5 ms. Fig. 5 shows as open circles the evolution of the contrast as we increase the interrogation time T . The contrast at short T is about 65%, which is coherent with the 80% efficiency of each of the two Raman π/2 pulses. When increasing T , the contrast decreases rapidly, which is due to the transverse inhomogeneity of the differential light shift induced on the hyperfine transition by the Gaussian profile of the IR laser used for transverse trapping [17]. This position dependent frequency shift induces an inhomogeneous broadening along the transverse direction, which is the main effect limiting the coherence lifetime in the lattice trap.

Many schemes have been proposed and demonstrated to cancel this source of inhomogeneous dephasing in an optical trap [18][19][20]. One particularly efficient method in our case is to add a low power laser beam, mode matched with the transverse trapping beam, and whose frequency is tuned between the two hypefine levels, as reported in [21]. This beam compensates the differential ligthshift induced by the transverse trapping light with a laser power of only a few tens of nW, so that decoherence due to photon scattering is negligible at the experiment's time scale. In practice, we use as a compensating beam a fraction of one of the two Raman beams, with an additional detuning of 80 MHz in order to prevent undesired Raman transitions. This beam is superposed with the IR laser (see Fig. 2), and its size, position and power are adjusted to optimize the contrast of the interferometer for long interrogation times of several hundreds of milliseconds, for which otherwise the contrast is zero. In our case, for an IR power of about 2 W, the differential light shift is compensated with a power of 12 nW. Fig. 5 displays as black squares the evolution of the contrast vs T with the compensating beam, and clearly shows the improvement of the lifetime of the coherence.

Finally, as a preliminary investigation, we evaluated the frequency sensitivity of the interferometer. This is motivated by the possibility to use this trapped atomic spatial interferometer for metrology applications such as gravimetry or short range forces measurement [22]. We locked the Raman lasers frequency difference to the center fringe of the interferometer, with a computer controlled servo integrator. Performing the measurement alternatively on the left and right transitions (∆m = ±3) and calculating the difference of the measured frequencies allows us to measure 6ν B while cancelling the frequency shifts of the hyperfine clock frequency, due to, for example, the quadratic zeeman effect and the differential lightshifts induced by the trapping lasers. For an interrogation time of T = 400 ms and a cycle time of T c = 1.4 s, the Allan standard deviation of the frequency difference decreases as 0.1 Hz.τ -1/2 with τ the integration time in seconds. This corresponds to a statistical uncertainty on the measurement of the Bloch frequency of 6 × 10 -5 in relative value after 1 s integration.

Our Wannier-Stark interferometer shows great potential for metrology applications. As an example, we plan to perform it close to the reflecting surface of the lattice, which would allow the measurement of short range forces (Casimir-Polder, short range modifications of gravity) [START_REF] Pereira | Proceedings of the 7 th symposium Frequency Standards and Metrology[END_REF]. The statistical uncertainty in the measurement of the Casimir-Polder potential, assuming the performance demonstrated here, would reach 1% for a distance of 5 µm and a measurement time of 1000 s.

The technique for controlled and coherent transport of atoms demonstrated in this work is unique in terms of versatility. The high resolution reachable by the Raman transitions (up to 1 Hz) suggests the possibility of selectively addressing one single lattice site, using for example the lightshift induced by a focused laser to lift the degeneracy between the transitions. Although demonstrated here for a thermal cloud, this technique is perfectly suitable for degenerate quantum gases. Besides, it also works with one photon transitions, as recently highlighted in [23]. All these features make this tool a potential candidate for the realization of quantum logic operations.

This research is carried on within the project iSense, which acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number: 250072. We also gratefully acknowledge support by Ville de Paris (Emergence(s) program) and IFRAF. G.T. thanks the Intercan network and the UFA-DFH for financial support.

Helpful discussions with A. Landragin, P. Lemonde, A. Clairon, S. Bize, M-C. Angonin and R. are greatfully acknowleged.

FIG. 1 :

 1 FIG. 1: Atoms in the first band of the lattice form a Wannier-Stark ladder of eigenstates. The Raman probe laser couples the ground to the excited hyperfine level in the different WS states separated by the Bloch frenquency.

FIG. 2 :

 2 FIG.2: Experimental setup for the optical trapping and Raman intersite transitions. The different beams are superposed using dichroic mirrors. The Raman beams are also superposed and one of them is retro-reflected to allow counterpropagating transitions.

FIG. 3 :

 3 FIG.3: Raman spectra for two different lattice depths, showing evidence of transitions between up 9 neighboring lattice sites, each having a different Rabi frequency according to equation 1. The excitation time is 10 ms, which is smaller than the duration of a π pulse for each transition. The peaks are separated by the Bloch frequency of our system νB ≈ 569 Hz.

FIG. 4 :

 4 FIG. 4: Calculation of the normalized Rabi frequencies for ∆m = ±1, ∆m = ±2 and ∆m = ±3 transitions, as a function of the lattice depth. Inset : Experimental Rabi oscillations on the transition ∆m = -3, for U l = 4ER and U l = 3ER.

FIG. 5 :

 5 FIG. 5: Contrast of the interferometer versus Ramsey interrogation time T . Open circles (resp. black squares) display the contrast without (resp. with) the light shift compensation beam.

* franck.pereira@obspm.fr


Tuija Tapaninen

Pertti J Neuvonen

MD Mikko Niemi
email: mikko.niemi@helsinki.fi

Rifampicin reduces the plasma concentrations and the renin-inhibiting effect of aliskiren

Keywords: Pharmacokinetics, Drug interaction, Aliskiren, Rifampicin, Multidrug resistance transporter 1 P-glycoprotein

come

Rifampicin reduces the plasma concentrations and the renin-inhibiting effect of aliskiren

INTRODUCTION

Aliskiren is a renin inhibitor, which belongs to a class of antihypertensive drugs acting on the renin-angiotensin-aldosterone system [1][2][3]. It has a low oral bioavailability of about 2-3%, probably mainly because of poor absorption [3][4][5]. The peak plasma aliskiren concentration (C max ) is reached within 1 to 3 h. Aliskiren is eliminated primarily in the unchanged form by biliary excretion into the feces and, to a lesser extent, by renal excretion of unchanged aliskiren (about 0.4% of oral dose) and by oxidative biotransformation (about 1.4% of oral dose), mainly via CYP3A4 [5]. The elimination half-life (t ½ ) of aliskiren is about 30-40 h.

Aliskiren is a substrate of the multidrug resistance transporter 1 (MDR1) Pglycoprotein, but not of breast cancer resistance protein or multidrug resistanceassociated protein 2 [6]. P-glycoprotein efflux transporter is expressed on the apical membrane of small intestinal enterocytes, hepatocytes, and proximal tubule cells and can reduce the intestinal absorption and enhance the elimination of its substrates [START_REF] Fromm | Importance of P-glycoprotein at blood-tissue barriers[END_REF]. In addition, P-glycoprotein is expressed at blood-tissue barriers, such as blood-brain barrier, protecting tissues from potentially toxic xenobiotics [START_REF] Fromm | Importance of P-glycoprotein at blood-tissue barriers[END_REF]. The hepatic uptake of aliskiren is thought to be mediated by organic anion transporting polypeptide 2B1 (OATP2B1) [6]. Ketoconazole (an inhibitor of both CYP3A4 and P-glycoprotein [START_REF] Jurima-Romet | Terfenadine metabolism in human liver. In vitro inhibition by macrolide antibiotics and azole antifungals[END_REF][START_REF] Olkkola | Midazolam should be avoided in patients receiving the systemic antimycotics ketoconazole or itraconazole[END_REF][START_REF] Wang | Interaction of common azole antifungals with P glycoprotein[END_REF][START_REF] Marzolini | Polymorphisms in human MDR1 (P-glycoprotein): recent advances and clinical relevance[END_REF]) and cyclosporine (a potent inhibitor of CYP3A4, P-glycoprotein, and OATP2B1 [12][13][14]) have raised the area under the plasma aliskiren concentration-time curve (AUC) 1.8-fold and 5-fold, respectively [6,15]. Rifampicin is an inducer of several drug-metabolizing enzymes (including CYP3A4) and of some drug transporters (such as P-glycoprotein), and can markedly reduce the plasma concentrations and efficacy of several CYP3A4 and P-glycoprotein substrates [16][17][18][19][20][21][22]. However, there are no published studies on the interactions of aliskiren with inducers of CYP3A4 and P-glycoprotein. Our aim was to study the possible effects of rifampicin on the pharmacokinetics and pharmacodynamics of aliskiren. The hypothesis was that rifampicin may reduce the plasma concentrations and the renin-inhibiting effect of aliskiren.

METHODS

Subjects

Twelve healthy white Finnish volunteers (7 women, 5 men) participated in the study after giving written informed consent. Their mean ± SD age was 21 ± 2 years (range, 20-25 years), mean height 174 ± 10 cm (range, 156-187 cm), and mean weight 66 ± 11 kg (range, 50-89 kg). Participants were ascertained to be healthy by medical history, physical examination, and routine laboratory tests. Subjects with a systolic blood pressure less than 110 mm Hg were not included in the study. None of the participants was on any continuous medication, including oral contraceptives, and none was a tobacco smoker.

Study design

The study protocol was approved by the Coordinating Ethics Committee of the Helsinki and Uusimaa Hospital District, and the National Agency for Medicines in Finland. In a randomized two-phase crossover study with a wash-out period of four weeks, the volunteers took 600 mg rifampicin (Rimapen; Orion Pharma, Espoo, Finland) or placebo once daily at 20.00 h for five days. On day six, after an overnight fast, a single oral dose of 150 mg aliskiren (Rasilez; Novartis, Horsham, Great Britain) was administered with 150 ml of water at 08.00 h. A standardized warm meal was served 4 h after the administration of aliskiren, and a standardized light meal after 7 h and 10 h.

The participants were under direct medical supervision for 12 h after aliskiren ingestion.

Fluids for intravenous infusion were available for immediate use in case of hypotension, but were not needed. Use of other drugs was prohibited for one week before and after,

Blood sampling and pharmacodynamic measurements

On the days of the administration of aliskiren, a forearm vein of each subject was cannulated with a plastic cannula and kept patent with an obturator. Timed blood samples for drug concentration measurements (5 or 10 ml each) were drawn into tubes containing ethylenediaminetetraacetic acid (EDTA) prior to and 0.5, 1, 2, 3, 4, 5, 7, 9, 12, 24, 34, 48, and 72 h after aliskiren administration. Blood samples for the determination of plasma renin activity (5 ml each) were drawn before the administration of aliskiren and 4 h and 24 h thereafter into chilled EDTA tubes, which were placed on ice immediately after sampling. Plasma was separated within 30 min. Urine was collected up to 12 h after aliskiren ingestion. Urine aliquots and plasma were stored at -70ºC until analysis. Plasma renin activity was measured using a commercially available radioimmunoassay method (RENCTK; DiaSorin, Saluggia, Italy). Systolic and diastolic blood pressures, and heart rate were measured twice (mean value was used in the calculations) from the forearm with an automatic oscillometric blood pressure monitor (Omron M5-I; Omron Healthcare Europe BV, Hoofddorp, The Netherlands), with the subject in a sitting position, prior to and 2, 4, 7, 9, 12, and 24 h after the administration of aliskiren. (LC/MS/MS) system (Sciex Division of MDS Inc, Toronto, Ontario, Canada) [5].

Determination of aliskiren concentrations

Acebutolol served as an internal standard. The limit of quantification of plasma aliskiren was 0.25 ng/ml, and the intra-day coefficient of variation (CV) was 1.2% at 2 ng/ml, 1.0% at 20 ng/ml, and 1.2% at 200 ng/ml (n=6). The calibration curve for plasma aliskiren was linear over the range 0.25-500 ng/ml (r>0.999, weighting 1/x). The limit of quantification of aliskiren in urine was 9 ng/ml, and the intra-day CV was 4.8% at 9 ng/ml, 4.5% at 120 ng/ml, and 3.4% at 1200 ng/ml (n=6). The calibration curve for urine aliskiren was linear over the range 9-1800 ng/ml (r>0.999, weighting 1/x).

Pharmacokinetics

The pharmacokinetics of aliskiren were characterized by C max , time to C max (t max ), t ½ , AUC 0-72 , AUC 0-∞ , the amount of aliskiren excreted into urine from 0 to 12 h (Ae), and the renal clearance (Cl renal ). Pharmacokinetic parameters were calculated with non- The pharmacodynamics of aliskiren were characterized by plasma renin activity at 4 h and 24 h after aliskiren intake, as well as average systolic and diastolic blood pressures, and heart rate. Average values of systolic and diastolic blood pressures and heart rate were calculated by dividing the area under the effect-time curve from 0 to 24 h by 24 h.

Statistical analysis

The results are expressed as geometric means (95% confidence interval), unless otherwise indicated. The C max , AUC, Ae, and Cl renal data were logarithmically transformed before statistical analysis. Statistical comparisons of the pharmacokinetic (other than t max ) and pharmacodynamic variables during the placebo and rifampicin phases were carried out using repeated-measures analysis of variance with treatment phase as a within-subjects factor and treatment sequence as a between-subjects factor.

The t max data were compared using the Wilcoxon signed rank test. The frequencies of individuals with a double-peak in aliskiren plasma concentration curve were compared between the phases using the McNemar test. Possible correlations between the pharmacokinetic variables of aliskiren and subject body weight were analyzed with the Pearson correlation coefficient. The data were analyzed using the statistical program SPSS 16.0 for Windows (SPSS, Chicago, IL, USA). Differences were considered statistically significant when P was <0.05. The number of subjects was estimated to be sufficient to detect a 40% difference in the AUC 0-∞ of aliskiren between the placebo and rifampicin phases, with a power of 80% (α-level 5%) [4].

RESULTS

Effect of rifampicin on aliskiren pharmacokinetics

Rifampicin markedly reduced the plasma concentrations of aliskiren (Fig. 1, Table 1).

The C max of aliskiren was reduced by 39% (P=0.017) and its AUC 0-∞ by 56% (P<0.001) by rifampicin. The Ae was decreased by 48% by rifampicin (P=0.001). Rifampicin had no significant effect on the t max , t ½ , or Cl renal of aliskiren. The extent of the interaction showed marked interindividual variability: decrease in aliskiren AUC 0-∞ ranged from 20% to 76% and the effect of rifampicin on the aliskiren C max ranged from a 78% decrease to a 9% increase (Fig. 2). Plasma aliskiren concentration showed a doublepeak in 11 out of 12 subjects during the placebo phase and 6 out of 12 subjects during the rifampicin phase (P=0.063). None of the pharmacokinetic variables of aliskiren correlated with subject body weight, during either the placebo or rifampicin phase (|r|<0.38, P>0.23).

Effect of rifampicin on aliskiren pharmacodynamics

Plasma renin activity 24 h after aliskiren intake was 61% higher during the rifampicin phase than during the placebo phase (P=0.008) (Fig. 3, Table 2). No significant difference existed in the systolic or diastolic blood pressure, or the heart rate between the rifampicin and placebo phases.

DISCUSSION

This study demonstrates that rifampicin markedly reduces the plasma concentrations of aliskiren. The C max and AUC 0-∞ of aliskiren were reduced by 39% and 56%, respectively. Marked interindividual variability was evident in the extent of the interaction. Plasma renin activity 24 h after aliskiren intake was 61% higher during the rifampicin phase than during the placebo phase. The interaction between rifampicin and aliskiren may result in a need to adjust aliskiren dose.

Aliskiren, a drug with a low oral bioavailability, is a substrate of P-glycoprotein efflux transporter and is slightly metabolized by CYP3A4 [3,5,6]. In the present study, rifampicin reduced the C max and AUC 0-∞ of aliskiren without affecting the t ½ or the Cl renal , indicating that rifampicin increased mainly the first-pass elimination of aliskiren with no or minimal effect on its systemic clearance. Rifampicin has a strong inducing effect on intestinal and hepatic CYP3A4 with a smaller, though significant, effect on Pglycoprotein [16][17][18][19][20][21][22]. It is noteworthy that intestinal CYP3A4 and P-glycoprotein act synergistically to reduce the oral bioavailability of their joint substrates [23].

Because aliskiren has a low hepatic extraction ratio (0.10) [24], induction of its hepatic elimination unlikely explains this interaction observed during the first-pass phase.

Moreover, significant change in the hepatic extraction of such a drug with a low hepatic extraction should result in a shortened elimination t ½ . In addition, plasma aliskiren concentration data showed double-peaks characteristic of entero-hepatic circulation, particularly during the placebo phase. This phenomenon appeared to be diminished during the rifampicin phase, consistent with reduced intestinal reabsorption of aliskiren due to induction of intestinal P-glycoprotein and CYP3A4. Taken together, the interaction between rifampicin and aliskiren is most likely caused by induction of Pglycoprotein-mediated efflux and CYP3A4-catalyzed biotransformation of aliskiren in the gut wall [16][17][18][19][20][21][22].

In this study, the renin-inhibiting effect of aliskiren was attenuated during the rifampicin phase. However, no differences in hemodynamic effects were observed between the rifampicin and placebo phases after a single dose of aliskiren in healthy volunteers. This can be explained by the delayed start of the blood pressure decreasing effect of aliskiren, which gradually reaches its maximum after four weeks treatment [15,25].

Moreover, the pharmacodynamic response to aliskiren in patients with hypertension may be distinct from normotensive healthy individuals [3,15]. On the other hand, the pharmacokinetics of aliskiren is similar in patients with hypertension and healthy individuals [3,15]. According to pharmacokinetic theory, the dose-interval AUC at steady state is equal to the AUC 0-∞ after a single dose [26]. However, the pharmacokinetics of aliskiren is slightly non-linear [3], interfering with the direct extrapolation of the present results to steady-state. Because aliskiren shows a doseresponse relationship in the range 150-300 mg, but no clear effect on blood pressure with lower doses [15, 25], it is possible that rifampicin reduces the efficacy of aliskiren in patients with hypertension. Similarly to rifampicin, a high-fat meal has reduced the AUC of aliskiren by about 70% [3]. To minimize variability due to the food effect, aliskiren is recommended to be taken once daily with a light meal or in a routine pattern with regard to meals [15,27]. In conclusion, rifampicin considerably reduces the plasma concentrations of aliskiren.

Clinicians should be aware of the possibility that rifampicin may reduce the antihypertensive efficacy of aliskiren.

F o r P e e r R e v i e w

A revised manuscript for European Journal of Clinical Pharmacology, January 2010 20

INTRODUCTION

Aliskiren is a renin inhibitor, which belongs to a class of antihypertensive drugs acting on the renin-angiotensin-aldosterone system [1][2][3]. It has a low oral bioavailability of about 2-3%, probably mainly because of poor absorption [3][4][5]. The peak plasma aliskiren concentration (C max ) is reached within 1 to 3 h. Aliskiren is eliminated primarily in the unchanged form by biliary excretion into the feces and, to a lesser extent, by renal excretion of unchanged aliskiren (about 0.4% of oral dose) and by oxidative biotransformation (about 1.4% of oral dose), mainly via CYP3A4 [5]. The elimination half-life (t ½ ) of aliskiren is about 30-40 h.

Aliskiren is a substrate of the multidrug resistance transporter 1 (MDR1) Pglycoprotein, but not of breast cancer resistance protein or multidrug resistanceassociated protein 2 [6]. P-glycoprotein efflux transporter is expressed on the apical membrane of small intestinal enterocytes, hepatocytes, and proximal tubule cells and can reduce the intestinal absorption and enhance the elimination of its substrates [START_REF] Fromm | Importance of P-glycoprotein at blood-tissue barriers[END_REF]. In addition, P-glycoprotein is expressed at blood-tissue barriers, such as blood-brain barrier, protecting tissues from potentially toxic xenobiotics [START_REF] Fromm | Importance of P-glycoprotein at blood-tissue barriers[END_REF]. The hepatic uptake of aliskiren is thought to be mediated by organic anion transporting polypeptide 2B1 (OATP2B1) [6]. Ketoconazole (an inhibitor of both CYP3A4 and P-glycoprotein [START_REF] Jurima-Romet | Terfenadine metabolism in human liver. In vitro inhibition by macrolide antibiotics and azole antifungals[END_REF][START_REF] Olkkola | Midazolam should be avoided in patients receiving the systemic antimycotics ketoconazole or itraconazole[END_REF][START_REF] Wang | Interaction of common azole antifungals with P glycoprotein[END_REF][START_REF] Marzolini | Polymorphisms in human MDR1 (P-glycoprotein): recent advances and clinical relevance[END_REF]) and cyclosporine (a potent inhibitor of CYP3A4, P-glycoprotein, and OATP2B1 Rifampicin is an inducer of several drug-metabolizing enzymes (including CYP3A4) and of some drug transporters (such as P-glycoprotein), and can markedly reduce the plasma concentrations and efficacy of several CYP3A4 and P-glycoprotein substrates [16][17][18][19][20][21][22]. However, there are no published studies on the interactions of aliskiren with inducers of CYP3A4 and P-glycoprotein. Our aim was to study the possible effects of rifampicin on the pharmacokinetics and pharmacodynamics of aliskiren. The hypothesis was that rifampicin may reduce the plasma concentrations and the renin-inhibiting effect of aliskiren.

METHODS

Subjects

Twelve healthy white Finnish volunteers (7 women, 5 men) participated in the study after giving written informed consent. Their mean ± SD age was 21 ± 2 years (range, 20-25 years), mean height 174 ± 10 cm (range, 156-187 cm), and mean weight 66 ± 11 kg (range, 50-89 kg). Participants were ascertained to be healthy by medical history, physical examination, and routine laboratory tests. Subjects with a systolic blood pressure less than 110 mm Hg were not included in the study. None of the participants was on any continuous medication, including oral contraceptives, and none was a tobacco smoker.

Study design

The study protocol was approved by the Coordinating Ethics Committee of the Helsinki and Uusimaa Hospital District, and the National Agency for Medicines in Finland. In a randomized two-phase crossover study with a wash-out period of four weeks, the volunteers took 600 mg rifampicin (Rimapen; Orion Pharma, Espoo, Finland) or placebo once daily at 20.00 h for five days. On day six, after an overnight fast, a single oral dose of 150 mg aliskiren (Rasilez; Novartis, Horsham, Great Britain) was administered with 150 ml of water at 08.00 h. A standardized warm meal was served 4 h after the administration of aliskiren, and a standardized light meal after 7 h and 10 h.

The participants were under direct medical supervision for 12 h after aliskiren ingestion.

Fluids for intravenous infusion were available for immediate use in case of hypotension, but were not needed. Use of other drugs was prohibited for one week before and after,

Blood sampling and pharmacodynamic measurements

On the days of the administration of aliskiren, a forearm vein of each subject was cannulated with a plastic cannula and kept patent with an obturator. Timed blood samples for drug concentration measurements (5 or 10 ml each) were drawn into tubes containing ethylenediaminetetraacetic acid (EDTA) prior to and 0.5, 1, 2, 3, 4, 5, 7, 9, 12, 24, 34, 48, and 72 h after aliskiren administration. Blood samples for the determination of plasma renin activity (5 ml each) were drawn before the administration of aliskiren and 4 h and 24 h thereafter into chilled EDTA tubes, which were placed on ice immediately after sampling. Plasma was separated within 30 min. Urine was collected up to 12 h after aliskiren ingestion. Urine aliquots and plasma were stored at -70ºC until analysis. Plasma renin activity was measured using a commercially available radioimmunoassay method (RENCTK; DiaSorin, Saluggia, Italy). Systolic and diastolic blood pressures, and heart rate were measured twice (mean value was used in the calculations) from the forearm with an automatic oscillometric blood pressure monitor (Omron M5-I; Omron Healthcare Europe BV, Hoofddorp, The Netherlands), with the subject in a sitting position, prior to and 2, 4, 7, 9, 12, and 24 h after the administration of aliskiren. (LC/MS/MS) system (Sciex Division of MDS Inc, Toronto, Ontario, Canada) [5].

Determination of aliskiren concentrations

Acebutolol served as an internal standard. The limit of quantification of plasma aliskiren was 0.25 ng/ml, and the intra-day coefficient of variation (CV) was 1.2% at 2 ng/ml, 1.0% at 20 ng/ml, and 1.2% at 200 ng/ml (n=6). The calibration curve for plasma aliskiren was linear over the range 0.25-500 ng/ml (r>0.999, weighting 1/x). The limit of quantification of aliskiren in urine was 9 ng/ml, and the intra-day CV was 4.8% at 9 ng/ml, 4.5% at 120 ng/ml, and 3.4% at 1200 ng/ml (n=6). The calibration curve for urine aliskiren was linear over the range 9-1800 ng/ml (r>0.999, weighting 1/x).

Pharmacokinetics

The pharmacokinetics of aliskiren were characterized by C max , time to C max (t max ), t ½ , AUC 0-72 , AUC 0-∞ , the amount of aliskiren excreted into urine from 0 to 12 h (Ae), and the renal clearance (Cl renal ). Pharmacokinetic parameters were calculated with non- The pharmacodynamics of aliskiren were characterized by plasma renin activity at 4 h and 24 h after aliskiren intake, as well as average systolic and diastolic blood pressures, and heart rate. Average values of systolic and diastolic blood pressures and heart rate were calculated by dividing the area under the effect-time curve from 0 to 24 h by 24 h.

Statistical analysis

The results are expressed as geometric means (95% confidence interval), unless otherwise indicated. The C max , AUC, Ae, and Cl renal data were logarithmically transformed before statistical analysis. Statistical comparisons of the pharmacokinetic (other than t max ) and pharmacodynamic variables during the placebo and rifampicin phases were carried out using repeated-measures analysis of variance with treatment phase as a within-subjects factor and treatment sequence as a between-subjects factor.

The t max data were compared using the Wilcoxon signed rank test. The frequencies of individuals with a double-peak in aliskiren plasma concentration curve were compared between the phases using the McNemar test. Possible correlations between the pharmacokinetic variables of aliskiren and subject body weight were analyzed with the Pearson correlation coefficient. The data were analyzed using the statistical program SPSS 16.0 for Windows (SPSS, Chicago, IL, USA). Differences were considered statistically significant when P was <0.05. The number of subjects was estimated to be sufficient to detect a 40% difference in the AUC 0-∞ of aliskiren between the placebo and rifampicin phases, with a power of 80% (α-level 5%) [4].

RESULTS

Effect of rifampicin on aliskiren pharmacokinetics

Rifampicin markedly reduced the plasma concentrations of aliskiren (Fig. 1, Table 1).

The C max of aliskiren was reduced by 39% (P=0.017) and its AUC 0-∞ by 56% (P<0.001) by rifampicin. The Ae was decreased by 48% by rifampicin (P=0.001). Rifampicin had no significant effect on the t max , t ½ , or Cl renal of aliskiren. The extent of the interaction showed marked interindividual variability: decrease in aliskiren AUC 0-∞ ranged from 20% to 76% and the effect of rifampicin on the aliskiren C max ranged from a 78% decrease to a 9% increase (Fig. 2). Plasma aliskiren concentration showed a doublepeak in 11 out of 12 subjects during the placebo phase and 6 out of 12 subjects during the rifampicin phase (P=0.063). None of the pharmacokinetic variables of aliskiren correlated with subject body weight, during either the placebo or rifampicin phase (|r|<0.38, P>0.23).

Effect of rifampicin on aliskiren pharmacodynamics

Plasma renin activity 24 h after aliskiren intake was 61% higher during the rifampicin phase than during the placebo phase (P=0.008) (Fig. 3, Table 2). No significant difference existed in the systolic or diastolic blood pressure, or the heart rate between the rifampicin and placebo phases.

DISCUSSION

This study demonstrates that rifampicin markedly reduces the plasma concentrations of aliskiren. The C max and AUC 0-∞ of aliskiren were reduced by 39% and 56%, respectively. Marked interindividual variability was evident in the extent of the interaction. Plasma renin activity 24 h after aliskiren intake was 61% higher during the rifampicin phase than during the placebo phase. The interaction between rifampicin and aliskiren may result in a need to adjust aliskiren dose.

Aliskiren, a drug with a low oral bioavailability, is a substrate of P-glycoprotein efflux transporter and is slightly metabolized by CYP3A4 [3,5,6]. In the present study, rifampicin reduced the C max and AUC 0-∞ of aliskiren without affecting the t ½ or the Cl renal , indicating that rifampicin increased mainly the first-pass elimination of aliskiren with no or minimal effect on its systemic clearance. Rifampicin has a strong inducing effect on intestinal and hepatic CYP3A4 with a smaller, though significant, effect on Pglycoprotein [16][17][18][19][20][21][22]. It is noteworthy that intestinal CYP3A4 and P-glycoprotein act synergistically to reduce the oral bioavailability of their joint substrates [23].

Because aliskiren has a low hepatic extraction ratio (0.10) [24], induction of its hepatic elimination unlikely explains this interaction observed during the first-pass phase.

Moreover, significant change in the hepatic extraction of such a drug with a low hepatic extraction should result in a shortened elimination t ½ . In addition, plasma aliskiren concentration data showed double-peaks characteristic of entero-hepatic circulation, particularly during the placebo phase. This phenomenon appeared to be diminished during the rifampicin phase, consistent with reduced intestinal reabsorption of aliskiren due to induction of intestinal P-glycoprotein and CYP3A4. Taken together, the interaction between rifampicin and aliskiren is most likely caused by induction of Pglycoprotein-mediated efflux and CYP3A4-catalyzed biotransformation of aliskiren in the gut wall [16][17][18][19][20][21][22].

In this study, the renin-inhibiting effect of aliskiren was attenuated during the rifampicin phase. However, no differences in hemodynamic effects were observed between the rifampicin and placebo phases after a single dose of aliskiren in healthy volunteers. This can be explained by the delayed start of the blood pressure decreasing effect of aliskiren, which gradually reaches its maximum after four weeks treatment [15,25].

Moreover, the pharmacodynamic response to aliskiren in patients with hypertension may be distinct from normotensive healthy individuals [3,15]. On the other hand, the pharmacokinetics of aliskiren is similar in patients with hypertension and healthy individuals [3,15]. According to pharmacokinetic theory, the dose-interval AUC at steady state is equal to the AUC 0-∞ after a single dose [26]. However, the pharmacokinetics of aliskiren is slightly non-linear [3], interfering with the direct extrapolation of the present results to steady-state. Because aliskiren shows a doseresponse relationship in the range 150-300 mg, but no clear effect on blood pressure with lower doses [15,25], it is possible that rifampicin reduces the efficacy of aliskiren in patients with hypertension. Similarly to rifampicin, a high-fat meal has reduced the AUC of aliskiren by about 70% [3]. To minimize variability due to the food effect, aliskiren is recommended to be taken once daily with a light meal or in a routine pattern with regard to meals [15,27].

Deleted: November Deleted: 09 Deleted: the Deleted: ¶ In addition to rifampicin, also other inducers of CYP3A4 and the Pglycoprotein may interact with aliskiren. The antiepileptic drugs carbamazepine and phenytoin are approximately similarly potent inducers of CYP3A4 as rifampicin [19,20,25], whereas St John's wort has a smaller, though significant, effect [26]. Carbamazepine and phenytoin have reduced the AUC of the orally given CYP3A4 probe substrate midazolam (not a substrate of the P-glycoprotein [27]) by 94% and rifampicin by 96-98% [19,20,25,28]. Phenytoin has reduced the AUC of oral digoxin (a substrate of the Pglycoprotein but not of CYP3A4 [21, [START_REF] Lacarelle | Metabolism of digoxin, digoxigenin digitoxosides and digoxigenin in human hepatocytes and liver microsomes[END_REF]30]) by 23% and rifampicin by 30% [21,[START_REF] Rameis | On the interaction between phenytoin and digoxin[END_REF], but carbamazepine treatment has not affected the oral bioavailability of digoxin [32]. St John's wort extract has reduced the AUC of oral midazolam by 52% and the AUC of oral digoxin by 25% [26,33]. ¶ Deleted: Aliskiren is approved for the treatment of hypertension at once daily oral doses of 150 mg and 300 mg, and its antihypertensive effect is attained in two weeks [15]. Blood pressure reduction with 75-mg aliskiren has not been consistently greater than with placebo [15,34,35]. Therefore, during continuous treatment it is probable that rifampicin reduces the antihypertensive effect of aliskiren. In conclusion, rifampicin considerably reduces the plasma concentrations of aliskiren.

Deleted

Clinicians should be aware of the possibility that rifampicin may reduce the antihypertensive efficacy of aliskiren.

F o r P e e r R e v i e w

A revised manuscript for European Journal of Clinical Pharmacology, January 2010 20

  for European Journal of Clinical Pharmacology, January 2010 4

  grapefruit products and alcohol for five days before and three days after aliskiren administration.

  Plasma and urine concentrations of aliskiren were quantified using an Applied Biosystems SCIEX API 2000 Q Trap liquid chromatography-tandem mass spectrometry

  compartmental methods using MK-Model, version 5.0 (Biosoft, Cambridge, United Kingdom). The terminal log-linear part of each concentration-time curve was identified visually. The elimination rate constant (k e ) was determined by linear regression analysis of the log-linear part of the plasma drug concentration-time curve. The t ½ was calculated by the equation t ½ = ln2/k e . The AUC values were calculated by a combination of the linear and log-linear trapezoidal rules with extrapolation to infinity, when appropriate, by division of the last measured concentration by k e . The Cl renal of aliskiren was calculated by the equation Cl renal = Ae/AUC 0-12 .

  for European Journal of Clinical Pharmacology, January 2010 11

  for European Journal of Clinical Pharmacology, January 2010 12

  Vaidyanathan S, Camenisch G, Schuetz H, Reynolds C, Yeh CM, Bizot MN, Dieterich HA, Howard D, Dole WP (2008) Pharmacokinetics of the oral direct renin inhibitor aliskiren in combination with digoxin, atorvastatin, and ketoconazole in healthy subjects: the role of P-glycoprotein in the disposition of aliskiren. J Clin Pharmacol 48 (11): 1323-1338 7 Fromm MF (2004) Importance of P-glycoprotein at blood-tissue barriers. Trends Pharmacol Sci 25 (8): 423-429 8 Jurima-Romet M, Crawford K, Cyr T, Inaba T (1994) Terfenadine metabolism in human liver. In vitro inhibition by macrolide antibiotics and azole antifungals. Drug Metab Dispos 22 (6): 849-857 9 Olkkola KT, Backman JT, Neuvonen PJ (1994) Midazolam should be avoided in patients receiving the systemic antimycotics ketoconazole or itraconazole. Clin Pharmacol Ther 55 (5): 481-485 10 Wang EJ, Lew K, Casciano CN, Clement RP, Johnson WW (2002) Interaction of common azole antifungals with P glycoprotein. Antimicrob Agents Chemother 46 (1): 160-165 11 Marzolini C, Paus E, Buclin T, Kim RB (2004) Polymorphisms in human MDR1 (P-glycoprotein): recent advances and clinical relevance. Clin Pharmacol Ther 75 (1): 13-33 P e e r R e v i e w A revised manuscript for European Journal of Clinical Pharmacology, January 2010 19

Fig. 1 Fig. 2 Fig. 3

 123 FIGURE LEGENDS

[

  12-14]) have raised the area under the plasma aliskiren concentration-time curve (AUC) 1.8-fold and 5-fold, respectively [6, 15]. for European Journal of Clinical Pharmacology, January 2010 4

  grapefruit products and alcohol for five days before and three days after aliskiren administration.

  Plasma and urine concentrations of aliskiren were quantified using an Applied Biosystems SCIEX API 2000 Q Trap liquid chromatography-tandem mass spectrometry

  compartmental methods using MK-Model, version 5.0 (Biosoft, Cambridge, United Kingdom). The terminal log-linear part of each concentration-time curve was identified visually. The elimination rate constant (k e ) was determined by linear regression analysis of the log-linear part of the plasma drug concentration-time curve. The t ½ was calculated by the equation t ½ = ln2/k e . The AUC values were calculated by a combination of the linear and log-linear trapezoidal rules with extrapolation to infinity, when appropriate, by division of the last measured concentration by k e . The Cl renal of aliskiren was calculated by the equation Cl renal = Ae/AUC 0-12 .

  mean values ± SD in the text and tables and, for clarity, as mean values ± SEM in the figures Deleted:

  for European Journal of Clinical Pharmacology, January 2010 10

  for European Journal of Clinical Pharmacology, January 2010 11

  by inducing its Pglycoprotein-mediated transport and CYP3A4-catalyzed biotransformation. The efficacy of aliskiren is likely to be reduced by rifampicin and other inducers of the P-glycoprotein and CYP3A4. ¶ P e e r R e v i e w A revised manuscript for European Journal of Clinical Pharmacology, January 2010 19

Fig. 1 Fig. 2 Fig. 3

 123 FIGURE LEGENDS

Table 1

 1 Pharmacokinetic variables of aliskiren in twelve healthy volunteers after a single 150 mg oral dose of aliskiren following a 5-day

	treatment with 600 mg rifampicin or placebo once daily
	Variable	Placebo phase	Rifampicin phase	Geometric mean ratio/	P value
		(control)		mean difference
				(95% CI)
	C max (ng/ml)	137 (83, 226)	83 (59, 118)	0.61 (0.41, 0.90)	0.017
	t max (h)	1 (0.5-5)	0.5 (0.5-5)		0.254
	t ½ (h)	28.4 ± 8.5	24.4 ± 7.4	-4.0 (-9.6, 1.6)*	0.143
	AUC 0-72 (ng•h/ml)	537 (349, 825)	244 (182, 325)	0.45 (0.36, 0.57)	<0.001
	AUC 0-∞ (ng•h/ml)	581 (376, 898)	258 (191, 349)	0.44 (0.35, 0.56)	<0.001
	Ae (mg)	0.576 (0.403, 0.822) 0.299 (0.227, 0.394)	0.52 (0.38, 0.72)	0.001
	Cl renal (l/h)	1.52 (1.21, 1.91)	1.63 (1.38, 1.93)	1.07 (0.91, 1.26)	0.347

Data are given as geometric mean (95% CI), t max data as median (range), and t ½ data as mean ± SD. CI, Confidence interval; C max , peak plasma concentration; t max , time to C max ; t ½ , elimination half-life; AUC 0-72 , area under the plasma concentration-time curve from 0 to 72 h; AUC 0-∞ , area under the plasma concentration-time curve from time 0 to infinity; Ae, amount excreted into urine within 12 h; Cl renal , renal clearance. *Mean difference (95% CI).

Table 2

 2 Pharmacodynamic variables of aliskiren in twelve healthy volunteers after a single 150 mg oral dose of aliskiren following a 5-day treatment with 600 mg rifampicin or placebo once daily

	Variable	Placebo phase	Rifampicin	Mean difference	P value
		(control)	phase	(95% CI)
	Plasma renin activity (ng/ml/h)
	Baseline	2.21 ± 1.95	2.20 ± 1.40	-0.01 (-1.04, 1.01) 0.980
	At 4 h	0.01 ± 0.04	0.19 ± 0.39	0.18 (-0.09, 0.45) 0.166
	At 24 h	1.27 ± 1.20	2.04 ± 1.11	0.77 (0.25, 1.29)	0.008
	Systolic blood pressure (mm Hg)
	Baseline	124 ± 15	127 ± 15	4 (-1, 8)	0.078
	Average 0-24h	126 ± 13	126 ± 16	0 (-4, 4)	0.847
	Average 0-24h /Baseline	1.02 ± 0.06	0.99 ± 0.03	-0.03 (-0.07, 0.01) 0.093
	Diastolic blood pressure (mm Hg)
	Baseline	77 ± 11	76 ± 9	0 (-5, 5)	0.926
	Average 0-24h	74 ± 10	73 ± 8	-1 (-6, 3)	0.482
	Average 0-24h /Baseline	0.97 ± 0.06	0.95 ± 0.06	-0.02 (-0.07, 0.04) 0.544
	Heart rate (1/min)
	Baseline	63 ± 13	64 ± 7	1 (-5, 7)	0.729
	Average 0-24h	70 ± 12	69 ± 9	0 (-5, 5)	0.888
	Average 0-24h /Baseline	1.11 ± 0.09	1.08 ± 0.07	-0.03 (-0.08, 0.02) 0.233
	Data are given as mean ± SD.
	CI, Confidence interval; Baseline, before the administration of aliskiren.

Table 1

 1 Pharmacokinetic variables of aliskiren in twelve healthy volunteers after a single 150 mg oral dose of aliskiren following a 5-day treatment with 600 mg rifampicin or placebo once daily

						Deleted: November
						Deleted: 09
						Deleted: Table 1 Pharmacokinetic
						variables of aliskiren in twelve healthy
						volunteers after a single 150 mg oral dose
						of aliskiren following a 5-day treatment
	Variable	Placebo phase	Rifampicin phase	Geometric mean ratio/	P value	with 600 mg rifampicin or placebo once daily ¶
		(control)		mean difference		Formatted Table Variable	... [3]
				(95% CI)		Formatted: Right
	C max (ng/ml)	137 (83, 226)	83 (59, 118)	0.61 (0.41, 0.90)	0.017	Formatted: Right
	t max (h)	1 (0.5-5)	0.5 (0.5-5)		0.254	Formatted: Right
	t ½ (h)	28.4 ± 8.5	24.4 ± 7.4	-4.0 (-9.6, 1.6)*	0.143	Formatted: Right
	AUC 0-72 (ng•h/ml)	537 (349, 825)	244 (182, 325)	0.45 (0.36, 0.57)	<0.001	Formatted: Right
	AUC 0-∞ (ng•h/ml)	581 (376, 898)	258 (191, 349)	0.44 (0.35, 0.56)	<0.001	Formatted: Right
	Ae (mg)	0.576 (0.403, 0.822) 0.299 (0.227, 0.394)	0.52 (0.38, 0.72)	0.001	Formatted: Right
	Cl renal (l/h)	1.52 (1.21, 1.91)	1.63 (1.38, 1.93)	1.07 (0.91, 1.26)	0.347	Formatted: Right
	Data are given as geometric mean (95% CI), t max data as median (range), and t ½ data as mean ± SD.
	CI, Confidence interval; C max , peak plasma concentration; t max , time to C max ; t ½ , elimination half-life; AUC 0-72 , area under the plasma
	concentration-time curve from 0 to 72 h; AUC 0-∞ , area under the plasma concentration-time curve from time 0 to infinity; Ae, amount
	excreted into urine within 12 h; Cl renal , renal clearance.

*Mean difference (95% CI).

Table 2

 2 Pharmacodynamic variables of aliskiren in twelve healthy volunteers after a single 150 mg oral dose of aliskiren following a 5-day treatment with 600 mg rifampicin or placebo once daily Data are given as mean ± SD. CI, Confidence interval; Baseline, before the administration of aliskiren.

	Variable	Placebo phase	Rifampicin	Mean difference	P value
		(control)	phase	(95% CI)
	Plasma renin activity (ng/ml/h)
	Baseline	2.21 ± 1.95	2.20 ± 1.40	-0.01 (-1.04, 1.01) 0.980
	At 4 h	0.01 ± 0.04	0.19 ± 0.39	0.18 (-0.09, 0.45) 0.166
	At 24 h	1.27 ± 1.20	2.04 ± 1.11	0.77 (0.25, 1.29)	0.008
	Systolic blood pressure (mm Hg)
	Baseline	124 ± 15	127 ± 15	4 (-1, 8)	0.078
	Average 0-24h	126 ± 13	126 ± 16	0 (-4, 4)	0.847
	Average 0-24h /Baseline	1.02 ± 0.06	0.99 ± 0.03	-0.03 (-0.07, 0.01) 0.093
	Diastolic blood pressure (mm Hg)
	Baseline	77 ± 11	76 ± 9	0 (-5, 5)	0.926
	Average 0-24h	74 ± 10	73 ± 8	-1 (-6, 3)	0.482
	Average 0-24h /Baseline	0.97 ± 0.06	0.95 ± 0.06	-0.02 (-0.07, 0.04) 0.544
	Heart rate (1/min)
	Baseline	63 ± 13	64 ± 7	1 (-5, 7)	0.729
	Average 0-24h	70 ± 12	69 ± 9	0 (-5, 5)	0.888
	Average 0-24h /Baseline	1.11 ± 0.09	1.08 ± 0.07	-0.03 (-0.08, 0.02) 0.233

Table 1

 1 Pharmacokinetic variables of aliskiren in twelve healthy volunteers after a single 150 mg oral dose of aliskiren following a 5-day treatment with 600 mg rifampicin or Data are given as mean ± SD, t max data as median (range). CI, Confidence interval; C max , peak plasma concentration; t max , time to C max ; t ½ , elimination half-life; AUC 0-72 , area under the plasma concentration-time curve from 0 to 72 h; AUC 0-∞ , area under the plasma concentration-time curve from time 0 to infinity; Ae, amount excreted into urine within 12 h; Cl renal , renal clearance.

	placebo once daily
	Variable	Placebo phase	Rifampicin	% of control	Geometric mean
		(control)	phase	(range)	mean differe
					(95% CI)
	C max (ng/ml)	179 ± 134	95 ± 55	53% (22-109%)	0.61 (0.41, 0
	t max (h)	1 (0.5-5)	0.5 (0.5-5)
	t ½ (h)	28.4 ± 8.5	24.4 ± 7.4	86% (52-131%)	-4.0 (-9.6, 1.
	AUC 0-72 (ng•h/ml)	661 ± 457	270 ± 138	41% (24-81%)	0.45 (0.36, 0
	AUC 0-∞ (ng•h/ml)	718 ± 501	288 ± 150	40% (24-80%)	0.44 (0.35, 0
	Ae (mg)	0.662 ± 0.365	0.326 ± 0.144	49% (27-104%)	0.52 (0.38, 0
	Cl renal (l/h)	1.60 ± 0.44	1.69 ± 0.45	106% (74-184%)	1.07 (0.91, 1

*Mean difference (95% CI).

European Journal of Clinical Pharmacology

ACKNOWLEDGEMENTS

We thank Kaisa Kurkinen for the drug concentration measurements, and Mikko Neuvonen, Eija Mäkinen-Pulli, Lisbet Partanen and Jouko Laitila for skilled assistance.

This study was supported by grants from the Helsinki University Central Hospital Research Fund (Helsinki, Finland) and the Sigrid Jusélius Foundation (Helsinki, Finland).

ACKNOWLEDGEMENTS

We thank Kaisa Kurkinen for the drug concentration measurements, and Mikko Neuvonen, Eija Mäkinen-Pulli, Lisbet Partanen and Jouko Laitila for skilled assistance.

This study was supported by grants from the Helsinki University Central Hospital Research Fund (Helsinki, Finland) and the Sigrid Jusélius Foundation (Helsinki, Finland).

COMPETING INTEREST

None to declare.

ABSTRACT

Purpose

This study aimed to investigate the effect of rifampicin, an inducer of CYP3A4 and Pglycoprotein, on the pharmacokinetics and pharmacodynamics of aliskiren, a renin inhibitor used in the treatment of hypertension.

Methods

In a randomized crossover study, twelve healthy volunteers took 600 mg rifampicin or placebo once daily for five days. On day six, they ingested a single 150 mg dose of aliskiren. Plasma aliskiren concentrations were measured up to 72 h and urine concentrations up to 12 h, pharmacodynamic variables were measured up to 24 h.

Results

Rifampicin reduced the peak plasma aliskiren concentration (C max ) by 39% (95% confidence interval 0.41, 0.90; P=0.017) and the area under the plasma aliskiren concentration-time curve (AUC 0-∞ ) by 56% (95% confidence interval 0.35, 0.56; P<0.001). Rifampicin had no significant effect on aliskiren elimination half-life (t ½ ) or its renal clearance (Cl renal ). Plasma renin activity 24 h after aliskiren intake was 61% higher during the rifampicin phase than during the placebo phase (P=0.008).

Conclusions

Rifampicin considerably reduces the plasma concentrations and the renin-inhibiting effect of aliskiren by decreasing its oral bioavailability.

KEYWORDS