Yannick Privat
email: yannick.privat@unistra.fr

Maïtine Bergounioux
email: maitine.bergounioux@univ-orleans.fr

Élie Bretin
email: elie.bretin@insa-lyon.fr

How to position sensors in thermo-acoustic tomography?

Keywords: wave equation, observability, shape optimization, calculus of variation, minimax problem, primal-dual algorithm AMS classification: 49J20, 35M33, 80A23, 93C20

Thermo-acoustic tomography is a non-invasive medical imaging technique: the object to be reconstructed is excited by an impulse, inducing inhomogeneous heating and thus tissue expansion. This creates an acoustic wave pressure that can be measured by sensors. The reconstruction of internal heterogeneities can then be achieved by solving an inverse problem, once the sound wave measurements are known outside the body. As the measured pressure intensity is expected to be low, a difficult problem is to position the sensors properly.

This article is devoted to determining the positions of the sensors in order to carry out the reconstruction procedure in an optimal way. We first introduce a model of optimal sensor position that involves a deviation function between the theoretical pressure and the measured pressure for a first series of sensor measurements, and a constant observable type function that describes the quality of reconstruction. We use it to determine an appropriate position of the sensors for a second set of measurements.

Far from providing an intrinsic solution to the general issue of positioning sensors, solving this problem makes it possible to obtain a new position of the sensors improving the quality of the reconstruction before obtaining a new series of measurements.

This model is analyzed mathematically: we study the existence problems and introduce a numerical algorithm to solve them. Finally, several 2D numerical simulations illustrate our approach.

Introduction

In this article, we focus on thermo-acoustic tomography, a non-invasive medical imaging technique that seems very promising because, compared to other imaging techniques such as X-ray tomography or emission tomography, it is based on simple and relatively inexpensive equipment. It is particularly well suited for low-density tissue imaging and is likely to become a major tool for mammography and even brain imaging. The development of non-invasive and non-ionizing imaging techniques is particularly important for the early detection of breast cancer in young patients, whose mammograms are generally of low contrast. The principle of this imaging process is very simple: the tissue to be visualized is irradiated by a pulse and this energy induces a heating process. If the pulse is an electromagnetic radio-frequency pulse, the technique is called thermo-acoustic tomography (TAT); if the pulse is a laser (the frequency is much higher), it is called photo-acoustic tomography (PAT). In any case, this creates a thermally induced pressure surge that propagates as a sound wave, which can be detected by sensors located outside the body to image. By detecting pressure waves, heterogeneities can be observed: this gives important information such as the position and/or size of tumors in breast cancer. For more details on the process and related work, we refer to [START_REF] Akhouayri | Quantitative thermoacoustic tomography with microwaves sources[END_REF][START_REF] Bergounioux | An optimal control problem in photoacoustic tomography[END_REF][START_REF] Bonnefond | Contributions à la tomographie thermoacoustique[END_REF][START_REF] Patch | Photo-and thermo-acoustic imaging introduction[END_REF] and the references contained therein. Basically, TAT and PAT are two hybrid techniques using electromagnetic waves as excitation (input) and acoustic waves as observation (output). Both techniques lead to similar misplaced reverse problems. The modeling of the direct problem does not lead to the same equations, since the physical background is not the same. However, the process can be described using two equations (or systems of equations):

• The first system of equations describes the generation of the heating process inside the body; in PAT, this system involves the fluence equation (the fluence rate is the average of the light intensity in all directions) which is a diffusion equation [START_REF] Bergounioux | An optimal control problem in photoacoustic tomography[END_REF] whereas in TAT, this equation is replaced by the Maxwell equations [START_REF] Akhouayri | Quantitative thermoacoustic tomography with microwaves sources[END_REF]. Temperature is described by the classical thermal equation, which can be neglected in PAT since the high speed of light implies that the thermal effect is almost instantaneous. In both cases, the resulting term is a source pressure wave p 0 at the time t = 0.

• The second equation models the behaviour of the acoustic wave once the source p 0 is known. It writes

       ∂ tt p(t, x) -div(c(x)∇p(t, x)) = 0 in (0, T ) × B, p(0, •) = p 0 , on B, ∂ t p(0, •) = 0 on B, p = 0 on (0, T ) × ∂B, (1.1)
where T > 0 is arbitrary and B is a given ball whose radius will be chosen appropriately in the sequel. Most articles on the subject have focused on this second equation to address inversion. Indeed, if measurements are made to determine p 0 , then the heterogeneities can be identified by quantitative estimates of two appropriate physical parameters: diffusion and absorption coefficients in PAT, or electrical sensitivity and conductivity in TAT.

In this article, we focus on the following question:

from the knowledge of a first series of measurements, how to locate the sensors before performing a second one, in a relevant way?

Such a question is related to the general problem of optimal sensor positioning. In the following, we focus on the pressure equation to obtain the best reconstruction of p 0 . The future analysis can be used interchangeably in both PAT and TAT frameworks. In the following, we always refer to the TAT process for readability reasons.

Our goal is therefore to determine the position of the sensors providing the best possible reconstruction. As a result, we do not focus on solving the inverse problem (where the sensors are fixed). Many articles on TAT/PAT reconstruction have been written in recent decades. For example, we refer to [START_REF] Akhouayri | Quantitative thermoacoustic tomography with microwaves sources[END_REF][START_REF] Arridge | Optical tomography in medical imaging[END_REF][START_REF] Bal | Quantitative thermo-acoustics and related problems[END_REF][START_REF] Bal | Inverse diffusion theory of photoacoustics[END_REF][START_REF] Bergounioux | An optimal control problem in photoacoustic tomography[END_REF][START_REF] Bonnefond | Contributions à la tomographie thermoacoustique[END_REF][START_REF] Patch | Photo-and thermo-acoustic imaging introduction[END_REF][START_REF] Xu | Photoacoustic imaging in biomedecine[END_REF] and to the book by Ammari et al. [START_REF] Ammari | An introduction to mathematics of emerging biomedical imaging[END_REF][START_REF] Ammari | Reconstruction of small inhomogeneities from boundary measurements[END_REF] or Scherzer et al. [START_REF] Scherzer | Variational Methods in Imaging[END_REF].

To our knowledge, the question of the optimal positioning of the sensors has not yet been addressed in the context of photoacoustic tomography. However, it can be noted that the search for the optimal position and/or shapes of the sensors for the wave/heat equation in a delimited domain has been studied in [START_REF] Privat | Optimal location of controllers for the onedimensional wave equation[END_REF][START_REF] Privat | Optimal observation of the one-dimensional wave equation[END_REF][START_REF] Privat | Complexity and regularity of maximal energy domains for the wave equation with fixed initial data[END_REF][START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF][START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF]. In these articles, it is assumed that, in a certain sense, a phenomenon of reflection of solutions at the boundary of the domain occurs. Such an assumption is not relevant for the application we are considering here and we propose an alternative approach. Also worth mentioning is [START_REF] Ammari | An inverse initial boundary value problem for the wave equation in the presence of imperfections of small volume[END_REF][START_REF] Ammari | Mathematical modeling in photoacoustic imaging of small absorbers[END_REF], where the issue of recovering small inhomogeneities of conductivity in a homogeneous background medium from dynamic boundary measurements over a part of the boundary and over a finite time interval is addressed. These problems differ from those studied here, since the proposed reconstruction algorithm depends on particular choices of initial data and boundary conditions. In the following, we basically look for the initial data and optimal sensor positions at the same time, which requires a dedicated method.

Another related problem is the optimization of electrode positions in electrical impedance tomography, which aims to retrieve information on electrical conductivity within a physical body from current and potential boundary measurements. The optimal electrode locations are determined so that, taking into account the measurement, the conditional conductivity density is as localized as possible (see for example [START_REF] Demidenko | On optimal current patterns for electrical impedance tomography[END_REF][START_REF] Garde | Convergence and regularization for monotonicitybased shape reconstruction in electrical impedance tomography[END_REF][START_REF] Hyvönen | Optimizing electrode positions in electrical impedance tomography[END_REF][START_REF] Neumayer | Optimal design of ect sensors using prior knowledge[END_REF][START_REF] Yan | Optimum design of electrode structure and parameters in electrical impedance tomography[END_REF]). Nevertheless, a huge difference between this work and the present paper lies in the physical model itself. The modeling and approaches studied in our paper are adapted to the wave equation model. Indeed, we will use a certain observability constant to define the functional cost, time inversion techniques to solve the resulting problem numerically, etc.

In the following, the determination of the optimal position or shape of the sensors will be addressed in two ways:

(i) First attempt: prescription of the total surface area of the sensors. We first choose to process sensors described by the characteristic function of a measurable subset whose Lebesgue measurement is prescribed. A possible disadvantage of such an approach is that the optimal solution may not exist or may have a high/infinite number of connected components (see problem (2.13)).

(ii) Second attempt: prescription of the maximum number of sensors. To avoid the emergence of overly complex solutions and to make problem modeling more realistic, we will examine a second point of view where a maximum number of N 0 of sensors (specifically connected components of the sensor set) is prescribed, in addition to the total surface mentioned above (see Problem (2.21)).

The document is organized as follows: Section 2 is dedicated to problem modeling. We justify here the point of view adopted. In the section 3, we analyze the problem of optimization, in particular the problems of existence and deduce the optimal conditions. Finally, the 4 section is devoted to describing a numerical algorithm and presenting the first promising numerical results.

Modeling the problem 2.1 The PDE (direct) model and the sensors set

Throughout this paper, we will use the notation 1 A as the characteristic function of a set A ⊂ IR d (d ≥ 1) which is the function equal to 1 on A and 0 elsewhere.

As explained in Section 1, the acoustic wave is assumed to solve the partial differential equation

       ∂ tt p(t, x) -div(c(x)∇p(t, x)) = 0 in (0, T ) × B, p(0, •) = p 0 , on B, ∂ t p(0, •) = 0 on B, p = 0 on (0, T ) × ∂B, (2.1)
where T > 0 is arbitrary and B ⊂ IR d is a given ball whose radius R is chosen in the sequel.

Let us also introduce Ω as a convex open subset of IR d , representing the body to image, which means in particular that supp(p 0 ) ⊂ Ω, (

where supp(p 0 ) denotes the support of p 0 . The convexity assumption on Ω is technical, and will be used to make the set of admissible designs well defined in the sequel. This will be commented later.

For this reason, we assume that B is large enough so that Ω ⊂ B; the set B \ Ω stands for the ambient media (water or air), where the wave propagates.

The function c ∈ L ∞ (IR d ) stands for the sound speed and satisfies

c(x) ≥ c 0 > 0 a.e. x ∈ IR d .
A typical choice for c is to assume it is piecewise constant, namely:

c = c 1 1 Ω + c 2 1 B\Ω , with (c 1 , c 2 ) ∈ (IR * + ) 2 , (2.3)
with c 1 , c 2 > 0. Recall that for every initial p 0 ∈ H 1 0 (Ω), there exists a unique solution p to System (2.1) and that solution satisfies p ∈ C 1 [0, T ], L 2 (B) ∩ C 0 [0, T ], H 1 0 (B) (see e.g. [START_REF] Dautray | Mathematical analysis and numerical methods for science and technology[END_REF][START_REF] Evans | Partial differential equations[END_REF]).

Remark 2.1. For (future) numerical purposes, we choose B as a bounded ball with radius R. However, R is large enough, then p vanishes on ∂B all along the recording process (i.e. for t T ). Precisely, there exists R > 0 large enough such that, for every p 0 ∈ H 1 0 (Ω), the solution of problem (2.1) coincides with the solution of

   ∂ tt p(t, x) -div(c(x)∇p(t, x)) = 0 in (0, T ) × IR d , p(0, •) = p 0 (•) on IR d , ∂ t p(0, •) = 0 on IR d , (2.4)
with the convention that the initial datum p 0 has been extended by 0 to the whole space IR d . One refers for instance to [START_REF] Evans | Partial differential equations[END_REF].

As mentioned in Section 1, the inverse problem of recovering p 0 ∈ H 1 0 (Ω) from p measurements on a set (0, T ) × Σ (where Σ is the position of the sensors) has been extensively studied over the past two decades. We refer to [START_REF] Ammari | An introduction to mathematics of emerging biomedical imaging[END_REF][START_REF] Ammari | Reconstruction of small inhomogeneities from boundary measurements[END_REF][START_REF] Bergounioux | An optimal control problem in photoacoustic tomography[END_REF][START_REF] Bonnefond | Contributions à la tomographie thermoacoustique[END_REF][START_REF] Patch | Photo-and thermo-acoustic imaging introduction[END_REF] and the references therein for more details. However, the point of view we adopt here is a variational one. Specifically, we do not use a reconstruction formula (exact or not) or a time inversion approach. to deal with the inverse problem. Following the philosophy of [START_REF] Akhouayri | Quantitative thermoacoustic tomography with microwaves sources[END_REF][START_REF] Bergounioux | An optimal control problem in photoacoustic tomography[END_REF][START_REF] Bonnefond | Contributions à la tomographie thermoacoustique[END_REF], we rather include the unknown parameter in a cost functional. As a result, the reconstruction of p 0 and the question of the position of the sensors will be addressed in terms of functional minimization.

The sensors set. To model the sensors optimal positioning, we first describe the class of admissible designs/sensors to be considered.

Let us endow the Lipschitz set ∂Ω with the usual (d-1)-dimensional Hausdorff measure H d-1 . In the sequel, we say by convention that Γ ⊂ ∂Ω is measurable whenever it is measurable for the Hausdorff measure H d-1 .

Introduce Σ ⊂ IR d as the subdomain of IR d occupied by sensors. Roughly speaking, we will assume that every connected component of Σ is located around the boundary of Ω and has a positive thickness ε. Precisely, we assume the existence of a measurable set Γ ⊂ ∂Ω such that

Σ = {s + µ ν(s), s ∈ Γ, µ ∈ [0, ε]} , (2.5)
where ν(s) denotes the outward unit normal to Ω at s (see Figure 2.1). The set Σ is thus supported by the annular ring

∂Ω := {s + µν(s), s ∈ ∂Ω, µ ∈ [0, ε]}.
thickness ". More precisely, we assume the existence of a measurable set Ä B⌦ such that ⌃ " ts `µ ⌫psq, s P , µ P r0, "su ,

where ⌫psq denotes the outward unit normal to ⌦ at s (see figure 1). The set ⌃ is thus supported by the annular ring Ä B⌦ :" ts `µ⌫psq, s P B⌦, µ P r0, "su. Remark 2.2. In this work, we have chosen to treat volumetric sensors: this will determine the choice of the term observability in the cost functional. Another relevant model could be that of boundary sensors, which would imply changing the choice of the above observability term in the cost functional.

When investigating the optimal position or shape of sensors without any restriction on the sensors domain measure, the solution is trivial and is given by Σ = ∂Ω (see e.g. [START_REF] Privat | Optimal location of controllers for the onedimensional wave equation[END_REF][START_REF] Privat | Optimal observation of the one-dimensional wave equation[END_REF][START_REF] Privat | Complexity and regularity of maximal energy domains for the wave equation with fixed initial data[END_REF][START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF][START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF]). This is not relevant for practical purposes and this is why we also assume in the sequel that the measure of the sensors domain is prescribed: the measurable subsets Γ of ∂Ω satisfy H d-1 (Γ) = LH d-1 (∂Ω), where L ∈ (0, 1) denotes some given real number. Let us introduce the two classes

V L = {a ∈ L ∞ (∂Ω), a ∈ {0, 1}, a.e. in ∂Ω, ∂Ω a(x)dH d-1 (x) = LH d-1 (∂Ω)}. (2.6)
of admissible subsets Σ ⊂ B characteristic functions and

U L = {a ∈ L ∞ ( ∂Ω), a(s + µν(s)) = X(s), a.e. (s, µ) ∈ ∂Ω × [0, ε], with X ∈ V L } (2.7) of admissible subsets Γ ⊂ ∂Ω characteristic functions.
Since Ω is assumed to be convex, the set U L well defined. Indeed, Ω has in particular a Lipschitz boundary and the normal outward vector ν(•) exists almost everywhere on ∂Ω. The convexity assumption implies that the map ∂Ω s + µν(s) → (s, µ) ∈ ∂Ω × [0, ε] is invertible according to the projection on a convex set theorem.

Criteria choices and optimization problems

In this section, we model the problem by describing the cost functional and set the optimization problems. Let p obs ∈ L 2 ((0, T )×Σ) denote the measured pressure: it is defined in (0, T )×Σ. We extend p obs by 0 to (0, T )×B and denote the obtained function similarly, with a slight abuse of notation. Hence, one has p obs = 1 Σ p obs so that p obs ∈ L 2 (0, T, L 2 (B)). For p 0 ∈ H 1 0 (Ω), we also introduce p [p 0 ] as the solution of Problem (2.1) for the initial datum p 0 .

The issue we are now addressing is as follows: given a first (series of ) measurement(s), how can we determine a relevant sensor position before performing a new (series of ) measurement(s)? To achieve this objective, we introduce an optimal design problem. Our approach can be split into two steps that we roughly describe.

• First step: determination of an initial pressure condition p 0 . Recall that, in the PDE model we consider (see Eq. (2.1)), the initial velocity is assumed to vanish identically in B. It is therefore enough to reconstruct the initial pressure. A first natural (naive) idea would be to consider the problem inf

p 0 ∈P 0 (Ω) A 1 (1 Σ , p 0 ) (2.8)
where

A 1 (1 Σ , p 0 ) := 1 2 T 0 B 1 Σ (x)(p [p 0 ] (t, x) -p obs (t, x)) 2 dx dt.
(2.9)

and P 0 (Ω) is a given functional space to be chosen adequately in the sequel. Unfortunately, the well-posedness of such least square problems, is in general not ensured.

In addition, due to the uncertainties of sensor measurements and the fact that Σ is a strict sub-domain of ∂Ω, there could be many initial pressures of p 0 leading to the observation p obs . Therefore, we decide to select an initial pressure function denoted p 0 by solving a penalized optimization problem in order to impose three kinds of physical constraints: we look for (i) positive pressure term, (ii) whose support is included in a fixed compact set K of B, and (iii) belonging to a functional space well-chosen to ensure the well-posedness of the problem and therefore good reconstruction properties. From a practical viewpoint, this will be done adding a penalization-regularization term denoted R(p 0 ) in the definition of the criterion. The choice of such a term will be introduced and commented in Section 3.1.

Solving the resulting problem (see its definition below) is a way to define an initial pressure function reconstructed (almost) everywhere in B and not only on Σ. This is the key point to address the optimal design problem in the second step.

• Second step: determination of the best position of sensors. Once an initial pressure p 0 has been determined with a given position of sensors, the new position will be obtained by solving the optimal design problem sup

1 Σ ∈U L A 2 (1 Σ , p 0 ),
where

A 2 (1 Σ , p 0 ) := T 0 B 1 Σ (x)∂ t p [ p 0 ] (t, x) 2 dx dt p 0 2 H 1 (Ω)
.

(2.10)

The functional to maximize represents the quality of the observation: we discuss and comment on such a choice below. In summary, we are looking for the position of the sensors allowing the best observation of the worst possible pressure p 0 leading to the observation p obs .

To summarize, this two step procedure leads to the following optimal design problem:

Optimal position of sensors (new version). Let p obs ∈ L 2 (Σ) and γ ∈ IR * + be a fixed parameter.

1. Computation of an initial pressure function p 0 (whenever it exists) by solving the problem inf

p 0 ∈P 0 (Ω) J 0 (1 Σ , p 0 ), (2.11)
where the cost-functional J 0 is defined by

J 0 (1 Σ , p 0 ) = A 1 (1 Σ , p 0 ) + γR(p 0 ), γ > 0 (2.12)
and the admissible set is

P 0 (Ω) = {p 0 ∈ L 2 (Ω) | supp(p 0 ) ⊂ K, p 0 0 a.e. in Ω and R(p 0 ) < +∞},
where R(p 0 ) is a penalization-regularization term whose choice will be made precise in Section 3.1.

2. Assuming that Problem (2.11) has a solution p 0 , determination of a new sensors position by solving sup

1 Σ ∈U L A 2 (1 Σ , p 0 ) (2.13)
Next, to deal with more realistic constraints, we introduce a modified optimal design problem, where then sensors set is assumed to be the union of connected components N 0 having the same length, N 0 being a given non-zero integer. For simplicity's sake, we will assume that the dimension of the space is d = 2 and that the boundary of the convex set Ω is smooth, say C 1 .. Let O denote any point of Ω and assume that Ω has as polar equation r = ρ(θ) in a given orthonormal basis of IR 2 centered at O, where ρ is a Lipschitz function of the one-dimensional torus T = IR/[0, 2π).

Let us make this sensors set precise. Let > 0 be such that N 0 < T ρ(θ) 2 + ρ (θ) 2 dθ. We choose to consider the set of sensors represented by Σ ⊂ ∂Ω associated to the domain Γ with 1 Γ ∈ V L through the formula (2.7)-(2.6), parametrized by a nondecreasing family (θ n ) n∈{1,...,N 0 } ∈ T N 0 such that Σ is associated to Γ by the relation

1 Σ (x) = 1 Γ (s + µν(s)) = 1 Γ (s), for a.e. x ∈ ∂Ω, s ∈ ∂Ω, µ ∈ [0, ε] (2.14)
for some 1 Γ ∈ V L , where

Γ = N 0 n=1 Γ n with Γ n = {(r cos θ, r sin θ) ∈ Γ | θ ∈ (θ n , θn )}, (2.15)
where θn is defined from θ n by the relation

θn θn ρ(s) 2 + ρ (s) 2 ds = .
(2.16)

Note that, since the mapping T θ → θn θn ρ(s) 2 + ρ (s) 2 ds is continuous and monotone increasing, it defines a bijection. This justifies the consistence of definition of θn by (2.16). Basically, the last equality requires that all sensors be represented by N 0 of almost identical connected components (up to isometrics).

To avoid the overlapping of sensors, we will also require that ∀n ∈ {1, . . . , N 0 -1}, θn θ n+1 .

(2.17)

It will be useful for the upcoming analysis to note that this last condition will rewrite ∀n ∈ {1, . . . , N 0 -1},

θ n+1 θn ρ(s) 2 + ρ (s) 2 ds . (2.18)
As a conclusion, we will restrict the set of admissible configurations to

U L,N 0 = {1 Σ ∈ U L | Σ associated to Γ by (2.14) with 1 Γ ∈ V L,N 0 }, (2.19)
where

V L,N 0 = {1 Γ ∈ V L | Γ
1 Σ ∈U L,N 0 A 2 (1 Σ , p 0 ), (2.21)
where U L,N 0 is given by (2.19).

In the section 4, we will discuss this method and illustrate it with several numerical results.

Comments on the terms A 1 and A 2 . We end this section with comments on the choice of the functional.

The term A 1 (1 Σ , p 0 ) is a least-square fidelity term, ensuring that the initial pressure p 0 makes the pressure p [p 0 ] (corresponding to the reconstructed image) as close as possible to the observed pressure p obs on the domain occupied by sensors. The term A 2 (1 Σ , p 0 ) is inspired by the notion of observability in the theory of control or inverse problems. Indeed, the equation (2.1) is said to be observable in time T on Σ if there is a positive constant C such that the inequality

C p 0 2 H 1 0 T 0 B 1 Σ (x) ∂ t p [p 0 ] (t, x) 2 dxdt, (2.22)
holds for every initial datum p 0 ∈ H 1 0 (Ω). The largest constant C = C T (1 Σ ) such that the inequality (2.22) holds is the so-called observability constant and writes

C T (1 Σ ) = inf p 0 ∈H 1 0 (B) T 0 B 1 Σ (x) ∂ t p [p 0 ] (t, x) 2 dx dt p 0 2 H 1 0 .
This constant C T (1 Σ ) gives an idea of the well-posed character of the p 0 reconstruction from measurements of p [p 0 ] on Σ. In short, the higher the observability constant, the better the quality of the reconstruction. The functional A 2 thus models the efficiency of a set of sensors occupying the Σ domain. It should also be noted that the choice of the observation variable 1 Σ ∂ t p [p 0 ] is determined by the fact that the sensors are generally piezoelectric microphones, which are supposed to record temporal pressure variations.

Finally, to understand the role of maximization with respect to the position of the sensors, we have illustrated a a bad situation on Figure 2.2; here a sensors position on a domain Σ 2 gives a less precise result than another position on a subdomain Σ 1 , since the observed pressure coincides only with the real one on the domain Σ 2 . Maximization with respect to design can then be interpreted as a functional worst case function, used to move the sensors to a position ignored by previous measurements and where the reconstructed pressure is far from that observed.

pressure p rp 0 s for the reconstructed image as close as possible to the observed pressure domain occupied by sensors.

he term A 2 p1 ⌃ , p 0 q is inspired by the notion of observability in control or inverse pr ry. Indeed, the equation ( 1) is said observable in time T on ⌃ whenever there exists a p tant C such that the inequality

C}p 0 } 2 H 1 0 § ª T 0 ª B 1 ⌃ pxq ˇˇB t p rp 0 s pt, xq ˇˇ2 dxdt,
s for every initial datum p 0 P H 1 0 p⌦q. The largest constant C " C T p1 ⌃ q such th uality [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF] holds is the so-called observability constant and writes

C T p1 ⌃ q " inf p 0 PH 1 0 pBq ≥ T 0 ≥ B 1 ⌃ pxq ˇˇB t p rp 0 s pt, xq ˇˇ2 dxdt }p 0 } 2 H 1 0 .
constant C T p1 ⌃ q provides an account of the well-posedness character of the inverse prob nstructing p 0 from measurements of p rp 0 s on ⌃. Roughly speaking, the largest the observ tant is, the best will by the reconstruction quality. he functional A 2 thus models the e ciency of a set of sensors occupying the domain ⌃. that the choice of the observation variable 1 ⌃ B t p rp 0 s is driven by the fact that sensors eneral of piezoelectric microphones, and are thus assumed to record time variations sure. inally, to understand the role of the maximization with respect to the location of sens illustrated a bad situation on Figure 2, where a location of sensors on a domain ⌃ 2 pro accurate result than another location on a subdomain ⌃ 1 , since the observed pressure co with the real pressure on the domain ⌃ 2 . The maximization with respect to the desi be interpreted as a "worst case" functional, used to move the sensors toward a location i he previous measures and where the reconstructed pressure is far from the observed on ark 4. Notice that, by energy conservation, one has for every t P r0, T s, ª

B B t ppt, xq 2 dx § ª B `Bt ppt, xq 2 `|rppt, xq| 2 ˘dx § ª ⌦ |rp 0 pxq| 2 dx " }p 0 } 2 H 1
0 p⌦q , therefore, we infer that A 2 p1 ⌃ , p 0 q P r0, T s for every p 0 P H 1 p⌦qzt0u. 3 Analysis of Problems (2.11) and (2.13)

Choice of the regularization term R(p 0 )

Let's add some comments to fully specify the model (via the cost functional). If we consider Problem (2.8), it is clear that the ill-posedness of this problem prevents a solution from being found (to highlight such a claim, one can easily build counterexamples based on gaussian beams). Therefore, it is necessary to define a coercive cost functional in relation to the functional space by adding a penalty-regularization term R(p 0 ).

• If we decide to choose L 2 (B), as initial pressure space, we ask for the weakest regularity. In that case R(p 0 ) = p 0 L 2 . This implies that p 0 may be small since the first term only involves "sparse" observation of p and is expected to be small even for small ε.

• We hope to recover the heterogeneities as images with sharp edges, which can be captured adequately by considering the functions of bounded variation. Indeed, the involvement of the first derivative via the total variation has a noise-reducing effect. However, to our knowledge, the solution of the wave equation with the BV data is not standard. For this reason, in the upcoming analysis, we prefer to process more regular data and choose a penalization functional that induces a H 1 -smoothing effect. Therefore, we expect to recover diffuse objects. To avoid a too hard regularization process, we also use the W 1.1 standard (which is the same as T V -one in this case).

• Finally, we don't have any compactness result to achieve the convergence process of subsequences. Therefore, we add a (small) viscosity penalization term which must rather be interpreted as a theoretical tool.

Eventually, we choose

R(p 0 ) = ∇p 0 L 1 + ε 2γ p 0 2 H 1 , (3.1)
so that the cost functional reads

J(p 0 ) := A 1 (1 Σ , p 0 ) + γ ∇p 0 L 1 + ε 2 p 0 2 H 1 , (3.2)
where ε γ and A 1 (p 0 ) = A 1 (1 Σ , p 0 ) is given by (2.9). Throughout this section, we omit the dependence with respect to Σ since this set is fixed. Hence, Problem (2.11) can be written as inf

p 0 ∈P K (Ω) J(p 0 ), (3.3)
where K is a fixed compact subset of Ω and

P K (Ω) = {p 0 ∈ H 1 0 Ω) | supp(p 0
) ⊂ K and p 0 0 a.e. in Ω}.

Note that P K (Ω) is convex and closed for the weak H 1 (Ω)-topology.

Remark 3.1. Typically, we choose K := {x ∈ Ω | d(x, ∂Ω) ≥ δ } for some positive parameter δ.

Analysis of Problem (3.3)

In the sequel we denote by bold letters functions that are defined on (0, T ) × B. Precisely, for any p ∈ H 1 0 (Ω) , we denote p the solution of (2.1) where p(0, •)=p.

Existence and uniqueness. Proof. Let (p n ) n∈IN be a minimizing sequence of (3.3). Therefore, (p n ) n∈IN is weakly convergent to some p 0 in H 1 (Ω) and W 1,1 (Ω) and strongly in L 2 (Ω). As P K (Ω) is H 1 (Ω) weakly closed then p 0 ∈ P K (Ω). Moreover, the sequence (p n ) n∈IN defined by p n := p [pn] strongly converges to

p 0 := p [p 0 ] in L ∞ [0, T ], L 2 (B) (see [16, Theorem 2 p. 567]).
According to the Lebesgue theorem, we infer that A 1 (p n ) → A 1 (p 0 ) as n → +∞. We end the proof with the lower semi-continuity of the norms in the regularizing term. Uniqueness is an easy consequence of the strict convexity of the H 1 -norm. Note that the p 0 2 H 1 term is used both for existence and uniqueness.

Optimality conditions. Let p * be the solution to (3.3) and (as before) p * the solution of (2.1) where p * (0, •) = p * and look for optimality conditions. For every p ∈ P K (Ω) we get 0 ∈ ∂J(p * ),

where ∂J(p * ) stands for the subdifferential of J at p * (see [START_REF] Ekeland | Convex Analysis and Variational Problems[END_REF] for example). Indeed, J is not Gâteaux-differentiable because of the TV term. This yields

-D[A 1 ](p * ) -ε(p * -∆p * ) ∈ γ∂T V (p * ). (3.4)
Recall that T V (p * ) = ∇p * L 1 since p * ∈ H 1 0 (Ω) but we use this notation for convenience. We first compute D[A 1 ](p * ) (the derivative with respect to p * ) by introducing an adjoint state. Let us define q * as the solution of

   ∂ tt q * (t, x)-div(c(x)∇q * (t, x)) = (p * (t, x)-p obs (t, x)))1 Σ (x) in (0, T ) × B, q * (T, •) = 0, ∂ t q * (T, •) = 0 on B, q * = 0 on (0, T ) × ∂B, (3.5)
For every p ∈ H 1 0 (Ω), a simple computation yields

D[A 1 ](p * ) • p = T 0 B 1 Σ (x)(p * (t, x) -p obs (t, x)) p(t, x)dx dt.
Indeed, by linearity, the derivative of p 0 → p [p 0 ] with respect to p 0 is p [p 0 ] . Using (3.5) and two integrations by parts, we get

D[A 1 ](p * ) • p = - B 1 Σ (x) q * (0, x)p(x) dx , so that D[A 1 ](p * ) = -1 Σ q * (0). Equation (3.4) writes 1 γ 1 Σ q * (0) -ε(p * -∆p * ) ∈ ∂T V (p * ). (3.6)
Finally, the first order optimality conditions are stated in the following result.

Theorem 3.2. Fix K, Σ, ε > 0 and γ > 0. The function p * ∈ H 1 0 (Ω) is the optimal solution to (3.3) if and only if equation (3.6) is satisfied with q * solution to (3.5) and p * solution to (2.1) with p * as initial condition.

The optimality condition above is sufficient by a convexity argument.

Remark 3.2. The computation of ∂T V (p * ) is standard in a finite dimensional setting, either using a primal-dual algorithm or performing an approximation of T V (p * ) .

Analysis of Problem (2.13)

We now focus on the optimal design problem (2.13). To perform the analysis, we give an equivalent formulation of A 2 (1 Σ , p 0 ) using the particular form of 1 Σ . Indeed, for all x ∈ Σ, we will say that Γ is a subset of ∂Ω associated to Σ ∈ U L whenever

1 Σ (x) = 1 Γ (s + µν(s)) = 1 Γ (s), for a.e. x ∈ ∂Ω, s ∈ ∂Ω, µ ∈ [0, ε] (3.7)
for some 1 Γ ∈ V L where 1 Γ is the characteristic function of a subset Γ of ∂Ω. Using Fubini-Tonelli-Lebesgue theorem, one has

A 2 (1 Σ , p 0 ) = 1 p 0 2 H 1 (Ω) T 0 B 1 Σ (x)∂ t p [p 0 ] (t, x) 2 dx dt = 1 p 0 2 H 1 (Ω) ∂Ω 1 Γ (s)ψ [p 0 ] (s) dH d-1 (3.8)
where

ψ [p 0 ] (s) = T 0 ε 0 ∂ t p [p 0 ] (t, s + µν(s)) 2 dµ dt (3.9)
Indeed, it follows directly from the regularity property of p [p 0 ] that ψ [p 0 ] ∈ L 1 (∂Ω). Next theorem gives an existence result: [START_REF] Bergounioux | An optimal control problem in photoacoustic tomography[END_REF]. Then, the optimal design problem (2.13) has at least one solution. Moreover, there exists a real number λ such that Σ * is a solution of (2.13) if and only if Σ * is associated to Γ * (in the sense of (3.7)) where

Theorem 3.3. Let p 0 ∈ H 1 (Ω) be a solution of Problem (2.
1 {ψ [ p 0 ] (s)>λ} 1 Γ * (s) 1 {ψ [ p 0 ] λ} (s),
for a.e. s ∈ ∂Ω.

(3.10)

Proof. The proof is a direct adaptation of [START_REF] Privat | Complexity and regularity of maximal energy domains for the wave equation with fixed initial data[END_REF]Theorem 1]. It is based directly on the fact that the functional A 2 rewrites as (3.8), as well as a standard argument of decreasing rearrangement. Another convexity argument can be used to get this result, observing that the functional Ã2 :

L ∞ (∂Ω, [0, 1]) ρ → ∂Ω ρ(s)ψ [p 0 ] (s) dH d-1
is continuous for the weak-star topology of L ∞ and that the set

C L = ρ ∈ L ∞ (∂Ω, [0, 1]) | ∂Ω ρ = LH d-1 (∂Ω)
is compact for this topology. It follows that the problem

inf ∂Ω ρ(s)ψ [p 0 ] (s) dH d-1 , ρ ∈ C L
has at least one solution. Moreover, one shows easily that the solution can be chosen among the extremal points of the convex set C L , by using the convexity of the mapping Ã2 .

Analysis of Problem (2.21)

This section is devoted to solving Problem (2.21) in the two-dimensional case, where the sensors set is the union of N 0 similar connected components (the definition of the sensors set has been made precise in (2.19)). Let us define .11) where ψ [ p 0 ] is given by (3.9).

f [ p 0 ] : T θ → ψ [ p 0 ] (ρ(θ) cos θ, ρ(θ) sin θ). ( 3
Theorem 3.4. Let p 0 ∈ H 1 (Ω) be a solution of Problem (2.11). Then, the optimal design problem (2.21) has at least a solution Σ N 0 , . Let Γ N 0 , (resp. (θ n ) n∈{1,...,N 0 } ∈ T N 0 ) be the associated set of ∂Ω in the sense of (2.14) (resp. the associated family of angles in the sense of (2.15)). Then,

1. if N 0 = 1, then, f [ p 0 ] (θ 1 ) = f [ p 0 ] ( θ1 ). 2. if N 0 2, let n ∈ {1, . . . , N 0 -1}. One has the following alternative: either θn = θ n+1 , or f [ p 0 ] (θ n ) = f [ p 0 ] ( θn ).
If one assumes furthermore that ρ belongs to C 2 (T), then one has

f [ p 0 ] ( θn ) ρ( θn ) 2 + ρ ( θn ) 2 - f [ p 0 ] (θ n ) ρ(θ k n 2 + ρ (θ n ) 2 0 whenever N 0 = n = 1 or N 2 2 and n ∈ {1, . . . , N 0 -1} is such that θn < θ n+1 .
Proof. We first assume, without loss of generality that N 0 2, the case where N 0 = 1 being easily inferred by adapting the following reasoning. In what follows, we will use several times that ρ(θ) > 0 for all θ ∈ T, which is a consequence of the fact that the point O (center of the considered orthonormal basis) belongs to the open set Ω.

First, using the same computations as those at the beginning of Section 3.3, we claim that Problem (2.21) can be recast as sup

(θ 1 ,...,θ N 0 )∈Θ L,N 0 J(θ 1 , . . . , θ N 0 ) (3.12)
where

J(θ 1 , . . . , θ N 0 ) = N 0 n=1 θn θn f [ p 0 ] (θ) ρ(θ) 2 + ρ (θ) 2 dθ, (3.13)
with f [ p 0 ] defined by (3.11), θn defined by (2.16) and

Θ L,N 0 = (θ 1 , . . . , θ N 0 ) ∈ T N 0 | θ n+1 θn ρ(s) 2 + ρ (s) 2 ds , n = 1, . . . , N 0 -1 . (3.14)
Existence of a solution to this problem is standard and follows immediately from both the compactness of Θ L,N 0 in T N 0 and the continuity of the functional. Let (θ 1 , . . . , θ N 0 ) ∈ Θ L,N 0 be a solution of the optimization problem (3.12). Let us assume the existence of n ∈ {1, . . . , N 0 -1} such that θn < θ n+1 , in other words such that

θ n+1 θn ρ(s) 2 + ρ (s) 2 ds > . (3.15)
Then, it follows from the Karush-Kuhn-Tucker theorem that ∂J ∂θn (θ 1 , . . . , θ N 0 ) = 0, which rewrites

ρ(θ n ) 2 + ρ (θ n ) 2 f [ p 0 ] (θ n ) = f [ p 0 ] ( θn ) ρ( θn ) 2 + ρ ( θn ) 2 ∂ θn ∂θ n (θ n ), simplifying into f [ p 0 ] (θ n ) = f [ p 0 ] ( θn ), (3.16)
by using that

∂ θn ∂θ n (θ n ) = ρ(θ n ) 2 + ρ (θ n ) 2 ρ( θn ) 2 + ρ ( θn ) 2 , (3.17)
according to the combination of (2.16) with the implicit functions theorem. Furthermore, assuming that ρ is a C 2 function, the necessary second order optimality conditions write

∂ 2 J ∂θ 2 n (θ 1 , . . . , θ N 0 ) 0, which comes to - ρ (θ n ) (ρ(θ n ) + ρ (θ n )) ρ(θ n ) 2 + ρ (θ n ) 2 f [ p 0 ] (θ n ) -f [ p 0 ] (θ n ) ρ(θ n ) 2 + ρ (θ n ) 2 + ∂ θn ∂θ n (θ n )f [ p 0 ] ( θn ) ρ(θ n ) 2 + ρ (θ n ) 2 + ρ (θ n ) (ρ(θ n ) + ρ (θ n )) ρ(θ n ) 2 + ρ (θ n ) 2 f [ p 0 ] ( θn ) 0 (3.18)
Combining this inequality with both the optimality condition (3.16) and the relation (3.17) yield the desired result.

Numerical simulations

We are now interested in the numerical resolution of the problems according to the strategy provided by (2.11)-(2.13). However, as explained above, we will make some simplifications of the model for its numerical implementation.

We assume here that the sound speed c(•) is constant equal to 1. Since the aim is to validate the methodology introduced in the section 2.2, this simplification allows us to calculate in a simple way the solution of the Fourier space wave equation and thus obtain a high resolution in time and space.

Minimizing J 0 functional (Problem (2.11)) is carried out using a gradient splitting descent. More precisely, we alternate explicit processing of A 1 (1 Σ , p 0 ) via a time reversal imaging method [START_REF] Ammari | Time reversal in attenuating acoustic media[END_REF][START_REF] Bretin | A time reversal algorithm in acoustic media with dirac measure approximations[END_REF][START_REF] Hristova | Time reversal in thermoacoustic tomography-an error estimate[END_REF][START_REF] Hristova | Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media[END_REF], with implicit processing of the term T V using the algorithm introduced by A. Chambolle in [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF]. In addition, in practice, we also apply the FISTA strategy [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF] to accelerate convergence.

With regard to determining the best position for the sensors Γ * , we calculate the energy function ψ [p 0 ] defined for all s ∈ ∂Ω by

ψ [p 0 ] (s) = T 0 ∂ t p [p 0 ] (t, s) 2 dt.
• For Problem (2.13) (continuous setting): We apply Theorem 3.3, which shows that Γ * should have the form

Γ * λ = {s ∈ ∂Ω | ψ [p 0 ] (s) λ}.
It is also a simplification since ψ [p 0 ] is only observed on ∂Ω and the optimal constant λ is determined by using a dichotomy approach, looking for λ so that the constraint

H d-1 (Γ * λ ) = LH d-1 (∂Ω)is satisfied.
• For Problem (2.21) (discrete setting, with a maximum number of connected components): although Theorem 3.4 provides a partial characterization of the solution, we have observed numerically the existence of many local optima. For this reason, we have chosen to set up a genetic algorithm procedure. Precisely, we solve Problem (3.12) by using the Matlab c function ga designed to solve a finite dimensional optimization problem with a genetic algorithm. In practice, it is not worth taking into account the constraint (3.15) since it is naturally satisfied to the optimum.

We recall several standard results about time reversal Imaging in Section (4.1). In addition, in section (4.2), we show how the gradient of the functional A 1 (1 Σ , p 0 ) can be calculated using a generalized time reversal imaging technique. We also present a classical discretization of the wave equation (4.3) and its integration into Fourier space. Finally, we show some numerical illustrations of our approach in section 4.4, which highlights the improved reconstruction of the source p 0 when optimizing the position of the sensors.

Time reversal imaging

Recall that p [p 0 ] denotes the solution of Problem (2.1) for the initial datum p 0 . If the given data p obs are complete on the boundary of Ω i.e

p obs (t, y) = p [p 0 ] (t, y) for all (t, y) ∈ [0, T ] × ∂Ω,
then, the reconstruction of the initial source p 0 from data g = p obs can be done following time reversal imaging method, by using that

p 0 (•) I[p obs ](•) = w(T, •),
where w is defined as the solution of the (backward) wave equation

     ∂ tt w(t, x) -∆w(t, x) = 0, (t, x) ∈ [0, T ] × Ω, w(0, x) = ∂ t w(0, x) = 0, x ∈ Ω, w(t, y) = p obs (y, T -t), t ∈ [0, T ].
More precisely, T is required to be sufficiently large to satisfy u(T, •) 0 and ∂ t u(T, •) 0 on Ω [START_REF] Hristova | Time reversal in thermoacoustic tomography-an error estimate[END_REF]. However, as explained in [START_REF] Ammari | Time reversal in attenuating acoustic media[END_REF], the discretization of this imaging functional requires data interpolation on the boundary of Ω: this introduces smoothing effects on the reconstructed image (identical to the use of the penalization term R(p 0 ) in Problem (2.8)). In practice, it is more efficient to use an approximation version reading

I[p obs ](x) = T 0 v s (T, x)ds,
where v s solves the wave equation

∂ 2 tt v s (t, x) -∆v s (t, x) = ∂ t δ {t=s} g(x, T -s) δ ∂Ω , (t, x) ∈ IR × IR d v s (t, x) = 0, ∂ t v s (t, x) = 0, x ∈ IR d , t < s.
Here, δ {t=s} denotes the time Dirac distribution at time t = s and δ ∂Ω is the surface Dirac measure on the manifold ∂Ω.

In particular, by using the so-called Helmholtz-Kirchhoff identity, it is proved in [3] that when Ω is close to a sphere with large radius in IR d , there holds p 0 (x) I[p obs ](x).

Another advantage of the modified time reversal imaging technique is its variational character. Indeed, recall that p [p 0 ] can be expressed as

p [p 0 ] (•) = ∂ t G(t, •) * p 0 ,
where * is the convolution product in space, G the temporal Green function obtained as the inverse Fourier transform of G ω

G(t, •) = F -1 t [G ω (•)](t),
where G ω denotes the outgoing fundamental solution to the Helmholtz operator -(∆+ω 2 ) in IR d , that is the distributional solution of the equation

(∆ + ω 2 )G ω (x) = -δ {x=0} x ∈ IR d
subject to the outgoing Sommerfeld radiation equation

lim |x|→∞ |x| d-1 2 ∂ ∂|x| -iω u(x) = 0.
The discrepancy functional A 1 defined by (2.9) can then be recast as

A 1 (1 Σ , p 0 ) = 1 2 T 0 B
1 Σ (y)((∂ t G(t, .) * p 0 (.))(y)p obs (t, y)) 2 dy dt.

Its Gâteaux-derivative with respect to the variable p 0 , say

dA 1 (1 Σ , p 0 ), h = lim τ 0 A 1 (1 Σ , p 0 + τ h) -A 1 (1 Σ , p 0 ) τ , writes dA 1 (1 Σ , p 0 ), h = Ω ∇A 1 (1 Σ , p 0 )(x)h(x) dx,
where ∇A 1 (1 Σ , p 0 ) is the gradient with respect to p 0 , identified to

∇A 1 (1 Σ , p 0 ) = T 0 ∂ t G(t, .) * (p [p 0 ] (t, •) -p obs (t, •))1 Σ (•) dt or ∇A 1 (1 Σ , p 0 ) = T 0 v s (T, •) ds, (4.1)
where v s solves

∂ 2 tt v s (t, x) -∆v s (t, x) = ∂ t δ {t=s} p [p 0 ] (T -s, •) -p obs (T -s, •) 1 Σ (•), v s (t, x) = 0, ∂ t v s (t, x) = 0, x ∈ IR d , t < s.
This claim follows from an straightforward adaptation of the proof of Theorem 3.2. In particular, the gradient ∇A 1 (1 Σ , p 0 ) corresponds to the modified time reversal imaging associated to the data p [p 0 ] -p obs , where the Dirac mass δ ∂Ω is replaced by the characteristic function 1 Σ and

∇A 1 (1 Σ , p 0 ) = I[p [p 0 ] -p obs ].

Solving Problem (2.11)

In this section, we focus on numerical algorithms to solve Problem (2.11). Because of the smoothing effects mentioned in Section 4.1, it is not worth to add a H 1 -penalization term from a practical point of view. This is why we eventually consider that

J 0 (1 Σ , p 0 ) = A 1 (1 Σ , p 0 ) + γT V (p 0 ).
A first idea is to use a gradient-iterative scheme with an implicit treatment of the T V norm, combined with an explicit treatment of A 1 (1 Σ , p 0 ). In this context, the simplest iterative scheme is the forward-backward algorithm, reading

p n+1 0 = (I + ηγ∂T V ) -1 (p n 0 -η∇A 1 (1 Σ , p n 0 )), n 0,
where p 0 = 0 and η is a given (small positive) descent step. The T V proximal operator prox ηT V [u] is defined by

prox ηT V [u] = (I + η∂T V ) -1 (u) = argmin v∈L 2 (B) 1 2η u -v 2 L 2 (B) + T V (v) .
is computed by using the dual approach introduced by Chambolle in [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF].

Finally, the gradient ∇A 1 (1 Σ , p 0 ) is computed via a time reversal imaging approach. Indeed, ∇A 1 (1 Σ , p 0 ) given by (4.1) can also be expressed (using the superposition principle) as

∇A 1 (1 Σ , p 0 ) = v s (T, •), with ∂ 2 tt v s -∆v s = F, (t, x) ∈ IR + × IR d , v s (0, x) = 0 = ∂ t v s (0, x) = 0, x ∈ IR d ,
where the right hand-side term is the measure On Figure 4.2, we give the result of the reconstruction procedure described in section 2.2 using partial data g = p obs with L = 0.3. To test its effectiveness, we also added 5% of noise on the data g. The position of sensors is plotted with white marks on each picture. Each line corresponds to a different choice of the source p 0 : we have plotted the source p 0 , the result of the reconstruction using the time reversal imaging p 0 0 and the reconstruction of the source p n 0 after n = 30 iterations. In particular, we observed that some information on p 0 are lost and in particular the discontinuities with normal directions that do not meet any sensor. • the energy function ψ u[p n 0 ] computed on Ω and the associated optimal position of sensors plotted with red marks.

F = ∂ t δ {t=s} p [p 0 ] (T -s, •) -p obs (T -s, •) 1 Σ .
• the reconstructed source p n 0 after n = 15 iterations after using the new position of sensors. Finally, as expected, the reconstruction of the source is much better by using the new position of sensors even if, the reconstruction remains unperfect.

We present now an illustration of the influence of the initial location of the sensors on our procedure. On Fig. 4.4, we present several numerical experiments where each line corresponds to a different value of the initial location. As with previous simulations, we observe that the reconstruction of the source is much better by using the new position of the sensors. As for choosing a good initial position, it is not an easy task, especially since we do not have any a priori information on the source. Let us illustrate the interest of using the term A 2 as a good reconstruction quality factor. On Figures 4.5 and 4.6, we first compute the optimal position of sensors, respectively in the continuous and discrete1 settings, using the true value of the source term p 0 . Second, using the new sensors position, we provide an estimate of the source p 0 by solving Problem (2.11). In each case, we observe that the source reconstruction is almost perfect.

Conclusion

This article is a first attempt to efficiently locate sensors in the highly sensitive context of thermo-acoustic tomography. While the first numerical results appear promising, we plan to investigate this issue further by examining the following elements

• other modeling choices. In particular, are there better choices of functional A 2 as quality factors for reconstruction (e.g. other observers)?

• how to choose more appropriately the term regularization in the optimization problem and study the sensitivity of the solution to this term?

• will the iterative scheme of successively estimating the source p 0 and then a new position of the sensors converge? In this case, can the limit be identified?

• can we analyze the relationships between the reconstruction parameters and the number of sensors used for the experiment?

• how to improve the optimization procedure, especially when it comes to a finite number of sensors for which we use an efficient but expensive genetic algorithm?

• could the reconstruction be improved if one is able to rotate the sensors freely around the target? Indeed, a first refinement of the method introduced in the present article could read as follows: fix N sensors. Compute the best location. Perform k -1 rotations to get kN measurements and go to next step to find the new best location with kN sensors. However, this issue needs to be more thorough and thoughtful, taking into account its practical implementation and possibly proposing a dedicated reconstruction algorithm.

More generally, we also plan to collaborate with physics researchers to conduct experiments and test our approach and method on real medical imaging data. We believe that the techniques developed in this article can be adapted to many other situations. Nevertheless, it is likely that several additional constraints on the sensor set must be taken into account, usually a restriction on the sensor position areas.

Figure 1 : 3 Figure 2 . 1 :

 1321 Figure 1: The set of sensors.

Figure 2 :

 2 Figure 2: A bad situation.

Figure 2 . 2 :

 22 Figure 2.2: A bad situation.

Theorem 3 . 1 .

 31 Fix ε > 0 and γ > 0. Then problem (3.3) has a unique solution p * .

Figure 4 . 1 :

 41 Figure 4.1: Source reconstruction using time reversal imaging I. Each line corresponds to a different choice of the source p 0 . Left: initial source p 0 ; middle: data g = p obs ; right: I[g]. The white dots corresponds to sensors positions.

Figure 4 . 2 :

 42 Figure 4.2: Minimization of J 0 w.r.t. (1 Σ , p 0 ). Each line corresponds to a different choice of the source p 0 . Left: initial source p 0 ; middle: reconstruction using time reversal imaging-I[g]; right: reconstructed source p n 0 after n = 30 iterations. The white dots correspond to sensors positions.

Figure 4 . 3 :

 43 Figure 4.3: Optimization of sensors position. Each line corresponds to a different choice of the source p 0 ; Left: reconstructed source p n 0 after n = 30 iterations by using initial positions of the sensors; middle: function ψ p n 0 defined by (3.9) on Ω and new position of sensors (red dots); right: reconstructed source p n 0 after n = 20 iterations by using the new position of sensors.

Figure 4 . 4 :

 44 Figure 4.4: Optimization of sensors position : influence of the initial position of sensors. Each line corresponds to a different choice of the initial location of the sensors ; Left: reconstructed source p n 0 after n = 30 iterations by using initial positions of the sensors; middle: function ψ p n 0 defined by (3.9) on Ω and new position of sensors (red dots); right: reconstructed source p n 0 after n = 20 iterations by using the new position of sensors.

Figure 4 . 5 :Figure 4 . 6 :

 4546 Figure 4.5: Optimization of sensors position. Each line corresponds to a different choice of the source p 0 . Left: function ψ [p 0 ] (x) defined by (3.9) on Ω and the best position of sensors (green dots); middle: reconstruction by using the time reversal imaging I[g]; right: reconstructed source p n 0 after n = 30 iterations by using the resulting optimal position of sensors.

Meaning that we consider a given number of sensors.

* Y. Privat was partially supported by the Project "Analysis and simulation of optimal shapes -application to lifesciences" of the Paris City Hall.

Time reversal Imaging, wave equation and discretization

Recall that the direct problem and the time reversal imaging approach only require to solve a Cauchy wave equation on the form

Comments on numerics. With regard to numerical discretization, all wave-like equations are solved in in the box B = [-D/2, D/2] 2 with periodic boundary conditions, where D is supposed to be large enough to prevent any reflection on the boundary. The numerical integrations of each equation are then performed exactly in Fourier space.

We recall that the Fourier truncation of a two-dimensional function

Now, about computation of wave equation solutions, remark that a generic wave equation

which can by simply integrated by solving the linear EDO system

.

Numerical experiments

All the numerical simulations of this section are done with the following set of parameters:

• the set Ω is a two-dimensional ball of radius 1

and the record time is T = 2;

• the set K is a two-dimensional ball of radius 0.85

• we use a regular time step discretization dt = T /2 10 and dx = D/2 9 .

• the thickness parameter ε is equal to 0.03.

• the T V -parameter γ = 0.01.

• the descent step η = 0.5.

To avoid inverse crime [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF][START_REF] Wirgin | The inverse crime[END_REF], we also use two different grids to compute the solution of the direct and inverse problem: the direct data g are then computed on a grid two times smaller, namely dx = D/2 10 .

On Figure 4.1, we provide a first experiment using ideal complete data g = p obs on the whole boundary ∂Ω as well as the time reversal imaging I(g) for three different values of the pressure p 0 . It is observed that in each case, the reconstructed source and the exact source are very close.


Impact of cerebral blood flow level on the fluctuations of resting-state BOLD fMRI in anesthetized rats

Sophie Achard (1), Guillaume Becq (1), Tarik Habet (2), Nora Collomb (2), Margaux Faucher (2), Chantal Delon-Martin (2), Véronique Coizet (2), and Emmanuel L Barbier (2)

(1) Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France

(2) Univ. Grenoble Alpes, INSERM, CHU Grenoble Alpes, GIN, Grenoble, France

Synopsis

The Blood Oxygen Level Dependent (BOLD) signal, used in resting-state functional Magnetic Resonance Imaging (fMRI), is tighly linked to Cerebral Blood Flow (CBF). This study evaluates the impact of the CBF on the lowfrequencies BOLD fluctuations and four physiological parameters during restingstate in 5 groups of rats (Wistar and Long-Evans) anesthetized with Isoflurane, Medetomidine, Etomidate and Urethane. It is shown that the CBF is not related to physiological parameters, and there exists a range of CBF values where the BOLD fluctations are sufficiently high for being used in any other analysis.

Introduction

In the context of preclinical imaging, one generally uses anesthesia to restrain the animal while in the MRI scanner. One can choose its anesthesia based on EEG measurements, which estimate the brain activity level. But anesthesia also modulates the cerebral blood flow (CBF) and thereby may modulate the fluctuations of the BOLD signals. It is thus of interest to search for the influence of CBF on BOLD signals at different frequencies. Also, as the anesthetic is changing the physiological parameters, it is important to evaluate the link between physiological measurements and CBF.

Materials and Methods

Groups: 5 groups of rats were considered. One group of Wistar rats, anesthetized with isoflurane (1%; n = 6), 4 groups of Long Evans rats anesthetized with isoflurane (1%; n = 6), Urethane (n = 7), Medetomidine (bolus of 0.24 mg / kg and infusion of 0.12 mg / kg / h; n = 6), and Etomidate (bolus of 10 mg / kg and infusion, n = 7). Data acquisition: Rats were imaged at 9.4 T with a volume transmit, 4-channel surface receive cross coil configuration (Paravision 6.0.1, Bruker, Ettlingen, Germany). After an anatomical T2w scan, cerebral blood flow (CBF) was mapped using pseudo continuous arterial spin labeling (pCASL) following inter-pulse phase optimization (Hirschler et al early view). Then, resting state fMRI was performed with a single shot echo-planar imaging (TE = 20 ms, TR = 500 ms, spatial resolution 0.47 x 0.47 x 1.00 mm, 9 slices, 3600 repetitions, duration: 30min). Eventually, a second CBF map was acquired. Heart rate and breath rates were monitored throughout the entire procedure. Data processing: Statistical Parametric Mapping program (SPM12, see 1) was used to co-register anatomical, CBF, and functional MRI data. CBF maps were obtained using homemade routines based on 2. CBF values and rs-fMRI time courses were computed for each of the 51 parcels of an homemade rat brain atlas. Briefly, the atlas was obtained by fusing two existing atlases proposed in [3 -4], and merging small regions. Mean CBF values were extracted by averaging the values obtained before and after the rs-fMRI acquisition. Each rs-fMRI time course was decomposed using a dyadic wavelet transform at different scales (detail or wavelet coefficients cD1 to cD7, in addition scaling or approximation coefficients (cA7) captures the polynomial trends contained in the signal.The 0.05 -0.10 Hz frequency band, cD4, is the one generally evaluated in resting-state fMRI. Standard deviations (std) of the wavelet coefficients are computed for each area and each individual in each group for quantifying the fluctuations of the BOLD signals.

1

General anesthetics depending on the drug modulate the CBF by up to one order of magnitude (Figure 1). For each anesthetic, the CBF values are homogeneous. Moreover, there is no evidence of a link between CBF and physiological parameters (breath rate, heart rate...) (Figure 2). Figure 3 shows the std of each wavelet coefficient time series (as a function of CBF, per area and per bin of 10 ml / 100 g / min). Distribution shapes seem comparable between scales and groups. Generally, the BOLD signals follow a trapezoidal shape with a plateau for CBF values between 20 and 150 ml / 100 g / min, and become null for higher CBF values. The amplitude of the plateaus are very similar, despite the broad range of CBF values (cf. Figure 1). The shapes for Iso-L and Iso-W are slightly different with larger transitions to and from plateau leading to a larger trapezoidal shape. For Iso-W, the plateau for cD4 BOLD std is reached for 80 ml / 100 g / min and decreases for CBF values greater than 150 ml / 100 g / min.

Discussion

For a large majority of anaesthetic agents, the amplitudes of the standard deviations of the resting state BOLD signals are not related to CBF. When applying wavelets, the fluctuations of the BOLD signals remain large. Wavelets have been shown to handle properly fractal properties of the BOLD signals as reported in 5, which is of particularly interest in the context of brain connectivity. It is shown here that wavelet decompositions are not altering the BOLD standard deviation, especially at low frequencies usually used for preprocessing BOLD signal as in 5. If all anesthetics show a trapezoidal shape of distribution with more or less transition steps, isoflurane seems to induced some relations with perfusion, especially for the Wistar strain. This is potentially due to the vasodilator effect of this molecule as noted in 6. For large values of CBF, BOLD fluctuations are decreasing.

Conclusion

For the anaesthetic agents evaluated in this study, dynamics of brain area signals are not related to CBF. However, isoflurane yields to reduced fluctuations of BOLD signals, for some brain areas with too low or too large values of CBF.

Acknowledgements

This work was partly funded by a grant from AGIR-PEPS, Université Grenoble Alpes-CNRS.

8 References

Figures

Figure 1 :

 1 Figure 1: Brain areas of the atlas (atlas), normalized CBF map and normalized FMRI (FMRI). CBF values are computed for each area. CBF distributions per group represented by boxplots for all areas and all individuals. Distributions of the standard deviation values of the wavelet detail coefficients cD4 ([0.06, 0.13[ Hz) obtained from the bold signal in each area (Bold cD4) versus CBF. Each color represents an individual inside a group.

Figure 2 :

 2 Figure 2: Mean CBF versus mean physiological values for each individual in each group.

1

  Members and collaborators of the Wellcome Trust Centre for Neuroimaging, SPM: Statistical parametric mapping, http://www.fil.ion.ucl.ac.uk/spm/ 2 Alsop DC, Detre JA, Golay X, Günther M, Hendrikse J, Hernandez-Garcia L, Lu H, MacIntosh BJ, Parkes LM, Smits M, van Osch MJ, Wang DJ, Wong EC, Zaharchuk G. Recommended implementation of arterial spin-labeled perfusion MRI for clinical applications: A consensus of the ISMRM perfusion study group and the European consortium for ASL in dementia. Magn Reson Med. 73(1):102-16, 2015. 3 P. A. Valdés-Hernandez et al., An in vivo MRI template set for morphometry, tissue segmentation and fMRI localization in rats, Frontiers in Neuroinformatics, 5(26):1-19, 2011. 4 E. A. Papp et al., Waxholm Space atlas of the Sprague Dawley rat brain, Neuroimage, 97: 374-386, 2014. 5 S. Achard et al., A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs, J. Neurosci., 26(1): 63-72, 2006. 6 K. S. Hendrich et al., Cerebral perfusion during anesthesia with fentanyl, isoflurane, or pentobarbital in normal rats studied by arterial spin-labeled MRI, Mag. Res. in Med., 46(1): 202-206, 2001.

 [0.25, 0.5[ [0.13, 0.25[ [0.06, 0.13[ [0.03, 0.06[ [0.02,


V Visentin

P Marq

S Bour

C Subra

D Prévot

N Morin

P Valet

M C Monje

F Nepveu

C Carpéné
email: carpene@toulouse.inserm.fr

Effect of prolonged treatment with tyramine on glucose tolerance in streptozotocin-induced diabetic rats

Keywords: Semicarbazide-sensitive amine oxidase, Monoamine oxidase, Diabetes, Insulin, Rats

The biogenic amine tyramine has been reported to stimulate in vitro glucose trans port in adipocytes, cardiomyocytes and skeletal muscle, and to improve in vivo glu cose utilization in rats. These effects were dependent on amine oxidation, since they were blocked by inhibitors of monoamine oxidase (MAO) and semicarbazide-sensi tive amine oxidase (SSAO). We thus tested in this work whether a prolonged treat ment with tyramine could improve glucose tolerance in streptozotocin-induced dia betic rats. First, tyramine content of standard rodent chow was determined by HPLC and daily tyramine intake of contrai rats was estimated to be around 26 µmol/kg body weight. Then, tyramine was administred during 3 weeks in strep tozotocin-induced diabetic rats at 29 µmol/kg by daily i.p. injection alone or togeth er with vanadate 0.02 µmol/kg. In another group of diabetic rats, tyramine was sub cutaneously delivered at 116 µmol/kg/day by osmotic minipumps. Ali tyramine treatments resulted in a decrease of the hyperglycemic responses to an i.p. glucose load. Adipocytes isolated from either untreated or treated diabetic rats were sensitive to the stimulation of glucose uptake by tyramine. However, diabetic animais receiv ing tyramine for three weeks did not recover from their hyperglycemia, hypoinsu linemia and glucosuria. These results show that the improvement of glucose tolerance induced by prolonged tyramine administration occurs in an insulin-depleted mode! and probably results from peripheral insulin-like actions of the oxidation of MAO/SSAO substrates, such as the stimulation of glucose uptake into adipocytes.

Amine oxidation is occuring in many tissues where it bas been considered as a way to scavenge exogenous or biogenic amines and to terminate the action of sev eral neurotransmitters [START_REF] Lyles | /nt[END_REF]. Two families of amine oxidases are involved in amine oxi dation: the F AD-dependent monoamine oxidases (MAO) and the copper-contain ing amine oxidase family, essentially rep resented by semicarbazide-sensitive amine oxidases (SSAO). We have recently reported that adipose and muscular tis sues, which are sensitive to insulin, regarding to the stimulation of glucose uptake, express substantial amount of MAO and SSAO (10). In keeping with this, several amines such as tyramine or benzylamine markedly stimulate glucose transport in adipocytes via their oxidation by MAO or SSAO (2,9). In fact, this stimulation of glucose transport takes place in a manner that partially resembles that of insulin, leading to the translocation of the insulin-sensitive glucose trans porter GLUT4 to the cell surface (16). Other insulin-like eff ects, have been reported for tyramine, which is a substrate for both MAO and SSAO: stimulation of adipose differentiation (5,8), inhibition of lipolysis (15), and, most importantly, in vivo acute improvement of glucose toler ance in conscious rats (10).

The present work aimed at investigat ing whether prolonged administration of tyramine could improve the impaired glu cose disposai of diabetic rats. Therefore, we have studied the capacity of chronical ly administred tyramine to modify the hyperglycemic response induced by glu cose load in rats previously rendered hyperglycemic and insulin-deficient by streptozotocin injection. First of all, we have estimated the amount of alimentary tyramine daily ingested by laboratory rats, by determining the amount of amines in the standard chow with an highperfor-mance liquid chromatographie (HPLC) method. Then, a dose approximatively equivalent to the estimated daily oral intake of tyramine, was daily i.p. injected during 3 weeks to streptozotocin-induced diabetic rats. Other groups of diabetic rats received vanadate at a dose ineffective on its own (0.02 µmol/kg), alone and in com bination with tyramine, since vanadate bas been reported to improve the insulin like effect of amine oxidase substrates (16) and the antidiabetic action of chronically administred benzylamine (1). A higher dose of tyramine (116 µmol/kg/day) was also chronically administred during three weeks via mini-osmotic pumps implanted in the dorsal region of streptozotocin dia betic rats.

W e report here that the hyperglycemia provocked by i.p. glucose load was reduced in tyramine-treated rats, when compared to untreated diabetic rats, tra ducing an overall increased glucose uti lization. Accordingly, tyramine and other amine oxidase substrates were able, in the presence of vanadate, to stimulate in vitro glucose uptake into adipocytes of diabetic rats. Although the prolonged treatments with tyramine did not completely correct the troubles induced by streptozotocin, such as reduced body weight gain, hyper glycemia and hypoinsulinemia, we pro pose that the improvement of glucose dis posai by MAO/SSAO substrates may be useful for the treatment of glucose intoler ance.

Materials and Methods

Chemicals.-Tyramine and amines used as standards for HPLC calibration, colla genase, cytochalasin B, fatty-acid-free bovine serum albumin, dansyl chloride, and other reagents were obtained from Sigma-Aldrich (Saint Quentin Fallavier, France). 2-[l,2-3 HJ-Deoxyglucose (2-DG, 26 Ci/mmol) was purchased from Perkin Elmer Life science Products (Boston, MA). HPLC analysis.-After derivatization with dansyl chloride, amines were sepa rated using a gradient elution and a reversed-phase HPLC method with a Waters HPLC system (Milford, MA, USA) coupled to fluorescence detection (Àex/em : 350/520 nm) as previously reported (14). A RP-C18 column (5 µm, 200 x 2 mm, 5 µm particle size, 200 A porosity) (Bishoff Chromatography, Leonberg, D) was used to separate the amines at 25 °C under a linear gradient elution (methanol/water at 0.3 ml/min). Retention times and peak areas were acquired and processed in a Waters Mille nium32 workstation. Amine-containing solutions were prepared from rat chow as following: pellets were crushed (2.5 g in 75 ml of water), mixed, and 75 ml of 0.4 M perchloric acid was added; the mixture was stirred for 30 min and centrifuged at 3600 rpm for 5 min. Samples were filtered through a 0.45 µm nylon filter (Millipore) before automated injection.

Animais and tissue sampling.-Male Wistar rats with free access to food and water were rendered hyperglycemic and insulin-deficient by single injection of streptozotocin ( 65 mg/kg) two weeks before prolonged tyramine treatments consisting in either daily i.p. injection of 29 µmol/kg or implantation in the dorsal region of osmotic minipump delivering 116 µmol/kg/d (Alzet, Palo Alto, CA). After three weeks of treament, animais were subjected to glucose tolerance tests. After sacrifice, the tissues removed were immediately used for adipocyte isolation and subsequent determinations of glucose uptake as previously described (4).

Rats were fasted during 6 hours before the tests. Blood samples were drawen from tail vein of conscious animais at the indicated times before and after glucose load (i.p. bolus of 2 g/kg at time 0). Blood glucose was immediately determined with Glucotrend II glucometer (Roche Diag nostics, Mannheim, D), as already detailed (10).

Statistical analysis.-Results are given as mean ± S.E.M. Statistical significance was assessed by use of Student's t-test. NS means no significant diff erence between the compared samples.

Results

Biogenic amine content of standard chow and daily tyramine intake in rat.

The pellets of the control chow given to rat (Global rodent diet, 2016 R, Harlan, France), containing 16 % protein and 4 % fat were treated as described in Materials and Methods and the content of the five biogenic amines mostly found in food or in animais (14) was determined by HPLC. The amine content (µg/g) was (mean ± SEM of three different determinations): tyramine, 45.3 ± 1.9; spermidine, 40.9 ± 3.2; spermine, 29.6 ± 2.3; putrescine, 14.8 ± 1.7; cadaverine 7.5 ± 0.8. Thus, tyramine was one of the major biogenic amines found in chow pellets while cadaverine was less abundant. Considering that Wis tar rats consumed, under our breeding conditions, approx. 20 g of pellets per day, the daily amount of ingested tyramine was estimated to be 26 µmol/kg body weight. We therefore decided to i.p. inject tyra mine at 29 µmol/kg/day in diabetic rats during three weeks. A fourfold larger dose (116 µmol/kg/day) was also tested via continuous s.c. administration by osmotic m1m-pumps.

Effect of repeated administration of tyramine and vanadate on glucose toler

ance in diabetic rats.lntraperitoneal glu cose tolerance tests (IPGTT) were con ducted at the end of the three-week pro longed tyramine treatment. The hyper glycemic response provocked by glucose load was markedly reduced in the group of diabetic rats chronically treated with tyramine 29 µmol/kg than in untreated diabetic animals: the peak increase of blood glucose occuring 15-30 min after glucose load was f!attened (Fig. 1 ). The chronic treatment with 0.02 µmol/kg, sodium orthovanadate did not diminish Influence of a repeated i.p. treatment with tyramine and/or vanadate on glucose tolerance test in male type 1 diabetic rats. After three weeks of treatment, rats were fasted dur ing 6 hours before glucose Joad (2 g/kg, i.p. bolus). Results are given as increase over blood glucose, which was, at cime 0 and in mg/100 ml: 386 ± 13 for untreated diabetic rats (closed squares), 513 ± 33 for vanadate-( closed circles ), 400 ± 28 for tyramine ( 4 mg/kg, open triangles), and 470 ± 22 for vanadate plus tyramine-group (closed triangles), respectively. Mean ± SEM of the number of rats given in paren theses. Different from untreated diabetic at '' p < 0.05.

the hyperglycemic response to IPGTT and was unable to improve the antihyper glycemic effect of chronically administred tyramine (Fig. 1 ). However, the daily administration of tyramine did not correct the reduced weight gain of diabetic ani mals since, after three weeks of treatment, the body weights were 221 ± 21, and 192 ± 8 g for untreated, and tyramine treated diabetic rats, vs 296 ± 15 g for nor momoglycemic controls (n = 5, p< 0.02). The low dose of vanadate was without any influence on weight gain and the rats treated daily with tyramine and/or vana date did not exhibited a normalization of their elevated blood glucose (not shown). A higher dose of tyramine was thus tested alone, without combination with vana date.

Influence of continuous tyramine administration on glucose tolerance and biological parameters of diabetic rats.

The hyperglycemic response of diabetic rats implanted with osmotic pumps deliv ering tyramine at 116 µmol/kg/day is shown in Fig. 2. A clear-cut decrease was found in the area under the curve (AUC) of the increase in blood glucose in the rats treated by prolonged tyramine adminis tration, when compared to untreated dia betic rats (arbitrary units were 13.0 ± 0.1 and 4.9 ± 0.1, n = 5, p< 0.01). According ly, the rats chronically treated with tyra mine exhibited a tendency to disminish their elevated fasting plasma glucose levels at the time of sacrifice (250.4 ± 18. 9 vs 328.4 ± 20.2 mg/100 ml, n = 5, p< 0.05) without returning to the levels of normo glycemic contrais (65.6 ± 3.4 mg/100 ml). However, chronic treatment with high dosage of tyramine did not correct other defects of streptozotocin-induced dia betes, such as reduced body weight (not shown), severe hypoinsulinemia (O. 9 ± 0.4 vs 0.7 ± 0.1 ng/ml, NS), increased plasma Influence of chronic tyramine treat ment on glucose uptake into adipocytes of diabetic rats.-To test whether the antihy perglycemic effect of tyramine observed during IPGTT could be explained by an increased peripheral glucose utilization in severely hypoinsulinic animals, we tested the insulin-like effects of tyramine on glu cose transport in adipocytes. Table I shows that tyramine was able to activate hexose uptake in white adipocytes, even in the absence of vanadate. lnterestingly, the fold increase over basal induced by 1 mM tyramine or by 100 nM insulin seemed to be greater in treated diabetic rats than either untreated diabetic rats or normo glycemic controls. Vanadate 0.1 mM did not modify basal or insulin-stimulated uptake but potentiated the effect of tyra mine in all the experimental groups.

Discussion

T aken together, our observations show that tyramine administration improves glucose tolerance in diabetic, insulin depleted rats, The antihyperglycemic properties of tyramine are likely the con sequence of a stimulation of hexose uptake in insulin-sensitive tissues, Even though we did not measure the variations of plasma insulin during IPGTT in the streptozotocin-treated rats, it can be sup posed that an indirect effect via stimula tion of insulin secretion was not involved in the antihyperglycemic action of tyra mine. A stimulation of glucose transport in peripheral tissues by tyramine, as evi denced in vitro on fat cells, is therefore expected to occur under in vivo condi tions. This insulin-like effect could be suf ficient to explain the increase in glucose tolerance, although an inhibition of hepat ic glucose output or a stimulation of insulin-secretion induced by tyramine could not be excluded since there is very high expression of MAO in liver (10) and since the role of MAO in insulin secretion is still unsolved in the endocrine pancreas (11 ). lt is important to note that the tyra mine-induced stimulation of glucose tran port we report here in adipocytes from diabetic rats has been previously oberved also in cardiomyocytes and skeletal mus cle from control rats (10). Tyramine could therefore be suspected to lower hyper glycemic response to glucose load by stimulating glucose uptake in all the insulin-sensitive tissues. Moreover, inhi bition of lipolysis is another insulin-like action of tyramine found on adipose cells (15) that may facilitate glucose disposai. The improvement of glucose disposai induced by a chronic treatment with tyra mine is in perfect agreement with our pre vious studies in which an acute adminis tration of tyramine, just before a glucose load, was found to exert antihyper glycemic properties (10). The present data also permit to assess that insulin-like effect of tyramine on glucose diposal per sists after repeated administration. This assessment, together with the blockade of tyramine effects by MAO and SSAO inhibitors (10), led to propose that the effect of tyramine is not a receptor-medi ated one, which could had been much more prone to desensitization, but is more likely a consequence of its metabolism by detoxifying enzymes.

The role of vanadate remains unclear since it was able to potentiate the acute eff ect of tyramine on the in vitro mode],

but not the chronic effect of the biogenic amine when tested in vivo. Although both doses of vanadate were ineff ective in IPGTT or in hexose uptake assays, they were peraphs not equivalent, when com pared to the respective tresholds of the diverse insulin mimicking actions of this transition metal. ln fact, we have already reported that an i.p. dose of vanadate at 25 µmol/kg is poorly effective in reducing per se the hyperglycemic response to IPGTT in rats. The daily dosage used for vanadate treatment in the present study (0.02 µmol/kg) can therefore be consid ered as being around one thousand fold lower than the treshold for acute in vivo antidiabetic eff ects of vanadate. Regarding stimulation of glucose uptake into rat adipocytes, the dose used to potentiate tyramine effect (0.1 mM) corresponded to the treshold for vanadate in our as well as in other studies ( 6 ). Therefore, the obser vation of a synergism between tyramine and vanadate only in isolated adipocytes, but not in vivo, could be explained by the fact that, even perhaps cumulative, only very low doses of vanadate were used for in vivo chronic administration. lnstead of increasing vanadate concen tration we chosen to increase tyramine dosage. Since it has been previously reported that, at the lower dose tested ( 4 mg/kg), an iv. bolus of tyramine induced a transient hypertensive effect that lasted less than 10 min in anaes thetized rats ( 10 ), we used a diff erent mode of administration: continous subcu taneous delivery via osmotic minipumps. With this mode of administration, we did not observe any behavioural change in the treated rats. It is important to note that both doses of tyramine tested in this study ( 4 and 16 mg/kg) were largely lower than the oral dose of 87 mg/kg reported to elic it an increase in blood pressure of 30 mm Hg in rat, which is a reference for clinical risk treshold (3). Moreover, the determi nation by HPLC of tyramine content in rodent chow allowed to make, at least to our knowledge, the first reported estima tion of the daily tyramine intake of a lab oratory rat: 26 µmol/kg. This corresponds to approx 1 mg of tyramine ingested/day by a rat of 250 g. Such an amount is found in many foods and beverages: for instance, the amount of tyramine contained in a liter of wine is 0.2-6 mg/1 [START_REF] Romero | [END_REF]. On the basis of our study, one can consider that it is sufficient to increase by two-to eight fold the spontaneous alimentary tyramine intake to obtain a significant effect on glu cose disposai. Indeed, the oral ingestion of tyramine led to an important, but not yet well defined, intestinal degradation. Therefore 29 µmoles of i.p. injected tyra mine led to an amount of biogenic amine in the peripheral tissues, including adipose depots, much more greater than brought by the dietary amine. In man, endogenous plasma tyramine concentration is around 10 nM under fasting conditions ( 4 ). This trace amount makes unlikely that only tyramine participates in a putative inter play between dietary amines and glucose metabolism. However, tyramine and other dietary amines share insulin-mimic king properties and we propose that their effects could be cumulative under in vivo conditions rather than competitive until the activity of amine oxidases in adipose tissues or in other peripheral organs does not reach saturation. Thus, it cannot be excluded that it is possible to reach with an oral load a sufficient amount of dietary amines capable to interfere with glucose metabolism as observed here with i.p. or s.c. administred tyramine. To support this view, it can be mentionned that mice fed a diet based on oat variety with a low polyamine content develop smaller lym phosarcoma tumours than animais fed a standard diet [START_REF] Tiburcio | COST 917 -Biogenically active amines in food[END_REF]. This shows that changes in the amount of dietary amines or polyamines can influence the organism maybe via changes in the absorbed amount of amines, escaping from degra dation by the intestinal barrier. At the end, the stimulating effects of chonic tyra mine challenge on glucose disposai we observed in a model of type I diabetes led us to propose that amine oxidation can constitute a rationale for therapeutic approach of diseases linked to glucose intolerance. Se ha descrito que la amina biogénica tira mina estimula in vitro el transporte de glucosa en adipocitos, cardiomiocitos y musculo esquelético y mejora la utilizaci6n de glucosa en la rata. Estos efectos son dependientes de la oxidaci6n de la tiramina por la monoamino oxidasa (MAO) y la amino-oxidasa sensible a semicarbazida (SSAO). En este trabajo, se estudia si un tratamiento cr6nico con tiramina aumenta la tolerancia a la glucosa en ratas dia béticas por estroptozotocina. El contenido en tiramina del alimento estandar para roedores se determin6 por HPLC y se estim6 que el con sumo diario de tiramina en ratas control era de unos 26 µmol/kg de peso corporal. Por tanto, se administr6 diariamente durante 3 semanas tiramina por vfa i.p. a la dosis de 29 µmol/kg, sola o con vanadato 0.02 µmol/kg a ratas dia béticas por estroptozotocina. Otro grupo reci bi6 tiramina por vfa subcutanea mediante minibombas osm6ticas liberando 116 µmol/ kg/dfa. Los tratamientos con tiramina han inducido disminuci6n de las respuestas hiper glucemiantes a la sobrecarga de glucosa. En adipocitos aislados de ratas diabéticas, tratadas o no, la tiramina estimula el transporte de glu cosa. Sin embargo, los animales diabéticos tra tados con tiramina no se recuperaron de su hiperglucemia, hipoinsulinemia y glucosuria. Nuestros datos sugieren que la mejora de la tolerancia a la glucosa inducida por el trata miento cr6nico con tiramina se observa en un modelo deficiente en insulina y probablemente se debe a acciones insulinomiméticas. Esta tambien indica que la administraci6n de sustra tos de MAO/SSAO podrfa constituir la base de nuevos tratamientos para mejorar la utiliza ci6n de la glucosa.

Palabras clave: Amino-oxidasa sensible a semicar bazide, Monoamino-oxidasa, Diabetes, Insulina, Rata.

  Fig. 1. Influence of a repeated i.p. treatment with tyramine and/or vanadate on glucose tolerance test inmale type 1 diabetic rats. After three weeks of treatment, rats were fasted dur ing 6 hours before glucose Joad (2 g/kg, i.p. bolus). Results are given as increase over blood glucose, which was, at cime 0 and in mg/100 ml: 386 ± 13 for untreated diabetic rats (closed squares), 513 ± 33 for vanadate-( closed circles ), 400 ± 28 for tyramine ( 4 mg/kg, open triangles), and 470 ± 22 for vanadate plus tyramine-group (closed triangles), respectively. Mean ± SEM of the number of rats given in paren theses. Different from untreated diabetic at '' p < 0.05.

Fig. 2 .

 2 Fig. 2. Influence of a continuous delivery of tyramine by osmotic pump on glucose tolerance test in diabetic rats.Male rats (6-hour fasted) received a glucose load (2 g/kg i.p., arrow). Blood glucose levels are given as increase over blood glucose, which was, at time 0: 412 ± 46 and 415 ± 50 mg/100 ml for untreated dia betic ( closed squares) and tyramine-treated rats (116 µmol/kg/d for 3 weeks, closed triangles), respectively. Different from untreated diabetic at ') < 0.05; ,,.,, p < 0.02.

  V. VISENTIN, P. MARQ, S. BOUR, C. SUBRA, D. PRÉVOT, N. MORIN, P. V ALET, M. C. MONJE, F. NEPVEU y C. CARPÉNÉ. Efecto del tratamiento prolonga do con tiramina sobre la tolerancia a la glucosa en ratas diabéticas. J. Physiol. Biochem., 59 (3 ), 225-232, 2003.

Table 1 .

 1 Stimulation of glucose uptake (increase over basal uptake) by insu/in , tyramine and vanadate in adipocytes from contrai or diabetic rats.

			Diabetics
	Grou p :	normo g lycemic	untreated	tyramine-treated
	lnsulin 100 nM	8.2 ± 1.1	10.2 ± 1.4	14.9 ± 2.9
	Tyramine 1 mM	2.1 ± 0.2	2.5 ± 0.4	3.1 ± 0.9
	Vanadate 0.1 mM	1.3 ± 0.2	1.0 ± 0.1	1.0 ± 0.3
	Vanadate + insulin	8.2 ± 1.0	11.6 ± 1.8	11.9 ± 3.6
	Vanadate + tyramine	7.1±1.0	7.3 ± 1.9	9.3 ± 4.0

Basal 2-DG uptake was 1.06 ± 0.17, 1.33 ± 0.27, and 1.35 ± 0.36 nmoles of 2-deoxyglucose/100 mg of lipids/1 O min in normoglycemic, untreated and tyramine-treated diabetics, respectively (n = 5, NS). Mean ± S.E.M. of live different determinations.

Acknowledgments

This work was partly supported by "Commu nauté de travail des Pyrénées", by European Union contract QLG1CT1999 00295 "TUNEUP" and by "Pole Aliment-Santé Midi Pyrénées".