Philippe Allain

Lise M'barek

Ratko Radakovic

Malika Noquet

Alexandre Laurent

Different aspects of emotional processes in apathy: Application of the French translated dimensional apathy scale

MEDIZINISCHE MIKROBIOLOGIE Wie der Malaria-Erreger

Plasmodium falciparum ist der Erreger der Malaria tropica, die gefährlichste Form der Infektionskrankheit. Die Parasiten werden von weiblichen Stechmücken der Gattung Anopheles auf den Menschen übertragen und vermehren sich in roten Blutkörperchen. Die Blutstadien des Erregers sind für die pathologischen Erscheinungen der Malaria verantwortlich und auch das Hauptziel der zellulären und humoralen Immunantwort. Allerdings ist die erworbene Immunität ineffizient, selbst nach wiederholter Infektion mit dem Parasiten. Die Fähigkeit, der Immunabwehr des Menschen zu entgehen, scheint auch der Grund zu sein, dass bislang keine wirksame Schutzimpfung gegen die Malaria entwickelt werden konnte. Die Mechanismen, mit denen P. falciparum die immunologischen Abwehrreaktionen seines Wirtes unterdrückt, waren bis heute weitgehend unbekannt. Nach neusten Erkenntnissen nutzt der Erreger dafür einen Regulationsmechanismus des Wirts, der Autoimmunerkrankungen vorbeugt.

Um Autoimmunreaktionen zu verhindern, exprimieren Immunzellen inhibierende Rezeptoren (Immunregulatoren), die für körpereigene Antigene (Selbst-Antigene) spezifisch sind. Einige Viren nutzen diese Selbst-Toleranz aus, indem sie Proteinmoleküle produzieren, die an diese inhibierenden Rezeptoren binden und so eine Immunantwort unterdrücken. Ein multinationales Forscherteam unter der Leitung der Abteilung für Immunchemie der Universität Osaka, Japan, untersuchte jetzt, ob P. falciparum in ähnlicher Weise inhibierende Rezeptoren von Immunzellen durch Expression von Proteinmolekülen auf der Oberfläche von infizierten Erythrocyten ausnutzt [START_REF]wahrscheinlich auch dafür verantwortlich, dass sich im Laufe einer Malaria-Infektion nur eine partielle Immunität entwickelt[END_REF]. Zunächst testeten die Forscher die Bindung von 13 menschlichen Immunregulatoren an Erythrocyten, die entweder mit dem P. falciparum Laborstamm 3D7 oder mit aus Malaria-Patienten isolierten Erregern infiziert waren. Nur für einen der untersuchten Rezeptoren, den Leukocyten-Immunglobulin-ähnlichen Rezeptor B1 (leucocyte immunoglobulin-like receptor B1, LILRB1),war bei einem Teil der infizierten Erythrocyten eine Bindung nachweisbar. LILRB1 interagierte auch mit Erythrocyten, die mit vier anderen P. falciparum Laborstämmen (CDC1, K1, FCR3 und Dd2) infiziert waren.

  Plasmodium falciparum die Immunabwehr seines Wirtes manipuliert Die natürlich erworbene Immunität gegen Malaria ist nur partiell und kann eine erneute Infektion nicht verhindern. Die zugrundeliegenden Mechanismen der Immunmanipulation waren bislang nicht bekannt. Jetzt wurde gezeigt, dass der Malaria-Erreger Plasmodium falciparum bestimmte Proteine auf der Zelloberfläche von infizierten Erythrocyten exprimiert, die an inhibierende Rezeptoren von Immunzellen binden. Dadurch werden die Immunzellen in ihrer Aktivität gehemmt und die Immunantwort gegen die Malaria-Erreger abgeschwächt.

  Abb. Schema zum Mechanismus der Immunmanipulation durch Plasmodium falciparum. Der Malaria-Erreger induziert die Expression von RIFIN-Proteinen auf der Zelloberfläche von infizierten Erythrocyten. Einige RIFIN-Proteine evolvierten zu Liganden für den inhibierenden Immunregulator LILRB1. Die Bindung von RIFIN an LILRB1 bewirkt eine Unterdrückung der Aktivität der Immunzelle. Dadurch kann der Malaria-Erreger der Immunabwehr seines menschlichen Wirtes entkommen. Dieser Mechanismus ist


Falcoz

J M Avellaneda
email: jean-marc.avellaneda@promes.cnrs.fr

F Bataille

A Toutant

G Flamant

Entropy generation minimization in a channel flow: Application to different advection-diffusion processes and boundary conditions

Keywords: Second law, Heat and mass transfer, Optimization, Variational analysis, Entropy minimization

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

Introduction

This study addresses the application of variational methods to improve heat and mass transfers in convective flows with optimal velocity fields as a result. In many industries and engineering applications, it is nec-5 essary to optimize heat or mass transfers in order to improve the technical or economic efficiency of machines, plants or processes. The intensification of heat transfer is sought in thermal power plants like concentrated solar [START_REF] Akbarzadeh | Heat transfer enhancement in parabolic trough collectors: A comprehensive review[END_REF][START_REF] Flamant | A new heat transfer fluid for concentrating solar systems: Particle flow in tubes, Energy Procedia[END_REF][START_REF] Soo Too | Enhancing heat transfer in air tubular absorbers for concentrated solar thermal applications[END_REF][START_REF] Avellaneda | DNS of turbulent low Mach channel flow under asymmetric high temperature gradient: Effect of thermal boundary condition on turbulence statistics[END_REF], nuclear [START_REF] Liu | Enhancement of heat transfer performance in nuclear fuel rod using nanofluids and surface roughness technique[END_REF] or geothermal [START_REF] Xu | A review on heat transfer and energy conversion in the enhanced geothermal systems with water/CO2 as working fluid[END_REF] ones, but also 10 in the process, automotive or aerospace industries and for cooling systems. Heat transfer enhancement is an innovative field of research [START_REF] Saha | Advances in Heat Transfer Enhancement[END_REF] that uses many techniques whether active (requiring an external energy input) such as the use of oscillating walls, or passive such as the ad-15 dition of mixing promoters in the flow channel [START_REF] Daguenet-Frick | Experimental analysis of the turbulent flow behav-870 ior of a textured surface proposed for asymmetric heat exchangers[END_REF] [START_REF] Colleoni | Optimization of winglet vortex generators combined with riblets for wall/fluid 875 heat exchange enhancement[END_REF] or the search for optimized transfer fluids. Convective mass transfer enhancement is also an area of particular interest for chemical processes [START_REF] Cai | Enhancement of CO2 absorption under taylor flow in the presence of fine 880 particles, Fluid Dynamics and Transport Phenomena[END_REF] and biotechnology [START_REF] Kraakman | Review of mass transfer aspects for biological gas treatment[END_REF][START_REF] Taherdazeh | Mass transfer enhancement in moving biofilm structures[END_REF] that take advantage of passive [START_REF] Aravind | Numerical study on convective mass transfer enhancement by lateral sweep vortex generators[END_REF][START_REF] Al Mers | Optimal 895 design study of cylindrical finned reactor for solar adsorption cooling machine working with activated carbonammonia pair[END_REF] or active 20 methods [START_REF] Saien | Liquid-liquid extraction inten-900 sification with magnetite nanofluid single drops under oscillating magnetic field[END_REF][START_REF] Elperin | Mass transfer during solute extraction 905 from a fluid sphere with internal circulation in the presence of alternating electric field[END_REF]. In both heat transfer and mass diffu-sion, the use of suspended fine particles [START_REF] Kordac | Mechanism of enhanced gas absorption in presence of fine solid particles. effect of molecular diffusivity on 910 mass transfer coefficient in stirred cell[END_REF] or nanofluids [START_REF] Ashrafmansouri | Mass transfer in nanofluids: A review[END_REF][START_REF] Bianco | Heat Transfer Enhancement with Nanofluids[END_REF] is also being studied to increase exchanges. The intensification of heat or mass transfers is often produced by the passive promotion or the active realization 25 of mixing [START_REF] Moradi | On the mixing enhancement in annular flows[END_REF] whose performance is dependent on the fluid flow pattern [START_REF] Winnemöller | Comparison of the mixing efficiency of different injector configurations[END_REF][START_REF] Aubin | Current methods for characterising mixing and flow in microchannels[END_REF][START_REF] Nathan | Impacts of a jets exit flow pattern on mixing and combustion performance[END_REF]. Therefore, searching for optimal flow patterns can be useful to improve heat or mass transfer in convective flows.

One way to address this issue is applying thermody-30 namic optimization by using the entropy generation rate as a criterion [START_REF] Bejan | A study of entropy generation in fundamental convective heat transfer[END_REF][START_REF] Narayan | Entropy generation minimization of combined heat and mass transfer devices[END_REF]. In convective flow heat and mass transfer, the total entropy generation rate is generally a sum of several terms. In the absence of radiative heat transfer [START_REF] Lou | Experimental and numerical analysis of radiative entropy generation in industrial and boiler furnaces[END_REF], chemical reaction [START_REF] Bouras | Entropy generation optimization in internal combustion engine[END_REF][START_REF] Bidi | A numerical evaluation of combustion in porous media by EGM (entropy generation minimization)[END_REF], cross effects such as Dufour or Soret ones [START_REF] Qayyum | Op-955 timization of entropy generation and dissipative nonlinear radiative Von Karman´s swirling flow with Soret and Dufour effects[END_REF] and dissipation related to electric or magnetic fields [START_REF] Bouabid | Heat 960 and mass transfer for Hartmann and Dufour´s effects on irreversibilities at double-diffusive natural convection in a square cavity[END_REF][START_REF] Mahmud | Thermodynamic 965 analysis of mixed convection in a channel with transverse hydromagnetic effect[END_REF], the entropy generation rate originates from viscous friction, on the one hand, and from the diffusion of heat or mass through finite temperature or mass concentration differences, on 40 the other hand. Acting on one of the two terms, for example by reducing the rate of heat conduction entropy generation, can lead to an increase in viscous friction entropy generation and it is necessary to seek the right compromise between two opposing effects. The total 45 generated entropy rate can be minimized in order to find this optimal trade-off [START_REF] Bejan | Entropy generation minimization: the new thermodynamics of finite-sized devices and finite-time processes[END_REF][START_REF] Bejan | Entropy generation minimization[END_REF][START_REF] Jankowski | Minimizing entropy generation in internal 975 flows by adjusting the shape of the cross-section[END_REF]. It is also possible to look for a minimum of entropy generation while fixing a constraint on the total viscous dissipation in the flow [START_REF] Li | A novel optimization approach to convective heat transfer enhancement for solar 980 receivers[END_REF][START_REF] Jia | An optimization approach to find the thermodynamic limit on convective mass transfer enhancement for a given viscous dissipation[END_REF], the latter being related to the pressure drop, which is a quantity to be controlled in industrial applications.

In the present study, the calculus of variations is used to minimize a functional objective constructed as a linear combination of the entropy generation rate main 55 contributor on the one hand and the viscous dissipation on the other hand [START_REF] Cao | Multi-objective optimization method for enhancing chemical reaction process[END_REF][START_REF] Cao | An optimization method to find the thermodynamic limit on enhancement of solar thermal decomposition of methane[END_REF][START_REF] Jia | A criterion beyond conservation equations for complex transport process modeling a case of Rayleigh-Bénard convection[END_REF][START_REF] Avellaneda | Similarities between heat and mass transfer enhancement in convective flow, using variational optimization technique[END_REF]. A weighting coefficient allowing to give more or less importance to the viscous dissipation during the optimization process is used. For each value of this weighting factor, the optimal velocity and scalar fields are obtained by varying a volume force field source term and the resulting improvement in entropy generation rate is calculated. The method is applied to heat transfer and mass diffusion cases in order to identify common behaviors or differences. In addition, the influence of the following boundary conditions is analyzed: the inlet velocity and the intensity of the incoming heat flow or minor chemical species injection. Depending on the value of the total viscous dissipation weighting coefficient, two different flow patterns appear and the robustness of the transition between these two regimes is analyzed.

Numerical method

Two types of diffusion processes in a convective flow are analyzed: heat transfer on the one hand and mass diffusion of a chemical minor species on the other hand. The corresponding equations are presented below.

Governing equations

We consider the incompressible steady-state twodimensional flow of a Newtonian fluid with constant 80 properties. The corresponding mass and momentum conservation equations are the following:

∇.V = 0 (1) ρV.∇V = -∇P + µ∇ 2 V + F (2)
In this set of equations, V is the velocity vector of the fluid, ρ its density, P its pressure, µ its dynamic viscosity and F is a volume force field that will be used in the optimization process described below: F can be considered as a virtual force field allowing the velocity field to be varied in order to examine all possible flow configurations and find the one that minimizes the objective functional (and respects the conservation equations and boundary conditions), without a priori on its pattern. The practical realization of the resulting optimal velocity field can be approached by passive means, such as baffles placed in the fluid path, or by the use of porous materials as described in [START_REF] Jia | An optimization approach to find the thermodynamic limit on convective mass transfer enhancement for a given viscous dissipation[END_REF] and [START_REF] Li | A novel optimization approach to convective heat transfer enhancement for solar 980 receivers[END_REF].

95

Depending on the diffusion process, the transported scalar is respectively the temperature (for heat transfer) or the mass fraction of the minor species (for mass transfer).

In the case of heat transfer, viscous heating and grav-100 ity are neglected and there is no source term in the energy equation nor any radiative exchange. The energy equation writes:

V.∇T = k ρC p ∇ 2 T ( 3
)
where T is the fluid temperature, k its thermal conductivity and C p its thermal capacity at constant pres-105 sure.

In the case of mass diffusion, a two species high dilution ideal solution is considered and the diffusion equation writes:

V.∇w 1 = D.∇ 2 w 1 (4)
where w 1 is the mass fraction of the minor species 110 and D is the diffusion coefficient. Eqs. 3 and 4 have similar mathematical form and differ in the diffusivity coefficient physical meaning and magnitude.

In the case of heat transfer, the local entropy generation rate (by unit of volume, which is expressed by the 115 triple prime superscript notation) can be calculated using the following expression, where Φ is the viscous dissipation function [START_REF] Bejan | Entropy generation minimization[END_REF] and u and v are the longitudinal and normal (to the heated wall) components of the velocity respectively:

120 Ṡ gen = k T 2 (∇T ) 2 + Φ T (5)
Φ = µ 2 ( ∂u ∂x ) 2 + ( ∂v ∂y ) 2 + ( ∂u ∂y + ∂v ∂x ) 2 (6)
Equation 5 is obtained by considering a finite-size control volume ∆x∆y∆z located at any position in the flow field and small enough so that the thermodynamic state inside the control volume may be regarded as homogeneous. The first and second laws 125 of thermodynamics as well as the fundamental relation du = T ds -Pd(1/ρ) (u and s being the specific internal energy and entropy respectively) are applied to this control volume considered as an open system to deduce the local equation Ṡ gen = -(1/T 2 )q + Φ/T , where q is by thermal conduction, the Fourier law q = -k∇T is applied and Eq. 5 is deduced, revealing two causes of local irreversibility: viscous (linked to the dynamic viscosity µ) and conductive (linked to the thermal conductivity k) 135 as underlined in [START_REF] Bejan | Entropy generation minimization[END_REF].

In the case of mass diffusion, the expression of the local entropy generation rate is more complex [START_REF] Carrington | Second law analysis of combined heat and mass transfer phenomena[END_REF] and takes into account the existence of two species in the fluid:

140 Ṡ gen = ρ 2 RD M1 M2 w 1 (1 -w 1 )c (∇w 1 ) 2 + Φ T (7)
In Eq. 7, R is the (molar) ideal gas constant, M1 and M2 are the molar masses of the minor species and of the solvent respectively and c is the total molar concentration in [mol.m -3 ], which is assumed to be constant for the high dilution ideal solution.

Variational problem

Since the aim is to minimize the entropy generation rate due to the diffusion phenomenon while taking into account the total viscous dissipation, it is reasonable to consider a linear combination of these two terms. The 150 objective functionals to be minimized are provided in Eqs. 8 and 9, corresponding to the case of heat transfer and mass diffusion respectively (Ω being the control volume domain):

J = Ω k T 2 (∇T ) 2 + W Φ Φ dΩ (8) J = Ω ρ 2 RD M1 M2 w 1 (1 -w 1 )c (∇w 1 ) 2 +W Φ Φ dΩ (9)
Viscous heating has been neglected in the heat equa-155 tion (Eq. 3) as the simulations are done in the case of very low values of the Brinkman number. This assumption must be changed for high Reynolds number flows or fluids with high viscosity (like oils or polymers, which can also be non-Newtonian). Moreover, although the rate of entropy generation in Eqs. 5 or 7 is the sum of two terms, we consider situations where the entropy generation rate due to viscous friction is negligible when compared to the heat (or mass) transfer part and the objective functional in Eqs. 8 or 9 takes consistently 165 into account the heat or mass transfer contributor only in the entropy generation term: Ṡ gen = (k/T 2 )(∇T ) 2 or Ṡ gen = (ρ 2 RD)/( M1 M2 w 1 (1w 1 )c)(∇w 1 ) 2 . The second term of the objective functional is W Φ Φ, the product of the weighting factor W Φ by the viscous dissipation Φ, which represents the pressure drop objective in the optimization problem. W Φ can be seen as a weighting factor in a multi-objective optimization problem. Minimizing the entropy generation rate and the pressure drop, while these two objectives are contradictory, does not lead to 175 a single solution. The use of a weighting factor allows to find a whole range of optimized trade-offs: with high values of W Φ , the emphasis is on the reduction of pressure drop (and more precisely of the total viscous dissipation Φ tot , which corresponds to the mechanical power 180 required to maintain the flow). Conversely, small values of W Φ correspond to a focus on minimizing the entropy generation rate in the channel. Each value of W Φ leads to an optimal velocity field that minimizes differently the two contradictory objectives pursued since they are 185 assigned different weighting factors.

In order to take into account the constraints expressed by Eqs. 1 and 3 or 4, Lagrange multipliers λ 1 and λ 2 (depending on the position) are introduced. Finally, the Lagrangian criteria to minimize are Eq. 10 for heat transfer 190 and Eq. 11 for mass diffusion.

J = Ω k T 2 (∇T ) 2 + W Φ Φ +λ 2 k ρC p ∇ 2 T -V.∇T + λ 1 ∇.V dΩ (10)
J = Ω ρ 2 RD M1 M2 w 1 (1 -w 1 )c (∇w 1 ) 2 + W Φ Φ +λ 2 D.∇ 2 w 1 -V.∇w 1 + λ 1 ∇.V dΩ (11)
The function to be minimized (the entropy generation rate and the viscous dissipation combined linearly in order to cope with a multi-objective optimization problem) is a functional (a function of functions, which is 195 also non-linear) and the solution sought is a set of functions of the position (the velocity vector field and the scalar field) and not a simple number or even a single vector or tensor. These solution fields must also comply with the conservation equations (also non-linear) and 200 the boundary conditions. One way to find the minimum of the functional could be to use iterative search algorithms (starting from an initial possible field and trying to get closer to the minimum by calculating gradients and penalty functions, for example) or methods based 205 on meta-heuristics (such as genetic approaches) [START_REF] Rao | Engineering Optimization[END_REF]. This implies to carry out a potentially high number of resolutions of the fluid flow equations and of evaluations of the objective functional. On the other hand, the tion problem into a system of differential equations [START_REF] Gelfand | Calculus of variations[END_REF] [44] (the Euler-Lagrange equations) and the optimization is therefore carried out mathematically and not numerically. One single numerical resolution of the resulting differential equation system is then carried out to find the velocity and scalar fields. In this method, the standard procedure for taking into account the equality constraints that must be respected by the solutions (here: the conservation equations) is done using Lagrange multipliers. The continuity equation and the energy conservation (or mass diffusion) equation are thus taken into account by two Lagrange multipliers. Given the presence of the volume force field F in the momentum equation, Eq. 2 is taken into account after obtaining the Euler-Lagrange equations. This procedure allows to find the expression of the volume force field F that minimizes the objective functional J : it is described in a synthetic way in the following paragraphs and an appendix at the end of the paper provides the details of the calculations.

Making the first variation of J with respect to u and v vanish and taking into account the momentum conservation equation (Eq. 2) gives the formula for the volume force field F in Eq. 12 and Eq. 13 for heat transfer and mass diffusion respectively.

F = λ 2 2W Φ ∇T + ρV.∇V (12)
F = λ 2 2W Φ ∇w 1 + ρV.∇V (13)
In addition, equaling to zero the first variation of J with respect to T leads to the transport equation of the λ 2 Lagrange multiplier in Eq. 14 and Eq. 15 for the heat transfer and mass diffusion case respectively.

∇. ρVλ 2 - -k C p ∇λ 2 = 2kρ T ∇. ∇T T (14) ∇.[ρVλ 2 -(-ρD)∇λ 2 ] = ρ 3 RD M1 M2 w 1 (1 -w 1 )c ∇. ∇w 1 w 1 (1 -w 1 ) + ∇.(∇w 1 ) w 1 (1 -w 1 ) (15)
The boundary conditions to be used for Eqs. 14 or 15 depend on those defined for the transported scalar (the temperature or the mass fraction of the minor species): if the scalar function is set at a boundary (Dirichlet condition), Eq. 16 applies. On the other hand, if the flux density of the scalar function is set at the boundary (Von Neumann condition), Eq. 17 or Eq. 18 are used for heat transfer and mass diffusion respectively.

λ 2 = 0 ( 16
)
∂λ 2 ∂n = 2ρC p T 2 ∂T ∂n ( 17
)
∂λ 2 ∂n = ρ 2 R M1 M2 c 2 w 1 (1 -w 1 ) ∂w 1 ∂n (18)
In these equations, ∂/∂n is the normal derivative to the boundary.

Finally, in the case of heat transfer, for each selected For a fixed force field (e.g. if F = 0 or any other value that may be non-uniform) and for given boundary conditions, Eqs. 1 and 2 determine a single velocity field 260 dependent on the selected field F, and Eq. 3 or 4 lead to the corresponding unique scalar field. These velocity and scalar fields determine the value of the total entropy generation rate and the total viscous dissipation. In this paper, the volume force field F is considered as an un-265 known of the optimization problem and the calculus of variations is used to find directly the expression (in Eqs. 12 or 13) of F that leads to the velocity and temperature fields minimizing the objective functional.

To summarize, in this paper, optimal theoretical ve-270 locity fields are obtained by minimizing the entropy generation rate for different values of the mechanical power required to maintain fluid flow (by varying the weighting factor W Φ ). These suggested velocity fields provide a target to be approached using passive physi-275 cal means like porous media or modifications of the geometry inside the channel. They also provide a better understanding of how flow structures are related to the minima of entropy generated and the values of the total viscous dissipation. The volume force field F is a vir-280 tual force allowing to vary the velocity field pattern in order to find the one that minimizes the objective functional.

Numerical model

The fluid is water (a pure substance when the heat is studied). The domain dimensions are 30 mm × 5 mm and a 1200 × 200 uniform mesh is used (Fig. 1). Mesh independence checks have been performed with 290 2400 × 400 and 3600 × 600 meshes and are described in section 3.3. The top and bottom plates are impermeable and the no-slip boundary condition is applied. The inlet velocity V in is uniform and constant and the gage pressure is set to zero at the outlet. In the case of 295 heat transfer, the inlet temperature is set to T in = 300 K and the walls are adiabatic except for the middle third of the lower plate where a uniform and constant surface heat flux density q is set to 15000 W.m -2 unless otherwise indicated. In the case of mass diffusion, the 300 inlet mass fraction of the minor species is set to a fixed value w 1,in = 0.01 unless otherwise stated and the middle third of the lower wall is subjected to a fixed production rate per unit area of the minor species ṁ1 set to 0.01 kg.m -2 .s -1 unless otherwise indicated.

305

The CFD code is ANSYS Fluent 15.0 with the SIM-PLE velocity-pressure coupling algorithm. UDF scripts have been developed to solve the additional transport equation (14 or 15) and to define the source terms (in particular the F volume force field that is applied in the 310 central region of the domain). Second-order discretization schemes are used for pressure and energy. Momentum and λ 2 equations are solved with a QUICK scheme.

In the present paper, 2D flows have been considered.

315

Two-dimensional simulations can be useful in the case of high aspect ratio cross-sections with a large transversal dimension with respect to the height of the channel (which are considered in the context of solar receivers in concentrating solar power plants, for exam-320 ple). The method discussed in this paper can be applied to three-dimensional flows while requiring significantly more computational resources. The analysis carried out in two dimensions enables to establish qualitative be-haviors that are likely to persist in three dimensions (for 325 laminar flows) and will facilitate identifying the work to be carried out for three-dimensional flows.

In total, about a hundred simulations were performed to simulate heat transfer or mass diffusion situations, with different values of the W Φ weighting coefficient 330 that spans across four orders of magnitudes. Moreover, several parameters have been varied in order to analyze their influence: the inlet velocity, the heat flux density (in the heat transfer case), the mass flux density and the inlet mass fraction of the minor species (in the mass dif-335 fusion case).

Results and discussion

Heat transfer optimization

In this section, we consider the heat transfer case where the channel is heated from the middle third of 340 its bottom plate with a constant and uniform heat flux density q . The mass diffusion case will be examined later in section 3.2. Depending on the importance of the weighting factor W Φ , the velocity (Fig. 2) and temperature (Fig. 345 3) fields exhibit different shapes corresponding to flow regimes more or less perturbed when compared to a reference simulation for which no optimization is activated (i.e. when the applied volume force field F is zero) [START_REF] Avellaneda | Similarities between heat and mass transfer enhancement in convective flow, using variational optimization technique[END_REF]. For very large values of W Φ (for example, when 350 W Φ = 10 7 K -1 ) these fields are very close to those obtained without optimization. For smaller W Φ values, like W Φ = 1.05 × 10 5 K -1 , the fields are modified without being disrupted: in particular, the flow is slightly pressed against the heated area. For even smaller W Φ values, for example W Φ = 1.047 × 10 5 K -1 , the temperature and velocity profiles are clearly perturbed compared to the situation without optimization: the fluid is still pushed towards the heated zone but a clockwise vortex also appears (Fig. 4) that tends to capture the heated fluid at the beginning of the heat exchange zone near the lower plate to move it up in the channel and then accelerate it, before pushing it towards the lower plate. This velocity field results in a thermal plume shape at the beginning of the heated zone. The presence of a vortex in the velocity field resulting from variational optimization is also observed in other physical situations, such as the improvement of chemical reaction [START_REF] Cao | An optimization method to find the thermodynamic limit on enhancement of solar thermal decomposition of methane[END_REF] or diffusion [START_REF] Jia | An optimization approach to find the thermodynamic limit on convective mass transfer enhancement for a given viscous dissipation[END_REF] processes and the optimization of heat transfer in a turbulent gas flow [START_REF] Li | A novel optimization approach to convective heat transfer enhancement for solar 980 receivers[END_REF]. Vortex generation is also a subject of numerical simulation [START_REF] Colleoni | Optimization of winglet vortex generators combined with riblets for wall/fluid 875 heat exchange enhancement[END_REF] and experimental [START_REF] Daguenet-Frick | Experimental analysis of the turbulent flow behav-870 ior of a textured surface proposed for asymmetric heat exchangers[END_REF] work aimed, for example, at associating and optimizing actuators and riblets to produce vortexes.

The transition from a low to a highly perturbed regime occurs around a critical value of W Φ that depends in particular on the Reynolds number. When Re = 10, the transition takes place around W Φ = 2.53 × 10 5 K -1 while for Re = 20 it occurs around W Φ = 1.05 × 10 5 K -1 . As can be seen in Figs. When the optimization is applied (i.e. the force field F is computed and used in the momentum equation), the total entropy generated in the channel is smaller than the case with no optimization (see Fig. 5 where the 390 two vertical bars materialize the critical W Φ values triggering the transition between the slightly to the highly perturbed velocity and temperature profiles). The re-

W φ [K -1 ]
No optimization: I s = 0% Re=10 Re=20

Figure 6: Improvement factor (Eq. 19) as a function of W Φ .

duction in entropy generated is all the more important as W Φ is small and the gain obtained compared to the 395 case without optimization can be estimated using an improvement factor I s defined in Eq. 19 and presented in Fig. 6. When Re = 20 and W Φ = 2.3 × 10 4 K -1 , I s ≈ 30% and the total entropy generation rate in the channel is about 70% of non-optimized case. For high 400 values of W Φ , decreasing this factor by one order of magnitude leads to a small (or even zero) marginal gain in entropy rate reduction. On the other hand, when W Φ approaches its critical value and below this point, the slope of the marginal gain is higher (around 8 mW.K -1 405 by order of magnitude of W Φ ). As can be seen in Fig. 7, the maximum temperature inside the channel decreases significantly as W Φ declines and for medium values of W Φ the curve slope increases when the Reynolds number decreases. The 410 velocity fields resulting from the optimization increase the homogeneity of the temperature field: the standard deviation of the temperature is reduced from 3.74 K to 3.32 K when W Φ changes from 2.09 × 10 8 K -1 to 1.047 × 10 5 K -1 at Re = 20. The entropy generation rate by viscous friction increases as W Φ decreases (Fig. 8) and so does the total viscous dissipation Φ tot (Fig. 9) and the maximum velocity within the flow (Fig. 10). The order of magnitude of the viscous friction entropy generation rate Ṡ gen, f 420 (between 1 nW.K -1 and 10 nW.K -1 , depending on the Reynolds number) is much lower than that of the heat conduction entropy generation rate Ṡ gen,c (10 mW.K -1 ), which is by far the main contributor to the total entropy generation rate Ṡ gen = Ṡ gen,c + Ṡ gen, f . The smaller W Φ , 425 the higher the velocities in the channel, leading to increased velocity gradients, viscous dissipation and gen- The objective functional defined in Eq. 8 decreases as W Φ gets smaller (Fig. 11). It is a linear combination of two terms, Ṡ gen,c ≈ Ṡ gen and Φ tot that evolve in opposite ways when W Φ varies (Fig. 12). Taking the case Re = 20 as an example, when W Φ falls by approximatively four orders of magnitude from 2.1 × 10 8 K -1 to 2.3 × 10 4 K -1 , the entropy generation rate by heat conduction decreases from 27 mW.K -1 to 19 mW.K -1 and the total viscous dissipation raises from 1.43 µW to 2.60 µW. Since the entropy generation by heat conduction and the total dissipation do not change their order of magnitude, the evolution of W Φ has the decisive effect and leads to a decrease of the objective functional. The boundary conditions have an effect on the opti-450 mized solutions. the influence of the heat flux density q and of the inlet velocity V in are discussed below. ]

I S = 1 - Ṡ gen, optimized Ṡ gen, no optimization (19)
W φ [K -1 ]
q"=15000 W.m -2 q"=10000 W.m -2 The higher the heat flux density, the higher the entropy generation rate by heat conduction and the higher the total entropy generation rate in the channel (Fig. 13).

455

The critical value of W Φ is lower for a reduced heat flux density: when Re = 20, changing the heat flux boundary condition from 15000 W.m -2 to 10000 W.m -2 leads to a fall of the critical W Φ from about 1 × 10 5 K -1 to 4 × 10 4 K -1 . This change in the critical W Φ values 460 directly affects the behavior of the entropy generation rate by viscous friction (Fig. 14) as well as the maximum velocity in the channel (Fig. 15): as long as W Φ ]

W φ [K -1 ]
q"=15000 W.m -2

q"=10000 W.m q"=15000 W.m -2 q"=10000 W.m -2 is large enough, these two quantities are almost independent of the value of the flow density applied to the 465 heated wall. On the other hand, as soon as W Φ falls below the highest critical value (here, the one corresponding to 15000 Wm -2 ), the curves are dissociated and the rate of generation of viscous entropy is higher in the case where q = 15000 Wm -2 compared to that where 470 q = 10000 Wm -2 . For a fixed W Φ value, a reduction in the heat flux density applied to the wall results in a reduction in the rate of entropy generation by heat conduction and therefore in the rate of total entropy generation (Fig. 16), 475 the conductive component still remaining largely dominant in the entropy generation mix when compared to the viscous component. Similarly, the maximum temperature reached inside the channel decreases with the imposed heat flow density (Fig. 17 cous entropy generation also decrease as the imposed heat flux density is reduced (while keeping Re and W Φ fixed). For high values of q , the flow regime is of the highly perturbed type when compared to the case with-485 out optimization, whereas for a small applied heat flux density, the flow regime is of the slightly or very slightly perturbed type. The transition between the two regimes, visible in Figs. [START_REF] Elperin | Mass transfer during solute extraction 905 from a fluid sphere with internal circulation in the presence of alternating electric field[END_REF] and 17 takes place for a critical value q critical , which is about 18000 W.m -2 for Re = 20 and 490 W Φ = 1.23 × 10 5 K -1 . Indeed, as observed during the examination of Figs. 13 to 15, a reduction in the heat flux density applied to the heated wall leads to a reduction in the critical value of W Φ . As a result, W Φ being set to a given value W Φ,0 (= 1.25 × 10 5 K -1 for ex-495 ample), reducing q from a value corresponding to a highly perturbed (24000 W.m -2 ) regime ends up making W Φ critical (q ), coincide with W Φ,0 , which induces the transition to a low perturbed flow regime where viscous entropy generation rates and maximum velocities are lower and stabler. If now W Φ is fixed (at 1.23 × 10 5 K -1 ) and q is also fixed (at 15000 W.m -2 ) while varying the inlet velocity V in between 6 mm.s -1 (Re ≈ 30) and 1 mm.s -1 (Re ≈ 5), one can observe that the entropy generation rate of vis-505 cous origin as well as the maximum velocity in the flow decrease with V in (Figs. 18 and19). However, the maximum velocity curve exhibits a significant disruption: a transition appears for a critical value V in,critical approximately equal to 3.2 mm.s -1 (Re = 15.8) between highly 510 perturbed flow regimes (when the inlet velocity is lower than V in,critical ) and slightly perturbed ones. Indeed, as already observed in Fig. 5, a reduction in the Reynolds number leads to an increase of the critical W Φ . If one sets W Φ to a fixed value W Φ,0 and reduces the inlet 515 velocity, there is a point where the increasing critical weighting parameter W Φ,critical (V in ) reaches the value W Φ,0 and triggers the transition from slightly to highly perturbed profiles.

At fixed W Φ and q , lowering the inlet velocity leads 520 to an increase of the maximum temperature in the channel (Fig. 19). The total entropy generation rate also growths as V in decreases with the exception of a behavior change for very small inlet velocities (when V in ≤ 2 mm.s -1 ) as exhibited in Fig. 18. 525

Mass transfer optimization

In this section, we consider the mass transfer case where at the middle third of the bottom plate a minor species diffuses at a constant and uniform mass production time rate by unit of surface ṁ1 . The objective is to First, one finds again the presence of slightly and highly perturbed flow regimes (Figs 20 and21) when compared to the non-optimized case, whose shapes are very similar to that of the velocity and transported scalar profiles of the heat transfer case (Figs. 2 and3), the 540 key scalar quantity being here the mass fraction of the chemical minor species whereas this was previously the temperature. The evolution of entropy generation and viscous dissipation quantities as a function of the weighting parameter W Φ are presented in Figures 22 to 27 by comparing the case of heat transfer with that of mass diffusion at Re = 10. A decrease in W Φ leads to a reduction in the total entropy generation rate (Fig. 22) and in the objective functional (Fig. 23), unlike the entropy generation rate by viscous dissipation (Fig. 24) and the total viscous dissipation (Fig. 25) that increase as W Φ decreases. These behaviors are quite similar to those observed previously in the case of heat transfer and it is the same for the relationship between the total entropy generation rate and the total viscous dissipation (Fig. 26). The entropic improvement factor I s defined in Eq. 19 becomes higher as W Φ gets lower

φ tot [W] W φ [K -1 ]
heat mass (Fig. 27). Whether it is a heat transfer case or a mass diffusion case, there is a critical W Φ value that triggers 560 the transition from the slightly perturbed regime (when W Φ is large) to the highly perturbed regime (for W Φ smaller than W Φ,critical ). The critical value of W Φ is about 2.5×10 5 K -1 for the heat transfer case and around 1.8 × 10 5 K -1 for the mass diffusion case. As a conclu-565 sion, without being strictly identical, the optimization of heat transfer and mass diffusion exhibit very similar behaviors.

If the value of W Φ is set to 1.67 × 10 5 K -1 at a fixed Reynolds number (Re = 10) and the mass production rate at the exchange section of the bottom part of the channel is varied, the total and viscous entropy generation rates have profiles similar to those of heat transfer: all entropy generation rates decrease as the input flux density decreases (Fig. 28). In addition, a crit-575 ical value of ṁ1 exists, for which the transition takes place between the very disturbed flow regime (for large values of ṁ1 ) and the slightly disturbed regime (below ṁ1,critical ≈ 9 × 10 -3 kg.m -2 .s -1 ). As ṁ1 gets lower, the maximum and average mass fraction of the minor species decrease as does the maximum velocity in the channel (Fig. 29), which exhibits the disruption at the critical value ṁ1,critical .

If now, for a fixed value of the weighting factor (W Φ = 1.67 × 10 5 K -1 ), the Reynolds number (Re = 585 10) and the bottom plate mass flux density are fixed (0.01 kg.m -2 .s -1 ), while varying the minor mass fraction w 1,in at the inlet from 10 -5 to 0.03, all velocity and mass fraction fields stay in the highly perturbed flow regime and there is no observed critical value of w 1,in 590 in the tested value range. A decrease in the inlet mass fraction of the minor species leads to an increase in all entropy generation rates whether it comes from mass diffusion or from viscous friction (Fig. 30). Correlatively, a lower value of w 1,in corresponds to a reduction 595 in the maximum and the average mass fraction of the minor species in the channel and to an increase of the maximum velocity (Fig. 31).

Robustness of the transition

In sections 3.1 and 3.2, the presence of a regime tran-600 sition between slightly and highly perturbed velocity and scalar fields has been pointed out, as the existence of critical values of the W Φ weighting parameter in the functional objective that trigger this transition. This has been observed for different Reynolds numbers, different 605 values of the physical diffusivity coefficients and different boundary conditions. It is also useful to examine the mesh independence of this regime transition, whether it depends on initial conditions and what could be its underlying cause.

610

Keeping the same physical domain size, two finer meshes have been tested for the heat transfer case when Re = 20 and q = 15 kW.m -2 : in addition to the base

[W.K -1 ] W φ [K -1 ]
Grid 1200x200 Grid 2400x400 Grid 3600x600

Figure 32: Total entropy generation rate as a function of W Φ , complemented by finer mesh simulations. Re = 20.

[W.K -1 ] W φ [K -1 ]
Grid 1200x200 Grid 2400x400 Grid 3600x600 Grid 1200x200 with smooth I.C. 1200×200 mesh, 2400×400 and 3600×600 meshes have been used and the corresponding results for the total en-615 tropy generation rate as a function of W Φ are presented in Fig. 32. The Ṡ gen curve has been extended to smaller values of W Φ (down to 10 3 K -1 ) and the corresponding extended part of the graph is consistent with the above results: the total entropy generation rate keeps declining 620 as W Φ decreases.

A zoom around the critical W Φ value is provided in Fig. 33. In the case of finer grids, the critical W Φ still exists but at a lower value while keeping the same order of magnitude (≈ 8.5 × 10 4 K -1 in place of 625 ≈ 1 × 10 5 K -1 ). Furthermore, tests have been performed with the 1200 × 200 mesh by approaching the critical value of W Φ step by step in a smooth manner, each simulation starting from the results of the converged previous one with a value of W Φ slightly lower (while all 630 the simulations presented above are started from the ho-mogeneous initial conditions where the temperature of the channel is set to T in and the longitudinal velocity is set to V in , the normal velocity being zero at the initial time of the simulation). The corresponding results are presented in Fig. 33: the existence of the critical W Φ is still confirmed at a value close to the one obtained with the finer grids (≈ 8.4 × 10 4 K -1 ). So, the existence and the approximate critical W Φ value are quite robust.

For the high values of W Φ , the product ΦW Φ is much higher than the diffusion entropy generation rate (by heat conduction or mass diffusion, depending on the type of simulation performed) and the second term of the objective functional is predominant. Conversely, for small values of W Φ , the first term of J in Eq. 8 or 9 is the majority. In the critical value area of W Φ , in which the change in flow regime takes place, the two terms composing the objective functional have close orders of magnitude: the difference in orders of magnitude, estimated by Log 10 (Φ tot W Φ / Ṡ gen,d ), where Ṡ gen,d is the entropy generation rate by diffusion of heat or mass, lies between 0.37 and 0.85 depending on the Reynolds number and the transfer type. The flow regime transition could therefore be related to the fact that the optimization program focuses on viscous dissipation when W Φ 655 is large and on entropy generation by heat conduction or mass diffusion when W Φ is small. In the second case, the optimized solution can rely on a more complex and intense velocity field to produce a thermal mixture, at the expense of an increase in viscous dissipation which effect on the objective criterion is negligible due to the small size of the W Φ factor. The transition from one regime to another would then be linked at least in part to the shift from an optimization focused on viscous dissipation to an optimization focused on entropy generation 665 by diffusion. When W Φ becomes very large, the first term in Eqs. 12 or 13 cancels and the resulting momentum conservation equation corresponds to that of Stokes flow ∇P = µ∇ 2 V, for which the advection term is negligible 670 when compared to the viscous term. This corresponds to low Reynolds number flows without optimization. If W Φ tends towards zero, the first term in Eqs. 12 or 13 becomes large and leads to a coupling between the pressure, the velocity and the temperature (or the mass fraction of the minor species) in the momentum conservation equation. The smaller W Φ , the more intense (in magnitude) and complex the velocity field becomes due to the presence of vortexes in particular. Exploring the behavior of optimized velocity fields for smaller and smaller values of W Φ requires working with finer meshes and demands more computing resources. In this paper, we went down as far as W Φ = 10 3 K -1 (Fig. 32) in order to stay within reasonable limits in terms of computation time, while observing a wide range of values 685 for this weighting parameter. Conversely, large values of the weighting factor (up to W Φ ≈ 10 8 K -1 ) allow to examine the effect of the optimization when the overwhelming emphasis is on reducing (or increasing as little as possible) the pressure drop, even if it means an 690 increase (or a smaller reduction) of the entropy generation rate in the channel.

A discontinuity appears at a precise critical value of the weighting factor. The criterion for identifying W Φ,critical is the jump in the maximum velocity within 695 the channel (Figs. [START_REF] Cai | Enhancement of CO2 absorption under taylor flow in the presence of fine 880 particles, Fluid Dynamics and Transport Phenomena[END_REF][START_REF] Saien | Liquid-liquid extraction inten-900 sification with magnetite nanofluid single drops under oscillating magnetic field[END_REF][START_REF] Kordac | Mechanism of enhanced gas absorption in presence of fine solid particles. effect of molecular diffusivity on 910 mass transfer coefficient in stirred cell[END_REF][START_REF] Bianco | Heat Transfer Enhancement with Nanofluids[END_REF][START_REF] Qayyum | Op-955 timization of entropy generation and dissipative nonlinear radiative Von Karman´s swirling flow with Soret and Dufour effects[END_REF]. Other physical quantities also exhibit a discontinuity for the same value of W Φ,critical . This is the case for the entropy generation rates (due to viscous friction in particular) and the total viscous dissipation (Figs. 5, 8, 9, 12, 13, 14, 16, 18, 22, 700 [START_REF] Bejan | A study of entropy generation in fundamental convective heat transfer[END_REF][START_REF] Narayan | Entropy generation minimization of combined heat and mass transfer devices[END_REF][START_REF] Lou | Experimental and numerical analysis of radiative entropy generation in industrial and boiler furnaces[END_REF][START_REF] Bidi | A numerical evaluation of combustion in porous media by EGM (entropy generation minimization)[END_REF]. Simultaneously, a change in the pattern of the velocity field occurs at W Φ = W Φ,critical as can be seen in Figs. 2 and3 or 20 and 21: if W Φ is higher than the critical value, the flow streamlines are a little closer to the lower plate above the heated zone but the 705 flow remains close to the one without optimization. We have labeled this configuration as "slightly perturbed" as opposed to the one that appears for W Φ below the critical value: in this case, a vortex is present and, as indicated previously, the velocity magnitude increases 710 sharply. We have referred to this second configuration as "highly perturbed". The present study shows that the critical value W Φ,critical depends on the boundary conditions and the Reynolds number in particular. The physical properties of the fluid may surely have an influence, 715 as this critical value seems to correspond to an approximate balance between the two terms of the objective functional (the entropy generation rate in the channel on the one hand and the weighted product W Φ times Φ, on the other hand). The optimizations carried out 720 in the present work have all been performed at constant physical properties in order to examine in detail the behavior of the method in this first case. This enables to start from initial solutions known in the literature [START_REF] Jia | An optimization approach to find the thermodynamic limit on convective mass transfer enhancement for a given viscous dissipation[END_REF] and then to vary the boundary conditions and the opti-725 mization parameters. However, the properties of fluids generally depend on temperature. In the case of heat transfer for real water, the dynamic viscosity varies significantly [START_REF] Zonta | Modulation of turbulence in forced convection by temperature-dependent viscosity[END_REF] over the temperature range observed in the simulations (from 300K to 335K depending on the 730 Reynolds number or the heat flux density provided -see Figs. 7 and17). The temperature dependence of viscosity can have an impact on the precise value of the critical weighting factor W Φ,critical that determines the transition between different flow regimes. Taking into 735 account thermo-dependent properties involves solving more complex fluid governing equations, as the temperature field would influence the flow dynamics due to the coupling between the energy and momentum equations. In addition, the variational problem is modified and the equations determining the force field F are made more complex. A detailed study of the influence of the characteristics of the fluid (including its viscosity and thermal conductivity and their dependence on temperature, in particular) is a research work in itself to be carried out as a continuation of the present work.

Conclusion

In the present paper, convective flow transfer is submitted to variational methods in order to find optimized velocity and scalar fields that minimize the entropy generation rate and the total viscous dissipation in a channel. This two-objective optimization is addressed by a linear combination of the two criteria to be minimized using a weighting factor allowing to control the relative importance of the total viscous dissipation in the objective functional. Heat transfer and mass diffusion cases are studied and the influence of the weighting factor on the resulting fields and the key physical quantities is analyzed. The velocity patterns suggested by the optimization program lead to a reduced total entropy generation rate in the channel, the relative improvement being larger as the weighting factor or the Reynolds number are lower. These patterns can be a reference for the design of enhanced exchangers. Improved scalar homogeneity (of the temperature or the minor species mass fraction, depending on the type of transfer) is observed as the scalar maximum value and standard deviation are lower while the weighting factor is reduced. This enhancement is achieved at the expense of an increase in the entropy generation rate by viscous friction, which stays nevertheless at a negligible level. The maximum velocity and the total viscous dissipation also increase when the weighting factor decreases. Two main flow regimes are observed, depending on the level of perturbation of the velocity and scalar fields. In highly perturbed regimes, a more complex flow is observed with a vortex and a displacement of the flow towards the diffusion entry area (the heated segment or the mass diffusion entry zone, depending on the transfer process). The transition from a slightly to a highly perturbed flow is triggered by a critical value of the weighting factor, the highest perturbation level corresponding to the lowest weighting factors. This critical value depends on the Reynolds number, the heat or mass transfer intensity at the exchange plate and on the inlet velocity. At iden-785 tical Reynolds number, the optimization of mass diffusion and heat transfer exhibit similar behaviors. The existence of the transition between two perturbation flow regimes and the corresponding critical value of the viscous dissipation weighting factor is confirmed with finer 790 meshes and smoother initial conditions.

Acknowledgement

Appendix

In this appendix, the transport equation for the λ 2 Lagrange multiplier (Eq. 14) and the volume force field F (Eq. 12) are found by applying the calculus of variations in the case of heat transfer optimization. The same 805 procedure can be used to obtain the corresponding equations in the case of mass diffusion (Eqs. 15 and 13).

The Lagrangian criterion to minimize writes:

J = Ω k T 2 (∇T ) 2 + W Φ Φ +λ 2 k ρC p ∇ 2 T -V.∇T + λ 1 ∇.V dΩ = Ω F J dΩ (20)
In this equation, the term to be integrated has been named F J and a compact notation is used in the fol-810 lowing calculations: the subscript (.) ,X stands for the derivative with respect to X and (.) ,XX stands for the second derivative (T ,xx means ∂ 2 T/∂x 2 and T 2 ,x means (∂T/∂x) 2 ). F J can be rewritten:

F J = k T 2 (T
For functional in Eq. 20 to have a minimum, its first variation (or differential) must cancel out and making this first variation of J with respect to the temperature vanish implies the following Euler-Lagrange equation [START_REF] Gelfand | Calculus of variations[END_REF] [44]:

∂F J ∂T - ∂
Injecting the above expressions into Eq. 22 and taking into account the continuity equation u ,x + v ,y = 0 leads to Eq. 24 that can be written in the form of a transport equation for the λ 2 scalar (Eq. 25, corresponding to Eq. 14).

∇. ρVλ 2 --k

C p ∇λ 2 = 2kρ T ∇. ∇T T (25)
The same method can be used to find the expression of the volume force field F by making the first variation of J vanish with respect to the longitudinal and wall-normal components of the velocity (u and v respectively) and by using the continuity equation, which leads

  145

Figure 1 :

 1 Figure 1: Domain characteristics.

Figure 2 :

 2 Figure 2: Velocity magnitude fields with no optimization, slightly and highly perturbed profiles (Re = 20).

Figure 3 :

 3 Figure 3: temperature fields with no optimization, slightly and highly perturbed profiles (Re = 20)

2 and 3 ,

 3 a small change in the W Φ value can lead to a significant change in flow: when Re = 20, reducing W Φ by

Figure 4 :

 4 Figure 4: Zoom of velocity vector and stream function fields when W Φ = 1.047 × 10 5 K -1 . The vertical lines materialize the abscissa x = 10 mm and x = 20 mm of the heated region. Re = 20.

Figure 5 :

 5 Figure 5: Entropy generation rate as a function of W Φ .

Figure 7 :

 7 Figure 7: Maximum temperature in the channel as a function of W Φ .

Figure 8 :

 8 Figure 8: Entropy generation rate due to viscous friction as a function of W Φ .

Figure 9 :

 9 Figure 9: Total viscous dissipation as a function of W Φ .

Figure 10 :

 10 Figure 10: Maximum velocity magnitude in the channel as a function of W Φ .

Figure 11 :

 11 Figure 11: Objective functional J as a function of W Φ .

Figure 12 :

 12 Figure 12: Total entropy generation rate as a function of the total dissipation.

Figure 13 :

 13 Figure 13: Total entropy generation rates as a function of W Φ for two values of the input heat flux. Re = 20.

- 2 Figure 14 :

 214 Figure 14: Total entropy generation rates by viscous friction as a function of W Φ for two values of the input heat flux.

Figure 15 :

 15 Figure 15: Maximum velocity magnitude in the channel as a function of W Φ for two values of the input heat flux.

Figure 16 :

 16 Figure 16: Total and viscous entropy generation rates as functions of the heat flux input from the bottom of the channel (Re = 20 and W Φ = 1.23 × 10 5 K -1 ).

Figure 17 :

 17 Figure 17: Maximum temperature and maximum velocity magnitude in the channel (Re = 20 and W Φ = 1.23 × 10 5 K -1 ).

Figure 18 :

 18 Figure 18: Total and viscous entropy generation rates as functions of V in (W Φ = 1.23 × 10 5 K -1 ).

Figure 19 :

 19 Figure 19: Maximum velocity magnitude and maximum scalar value in the channel as functions of V in (W Φ = 1.23 × 10 5 K -1 ).

  530verify if the behaviors described in section 3.1 are similar in a different advection-diffusion phenomenon but with close mathematical model. The comparison between heat and mass transfer simulations is done at the same Reynolds number Re = 10.

  535

Figure 20 :

 20 Figure 20: Velocity magnitude with no optimization, slightly and highly perturbed profiles (Re = 10).

Figure 21 :

 21 Figure 21: Mass fraction of the minor species with no optimization, slightly and highly perturbed profiles (Re = 10).

Figure 22 :

 22 Figure 22: Total entropy generation rate as a function of W Φ .

Figure 23 :

 23 Figure 23: Objective functional J as a function of W Φ .

Figure 25 :

 25 Figure 25: Total viscous dissipation as a function of W Φ .

Figure 27 :

 27 Figure 27: Improvement factor as a function of W Φ .

Figure 28 :

 28 Figure 28: Total and viscous entropy generation rates as functions of the diffusion intensity from the bottom plate (W Φ = 1.67 × 10 5 K -1 and Re = 10).

Figure 29 :

 29 Figure 29: Maximum and average mass fraction of the minor species and maximum velocity as functions of the diffusion intensity from the bottom plate (W Φ = 1.67 × 10 5 K -1 and Re = 10).

Figure 30 :

 30 Figure 30: Total and viscous entropy generation rates in the channel as functions the inlet mass fraction of the minor species (W Φ = 1.67× 10 5 K -1 and Re = 10).

Figure 31 :

 31 Figure 31: Maximum velocity magnitude in the channel as functions the inlet mass fraction of the minor species (W Φ = 1.67 × 10 5 K -1 and Re = 10).

Figure 33 :

 33 Figure 33: Total entropy generation rate as a function of W Φ -zoom on transition zone showing the influence of the grid resolution and the initial conditions (I.C.). Re = 20.

  This work was supported by French Investments for the future ("Investissements d'Avenir") programme managed by the National Agency for Research (ANR) 795 under contract ANR-10-LABX-22-01 (labex SOL-STICE) and has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 654663, SOLPART project.

  800

Fφq

  (molarity) [mol.m -3 ] Mi Molar mass of chemical species number i [kg.mol -1 ] R Ideal gas constant (molar) [J.K -1 .mol -1 ] Volume force vector by unit of volume [N.m -3 ] V Velocity vector [m.s -1 ] ṁ1 Mass production time rate by unit of surface of the chemical minor species (mass flux density) [kg.m -2 .s -1 ] Ṡ gen Local entropy generation rate [W.K -1 .m -3 ] (eq. 5 or 7) Ṡ gen,d Entropy generation rate from the diffusion process (heat conduction or mass diffusion) in the channel [W.K -1 ] Ṡ gen, f Viscous friction entropy generation rate in the channel [W.K -1 ] Ṡ gen Total entropy generation rate in the channel [W.K -1 ] Viscous dissipation function [W.m -3 ] (eq. 6) Φ tot Total viscous dissipation in the channel [W] ρ Density [kg.m -3 ]C p Thermal capacity [J.kg -1 .K -1 ] D Diffusion coefficient [m 2 .s -1 ] Heat flux density [W.m -2 ]ReReynolds number (based upon the channel height)T in Inlet temperature [K] u, vLongitudinal and normal components of the ve-1060 locity [m.s -1 ] U in Inlet (longitudinal) velocity [m.s -1 ] w i Mass fraction of chemical species number i [kg.kg -1 ] w 1,in Inlet mass fraction of chemical minor species 1065 [kg.kg -1 ] W Φ Weighting factor in objective functional and Lagrangian criterion [K -1 ]

  T 2 ,x -λ 2 u ,xuλ 2 ,x T 2 ,y -λ 2 v ,yvλ 2 ,y

		+	∂ 2 ∂x 2	∂x ∂F J ∂T ,xx	∂F J ∂T ,x + ∂ 2 ∂y 2 -∂T ,yy ∂ ∂F J ∂y ∂T ,y = 0 ∂F J	(22)
	820	Each term of Eq. 22 writes:
				∂F J ∂T	=	-2k T 3 (T 2 ,x + T 2 ,y )
		∂ ∂x T 2 ∂ ∂F J ∂T ,x = 2k 4k T 3 T ,xx -∂y ∂F J ∂T ,y = 2k 4k T 3 T ,yy -T 2 ∂ 2 ∂x 2	∂F J ∂T ,xx	=	k ρC p	λ 2 ,xx
					∂ 2 ∂y 2	∂F J ∂T ,yy	=	k ρC p	λ 2 ,yy

  2 ,x + T 2 ,y ) -2k T 2 (T ,xx + T ,yy ) + uλ 2 ,x + uλ 2 ,y ,xx + λ 2 ,yy )

	+	k ρC p	(λ 2

the heat flux vector. The heat transfer being carried out

to:

These two equations can be combined into a more compact vector equation (Eq. 27) that can be compared with the momentum equation (Eq. 28), which leads [START_REF] Jia | An optimization approach to find the thermodynamic limit on convective mass transfer enhancement for a given viscous dissipation[END_REF] to identify ∇P (Eq. 29) and F (Eq. 30), the expression 835 searched for the volume force.

-λ 2 2WΦ

∇T -

7. References


Sébastien Salas
email: sebastien.salas@ap-hm.fr

Keywords: Tumeurs desmoïdes, Fibromatose, oncogenèse, biologie, voies de signalisation Desmoid tumors, Fibromatosis, oncogenesis, biology, signaling pathways

Les tumeurs desmoïdes dérivent de cellules souches mésenchymateuses et leur pathogénie est fortement liée à la cascade Wingless/Wnt où la dérégulation de la β-caténine joue un rôle majeur. Une mutation du gène CTNNB1 codant la β-caténine est retrouvée dans la majorité des cas de tumeurs desmoïdes sporadiques et des mutations constitutionnelles du gène APC ont été décrites dans les formes héréditaires chez les patients atteints de polypose adénomateuse familiale (FAP). Les oestrogènes, pourraient jouer également un rôle dans la pathogénie et cela est à la base de l'utilisation de l'hormonothérapie. D'autres voies de signalisation ont été impliquées dans le développement des tumeurs desmoïdes comme les voies Notch, Hedgehog, JAK/STAT, PI3 Kinase/AKT et mTOR. Les métalloprotéases sont exprimées dans les tumeurs desmoïdes et jouent un rôle dans la capacité d'invasion. Le TGFß, en tant que facteur de croissance, stimule l'activité transcriptionnelle de la β-caténine. De futures études devront s'attacher à mieux décrire et comprendre l'environnement immunitaire des tumeurs desmoïdes. Une des difficultés majeures pour l'étude expérimentale des tumeurs desmoïdes est la quasi-absence de modèle pré-clinique que ce soit in vitro ou in vivo. C'est en partie pour cela que les interactions entre les différentes voies de signalisations ici présentées et leurs conséquences dans le développement des tumeurs desmoïdes sont encore mal connues.

Les tumeurs desmoïdes, tumeurs rares à prolifération fibroblastique et myofibroblastique, dérivent de cellules souches mésenchymateuses et leur pathogénie est fortement liée à la cascade Wingless/Wnt où la dérégulation de la β-caténine joue un rôle majeur. On distingue deux types de tumeurs desmoïdes : les tumeurs desmoïdes sporadiques et les tumeurs desmoïdes héréditaires. Les tumeurs desmoïdes sont considérées comme étant une prolifération monoclonale, puisqu'elles proviennent d'une seule cellule progénitrice [START_REF] Alman | Aggressive fibromatosis (desmoid tumor) is a monoclonal disorder[END_REF]. Les causes de tumeur desmoïde sporadique ne sont pas clairement identifiées par définition mais une mutation du gène CTNNB1 codant la βcaténine y est retrouvée dans près de 85 % des cas. Plusieurs études suggèrent des corrélations entre un traumatisme ou une imprégnation en hormones sexuelles, en particulier des oestrogènes et le développement des tumeurs desmoïdes [START_REF] Lopez | Problems in diagnosis and management of desmoid tumors[END_REF][START_REF] Hunt | Principles in the management of extra-abdominal desmoids[END_REF][START_REF] Urist | Trauma and neoplasm; report of a case of desmoid tumor following simple fracture of the radius and ulna[END_REF][START_REF] Reitamo | The desmoid syndrome. New aspects in the cause, pathogenesis and treatment of the desmoid tumor[END_REF]. Le type héréditaire se développe chez les patients atteints de polypose adénomateuse familiale (FAP) et cause le plus souvent des tumeurs desmoïdes intra-abdominales. Des mutations constitutionnelles du gène APC ont été décrites dans cette forme qui peut alors rentrer dans le cadre d'un syndrome de Gardner. Il s'agit d'une maladie autosomique dominante associée à la formation de centaines de polypes du côlon qui peuvent se transformer en tumeurs colorectales malignes avec le temps [START_REF] De Marchis | Desmoid Tumors in Familial Adenomatous Polyposis[END_REF][START_REF] Lips | The role of APC and beta-catenin in the aetiology of aggressive fibromatosis (desmoid tumors)[END_REF]. Le taux cumulé de tumeurs desmoïdes chez les patients FAP est de 20,6 % à 60 ans [START_REF] Bertario | Genotype and phenotype factors as determinants of desmoid tumors in patients with familial adenomatous polyposis[END_REF].

La voie de signalisation Wnt/β-Caténine

Cette voie est cruciale dans la pathogenèse des tumeurs desmoïdes. La grande majorité (environ 85 %) des tumeurs desmoïdes présente une mutation dans l'exon 3 du gène CTNNB1 [START_REF] Li | TBL1-TBLR1 and beta-catenin recruit each other to Wnt target-gene promoter for transcription activation and oncogenesis[END_REF][START_REF] Salas | Molecular characterization by array comparative genomic hybridization and DNA sequencing of 194 desmoid tumors[END_REF][START_REF] Crago | Near universal detection of alterations in CTNNB1 and Wnt pathway regulators in desmoid-type fibromatosis by wholeexome sequencing and genomic analysis[END_REF]. Moins fréquemment, des mutations APC perte de fonction sont observées, le plus souvent dans un contexte de FAP [START_REF] Lips | The role of APC and beta-catenin in the aetiology of aggressive fibromatosis (desmoid tumors)[END_REF]. Les mutations de ces deux gènes sont mutuellement exclusives et activent la même voie de signalisation Wnt/β-Caténine. La β-caténine régule l'adhésion cellulaire et la transcription cellulaire. Dans les tumeurs desmoïdes, il y a une stabilisation anormale et une accumulation nucléaire de la β-caténine.

Dans le noyau, elle opère comme activateur transcriptionnel en étant liée aux membres de la famille des TCF/LEF et éventuellement à d'autres coactivateurs de la transcription des gènes cibles de la voie Wnt/APC/β-caténine, dont certains facteurs prolifératifs, tels que S100A4 ou CTHRC1 [START_REF] Li | TBL1-TBLR1 and beta-catenin recruit each other to Wnt target-gene promoter for transcription activation and oncogenesis[END_REF]. Trois mutations principales ont été décrites au niveau de l'exon 3 de CTNNB1, T41A, S45F et S45P [START_REF] Salas | Molecular characterization by array comparative genomic hybridization and DNA sequencing of 194 desmoid tumors[END_REF][START_REF] Crago | Near universal detection of alterations in CTNNB1 and Wnt pathway regulators in desmoid-type fibromatosis by wholeexome sequencing and genomic analysis[END_REF]. Ces mutations affectent le site d'interaction APC /β-Caténine et modifient la capacité de APC à entraîner la dégradation de la β-Caténine par le protéasome [START_REF] Li | TBL1-TBLR1 and beta-catenin recruit each other to Wnt target-gene promoter for transcription activation and oncogenesis[END_REF]. Cette accumulation nucléaire de la β-caténine peut alors être déterminée par immunohistochimie (IHC), et peut servir d'outil diagnostic permettant de différencier les tumeurs desmoïdes des autres tumeurs mésenchymateuses [START_REF] Ng | Nuclear beta-catenin in mesenchymal tumors[END_REF]. Les tumeurs desmoïdes n'ayant pas de mutation CTNNB1 dans l'exon 3, classées alors de type sauvage (WT), représentaient environ 15 % de toutes les tumeurs desmoïdes. Ce nombre a grandement diminué au cours des cinq dernières années depuis que les nouvelles générations de séquenceur permettent un séquençage du génome complet de la tumeur [START_REF] Crago | Near universal detection of alterations in CTNNB1 and Wnt pathway regulators in desmoid-type fibromatosis by wholeexome sequencing and genomic analysis[END_REF][START_REF] Colombo | Novel intra-genic large deletions of CTNNB1 gene identified in WT desmoid-type fibromatosis[END_REF]. Nous savons maintenant que la proportion des tumeurs desmoïdes portant une mutation dans un des acteurs de la voie de signalisation Wnt/β-caténine est très proche des 100 %.

Le nombre de récidive est plus élevé dans les tumeurs mutées S45F par rapport aux autres tumeurs mutées T41A et WT [START_REF] Lazar | Specific mutations in the betacatenin gene (CTNNB1) correlate with local recurrence in sporadic desmoid tumors[END_REF][START_REF] Colombo | CTNNB1 45F mutation is a molecular prognosticator of increased postoperative primary desmoid tumor recurrence: an independent, multicenter validation study[END_REF][START_REF] Van Broekhoven | Prognostic value of CTNNB1 gene mutation in primary sporadic aggressive fibromatosis[END_REF]. Ces résultats sont débattus et certaines études font état de résultats contradictoires [START_REF] Dômont | High frequency of beta-catenin heterozygous mutations in extra-abdominal fibromatosis: a potential molecular tool for disease management[END_REF][START_REF] Mullen | β-Catenin mutation status and outcomes in sporadic desmoid tumors[END_REF] et aucune différence d'expression de gènes cibles de la voie Wnt/ β-caténine n'a été cependant observée dans les tumeurs desmoïdes en fonction du type de mutation. D'autres études sont donc nécessaires pour confirmer ou infirmer ces données et identifier le potentiel mécanisme expliquant les différences d'évolutivité en fonction des différents types de mutation de CTNNB1 [START_REF] Timbergen | Wnt targets genes are not differentially expressed in desmoid tumors bearing different activating β-catenin mutations[END_REF].

Le rôle des oestrogènes

Le rôle des hormones sexuelles, en particulier des oestrogènes, est principalement basé sur des observations cliniques, et ce potentiel rôle a été à la base du traitement systémique des tumeurs desmoïdes par hormonothérapie. Les tumeurs desmoïdes surviennent parfois dans un contexte de grossesse [START_REF] Penel | Surgical versus non-surgical approach in primary desmoid-type fibromatosis patients: A nationwide prospective cohort from the French Sarcoma Group[END_REF]. Il a été également rapporté des cas d'accélération de la croissance tumorale pendant la grossesse ou pendant la prise de contraceptifs oraux et des cas de régression à la ménopause [START_REF] Reitamo | The desmoid syndrome. New aspects in the cause, pathogenesis and treatment of the desmoid tumor[END_REF][START_REF] Häyry | The desmoid tumor. II. Analysis of factors possibly contributing to the etiology and growth behavior[END_REF]. Les tumeurs desmoïdes expriment le récepteur des oestrogènes ER-ß et non le ER-a [START_REF] Deyrup | Estrogen receptor-beta expression in extraabdominal fibromatoses: an analysis of 40 cases[END_REF]. Le tamoxifène et le toremifène ont alors été considérés comme des traitements potentiels. Le toremifène n'a pas d'influence sur la prolifération cellulaire mais diminue l'expression de GAG, du TGFß1, du collagène et de la fibronectine et diminue l'affinité pour les récepteurs de type I pour TGFb1 et de type II pour 125I-TGF-ß1 [START_REF] Locci | Desmoid and fibroma tumors differently respond to TGFbeta(1) stimulus and ECM macromolecule accumulation[END_REF][START_REF] Stabellini | Toremifene decreases type I, type II and increases type III receptors in desmoid and fibroma and inhibits TGFbeta1 binding in desmoid fibroblasts[END_REF]. Récemment, il a été montré par IHC que l'expression de ER-β et de la cycline D1 était associée à la prolifération et qu'une forte expression pouvait être corrélée à la récidive [START_REF] Santti | Estrogen receptor beta expression correlates with proliferation in desmoid tumors[END_REF].

Les autres voies de signalisation impliquées dans le développement des tumeurs desmoïdes La voie de signalisation Notch

Durant le développement tissulaire, la voie Notch est impliquée dans la prolifération cellulaire, la différenciation, la survie cellulaire, la neurogenèse et l'homéostasie cellulaire [START_REF] Artavanis-Tsakonas | Notch signaling: cell fate control and signal integration in development[END_REF]. Peu d'études se sont penchées sur le rôle de la voie Notch dans les tumeurs desmoïdes, cependant, il a été démontré que les tumeurs desmoïdes expriment NOTCH1 et une de ses cibles avale, HES1 [START_REF] Carothers | Mesenchymal stromal cell mutations and wound healing contribute to the etiology of desmoid tumors[END_REF]. Dans un essai de phase 1, cinq patients sur sept ont présenté une réponse partielle après avoir été traités par un inhibiteur de γ-secrétase par voie orale (PF-03084014), empêchant le clivage des récepteurs Notch à la surface cellulaire [START_REF] Messersmith | A Phase I, dose-finding study in patients with advanced solid malignancies of the oral γ-secretase inhibitor PF-03084014[END_REF]. Cela a donné lieu à une étude in vitro réalisée par Shang et al. qui ont démontré, par IHC, que la surexpression de HES1 était nucléaire dans les tumeurs desmoïdes, spécifiquement par rapport aux tissus cicatriciels [START_REF] Shang | Targeting the Notch pathway: A potential therapeutic approach for desmoid tumors[END_REF]. À noter que l'expression de NOTCH1 et HES1 est également retrouvée in vivo dans cette étude. L'inhibition de cette voie diminue, proportionnellement à la dose de PF-03084014 administrée, l'expression de NICD (domaine intracellulaire du récepteur Notch normalement transloqué dans le noyau où il agit comme facteur de transcription) et de HES1 dans des cellules de tumeurs desmoïdes. Ainsi, l'inhibition de la voie de signalisation Notch entraîne un arrêt du cycle cellulaire G1 en diminuant la prolifération, la migration et l'invasion [START_REF] Shang | Targeting the Notch pathway: A potential therapeutic approach for desmoid tumors[END_REF].

La voie de signalisation Hedgehog

L'activation de la voie de signalisation Hedgehog maintient les cellules progénitrices dans un état plus indifférencié avec des capacités de prolifération plus importante. Ghanbari et al. ont montré que les gènes cibles de cette voie comme GLI1, PTCH1, et Hedgehog interacting protein (HHIP) sont surexprimé dans des échantillons de tumeurs desmoïdes par rapport aux tissus environnants [START_REF] Ghanbari-Azarnier | Targeting stem cell behavior in desmoid tumors (aggressive fibromatosis) by inhibiting hedgehog signaling[END_REF]. Il est alors envisagé que cette voie participe à la prolifération des cellules de tumeurs desmoïdes compte tenue de l'origine mésenchymateuses de ces cellules. De plus, il a été observé une surexpression de ces mêmes cibles, Gli1, Gli2 et Ptch1 dans les tumeurs desmoïdes de souris (Apc+/1638N) par rapport au tissu normal. Ainsi, in vivo, une modification post-traductionnelle de ces cibles induites par le Triparanol (inhibiteur pharmacologique) conduit à une réduction du volume tumoral chez les souris Apc+/1638N. Dans des modèles murins Apc+/1638N et Gli2+/-le développement tumoral est également moins important [START_REF] Ghanbari-Azarnier | Targeting stem cell behavior in desmoid tumors (aggressive fibromatosis) by inhibiting hedgehog signaling[END_REF][START_REF] Bi | Triparanol suppresses human tumor growth in vitro and in vivo[END_REF].

La voie de signalisation JAK/STAT

Les ligands notamment de la famille de l'IFN-β entraînent l'activation des tyrosines kinases JAK. Celle-ci passe alors par une transphosphorylation des récepteurs multiplexés et entraîne des phosphorylations avales en cascade. Les récepteurs ainsi phosphorylés peuvent alors servir de sites d'amarrage pour les facteurs de transcription localisé dans le cytoplasme (STAT). Ces STAT deviennent phosphorylés par les JAKs, dimérisent et sont alors transloqués dans les noyaux où ils activent ou répriment la transcription de gènes cibles. Dans les tumeurs desmoïdes humaines et dans des modèles murins de tumeurs desmoïdes (Apc/Apc1638N), il a été relevé une réponse accrue à l'IFN-β suggérant que la voie de signalisation JAK/STAT est activée [START_REF] Tjandra | IFN-{beta} signaling positively regulates tumorigenesis in aggressive fibromatosis, potentially by modulating mesenchymal progenitors[END_REF]. En accord avec ces données précliniques, des cas de régression de tumeurs desmoïdes après traitement par IFN ont été rapportés [START_REF] Fernberg | Interferon-induced remission in aggressive fibromatosis of the lower extremity[END_REF][START_REF] Geurs | Regression of a great abdominal desmoid tumor by interferon alpha[END_REF][START_REF] Ariën | Treatment of Aggressive Pelvic Fibromatosis With Interferon[END_REF][START_REF] Raguse | Interferon-induced remission of rapidly growing aggressive fibromatosis in the temporal fossa[END_REF] et une étude rétrospective a évalué l'effet chez 13 patients de l'IFN-alpha plus trétinoïne avec des résultats encourageants [START_REF] Leithner | Treatment of extraabdominal desmoid tumors with interferon-alpha with or without tretinoin[END_REF].

La voie de signalisation PI3 Kinase/AKT et mTOR

La relation entre les tumeurs desmoïdes et cette voie a été peu étudiée. Si une surexpression de la ß-caténine et du PDGFR-ß a bien été retrouvée par IHC dans une série de 29 échantillons, l'absence d'expression de PDGFR-alpha et de phospho-Ser-473 AKT suggère que cette voie est inactivée [START_REF] Dufresne | Identification of biological factors predictive of response to imatinib mesylate in aggressive fibromatosis[END_REF]. Meazza et al. ont montré qu'une partie des tumeurs desmoïdes pédiatriques présentait une mutation E17K AKT1 (31 %) mais cette mutation n'est pas retrouvée chez l'adulte [START_REF] Meazza | AKT1 and BRAF mutations in pediatric aggressive fibromatosis[END_REF]. La survie sans récidive est meilleure pour les patients présentant une mutation E17K AKT1. Récemment, Rosenberg et al. ont mis en évidence un effet antitumoral du sorafenib, un inhibiteur multikinase (ex. VEGFR, c-Kit et PDGFR). Or, le sorafenib entraîne une diminution de la prolifération et de l'invasion de façon dose dépendante à travers les voix Ras/MEK/ERK et PI3 kinase/AKT/mTOR. Aucun effet synergique ou additif sur la prolifération cellulaire n'a été observé entre le sorafenib et l'évérolimus [START_REF] Rosenberg | Sorafenib inhibits proliferation and invasion in desmoid-derived cells by targeting Ras/MEK/ERK and PI3K/Akt/mTOR pathways[END_REF].

L'efficacité de l'imatinib, inhibiteur de KIT et de PDGFR, a été évaluée dans plusieurs études cliniques chez des patients atteints de tumeurs desmoïdes mais ces données doivent être interprétées avec prudence car en outre les cibles de l'imatinib restent difficiles à isoler dans les tumeurs desmoïdes [START_REF] Penel | Imatinib for progressive and recurrent aggressive fibromatosis (desmoid tumors): an FNCLCC/French Sarcoma Group phase II trial with a long-term follow-up[END_REF][START_REF] Chugh | Efficacy of imatinib in aggressive fibromatosis: Results of a phase II multicenter Sarcoma Alliance for Research through Collaboration (SARC) trial[END_REF][START_REF] Heinrich | Clinical and molecular studies of the effect of imatinib on advanced aggressive fibromatosis (desmoid tumor)[END_REF][START_REF] Heinrich | Phase II, open-label study evaluating the activity of imatinib in treating life-threatening malignancies known to be associated with imatinib-sensitive tyrosine kinases[END_REF][START_REF] Kasper | Imatinib induces sustained progression arrest in RECIST progressive desmoid tumours: Final results of a phase II study of the German Interdisciplinary Sarcoma Group (GISG)[END_REF][START_REF] George | Selective kinase inhibition with daily imatinib intensifies toxicity of chemotherapy in patients with solid tumours[END_REF]. Des essais cliniques ont également été menés pour évaluer l'effet du sunitinib et du sorafenib avec notamment récemment la publication d'une phase 3 versus placebo [START_REF] George | Multicenter phase II trial of sunitinib in the treatment of nongastrointestinal stromal tumor sarcomas[END_REF][START_REF] Jo | A prospective multicenter phase II study of sunitinib in patients with advanced aggressive fibromatosis[END_REF][START_REF] Gounder | Sorafenib for Advanced and Refractory Desmoid Tumors[END_REF]. Les mécanismes biologiques sous-jacents à l'activité du sorafenib dans les tumeurs desmoïdes demeurent inconnus parce que le sorafenib cible de multiples tyrosine kinases, ce qui lui permet d'affecter de multiples voies de signalisation. Ces études ont mené également à évaluer l'utilisation d'un autre TKI, le pazopanib, qui devient alors une alternative de prise en charge ((DESMOPAZ). ClinicalTrial. gov: https://clinicaltrials).

Le rôle des métalloprotéases

Les métalloprotéases (MMP-3, MMP-7 et MMP-9) sont exprimées dans les tumeurs desmoïdes et jouent un rôle dans l'« invasivité » de la tumeur desmoïde [START_REF] Kong | Matrix metalloproteinase activity modulates tumor size, cell motility, and cell invasiveness in murine aggressive fibromatosis[END_REF][START_REF] Denys | Invasion and MMP expression profile in desmoid tumours[END_REF]. En effet, l'inhibition des MMP diminue l'invasion et la motilité cellulaire et MMP7, surexprimé dans les tumeurs desmoïdes mutées CTNNB1 par rapport au WT, pourrait être associée au pronostic des tumeurs desmoïdes, comme cela a été montré dans le cancer du pancréas [START_REF] Matono | Correlation between beta-catenin widespread nuclear expression and matrix metalloproteinase-7 overexpression in sporadic desmoid tumors[END_REF][START_REF] Jones | Comprehensive analysis of matrix metalloproteinase and tissue inhibitor expression in pancreatic cancer: increased expression of matrix metalloproteinase-7 predicts poor survival[END_REF]. En accord avec ces données, l'ilomastat, un inhibiteur des MMP, entraîne une diminution de la migration des cellules de tumeurs desmoïdes humaines et murines [START_REF] Kong | Matrix metalloproteinase activity modulates tumor size, cell motility, and cell invasiveness in murine aggressive fibromatosis[END_REF][START_REF] Denys | Invasion and MMP expression profile in desmoid tumours[END_REF].

Le rôle du TGF-ß dans les tumeurs desmoïdes et autres facteurs de croissance

Le TGF-ß, en tant que facteur de croissance, stimule l'activité transcriptionnelle de la beta-caténine. La voie de signalisation du TGF-ß a alors été étudiée en comparant les tumeurs desmoïdes à du tissu cicatriciel hypertrophique et à du tissu fibreux d'échantillons humains. Non exprimé dans le tissu fibreux, elle est activée à des niveaux similaires dans les tumeurs desmoïdes et le tissu cicatriciel. SMAD2/3 phosphorylé, en aval du récepteur dans la voie de signalisation de TGF-ß, était exprimé dans la majorité des cas de tumeurs desmoïdes mais dans une minorité de tissus cicatriciels et pas du tout dans les tissus fibreux [START_REF] Amini Nik | TGF-beta modulates beta-Catenin stability and signaling in mesenchymal proliferations[END_REF]. Diverses études ont porté sur l'effet de la stimulation des lignées cellulaires de tumeurs desmoïdes par du TFG-ß ou de l'EGF. Cela entraîne des régulations transcriptionnelles de divers gènes cibles comme SMAD4 et CTNNB1, et une augmentation des glycosaminoglycanes et du collagène. De plus, l'EGF accroît la motilité des cellules des tumeurs desmoïdes [START_REF] Locci | Desmoid and fibroma tumors differently respond to TGFbeta(1) stimulus and ECM macromolecule accumulation[END_REF][START_REF] Amini Nik | TGF-beta modulates beta-Catenin stability and signaling in mesenchymal proliferations[END_REF][START_REF] Trang | Potential for functional redundancy in EGF and TGFalpha signaling in desmoid cells: a cDNA microarray analysis[END_REF][START_REF] Joyner | Desmoid cell motility is induced in vitro by rhEGF[END_REF].

Récemment, il a été montré que TGF-β1 stimule la croissance cellulaire des cellules de tumeurs desmoïdes indépendamment de la présence d'une mutation de CTNNB1. Le TGF-β1 agit à travers la voie canonique SMAD. Ainsi, la phosphorylation de SMAD3 est cruciale pour la croissance cellulaire dépendante du TGF-β1. De plus, il a été montré que la confluence des cellules est un élément déterminant pour le TGF-β1 puisqu'elle induit la différenciation du myofibroblaste. Ceci implique que les communications intercellulaires ont un rôle important sur le comportement du myofibroblaste des tumeurs desmoïdes [START_REF] Enzo | Growth rate and myofibroblast differentiation of desmoid fibroblast-like cells are modulated by TGF-β signaling[END_REF]. Plusieurs études montrent également une surexpression du PDGF et de ses récepteurs (PDGFa, PDGFRa, PDGFß, et PDGFRß) dans les tumeurs fibreuses dont les tumeurs desmoïdes [START_REF] Liegl | Immunohistochemical and mutational analysis of PDGF and PDGFR in desmoid tumours: is there a role for tyrosine kinase inhibitors in c-kit-negative desmoid tumours?[END_REF][START_REF] Gibson | Platelet-derived growth factor receptors and ligands are upregulated in paediatric fibromatoses[END_REF].

Signature moléculaire et risque de rechute

Salas et al [START_REF] Salas | Gene Expression Profiling of Desmoid Tumors by cDNA Microarrays and Correlation with Progression-Free Survival[END_REF] ont identifié une signature moléculaire de 36 gènes capable de prédire la rechute avec une valeur prédictive positive de 78 %. Les deux top gènes surexprimés dans le groupe de patients qui ont récidivé sont FECH (ferrochelatase"), impliqué dans la biosynthèse de l'hème et STOML2 (stomatine Like proteine 2) impliqué dans l'activité mitochondriale. Le principal gène surexprimé dans le groupe de patient non récidivant était TRIP6 impliqué dans l'adhésion cellulaire.

Immunité et tumeurs desmoïdes

Modèle pré-clinique

Une des difficultés majeures pour l'étude expérimentale des tumeurs desmoïdes est la quasi absence de modèle pré-clinique que ce soit in vitro ou in vivo chez l'animal [START_REF] Van Broekhoven | Time trends in the incidence and treatment of extra-abdominal and abdominal aggressive fibromatosis: a population-based study[END_REF]. La mise en culture de cellules tumorales obtenues après chirurgie entraîne inévitablement une prolifération de fibroblastes normaux et la perte concomitante des cellules tumorales.

Même lorsque l'on obtient une lignée cellulaire primaire, les cellules tumorales atteignent alors la sénescence après plusieurs passages.

Les modèles cellulaires et animaux exprimant CTNNB1 muté sont difficiles à produire et le modèle de souris Apc+/Apc1638N bien qu'utile comme modèle est peut-être trop éloigné des tumeurs desmoïdes non lié à APC (90 % des tumeurs desmoïdes) pour être un modèle fiable [START_REF] Smits | Apc1638N: a mouse model for familial adenomatous polyposis-associated desmoid tumors and cutaneous cysts[END_REF]. Aucun modèle animal de tumeur desmoïde basé sur des mutations spécifiques de la β-Caténine n'est, à notre connaissance, publié mais un modèle Xenopus tropicalis APC muté a été récemment mis au point [START_REF] Naert | TALENs and CRISPR/Cas9 fuel genetically engineered clinically relevant Xenopus tropicalis tumor models[END_REF].

Conclusion

Les interactions entre les différentes voies de signalisations ici présentées sont encore mal connues et extrêmement complexes. L'efficacité thérapeutique optimale sera sûrement obtenue en ciblant de façon combinée les principales voies impliquées dans la tumorigénèse des tumeurs desmoïdes. Cependant l'évolutivité aberrante des tumeurs desmoïdes et probablement la variabilité au cours du temps des processus biologiques rend encore plus complexe l'étude de ces tumeurs et cela d'autant plus qu'il existe un manque de modèle proche du modèle humain. Les études futures devraient néanmoins se pencher sur l'inhibition thérapeutique de la voie de signalisation Wnt/β-caténine, la seule décrite ici comme activer par une altération génétique partagée par toutes les cellules cancéreuses composant les tumeurs desmoïdes. Bien entendu, comme dans l'ensemble de cancers, l'immunothérapie présente également un espoir thérapeutique pour les patients porteurs de tumeurs desmoïdes mais pour cela des études devront tout d'abord s'attacher à mieux décrire et comprendre l'environnement immunitaire des tumeurs desmoïdes.

Liens d'intérêts

The author(s) declare that they have no competing interests.

Références

  Dans une étude rapportée par Colombo et al. les groupes de gènes « Inflammation-Réponse immunitaire humorale » et « Liaison à l'antigène » sont significativement représentatifs des tumeurs desmoïdes porteuses de la mutation de la β-caténine T41A. Il a été rapporté par ailleurs un faible nombre de lymphocytes T et de macrophages infiltrant, sur un faible nombre d'échantillons de tumeurs desmoïdes. Enfin l'expression de PD-1/PD-L1 est faible voire absente [61].