Mohammed Benmir
email: med.benmir@gmail.com

Salma Chabbar

Rajae Aboulaich

Nabil Ismaili

A Hybrid Model of Tumor Growth Under Chemotherapy Medicine

Keywords: Hybrid tumor model, PDE model, Multi-agents simulation, Chemotherapy medicine

In this work we propose a hybrid model to simulate the evolution of a solid tumor. We consider a tumor under treatment and we use agent-based modeling to simulate the cells behavior while the diffusion of nutrients and drugs in the tumor environment is described using partial differential equations. The proposed model is implemented using NetLogo software and the partial differential equations are solved using a finite difference method and implemented using Python. The tests carried out give encouraging results and show that the number of tumor cells decreases under the effect of the chemotherapy and the size of the tumor decreases as well. This study offers an easy-to-use decision support tool for oncologists and clinicians.

I INTRODUCTION

Cancer is a major public health problem and one of the leading causes of death in the world. It is imperative to find promising clinical approaches to fight cancer based on new or existing therapies such as surgery, radiation therapy, hormone therapy, immunotherapy, cytotoxic chemotherapy and targeted therapy. These treatments depend on the type, location and grade of cancer, as well as the person's health and wishes. Chemotherapy is the use of drugs to treat a disease. Specifically, chemotherapy generally refers to the destruction of cancer cells using powerful chemicals. The division and death of cells is regulated in an orderly way in the human body. However, in cancer, cell division and growth are out of control. As a result, damaged cells begin to occupy more and more space in a part of the body and push out useful healthy cells. This is how a tumor develops in that part of the body. The fight against these cancerous cells and the modification of their mode of production and accumulation is therefore a crucial question for medical science. Scientists have developed different methods for the treatment of cancer. The use of less invasive methods has always played an essential role in the treatment of patients. Chemotherapy, as a method of treating cancer, involves applying drugs that affect the ability of cancer cells to divide and reproduce. The drug weakens cancer cells and destroys them by applying it directly to the cancer site or by passing it through the bloodstream [START_REF] Byrne | Modelling the response of vascular tumours to chemotherapy: a multiscale approach[END_REF][START_REF] Hinow | A spatial model of tumor-host interaction: application of chemotherapy[END_REF][START_REF] Alarcón | Multiscale modelling of tumour growth and therapy: the influence of vessel normalisation on chemotherapy[END_REF]. Mathematical modeling is a powerful tool for testing hypotheses, confirming experiments and simulating the dynamics of complex systems. As well as helping to understand the mechanistic foundations of dynamic and complex systems. In the field of oncology in particular, these models can be calibrated using experimental or clinical data. Competing hypotheses of tumor progression can be combined and treatment options thoroughly analyzed before clinical intervention [START_REF] Kourou | Machine learning applications in cancer prognosis and prediction[END_REF][START_REF] Menden | Machine learning prediction of cancer cell sensitivity to drugs based on genomic and chemical properties[END_REF][START_REF] Enriquez-Navas | Exploiting evolutionary principles to prolong tumor control in preclinical models of breast cancer[END_REF][START_REF] Welter | Computational model for tumor oxygenation applied to clinical data on breast tumor hemoglobin concentrations suggests vascular dilatation and compression[END_REF]. Quantitative modeling techniques are numerous, and an increasing number of theoretical approaches are successfully applied to cancer biology.

Tumor growth models can be categorized into continuous models, discrete models, and hybrid models, which combine the strengths of discrete and continuous approaches. Continuous models are based on ordinary differential equations (ODE) or partial differential equations (PDE) to describe the tumor growth phenomenon [START_REF] Friedman | Asymptotic stability for a free boundary problem arising in a tumor model[END_REF][START_REF] Zhang | Boundary element method for a free boundary problem modeling three dimensional tumor growth[END_REF]. PDE models account for greater complexity than ODE models. By using cell densities and nutrient concentrations as state variables, PDE models can be used to analyze various spatio-temporal phenomena. However, in some situations, individual and agent-based modeling (ABM) approaches, which treat the cells as discrete objects with predefined interaction rules, may offer an improvement over differential equation methods. The ABM approach provides a natural description of a system as a form of computational science. ABM agents interact and influence each other, make individual decisions, learn from their experiences and adapt their behaviors. They are therefore better adapted to their environment. A number of ABM simulation packages have been developed and applied to cancer research, among them [START_REF] Rejniak | Hybrid models of tumor growth[END_REF][START_REF] Ribba | A model of vascular tumour growth in mice combining longitudinal tumour size data with histological biomarkers[END_REF][START_REF] Wilensky | An introduction to agent-based modeling: modeling natural, social, and engineered complex systems with NetLogo[END_REF]. These packages have facilitated the overall process of developing an agent-based cancer model and also enabled computational oncologists to focus their time and energy on the specific cancer problems that interest them. The solid tumors develop in two distinct phases: the avascular phase and the vascular phase. Modeling the avascular phase of tumor growth and the vasculature of tumor angiogenesis has been covered widely in literature see [START_REF] Chaplain | Mathematical modeling of tumor-induced angiogenesis[END_REF][START_REF] Martins | Multiscale models for the growth of avascular tumors[END_REF][START_REF] Roose | Mathematical models of avascular tumor growth[END_REF][START_REF] Sleeman | Partial differential equations of chemotaxis and angiogenesis[END_REF][START_REF] Welter | Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth[END_REF][START_REF] Welter | Interstitial fluid flow and drug delivery in vascularized tumors: A computational model[END_REF]. This work follows the work in [START_REF] Chabbar | Simulating tumor growth using mathematical and agent-based modeling[END_REF], using an analogous approach, we propose a more complete model by considering a treatment by chemotherapy in the multi-agent environment.

This paper is organized as follows. In section 2, we present the diffusion-reaction equations for the oxygen and chemotherapeutic drug concentrations. Section 3 provides an overview of the hybrid tumor growth model under chemotherapy medicine using ABM and PDEs. The cell behavior is described using a multi-agent model in NetLogo where the oxygen and the drug treatment is diffused through the agent environment using the continuous diffusion equations to determine whether the environment is conducive to tumor cell survival and proliferation. In section 4, we present the results of the simulations that illustrate the impact of chemotherapy on tumor growth. Finally, in Section 5, we discuss our results and future research directions.

II DIFFUSION-REACTION MODEL FOR TEMPORAL-SPATIAL EVOLUTION OF OXYGEN AND CHEMOTHERAPEUTIC DRUG CONCENTRATIONS

To simulate the temporal-spatial evolution of oxygen and drug concentrations in tumor systems, we employ the following diffusion-reaction models.

Oxygen diffusion equation

To survive and multiply, the cells need nutrients and especially oxygen. Hypoxia occurs when a cell is deprived of oxygen. Severe hypoxia can lead to cell death [START_REF] Chabbar | Simulating tumor growth using mathematical and agent-based modeling[END_REF][START_REF] Shimizu | Induction of apoptosis as well as necrosis by hypoxia and predominant prevention of apoptosis by Bcl-2 and Bcl-XL[END_REF]. When faced with an unfavorable environment, a cell can transition to a quiescent resting phase where it stops multiplying and reduces its nutrients needs to survive waiting for the environmental conditions to become favorable again [START_REF] Chabbar | Simulating tumor growth using mathematical and agent-based modeling[END_REF][START_REF] Meijer | Le cycle de division cellulaire et sa régulation[END_REF]. Oxygen is therefore an important parameter in the dynamics of cells. The equation for the diffusion of the oxygen concentration is based on the one proposed in [START_REF] Chabbar | Simulating tumor growth using mathematical and agent-based modeling[END_REF].

     ∂ t C -∆C = α C B -C -α cons C ∂ n C = 0 C t = 0, x, y = C 0 x, y .
where C B is the oxygen concentration in the vascular system, α C B -C is the rate of oxygen concentration supplied by the vascular system and α cons is the consumption rate of oxygen by the cells, which depends on the cell type such that quiescent cells have a lower oxygen consumption than other cells.

Chemotherapeutic drug diffusion equation

Chemotherapy drugs do not distinguish between healthy and cancerous cells. They can be divided into three classes. Class I drugs are non-phase specific and affect both quiescent and proliferating cells; class II drugs are phase specific and target only cells in a given phase; and class III drugs kill only proliferating cells [START_REF] Hinow | A spatial model of tumor-host interaction: application of chemotherapy[END_REF]. In our work we treat the third class of drugs that kill only proliferating cells. The diffusion equation of the drugs in the cell environment is considered similar to that of oxygen as proposed in [START_REF] Byrne | Modelling the response of vascular tumours to chemotherapy: a multiscale approach[END_REF].

     ∂ t φ -∆φ = β φ B -φ -β cons φ ∂ n φ = 0 φ t = 0, x, y = φ 0 x, y .
where φ B is the drug concentration in the vascular system, β φ B -φ is the rate of drug concentration supplied by the vascular system and β cons is the consumption rate of drug by the cells, which also depends on the cell type such that quiescent cells have a lower drug consumption than other cells [START_REF] Byrne | Modelling the response of vascular tumours to chemotherapy: a multiscale approach[END_REF].

III HYBRID AGENT-BASED / PDE MODEL

In this section, a discrete cell population model based on the one in [START_REF] Chabbar | Simulating tumor growth using mathematical and agent-based modeling[END_REF] where the cell behavior is described using ABM and combined with PDEs to describe the operating environment of the cells is presented. A schematic overview of the resulting hybrid tumor model is given in Figure 1. The governing rules for cell behavior are modeled using NetLogo software: a programming language and modeling environment for agent-based systems. The oxygen and chemotherapeutic drug equations are solved using a finite difference method in Python. The interaction between Python and NetLogo is ensured by an extension implemented in NetLogo which allows the exchange of data between the two programs.

The biological rules governing the cell behavior in Netlogo according to their environmental conditions are presented below.

Agent-based model

The tumor environment Ω in NetLogo is modeled as a square grid divided into patches where the agents move freely: Ω = k P atch k . The evolution of the tumor is followed in a time interval [0, T ]. Each agent represents a cell. Three different cellular states are defined: the healthy state (S), the proliferative cancer state (P) and the quiescent state (Q). Necrotic dead cells are discarded. Oxygen and chemotherapeutic drug are diffused through Ω, each P atch k has its own level of oxygen and drug concentrations. The cells move freely within the patches and meet the biological criteria listed in the paragraph below.

Cellular automata rules

At each time iteration, two random numbers p 1 and p 2 in [0, 1] are generated and the following rules are applied to each cell:

• A normal healthy cell S can divide at a rate λ S if : p 1 < λ S and if, in the patch where it is located, there is sufficient oxygen and there is not a lethal concentration of drug.

• Healthy cells mitosis spawns a new healthy cell.

• S cell can mutate into P cell with a probability p mut if p 2 < p mut .

Its mitosis rate is then multiplied by 10 to emphasize the uncontrolled proliferation of tumor cells. • The conditions for the mitosis of a P cell are the same as for the S cell.

• P cells mitosis spawns a new P cell.

• Each of S and P cells need a level of oxygen C S,P in order to survive and proliferate.

When it is in hypoxia:

1. If it is a S cell: the cell dies and it will be destroyed.

2. If it is a P cell: to model the uncontrollable nature of cancer cells, the P cell attempts to survive the unfavorable environment by transitioning to the quiescent state Q. It therefore stops dividing and its oxygen requirement is then reduced to C Q to enable it to survive. • Q cell can revert to being P cell if the environment becomes favorable in oxygen.

• A cell that remains quiescent for more than T max iterations in a row will die of necrosis and will be destroyed. • The class III drugs used only target multiplying cells and only have an effect on the proliferating cells when the lethal dose is reached. Therefore, they do not affect Q cells. • If the concentration of drug in the patch reaches the drug's threshold φ * : if it is a S or P cell, the cell dies while a Q cell is not affected.

Hybrid model implementation

The hybrid model makes use of the results of the oxygen and the chemotherapeutic drug diffusion equations in the agent-based model defined through the cellular automata rules. The model is similar to the one used in [START_REF] Chabbar | Simulating tumor growth using mathematical and agent-based modeling[END_REF], but under chemotherapy and runs as follows :

-The first step is the initialization of the model on NetLogo: 1. N healthy cells are created in the environment. 2. Each P atch k is supplied with its initial oxygen concentration C 0 k and initial drug concentration φ 0 k . 3. A specific P atch B is selected to hold the blood vessel supplying oxygen and drug to the environment -At each time iteration t:

1. The oxygen and the drug concentrations C t k and φ t k respectively, are collected from the patches and sent to Python as the initial conditions. 2. ∂ t C -∆C = 0 and ∂ t φ -∆φ = 0 are solved in Python using a finite difference method. 3. The new oxygen and the new drug concentration levels C t+1 k and φ t+1 k respectively, are sent back to NetLogo and redistributed to the patches. 4. The environment is supplied with oxygen C B from the vascular system through P atch B . 5. If the drug is administered, the environment is supplied with φ B from the vascular system through P atch B . 6. The cells interact with their environment according to the cellular automata rules. 7. The oxygen and the drug levels in each patch are reduced by the amounts consumed by the cells.

IV SIMULATIONS

For the simulations, we assign different colors to each cell depending on its type and to the background of the patches depending on their oxygen and drug levels. The S cells are blue, the P cells red and the Q cells brown. Patches have a gray shade proportional to their oxygen and drug concentration. A patch filled with oxygen or medication is white, while a patch with insufficient oxygen or a low level of medication is black. We investigate how the dynamics of the normal and cancer cells change when a chemotherapeutic drug is introduced. We assume that the drug is continuously administered into the blood vessel. It diffuses through the tissue and is absorbed by all types of cells. The initial setup used to test the response of our virtual tumor to different drugs is shown in Figure 2 and the numerical values of the parameters used are given in Table 1. The simulation results presented in Figure 3 show that the drug class used is not very effective on the tumor modeled. The drug fails to completely treat the tumor. The proliferating cancer cells were eliminated but the quiescent cells survived the treatment. The patch that contains the vasculature is clearly visible in the middle of the environment with a white color. Before the drug level in the patches reaches the critical threshold, the cells furthest from the center and thus from the oxygen source have either died of hypoxia or gone into quiescence. The healthy and cancerous cells close to the center managed to survive. Once the drug levels increased and reached the critical threshold, the S and P cells close to the center, therefore close to the source of the diffusion of the drug, whether they are healthy or cancerous, were killed. As for the quiescent cells, they have survived the treatment. We tested different scenarios by changing the concentration of drug administered φ B and we noticed that the effectiveness of the treatment does not increase much. The quiescent cells are still present at the end of the simulations although the higher the dose, the further they are from the center. This is explained by the fact that the modeled tumor is still in the avascular stage. The vascularization present is not sufficient to supply the tissue with oxygen, which results in a large number of quiescent cells that are not affected by the class of drugs used.

Symbol

V CONCLUSION AND REFERENCES

Conclusion

The aim of this work was to present a hybrid method to model tumor growth under chemotherapy medicine. We used agent-based modeling to simulate the behavior of cells, while the diffusion of oxygen and chemotherapeutic drug in the tumor environment is described using diffusion-reaction equations. The simulations show that the tumor cell population generally grows exponentially when not exposed to anti-cancer drug or low drug dose, while tumor cell growth is contained with high drug dose. However, to effectively treat the tumor, a drug that is appropriate to its stage of development must be used.

In our case, to improve the effectiveness of the treatment we can either change the type of drug used or include in, our model, the phenomenon of angiogenesis which allows the tumor to create its own vasculature and enter the vascular stage. The hybrid model developed can also be extended to take into account some important constraints like the effect of glucose consumption and hydrogen ion production on tumor growth. The extracellular matrix, cell-cell adhesion and cell movement can also be included in the model as additional environmental parameters to predict the survival probability of the abnormal cells.

Figure 1 :

 1 Figure 1: Schematic diagram of the hybrid model.

Figure 2 :

 2 Figure 2: (Left) Initialization using NetLogo: S = 1000, p mut = 0.01 and λ S = 0.001. (Right) The distribution of the oxygen concentration (upper panel), and of the drug concentration (lower panel).

Figure 3 :

 3 Figure 3: Evolution of the cells, oxygen and drugs distributions at different iterations with φ B = 10µM/iteration (From the top to the low at iterations suivantes: t = 30, 60 and 97 iterations).

Table 1 :

 1 Summary of the parameters used in the tumor growth process

		Value	Unit
	φ *	0.20	µM/iteration/cell
	[C S , C P , C Q ] [0.0157, 0.057, 0.0032] µM/iteration/cell
	[φ S , φ P , φ Q ]	[0.034, 0.034, 0.007]	µM/iteration/cell
	T max	20	iterations


domain: spi info phys math

text:
Antoine Manier

Julien Moras

Jean-Christophe Michelin

Hélène Piet-Lahanier

Railway lidar semantic segmentation with axially symmetrical convolutional learning


Jean-Baptiste Gorce

Eric Falcon

Statistical Equilibrium of Large Scales in three-dimensional Hydrodynamic Turbulence

We investigate experimentally three-dimensional (3D) hydrodynamic turbulence at scales larger than the forcing scale. We manage to perform a scale separation between the forcing scale and the container size by injecting energy into the fluid using centimetric magnetic particles. We measure the statistics of the fluid velocity field at scales larger than the forcing scale (energy spectra, velocity distributions, and energy flux spectrum). In particular, we show that the large-scale dynamics are in statistical equilibrium and can be described with an effective temperature, although not isolated from the turbulent Kolmogorov cascade. In the large-scale domain, the energy flux is zero on average but exhibits intense temporal fluctuations. Our work paves the way to use equilibrium statistical mechanics to describe the large-scale properties of 3D turbulent flows.

Introduction.-Three-dimensional (3D) hydrodynamics turbulence has been extensively studied to characterize the energy transfers in the inertial range, the interval between the energy injection scale and the small (dissipative) scale [START_REF] Richardson | Weather Prediction by Numerical Process[END_REF][START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF][START_REF] Kolmogorov | Dissipation of energy in the locally isotropic turbulence[END_REF][START_REF] Frisch | Turbulence: The Legacy of A. N. Kolmogorov[END_REF][START_REF]Pope Turbulent flows[END_REF]. While they control many properties of 3D turbulent flows, e.g., mixing in industrial flows, or transport of tracers in geophysical and astrophysical turbulent flows [START_REF]Moffatt Magnetic Field Generation in Electrically Conducting Fluids[END_REF], the large-scale properties of turbulence, the scales larger than the forcing scale, have been less investigated. Indeed, in most experiments and direct numerical simulations (DNS), 3D turbulent flows are forced at a scale comparable to the container size to study the turbulent energy cascade within the inertial range. However, it has been conjectured that the large-scale modes of turbulent flows possess the same energy and are in a statistical stationary equilibrium regime [START_REF] Frisch | Turbulence: The Legacy of A. N. Kolmogorov[END_REF][START_REF] Burgers | On the application of statistical mechanics to the theory of turbulent fluid motion[END_REF][START_REF] Hopf | Statistical hydromechanics and functional calculus[END_REF][START_REF] Lee | On some statistical properties of hydrodynamical and magneto-hydrodynamical fields[END_REF][START_REF] Kraichnan | Helical turbulence and absolute equilibrium[END_REF]. This equipartition regime, also called thermal equilibrium, would occur if no mean energy flux is transferred from the forcing scale to the large scales. Such a statistical equilibrium is difficult to observe in most experimental systems and numerical simulations because there is no scale separation between the forcing scale and the container size. Numerical simulations have recently confirmed the statistical equilibrium in 3D forced turbulent flows for the spectrally truncated Navier-Stokes equations [START_REF] Alexakis | Energy fluxes in quasi-equilibrium flows[END_REF] and the truncated Euler equation [START_REF] Cichowlas | Effective dissipation and turbulence in spectrally truncated Euler flows[END_REF][START_REF] Dallas | Statistical Equilibria of Large Scales in dissipative Hydrodynamic Turbulence[END_REF][START_REF] Alexakis | On the thermal equilibrium state of large-scale flows[END_REF][START_REF] Verma | Chatterjee[END_REF], but experimental evidence of this regime remains elusive.

Here, we generate 3D hydrodynamic turbulence using centimetric magnetic particles immersed in a large fluid reservoir. This method provides a wide interval between the energy injection scale and the container size. We observe a statistical equilibrium regime in this large-scale interval while a turbulent cascade develops in the inertial range. We also show that the effective temperature of the statistical equilibrium regime is related to the injection of energy. Note that large-scale structures in decaying 3D turbulence have been investigated [START_REF] Batchelor | The large-scale structure of homogeneous turbulence[END_REF][START_REF] Saffman | The large-scale structure of homogeneous turbulence[END_REF][18][START_REF] Lesieur | Turbulence in fluids[END_REF][START_REF] Krogstad | Is grid turbulence Saffman turbulence?[END_REF][START_REF] Yoshimatsu | No return to reflection symmetry in freely decaying homogeneous turbulence[END_REF], but are different from the stationary (forcing) case [START_REF] Alexakis | On the thermal equilibrium state of large-scale flows[END_REF]. Other turbulent systems also exhibit large-scale statistical equilibrium, e.g., in wave turbulence with no inverse cascade, such as capillary waves [START_REF] Michel | Observation of Thermal Equilibrium in Capillary Wave Turbulence[END_REF][START_REF] Balkovsky | Large-scale properties of wave turbulence[END_REF][START_REF] Abdurakhimov | Bidirectional energy cascade in surface capillary waves[END_REF], bending waves in mechanical plates [START_REF] Miquel | Low-frequency spectra of bending wave turbulence[END_REF], or optical waves [START_REF] Connaughton | Condensation of Classical Nonlinear Waves[END_REF]. Con-versely, the presence of an inverse cascade implies that two-dimensional hydrodynamic turbulence [START_REF] Kraichnan | Inertial Ranges in Two-Dimensional Turbulence[END_REF], gravity wave turbulence [START_REF] Falcon | Saturation of the Inverse Cascade in Surface Gravity-Wave Turbulence[END_REF], or acoustic wave turbulence in superfluid [START_REF] Ganshin | Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium[END_REF] do not exhibit a statistical equilibrium regime.

Theoretical backgrounds.-In the case of nonhelical, incompressible, inviscid, and force-free turbulent flows, Kraichnan [START_REF] Kraichnan | Helical turbulence and absolute equilibrium[END_REF] derived the statistical equilibrium energy spectrum E T (k) for low wavenumbers k

E T (k) = 4παk 2 α 2 -β 2 k 2 .
(

α, β are determined by the total energy and the helicity of the system. This result referred to as absolute equilibrium, is related to classical equilibrium statistical mechanics and is equivalent to the equipartition of the total kinetic energy among the large-scale Fourier modes [START_REF] Hopf | Statistical hydromechanics and functional calculus[END_REF][START_REF] Lee | On some statistical properties of hydrodynamical and magneto-hydrodynamical fields[END_REF][START_REF] Lesieur | Turbulence in fluids[END_REF][START_REF] Rose | Fully developed turbulence and statistical mechanics[END_REF]. One can also say that the large-scale equipartition implies that the spectral energy density per unit mass E T (k)dk is equal to the number of modes times the energy per mode per mass, i.e.,

E T (k)dk = 4πk B T ρ k 2 dk (2)
with k B the Boltzmann constant, ρ the fluid density, and T a temperature in a classical thermodynamic equilibrium sense. Therefore, one obtains the expression which is equivalent to Eq. ( 1) when βk ≪ α. Note that deviations from Eq. ( 1) are expected for a broadband spectral forcing instead of a narrow one [START_REF] Alexakis | On the thermal equilibrium state of large-scale flows[END_REF] or in the case of anisotropic turbulence [START_REF] Thalabard | A statistical mechanics framework for the large-scale structure of turbulent von Kármán flows[END_REF]. By assuming high-Reynolds number isotropic turbulence, the energy spectrum in the inertial range (i.e., for high k) is given by the Kolmogorov spectrum [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF]

E K (k) = C K ǫ 2/3 k -5/3
(3)

with C K ≃ 1.6 the experimentally measured Kolmogorov constant [START_REF] Saddoughi | Local isotropy in turbulent boundary layers at high Reynolds number[END_REF], and ǫ the rate of energy dissipation per unit mass. In the stationary regime, ǫ is constant and equal to the energy flux transferred from the forcing scale to the dissipation scale. Therefore, ǫ is also equal to the energy injection rate in the stationary regime.

Typeset by REVT E X Experimental setup.-We inject energy homogeneously into the fluid using small magnetic particles. The nonlinear transfer of energy and turbulent cascade towards small scales (inertial range) have been characterized using this method [START_REF] Cazaubiel | Three-dimensional turbulence generated homogeneously by magnetic particles[END_REF]. To measure the large-scale properties of turbulence, we scaled up this experimental system. A plexiglass square container of length L = 32 cm and height h = 22 cm is filled with water (22.5 L) and sealed by a transparent lid. This fluid container sits between a pair of Helmholtz coils (0.49 m inner diameter and 1 m outer diameter). The pair of coils is powered by a sinusoidal current (Itech IT7815 AC 15 kW power supply) and generates a vertical oscillating magnetic field B(t) with an amplitude B ∈ [0, 360] G and a frequency F ∈ [0, 25] Hz. This AC magnetic field is homogeneous within all the volume of the fluid container (5% accuracy) and transfers kinetic energy to N neodymium magnets encapsulated in plexiglass shells (1 cm), which are immersed in the fluid (N ∈ [50, 450]). The volume fraction of the magnetic particles is smaller than 1.5%. The kinetic energy of the magnetic particles is then transferred to the surrounding fluid randomly in both space and time (see [START_REF] Falcon | Equation of state of a granular gas homogeneously driven by particle rotations[END_REF][START_REF] Falcon | Dissipated power within a turbulent flow forced homogeneously by magnetic particles[END_REF][START_REF] Cazaubiel | Three-dimensional turbulence generated homogeneously by magnetic particles[END_REF] and movies in the Supplemental Material [36,37]). The forcing scale is estimated to be 5 cm. It corresponds to the integral scale L i defined as the abscissa of the maximum of the energy spectrum (see below). Note that L i cannot be accurately computed from the autocorrelation function of the velocity field since the container size L is not eight times larger than the integral scale [START_REF]Pope Turbulent flows[END_REF][START_REF] Cazaubiel | Three-dimensional turbulence generated homogeneously by magnetic particles[END_REF][START_REF] O'neill | Autocorrelation functions and the determination of integral length with reference to experimental and numerical data[END_REF]. The fluid velocity is measured locally by nonintrusive Laser Doppler Velocimetry (LDV Dantec Flow Explorer 1D) with a sampling frequency of 250 Hz. We perform Particle Image Velocimetry (PIV) [START_REF] Thielicke | PIVlab-towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB[END_REF] to measure the fluid velocity field in a horizontal xy plane (32×32 cm 2 ). The fluid is seeded with Polyamide fluid tracers (50 µm) illuminated by a horizontal laser sheet and a high-speed camera (Phantom V1840, 2048 × 1952 pixels 2 at 200 fps), located on the top of the fluid container, records time series of images. The mean fluid velocity is smaller than the standard deviation of the velocity fluctuations σ u (< 10%), such that one can assume that there is no mean flow. The isotropy of the velocity field is also checked for different values of N [START_REF]The measured isotropy ratios, σu/σv x and σu/σv y , are smaller than 7% regardless of N[END_REF]. Typical values of the turbulent flow are the following: the dissipation rate ǫ is 3×10 -4 m 2 /s 3 , the Reynolds number at the integral scale L i ≃ 5 cm is 650, and the Reynolds number at the Taylor scale L λ ≃ 7.6 mm is Re λ = 100. We have Re λ ∈ [56, 100] when changing the experimental parameters (F , N , or B).

Spatial power spectrum.-The longitudinal and transverse horizontal fluid velocities are defined as u(x, t) and v(x, t). We first measure the longitudinal E uu (k x ) and transverse E vv (k x ) spectra (Fig. 1). The inertial range is consistent with the Kolmogorov prediction over a decade. The power spectra are proportional to k sional (1D) power spectra is equal to E vv (k x )/E uu (k x ) = 4/3 [START_REF]Pope Turbulent flows[END_REF] (black lines in Fig. 1). In the case of isotropic turbulence, the 3D power spectrum E(k) is derived from the longitudinal and transverse spectra [START_REF]Pope Turbulent flows[END_REF]18]

E(k) = -k d dk 1 2 E uu (k x ) + E vv (k x ) . (4)
The energy spectrum E(k) is shown in Fig. 1. A k 2 power law is observed in the energy spectra at lower wavenumbers, illustrating the statistical equilibrium regime, while a k -5/3 power law is observed at higher wavenumbers, indicating a direct energy cascade in the inertial range.

In between these regimes, the wavenumbers close to the value k i = 2π/L i suggest that the statistical equilibrium state and the out-of-equilibrium one interact with each other (see below). Effective temperature.-Both k 2 and k -5/3 power laws are consistently observed in the energy spectra when increasing the energy injection rate ǫ (inset of Fig. 1). For each 3D spectrum E(k), we compute the effective temperature by integrating both members of Eq. ( 2) in the large-scale interval k ∈

[k L = 2π/L, k i = 2π/L i ] T exp = 3ρ 32π 4 k B L 3 L 3 i L 3 -L 3 i ki kL E(k)dk . (5)
FIG. 2. Effective temperature Texp of the statistical equilibrium regime of the large scales for different energy injection rates ǫ. The energy injection rate ǫ is measured using

ǫ = 2ν ∞ 2π/L k 2 E(k)dk.
Texp is measured from the 3D spectra shown in Fig. 1 using Eqs. (2) (•) and ( 5) (×). The solid dashed line corresponds to Eq. ( 6) using ki/2π = 13.3 m -1 . Insets: Structure functions S2(r) (top) and S3(r)/(4ǫ/5) (bottom) for different ǫ (same colors as in the inset of Fig. 1). Solid lines correspond to r 2/3 and -r, respectively.

T exp is shown in Fig. 2 as a function of the energy injection rate ǫ (×). One can also estimate the temperature by fitting directly the experimental 3D spectra with Eq. ( 2), which leads to similar values of the temperature (• in Fig. 2). T exp is found to be 13 orders of magnitude higher than the room temperature and proportional to ǫ 2/3 (Fig. 2). This result is explained by equating Eqs. ( 2) and (3), which gives the relationship

T = ρC K 4πk B k -11/3 i ǫ 2/3 . (6)
Structure functions.-The velocity increments at a distance r, S i (r) = [v(x + r) -v(x)] i are now computed from the PIV measurements. The insets of Fig. 2 show that S 2 (r) ∼ (ǫr) 2/3 and S 3 (r) = -4ǫr/5 in the inertial range, as predicted theoretically [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF][START_REF] Kolmogorov | Dissipation of energy in the locally isotropic turbulence[END_REF][START_REF]Pope Turbulent flows[END_REF]. For large scales (i.e., r > 0.1 m), S 2 (r) and S 3 (r) are found to be roughly independent of r, except when r ≃ L. We also found that S 2 (r) ≃ 2σ 2 u , suggesting that the velocities are uncorrelated at long distances, as expected [18].

Velocity probability distribution.-The probability distribution functions (PDF) of the velocity field (Fig. 3) are found to be strongly non-Gaussian probably because the PDFs of the Lagrangian magnetic particle velocity are stretched exponentials (see Supp. Mat. [36]). However, we show that the large-scale modes are normally distributed by applying a spatial low-pass filter to the FIG. 3. Probability density functions (PDF) of the normalized fluid velocity fluctuations u/ u 2 of (♦) large-scale modes, ( ) all modes, and (•) small-scale modes for σu = 0.6 cm/s. The cutoff value of the filter is equal to k/2π = 9.4 (m -1 ). The black dashed line represents a Gaussian distribution. Inset: Kurtosis (K = u 4 / u 2 2 ) of the large-scale modes as a function of ǫ (σu ∈ [0.6, 1.3] cm/s).

velocity field, confirming that the large-scale modes have reached a statistical equilibrium. The Kurtosis of the low-pass filtered velocity distribution is equal to 3 (inset Fig. 3). The shape of the PDF of the low-pass filtered velocities is also independent of the energy injection rate ǫ, which is illustrated by the constant value of the kurtosis (inset Fig. 3). High-pass filtering of the velocity field also shows that the non-Gaussianity of the PDFs is reminiscent of the magnetic particle velocity one.

Mean energy flux.-Measuring the energy flux is essential to understanding the dynamics of turbulent phenomena [START_REF] Verma | Variable energy flux in turbulence[END_REF]. We compute the time-averaged energy flux spectrum Π(k) from the expression Π(k

, t) = v < k • [v < k • ∇v > k ] r + v < k • [v > k • ∇v > k ] r [4], where v < k (r) ≡ k 0 v(k ′ )e ik ′ •r dk ′ is the low-filtered velocity field at the wavenumber k, v > k (r) ≡ ∞ k v(k ′ )e ik ′ •
r dk ′ the high-filtered one, and v is the Fourier transform of the velocity field v. The zero-mean energy flux measured at low wavenumbers confirms the statistical equilibrium regime (Fig. 4). The interval in which the mean energy flux is zero corresponds to the same interval in which the k 2 power law of the energy spectrum is observed. In the inertial range, the energy flux is positive and implies a direct energy cascade towards high wavenumbers, corresponding to the turbulent cascade shown in Fig. 1. For wavenumbers k/2π higher than 100 m -1 , the energy flux strongly decreases, which is consistent with Fig. 1.

Energy flux fluctuations.-Although no energy cascades within the large scales in equipartition, intense temporal fluctuations of the energy flux Π(k, t) are observed (bottom inset of Fig. 4 -see also Supp. Mat. [36]). This highlights that the large-scale domain is not isolated from the inertial range. Within the large-scale interval, the energy flux follows a Gaussian distribution (bottom inset of Fig. 4), whose standard deviation

σ Π (k) = Π(k, t) 2 1/2
is proportional to k 2 (top inset), similarly to the energy spectrum E(k). We can thus infer that σ Π (k)/E(k) ∼ k 0 for k < k i . The damped fluctuations of zero-mean energy flux observed at low wavenumbers (top inset Fig. 4) have also been reported in DNS [START_REF] Alexakis | Energy fluxes in quasi-equilibrium flows[END_REF][START_REF] Dallas | Statistical Equilibria of Large Scales in dissipative Hydrodynamic Turbulence[END_REF]. The top inset of Fig. 4 also shows that these fluctuations are intense within the direct cascade but are strongly damped by viscous dissipation at high wavenumbers.

Temporal power spectrum.-We now measure the temporal spectrum E u (f ) of the horizontal velocity u(t) (Fig. 5). The signal is recorded for T = 5 hours to converge the statistics at low frequencies (f < f i ), which represent the large-scale modes. To avoid a significant increase in the fluid temperature, we repeatedly performed LDV measurements for 100 s and then we let the fluid cool down for 10 min. The signal u(t) is shown in the inset of Fig. 5, with a low-pass filtered signal (black) to emphasize the slow modes of the temporal signal. The frequency spectrum shown in Fig. 5 is proportional to f -5/3 at high frequencies (f > f i ). This is consistent with the k -5/3 power law observed in the unidimensional FIG. 5. Temporal power spectrum density of the horizontal velocity Eu(f ). The dashed line represents a f 0 power law and the dot-dashed line represents a f -5/3 power law. The forcing parameters are identical to Fig. 1. fi indicates the beginning of the inertial range, i.e., the typical correlation frequency of the flow. Inset: Horizontal velocity u(t) low-pass filtered at 2 Hz (blue), 0.2 Hz (red), and 0.02 Hz (black), as illustrated by the colored arrows in the main figure .   energy spectrum, which implies a direct energy cascade in the inertial range (Fig. This power law was preby the Tennekes' model (large-scale advection of turbulent eddies) in isotropic turbulence without mean flow [START_REF] Cazaubiel | Three-dimensional turbulence generated homogeneously by magnetic particles[END_REF][START_REF] Tennekes | Eulerian and lagrangian time microscales in isotropic turbulence[END_REF]. At low frequencies (f < f i ), the frequency spectrum is found to be almost flat f 0 , implying that large scales are uncorrelated. This is similar to the unidimensional spatial spectrum E uu (k) ∼ k 0 at large scales (Fig. 1), suggesting that we observe the statistical equilibrium regime at low frequencies.

Conclusion.-We have experimentally shown that the large-scale dynamics in forced dissipative 3D hydrodynamic turbulence are in agreement with the statistical equilibrium prediction. This system is a remarkable example in which the large scales are in statistical equilibrium, while smaller scales are in an out-of-equilibrium stationary regime. A direct consequence of this experimental validation is that simulations leading to a statistical equilibrium regime, such as those of the truncated Euler equation [START_REF] Dallas | Statistical Equilibria of Large Scales in dissipative Hydrodynamic Turbulence[END_REF], could provide a new tool to efficiently simulate the large-scale dynamics of 3D turbulent flows in various fields. Our findings also pave the way to possibly use concepts of equilibrium statistical mechanics (such as fluctuation-dissipation and fluctuation theorems) for large-scale turbulent flows. It can help better understand the interactions between the degrees of freedom at equilibrium (large scales) with out-of-equilibrium structures (small scales), which are essential when study-ing turbulent phenomena. In the future, we will explore the transient regimes of the equilibrium regime of large scales, called the thermalization processes, by measuring the growth and decay of turbulence [START_REF] Saffman | The large-scale structure of homogeneous turbulence[END_REF][START_REF] Krogstad | Is grid turbulence Saffman turbulence?[END_REF]. More generally, better identifying the mechanisms governing large-scale properties of turbulent flows such as statistical equilibrium, condensation, or inverse cascade, is of primary interest in 3D turbulence [START_REF] Dallas | Statistical Equilibria of Large Scales in dissipative Hydrodynamic Turbulence[END_REF][START_REF] Falcon | Dissipated power within a turbulent flow forced homogeneously by magnetic particles[END_REF][START_REF] Alexakis | Cascades and transitions in turbulent flows[END_REF][START_REF] Xia | Upscale energy transfer in thick turbulent fluid layers[END_REF], wave turbulence [START_REF] Michel | Observation of Thermal Equilibrium in Capillary Wave Turbulence[END_REF][START_REF] Miquel | Low-frequency spectra of bending wave turbulence[END_REF][START_REF] Falcon | Saturation of the Inverse Cascade in Surface Gravity-Wave Turbulence[END_REF], and climate modeling [START_REF] Shaw | Particle-turbulence interactions in atmospheric clouds[END_REF]. S3: Probability density functions of the magnetic particle velocity for different volume fractions φ in a quasi-2D cell (15 cm × 8 cm × 1.4 cm). v x is the horizontal velocity and v z is the vertical velocity of the magnetic particles. Solid lines display the best fits as exp[(-v i /σ vi ) 3/2 ] independent of φ. F = 50 Hz, B = 162 G. Note that such a typical nonGaussian PDF of the particle velocity was reported in the air (i.e., for a 3D inelastic "granular gas") using this forcing [START_REF] Falcon | Equation of state of a granular gas homogeneously driven by particle rotations[END_REF], or for a granular gas forced by a container boundary [START_REF] Rouyer | Velocity distributions in homogeneous granular fluids: the free and the heated case[END_REF].

-5/ 3 x

 3 FIG.1. 3D power spectrum density E(k) (green) derived from the 1D spectrum of the longitudinal velocity Euu(kx) (red), and transverse velocity Evv(kx) (blue), (Eq. (4)). Dashed line: k 2 power law illustrating the large-scale statistical equilibrium regime. Dot-dashed line: k -5/3 power law illustrating the inertial range of the turbulent cascade. The vertical dashed line corresponds to the inverse of the integral scale ki/2π = 1/Li and separates the large-scale domain (k < ki) from the inertial range (k > ki). The PIV measurements are performed at F = 20 Hz, B = 290 G and N = 55. Inset: Power spectrum densities at different ǫ ∈ [1.1, 3.2] × 10 -4 m 2 /s 3 .

FIG. 4 .

 4 FIG. 4. Time-averaged energy flux spectrum Π(k). At large scales (k < ki), zero-mean energy flux is measured (equipartition). In the inertial range (k > ki), the energy flux is positive and implies a direct cascade of energy. Insets: (top) Standard deviation of the energy flux spectrum σΠ(k). (bottom): PDFs of the fluctuations of the energy flux Π/σΠ for three values of the wavenumber k (colored bullets). The green line represents a Gaussian distribution.

  FIG. S2: Intense temporal fluctuations of the energy flux Π(k, t) at different times (thin gray lines) compared to the mean energy flux Π(k) (thick black line) measured in our experiment. The vertical dashed line separates the large-scale region from the inertial range domain.

We thank A. Cazaubiel, J.-C. Bacri, and S. Fauve for fruitful discussions. We thank P. Delor, A. Di Palma, Y. Le Goas, and V. Leroy for technical help. This work was supported by the French National Research Agency (ANR DYSTURB project No. ANR-17-CE30-0004) and by the Simons Foundation MPS N o 651463-Wave Turbulence.

Supplemental Material of "Statistical equilibrium of large scales in three-dimensional hydrodynamic turbulence"

Jean-Baptiste Gorce and Eric Falcon

Université Paris Cité, CNRS, MSC Laboratory, UMR 7057, F-75 013 Paris, France

In this Supplemental Material, we present movies (Sec. I) and an image (Sec. II) of the flow using fluid tracer trajectories, the temporal fluctuations of the energy flux (Sec. III), movies of the magnetic particle trajectories (Sec. IV), and the probability distribution of the magnetic particle velocity (Sec. V).

I. MOVIES

Movies


Thomas Dupont
email: thomas.dupont@etsmtl.ca

Philippe Leclaire
email: philippe.leclaire@u-bourgogne.fr

Olga Umnova
email: o.umnova@salford.ac.uk

Raymond Panneton
email: raymond.panneton@usherbrooke.ca

Perforated materials with periodically distributed annular cavities for low frequency acoustic absorption

A microstructure design of thin low-frequency resonant acoustic absorbers is proposed. A perforated material is studied, in which the main perforations are connected to a collection of periodically spaced very thin annular dead-end pores with respect to the lateral size, these absorbers are called multi-pancake materials. It has been shown in the past that the acoustical properties of perforated materials can be described by the classical theory of porous media. With the help of this model in the rigid frame approximation, it is shown that at low frequencies, the periodic array of annular dead-end pores increases the effective compressibility without modifying the effective dynamic density. Due to this effect, the first absorption peak appears at much lower frequency, compared to that of the perforated structure without dead-end pores. A transfer matrix approach is proposed to model and optimize the absorber. Prototypes have been 3D printed and tested for sound absorption and transmission loss. New materials capable of producing absorption peaks at a few hundred Hz for an overall material thickness of a few cm were designed. A good agreement between the data and the model predictions is demonstrated. New designs for broader absorption and under higher sound pressures are investigated.

INTRODUCTION

The development of thin acoustic materials for low frequency absorption is an important challenge for the industry and is required in many applications, e.g. the acoustic liners for turbo fan engine, passenger compartments, or the acoustic treatments of buildings. To improve the absorption at low frequencies, systems involving Micro Perforated Plates (MPP) with extended tubes [START_REF] Li | Enhancing the low frequency sound absorption of a perforated panel by parallel-arranged extended tubes[END_REF][START_REF] Simon | Long Elastic Open Neck Acoustic Resonator for low frequency absorption[END_REF] or periodic arrays of resonators [START_REF] Lagarrigue | Absorption of sound by porous layers with embedded periodic array of resonant inclusions[END_REF] were used. Boutin 4 proposed to use the homogenization method to model the acoustic of porous media with inner resonators. Based on Bradley's works [START_REF] Bradley | Time harmonic Bloch wave propagation in periodic waveguides. Part I. Theory[END_REF] , Leclaire et al. [START_REF] Leclaire | Acoustical properties of air-saturated porous material with periodically distributed dead-end pores[END_REF] studied a structured perforated material containing periodically spaced dead-end (DE) pores (Fig. 1). It has been observed that the presence of the periodic lattice of dead-end pores leads to the appearance of absorption peaks at low frequencies and at high frequencies (bandgap or stop bands for the sound transmission). Similar observations on other sound absorbing systems were reported by Groby et al. [START_REF] Groby | The use of slow waves to design simple sound absorbing materials[END_REF] .

In reference [6], it has been proposed to model the resonant material with periodically spaced dead-end cavities by a transfer matrix approach or using low frequency asymptotic expansions. The presence of the network of DE pores has the effect, at low frequencies, of increasing the effective compressibility of the material without changing its effective dynamic density. This induces a decrease of the effective celerity of the material [START_REF] Groby | The use of slow waves to design simple sound absorbing materials[END_REF] . Due to this effect, the presence of a periodic network of pores significantly reduces the resonance frequencies of the material. This leads to the appearance of the absorption peaks at much lower frequencies without increasing the thickness of the resonant material. The model has been validated by numerical simulation and by comparing its predictions with measurements on prototype samples [START_REF] Leclaire | Acoustical properties of air-saturated porous material with periodically distributed dead-end pores[END_REF] .

Based on this work, we propose a new type of microstructure design for low frequency resonant materials. In order to enhance the thermal effects at low frequencies and thus to decrease the first resonance frequency, a perforated material with a periodic lattice of very thin annular cavities (or ring, "pancake" or "mille-feuille" cavities with ratios of radii to thicknesses of 40 or more for example). The proposed microstructure is directly inspired from the design of heat exchangers. A patent application has been filed on the thin multi-pancake resonant materials [START_REF] Leclaire | Résonateur acoustique de faible épaisseur de type mille-feuille perforé pour l'absorption ou le rayonnement acoustique très basses fréquences (Low thickness perforated mille-feuille acoustic resonator for absorbing or radiating very low acoustic frequencies)[END_REF] . The determination of the pancake cavity impedance is inspired from the work by Dickey and Selamet 9 on the acoustic behavior of a Helmholtz resonator with a "pancake" shape. A transfer matrix approach is proposed to model and optimize the absorber. Prototypes have been 3D printed and tested for sound absorption and transmission loss.

This paper is drawn from the cited article [10]. Complementary information is provided in this reference [START_REF] Dupont | A microstructure material design for low frequency sound absorption[END_REF] .

Fig. 1 -Main pore (with a cross-sectional area Amp) with periodically arranged dead-end pores, n = 4 identical dead-end pores with cross-section area Ade and length d per period h. The dead-end pores are located at "nodes".

PROPOSED MICROSTRUCTURE DESIGN OF THE RESONANT MATERIAL

Low frequency approximation of the effective properties of a perforated plate with

periodically spaced short dead-ends (Fig. 1)

When the wavelength 𝜆 is much greater than the spacing h and the length d of the dead-ends (𝜆 ≫ ℎ, 𝑑), low frequency approximations for the effective density 𝜌 𝑒 and compressibility 𝐶 𝑒 of the air inside a perforated plate with periodically spaced short dead-ends were given as [START_REF] Leclaire | Acoustical properties of air-saturated porous material with periodically distributed dead-end pores[END_REF] :

𝜌 𝑒 = 𝜌 𝑚𝑝 (1)
𝐶 𝑒 = 𝐶 𝑚𝑝 + 𝐶 𝑑𝑒 𝑛𝐴 𝑑𝑒 𝐴 𝑚𝑝 𝑑 ℎ = 𝐶 𝑚𝑝 + 𝐶 𝑑𝑒 𝑉 𝑑𝑒 𝑉 𝑚𝑝 (2)
where 𝜌 𝑚𝑝 , 𝐶 𝑚𝑝 and 𝐶 𝑑𝑒 are respectively the density of the air in the main perforation, its compressibility and the compressibility of air in the dead-ends. n is the number of dead-ends at each node. 𝐴 𝑚𝑝 and 𝐴 𝑑𝑒 are respectively the cross sections of the main pore and of the dead-end pores while 𝑉 𝑚𝑝 and 𝑉 𝑑𝑒 are their respective volumes. These equations show that a periodic array of DE pores at low frequencies does not modify the effective density of air in the main pore but it increases significantly its effective compressibility. That means that the periodic array of DE pores has a significant influence on thermal effects occurring in the thermal boundary layers near the DE pore walls. The effective compressibility (inverse of the bulk modulus) is associated with the thermal characteristic length [START_REF] Champoux | Dynamic tortuosity and bulk modulus in air-saturated porous media[END_REF] .

The low frequency approximation of the effective sound speed is given by:

𝑐 𝑒 = 1 𝑅𝑒√𝜌 𝑒 𝐶 𝑒 𝜆≫ℎ,𝑑 → 1 𝑅𝑒√𝜌 𝑒 (𝐶 𝑚𝑝 + 𝐶 𝑑𝑒 𝑉 𝑑𝑒 𝑉 𝑚𝑝 ) (3)
It is clear from this equation, that periodic array of DE pores decreases sound speed in the material. If we consider a quarter-wavelength thickness L material with straight main pores only (without lateral dead-end), the first resonance frequency is given by:

𝑓 𝑟1 = 𝑐 𝑚𝑝 4𝐿 (4)
where 𝑐 𝑚𝑝 is the effective sound speed in the main pore. If a periodic array of lateral DE pores is added, the effective sound speed ce is given by equation ( 4), and the first resonance frequency is modified as follows:

𝑓 𝑟1 * = 1 4𝐿 1 𝑅𝑒√𝜌 𝑒 (𝐶 𝑚𝑝 + 𝐶 𝑑𝑒 𝑉 𝑑𝑒 𝑉 𝑚𝑝 ) (5)
Thus, periodic arrays of DE pores can reduce the first resonance frequency significantly.

Proposed resonator design

According to Eqs. ( 5) and ( 2), the resonance frequency fr1 can be decreased by increasing the ratio 𝑛𝐴 𝑑𝑒 𝑑 𝐴 𝑚𝑝 ℎ ⁄ . Central perforations should bear as many dead-end pores as possible at each node, the dead-ends pores should have large volumes and the periodic distance should be small. The different conditions found are not all compatible with one another (large DE volumes require large cross sections and long dead-ends) and with a sufficient perforation rate on the surface of the plate. Furthermore, the parameters involved in the ratio will have an effect on the compressibilities 𝐶 𝑚𝑝 and 𝐶 𝑑𝑒 through the thermal characteristic length [START_REF] Dupont | A microstructure material design for low frequency sound absorption[END_REF] . In a first approach, however, it is thought that the compressibilities will have less effect than this ratio; it therefore provides us with a simple design tool. The proposed microstructure is also inspired from the design of heat exchangers. The proposed design is shown on Figure 2. This "perforated pancake absorber" can be built using 3D printing technology resin (stereo-lithography process -Figure 2).

Fig. 2 -Perforated material with multiple 'pancake' cavities (sample A). The full sample is shown on the left and its longitudinal cross section is in the center. On the right a 3D printing sample which has been made using a transparent is shown. The parameters of this sample are given in Table 1 (sample thickness sample L = 31 mm and external diameter dsamp = 29 mm).

Lumped parameter model with Transfer Matrix formulation

The lump element approach is used here to account for the pancakes cavities located in parallel to the main pore. First, we consider a single cell with thickness h (period) composed of a main pore and a pancake cavity (see Figure 3). Fig. 3 -One single periodic cell composed of two halves of the main pore, a common pore and a "pancake" cavity. The common pore corresponds to the common volume between the main pore and the "pancake" cavity.

The common pore corresponds to the common volume between the main pore and the pancake cavity. Thus the single periodic cell is split up in two half main pores (with a thickness hmp/2), a common pore and a pancake cavity (with a thickness hde).

The two halves of the main pores are identified as effective fluids with parameters (wavenumber kmp, characteristic impedance Zmp) given by the Johnson-Champoux-Allard (JCA) model [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials 2nd[END_REF] for porous media. It is assumed that Re(kmpdmp) << 1 so that the waves inside the main pores can be considered as plane waves. The transfer matrix of a main pore for a thickness hmp/2 is given by:

           ) 2 / cos( ) 2 / sin( ) 2 / sin( ) 2 / cos( ) 2 / ( mp mp mp mp mp mp mp mp mp mp h mp h k h k Z j h k jZ h k mp T (6)
Time dependence e jωt is assumed throughout. The pancake cell is composed of a common pore and a truncated pancake cavity, the two elements are separated by a virtual wall. It is proposed to consider the common pore as an element without thermo-viscous losses. Thus, the transfer matrix of this common element for a thickness hde / 2 is given by:

           ) 2 / cos( ) 2 / sin( ) 2 / sin( ) 2 / cos( 0 0 0 0 0 0 ) 2 / ( de de de de h com h k h k Z j h k jZ h k de T (7)
with (k0, Z0) are the wavenumber and characteristic impedance of air.

Then the pancake cavity is considered as a side branch (same as a DE pore) which is represented by a surface impedance ZS,de. It is proposed to use the "pancake" cavity surface impedance expression proposed by Dickey and Selamet [9], valid for circular cross section, to model a Helmholtz resonator (one open-closed cavity) with a small cavity length-to-diameter ratio:

                 ) ( ) ( / ) ( ) ( ) ( ) ( / ) ( ) ( ) 2 ( 1 ) 2 ( 1 ) 1 ( 1 ) 1 ( 1 ) 2 ( 0 ) 2 ( 1 ) 1 ( 1
r k H r k H r k H r k H r k H r k H r k H r k H jZ Z ( 8
)
with 𝐻 𝑖 (𝑛) (𝑘𝑟) the Hankel function of i th order and of n th kind of the wavenumber k and of the radial position r relatively to the center. The effective parameters (kde, Zde) of the air in the pancake cavity are calculated using JCA model [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials 2nd[END_REF] . It is assumed here that hde << rde << λ, so the expressions for slit geometry are used (Table 2). At the center of the common pore, the admittance relation is applied. The transfer matrix of the pancake cavity (as side branch) is given by:

           1 1 0 1 , mp de de S de A A Z T (9)
with 2 mp mp r A   the cross-section area of the main pore and Ade = 2πrmphde the area of the virtual wall separating the common pore and the pancake cavity.

The transfer matrix of the periodic cell with thickness h = hmp + hde is given by:

) 2 / ( ) 2 / ( ) 2 / ( ) 2 / ( mp de de mp h mp h com de h com h mp cell T T T T T T  (10)
Our material is constituted of a series of N pancake cavities. The transfer matrix is given by:

  22 N samp end cell end  T T T T (11)
with T 𝑒𝑛𝑑1 and T 𝑒𝑛𝑑2 the transfer matrix which includes the pore end effect and the contraction effect [START_REF] Dupont | A microstructure material design for low frequency sound absorption[END_REF] . The normal sound absorption coefficient of the absorber (for a rigidly backed sample) and its normal sound transmission loss (for the configuration which the sample is coupled to two semiinfinite regions) are then calculated using tsamp,ij of eq. 11 :

     (13)

NUMERICAL APPROACH AND EXPERIMENTAL DETAILS

It is proposed to compare the semi analytical results with those obtained numerically in a virtual tube. The virtual measurements are obtained with a three dimensional acoustical FEM simulations using COMSOL Multiphysics software. A three-microphone method [START_REF] Salissou | Complement to standard method for measuring normal incidence with three microphones[END_REF] is used to get the virtual measurements. The elements of the absorbers were modeled as those filled with effective fluids given by JCA model as outlined in Table 2. The details of the numerical approach are given in reference [10].

For the experimental measurements, an acoustic tube with cross section diameter with dtube = 29 mm is used. The frequency range is between 100 Hz to 6400 Hz. The four-microphone method [START_REF] Astm E | Standard test method for measurement of normal incidence sound transmission of acoustical materials based on the transfer matrix method[END_REF]15 is used. Several samples have been tested. The results have been averaged over several measurements. Then the results are presented with the mean value 𝑥̅ and the expanded uncertainty with a coverage factor h = 1: 𝑥 = 𝑥̅ ± ℎ𝜎, with 𝜎 the standard deviation.

The absorption coefficient results are presented for the configuration in which the sample is backed by a rigid wall. The transmission loss results are presented for configuration in which the sample is coupled to two semi-infinite regions. The complete experimental details are given in reference [10].

ANALYTICAL, NUMERICAL AND EXPERIMENTAL RESULTS

Material parameters for the models

One prototype (sample A) is presented in this article, a second prototype (the sample B) has been presented in reference [10]. Both prototypes are designed to be tested in the impedance tube. An external diameters for sample A is proposed; dsamp = 29 mm, corresponding to the acoustic tubes cross-section diameter. The samples have a thickness of L= 31 mm, Table 1 summarizes the sample parameters. The effective fluid parameters of the main pore are given in Table 2.

Results

The measured and simulated normal incidence sound absorption coefficient of the rigidly backed sample A and the normal sound transmission loss of sample A (in transmission configuration) are presented in Figure 4. The first absorption peak is high (close to 100%) and its frequency is around 550 Hz. This frequency is equal to the first resonance frequency of a perforated material without pancake cavities (quarter-wavelength tube resonance) with a thickness L*= 4.8 L = 148 mm. Thus, for these samples, adding periodic array of pancake cavities allows reduction in thickness by a factor of 4.8. The first resonance of the perforated sample without pancake cavities with same thickness L= 31 mm is fres* = 2578 Hz. The agreement between the models and the experiments is very good for the first absorption peak (see Table 3 for the values of the frequency and the amplitude of the first and the second sound absorption peaks).

Fig. 4 -Normal sound absorption coefficient (abs. configuration: sample backed by a rigid wall) and normal sound transmission loss (transmission configuration) of the sample A (perforated multi-pancake cavity with 29 mm external diameter). The three curves correspond to: results measured in the impedance tube (---), results simulated by Finite Element approach with rigid frame assumption (-.-.-), results simulated by TMM approach (---).

For the second and the third absorption peaks, the models (TMM and numerical for rigid frame) predict a slightly smaller frequency shift than observed and overestimate the amplitudes. Above 2500 Hz, both models fail to correctly predict the acoustic behavior of the tested sample. The discrepancies are thought to be caused by several effects including the resonance effect due to elasticity of the material and stopband effects occurring in certain frequency bands. The agreement between the present TMM model and the numerical (FE) model is good in the studied frequency range. However, a noticeable difference is found on the absorption peak frequencies and amplitudes (see Table 3), especially above 3000 Hz. This discrepancy could be explained by the fact that plane wave assumption has been considered in the TMM model. Moreover, due to periodicity of the pore structure, it can be observed that the models predict a stopband above 4700 Hz (the stopband corresponds to absorption coefficient dropping to zero and the transmission loss raising to high values). However, the analysis of the transmission loss shows that there are two stopband effects; one is between 3500 Hz and 4200 Hz and the second is above 5500 Hz. To attempt to explain the two observed experimental stop bands, a numerical model with elastic frame assumption was used (COMSOL Multiphysics software with vibroacoustic option). Mechanical parameters of the Acrylonitrile Butadiene Styrene (ABS); with the density ρs = 1040 kg/m 3 , the Young's modulus E = 2275×10 [START_REF] Leclaire | Acoustical properties of air-saturated porous material with periodically distributed dead-end pores[END_REF] Pa, and the Poisson's ratio ν = 0.35 were used in the model. The results are shown on Figure 5. Accounting for frame elasticity leads to appearance of low absorption coefficient in the frequency range between 3.5 kHz and 4.2 kHz. It would appear that there would therefore be two bandgap effects; the first one corresponding to a vibrational bandgap effect (vibrational resonance of the membrane in periodic array arrangement [START_REF] Lee | Acoustic metamaterial with negative density[END_REF] ) and the second one corresponding to an acoustic bandgap effect (acoustic resonance of the pancake cavities in periodic array arrangement). Indeed the first natural frequency of flexural vibrations of a circular ABS plate (similar to the wall between two pancake cavities) is found [START_REF] Zagrai | A soft table for the natural frequencies and modal parameters of uniform circular plates with elastic edge support[END_REF] in vacuo condition between 2149 Hz (with "simply supported" boundary conditions) and 4383 Hz (with "clamped" boundary conditions). The real boundary conditions for the wall of pancake cavities are between "simply supported" and "clamped". Thus the first natural frequency of flexural vibrations is included in the first bandgap frequency bandwidth. Moreover the first acoustic resonance of the pancake cavity is 5239 Hz; this frequency is included in the second bandgap frequency bandwidth. It can be noticed that similar two bandgap effects have been observed for an acoustic metamaterial composed of a periodically arranged Helmholtz resonators and membranes [START_REF] Seo | Acoustic metamaterial exhibiting four different sign combinations of density and modulus[END_REF] .

.).

Note that in reference [10] the prototype B was presented, the diameter was increased (with an external diameter dsamp = 44.4 mm, the pancake cavity diameter is dde = 40 mm) in order to increase the pancake cavity volume and therefore decrease the first absorption peak (see Eq. 5). The first absorption peak is observed around 392 Hz, which means that adding a periodic array of pancake cavities allows the reduction in thickness by a factor of 6.6 compared to simply perforated sample (quarter-wavelength tube), moreover the amplitude of the absorption peak is preserved.

CONCLUSIONS

Based on the low frequency approximation of effective parameters of a structured perforated material containing periodically spaced dead-end pores, a microstructure design for low frequency sound absorber materials was proposed in this work. It corresponds to a perforated material with a periodic lattice of internal "pancake" cavities. Sound absorber prototypes have been made and tested in the impedance tube. The periodic lattice of "pancake" cavities allows decreasing significantly the frequencies of the absorption peaks without increasing the thickness of the sample. Moreover two stopband effects have been observed for these materials at higher frequencies: the first stopband is due to the acoustic resonance of the periodic pancake cavities and the second to the frame resonance of the sample.

Future works are concerned with the optimization of the microstructure to improve the low frequency absorption properties. Studies are in progress in order to enlarge the absorber frequency bandwidth (especially at mid and high frequencies) and to optimize the frequency value of the first absorption peak (lower as possible).

Fig. 5 -

 5 Fig. 5 -Normal sound absorption coefficient (abs. configuration: sample backed by a rigid wall) and normal sound transmission loss (trans. configuration) of the sample A (perforated multipancake cavity with 29 mm external diameter). The curves correspond to: results measured in the impedance tube (---), results simulated by Finite Element approach with rigid frame assumption (-.-.-), results simulated by Finite Element vibro-acoustic approach (with elastic frame) (. . .).

Table 1 -

 1 Parameters of the multi-pancakes samples.

	Sample A Sample B 10	dsamp (mm) 29 44.4	dmp (mm) 4 4	dde (mm) 26 40	hmp (mm) 1 1	hde (mm) 1 1	N 15 15	L (mm) 31 31

Table 2 -

 2 Johnson-Champoux-Allard (JCA) parameters of the main pore and pancake cavity, with η the dynamic viscosity of air.

	Main pore (circular pore) "Pancake" cavity (slit)	Viscous length Λ (mm) rmp hde	Thermal length Λ' (μm) rmp hde	Tortuosity α∞ 1 1	Static air flow resistivity σ (Pa•s/m²) 8𝜂 (𝑟 𝑚𝑝 2 𝜙) ⁄ 12𝜂 (ℎ 𝑑𝑒 2 𝜙) ⁄	Open porosity  (%) 1 1

Table 3 -

 3 Mean frequency values and mean amplitudes of the first and the second absorption peaks for sample A. The results are given for the experimental approach, the present model and the two FEM models (with rigid frame assumption and with elastic frame assumption).

		Frequency (Hz)	Amplitude (%)	Frequency (Hz)	Amplitude (%)
		1 st peak	1 st peak	2 nd peak	2 nd peak
	Experimental Present model (TMM) FEM model (rigid frame) FEM model (elastic frame)	550 560 580 580	99.7 97.8 91.9 88.7	1603 1634 1680 1680	83.8 95.8 98.4 98.8


Arnaud Lehner

Daniel Côme

Françoise Corbineau

Oligosaccharide metabolism in wheat embryo as possible indicator of sprouting susceptibility

CONCLUSION

 Germination was associated with a strong decrease in raffinose, but not in sucrose, thus leading to a sharp reduction in the raffinose/sucrose ratio value in the embryo of both dormant and non dormant seeds.

 Dormant seeds did not germinate at 30°C, and their embryo did not show any change in raffinose content nor in the raffinose/sucrose ratio.

 Our results demonstrate that the raffinose/sucrose ratio in the embryo might be a good criterion for detecting that sprouting occurred, even though any visible symptom is observed.

INTRODUCTION

Dormancy of cereal seeds, which corresponds to their inability to germinate at relatively high temperatures, generally above 20°C, prevents germination during seed development on the mother-plant. Lack of dormancy results in preharvest sprouting during adverse weather conditions (cold temperature and high rainfall), and induces loss in grain and flour quality due to synthesis of enzymes during germination, notably α-amylase (Meredith and Pomeranz, 1985).

The aims of the present work were to investigate whether soluble sugar metabolism in the embryo is an early indicator of sprouting injury, and whether it is correlated with dormancy in wheat (Triticum aestivum L., cv Charger).

Soluble sugar contents of the embryo were measured by HPLC in dry freshly harvested (dormant, D) seeds and seeds stored dry for 5-6 months at 20°C (non dormant, ND), and in both types of seeds incubated in water or in the presence of polyethylene glycol (PEG) (-2.5 MPa) for various durations at 20°C and 30°C.

PROTOCOLES

Seed imbibition : Seeds were placed on a layer of cotton wool moistened with distilled water or a PEG solution (-2.5 MPa) for 5, 8, 14 and 24 h at 20 and 30°C. At the end of the imbibition periods, embryos were immediatly isolated and fixed in liquid nitrogen before sugar extraction.

Sugar extraction : Soluble sugars were extracted according to Black et al. (1996). Ten isolated embryos were ground at room temperature in a mortar in 1 ml 80% aqueous ethanol containing melezitose (1.5 mg ml -1 ) as internal standard, and the extract was heated for 30 min at 80°C. After centrifugation for 15 min at 14 000 g and removal of the supernatant, the pellets were resuspended and reextracted twice, in 0.5 ml and then 0.35 ml 80% aqueous ethanol, at room temperature and centrifuged again. The three supernatants were combined and reduced to dryness in a centrifugal evaporator (RC 10-22, Jouan, France). The dry extracts were dissolved in 100 µl of ultra-pure water and then filtered through acetate filter (0.45 µm pore size, Nalgene, USA) before being analysed by HPLC.

Sugar determination : Ten µl samples of sugar extracts were injected onto a Spherisorb-NH2 column (Thermo Separation Products, France) and eluted with 70/30 (v/v) acetonitrile/H 2 O at a flow rate of 1 ml min -1 using a Spectra Physics 8700 pump. The eluents were analysed with a differential refractometer (Spectra Physics 8430, France), and the peak areas were integrated by a Spectra Physics 4290 integrator. Sucrose and raffinose were identified by co-elution with standards (Sigma, USA). The results are expressed as mg of sugars g -1 DW of embryos and correspond to the means of 3 measurements  SD.

OLIGOSACCHARIDE METABOLISM IN WHEAT EMBRYO AS A POSSIBLE INDICATOR OF SPROUTING INJURY

 Changes in sugar contents during imbibition on water

In the embryos of dry dormant and non dormant seeds, soluble sugars were mainly sucrose (73.0 mg g -1 DW) and raffinose (32.2 mg g -1 DW) (Fig. 1). Germination of dormant seeds at 20°C and non dormant ones at 20 and 30°C resulted in a decrease in raffinose as early as 8 h of incubation, and in sucrose after 24 h, i.e. after radicle protrusion (Figs 1A and 1C). Consequently, the raffinose/sucrose ratio declined rapidly (Figs 1B and 1D). In dormant seeds, which did not germinate at 30°C, soluble sugars remained constant at their initial levels (Fig. 1A), and the raffinose/sucrose ratio value was at least 0.45 within 24 h of imbibition (Fig. 1B).

 Changes in sugar contents during imbibition on PEG

Raffinose content decreased in the embryo of non dormant seeds incubated at 30°C for 24 h on a PEG solution at -2.5 MPa (Fig. 2C), even though no germination occurred during this time (data not shown). In contrast, raffinose did not significantly change within 24 h of incubation of dormant seeds at 20 and 30°C (Fig. 2A), and of non dormant ones at 20°C (Fig. 2B).

Sucrose content of the embryo remained constant in dormant seeds incubated at 30°C (Fig. 2A), whereas it slightly increased in the embryo of dormant seeds placed at 20°C (Fig. 2A) and in that of non dormant ones placed at 20 and 30°C (Fig. 2B).

The raffinose/sucrose ratio value remained then high in the embryo of dormant seeds at 20 and 30°C (Fig. 2C) and in that of non dormant ones at 20°C (Fig. 2D), but it strongly declined after 8 h of incubation in the case of non dormant seeds at 30°C, reaching 0.1 at 24 h (Fig. 2D).

Figure 1 -Figure 2 -

 12 Figure 1 -Changes in sucrose () and raffinose () contents (A, C) and in the raffinose/sucrose ratio (B, D) in embryos isolated from dormant (D) and non dormant (ND) seeds imbibed on distilled water at 20 and 30°C. Arrows indicate the beginning of germination. Means of 3 measurements ± SD. Where no bars are shown, the spread of SD is less than the size of the symbols.

  A. LEHNER, D. CÔME and F. CORBINEAU Physiologie Végétale Appliquée, EA 2388 « Physiologie des semences », Université Pierre et Marie Curie, 3, rue Galilée, 94200 Ivry-sur-Seine, France. Email : corbi@ccr.jussieu.fr, http://lpva.snv.jussieu.fr

	Table 1-Germination percentages obtained after 8, 24 and 48 h with dormant (D) and non dormant
	(ND) seeds placed on water at 20 and 30°C. Means of 4 replicates ± SD.
	Imbibition time (h)		Germination (%) at
		20°C		30°C
		D	ND	D	ND
	8	0	0	0	6 ± 1
	24	35 ± 5	62 ± 3	1 ± 1	89 ± 1
	48	59 ± 2	99 ± 1	7 ± 4	99 ± 1


Arnaud Lehner

Norbert Mamadou

Daniel Côme

Christophe Bailly

François Corbineau

Ageing of wheat grains : Two (or more) ways to die?

INTRODUCTION :

During storage of seeds, ageing leads to a progressive decrease in their vigour, and then in a loss of their viability. Seed deterioration is associated with numerous cellular and biochemical alterations including loss of membrane integrity, impairment of RNA and protein synthesis, and DNA degradation. However, reactive oxygen species accumulation and lipid peroxidation are often invoked to be the major causes of seed deterioration (Priestley, 1986). Soluble sugars might also be involved in seed storability (Obendorf, 1997). Soluble sugar contents and antioxidant enzyme activities were measured in the embryo of wheat grains throughout their storage in 2 conditions (45°C and 100% RH and 30°C and 75% RH) in order to investigate whether grain deterioration during ageing was related to lipid peroxidation resulting in a decrease in the efficiency of the antioxidant defence system, and in changes in sugar metabolism.

RESULTS:

 Effect of ageing treatments on seed germination Ageing at 45°C and 100% RH (Fig. 1A) and at 30°C and 75% RH (Fig. 1B) was associated with a decrease in seed vigour, evaluated by T 50 (Fig. 1, curve 1) and then in seed viability (Fig. 1, curve 2). The longer the treatment and the higher the temperature and the RH, the stronger the reduction of seed viability was. The half viability period was only 6 days at 45°C and 100% RH (Fig. 1A), as against about 3.75 months at 30°C and 75% RH (Fig. 1B).

 H 2 O 2 content and lipid peroxidation H 2 O 2 content remained close to 3.1-3.4 µmol g -1 DW during ageing at 30°C and 75% RH. In contrast, at 45°C and 100% RH, H 2 O 2 content decreased during the first 3 days reaching 1.04 µmol g -1 DW and then increased regularly up to 5.54 µmol g -1 DW after 6 days of ageing (Fig. 2). MDA content of the embryo did not change significantly during both ageing treatment (Fig. 2).

 Antioxidant enzyme activities

When seed are aged at 30°C and 75% RH, there is no clear change in CAT, SOD and GR activities (Fig. 2, right hand panel). In contrast, at 45°C and 100% RH (Fig. 2, left hand panel), loss of viability was associated with a strong decrease in CAT and SOD activities, whereas after 6 days of treatment GR activity increased more than 4 fold compared to the control.

 Sucrose and raffinose contents

At 30°C and 75% RH, loss of viability was associated with an increase in both sucrose and raffinose. Consequently the Ra/Su ratio remained constant at about 0.6-0.9 while seeds were dying (Fig. 3B). When ageing was performed at 45°C and 100% RH, loss of viability was associated with a strong decline in sucrose content, when raffinose content remained constant (Fig. 3A). The Ra/Su ratio increased then progressively to reach 2.5 after 6 days of treatment as against 0.7 for the non-aged control seeds (Fig. 3A).

CONCLUSION :

➔ As expected, wheat seeds were more sensitive to accelerated ageing performed at 45°C and 100% RH (i.e. when the water content of the embryo was 50-60% DW) rather than when ageing was performed at 30°C and 75% RH (i.e. When the water content of the embryo was around 18% RH).

➔ Loss of grain viability did not seem to be associated with lipid peroxidation. At 45°C and 100% RH, loss of viability might also be attributed to H 2 O 2 production without MDA accumulation.

➔ Loss of viability was concomitant to a reduction of the potential efficiency of the antioxidant defence system during ageing at 45°C and 100% RH.

➔ Changes in soluble sugars contents were not good markers of loss of seed vigour.

➔ Our results suggested that many different biochemical mechanisms are involved in grain death and that using one type of ageing is not enough to understand and to predict seed storability.

Fig. 2 .

 2 Fig. 2. Changes in H 2 O 2 and MDA contents and in CAT, SOD and GR activities in embryos isolated from grains aged for various durations at 45°C and 100% RH (left hand panel) or at 30°C and 75% RH (right hand panel). Black curves represent seed viability. Means of 3 measurements ± SD.

  Fig.3. Changes in sucrose (◼) and raffinose (◼) contents and in the Ra/Su ratio (black curve) in embryos isolated from grains after various durations at 45°C and 100% RH (A) and at 30ºC and 75% RH (B). Means of 3 measurements ± SD.