Frédéric Jacquemin

Sylvain Fréour

Ronald Guillén

A self-consistent approach for transient hygroscopic stresses and moisture expansion coefficients of fiber-reinforced composites

come

A self-consistent approach for transient hygroscopic stresses and moisture expansion coefficients of fiber-reinforced composites

INTRODUCTION

The increasing deployment of composite laminates in engineering applications, because of their high strength-to-weight ratio as well as corrosion and fatigue resistance, has given greater attention to the long term behaviour of composite components in hygroscopic environments. Composite materials are more and more common alternative to traditional materials such as metals and polymers. Now, contrary to metallic materials or metal matrix composites, carbon/epoxy composites can absorb significant amount of water. Moreover, the moisture contents of the epoxy resin and the carbon fiber are strongly different, because carbon fibers are generally considered not to absorb water. In addition, carbon fibers and epoxy resin are characterized by heterogeneous Coefficients of Moisture Expansion (CME). As a consequence, internal stresses rise from hygroscopic loading of composites. Now, the knowledge of internal stresses is necessary to predict a possible damage occurrence in the material during its manufacturing process or service life. Numerous papers available in the literature deal with this question, using Finite Elements Analysis or Continuum Mechanics based formalisms. These methods allow the calculation of the macroscopic stresses in each ply constituting the composites. But, they do not provide information on the local mechanical states in the fiber and matrix of a given ply. The present work is precisely focused on the study of the internal stresses in the constituents of the ply. In order to reach this goal, scale transition models are required. The classical elastic selfconsistent scale transition model is extended in order to predict the multi-scale hygro-elastic behaviour of composite materials. The formalism described takes into account heterogeneities of moisture content between matrix and fiber. Constitutive laws are given for both the macroscopic elastic properties and the CME of the ply.

PREDICTION OF THE HOMOGENIZED HYGRO-ELASTIC PROPERTIES

Self-Consistent estimates for the hygro-elastic behaviour of composites

Self-Consistent models based on the mathematical formalism proposed by Kröner [START_REF] Kröner | Berechnung der elastischen Konstanten des Vielkristalls aus des Konstanten des Einkristalls[END_REF] constitute a reliable method to predict the micro-mechanical behaviour of heterogeneous materials. The model was extended by Hutchinson [START_REF] Hutchinson | Elastic-plastic behaviour of polycrystalline metals and composites[END_REF] to thermo-elastic cases. In the present paper, the classical model is improved in order to take into account stresses and strains due to moisture in carbon fiber-reinforced polymer matrix composites. In the model, ellipsoidal inclusions are embedded in an infinite medium, called Homogeneous Effective Medium or HEM. HEM is assumed to behave as one particular ply: its hygro-elastic properties come from homogenization operations performed on the hygro-elastic constants of the matrix and fibers constituting the considered ply. The material is investigated at two different scales for the needs of micro-mechanical modeling:

-the average behaviour of a ply, represented by the HEM, defines the macroscopic scale of the model. It is denoted by the superscript I . -the properties and mechanical states of the matrix and fiber are respectively indicated by the superscripts m and f . Within this approach, the hygroscopic dilatation generated by a moisture content increment ∆C I is treated as a transformation strain. The macroscopic stresses σ Ι are given by: ( ) I ∆C :

I I I I L β ε σ - = ( 1
)
where L is the elasticity tensor, ε the strains due to elastic and hygroscopic loads and β the coefficients of moisture expansion. At microscopic scale, similar equations can be written:

( ) α ∆C : α α α α β ε σ - = L (2)
where α replaces the appropriate superscript (i.e. f or m ) for the carbon fiber or polymer matrix. One should notice that, contrary to the thermo-elastic self-consistent model, where the temperature increment ∆T is generally considered homogeneous at every scale, the moisture content increment is often strongly different in the fiber and the matrix.

In a fundamental work [START_REF] Eshelby | The determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems[END_REF], Eshelby demonstrated that when the elementary inclusions (here the matrix and the fiber) were assumed to have ellipsoidal shapes, stresses and strains were related by the following scale transition relation:

( )

I I I I R L ε ε σ σ α α - - = - : : (3)
R I is the reaction tensor. It satisfies:

( )

1 1 1 : : - - -       - = - = I I I I esh I esh I E E L S S I R (4)
In (4), I stands for the fourth order identity tensor. Hill's tensor E I expresses the dependence of reaction tensor on the morphology assumed for the matrix and its reinforcements [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF]. It can be expressed as a function of Eshelby tensor, through ( )

1 - = I I I esh E L :S .
In another main paper, Hill demonstrated the following useful volume average equations [START_REF] Hill | The essential structure of constitutive laws for metals composites and polycrystals[END_REF]:

m f, α m f, α I I ε ε σ σ α α = = = = (5)
Using (1-5), we easily obtain a linear relation between microscopic and macroscopic fields:

( ) ( ) ( ) [ ] m f, α I α 1 m f, α
L L R L L R L L R L L β β ε ε α α α α (6)
Since this relation must be satisfied for any hygro-mechanical state {ε Ι , ∆C I , ∆C f , ∆C m }, the first term of the right member of (6) must be equal to I, while the second term must be null.

Then, we obtain the very classical self-consistent estimate for the macroscopic elastic stiffness [START_REF] Tsai | Introduction to composite materials[END_REF] and the homogenized CME for the composite (8):

L R L L L R L L β β α α α α (8)
L I , β Ι and ∆C I being constant over the microscopic elements, we may extract them from the summation:

( ) ( ) I m f, α α 1 1 m f, α 1 ∆C : ∆C : : : : : I I α α I I α I I α β L β L R L L R L L = + + = - - = - (9)
Finally, we obtain an explicit form for the macroscopic CME:

( ) ( ) m f, α α 1 1 m f, α 1 1 I ∆C : : : : : ∆C 1 = - - = - - + + = α α α α β β L R L L R L L L I I I I I I (10)
Since the increment of moisture concentration in fiber is usually null, the introduction of this additional assumption (i.e. ∆C f = 0) involves the following simplifications:

( ) ( ) m m I I m I I I I L R L L R L L L β β α : : : : : ∆C ∆C v 1 1 m f, α 1 1 I m m - - = - - + + = (11)
where v m stands for the volume fraction of matrix in the considered ply (v m + v f = 1).

When the equilibrium state is reached, the maximum moisture content of the neat resin may be estimated from the maximum moisture content of the composite. By assuming that the fibers do not absorb any moisture, ∆C I and ∆C m are related by the expression given by Loos and Springer [START_REF] Loos | Moisture absorption of graphite -epoxy composition immersed in liquids and in humid air, Environmental effects on composite materials[END_REF]:

I m m m m m I ρ ρ v ∆C W ∆C ∆C = = (12)
where W m is the weight fraction (percent) of the resin in the composite, ρ I and ρ m are respectively the composite and resin densities.

Actually, the theoretical relation (12) of Loos and Springer can be affected by a factor γ (γ≥1). γ stands for the ratio between mass water in the composite and the matrix. An imperfect adhesion in fiber-reinforced composites or a porosity of the matrix plays an important role in the diffusion process. Taking into account this particularity leads to the following modifications for the relation (12):

I m m m m m I ρ ρ v ∆C W ∆C ∆C γ = γ = (13)
Introducing ( 13) in [START_REF] Tsai | Composite design[END_REF], the macroscopic CME are then expressed:

( ) ( ) m m 1 I I m 1 1 I I 1 I m I I : : : : : m f, α β β L R L L R L L L - - - α - + + γρ ρ = = (14)

Tsai-Hahn model for the estimation of the macroscopic CME

Usually, identical models are used for the coefficients of thermal expansion and coefficients of moisture expansion. Thus, these models do not take into account a jump in moisture content between the constituents. On the opposite, the Tsai-Hahn model [START_REF] Tsai | Introduction to composite materials[END_REF], which introduces different moisture contents in the fiber and the matrix, constitutes an interesting assessment of the validity of SC estimates for CME. Introducing the ratio between mass water in the composite and the matrix and assuming that the fibers do not absorb any moisture, the Tsai-Hahn estimation of macroscopic CME is:

       + - γ + = γ + = I 11 m m f f m I m m I 22 m I m m f f m m I 11 β ) ν v ν (v ρ ρ β ) ν (1 β ρ ρ ) E v E v ( β E β (15)

MACROSCOPIC ANALYSIS

Moisture concentration

Consider an initially dry, uni-directionally reinforced composite pipe, whose inner and outer radii are a and b respectively, and let the laminate be exposed to an ambient fluid with boundary concentration c 0 . The macroscopic moisture concentration, c I (r,t), is solution of the following system with Fick's equation ( 16), where D I is the transverse diffusion coefficient of the composite. Boundary and initial conditions are described in (17):

      ∂ ∂ + ∂ ∂ = ∂ ∂ r c r 1 r c D t c I 2 I 2 I I , a<r<b (16)
     = = = 0 ) 0 , r ( c c ) t , b ( c and c ) t , a ( c I 0 I 0 I (17)
Applying the Laplace transform to the latter system and using the residue theory to express the solution in time space [START_REF] Crank | The mathematics of diffusion[END_REF], we finally obtain the macroscopic moisture concentration:

{ } ) r ( Y B ) r ( J A ) ( ) exp( 2 c ) , r ( c m 0 m m 0 m 1 m m u m m 2 0 I ω + ω ω ∆′ ω τ ω - + = τ ∑ ∞ = (

Macroscopic stresses

At the initial time, let us assume that the pipe is stress free. The hygroelastic stresses for every ply at any time are calculated by using the classical equations of solid mechanics: constitutive laws of hygroelastic orthotropic materials, strain-displacement relationship, compatibility and equilibrium equations and boundary conditions [START_REF] Jacquemin | A closed-form solution for the internal stresses in thick composite cylinders induced by cyclical environmental conditions[END_REF].

Introducing a characteristic modulus 0 L , we introduce the following dimensionless variables:

. b / ) w , v , u ( ) w , v , u ( , /L , L / I I I I I I 0 0 = = = I I L L Ι Ι σ σ
The axial and circumferential components of the displacement field of the i th ply, respectively

) r , x ( u I and ) r , x ( v I
, are then expressed:

       = = . constants are R , R r x R ) r , x ( v x R ) r , x ( u 2 1 2 I 1 I (19)
The radial component of the displacement field I w satisfies the following equation: It is shown that the general solution of equation (20) writes as the sum of a solution of the homogeneous equation and of a particular solution [START_REF] Jacquemin | A closed-form solution for the internal stresses in thick composite cylinders induced by cyclical environmental conditions[END_REF].

[ { } ] ) r ln( r ] ) 1 ) 3 k 2 (( r ) 3 k 2 ( 2 ) 1 ) 3 k 2 (( r ) r ln( [ )! 1 k ( ! k ) ( ) 2 1 ( (-1) 2 ) 1 ) 3 k 2 (( r )] 2 k ( ) 1 k ( ) 2 1 ln( 2 [ )! 1 k ( ! k ) ( ) 2 1 ( (-1) B ) 1 ) 3 k 2 (( r )! 1 k ( ! k ) ( ) 2 1 ( ) 1 ( A L K ) ( ) 2exp(- r R r R ) r ( w 2 2 3 k 2 2 3 k 2 0 k 2 k 2 m 1 k 2 k 2 ) 3 k 2 ( m 0 k 2 k 2 m 1 k 2 k i 0 k 2 3 k 2 2 k 2 m 1 k 2 k i 1 m I 22 1 m u m 2 m 4 3 I - - + + - - + + ω + - + + ψ - + ψ - ω + ω π + - + + ω - ω ∆′ ω τ ω - + = + + ∞ = + + + ∞ = + + ∞ = + + + ∞ = ∑ ∑ ∑ ∑
Finally, the displacement field depends on four constants to be determined : R i for i=1..4. These four constants result from the following conditions: global force balance of the cylinder; nullity of the normal stress on the two lateral surfaces.

EXAMPLE

Evaluation of the overall CME of a ply in a T300/5208 composite using Self-Consistent model

The predictive integral form (14) demonstrated for the homogeneous equivalent CME of a ply, according to the self-consistent scale transition formalism developed in the present work, was checked in the case of a T300/5208 lamina. Table 1 presents typical data for the properties of the T300 carbon fibers and N5208 epoxy matrix and the mechanical properties of the T300/5208 composite, deduced from SC computations. CME for the constituents are given in Table 2.

"Table 1. Mechanical properties." 1 E [GPa] 3 E , 2 E [GPa] 13 , 12 ν ν 23 G [GPa] 12 G [GPa]
T300 fibers [ T300 fibers [START_REF] Soden | Lamina properties lay-up configurations and loading conditions for a range of fiber-reinforced composite laminates[END_REF] 0 0 N 5208 epoxy matrix [START_REF] Tsai | Composite design[END_REF] 0.6 0.6

Calculations have been performed for several moisture content ratios varying from 1 to the maximum moisture content ratio. In the studied case, the ratio between composite and resin densities being 1.33, the maximum moisture content ratio given by ( 12 Computations show that SC formalism achieves to reproduce a strongly anisotropic hygroscopic behaviour at macroscopic scale: the longitudinal CME of the ply remains times weaker than the one of epoxy matrix, whatever the assumed moisture content ratio. On the opposite, transverse macroscopic CME can be either higher or weaker than resin CME. As predicted by Tsai-Hahn forms, the CME calculated owing to SC model can be stronger in the ply than in the epoxy matrix, especially for high values of ∆C m /∆C I . This corresponds to quasi-perfect materials, i.e. non porous matrix and no adhesion defects at its interface with the fiber. Transverse macroscopic CME significantly increases with moisture content ratio. In the studied case, T300/5208 CME reaches the values observed in the resin for ∆C m /∆C I ≅ 2.

Local and macroscopic stress states

Macroscopic stress state

A thin uni-directionally reinforced composite pipe, with thickness 4 mm, initially dry then exposed to an ambient fluid, made up of T300/5208 carbon-epoxy plies, is considered. The hygroscopic properties are presented in Table 3.

"Table 3. Hygroscopic properties for T300/5208 composite [START_REF] Loos | Moisture absorption of graphite -epoxy composition immersed in liquids and in humid air, Environmental effects on composite materials[END_REF]."

Diffusion coefficient 1.5 × 10 -7 mm 2 /s Boundary concentration c 0 =1.5 % Fig. 2 shows the time-dependent concentration profiles, resulting from the application of a boundary concentration c 0 , as a function of the normalized radial distance from the inner radius r dim . At the beginning of the diffusion process, important concentration gradients occur near the external surfaces. As time advances, the concentration increases and finally reaches its uniform equilibrium level (noticed perm in the caption). Fig. 3 shows the transverse macroscopic stresses for different moisture content ratios. As time advances, the transverse stress in the central ply increases, attaining its maximum value after a short time interval, then decreasing gradually as the concentration distribution (Fig. 2) approaches its uniform equilibrium level. The maximum transverse stress, reached in the central ply, is growing function of the moisture content ratio. The transverse stresses for a moisture content ratio of 2 (Fig. 3a) reach positive values which are about 70 % of the transverse tensile strength Y (Table 4). For the maximum moisture content ratio, the transverse stress exceeds the macroscopic tensile strength in the transverse direction. The transient analysis emphasizes the existence of maximum stresses in the first times. In the inner ply, the macroscopic transverse stress is not constant. Fig. 3b shows the average transverse macroscopic stresses in the inner ply. The average transverse stresses remain negative whatever the moisture content ratio. For the maximum moisture content ratio, the average transverse stresses do not reach the compressive strength in the transverse direction Y´. Finally, a macroscopic tensile failure occurs in the central ply.

f f f L ε σ : = (21)
Using ( 21) in [START_REF] Kröner | Berechnung der elastischen Konstanten des Vielkristalls aus des Konstanten des Einkristalls[END_REF][START_REF] Eshelby | The determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems[END_REF] we obtain the following scale transition expression for the strains in fibers: ( ) ( )

I I I I I I f f R L R L L ε σ ε : : : : 1 + + = - (22)
The use of ( 22) implies the knowledge of the macroscopic stresses and strains in the ply. This especially requires the measurement or calculation of the CME of the composite material. If these conditions are satisfied, the local mechanical states in the epoxy matrix are provided by Hill's strains and stresses average laws (5): The transverse stresses in the matrix and fiber, corresponding to the macroscopic stresses in the central ply, for the different moisture content ratios are respectively plotted in Figs. 4a and4b.

       - = - = f m f I m m f m f I m m v v v 1 v v v 1 ε ε ε σ σ σ (
- The transverse stresses in the matrix and fiber are growing functions of the moisture content ratio. For the maximum moisture content ratio, the transverse stress in the matrix (Fig. 4a) varies with time from tensile values superior to 30 MPa (first stages) to compressive values around 130 MPa (corresponding to the permanent state).

The transverse stresses in the matrix and fiber, corresponding to the average macroscopic stress on the inner ply, for the different moisture content ratios are respectively plotted in Figs. 5a and5b. The transverse stress in the matrix remains negative whatever the moisture content ratio. The maximum transverse compressive stress is reached at the first time and for a ratio of 3.33; the stress values could reach the compressive strength of the matrix while macroscopically a transverse tensile failure occurs in the central ply. The stress in the fiber is initially compressive and becomes tensile as time advances.

CONCLUSIONS

A hygro-elastic self-consistent model for unidirectional fiber-reinforced composites subjected to transient moisture concentration has been investigated in this paper. The moisture contents in the matrix and fiber are strongly different. Accordingly the classical self-consistent approach is extended in order to predict the macroscopic elastic properties and the Coefficients of Moisture Expansion (CME). CME predicted have been compared to Tsai-Hahn estimates. An imperfect adhesion in fiber-reinforced composites and a porous matrix induce various ratio between the moisture content of the neat resin and the moisture content of the composite. The macroscopic (ply) and local (fiber and matrix) internal stress states are evaluated for various moisture content ratios between the matrix and the ply. The macroscopic stresses are calculated by using continuum mechanics formalisms and the local stresses derived from the scale transition model. It is shown, for maximum moisture content ratio, corresponding to a perfect adhesion between the fiber and matrix and a non porous matrix, that a macroscopic tensile failure occurs in the central ply of the composite pipe considered. On the contrary, scale transition model enlightens a possible damage due to compressive stresses in the matrix for the inner ply. This demonstrates the complementarities between continuum mechanics and scale transition modeling for the study of heterogeneous materials such as composites.

  s functions of order zero, u ∆ is the determinant of 2*2 matrix [ ] a . m A and m B are determinants of matrices deduced from [ ] a by substituting respectively column 1 and 2 by the constant vector { } is the derivative of u ∆ with respect to ω calculated for m ω the m th positive root of u ∆ . r and τ are defined by the relations r/b r = and 2 I t)/b (D = τ .Furthermore, the elements of [ ] a and { }

Fig. 1 .

 1 Fig. 1. Longitudinal and transverse CME estimated for T300/5208 composite. Comparison between SC and Tsai-Hahn models

Fig. 2 .

 2 Fig. 2. T300/5208 concentration profiles

Fig. 3 .

 3 Fig. 3. T300/5208 Transverse macroscopic stresses: a) central ply, b) inner ply

  23)Forms (21-23) depend on neither the local CME of the matrix nor its moisture content.

Fig. 4 .

 4 Fig. 4. a) Transverse stresses in N5208 matrix (central ply), b) Transverse stresses in T300 fiber (central ply)

Fig. 5 .

 5 Fig. 5. a) Transverse stresses in N5208 matrix (inner ply), b) Transverse stresses in T300 fiber (inner ply)

Table 2 .

 2 CME for the components of T300/5208 composite."

	10]	230	15	0.2		7	15
	N 5208 epoxy matrix [11]	4.5	4.5	0.4		1.6	1.6
	T300/5208 (SC)	139.6	9.8	0.28		3.5	6.4
	"11 β	β	22	,	β	33

Table 4 .

 4 Macroscopic failure stresses in the longitudinal, transverse and normal directions[START_REF] Tsai | Composite design[END_REF]."

	Strengths [MPa]	X, X´	Y, Z	Y´, Z´	S
	T300/5208	1500	40	246	68
	4.2.2 Local stress states
	By assuming that the fibers do not absorb any moisture, the local stresses-strains relation in
	reinforcements (2) becomes:


L Batista

T Bastogne

A Delaunois

J.-P Valentin

F Atienzar

A novel statistical signal processing method to estimate effects of compounds on contractility of cardiomyocytes using impedance assays

Keywords: signal processing, biostatistics, high-content screening, impedance, label-free technology, cardiomyocyte, cardiotoxicity

Label free methods such as cell impedance assays are in vitro tests increasingly used in drug development and producing large and high-content data files. Since the current commercial software are not suited for fully automated analysis, there is a need to develop validated and rapid solutions to extract relevant information for biologists. This need is particularly obvious in the case of impedance signals analysis from cardiomyocytes. The proposed solution is based on three main steps. The first one consists in calculating five indices informing about the time variations of frequency (F), amplitude (A), shape (S) of beatings, trends (T) of the cardiomyocyte dependent on spreading, viability and attachment as well as irregularity (I) of the contractility. In a second phase, two summary statistics are proposed to test the concentration effect of drugs on the five FASTI indices. Results of the statistical tests are finally aggregated in a cardio-effect grade to compare the tested molecules in a cardio-impact scale graduated from 0 (no influence) to 10 (highly disturbed effects in cardiomyocytes). This innovative approach was tested using in vitro data obtained from cell impedance analysis of three known molecules (2 cardiotoxic and 1 noncardiotoxic compounds). Results have clearly shown the ability of the proposed approach to identify significant effects on the contractility of cardiomyocytes. This solution speeds up the analysis of cardiomyocyte impedance data, takes into account all the kinetic data generated and is now available for biologists on a web-platform: i-Cardio TM developed by CYBERnano TM .

Introduction

The CiPA 1 initiative [START_REF] Cavero | Comprehensive in vitro Proarrhythmia Assay, a novel in vitro/in silico paradigm to detect ventricular proarrhythmic liability: a visionary 21st century initiative[END_REF][START_REF] Fermini | A New Perspective in the Field of Cardiac Safety Testing through the Comprehensive In Vitro Proarrhythmia Assay Paradigm[END_REF] aims at developing a new approach to assess the proarrhythmic potential of new chemical entities. So far, cardiovascular safety assessment was mainly focused on QT interval prolongation, which is used as a marker for predicting the risk of a compound to cause potentially fatal ventricular cardiac arrhythmia called Torsade de Pointes (TdP). The current preclinical safety guideline (ICH 2 S7B) relies on preclinical electrophysiology tests against hERG (human-ether-à-go-go Related Gene) and an in vivo QT measurement. Nevertheless, this approach may increase the number of false positive drugs. In addition, to increase the likelihood of success, an effective derisking strategy should evaluate proarrhythmia liability, hemodynamic cardiac contractility assessment, taking into account both functional and structural aspects of cardiotoxicity. All this situation has motivated ICH to reconsider approaches to the evaluation of drug-induced cardiac injuries. To that aim, in silico action potential simulations based on the O'Hara-Rudy model [START_REF] O'hara | Simulation of the undiseased human cardiac ventricular action potential: Model formulation and experimental validation[END_REF] have been integrated in the CiPA (Comprehensive In Vitro Proarrhythmia Assay) effort to better assess human cardiotoxicity risks. A second main component of the CiPA project relies on stem cell-derived human myocyte studies. Two technologies were suggested for evaluation, namely the multielectrode array (MEA) and the voltage-sensing optical (VSO) platforms. However, other emerging techniques exist, such as the cardiomyocyte impedance analysis, which proved to be useful to identify cardiotoxic compounds [START_REF] Guo | Microarchitectural design space exploration made fast[END_REF].

In this context, in the present paper, we evaluated the real-time cell analysis (RTCA) platform, measuring cell impedance, to assess effects on contractility of cardiomyocytes exposed to a small panel of drugs. By analogy, this approach may be compared with the detection and characterization of abnormalities in ECG signals. Mahmoud et al. [START_REF] Mahmoud | Time-frequency analysis of normal and abnormal biological signals[END_REF] reported that the time-frequency distribution is employed to reveal the exact multicomponent structure of surface electromyogram with the presence of ECG noise and abnormal cardiac signals. [START_REF] Martis | Computer aided diagnosis of atrial arrhythmia using dimensionality reduction methods on transform domain representation[END_REF] compared four methods to diagnose ECG abnormalities [START_REF] Martis | Computer aided diagnosis of atrial arrhythmia using dimensionality reduction methods on transform domain representation[END_REF]. They associate two different transforms (wavelet vs cosinus) with statistical techniques of reduction such as the Principal Component Analysis followed by clustering analysis involving K-nearest neighbor, decision tree and artificial neural networks. Since that time, a large spectrum of methods have been tested such as sequential Bayesian methods [START_REF] Lin | Sequential beat-to-beat P and T wave delineation and waveform estimation in ECG signals: Block Gibbs sampler and marginalized particle filter[END_REF], multiscale energy and eigenspace approach [START_REF] Sharma | Multiscale Energy and Eigenspace Approach to Detection and Localization of Myocardial Infarction[END_REF], multiresolution time-dependent entropy method [START_REF] Farashi | A multiresolution time-dependent entropy method for QRS complex detection[END_REF], signal decomposition model-based Bayesian approach [START_REF] Roonizi | A Signal Decomposition Model-Based Bayesian Framework for ECG Components Separation[END_REF] and continuous wavelet transform [START_REF] Yochum | Automatic detection of P, QRS and T patterns in 12 leads ECG signal based on CWT[END_REF]. However, several main differences may be emphasized between the ECG and cardiomyocyte impedance analysis. Firstly, PQRST waves do not exist in CI (cell index) signals and their beating shape may change completely depending on the drugs used [START_REF] Xi | Functional Cardiotoxicity Profiling and Screening Using the xCELLigence RTCA Cardio System[END_REF][START_REF] Jonsson | Impedance-Based Detection of Beating Rhythm and Proarrhythmic Effects of Compounds on Stem Cell-Derived Cardiomyocytes[END_REF]. Secondly, conversely to usual ECG that are recorded during about 1 Comprehensive In Vitro Proarrhythmia Assay 2 International Conference on Harmonisation 10s only, CI responses can be measured during one week of experimentation and generate larger data files to be processed.

MEA, VSO and RTCA platforms are increasingly used for drug discovery applications. In such a context, large number of molecules can be tested on cardiomyocytes and several parameters can be analyzed to assess cardiotoxicity risks. This computational task is time-consuming and user dependent. Our objective was to develop an automatic, reliable and fast method to compare the effects of tested drugs on cell impedance (CI) responses in cardiomyocytes. To address this issue, we propose a triple-stage method. The first step is based on the estimation of five time-variant parameters of CI responses. In a second phase, a concentration-effect modeling of the five indices is performed and statistical tests are used to assess the relevance of all variations in comparison with the control group. Finally, a score of critical effects is proposed to compare the alteration effects on the CI responses of tested molecules in a new cardio-impact scale. The second main intended purpose of this article is to assess the practical relevance of this new approach applied to real cardiomyocyte impedance data.

This paper is organized as follows. In section 2, the cardiomyocyte impedance analysis is introduced. Section 3 describes the methodology and the impedance data corresponding to three molecules are presented in section 4. Finally, as a proof of concept, results are discussed with a set of two cardiotoxic and one noncardiotoxic compounds to demonstrate the potential of the developed approach to analyze impedance data in a rapid and reliable way.

Cardiomyocyte Impedance assays and summary of Material & Methods

The use of electrical impedance to measure cellular processes was first reported by Giaever in 1984, [START_REF] Giaever | Monitoring fibroblast behavior in tissue culture with an applied electric field[END_REF][START_REF] Giaever | Micromotion of mammalian cells measured electrically[END_REF]. The xCELLigence platform generating impedance data has been identified as a powerful and reliable tool in drug discovery for toxicity and pharmacology studies [START_REF] Abassi | Kinetic cell-based morphological screening: Prediction of mechanism of compound action and off-target effects[END_REF][START_REF] Atienzar | Evaluation of impedance-based label-free technology as a tool for pharmacology and toxicology investigations[END_REF]. All the impedancebased cell analyzers use the fact that cells have an electrical insulating membrane. This property comes from its biological composition. Indeed, the cell membranes are made of phospholipid bilayer, and these lipids act as electrical insulators, [START_REF] Wegener | Electric cell-substrate impedance sensing (ecis) as a noninvasive means to monitor the kinetics of cell spreading to artificial surfaces[END_REF][START_REF] Xiao | Assessment of cytotoxicity using electric cell-substrate impedance sensing: Concentration and time response function approach[END_REF]. Consequently, when an electrical current generated by electrodes at the bottom of a well flows into a ion solution, the whole well behaves as a variable impedance. The most cells fix on the substrat at the bottom of the wells, the most it decreases the current exchange area, increasing the impedance [START_REF] Wegener | Electric cell-substrate impedance sensing (ecis) as a noninvasive means to monitor the kinetics of cell spreading to artificial surfaces[END_REF][START_REF] Atienzar | The Use of Real-Time Cell Analyzer Technology in Drug Discovery: Defining Optimal Cell Culture Conditions and Assay Reproducibility with Different Adherent Cellular Models[END_REF]. This electrical impedance area is modified by multiple parameters including the number of cells, their size, the way cells communicate and spread, as well as their substrate adhesion. The data acquisition system ( RTCA cardio) allows to get one numerical value of the cell impedance every 13ms. The impedance value is finally transformed into dimensionless quantity entitled Cell-index (CI ) [START_REF] Abassi | Label-free, real-time monitoring of ige-mediated mast cell activation on microelectronic cell sensor arrays[END_REF][START_REF] Atienza | Dynamic monitoring of cell adhesion and spreading on microelectronic sensor arrays[END_REF][START_REF] Xing | Dynamic monitoring of cytotoxicity on microelectronic sensors[END_REF]. These CI data generated by this technology provide two types of information: the viability of cardiomyocytes by examining the pattern of the impedance signal trend and the cardiomyocytes contractility by analyzing the periodic pattern of the impedance signals over short periods called sweeps. As illustrated in Figure 1, the cell impedance sensing is sensitive enough to monitor cell spreading and clustering as well as beating of the cardiomyocytes. The spreading of the cells correspond to the increase in CI while the beating corresponds to the contraction of the cardiomyocytes. Those two phenomenons differ by amplitude and time constant. Indeed, the spreading is 90 a slow and high amplitude process while the cardiac activity is a fast and slow amplitude process.

The measurement profile is recorded by a station on each well of a 96 wells plate. The data acquisition was split up into sweeps of [START_REF] Atienzar | The Use of Real-Time Cell Analyzer Technology in Drug Discovery: Defining Optimal Cell Culture Conditions and Assay Reproducibility with Different Adherent Cellular Models[END_REF] Mouse cardiomyocytes (inducible pluripotent stem cells) were purchased from Axiogenesis (Cologne, Germany). Famotidine (non-cardiotoxic drug) and Doxorubicine (cardiotoxic drug in humans) were ordered from Sigma-Aldrich (Saint-Louis, MO, USA), and Sunitinib (cardiotoxic drug in human) from VWR (Radnor, PA, USA). The mouse cardiomyocytes were thawed based upon Axiogenesis standard protocol and seeded on RTCA cardio E-plates at a density of 40 • 10 3 cells/well. To quantify cell status based on the measured cell-electrode impedance, a parameter termed cell index (CI) is derived. This index is a time and sampling frequency dependent quantity described as follows:

CI(t k , f i ) = max i=1,2...k R cell (t k , f i ) R bg (f i ) -1 (1)
where R bg and R cell are the resistances measured at time t k and frequency f i without and in presence of cells respectively [START_REF] Abassi | Label-free, real-time monitoring of ige-mediated mast cell activation on microelectronic cell sensor arrays[END_REF][START_REF] Atienza | Dynamic monitoring of cell adhesion and spreading on microelectronic sensor arrays[END_REF][START_REF] Xing | Dynamic monitoring of cytotoxicity on microelectronic sensors[END_REF]. N is the number of the frequency points at which the impedance is measured. For more details please refer to Atienzar et al. ( 2013) [START_REF] Atienzar | Evaluation of impedance-based label-free technology as a tool for pharmacology and toxicology investigations[END_REF]. Cell index were monitored for a few days (usually 2 to 3 days) until all parameters measured were sufficiently stable (cell index, beating, amplitude, frequency, etc.). E-plates (Roche Diagnostics, Vilvoorde, Belgium) were coated, according to the manufacturer's recommendations. Fibronectin (coating) was purchased from Sigma-Aldrich (Saint-Louis, MO, USA). Final % dimethyl sulfoxide (DMSO) in each well was 0.1%. Fresh concentrated stock solutions were prepared in DMSO immediately before first use and then kept at -20 until compound addition (usually from t=0 up to t = 2 to 3 days), every minute after compounds addition for the first 30 min and every 15 min for the last 23.5h. 20 s sweeps were used to collect cell index data at all steps described previously. Data analysis was performed using the algorithms developed by Cybernano and presented in this manuscript.

Cardio-Effect Estimation Method

An overview of the cardio-effect estimation method proposed in this article is presented in Figure 2. This approach is composed of four main steps developed in the next sections. The main output results generated by each computational step are indicated on the right part of the figure.

Morphological characterization of the CI response alteration

A behavioral characterization of the sweep signals is addressed in this section to detect and estimate the main changes in the CI responses caused by the tested molecules. The beating shape depends on cell environment and many biological factors such as the cell line, cell viability, medium type, compound and its concentration. As a consequence many different beating patterns can be observed as already emphasized by Jonsson et al. in [START_REF] Jonsson | Impedance-Based Detection of Beating Rhythm and Proarrhythmic Effects of Compounds on Stem Cell-Derived Cardiomyocytes[END_REF] and Xi et al. in [START_REF] Xi | Functional Cardiotoxicity Profiling and Screening Using the xCELLigence RTCA Cardio System[END_REF]. In this context, no model-based approach has been used due to the difficulty to find out a relevant and general model structure able to accurately fit all the possible shapes. An alternative solution consists in estimating parameters of the beating shape directly from the measured signals. In order to integrate all time dependent information, we propose to estimate for each sweep five time-indices, denoted FASTI thereafter :

X nn (t); t 2 [0, ..., T 1] E(d); d 2 [1, ..., D] ( H 0 : (j) X,d = 0 H 1 : (j) X,d 6 = 0
• Frequency (F): estimate of the mean beating rate per minute in each sweep;

• Amplitude (A): average peak-to-trough magnitude in each sweep;

• Shape (S): score characterizing the shape of the mean beating pattern in each sweep;

• Trend (T): value obtained from the cell index trend signal informing about the cardiomyocyte viability status, spreading, ... Famotidine (negative control compound) and Sunitinib (positive control compound) respectively. The FASTI responses are correlated in the three examined cases but this correlation is not systematic and not been observed on other compounds. Moreover, some technical outliers such as peaks may intensify this effect.

Frequency

The Frequency characteristics informs about the evolution of the beating rate (bpm) over the experimentation period [0; T -1], where T is the number of cardio sweeps. One usual technique used to estimate F (t) is based on the detection of peaks. However, this approach is very sensitive to the noise and to avoid this drawback we propose to compute the maximum peak from the power spectral density (DSP). The latter is estimated by the Welch estimator based on the periodogram with windowing and averaging in order to reduce the variance of the estimates:

F (t) = arg max S W (f, y t ) (2)
S W (f, y t ) = 1 P P -1 p=0 1 ||w|| 2 M -1 m=0 w(m)y t (pS + m)e -j2πf m 2 , (3)
where w ∈ R M is a hamming window, M is the size of the window, and S/M represents the window overlapping. We choose M = 192 to get enough information in each window, and S = 96 to have 50% of overlaying. p ∈ [0, ..., P ] with P the total number of windows so that pS + M ≤ K, with P = 14.

Amplitude

As the frequency, the peak-to-trough amplitude of the beating signal is a complementary parameter characterizing the cardiac activity. This feature is determined by subtracting the minimum value of the sweep to the maximum one:

A(t) = max(y t (k)) -min(y t (k)). (4)
In practice, each sweep usually contains more than 20 beating waves. To avoid outliers on the estimation of the response peaks, we build a histogram of the cell index values and estimate min(y t (k)) and max(y t (k)) as the 3-rd and 97-th percentiles.

Shape

When the characteristics F (t), A(t) and T (t) of two distinct wells are identical, their beating pattern can still be different in term of shape. While ECG are complex signals due to the heart structure, the cell index signal of cardiomyocyte monolayer cultures are usually simpler with generally one main peak. Our proposal is to capture the sharpness of the signal computed as:

S(t) = 1 - 2 A(t) min T (t) -min(y t (k)), max(y t (k)) -T (t) . (5)
When y t (k) is periodic and symmetric (w.r.t its mean value) such as a sinusoidal signal, S(t) tends to 0. On the contrary, in the absence of peaks (flat signal), S(t) converges to 1. Otherwise, when y t (k) is not a symmetric beating pattern its value is between 0 and 1. The more symmetric the signal, the more S(t) is 180 closed to 0. As with the A index, a robust estimation procedure is implemented to avoid any outlier on the computation of min(y t (k)) and max(y t (k)).

Trend

The viability of cardiomyocytes is derived from the CI data. At time t, the trend (T ) is estimated by the empirical mean value of cell index over the t-th sweep :

T (t) = 1 K K-1 k=0 y t (k). (6)

Irregularity

The Irregularity time-characteristics I(t) for the t th recorded sweep is a discrete variable taking three possible values: {0, 1, 2}. If the beating pattern is regular all along the sweep I(t) = 0 but if it is drastically varying during the sweep time range then I(t) = 2. The intermediate value I(t) = 1 is associated with a moderate change of the beating shape. The computation of I(t) relies on the mean energy of the measured signal:

E(k) = k+P -1 n=k |y t (n)| 2 , ( 7
)
where k ∈ [0, ..., K -P -1] and P is the number of time points in a beating period. Therefore, by considering a sliding window calculation, E(k) becomes a new signal providing information about the variation of the mean energy within the sweep range. For a noise-free periodic signal, E(k) is constant but in practice the CI values are noisy and E(k) can be described as follows:

E(k) = a + e, e ∼ N (0, σ e ), (8)
where the parameter a denotes the mean energy of the current sweep. From 185 the likelihood function and the Fisher information of this model, we are able to estimate a and σ e and their corresponding standard errors. Given those estimates, the Table 2 describes the algorithm to compute I(t). If one of the two variation coefficients cv a and cv σe are too large it implies that E(k) is likely not a constant, i.e. y t is irregular. A second irregularity index used to compute I(t) is related to the noise-signal ratio R. Three example of irregularity values are presented in Figure 6. They show the ability of this index to detect different levels of beating irregularity within a cardio sweep.

No.

Step

1 compute R = σe /â 2 compute cv a = σa /â 3 compute cv σe = σσe /σ e 4 if (cv a > 0.05) or (cv σe > 0.05) or (R > s 2 ) I(t) = 2 5 else if s 2 ≤ R < s 1 I(t) = 1 6 else I(t) = 0 end if

Normalization

For clarity, X(t) is used thereafter to refer to one the four indices: F (t), A(t), S(t), T (t) because they received the same processing. Nevertheless, I(t) is treated separately. To reduce the potential effects of uncontrolled variations between the wells and the plates, a double-step normalization process is applied to the previous time-indices.

Time normalization

The first normalization consists in fixing to zero all the indices at time t m , i.e.: the time instant just before drug administration.

X n (t) = X(t) -X(t m -1), ( 9
)
with t ∈ [t m , • • • , t m +∆t]
. t m is selected based on the stability of the parameters usually obtained two days after cardiomyocyte seeding (time usually required to have a stabilization of the parameters). The study time frame was fixed to ∆t = 24h.

Normalization

Since the goal of the study is to compare the drug effect with a control agent (vehicle compound), indices are centered and reduced as follows: where µ c (t) and σ c (t) are the empirical mean and standard-deviation of the control group.

X nn (t) = X n (t) -µ c (t) σ c (t) (10)

Concentration-effect Analysis

To analyze how the normalized FASTI indices are changing with respect to the compound concentrations, a summary of the information contained in those meta-signals was performed. To that aim, we calculate two statistical summaries of X nn , noted S X,1 and S X,2 . S X,1 is the root mean square value of X nn :

S X,1 = 1 T T -1 t=0 X nn (t) 2 . ( 11
)
When S X,1 is near zero, X nn is close to the reference profile (control). The second component, S X,2 , corresponds to the empirical standard deviation of the innovation term:

S X,2 = 1 T -1 T -2 t=0 (X nn (t + 1) -X nn (t)) 2 . ( 12
)

Effect of Test Compounds on Cardiomyocytes

In this section we present experimental results of the proposed statistical signal processing method applied to in vitro data obtained from cell impedance analysis. The objective was to assess the suitability of the proposed methodology to correctly discriminate the effects of Famotidine (non-cardiotoxic drug), Sunitinib and Doxorubicine (cardiotoxic drug in humans).

The global effect scores (E(d)), Hill model response (EC 50 estimation) and the ten FASTI summary statistics (S X,1 and S X,2 in section 3.3) are presented in Figures 7,8 and9 for the three tested compounds, respectively. Maximum therapeutic plasma concentrations of Sunitinib and Doxorubicin were obtained from Pharmapendium3 .

For Famotidine, we clearly observe no significant effect on the cardiomyocytes up to 100 µM equivalent to ca. 50 fold the maximum concentration in human blood (C max ). In humans, Famotidine (0.5 mg kg -1 ) has a C max value of 0.64 µg mL -1 (i.e. 1.90 µM) [START_REF] James | Pharmacokinetics and pharmacodynamics of famotidine in infants[END_REF]. The global effect score remains under 1 in the tested concentration range and no significant variation of any FASTI index was detected. Some red dots appear in low concentration (100 µM) of Famotidine for S A,1 but those not significant results can be ignored as not reproduced at higher concentrations. Although some cases of QT prolongation have been reported in humans (not in animals) [START_REF] Vargas | Evaluation of drug-induced QT interval prolongation in animal and human studies: a literature review of concordance[END_REF], Famotidine does not inhibit hERG nor increases cardiac action potential in vitro [START_REF] Nakamura | Famotidine neither affects action potential parameters nor inhibits human ether-a-go-go-related gene (hERG) K+ current[END_REF]. Moreover, it does not affect cardiac contractility or heart rate in animals or in humans [START_REF] Melzer | Famotidine has no effect on cardiac performance and liver blood flow in the rat[END_REF]. Therefore, the low global effect score well reflects the overall safe cardiac properties of Famotidine.

In the second example, Sunitinib was classified as a cardiotoxic drug. Its global score and FASTI indices show a significant alteration of the beating activity with a median effect concentration estimated to be EC 50 =1.3 µM. The main cardiac effect of sunitinib is hypertension and left ventricular dysfunction, which has been reported in up to 12% metastatic carcinoma patients treated with this drug [START_REF] Kollmannsberger | Sunitinib therapy for metastatic renal cell carcinoma: recommendations for management of side effects[END_REF]. Bradycardia or QT prolongation are more rarely observed. C max of Sunitinib has been reported to be 0.253 µM in humans (Guo et al. [START_REF] Guo | Microarchitectural design space exploration made fast[END_REF]). The ratio between the EC 50 and C max is only ca. 5 fold (1.3/0.253).

The last example deals with doxorubicin, an anthracycline drug widely used to treat a variety of tumours. Doxorubicin is a structural cardiotoxicant causing QT prolongation without acute cardiac ion channel such as hERG liability [START_REF] Ducroq | Dexrazoxane protects the heart from acute doxorubicin-induced qt prolongation: a key role for iks[END_REF][START_REF] Guo | Refining the human ipsc-cardiomyocyte arrhythmic risk assessment model[END_REF]. It was recently proposed that doxorubicin-induced QT prolongation could be due to the direct inhibition of IKs (a slowly activating component of delayed rectifier K+ channel) [START_REF] Ducroq | Dexrazoxane protects the heart from acute doxorubicin-induced qt prolongation: a key role for iks[END_REF]. This examples illustrates the danger of neglecting I(Ks) in favor of hERG screening alone, for early preclinical testing for possible induction of torsade de pointes [START_REF] Ducroq | Dexrazoxane protects the heart from acute doxorubicin-induced qt prolongation: a key role for iks[END_REF]. Doxorubicin has a maximum therapeutic plasma concentration of 15.34 µM [START_REF] Guo | Refining the human ipsc-cardiomyocyte arrhythmic risk assessment model[END_REF][START_REF] Pointon | Phenotypic profiling of structural cardiotoxins in vitro reveals dependency on multiple mechanisms of toxicity[END_REF]. In the present study, a EC 50 of 9.7 µM is reported in mouse cardiomyocytes, which is even below the clinical efficacious plasma concentration of doxorubicin. In other words, these results suggest that doxorubicin has the potential to induce cardiotoxicity at the efficacious concentration in human.

Conclusion

In the context of the CiPA initiative, this paper addresses the identification problem of new biosignal markers able to predict the risk of molecules to potentially disturb the contractility of cardiomyocytes. The proposed and developed approach deals with a statistical signal processing method that provides to biologists three levels of information.

It firstly computes five indices informing about the time variations of irregularity, frequency, amplitude and shape of beatings but also an estimation of the cardiomyocyte viability. That first step allows to speed up the analysis of cardiomyocyte RTCA data. Indeed, instead of analyzing individual time points, the FASTI indices bring a global overview of the cardiomyocyte behavioral characteristics and allow to quickly identify subtle changes over time.

In a second step, two summary statistics of the FASTI indices are computed over all the experimental time range and are used to test dose response relationship. Those two additional attributes provide clarifications about the threshold concentration effects of each molecule. The Student's t-test is used to detect the statistical significance of each concentration effect with respect to a given control.

A global cardio-effect score is finally proposed to compare the alteration effects of the tested pharmaceutical agents on the cardiomyocyte contractility. To this aim, we have proposed a new cardio-effect scale graduated from 0 (no influence) to 10 (critical effect). This approach allows thus to compare different compounds on the same diagram.

To assess its practical relevance, the proposed approach was applied to the cardiomyocyte impedance data of three known compounds. Compared to recent results, these preliminary results have corroborated the ability of the proposed method to efficiently discriminate the effects of the tested drugs. The main advantage of this approach is to provide rapid and reliable analysis of complex signals. The complete approach was implemented on a web-platform: i-Cardio TM 4 and is now available for all users of the RTCA technologies.

Finally, to confirm the validity of the proposed approach, the next step will evaluate this new method to a larger batch of molecules. In particular, we will address the potential correlations between the results provided by such datadriven modeling approaches and the underlying biological phenomena associated with cardiotoxicity.

Figure 1 :

 1 Figure 1: Real-time measurement of cardiomyocyte impedance responses. The trend response is measured the cell index (arbitrary unit) with a sampling period of about 5mn. This increasing trend signal provides information about the cardiomyocyte attachment and spreading during the experimentation period. Zooming into the cell index response allows to monitor the contraction of the cardiomyocytes. At each time point t i , a cardio-sweep is measured during 20s with a sampling rate of 77.35Hz to observe the cardiomyocyte beating pattern. k is the time index used in each cardio-sweep.

  ); X 2 {F, A, S, T, I}; t 2 [0, ...,T 1]

Figure 2 :

 2 Figure 2: Overview of the method used to analyze the CI data. The proposed method is composed of four main steps and provides three major results. The FASTI indices answer two questions: when do the CI responses change and which parameters are altered? The concentration-effect analysis informs about which compound concentration significantly affect the contractibility parameters and viability trend. Finally, a global score is computed to compare the effects of each tested compound on the cardiomyocyte contractility.

•

  Irregularity (I): score indicating how much the beat pattern is changing within each sweep. Those five characteristics are calculated over the time range: t ∈ [0, ..., T -1].For instance, Figures3, 4 and 5show the time evolution of the FASTI indices for the cardiomyocytes exposed to solvent (control condition) and to two molecules:

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Overview of FASTI indices over time from cardiomyocytes exposed to solvent (control condition, 0.1% DMSO). Except for some rare events due to technical artefacts, the FASTI indices are stable, no significant change and no irregularity are detected in the 24h timeframe. The black vertical line indicates the time of the DMSO addition. Data are presented for six wells.

Figure 6 :

 6 Figure 6: Examples of cardiomyocytes beating exposed to 33nM of Sunitinib (cardiotoxic drug). The beating profiles were obtained at three different time instants after its administration. They illustrate three distinct values of irregularity. A: I = 0 at t = 65.79h (constant beating rate during the whole sweep); B: I = 1 at t = 72.93h (prolongation of the beating period at the beginning of the sweep but the other peak-to-peak periods are constant); C: I = 2 at t = 124.41h (the beating rate is inconstant during the sweep.

Figure 7 :

 7 Figure 7: Summary of concentration-effects induced by Famotidine on mouse cardiomyocytes. A. Global effect scores. Minimum score (0) means that the FASTI parameters are not significantly different compared to control data (cardiomyocytes exposed to 0.1% DMSO). A score of 10 corresponds to severe effects often manifested by a beating arrest of the cardiomyocytes. Highest score for Famotidine is 1. B. Estimated EC 50 value (no data reported up to 100 µM). C. FASTI indices: blue dots correspond to control values, green dots indicate no significant change w.r.t. control and red dots point out significant changes compared to control data. Each time red dots are detected, the global effect score is increased by one unit.

Figure 8 :

 8 Figure 8: Summary of concentration-effects induced by Sunitinib on mouse cardiomyocytes. A. Global effect scores. Minimum score (0) means that the FASTI parameters are not significantly different compared to control data (cardiomyocytes exposed to 0.1% DMSO). A score of 10 corresponds to severe effects often manifested by a beating arrest of the cardiomyocytes. Highest score for Sunitinib is 10. B. EC 50 value is estimated between 1 µM and 3 µM. Since the variation of the global score is abrupt, the EC 50 estimated value provided by the Hill model (1.3 µM) is uncertain. C. FASTI indices: blue dots correspond to control values, green dots indicate no significant change w.r.t. control and red dots point out significant changes compared to control data. Each time red dots are detected, the global effect score is increased by one unit.

Figure 9 :

 9 Figure 9: Summary of concentration-effects induced by Doxorubicine on mouse cardiomyocytes. A. Global effect scores. Minimum score (0) means that the FASTI parameters are not significantly different compared to control data (cardiomyocytes exposed to 0.1% DMSO). A score of 10 corresponds to severe effects often manifested by a beating arrest of the cardiomyocytes. Highest score for Doxorubicine is 8. B. Estimated EC 50 values: 9.7 µM for Doxorubicine. C. FASTI indices: blue dots correspond to control values, green dots indicate no significant change w.r.t. control and red dots point out significant changes compared to control data. Each time red dots are detected, the global effect score is increased by one unit.

Table 1 :

 1 secondes recorded every 5 minutes. In each sweep, the sampling rate is equal to 77.35 Hz, resulting Description of the parameters and variables used in the proposed statistical signal processing method

	95
	in a K = 1547 data-points by sweep. In other terms for control cardiomyocytes,
	each sweep contains around 28 beats and for a five-day measurement campaign,
	there are about 1440 sweeps to be processed for just one tested well. All the
	variables and parameters are listed in Table 1.
	Symb.	Legend
	moment of measurement of the (t + 1)-th sweep k ∈ [0, ..., K -1] moment of CI measurement within each sweep t ∈ [0, ..., T -1] y t (k) (k + 1)-th CI value in the t-th sweep
	T	number of recorded sweeps
	K	number of time points in a sweep signal
	P	number of time points in a beating period
	t m	time of drug administration
	d ∈ [1, ..., D] D	index of the drug concentration number of concentration (dose) levels
	r ∈ [1, ..., R] R	index of the replicated well number of assay replications
	∆t	time range of analysis after compound addition
	i ∈ [F, A, S, T, I] FASTI index F (t) beating mean frequency characteristic of the t-th sweep
	A(t)	beating mean amplitude characteristic of the t-th sweep
	S(t)	beating pattern characteristic of the t-th sweep
	T (t)	t th value of the CI trend, mean value of the t-th sweep
	I(t)	beating irregularity feature of the t-th sweep

Table 2 :

 2 Algorithm used for the computation of the irregularity characteristic associated with the t th sweep. s 1 ands 2 are two thresholds chosen by the user.

https://www.pharmapendium.com

https://www.cybernano.eu/i-cardio

In other terms, S X,2 estimates the mean magnitude of the abrupt and chaotic variations of X nn . When S X,2 is close to zero, the profile of X nn (t) is regular and its value is increasing when X nn (t) becomes instable. For the irregularity characteristics I(t), we use the proportion of irregularity of type 1 for S I,1 and the total proportion of irregularity, that is type 1 and type 2, for S I,2 .

At this step, for each well, we have a total 10 statistics, i.e. 2 statistics for 5 indices. Now, we can group them by concentration for each tested compound. Each condition relative to the compounds is repeated in three different wells of the same plate. In such a context, statistical tests can be applied for each concentration level to decide whether significant differences exist between the tested compounds and the control group. An ANOVA test is used and the latter relies on the following model:

where

X,0 is the mean response of S X,j (0, r) for the null concentration (d = 0) and β (j) X,d is the mean additive effect of the dose d on S X,j . Once the parameters estimated, a T-test is performed and its p-value is used to decide whether or not the effect β (j) X,d of the concentration level d is significantly different from 0. In this application, the significance threshold is fixed to p = 0.01. For the I index, a χ 2 -test is applied to assess the effect of each concentration level in comparison with the null-concentration (vehicle) sample.

Effect scoring and modeling

Based on the results of the previous statistical tests, a score of critical effect E(d) is calculated for each concentration d of the tested compound on the cell impedance responses. E(d) is defined as follows:

where I = {F, A, S, T, I}, 1 is the index function and p X,j,d is the p-value associated with the statistical test of significance for the statistics S X,j . E ranges from 0 (no effect) to 10 (strong effect). If the concentration effect is significant, a median-effect concentration (EC 50 ) is estimated by fitting a Hill model to the values of E(d) [START_REF] Sebaugh | Guidelines for accurate ec50/ic50 estimation[END_REF]. This median-effect concentration will be used to compare the effects of compounds on the cardiomyocyte contractility. Nevertheless, when the score variation is abrupt, e.g. from 0 to 10 between two successive tested concentrations, the EC 50 estimation based on the Hill model may be too uncertain. In that case, the experimenter is advised to test new concentrations within the appropriate variation range.