T Bastogne
email: thierry.bastogne@cran.uhp-nancy.fr

A Samson

S Mézières-Wantz

P Vallois

S Pinel

M Barberi-Heyob

System identification of tumor growth described by a mixed effects model

Keywords: System identification, Biomedical systems, Statistical inference, Variability, Uncertain dynamic systems

System identification of treated tumor growth is addressed in this paper. Three main difficulties are examined: (i) the determination of a suited dynamical model structure (modeling problem), (ii) the inter-individual variability of the therapeutic responses (population identification problem or longitudinal data analysis) and (iii) the effects of some categorical factors on the model parameters. To solve these problems, a mixed effect model of tumor growth, a two step identification approach and an estimation algorithm based on expectation maximization, are proposed and applied to in vivo data. A double effect of treatments on the tumor volume responses is pointed out.

INTRODUCTION

New cancer therapies cannot be developed efficiently unless it is possible to predict the therapeutic effects on the tumor growth, in a reliable, robust and reproducible manner. Reliability, robustness and reproducibility of the therapeutic outcomes can be expressed as control problems. In de [START_REF] De Pillis | A mathematical tumor model with immune resistance and drug therapy: an optimal control approach[END_REF]; [START_REF] Tervo | Optimal control model for radiation therapy inverse planning applying the boltzmann transport equation[END_REF]; [START_REF] Dua | Optimal delivery of chemotherapeutic agents in cancer[END_REF], control theory was applied to cancer therapies. These control strategies require to consider mathematical modeling of tumor growth. Such modeling studies were developed in [START_REF] Fischer | Mathematical simulation of radiation theory of solid tumors. I. Calculations[END_REF]; [START_REF] Thompson | Thermodynamics and Kinetics of Biological Processes[END_REF]; [START_REF] Adam | A survey of Models for Tumor-Immune System Dynamics[END_REF]; [START_REF] Drasdo | A single-cell-based model of tumor growth in vitro: monolayers and spheroids[END_REF]. The growth models proposed were most of time kinetic and not dynamic. The effect of input variables on the growth response, like the damage caused by the treatment, was not explicitly taken into account. In this control framework, the mathematical model of the system must be simple enough so that it can be analyzed with available mathematical techniques, and accurate enough to describe the important aspects of the relevant dynamical behavior. Moreover, size of the tumor growth data sets is generally small which reasonably limits the model complexity. The first problem addressed herein is thus to determine a dynamic model structure able to correctly fit dynamics of tumors and simple enough to be used in a control perspective.

The second main issue is the inter-individual variability of treatment responses. Indeed, in vitro as well as in vivo experiments are always repeated to assess the reproductiveness degree of the experimental responses. Quantifying the inter-individual variability of the model parameters is a crucial point, in particular in the analysis of the biological model. The problem is not to identify one biological system but a batch of similar biological systems. The associated inference problem is based on longitudinal data, i.e. data observed over subjects as well as time. A first modeling approach consists in estimating as many models as the number of subjects, and then to statistically compute the empirical distribution of the model parameters. The main drawback of this approach is to increase the parameterto-observation ratio. Another approach is to use mixed effects models. The latter have been largely employed for the identification of static systems but their use in dynamic system identification problems is much more recent [START_REF] Lindstrom | Nonlinear mixed-effects models for repeated measures data[END_REF]; [START_REF] Samson | Extension of the SAEM algorithm to left-censored data in nonlinear mixed-effects model: application to hiv dynamics model[END_REF][START_REF] Samson | The SAEM algorithm for group comparison tests in longitudinal data analysis based on non-linear mixed-effects model[END_REF]).

A third typical problem in experimental biology is to assess the effects of biological factors on the model parameters. In other terms, that consists to identify some deterministic causes of the response variability and their influence degree. Some of these factors can be continuous or categorical. In this application, the treatment mode (radiotherapy and combined chemotherapy-radiotherpy) is considered as a categorical factor.

In this paper, the system identification of treated tumor growth described by a mixed model from in vivo longitudinal data sets is proposed. The analysis of the mixed model is performed using the Monolix1 software. A basic model structure of tumor growth based on cell population dynamics, completed by a treatment function, is proposed in section 2. The mixed effect model is presented in section 3. The experimental setup and estimation data sets are described in section 4. The SAEM algorithm used for the identification of mixed effects models is presented in section 5. Identification results are then shown and a discussion is held in section 6. Conclusions and perspectives of this work are drawn in section 7.

DETERMINISTIC TUMOR GROWTH MODELING

Non treated tumor: an autonomous system

A tumor can be regarded as a population of cancer cells. Its growth may then be described by a model of cell population dynamics defined by

x(t) = x n (t) -x m (t) (1) ẋn (t) = f n x(t) (2) ẋm (t) = f m x(t), u(t) (3) x n (0) = x 0 and x m (0) = 0.
(4) f n (•), f m (•) are two mathematical functions associated with usual growth models presented in Walter and [START_REF] Lamprecht | [END_REF] and [START_REF] Thompson | Thermodynamics and Kinetics of Biological Processes[END_REF]. One of the most commonly used growth model is probably the Malthusian model, sometimes called the simple exponential growth model, in which reproduction and mortality are supposed to be proportional to the total number of cells (x) in the tumor. In this case, reproduction and mortality dynamics are defined by ẋn (t) = ax(t) (5) ẋm (t) = bx(t).

(6) The model output variable v is the tumor volume and is defined as proportional to the number of cancer cells in the tumor, v(t) = γ x(t). Solution of these differential equations is given by v(t) = v 0 e α t = f 0 (t) ∀t ≥ 0 (8) where v 0 = γ x(0) is the initial volume of the tumor. This equation describes the autonomous or natural mode of a (non treated) tumor growth. The parameter α = ab is related to the mean frequency of the cancer cell cycle, and is defined positive. In other terms, a tumor is an unstable system. The vector of unknown parameters for this model is Θ = (v 0 , α).

Treated tumor: a controlled system

The anti-cancer treatment effect on the tumor volume is described by a dynamical function added to (8) to give

v(s) = v 0 (s -α) - ν s (s + ν) e -τ s u(s), ( 9
)
where s is the Laplace variable. The first part of this equation describes the natural behavior of the tumor (equation ( 8)) while the second one corresponds to the forced mode (treatment effect). ν is a positive parameter related to the damage propagation velocity associated with the anti-cancer treatment. τ denotes a time delay between the end of the treatment and the beginning of its effects on the tumor size. u(s) represents the undergone damage variable. In practice, u is not exactly an input variable since it cannot be directly controlled. Indeed as shown in Figure 1, u is the consequence (i.e. the output variable) of another process (radiation process for the radiotherapy, chemical process for the chemotherapy and photoreaction process for the photodynamic therapy) which is controlled by treatment factors (essentially doses). This preliminary process is not considered in this study since our objective is to model the general behavior of a tumor after any kind of treatment. Moreover, the damage variable u cannot be measured online, i.e. without stopping the tumor growth (mouse sacrifice), which makes the tumor growth modeling from experimental data a blind system identification problem. However, one realistic assumption about u can be stated. In practice, damage induced by the treatment are caused during a short period of time (typically <1h) compared to the response time of the tumor growth dynamics (about several ten-day periods). As a consequence, u(t) may be approached by a Dirac pulse, u(t) = u 0 δ(t) where u 0 denotes the magnitude of tumor damage caused by the treatment. After application of the inverse Laplace transform to (9) with u(s) = u 0 , we get 12)

v(t) = f 0 (t) -f u (t), ( 10
) with: f 0 (t) = v 0 e α t H(t) (11) f u (t) = u 0 1 -e -ν(t-τ ) H(t -τ ). (
H(t) = t -∞ δ(z)
dz is the Heaviside step function. Hereafter, the value of the explained volume v i (t j ) of the subject (mouse tumor) i observed at time t j will be noted v ij to simplify reading. Θ = (v 0 u 0 α ν τ ) is the vector of unknown model parameters.

MIXED EFFECTS MODEL

In practice, i.e. in in vivo context, uncertainty about the tumour growth is large and comes from several sources. Main causes are probably measurement errors, modeling approximations and biological variability, i.e. interindividual and uncontrolled variations of the response. Indeed, one particularity in experimental biology is the repetition of experiments on similar subjects in order to assess reproducibility of results. In this context, intersubject variations of responses are unavoidable and system identification becomes an inference problem from longitudinal data. The inference problem is not based on time series only but on longitudinal data (observed over time as well as over subjects). This particular issue can be solved by using mixed effects models. In our case, the mixed model of the tumor growth is given by:

y ij = v ij (Θ i ) + a e ij with e ij ∼ i.i.d. N (0, 1) (13)
where y ij ∈ R is the j th observation (measured volume) of subject i, 1 ≤ i ≤ r with r the number of subjects (repeated in vivo experiments), and 1 ≤ j ≤ n i with n i the number of observations (time points) of subject i. As previously presented, v ij is the j th model output value (defined in ( 10)) of subject i. e ij is the within-group error.

To deal with the inter-individual variability, the vector Θ i of individual random parameters is decomposed in fixed and random effects: 14) λ is an unknown vector of length d, composed of the nominal or mean values of the model parameters. β is an unknown vector of coefficients. c i is a categorical covariate associated with the treatment mode, that takes the value 1 for radiotherapy and 2 for a combined chemotherapyradiotherapy. Fixed effects are defined in µ = (λ, β). η i denotes a vector of random effects. It is supposed that e ij and η ij are mutually independent. Ω is the covariance matrix of the random effects and is defined herein as a diagonal matrix. Accordingly, the l th individual parameter θ il of subject i can be decomposed as

Θ i = λ + β c i + η i with ln(η i ) ∼ i.i.d. N (0, Ω) (
θ il = λ l + β l c i + η il with ln(η il ) ∼ i.i.d. N (0, ω 2 l ) (15)
where ω 2 l = Ω(l, l). Finally, the model hyper-parameters to estimate from experimental data are gathered in ψ = (µ, Ω, a).

ESTIMATION METHOD BY THE SAEM ALGORITHM

The parameter estimation of nonlinear mixed model is complex: the likelihood has no closed form because of the nonlinearity of the regression function in the individual parameters. Let us denote y = (y ij ) 1≤i≤r,1≤j≤ni the vector of whole data. The Expectation-Maximization (EM) algorithm is a classical approach to estimate parameters of models with non-observed or incomplete data, [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]. For non-linear mixed models, the non-observed vector is the individual parameter vector Θ = (Θ 1 , . . . , Θ r ) and the complete data of the model is (y, Θ). Let us define the function

Q(ψ|ψ ′ ) = E(L c (y, Θ; ψ)|y; ψ ′ ),
where L c (y, Θ; ψ) is the log-likelihood of the complete data. At the m-th iteration of the EM algorithm, the E step is the evaluation of

Q m (ψ) = Q(ψ | ψ m )
, whereas the M step updates ψ m by maximizing Q m (ψ). For cases in which the E step has no analytic form, [START_REF] Delyon | Convergence of a stochastic approximation version of the EM algorithm[END_REF] introduce a stochastic version of the EM algorithm that evaluates the integral Q m (ψ) by a stochastic approximation procedure.

The authors prove the convergence of this SAEM algorithm under general conditions if L c (y, Θ; ψ) belongs to a regular curved exponential family: L c (y, Θ; ψ) = -Λ(ψ) + S(y, Θ), Φ(ψ) , where ., . is the scalar product and S(y, Θ) is the minimal sufficient statistic of the model. The E step is then divided into a simulation step (S step) of the nonobserved data Θ (m) under the conditional distribution p(Θ|y; ψ m ) and a stochastic approximation step (SA step) of E S(y, Θ)| ψ m : 16) using (γ m ) m≥0 a sequence of positive numbers decreasing to 0. The M step is thus the update of the estimate ψ m :

s m+1 = s m + γ m (S(y, Θ (m) ) -s m ), (
ψ m+1 = arg max ψ (-Λ(ψ) + s m+1 , Φ(ψ) ) .
For non-linear mixed models, the simulation step is complex because the posterior distribution p(Θ|y; ψ) has no analytical form. In this case, a Monte Carlo Markov Chain is used to simulate Θ. At the m-th iteration, the S step is the simulation of Θ (m) with use of a Metropolis-Hastings algorithm which constructs a Markov Chain, with p(Θ|y; ψ m ) as the unique stationary distribution (see [START_REF] Kuhn | Maximum likelihood estimation in nonlinear mixed effects models[END_REF] for more details). [START_REF] Kuhn | Maximum likelihood estimation in nonlinear mixed effects models[END_REF] prove that under general hypotheses, the sequence ( ψ m ) m≥0 obtained by the SAEM-MCMC algorithm converges almost surely towards a (local) maximum of the likelihood. This algorithm is implemented in the Monolix software.

EXPERIMENTAL SETUP

Output variable

In practice, the tumor volume is estimated from the measurement by an electronic caliper of two distances d 1 and d 2 which are supposed to be the small and large axis of an ellipse respectively. The depth axis of the ellipsoid tumor is not accessible to the experimenter. The estimation formula of the tumor volume is then given by

y = d 2 d 2 1 2 with d 1 < d 2 . ( 17
)

Material and method

Female nude mice were used for tumor implantation. The model of human malignant glioma was obtained using U87 cells, xenografted in mice. Treatments started when tumors reached a volume of 250 ± 50 mm 3 and were based on the combination of topotecan (daily intraperitoneal injection of 0.15µg/g, 5 days/week) with ionising radiation (2 Gy/day, 5 days/week) delivered for four weeks. The et al. (2006). Measurements were carried out until tumors reached a size of 2cm 3 , at which time mice were sacrificed by cervical dislocation.

In this study, the system identification of anticancer responses is decomposed into two successive parts. In a first step, the autonomous growth model (f 0 , see ( 8)), composed of two parameters v 0 and α, is identified from the control data set. No covariate is included in this case.

Secondly, parameters or more exactly hyper-parameters of the treated tumor model (f 0 and f u , see ( 10)) are estimated from the treated data set. The treatment covariate (radiotherapy / combined therapy) is then included. The two identification steps were carried out with Monolix.

In vivo data sets

In vivo kinetic responses of the tumor volume for the control and the two treated groups are presented in Figure 2. The control data set contained 15 kinetic responses while the treated data sets were composed of 7 responses.

IDENTIFICATION RESULTS

Identification of the natural tumor growth

Identification results of the natural tumor growth, obtained from the control data set, are presented in Table 2.

s.e and r.s.e mean the empirical standard deviation and the empirical variation coefficient respectively. ω v0 and ω α are two diagonal elements of the covariance matrix Ω, defined in (14). Figure 3 presents a scatterplot of the observed values with the median predicted response and its 90% prediction interval. The prediction interval is computed with the population parameters in the sample. Some estimated responses using the population parameters and the individual parameters are exhibited in Figure 4 and compared with measured responses. Figure 5 displays the empirical distributions of the population and individual weighted residuals (PWRES and IWRES respectively). Quantilequantile plots are also indicated to assess the gaussian assumption about the output residual (see equation ( 13)). 3. β are the coefficients of the treatment covariate. Note in bold text in Table 3 that uncertainty about u 0 and ν is too large to consider these estimates as relevant. This result mainly puts into question the choice of the model structure and particularly the dynamical part associated with the treatment. Four estimated responses using the population parameters and the individual parameters are exhibited in Figure 6 and compared with measured responses. It is shown that the delayed treatment effect is correctly fitted. Figure 7 displays the empirical distributions of the population and individual weighted residuals and the quantile-quantile plots. As for the control group, there is no significant fact to reject the Gaussian hypothesis about the output residual distribution. Figure 8 displays the empirical distributions of four model parameters (v 0 , α, u 0 , τ ) for the two treated groups.

DISCUSSION

Three main points emerge from these estimation results:

• the intracellular damages caused by the anticancer treatment cannot be measured on-line, i.e. without a mouse sacrifice before the end of the experiment. Estimating u 0 from in vivo data allows to quantify the damage amount. According to Figure 8 and the empirical distribution of u 0 , alterations caused by the combined-therapy are more important than the

Fig. 1 .

 1 Fig. 1. Two consecutive phenomena: therapy and tumor growth

Fig. 2 .

 2 Fig. 2. Longitudinal data sets: tumor volume responses, (a) the control group, (b) radiotherapy group, (c) combined chemotherapy-radiotherapy group complete biological and medical protocol is defined in Pinelet al. (2006). Measurements were carried out until tumors reached a size of 2cm 3 , at which time mice were sacrificed by cervical dislocation.

Fig. 4 .

 4 Fig. 3. Predicted response and 90% uncertainty interval (control group)

Fig. 5 .

 5 Fig. 5. Empirical distributions of the population and individual weighted residuals and quantile-quantile plots (control group)

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Measured (+) and estimated responses using the population parameters (-) and the individual parameters (• -•) (treated groups)

Table 1 .

 1 Main notations

	Symb. Description
	t	time
	v(t)	model output (explained volume)
	y(t)	measured volume of the tumor
	x(t)	number of cancer cells of the tumor
	xn(t)	number of cells born during [0; t]
	xm(t)	number of dead cells during [0; t]
	u(t)	undergone damage induced by the treatment
	δ(t)	Dirac delta function (impulse)
	x 0	initial number of cells (in the graft)
	d	number of parameters
	Θ	vector of the model parameters in R d
	n	number of observation
	r	number of repeated experiments

Table 2 .

 2 Estimation results (control group)

	parameter	estimate	s.e.	r.s.e.(%)
	v 0		339		19	6
	α		0.121		0.0038	3
	ωv 0		0.193		0.038	20
	ωα		0.0575	0.039	69
	a		130		8.5	7
	3000
	2500	observed data
		Prediction interval 90%
		Predicted median
	2000
	1500
	1000
	500
	0
	0	2	4	6	8	10	12	14

Table 3 .

 3 Estimation results (treated group)

	parameter	est. value	s.e.	r.s.e.(%)
	v 0	386	34	9
	βv 0 (CAT 2)	-0.293	0.13	44
	α	0.0578	0.004	7
	βα (CAT 2)	0.732	0.096	13
	u 0	1.6e+04	1.4e+05	893
	βu 0 (CAT 2)	-159	6.4e+03	4.02e+03
	ν	0.00774	0.071	916
	τ	21.4	1.3	6
	βτ (CAT 2)	-0.51	0.089	17
	ωv 0	0.18	0.046	25
	ωα	0.099	0.037	38
	ωu 0	12.9	3.4e+07	2.61e+08
	ων	0.442	4.1	930
	ωτ	0.129	0.036	28
	a	118	7.8	7

Monolix (MOdèles NOn LInéaires à effets miXtes) is a software dedicated to the analysis of nonlinear mixed effects models. It is designed for academic and industrial researchers or engineers, concerned with the analysis of longitudinal data in various fields. Monolix is developed by the Monolix group and is implemented in Matlab c . http://www.monolix.org


domain: sde chim

text:
Jing Ma

Clement Trellu
email: clémenttrellu@univ-eiffel.fr

Ekaterina Skolotneva

Nihal Oturan

Mehmet A Oturan

Semyon Mareev
email: semyonmareev@bk.ru

Investigationg the reactivity of TiOx and BDD anodes for electrooxidation of organic pollutants by experimental and modeling approaches

Keywords: Anodic oxidation, Hydroxyl radical, Modeling, Paracetamol, Terephthalic acid, Mass transport

The reactivity of two non-active anodes (BDD and TiOx) for the removal of organic pollutants from water was investigated and compared. A mathematical model for electro-oxidation of organic compounds was developed by considering (i) an analytical expression of the spatial distribution of  OH concentration at non-active anodes, (ii) mass transport through the diffusion boundary layer and (iii) both direct electron transfer and  OH-mediated oxidation for degradation of organic pollutants. The model was calibrated from experimental data using, for example, terephthalic acid as  OH probe molecule or ethanol as  OH quencher during paracetamol degradation. Calculation of the fraction of current ascribed to the different possible reactions as well as a sensitivity analysis allowed for highlighting the key mechanisms and limiting phenomena. Degradation of organics occurs almost exclusively through  OH-mediated oxidation under mass transport limitation and for sufficiently high reaction rate constants. Improvement of degradation kinetics strongly depends on mass transport enhancement of organic compounds to the reaction zone at the electrode surface. Competition between the mother molecule and degradation by-products for reaction with  OH did not significantly affect degradation kinetics under mass transport limitation. Direct electron transfer can potentially become important in presence of large amounts of  OH scavengers or for organic compounds that react slowly with  OH (k<10 5 m 3 mol -1 s -1 ). Resuls also showed that the implementation of a suitable  OH quenching experiment required the use of a concentration of ethanol of at least 100 mM for experiments performed at 5 mA cm -2 and with a concentration of target pollutant of 0.1 mM.

Introduction

Anodic oxidation (AO) is an electrochemical advanced oxidation process (EAOP) that is increasingly recognized as a promising next-generation technology for the treatment of contaminated effluents containing recalcitrant organic pollutants [START_REF] Sirés | Electrochemical advanced oxidation processes: today and tomorrow[END_REF][START_REF] Radjenovic | Challenges and Opportunities for Electrochemical Processes as Next-Generation Technologies for the Treatment of Contaminated Water[END_REF][START_REF] Chaplin | Critical review of electrochemical advanced oxidation processes for water treatment applications[END_REF][START_REF] Martínez-Huitle | Single and Coupled Electrochemical Processes and Reactors for the Abatement of Organic Water Pollutants: A Critical Review[END_REF][START_REF] Ganiyu | Renewable energies driven electrochemical wastewater/soil decontamination technologies: A critical review of fundamental concepts and applications[END_REF]. This process is based on the generation of a large amount of highly reactive hydroxyl radicals (  OH) from water discharge at the surface of suitable electrode materials [START_REF] Comninellis | Electrocatalysis in the electrochemical conversion/combustion of organic pollutants for waste-water treatment[END_REF][START_REF] Panizza | Direct and mediated anodic oxidation of organic pollutants[END_REF][START_REF] Karim | Boron-doped diamond electrodes for the mineralization of organic pollutants in the real wastewater[END_REF][START_REF] Oturan | Outstanding performances of the BDD film anode in electro-Fenton process: Applications and comparative performance[END_REF]. The degradation of organic compounds through  OH-mediated oxidation is the key phenomenon for the non-selectivity of the process.

Direct electron transfer (DET) at the anode surface can also participate in the degradation of organic pollutants. Thus, AO is able to achieve the complete degradation and mineralization of a large range of organic pollutants [START_REF] Brillas | Decontamination of wastewaters containing synthetic organic dyes by electrochemical methods. An updated review[END_REF][START_REF] Oturan | Unprecedented total mineralization of atrazine and cyanuric acid by anodic oxidation and electro-Fenton with a boron-doped diamond anode[END_REF][START_REF] Oturan | Occurrence and Removal of Organic Micropollutants in Landfill Leachates Treated by Electrochemical Advanced Oxidation Processes[END_REF][START_REF] Trellu | Anodic oxidation of surfactants and organic compounds entrapped in micelles -Selective degradation mechanisms and soil washing solution reuse[END_REF][START_REF] Trellu | Comparative study on the removal of humic acids from drinking water by anodic oxidation and electro-Fenton processes: Mineralization efficiency and modelling[END_REF][START_REF] Anglada | Laboratory and pilot plant scale study on the electrochemical oxidation of landfill leachate[END_REF][START_REF] Groenen-Serrano | Role of Hydroxyl Radicals During the Competitive Electrooxidation of Organic Compounds on a Boron-Doped Diamond Anode[END_REF][START_REF] Panizza | Anodic oxidation of 2naphthol at boron-doped diamond electrodes[END_REF][START_REF] Zhuo | Efficient Electrochemical Oxidation of Perfluorooctanoate Using a Ti/SnO2-Sb-Bi Anode[END_REF][START_REF] Cotillas | Degradation of dye Procion Red MX-5B by electrolytic and electro-irradiated technologies using diamond electrodes[END_REF][START_REF] Radjenovic | Facing the Challenge of Poly-and Perfluoroalkyl Substances in Water: Is Electrochemical Oxidation the Answer?[END_REF].

Anode materials have been classified in two main classes: active materials with low overpotential for oxygen evolution reaction (OER) and non-active materials with high overpotential for OER [START_REF] Martínez-Huitle | Single and Coupled Electrochemical Processes and Reactors for the Abatement of Organic Water Pollutants: A Critical Review[END_REF][START_REF] Panizza | Direct and mediated anodic oxidation of organic pollutants[END_REF]. The latter ones are the most suitable for the accumulation of  OH at the anode surface. Among the variety of anode materials tested in the literature, the boron-doped diamond (BDD) appeared as the gold standard for application in the AO process [START_REF] Martínez-Huitle | Single and Coupled Electrochemical Processes and Reactors for the Abatement of Organic Water Pollutants: A Critical Review[END_REF][START_REF] Panizza | Direct and mediated anodic oxidation of organic pollutants[END_REF][START_REF] Karim | Boron-doped diamond electrodes for the mineralization of organic pollutants in the real wastewater[END_REF][START_REF] Oturan | Outstanding performances of the BDD film anode in electro-Fenton process: Applications and comparative performance[END_REF][START_REF] Nidheesh | Environmental Applications of Boron-Doped Diamond Electrodes: 1. Applications in Water and Wastewater Treatment[END_REF].

However, this material is expensive and several researches are focused on the development of alternative non-active materials with lower production costs. Particularly, sub-stoichiometric titanium oxides (TiOx) appeared as a promising material. Several recent papers have highlighted the capacity of this material to degrade and mineralize various organic pollutants [START_REF] Guo | Development and characterization of ultrafiltration TiO2 Magnéli phase reactive electrochemical membranes[END_REF][START_REF] Zaky | Porous Substoichiometric TiO 2 Anodes as Reactive Electrochemical Membranes for Water Treatment[END_REF][START_REF] Trellu | Mineralization of organic pollutants by anodic oxidation using reactive electrochemical membrane synthesized from carbothermal reduction of TiO2[END_REF][START_REF] Trellu | Integration of sub-stoichiometric titanium oxide reactive electrochemical membrane as anode in the electro-Fenton process[END_REF][START_REF] Ganiyu | Sub-stoichiometric titanium oxide (Ti4O7) as a suitable ceramic anode for electrooxidation of organic pollutants: A case study of kinetics, mineralization and toxicity assessment of amoxicillin[END_REF][START_REF] Le | Energy-Efficient Electrochemical Oxidation of Perfluoroalkyl Substances Using a Ti4O7 Reactive Electrochemical Membrane Anode[END_REF][START_REF] Oturan | Sub-stoichiometric titanium oxide as a new anode material for electro-Fenton process: Application to electrocatalytic destruction of antibiotic amoxicillin[END_REF].

The effectiveness of the AO process depends on (i) the reactivity of the anode material and (ii) mass transport conditions of organic compounds from the bulk to the reaction zone in the vicinity of the electrode surface. A crucial issue is related to the short lifetime of  OH. They are present only in a thin boundary layer (<1 μm) at the anode surface, which constitutes the reaction zone [START_REF] Kapałka | The importance of electrode material in environmental electrochemistry: Formation and reactivity of free hydroxyl radicals on borondoped diamond electrodes[END_REF][START_REF] Kapałka | Kinetic modelling of the electrochemical mineralization of organic pollutants for wastewater treatment[END_REF]. AO can be operated either under mass transport or current/kinetic control, depending on (i) convective-diffusive transport of pollutants from the solution to the reaction zone and (ii) oxidation rate in the reaction zone [START_REF] Martínez-Huitle | Single and Coupled Electrochemical Processes and Reactors for the Abatement of Organic Water Pollutants: A Critical Review[END_REF][START_REF] Trellu | Electro-oxidation of organic pollutants by reactive electrochemical membranes[END_REF].

The development of suitable electrode materials requires a better understanding of the complex phenomena occurring at the electrode/solution interface, including mass transport,  OH-mediated oxidation and DET. Quantitative description of these phenomena and determination of the optimal parameters for AO requires the combination of an experimental approach with an adequate mathematical model. Several milestone contributions have been given in the literature.

First, [START_REF] Comninellis | Electrocatalysis in the electrochemical conversion/combustion of organic pollutants for waste-water treatment[END_REF] proposed a relation for the determination of the instantaneous current efficiency depending on the nature of the electrode material (active or non-active), the concentration of organic compounds and rate constants for oxidation of organic compounds and O2 evolution [START_REF] Comninellis | Electrocatalysis in the electrochemical conversion/combustion of organic pollutants for waste-water treatment[END_REF]. Then, [START_REF] Simond | Theoretical model for the anodic oxidation of organics on metal oxide electrodes[END_REF] have taken into account diffusion limitation [START_REF] Simond | Theoretical model for the anodic oxidation of organics on metal oxide electrodes[END_REF]. [START_REF] Panizza | Anodic oxidation of 2naphthol at boron-doped diamond electrodes[END_REF] have also presented a macroscopic model describing the evolution of the chemical oxygen demand (COD) according to the average mass transfer coefficient (km) and current density [START_REF] Panizza | Anodic oxidation of 2naphthol at boron-doped diamond electrodes[END_REF]. This model supposes that oxidation of organic compounds is non-selective and very fast. [START_REF] Scialdone | Electrochemical incineration of oxalic acid at boron doped diamond anodes: Role of operative parameters[END_REF][START_REF] Scialdone | Electrochemical oxidation of organics in water: Role of operative parameters in the absence and in the presence of NaCl[END_REF] have further developed these approaches by taking into consideration the oxidation pathway through DET [START_REF] Scialdone | Electrochemical incineration of oxalic acid at boron doped diamond anodes: Role of operative parameters[END_REF] and the possibility of a limitation from the oxidation rate at the anode [START_REF] Scialdone | Electrochemical oxidation of organic pollutants in water at metal oxide electrodes: A simple theoretical model including direct and indirect oxidation processes at the anodic surface[END_REF]. [START_REF] Lan | An experimental and modelling study of the electrochemical oxidation of pharmaceuticals using a boron-doped diamond anode[END_REF] have also worked on the consideration of direct electro-oxidation at the anode surface [START_REF] Lan | An experimental and modelling study of the electrochemical oxidation of pharmaceuticals using a boron-doped diamond anode[END_REF]. [START_REF] Cañizares | Modeling of Wastewater Electro-oxidation Processes Part I. General Description and Application to Inactive Electrodes[END_REF] proposed another approach based on reactor-level description of the process [START_REF] Cañizares | Modeling of Wastewater Electro-oxidation Processes Part I. General Description and Application to Inactive Electrodes[END_REF]. [START_REF] Mascia | Kinetics of the electrochemical oxidation of organic compounds at BDD anodes: modelling of surface reactions[END_REF] have presented a 1-D model allowing the description of the concentration profile of organics according to the distance from the electrode [START_REF] Mascia | Kinetics of the electrochemical oxidation of organic compounds at BDD anodes: modelling of surface reactions[END_REF]. While [START_REF] Mascia | Kinetics of the electrochemical oxidation of organic compounds at BDD anodes: modelling of surface reactions[END_REF] used lumped parameters for the description of the concentration profile of  OH, Kapalka et al. ( 2009) have then detailed the evolution of  OH at the surface of non-active anode in order to propose an analytical expression for the spatial distribution of  OH concentration [START_REF] Kapałka | The importance of electrode material in environmental electrochemistry: Formation and reactivity of free hydroxyl radicals on borondoped diamond electrodes[END_REF]. This approach allowed for a more precise calculation of the thickness of the reaction zone. Several authors have also proposed different ways to take into consideration other electrogenerated oxidants (active chlorine, sulfate radicals), which might play an important role depending on both nature and concentration of inorganic salts in the medium and reactivity of target compounds [START_REF] Lan | An experimental and modelling study of the electrochemical oxidation of pharmaceuticals using a boron-doped diamond anode[END_REF][START_REF] Scialdone | Electrochemical oxidation of organics in water: Role of operative parameters in the absence and in the presence of NaCl[END_REF][START_REF] Mascia | Electrochemical treatment of phenolic waters in presence of chloride with boron-doped diamond (BDD) anodes: Experimental study and mathematical model[END_REF][START_REF] Mousset | Charge transfer/mass transport competition in advanced hybrid electrocatalytic wastewater treatment: Development of a new current efficiency relation[END_REF].

Finally, [START_REF] Misal | Modeling electrochemical oxidation and reduction of sulfamethoxazole using electrocatalytic reactive electrochemical membranes[END_REF] and [START_REF] Skolotneva | A 2D Convection-Diffusion Model of Anodic Oxidation of Organic Compounds Mediated by Hydroxyl Radicals Using Porous Reactive Electrochemical Membrane[END_REF] also developed models for electrooxidation of organic compounds on porous electrodes in flow-through configuration [START_REF] Misal | Modeling electrochemical oxidation and reduction of sulfamethoxazole using electrocatalytic reactive electrochemical membranes[END_REF][START_REF] Skolotneva | A 2D Convection-Diffusion Model of Anodic Oxidation of Organic Compounds Mediated by Hydroxyl Radicals Using Porous Reactive Electrochemical Membrane[END_REF].

Based on these previous works, a 1D model representing an AO mixed-tank reactor with either TiOx or BDD plate anode operated under galvanostatic conditions was described in this study.

The model took into consideration (i) convection/diffusion mass transport of organic compounds (ii) reaction of organics through DET and (iii)  OH-mediated oxidation of organics. This theoretical work was combined with a detailed experimental approach for the calibration of model parameters. Probe molecules and quenchers were used for determination and discussion of the role of  OH-mediated oxidation and DET. This works provides new insights on (i) the predominance of each mechanism in the degradation of organic pollutants, (ii) the identification of key parameters influencing the reactivity of BDD and TiOx plate electrodes, (iii) competition phenomena between initial molecules and by-products and (iv) the assessment of the suitability of quenching experiments.

Material and methods

Chemicals

All chemicals used in this study were of reagent grade. Terephthalic acid (TA), paracetamol (PCT), sodium sulfate, sodium perchlorate and ethanol were purchased from Acros Organics, Sigma Aldrich, Honeywell Fluka, Alfa Aesar and Chimie-Plus Laboratoires, respectively.

Solvents used for analysis were HPLC grade and purchased from Riedel-de-Haën. Solutions were prepared using ultrapure water (18.2 MΩ cm) from a Millipore Milli-Q system (Molsheim, France).

Electrode Materials

BDD plate (DIACHEM ® ) was obtained from CONDIAS (Germany), while Ti/TiOx plate was supplied by Saint Gobain Research Provence. Dimensions of both plate electrodes were 8 x 4 cm. A carbon felt with the same dimensions was used as a counter electrode. Ti/TiOx plate was synthesized using plasma deposition on a Ti substrate, as detailed in a previous publication from our group [START_REF] Ganiyu | Sub-stoichiometric titanium oxide (Ti4O7) as a suitable ceramic anode for electrooxidation of organic pollutants: A case study of kinetics, mineralization and toxicity assessment of amoxicillin[END_REF]. A homogeneous coating was obtained on the surface of the Ti substrate as it can be observed from scanning electron microscopy (Fig. SI-1). The cross section view highlights that the thickness of the TiOx coating was around 300 µm. XRD data are provided in Fig. SI-2.

The TiOx coating was made of a mixture of TiOx phases, including Ti4O7 as the main phase.

Preliminary tests showed that the electrode was stable in the range of operating conditions used in this study. Reproducible results were obtained during seven successive degradation kinetics.

(Fig. SI-3).

Experimental oxidation of model compounds

Electrochemical oxidation of model compounds was carried out using an undivided electrochemical cell containing 300 mL of solution continuously mixed under magnetic agitation (750 rpm) (Fig. SI-4). The cell was operated under galvanostatic mode with an applied current density of 5 mA cm -2 . Sodium sulfate (0.05 M) was used as a supporting electrolyte. The anode and the cathode were placed in parallel with an inter-electrode gap of 2.5 cm. The geometric surface area of electrodes immersed in the solution was 23.6 cm 2 .

A probe molecule was used in order to assess the reactivity of electrodes for  OH formation. TA (0.1 mM) was selected as the optimal compound, as suggested by Chaplin et al. [START_REF] Jing | Mechanistic Study of the Validity of Using Hydroxyl Radical Probes To Characterize Electrochemical Advanced Oxidation Processes[END_REF] and other authors [START_REF] Xie | Flow anodic oxidation: Towards highefficiency removal of aqueous contaminants by adsorbed hydroxyl radicals at 1.5 V vs SHE[END_REF] in order to avoid false detection of  OH in electrolytic systems.

Another set of experiments was performed using PCT (0.1 mM) as a model compound.

Quenching experiments were performed in order to assess the influence of different reaction mechanisms for the degradation of PCT [START_REF] Cai | Extremely efficient electrochemical degradation of organic pollutants with co-generation of hydroxyl and sulfate radicals on Blue-TiO2 nanotubes anode[END_REF][START_REF] Zhou | Investigations of diatrizoate degradation by photo-activated persulfate[END_REF][START_REF] Anipsitakis | Radical Generation by the Interaction of Transition Metals with Common Oxidants[END_REF]. Ethanol was added for quenching the influence of both  OH and SO4 -. The molar ratio between ethanol and РСТ was 1000:1. Tert-butyl alcohol (TBA) was also used for quenching only the influence of  OH in order to potentially highlight the role of sulfate radicals. For all the experiments, average values and standard deviations related to the evolution of the concentration of organic compounds were obtained from triplicate experiments.

Linear sweep voltammetry (LSV)

A similar electrochemical cell was used for linear sweep voltammetry (LSV). LSV was performed in 50 mM Na2SO4 at 10 mV s -1 using stainless steel as counter electrode and an Ag/AgCl (3 M KCl) electrode as reference electrode. LSV curves were recorded from OCP (open circuit potential) to 3.8 V. LSV data were corrected according to the uncompensated resistance of the system.

Mass transport

The average mass transport coefficient (km) was obtained by using the limiting current density method [START_REF] Guo | Development and characterization of ultrafiltration TiO2 Magnéli phase reactive electrochemical membranes[END_REF][START_REF] Cañizares | Measurement of Mass-Transfer Coefficients by an Electrochemical Technique[END_REF]. The limiting current (jlim) for oxidation of 5 mM Fe(CN)6 4-was measured through a CV scan in a similar electrochemical setup (using 100 mM NaClO4 as supporting electrolyte, stainless steel as counter electrode and an Ag/AgCl (3 M KCl) as reference electrode.

Excessive (20 mM) Fe(CN)6 3-was added in order to avoid any limitation from reactions at the cathode. Mass transfer coefficient was calculated according to Eq. 1.

𝑘 =

(Eq. 1)

Where z is the number of electrons involved in anodic oxidation (z =1), F is Faraday's constant, A is applied geometric surface area (m 2 ) and Cb is the concentration of Fe(CN)6 4-in bulk solution (mol m -3 ).

Quantification of organic compounds

Concentration of PCT was monitored by a Thermo Scientific UltiMate 3000 UHPLC system. An Hypersil Gold column (100×2.1 mm, 1.9 µm) was set in a thermostated compartment at 40 °C.

The system was also equipped with an RS pump and a diode array detector (at 243 nm). The mobile phase was a mixture of methanol (15%) and phosphate buffer (5 mM Na2HPO4 and 7.5 mM H3PO4) (85%) with a flow rate of 0.1 mL/min.

The concentration of TA was measured by a Hitachi Elite LaChrom HPLC system equipped with a Purospher star RP-18e (5µm) column (set in a column oven at 40 °C), a UV detector L-2400 set at 254 nm and a pump L-2130. The mobile phase was a mixture of methanol (40%) and water containing 0.1% formic acid (60%) at a flow rate of 0.8 mL/min.

A TOC analyzer (Shimadzu TOC-L) was used for quantification of the mineralization yield of the treated solutions.

Calculations

Problem formulation and mathematical modeling is detailed in Supplementary Information (SI) file (See please Text SI-1). In short, the analytical expression for the spatial distribution of  OH concentration was calculated according to Kapalka et al. (2009) [START_REF] Kapałka | The importance of electrode material in environmental electrochemistry: Formation and reactivity of free hydroxyl radicals on borondoped diamond electrodes[END_REF], thus allowing to highlight the reaction zone at the electrode surface. Mass transport of organic compounds was considered to be limited by the diffusion boundary layer. Both  OH-mediated oxidation and DET were taken into consideration for degradation of organic compounds.

The mathematical problem was solved numerically by finite element method using the Comsol Multiphysics 5.6 commercial software package. Reliability of the calibration method was assessed through the calculation of three statistical criterions used for process-oriented models, including root mean square error (RMSE), modelling efficiency (ME) and index of agreement (IoA) [START_REF] Janssen | Calibration of process-oriented models[END_REF]. These values are given in

Results and Discussion

Experimental determination of mass transport conditions

Mass transport conditions of dissolved compounds from the bulk to the electrode surface was characterized in the mixed-tank reactor by using the limiting current density technique in the ferro-ferricyanide system [START_REF] Cañizares | Measurement of Mass-Transfer Coefficients by an Electrochemical Technique[END_REF]. A current plateau was clearly observed using BDD electrode (Fig. SI-6), thus allowing for determination of the mass transport coefficient from Eq. 1. This coefficient was found to be km = 2.85 (± 0.31) x 10 -5 m s -1 . By taking into account a diffusion coefficient of D = 7.26 x 10 -10 m 2 s -1 , the thickness of the diffusion boundary layer (δ) was determined as 25.5 (± 2.3) µm using Eq. 2. These values are typical for plate electrodes implemented in mixed-tank reactors with high stirring rate [START_REF] Trellu | Electro-oxidation of organic pollutants by reactive electrochemical membranes[END_REF].

𝛿 = (Eq. 2)

From this value of km, it is possible to estimate the limiting current density according to Eq. 3, assuming a fast and non-selective reaction of organic compounds in the reaction zone [START_REF] Panizza | Anodic oxidation of 2naphthol at boron-doped diamond electrodes[END_REF]. For the treatment of 0.1 M of TA or PCT, the limiting current density was calculated as 0.82 and 0.93 mA cm -2 , respectively. Thus, experiments performed in this study at j = 5 mA cm -2 were performed under strong mass transport limitation (j > jlim).

𝑗 = 𝑛 𝐹 𝑘 𝐶 (Eq. 3)
where jlim is the limiting current density (A m -2 ), n is the number of electrons involved in mineralization of organic compounds (e.g., 30 for TA and 34 for PCT), F is the Faraday constant (96 485 C mol -1 ) and C is the concentration of organic compounds in the solution (mol m -3 ).

Oxygen evolution reaction

LSV (Fig. 1) allowed for characterization of OER. High potential for OER was observed for both materials by using the tangent method, i.e., 2.21 and 2.75 V vs Ag/AgCl for BDD and TiOx, respectively. At these potentials, it is thermodynamically possible to form  OH from water oxidation since the thermodynamic potential of  OH is 2.13 V vs Ag/AgCl [START_REF] Kapałka | The importance of electrode material in environmental electrochemistry: Formation and reactivity of free hydroxyl radicals on borondoped diamond electrodes[END_REF]. As already described in the literature, the complex mechanisms of OER at the surface of non-active anodes can be simplified in several successive reactions including first (i) water discharge and formation of  OH (Eq. 4) and (ii) reaction of  OH with each other for formation of H2O2 (waste reaction for  OH) (Eq. 5) [START_REF] Kapałka | The importance of electrode material in environmental electrochemistry: Formation and reactivity of free hydroxyl radicals on borondoped diamond electrodes[END_REF][START_REF] Michaud | Electrochemical oxidation of water on synthetic boron-doped diamond thin film anodes[END_REF].

H2O   OH + H + + e - (Eq. 4)

2  OH  H2O2 (Eq. 5)
Weak adsorption of  OH at electrode surface (physisorption) is usually considered for this kind of electrode, therefore,  OH were considered as quasi-free. Reaction of  OH with the electrolyte and reaction with the electrode material (corrosion) were neglected. Thus, parameters related to Eq. 4 were calibrated from LSV data (Fig. 1). The Butler-Volmer equation used to describe this reaction is given in SI (Eq. 10, SI) . The value of formal potential was obtained by the tangent method. Standard methodology was used for evaluation of kinetic parameters by plotting data in Tafel coordinate as described in the literature [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF]. Values are reported in the recapitulative table in SI (Table SI-2). The fitting was low for VA < 2.2 V vs Ag/AgCl (i.e. I < 50 mA) in the case of the BDD anode. As reported in the literature, it might be ascribed to the presence of surface redox couples or functional groups, which act as a barrier for OER on BDD anode [START_REF] Kapałka | Determination of the Tafel slope for oxygen evolution on boron-doped diamond electrodes[END_REF].

However, the anodic oxidation process was then applied in the range of values of VA and I for which the fitting was optimal. It can be considered that further oxidation of H2O2 lead to the formation of O2 (Eqs. 6 -7) [START_REF] Kapałka | The importance of electrode material in environmental electrochemistry: Formation and reactivity of free hydroxyl radicals on borondoped diamond electrodes[END_REF][START_REF] Michaud | Electrochemical oxidation of water on synthetic boron-doped diamond thin film anodes[END_REF]. Thus, in the next sections of the manuscript, the fraction of current spent for O2 evolution was calculated from the amount of  OH reacting between each other (dimerization) (Eq. 5).

Formation of O2 can also arise from oxidation of  OH to atomic oxygen followed by formation of O2 or O3 (Eqs. 8 -10). For sake of simplicity, these reactions were not taken into consideration in the model. The generation of  OH was therefore considered as ideal. Thus, in the following sections, the calculated amount of  OH available for reaction with target compounds must be considered as an upper limit.

H2O2  O2 + 2 H + + 2 e - (Eq. 6)

H2O2 + 2  OH  O2 + 2 H2O (Eq. 7)

 OH  O  + H + + e - (Eq. 8)
O  + O2  O3 (Eq. 9)

2 O   O2 (Eq. 10)

Hydroxyl radical mediated oxidation of a probe molecule (TA)

In a solution containing organic compounds,  OH-mediated oxidation of organic compounds represents an important pathway for disappearance of  OH. In fact, reaction rates between  OH and organic compounds are usually in the same range (k = 10 5 -10 7 m 3 mol -1 s -1 ) [START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF] than reaction rate of  OH between each other (Eq. 5) (k = 5.5 x 10 6 m 3 mol -1 s -1 ) [START_REF] Buxton | Critical-Review of Rate Constants for Reactions of Hydrated Electrons, Hydrogen-Atoms and Hydroxyl Radicals (.oh/.o-) in Aqueous-Solution[END_REF]. There is a possible competition between the oxidation of organic compounds and this waste reaction (Eq.

5).

The capacity of BDD and Ti/TiOx electrode for  OH-mediated oxidation of organic compounds was assessed using TA as probe molecule since it is strongly refractory to DET. It was observed that the degradation kinetic of TA followed a pseudo-first order degradation kinetic (R 2 = 0.99) with a rate constant of 0.009 min -1 and 0.017 min -1 for Ti/TiOx and BDD anode, respectively.

For modeling, the reaction rate of TA with  OH was taken from literature (kTA = 3.3 x 10 6 m 3 mol -1 s -1 ) [START_REF] Fang | OH radical formation by ultrasound in aqueous solutions Part I: the chemistry underlying the terephthalate dosimeter[END_REF]. Diffusion coefficient of TA was taken as equal to the one of PCT (the only one available in the literature). The formation rate of  OH was calculated as an upper limit as described in sub-section 4.2.  OH-mediated oxidation was the only reaction pathway taken into consideration. Reaction of both TA and degradation by-products was taken into consideration.

However, as described in sub-section 4.6 (sensitivity analysis), reaction of  OH with by-products do not affect degradation kinetics of the initial compound during these electro-oxidation experiments performed under conditions of mass transport limitation. Thus, the thickness of the diffusion boundary layer (δ) was the only key parameter calibrated for fitting the experimental data of TA degradation kinetics with BDD and TiOx electrodes (Fig. 2). The obtained values of δ were 19 and 33 µm for BDD and TiOx, respectively. It corresponds to mass transport coefficients of TA (km) of 3.42 x 10 -5 and 1.97 x 10 -5 m s -1 for BDD and TiOx, respectively.

The calibrated value of δ was 25% lower for electro-oxidation of TA at BDD anode, compared to δ obtained in the ferro-ferricyanide system (i.e higher km during electro-oxidation of TA). As the generation of  OH available for reaction with organic compounds was calculated as an upper limit (sub-section 4.2), the calibration of more favorable mass transport conditions (lower δ)

could not come from a compensation of an underestimation of the calculated amount of  OH.

Therefore, lower value of δ could be ascribed to different operating conditions during electrooxidation of TA, which modify the local hydrodynamic conditions at the anode surface. TA electro-oxidation occured in the potential region of water oxidation, while the ferro-ferricyanide system was operated at lower anodic potential. Therefore, formation of O2 gas bubbles might strongly modify local hydrodynamic conditions at the electrode/solution interface. These results highlight that these local turbulences might enhance mass transport conditions in this configuration. Such local turbulences might also be influenced by surface roughness of the electrode.

The calibrated value of δ with TiOx anode was 42% higher than the value obtained with BDD anode (i.e lower km during electro-oxidation of TA at TiOx anode). It was even 29% higher than the value obtained in the ferro-ferricyanide system with BDD anode. The calculated profile of concentration of  OH at electrode surface is presented in Fig. 3B and SI-5B. It followed a similar trend than previously reported by Kapalka et al. ( 2009) [START_REF] Kapałka | The importance of electrode material in environmental electrochemistry: Formation and reactivity of free hydroxyl radicals on borondoped diamond electrodes[END_REF]. The calculated reaction layer thickness was about 0.8 µm with a maximum surface concentration of  OH around 20 µM. This concentration profile is usually affected by the concentration of organic compounds that can react with  OH. However, under the conditions of mass transport limitation of this study (sub-section 3.1), the concentration profile was not significantly affected as it can be seen from the absence of significant evolution during the treatment (i.e., decrease of concentration of organic compounds).

The evolution of the concentration of TA as a function of electrode distance can be divided into 3 main sections, including (i) the bulk solution where the concentration was considered as homogeneous (x>19 µm for BDD), (ii) the diffusion boundary layer where the concentration followed a linear decrease according to diffusion conditions (0.8<x<19 µm for BDD) and (iii) the reaction zone (x<0.8 µm) where organic compounds react with  OH (Fig. 3A for BDD and Calculations allowed the estimation of the fraction of current spent on TA degradation from  OH-mediated oxidation (Fig. 4). It started from 0.67% and 0.39% for BDD and TiOx, respectively. Then, these values continuously decreased during the treatment because of the decrease of the concentration of TA. The lower value obtained from TiOx compared to BDD was ascribed from the model to the lower calibrated valued of km that reduces the amount of TA in the reaction zone where  OH are accumulated (comparison of Fig. 3B and SI-5B). However, as discussed above, it might also be ascribed to a lower concentration of  OH in the reaction zone since the formation of  OH is considered as ideal in the model.

The mineralization rate of by products (BP) is a complex phenomenon depending on the generation of various intermediates. For sake of simplicity, it was quantified by calibrating the global reaction rate of degradation by-products (kBP, Eq. 14, SI)) from experimental data of TOC removal. Non-selective mineralization of by-products was considered [START_REF] Panizza | Direct and mediated anodic oxidation of organic pollutants[END_REF]. The diffusion coefficient of by-products was taken as equal to the one of TA. The total amount of  OH consumed per molecule of by-product mineralized was calculated according to the equation given in SI (Eq. 5, SI). A higher value of current was spent on mineralization of by-products as the full mineralization of TA involves 30 electrons (instead of only 1 for initial degradation step). However, the main part of current was spent on waste reactions (O2 evolution) (Fig. 4). In fact, electro-oxidation was operated under mass transport limitation, meaning that the amount of  OH was in large excess compared to the amount of organic compounds. The lower the concentration of organic compounds in the solution, the higher the level of waste reactions was.

The mineralization current efficiency (i.e., fraction of current used in reaction with organic compounds) was initially at 11% and 1.8% and decreased to 3.8% and 1.4% after 2 hours of treatment in the case of BDD and TiOx, respectively. Table 1 provides a comparison of the efficiency (in terms of degradation kinetic and mineralization rate) achieved in this study with results previously reported in the literature. It appears that these results were in the same range than data reported in the literature for configurations operated under mass transport limitation.

However, it is difficult to compare rigorously such data since several operating parameters influence the effectiveness of the process (pollutant concentration, current density, ratio between electrode surface (S) and reactor volume (V)). For example, the influence of the S/V ratio is further detailed in section 4.9 using a sensitivity analysis of the model. It is interesting to notice that the difference of effectiveness of BDD and TiOx was higher for mineralization of TA than for  OH-mediated degradation of TA. For example, TA degradation yield after 1 h of treatment was 60 and 37% for BDD and TiOx, respectively, while TOC removal after 2 h of treatment was (65 ± 3)% and (21 ± 5)% for BDD and TiOx, respectively.

These results might be explained by (i) a better electrokinetic behavior of BDD for oxidation by DET of by-products reacting slowly with  OH (e.g. carboxylic acids) and/or (ii) a less favourable electrokinetic behavior of TiOx for accumulation of  OH at its surface (e.g. further evolution of  OH through Eqs. 8-10)

during TA oxidation at t = 10 (blue), 100 (red), 1000 (pink), 3600 (yellow), 7200 (green) s. not considering SO4 --mediated oxidation for this specific case. Only DET and  OH-mediated oxidation were considered for modeling. Diffusion coefficient of PCT was obtained from the literature, while the one of ethanol was considered as equal to the one of PCT. Reaction rate constants for mineralization of ethanol by  OH and degradation of PCT with  OH were obtained from the literature. The thickness of the diffusion layer was taken from values calibrated in section 4.3. Conditions for  OH generation were kept as descrived in section 4.2. Thus, all conditions for  OH-based reactions were considered as established. Mineralization of PCT byproducts was calibrated in the following experiments without quenchers. However, as mentioned in section 4.3 and detailed in section 4.6, mineralization of PCT by-products was not a parameter affecting degradation kinetic of the initial compound. Thus, experimental data obtained for PCT oxidation with ethanol as quencher (Fig. 5A) were used for calibration of the formal potential related to DET of PCT (Eq. 16, SI) at the surface of BDD and TiOx electrodes.

Faster removal of PCT was obtained with BDD anode compared to TiOx. It was necessary to consider a more favorable reactivity of PCT for DET at BDD anode (E °PCT = 2.38 and 3.30 V vs Ag/AgCl for BDD and TiOx, respectively) in order to fit with the experimental data. Such better electrokinetic behavior of BDD for DET indicate that it might also be an important parameter to take into consideration for explaining the faster mineralization rates obtained with BDD compared to TiOx. Calculations showed that using a concentration of 100 mM of ethanol was suitable for quenching  OH. In such conditions with a high concentration of ethanol, the system was operated under current limitation (j<jlim). Ethanol provided a huge pathway for the disappearance of  OH and the reaction layer thickness decreased to 5 nm (160 times less than without ethanol) with a maximum surface concentration of  OH around 0.2 µM (100 times lower than without ethanol) (Fig. SI-9). This presence of ethanol strongly decreased the possibility of PCT to react with  OH.

At time t = 0 with the BDD anode, the calculated fraction of current spent for PCT degradation by  OH-mediated oxidation was only 0.02% while it was 0.33% for PCT degradation by DET (Fig. 5B). After 2 hours of treatment, it was calculated that only 6.1% of the degradation yield of PCT was ascribed to  OH (Fig. . A similar trend was obtained with TiOx anode (Fig. . The fraction of PCT removed by  OH (11.0%) was slightly higher as it was calibrated that the reaction of PCT through DET is less favorable at the TiOx anode.

Further calculations were performed in order to assess the relevance of using different concentration of ethanol in the case of the BDD electrode. Using a concentration of 10 mM would increase the fraction of current spent on  OH-mediated oxidation of PCT from 0.02 to 0.17% (at time t = 0). On the contrary, the fraction of current spent on PCT oxidation by DET would decrease from 0.33 to 0.25% (i.e., in the same ordre of magnitude than for  OH-mediated oxidation). Thus, decreasing the concentration of ethanol to 10 mM would not allow for reaching sufficient quenching effect, even if it corresponds to a high ratio between ethanol and the target pollutant (10/0.1 = 100).

Oxidation of PCT without quencher

A similar experiment was then performed without the addition of any  OH scavenger. Only DET and  OH-mediated oxidation were considered for modeling. Diffusion coefficient and reaction rate constants of PCT with  OH was obtained from the literature, while the one of ethanol was considered as equal to the one of PCT. The thickness of the diffusion layer was taken from values calibrated in section 4.3. Conditions for  OH generation were kept as descrived in section 4.2. Conditions for oxidation of PCT by DET were taken from the values calibrated in section 4.4. Thus, all model parameters were taken from previous experiments (or from literature) and good agreement was observed between experimental and model values obtained for the degradation kinetic of PCT (Fig. 6A). Besides, the global reaction rate of degradation byproducts was calibrated from experimental data of TOC removal. As mentioned in previous sections and detailed in section 4.6, mineralization of by-products was not a phenomenon affecting degradation kinetic of the initial compound. The diffusion coefficient of by-products was taken as equal to PCT. The total amount of  OH consumed per molecule of by-product mineralized was calculated according to the equation given in SI (Eq. 8, SI).

As observed previously, PCT can react through both DET and  OH-mediated oxidation.

However, calculations highlighted that PCT was almost exclusively degraded through  OHmediated oxidation in these conditions (i.e., absence of quenchers, mass transport limitation).

With BDD anode, at time t = 0, 0.68% of the current was spent on degradation of PCT by  OH while it was only 0.0022% for DET (Fig. 6B). After 2 h of treatment, it was calculated that more than 99.99% of PCT was removed by  OH-mediated oxidation (Fig. . Similar trend was obtained with TiOx anode (Fig. . In fact, PCT is able to react rapidly in the reaction zone where  OH are accumulated and a very low amount of PCT reaches the electrode surface (x = 0) for reacting through DET. This phenomenon was clearly emphasized in Fig. 7 for BDD and Fig. SI-13 for TiOx. PCT molecules are able to reach the electrode surface for reaction through DET only when ethanol was added as quencher of  OH, while concentration of PCT was 0 at the electrode surface in absence of  OH quencher. All PCT molecules are able to react very fast with  OH in the reaction zone when the process is operated under mass transport limitation.

These results emphasized that different degradation mechanisms and pathways might happen depending on operating conditions (presence of radical scavengers, mass transport vs current limitation).

As observed with TA, a more important difference was observed between TiOx and BDD for mineralization rate, compared to degradation rate. For example, the mineralization yield of PCT after 2 h of treatment was 14 and 66% for TiOx and BDD, respectively, while the degradation yield after 1 h was 44 and 67% for TiOx and BDD, respectively. As explained above, it might be related to the better electrokinetic behavior of BDD for mineralization of by-products more refractory to  OH and/or by the evolution at the TiOx electrode of a greater amount of  OH through waste reactions of Eqs. 8-10. The next objective was to study the influence of target compound reactivity with  OH. The first case studied was that of a molecule strongly refractory to DET such as TA. All model parameters were taken from the sub-section 4.3 on TA degradation and the influence of the value of the reaction rate constant with  OH (kTA) was tested (Fig. 9A).

There was not any significant effect of kTA in the range for values above 10 6 m 3 mol -1 s -1 . In this range of values,  OH generated in the reaction zone were sufficient to oxidize all target compounds under mass transport control (i.e., the concentration of target compound at electrode surface, x = 0, was always 0). Therefore, the effectiveness of the process depends only on mass transport conditions. The degradation kinetic starts decreasing only from kTA = 10 5 m 3 mol -1 s -1 and further strongly drops at 10 4 and 10 3 m 3 mol -1 s -1 . In this range of values, the degradation of the target compound is kinetically limited by the reaction with  OH. The amount of  OH in the reaction zone is not sufficient to oxidize all target compounds because the reaction kinetic is too slow. Increasing the current density (and therefore the amount of generated  OH) would counteract such phenomenon for increasing degradation kinetic. Thus, even under mass transport limitation according to Eq. 3, the current intensity would also play a role on the removal of the most recalcitrant compounds.

The second case studied was that of a molecule that can potentially react through both DET and

 OH mediated oxidation such as PCT. All model parameters were taken from the sub-sections 4.4 and 4.5 on PCT degradation. The influence of the value of the reaction rate constant with  OH (kPCT) was tested (Fig. 9B). As in the first case and for the same reason, there was not any significant effect of kPCT in the range 10 6 -10 8 m 3 mol -1 s -1 . Then, the degradation kinetics of the target compound decreased for lower values but a much lower decrease was observed compared to the first case. In fact, target compounds that do not fully react in the reaction zone with  OH can reach the electrode surface and react by DET. In such case, DET would participate to counteract the slow reaction with  OH. These calculations show that electrokinetic parameters related to DET might play a key role on the removal of organic compounds that are the most recalcitrant to reaction with  OH.

For example, oxalic acid is a well-known end by-products of degradation of organics by AOPs/EAOPs. It is difficult to remove because of its slow reaction with  OH (kOA = 1.4 x 10 3 m 3 mol -1 s -1 ). With such a reaction rate constant, its reactivity through DET might make important differences. It is the reason why DET is also an important phenomenon to take into consideration for comparing the effectiveness of non-active electrodes for achieving complete mineralization of organic pollutants.

values of reaction rate constant of TA with  OH, kTA = 3.3 x 10 3 (blue), 3.3 x 10 4 (pink), 3.

Sensitivity analysis: influence of the ratio S/V

As observed in Table 1, it is difficult to compare data obtained from the literature since they have been obtained in different operating conditions. When the anodic oxidation process is applied under mass transport limitation, one crucial parameter is for example the ratio between the geometric surface area of the electrode and the volume of the treated solution (S/V). Results presented in Fig. 11A presents the influence of this ratio on the degradation kinetic of PCT using BDD anode, as calculated from the model. The calculated value of kapp is strongly improved by simply increasing this ratio. Fig. 11B shows the linear correlation obtained between kapp and the

Conclusions

Combination of both experimental and modeling approaches allowed for better understanding the reactivity of non-active electrodes such as BDD and TiOx. A mathematical model taking into account both  OH-mediated oxidation and DET was developed and calibrated using experimental data. The fraction of current spent in the different possible reactions was calculated and allowed for discussion of the predominant reaction mechanism accoding to the experimental configuration.

Results showed that organic compounds are fully oxidized by  OH when (i) the process is operated under mass transport control and (ii) the reaction rate constant with  OH is sufficiently high. Moreover, none significant competition between the mother molecule and degradation byproducts was observed under mass transport limitation. The role of DET become potentially crucial for the removal of organic compounds with low reaction rates with  OH (below k < 10 5 m 3 mol -1 s -1 ) and/or in presence of large amounts of  OH scavengers. Thus, electrokinetic parameters related to DET might also play a key role for achieving complete mineralization of organic compounds and for application of the process under current limitation.

The use of radical quenchers is currently widely used in the litterature for identification of reactive species during oxidation processes. The results obtained in this study showed that a concentration of 100 mM of ethanol was required to ensure that  OH-mediated oxidation was not significant at 5 mA cm -2 and for a concentration of target compound of 0.1 mM.

Finally, degradation kinetics obtained with TiOx electrodes were close to that obtained with BDD, even for a  OH probe molecule such as TA. However, the mineralization power of TiOx was lower; further investigations are required to clearly identify the underlying reasons.

Particularly, further investigations are required to better understand the different mechanisms related to OER and formation of  OH at the surface of these anodes.

Figure 1 -

 1 Figure 1 -Linear scanning voltammetry (10 mV s -1 ) for characterization of oxygen evolution

  It might be explained by less favorable local mass transport conditions, related to physical properties of the electrode surface (e.g., hydrophobicity, local stagnant micro-volumes associated to different surface morphology / roughness). The calculated value of δ is a global parameter that averages heterogeneous mass transport conditions in the vicinity of the electrode surface. Another explanation might come from an overestimation of the calculated amount of  OH available for reaction with organic compounds. The higher calibrated value of δ (i.e. less favorable mass transport conditions) might be the result of the compensation of this overestimation. The TiOx surface might further promote direct evolution of  OH to O2 through Eqs. 8-10. Most probably, both phenomena (different mass transport conditions and different reactivity) might influence the lower TA degradation kinetic with TiOx electrode. Suitable comparison of the reactivity of both electrodes would require to compare two electrodes with very similar physical properties (morphology / roughness, hydrophobicity).

Figure 2 -

 2 Figure 2 -Experimental (dots) vs model (lines) data for terephthalic acid degradation on BDD

Fig. SI -

 SI Fig. SI-7A for TiOx). Calculations showed that TA concentration was 0 at the surface of both

Figure 4 -

 4 Figure 4 -Fractions of current spent on O2 evolution (blue line), terephthalic acid (TA)

Figure 5 -

 5 Figure 5 -(a) Experimental (dots) vs model (lines) data for paracetamol (PCT) degradation on

Figure 6 -

 6 Figure 6 -(a) Experimental (dots) vs model (lines) data for paracetamol (PCT) degradation on

Figure 7 - 4 . 7

 747 Figure 7 -Concentration profiles of PCT during the PCT oxidation on BDD electrode in the

3 x 10 5 (

 5 green), 3.3 x 10 6 (yellow), 3.3 x 10 7 (red) m 3 mol -1 s -1 . (b) Degradation kinetics of paracetamol (PCT) on BDD electrode at different values of reaction rate constant of PCT with OH, kPCT = 10 3 (blue), 10 4 (red), 10 5 (pink), 10 6 (yellow), 10 7 (green) m 3 mol -1 s -1 .4.8 Sensitivity analysis: influence of mass transport coefficientUnder mass transport limitation, mass transport conditions obviously influence strongly the degradation kinetic of the target compound. As observed in Fig.10, the change of the mass transport coefficient by one order of magnitude has a huge effect on degradation kinetic. It is the reason why a great attention is currently given to the development of porous electrodes that can be implemented under flow-through configuration. Using electrodes with pore size lower than the diffusion boundary layers allows for reducing adverse effects from diffusion phenomena. Thus, mass transport can be effectively improved by increasing the convective flux through the porous electrode.

Figure 10 -

 10 Figure 10 -Degradation kinetics of PCT on BDD electrode at different values of mass transport

  V. The value of kapp (in min -1 ) was directly proportional to the ratio S/V (in m -1 ). This value of the S/V ratio should be further taken into consideration for the comparison of results obtained in different studies.

Figure 11 -

 11 Figure 11 -Influence of the ratio between electrode surface (S) and solution volume (V) on the

  Table SI-1. The values of ME and IoA were always higher than 0.97 and 0.993, respectively. A list of all model parameters is provided in Table SI-2. Parameters were obtained either from the literature or from experimental data, as indicated in Table SI-2. The methodology for calibration of unknown parameters is explained in the results and discussion section.

The calculations of fractions of current spent on the different possible reactions are detailed in Text SI-2 and SI-3.

Table 1 -

 1 Comparison of the effectiveness of the anodic oxidation process in undivided mixedtank reactor. The mineralization current efficiency corresponds to the indicated TOC removal.

	Anode	Pollutant [C] -mM	Current mA cm -2	Selectrode/Vreactor m -1	TOC removal %	MCE 2 %	kapp 3 min -1	Ref
	Ti/TiOx	Amoxicillin 0.1	5	10.4	40	5	0.03	[26]

1 p-aminosalcilyc acid ; 2 mineralization current efficiency ; 3 pseudo-first order kinetic rate constant

Acknowledgments

Jing Ma thanks China Scholarship Council for supporting her PhD work in Université Gustave Eiffel. Authors acknowledge Saint-Gobain Research Provence for providing TiOx electrodes. This research was partially funded by the Ministry of Education and Science of the Russian Federation (project No. FZEN-2020-0022).

As soon as the electro-oxidation starts, the solution contains the initial target molecule as well as the degradation by-products. Therefore, there is a potential competition between these different compounds for reaction with  OH. The objective of this section was to study whether this competition might have a significant effect on degradation kinetic of the target compound.

For this purpose, the influence of two characteristics of by-products was tested, including (i) the reaction rate constant with  OH (kBP) and (ii) the diffusion coefficient of by-products, which might affect mass transport conditions. Interestingly, calculations showed that the variation of these values did not affect significantly PCT degradation kinetics (Fig. 8A and8B). It was previously observed that a higher amount of current is actually spent on mineralization of byproducts compared to initial degradation of PCT. However, under mass transport limitation, it still represents a low fraction of the total current provided (e.g. less than 4.2% after 2 h of treatment in the conditions of Fig. 6). Thus, competition effects are not significant since the reaction rate of  OH with by-products does not significantly modify the amount of  OH available for reaction with PCT.

In this study, mineralization of by-products was considered to occur only through  OH-mediated oxidation. Actually, DET could also participate. Therefore, the amount of hydroxyl radicals used for mineralization is overestimated with this model. Despite this overestimation, this study highlighted that the consumption of  OH by reaction with by-products does not affect the degradation kinetic of TA at BDD anode when operating AO under mass transport limitation.