domain: chim sde

text:
Nicolas Mazzella
email: nicolas.mazzella@inrae.fr

Marion Bernard

Robin Guibal

Sebastien Boutry

Sophie Lissalde

Gilles Guibaud

Proposal of a new empirical model with flow velocity to improve timeweighted average concentration estimates from the Polar Organic Chemical Integrative Samplers

Keywords: Passive sampling, pesticides, sampling rates, calibration, kinetic model, accuracy

It is now widely recognized that the sampling rate of Polar Organic Chemical Integrative Samplers (POCIS) is significantly affected by flow velocity, which can cause a consequent bias when determining time-weighted average concentrations (TWAC). We already observed the desorption of deisopropylatrazine (DIA) over time when added to the receiving phase of a POCIS. This desorption rate was particularly influenced by flow velocity, in an agitated water environment in situ. In the method presented here, we calibrated 30 pesticides under controlled laboratory conditions, varying the flow velocity over four levels. We simultaneously studied the desorption rate of DIA-d5 (a deuterated form of DIA) over time. An empirical model based on a power law involving flow velocity was used to process the information from the accumulation kinetics of the compounds of interest and elimination of DIA-d5. This type of model makes it possible to consider the effect of this crucial factor on exchange kinetics, and then to obtain more accurate TWACs with reduced bias and more acceptable dispersion of results.

Introduction

Sampling rate (Rs) is a critical parameter when using passive samplers to estimate a time-weighted average concentration (TWAC). Rs can be represented by the volume of water purified by a passive sampler per unit time for a given compound [START_REF] Alvarez | Development of a passive, in situ, integrative sampler for hydrophilic organic contaminants in aquatic environments[END_REF]). This parameter must be determined in a prior calibration step. However, transferring such sampling rates to the field can lead to uncertainties about the results because experimental conditions in the laboratory generally differ from those in the field. Indeed, Rs depends on a number of environmental factors such as turbulence, temperature, salinity, organic and suspended matter content, and biofouling [START_REF] Söderström | Strategies for monitoring the emerging polar organic contaminants in water with emphasis on integrative passive sampling[END_REF][START_REF] Vrana | Passive sampling techniques for monitoring pollutants in water[END_REF]. Among these factors, flow velocity is known to play a key role in water boundary layer thickness, which can cause the associated sampling rate to vary quite significantly, resulting in large biases up to factors of 3-4 [START_REF] Mazzella | Evaluation of the Use of Performance Reference Compounds in an Oasis-HLB Adsorbent Based Passive Sampler for Improving Water Concentration Estimates of Polar Herbicides in Freshwater[END_REF]. Recently, diffusive passive samplers have been developed to overcome the effects of flow velocity on the uptake of chemicals, including the technologies organic Diffusive Gradients in Thin-films (o-DGT) and Microporous Polyethylene Tube (MPT) (Fauvelle et al. 2017b, Guibal et al. 2019). In these cases, the strategy consists in increasing the membrane resistance to limit the consideration of the water boundary layer (WBL) thickness relative to the flow velocity variations. This approach is however limited at very low flow velocities (≈1 cm s -1 ) (Fauvelle et al. 2017a) or, for o-DGT, requires different diffusion gel thicknesses to explicitly evaluate the WBL contribution [START_REF] Bonnaud | Calibration comparison between two passive samplers -o-DGT and POCIS-for 109 hydrophilic emerging and priority organic compounds[END_REF][START_REF] Bonnaud | Determination of diffusion coefficients in agarose and polyacrylamide gels for 112 organic chemicals for passive sampling by organic Diffusive Gradients in Thin films (o-DGT)[END_REF][START_REF] Challis | Development and Calibration of an Organic-Diffusive Gradients in Thin Films Aquatic Passive Sampler for a Diverse Suite of Polar Organic Contaminants[END_REF]. With a Polar Organic Chemical Integrative Sampler (POCIS) configuration, another solution for correcting such flow velocity induced biases is Performance Reference Compounds (PRCs). The rate of desorption (ke) of a PRC relative to the flow velocity must be correlated with the simultaneous uptake of the pollutant(s). This is equivalent to assuming that the exchanges within the sampler are isotropic, regardless of the compound of interest. An Rs in situ can then be estimated for each molecule by extrapolating the ke of the PRC observed under given flow conditions. The correction factor, known as the environmental adjustment factor (EAF), can then be determined from the ratio of ke obtained in the laboratory (ke calibration) and in the field (ke in situ).

For POCIS, we previously showed that desisopropylatrazine-d5 (DIA-d5) had a satisfactory desorption potential and checked the isotropy for this single compound [START_REF] Mazzella | Determination of kinetic and equilibrium regimes in the operation of polar organic chemical integrative samplers: Application to the passive sampling of the polar herbicides in aquatic environments[END_REF]. The application of an EAF made it possible to obtain quite accurate field estimates of Rs in situ for some polar substances [START_REF] Lissalde | Liquid chromatography coupled with tandem mass spectrometry method for thirty-three pesticides in natural water and comparison of performance between classical solid phase extraction and passive sampling approaches[END_REF], Lissalde et al. 2014). However, this empirical approach included phenomena of adsorption of molecules in the membrane [START_REF] Vermeirssen | Transfer Kinetics of Polar Organic Compounds over Polyethersulfone Membranes in the Passive Samplers Pocis and Chemcatcher[END_REF], their diffusion in the micro-and macroporosity of the receiving phase and, finally, their binding to one of the two interaction sites (i.e., hydrophilic or hydrophobic) within the same adsorbent [START_REF] Bäuerlein | Sorption Behavior of Charged and Neutral Polar Organic Compounds on Solid Phase Extraction Materials: Which Functional Group Governs Sorption[END_REF]. Furthermore, highly polar compounds that can be desorbed and used as PRCs, and more hydrophobic molecules that are more strongly retained in the receiving phase, are likely to be influenced in different ways by all of these processes [START_REF] Harman | The Challenge of Exposure Correction for Polar Passive Samplers-The PRC and the POCIS[END_REF]. As a result, while the PRC approach with an EAF correction provided satisfactory results for POCIS with some triazine, phenylurea, and chloroacetanilide herbicides [START_REF] Dalton | Atrazine contamination at the watershed scale and environmental factors affecting sampling rates of the polar organic chemical integrative sampler (POCIS)[END_REF], Lissalde et al. 2014[START_REF] Mazzella | Evaluation of the Use of Performance Reference Compounds in an Oasis-HLB Adsorbent Based Passive Sampler for Improving Water Concentration Estimates of Polar Herbicides in Freshwater[END_REF], it has not yet been possible to generalize it to other substances because we do not know enough about the mechanisms at work. The approach also requires compliance with the "80/20 rule" [START_REF] Booij | An Improved Method for Estimating in Situ Sampling Rates of Nonpolar Passive Samplers[END_REF]. In other words, the PRC desorption rate after field exposure must be between 20 and 80% of the initial content of this compound, when the receiving phase is spiked. Thus, if the PRC used has very little desorbed (i.e., residuals after exposure >80%) or, on the contrary, has been completely eliminated (i.e., residuals after exposure <20%), then we cannot apply the EAF-based method. Nevertheless, these are only the most severe and problematic cases of bias, indicating that flow velocity was either extremely low or much greater than in the laboratory, during both calibration and Rs determination steps.

In this study, a POCIS calibration was performed with some selected moderately polar pesticides (i.e., log Kow between 2 and 4) according to increasing flow velocities. The relationship between sampling rates and simultaneous desorption rate of DIA-d5 was investigated using a power-law relationship with water flow velocity as parameter. Based on these findings, we propose a model that accounts for the effects of such a factor more explicitly.

We assessed the improvements offered by this new method by using it to reprocess previous data. It avoids, for instance, the main limitations of the earlier EAF approach (i.e., the unproven hypothesis of isotropic exchange for all analytes and the "80/20 rule" constraint). In addition, it appears possible to obtain a lower confidence interval associated with the variability of Rs as a function of the estimated flow rate, now ranging from 47 to 61%, compared with the over 200% previously obtained empirically.

Material and methods

Chemicals and reagents

The solvents (methanol, acetonitrile and ethyl acetate -HPLC grade) were obtained from Biosolve (Biosolve SARL, Dieuze, France). Ammonium acetate was purchased from Fluka (Sigma Aldricht, Schnelldorf, Germany). Ultrapure water (UPW, resistivity > 18 M) was produced by a Synergy UV system from Millipore (Billerica, MA, USA). Analytical standards, DIA-d5 and internal standards (atrazine-d5, carbofuran-d3, DEA-d6, diuron-d6, methomyl-d3, metolachlor-d6, pyrimicarb-d6 and tebuconazol-d6) of >95.5% purity were purchased from Dr. Ehrenstorfer GmbH (Augdburg, Germany). Individual stock solutions were prepared in acetonitrile (100 mg L -1 ) and stored at -18°C for no more than six months. A working solution containing each of the analytical standards was prepared by dilution of the individual stock solutions in acetonitrile (1 mg L -1 ). This was stored at -18°C for six months.

Sampler preparation and analysis

The passive sampler configuration used was a POCIS [START_REF] Alvarez | Development of a passive, in situ, integrative sampler for hydrophilic organic contaminants in aquatic environments[END_REF] containing 200 mg of Oasis® HLB sorbent (30-µm particle size, 810 m² g -1 , divinylbenzene-Nvinylpyrrolidone, Waters, France) sandwiched between two microporous polyethersulfone membranes of 90 mm diameter and 0.1 µm pore size (PALL®, VWR, France). Two support stainless discs were used to maintain the membrane-sorbent-membrane layers (inner diameter 5.4 cm, outer diameter 10.2 cm). Prior to assembling the POCIS device, the Oasis® HLB sorbent was spiked with DIA-d5 at approximately 4 µg g -1 [START_REF] Mazzella | Evaluation of the Use of Performance Reference Compounds in an Oasis-HLB Adsorbent Based Passive Sampler for Improving Water Concentration Estimates of Polar Herbicides in Freshwater[END_REF], with methanol as the spiking diluent [START_REF] Belles | Development of the performance reference compound approach for the calibration of "polar organic chemical integrative sampler" (POCIS)[END_REF]. Each spiked POCIS exposed during the calibration step was associated with a triplicate of "references" (3 mL empty polypropylene SPE tubes and polyethylene frits containing the same sorbent). These references were used to determine both the initial spike concentration (C(0)) and homogeneity. After exposure, the POCIS were kept at 4°C for 24 h, then treated and analyzed with the references by HPLC-ESI-MS/MS as described in previous studies [START_REF] Bernard | Lab-scale investigation of the ability of Polar Organic Chemical Integrative Sampler to catch short pesticide contamination peaks[END_REF][START_REF] Lissalde | Liquid chromatography coupled with tandem mass spectrometry method for thirty-three pesticides in natural water and comparison of performance between classical solid phase extraction and passive sampling approaches[END_REF]. Briefly, quantitations were done using internal calibration curves ranging from 1 to 100 µg L -1 . For quality assurance and quality control, procedural blanks and two calibration points (i.e., 5 and 25 µg L -1 ) were periodically injected every 10 samples. In addition, recoveries from spiked POCIS were checked and typically observed to be between 80 ± 20%, as reported in earlier studies [START_REF] Lissalde | Liquid chromatography coupled with tandem mass spectrometry method for thirty-three pesticides in natural water and comparison of performance between classical solid phase extraction and passive sampling approaches[END_REF][START_REF] Mazzella | Determination of kinetic and equilibrium regimes in the operation of polar organic chemical integrative samplers: Application to the passive sampling of the polar herbicides in aquatic environments[END_REF]).

Calibration of POCIS under different flow velocities

To calculate average concentrations in the aquatic environment from those measured in the POCIS HLB receiving phase, one must determine the accumulation kinetic constants associated with each target substance. To do this, POCIS were exposed in channels filled with tap water (500 L) initially spiked with a selection of contaminants at a concentration of 0.5 µg L -1 (see list of analytes in Table 1). Then, 14 POCIS were exposed to the following flow velocities: V1 = 2.7 ± 0.9 cm s -1 ; V2 = 7.2 ± 1.0 cm s -1 ; V3 = 19.9 ± 3.3 cm s -1 . To maintain contaminant concentration constant in the medium over the 21 days of calibration, a daily renewal rate of 15% of the water volume was applied using multi-channel peristaltic pumps (75 L d -1 ). To compensate for this renewal and maintain the pesticides at a nominal concentration around 0.2 µg L -1 (data are provided in an Excel file in the supp. mat.), a constant supply of contaminants was provided by a syringe pump containing the spiking solution (10 mg L -1 ), at a flow rate of 3 mL d -1 . A 100-L tank, subject to the same spiking and renewal conditions as the channels, was used to expose the POCIS to near-zero flow velocity: V0 = 0.3 ± 0.2 cm.s -1 (Figure S 1). Further details on the experimental setup can be found in Guibal et al. [START_REF] Guibal | Experimental Estimation of 44 Pharmaceutical Polar Organic Chemical Integrative Sampler Sampling Rates in an Artificial River under Various Flow Conditions[END_REF]). The POCIS were removed in duplicate at different times (1, 3, 5, 7, 10, 15, and 21 days), and replaced with "blanks" (i.e., non-analyzed POCIS) to maintain a consistent sampling pressure on the experimental system. Regular (t = 0, 2, 5, 7, 12, 14, 16, 19, and 21 days) water sampling was carried out to monitor channel concentration over the 21-day calibration period.

Analyses of both POCIS and spot water samples were performed by HPLC-MS/MS as described elsewhere [START_REF] Lissalde | Liquid chromatography coupled with tandem mass spectrometry method for thirty-three pesticides in natural water and comparison of performance between classical solid phase extraction and passive sampling approaches[END_REF]. POCIS extracts were typically diluted 1:3 prior to analyses.

For each flow velocity, the kinetic constants were determined by plotting the concertation factor (CFHLB, L g -1 ) against time (d), where CFHLB is determined with Equation 1:

Equation 1 𝐶𝐹 𝐻𝐿𝐵 = 𝐶 𝑃𝑂𝐶𝐼𝑆 𝐶 𝑤 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑
where CPOCIS is the amount of analyte trapped in the POCIS phase (µg g -1 ), and Cw weighted concentration in water (µg L -1 ) corresponds to POCIS exposure time as in previous studies [START_REF] Bernard | Lab-scale investigation of the ability of Polar Organic Chemical Integrative Sampler to catch short pesticide contamination peaks[END_REF][START_REF] Fauvelle | Use of Mixed-Mode Ion Exchange Sorbent for the Passive Sampling of Organic Acids by Polar Organic Chemical Integrative Sampler (POCIS)[END_REF][START_REF] Lissalde | Liquid chromatography coupled with tandem mass spectrometry method for thirty-three pesticides in natural water and comparison of performance between classical solid phase extraction and passive sampling approaches[END_REF][START_REF] Mazzella | Determination of kinetic and equilibrium regimes in the operation of polar organic chemical integrative samplers: Application to the passive sampling of the polar herbicides in aquatic environments[END_REF]. Cw weighted over consecutive periods (i.e., 0-2 d, 2-12 d or 0-21 d) are provided as supplementary material (SI_accumulation and water conc. Excel file).

If the analyte follows linear accumulation kinetics (Figure S 2), a linear model (LM) can be used to determine the kinetic constants as follows:

Equation 2

𝐶𝐹 𝐻𝐿𝐵 = 𝑘 𝑢 • 𝑡 = 𝑅 𝑠 • 𝑡 𝑀 𝑃𝑂𝐶𝐼𝑆
where ku is the kinetic constant for analyte uptake (L d -1 g -1 ), t the exposure time (d), Rs the sampling rate in L d -1 , and MPOCIS the mass of HLB sorbent (g) in the POCIS.

For DIA-d5, which we used as a performance reference compound, the analyte followed firstorder elimination kinetics, and a nonlinear least-squares model (NLS) can be used to determine the kinetic constants as follows:

Equation 3

𝐶 (𝑡) = 𝐶 (0) • 𝑒 -𝑘 𝑒 𝑡
where C(t) is the concentration of PRC measured in the HLB sorbent after an exposure time t, and C(0) the initial concentration of DIA-d5 within the same sorbent. From this relationship, we can then deduce the exchange constant ke (d -1 ).

Theory and modeling

In the previous studies [START_REF] Booij | POCIS Calibration for Organic Compound Sampling in Small Headwater Streams[END_REF][START_REF] Vermeirssen | Transfer Kinetics of Polar Organic Compounds over Polyethersulfone Membranes in the Passive Samplers Pocis and Chemcatcher[END_REF], flow effects on Rs were empirically modeled by a power-law relationship with flow velocity as follows: Equation 4𝑅 𝑠 = 𝑏 • 𝑈 𝑚 + 𝑐 Furthermore, the sum of the resistances for transport through the water boundary layer (WBL), membrane and sorbent equals the overall resistance to mass transfer (1/ko) [START_REF] Alvarez | Comprehensive Analytical Chemistry[END_REF], Fauvelle et al. 2017a): Equation 51

𝑘 𝑜 = 𝐴 𝑅 𝑠 = 1 𝑘𝑤 + 𝐴 𝑅 𝑠,𝑚𝑎𝑥
where 1/kw is the resistance associated with the WBL thickness of the water-membrane interface and A/Rs, max is the residual resistance (i.e., 1/kw→0) associated with the membrane thickness and/or the diffusion in the interstitial water surrounding the sorbent particles. A corresponds to the effective exchanging surface area of a POCIS. This value was previously found to be between 11.2 and 16.9 cm² depending on the sorbent mass used as the receiving phase [START_REF] Fauvelle | Optimization of the polar organic chemical integrative sampler for the sampling of acidic and polar herbicides[END_REF]. In fact, when the device is immersed and the receiving phase is fully wetted the sorbent is seen to rapidly sink to the bottom of the POCIS. This generally results in a smaller effective surface area than expected as the sorbent then only covers one third of the total membrane surface area. Thus, in the present study, an average value of A = 15 cm² (average value typically observed in our laboratory when calibrating POCIS with 5.4 cm inner diameter and 200 mg of receiving phase) was systematically considered.

For the method proposed here, we adopted a similar approach to the previous studies mentioned, while taking into account the mass transfer resistance, 1/ko (d cm -1 ) during the uptake regime for chemicals that we calibrated at different flow velocities (U; cm s -1 ). If we also consider m ≈ 0.5, as suggested by Booij and Chen (Booij and Chen 2018) for atrazine uptake with POCIS, we can obtain the following model: Equation 61

𝑘 𝑜 = 𝑏 √𝑈 + 𝑐
where b and c are expressed in d cm -1/2 s -1/2 and d cm -1 , respectively.

Results and discussion

Nonlinear modeling for flow velocity-mass transfer relationships

As shown in Figure 1 (a), the fitting of Equation 6 considering all the Rs data results in a power-law curve indicating a rapid decrease of the resistance as the flow velocity increases.

The b and c parameters adjusted for all substances are given in Table 1. It is also possible to fit a more general curve considering all 92 observations, including every compound, and thus to obtain the global parameters b = 0.100 ± 0.002 and c = 0.019 ± 0.004. The quality of this global fit can be judged by the R² = 0.967 and distribution of the residuals, which appears acceptable

(Figure S 3
). The residuals are, on the whole, rather uniformly distributed around zero.

However, there is some heteroscedasticity with the dispersion of the residuals increasing for V0. This can be explained by the higher uncertainty in the estimation of Rs, which was then used to derive 1/ko, due to the boundary layer control when the flow velocity is very low. In other words, in this particular condition, in Equation 5, the 1/kw term predominates over the where Ksw corresponds to the HLB-water sorption coefficient in the case of DIA uptake. During the calibration, the uptake of the DIA followed nonlinear kinetics, as previously observed [START_REF] Bernard | Lab-scale investigation of the ability of Polar Organic Chemical Integrative Sampler to catch short pesticide contamination peaks[END_REF][START_REF] Fauvelle | Use of Mixed-Mode Ion Exchange Sorbent for the Passive Sampling of Organic Acids by Polar Organic Chemical Integrative Sampler (POCIS)[END_REF][START_REF] Mazzella | Determination of kinetic and equilibrium regimes in the operation of polar organic chemical integrative samplers: Application to the passive sampling of the polar herbicides in aquatic environments[END_REF], and an average value of Ksw = 30370 ± 3600 mL g -1 was observed (raw data are provided in an Excel file in the supp. mat.).

In the same way as with the uptake data of the calibrated compounds (see above), when the desorption of DIA-d5 was treated with Equation 7, it produced the curve shown the Figure 1 (b). The corresponding coefficients b and c for DIA-d5 are reported at the bottom of Table 1.

Another way of representing modeling of this kind (Equation 6or its application to DIA-d5 only in Equation 7) is to transform the x and y values into a logarithmic scale. For example, by considering log0.5 for the flow velocity because this factor evolves according to its square root, and, in a more conventional way, the decimal log of 1/k0. In this case, we obtain a linear form described by Equation 8 (Figure S 4). This leads to the same estimations of coefficients b and c, but with the benefit of making an F-test possible (Table S 2) to check whether the variance is due to the model fitting or experimental random error. Equation 8𝑙𝑜𝑔 10 (

1 𝑘 0 ) = 𝑏 1 × 𝑙𝑜𝑔 0.5 (𝑈) + 𝐶 1
In this relationship, parameters b1 and c1 differ from b and c defined earlier because the variables underwent a logarithmic transformation.

A practical limitation of the PRC approach with POCIS has generally been reported because of the "80/20 rule". For instance, [START_REF] Buzier | Limitation of flow effect on passive sampling accuracy using POCIS with the PRC approach or o-DGT: A pilot-scale evaluation for pharmaceutical compounds[END_REF] mention that PRC correction efficiency increased with the deployment time, but significant desorption of DIA-d5 was observed under quiescent conditions during at least 15 and 21 days of exposure. Other compounds, such as cyanazine, have been studied as PRCs [START_REF] Belles | Development of the performance reference compound approach for the calibration of "polar organic chemical integrative sampler" (POCIS)[END_REF]. Although the elimination rates of cyanazine and DIA were different in this earlier study, flow velocity appeared affect both in the same way. The authors concluded that, given their different desorption rates, it would be possible to use these two compounds as PRCs simultaneously,

particularly by applying a model based on an environmental adjustment factor (Equation 9).

The latter necessarily requires compliance with the "80/20 rule". In practice, one of the two PRCs would have completely desorbed, and thus be unusable, while the other one would lose between 20 and 80% of its initial quantity. The model we propose is fitted with the desorption of DIA-d5, although a similar approach with another PRC, such as cyanazine, would probably result in an analogous outcome (i.e., the resistance to mass transfer would drop with the velocity of flow according to a power-law relationship). Actually, in the model proposed here, this restriction (i.e., the need for a PRC desorption rate > 20%) is no longer required. In the specific case where C(t)≈C(0) (and then ke→0; Equation 3) after a given period of in situ deployment (typically ≥ 2 weeks), we would then have an estimate of U < 1 cm s -1 (application of Equation 7). The resulting Rs estimates for the analytes of interest would be close to the minimum, i.e., around 100 mL d -1 . Such estimates would be considerably lower than Rs values typically obtained during laboratory calibration (usually 200-400 mL d -1 ), with a U between 2 to 10 cm s -1 [START_REF] Ahrens | Characterization of five passive sampling devices for monitoring of pesticides in water[END_REF][START_REF] Bernard | Multivariate Tiered Approach To Highlight the Link between Large-Scale Integrated Pesticide Concentrations from Polar Organic Chemical Integrative Samplers and Watershed Land Uses[END_REF][START_REF] Fauvelle | Use of Mixed-Mode Ion Exchange Sorbent for the Passive Sampling of Organic Acids by Polar Organic Chemical Integrative Sampler (POCIS)[END_REF][START_REF] Lissalde | Liquid chromatography coupled with tandem mass spectrometry method for thirty-three pesticides in natural water and comparison of performance between classical solid phase extraction and passive sampling approaches[END_REF][START_REF] Morin | Determination of uptake kinetics and sampling rates for 56 organic micropollutants using "pharmaceutical[END_REF][START_REF] Morin | Kinetic accumulation processes and models for 43 micropollutants in "pharmaceutical[END_REF][START_REF] Poulier | Can POCIS be used in Water Framework Directive (2000/60/EC) monitoring networks? A study focusing on pesticides in a French agricultural watershed[END_REF]. Equation 9𝑅 𝑠-𝑎𝑛𝑎𝑙𝑦𝑡𝑒 (𝑖𝑛 𝑠𝑖𝑡𝑢) = 𝑅 𝑠-𝑎𝑛𝑎𝑙𝑦𝑡𝑒 (𝑐𝑎𝑙𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛)

𝑘 𝑒-𝑃𝑅𝐶 (𝑐𝑎𝑙𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛)

× 𝑘 𝑒-𝑃𝑅𝐶 (𝑖𝑛 𝑠𝑖𝑡𝑢)

Figure 2 shows that b and c values are unlikely to follow a normal distribution. We therefore plotted the medians in red together with their percentiles, covering 65% of the data (equivalent to ± 1 standard deviation), and then 95% of the data. Looking first at the values in the red areas only, we see a dispersion of b accordingly to the 95% percentiles, ranging from 0.07 to 0.46 around a median of 0.127 d cm -1/2 s -1/2 . If we consider 65% of the data, the range is indeed narrower, between 0.08 and 0.23 d cm -1/2 s -1/2 , revealing the higher deviations for the compounds fluzilazole, boscalid, chlorfenvinphos and linuron. In other words, most of the analytes have b parameters that converge towards the central value defined by the median. The mean was also fairly close, estimated at 0.161 d cm -1/2 s -1/2 . For the c parameter, and again for the values delimited by the red areas, the dispersion around the median (0.005 d cm -1 ) appeared to be higher, even when only 65% of the data were considered (0.000-0.018 d cm -1 ). This is primarily because many estimates of the c parameter tended towards 0 (11 compounds out of 30) after applying Equation 6 modeling. However, the mean of c (0.008 d cm -1 ) remained close to the median, and of the same order of magnitude.

In summary, for the majority of the analytes studied here, it might be feasible to estimate parameters b and c through the use of values that fall around the median (or mean). Considering such "generic" constants for b and c, an attempt would then be made to pinpoint the degree of error and specific potential biases that may result.

Model output and validation steps

We calculated flow velocity (U calculated) from the elimination rate of DIA-d5 using Equation 7. This flow velocity estimate was then used to calculate ko calculated (Equation 6).

Finally, Rs was deduced by multiplying ko calculated with effective surface area (i.e., 15 cm²). In Figure 3, we plotted the 1/ko obtained experimentally during calibration under quiescent conditions (i.e., V0) against the 1/ko calculated from the DIA-d5 elimination rate. The alignment on the y = x line, and fact that all projected points are within the ± 20% interval, indicate a good equivalence between the 1/ko data obtained with the two approaches. The same exercise can be carried out for the highest flow velocity applied during POCIS calibration (i.e., V3). Figure 4 (a) shows a similar comparison and agreement between experimental and recalculated 1/ko, although there are significant deviations for fluzilazole, boscalid, chlorfenvinphos and linuron in this case. It is also noteworthy that flusilazole, boscalid, chlorfenvinphos and linuron have higher b values than the other compounds (Table 1 and Figure 2). It is expected that b is fairly independent of compound, since kw increases only weakly with increasing Dw, as reported by [START_REF] Booij | Review of atrazine sampling by polar organic chemical integrative samplers and Chemcatcher[END_REF] in the case of atrazine sampling by POCIS. For these four chemicals, a significant bias of about +82% may appear at high flow rates ≥ 20 cm s -1 . With the exception of linuron, these are among the most hydrophobic compounds on the list, with log Kow between 3.8 and 4.9. Based on the compounds calibrated here, it is therefore possible that this approach is limited to moderately polar chemicals with log Kow essentially between 2.1 and 3.8 (Table 1). Such a result seems to be supported by the results of a two-way ANOVA (Table S 1), considering the factors Dw (i.e., diffusion in water at 25°C) and log Kow. This analysis revealed that coefficient c, which could be associated with diffusion across the membrane, may partly depend on log Kow. However, unlike for passive samplers dedicated to hydrophobic compounds, Kow alone remains a poor predictor of Ksw and Kmw for passive samplers with hydrophilic microporous membranes and sorbent [START_REF] Bonnaud | Calibration comparison between two passive samplers -o-DGT and POCIS-for 109 hydrophilic emerging and priority organic compounds[END_REF]. The difference could indicate that this type of analyte may interact with the membrane matrix or surface during transfer through the water-filled pores. In other words, the porous media mass transport equations proposed by [START_REF] Booij | A method to account for the effect of hydrodynamics on polar organic compound uptake by passive samplers[END_REF] are probably unsuitable for such chemicals, unlike atrazine (log Kow estimated at 2.6).

Parts (b) and (c) of Figure 4 summarize the distribution of experimental and recalculated

Rs for the two boundary flowing conditions (i.e., V0 and V3). It appears that both types of Rs values would be around 75 mL d -1 for a quiescent medium (and vary between 30 and 120 mL d -1 according to the observation confidence interval), regardless of the diffusion in water (Dw)

or polarity (log Kow) of the substances considered (Table 1). The same appears to be valid for a highly agitated medium, with a mean Rs of 400 mL d -1 (varying between 200 and 600 mL d -1

according to the observation confidence interval). Consequently, for a flow velocity very close to zero, we can assume that the sampling rates can vary around a central value with a dispersion of ±61 % (implying a relative standard deviation of about 30.5 % and with a t-value of 2 for a confidence interval of 95 %).. The resulting enlargement for the relative standard deviation of Rs (i.e. 61%) seems significantly lower than what we were able to ascertain empirically by [START_REF] Poulier | Can POCIS be used in Water Framework Directive (2000/60/EC) monitoring networks? A study focusing on pesticides in a French agricultural watershed[END_REF]. Actually, in this earlier study, an enlargement factor of two-fols (i.e. from -50% to +100% Rs amplitude) was typically used with an Rs obtained from laboratory calibration to extrapolate to field conditions with different flow regimes.

We applied our approach to previously published laboratory and in situ calibration data.

Table 2 shows the current velocities (U estimated) recalculated using Equation 7and the coefficients from Table 1, as well as the ke reported for DIA-d5 in these previous studies. The resulting recalculated flow velocities were in good agreement with the values previously measured (U measured) with an electromagnetic flow meter. Table 3 shows the Rs estimated for some pesticides (with log Kow between 2.5 and 3.5, and a mean log Kow of 3.125) using Equation 6and the parameters b and c from the present calibration. Under these conditions (i.e., a "specific model"), an estimate of mean Rs of 308 mL d -1 was obtained, for example, in contrast to a mean Rs of 280 mL d -1 determined experimentally elsewhere [START_REF] Fauvelle | Use of Mixed-Mode Ion Exchange Sorbent for the Passive Sampling of Organic Acids by Polar Organic Chemical Integrative Sampler (POCIS)[END_REF]) (i.e., a relative deviation of 10% on average). A similar comparison with two other POCIS calibration datasets [START_REF] Lissalde | Liquid chromatography coupled with tandem mass spectrometry method for thirty-three pesticides in natural water and comparison of performance between classical solid phase extraction and passive sampling approaches[END_REF][START_REF] Mazzella | Determination of kinetic and equilibrium regimes in the operation of polar organic chemical integrative samplers: Application to the passive sampling of the polar herbicides in aquatic environments[END_REF]) indicated a bias that is also fairly low, ranging from 10 to 25%. A larger discrepancy can be observed in in situ data (70% bias), but must be put into perspective considering that only atrazine was detected during this field study [START_REF] Mazzella | Evaluation of the Use of Performance Reference Compounds in an Oasis-HLB Adsorbent Based Passive Sampler for Improving Water Concentration Estimates of Polar Herbicides in Freshwater[END_REF]. The exercise was also carried out considering the "generic" coefficients with b' = 0.100 d cm -1/2 s -1/2 and c' = 0.019 d cm -1 (i.e., the "general model"), resulting in quite acceptable deviations, rather close to the biases observed with the "specific models" fitted for each compound reported here.

Finally, the previously defined median was in good agreement with the mean of b' = 0.100 ± 0.004 (95% confidence interval limits) when we plotted the values of the "generic" b' and c' parameters on Figure 2 (dashed lines and blue areas). The discrepancy appears to be larger for c' = 0.019 ± 0.008, but the red area, which contains 95% of the c values, and the blue area, which represents the confidence interval associated with the mean value of c', overlap significantly. This makes the differences non-significant, so parameters b' and c' could be used when applying Equation 6 to estimate Rs, taking into account the influence of the current velocity by the additional use of Equation 7. In this case, we would have to accept biases of 10 to 25% in stirred conditions (i.e., > 2-3 cm s -1 ) and up to 70% in quiescent conditions. In light of this possibility, a template containing the two generic coefficients b' and c' is provided in the supplementary material (see file "SI_Rs and Ksw data and template.xlsx).

Conclusion

In this study, a POCIS calibration was performed with some moderately polar pesticides (i.e., log Kow between 2 and 4) as flow velocities increased. A power-law relationship with water flow velocity as a parameter was used to investigate the relationship between sampling rates and simultaneous desorption rate of DIA-d5. Based on these findings, we proposed a model that explicitly accounts for the effects of such a factor. We evaluated the benefits of this new method by reprocessing previous data with it. This approach also cuts out the main limitations of the previous EAF approach, such as the unproven hypothesis of isotropic exchange for all analytes and the "80/20 rule" constraint. Furthermore, it appears that a lower confidence interval associated with the variability of Rs as a function of flow rate.

Figures and tables

Figure 1. The red line corresponds to the median values for all data and the red shaded areas correspond to the percentiles covering 65% and 95% of the data, respectively. Blue dashed lines correspond to the means obtained from the "global" model fitting (Table 3). Blue shaded area corresponds to the 95% confidence interval associated with this model output.

A

  /Rs,max term, reflecting a stronger influence of the WBL (and the diffusion of the compounds in it) as U decreases. Conversely, for U→∞, the 1/kw term becomes almost negligible and the contribution of A/Rs,max is more likely to be estimated. Finally, for all observations, five values lie outside the ±2 standard deviation interval typically representing a 95 % confidence interval.[START_REF] Lissalde | Polar organic chemical integrative samplers for pesticides monitoring: Impacts of field exposure conditions[END_REF] observed that in situ ke of DIA-d5 (Equation3) increased with increasing flow velocity. In such a desorption regime (i.e., DIA-d5), we obtain a

  Figure 1. Variation of mass transfer resistance vs. flow velocity: a) for all chemicals during a linear uptake only, and the fitting of Equation 6 with average values, b) for DIA-d5 only during the desorption process (standard deviations on both 1/ko estimates of DIA-d5 and the flow velocity measurements are reported in this lower panel).

Figure 2 .

 2 Figure 2. Distributions of parameters b and c for each analyte from Table1. The red line corresponds to the median values for all data and the red shaded areas correspond to the percentiles covering 65% and 95% of the data, respectively. Blue dashed lines correspond to the means obtained from the "global" model fitting (Table3). Blue shaded area corresponds to the 95% confidence interval associated with this model output.

Figure 3 .

 3 Figure 3. Correlation (y = x with ± 20 % interval limits indicated by the dotted lines) between 1/kO experimental data (y-axis) at V0 flow condition and 1/kO recalculated estimates with Equation 6 (x-axis).

Figure 4 .

 4 Figure 4. (a) Correlation (y = x with ± 20 % interval limits indicated by the dotted lines) between 1/kO experimental data (y-axis) at V3 flow condition and 1/kO recalculated estimates with Equation 6 (x-axis). (b) Rs experimental and estimated data (V0 in blue and V3 in brown) as a function of Dw, (c) Rs experimental and estimated data (V0 in blue and V3 in brown) as a function of log Kow. Open and filled circles corresponds to experimental and estimated values, respectively. The brown and blue solid lines represent the mean values of the data, respectively.

Figure S 2 .

 2 Figure S 2. Example of metolachlor linear uptake obtained during laboratory calibration of POCIS and associated with the four flow velocities (V0 to V3).

Figure S 3 .

 3 Figure S 3. Plotting of residuals for the fitting of Equation 6. The grey horizontal dashed lines correspond to the 95 % confidence interval limits.

Figure S 4 .

 4 Figure S 4. Logarithmic transformation (Equation 8) of x and y values (a), and corresponding plotting of residuals (b). The grey horizontal dashed lines (b panel) correspond to the 95 % confidence interval limits.

Table 1 .

 1 b (d cm -1/2 s -1/2 ) and c (d cm -1 ) parameters obtained for each compound following a linear uptake over 21 days, and R² of fitted linear models. b and c parameters deduced from DIA-d5 desorption curve are provided as well. The log Kow and Dw values are also summarized. Estimated with https://www3.epa.gov/ceampubl/learn2model/part-two/onsite/estdiffusion-ext.html Table2. ke values and flow velocities either measured or calculated (with DIA-d5 desorption rates) from previous published works.

	Chemicals	b	c	R²	RMSE * log Kow + Dw (×10 -6 cm s -1 ) §
	Acetochlor	0.112 (0.006)	0.004 (0.005)	0.994	0.006	3.5	4.57
	Atrazine	0.074 (0.002)	0.012 (0.002)	0.996	0.004	2.6	5.31
	Carbaryl	0.128 (0.007)	0.004 (0.007)	0.980	0.015	2.4	5.85
	Carbofuran	0.115 (0.005)	0.019 (0.005)	0.983	0.013	2.1	5.05
	Dimetomorph	0.155 (0.009)	0.001 (0.000)	0.973	0.022	3.3	3.79
	Isoproturon	0.102 (0.003)	0.011 (0.003)	0.992	0.008	2.6	5.31
	Pyrimicarb	0.069 (0.001)	0.021 (0.001)	0.999	0.002	1.8	5.08
	Therbuthylazine 0.090 (0.004)	0.002 (0.004)	0.979	0.011	3.4	4.43
	Alachlor	0.088 (0.002)	0.011 (0.002)	0.985	0.012	3.6	4.52
	Chlortoluron	0.161 (0.006)	0.001 (0.000)	0.988	0.015	2.4	5.25
	DCPMU	0.145 (0.004)	0.008 (0.004)	0.996	0.008	2.1	5.63
	DCPU	0.123 (0.004)	0.006 (0.004)	0.991	0.010	2.1	5.98
	Diuron	0.174 (0.008)	0.001 (0.000)	0.990	0.015	2.5	4.44
	IPPMU	0.089 (0.003)	0.013 (0.002)	0.992	0.007	2.4	5.37
	IPPU	0.074 (0.003)	0.031 (0.003)	0.999	0.001	2.1	5.66
	Linuron	0.502 (0.024)	0.000 (0.000)	0.987	0.049	3.0	5.28
	Metazachlore	0.067 (0.002)	0.012 (0.002)	0.992	0.005	3.0	4.55
	Metolachlore	0.082 (0.003)	0.009 (0.003)	0.986	0.008	3.5	4.40
	Cybutryne	0.165 (0.008)	0.001 (0.000)	0.986	0.017	2.9	4.82
	Flusilazole	0.333 (0.032)	0.000 (0.000)	0.975	0.046	4.7	4.56
	Tebuconazole	0.156 (0.007)	0.001 (0.000)	0.986	0.016	3.7	4.19
	Cyproconazole	0.068 (0.002)	0.009 (0.002)	0.990	0.006	2.9	4.39
	Epoxiconazole	0.171 (0.008)	0.001 (0.000)	0.990	0.015	3.7	4.33
	Spiroxamine	0.126 (0.010)	0.040 (0.010)	0.991	0.010	2.5	4.42
	Flurtamone	0.155 (0.013)	0.000 (0.000)	0.973	0.022	4.6	4.30
	Thiodicarb	0.175 (0.007)	0.017 (0.007)	0.992	0.013	1.6	4.20
	Prometryn	0.138 (0.005)	0.003 (0.005)	0.985	0.014	3.5	5.19
	Dimethanamid	0.082 (0.003)	0.009 (0.003)	0.989	0.007	2.9	4.58
	Chlorfenvinphos 0.485 (0.033)	0.000 (0.000)	0.972	0.070	3.8	4.56
	Boscalid	0.438 (0.028)	0.000 (0.000)	0.979	0.054	4.9	5.10
	DIA-d5	0.078 (0.002)	0.004 (0.001)	0.804	0.027	1.1	6.33

* Root-mean-square error.

+ Information from the Pesticide Properties DataBase -PPDB (http://sitem.herts.ac.uk/aeru/ppdb/en/atoz.htm) and Ineris (https://substances.ineris.fr/fr/). §

Table 3 .

 3 Sampling rates reported in some previous laboratory or field POCIS calibrations. Lab 1, 2 and 3 and In situ corresponds to the respective reference indicated in Table2. Bias (%) between experimental and recalculated Rs are reported as well.

	Sampling rates (mL.d -1 )

Table S 2

 S . F-test results for the fitting of Equation8.

		Sum of Squares	df	Mean Square	F	p
	X=log0.5(U)	10.61	1	10.6075	416	< .001
	Residuals	3.01	118	0.0255
	Note. Type 3 sum of squares.

* Relative deviation between calculated mean Rs (with either specific or generic b and c parameters) and experimental mean Rs.

Acknowledgments

This study was financed by the Adour-Garonne French Water Agency. Authors would like to thank Gwilherm Jan and Sylvia Moreira from Inrae, and Julie Leblanc, Patrice Fondanèche and Karine Cleries from University of Limoges for technical and laboratory assistance.

Supplementary material


Afef Awadid
email: afef.awadid@irt-systemx.fr

Boris Robert
email: boris.robert@irt-saintexupery.com

Benoît Langlois
email: benoit.langlois@thalesgroup.com

MBSE to Support Engineering of Trustworthy AI-Based Critical Systems

Keywords: Engineering Processes, Methodological Guidelines, Trustworthiness Environment, MBSE, Arcadia Method, AI-Based Critical Systems, Machine Learning

Because of the multidisciplinary nature of the engineering of a critical system and the inherent uncertainties and risks involved by Artificial Intelligence (AI), the overall engineering lifecycle of an AI-based critical system requires the support of sound processes, methods, and tools. To tackle this issue, the Confiance.ai research program intends to provide a methodological end-to-end engineering approach and a set of relevant tools. Against this background, an MBSE approach is proposed to establish the methodological guidelines and to structure a tooled workbench consistently. In this approach, the system of interest is referred to as the "Trustworthiness Environment" (i.e. the Confiance.ai workbench). The approach is an adaptation of the Arcadia method and hence built around four perspectives: Operational Analysis (the engineering methods and processes: the operational need around the Trustworthiness Environment), System Analysis (the functions of the Trustworthiness Environment), Logical Architecture and Physical Architecture (abstract and concrete resources of the Trustworthiness Environment). Given the current progress of the Confiance.ai program, this paper focuses particularly on the Operational Analysis, leading to the modeling of engineering activities and processes. The approach is illustrated with an example of a machine learning model robustness evaluation process.

INTRODUCTION

The use of Artificial Intelligence (AI) techniques is becoming increasingly popular in various applications [START_REF] Miglani | Deep learning models for traffic flow prediction in autonomous vehicles: A review, solutions, and challenges[END_REF], as the technologies mature and become more affordable [START_REF] Boardman | An exploration of maintaining human control in AI enabled systems and the challenges of achieving it[END_REF]. These techniques could be physically embodied as in the case of safetycritical systems such as electricity grids or on-board aircraft networks or exist only as software agents that autonomously process data at speeds or for durations that humans are not capable of.

Applying AI techniques can confer a competitive advantage to industry by providing not only high value-added products and services but also support to decision-makers [START_REF] Mattioli | AI engineering to deploy reliable AI in industry[END_REF]. In this sense, production efficiency, product quality, and service level will be improved by AI [START_REF] Li | Applications of artificial intelligence in intelligent manufacturing: a review[END_REF]. a https://orcid.org/0000-0001-7525-613X However, while AI has much potential for innovative applications, it raises several concerns such as security and safety [START_REF] El-Sherif | Telehealth and Artificial Intelligence insights into healthcare during the COVID-19 pandemic[END_REF]. These concerns are even more salient when it comes to critical systems.

AI-based critical systems are defined as systems containing at least one subsystem or component using AI technology (the most representative today being Machine Learning), alongside traditional software components, and whose failure leads to unacceptable circumstances such as loss of human lives [START_REF] Mwadulo | Suitability of agile methods for safety-critical systems development: a survey of literature[END_REF]. The engineering of such systems is regarded as a multi-engineering process that addresses conventional engineering disciplines (i.e., data and knowledge engineering, algorithm engineering, system and software engineering, safety and cybersecurity engineering, and cognitive engineering) with respect to the effects induced by the use of AI [START_REF] Adedjouma | Towards the engineering of trustworthy AI applications for critical systems-The Confiance[END_REF].

Given this multidisciplinary process for engineering a critical system and the inherent uncertainties and risks involved by AI, the overall engineering lifecycle of a trustworthy AI-based critical system requires the support of sound processes, methods, and tools [START_REF] Mattioli | AI engineering to deploy reliable AI in industry[END_REF].

To address this issue, the Confiance.ai research program2 aims to provide a methodological end-toend engineering approach and a set of tools consistent with this approach. Considering the complexity and heterogeneity of the intended outcome, we have applied a Model-Based Systems Engineering (MBSE) approach allowing to establish and formalize the methodological guidelines and to structure a tooled workbench consistent with these guidelines.

This paper therefore attempts to answer the research question of how to support the engineering of trustworthy AI-based critical systems with the help of the modeling of methodological guidelines.

The rest of the paper is organized as follows. Section 2 introduces the context and motivation of this work. The proposed MBSE approach for supporting the engineering of trustworthy AI-based critical systems is presented in Section 3. In Section 4, the utility of the approach is illustrated with an example of formalized engineering activities and processes (methodological guidelines) for AI-based systems. The paper is rounded off with some general conclusions and plans for future works in Section 5.

CONTEXT AND MOTIVATION

This section presents the overall problem that the Confiance.ai research program aims to tackle (Section 2.1), the ambition of an integrated solution developed by Confiance.ai (Section 2.2), and the more specific problem that the work presented in this paper aims to help solve (Section 2.3).

Trustworthiness in AI-based Systems

An AI-based system integrates at least one subsystem or component using AI technology. In our work we have reduced AI to Machine Learning, since it is the AI technique mostly considered currently by industries for integration in their systems. Such integration has consequences on the development of the system, for example:

 The benefit of enhancing the system with more autonomy;  The drawback of unpredictability, which deters reliability;  The legal and moral obligation to respect duties such as the AI Act, explainability, and the place of humans at the center of decision-making;  The evolutivity of the system in an everchanging environment to be periodically reconsidered, for instance through monitoring that can provide feedback to the learning process. A key value that must be introduced before deploying an AI-based system, especially for critical systems, is trustworthiness. Trustworthiness is not reduced to predictability. It includes all the AI-related quality attributes of an AI-based system [START_REF] Mattioli | AI engineering to deploy reliable AI in industry[END_REF]. Moreover, trustworthiness must be considered along all the steps the engineering cycle of a critical system (as early as the business/mission analyses, then during specification, architecture, development, implementation, integration, verification, validation, qualification, deployment), and at each systemic level (system, component, ML model, data…).

To make sure that trustworthiness is considered at each of these steps and levels, it is necessary to have a consistent end-to-end engineering approach that considers the trustworthiness properties through dedicated engineering activities and processes (to identify them, specify them, allocate them, develop them by construction, evaluate them, etc.), using relevant methods and tools.

Confiance.ai's Trustworthiness Environment

To support the French industry on this matter of trustworthiness in AI-based systems, the Confiance.ai research program aims to develop a set of consistent methods and tools addressing, for example, AI robustness, AI monitoring, AI explainability, AI embeddability, and so on. These methods and tools shall not be disparate standalone objects. On the contrary, they shall be consistently integrated in an end-to-end approach that covers all the necessary engineering steps.

To this end, the Confiance.ai research program provides a tooled workbench called the Trustworthiness Environment, that integrates the software tools developed within Confiance.ai. The use of this workbench will be consistent with an end-to-end engineering method that integrates the local methods elaborated for each Confiance.ai's research topic.

The focus of this paper is not to present all the local topic-specific answers provided by Confiance.ai (how to develop an ML model robust by design, how to make an ML model explainable, how to embed an ML model, etc.), but to propose an MBSE-based approach that allows the complex methodological integration necessary to build an end-to-end engineering method for critical AI-based systems.

Lack of Formalization of AI Engineering

Classical systems engineering and software engineering (i.e. for systems and software that do not embed AI components) are already widely normalized: at a generic high-level through standards that are not new (for example ISO/IEC/IEEE 15288 and ISO/IEC/IEEE 12207), and at a more specific low-level through domain-specific standards and company internal methods & processes. This is not yet the case for AI, more particularly for Machine Learning (ML) technologies that are the main focus of Confiance.ai. There is no detailed stepby-step guide yet for the development of a critical system that embeds ML technology. There are two main reasons for this. Firstly, Systems Engineering and ML engineering are two separate engineering domains with different cultures and the connection between them is currently relatively weak. Secondly, ML engineering processes are not yet formally described. This is, in part, due to the fact that this technology is still evolving rapidly.

To facilitate the integration of ML into critical systems, it is essential to provide a comprehensive description of ML engineering processes and establish connections with traditional systems/software engineering processes. If required, modifications or adaptations to the existing engineering practices should be made to ensure a seamless integration of ML technology.

There is such ongoing work at standardization level: ISO/IEC DIS 5338 for example, which completes ISO/IEC/IEEE 15288 and ISO/IEC/IEEE 12207 for AI-specific matters, or AS 6983 from SAE. We take into account the available drafts of these future standards to structure our approach, and thus make sure that our end-to-end engineering method will, by construction, be consistent with them. However, these standards are, by nature, generic, and our ambition is to go deeper at low-level engineering activities to provide concrete solutions to engineers involved in the development of ML-based critical systems.

MBSE TO SUPPORT ENGINEERING OF TRUSTWORTHY AI-BASED CRITICAL SYSTEMS

This section focuses on the chosen MBSE method/tool used as bases for our approach (Section 3.1), the specialization of the MBSE approach to our goal related to Confiance.ai's Trustworthiness Environment (Section 3.2), and the work strategy used to gather the necessary inputs for elaborating our end-to-end engineering method (Section 3.3).

Choice of ARCADIA/Capella as bases for MBSE method/tool

Several methods and tools are available for formalization and modeling (e.g. BPMN methods and tools for the modeling of processes). However, our ambition is not only to model a set of engineering activities and processes for critical AI-based systems, but also to specify and structure a tooled workbench, the Trustworthiness Environment, consistent with these engineering activities and processes. Therefore, we decided to base our modeling approach on the MBSE method/tool couple ARCADIA/Capella, by adapting the definition of the four ARCADIA/Capella perspectives in the following way, as shown in Figure 1:  Operational Analysis: the operational need around the Trustworthiness Environment, that is to say, the engineering methods and processes that the Trustworthiness Environment shall support.  System Analysis: the functions/services that the Trustworthiness Environment shall provide in order to support the engineering processes defined in the Operational Analysis perspective.  Logical Architecture: the abstract resources to be used by the Trustworthiness Environment.  Physical Architecture: the concrete resources to be used by the Trustworthiness Environment (mainly, the software tools developed by Confian.ce.ai to adress AI trustworthiness).

As an MBSE tool, Capella was chosen also because we knew that the Viewpoints needed for our modeling (cf. next Section 3.2) would not necessarily be natively consistent with any method/tool, and given the openness of Capella, we were confident that we would be able to have Capella adapted to our need.

Specialization of the MBSE approach for the Trustworthy Environment

In order to explain how an MBSE approach can be applied to our need to model the Trustworthiness Environment, it is useful to set the key concepts contextualizing such application. To do so, we are inspired by the standard-based conceptual model of reference architecture [START_REF] Awadid | Reference Architectures for Cyber-Physical Systems: Towards a Standard-Based Comparative Framework[END_REF] defined in accordance with the ISO/IEC/IEEE 42020 standard (ISO/IEC/IEEE 42020, 2019). The resulting conceptual model is presented in Figure 2. In this conceptual model, the concepts of "Stakeholder", "Concern", "Viewpoint" and "View" are defined by the ISO/IEC/IEEE/DIS 42010 standard (ISO/IEC/IEEE/DIS 42010, 2020) as follows. In our context, it refers to the engineering roles that will use the Trustworthiness Environment.

Concern: "a matter of relevance or importance to one or more stakeholders regarding an entity of interest". For our approach, it refers to the expectations of the engineers regarding the support to their activities that the Trustworthy Environment shall provide.

Viewpoint: "conventions for the creation, interpretation, and use of an architecture view to frame one or more concerns". We had to build our own meta-model to establish the concepts and relationships between concepts necessary to produce our Views.

View: "information item comprising part of an architecture description that expresses the architecture of an entity of interest, and that is governed by an architecture viewpoint". The Views of our modeling approach need to have the expressivity necessary to describe activities, processes, tools supporting the engineering of critical ML-based systems.

Perspective: this term is specific to ARCADIA/Capella, it refers to each analysis/architecture phase of the ARCADIA method, as shown in Figure 1. Since we based our MBSE approach on ARCADIA/Capella, our Views are captured in an ARCADIA/Capella perspective.

Figure 3 3 above shows the meta-model of one of the main Viewpoints of our approach, that governs the Views describing the engineering activities and processes. Concepts and their relationships were defined in a tool-agnostic way. Then, the concepts of this meta-model were mapped to objects of Capella's Operational Analysis perspective (see Table 1).

Since Capella had restrictions regarding the implementation of these concepts and their relations ships, the Confiance.ai program had a Capella plugin developed in order to have our Viewpoints matched.

Therefore, the modeling approach we used is not exactly the ARCADIA method and the Capella tool, it is an adaptation of ARCADIA/Capella fitting our modeling need.

Work method

In order to build the Operational Analysis perspective of our model, that is to say, formalize the engineering activities and processes for critical ML-based systems, two strategies are combined:

 Top-down, drafts of standards that address the subject of ML-based systems (ISO/IEC DIS 5338, AS 6983…) are taken into account, in order to provide the overall structure for our end-to-end engineering approach. Figure 4 below shows the main engineering steps of our end-to-end method, organized according to the traditional "V" cycle ("W" cycle at ML model level).  Bottom-up, the various research works of the Confiance.ai program are analyzed (produced documents, interviews of researchers) in order to identify the methods and tools that zoom in the ML-specific details of one of the main engineering steps, with the condition of being mature enough for integration in the end-to-end engineering method.

EXAMPLE OF A FORMALIZED ENGINEERING PROCESS

Once an ML model has been developed (built, configured, trained), it needs to be evaluated, to make sure that all the required properties are reached. One of the important trustworthiness attributes to be considered is the robustness. To evaluate this aspect, the Confiance.ai has explored two techniques. The first one uses testing: the ML model is executed with a perturbed dataset, and the deviation of the model from its nominal behavior is measured. The second ones uses formal verification: model robustness, expressed by formal properties, is mathematically verified by accessing the internal structure of the model. Given the cost of formal verification, Confiance.ai recommends to start by robustness testing, and to perform robustness formal verification only once testing has shown satisfactory results. This is illustrated by the following Figure 5. Given the size of the Capella diagrams, the detail of each diagram may be difficult to read, but the detail of each modelled process is not the point of this paper. The point is rather to show that MBSE enabled the formalized construction of a complex and comprehensive multi-level workflow supporting the engineering of critical AI-based systems.

The first diagram at the top of Figure 5 is the modelling of the full engineering cycle (equivalent to the Figure 4 previously shown).

The "pink zoom" of Figure 5 expands the content of the "Evaluate the trained ML Model" first-level engineering activity and focuses on a key trustworthiness attribute: robustness, thus showing a second diagram that describes the "Evaluate robustness of the trained ML Model" second-level engineering activity.

The "green zoom" of Figure 5 expands the content of the first phase of this second-level engineering activity, thus showing a third diagram that describes the "Evaluate robustness of the trained ML Model with sampling and perturbation" thirdlevel activity.

This third diagram contains the lowest-level engineering activities, directly corresponding to methods and tools developed or recommended by the Confiance.ai research team having the expertise in model robustness test.

Such "zooms" are possible for any step of the overall engineering cycle. The engineering of MLbased critical systems is thus supported. Going through the whole cycle, from system to component then to ML model and data, from specification to development then to IVV, engineers can zoom to the engineering activities specific to ML-based systems, for the required trustworthiness aspects (safety, robustness, explainability, embeddability…), and be assisted by the methods and tools associated to these activities.

CONCLUSIONS AND FUTURE WORKS

This paper proposed an MBSE approach (based on a modification of ARCADIA/Capella) to support the complex engineering of trustworthy AI-based critical systems.

The system of interest of our modelling is the Trustworthy Environment, the tooled workbench to be delivered by the Confiance.ai research program. At our current stage of progress, we are primarily focused on the "Operational Analysis" perspective of the proposed modeling approach. This involves identifying and formalizing the activities and processes required for engineering an AI-based critical system, with the ambition to obtain in this way an applicable end-to-end engineering method. To do so, we rely, on one hand, for higher-level engineering activities and processes (structure of our approach), on in-work standards such as ISO 5338 and AS 6983, and on the other hand, for lower-level engineering activities and processes (details of our approach), on the expertise of the various research teams of Confiance.ai.

Several future works are planned. First, we need to consolidate and complete the approach along the full engineering cycle. Currently, not all engineering steps are covered yet.

Second, Confiance.ai intends to publish the obtained end-to-end engineering method through a website. This entails work to make our modeling as graphic and easily navigable as possible.

Third, the obtained end-to-end engineering approach needs to be evaluated against use cases. Each specific method integrated in our approach has already been locally, on its own, evaluated against a use case. What remains to be done is evaluating portions of our obtained end-to-end engineering method, i.e. successions of engineering activities and processes, involving different methods and tools on a same use case.

Fourth, the modeling approach has to be continued by the System Analysis (functional specification of the Trustworthy Environment) and by the Logical and Physical Architecture (architecture of the Trustworthy Environment). Confiance.ai's Trustworthy Environment is already under construction, by collecting and integrating all software tools developed by the various teams of the research program. However, having a full modeling from Operational Analysis to Physical Architecture will ensure full consistency and traceability between the methodological guidelines described in our Operational Analysis and the relevant tools to be considered in the Physical Architecture.

Figure 1 :

 1 Figure 1: The ARCADIA modeling pyramid applied to Confiance.ai's Trustworthy Environment

Figure

  Figure 2: Conceptual model for the description of the Trustworthiness Environment

Figure 4 :

 4 Figure 4: End-to-end engineering cycle for ML-based systems

Figure 5 :

 5 Figure 5: Zoom on the evaluation of ML model robustness by test of its response upon perturbation of input data

Table 1 :

 1 : "Engineering Activities for trustable AI" Viewpoint Correspondence with Capella objects Figure

	Meta-model concept	Capella object
	Engineering Process	Operational Process
	Process Sequence Flow	Sequence Link
	Process Activity	Operational Process
		Involvement Operational
		Activity
	Process Item Flow	Operational Process
		Involvement Link
	Engineering Role	Role
	Engineering Activity	Operational Activity
	Engineering Exchange	Operational Interaction

https://www.confiance.ai/

ACKNOWLEDGMENTS

This work has been supported by the French government under the "France 2030" program.


Emilie Telle

Xavier Saloppé

Anne - Karanakov

Lyse

Eline Lefebvre

Thierry Pham

Hoang

Quality of Life (QoL) among forensic inpatients with (IDS) or without Dissociative Syndromes (IWDS): Comparison of Self-Report and Proxy Assessments

Results

First, all inpatients perceive a better QoL than care managers for majority of domains. The results highlight a majority of moderate ICC on physical health, respect, humanity, relationships, justice, recidivism programme and dignity areas. But ICC values are lower for IDSs than IWDSs. Similar pattern is found on a group-level, with IDCs perceiving a better QoL for specific domains and IWDCs for more generic domains than their referring health professionals' perception.

Discussion

Method

Objectives Assessing QoL of forensic inpatients through a holistic approach with generic and forensic-specific instruments to better understand issues related to their life context, and specifically in IDSs due to their frequent exclusion in literature. Two sub-objectives follow from this: o Assessing difference in perception of QoL between IDSs and IWDSs; o Assessing level of agreement in perception of QoL between forensic inpatients (IDSs and IWDSs) and referring health professionals to determine areas of agreement and disagreement to be improved in their institutional care. Specific questionnaire for long-term forensic psychiatric inpatients, consisting of 144 items assessed with 100 mm visual analogue scales (VAS), on which patients indicated their level of agreement. For the purpose of this study, we selected domains particularly associated with the daily life of patients not covered by other instruments (see Table 1).

Study was approved by research ethics committee of the Regional Psychiatric Centre "Les Marronniers". Study is part of the HPS's systematized data collection (socio-demographic, diagnostic (MINI; SCID-II) and criminological) and part of a larger project supported by CRDS assessing perception of patients' QoL (CRDS, 2020). Participants were individually assessed at least one month after their admission and their referring health professionals completed same questionnaires and provided a consensus assessment within 15 days.

Analyses were performed with SPSS 25.0 (IBM Corp., 2017). Three levels of analyses for agreement levels between referring healthcare professionals and total sample, and patients' group (IDSs and IWDSs). First, paired-group comparisons were made but not reported in the present poster (Wilcoxon test).

Then, intra-class coefficients (ICCs) were performed to measure level of individual agreement between healthcare professional and the patient (see

Introduction

QoL is a multidimensional concept combining objective and subjective measures (Bouman & Bulten, 2009;Thibaut & Pham, 2018). It promotes recovery by considering patients' perceptions without obscuring psychiatric symptoms. It enables rehumanisation, empowerment and involvement despite constraints such as complexity of psychopathology and recidivism (Vorstenbosch, & Castelletti, 2020). Use of generic instruments highlights individual's general representation whereas specific instruments focuses forensic hospitalization. A better QoL perception acts on recidivism risk reduction and therapeutic adherence (Bayliss et al., 2019). But low agreement between self-and hetero-assessment are highlighted (Jardon et al., 2014). There is more agreement found on objective and quantifiable domains than subjective and unobservable (Schel et al., 2015;Vorstenbosch & Castelletti, 2020). IDS are frequently excluded from research due to associated cognitive distortions which interact with assessments objectivity (Bouman et al., 2008;Jardon et al., 2014). Existing literature, mainly relying on a psychotic population, highlights a general deterioration of their QoL (Llorca & Gorwood, 2016) compared to other populations, specifically on social relationships, sexuality and leave domains (Saloppé & Pham, 2006b;Schel et al., 2015). It is essential for health professionals to target specific needs of each patient. Few studies assess degree of agreement between forensic inpatients and their referring caregivers (Schel et al., 2015;Vorstenbosch & Castelletti, 2020). These needs may differ significantly from those intended by the professionals.

To our knowledge, no study considering all these aspects on QoL has been conducted in French-speaking Belgian forensic inpatients.

Main results show that referring professionals tend to worsen QoL of their referred inpatients, especially with regard to IDSs, whether in generic or specific areas (Vorstenbosch & Castelletti, 2020). Of the 27 QoL aspects investigated, 16 agreements were found between professionals and total sample, the majority being moderate. Difficulty of hetero-assessment is exacerbated with IDSs. Number of agreements obtained between IWDSs and professionals is higher than with IDSs (12>10). A majority of areas remain without agreement in both groups. Domains with no or low agreement should be areas for vigilance and communication with professionals and constitute needs to be adjust for therapeutic treatment in line with the principles of the Good Lives Model (GLM; Ward & Beech, 2006). Moreover, possibility for patients to express themselves openly on their general and specific perception of QoL aims to strengthen therapeutic alliance, promotes rehabilitation and reduces institutional tensions (da Costa et al., 2020). Some future perspectives would be to: increase the sample size to be more representative of the French-speaking Belgian forensic inpatients population; take into account the variability of perception according to patient or professional's gender; involve professionals in the research to increase the number of consensual opinions; assess severity of DS using specific measures to this symptomatology; take into account degrees of relational and emotional involvement which influence professionals' QoL perception of their referring patient.

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  Population 42 consensual opinions between male forensic inpatients not guilty for reason of insanity (NGRI) (Law relating to detention, 2014) in a High-Risk Security Forensic Hospital and their care managers (psychiatric nursing staff). Based on the total sample, two groups were formed based on Axis I diagnosis (MINI; Sheehan et al., 1998) : o 16 IDSs (Mean Age = 52.08; SD = 10.83; Mean Length of Stay = 12.00; SD =7.85); o 26 IWDSs (Mean Age = 51.32; SD = 11.90; Mean Length of Stay = 12.75; SD = 9.56). Data analysis Measures World Health Organization Quality Of Life-Bref (WHOQOL-Bref; Whoqol Group, 1998) Generic, multidimensional questionnaire of 26 items assessed on a 5-point Likert scale, divided into 4 factors (see

Table 1

 1

).

Measuring Quality of Prison Life Adapted for Forensic Psychiatry

(MQPL-Forensic; Wong et al., 2008)

Specific questionnaire to forensic psychiatric environment (adapted by CRDS) of 69 items assessed on a 5-point Likert scale, divided into 15 dimensions (see Table

1

). Forensic Inpatient Quality of Life Questionnaire

(FQL; Vorstenbosch et al., 2014)

Table 1). ICCs are interpreted according to standards of Koo and Li (2016): <.50 = low; between .50 and .75 = moderate; between .75 and .90 = good; >.90 = excellent. Procedure