Blerina Gkotse
email: blerina.gkotse@cern.chwww

Georgi Gorine

Pierre Jouvelot

Federico Ravotti

Giuseppe Pezzullo

Isidre Mateu

Software Upgrades of Beam and Irradiation Test Infrastructures in AIDA-2020

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

AIDA-2020

AIDA-2020 is an EU-funded project about Advanced European Infrastructures for Detectors at Accelerators involving about 60 institutes in 24 EU countries. Within this project, the Work Package 15 (WP15) deals with beam and irradiation test facilities, infrastructures dedicated to the qualification of particle detectors, materials, and components prior to their installation in High-Energy Physics (HEP) experiments. In this paper, we present the contributions of our team at CERN to the WP15 activities.

These activities involve online databases of irradiation and test beam facilities, a dedicated web application for the data management of the CERN proton irradiation facility (IRRAD) and an ontology for irradiation experiments and their data management (IEDM).

AIDA

  for Detectors at Accelerators (AIDA-2020) is an EU-funded project that unites important European research infrastructures in the field of detector development and testing. Map of participant countries involved in AIDA-2020 CMS Pixel Detector View of the ATLAS calorimeters from below Beam and irradiation test facilities: Infrastructures for the qualification of Particle detectors, materials, and components prior to their installation in High-Energy Physics (HEP) experiments such as those performed at CERN. and test beam facilities: Online web application containing information about irradiation and test beam facilities. http://cern.ch/irradiation-facilities IRRAD Data Manager (IDM): A data management web application used in the Proton Irradiation Facility at CERN (IRRAD) Experiments, samples, users and dosimeters registration  Real-time follow-up of irradiation experiments  Computation of proton interaction parameters  Display and archive of dosimetry results  User Interface preferences customization  History and details of past experiments (with user permission) Online documentation with Widoco http://cern.ch/iedm Ontology representation with Protégé https://gitlab.cern.ch/bgkotse/iedm Graph representation of some core IEDM entities and properties IEDM includes concepts of data management of irradiation experiments extending classes from the Ontology of Scientific Experiments (EXPO), the Units of Measure ontology (OM) and the Friend-of-a-Friend ontology (FOAF).http://aida2020.web.cern.ch/content/newsletter


Quentin Schoen
email: quentin.schoen@mines-albi.fr

Franckfontanili
email: franck.fontanili@mines-albi.fr

Matthieu Lauras
email: matthieu.lauras@mines-albi.fr

Sébastien Truptil
email: sebastien.truptil@mines-albi.fr

Improving parcels transportation performance by introducing a hitchhiker parcel model

Keywords: Physical Internet, Transportation, Monitoring, Simulation I

Thanks to technological opportunities on data acquisition, broadcast and treatment in real time, new logistics opportunities are emerging quickly. These opportunities are about to change deeply the way we transport parcels through hyper-connected and open networks. In order to monitor the logistics flows on these routes and hubs, new ways of thinking are developed, like the Physical Internet paradigm. Because the current transport management systems should not be able to deal with this complex and open environment, we have to design new ways of monitoring parcels on the network. We describe here the \emph{hitchhiker parcel model}, a new paradigm allowing each parcels to take advantage of the local opportunities and beware the risks. The targeted organization described and the underlying questions have to be evaluated through performance indicators. The first experiments presented here aims to give first insights about the model evaluation and its relevance.

INTRODUCTION

If we take a look to expected logistics trends [START_REF]Logistics trend radar[END_REF] [2] [START_REF] Savelsbergh | 50th anniversary invited article-city logistics: Challenges and opportunities[END_REF] [4], it appears that the way we manufacture, store, distribute and use objects are about to change deeply. In fact, new technological opportunities like Internet of Things (IoT), cloud computing and data treatment abilities are enabling and fostering a relevant hyper-connection of any object to the World Wide Web.

This ability to equip any physical object with sensors (motion, temperature, pressure, location, etc.) and connect it easily and quickly to the Internet opens new opportunities in the logistics field. In particular for transport management, knowing the status and location of any actors and objects while they are spread in a region/country might be relevant to make appropriate decisions and meet new challenges. These technological opportunities are fostering an "anytime, anything, anywhere delivery" model [START_REF]Logistics trend radar[END_REF] in order to satisfy the customers' needs. This model implies a network complexification and require from carriers agility. As a consequence, logistics activities breakdown between stakeholders and servicisation try to offer solutions to link the producer to the customer in effective, agile and efficient way, wherever they are and whenever the order is made. These trends are already visible in our everyday life with resource sharing platforms (car and flat sharing, food delivery) and new delivery hubs as smart-lockers [START_REF] Faugere | Hyperconnected pickup & delivery locker networks[END_REF].

Despite these new trends, abilities and tools to manage the supply chain appear no longer adapted. In one of the first article presenting the Physical Internet paradigm, \cite{montreuil_toward_2011} illustrates some problems the \emph{current} logistics organisation is facing with on social, economical and environmental aspects. Air and packaging shipping, empty travel of vehicles and containers, carriers poor social working conditions, storage and transportation facilities misused and/or under-utilized are some of the symptoms presented in this paper. Meantime, despite the use of Transport Management Systems (TMS) by companies, we already observe lacks in the ability to deal with parcels in agile, effective and efficient manner [START_REF] Montreuil | Toward a physical internet: meeting the global logistics sustainability grand challenge[END_REF]. These centralized and deterministic systems were not designed initially to deal with the upcoming network complexity and parcels traceability while numerous actors are successively in charge with different logistics means (vehicles, hubs, storage facilities, etc.). This leads, among other issues, to the symptoms described above, and it should be worse in few years if do not try to re-design these systems.

In order to answer the parcels transportation monitoring challenge, the Physical Internet paradigm [START_REF] Montreuil | Toward a physical internet: meeting the global logistics sustainability grand challenge[END_REF] makes a list of disruptive recommendations and assumptions about the physical and informational standardisation of the supply chain. On the physical aspect it considers the encapsulation of the products in world-standard smart green modular containers, handled, sorted, stored and transported through adapted facilities. On an informational aspect it considers an hyper-connected world in which any object (container, vehicle, hub) may collect, treat, broadcast and receive data about its environment and/or state. In this situation, 3 types of monitoring are proposed:

1. A centralised monitoring: objects do not barely make decisions but relays informations to a agent 2. A pre-planned and de-centralised monitoring: the containers follow one of the several itineraries planned before its departure 3. A fully de-centralised monitoring: objects make decisions about themselves autonomously

The first proposition looks like the current organisation with a higher visibility of the field situation in real time. The second one tries to take advantage of parcels and tags new abilities in order to decentralised the parcels routing decisions, following one of the pre-determined schedules. The third one allows a full autonomy of the parcels, depending on the real time surrounding environments.

These three monitoring types may be relevant, depending on the situation and the maturity level of the interacting objects. However, a centralized system may not be truly relevant to deal with specific parcels and amounts of local transportation opportunities and breakdowns between stakeholders. Secondly, considering the expected network complexity in terms of nodes and stakeholders diversity and quantity on one hand, and on another hand, the agility need in this almost unpredictable system, the second proposition with schedules to follow appears a bit tight. For these reasons, trying to take full advantage of IoT expected capabilities in term of data exchange and treatment in real time, we think relevant to consider in the following parts the third parcel monitoring system.

As a consequence, in this paper we address the following research question: "Does a parcel decentralised monitoring system, based on smart and opportunistic parcels, increase the whole performance of the logistics network?". To answer that, we first describe the TMS evolution we think necessary to monitor the upcoming supply chain. Then, we describe the hitchhiker parcel model, a new paradigm we think relevant to deal with it. After that, we define a targeted organisation and some KPIs we should used to allow the parcel to find its way and make its decision itself. Next, we propose an experience plan and present the first experiments. Finally, we discuss this model and give perspectives to pursue these experiments.

II. BACKGROUND AND RESEARCH QUESTIONS

In this part, we describe in which extent the current TMS should evolve in order to pursue opportunities offered by the expected logistic environment. The first part gives on overview of the logistics opportunities and trends whereas the second one is focused on the current TMS approach and the required evolutions. In the third part we describe the hitchhiker parcel model we propose and address the research question.

A. New opportunities of the expected logistics network

An hyper-connected network: For Small and Medium Enterprise (SME) it may be expensive and it represents an important risk to buy a software to deal with their supply chain. For several years, the development of Software as a Service (SaaS) model is allowing a broader access to IT systems [7] [8], giving to SMEs the opportunity to choose precisely the functionalities they want [START_REF] Harris | ICT in multimodal transport and technological trends: Unleashing potential for the future[END_REF] and take full advantage of cloud computing. In addition, the SaaS provider takes care of the infrastructure (hardware, network, updates) to support the software. Thus, a plug-and-play approach allows the SMEs to focus on their process and core system rather than on the software. Moreover, because of this approach all the software units must be easily connected to other ones, which implies interoperability and foster data exchange. Little by little, this model creates an informational network through companies and customers, each one taking advantages of web services through Application Programming Interface (API).

Combined with the cloud computing and computing power increase, we assist for few years to the use of smart, small and affordable devices allowing people and objects to be connected anywhere at anytime [START_REF] Harris | ICT in multimodal transport and technological trends: Unleashing potential for the future[END_REF]. The Internet of Things (IoT) development allows easier and more reliable data collection and broadcast from the field to a decision support system [START_REF] Porter | How smart, connected products are transforming competition[END_REF]. On a transportation point of view it means an ability to know the location and state of any parcel and vehicle any time. Considering that we are able to treat in real time these amount of data through Big Data technologies [START_REF] Wamba | How 'big data' can make big impact: Findings from a systematic review and a longitudinal case study[END_REF], the supply chain management should be deeply modified thanks to the real time understanding of the field situation.

This hyper-connection of the objects should allow the firm to reach the objective of "any-time, anywhere delivery model" [START_REF]Logistics trend radar[END_REF]. In fact, we are facing with a distribution customization based on the consumers desires, which implies to build a network agile and reactive enough to fit with the consumer lifestyle. In order to reach this objective, the number of potential paths for a parcel between the firm and a consumer increase and the transshipments must not be a problem. The last mile delivery, the reverse logistic management, and the implementation of smart lockers [START_REF] Faugere | Hyperconnected pickup & delivery locker networks[END_REF] and shared warehouse are examples of facilities which should create a strong and agile network.

An open logistics network: In order to take advantage of these opportunities, we assist for some years to a splitting of logistic activities, each stakeholders being responsible of its own perimeter. Thus, to deal with this complexity, companies are requiring services from 3-4-5PL providers [START_REF] Vasiliauskas | Principle and benefits of third party logistics approach when managing logistics supply chain[END_REF] which aim to deal with their logistics activities (storage, transportation, product flow and supply chain management, optimization, etc.). Meantime, people are taking advantages of new services like shared car, shared flat, on demand transport, through new sharing economy platforms \cite{hamari_sharing_2016} which aim to link individuals needs with available abilities (Uber, Deliveroo, AirBnB, Blablacar, etc.). In both case, logistics activities are more and more considered as services offered by specialist companies and individuals as it is the case for SaaS.

This new way of thinking imply more collaboration and interfaces when a parcel needs to be transported from point A to point B. In fact, one actor may be responsible for the first transportation, a second one for the temporary storage on a transshipment hub, a third one for the last mile delivery and a forth one was responsible of real time data collection and traceability. Moreover all this transportation steps may be achieved by different transportation means like autonomous cars [START_REF] Van Meldert | Introducing autonomous vehicles in logistics: a review from a broad perspective[END_REF], platooning trucks systems [START_REF] Chan | Cooperative control of SARTRE automated platoon vehicles[END_REF], bicycle, drones, smart lockers [START_REF] Faugere | Hyperconnected pickup & delivery locker networks[END_REF] and shared [START_REF] Franklin | Shared warehousessharing risks and increasing eco-efficiency[END_REF] autonomous hubs. These opportunities imply an ability for Figure 1. TMS evolution parcels to be suitable for this automation and multi-modality which should contribute to improve the performance. Indeed, as explain by [START_REF] Harris | ICT in multimodal transport and technological trends: Unleashing potential for the future[END_REF] the multimodality adds constraints on the package, the transport and storage conditions but allows to combine the advantages of each transportation mean used (flexibility of road haulage, high capacity of trains, lower costs of boats). In order to reach this physical intermodality, [START_REF] Montreuil | Toward a physical internet: meeting the global logistics sustainability grand challenge[END_REF] wants to create and use standardise and smart Picontainers.

In this context of logistics activities servicization, splitting of responsibilities between several stakeholders, and use of different means more and more autonomous all the supply chain along, the logistics network appears more and more open and flexible. However, if we compare these expected needs and the TMS abilities, it seems that there is a gap to fill if we want tools able to deal with products all along the supply chain.

TMS origins: The Transport Management Systems were initially created in 90s to provide "an enabling tool for the safe and efficient operation of Freight Transport systems [START_REF] Giannopoulos | The application of information and communication technologies in transport[END_REF]. According to [START_REF] Zeimpekis | Urban dynamic real-time distribution services: Insights from SMEs[END_REF] the major functionalities were:

Following the vehicles location in real time and provide KPI about them Providing a delivery proof Monitoring the load through sensors and data collection (temperature, remaining capacity, etc.) In 2000s, the logistics network and activities became more complex, involving more and more actors responsible of specific perimeters. Because of this complexity, the interconnectivity between IT systems was not as efficient and relevant as it should have to be as [START_REF] Giannopoulos | The application of information and communication technologies in transport[END_REF] foresaw it. In order to solve these issues TMS functionalities and software components are becoming more interoperable for few years through plug-and-play components, available on SaaS platforms.

Current TMS functionnalities: As a consequence, the current TMS are most of the time split in components "activated" or not, depending on the company needs: the carriers may need transportation means and drivers management module whereas the companies which need their goods to be transported by subcontractors may need freight and vehicle exchange platform and real time traceability. This structure offers more flexibility and allows even SME to access this kind of service.

The current most common functionalities are represented in Figure 1 [19]: form edition for each entity to be transported, vehicle rounds planning and optimizing, real time vehicle monitoring, vehicle and container track and trace of each transportation step, freight and vehicle exchange platform, transport quality management, transportation means management, finances and billing activities, etc.

Current TMS approach and limitation: One of the main objective of the current TMS is to design in real time the most effective and efficient transportation rounds, minimizing the cost and maximizing the vehicle load. In addition, following the vehicle status and trailer/goods vehicle we consider that we follow the containers status. Nevertheless, it appears that this system does not prevent to ship a lot of air packaging [START_REF] Montreuil | Toward a physical internet: meeting the global logistics sustainability grand challenge[END_REF] and multi-modality is still hardly considered because the current practices may slow down the delivery and endanger the parcel. The TMS organisation with rounds and vehicles monitoring does not foster agility [START_REF] Barthe-Delanoe | A platform for event-driven agility of processes: A delivery context use-case[END_REF] in case of unexpected event requiring products transported detour and real time decision. The whole logistics network is split between numerous competitive stakeholders networks with their own means which implies redundancy and inefficiency. Finally, this organisation and vehicle monitoring is often not sufficient for sensitive products monitoring which require more precision about their environment (temperature, shocks, direction, etc.).

Considering the expected logistics environment described before, these current limitations should have a strong impact, preventing to monitor in a relevant way the parcel on the upcoming network.

B. Expected failures and key elements to improve

Taking into account the previous sections, we summarize here some key elements we should work on in Figure 1 [START_REF] Schoen | A new approach for supply chain management monitoring systems adapted to crisis[END_REF]:

Parcels monitoring: Following in real time each parcel status and location. Ability to make decision about them considering all their constraints and their environment.

Vehicles and hubs monitoring: Taking advantage of any storage and load displacement capability in order to improve the whole network effectiveness and efficiency. Automatic parcel assignment in the available means: Defining rules based on KPIs and compatibility between parcels and vehicles/hubs in order to assign automatically the first ones to the second ones. Considering the expected capabilities of data collection and treatment technologies, on one hand, and on the other hand, the major logistics trends described above, these TMS evolution appear feasible and relevant.

C. The hitchhiker parcel model

Description: In this hyper-connected and open environment, it appears to us (cf. Introduction) that a decentralised and opportunistic model to monitor parcels would be relevant. In this situation, each parcel may be compared with a hitchhiker. In fact, it takes advantages of the opportunities and beware the risks, without knowing in advance its itinerary to reach the recipient, as a hitchhiker during its trip. Analysing its environment and knowing its products constraints, each parcel should be able to make relevant decisions about vehicles to take and hubs to be stored in, waiting for another displacement opportunity. As it is anticipated in the future by [START_REF] Montreuil | Toward a physical internet: meeting the global logistics sustainability grand challenge[END_REF], we consider distributed multi-segment intermodal transport as the major transport organisation. This means that on the whole network vehicles are mostly similar to shuttles between hubs able to deal with different vehicle types.

In our study, considering the collaborative economy and platforms which already emerged, we think relevant to take also into account the vehicles that did not initially planned to carry parcels. The vehicles flow, regardless of their type, is considered as load capacity displacement. Any vehicle, in particular private ones, passing nearby the hub a parcel is stored in, planning to pass close to another one considered interesting by the parcel to get closer to its destination, is susceptible to carry it. To summarize, we consider the parcels displacement needs on one hand, on the other hand, we consider the displacement and storage means susceptible to meet the parcels needs.

Decision making: Currently each carrier takes in charge the parcel he has been ordered for. In the hitchhiker parcel model the parcel decides itself whether or not a transportation availability is relevant for it. Thus, in this context we have to answer the following questions:

1. How does a parcel decide that a transportation opportunity is relevant for it? 2. When a vehicle offers a transportation opportunity to parcels, how does the vehicle decide the ones it takes in charge if the interested parcels number / volume / weight exceeds the vehicle capacity? Our goal is to define a new way of monitoring the parcels in the upcoming logistics environment, because we expect the current monitoring systems inappropriate to deal with it. To answer this challenge, we make the assumption that the Physical Internet opportunities and the hitchhiker parcel model are adapted. As a consequence, in order to evaluate the relevance of our model, we have to answer the following research question: "Does a parcel decentralized monitoring system, based on smart and opportunistic parcels, increase the whole performance of the logistics network?

To answer it, we have to describe the targeted organization and KPIs that parcels, vehicles and hubs may use to make decisions. In fact, depending on KPIs and decision making algorithms used, the parcels behavior will change, so will the global logistics network performance. Experiments in section IV are designed to answer the research question, evaluating the performance of this model.

III. PROPOSAL

A. The hitchhiker parcel and sensitivity

In the described hitchhiker parcel model, each parcel must be able to track physical data about its environment, exchange them with the surrounding elements (vehicles, hubs), interpret them and make decisions based on these treatments by itself.

As a hitchhiker, each parcel may be different and more or less affected by its environment and unwanted events like delay or unsuitable transportation conditions. The Merriam-Webster dictionary define sensitivity as: the quality or state of being sensitive: such as the capacity of being easily hurt Thus, in order to distinct the parcels we define a sensitivity factor si which aims to make a difference between a parcel we should monitor strictly and another one which may not be affected as much by an unexpected and/or unwanted event. This factor is set arbitrarily between 1 (ie. low sensitivity) and 10 (ie. high sensitivity) and depends on the parcel internal factors (product fragility, scarcity, dangerous nature, cost, thermal/mechanical/chemical/... insulating capacity of the container, etc.) and external ones (impacts of a delivery issue on the parcel recipient). In the further KPIs description we use this factor dimensionless to discriminate the parcels and set a priority between them. -T1: Opens the package and take the products In case of transshipment, a second carrier repeats the same process with parcels left on this site by the first one. In any case, we do not consider in the whole paper the steps achieved by the sender nor the recipient because it does not belongs to the transportation steps.

In our proposal, the key questions (cf. II-C) are asked in carrier's task T2 when several options are possible for the parcel.

Organizational model: We describe in the figure 2 the organizational model with BPMN language. We make the assumption that vehicles arrive empty in the loading area of the departure site and unload all the parcels they have transported in the destination hub. As we see it on the diagram, we can split the decision process in 3 steps:

1. Arrived vehicle broadcast of its identity and route to the parcels on the site 2. Parcels KPI calculation and decision on the willingness to take advantage of this transportation opportunity 3. Classification of the voluntary parcels in case of vehicle capacity issue We observe the data exchange between the vehicle and parcels once a new opportunity appears. The decision of being volunteer for this transportation opportunity is calculated considering the vehicle parameters, the itinerary parameters (edge and destination hub) and the parcel parameters (state, destination, sensitivity factor, etc.).

C. KPIs to make a decision

We describe in the following parts some KPIs which appear to us relevant for a parcel to decide whether or not a vehicle represents an opportunity and to classify the volunteer parcels. We make the assumption that any parcel must reduce as much as possible the time spent between the sender and the recipient. In fact, during transportation and transshipment steps parcels are expected by their recipients and exposed to risks that may injured products inside, in particular if the sensitivity factor si is high.

In the following parts, for each parcel, we settle the first step (n=1) when a vehicle leaves the sender's hub, and the last one (n=2N+2) when a vehicle stops in the parcel's recipient hub, considering N as the number of transshipment hubs.

Real time and step sensitivity: In order to take into account the sensitivity of each parcel in the shipments monitoring, we define in (1) the real time sensitivity for each parcel i, just after the step n, which means between .

(

) 1
Based on (1) we define in (2), the step sensitivity of parcel i between steps n and n+1.

(2)

Based on historical values registered by former parcels on the same place, if a parcel is able to estimate the time it will spend between two steps (loading -unloading or unloading -loading) it is able to estimate in advance its potential future step sensitivity.

Cumulative real time and step sensitivity: Based on the former equations we can add up the step sensitivities all the transportation steps along in order to compare the parcels, whatever their previous itinerary were. This is possible if parcels considered crossed at least one transshipment site, ie. N>1. Thus, we define the cumulative real time sensitivity of each parcel i between its departure from the sender site and any given moment in (3):

(3) Based on (3) we define in (4), the cumulative step sensitivity of parcel i considered at step n.

(4)

As a consequence, (3) may be simplified with (5): Equation ( 5) registers the transportation duration of each parcel so far and take into account their sensitivity to compare them. The highest this value is, the fastest the parcel should be transported to its recipient. However, we do not know if the situation is urgent or not, the parcel may have plenty of time to reached the recipient.

Maximum transit time: In order to fill this gap, we define for each parcel i. This is the difference between the latest arrival time (LA) and the earliest departure time (ED. Thanks to we define as the portion of available time consumed in real time} in (6) which add the time consumed between the earliest departure and the first care, and the time consumed in transportation step(s) so far. [START_REF] Montreuil | Toward a physical internet: meeting the global logistics sustainability grand challenge[END_REF] If their is no transshipment, ( 6) is simplified with ( 7).

Thus, this indicator is useful to detect the parcels which should be transported quickly in order to respect their latest arrival time. However, this indicator does not take into account the network ability to forward the parcel on time. In fact, even if = 0.8, if this parcel is very close to its recipient it might be more relevant to make it wait, compared with another one with = 0.5 but very far from the recipient. NB: The sensitivity factor is not part of these equations not to make them more complex but it could be a perspective.

Network conveyance capacity:

Considering the hitchhiker parcel model we do not know in advance edges and hubs a parcel will pass through to reach its destination. Nevertheless, we can use the shared historical data about previous parcels which pass through these links. In fact, thanks to these data we may predict how much time will be necessary to leave a hub and reach another one, regardless the itinerary. We define as the average time for a y type parcel to travel from step n to step m in a vehicle type z. is defined between two steps which implies that it is a discrete variable. To define , the network conveyance capacity, we use to evaluate the time already consumed and the time that should be consumed to reach the recipient site in [START_REF] Harris | ICT in multimodal transport and technological trends: Unleashing potential for the future[END_REF]. In this specific case, the step m is the recipient. If Λ << 1, apparently the parcel will arrive on time to its destination If Λ ≈ 1, apparently the parcel should arrive on time to its destination but the probability is lower than the first one If Λ >> 1$, apparently the parcel will be late to reach its destination, an alternative solution should be found to avoid this. This indicator appears as a good one to evaluate in real time a parcel status. Combined with and it allows the parcel and the vehicle to evaluate the urgency of parcel to be taken in charge.

In the organisation we described so far, parcels may go through several hubs to reach their destination. In order to determine if a hub is relevant for a parcel to get closer to the recipient hub we define 4 steps: n is the potential loading step of a parcel $i$ in its current hub v is the step of delivery on the temporary hub v+1 is the loading step on the temporary hub m is the delivery step of a parcel i on its recipient hub With these steps we define , the convergence rate of type y parcel which is offered to go on site v to reach its destination m in [START_REF] Porter | How smart, connected products are transforming competition[END_REF]. [START_REF] Porter | How smart, connected products are transforming competition[END_REF] Equation ( 10) compares two options:

1. The estimated time to leave from the current site to the vehicle destination added to the time spent there and the time to go from there to the parcel destination 2. The average time to leave from the current site and go the destination site It is important to understand that all the are considered regardless to the itinerary.

allows us to evaluate if a vehicle destination is relevant for a parcel. Once again, this indicator is useful if compared to 1:

If Cv << 1, it is really interesting to take this transportation because it shortens the time to reach the destination compared with a direct path If Cv = 1, the vehicle destination may be the parcel destination. If it is not, the detour is not disadvantageous from a time perspective (it might be less interesting on a cost or risk exposure perspective).

If Cv >> 1, the vehicle destination represents a detour time consuming, compared with an average lift from the current site to the parcel destination site.

Based on this, we can imagine a Cv maximum level, above which a parcel does not even consider the transportation opportunity because the vehicle destination would represent a too-long detour. In order to answer our research question, we define an experimental plan to measure in what extent the hitchhiker parcel model improves the performance of the whole expected network. The main idea is to compare current optimal rounds organisation with our model, modifying and improving the KPIs used and the decision making algorithms. We present here the first experiment we led to measure in a simple use case if our model may be relevant. A second experiment is led in order to improve the effectiveness and discuss the targeted organisation. We built from scratch a discrete-event simulation model runned by Witness© in order to lead these experiments. This simulation platform is relevant because the number of parameters (about vehicles, parcels, hubs, routes) we can play with is high. In addition, we need to understand precisely, step by step, what would happen with each decision making algorithms.

A. Description of the experiment and assumptions

Being the first experiment we run on this model we first make it simple to be able to analyse the results and consistent with a real use case. In fact, the situation described is based the blood supply chain [START_REF] Schoen | Tracking in real time the blood products transportations to make good decisions[END_REF] where a major hub (the production plant) needs to send different products to distribution centres. These distribution centres are meantime collection centres of raw materials. This organization is based on the well-known milk-run problem [START_REF] Brar | Milk run logistics: Literature review and directions[END_REF].

The network is made of 4 sites, 1 major and 3 secondary ones. The transport commonly used in this situation is a pick-up and delivery rounds, leaving first from the major site and visiting all the secondary sites one after the other. If a parcel needs to be transported from a secondary site to another one, it might needs to be stored temporarily on the major site before the second same round begins.

The figure 3 represents this network. First, we define the required time to cross each edge (yellow continuous lines). Secondly we define the average time spends by a parcel on a hub in the hitchhiker parcel model (grey circles on hubs). Considering a vehicle stays 2 time units on a hub (yellow circles on hubs) and might be anywhere when a parcel is ready to go, we use the average between the half of a return trip to its both surrounding site. For instance, the calculation for 8 on h1200 is: ((6+2+6)/2)+((8+2+8)/2)/2. The yellow dotted lines define the time to go from one hub to the other one but does not represent a passable edge. The average time on them is the average between both paths in the hitchhiker parcel model. For instance, the calculation for 28 between h1100 and h1300 is: ((6+8+8)+(12+12+10))/2. Each edge may be used both directions. We consider just one type of vehicle, the same for both cases (rounds and hitchhiker parcel model), with the same speed and an infinite capacity.

We consider that all parcels are the same in term of weight, dimensions and maximum transportation duration (earliest departure ED = 0). The only differences are the sender, the recipient and the sensibility factor, depending on the parcel type. We consider that the major hub h1100 sends 4 parcels to each hub. Meantime, each site sends 3 parcels. We summarize these parcels sending in table I.

B. Round description

In order to compare the effectiveness of both models, rounds and hitchhiker parcel model, we begin to design a simple pick up and delivery (P&D) round. We consider that all the parcels are available on each site at the beginning (t=0) and the vehicle arrives at t=3, because the round is well designed to suit the needs. As define hereafter the vehicle arrives on hub h1100 (t=3), it loads all the parcels, leaves the hub (t=5), arrives on hub h1200 (t=11), unload the parcels whose the recipient is this site and loads the parcels addressed to hub h1100. Then, it leaves the hub (t=13), etc.

C. The hitchhiker parcel model, first experiment

Experiment: With parcels described in Table I we test here the model defined above, considering as a decision on the willingness to take advantage of a transportation opportunity: Cv ≤ 1. In this case the parcel is eligible to the transport, in the other case it stays on hub waiting for another opportunity. We checked before that this rule does not prevent a parcel to reach its destination thanks to at least one specific way. We used 4 vehicles equivalent to shuttles, each one responsible of an edge. When the experiment begins, each of them leaves a hub empty, travelling to the next one.

We gather on Table II the results and a comparison with the round. KPIs used to compare the scenarios are the cumulative step sensitivity Ω and the portion of available time consumed Ψ. The first one indicates for each parcel how long was the trip and the second one indicates when it took place.

Analysis: As we can see, Ψhh > Ψr for parcels c01 to c04 because even if these 4 parcels took the same time to travel from h1100 to h1200 (Ωhh = Ωr) with both organisations, the round took them first. Secondly, Ωhh is very high for parcels c05 to c08. This implies they waited a long time on hub h1200 to go from h1100 to h1300, after travelling with c01 to c04.

On the contrary, all the other parcels took less time with the hitchhiker parcel model than with the round Ωhh < Ωr). Parcels c09 to c12 either stayed all the round along transported in the vehicle waiting for h1400, either took the vehicle which go back and forth from h1100 to h1400. The second option takes less time than the first one. Conclusions are the same for c13 to c15, the travel is shorter taking directly the vehicle between h1100 to h1200 than waiting into the vehicle to reach h1100. Parcels c16 to c18 are an example of the parcel selection rule based on Cv. In fact, the travel is shorter passing by h1200 than h1400. Thus, thanks to a vehicle available at the right hub at the right moment these parcels took less time to go from h1300 to h1100 with the hitchhiker parcel model and arrived before the round (Ψhh < Ψr).

Finally, parcels c19 to c21 arrived before the round with the hitchhiker parcel model but consumed a very long time (0.95). This is explainable by the fact that the vehicle initial conditions (leaving h1400 empty) are not optimal for these parcels. As for the round this is logical, they waited for the vehicle to arrive from h1300.

Conclusions: As we can see, the initial conditions have a major impact on the results. In fact, in the hitchhiker parcel model experiment, the vehicle synchronisation is essential not to make parcels waiting too much time on a hub. In a larger extent, the highest is the frequency of vehicle visits on a site, the best effective this organisation will be. Moreover, as the parcels c16 to c18 demonstrate it, passing by a hub with a transshipment may be more efficient than waiting in a hub a round to come. In fact, as for the parcels leaving from h1200 and h1400, the frequent vehicle visits in both direction creates several opportunities to reach hub h1100. Finally, to make this system efficient, it appears necessary to share the vehicles with other company needs and to consider private vehicles to fill them as much as possible . In fact, whereas the round travelled 36 time units, the hitch-hiking vehicles consumed 214 time units.

D. The hitchhiker parcel model, second experiment

Experiment: In this second experiment, considering that frequency and initial conditions are key elements for the hitchhiker parcel model to be effective, we change them. The major improvement are the initial conditions, considering that the 4 vehicles are similar to the round, arriving on a site after 3 time units and offering a load capacity displacement to another hub. We do not change anything else in the hitchhiker parcel model experiment. Nothing is changed in the round experiment.

Analysis: Parcels c01 to c04 obtain logically the same results with both organisation. A vehicle arrives (t=3), leaves h1100 to reach h1200 and unload parcels. Parcels c05 to c08 need more time with the hitchhiker parcel model compared with the round (Ωhh > Ωr), waiting on h1200 but the synchronisation is far better than before. In fact, in total these 4 parcels use 532 time units versus 988 in the first experiment. In addition, Ψhh as decreased from 1.62 to 0.79 which confirms that they are cared fastest than in the first experiment. Parcels c09 to c12 require logically the same time than in the first experiment, as it is the case for parcels c13 to c15 and c19 to c21. Indeed they just need to be transported one time directly to h1100, thus Ωhh does not change, compared with the first experiment. Finally, we notice that Ψhh raises a bit, compared with the first experiment for parcels c09 to c18. This is explained by the initial conditions change, vehicles arriving 5 time units later.

Conclusion: This second experiment allow us to conclude that in this very simple case initial conditions and vehicle hub visits frequency are key elements. The results are presented in Table IV. These encouraging results confirm that the model we present in this paper may be relevant in the expected logistics environment. The vehicle synchronisation is essential and have a direct impact on the network capacity to dispatch parcels. Indeed, contrary to the first experiment, vehicles in the hitchhiker parcel model uses 92 time units instead of 214. Hubs with the highest vehicle opportunities will be apparently preferred to less visited ones. Another option could be to signal to any vehicle around a hub that a parcel just arrived on it and broadcast its parameters. This would allow parcels not just to wait for a vehicle but inform and encourage the transportation means to pay attention to this need.

V. CONCLUSION

As we demonstrate it, the upcoming logistics environment is about to offer new opportunities the current management systems should not be able to seize. In order to meet the upcoming logistics challenges offered by new technologies, required by the ineffectiveness of the current system and customers wishes, new way of thinking like the Physical Internet emerged. Based on these assumption, our model aims to provide a original way of monitoring parcels on an open and hyper-connected network. The hitchhiker parcel model we built and evaluated is the first step before other experiments. The first ones we present here prove that on a simple case, this model may be relevant to carry parcels on time. In addition, as we presented it, this decentralised system based on smart and opportunistic parcels could increase the whole performance of the network. In this simple case, compared with the round organisation, parcels were transported less time and delivered in average sooner. However, the efficiency must be improved, considering crowd-sourcing transportation and mixing all the parcels to raise the vehicles fill rate. In fact, the first experiments lead to an important increase of the time units consumed by vehicles.

This model have to be improved and its effectiveness confirmed by broader and more complex experiments. This future research works will compare the current situation with the expected one considering for example:

Vehicles capacity issues A bigger network with more hubs, edges and vehicles Cost relevant indicators to monitor the parcel which owns a pot Risks relevant indicators to monitor the parcel through secure hubs and edges New parcel choices algorithms combining time, cost and risks indicators Unwanted event during transportation steps and parcel surveillance to measure agility capabilities Ephemeral hubs settled dynamically by vehicles exchanging parcels without long term storage facilities Etc. Future paper addressing these parameters and limitations should allow a better understanding of this kind of model capabilities and relevant implementation perimeters.

Figure 2 .

 2 Figure 2. Parcel transportation: Expected scenario

( 7 )

 7 Finally, we can calculate the portion of available time consumed on step n} with (8).

Figure 3 .

 3 Figure 3. Experimental network required time

TABLE I .

 I PARCELS PARAMETERS DESCRIPTION

TABLE II .

 II FIRST EXPERIMENTAL RESULTS

	Id	Ωhh	Ωr	Ωr -Ωhh	Ψh	Ψr	Ψr -Ψr
	c01	18	18	0	0.52	0.26	-0.26
	c02	30	30	0	0.52	0.26	-0.26
	c03	54	54	0	0.52	0.26	-0.26
	c04	12	12	0	0.52	0.26	-0.26
	c05	156	48	-108	1.62	0.5	-1.12
	c06	260	80	-108	1.62	0.5	-1.12
	c07	468	144	-324	1.62	0.5	-1.12
	c08	104	32	-72	1.62	0.5	-1.12
	c09	36	84	48	0.62	0.79	0.17
	c10	60	140	80	0.62	0.79	0.17
	c11	108	252	144	0.62	0.79	0.17
	c12	24	56	32	0.62	0.79	0.17
	c13	30	170	140	0.33	1.12	0.79
	c14	30	170	140	0.33	1.12	0.79
	c15	54	306	252	0.33	1.12	0.79
	c16	100	120	20	0.71	1.12	0.4
	c17	100	120	20	0.71	1.12	0.4
	c18	180	216	36	0.71	1.12	0.4
	c19	60	60	0	0.95	1.12	0.17
	c20	60	60	0	0.95	1.12	0.17
	c21	108	108	0	0.95	1.12	0.17

TABLE III

 III

			.	SECOND EXPERIMENTAL RESULTS
	Id	Ωhh	Ωr	Ωr -Ωhh	Ψh	Ψr	Ψr -Ψr
	c01	18	18	0	0.26	0.26	0
	c02	30	30	0	0.26	0.26	0
	c03	54	54	0	0.26	0.26	0
	c04	12	12	0	0.26	0.26	0
	c05	84	48	-36	0.79	0.5	-0.29
	c06	140	80	-60	0.79	0.5	-0.29
	c07	252	144	-108	0.79	0.5	-0.29
	c08	56	32	-24	0.79	0.5	-0.29
	c09	36	84	48	0.74	0.79	0.05
	c10	60	140	80	0.74	0.79	0.05
	c11	108	252	144	0.74	0.79	0.05
	c12	24	56	32	0.74	0.79	0.05
	c13	30	170	140	0.45	1.12	0.67
	c14	30	170	140	0.45	1.12	0.67
	c15	54	306	252	0.45	1.12	0.67
	c16	100	120	20	0.83	1.12	0.29
	c17	100	120	20	0.83	1.12	0.29
	c18	180	216	36	0.83	1.12	0.29
	c19	60	60	0	0.41	1.12	0.71
	c20	60	60	0	0.41	1.12	0.71
	c21	108	108	0	0.41	1.12	0.71

TABLE IV

 IV

	.	EXPERIMENTS COMPARISONS
	Experiment	Σ Ωhh	Ψhh	Σ Ωr	Ψr
	1	2052	0.812	2280	0.774
	2	1596	0.582	2280	0.774


Aude Ratier

Christelle Lopes
email: christelle.lopes@univ-lyon1.fr

Pierre Labadie

Hélène Budzinski

Nicolas Delorme

Herve Queau

Laurent Peluhet

Olivier Geffard

Marc Babut

Hervé Quéau

A Bayesian framework for estimating parameters of a generic toxicokinetic model for the bioaccumulation of organic chemicals by benthic invertebrates: proof of concept with PCB153 and two freshwater species

Keywords:

  A Bayesian framework for estimating parameters of a generic toxicokinetic model for the bioaccumulation of organic chemicals by benthic invertebrates: Proof of concept with PCB153 and two freshwater species

Introduction

In environmental risk assessment (ERA), models based on toxicokinetic (TK) approaches are widely recognized as providing diagnostics (models for understanding) and prognostics (models for prediction), sometimes used by decision-makers (e.g., [START_REF] Pavan | Review of QSAR Models for Bioconcentration[END_REF][START_REF] Epa | Approaches for the Application of Physiologically Based Pharmacokinetic (PBPK) Models and Supporting Data in Risk Assessment[END_REF][START_REF] Ipcs | Characterization and Application of physiologically Based Pharmacokinetic Models in Risk Assessment[END_REF]. TK models are relevant and widely used to predict chemical concentrations in biological organisms from those to which they are exposed in their environment. This process, also called bioaccumulation, depends on environmental conditions (temperature, light, food availability), contaminant properties (octanol-water partition coefficient, water solubility, dissociation and volatilization, sorption in sediment) [START_REF] Mamy | Prediction of the fate of organic compounds in the environment from their molecular properties: A review[END_REF] and biological characteristics of the species (life traits, diet, lifecycle). TK models describe the process of bioaccumulation as the net balance between the uptake of contaminants from different sources (water, diet) and their elimination by different processes (excretion, growth and/or biotransformation) [START_REF] Mackay | Bioaccumulation of persistent organic chemicals: Mechanisms and models[END_REF]. Different TK models have been proposed, such as compartmental models and physiologically based toxicokinetic (PBTK) models [START_REF] Grech | Toxicokinetic models and related tools in environmental risk assessment of chemicals[END_REF][START_REF] Landrum | Toxicokinetics in aquatic systems: Model comparisons and use in hazard assessment[END_REF]. Compartmental models describe toxicant fluxes between compartments, which may or may not have a physiologic or anatomic meaning. A PBTK model subdivides the body into compartments representing real tissues or organs connected through a fluid, usually blood [START_REF] Bois | Modeling Pharmacokinetics[END_REF]. For aquatic invertebrates, compartmental models were originally developed for metals and then for some organic contaminants where the organism is often considered as a single compartment.

For soluble contaminants, it is usually assumed that the water column is the main exposure and uptake route. Nevertheless, dietary uptake is of greater importance for hydrophobic contaminants due to their high adsorption on organic matter or food [START_REF] Gross-Sorokin | Uptake and depuration of 4-nonylphenol by the benthic invertebrate Gammarus pulex: How important is feeding rate?[END_REF]. Moreover, it has been shown for several aquatic invertebrates, such as the insect Chironomus riparius, that exposure to sediment cannot be ignored [START_REF] Leppänen | Relative importance of ingested sediment and pore water as bioaccumulation routes for pyrene to oligochaete (Lumbriculus variegatus, Muller)[END_REF][START_REF] Sidney | Trait-based modelling of bioaccumulation by freshwater benthic invertebrates[END_REF]. The importance of dietary uptake for aquatic invertebrates has been increasingly assessed in recent years [START_REF] Ashauer | Toxicokinetic and toxicodynamic modeling explains carry-over toxicity from exposure to diazinon by slow organism recovery[END_REF][START_REF] Carrasco-Navarro | Bioconcentration, biotransformation and elimination of pyrene in the arctic crustacean Gammarus setosus (Amphipoda) at two temperatures[END_REF][START_REF] Englert | Relative importance of dietary uptake and waterborne exposure for a leaf-shredding amphipod exposed to thiacloprid-contaminated leaves[END_REF][START_REF] Miller | Uptake, biotransformation and elimination of selected pharmaceuticals in a freshwater invertebrate measured using liquid chromatography tandem mass spectrometry[END_REF][START_REF] Rösch | Mechanistic Understanding of the Synergistic Potential of Azole Fungicides in the Aquatic Invertebrate Gammarus pulex[END_REF]. Chironomus sp. and Gammarus sp. are freshwater invertebrates widely used in ecotoxicology, due to their widespread presence throughout the Northern hemisphere and their capacity to accumulate various organic and inorganic contaminants [START_REF] Amiard | Comparative study of the patterns of bioaccumulation of essential (Cu, Zn) and non-essential (Cd, Pb) trace metals in various estuarine and coastal organisms[END_REF]Ashauer, 2012;[START_REF] Bertin | Bioaccumulation of perfluoroalkyl compounds in midge (Chironomus riparius) larvae exposed to sediment[END_REF][START_REF] Lebrun | Waterborne nickel bioaccumulation in Gammarus pulex: Comparison of mechanistic models and influence of water cationic composition[END_REF][START_REF] López-Doval | Effects of indomethacin and propranolol on Chironomus riparius and Physella (Costatella) acuta[END_REF][START_REF] Lydy | Effects of temperature on the toxicity of Mparathion, chlorpyrifos, and pentachlorobenzene to Chironomus tentans[END_REF]. These organisms are also an important food source for fish, amphibians and birds [START_REF] Macneil | The dynamics of predation on Gammarus spp. (Crustacea: Amphipoda)[END_REF], hence the transfer of contaminants within the aquatic food web.

Nowadays, TK model parameters are often determined from short-term exposures of organisms under controlled laboratory conditions [START_REF] Ashauer | Toxicokinetic and toxicodynamic modeling explains carry-over toxicity from exposure to diazinon by slow organism recovery[END_REF]. OECD Guideline 315 (OECD, 2008) suggests two methods to estimate the uptake and elimination rates. The most frequently used sequential method estimates the elimination rate using nonlinear regression depuration data which is then fixed to estimate the uptake rate with the uptake data. As elimination also occurs during the accumulation phase, separately estimating parameters that are linked does not allow taking into account their correlation on uncertainty. However, the precision of parameter estimates is a relevant point to strengthen environmental assessments [START_REF] Lin | A Bayesian approach to parameter estimation for a crayfish (Procambarus spp.) bioaccumulation model[END_REF][START_REF] Richards | Temperature-dependent bioaccumulation of copper in an estuarine oyster[END_REF]. The simultaneous method estimates both the uptake and elimination rates together, and is considered a potentially more reliable and more realistic model. Only recent studies [START_REF] Ashauer | Toxicokinetic and toxicodynamic modeling explains carry-over toxicity from exposure to diazinon by slow organism recovery[END_REF][START_REF] Miller | Assessing the Reliability of Uptake and Elimination Kinetics Modelling Approaches for Estimating Bioconcentration Factors in the Freshwater Invertebrate, Gammarus Pulex[END_REF][START_REF] Miller | Uptake, biotransformation and elimination of selected pharmaceuticals in a freshwater invertebrate measured using liquid chromatography tandem mass spectrometry[END_REF] have applied simultaneous methods for parameter approximation. Sequential or simultaneous methods can be deployed in a frequentist or Bayesian approach. Apart from the problem of the correlation between all the model's parameters, the frequentist approach (sequential or simultaneous) cannot simultaneously use different kinds of data (e.g., bioaccumulation and growth data) to estimate common parameters. Bayesian inference bypasses these limits by estimating all model parameters from all kinds of data [START_REF] Gelman | Bayesian Data Analysis[END_REF] and thus provides a more comprehensive approach and a better quantification of uncertainty in parameter estimates as well as a better consideration of variability in model predictions [START_REF] Bernillon | Statistical issues in toxicokinetic modeling: A Bayesian perspective[END_REF]. The application of Bayesian inference to TK models in aquatic invertebrates remains limited [START_REF] Lin | A Bayesian approach to parameter estimation for a crayfish (Procambarus spp.) bioaccumulation model[END_REF].

The aim of this study was to propose a Bayesian framework to simultaneously estimate all the parameters of a generic TK model from uptake and elimination data together. As a first development step, we applied this concept to two aquatic invertebrate species. The resulting joint posterior distribution giving the probability distribution of all parameters together will enable a more accurate assessment of uncertainty around estimates and thus TK model predictions. We illustrate our approach with the bioaccumulation of the well-known contaminant PCB153 by two freshwater benthic invertebrate species with different life traits: the Diptera C. riparius and the amphipod crustacean Gammarus fossarum.

Model

Generic TK model

A first-order kinetic bioaccumulation model that accounts for different uptake pathways and elimination processes can be expressed as follows (Eq. ( 1)):

dC(t) dt = ∑ (k i ×C i (𝑡𝑡)) n i=1 -∑ (k j ×C(t)) p j=1 (1)
where C(t) is the contaminant concentration at time t (days) in the whole organism (ng gorg -1 where the mass of the organism is expressed in wet weight (ww)), n is the number of contamination sources, ki the uptake rate from the contamination source i, Ci(t) the contaminant concentration in the contamination source i at time t (days), p the number of elimination processes and kj the elimination rate related to process j.

If we consider that Ci(t) is constant over time, which is appropriate in laboratory conditions, Eq. ( 1) can be analytically integrated by distinguishing uptake (Eq. ( 2)) from elimination phases (Eq. ( 3)):

⎩ ⎪ ⎨ ⎪ ⎧ C(t)= ∑ (k i ×C i ) n i=1 ∑ k j p j=1 + �C 0 - ∑ (k i ×C i ) n i=1 ∑ k j p j=1 � × e -�∑ k j p j=1 �×t for 0 < t < t c
(2)

C(t)= ∑ (k i ×C i ) n i=1 ∑ k j p j=1 × e -�∑ k j p j=1 �×(t-t c ) + �C 0 - ∑ (k i ×C i ) n i=1 ∑ k j p j=1
� × e -�∑ k j p j=1 �×t for t > t c

(3) where C0 is the contaminant concentration in the whole organism at the beginning of exposure (ng gorg -1 ) and tc the time at the end of the uptake phase (days).

Application of the model to the two species studied

Chironomids

For chironomids, we consider that the exposure sources are water (respiration) and sediment (ingestion), while elimination occurs due to excretion and growth dilution; Eq. (1) can thus be rewritten as follows (Eq. ( 4)):

dC(t) dt = k w ×C w +k s ×C s -�𝑘𝑘 𝑒𝑒 +k g � ×C(t) (4)
where kw is the uptake rate from the water (L gorg -1 d -1 ), Cw the contaminant concentration in water (ng L -1 ), ks the uptake rate from the sediment (gsed gorg -1 d -1 ), Cs the contaminant concentration in sediment (ng gsed -1 dw), ke the elimination rate (d -1 ) and kg the growth rate (d -

1

).

The chironomid growth rate is obtained from the von Bertalanffy growth equation (Eq. ( 5)), one of the most widely used models for describing the growth of benthic invertebrates [START_REF] Von Bertalanffy | A quantitative theory of organic growth (inquires on growth laws. II)[END_REF]K. Nakamura, 1973):

L(t) = L max -(L max -L 0 )×e (-𝑘𝑘 𝑔𝑔 ×𝑡𝑡) (5)
where L(t) is the chironomid size (mm) at time t (d), Lmax is the asymptotic size (mm), L0 is the size at birth (mm) and kg is the growth rate (d -1 ).

Given that, for highly hydrophobic compounds, contamination from water could be restricted (kw=0), a sub-model accounting for sediment as the only contamination source was also considered:

dC(t) dt = k s ×C s -(k e +k g )×C(t) (6)

Gammarids

Gammarids feed on detritus such as litter [START_REF] Forrow | Toward a mechanistic understanding of contaminantinduced changes in detritus processing in streams: Direct and indirect effects on detritivore feeding[END_REF]. As a consequence, exposure to chemicals could occur from water, litter (leaves) and sediment consumption, since sediment particles deposit on the surface of leaves when gammarids forage [START_REF] Bertin | Potential exposure routes and accumulation kinetics for poly-and perfluorinated alkyl compounds for a freshwater amphipod: Gammarus spp. (Crustacea)[END_REF]. Furthermore, we assumed that gammarids would not grow during the experiment, as shown by [START_REF] Galic | Effects of temperature on the performance of a freshwater amphipod[END_REF], for the adult size considered here. As a result, Eq. ( 1) can be rewritten as follows (Eq. ( 7)):

dC(t) dt = k w ×C w +k s ×C s +k l ×C 𝑙𝑙 -𝑘𝑘 𝑒𝑒 ×C(t) (7)
where kl is the uptake rate from the leaves (ng gorg -1 d -1 ) and Cl the contaminant concentration in leaves (ng g -1 ).

According to the experimental conditions, several sub-models could be considered and tested according to several hypotheses on the exposure routes (kw=0 and/or ks=0 and/or kl=0, Table S1).

Materials and methods

Chemicals, reagents and quality control

Solid 2,2′, 4,4′, 5,5′ hexachlorobiphenyl (PCB153) was purchased from Sigma-Aldrich (St Quentin-Fallavier, France). A working solution was prepared in acetone at 1.01 g L -1 for the contamination of sediment and leaves. The native SRM2262 solution and the internal standard PCB198 (99%) used for the quantification of PCB153 were provided by LGC Promochem (NIST) and Ultra Scientific, respectively. Native PCB153 recovery was determined using spiked samples for water (89 ± 1%), technical sand (75 ± 11%), sediment (NIST SRM1941b

Organics in Marine Sediment, 64 ± 8%) and fish (NIST SRM 1947 Lake Michigan Fish Tissue, 65 ± 12%) reference materials. The limit of detection (LoD) was determined as the concentration with a signal-to-noise ratio of 3 (LoD = 0.003 ng L -1 for water, between 0.003 and 0.040 ng g -1 dw for sediment, leaves and organisms). Replicate procedural blanks (n = 13) were analyzed for each series of samples where the PCB153 concentration was always below the LoD.

Matrix spiking

Sediment spiking

In January and March 2017, 60 L of natural sediment was collected from a watercourse in the Miribel-Jonage nature park (Vaulx-en-Velin, eastern central France near Lyon, 4°59′27″E

and 45°47′55″N for the chironomid experiment and 5°00′51″E and 45°79′71″N for the gammarid experiment). The sediment was collected using a manual dredger, sieved at 2 mm, pooled in a polypropylene (PP) jar, and stored at 4 ± 2 °C until they were used. For chironomid and gammarid experiments, respectively, the sediment was characterized by a water content of 67.9 and 51.0%, a particulate organic carbon content of 11.9 and 2.87% on dry weight matter, and 0.370 and 0.250% particulate nitrogen content. The sediment was homogenized and mixed with mechanical action (paint propeller connected to an electric drill) for 20 min.

Then 1.2 L of sediment was added in 20 and nine Pyrex bottles (2 L) for the chironomid and gammarid experiments, respectively. Each bottle was spiked with a solution of PCB153 in an acetone carrier at a nominal concentration of 100 and 50 ng g -1 dry weight (dw) for chironomid and gammarid experiments, respectively. The amount of carrier added to the sediment was minimal (0.07 and 0.08 µL/gsed for chironomid and gammarid experiments, respectively). Then each bottle was rotated for 24 h (chironomids) to 72 h (gammarids) at 15 revolutions per minute (rpm) at room temperature (21 °C). After 48 h storage at 4 °C, contaminated sediment in each bottle was transferred in a polypropylene jar and homogenized and restless with mechanical action for 20 min before being added to aquaria.

Leaf spiking

Alder leaves (Alnus glutinosa) were collected in November 2016 and stored in plastic boxes. Prior to exposing gammarids to PCB153, the alder leaves were placed in a bucket filled with several liters of ground water for 7 days at 21 °C. The water was renewed every 2 days to remove the exudates from the leaves. Several batches of 5 gdw leaves were placed in Pyrex bottles (2 L) containing 1 L of groundwater and were spiked at a nominal concentration of wetweight basis 50 ngPCB153 kg -1 leaf,ww with a solution of 5 µg L -1 of PCB153 in acetone. These batches were rotated for 72 h at 10 rpm at room temperature (21 °C). Then leaves were rinsed with ground water for 3 days in Pyrex bottles before being placed in aquaria.

Organism exposure to PCB153

Chironomid exposure

A total of seven aquaria (38-20-24.5 cm in polystyrene) were prepared with 3 L of homogenized spiked sediment and 15 L of groundwater. Each aquarium was allowed to settle for 10 days before introducing the chironomids. A control aquarium was prepared in the same way with reagent control sediment (Fig. S1).

Benthic invertebrate C. riparius were obtained from laboratory cultures carried out according to standard methods (OECD, 2004;[START_REF] Afnor | Determination of the Toxicity of Fresh Water Sediments to Chironomous Riparius -Part 1: Natural Sediments Water Quality[END_REF]. A total of 400 fourth-instar larvae (7-day-old larvae post-hatching, L4) were added to each aquarium. Chironomids were exposed to spiked sediment for 7 days at 21 ± 0.2 °C in aerated and static water. A 16:8-h light:dark cycle was maintained throughout the experiment. Larvae were fed every day with 400 mg commercial food Tetramin® per aquarium. The water quality parameters were monitored and are presented in Annex S1. Chironomid survival, length and wet weight were determined at each sampling time. To determine the total length, ten larvae were photographed using an IEEE 1394 Digital CCD camera (F2, FOculus, Germany) mounted on an Olympus BX51 light and SZX12 stereo zoom microscopes at low magnifications. The mean lengths were determined using digital image analysis software (SigmaScan Pro software).

Gammarid exposure

Three weeks before the start of the experiment, about 3,000 male gammarids were collected with a hand net at a reference site (Saint-Maurice de Rémens, France, 5°26′22″E -45°95′79″N). Gammarids were brought to the laboratory and acclimated 3 weeks in aquaria with continuously renewed groundwater under constant aeration, a 16:8-h light:dark photoperiod was maintained and the temperature was kept at 12 °C. Organisms were fed ad libitum with alder leaves. Only male gammarids (11.4 ± 0.9 mm) were selected, in order to eliminate potential biases due to neonate release by females.

Gammarid experiments were composed of two phases: uptake and elimination. For the uptake phase, different exposure routes were tested: gammarids were exposed to spiked leaves (E1 condition) or to spiked sediment (E2 condition) for 7 days at 12 ± 0.2 °C under a 16:8-h light:dark cycle maintained throughout the experiment. A third condition (E3) was tested, similar to E2 but without organisms, in order to determine whether there was a contamination transfer from sediment to leaves (Fig. S1). The overlying water was renewed daily under constant aeration. A control aquarium for E2 was prepared with homogenized reagent sediment (0.08 µL of acetone/g sed).

At the beginning of the experiment, 300 individuals were added per aquarium (test and control). Gammarids exposed to the contaminated leaves (E1) were distributed in three aquariums each containing 15 L of groundwater and one batch of previously spiked leaves per aquarium. Organisms exposed to spiked sediment (E2) were distributed in three aquariums each containing 3 L of spiked sediment, 15 L of groundwater and one batch of previously clean re-hydrated leaves. After 7 days of exposure, gammarid survival for each condition was determined and the organisms were transferred to a clean medium: 270 organisms per aquarium with groundwater for the E1 condition or with clean sediment and groundwater for the E2 condition. During depuration, gammarids were fed with clean leaves (5 g dw). The water quality parameters were monitored and are presented in Annex S1. Gammarid survival, length and wet weight were determined at the end of the uptake and elimination phases.

Sample collection

The overlying water (OW) was sampled in 1-L polyethylene (PE) bottles. Subsamples from sediment were deposited in 180-mL PE tubes. Organisms were collected in 50-mL Falcon® tubes. Every day, chironomids were collected by sieving sediment at 500 µm. The OW and sediment samples were collected at days 0, 4 and 7. The OW, sediment and organism control samples were collected at days 0 and 7. At days 0, 1, 2, 4, 7, 8, 9, 11 and 14, 90 gammarids were collected for each condition (E1 and E2). The OW, sediment, leaf and organism control samples were collected at days 0, 7 and 14. All samples were stored at -21°C and lyophilized at -65°C (Christ-Alpha 1-4LD, Bioblock Scientific) under a pressure of 0.050 mbar for 48 h for organisms, 72 h for leaves and 7 days for sediment.

PCB153 analyses

Water samples were filtered using 47 mm GF/F glass microfiber filters (Whatman®), and approximately 10 mL of filtered samples were extracted using a SPME procedure.

Approximately 0.5 g of sediment samples and 0.2 g of leaves and organisms were extracted by microwave-assisted extraction (Milestone SRL, Sorisole, Italy) with 12 mL of DCM at 80 °C for 15 min. The extracts were filtered, concentrated under nitrogen flow at 40 °C and cleaned up on columns containing activated copper and acidified silica gel (40% H2SO4 w/w) previously conditioned with 5 mL of pentane; after extract loading, PCB153 was eluted with three times 5 mL of a pentane/DCM (90/10, v/v). The eluate was further concentrated, solvent exchanged to isooctane and taken to a final volume of 100 μL.

PCB153 was analyzed using 6890N Agilent Technologies gas chromatography (Massy, France) connected to an electron capture detector (ECD). Analytes were injected

(1 µL) in pulsed splitless mode and separated with a J&W HP-5MS column (5% phenyl -95% methylpolysiloxane; 60 m × 0.25 mm × 0.1 μm). Helium (vector gas, 1.3 mL min -1 ) and nitrogen (auxiliary gas, 60.0 mL min -1 ) were used. The injector temperature was set at 280 °C and the detector temperature at 300 °C. The kiln temperature program was: 90 °C for 2 min (80 °C for water samples), 15 °C min -1 to 178 °C (20 °C min -1 to 190°C), 2 °C min -1 to 230 °C (210 °C), 30 °C min -1 (15 °C min -1 ) to 300 °C, 300 °C for 3.8 min (5 min). PCB153 was quantified relative to internal standard (PCB198, 9.6-11.8 ng).

Data analysis

Significant differences were considered according to the Wilcoxon test with an α risk of 5%.

Graphical representations were made with the statistical software R (version 3.3.3, R Core Team, 2017).

4. Link between model and data: Bayesian inference

Stochasticity

For both chironomids and gammarids, we assumed a gaussian distribution of the contaminant concentration in the organism:

Cobs(t)~N(C(t), 1/σ²) (8)
where N stands for the Normal law, Cobs(t) corresponds to the contaminant concentration in the organism at time t measured during the experiments, C(t) is the contaminant concentration at time t predicted by the model and σ is the standard deviation of contaminant concentration in the organism.

For chironomid size, we also assumed a normal distribution:

Lobs(t)~N(L(t), 1/σL ² ) (9)
where Lobs(t) is the length measured at time t during the experiments, L(t) is the length predicted by the von Bertalanffy model at time t, and σL is the standard deviation of organism size. and gammarids (c), which symbolize the deterministic links between parameters and variables for the complete generic (Eq. ( 1)), chironomids (Eq. ( 4)) and gammarids (Eq. ( 7)) models and the stochastic links between the observed and predicted data.

Graphical representation

Definition of priors

Before conducting an experimental study, a prior distribution is defined for each parameter according to information available from the literature and/or previous experiments.

Depending on the source and the conditions that the information comes from, informative, semi-informative or noninformative prior distribution can be used. If a parameter was already estimated in previous studies or if previous data are available, a normal prior distribution can be used (with the mean value estimated and the precision with a standard deviation twice the value estimated to take into account the potential differences in experimental conditions).

However, if no information is available but an order of magnitude is (positive only, for example), it is possible to apply a weakly informative prior, as a uniform distribution. If any information is available on the order of magnitude of a parameter, its prior can be defined on the decimal logarithm scale in order to give the same probability to lower or higher estimates. For variance parameters, we use a noninformative (0.001, 0.001) gamma prior, as is usually done [START_REF] Lambert | How vague is vague? A simulation study of the impact of the use of vague prior distributions in MCMC using WinBUGS[END_REF][START_REF] Richards | Temperature-dependent bioaccumulation of copper in an estuarine oyster[END_REF].

For chironomids, priors were defined from the values given by [START_REF] Schuler | Toxicokinetics of sedimentsorbed benzo[a]pyrene and hexachlorobiphenyl using the freshwater invertebrates Hyalella azteca, Chironomus tentans, and Lumbriculus variegatus[END_REF] where Chironomus tentans were exposed to PCB153-spiked sediment. They found a mean ks value of 0.054 gsed. gorg -1 .h -1 and a mean ke value of 0.011 h -1 . We thus assumed a log10-normal (0.113, 5) prior for ks (in gsed gorg -1 d -1 ) and a log10-normal (-0.578, 5) prior for ke (in d -1 ). For kw, we assumed a log10-uniform (-5, 2) prior. We used a log10-normal (0.236, 2) prior for kg (in d -

1

) according to [START_REF] Péry | A modeling approach to link food availability, growth, emergence, and reproduction for the midge Chironomus riparius[END_REF], corresponding to a growth rate of 1.72 d -1 . Priors for L0 and Lmax were assumed to follow log10-normal distributions and were set, respectively, at (0.778, 0.64; in days) corresponding to an initial size of 6 mm, and (1.056, 0.64; in days) corresponding to a mean limit size of 11.4 mm [START_REF] Péry | A modeling approach to link food availability, growth, emergence, and reproduction for the midge Chironomus riparius[END_REF]. We assumed a non-informative (0.001, 0.001) gamma prior for the precision.

For gammarids, little information was available in the literature. As a consequence, we used a non-informative (-5, 2) log10-uniform prior for uptake and elimination rate constants (ks, kl, kw, ke). We also assumed a non-informative (0.001, 0.001) gamma prior for precision.

Implementation of the model -MCMC simulations

Model computation was performed with JAGS and R software (R Core Team, 2017;

Plummer, 2016). The models were fitted to bioaccumulation data using Bayesian inference via Markov Chain Monte Carlo (MCMC) sampling. For each model tested, we started by running a short sampling (5,000 iterations after a burn-in phase of 10,000 iterations) using the Raftery and Lewis (1992) method to set the necessary thinning and number of iterations to reach an accurate estimation of each model parameter.

For chironomids, MCMC sampling was based on 150,000 iterations for three chains after discarding the first 5,000 iterations. Samples from every 40 th iteration were stored to reduce autocorrelation. For gammarids, 26,000 iterations were done after discarding the first 10,000 iterations. Samples from every seventh iteration were stored to reduce autocorrelation in the sample.

To monitor the convergence of the chains, we used a visual inspection as well as the Gelman criterion [START_REF] Gelman | Bayesian Data Analysis[END_REF]. The R codes are available in supporting information (Annex S2).

Posterior distributions and relevance of model predictions

From the joint posterior distribution, we can obtain the marginal posterior distribution for each parameter, which can be summarized by the mean or median and standard deviation.

The accuracy of model parameter estimation can be visualized by comparing prior and posterior distributions: a thin posterior distribution reflects that the data contributed enough information to precisely estimate parameters.

To check the relevance of model predictions, we represent, for each experiment, observed data superimposed on the model simulated with the median of the posterior distribution for each parameter and the 95% credibility band of the predicted data considering parameter uncertainties and stochasticity. To obtain the 95% credibility band, the predicted data were simulated with the model for each MCMC iteration and the stochastic model considered for observed data (Eqs. ( 8) and ( 9)).

Model comparisons

Several hypotheses were considered and tested for chironomids (Eqs. ( 4) and ( 6)) and for gammarids (Table S1) according to the experimental data available. To compare the different sub-models fitted, we analyzed the precision of each parameter estimation and the relevance of model predictions through the deviance information criterion (DIC), a Bayesian measurement that weighs the quality of model fit with its complexity. Sub-models with lower DIC values are expected to effectively balance between predictive capacity and complexity [START_REF] Spiegelhalter | Bayesian measures of model complexity and fit[END_REF].

For Bayesian inference, JAGS 4.2.0 for Windows and the rjags package for R software were used.

Results

Chironomus riparius

Sediment, water and chironomid contamination

The PCB153 concentrations in control and spiked sediment at day 0 were, respectively, 1.01 ± 0.25 and 83.3 ± 20.8 ng gsed -1 versus 100 ng gsed -1 (dw) expected. At the end of the experiment, the PCB153 concentration in spiked sediment was 89.6 ± 22.4 ng gsed -1 (dw). No significant difference was observed for the concentrations in the sediment monitored during the experiment (p-value = 0.333).

At the beginning of the experiment, the PCB153 concentrations in water were 0.077 ± 0.009 ng L -1 and 0.230 ± 0.025 ng L -1 in the control and exposed aquaria, respectively, while they were 0.259 ± 0.028 ng L -1 and 3.85 ± 0.420 ng L -1 , respectively, at the end of the experiment.

PCB153 concentrations in C. riparius exposed to spiked sediment increased from 0.089 ± 0.031 to 142 ± 50.0 ng gorg -1 in 7 days.

Chironomid survival and growth

An acceptable survival rate was observed during the experiment for the test condition (88%) and for the control (76%). No adverse effect of spiked sediment on chironomid growth was observed (p-value = 0.248).

Parameter estimates

Two models were fitted to uptake data: Equation ( 4), which considers water and sediment exposure routes, and Equation ( 6), which considers sediment only. Similar marginal posterior distributions were obtained for each parameter with both models (Table S2, Fig. S2).

Since the DICs for both models were similar (87.66 for Eq. ( 4) and 87.71 for Eq. ( 6)), we selected the most parsimonious one (with the fewest parameters), Eq. ( 6), which accounts for exposure from sediment only.

The inference process quickly converged, and thin posterior distributions were obtained for each parameter, meaning that data contribute sufficient information to accurately estimate model parameters (Fig. S2). Median values and 95% credibility intervals for each parameter were estimated and are summarized in Table 1.

Model predictions

The model predictions fit the bioaccumulation and growth data well (Fig. 2). For bioaccumulation (Fig. 2a), two of the eight measurements were outside the 95% credibility band of the predicted data, and for growth 23 of the 80 measurements (Fig. 2b).

Gammarus fossarum

Leaf, sediment, water and gammarid contamination

PCB153 concentrations in the different matrices (water, leaves and sediment) are summarized in Table 2. There was no difference among PCB153 concentrations in sediment throughout the exposure phase (p-value = 0.939). Leaves added to spiked sediment without gammarids (E3 condition) were significantly less contaminated than leaves in spiked sediment with gammarids (E2 condition) (p-value = 0.042).

As shown in Figure 3, we obtained similar PCB153 concentrations in gammarids over time for the two conditions tested (E1 and E2). PCB153 concentrations in G. fossarum exposed to spiked leaves (E1) and sediment (E2) increased from an initial concentration of 0.320 ± 0.110 to 10.9 ± 3.80 and 11.2 ± 3.93 ng gorg -1 , respectively, at the end of uptake phase. When gammarids were transferred into a clean media, the E1 and E2 PCB153 concentration in the organisms decreased to 4.71 ± 1.65 and 6.41 ± 2.24 ng gorg -1, respectively, at day 14.

Growth and survival

High survival rates were observed during the experiments. Survival after uptake and elimination periods were, respectively, 93 and 91% in control aquaria, 96 and 97% for E1 and 96 and 94% for E2.

Model parameters

Several models were fitted according to the exposure routes considered (Table S1).

The models with similar lowest DIC values were those corresponding to hypotheses #3, 4, 6 and 7 in Table S1 (DIC values are reported in Table S2 in the SI). Similar model parameter estimations and marginal posterior distributions for each parameter were obtained for hypotheses #3, 4 and 6 (Table S2, Fig. S3). However, when the water exposure route was considered (hypothesis #7), marginal posterior distributions displayed two peaks for uptake rate constants kl and ks (Fig. S3), and the median values estimated for kl and ks considerably decreased (Table S2). We concluded accordingly that considering water as a contamination source did not contribute relevant information to accurately estimate model parameters. This conclusion is consistent with the fact that PCB153 concentrations measured in leaves and sediment were around 10,000 times higher than the concentration in water, in accordance with the hydrophobic character of this substance. Consequently, the model corresponding to hypothesis #6 was also eliminated. Between hypotheses #3 and 4, both considering contamination from leaves and sediment, we decided to keep the model corresponding to hypothesis #3. Indeed, this model is simpler than the one corresponding to hypothesis #4, in that it ignores that PCB153 transfers from sediment to leaves, while DICs are similar.

The inference process for the gammarid model corresponding to hypothesis #3 quickly converged, and thin posterior distributions were obtained for all parameters (Fig. S3). Median values and 95% credibility intervals for each parameter were estimated and are summarized in Table 1. The median estimation of k s was around five times higher than that of kl.

Model predictions

The model predictions fit well with uptake and elimination data (Fig. 3). For bioaccumulation in the E1 condition, two of the 11 measurements were out the 95% credibility band of the predicted data, and two of the 10 measures for the E2 condition.

Discussion

Robustness of Bayesian inference

We used Bayesian inference to simultaneously estimate all the model parameters together to enable a more accurate assessment of uncertainty around model predictions. One of the major advantages of the Bayesian analysis is the possibility of using data from different experiments simultaneously and even different types of data (e.g., uptake and elimination data, or bioaccumulation and growth data) to estimate parameters common to the different models.

This was illustrated here by estimating the elimination rate from uptake and elimination data for gammarids (Fig. 3, Table 1), as well as by estimating the chironomid growth rate from bioaccumulation and growth data (Fig. 2, Table 1).

To evaluate the model performance, we used four criteria: (i) marginal posterior distributions for each model parameter, which generally provides more information compared to prior distributions; (ii) goodness of fit to experimental data; (iii) DIC value; and (iv) the principle of parsimony.

For both chironomid and gammarid models, thin posterior distributions were obtained, meaning that data provided enough information to estimate parameters precisely (Fig. S2-a and S3-a). The simultaneous estimate of model parameters assesses the uncertainty on parameters considering the correlation between them. This is an improvement compared to the common approach in TK modeling, where model parameters are often estimated sequentially, without accounting for autocorrelation or confounding factors. The existing knowledge on parameter values derived from the literature can also be accounted for, through the definition of prior distributions, which is not the case in the common approaches. Moreover, in the common framework it is not easy to consider the overall uncertainty of predictions, as recommended in the context of environmental risk assessments [START_REF] Lin | A Bayesian approach to parameter estimation for a crayfish (Procambarus spp.) bioaccumulation model[END_REF].

For both species, model predictions fit well with experimental data. The measurements out the 95% credibility band of the predicted data for both species could indeed be due to the low number of data points and their proximity. For chironomids, [START_REF] Schuler | Toxicokinetics of sedimentsorbed benzo[a]pyrene and hexachlorobiphenyl using the freshwater invertebrates Hyalella azteca, Chironomus tentans, and Lumbriculus variegatus[END_REF] obtained a ks mean value of 1.30 ± 0.014 gsed gorg -1 d -1 , and a ke mean value of 0.264 ± 0.006 d -1 for Chironomus tentans exposed to PCB153-spiked sediment. Here, we obtained lower values, especially for ks. This could hypothetically be due to (i) the difference between the species tested, (ii) the difference in experimental conditions (shorter exposure duration and lower sediment concentration in [START_REF] Schuler | Toxicokinetics of sedimentsorbed benzo[a]pyrene and hexachlorobiphenyl using the freshwater invertebrates Hyalella azteca, Chironomus tentans, and Lumbriculus variegatus[END_REF]) and (iii) growth dilution, which was ignored in [START_REF] Schuler | Toxicokinetics of sedimentsorbed benzo[a]pyrene and hexachlorobiphenyl using the freshwater invertebrates Hyalella azteca, Chironomus tentans, and Lumbriculus variegatus[END_REF]. The literature had demonstrated the importance of considering the growth rate for fourth-instar larvae of chironomids [START_REF] Péry | A modeling approach to link food availability, growth, emergence, and reproduction for the midge Chironomus riparius[END_REF][START_REF] Bertin | Bioaccumulation of perfluoroalkyl compounds in midge (Chironomus riparius) larvae exposed to sediment[END_REF]. In the present study we estimated this model parameter at kg = 0.123 [0.029-0.356] d -1 ; in the literature, the values vary between 0.355 and 1.72 [START_REF] Péry | A modeling approach to link food availability, growth, emergence, and reproduction for the midge Chironomus riparius[END_REF][START_REF] Bertin | Bioaccumulation of perfluoroalkyl compounds in midge (Chironomus riparius) larvae exposed to sediment[END_REF]. Furthermore, [START_REF] Watts | A comparative study of Chironomus riparius meigen and Chironomus tentans fabricius (Diptera:Chironomidae) in aquatic toxicity tests[END_REF] observed that fourth-instar larvae of C. tentans were much larger than those of C. riparius. To our knowledge, this study is the first reporting the bioaccumulation of PCB153 from several potential food sources for gammarids. Previous studies with gammarids examined PCB uptake from water exposure [START_REF] Sanders | Biological magnification of a polychlorinated biphenyl (Aroclor® 1254) from water by aquatic invertebrates[END_REF][START_REF] Lynch | Availability of a hexachlorobiphenyl isomer to benthic amphipods from experimentally contaminated natural sediments[END_REF], whereas accumulation through diet remains unexplored [START_REF] Pinkney | Uptake and Retention of14C-Aroclor® 1254 in the Amphipod, Gammarus tigrinus, Fed Contaminated Fungus, Fusarium oxysporum[END_REF].

Statistical model selection is commonly based on the parsimony principle, by which hypotheses should be kept as simple as possible. The idea is that by adding parameters to a model we could improve the fit to some degree, but at the same time parameter estimates worsen because there is less information available per parameter. In addition, the computations typically require more time by adding parameters. In this study, we applied this principle when similar DICs were obtained, so the models with the least parameters were selected.

Modeling and biological implications

One advantage of the generic model developed in this study is that it allows us to consider each uptake or elimination route as a module to add according to the availability of the data measured. The different models tested can then determine which routes are the most important in the accumulation of contaminant by the organism. [START_REF] Sidney | Trait-based modelling of bioaccumulation by freshwater benthic invertebrates[END_REF] showed that for hydrophobic PCBs, particle ingestion was the dominant uptake route whatever species was tested (C. riparius, Hyalella azteca, Lumbriculus variegatus and Sphaerium corneum).

This is consistent with our results for chironomids and gammarids, which confirmed that sediment is the major PCB contamination source.

The model considering the water and sediment exposure route (Eq. ( 4)) gave similar results to the model considering only the sediment exposure route (Eq. ( 6)) (i.e., similar marginal posterior distributions for ks and ke, predicted concentrations and DIC values). As a consequence, we concluded that not considering water exposure in the chironomid model contributes no more information and confirms that sediment is the major exposure route.

For gammarids, several hypotheses were tested, and the most parsimonious model was the one that did not account for the water exposure route (hypothesis #3). Furthermore, considering or not the transfer of PCB153 from sediment to leaves in E2 condition did not change the value and the precision of the parameters (hypothesis #4 and #3, Table S2),

demonstrating that these data do not contribute additional valuable information. Indeed, previous experiments with perfluorinated alkyl compounds hypothesized that due to gammarid activity, suspended particles of contaminated sediment were deposited on the surface of leaves and were then ingested by gammarids [START_REF] Bertin | Potential exposure routes and accumulation kinetics for poly-and perfluorinated alkyl compounds for a freshwater amphipod: Gammarus spp. (Crustacea)[END_REF]. These results confirm that sediment is the major contamination source.

Species comparison

Model parameter estimates are therefore consistent with the life traits of these two species:

an uptake rate from sediment approximately ten times higher for chironomids (living in the sediment) than gammarids (living at the surface of sediment) and similar elimination rate constants (Fig S4).

Model limitations and implications for future use

Two limits could be highlighted in the use of the Bayesian approach in TK models: (i) the complexity of computation and (ii) the choice of prior distributions. Due to its complexity, the use of the Bayesian approach is limited in TK models. However, for invertebrates, TK models could be considered simpler and its Bayesian computation also stem from the organism being considered a unique compartment. The majority of the problems encountered in the Bayesian approach have resulted from the choice of prior distributions. It is crucial to define prior distributions according to previous data similar to the experiment rather than to consider prior distributions selected in this paper as "generic." Nevertheless, considering previous data (a priori) in the TK model could be useful to limit the cost and the number of experiments.

Further investigations will apply this Bayesian framework to more complex processes, including biotransformation or concentration dependency, where the frequentist approach could have limitations. Moreover, calculating prediction intervals has an advantage in that they can also be calculated around simulations with fluctuating input concentrations, even for scenarios that differ from the calibration experiments, and in that the correlation among parameters is accounted for [START_REF] Ashauer | Toxicokinetic and toxicodynamic modeling explains carry-over toxicity from exposure to diazinon by slow organism recovery[END_REF]. In particular for European regulations, this approach could make it possible to predict the concentration in the biota because the concentration in sediment is known, and uncertainties are thus accounted for.

Conclusions and perspectives

In this paper, we proposed a generic TK model in a Bayesian framework to estimate toxicokinetic parameters. This approach could be useful in order to calculate a more accurate estimation of prediction uncertainty. We demonstrated the ability of Bayesian analysis to simultaneously estimate model parameters considering several exposure routes from a PCB153 environmental exposure experiment under controlled conditions in two invertebrate species, C. riparius and G. fossarum. From a methodological point of view, this paper illustrates that considering the correlation between parameters and different types of data is a real added value for TK modeling. We demonstrated the ability of a generic TK model to consider uptake and elimination routes as modules that can be added, depending to the availability of measured data. From an ecotoxicological point of view, we showed differences in PCB153 bioaccumulation between chironomids and gammarids, which could be explained by the different life traits of these two organisms. We also confirmed that sediment is the major route of exposure for invertebrates exposed to highly hydrophobic organic contaminants. Further investigations will apply this Bayesian framework to other benthic invertebrate species, and other organic contaminants, so as to address more complex processes, including biotransformation.

Figure 1

 1 Figure 1 represents the directed acyclic graphs (DAGs) for generic (a) chironomids (b)

Figure 1 .

 1 Figure 1. Directed acyclic graph (DAG) for (a) generic (b) chironomid and (c) gammarid models. Observed variables, such as the contaminant concentration in organisms, sediment and leaves, are represented by rectangle nodes. Model parameters and variables are represented by circular nodes. Dotted arrows represent deterministic links (Eqs. (1), (4) and (7)), while solid arrows represent stochastic links between predicted and observed data.

Figure 2 .

 2 Figure 2. Observed data (dots) and model predictions (solid and dashed lines) for (a) PCB153concentrations (ng gorg -1 ) in chironomids (Eq. (6)) and (b) chironomid size (mm) (Eq. (5)) from days 0 to 7. The observed data are single values (n=1). On Figure2-b, mean sizes at each

Figure 3 .

 3 Figure 3. Measured (dots and triangles) and predicted (solid and dashed curves, from Eqs.(12) and (14)) PCB153 concentrations (ng gorg -1 ) in gammarids during the uptake (days 0-7) and the elimination (days 7-14) phases (separated with the black dashed vertical line). The measured data are single values (n=1).

Table 1 .

 1 Summary of the toxicokinetic model parameters: priors, median and percentile values determining the 95% credibility interval.

	Organisms	Parameters	Priors	Median	Percentiles
					2.5%	97.5%
	Chironomus	ks	Log10-normal	0.473	0.359	0.804
	riparius		(0.113, 5)
		ke	Log10-normal (-0.578, 5)	0.121	0.041	0.392
		kg	Log10-normal
				0.123	0.029	0.356
			(0.236, 2)
		L0	Log10-normal
				6.75	4.85	8.71
			(0.778, 0.64)
		Lmax	Log10-normal
				18.5	13.3	40.7
			(1.056, 0.64)
		σ	Gamma (0.001, 0.001)	11.8	6.53	27.5
		σL	Gamma (0.001, 0.001)	1.12	0.672	2.18
	Gammarus	kl	Log10-unif (-5, 2)	0.013	0.010	0.016
	fossarum	ks	Log10-unif (-5, 2)	0.071	0.057	0.087
		ke	Log10-unif (-5, 2)	0.178	0.131	0.229
		σ	Gamma (0.001, 0.001)	1.48	1.11	2.12

Table 2 .

 2 Summary of PCB153 concentrations in water, leaves and sediment according to gammarid experiments (control, E1, E2 and E3). Standard deviations (sd) are calculated on analytical replicates (n=2 when sd are given, except for leaves in E3 condition at day 7 and for sediment in E2 condition at day 0 where n=3).

		Water (ng L -1 )	Leaves (ng gleaves -1 )	Sediment (ng gsed -1 )
	Days	0	7	14	0	7	14	0	7	14
	Control	0.10	-	0.10	1.46 ± 0.18	1.13	1.09	0.29 ± 0.22	-	0.21 ± 0.11
	E1 a	0.30 0.10 0.10	260 ± 6.07	245 ± 30.1	3.86	-	-	-
	E2 b	0.10 0.20 0.00	1.46 ± 0.18	44.9 ± 12.7	1.72 ± 0.01	56.6 ± 8.68	30.0 ± 3.12	0.23 ± 0.12
	E3 c	0.10 0.80	-	1.46 ± 0.18	5.52 ± 0.87	-	56.6 ± 8.68	45.1 ± 1.44	-

a spiked leaves; b spiked sediment; c spiked sediment without gammarids

Acknowledgments

This study was supported by the French Agency for Biodiversity. The authors are thankful to Grand Parc Miribel-Jonage for providing the sediment used in experiments. The authors are also grateful to Marie-Laure Delignette-Muller (LBBE, UMR CNRS 5558, France) for Bayesian inference discussions as well as Myriam Arhror and Matthieu Masson (Irstea Lyon, France) for sediment characteristic analysis. The authors are also grateful to anonymous reviewers and to Ms. L. Northrup for editing the manuscript.