David Sferruzza
email: david.sferruzza@ls2n.fr

Jérôme Rocheteau
email: jerome.rocheteau@icam.fr

Christian Attiogbé
email: christian.attiogbe@ls2n.fr

Arnaud Lanoix
email: arnaud.lanoix@ls2n.fr

Extending OpenAPI 3.0 to Build Web Services from their Specification

Keywords: Software Engineering, Web Applications, Web Services, Model-Driven Engineering, OpenAPI 3.0

Web services are meant to be used by other programs. Developers (or other programs) need to understand how to interact with them, which means documentation is crucial. Some standards like OpenAPI define ways to document web services and target both humans and programs. Many tools can be used to help developers to work in a forward engineering process: they use hand-written OpenAPI models as input and automatically generate a skeleton of a working application, for example. However, this approach is not suitable to generate working applications if several evolutions occur over time, which often results in a misalignment between the OpenAPI model and the web services implementation. Here we show how we extend the OpenAPI 3.0 specification to allow building actual web services using a Model-Driven Engineering (MDE) approach. We extend the SWSG tool to make it possible to generate code from an extended OpenAPI model. This leverages a MDE approach to build web services from a model while benefiting from OpenAPI 3.0 tooling and ecosystem.

INTRODUCTION

Context. Web services often aim to back various web or mobile applications which deliver a service to end-users. Lots of different processes and tools can be used to build web services. Some of them fit well for rapid-prototyping whereas others were made with production-readiness in mind. The first must allow fast iteration loops while progressively building from a high-level to a low-level, and the latter must promote good stability, maintainability and performance. Both approaches can be interesting depending on the context; and some of them can fit both needs. For example, SWSG (Sferruzza et al. 2018), a framework to generate consistent web services for both prototyping and production, was developed in the context of Startup Palace (a web company). At the same time, it is important to design web services in a way that makes them actually usable. This implies providing a good documentation that can be used by developers writing consumer applications or by these consumer applications if they can adapt their behavior dynamically. While this kind of documentation can take many forms, some standards such as OpenAPI1 and RAML2 are widely used by the industry. These standards define formats of specifications that describe HTTP APIs of web services and are the base of ecosystems of tools. This paper focuses on OpenAPI 3.0 because it is the standard used at Startup Palace. Tools around OpenAPI can be used in two main ways. First, with a forward engineering process, developers create manually an OpenAPI model, and refine it using various tools. For example: one-time generation of an implementation skeleton in a given technology. Second, with a reverse engineering process, a tool is used to extract an OpenAPI model from a working implementation (which can be enhanced by annotations).

Motivation. Web companies like Startup Palace would benefit from using the first process because its top-down approach fits well in the activity of helping startups to launch their products step-by-step. While the second process is useful to document existing web services, it cannot leverage the benefits of building web service from a high-level (model) to a low-level (implementation). Yet [START_REF] Deremer | Programming-inthe Large versus Programming-in-the-Small[END_REF] shows the advantages of using different languages for programming-in-the-large and for programmingin-the-small. The first process makes it possible to partially leverage these advantages, but lacks the ability to keep the OpenAPI model and the implementation aligned throughout the life of the project. Indeed, implementation is often obtained by a projection of the model that is then manually modified to implement business logic. Any following model evolution needs to be projected again which would require manual modifications to be re-applied from the beginning. This work is intended to go beyond this limitation.

Contribution. We extend the approach presented in (Sferruzza et al. 2018). It consisted of a meta-model designed to express a high-level representation of web services in order to provide support such as visualization, verification and code generation. We add the possibility to better specify service parameters, improve the type system and make re-usability easier by adding a mechanism to bind variables in component definitions and in their equivalents in instances' contexts.

We introduce extensions to OpenAPI 3.0 (Sferruzza 2018); we take advantage of them to make it possible to add the various information of our models of web services into any OpenAPI model. This allows us to merge our meta-model with one supported by the industry and to polish it by removing parts that already exist in OpenAPI while benefiting mutually of our SWSG tool or of any tool designed for OpenAPI 3.0.

We propose a prototype 3 to build consistent web services from an extended OpenAPI 3.0 model. This tool is an enhanced version of the tool we proposed in (Sferruzza et al. 2018), that could (i) check model consistency and (ii) generate source code of working web services from a given valid model. We improve model consistency verification and code generation, according to the recent evolutions of the meta-model. Then we add support for OpenAPI 3.0 models that use our extensions. The prototype converts these models to models that are compliant with our meta-model so they can benefit of our tooling.

The article is structured as follows. Section 2 presents related work. Section 3 shows, through an example project, how OpenAPI is usually used in the industry. Section 4 defines extensions to the OpenAPI 3.0 specification. Section 5 improves the meta-model presented in (Sferruzza et al. 2018). Section 6 presents a new process to use these extensions in 3 https://gitlab.startup-palace.com/research/swsg SWSG and automatically generate web services from an OpenAPI model. Section 7 re-runs the example in Section 3 using the newly introduced tools. Finally, Section 8 concludes the article with some lessons and future work.

RELATED WORK

The use of MDE for development and automatic generation of web services or web applications is not a new topic [START_REF] Bernardi | M3D: A Tool for the Model Driven Development of Web Applications[END_REF][START_REF] Bernardi | Automated Development of Constraint-Driven Web Applications[END_REF][START_REF] Scheidgen | Metamodeling vs Metaprogramming: A Case Study on Developing Client Libraries for REST APIs[END_REF]. Indeed, this work is itself built on top of the approach of SWSG (Sferruzza et al. 2018) and REIFIER [START_REF] Rocheteau | Reifier: Model-Driven Engineering of Component-Based and Service-Oriented JEE Applications[END_REF].

SWSG shares the meta-modeling approach with tools such as M3D (introduced in [START_REF] Bernardi | M3D: A Tool for the Model Driven Development of Web Applications[END_REF]) and extended in [START_REF] Bernardi | Automated Development of Constraint-Driven Web Applications[END_REF]) that also focus on building web services using MDE. One of the main differences between SWSG and M3D is that SWSG was developed with a focus on design-time support. For example, it allows to automatically verify properties about the structural consistency of models. Even if SWSG is definitely related to existing standards such as BPEL [START_REF] Fu | Analysis of Interacting BPEL Web Services[END_REF] or WSDL, our approach differs on several aspects. First, we want to avoid the shortcomings described in [START_REF] Gronmo | Model-Driven Web Services Development[END_REF], that is WSDL models contain too much technical details and are difficult to understand for humans. Indeed, our meta-model is simpler and less expressive than WSDL or BPEL. Second, this allows SWSG to provide more support to users (the balance between flexibility and support is discussed in (van der Aalst, Pesic, and Schonenberg 2009)). Finally, SWSG now relies on OpenAPI.

OpenAPI is an active and growing industry standard that is also involved in various research areas. In [START_REF] Cao | Automated Generation of REST API Specification from Plain HTML Documentation[END_REF], it was chosen for its popularity over WADL and other industry standards in order to automatically transform plain HTML documentations of web services to a machine-readable format. Moreover, [START_REF] Cremaschi | Toward Automatic Semantic API Descriptions to Support Services Composition[END_REF] provides a great state of the art of service description formats (which seems to be an updated version of the work proposed in (Tsouroplis et al. 2015, §3.2)) that brings out OpenAPI as the most promising choice at the moment and enriches it with semantic annotations. [START_REF] Schwichtenberg | From Open API to Semantic Specifications and Code Adapters[END_REF] shows an approach that is agnostic to service description formats but uses OpenAPI in the article. The popularity of OpenAPI is also highlighted by its use in other do-mains. For example [START_REF] Willighagen | Automatic OpenAPI to Bio.Tools Conversion[END_REF] describes a case where OpenAPI 2.0 is used in combination of other tools from the life sciences community and points out that the specification extension mechanism of OpenAPI 3.0 (that we use in this article) might be an interesting opportunity of improvement. Another example in the telecommunication domain is presented in [START_REF] Pugaczewski | Software Engineering Methodology for Development of APIs for Network Management Using the MEF LSO Framework[END_REF]) which provides a section to emphasizes the trade-offs of Model-Driven Engineering and argues that they can be overcomed "with increased investment in the tools that support the development process"; we share the same vision.

A COMMON USAGE OF OpenAPI

The OpenAPI Specification defines a standard to express interfaces to HTTP APIs in a language-agnostic way. It aims at allowing "both humans and computers to discover and understand the capabilities of the service without access to source code, documentation, or through network traffic inspection": that is a metamodel. As in MDE, the point of having meta-models is to have tools that can rely on them in order to safely manipulate models and offer support to developers. Indeed, an ecosystem of tools was developed around OpenAPI by various actors. Such tools have several purposes, including but not limited to: providing an interactive graphical user interface from a model 45 , generating functional tests from a model6 or generating a model from an annotated implementation7 .

To ease the development of these tools, official examples of OpenAPI models are shipped with the specification. In this article, we focus on the Petstore example8 . It describes a simple application that exposes 4 services to list, show, add and remove data records representing animals. We assume that we are in the context of a company such as Startup Palace; that means the goal is to develop these web services in order for them to be consumed by user interface applications; for example, desktop and mobile applications for the owner of the store.

The common top-down development process follows. First, developers make several iterations on writing an OpenAPI model. This model must match the functional specifications and describe web services that are fully exploitable by consumer applications. For example, the description of one of the Petstore services is shown in Listing 1. When this model is stable enough, developers can use it as a specification to start building web services and consumer applications. There are some kinds of tools that can take the OpenAPI model as input and help to build compliant web services. For example, by generating a skeleton of an application using a given technological stack9 , or by generating automated tests that can be used to check if the web services are compliant 10 . But these tools have a major lack: they require humans to update the OpenAPI model whenever the web services evolve. This is a very common situation: specifications must evolve either because business requirements have changed or new constraints have been discovered while developing. Even if some tools can mitigate this issue, it is likely that developers will eventually stop maintaining the OpenAPI model after the web services reach production, making the two diverge over time. In long-term projects, this means giving up on every advantage provided by OpenAPI and its MDE approach.

To fix this shortcoming, we present in Section 6 an improved version of this process that leverages a metamodel of web services we introduced in (Sferruzza et al. 2018). In order for this new approach to be feasible, we first extend the OpenAPI 3.0 Specification in Section 4 and improve our meta-model in Section 5.

EXTENDING OpenAPI 3.0

The OpenAPI Specification describes a meta-model to express an interface to web services. Therefore, it does not describe how web services are implemented. To make our approach possible, we propose a way to merge OpenAPI with our meta-model of web services introduced in (Sferruzza et al. 2018).

To preserve tools compatibility we make use of Specification Extensions, as defined in OpenAPI 3.0. This mechanism allows to add data to models without breaking their compliance to the specification or their ability to be used by tools designed to be compatible with it. In (Sferruzza 2018) we present our extensions to OpenAPI 3.0. The following paragraphs discuss the nature of these extensions for each aspect of our meta-model.

Because an OpenAPI model can be seen as a tree (before references are resolved) that has the OpenAPI object as root, we make use of the notion of paths. The OpenAPI > components path designate the child of the root named components. Sub-children are separated using a > symbol; the OpenAPI > components > requestBodies designates the requestBodies child of the components child of the root.

Data Model. An OpenAPI 3.0 model can contain a set of Schema objects 11 . A Schema object defines a data type for input or output data, which can then be referenced from elsewhere in the model. This mechanism provides more expressiveness than ours and was made for the same purpose. Thus it is a good fit for the entities of our meta-model.

Processes. On purpose, there are no equivalent of our component system in OpenAPI 3.0, because it describes processes that are internal to the web services which is out of OpenAPI's scope. Accordingly we add two properties in the Components object of OpenAPI. They contain sets of atomic and composite component definitions 12 . Exact schemas of these components follow on from their definition in our meta-model.

Services. Describing services is the main feature of OpenAPI. As such they can be specified in a quite expressive way. Yet the service meta-model in OpenAPI is not a superset of ours because of two lacks that require it to be extended. First, each service must be associated to a component instance that describes its 11 In the OpenAPI > components > schemas path. 12 Their paths are OpenAPI > components > x-swsg-ac and OpenAPI > components > x-swsg-cc.

behavior 13 . Second, if a service has a requestBody property (that describes the type of the data required in the request body), the RequestBody object must contain a variable name14 . When generating the web services code, this is used to include the request body contents in a variable of the execution context.

EXTENDING A META-MODEL OF WEB SERVICES

Notations. A tuple T is a product type between n types T 1 to T n , with n ≥ 2. It is denoted

T ≡ T 1 × . . . × T n . A value t of type T is written as t = (t 1 , . . . , t n ) where t 1 ∈ T 1 , . . . , t n ∈ T n .
A record R is a tuple with labeled elements. It is denoted R ≡ label 1 : T 1 , . . . , label n : T n . It is syntactic sugar over a tuple T 1 × . . . × T n and n functions label 1 : R → T 1 , . . . , label n : R → T n . A value r of type R is written as r = (t 1 , . . . , t n ) where t 1 ∈ T 1 , . . . , t n ∈ T n . Associated functions can also be written r.label 1 , . . . , r.label n . P(T ) is the type of sets of elements of type T .

In (Sferruzza et al. 2018) we presented a metamodel of web services. This meta-model has a voluntarily simple design in order to provide two advantages: (i) to give developers good abstractions to write reusable code while giving them a good flexibility and (ii) to allow tools to provide support to developers, such as design-time consistency verification. It is a combination of three elements: a set of entities that stands for the data model, a set of components that stands for the process model and an ordered list of services that exposes components to the outer world. It is defined by M ≡ entities : P(E), components : P(C), services : List(S)15 . We identified some design issues and we propose evolutions to fix them.

Weakness of Type System. Some parts of our metamodel describe a type system. This type system is used to define entities' attributes and components' contracts. Valid types are defined by the union of a set of predefined scalar types (such a String or Boolean) and the parametrized type EntityRe f (E) where E is the name of an entity declared in the current model. In order to deal with realistic use cases, we add two new parametrized types: SeqOf(T). Represents an ordered list of elements of any valid type T .

OptionOf(T').

Represents an optional value of type T , that is every value of type T plus a special null value that represent the absence of value.

From now on, E denotes the entities that make use of this enhanced type system.

Restricted Component Reusability. When defining an atomic component in a model, one must provide its contract: three sets that contain variables required on the execution context, variables that will be added and variables that will be removed. The names of these variables are then used in the component implementation. A problem appears when instantiating such a component (for example in a composite component definition): variables in the component's contract must have the same name in every execution context it is instantiated in. For example, it is possible to instantiate an atomic component c 1 = ("atomic", / 0, / 0, {("v1", String)}, / 0) twice in a row in a composite component c 2 = ("composite", / 0, [("atomic", / 0), ("atomic", / 0)]). But, because this component adds a variable to the execution context, this will violate a verification rule. Indeed the second instance cannot add an already defined variable to the execution context. To fix this, we introduce an alias property of type P( source : String, target : String ) to the component instance record (previously denoted by CI ≡ component : C, bindings : P( param : V, argument : Term ) ). The source name designates a variable in the component definition, and target is the name it should take in this instance's context only. This makes it possible to define and implement components independently from their instantiation contexts. This results in CI ≡ component : C , bindings : P( param : V, argument : Term ), aliases : P( source : String, target : String ) where CI is the updated component instances and C the set of components that uses them. Both of them use the enhanced type system. Using this mechanism, c 2 becomes c 2 = ("composite", / 0, [("atomic", / 0, / 0), ("atomic", / 0, {("v1", "v2")})]); there won't be a verification error anymore, the resulting execution context will contain v1 and v2 and the definition of c 1 remains the same.

Unexpressive Service Parameters. In a service definition, it is possible to specify parameters that will be extracted from the HTTP request's URL at runtime. The resulting variables are then injected in the execution context so that components can access them. We change the type of the params attribute of the service record from P(V ) to P( location : L, variable : V ) where L ≡ {Query, Header, Path, Cookie, Body}. This makes it possible to extract parameters' values from various parts of the HTTP request which is often necessary in real world services. Now on, S denotes the services that integrate these modifications and make use of CI component instances and the enhanced type system.

The resulting meta-model M ≡ entities : P(E ), components : P(C ), services : List(S ) is defined by the BNF grammar in Figure 1.

AN EXTENSION OF SWSG TO BUILD WEB SERVICES

To support the approach of automatically building web services from an OpenAPI model, we extend the SWSG tool introduced in (Sferruzza et al. 2018). First, its internal meta-model and the corresponding concrete syntax are updated to match the changes presented in Section 5. Second, an OpenAPI 3.0 parser is implemented; it is designed to work with the extensions presented in Section 4. Finally, a custom model-to-model transformation is implemented in order to convert extended OpenAPI models to SWSG models. The process that follows on from these improvements is illustrated by Figure 2; it is an extended version of (Sferruzza et al. 2018, Figure 2).

Transformation of OpenAPI Models. Transforming extended OpenAPI models to SWSG models is quite straightforward, except for schemas/types. OpenAPI defines some primitive types and relies on a modified version of the JSON Schema Specification16 for complex types. There are two issues. First, the JSON Schema Specification is more expressive than SWSG's type system. For example, it allows to define refined types, e.g. to add a minimum length to a string. Second, it supports both literal and referenced definition of attributes' types. Moreover, references are quite expressive and can target many places in the OpenAPI main document or even in another one.

In comparison, SWSG only supports literal primitive types or references to other entities.

While the second issue is more an engineering problem, the first would require to improve SWSG's type system in order for it to support expressing every possible OpenAPI type. Yet, we choose not to address them for now because they are not essential to test and validate our approach. Therefore our prototype might return errors when working with some OpenAPI models that contain unsupported types or references in schemas.

Code Generation. The process defined by Figure 2 is generic: it does not rely on a specific language or technology. Yet the language and technologies used to implement atomic components must be identical or compatible with those of the code generation target. Because we experiment in Startup Palace's context, our prototype targets the PHP programming language 17 with the Laravel web framework 18 , which is a common tool stack. 17 https://php.net/ 18 https://laravel.com/ Code generation is very similar to the one presented in (Sferruzza et al. 2018). The generated code is not meant to be manually edited, but can be easily integrated in a manually-developed Laravel application. Because this MDE approach was designed to allow shallow consistency verification, most inconsistencies in the model are caught at compile-time. This does not prevent developers to create flawed applications, as they have full control on the atomic components implementations. Indeed this flexibility comes at the cost of a bit of support.

CASE STUDY AND ASSESSMENT

We derive a new process from the common process explained in Section 3.

Step 1: design a stable OpenAPI model.

Step 2: developers need to design SWSG com-ponents that will implement their services and use the extensions we introduced in Section 4 to write them inside the OpenAPI model. Every atomic component defined in the model must be provided with an implementation.

Step 3: SWSG can check the model and generate working web services if the verification is successful.

We re-run the example presented in Section 3 with this new process. The example service defined by Listing 1 is a part of a standard OpenAPI model. We need a component that will handle the response generation when this service will receive requests. We create a composite component called FindPet and reference it from the service, as shown in Listing 2. This composite component has two children that are atomic components. The first takes an ID as input, uses it to query the database and adds the Pet result to the context. The second takes a Pet, serializes it in JSON and put it in an HTTP response. These three components are defined in Listing 3. Implementations written for every atomic components. Listing 4 shows the implementation of the GetPetById component as an example 19 .

/pets/{id}: get:

x-swsg-ci: component: FindPet

Listing 2: Instantiating a Component from a Service.

We get a PreconditionError when we run SWSG on these inputs. This verification error indicates that a component's precondition is not fulfilled in a given instantiation context. In the current case, we learn that the GetPetById misses a string named id when instantiated by the FindPet component in the GET /pet/{id} service. Indeed, we voluntarily introduced an error in Listing 3: the GetPetById component is given an integer (by the service) whereas it requires a string. In this particular example, it should require an integer id variable in order to be consistent with the service parameter. Nevertheless, in more complex projects, this component might have been used inside several other composite components and services. Thus, it might not be a good solution to just change the component's definition because it might break other workflows. This is the kind of mistakes 19 The PHP class in Listing 4 depends on the Component interface and on the Ctx and Params classes. They are defined in code output by the code generator and are just implementation details of the SWSG specification in this specific code generator. Different code generators could require different constraints on implementations of atomic components.

components:

x-swsg-cc:

-name: FindPet components:

-component: GetPetById -component: RenderPet x-swsg-ac:

-name: RenderPet pre:

-name: pet type: entity: Pet -name: GetPetById pre:

-name: id type: String add:

-name: pet type: entity: Pet

Listing 3: Components in the SWSG Petstore Example. SWSG can prevent us to make: because they are reported very early at compile-time, instead of runtime which is too late. Developers can study the problem and decide if they have to build a better implementation or if the process model was badly designed. With the contributions of this article, the decision can even escalate to whether the OpenAPI design has flaws or not, because SWSG guarantes that the process model and the implementations are aligned with the OpenAPI model altogether.

The model and code of this case study are available in the repository of SWSG20 .

We refined the meta-model presented in (Sferruzza et al. 2018) in order to improve its expressiveness and fix shortcomings with the reusability of some elements it allows to define. We also introduced extensions to the OpenAPI 3.0 Specification and merged the previous meta-model into OpenAPI. This allowed web services developers to define implementations of web services using MDE, starting from the corresponding high-level contract as expressed by a standard OpenAPI model. Evolutions were made to the SWSG tool so that it could support and automate this process. In addition to the previous advantages of SWSG, this made easier to write a model of web services and to keep it aligned with the implementation.

Even if the type system was improved, it could be improved further and allow subtyping in component preconditions, for example. Model composition is theoretically handled by OpenAPI but not currently supported by SWSG. The whole approach still needs better evaluation on more realistic (by nature and by size) case studies. Finally, continuous evaluation remains beneficial, and SWSG could still be improved to be able to automatically check the compliance of atomic components to their contract in the model. This could allow developers to focus on less tedious work that adds direct value to their products.

Figure 2 :

 2 Figure 1: BNF Grammar of the extended Meta-Model of Web Services.

  Implementation of the GetPetById Atomic Component.

https://github.com/OAI/OpenAPI-Specification/blob/ master/versions/3.0.1.md

https://raml.org/

https://github.com/swagger-api/swagger-editor

https://github.com/swagger-api/swagger-UI

https://github.com/apiaryio/dredd

https://github.com/vanderlee/PHPSwaggerGen

https://github.com/OAI/OpenAPI-Specification/blob/3. 0.1/examples/v3.0/petstore-expanded.yaml

https://github.com/pmlopes/slush-vertx

https://github.com/apiaryio/dredd

In the x-swsg-ci property of the service.

In a x-swsg-name property.

List(T ) designates a list of elements of type T ; it is better introduced in(Sferruzza et al. 2018, §2.1).

https://tools.ietf.org/html/draft-wright-json-schema-

https://gitlab.startup-palace.com/research/swsg/tree/ master/examples/petstore


E Suard

R Clément

Y Fayolle

M Alliet

C Albasi

S Gillot

Electrical resistivity tomography used to characterize bubble distribution in complex aerated reactors: Development of the method and application to a semi-industrial MBR in operation

A numerical procedure is proposed to define ERT methodology for reactors with complex geometry.

• ERT was used to study the gas phase distribution in an aerated plant scale-MBR filled with activated sludge.

• ERT is adapted to bubble dispersion characterization in membrane modules with activated sludge.

• The bubble dispersion over the mem- brane module surface depends on the air flow and MLSS concentration.

Introduction

Membrane bioreactors (MBRs) are biological wastewater treatment processes, composed of an activated sludge reactor coupled with membrane filtration. Their use has continued to expand for the last two decades given their advantages such as high effluent quality and compactness [START_REF] Krzeminski | Membrane bioreactorsa review on recent developments in energy reduction, fouling control, novel configurations, LCA and market prospects[END_REF]. However, fouling control of membranes remains a critical issue at the industrial scale [START_REF] Akhondi | The performance and fouling control of submerged hollow fiber (HF) systems: a review[END_REF]. Among the methods employed to prevent fouling, coarse bubble aeration limits deposition of particles on the membrane's surface and reduces fouling. In this way, a sustainable filtration flux can be maintained, but this air injection generates significant operating costs [START_REF] Krzeminski | Membrane bioreactorsa review on recent developments in energy reduction, fouling control, novel configurations, LCA and market prospects[END_REF]. Consequently, recent research has focused on membrane aeration optimization and more specifically on characterizing the hydrodynamics in such reactors. The main objective has been to gain insight into mechanistic impacts of the aeration strategy on fouling limitation such as turbulence and back transport, and also fiber movement and local shear stress [START_REF] Braak | Aeration and hydrodynamics in submerged membrane bioreactors[END_REF]. Such studies are required to identify processes but also to develop experimental databases needed to build models. Particle image velocimetry (PIV) and high-speed cameras have been used to describe fiber movement and bubble distribution at several gas flow rates at the laboratory scale [START_REF] Wicaksana | Fibre movement induced by bubbling using submerged hollow fibre membranes[END_REF][START_REF] Yeo | The relationship between performance of submerged hollow fibers and bubble-induced phenomena examined by particle image velocimetry[END_REF] and the semi-industrial scale [START_REF] Duc | Local hydrodynamic investigation of the aeration in a submerged hollow fibre membranes cassette[END_REF][START_REF] Fulton | Distribution of surface shear forces and bubble characteristics in full-scale gas sparged submerged hollow fiber membrane modules[END_REF][START_REF] Liu | Experimental investigation of hydrodynamic behavior in a real membrane bio-reactor unit[END_REF]. The electrochemical method, using an electrolyte solution and shear probes embedded in test fibers [START_REF] Fulton | Distribution of surface shear forces and bubble characteristics in full-scale gas sparged submerged hollow fiber membrane modules[END_REF][START_REF] Chan | Shear profiles inside gas sparged submerged hollow fiber membrane modules[END_REF][START_REF] Ye | Pulse bubble sparging for the control of hydraulically reversible fouling in submerged hollow fiber membrane systems[END_REF] and PIV [START_REF] Yeo | The relationship between performance of submerged hollow fibers and bubble-induced phenomena examined by particle image velocimetry[END_REF], have been used to monitor shear stress at the membrane's surface. However, the application of such methods requires specific fluid characteristics such as transparency or electrochemical properties and cannot be operated in presence of activated sludge. On the other hand, the impact of rheology on bubble size, shape and preferential flow path have been reported [START_REF] Wicaksana | Fibre movement induced by bubbling using submerged hollow fibre membranes[END_REF][START_REF] Ratkovich | Investigation of the effect of viscosity on slug flow in airlift tubular membranes in search of a sludge surrogate[END_REF] and should not be neglected. Braak et al. [START_REF] Braak | Impact of Aeration on Mixed Liquor in Submerged-Membrane Bioreactors for Wastewater Treatment[END_REF] compared hydrodynamics between air/water and air/mixed liquor flows. They found velocities lower in the activated sludge than in the water for the air flow rates tested and shear stresses were one order of magnitude higher in activated sludge than in water. Review papers have also mentioned the importance of activated sludge rheology on hydrodynamics [START_REF] Ratkovich | Activated sludge rheology: a critical review on data collection and modelling[END_REF][START_REF] Drews | Membrane fouling in membrane bioreactors-Characterisation, contradictions, cause and cures[END_REF] and the scale-up problems transferring labscale experimental observations to large-scale MBRs [START_REF] Drews | Membrane fouling in membrane bioreactors-Characterisation, contradictions, cause and cures[END_REF][START_REF] Buetehorn | Limitations for transferring lab-scale microfiltration results to large-scale membrane bioreactor (MBR) processes[END_REF].

Alternative investigation methods must be developed to characterize gas-liquid hydrodynamics in MBRs under representative operating conditions in terms of reactor size, fiber density and fluid properties.

Alternatively, electrical resistivity tomography (ERT) could overcome these limitations. Through current injections and voltage measurements, ERT allows the reconstruction of conductivity distributions as well as electrical impedance resistivity (EIT). The main difference between those two techniques being how the current is injected (alternating current for EIT and direct current for ERT). This technique has been widely applied in the field of conventional chemical engineering and is well adapted to study phase distribution and mixing performance considering two or three phases with different conductivities [START_REF] Sharifi | Electrical resistance tomography (ERT) applications to chemical engineering[END_REF]. Geometries studied range from pipe with a diameter of 35 mm to mixing vessels with a diameter of 1.5 m. Generally, the authors emphasized the need to widen the scope of media studied, which was mostly water for the liquid phase even though some papers dealt with xantham gum [START_REF] Fransolet | Analysis of gas holdup in bubble columns with non-Newtonian fluid using electrical resistance tomography and dynamic gas disengagement technique[END_REF][START_REF] Pakzad | Using computational fluid dynamics modeling to study the mixing of pseudoplastic fluids with a Scaba 6SRGT impeller[END_REF]. Sharifi and Young [START_REF] Sharifi | Electrical resistance tomography (ERT) applications to chemical engineering[END_REF] advised to investigate fluids common in the food industry but also wastewater. Consequently, the electrical tomography measurement seems to be a convenient candidate to gain insight into gas distribution inside a MBR for different aeration strategies.

More recent studies have used EIT to analyze mixing performance and gas hold up inside a bubble column containing activated sludge [START_REF] Babaei | Analysis of gas phase characteristics and mixing performance in an activated sludge bioreactor using electrical resistance tomography[END_REF][START_REF] Babaei | The use of electrical resistance tomography for the characterization of gas holdup inside a bubble column bioreactor containing activated sludge[END_REF]. Most of these studies used electrode rings composed of 16 electrodes and implemented the adjacent electrode pair strategy. The EIT measurements were taken using a frequency current of 9.6 kHz in most cases. However, the authors identified some gaps in the literature: ERT and EIT could be used as monitoring tools at industrial scale but some developments to improve accuracy, measurement frequency and robustness were still needed [START_REF] Sharifi | Electrical resistance tomography (ERT) applications to chemical engineering[END_REF]. In most cases, the ERT and EIT are applied to simple reactor configuration with similar experimental procedure in terms of amount of implemented electrodes and applied electrode arrays. However, in case of complex reactor geometries, such as MBR with presence of membranes, the experimental methodology of ERT and EIT must be thoroughly adapted and validated considering specific geometries. with only one aerated membrane module and three modules in filtration (Experiment E), the SAD m was in the 0.14-0.18 Nm 3 /h/m 2 membrane area range in order to keep the same air flow rate per module. For each aeration strategy, a preliminary reference measurement without aeration was performed.

Temperature (T), conductivity (σ) and MLSS concentration ([MLSS]) were regularly measured during the experiment. The main values of these parameters are presented in Table 1. Activated sludge samples were collected during the overall period of the semi-industrial pilot monitoring for measurements of particle size distribution and rheology, using laser granulometer (Malvern MASTERSIZER 3000) and tubular rheometer [START_REF] Durán | Impact of suspended solids on the activated sludge non-newtonian behaviour and on oxygen transfer in a bubble column[END_REF], respectively.

ERT method and applied general methodology

The apparent electrical resistivity (Eq.1) is determined from a measurement of potential difference between two electrodes M and N (called potential electrodes), resulting from a current injection between two electrodes A and B (called injection electrodes). The combination of these four electrodes (two injection and two potential electrodes) is called a quadripole. The resulting electrical resistivity is called apparent and it differs from the true electrical resistivity due to studied media heterogeneity. Also, an inversion step is needed for image reconstruction of interpreted electrical resistivity (ER) distribution (see II.2.2).

= ρ k V I X∆ app MN AB (1)
with ρ app the apparent electrical resistivity (Ω.m), ΔV MN the electrical potential difference measured between electrodes M and N (V), I AB the intensity of the injected current between electrodes A and B (A), and k the geometric factor (m).

The geometric factor k depends on the position and the distance between the four electrodes composing the quadripole and the geometry of the reactor. This geometric factor k, specific to each quadripole, could be determined by two methods [START_REF] Clement | COMSOL Multiphysics Modelling for Measurement Device of Electrical Resistivity in Laboratory Test Cell[END_REF]: (i) from the experimental apparent ER data set on homogeneous medium (in our study, the measurements were done for the semi-industrial pilot filled with tap water and without membrane) or (ii) from numerical modeling (as indicated in Section 2.3.2) of a case considering an homogeneous ER distribution on the given surface.

The proposed general methodology, presented in Fig. 2, stems from a classical geophysical approach described in Radulescu et al. [START_REF] Radulescu | Time-lapse electrical resistivity anomalies due to contaminant transport around landfills[END_REF], which provides robust ERT measurements [START_REF] Clement | Improvement of electrical resistivity tomography for leachate injection monitoring[END_REF] and aims at characterizing the bubble dispersion in a filtration compartment.

First, a numerical study is conducted to assess the ability of the ERT method to recover precise values of electrical resistivity (ER) in theoretical ER distributions. To this purpose, a simplified geometry is created to model the bioreactor outlines. Different scenarios are conceptualized and consist in electrical resistivity values which are attributed to each element in order to represent the gas-liquid mixture within the membranes or surrounding them. Several theoretical ER distributions are then obtained.

To limit acquisition time, the number of electrodes should be restricted and the adapted sequences of quadripoles should be compatible With: T = Temperature (°C), σ = conductivity (mS.cm -1 ), SAD m = specific aeration demand, the ratio between the air flow rate and membrane area in filtration (Nm 3 /h/m 2 ).

In near-surface geophysics, a substantial literature uses the ERT method to characterize the ground and hydrodynamic transfers that may occur in the soil at a large scale [START_REF] Audebert | Time-lapse ERT interpretation methodology for leachate injection monitoring based on multiple inversions and a clustering strategy (MICS)[END_REF][START_REF] Chambers | River terrace sand and gravel deposit reserve estimation using three-dimensional electrical resistivity tomography for bedrock surface detection[END_REF][START_REF] Clement | Influence of shallow infiltration on time-lapse ERT: Experience of advanced interpretation[END_REF]. However, for geoscience applications, measurements are usually composed of a large number of quadripoles and the measurement may last several hours, in accordance with systems dynamics. It is therefore necessary to adapt the measurement sequence to obtain a short acquisition time, more appropriate to the study of aeration strategy impact on bubble distribution. Nevertheless, an effort should be made to conserve an efficient image reconstruction. For such application, the design of electrodes configuration can be supported using numerical study to guarantee a precise description of the evolution of studied systems [START_REF] Radulescu | Time-lapse electrical resistivity anomalies due to contaminant transport around landfills[END_REF].

This paper is original in that it adapts direct current electrical resistivity tomography (ERT) to study gas dispersion in a semi-industrial MBR (filtration compartment: 2 m 3 pilot) filled with activated sludge. To this aim, a classical geophysical methodology based on numerical study [START_REF] Radulescu | Time-lapse electrical resistivity anomalies due to contaminant transport around landfills[END_REF] was followed to adapt the ERT method to a complex and semi-industrial scale aerated reactor. The adaptation of the method as well as its validation through numerical study of theoretical cases is presented in a first part. Then the validated ERT method was implemented on the filtration compartment to observe bubble dispersion for activated sludge at several mixed liquor suspended solid concentrations and its potential to detect a liquid phase influence on bubble dispersion is experimentally evaluated in the second part of this paper.

Materials and methods

Semi-industrial filtration compartment (pilot)

Pilot characteristics

A semi-industrial PVC reactor (V liquid = 2.0 m 3 , L × l × h liquid (m) = 1.23 × 0.70 × 2.40, Fig. 1a) was equipped with three hollow fiber membrane modules (KMS Puron, PSH 34, Fig. 1b, filtration surface = 3 × 34 m 2 = 102 m 2 ). Each membrane module consists of nine fiber bundles, as illustrated in Fig. 1b.

The reactor was continuously filled with activated sludge (mixed liquor suspended solid (MLSS) concentration = 6 and 10 g/L) or clear water, and filtration occurred at a liquid flow rate of 1.3 m 3 /h. Fresh activated sludge was sampled from one of the reactors of the Seine Aval water resource recovery facility (WRRF) treating rejection water from the sludge treatment. A complementary experiment was conducted at a MLSS concentration of 3 g/L (obtained by diluting activated sludge with tap water). For this specific concentration, measurements were taken without filtration or activated sludge filling in order to maintain the MLSS concentration in the bioreactor.

Aeration conditions

The injectors of coarse bubbles are located at the bottom center of each fiber bundle. The operating conditions of the semi-industrial MBR are described in Table 1. For experiments A to D, air flow rates were chosen in the range of advised industrial SAD m (specific aeration demand, the ratio between the air flow rate and membrane area in filtration) between 0.27 and 0.60 Nm 3 /h/m 2 membrane area . For experiment

Table 1

Operating conditions in the semi-industrial MBR.

with multichannel acquisition. Considering these technical constraints, two different electrode arrays are selected to build two sequences of quadripoles (detailed in Section 2.3.4). These sequences are then tested to adapt the ERT method to the complex geometry and dynamics of the semi-industrial MBR. Through forward modeling with these sequences, theoretical apparent ER data sets are generated. These apparent ER data sets are then inverted to obtain interpreted ER distribution which corresponds to the resistivity distribution all over the surface of the reactor. The resulting interpreted ER distributions are finally compared with the theoretical ER distributions to evaluate the relevance of the chosen inversion parameters and electrode arrays used for building the sequences of quadripoles. Following these numerical developments, experiments are conducted on a semi-industrial filtration compartment filled with water or activated sludge to obtain gas distribution for several operating conditions.

ERT numerical study

Definition of theoretical ER distributions: conceptualized scenarios

The numerical study is performed to estimate the ability of the ERT method to recover theoretical ER distributions in a large and complex system.

First, a simplified geometry was defined using the COMSOL software to mimic the characteristics of the membrane bioreactor. A first block (Block 1 -L × l × h (m) = 1.2 × 0.7 × 2.4) was drawn to represent the experimental reactor volume filled with activated sludge. A second block at the center of the reactor volume was used to represent the zone occupied by the membrane modules (Block 2 -L × l × h (m) = 0.83 × 0.3 × 2.4) and was divided into 27 (9 × 3) volumes to represent the location of each fiber bundle. This geometry allows the generation of several theoretical ER distributions including zones with different electrical resistivity values while keeping the same mesh for each simulation. The mesh size of the geometry was reduced iteratively until variations between computed values were lower than 0.05%. To limit calculation time, the less refined meshing validating this condition was selected for forward modeling.

Then an ER value was attributed to each geometrical block to simulate several distributions of electrical resistivity inside the membrane module zone and to evaluate the sensitivity of ERT inversion for each theoretical ER distribution. For each scenario, an ER value (2.3 Ω.m) was attributed to Block 1 in each case to represent the liquid phase. Since ER is the inverse of conductivity, this value was determined from conductivity measurement previously performed on the activated sludge using a conductivity probe (WTW, TetraCon® 325).

The electrical resistivity of the individual elements in Block 2 was set at 4.05, 4.15, 4.25 or 4.46 Ω.m to simulate zones of the membrane module with and without aeration for different gas hold-up values. In first approximation, according to Maxwell equations, positive ER variations of 2.5, 5 and 10% limited on Block 2 correspond to scenarios with local gas hold up of 1.6, 3.2 and 6.3%, respectively. These values are in the range of global gas hold up estimated for the aeration strategy considered, in the semi-industrial pilot filled with activated sludge.

Cross sections of theoretical ER distributions are presented in Fig. 3. The theoretical ER distribution used as a reference (scenario for initial state without aeration) is shown in Fig. 3a and is supposed to represent the zone of the membrane modules without aeration, surrounded by sludge. An ER value is attributed to the whole Block 2 for each theoretical ER distribution presented in Fig. 3b, to stand for homogeneous ER distribution within the membrane module zone. Fig. 3c-e presents ER distributions with heterogeneous ER distributions inside the membrane module zone corresponding to a heterogeneous dispersion of bubbles in the fiber bundles. Small centered and off-center heterogeneities of different sizes are presented in Fig. 3c and3d, respectively, to evaluate the size detection limit and its sensitivity to distance from the electrodes. Fig. 3e presents theoretical ER distributions created to evaluate the ability of the method to detect different ER values for heterogeneous ER distributions inside the membrane module zone (Block 2).

Forward modeling

The forward modeling is based on the numerical modeling of the potential difference induced by the injected direct current for every selected quadripole.

The governing equation for ERT forward modeling can be derived from Maxwell's equations. Technically ERT is based on a very-lowfrequency alternating current (2-5 Hz) to limit polarization effects. In this case, the inductive and capacitive effects in Maxwell's equation could be ignored. Under static conditions, the electric field can be represented by the negative gradient of the electric potential V (Eq.2).

= -∇ E V (2)
with E the electric field (V/m) and ∇V the gradient of electric potential (V/m).

In conductive media, to simulate stationary electric current density, the stationary equation of continuity and Ohm's law are considered. In a stationary coordinate system, the point form of Ohm's law states that:

= + σ J E Je (3)
with J is the current density (A/m 2 ) equal to the current I per unit of cross-sectional area S (A), Je is the externally current density (in A/m 2 ), E is the electric field intensity and σ is the electrical conductivity (A/Vm or S/m).

The static form of the equation of continuity then states

∇ = -∇ ∇ - = σ . J . ( V Je) 0 (4)
The generalized form of the current sources (Q j ) can be written as follows:

-∇ ∇ - = σ . ( V Je) Q j ( 5
)
with Qj is the generalized form of the current sources Considering the semi-industrial pilot geometry, an insulating boundary condition was applied at the contact area between the homogenous media studied and the semi-industrial pilot, expressed by the Neumann condition. This equation specifies that there is no current flowing across the boundary.

= n. J 0

The modeling was performed using COMSOL Multiphysics 5.2a combined with Matlab 2015. With COMSOL, the simplified geometry of the semi-industrial reactor is taken into account and the electrical field distributions can be modeled using full 3D modeling. Electrical field distributions are built using the AC/DC module (quasi-stationary electromagnetic field) to evaluate the potential difference induced by the injected DC current and the electrical boundary conditions selected as infinite or insulating surfaces. This step is made automatically to calculate the electrical potential for every quadripole and every electrical resistivity distribution using F3D-lab Matlab-code (Forward 3D Laboratory) developed by Clement et al. [START_REF] Clement | Contribution of 3-D time-lapse ERT to the study of leachate recirculation in a landfill[END_REF]. The script is available from the authors on request. To achieve a realistic data set reflecting the properties of a field survey, apparent electrical resistivity was contaminated with 3% random voltage-dependent noise to simulate a low-noise acquisition.

For more information, the elementary theory of direct current (DC) conductive media modeling can be found in Clement & Moreau [START_REF] Clement | How should an electrical resistivity tomography laboratory test cell be designed? Numerical investigation of error on electrical resistivity measurement[END_REF].

Image reconstruction of interpreted ER distribution from apparent ER data set using inversion procedure

The inversion procedure consists of reconstructing the interpreted electrical resistivity (ER) distribution from the apparent ER data sets.

First, the studied cross-section is divided in a finite amount of elements. In this study, the inversion meshing was refined around the electrodes (minimum cell size = 0.22 cm 2 ) and the maximum cell size of the cross-section was 2.6 cm 2 .

To start, the inversion is initialized by applying an ER distribution (usually a median value of the apparent electrical resistivity) to all cells. Then, an optimization method is used to iteratively change the electrical resistivity of each cell to minimize the difference between the measured and calculated (through forward modeling) apparent ER value (Fig. 2).

In this study, the apparent ER data sets were inverted using a "set as reference" time-lapse procedure used in previous ones [START_REF] Loke | Time-lapse resistivity imaging inversion[END_REF]. For this time-lapse inversion procedure, the apparent ER data sets for scenarios without gas injection were classically inverted from a homogeneous ER distribution as initial ER distribution. Then, the previously obtained interpreted ER distributions were set as the initial ER for the inversion of the apparent ER data sets for scenarios with gas injection. This procedure is common for most time-lapse surveys, as proposed by Loke [START_REF] Loke | Time-lapse resistivity imaging inversion[END_REF], et al. [START_REF] Day-Lewis | Time-lapse imaging of salinetracer transport in fractured rock using difference-attenuation radar tomography[END_REF] and Vesnaver et al. [START_REF] Vesnaver | Time-lapse tomography[END_REF].

= ∑ = - rRMS N i N x x x 1 |( | | | data i model i data i , , 2 , 2 (7)
where x data,i is the measured apparent ER, x model,i the calculated (through forward modeling) apparent ER and N the number of quadripoles.

In the following, results of interpreted ER distributions are presented as Δρ distributions. Δρ is the percentage of ER variation with respect to the ER of the interpreted ER distribution of scenario without gas injection, according to Eq. [START_REF] Liu | Experimental investigation of hydrodynamic behavior in a real membrane bio-reactor unit[END_REF].

⎜ ⎟ = ⎛ ⎝ -⎞ ⎠ ρ ρ ρ ∆ 1 . 100(%) f i (8)
where ρ f and ρ i (in Ω.m) are the ER of the interpreted ER distribution with and without gas injection respectively.

Tested electrode arrays

An electrode array consists of injection (AB) and measurement (MN) dipoles. For each array, the resulting surface investigated depends on the spacing between the current injection electrodes and the distance between the two dipoles. For this study, two electrode arrays were selected to build two sequences of quadripoles, i.e. two lists of associated injection (AB) and measurement (MN) dipoles. In Fig. 4, the 24 electrode locations are indicated by e i (with i = ). This number of electrodes was selected from the numerical study, considering a sensitivity analysis of the results to this electrode number. Electrode arrays with 16 electrodes are less efficient to recover theoretical ER distributions than arrays with 24 electrodes, in the center zone of the media corresponding to membrane zone in particular (results not detailed here). As a consequence and to have a short acquisition time in accordance with the bubble dynamics, the number of electrodes was restricted to 24 per cross section (and corresponds to 27.9 electrodes/ m 2 of the studied section of the reactor).

The first sequence of quadripoles (DD sequence, principle illustrated in Fig. 4a) was based on the dipole-dipole array [START_REF] Loke | Electrical imaging surveys for environmental and engineering studies: A Practical Guide to 2D and 3D Surveys[END_REF], which corresponds to the classical electrode array employed in geophysics studies. The measurement (MN) and injection (AB) dipoles are separated and the space between them increased gradually. If injection (AB) and measurement (MN) dipoles consist in two adjacent electrodes (e i and e i+1 ), the DD sequence is equivalent to the adjacent electrode pair strategy mentioned in papers examining EIT measurements applied to bubble columns [START_REF] Babaei | The use of electrical resistance tomography for the characterization of gas holdup inside a bubble column bioreactor containing activated sludge[END_REF][START_REF] Fransolet | Analysis of electrical resistance tomography measurements obtained on a bubble column[END_REF][START_REF] Jin | Measurement of gas holdup profiles in a gas liquid cocurrent bubble column using electrical resistance tomography[END_REF]. The sequence of quadripoles must be adapted for the specific constraints related to the reactor design studied (rectangular shape of the bioreactor, electrode disposition). In geophysics, the number of electrodes between measurement and injection dipoles is generally kept lower than 6 to optimize the signal-to-noise ratio due to a large distance between two dipoles [START_REF] Loke | Electrical imaging surveys for environmental and engineering studies: A Practical Guide to 2D and 3D Surveys[END_REF]. This was experimentally confirmed as distant injection and measurement dipoles led to more extreme apparent electrical resistivity than other quadripoles, generating artifacts after the inversion step. Therefore, measurement dipole shifting was centered on the injection dipole with a number of electrodes between AB and MN lower than 5 (Fig. 4). This adapted sequence of quadripoles is composed with 300 quadripoles. Injection electrodes A and B are e i and e i+2 , respectively, for the 240 first quadripoles where the injection dipole takes all possible locations (24) while M and N are two consecutive electrodes. The last 60 quadripoles concern 6 combinations of consecutive injection electrodes (e i and e i+1 ) regularly distributed around geometry to take into account variations closer to the wall as the depth of investigation increases with increasing spacing between electrodes. Not all 24 locations were considered for consecutive injection electrodes to limit the size of the array and therefore the acquisition time (resulting from analysis not detailed here).

The second sequence of quadripoles (GD, sequence, principle illustrated in Fig. 4.b) is based on the gradient array [START_REF] Loke | Electrical imaging surveys for environmental and engineering studies: A Practical Guide to 2D and 3D Surveys[END_REF]. Twelve combinations of injection dipoles (opposite position of injection electrodes A and B from e 1 -e 13 to e 12 -e 24 , respectively) are considered. Measurement dipole MN is located between the two electrodes of the injection dipole (AB) and takes all the available positions between them (20 To perform 2D time-lapse inversions, BERT software [START_REF] Gunther | Three-dimensional modelling and inversion of dc resistivity data incorporating topography -II. Inversion[END_REF] was used with the following parameters: a Gauss-Newton scheme with global regularization was used for the minimization with anisotropic smoothness constraints (horizontal preferential direction), an option to recalculate the Jacobian matrix at each iteration step and a regularization parameter (λ) of 5. Among inversion parameters, the value of the regularization parameter has an influence on the interpreted ER distributions. Low values of λ will produce a highly structured model with huge parameter contrasts and high values of λ will provide smooth ER models [START_REF] Gunther | Inversion Methods and Resolution Analysis for the 2D/3D Reconstruction of Resistivity Structures from DC Measurements[END_REF]. The inversion parameters were optimized and set during the numerical study for the chosen meshing, leading to interpreted ER distributions close to the theoretical ER distributions (optimization not presented here). Consequently, the same inversion meshing and parameters were used for the inversion of the experimental data sets.

The relative root mean square error (rRMS, Eq. ( 7) was used to determine the misfit between the interpreted data set and the observed data set. Fig. 5. Initial interpreted ER models and Δρ distributions obtained from numerical simulationsrrms corresponds to relative root mean square error between the modeled and the theoretical apparent ER distribution data sets.

The inversion of experimental apparent ER data sets was done following the inversion procedure described in Section 2.3.3, applying the same inversion meshing and optimized parameters for the inversion as determined during the numerical study.

Results and discussion

Numerical study

The numerical study was performed in order to evaluate the ability of ERT to recover theoretical ER distributions in conditions equivalent to experimental ones and to validate the efficiency of the sequences of quadripoles presented in Section 2.3.4. The interpreted ER distribution A1d shows a more uniform profile than A1g in the zone corresponding to the membrane location. The obtained results for ER and Δρ distributions A1 to E3 replicate each theoretical variation of ER in terms of value and position within the studied surface. However, two remarks should be made. First, the imposed values of resistivity are retrieved when the zone with modified resistivity is large enough (cases B and E2) and calculated values are much smaller than fixed one for small zones (case C1 for example). Secondly, positive Δρ are observed around the delimited theoretical zones, which indicate that the method employed induces a slight diffusion of the information, especially for zones located far from the electrodes.

Validation and comparison of sequences of quadripoles

Fig. 6 presents the vertical Δρ distributions at the center of the cross section for Δρ distributions B3 and D3, as example. A smoothness coefficient is determined, as illustrated in Fig. 6, to estimate the Δρ spreading due to the inversion procedure of the ERT method for each Δρ distribution. Separations between individual elements in block 2 are emphasized by dashed lines in Fig. 6. The smoothness coefficient is defined as the ratio between the lengths of the heterogeneity (defined by Δρ > 0) for theoretical and interpreted Δρ distributions, determined from the theoretical center of the heterogeneity along both axes. Therefore, for a smoothness coefficient of 1, the limit of the heterogeneity is efficiently recovered in the numerical interpreted ER Fig. 6. Vertical Δρ distribution at the center of the cross section for Δρ distributions B3 and D3; illustration of the smoothness coefficient for case D3d. positions). This electrode array, because it considers opposite positions of injection electrodes, is expected to offer better measurement efficiency in the center part of the studied section, ie the zone corresponding to membrane zone and air injection. Each measurement dipole consists of two consecutive electrodes. A total of 240 quadripoles are considered in this case.

Experimental study: ERT measurements

The electrical imaging survey is carried out using a distribution of electrodes around the reactor. Electrodes were connected to a Syscal Pro (IRIS Instrument, Orléans, France) resistivity meter.

The resistivity meter allows the automatic selection of the quadripoles around the reactor, according to a sequence defined by the user. The multichannel resistivity meter allows to measure the potential resulting from ten potential dipoles during a pulse of current injection between injection electrodes. Consequently, ten quadripoles are measured for one injection current pulse. The injection of current lasts between 150 ms and 8 s. The choice of the injection time value depends on the chargeability of the studied media (i.e., if a medium has the ability to behave like a capacitor, the injection time should be extended).

The stainless steel electrodes were embedded in the reactor walls 0.42 m, 1.17 m and 1.92 m from the bottom (Fig. 2c). According to Clement and Moreau [START_REF] Clement | How should an electrical resistivity tomography laboratory test cell be designed? Numerical investigation of error on electrical resistivity measurement[END_REF], the impact of electrode diameter is considered negligible in the 6-12 mm range, which is the case for the electrodes embedded in the semi-industrial pilot walls (diameter = 8 mm). As determined during the numerical study, the number of electrodes was restricted to 24 per cross section to have a short acquisition time in accordance with the study of bubble dynamics. Several injection times were tested to validate the use of a short time. An injection time of 250 ms was used in the experiments presented in this article, resulting in ERT measurements of total duration around 40 s.

To assess data quality, measurements were carried out in clear water without air injection in the normal and the reciprocal mode [START_REF] Koestel | Quantitative imaging of solute transport in an unsaturated and undisturbed soil monolith with 3-D ERT and TDR[END_REF][START_REF] Labrecque | The effects of noise on Occam's inversion of resistivity tomography data[END_REF], where current and potential dipoles were switched. According to this methodology, quadripoles associated with apparent electrical resistivity presenting a difference greater than 5% between normal and reciprocal modes were discarded before the inversion step. Among the 300 initial selected quadripoles for the DD sequence, 280 remained for the inversion step. For each measurement, it results in an apparent electrical resistivity data sets composed of 280 individual apparent electrical resistivities.

The main identified errors are: (i) data diffusion far from electrodes, which is more pronounced for GD sequence than DD one and (ii) artifact generation close to high ER difference with initial. As expected, the numerical study allowed the evaluation and comparison of the two selected sequences of quadripole and offer information for data quality estimation for further experimental application of the ERT technique. Considering the obtained results, the DD sequence will be used for experimental ERT measurements presented in the next section.

Coupled to ERT measurement, such numerical approach is essential to optimize the experimental design in terms of electrode positions, quadripole configuration and sequences for reactor with complex internal geometries. In addition, such approach could be used in nearly future for the evaluation and application of ERT or EIT to full scale industrial reactors with complex geometries.

Experimental ERT measurements on the semi-industrial MBR

Activated sludge characteristics

Table 4 presents the operating conditions and the activated sludge characteristics during the ERT measurements. The shear rate in the reactor is estimated from the superficial gas and the activated sludge rheological parameters using the equation proposed by Sanchez-Perez et al. [START_REF] Sánchez Pérez | Shear rate in stirred tank and bubble column bioreactors[END_REF]. Concerning the activated sludge rheology, the obtained rheograms (not shown) confirms the non-newtonian behavior of the activated sludge for the different MLSS concentration as expected [START_REF] Seyssiecq | In situ rheological characterisation of wastewater sludge: Comparison of stirred bioreactor and pipe flow configurations[END_REF][START_REF] Durán | Impact of suspended solids on the activated sludge non-newtonian behaviour and on oxygen transfer in a bubble column[END_REF]. The activated sludge viscosity for the MLSS concentration of 3 g L -1 is calculated from the empirical model proposed by Duran et al. [START_REF] Durán | Impact of suspended solids on the activated sludge non-newtonian behaviour and on oxygen transfer in a bubble column[END_REF] for sludge from conventional activated sludge plant. In Table 4, the activated sludge viscosity was expressed at the temperature of the activated sludge measured during the experiments and for the estimated shear rate. These results illustrate the decrease in AS viscosity with an increase in the estimated mean shear rate in the reactor (associated to an increase in the air flow rate). In a same way, the AS viscosity increases with an increase in the MLSS concentration, from 3 to 10 g L -1 . For the different operating conditions, the apparent viscosity of the activated sludge varies from 1.3 to 4.2 mPa s -1 . As a complement, the activated sludge mean diameter (d 50 ) was regularly measured during the monitoring of the semi-industrial MBR. During the overall period, the measured d 50 varies from 39 to 64 µm with a mean value of 52 µm.

Experimental characterization of bubble dispersion using ERT:

Evaluation of the method and study of the impact of air flow rate and MLSS concentration Fig. 7a-e presents the experimental results under the operating conditions described in Table 1 for the 1.17 m cross section. Locations of the aerated membrane modules are represented by rectangular distribution. As a contrary, a value far from 1 indicates a poor recovery of the limit of the heterogeneity. Four values are available -D -x , D +x and D -y , D +y for the X-and Y-axes, respectivelyto study the symmetrical or asymmetrical character of the spread depending on the heterogeneity location. These coefficients for most Δρ distributions illustrated in Fig. 5 are presented in Table 2.

As seen in Fig. 5 and Table 2, the diffusion is less significant when the heterogeneity is close to the electrodes. For example, the smoothness coefficient for Δρ distribution D1d ranges from 2.7 on the electrode side (D -x ) to 5 towards the center of the cross section (D +x ) for the DD sequence, which is similar to the GD sequence (Δρ distribution D1g). Moreover, in case of small-centered heterogeneity (C1d and C1g), smoothness coefficients are higher in the X-direction than in the Y-direction, which supports the hypothesis of a relationship between diffusion and the distance to the electrodes since X is longer than Y for the cross section studied here.

Fig. 6 also presents Δρ values along the Y-axis for Δρ distributions B3d and B3g. These two distributions are the results for a theoretical ER distribution with a 10% ER difference for Block 2 with respect to the initial theoretical ER distribution (see Section 2.3.1). Maximum Δρ are closer to the modeled 10% difference in these two Δρ distributions compared with Δρ distributions D3d and D3g. As illustrated in Fig. 5 by the color used to represent the ER variations, the higher the smoothness coefficient, the lower the Δρ obtained. This way, the mean Δρ of the extended zone containing the ER difference is similar to the mean Δρ of the theoretical zone of the theoretical distribution.

For most Δρ distributions (Fig. 5), the electrical resistivity variations and single heterogeneity detection are similar for the two sequences, as is the smoothness coefficient (Table 2). However, the smoothness coefficient is higher for the GD sequence than for the DD sequence for small heterogeneity detection (Δρ distributions C1 or D3). The GD sequence appears to be less sensitive in areas distant from the electrodes than the DD sequence, which accentuates the diffusion of the information recovered by ERT inversion. It is consistent with observations in Δρ distributions E2g and E3g illustrated in Fig. 5, where the different heterogeneities are less distinguished than for the Δρ distributions obtained with the DD sequence (E2d and E3d).

Artifact generation

All inversion tools used for reconstructing images are known for artifact generation and information loss because the inversion may be ill-posed problem [START_REF] Clement | Improvement of electrical resistivity tomography for leachate injection monitoring[END_REF]. As observed in Fig. 5 and Fig. 6, negative Δρ appear next to heterogeneities. Similar heterogeneities have been observed in geophysical studies [START_REF] Clement | Improvement of electrical resistivity tomography for leachate injection monitoring[END_REF]. The values of negative Δρ seem related to ER differences: the higher the theoretical ER difference (with respect to the initial ER distribution) is, the higher the relative decrease in electrical resistivity next to the heterogeneity is, as observed when comparing Δρ distribution B1d with Δρ distribution B3d. However, for the experimental conditions studied, the proposed hypothesis postulates that there should not be any negative Δρ since gas injection induces an ER increase. This way, artifacts will be easily detected.

Conclusions of the numerical study

The potential of ERT for phase distribution analysis at large scale was numerically investigated. Heterogeneities were clearly detected for each theoretical ER distribution. For all conceptualized scenarios, the ERT is efficient to detect the evolution of electrical resistivity in terms of absolute value but also in terms of position within the studied surface. In addition, the ERT could be considered as highly sensitive to the evolution of local electrical resistivity, considering the small variations tested during the numerical study (to mimic values of local gas hold-up from 1.6 to 6.3%, consistent with previous studies in small bubbles columns or mixing vessel [START_REF] Jin | Analysis of bubble behaviors in bubble columns using electrical resistance tomography[END_REF][START_REF] Hashemi | Experimental investigation of the bubble behaviour in an aerated coaxial mixing vessel through electrical resistance tomography (ERT)[END_REF][START_REF] Babaei | The use of electrical resistance tomography for the characterization of gas holdup inside a bubble column bioreactor containing activated sludge[END_REF]). However, some errors mainly generated by ERT inversion for different simulated configurations are identified during this numerical study. The main ERT potential errors are summarized in Table 3.

shapes.

The results of the initial experiments (without gas injection, Column 1, Fig. 7) are presented in terms of the ratio between the electrical resistivities obtained using the ERT method and the electrical resistivity of activated sludge (or water), using a conductivity probe. Without aeration, the zone occupied by membrane modules is experimentally detected at each MLSS concentration and conductivity condition (Fig. 7, Column 1). Slight discrepancies are noted depending on the MLSS concentration and could be attributed to fiber arrangement, which is heterogeneous as mentioned in Buetehorn et al. [START_REF] Buetehorn | CFD simulation of single-and multi-phase flows through submerged membrane units with irregular fiber arrangement[END_REF] or possibly activated sludge properties such as floc presence, leading to slightly different configurations of fibers surrounded by sludge. Moreover, the membrane module zone seems better detected at high conductivity (low resistivity) since the results are expressed in ratio in order to facilitate the comparison of the different cases.

On Fig. 7 (Columns 2 to 4), the results are expressed in ER percentage of variation Δρ (with respect to the initial measurements without aeration). Variations in the range [-2.5%, 2.5%] are considered as the noise resulting from experimental measurement and therefore are displayed in grey.

As for numerical study results, artifacts can be observed close to zone of positive Δρ. These are more pronounced experimentally and could be related to higher Δρ values than the simulated Δρ values (45% vs 15% ER difference with initial ER, respectively). A complementary simulation was conducted by setting the electrical resistivity value of Block 2 at 6 Ω.m (results not shown here). As for experimental measurement, such artifacts (Δρ of -10%) appeared on the resulting interpreted ER model and could be attributed to inversion, which is in agreement with the previous hypothesis.

Concerning clean water Δρ distributions, a positive Δρ is observed around the zone occupied by membrane modules at high air flow rates, which is not in agreement with visual observation during experiments. A possible explanation could be a clean water conductivity decrease during the experiments (from 735 µS/cm to 694 µS/cm (6%)). For activated sludge experiments, the conductivity variation remained lower than 2%.

For all operating conditions (air flow rate and MLSS concentration), the gas is mainly localized in the membrane zone and only slight dispersion is measured around this zone. During their rising, the bubbles are confined in the membrane zone and could contribute to the fouling mitigation of the membrane by generating shear stress at the membrane surface and local movement of membrane [START_REF] Braak | Aeration and hydrodynamics in submerged membrane bioreactors[END_REF]. However, the homogeneity of the local bubble dispersion mainly depends in the MLSS concentration and the injected air flow rate.

For all MLSS concentrations, an increase in the gas flow rate induces a rise in the relative increase in the Δρ within the membrane zone. This trend is similar to the gas hold-up changes observed in activated sludge in Babaei et al. [START_REF] Babaei | The use of electrical resistance tomography for the characterization of gas holdup inside a bubble column bioreactor containing activated sludge[END_REF] and is consistent with the nonconductive property of the gas phase. Maximum Δρ values are closed to 45% and higher than simulated values during the numerical study. This order of magnitude could highlight that experimental local gas hold-up is higher than maximum value considered during the numerical study (maximum value of 6.3%, corresponding to an ER variation of 10%). However, for similar gas flow rates, Δρ are higher within the membrane zone for clean water than activated sludge experiments. This second observation is in agreement with the results obtained in bubble columns filled with hardwood pulp suspension [START_REF] Ishkintana | Gas holdup in pulp fibre suspensions: Gas voidage profiles in a batch-operated sparged tower[END_REF], xanthan gum [START_REF] Fransolet | Analysis of gas holdup in bubble columns with non-Newtonian fluid using electrical resistance tomography and dynamic gas disengagement technique[END_REF] and activated sludge [START_REF] Babaei | The use of electrical resistance tomography for the characterization of gas holdup inside a bubble column bioreactor containing activated sludge[END_REF][START_REF] Durán | Impact of suspended solids on the activated sludge non-newtonian behaviour and on oxygen transfer in a bubble column[END_REF], where gas hold-up is lower in these fluids than in clear water. This decrease in the gas hold-up could be linked to an increase in the bubble size and in the associated bubble rise velocity, as observed by Babaei et al. [START_REF] Babaei | Analysis of gas phase characteristics and mixing performance in an activated sludge bioreactor using electrical resistance tomography[END_REF].

For all gas flow rates, the Δρ measured for clean water and lower MLSS concentration (3 g/L) experiments highlight that bubble distribution is quite homogeneous throughout the membrane module zone (Fig. 7). For higher MLSS concentrations (6 and 10 g/L) and low/ average gas flow rates, only some parts of the membrane module zone present positive Δρ, which could illustrate preferential bubble pathways. The bubble distribution over membrane modules is more homogeneous for high air flow rates. The evolving bubble pathways and distribution over the membrane module with the air flow rate could be related to the specific plastic or pseudo-plastic behavior of the activated sludge [START_REF] Braak | Aeration and hydrodynamics in submerged membrane bioreactors[END_REF]. The heterogeneous distribution of the bubbles within the membrane zone is particularly pronounced for the operating conditions with the highest apparent viscosity of the liquid (from 1.7 to 4.2 mPa s -1 ). These results confirm how the presence of the solid in sludge can participate in the production of heterogeneity; the ER distributions may help determine the extent of this phenomenon and its dependency to the sludge properties. Moreover, such observation could be useful for aeration management of MBR and the optimization of the energy consumption of MBR process at full-scale by fixing the air flow rate considering the rheological properties of the activated sludge.

For one aerated membrane module experiment (Fig. 7e), the positive Δρ values are located more around the aerated membrane module With: T = Temperature (°C), σ = conductivity (mS.cm -1 ), SAD m = specific aeration demand, the ratio between the air flow rate and membrane area in filtration (Nm 3 /h/m 2 ), G the estimated shear rate using equation of Sánchez Pérez et al. [START_REF] Sánchez Pérez | Shear rate in stirred tank and bubble column bioreactors[END_REF] (s -1 ), μ app the apparent viscosity at temperature T and for the associated estimated shear rate (mPa s -1 ).

than in previous experiments. These observations confirm that heterogeneities are detected by the proposed adapted ERT method in complex systems.

Conclusions

The objective of this work was to adapt and apply ERT to bubble dispersion characterization in a semi-industrial membrane tank filled with clear water and activated sludge.

-First, a numerical approach was proposed to define and validate the ERT methodology for the study of bubble dispersion in large-scale reactors with complex geometries. This modeling approach demonstrated the ability of the ERT method to observe gas distribution patterns for low theoretical gas hold-up in the membrane zone (from 1.6 to 6.3%). The proposed numerical study could be very useful for the development of ERT at industrial scale. -Then, the defined ERT method was applied to the bubble dispersion characterization in a semi-industrial membrane bioreactor filled with clear water and activated sludge. The membrane module zone was well described by the measurements whatever the MLSS concentration in the case of initial experiments without aeration. Gas dispersion was more homogeneous for low MLSS concentrations (3 g/L) at low gas flow rates (SAD m = 0.29 Nm 3 /h/m 2 ). At 6 g/L and 10 g/L, a high air flow rate is required for the bubbles to be distributed on the whole membrane zone (SAD m = 0.46 Nm 3 /h/ m 2 ). This result is consistent with the non-newtonian behavior of sludge. -With current technology, the methodology presented in this paper was successfully used to adapt the ERT method and apply it to a semi-industrial MBR filled with activated sludge. The inversion parameters chosen and the sequence of quadripoles adapted presented here show promising results for observing bubble dispersion.

Other inversion methods exist and could be investigated in further research. Moreover, either increasing the number of electrodes or adding electrodes at the center of the bioreactorsince in this study, the diffusive character of the distribution is the highest in this locationcould improve the precision of the ERT measurement results obtained.

Fig. 1 .

 1 Fig. 1. a) Semi-industrial MBR, b) KMS Puron membrane module, c) pilot geometry and electrode locations (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2 .

 2 Fig. 2. General methodology of the numerical study and the experimental one.

Fig. 3 .

 3 Fig. 3. Horizontal cross sections of theoretical ER distributions.

Fig. 4 .

 4 Fig. 4. Adapted sequences based on classical electrode arrays, (a) dipole-dipole (DD) and (b) gradient (GD), with e i are electrode locations (i = [1-24]).

  Fig. 5a presents the interpreted ER model obtained from the numerical study based on theoretical ER distribution A1 for the sequences of quadripoles based on DD and GD arrays. Fig. 5b-e presents the results as Δρ distributions obtained from the theoretical ER distributions B1 to E3. Δρ distributions are presented for the two sequences of quadripoles (GD [g] and DD [d]). Theoretical locations of positive Δρ are indicated by rectangular shapes added to each figure.

Table 2

 2 Calculated smoothness coefficients (Y = yes, N = no).

	Sequence	Distribution	D -x	D +x	D -y	D +y	X-centered	Y-centered
	DD	B1d	1.10	1.15	1.34	1.36	Y	Y
		B2d	1.10	1.15	1.34	1.36	Y	Y
		B3d	1.10	1.15	1.34	1.35	Y	Y
	GD	B1g	1.15	1.16	1.35	1.36	Y	Y
		B2g	1.15	1.16	1.35	1.36	Y	Y
		B3g	1.15	1.16	1.34	1.35	Y	Y
	DD	C1d	6.1	5.7	3.5	3.2	Y	Y
		C2d	5.9	5.5	1.4	1.5	Y	Y
	GD	C1g	6.8	6.8	4.4	4.6	Y	Y
		C2g	6.1	5.7	1.7	1.7	Y	Y
	DD	D1d	2.7	5.0	2.8	2.7	N	Y
		D2d	2.5	4.2	1.4	1.6	N	Y
		D3d	1.3	1.3	2.1	3.2	Y	N
	GD	D1g	2.9	4.9	2.7	2.8	N	Y
		D2g	2.7	4.4	1.4	1.5	N	Y
		D3g	1.3	1.1	2.1	3.7	Y	N

Table 3

 3 Main observed ERT errors for DD and GD sequences (+ = Minor error (less than 5%)/+++ = Major error (More than 20%).

	Generated error

Table 4

 4 Operating conditions and activated sludge characteristics during ERT measurements. (mS.cm -1 ) G (s -1 ) μ app (mPa.s -1 )

	MLSS (g L -1 ) T (°C) σ 0 SAD m (Nm 3 /h/ m 2 ) 0.29 24 0.7	304	1.02
		0.38			349	1.02
		0.52			408	1.02
	3	0.29	27	2.2	265	1.35
		0.41			320	1.31
		0.46			341	1.29
	6	0.14	33	4.4	143	2.22
		0.18			167	2.10
		0.29	35	4.5	229	1.81
		0.41			283	1.67
		0.47			308	1.62
	10	0.29	33	4.4	147	4.37
		0.42			194	3.64
		0.54			234	3.22

Acknowledgements

Authors thank the MOCOPEE program (www.mocopee.com) and Irstea for financial support. The authors wish to thank the SIAAP staff of Fig. 7. Interpreted ER models (Initial -Membrane without aeration) and Δρ distributions for different SAD m and MLSS concentrations: a) 3 aerated membrane modules -SAD m = 0, 0.29, 0.38, 0.52 Nm 3 /h/m 2 [clean water, MLSS = 0 g/L], b) 3 aerated membrane modules -SAD m = 0, 0.29, 0.41, 0.46 Nm 3 /h/m 2 [sludge, MLSS = 3 g/L], c) 3 aerated membrane modules -SAD m = 0, 0.29, 0.41, 0.47 Nm 3 /h/m 2 [sludge, MLSS = 6 g/L], d) 3 aerated membrane modules -SAD m = 0, 0.29, 0.42, 0.54 Nm 3 /h/m 2 [sludge, MLSS = 10 g/L], e) 1 aerated membrane module -SAD m = 0.14, 0.18 Nm 3 /h/m 2 [sludge, MLSS = 6 g/L] -rrms corresponds to relative root mean square error between the modeled and the experimental apparent ER distribution data sets.


Rémy Gauthier

Hélène Follet

Max Langer

Evelyne Gineyts

Cécile Olivier

Pierre-Jean Gouttenoire

Lukas Helfen

Sylvain Meille

Jérôme Chevalier

Frédéric Rongiéras

Françoise Peyrin

David Mitton

Human cortical bone crack propagation mechanisms: what about the influence of loading rate?

Introduction:

A fracture is estimated every three seconds in the world, leading to an increased risk of impairment or even mortality in the elderly. Cortical bone biomarker for fracture mechanisms are well documented under quasi-static loading condition. It has been showed that, under a quasi-static condition, structural features such as the porosity network from the lacunar to the Haversian system, or even collagen remodeling properties are involved in bone crack propagation. However the biomechanical knowledge of bone fracture in a fall configuration remains poorly documented. The aim of the current study is thus to collect biomechanical, morphometric and biochemical parameters to investigate human cortical bone crack propagation mechanisms considering different loading conditions and different anatomical locations.

Materials and methods:

Paired human femoral diaphysis, femoral neck and radial diaphysis were extracted from 8 female donors (70 ± 14 y.o., min: 50 y.o., max: 91 y.o.). Toughness experiments were performed on cortical bone samples considering two loading rates: a quasi-static standard condition, and a condition representative of a fall. The mechanically tested specimen were then used to perform synchrotron radiation micro-computed tomography imaging (voxel = [0.7 µm] 3 ) in order to quantify the Haversian and lacunar systems. Microcracking mechanism during crack propagation was also investigated. Adjacent cortical bone specimens were finally used to measure the concentration of immature and mature enzymatic cross-links by a high performance liquid chromatography. The CX ratio (CX = [Immature] / [Mature]) was analyzed as it gives information on the collagen cross-links maturation.

Results and discussion:

Under quasi-static loading, collagen maturation appears to have an influence on the elastic contribution of bone behavior whereas the lacunar and Haversian systems are involved on the non-linear contribution, which represents the main contribution of bone global mechanical response. If bone porosity appears to be a major determinant under a standard loading condition, it is less involved in crack propagation mechanisms under fall-like condition. Crack propagation barriers seem less efficient in this critical loading condition. Moreover bone is less able to undergo microcracking in the case of a fall, meaning that this toughening mechanism is also less efficient. This study provides new information on human cortical bone crack propagation mechanisms under quasi-static and fall-like loading condition at different length scale, on different paired anatomical locations considering both weight-and non-weight bearing bones.

Acknowledgment:

This work was done in the framework of LabEx PRIMES (ANR-11-LABX-0063). This study was partly funded by the Région Rhône-Alpes and by the ANR project MULTIPS (ANR-13-BS09-0006). The authors also want to thank the Fédération INGE'LYSE for its financial support.


Rémy Gauthier

Hélène Follet

Max Langer

Evelyne Gineyts

Cécile Olivier

Pierre-Jean Gouttenoire

Lukas Helfen

Sylvain Meille

Jérôme Chevalier

Frédéric Rongiéras

Françoise Peyrin

David Mitton

What is the influence of strain rate on human cortical bone crack propagation mechanisms?

Introduction

Osteoporosis remains today a major health problem. Cortical bone fracture mechanisms are well documented under quasi-static loading condition. However the influence of strain rate on fracture needs to be better understood as bone fractures mainly occur after a fall from a standing height. The fracture process in human cortical bone is complex, and involves the overall organization of the tissue. Previous results showed that the whole porosity network, from the Haversian system to the lacunar network, as well as the collagen cross-links maturation, might be involved in the crack propagation mechanisms in cortical bone. Thus, the aim of the current study is to collect biomechanical, morphometric and biochemical parameters to make an overall characterization of human cortical bone crack propagation mechanisms considering two different loading conditions.

Materials and methods

In that objective, paired human femoral diaphysis, femoral neck and radial diaphysis were extracted from 8 female donors (70 ± 14 y.o., min: 50 y.o., max: 91 y.o.). Toughness experiments were performed on cortical bone samples. Two loading rates were considered for each bone: a quasi-static standard condition, and a condition representative of a fall. The mechanically tested specimen were then used to perform synchrotron radiation microcomputed tomography imaging (voxel = [0.7 µm]

3 ) in order quantify the Haversian and lacunar systems, and to evaluate the formation of micro-cracks during crack propagation. Adjacent cortical bone specimens were finally used to measure the concentration of immature and mature enzymatic cross-links by a high performance liquid chromatography. The CX ratio (CX = [Immature] / [Mature]) was analyzed as it gives information on the collagen cross-links maturation.

Results and discussion

Results showed that if collagen cross-links are involved in the elastic contribution of the fracture process, the plastic one is mainly governed by the porosity network associated with micro-cracking or crack deflection toughening mechanisms under a quasi-static loading. Under a fall-like condition, the cortical porosity is less involved in crack propagation mechanisms. Crack propagation barriers seem less efficient in this critical loading condition. This study provides new information on human cortical bone crack propagation mechanisms under quasi-static and fall-like loading condition, on different paired anatomical locations.

Acknowledgment

This work was done in the framework of LabEx PRIMES (ANR-11-LABX-0063). This study was partly funded by the Région Rhône-Alpes and by the ANR project MULTIPS (ANR-13-BS09-0006). The authors also want to thank the Fédération INGE'LYSE for its financial support.