Léa Duchesne
email: ljmduchesne@gmail.com

Karine Lacombe

Keywords: biosensors, lab-on-a-chip, microfluidics, nanotechnology, point-of-care tests, synthetic biology

According to the Global Burden of Diseases, chronic viral hepatitis B and C are one of the most challenging global health conditions that rank among the first causes of morbidity and mortality worldwide. Low and middle income countries are particularly affected by the health burden associated with HBV or HCV infection. One major gap in efficiently addressing the issue of viral hepatitis is universal screening. However, the costs and chronic lack of human resources for using traditional screening strategies based on serology and molecular biology preclude any scaling-up. Point-of-care tests have been deemed a powerful potential solution to fill the current diagnostics gap in lowresource and decentralized settings. Despite high interest resulting from their development in recent years, very few point-of-care devices have reached the market. Scaling down and automating all testing steps in one single device (e.g. sample preparation, detection and readout) is indeed challenging. But innovations in multiple disciplines such as nanotechnologies, microfluidics, biosensors and synthetic biology have led to the creation of chip-sized laboratory systems called "lab-on-a-chip" devices. This review aims to explain how these innovations can overcome technological barriers that usually arise for each testing step while developing integrated point-of-care tests. Point-of-care test prototypes rarely meet the requirements for mass production, which also hinders their large-scale production. In addition to logistical hurdles, legal and economic constraints specific to the commercialization of in vitro diagnostics, which have also participated in the low transfer of innovative point-of-care tests to the field, are discussed.

Introduction

In 2015, the Global Burden of Diseases study has ranked death from chronic viral hepatitis and associated conditions such as cirrhosis and liver cancer among the top 20 causes of death worldwide, with a steep increase between 1990 and 2015 (1). The World Health Organization (WHO) has recently taken steps in tackling the epidemic of hepatitis B and C by issuing guidelines for care and management [START_REF]Guidelines for the screening, care and treatment of persons with chronic hepatitis C infection[END_REF][START_REF]Guidelines for the prevention, care and treatment of persons with chronic hepatitis B infection[END_REF] and editing a strong statement towards the global elimination of both epidemics by year 2030 [START_REF]Global hepatitis report[END_REF]. The most affected countries in the fight against viral hepatitis are the low and middle income countries that have combined for years a global political inertia around hepatitis and structural barriers to screening (high costs, extreme centralization of technical and human resources) [START_REF] Lemoine | Reducing the neglected burden of viral hepatitis in Africa: strategies for a global approach[END_REF]. Indeed, conventional laboratory techniques require continuous access to electricity and water, refrigeration capacities and trained human resources able to perform and interpret complex assays, and are therefore restricted to major cities.

"Point-of-care tests" (POCTs) are portable devices meant to perform diagnostic assays at or near the site of patient care. Due to their small size and their fast time-to-answer, they could bring instantaneous diagnostic and monitoring capacities to decentralized settings. However, to make a real impact, POCTs not only need to be compact but also affordable, sensitive, specific, user-friendly, rapid, equipment-free and delivered to the populations who need them, as summarized in the "ASSURED" criteria created by the WHO [START_REF] Tdr | Mapping the landscape of diagnostics for sexually transmitted infections[END_REF]. Unfortunately, few POCTs currently on the market meet those requirements.

Regarding viral hepatitis diagnosis, the vast majority of commercialized POCTs are lateral flow immunoassays (LFIAs). Although portable and very easy to use, they often lack sensitivity and specificity, which limits their reliability and adoption in the field. The very few POCTs able to perform molecular assays feature bench-top formats that are not adapted to POC contexts because they require a power supply and minimally trained personnel and are expensive.

More generally, POCT development has been hindered by technical difficulties in scaling down, automating and integrating all testing steps-namely sample pre-treatment, amplification if needed, detection and readout of the result for interpretation-into a single "sample-to-answer" device.

Research in diagnostics development has focused mainly on miniaturizing and improving sensing mechanisms, resulting in the creation of very sensitive devices that necessitate the addition of bulky external equipment to perform sample preparation or to report test results. However, advances in many different scientific and technological disciplines over the last decade, such as micro and nanomaterials, bio-engineering and microfluidics, have enabled the creation of integrated devices, called lab-on-a-chip (LOC) [START_REF] Streets | Chip in a lab: Microfluidics for next generation life science research[END_REF]. When applied to clinical diagnostics, these automated laboratories miniaturized onto a chip can provide solutions to POCT development barriers and meet the ASSURED criteria.

This review aims to give an overview of the main difficulties encountered for scaling down each testing step and innovative techniques developed to overcome those difficulties in the field of viral hepatitis. Fig. 1 summarizes the main challenges and advances discussed here. In addition to these technical aspects, challenges related to the translation from the laboratory to the field of these technologies are also discussed.

Sample preparation

Sample preparation ensures the quality of analysis in complex matrix, such as blood or urine, by separating the target analyte from compounds that could cause cross-reactivity or interfere with the detection technique. Since biomarkers are present in particularly low molar concentration, concentration or amplification of the target are also often necessary. Conventional techniques for sample preparation are expensive and time-consuming. Research to simplify and automate sample preparation has lagged considerably compared to the development of sensing mechanisms suitable for POC applications. As a result, it remains one of the major challenges for POCT development.

Microfluidics systems aim to process and manipulate small volume of fluids automatically through specifically designed micro-or nano-channels and compartments. The unique physical and chemical properties presented by fluids at these scales have been exploited to create miniaturized sample preparation devices based on mechanical, chemical and affinity-based techniques [START_REF] Cui | Microfluidic Sample Preparation for Medical Diagnostics[END_REF].

Mechanical techniques

Separation of the target from the matrix can be performed using mechanical techniques, the most common being centrifugation and filtration. Some ingenious centrifuges based on daily objects, such as an egg beater or a salad spinner [START_REF] Brown | A Hand-Powered, Portable, Low-Cost Centrifuge for Diagnosing Anemia in Low-Resource Settings[END_REF], were developed to reduce costs, complexity and infrastructure requirements associated with these techniques. Although these devices are a first step to bringing sample preparation to the field, manual manipulations are still needed to transfer the extraction product to the analytical device, thereby increasing contamination risks. Microfluidic centrifugal devices, also known as lab-on-a-CD (LOD) because their microfluidic system is implemented on a rotating disc, can use the pseudo-forces created by the rotation to lyse cells, sort particles or mix fluids, and integrate other functions such as detection [START_REF] Kong | Lab-on-a-CD: A Fully Integrated Molecular Diagnostic System[END_REF]. Porous membranes made of fluidic components, such as pillars or holes, of sizes similar to target analytes can be used to filtrate cells or even single molecules according to their size or density. For example, Sang et al. were able to successfully separate plasma and HIV virus from whole blood samples by means of micrometer-scale pores [START_REF] Wang | Simple filter microchip for rapid separation of plasma and viruses from whole blood[END_REF]. These filters are precise and easy to manufacture and combine with detection devices but susceptible to clogging.

Other systems exploit the hydrodynamic properties of microfluidics to precisely control the flow rate and specifically sort compounds, through precise geometrical design of channels [START_REF] Karimi | Hydrodynamic mechanisms of cell and particle trapping in microfluidics[END_REF].

Chemical techniques

Sample preparation can also be performed using chemical reagents. Chemical protocols usually involve multiple steps that must be executed in a specific order and with precise timing, which are difficult to automate in miniaturized devices. Chemically modified papers able to collect, store and extract nucleic acids (NAs) from a blood sample have already been commercialized. Although very efficient, most of them still require manual interventions, such as adding reagents or washing steps, which decrease their value for potential POC utilization. The addition and mixing of reagents can be automated by using chambers containing pre-loaded reagents and designed to produce a turbulent flow while the sample flows through it. Droplet-generating microfluidic systems can deliver fluids in a complex, automated and specific sequence [START_REF] Tian | Distance-based microfluidic quantitative detection methods for point-of-care testing[END_REF].

Affinity-based techniques

Affinity-based mechanisms use a probe immobilized on a surface to specifically capture the target biomarker. As many detection methods also recognize their target by means of a probe, affinity-based separation techniques have great potential to integrate both operations into the same module.

Antibodies are the most broadly used affinity probes, due to their high selectivity and numerous pathogens they can target. However, using them in POCTs presents several drawbacks. First, they have limited stability in environmental conditions common in POC settings, such as extreme temperature.

Additionally, antibodies production methods are very well-known but expensive and complex.

Similarly, nucleic acid probes fabrication remains expensive. Therefore, alternative affinity tools, such as engineered binding proteins, aptamers or molecular imprinted polymers (MIPs), which are more versatile, cheaper and easier to produce, have been developed [START_REF] Ruigrok | Alternative affinity tools: more attractive than antibodies?[END_REF].

These affinity tools can be combined to nanoparticles (NPs) with magnetic or electric properties when used for electro-kinetics separation. By applying an electric field to the matrix, nanoparticle beads functionalized with an affinity tool can make the targets migrate to a specific area of the device for detection. For instance, a magnetic bead-based separation technique integrated into a small microfluidic chip could efficiently separate influenza A virus from whole blood, serum and saliva samples within 15 minutes [START_REF] Lien | Rapid detection of influenza A virus infection utilizing an immunomagnetic bead-based microfluidic system[END_REF].

Detection

The detection threshold for many POCTs currently on the market is too high compared to biomarker concentrations in raw samples. Various strategies have been adopted to improve test sensitivity. Firstly, a precise target isolation and concentration step can be implemented, as previously shown. Secondly, the target or its signal can be amplified to reach the low limit of detection (LLOD) of the assay. This process is particularly necessary for NA detection but miniaturizing it has been a major challenge because of its complexity [START_REF] Wu | Extraction, amplification and detection of DNA in microfluidic chip-based assays[END_REF]. Finally, ensuring the robustness and selectivity while developing sensing methods is crucial to overcome matrix effect.

Nucleic acid amplification-based techniques

Amplification by polymerase chain reaction (PCR) is the most commonly method used to duplicate small amounts of NA. Based on temperature-dependent reactions, processing PCR requires a thermocycler, a bulky and expensive piece of equipment, to control heating cycles. Mini thermocyclers with rechargeable batteries, such as the miniPCR [START_REF] Guevara | A simple, economical protocol for DNA extraction and amplification where there is no lab[END_REF], are now available on the market, or they can be found through open-source projects [START_REF]Thermocycler for PCR[END_REF]. However, because of their price (between US $200 and $650) and need for minimally skilled technicians, they are more suitable for small laboratories than for POC settings. Isothermal amplification methods were developed to reduce the complexity of heat control. However, they still require a minimal heating process and, therefore, a source of energy, a scarcity in low-resource or decentralized settings.

To adapt this need to POC settings, research teams used miniaturized sources of electricity, such as commercial heat packs [START_REF] Yeh | Self-powered integrated microfluidic point-of-care low-cost enabling (SIMPLE) chip[END_REF]. Others developed electricity-free devices, generating power or heat through chemical reaction or paper-based batteries. For example, the non-instrumented nucleic acid amplification (NINA) platform is a thermos-sized device incorporating a heater, based on an exothermic chemical reaction. The materials necessary to construct the heater are widely available and cost $0.60 per test. NINA was easily combined with an HIV lateral flow assay (LFA) and showed performance similar to commercialized PCRs [START_REF] Labarre | A Simple, Inexpensive Device for Nucleic Acid Amplification without Electricity-Toward Instrument-Free Molecular Diagnostics in Low-Resource Settings[END_REF].

Microfluidics' small size enables quick heat transfers, which offer great opportunities in scaling down the PCR process. Although the integration of all the complex NA amplification-based detection steps into one single device remains challenging, advances have been made in this domain [START_REF] Su | Microfluidic platform towards point-of-care diagnostics in infectious diseases[END_REF].

However, the costs of primer production and licensing fees associated to the majority of isothermal amplification reactions could hinder the adoption of such technologies for POCTs targeting developing countries.

Micro and nano-scale sensors

Biosensors are compact sensing devices made of a biological recognition element, immobilized on the surface of a transducer. The transducer converts the physical or chemical variation produced at its surface by the interaction between the recognition element and the target into a measurable or observable reporting signal proportional to the target concentration. Since the whole detection process occurs at once in the same module, without requiring additional steps like reagent addition or washing, biosensors have a fast time-to-answer. In addition, due to their very small size, they can perform with smaller sample volumes than in conventional assays, reducing the quantity of reagents needed and thus lowering assay costs. They have therefore been extensively studied to create POCTs [START_REF] Bhalla | Introduction to biosensors[END_REF]. Major impediments to wide adoption of biosensors in POCTs include mixed sensitivity and specificity in raw samples, complexity of production and need for external analyzer to interpret the signal [START_REF] Vigneshvar | Recent Advances in Biosensor Technology for Potential Applications -An Overview[END_REF].

Biosensors can be classified according to the nature of the reaction their transducer can detect (electrochemical, optical or mechanic) or their detection strategy. Labeled assays work as traditional sandwiched assays where the target is caught between the recognition element and a detector element functionalized with a signalling tag. For label-free assays, the transducer directly senses the changes produced at its surface by the bio-recognition process [START_REF] Luka | Microfluidics Integrated Biosensors: A Leading Technology towards Lab-on-a-Chip and Sensing Applications[END_REF]. The detection and signalling properties of a transducer or a tag are determined by the characteristics of the material they are made of. The development of nano-materials with novel signaling properties has enabled the creation of innovative ultra-sensitive detection methods [START_REF] Sang | Progress of new label-free techniques for biosensors: a review[END_REF]. Carbon nanotubes (CNTs), NPs and grapheme have undergone the most study. Among NPs, gold NPs are a preferred tool for labelling due to its biocompatibility, surface-tovolume ratio and ease to functionalize with biological affinity compounds. In addition, its ability to change color in response to a variation in the environment composition makes it a powerful signal enhancer. The high electron transfer capacities of CNTs and graphene make them able to greatly amplify the detection signal and are therefore effective transducers. For instance, they have been used as electrodes in many biosensor prototypes [START_REF] Syedmoradi | Point of care testing: The impact of nanotechnology[END_REF]. All these materials can also be coated on the sensing surface or inserted inside it to enhance the detection signal [START_REF] Wei | DNA diagnostics: nanotechnology-enhanced electrochemical detection of nucleic acids[END_REF]. Biosensor and nanomaterial-based technologies applied to HBV and HCV detection have been reviewed elsewhere [START_REF] Yao | Biosensors for hepatitis B virus detection[END_REF][START_REF] Uliana | Diagnostic tests for hepatitis C: recent trends in electrochemical immunosensor and genosensor analysis[END_REF][START_REF] Yildiz | Recent advances in micro/nanotechnologies for global control of hepatitis B infection[END_REF]. Table 1 presents a non-exhaustive list of the most recently developed viral hepatitis biosensors and their characteristics regarding detection and their level of integration in a single device.

Mass spectrometry (MS), although not based on a biological recognition element, is worth mentioning as a potential future micro-sensor for POCTs. Its mechanism, based on the direct identification of molecules or nucleic acid sequences of interest through measurement of their mass after ionization of the sample, could remove the need for sample preparation. Recent advances made to incorporate it into microfluidic structures open up opportunities for miniaturizing MS analysis systems [START_REF] Wang | Microfluidics-to-mass spectrometry: a review of coupling methods and applications[END_REF].

Synthetic biology-based biosensors

Synthetic biology aims to reprogram natural mechanisms of biological systems to meet human needs. Yet, many biological systems possess very sensitive machineries to sense environmental changes and adapt their metabolism to them. Therefore, synthetic biology researchers have engineered them to create detection tools.

Synthetic RNA regulators

Riboregulators are engineered RNAs capable of regulating expression of their own gene or of a target gene, in response to a change in their environment, at the translational or transcriptional level. Thus, they can be used as sensors regulating the expression of a measurable protein to report the presence or the absence of a target RNA. A new design of a translational riboregulator that can be adapted to any target RNA called "toehold switches" was developed [START_REF] Green | Toehold switches: de-novo-designed regulators of gene expression[END_REF]. Pardee et al. used this technology to construct a sensor able to detect Zika virus RNA in concentrations as little as 3Fm in 30 minutes [START_REF] Pardee | Rapid, Low-Cost Detection of Zika Virus Using Programmable Biomolecular Components[END_REF]. They created a fast toehold switch-based sensor development protocol, from the design stage to prototype manufacturing, which was completed in 5 days at a very low investment cost ($20/prototype) and a final cost of $0.10-$1.00 per test. Isothermal amplification of NA was necessary to reach clinically relevant sensitivity. Future research about how to combine this technology and simplified amplification techniques will therefore be needed to fashion an operational POCT. Up to now, synthetic biology-based sensing mechanisms could only be engineered in cell-systems, but a technique developed recently has made transcription and translation (TX-TL) reactions in vitro possible [START_REF] Shin | An E. coli cell-free expression toolbox: application to synthetic gene circuits and artificial cells[END_REF]. This TX-TL system was used to transfer the Zika biosensor onto filter paper.

RNA regulators acting at the transcriptional level, like the small transcription activating RNAs (STARs), could also be used to detect RNA as well as DNA. However, to our knowledge, this technology has still not been applied to diagnostics.

CRISPR-based biosensor

The CRISPR/Cas9 is an editing genome tool inspired from a bacterial defense system. CRISPR, the acronym for "clustered regularly interspaced short palindromic repeats," refers to repeated sequences corresponding to viral sequences situated into bacterial genomes. An endonuclease called cas9 uses the translated RNAs from CRISPR sequences to bind and then cut the viral genome. By replacing bacterial CRISPR with synthetic guide RNAs targeting a specific DNA sequence, researchers transformed this system into an editing genome tool. So far, it has been primarily used as molecular scissors to inhibit genes, by shortening them, or to add DNA sequences in a specific locus.

A CRISPR-based nucleic acid detection tool was recently developed, the SHERLOCK ("specific high sensitivity enzymatic reporter unlocking") platform [START_REF] Gootenberg | Nucleic acid detection with CRISPR-Cas13a/C2c2[END_REF], a modified version of CRISPR/Cas9. In this system, CRISPR RNAs were designed to target RNA instead of DNA. It also includes a different ribonuclease, the Cas13a, whose specificity is to cut any nearby RNAs when it binds to its target. Placed in contact with fluorescent RNAs, the CRISPR/Cas13a system can therefore report the presence of a target. Combined with isothermal NA amplification, SHERLOCK detected attomolar (10 -18 ) concentrations of NA in serum or urine. Moreover, it was specific enough to distinguish virus sequences differing by a single-base mismatch.

Metagenomics

The vast majority of NA detection techniques require that NA sequences be designed specific to the target. This phase is complex, time-consuming and expensive. Recent advances in metagenomics, a method to directly sequence all NA present in raw samples, could facilitate NA detection. Until now, this technique was too expensive and lengthy to be used in diagnostics. Moreover, data analysis must be conducted with computational software by high-skilled personnel. The decreasing cost and shorter turn-around time of metagenomics have heightened interest in developing and facilitating diagnostic metagenomics-based diagnostics. Thus, Taxonomer, a fast and user-friendly web-based metagenomic sequence analysis tool was developed [START_REF] Flygare | Taxonomer: an interactive metagenomics analysis portal for universal pathogen detection and host mRNA expression profiling[END_REF]. Furthermore, mobile sequencers are now available. For example, the MinIONs device is the size of a pen drive and was able to read the genome of Ebola viruses from 14 patients in only 48 hours [START_REF] Hayden | Pint-sized DNA sequencer impresses first users[END_REF].

Micro-sensors are powerful and versatile solutions to simplify detection. One of the major remaining challenges in designing them is the difficulty in suppressing background noise and intrinsic variability. In addition to increasing selectivity and sensitivity of detection strategies, efforts have been made to create built-in noise correction, by using a ratiometric signal for example [START_REF] Jin | Fabrication strategies, sensing modes and analytical applications of ratiometric electrochemical biosensors[END_REF]. Another challenge concerns the conversion of micro-sensors signals into a readable output for the end-user. This process often necessitates expensive and bulky reporting systems sensitive enough to measure a signal in small reaction volumes. An enhanced reporting signal such as those reviewed in this section can help to reduce the inherent complexity of these systems.

Integration

Ideally, untrained individuals should be able to use POCTs in field settings where sample contamination risks are high. Designing devices able to process all the assay procedures without external manipulations is therefore essential. Yet, most proposed solutions to scale down these different steps have been developed separately, resulting in devices that are very efficient but only address one specific hurdle and lack sufficient compatibility with other devices. This lack of synergy and flexibility limits the potential impact these innovations may have in broadening diagnosis access.

The case of viral hepatitis perfectly illustrates this phenomenon. Indeed, as shown in Table 1, although many sensitive innovative and miniaturized sensing mechanisms have been developed for the diagnosis of viral hepatitis, very few of them include all testing steps or are integrated in a fully operational device. However, promising solutions have been developed for other infectious diseases and could be adapted for diagnosis of viral hepatitis.

In recent years, combining microfluidics and biosensors has enabled the creation of on-chip miniaturized laboratory systems called "lab-on-a-chip" (LOC), bringing new opportunities for developing integrated diagnostic devices. The challenges of designing such devices lie not only in integration but also in meeting POCT requirements, namely low-cost and ease of production, process automation and a portable results-reporting system.

Substrates

Chip material selection is a key step that impacts the complexity of microfluidics networks that can be implemented in the chip, ease of manufacturing and price of the whole LOC. Yet, this must be balanced with the need for the material to be compatible with the sample, sensing mode and reagent characteristics. The first microfluidics chips were principally made of glass, silicon and especially polydimethylsiloxane (PDMS), an elastomer. Despite its many advantages for LOC conception-such as transparency, low-cost pilot scale manufacturing and ability to support thin channels and external elements-PDMS-based LOCs have been poorly commercialized due to high costs of large-scale production. Therefore, alternative materials adapted for POCT mass production were investigated, and two candidates appear to be particularly promising: thermoplastics and paper.

Mass production methods using thermoplastics, another kind of polymers, are less expensive than using PDMS and rely on replicating a mold. Producing the molds can be time-consuming, expensive and therefore unsuitable for prototyping, but direct fabrication techniques have been recently developed. Thermoplastics can support many different assays due to its compatibility with a wide variety of chemicals. Moreover, thermoplastic chips allow for the insertion of a high concentration of channels and compartments and thus have the capacity to generate complex designs, such as LODs.

Paper is an attractive choice for large-scale POCT fabrication due to its low-cost, capillarity properties, portability, worldwide availability, well-known manufacturing processes and compatibility with a wide range of chemical and biological components. Moreover, paper can drive fluids by capillarity, removing the need for external pumps or moving parts. Methods to insert channels defined by hydrophilic barriers or to create multiple-layered structures on paper have been developed, giving rise to microfluidic paper-based analytical devices (µPADs) able to support more complex assays than the traditional LFAs or dipstick assays. For instance, plasma separation and simultaneous detection of aspartate aminotransferase and alanine aminotransferase could be accurately performed in serum and whole blood by a three-dimensional (3D) µPAD created by stacking patterned paper and wax printing of channels [START_REF] Pollock | A paper-based multiplexed transaminase test for low-cost, point-of-care liver function testing[END_REF]. In 15 minutes, it provided a colorimetric output with an intensity proportional to the target concentrations. Such semi-quantitative measurements are common in paper-based assays but are sensitive to intra-rater discrepancies and are not sufficient when precise monitoring is needed. The creation of simple and cost-effective methods to print electrodes on µPADs has expanded their use in quantitative analysis of biomarkers. Thus, a microfluidic paper-based electrochemical device (µPED), tested in mouse serum, was able to detect HIV p24 core antigen and HCV core antigen with limits of detection of 300 pg/mL and 750 pg/mL, respectively [START_REF] Zhao | A portable paper-based microfluidic platform for multiplexed electrochemical detection of human immunodeficiency virus and hepatitis C virus antibodies in serum[END_REF].

Many features of POCTs are determined by their substrate, including their potential for large-scale manufacturing. Material selection must therefore be done at the early stages of POCT design while taking into account all the final device requirements.

Automation

The first LOCs required external devices for fluid control, sample loading and reagents addition and, therefore, manipulations by the user. To be used as POCTs, LOCs must be automated as much as possible.

To prevent reagent manipulations while testing, printing methods were developed to pre-pattern them in a stabilized form on the chip [START_REF] Byrnes | Progress in the development of paper-based diagnostics for lowresource point-of-care settings[END_REF]. Since POC settings lack storage or refrigeration capacity, diagnostic tests must be treated in order to ensure long-term conservation of reagents without efficacy loss. Solid or liquid chemicals can be conserved in a solid form by lyophilisation and then rehydrated by the sample itself. Freeze-drying was used to store the Zika toehold switch biosensor on paper, as previously cited [START_REF] Pardee | Rapid, Low-Cost Detection of Zika Virus Using Programmable Biomolecular Components[END_REF]. Encapsulation of reagents in natural polysaccharide was also developed to store chemicals on chips [START_REF] Jahanshahi-Anbuhi | Pullulan encapsulation of labile biomolecules to give stable bioassay tablets[END_REF]. Storage of reagents in a liquid form, which was a challenge until now, was performed using micro pouches [START_REF] Mauk | Microfluidic 'Pouch' Chips for Immunoassays and Nucleic Acid Amplification Tests[END_REF] or bio-inert glass ampoules.

Microfluidics systems are used to control fluidic transport through pressure-driven flow generated with pumps or pressure generators. Integration of such equipment directly into the device implies diminishing its portability or creating additional costs for miniaturization. Different chip designs enable equipment-free flow generation. Paper-based assays, for example, draw the sample through capillarity, with no external device. LODs use the various forces created by its rotation to drive flow between its different modules. A vacuum battery, able to suck liquids by releasing a vacuum that is pre-stored in PDMS pouches, was inserted into a fully integrated diagnostic microfluidic device [START_REF] Yeh | Self-powered integrated microfluidic point-of-care low-cost enabling (SIMPLE) chip[END_REF].

Automation of flow control may also involve the insertion of micro-valves into the device in order to compartmentalize the different operations performed within it. Valves also increase the complexity of manufacturing and therefore raise production costs, making them less desirable for POTC use. Passive valves regulated by fluid flow have been developed to simplify their design and insertion.

End-user interface

When not visible to the naked eye or when used for quantitative analysis, the signal produced by a sensor must be converted into an interpretable output. Reporting equipment-a microscope or spectrometers, for instance-often adds costs and complexity to test devices, counteracting the advantages of miniaturization.

Cellphones provide a portable, user-friendly and universally available processing unit.

Cellphone-based reporting systems have therefore been developed [START_REF] Lopez-Barbosa | The future point-of-care detection of disease and its data capture and handling[END_REF]. An initial strategy consists of on-site analysis using a cellphone camera and built-in software able to analyze the photo's features. This strategy gives an instantaneous result but is limited to assays with a visual output. Additionally, it requires the use of smartphones, which are not available in all countries. Therefore, compact assay analyzers using cellphones to display results were created, usually from open-source hardware called Arduino enabling a low fabrication cost [START_REF] Grinias | An Inexpensive, Open-Source USB Arduino Data Acquisition Device for Chemical Instrumentation[END_REF]. Other strategies are based on off-site analysis where the data collected by the cellphone camera or the analyzer are sent by text or e-mail to a trained operator who sends back an interpretation of the result. In the same vein, automated analysis by a cloud-based software was successfully developed [START_REF] Selck | Increased Robustness of Single-Molecule Counting with Microfluidics, Digital Isothermal Amplification, and a Mobile Phone versus Real-Time Kinetic Measurements[END_REF]. Most of these strategies are designed for a specific kind of analysis and a wireless or Internet network. To overcome these requirements, a team developed a device able to analyze almost any electrochemical reaction that is compatible with any cellphone and network [START_REF] Nemiroski | Universal mobile electrochemical detector designed for use in resource-limited applications[END_REF].

Open platforms

Diagnostic tests are usually drawn up anew according to the characteristics of a specific issue. It has been argued that creating reprogrammable basal platforms could save both development costs and time [START_REF] Patou | Evolvable Smartphone-Based Platforms for Point-of-Care In-Vitro Diagnostics Applications[END_REF]. Conception of such platforms is challenging because they must be able to support different samples, reagents and sensing modes.

An initial approach consists of designing devices able to run one specific analytical technique that works across different pathogens or assays. The open polyvalent platforms (OPPs) are bench-top devices conceived so that their instruments and reagents can be provided by different suppliers, allowing thus to reduce test prices through competition. Recently, a real-time PCR OPPs showed great performance in quantifying HBV DNA in the field, even with a generic assay [START_REF] Kania | Validation and scale-up of Hepatitis B viral load on polyvalent open PCR platforms in West African and South East Asian countries. A study of the AC12 working group (ANRS 12327)[END_REF]. This concept of design versatility has been applied to more portative prototypes. For example, a fully automated and integrated reverse transcription PCR-based device could be easily adapted to different pathogens (bacterial B. cereus, HIV) just by changing the reagents loaded in the chip [START_REF] Chen | An integrated, self-contained microfluidic cassette for isolation, amplification, and detection of nucleic acids[END_REF].

A second approach aims to create fluidic platforms whose design can be tailored to any type of analysis. Based on this idea, a design protocol for a programmable chemistry LOC, inspired by a music box, intended for resource-limited settings was recently published [START_REF] Korir | Punch card programmable microfluidics[END_REF]. It combines a silicon microfluidic chip containing reagents to a hand-cranked spinning wheel with pins on a punch card.

When a pin hits a hole, pumps or valves are activated in the chip, releasing droplets of fluids that were previously loaded onto the chip. Cards can be punched to create particular fluid delivery patterns. The silicon chip can also be modified to hold a specific number of reagent or mixing chambers. The chip is made of inexpensive materials (less than $5 overall) and can be re-used after washing and refilling of reagents.

Using basal platform designs could enable the creation of standards in LOC-based POCT production, which have been pointed out as a potential leverage for scaling up their commercialization [START_REF] Mohammed | Lab-on-a-chip or Chip-in-a-lab: Challenges of Commercialization Lost in Translation[END_REF]. However, standardization can also be considered as a drag on innovation. Continuing research is essential and, as there is still no one-size-fits-all POCT design, using a model such as these could prevent teams from having to start from scratch and ease the creation process of integrated devices.

Translation of micro-diagnostics to clinics

Despite the clear added value LOCs offer in overcoming current POCT limitations, their translation from research laboratories to concrete production and adoption in the field have been almost inexistent in the last decade.

The gap between research objectives and the requirements of the industrialization process, identified as one of the greatest bottlenecks to translation, is gradually closing. Manufacturing includes four major steps: design, fabrication, back-end processes and quality control. Regarding the design phase, LOC technologies have developed and now possess a range of tools broad enough to overcome almost any technical challenge. Additionally, awareness of the need to design devices from the very start that encapsulate all testing functionalities and have easy and low-cost assembly is rising. Although back-end processes and quality control only account for 20% of prototype production costs, these costs can reach up to 80% during mass production [START_REF] Becker | It's the economy[END_REF]. However, most proof-of-concept published today does not consider this difference. Substrate, sensor and design selection should be carefully considered to minimize device-to-device variability and ease reproducibility. But grants for "basic research" provide little incentive to develop such practical applications. Involving researchers in management sciences and process engineering when developing LOC-based POCTs could ensure appropriate design and simultaneously broaden the number of potential funding resources. Moreover, Sackmann et al. highlighted that LOC studies are published primarily in engineering journals and therefore do not reach clinicians or biologists, although they are the most likely potential end-users [START_REF] Sackmann | The present and future role of microfluidics in biomedical research[END_REF]. To ensure future clinical adoption, information about such technologies should be made more widely available to clinical laboratories and practitioners, who can even participate in the design and evaluation process.

Prior to their commercialization, "in vitro diagnostics" (IVD) must undergo a regulatory approval process in each country where the device will be marketed. Although these processes aim to ensure public safety, their pathways are generally burdensome, opaque and variable between countries. In most cases, National Regulatory Authorities (NRAs) request trials to be conducted in the field to prove the device's efficiency and relevance in application contexts. Such studies require substantial funding and also access to local clinical infrastructures. Complexity, duplication of proofs, costs, lack of harmonization and delayed market access may discourage potential investors from commercializing their product in all potential countries of application and contribute to limiting the availability of diagnostics worldwide. Initiatives have been launched to streamline harmonization of regulatory processes for various geopolitical scales, such as the International Medical Device Regulators Forum and the regional Harmonization Working Parties in Africa, Latin America and Asia [START_REF] Mcnerney | Diagnostics for Developing Countries[END_REF]. Similar organisms created to enhance harmonization of drug regulations yielded promising results, such as the harmonized medicines registration system established by the East African Community (EAC) [START_REF] Rugera | Regulation of medical diagnostics and medical devices in the East African community partner states[END_REF]. On the other hand, the WHO found that only 58% of its members possess legal procedures for IVD approval and that the majority of them are incomplete (57). Most countries lack the capacity to apply them, with consequent high levels of poor-quality diagnostics and public health implications. Regional organization of IVD regulations could prevent developers from duplicating trials and enable countries to share their resources.

While POCTs for LRS must be low-cost, their development requires substantial initial investment. Therefore, depending on the market size, return on investment might be low. As a result, POCTs are considered a risky market and lead investors to wait for a "killer application" involving lower risk investments to emerge, although the existence of such an application for LOC-based diagnostics has been questioned [START_REF] Caicedo | Microfluidics: The Challenge Is to Bridge the Gap Instead of Looking for a 'Killer App[END_REF]. This situation has raised the issue of alternative business strategies for POCT marketing. For instance, Diagnostics For All (DFA), a non-profit research company that can receive academic or foundation grants, has partnered with a for-profit subsidiary firm, Paper Diagnostics, which licenses DFA innovations, giving DFA a stable source of revenue through royalties [START_REF] Mace | A unique approach to business strategy as a means to enable change in global healthcare: a case study[END_REF].

Beyond technical barriers, for which solutions are increasingly coming, translating POCTs to the field is now principally hindered by the lack of links between the different worlds involved in POCT development. Multidisciplinary collaboration, including basic researchers, clinicians, management and industry sciences from the very beginning, should therefore be strengthened to accelerate and ease POCT adoption in the field.

Figures

  Figures

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 1 Fig.1.

  Fig.1.

Disclosures

Authors have no conflict of interest to declare.


Raphael Ducatez

A forward-backward random process for the spectrum of 1D Anderson operators

We give a new expression for the law of the eigenvalues of the discrete Anderson model on the finite interval [0, N ], in terms of two random processes starting at both ends of the interval. Using this formula, we deduce that the tail of the eigenvectors behaves approximately like exp(σB |n-k| -γ |n-k| 4 ) where Bs is the Brownian motion and k is uniformly chosen in [0, N ] independently of Bs. A similar result has recently been shown by B. Rifkind and B. Virag in the critical case, that is, when the random potential is multiplied by a factor 1 √ N

We are interested in the one dimensional discrete Anderson model on a finite domain [0, N ]. This model is very classical and has been studied extensively since the 70s. See for example the monograph of Carmona Lacroix [START_REF] Carmona | Spectral theory of random Schrödinger operators[END_REF]. Compared to higher dimensions case, it can be considered as a solved problem. However new approaches can always shed new light on this famous system.

The usual approach to tackle this system is the transfer matrix framework. The eigenvectors of the random Schrödinger operator satisfy a recursive relation of order 2, u n+2 = (V n+1 -λ)u n+1 -u n , which can be written in a matrix form. Using this relation, one can obtain an eigenvector everywhere on [0, N ] from the product of the transfer matrices applied to the boundary values. The advantage of such a formulation is that one can then use the very powerful results for random matrices product and from ergodic theory such as the Oseledets theorem.

In the historical approach of Kunz and Souillard [START_REF] Kunz | Sur le spectre des opérateurs aux différences finies aléatoires[END_REF] or in the proof from the book [START_REF] Cycon | Schrödinger operators: With application to quantum mechanics and global geometry[END_REF] a change of variables is used to deal with the conditional probability of the potential V with a fixed eigenvalue λ. In this short note, we propose another calculation of this conditional probability. We define a random variable k whose random law is close to the uniform law on [0, N ]. This variable splits the interval [0, N ] into two part [0, k] and [k, n]. On the left part, the matrices product is made from left to right. On the right part, the matrices product is made from right to left. And far from the cut, the laws of the matrices are very close to be independent.

The main interest of our approach is that the connection with the theorems for products of random matrices is more transparent in this setup. From this formula we can recover several known results. Relying on the positivity of the Lyapunov exponent, the formula can be used as a new proof of exponential Anderson localization of eigenvectors where the center of localization is uniformly distributed on [0, N ]. Moreover, because it gives a explicit random law, we can go beyond the exponential decay of the eigenvectors far from the center of localization and give an explicit law for their tail.

In the first section, we detail the model and we state our result. Then we give some applications of our theorem in the second section. In particular, we write an asymptotic result similar to the result of Rifkind and Virag in [START_REF] Rifkind | Eigenvectors of the critical 1-dimensional random schroedinger operator[END_REF]. In Section 3, we finally give the proof of the theorem.

Model and main result

We consider the discrete one dimensional Anderson model [START_REF] Anderson | Absence of diffusion in certain random lattices[END_REF] defined on [1, N ] through the operator

H (N ) = -∆ (N ) + V (N ) ω .
Here V

(N ) ω is a random iid potential and

∆ (N ) (x, y) = 1 if |x -y| = 1 0 otherwise
is the usual discrete Laplacian. Hence H is just the N × N symmetric matrix

H (N ) =          V 1 1 1 . . . . . . . . . . . . . . . 1 1 V N          .
We make the following assumption:

(H1) The random law of V ω is absolutely continuous with respect to the Lebesgue measure.

Transfer matrices

Transfer matrices have been one of the main tool to study the 1D Anderson model. One is interested in the eigenvectors, (Hu) n = λu n , which satisfy the recurrence relation

∀n ∈ [0, N ], -u n+1 + (V ω -λ)u n -u n-1 = 0, (1)
with u -1 = u N +1 = 0 such that the formula is valid for n = 0 and n = N . This can be written with transfer matrices

u n+1 u n = T (v ω (n) -λ) u n u n-1
where ∀x, T (x) =

x -1 1 0 .

We can then write the matrix product

M n (λ) = n k=1 T (v ω (k) -λ)
and we have

u n+1 u n = M n (λ) 1 0 .
The parameter λ is an eigenvalue if and only if there exist c ∈ R such that

M N (λ) 1 0 = 0 c ,
the condition u N +1 = 0 is then satisfied.

It will be convenient to denote the vector u n+1 u n as a complex number in the fashion

u n+1 + iu n = z n = r n e iφn
where r n ∈ R + and φ n ∈ R/2πZ. We also introduce the lifting of φ k , which we denote by θ k . This is just a discrete version of the continuous lifting from R/2πZ to R into the discrete case. It is defined recursively by

θ k = θ 0 = 0 φ k [2π] ∀k ∈ [0, N ] and θ k - π 2 ≤ θ k+1 < θ k + 3π 2 .
It can be seen that φ k+1 does not depend on r k+1 but only on φ k and T λ (v k -λ).

Therefore, for simplicity of notation, we use the same notation T for the operator on R/2πZ:

φ k+1 = T λ (v k )φ k .
Note that it is possible to recover r k from φ 0 , φ 1 , ..., φ N with the formula

r k+1 r k = r k+1 u k+1 u k+1 r k = cos φ k sin φ k+1 .
For this reason, in the rest of the paper we focus mostly on (φ k ) k=0...N . We note F(λ) = (φ k ) k=0...N which has been constructed from the recursive formula φ k+1 = T λ (v k )φ k and φ 0 = 0. And for λ an eigenvalue, we note Ph(λ) = (φ k ) k=0...N the phase of the corresponding eigenvector. Note that it is equal to

F(λ) with the condition φ N = π 2 [π].

Forward and backward processes

In this subsection, we define two natural random laws on the chain X = (φ k ) k=0,..,N . The first one is the Markov chain starting from φ 0 with an initial law µ f defined on S 1 and transition law φ k → φ k+1 = T (v k )φ k with a random measure ν for v k . We call it the forward process. The second one is the Markov chain starting from φ N with an initial law µ b and transition law

φ k → φ k-1 = T -1 λ (v k-1
)φ k with a random measure ν for v k and we call it the backward process. Then we introduce a cut in [0, N ], and we can define the random law product between these two processes which we call the forwardbackward process.

For a proper definition we use test functions on R N +1 which are bounded and continuous.

Definition 1 (Forward and backward processes). The probability P f on R N +1 , defined by

P f (F ) = ˆ• • • ˆdµ f (φ 0 )dν(v 1 ) • • • dν(v n )F (X)
for any test function F , is called the forward process. Similarly, the probability P b on R N +1 defined by

P b (F ) = ˆ• • • ˆdν(v 1 )...dν(v N )dµ b (φ N )F (X)
for any test function F , is called the backward process.

Remark 2. If we introduce ξ 0,n :

φ 0 → φ f n = n-1 k=0 T (v k )φ 0 and if for almost surely any v 1 , v 2 , ..., v n , µ b
and the push measure ξ(µ f ) are equivalent measures, then we remark that for any F :

P b (F ) = ˆ• • • ˆdν(v 1 )...dν(v n )dµ f (X 0 ) dµ b (X n ) dξ(µ f (X 0 )) v1,...,vn F (X) = P f F dµ b (X n ) dξ(µ f (X 0 )) v1,...,vn . Definition 3 (Forward-Backward process). For k ∈ [0, N ], we define P f,0..k ⊗ P b,k+1,...,N a forward process for X f = φ f 0 , φ f 1 ..., φ f k with φ f 0 = 0, (µ f = δ 0 ) and a backward process for X b = φ b N , φ b N -1 ..., φ b k , with φ b N = π 2 (µ b = δ π 2 )
which are independent from each other.

Main result

We are now ready to state the main theorem of our paper.

Theorem 4 (Law of the spectrum of the 1D Anderson model). For any test function G(λ, X), we have

E λ∈σ(H),X=Ph(λ) G(λ, X) = ˆR dλ N k=1 E P f,1..k ⊗P b,k+1,...,N G(λ, X)δ φ f k -φ b k [π] sin 2 (φ f k ) (2)
that we can rewrite as

E 1 N λ∈σ(H),X=Ph(λ) G(λ, X) = ˆR ρ(λ)dλ 1 N N k=1 E P f,1..k ⊗P b,k+1,...,N G(λ, X) δ φ f k -φ b k [π] sin 2 (φ f k ) ρ(λ) (3)
with ρ(λ) the density of state.

Recall that P h(λ) is the phase of the eigenvector corresponding to the eigenvalue λ.

This formula is to be understood as follows. One chooses k randomly in [1, N ] which splits the segment into two parts [1, k] and [k, N ]. On the left, we obtain a forward process, on the right, we obtain a backward process. The choice of k is not exactly uniform on [0, N ] because of the condition

δ φ f k -φ b k sin 2 (φ k ).
However, for large N , and for any k ≤ N not too close to 0 or N , the laws of φ f k and φ b k are very close to their invariant measure and then do not depend on k. Therefore the law of k becomes close to the uniform.

There is still a dependence between the two processes at the connection between the forward and backward processes. However, because of the mixing property of the matrix product, the correlations decay exponentially fast away from the cut k.

We recall that a stationary process

X k is called (α n ) n∈N -mixing if ∀k, max A,B |P(X k ∈ A, X k+n ∈ B) -P(X k ∈ A)P(X k+n ∈ B)| ≤ α n
The following is a well known result.

Proposition 5. There exists a constant C > 0 and 0 < κ < 1 such that the process φ k is (Cκ n ) n∈N -mixing.

For a proof, see [3, proposition IV.3.12].

Applications

We present here three application of our result. The first one is a formula for the integrated density of states. The second one is about the form of the tails of the eigenvectors. We then finish with a temperature profile from [START_REF] De Roeck | Step Density Profiles in Localized Chains[END_REF].

A formula for the integrated density of states

The following equality can be found as well in [START_REF] Carmona | Spectral theory of random Schrödinger operators[END_REF] (proposition VIII.3.10 and problem VIII.6.8).

Proposition 6. For λ ∈ R, let µ λ (dφ) = m λ (φ)dφ be the T λ -invariant measure on R/Z. The density of states

dN (λ) = lim N →∞ 1 N # σ(H) ∩ [λ, λ + dλ] is given by dN (λ) dλ = ˆR/2πZ sin 2 (φ)m λ (φ)m λ π 2 -φ dφ.
Proof. We apply our formula (3) in Theorem 4. We choose G(s, X) = G(s) (that does not depend on X) and recognize ´R G(λ)ρ(λ)dλ. More precisely,

1 N E λ∈σ(H N ) G(λ) = ˆG(λ)dλ 1 N k E P f,1..k ⊗P b,k+1,...,N [δ φ f k -φ b k sin 2 (φ k )] = ˆG(λ)dλ 1 N k ˆφ ρ k,λ (φ)ρ N -k,λ ( π 2 -φ) sin 2 (φ k )]
where ρ k,λ and ρ N -k,λ are the density probabilities of the angles of M k (λ) 1 0

and M N -k (λ) 1 0 . We can then conclude using that ρ k,λ → m λ and ρ N -k,λ → m λ when k → ∞ and N -k → ∞.

Brownian and drift for the eigenvectors

It is well known since the work of Carmona-Klein-Martinelli [START_REF] Carmona | Anderson localization for Bernoulli and other singular potentials[END_REF], Goldsheild-Molchanov-Pastur [START_REF] Goldsheid | A random homogeneous Schrödinger operator has a pure point spectrum[END_REF] and Kunz-Souillard [START_REF] Kunz | Sur le spectre des opérateurs aux différences finies aléatoires[END_REF] that the eigenvectors are localized and decay exponentially from the center of localization. An exact form of the eigenvectors has been recently proven in the critical case where V is replaced by V √ N in [START_REF] Rifkind | Eigenvectors of the critical 1-dimensional random schroedinger operator[END_REF]. The authors proved that the eigenvectors in the bulk have the form e σ B |t-u| 2 -γ|t-u| . We claim using our formula of Theorem 4 that a similar result holds for the tails of the eigenvectors in the non critical case.

For the reader's convenience we recall the heuristics of the following classical results. One can write any product of random matrices

M N = N i=1 T i as log( M N ) = log N i=1 M i M i-1 = N i=1 log T i M i-1 M i-1
In the case when T i are iid and there are some strong mixing property on Mi-1 Mi-1 , the terms

Y i = log T i Mi-1 Mi-1
should behave like iid random variables. One can then prove the strong law of large numbers, the central limit theorem, and Donsker's theorem. See the paper of Le Page [START_REF] Page | Théoremes limites pour les produits de matrices aléatoires[END_REF] for these results. One therefore defines a "mean", a "variance" and a "random walk" as follows.

Definition 7. The Lyapunov exponent is γ(λ) := lim N →∞ 1 N E log M N (λ) .
The limit variance is

σ 2 (λ) = lim N →∞ 1 N E (log M N (λ) -γ(λ)N ) 2 .
The random walk is

S n = 1 σ(λ) log M n (λ) -γ(λ)n .
and we consider its rescaling

W N (t) = 1 √ N S N t .
Finally, we denote by W the Wiener measure.

Theorem 8 (Limit theorem for products of random matrices). We have the following:

• γ(λ) > 0 and lim N →∞

1 N log M N (λ) = γ, almost surely; • σ 2 (λ) > 0; • W N → W in law.
We refer to [9, Theorems 2 and 3] for the proof of Theorem 8. We recover then the form of Brownian with drift and both on the right hand side and the left hand side of the cut. For λ an eigenvalue, and r k e iφ k the corresponding eigenvector, we note q λ k = log(r k ). For scaling, we set q λ (s) = q λ N s /N .

Proposition 9 (Tail of eigenvectors). 1) Choosing λ (N ) uniformly in σ(H (N ) ), we have the following convergence in law

(λ (N ) , q λ (N ) s ) → N ∞ ( λ, -|γ( λ)(s -x)|)
where λ is a random variable with law the limiting density of state ρ and x an independent variable on [0, 1] with uniform law.

2) There exists a sequence of random variables {x (N ) } with uniform law on [0, 1] such that

(λ (N ) , √ N q λ (N ) s + |γ(λ N )(s -x (N ) )| ) → N ∞ ( λ, σ( λ)W s-x )
where W is the Wiener measure.

The first statement is the very classical result of Anderson localization for the one dimensional model. The eigenvectors decay exponentially from their center of localization and this center is chosen uniformly on the domain. The second statement says that the typical deviation from the decay is the exponential of a Brownian (see Figure 1 for an illustration).

Rifkind and Virag [START_REF] Rifkind | Eigenvectors of the critical 1-dimensional random schroedinger operator[END_REF] studied the large eigenvectors of the one dimensional Anderson model in the continuous case where the potential is a white noise. It is the limit of the discrete model in the "critical regime" where the potential is scaled like

V (N ) ω = 1 √ N V ω .
In this regime, one cannot speak of localization because the length of the decay is as large as the size of the domain. However they proved the exact law of the form of the eigenvectors

q λ s = -|γ(λ)(s -x)| + σ(λ)W s-x .
To make the connection with our previous proposition, one can actually show that for V ω = v ω , with E(v 2 ω ) = σ 2 , in the limit → 0 and |λ| < 2, we have

γ(λ) = σ 2 4 -λ 2 2 and σ(λ) 2 2 = σ 2 4 -λ 2 2 .
Proof of Proposition 9. If in our formula (3) the term

δ φ f k -φ b k
were not there, then the forward and the backward processes would be completely independent. Our proposition would have then immediately followed from Theorem 8, under the conditions that r k obtained by the forward process and the r k obtained by the backward process are the same, and that the normalization

N n=1 |u n | 2 = 1
holds. The latter becomes in the limit sup q λ s = 0. Therefore we only have to check that the little perturbation around the cut k has no influence. We fix φ. Conditionally of φ b k = φ and φ f k = φ the forward and backward processes are independent. The results of Theorem 8 are true asymptotically with probability 1. Therefore for any φ in a set of full Lebesgue measure in S 1 the results of Theorem 8 are true conditionally of φ b k = φ and φ f k = φ.

A temperature profile

We will use our result to explain some numerical observations which have been made in [START_REF] De Roeck | Step Density Profiles in Localized Chains[END_REF]. In this article, the authors are interested in the temperature profile of a disordered chain connected to two thermal baths of temperatures T 0 and T N at the boundary 0 and N . According to [START_REF] De Roeck | Step Density Profiles in Localized Chains[END_REF], the temperature T (x) at site x is expected to be given in a certain limit by

T (x) = λ∈σ(H) |ψ λ (x)| 2 T 0 |ψ λ (0)| 2 |ψ λ (0)| 2 + |ψ λ (N )| 2 + T N |ψ λ (N )| 2 |ψ λ (0)| 2 + |ψ λ (N )| 2 (4)
where H is our one-dimensional random Schrödinger operator and ψ λ are its eigenvectors.

We prove that T converge to a step function where the transition from T 0 and T N happens in a neibourghood of x = N 2 at a scale √ N . This has been observed numerically in [START_REF] De Roeck | Step Density Profiles in Localized Chains[END_REF].

Proposition 10. For N large enough we have

lim N →∞ E T ( √ N x + N 2 ) = T 0 + (T N -T 0 ) ˆR P N (0, 1) ≤ 2γ(λ) σ(λ) x dN (λ)
where dN (λ) is the integrated density of states, γ(λ) is the Lyapunov exponent and σ(λ) is the limit variance.

The Lyapunov exponent is positive, continuous and so is bounded below on the support of σ(H). The variance σ(λ) is bounded as well. Therefore, uniformly in λ, P(N (0, 1) ≤ σ(λ) 2γ(λ) x) → 0 for x → -∞ and P(N (0, 1)

≤ σ(λ) 2γ(λ) x) → 1 for x → ∞. We have then T (y) = T 0 for N 2 -y √ N and T (y) = T N for y -N 2 √
N . This is the step function numerically observed in [START_REF] De Roeck | Step Density Profiles in Localized Chains[END_REF].

Proof. We use our formula and write:

E(T (x)) = k∈[0,N ] ˆR dλE P f,1..k ⊗P b,k+1,...,N |ψ λ (x)| 2 T 0 |ψ λ (0)| 2 |ψ λ (0)| 2 + |ψ λ (N )| 2 + T N |ψ λ (N )| 2 |ψ λ (0)| 2 + |ψ λ (N )| 2 δ φ f k -φ b k sin 2 (φ k )
With the notation of Proposition 9, we write

T 0 |ψ λ (0)| 2 |ψ λ (0)| 2 + |ψ λ (N )| 2 + T N |ψ λ (N )| 2 |ψ λ (0)| 2 + |ψ λ (N )| 2 = T 0 e N q λ (N ) 0 e N q λ (N ) 0 + e N q λ (N ) 1 + T 1 e N q λ (N ) 1 e N q λ (N ) 0 + e N q λ (N ) 1 .
Therefore in the limit N → ∞, this converges to T 0 for q λ 0 > q λ 1 and T 1 for q λ 0 < q λ 1 . We have then at the limit a Bernoulli T int with parameter given by Proposition 9:

T int =    T 0 with probability P N (0, 1) ≤ (2k-N )γ(λ) √ N σ(λ)
, T N with probability P N (0, 1)

≥ (2k-N )γ(λ) √ N σ(λ)
.

In order to conclude, we recall that the whole mass of |ψ λ | 2 is around a few number of sites around k so

E(T (x)) = ˆR dλ k∈[x-α(N ),x+α(N )] E P f,1..k ⊗P b,k+1,...,N |ψ λ (x)| 2 T 0 |ψ λ (0)| 2 |ψ λ (0)| 2 + |ψ λ (N )| 2 + T N |ψ λ (N )| 2 |ψ λ (0)| 2 + |ψ λ (N )| 2 δ φ f k -φ b k sin 2 (φ k ) + O e -γ(λ)α(N ) ,
where we have chosen α(N ) such that √ N α(N ) 1. Moreover for large N ,

P(N (0, 1) ≥ (2x -N )γ(λ) √ N σ(λ) ) = P(N (0, 1) ≥ (2k -N )γ(λ) √ N σ(λ) ) + o(1),
and we have then

E(T (x)) = ˆR dλ k∈[x-α(N ),x+α(N )] E P f,1..k ⊗P b,k+1,...,N |ψ λ (x)| 2 T 0 + (T N -T 0 )P N (0, 1) ≥ (2x -N )γ(λ) √ N σ(λ) δ φ f k -φ b k sin 2 (φ k ) + o(1)
Finally we use the following formula, for x not close to the edges

k∈[0,N ] E P f,1..k ⊗P b,k+1,...,N [|ψ λ (x)| 2 δ φ f k -φ b k sin 2 (φ k )] = dN (λ) dλ + o(1). Indeed, for any A Borel set of R, ˆR 1 A (λ) dN (λ) dλ dλ = lim 1 N E(T r(1 A (H))) = 1 N x E λ∈A∩σ(H) |ψ λ (x)| 2 = ˆ1A (λ)dλ 1 N x k∈[0,N ] E P f,1..k ⊗P b,k+1,...,N |ψ λ (x)| 2 δ φ f k -φ b k sin 2 (φ k ) .
We then note that the left term is asymptotically independent of x for x not close to the edges. Therefore

ˆR 1 A (λ) dN (λ) dλ dλ = ˆ1A (λ)dλ k∈[0,N ] E P f,1..k ⊗P b,k+1,...,N [|ψ λ (x)| 2 δ φ f k -φ b k sin 2 (φ k )] + o(1).
The proposition then follows, namely we have

E(T (x)) = T 0 + ˆR(T N -T 0 )P N (0, 1) ≥ (2x -N )γ(λ) √ N σ(λ) dN (λ) dλ dλ + o(1)
as we wanted.

Periodic boundary conditions

We have tried to obtain a similar result for periodic boundary conditions. With the multiscale analysis tools [START_REF] Fröhlich | Absence of diffusion in the anderson tight binding model for large disorder or low energy[END_REF], one has the exponential decay from the center of localization. But it would be also interesting to have an interpretation with forward backward process in this case.

In the critical regime, one would expect the form of the eigenvectors to be like e F (s) , on R/2πZ with F (s) = -γ min(|s -u|, |s -u + π|) + σ Bs-u with u uniformly chosen on [0, 2π] and B a Brownian bridge. So far we have not been able to prove this statement rigorously., but our intuition seems to be confirmed by numerical simulations (see Figure 2).

Remark. The condition u -1 = u N +1 = 0 in the Dirichlet case has to be replaced by Tr(M N (λ)) = 2. Indeed, let u n be an eigenvector of eigenvalue λ and z = G(λ, X) .

Remark 11. θ N : λ → θ N (λ) is continuous and strictly increasing (see the calculations below).

For finite N , the inverse function θ -1 N is continuous, so are G, X. We can therefore write

Figure 1 :

 1 Figure 1: A realization of log M n (λ) for N = 1000, v ω uniform on [0, 1] with Dirichlet boundary conditions. we add a fit of the form |γ(λ)(s -x)|.

  periodic boundary conditions mean M N (λ)z = z. So 1 is an eigenvalue of M N (λ). Therefore 1 is a solution of x 2 -T r(M N (λ))x + 1 = 0 and so Tr(M N (λ)) = 2.

Figure 2 :

 2 Figure 2: A realization of log M n (λ) with periodic boundary conditions for N = 3000, v ω uniform on [0, 0.3]. We add a fit of the form -γ min(|s -u|, |su + π|).

  The rest follows from a change of variable. Let us introduce I = π 2 + ∪ n∈Z [πn -, πn + ] and P (G):

Acknowledgement: We would like to thank Mathieu Lewin for his encouragement, his interest and his relevant comments. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement MDFT No 725528).

Then

In this formula appears the term dφ N dφ k | vω(N ),...,vω(k+1) . It is this term that changes the law from a forward process to a backward process. We then calcu-

and thus

We carry on the calculation,

We artificially add a variable φ as follows:

Then we use Remark 2 and get

where P f,1..k ⊗ P u b,k+1,...,N is the forward-backward process with µ b the uniform law on S 1 . We can then conclude, by taking the limit 1 2 1 φ N ∈I /2πZ dφ N → δ 0 .


Mathieu Grandemange

Olivier Cadot

Marc Gohlke

Reflectional symmetry breaking of the separated flow over three-dimensional bluff bodies

Experimental observation of a permanent reflectional symmetry breaking (RSB) is reported for a laminar three-dimensional wake. Based on flow visualizations, a first bifurcation from the trivial steady symmetric state to a steady RSB state is characterized at Re = 340. The RSB state becomes unsteady after a second bifurcation at Re = 410. It is found that this RSB regime is persistent at large Reynolds numbers and is responsible for a bistable turbulent wake.

Many industrial flows are produced by the motion of bluff bodies with a geometry of reflectional symmetry. The simplest cases that are frequently used can be found in the transport industries, especially for ground vehicles such as trains, cars, or lorries. The bluffness associated with the functional blunt shape provokes a massive flow separation and consequently a complex wake dynamics. Recently, bistability properties of such three-dimensional turbulent wakes have been reported in very distinctive geometries [1,2]: Each equiprobable state breaks the reflectional symmetry of the body but leads together to a statistically symmetric flow. This phenomenon reminds of symmetry breaking found in closed cell geometry [3] and may be a general property of turbulent flows. On the other hand, reflectional symmetry breaking (RSB) has been recently reported numerically in the bifurcation scenario of a disk wake at low Reynolds numbers [4,5]. A first bifurcation to a stationary mode also observed experimentally [6][7][8] selects a planar symmetry. Then the RSB appears as a Hopf bifurcation at Re = 123.7 from the development of two counter-rotating spiral modes, in addition to the previous stationary mode leading to a periodic shedding. The reflectional symmetry is finally restored for Reynolds number over 143.7 with the restabilization of the stationary mode. However, experimental evidence of this RSB is lacking, probably because of the small magnitude of asymmetry [4,5].

To our knowledge, experiments about bifurcations and symmetry breaking of the flow around three-dimensional (3D) bluff bodies with reflectional symmetry are needed. The present Rapid Communication investigates the flow around a basic body that is generally used to mimic car aerodynamics at Reynolds numbers ranging from 10 4 in laboratory experiments to 10 7 in real scale. It is shown, by means of flow visualizations, that this simple geometry provokes at low Reynolds numbers a bifurcation in the flow solution to a state that breaks the reflectional symmetry. The corresponding regime is persistent at large Reynolds numbers and responsible for the bistable property of the turbulent wake.

The three-dimensional body is described in Fig. 1.T h e Reynolds number Re = UH ν is based on the height H = 26 mm of the body, the main flow velocity U 0 , and the kinematic viscosity ν of the working fluid. The width W = 1.35H and the length L = 3.625H are set to reproduce a standard car [9].

First, the body is placed in a low speed hydrodynamics tunnel at a ground clearance C = 0.6H . Two dye injectors located behind the two backward supports are used to visualize the recirculating flow that forms at the trailing edge (see Fig. 1). The Reynolds number varies between 260 and 1300, adjusting abruptly the main flow velocity in the range 1 cm s -1 <U 0 < 5cms -1 ; the precision due to the accuracy of reading the rotameter is Re =±10. The dye is continuously injected with a syringe in such a way that the dye velocity at the exit of the injector never exceeds 10% of the main velocity. The tunnel has transparent walls, and it is also designed to visualize the wake in the flow direction (see Ref. [10]f o ra full description of the tunnel). Pictures are taken once the dye has filled the recirculating bubble. In the following, z W , which denotes the mean position of the wake in the z direction, is used as a parameter of order. It is measured as the mean position of the dye trail downstream from the recirculation bubble by image analysis of the top view picture, and its precision is 0.02W .

At the lowest Reynolds numbers, a steady symmetric regime (SS) is observed as depicted in Fig. 2(a) for Re = 310. The flow preserves the planar symmetry of the geometry and the presence of the ground induces a top-bottom asymmetry of the wake. This state remains stable up to Re = 340.

Past this critical Reynolds number, the wake starts to oscillate periodically in the plane of symmetry, as shown in Fig. 2(b). The structure of this unsteady symmetric regime (US) perfectly preserves the reflectional symmetry of the body, as can be stated by the corresponding top and rear views. It is associated with alternative loops that are shed from the top and bottom shear layers. However, this US regime is not permanent. Indeed, keeping the Reynolds number constant, this periodic shedding is progressively moving off the reflectional plane of symmetry: z W varies slowly around 0 with a characteristic time evolution of 1 min, i.e., roughly ten shedding periods. Then the wake selects randomly one orientation and stabilizes to an asymmetric position, typically after a duration of 10 min. There, the oscillations get attenuated and the flow reaches a steady asymmetric regime (SA) denoted as + ordepending on the sign of z W , which indicates the wake shift in the z direction. Figure 2(c) describes this steady state SA + that breaks the reflectional symmetry for Re = 365. Naturally, both SA + and SA -can be observed by repeating the Reynolds number increase from the SS regime without any intervention on the setup. This regime was stable over 2 h of observation; it is considered steady in the limit of the visualization means and water tunnel stability.

Eventually, when increasing the Reynolds number over 410, the SA wake starts to oscillate again as visible in Fig. 2(d) and leads to an unsteady asymmetric regime (UA). The flow preserves its orientation so that SA + turns into UA + and similarly SA -becomes UA -. Hence, Figs. 2(c) and 2(d) were obtained for two different experiments of the Reynolds number increase. The top view in Fig. 2(d) shows that the oscillations are concentrated only on one side of the body but its dynamics is complex. The unsteadiness is associated with oscillations in both the y and the z directions. It is then probable that two different frequencies coexist, as observed in the turbulent wake of elliptical and rectangular flat plates [11]; one is associated with the interaction of the top-bottom shear layers and the other with the interaction of the lateral ones. However, the setup does not allow a reasonably precise measurement of the oscillation amplitudes and frequencies in the cross-flow plane. Eventually, the spontaneous migration between the regimes UA + and UA - was not observed but can be induced by energetic perturbations as short time suppression of the free flow. a decrease of the Reynolds number, the flow goes fromU AtoSAatRe= 410 and then directly to the SS state as soon as the Reynolds number gets smaller than 340, so that the US regime is never observed. This behavior is summarized through the typical evolution of the wake shift z W in Fig. 3.

It is worth mentioning that during the transition from SS to SA, the US regime is always reported. This tends to indicate that imperfections of the shedding of the US regime are needed to initiate slight asymmetries, and this transient regime then loses slowly its intensity while the wake continuously moves off axis to eventually reach the SA regime.

From these observations, the bifurcation scenario presented in Fig. 4 is proposed to classify the different stable regimes: SS for Re < 340, SA for 340 < Re < 410, and UA for Re > 410. The US regime is displayed as an unstable state as it is transient and only visible between SS and SA. The evolution of the parameter of order with Re confirms the presence of a pitchfork bifurcation at Re = 340.

The same geometry is now placed in a wind tunnel at a free-stream velocity of U T 0 = 20 m s -1 . The flow is now turbulent and the associated Reynolds number is 9.5 × 10 4 . The ground clearance is set at 0.2H to approach the reference experiments of Ahmed et al. [9]. Besides, as the effect of the viscous forces is strongly attenuated compared to the precedent experiments, the intensities of the underbody flow are comparable despite the ground clearance difference. Stereo-particle image velocimetry (PIV) is performed in the wake at x = 1.5W and used to locate the barycenter of momentum deficiency in this cross-flow plane. A snapshot displayed in Fig. 5(a) presents the turbulent wake off the reflectional plane of symmetry, located at y T W = 0.40W and z T W =-0.15W .

The analysis of the 2000 velocity fields recorded at 10 Hz leads to the time evolution of the z T W presented in Fig. 5(b). The probability density function clearly shows the presence of two preferred positions z T W ≈±0.1W : The wake is either oriented to the positive or negative z direction. Conditional averaging on z T W enables the asymmetric topologies to be obtained; similar wake offsets in the z direction are observed in the z T W < 0 asymmetric state displayed in Fig. 5(c) and in the UA -regime shown in Fig. 2(d). Topology shifts are observed after a typical duration of 5 s, which is long compared to the expected shear layer dynamics at a Strouhal number of around 0.2, corresponding roughly to 50 Hz in these experiments. These shifts seem to occur randomly so that the mean flow statistically retrieves the symmetry of the setup.

As a conclusion, this Rapid Communication reports experimental evidence of the reflectional symmetry breaking (RSB) in three-dimensional laminar wakes: The bifurcation scenario of the wake presents asymmetric permanent regimes. The SA and UA regimes have equal probability of being shifted toward the z>0o rz<0 regions. The transient US regime seems necessary to initiate slight asymmetries of the flow to reach the permanent SA or UA regimes. Further work improving the measurement technique could clarify the amplitude and frequencies of the oscillations in the y and z directions. It could also confirm whether both oscillations emerge simultaneously and whether the transition from SA to UA is a Hopf bifurcation.

Besides, it is found that the bifurcation scenario strongly depends on the ground clearance: The steady RSB regime is not reported for C<0.4H . It may on the topbottom recirculation balance: A particular three-dimensional arrangement of the different shear is certainly needed and remains to be clarified. The condition of the appearance of RSB regimes is of fundamental interest as it may induce a significant side force and then additional drag.

Signatures of these permanent symmetry breaking regimes are also found at larger Reynolds numbers. If spontaneous migrations between UA + and UA -are unlikely to happen in laminar regimes, the energetic large scale structures of turbulence should certainly induce natural migrations in a random way. While the RSB is clearly evidenced for wakes of ground vehicles by the present work and by the experiments of Lawson et al. [1], we believe it to be much more general and observed in other types of wake flows. For instance, some asymmetry in the hull wake behind a container ship has been reported [12] despite huge efforts by the authors to respect the reflectional symmetry.

Eventually, these results point out the limits of the common procedure consisting in implementing the instrumentation of a blunt geometry based on its reflectional symmetries. In terms of three-dimensional unsteady simulations, reduced physical times of the calculation may also prevent the flow from reaching permanent asymmetric regimes.

FIG. 1 .

 1 FIG. 1. Geometry of the model: (a) Side view and (b) rear view. For the low Reynolds number experiment, fluorescent dye is injected through two pipes behind the backward supports.

FIG. 2 .

 2 FIG. 2. Flow visualizations at the Reynolds numbers (a) Re = 310, (b) and (c) Re = 365, and (d) Re = 415. For each Reynolds number: Top picture, top view; bottom picture, side view; right picture, rear view. The flow comes from the left for both top and side views. Regimes observed in (a), (c), and (d) are permanent states while (b) is a transient state. Observations in (c) and (d) have been obtained from different experiments of Re increase in order to show both states for which z W < 0andz W > 0.

FIG. 3 .

 3 FIG. 3. (Color online) Typical evolutions of the wake positions z W during a Reynolds number increase from 320 to 370 followed by a decrease back to 320: Continuous lines, permanent states; dashed lines, transient states.

FIG. 4 .

 4 FIG. 4. (Color online)Bifurcation scenario of the wake: Black and gray lines are steady and unsteady regimes, respectively, continuous and dashed lines are stable and unstable regimes, respectively, and crosses are experimental measurements and error bars.

FIG. 5 .

 5 FIG. 5. (Color online) (a) Instantaneous streamwise velocity in the turbulent wake at x = 1.5W , and the cross locates the barycenter of momentum deficiency at y T W = 0.40W and z T W =-0.15W . (b) Time evolution of the turbulent wake position z T W from 2000 snapshots recorded at 10 Hz (left) and its associated probability density function (right). (c) Mean streamlines are colored by streamwise velocity at x = 1.5W ; velocities are averaged from the snapshots where z T W < 0.

The authors are indebted to R. Godoy Diana, V. Raspa, and B. Thiria from ESPCI (Paris, France) for lending their low speed hydrodynamics tunnel. The work benefited from useful discussions with R. Monchaux.