Anais Lavoisier

Timpe Carsten

Kuehl

Peter

Wagner

Leonie

Tournier

Carole

Marco Ramaioli

Alejandro Avila
email: alejandro.avila-sierra@inrae.fr

Paediatric Solid Oral Dosage Forms for Combination Products: Improving In Vitro Swallowability of Minitablets Using Binary Mixtures with Pellets

Keywords: Soft robotics, swallowing, pellets, minitablet, binary mixture, flexible solid oral dosage form, combination products Paediatric Soft Robotic Tongue MT: Minitablet BM: Binary mixture v.f.: Volume fraction IDDSI: International Dysphagia Diet Standardisation Initiative

There is a growing interest in determining and improving the acceptability of solid oral dosage forms (SODF) in paediatric populations. Combination products can be an interesting solution to combine two active pharmaceutical ingredients. This study shows that binary mixture of paediatric SODF could enhance formulation swallowability and palatability.

Using the Paediatric Soft Robotic Tongue (PSRT)an in vitro device inspired by the anatomy and physiology of 2-year-old childrenwe investigated the oral phase of swallowing of multi-particulate formulations, i.e., pellets (350 and 700 µm particles), minitablets (MTs, 1.8 mm), and their binary mixtures (BM), by measuring the oral swallowing time, the fraction of particles swallowed, and the post-swallow residues. The effects of the feeding method, bolus volume, type of carrier, particle size, and particle volume fraction on pellets swallowability were also systematically studied.

The flowing ability of the carriers was affected by the addition of pellets: shear viscosity generally increased, yet little effect was observed on their shear thinning behaviour. Xanthan gum-based solutions achieved excellent swallowing levels for pellets (≥ 75 %), especially at high concentrations, i.e., 0.5 and 1%, favouring pellets transport without affecting postswallow residues. Particle swallowability does not appear to be influenced by pellet size but increasing particle volume fraction (v.f.) above 10% reduced the percentage of particles swallowed. At v.f. 0.4, pellets were easier to swallow (+ 13.1%) than MTs, being the administration method used strongly dependent of the characteristics of the multi-particulate formulation under study. To enhance MT swallowability, a novel approach involving MTpellets binary mixtures was explored, whereby the maximum solid packing fraction of the formulation was increased. Mixing MTs with only 24% of pellets improved particle swallowability, achieving swallowing levels similar to those of pellets alone. Therefore, a combination of SODF, i.e., MTs and pellets, improved particle swallowability, opening the door to new ways of modulating product palatability, particularly attractive for combination products.

Introduction

Difficulty with swallowing solid pharmaceutical formulations is a barrier that must be overcome across a broad range of ages, including young children [START_REF] Vandenberg | Interventions to Improve Pediatric Ability to Swallow Solid Oral Medications: A Systematic Review[END_REF].

Developing drugs for paediatric use is a complex and tedious process but essential as children need age-appropriate dosage forms of high quality, ethically researched, and appropriately authorised [START_REF] Mistry | Evidence of Acceptability of Oral Paediatric Medicines: A Review[END_REF][START_REF] Ternik | Assessment of Swallowability and Palatability of Oral Dosage Forms in Children: Report from an M-CERSI Pediatric Formulation Workshop[END_REF][START_REF] Wollmer | Review of paediatric gastrointestinal physiology relevant to the absorption of orally administered medicines[END_REF]. According to the European Medicines Agency (EMA), acceptability assessment must be part of the Paediatric Pharmaceutical Development Program of new formulations and described in the Paediatric Investigation Plan (PIP) (Committee for Medical Products for Human Use and Paediatric Committee, 2013).

The use of liquid oral pharmaceutical formulations has been traditionally preferred for paediatric use, yet major limitations have been reported (e.g., stability issues, regular use of excipients that may be harmful to children, and high storage and transportation costs) [START_REF] Orubu | A survey of caregivers of Nigerian children less than 6 years of age to determine the experience and perception of acceptability of oral solid dosage forms[END_REF]. The provision of flexible solid oral dosage forms (SODF) as an alternative has been proposed (WHO, 2012;[START_REF] Wollmer | Review of paediatric gastrointestinal physiology relevant to the absorption of orally administered medicines[END_REF], which allow for a flexible dosing regime and the use of safe excipients that ensure both high acceptability and adequate bioavailability.

When it comes to multi-particle oral dosage forms, size and shape, taste and aftertaste, dose volume, and ease of administration are generally considered critical drivers of acceptability [START_REF] Kozarewicz | Regulatory Perspectives on Acceptability Testing of Dosage Forms in Children[END_REF]. Children under 6 years old should not take tablets and capsules due to swallowing difficulties and choking risks (EMA 2006;[START_REF] Yeung | When do children convert from liquid antiretroviral to solid formulations?[END_REF] but minitablets (≤ 4 mm in diameter) are currently considered suitable for children between 2 and 5 years old [START_REF] Mistry | Evidence of Acceptability of Oral Paediatric Medicines: A Review[END_REF]. A number of clinical studies using placebos have demonstrated that young children (aged between 6 months and 5 years old) are able to swallow multiple minitablets simultaneously [START_REF] Klingmann | Acceptability of Mini-Tablets in Young Children: Results from Three Prospective Crossover Studies[END_REF][START_REF] Klingmann | Acceptability of Multiple Uncoated Minitablets in Infants and Toddlers: A Randomized Controlled Trial[END_REF][START_REF] Kluk | Can Preschool-Aged Children Swallow Several Minitablets at a Time? Results from a Clinical Pilot Study[END_REF] and tend to prefer this format over alternative formulations like powders, suspensions, or syrups [START_REF] Klingmann | Favorable Acceptance of Mini-Tablets Compared with Syrup: A Randomized Controlled Trial in Infants and Preschool Children[END_REF][START_REF] Klingmann | Acceptability of Uncoated Mini-Tablets in Neonates-A Randomized Controlled Trial[END_REF][START_REF] Klingmann | Acceptability of Mini-Tablets in Young Children: Results from Three Prospective Crossover Studies[END_REF][START_REF] Musiime | The Pharmacokinetics and Acceptability of Lopinavir/Ritonavir Minitab Sprinkles, Tablets, and Syrups in African HIV-Infected Children[END_REF][START_REF] Riet-Nales | Acceptability of Different Oral Formulations in Infants and Preschool Children[END_REF][START_REF] Spomer | Acceptance of Uncoated Mini-Tablets in Young Children: Results from a Prospective Exploratory Cross-over Study[END_REF][START_REF] Thomson | Minitablets: New Modality to Deliver Medicines to Preschool-Aged Children[END_REF]). On the other hand, pellets or beads (< 1.5 mm) could be also suitable for the administration of medicines to adults and children (4year-old and above) (Lopez et al., 2018a), being these sprinkle forms also preferred over traditional formulations like syrup [START_REF] Cloyd | Comparison of sprinkle versus syrup formulations of valproate for bioavailability, tolerance, and preference[END_REF][START_REF] Motte | Acceptability and tolerance of sodium valproate, a new sustained-action granule formulation, in monotherapy for epileptic children from 3 years old[END_REF][START_REF] Verrotti | Effects of the abrupt switch from solution to modified-release granule formulation of valproate[END_REF]. Recently, [START_REF] Miyazaki | Comparative acceptability of minitablets, fine granules, and liquid formulations in young children: An exploratory randomized crossover study[END_REF] has investigated and compared the acceptability and swallowability of minitablets (2 mm), pellets (mean diameter 263 ± 108 μm) and liquid formulations (syrup) among sixty-five Japanese children distributed in different age groups (2 ≤ 4, 4 ≤ 7, and 7 ≤ 9 years old). They showed better acceptability and swallowability of pellets and liquid formulations than of minitablets. An important practical limitation of minitablets and pellets, especially of the latter, is that large volumes would be often required, affecting palatability. Lopez et al. (2018a) reported a negative grittiness perception by 60% of children and 51% of adults using pellets of sizes ranging between 200-1000 µm at a volume fraction of ~ 0.12 in spring water.

Several in vitro models to study swallowability have been proposed in the literature as summarised by [START_REF] Marconati | A Review of the Approaches to Predict the Ease of Swallowing and Post-Swallow Residues[END_REF]. Some were used in the past to further our understanding of the effect on the oral phase of swallowing of the size and shape of SODF and of the carrier rheology (Marconati et al., 2018a;2018b). More recently, a novel in vitro model inspired specifically by the anatomy and physiology of 2-year-old childrenthe Paediatric Soft Robotic Tongue (PSRT)has been proposed [START_REF] Lavoisier | A novel soft robotic pediatric in vitro swallowing device to gain insights into the swallowability of mini-tablets[END_REF]. This soft robotic device has been used to demonstrate that a particle volume fraction up to 0.2 did not significantly deteriorate in vitro swallowability of minitablets (2.5 mm), however, both the way minitablets were administered in the carrier and the type of carrier did. Semi-solid foods with a shear viscosity of at least ~ 150 mPa.s at a shear rate of 50 s -1 and intermediate yield stresses (2-5 Pa) may facilitate drug transport (minitablets 2.5 mm) without affecting either bolus velocity or post swallow residues [START_REF] Lavoisier | A novel soft robotic pediatric in vitro swallowing device to gain insights into the swallowability of mini-tablets[END_REF]. However, the rheological properties of the carriers may be affected by the addition of multi-particulates, altering both carrier consistency and shear thinning behaviour [START_REF] Mueller | The Rheology of Suspensions of Solid Particles[END_REF], and therefore changing the dynamics of swallowing (Marconati et al., 2018b).

Pharmaceutical forms containing a single active pharmaceutical ingredient (API) are easier to manufacture since physicochemical interactions between APIs can be avoided. However, fixed-dose combination (FDC) -a combination of drugs, with at least two APIs, in a single dose formis especially attractive for enhancing compliance in poly-medicated patients [START_REF] Fernández-García | Oral Fixed-Dose Combination Pharmaceutical Products: Industrial Manufacturing Versus Personalized 3D Printing[END_REF]; it is well known and widely used in antimicrobial and antiretroviral therapies. A recent review covering in detail a range of commercialised FDCs and their associated manufacture challenges can be found in [START_REF] Fernández-García | Oral Fixed-Dose Combination Pharmaceutical Products: Industrial Manufacturing Versus Personalized 3D Printing[END_REF].

FDCs have several advantages over monotherapy, including the modification of pharmacokinetic and pharmacodynamic profiles of the drug [START_REF] Schlosser | Fixed-dose and fixed-ratio combination therapies in type 2 diabetes[END_REF][START_REF] Moriarty | Fixed-dose combination antihypertensives and risk of medication errors[END_REF], as well as the reduction of manufacturing (e.g., fewer excipients), packaging, and distribution costs [START_REF] Sadia | From 'fixed dose combinations' to 'a dynamic dose combiner': 3D printed bi-layer antihypertensive tablets[END_REF].

The behaviour of granular systems is strongly linked to the packing fraction [START_REF] Pillitteri | How size ratio and segregation affect the packing of binary granular mixtures[END_REF], being it conditioned by the mechanical and geometrical properties of the particles [START_REF] Wiącek | Geometrical parameters of binary granular mixtures with size ratio and volume fraction: experiments and DEM simulations[END_REF]. Mixing particles of differing size and volume fraction is a common way to increase the maximum packing density of a granular system. For instance, in binary mixtures formed by small and large particles, small particles may fit into the voids left by the large ones, allowing rearrangements, altering flowing and jamming behaviours without changing its volume [START_REF] Wiącek | Geometrical parameters of binary granular mixtures with size ratio and volume fraction: experiments and DEM simulations[END_REF][START_REF] Ikeda | Unified study of glass and jamming rheology in soft particle systems[END_REF][START_REF] Haxton | Universal jamming phase diagram in the hard-sphere limit[END_REF][START_REF] Song | A phase diagram for jammed matter[END_REF]. We hypothesised that this fundamental knowledge can be applied to improving multi-particulate pharmaceutical formulations and that the combination of different SODF sizes may enhance the maximum packing fraction and reduce the suspension viscosity, thus facilitating the swallowing process (see hypothesis illustration in Figure 1). This could in turn reduce other unpleasant sensations associated to palatability (e.g., grittiness) that large volumes of small formulations like pellets may cause. Additionally, this approach could favour the formulation of combination products, where SODF coating would avoid interactions, allowing the API dosage to be adjusted accordingly.

Using the Paediatric Soft Robotic Tongue (PSRT) model developed by [START_REF] Lavoisier | A novel soft robotic pediatric in vitro swallowing device to gain insights into the swallowability of mini-tablets[END_REF], this study investigates the oral phase of swallowing of solid multi-particulate oral formulations (i.e., pellets, minitablets, and their binary mixture) by monitoring swallowing time, number of particles swallowed, and the amount of post-swallow residues. The effects of the feeding method, bolus volume, type of SODF carrier, particle size, and particle volume fraction on pellets swallowability were also systematically studied.

Materials & Methods

Carriers

Two semi-solid food carriers, apple puree ("Pomme nature", 100 g, Andros France SNC, Biars-sur-Cère, France) and stirred yoghurt ("Velouté yaourt nature brassé", 125 g, Danone SA, Paris, France) were considered. Each carrier was diluted with artificial saliva [START_REF] Brodkorb | INFOGEST Static in Vitro Simulation of Gastrointestinal Food Digestion[END_REF][START_REF] Minekus | A Standardised Static in Vitro Digestion Method Suitable for Foodan International Consensus[END_REF] at a dilution ratio (carrier: artificial saliva) of 5:1 to mimic saliva incorporation during bolus formation in the mouth before swallowing [START_REF] Lavoisier | A novel soft robotic pediatric in vitro swallowing device to gain insights into the swallowability of mini-tablets[END_REF]. Xanthan gum solutions in mineral water (Vittel) at three different concentrations (0.25, 0.5, and 1 % w/v; 43708, xanthan from Xanthomonas campestris, Sigma-Aldrich, St. Louis, MO, USA) were used as carriers of reference; Xanthan gum is commonly used as thickeners for dysphagia (swallowing disorders) management.

Solid oral multiparticulate formulations

Placebo formulations, i.e., minitablets (1.8, 2.3, 2.5, and 3 mm in diameter) and pellets (Cellets®; 350 and 700 µm in diameter) were provided by Roche Pharma (Basel, CH) (Figure 2a). All formulations were coated with an ethylcellulose-based coating (Surelease® E-7-7050, Colorcon, Darford, Kent, UK) to avoid swelling and solubilisation during testing. 3)) in a scenario where MTs dominate and pellets fill the voids formed between MTs; (C) feeding methods used: (1) pellets were spread on top of, or (2) fully mixed with the carrier. Images are not in scale.

Packing fraction of binary mixtures

Krieger and Dougherty [START_REF] Krieger | A Mechanism for Non-Newtonian Flow in Suspensions of Rigid Spheres[END_REF] proposed a semi-empirical correlation for effective viscosity of solid-liquid suspensions:

Eq. (1)

Where η is dynamic viscosity of the suspension, η o is the dynamic viscosity of the fluid phase of the suspension, φ the fraction in volume of the disperse solid phase of the suspension, φ m the maximum packing fraction of particles in the disperse solid phase, and [η] the intrinsic viscosity, which depends on both the specific viscosity (ratio between the viscosity of the suspension and the dispersion liquid) and the volume fraction of solids.

The behaviour of granular systems, e.g., pharmaceutical formulations, is strongly linked to the packing fraction φ, being defined as the ratio between the true volume V and the apparent volume V a of a granular assembly [START_REF] Pillitteri | How size ratio and segregation affect the packing of binary granular mixtures[END_REF]. Voids left by the grains allow rearrangements in the structure, triggering the φ both flowing and jamming behaviours [START_REF] Ikeda | Unified study of glass and jamming rheology in soft particle systems[END_REF][START_REF] Haxton | Universal jamming phase diagram in the hard-sphere limit[END_REF][START_REF] Song | A phase diagram for jammed matter[END_REF]. In granular materials, binary mixtures are systems made of particles with two different sizes. For the case of spherical particles, one can define the size ratio (α) as a control parameter:

Eq. (2)

Where R MT and R Cs are the radii of large and small particles, respectively. We considered that both MTs and pellets are fully spherical particles, thus simplifying MT geometry. Large size ratios are known to reach higher packing fraction values than monodisperse systems [START_REF] Pillitteri | How size ratio and segregation affect the packing of binary granular mixtures[END_REF]. When the size ratio is large enough, small particles naturally percolate due to gravity. When the size ratio is higher than α p = 6.46) [START_REF] Soddy | The Kiss Precise[END_REF], gravity systematically leads to percolation of small particles through an ordered porous medium made by large ones. The minimum pore size between three kissing large particles (Figure 2b) can be defined as:

Eq. (3)

Any small particle with a radius R Cs < R MT /α p can then percolate.

The model used for calculating the maximum packing fraction of our binary mixtures formed by MTs and pellets is described by [START_REF] Pillitteri | How size ratio and segregation affect the packing of binary granular mixtures[END_REF], which considers granular segregation (further information of the model and its implementation are shown in SI). The main parameters of the model are the volume fraction (f) of small particles, i.e., pellets, and the particle size ratio (α) defined in Eq. 2. The volume fraction of small particles is then defined as:

Eq. ( 4)

Where V MT and V Cs are respectively the true volumes of minitablets and pellets.

Feeding conditions

The volume of food typically swallowed by 2-year-old children is 5 mL [START_REF] Jones | Volume of a Swallow[END_REF][START_REF] Vaiman | Surface Electromyographic Studies of Swallowing in Normal Children, Age 4-12 Years[END_REF]. The different feeding conditions used in this study are presented in Table 1. Two different scenarios were considered (Figure 2c) for SODF administration: (i)

SODF were spread on the top of the carrier, or (ii) fully mixed with the carrier. The different SODF fractions and bolus mass used in this work are presented in Table 2. For all cases, an initial swallow with the carrier under study was performed to lubricate the in vitro oral cavity.

Table 1. Feeding conditions used in the present study.

Type of feeding Experimental details

Method 1:

Carrier + SODF + Carrier 2.5 mL of the carrier was fed first, then the SODF were spread on top of another 2.5 mL of the carrier, fed from the bottom part and then swallowed. SODF were not covered by the carrier. A syringe was used for carrier administration into the oral cavity.

Method 2:

SDOF mixed with carrier SODF were mixed completely with the carrier. Bolus administration into the oral cavity was performed directly from the top holes. Bolus and carrier volume according to the detailed in Table 2.

Rheological measurements

The rheological properties of the carriers described in section 2.1 and of their mixtures with pellets (particle volume fraction of 0.2) were assessed with a Modular Compact Rheometer 301 (Anton Paar GmbH, Graz, Austria), using concentric cylinder geometry (CC27) of 1 mm gap. The mixtures were prepared just before testing. The shear viscosity of the carriers and mixtures was evaluated by steady shear tests in a range of shear rates between 5 and 500 reciprocal seconds, at room temperature (20°C). The yield stress of the carriers was estimated through steady stress tests by increasing the shear stress from 0.01 to 30 Pa.

Concerning the mixtures with pellets, in order to avoid pellets swelling due to the relative long time required for the steady stress test (~20 minutes), the yield stress (τ o ) of the mixtures with pellets was estimated from flow curves using the Herschel-Bulkley equation (Eq. 5).

Eq. (5)

Where τ and γ are the shear stress (Pa) and the shear rate (s -1 ), respectively. K is the consistency index (Pa s), and n is the flow behaviour index (dimensionless). The yield stress of the carriers was also estimated using the HB model for comparison.

IDDSI flow test

The International Dysphagia Diet Standardisation Initiative (IDDSI) flow test was performed at room temperature to evaluate the IDDSI level of each liquid carrier [START_REF] Iddsi | Complete IDDSI Framework detailed definitions 2[END_REF]. A standard Luer Slip Tip syringe was filled up to the 10 mL with the sample of interest. Then, the liquid is allowed to flow for 10 seconds. Based on the remaining volume left in the syringe, liquid samples are categorised in four levels of increasing thickness consistency: Level 0 (< 1 mL left); Level 1 (1-4 mL); Level 2 (4-8 mL); Level 3 (> 8 mL);

Level 4 (10 mL).

Sedimentation condition of pellets

The sedimentation condition of pellets in the carrier was estimated using the Stokes' law for a single sphere motion and sedimentation in a yield stress Herschel-Bulkley fluid [START_REF] Coussot | Rhéophysique, la matière dans tous ses états[END_REF] described as:

Eq. ( 6)

With . The symbols are the pellet radii and the yield stress of the carrier, respectively. No sedimentation occurs if d pellet < d critic , being d critic defined by: Eq. ( 7)

Where g is the acceleration due to gravity (9.81 m.s -2 ) at the Earth's surface, ρ c and ρ the density of the particle, i.e., pellet, and of the carrier respectively. Particle density was determined by weighing the volume occupied by the formulation of interest.

In vitro swallowing with the Paediatric Soft Robotic Tongue (PSRT)

A novel Paediatric Soft Robotic Tongue (PSRT) (see [START_REF] Lavoisier | A novel soft robotic pediatric in vitro swallowing device to gain insights into the swallowability of mini-tablets[END_REF] for full description) inspired on the oral anatomy and physiology of 2-year-old children was used to investigate the swallowability of multi-particulate paediatric oral formulations. The top of the oral cavity was pierced with 4 holes to locate pressure transducers for recording the dynamic evolution of palatal pressure during in vitro swallowing. The first and second holes were used as feeding holes for feeding method 1 and 2 respectively (Table 1). The feeding holes were sealed by rubber plug before triggering the swallow. The maximal pressure applied against the palate was fixed at 8 kPa for mimicking the swallowing of 2-year-old children [START_REF] Lavoisier | A novel soft robotic pediatric in vitro swallowing device to gain insights into the swallowability of mini-tablets[END_REF]. The target inflation pressures for the anterior and posterior chambers of the tongue were therefore adjusted (set at 20-25 kPa and 12-15 kPa, respectively). There was no delay between the deflation of the posterior chamber and the inflation of the anterior chamber.

At least three repeatitions were performed per swallowing test (Tables 1 and2).

Variables measured

A high-speed camera (acA2040-120um, Basler, Germany) was used to record the bolus transit at 150 frames per seconds. The time required for the bolus front to exit the oral cavity is called time bolus front out (t FO ).

The number of particles successfully swallowed during the in vitro swallowing test was also monitored. The bolus propelled out of the PSRT was collected in a pre-weighed aluminium container for being later placed in an oven at 105°C overnight to determine dry matter content. The number of SODF swallowed was calculated from the known dry matter content of the carrier.

The water fraction of the carrier of interest (X Carrier ) is calculated as:

Eq. (8) = /
Where and are the water mass in the carrier and the total mass of carrier, respectively. Then, quantifying the mass of water lost during drying ( ), the mass of the carrier swallowed can be calculated as:

Eq. ( 9)

The mass of SODF swallowed is:

Eq. ( 10
)
Where is the mass of bolus fed (in) or swallowed (out), is the mass of carrier fed (in) or swallowed (out), and , the mass of SODF introduced (in) or swallowed (out).

The amount of carrier residues left in the oral cavity was calculated as:

Eq. ( 11)

Eq. ( 12)

The mass of SODF left in the oral cavity after swallowing can be estimated as:

Eq. ( 13)

Statistical analyses

One-way analysis of variance (ANOVA) [START_REF] Gelman | Analysis of variancewhy it is more important than ever[END_REF] was applied to identify statistical differences between the means of two or more groups. The statistical analysis was completed using Fisher's least significant difference test (LSD) (Salkind, 2010) for pairwise comparisons of a target group of data.

Results & Discussion

Rheological properties of the carriers

As highlighted in the introduction, acceptability of multi-particulate medicines potentially depends on the rheology of the suspending liquid carrier [START_REF] Lopez | Effect of Formulation Variables on Oral Grittiness and Preferences of Multiparticulate Formulations in Adult Volunteers[END_REF]. In the following sections, the rheological properties of the carriers and their mixtures with pellets have been carefully investigated as they may compromise critical quality attributes such as suspendability, palatability (i.e., appearance, taste, and mouthfeel) and swallowability.

IDDSI flow test

Of the five carriers selected (section 2.1), xanthan gum-based solutions (XG) were used as a reference. The consistency of each carrier was first qualitatively evaluated according to the International Dysphagia Diet Standardisation Initiative (IDDSI) framework (IDDSI, 2019), being the carriers classified as: XG 0.25%, Level 1; XG 0.5% and yoghurt diluted with simulated salivary fluid (SSF), Level 2; and XG 1% and apple puree diluted with SSF, Level 3. The volume left in the syringe for the five the carrier used is reported in Table 3. The five carriers tested, therefore, show a wide variety of liquid thicknesses, ranging from slightly thick (L1) to moderately thick (L3).

Shear viscosity of both carriers and their mixtures with pellets

The flow curves obtained in steady shear (section 2.5) at 20°C are presented in Figure 3.

In general, all carriers showed shear thinning behaviour across the range of shear rates studied

( 5-500 s -1 ), decreasing the apparent viscosity as increased. At low (< 50 s -1 ), apple puree diluted with artificial saliva (AP) and XG 1 % showed similar rheological behaviour, same as happened for both diluted yoghurt (Y) and XG 0.5 %, whilst XG 0.25 % clearly showed the lowest shear viscosity of the five carriers tested. As increased (> 50 s -1 ), the shear thinning behaviour of the food carriers, i.e., AP and Y, was less marked than those of XG carriers, implying significant viscosity differences compared to the liquids of reference at high .

The inclusion of multiparticles is expected to produce an increase in both consistency and shear thinning behaviour [START_REF] Mueller | The Rheology of Suspensions of Solid Particles[END_REF]Marconati et al., 2018b). When pellets were added to the carrier and fully mixed, some rheological differences were observed as a function of : AP+pellets shows a higher shear viscosity than XG 1% across the full range studied, whilst Y+pellets is less shear thinning, when compared to Y alone. Despite Y+pellets and XG0.5%+pellets having similar viscosities at low , the viscosity of Y+pellets changed considerably depending on the applied: between 40-50 s -1 , it shows a similar viscosity to XG1%+pellets, while at higher (300-500 s -1 ) it behaves more like AP+pellets.

Viscosities at = 50 s -1 have been found to correlate with the initial thickness perception and are commonly used as a reference value for comparison between samples [START_REF] Stokes | Oral processing, texture and mouthfeel: From rheology to tribology and beyond[END_REF][START_REF] Chen | Food Oral Processing: Fundamentals of Eating and Sensory Perception Hoboken[END_REF][START_REF] Chen | Food oral processing -A review[END_REF]. Shear viscosity data at 50 s -1 and 20°C for both the carriers and their mixtures with pellets are listed in Table 3. Formulations were also classified according to the Japanese consistency scale as a function of shear viscosity for thickened drinks [START_REF] Cichero | The Need for International Terminology and Definitions for Texture-Modified Foods and Thickened Liquids Used in Dysphagia Management: Foundations of a Global Initiative[END_REF]) using a blue-based colour scale of differing intensities.

The most marked differences are found for AP and XG 0.25%, showing the highest and the lowest shear viscosities, respectively. AP is the only carrier which could be classified as over extremely thick at room temperature (587 ± 14 mPa.s), followed closely by XG 1% as extremely thick (460 ± 59 mPa.s). On the other hand, Y and XG 0.5% showed medium viscosity data (254 ± 11 mPa.s and 167 ± 1 mPa.s, respectively) and could be classified as moderately thick liquids, whilst XG 0.25%, the carrier with the lowest shear viscosity at room temperature, could be considered as mildly thick. The addition of pellets increased shear viscosity of the formulation, and subsequently, the consistency level. Two carriers+pellets remained in the same consistency scale compared to the properties of the carrier alone, i.e., XG 0.25% and AP, being categorised as mildly and over extremely thick liquids, respectively.

On the other hand, XG0.5%+pellets and XG1%+pellets increased a level of consistency, whilst Y+pellets increased two consistency levels from moderately to over extremely thick (same consistency level that AP and XG 1%). The most drastic increase in viscosity at 50 s -1 was observed for both AP+pellets (+741 mPa.s) and Y+pellets (+481 mPa.s). Table 3. IDDSI level and volume left in the syringe for the five the carrier used. Shear viscosity data (Figure 3) of carriers and their mixtures with pellets at a shear rate of 50 s -1 and 20°C. Average data of at least two repeats. Consistency scale according to the scale reported in [START_REF] Cichero | The Need for International Terminology and Definitions for Texture-Modified Foods and Thickened Liquids Used in Dysphagia Management: Foundations of a Global Initiative[END_REF] for thickened drinks in Japan.

Yield stress of carriers and their mixtures with pellets

Yield stress is representative of the level of the internal structure of carriers which must be overcome before flowing can occur [START_REF] Zargaraan | Rheological Aspects of Dysphagia-Oriented Food Products: A Mini Review[END_REF]. It provides valuable information about the effort needed to swallow a bolus [START_REF] Malouh | Are Medication Swallowing Lubricants Suitable for Use in Dysphagia? Consistency, Viscosity, Texture, and Application of the International Dysphagia Diet Standardization Initiative (IDDSI) Framework[END_REF] and the ability for suspending particles, limiting sedimentation. As detailed in section 2.5, yield stress data of carriers was determined by the steady stress test, whilst the yield stress of the mixtures with pellets was estimated from the flow curves using the Herschel-Bulkley (HB) equation (Eq. 5)

to avoid particle swelling and solubilisation. Yield stress values are listed in Table 4.

The highest and lowest yield stresses were measured for XG 1% (12.2 ± 0.9 Pa) and XG 0.25 % (0.9 ± 0.2 Pa), respectively. The rest of carriers tested showed intermediate yield stresses of 4.5 Pa, 4.1 Pa and 2.4 Pa for AP, XG 0.5%, and Y (Table 4). When pellets were added, the yield stress increased drastically for most cases, except for yoghurt, for which it seems to decrease in the presence of the particles, being now the formulation with the lowest yield stress. This fact is likely related to a signature of a shear-dependent microstructure due to the nonlinear behaviour of the interstitial fluid, which makes these materials containing particles different from suspensions in Newtonian media [START_REF] Ovarlez | Flows of suspensions of particles in yield stress fluids[END_REF]. Further information of the fitting model can be found in the SI, section 2.

Sedimentation of pellets

The sedimentation of pellets in the carriers of interest was estimated according to the theory of a single sphere motion in a yield stress Herschel-Bulkley fluid (section 2.7), where no sedimentation occurs if d pellet < d critic . The d crit is shown in Table 4 as a function of the type of carrier used.

In general, the size of the pellets (i.e., 350 or 700 µm) used is much lower that the critical diameter estimated for sedimentation. Therefore, pellets should remain suspended in the carrier during swallowing experiments. This estimation agrees with previous works (Lopez et al., 2018b), reporting that XG hydrogels at 0.25% w/v (η = 70 mPa.s at = 50 s -1 ) would be sufficient to maintain pellets (< 1000 μm in diameter) in suspension for at least 10-15 min. Table 4. Yield stress of both the carriers and of their mixtures with pellets of 350 µm in diameter at a volume fraction of 0.2 and 20°C. For the carriers, yield stress was determined using both the steady stress method and estimated by the HB model (section 2.5). For their mixtures with pellets, yield stress was estimated using the HB model. CC27 geometry was used for characterisation. Average data of at least two repeats. Density of the carrier at 20°C and sedimentation diameter calculated according to the detailed in section 2.7 are also shown.

In vitro swallowing of size-reduced multiparticle solid oral formulations

Effect of the administration method on pellets swallowability

The effect of the administration method on in vitro swallowing of pellets (350 µm in diameter) has been investigated using XG 1% (5 mL) and a particle volume fraction (v.f.) of 0.15 (Table 2). Two different scenarios were considered (Figure 2c):

(i) pellets were placed on top of the carrier (ii) pellets were distributed in the carrier matrix before feeding.

In the first case, pellets were placed at the top part of the carrier-air interface without apparent distribution through the carrier, reducing the percentage of pellets swallowed to only ~10%

(Figure 4). In real conditions, this could also lead to an unpleasant sensation due to the large number of particles of small size in contact with the palate. In contrast, the 2 nd administration method led to a uniform distribution of pellets on the carrier matrix, without apparent segregation of the particles once they were introduced into the in vitro oral cavity, favouring the swallowability of the pharmaceutical formulations (ca. 80% of pellets swallowed; Figure 4). In terms of bolus velocity and residues, no significant differences were found between the two feeding approaches. Multi-particulates dispersed in polymeric hydrogels tend to be easier to swallow for adults than multi-particulates dispersed in water, reducing the feeling of particles left in the mouth after swallowing [START_REF] Marconati | A Review of the Approaches to Predict the Ease of Swallowing and Post-Swallow Residues[END_REF][START_REF] Kluk | Application Properties of Oral Gels as Media for Administration of Minitablets and Pellets to Paediatric Patients[END_REF]. A previous in vitro work has shown that, in order to favour the swallowing process, minitablets (2.5 mm) should be placed on top of a spoon full of carrier rather than mixed with the carrier in its container [START_REF] Lavoisier | A novel soft robotic pediatric in vitro swallowing device to gain insights into the swallowability of mini-tablets[END_REF]. According to these results, the administration method used will strongly depend on the characteristics of the multi-particulate formulation under study (i.e., mini-tablets vs pellets).

The feeding method considered for further testing using pellets is: pellets and carrier mixed before feeding, keeping a total volume of the bolus constant at 5 mL. This agrees with a real scenario where pellets are poured into a commercial food product (i.e., yoghurt) and mixed homogenously before administration, taking then a certain amount of product with a teaspoon. A practical limitation of this scenario would be that the child should eat all the food to receive the right amount of drug.

Effect of the type of semi-solid carrier on pellets swallowability

The effect of the type of semi-solid food carrier on the in vitro swallowability of pellets was investigated using 350 µm pellets at v.f. = 0.2. Results are shown in Figure 5. In general, all the swallowing parameters evaluated (time of bolus front out, percentage of pellets swallowed, and percentage of both bolus and carrier residues) were affected by the type of carrier used. Regarding the percentage of pellets swallowed (Figure 5a), less particles were successfully swallowed with yoghurt and apple puree under the tested conditions, whilst xanthan gum-based solutions reached excellent success rates (≥ 75 %), especially at high concentrations, i.e., 0.5 and 1%. The addition of particles to the carrier increased its shear viscosity (cf. section 3.1.2), a fact that at first instance could favour the swallowing of pellets when using carriers of low viscosity and low yield stress. However, when the shear viscosity of formulations was above 300 mPa.s at high (i.e., 330 s -1 ) pellets swallowability decreased (Figure 6). This scenario seems to agree with previous in vivo studies where, in terms of the vehicle used, adult participants preferred samples with low and middle-range consistencies (XG 0.25-0.5%) over the extremely thick ones such as XG 1% (despite the good swallowing behaviour of this formulation), as those carriers achieved a good balance by "concealing the presence of particles in the mouth" but not being too thick, which had a detrimental impact on mouthfeel (Lopez et al., 2018b); they used a pellets volume fraction of ca. 0.06, a lower particle fraction than the one considered in this work. On the other hand, when pellets were added to the carriers in the current study, the formulation complexity of food products seems to lead to more complex interactions than the carriers of reference; for instance, the viscosity of diluted yoghurt and pellets, in particular, would be highly affected by the shear applied during swallowing (section 3.1.2).

Regarding the other swallowing variables investigated, bolus velocity seems to be related to both carrier rheology (cf. Figure 4) and the position of the bolus before triggering the swallow (see SI, section 3). Bolus containing carriers with low shear viscosity and low yield stress (i.e., XG 0.25%, 0.5% and diluted yoghurt) were distributed along the artificial tongue and showed shorter swallowing times, compared to bolus containing AP and XG 1%

(products with high shear viscosity and high yield stress), that were positioned closer to the artificial tongue's tip (see SI, section 3). The three XG solutions tend to reach lower levels of post-swallow residues after swallowing than food-based carriers (Figure 5c), but data variability led to no marked differences.

Even if XG solutions appeared to be better transporters than the other carriers, we assumed that children would prefer to consume food products rather than thickened liquids as reported for adults and older people with swallowing difficulties [START_REF] Lopez | Effect of Formulation Variables on Oral Grittiness and Preferences of Multiparticulate Formulations in Adult Volunteers[END_REF][START_REF] O'leary | Viscosity and Non-Newtonian Features of Thickened Fluids Used for Dysphagia Therapy[END_REF]. Therefore, diluted yoghurt was selected for further testing against apple puree as it showed (i) a faster bolus front out and (ii) a slightly higher percentage of pellets swallowed (+ 3.4%)., and (iii) could also get a better acceptability by children (e.g., masking power) for administration of the pharmaceutical formulations [START_REF] Morel | Selecting the food matrix with the highest masking power for delivering Recommendations for children intake. Formulating Better Medicines for Children, EuPFi -European Paediatric Formulation Initiative[END_REF].

Effect of the size of pellets on in vitro swallowing

Previous in vivo studies in adults showed that swallowing of pellets using XG solutions (0.25-1%) was more difficult when increasing particle size (from 200 to 700 µm) (Lopez et al., 2018a(Lopez et al., , 2018b;;Marconati et al., 2018b). Here, the effect of the pellet size on in vitro swallowing was investigated using diluted yoghurt as a carrier at v.f. = 0.2. Results are presented in Figure 7(a-c).

In general, no significant differences were found between the two sizes of pellets considered, 350 and 700 µm. However, the marked rheological change of the diluted yoghurt in presence of pellets (section 3.1) could be masking the effect of the pellet size on the swallowing parameters examined. Despite the size of the pellets seems not to affect the swallowing parameters studied, it is important to bear in mind that particle size affects oral grittiness [START_REF] Kimura | Effect of granule properties on rough mouth feel and palatability of orally disintegrating tablets[END_REF][START_REF] Mishra | Investigation of Organoleptic Characteristics in the Development of Soft Chews of Calcium Carbonate as Mineral Supplement[END_REF][START_REF] Engelen | Oral size perception of particles: effect of size, type, viscosity and method[END_REF][START_REF] Imai | Oral Perception of Grittiness: Effect of Particle Size and Concentration of the Dispersed Particles and the Dispersion Medium[END_REF], where small multi-particulates are considered less gritty and more pleasant than large ones [START_REF] Lopez | Effect of Formulation Variables on Oral Grittiness and Preferences of Multiparticulate Formulations in Adult Volunteers[END_REF]2018a), and the feeling of residual particles in the mouth (Marconati et al., 2018b).

Effect of the particle volume fraction on pellets swallowability

As discussed in the introduction, large volumes of multi-particulates are often required in practice, and it is still unclear the number of particles that could be administered maintaining acceptability. In this section, the effect of the particle volume fraction (v.f. = 0.1-0.4) on in vitro pellets swallowability was investigated using diluted yoghurt as carrier, keeping a total bolus volume of 5 mL. Results are presented in Figure 7(d-f).

All swallowing parameters studied were affected by the percentage of pellets added to the carrier. Overall, an v.f. increase resulted in a marked decrease of both the percentage of pellets swallowed and the bolus velocity (Figure 7d and 7e, respectively). At v.f. 0.1 and 0.2, there were not significant differences between bolus FO. However, the time for the oral phase of swallowing doubled as v.f. increased from 0.1 to 0.4. The percentage of pellets successfully swallowed decreased linearly by 1/3 as v.f. increased from 0.1 to 0.4, resulting in a swallowing success rate around 50 % at v.f. 0.4. On the other hand, post-swallow residues seem to increase as the volume fraction increased (Figure 7f), however, differences were not statistically significant.

Effect of the bolus volume on pellets swallowability

The effect of bolus volume (i.e., 5 mL, 6.25 mL, and 8.34 mL; Table 2) at two different v.f., 0.2 and 0.4 (Figure 7(d-i)) on pellets swallowability was investigated using pellets of 350 µm in diameter and diluted yoghurt (5 mL) as carrier. Both carrier and pellets were mixed before feeding into the in vitro oral cavity (section 2.4). According to these results, no significant differences between swallowing variables were found. Therefore, the volume of the bolus in the oral cavity (or the filling level) under the range of conditions tested did not affect the swallowability of pellets.

In vitro swallowing comparison between pellets and minitablets

The effects of the feeding method, type of carrier, particle size, and particle volume fraction on minitablets swallowability have been reported in previous work [START_REF] Lavoisier | A novel soft robotic pediatric in vitro swallowing device to gain insights into the swallowability of mini-tablets[END_REF]. In this section, the effect of the type of SODF used, i.e., MTs 1.8 mm or pellets 350 µm in diameter, on in vitro swallowing was investigated at v.f. = 0.4, using diluted yoghurt as a carrier. The total volume of the bolus was kept constant at 5 mL. Both the carrier and the SODF of interest were fully mixed before feeding (section 2.4). Results are shown in Figure 7(j-l). In terms of the percentage of particles successfully swallowed, pellets led to better outcomes than MTs (+ 13.1%; Figure 7j). However, no differences were found between MTs and pellets for the other swallowing variables studied (bolus FO and post-swallow residues)

under the conditions tested.

In vitro swallowing of a novel binary mixture formed by minitablets and pellets

Combining MTs and pellets might enhance their swallowability by increasing the maximum packing fraction of the solid phase which may help particle flow during swallowing. The size ratio and the minimum pore size (size of the voids formed by MTs), in air, of the possible binary mixtures (BM) formed by both MTs and pellets were calculated as detailed in section 2.3 and results are shown in Table 5. The most favourable packing conditions would be at the highest size ratio, which corresponds to the mixture formed by MT of 3 mm and pellets of 350 µm. However, avoiding pellets flow throughout the voids formed by the MTs (i.e., particle segregation) is desired to favour packing. Therefore, the minimum pore size was also taken into consideration. According to Table 5, the optimum binary mixture should be the one formed by MTs of 1.8 mm and pellets of 350 µm in diameter. This SODF combination was then selected for further testing. Using the theoretical model implemented in the SI (section 1) proposed by [START_REF] Pillitteri | How size ratio and segregation affect the packing of binary granular mixtures[END_REF], the maximum packing fraction (η max ) of the binary mixtures formed by the MT (1.8 mm) and pellets (350 µm) occurs at a value of f 0.24, being f the fraction of pellets. Therefore, 24 % of pellets should be added to the binary formulation to favour flowing of the particles during swallowing.

Keeping a total bolus volume of 5 mL, the in vitro swallowability of the binary mixture was tested using diluted yoghurt as carrier at a v.f. of 0.4. Swallowing results are shown in Figure 7(j-l). The binary mixture improved particle swallowability compared to MTs (ca. +17%) but did not outperform pellets significantly (+3.6%). On the other hand, no significant differences were found in terms of bolus velocity and post-swallow residues. However, the post-swallow residues seem to be slightly greater for MTs than for both BM and pellets (Figures 7k and7l), decreasing the percentage of residues as the proportion of pellets increases.

Therefore, our initial hypothesis was confirmed, as by adding 24% pellets to a binary granular formulation in which minitablets were the dominant granular system, swallowability was improved, reaching swallowing levels to those reported by small particles like pellets.

Limitations and perspectives for future studies

One limitation of this study concerns the oral salivary lubrication, not considered here.

Regarding the SODF tested, MTs were considered fully spherical for calculations. Moreover, among the several possible binary mixtures, only the mixture at the maximum solid packing fraction has been tested. Therefore, future research could be conducted considering these aspects, exploring the impact that a more realistic lubrication may have on the swallowing process of a wider variety of SODF mixtures.

Conclusions

The Paediatric Soft Robotic Tongue (PSRT)an in vitro device inspired by the anatomy and physiology of 2-year-old childrenwas used to investigate the oral phase of swallowing of multiparticle solid oral dosage forms (SODFs), i.e., pellets (350 and 700 µm particles), minitablets (MTs, 1.8 mm), and their binary mixtures (BM). Swallowing time, fraction of particles swallowed, and the amount of post-swallow residues were measured. A systematic study was also conducted to determine how the feeding method, bolus volume, type of carrier, particle size, and particle volume fraction affect pellets swallowability.

The ability of the carriers to flow was significantly affected by the addition of pellets:

shear viscosity generally increased, yet little effect was observed on their shear thinning behaviour. Xanthan gum-based solutions achieved excellent success rates (≥ 75 %), especially at high concentrations, i.e., 0.5 and 1%, favouring pellets transport without affecting postswallow residues. Despite the size of the pellets (350 and 700 µm) not having a significant impact on particle swallowability, the volume fraction (v.f.) of particles used above 10% did;

the number of particles swallowed decreased as the volume fraction increased. At v.f. 0.4, pellets were easier to swallow (+ 13.1%) than MTs, while the administration method used strongly depended on the characteristics of the multi-particulate formulation under study. To increase MT swallowability, a novel approach involving MT-pellets binary mixtures was explored, which increased the maximum solid packing fraction of the formulation. Our results confirm that by adding 24% pellets to MTs, SODF swallowability was improved, reaching swallowing levels similar to pellets. Such flexible formulations are therefore easier to swallow in vitro and should be explored for combination products.

Figure 1 .

 1 Figure 1. Hypothesis illustration. Multi-particulate pharmaceutical formulations: minitablets, pellets, and their high packing density binary mixture. On the right, the in vitro oral phase of swallowing of a representative carrier is illustrated.

Figure 2 .

 2 Figure 2. Schematic representation of the paediatric solid oral formulations tested: (A) shape and proportions of minitablets (MTs), frontal (left) and lateral (right) view, and pellets, L MT , d MT , and d Ps refer respectively to the length and diameter of the cylindrical part of MTs, and to the diameter of pellets; (B) sketch of binary mixtures for different size ratios (α) (increasing α from sketch (1) to (3)) in a scenario where MTs dominate and pellets fill the voids formed between MTs; (C) feeding methods used: (1) pellets were spread on top of, or (2) fully mixed with the carrier. Images are not in scale.

Figure 3 .

 3 Figure 3. Shear viscosity data of the carriers tested along their mixtures with pellets (350 µm in diameter at a particle volume fraction of 0.2) as a function of shear rate: (a) semi-solid food carriers, apple puree and yoghurt, and (b) the xanthan gum solutions at three different concentrations (0.25, 0.5, and 1 % w/v). The experiments were carried out at 20°C. The geometry used was a concentric cylinder geometry (CC27) of 1 mm gap for all samples. Average data of at least two repeats.

Figure 4 .

 4 Figure 4. Effect of the feeding method on in vitro swallowing. Method 1: pellets were spread on top of the carrier; Method 2: pellets were fully mixed with the carrier of interest. The swallowing variables studied were: (a) the percentage of pellets swallowed, (b) the time of bolus front out, and (c) the percentage of bolus residues. The carrier used was XG 1% (volume 5 mL), pellets of 350 µm in diameter, and a particle volume fraction of 0.15. Average data of at least three repeats. Red letters indicate significant differences (p < 0.05) between data groups.

Figure 5 .

 5 Figure 5. Effect of the type of carrier used on the in vitro swallowability of pellets of 350 µm in diameter: (a) percentage of pellets swallowed, (b) time of bolus front out (s), and (c) percentage of post-swallow residues. The carrier tested were diluted apple puree, diluted yoghurt, XG 0.25%, XG 0.5%, and XG 1%. Average data of at least three repeats. Red letters indicate significant differences (p < 0.05) between data groups.

Figure 6 .

 6 Figure 6. Percentage of pellets (350 µm) swallowed as a function of the shear viscosity values of the carrier-pellets formulations tested at two different shear rate, 50 s -1 (red) and 330 s -1 (blue). Average data of at least two repetitions. Lines are a guide to the eye.

Figure 7 .

 7 Figure 7. Effect of size (a-c), particle volume fraction (d-f) and bolus volume (g-i) of pellets, and of the type of SODF used (j-l) on the in vitro swallowing of SODF. The swallowing variables investigated are: (a, d, g, j) the percentage of the particles swallowed, (b, e, h, k) the time of bolus front out (s), and (c, f, i, l) the percentage of bolus post-swallow residues. Average data of at least three repeats. Red letters indicate significant differences (p < 0.05) between data groups. Abbreviations are: (v.f.) particle volume fraction, (MT) minitablet, and (BM) binary mixture formed by MTs and pellets as the detailed in section 2.3, where f defines the fraction of pellets.

Table 2 .

 2 SODF fractions, sizes, and bolus masses used in the present study.

	SODF volume fraction	Pellets size (µm)	SODF mass (gr)	Total bolus volume
				(mL)
	0.10	350	0.65	5.00
	0.20	350/700	1.30	5.00
	0.30	350	1.95	5.00
	0.40	350	2.60	5.00
	0.15	350	1.15	5.89
	0.20	350	1.62	6.25
	0.40	350	4.34	8.34

Table 5 .

 5 Size ratio and minimum pore size (size of the voids formed by minitablet), in air, for the possible minitablets-pellets combinations. The optimum binary mixture selected for further testing is marked in bold. The pore sizes leading to percolation of the binary mixtures are highlighted in grey.

	MT / mm	Pellet / µm	α	Pore size / µm
	1.8	700	2.6	278.6
	2.3	700	3.3	356.0
	2.5	700	3.6	387.0
	3.0	700	4.3	464.4
	1.8	350	5.1	278.6
	2.3	350	6.6	356.0
	2.5	350	7.1	387.0
	3.0	350	8.6	464.4

Acknowledgements

This study was funded by F. Hoffmann-La Roche AG


domain: sdv sde

text:
Etienne Bourdon

PhD Charlotte Havreng-Théry

MD Carmelo Lafuente

MD Joël Belmin
email: j.belmin@aphp.fr

Etienne Bourdon Msc

Effect of the Physical Environment on Health and Well-Being of Nursing Homes Residents: A Scoping Review

Keywords: physical environment, nursing homes, residents' health, residents 'well-being, systematic review

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Brief summary

 59 articles evaluated relations between general architecture, sensory environment, dining rooms, gardens and NH resident's health  It demonstrates that many features of the physical environment have a +/-influence on residents health and well-being  Fine tuning of artificial lights, increase access to natural light, reduction of sound are clear beneficial to resident health  Easy access to gardens, resident's centred interior renovation, enhanced dining environment demonstrate positive effect

This research did not receive any funding from agencies in the public, commercial or non-profit sectors


domain: scco sdv

text:
Etienne Bourdon

Enriched gardens improve cognition and independence

of nursing home residents with dementia: a pilot controlled trial

Enriched gardens improve cognition and independence of nursing home residents with dementia: a pilot controlled trial BOURDON E. 1.2 , BELMIN J. 2.3 (1) Sorbonne Université Paris Nord UR 3412 LEPS, Paris, France (2) AP-HP , Hôpital Charles Foix, Ivry sur Seine France (3) Sorbonne Université, Paris, France

Background

Design & purpose

Results § Dementia is a major issue worldwide § Considerable efforts were made to design therapeutic mediation tools and evaluate their benefits on the health of patients.

Methods § 4 NH with separated access to one CSG and one EG Participants were residents with dementia, independent for walking and with no severe dementia or behavioural troubles. § 3 groups based on the proximity to the CSG or EG for the first 2 groups and without easy access for the control group.

  § Interventions: staff members frequently invited residents to visit the EG or the CSG depending on their group allocation. § 12 enrichment modules were installed in the EG that stimulated cognitive, independence and walking/balance functions. § Cognitive function (MMSE), independence for activities of daily living (ADL) and risk of falls (unipodal stance and Timed up and go -(TUG)) were assessed at baseline and after 6 months. § Multicentre cluster-controlled pilot trial § Evaluate the benefits of the frequentation of an Enriched garden (EG) vs a Conventional sensory garden (CSG) by nursing home (NH) residents Discussion & Conclusion § The 120 participants were 81•0 +/-3•5 years old and comprised of 83 women. § Patients characteristics were not significantly different between the 3 groups. § Participants invited to visit the EG group, 6 months changes in MMSE showed improved compared to other groups Changes in ADL, TUG and unipodal stance were significantly improved in the group visiting the EG as compared to other groups, which indicates better functioning. § EGs offer a new approach to therapeutic mediation for residents of nursing homes with dementia


Matthieu Blanke

Marc Lelarge

Online greedy identification of linear dynamical systems

This work addresses the problem of exploration in an unknown environment. For multi-input multi-output, linear time-invariant dynamical systems, we use an experimental design framework and introduce an online greedy policy where the control maximizes the information of the next step. We evaluate our approach experimentally and compare it with more elaborate gradient-based methods. In a setting with a limited number of observations, our algorithm has low complexity and shows competitive performances. 1

I. INTRODUCTION

System identification is a problem of great relevance in many fields such as econometrics, robotics, aeronautics, mechanical engineering or reinforcement learning [START_REF] Ljung | System identification[END_REF]- [START_REF] Thomas | Model-based reinforcement learning: A survey[END_REF]. The task consists in estimating the parameters of an unknown system by sampling data from it. One is often interested in identifying the system with as few samples as possible, hence motivating the active identification paradigm: the controller wants to choose the inputs that yield maximally informative data. We focus on multi-input multi-output (MIMO) linear time-invariant (LTI) dynamical systems, which can model for example the motion of an aircraft system over time near an equilibrium position. Let A ∈ R d×d and B ∈ R d×m be two matrices; we consider the following discrete-time dynamics:

x 0 = 0,

x t+1 = Ax t + Bu t + w t , 0 ≤ t ≤ T -1, (1)
where x t ∈ R d is the state vector, w t ∼ N (0, σ 2 I d ) is a normally distributed isotropic noise with known variance σ 2 and the control variables u t ∈ R m are chosen by the controller with the following power constraint:

1 T T -1 t=0 u t 2 ≤ γ 2 . ( 2
)
The parameters of the system are (A B) := θ ∈ R d×q (q = d + m). We note (A B ) := θ the parameters of the real system which are unknown initially and are to be identified. It may happen that the controller knows B in advance, in which case θ = A and q = m. At time T , the controller returns an estimate of θ from the observed trajectory (x t ) 0≤t≤T . The goal of active system identification is to choose inputs (u t ) that make the trajectory as informative as

This work was partially supported by the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR19-P3IA-0001 (PRAIRIE 3IA Institute).

M. Blanke and M. Lelarge are with INRIA, DI/ENS, PSL Research University, Paris, France. matthieu.blanke@inria.fr, marc.lelarge@inria.fr possible for the estimation of θ , with input u t being chosen at time t using the past observations. We provide a formal mathematical formulation of this objective in Section I-D.

In practice, one must face two main limitations in the identification of a LTI dynamical system. First, the number of observations T is small. Indeed systems have complex dynamics in general and can only be approximated by LTI systems on short time scales. Moreover, running an experiment on the real system can be costly (think of an aircraft test flight for instance), so the controller wants to estimate the parameters with as few samples as possible. Second, our identification algorithm needs to take decisions as fast as possible in order to run realistically online. Hence, we attach great importance to the computational time C of the control design, which should be reasonably small for practical applications. We have in this work a critical eye on these two limitations and we seek an exploration algorithm that runs fast and is sample efficient in a regime with a limited number of observations.

A. Contributions

To the best of our knowledge, system identification guarantees are only available in the large T limit, making the hypothesis of linear time-invariant dynamics quite unlikely. Instead, we explore in this work a practical setting for linear system identification with strong limitations on the number of interactions with the real system and on the computational resources used for input design and estimation. Using a framework based on experimental design, we propose an online greedy algorithm requiring minimal resources. The resulting policy gives a control that maximizes the amount of information collected the next step. We show empirically that for short interactions with the system, this simple approach can actually outperform more sophisticated gradient-based methods. We study the computational complexity of our algorithm and compare its performance with gradient-based approaches and with an oracle that we have designed, both on average and on real-life dynamical systems.

B. Related work

System identification is a primary area of control theory [START_REF] Ljung | System identification[END_REF], [START_REF] Gevers | Optimal Experiment Design for Open and Closed-loop System Identification[END_REF]. It has been widely studied in the field of experimental design. The question of choosing the most informative input can be tackled in the framework of classical experimental design [START_REF] Fedorov | Theory of Optimal Experiments[END_REF], [START_REF] Pukelsheim | Optimal design of experiments[END_REF]. For the particular case of LTI dynamical systems, optimal design approaches have provided results for single-input single-output (SISO) systems [START_REF] Goodwin | Dynamic System Identification: Experiment Design and Data Analysis[END_REF], [START_REF] Keviczky | Design of Experiments" for the Identification of Linear Dynamic Systems[END_REF] or MIMO systems in the frequency domain or with randomized inputs [START_REF] Mehra | Optimal inputs for linear system identification[END_REF]. More recently, system identification has received considerable attention in the machine learning community, with the aim of obtaining finite-time bounds on the estimation error [START_REF] Jedra | Finite-time identification of stable linear systems optimality of the least-squares estimator[END_REF]- [START_REF] Tsiamis | Finite sample analysis of stochastic system identification[END_REF]. The issue of designing optimal inputs is tackled in [START_REF] Wagenmaker | Active learning for identification of linear dynamical systems[END_REF], [START_REF] Andrew J Wagenmaker | Taskoptimal exploration in linear dynamical systems[END_REF]. The authors derive an asymptotically optimal algorithm (as the number of observations tends to infinity) by maximizing an optimal design objective in the frequency domain. The inputs are played and updated in epochs with exponentially growing length.

C. Notations and assumptions

In the rest of this work, we note θ = (A B ) the unknown parameters underlying the dynamics. We define a policy π : (x 1:t , u 0:t-1 ) → u t as a mapping between the past trajectory and the future input. The set of policies satisfying the power constraint (2) is noted Π γ . We denote τ = (x 1:T , u 0:T -1 ) a trajectory, and we extend this notation to τ (π, T ) when the trajectory is obtained using a policy π until time T . We denote E θ the average for a dynamical system given by (1) with fixed (A B) = θ knowing the past trajectory, where the expectation is taken over the randomness of the noise w t and possibly from the policy inducing the control u t . We assume that the pair (A , B ) is controllable: the matrix R = (B A B . . . A d-1 B ) has rank d.

D. Adaptive identification

Let us fix an estimator θ : τ → θ(τ ) ∈ R d×q , yielding an estimate of the parameters from a given trajectory. Our objective is to find a policy π ∈ Π γ whose resulting trajectory τ (π, T ) gives a good estimate θ(τ ) for θ . We measure this performance by the mean squared error of the parameters:

MSE(π) = 1 2 E θ θ τ (π, T ) -θ 2 F . (3)
An optimal policy π would be one minimizing the MSE (3). Of course, this quantity depends on θ the true unknown parameter of the system through the dynamics (1), so finding π by computing and minimizing (3) is not possible. However, we can adaptively compute an approximation of the MSE by estimating θ sequentially, as follows. Definition 1 (Adaptive system identification): Given an estimate θi of θ , the next sequence of inputs can be chosen to minimize a cost function F approximating the MSE (3), using θi as an approximation of θ . Then, these inputs are played and θ is re-estimated with the resulting trajectory yielding an estimate θi+1 , and so on. We call planning the computational process of choosing the next inputs by optimizing the cost F . As observations are collected, the estimation of θ gets more and more accurate, hence the planning functional becomes closer and closer to the MSE, yielding an increasingly accurate policy with each iteration.

Adaptive identification is summarized in Algorithm 1, which takes as inputs a first guess for the parameters to estimate θ 0 , the parameters σ and γ of the problem, a planning schedule {t 0 , = 0, t 1 , . . . , t n-1 , t n = T } (n is the number of planning iterations, i.e. the number of times the policy is updated between times 0 and T ), a cost functional F and an estimator θ. An adaptive identification algorithm is Algorithm 1 Sequential system identification inputs initial guess θ 0 , noise variance σ 2 , power γ 2 , cost functional F , estimator θ, planning schedule {t i } 0≤i≤n ininitialize π 0 returns random inputs output final estimate θ T for 0 ≤ i ≤ n -1 do run the true system t i+1t i steps with inputs u t = π i (x 1:t , u 1:t-1 )

θ i = θ(x 1:ti , u 1:ti-1 ) estimation π i solves min π∈Πγ F (π; θ i , t i+1 ) planning end for
hence determined by a triplet ( θ, F, {t i }). A natural estimator is the least squares estimator θ = θLS which we define in Section II-A. In the rest of this work, we set θ = θLS . Example 1 (Random policy): A naive strategy for system identification consists in playing random inputs with maximal power at each time step. This corresponds to the choice t i = i and π i returning u t ∼ N 0, γ 2 m I m . Example 2 (Task-optimal pure exploration): In [START_REF] Andrew J Wagenmaker | Taskoptimal exploration in linear dynamical systems[END_REF], the authors propose for F an A-optimal design functional (see Section II-B for details) and show that it approximates the MSE (3) in the long time limit at an optimal rate when T → +∞. Their identification algorithm works with planning intervals of exponentially growing length t i = 2 i × T 0 for some initial duration T 0 .

Example 3 (Oracle): An oracle is a controller who is assumed to choose its policy with the knowledge of the true parameter θ . Hence, it can perform one single, offline optimization of F (π; θ , T ) = MSE(π) over {t i } = {0, T }. By definition, the inputs chosen by the oracle are the optimal inputs for our mean squared error system identification problem.

II. BACKGROUND

It is convenient to describe the structure of the state as a function of the inputs and noise. By integrating the dynamics (1), we obtain the following result.

Proposition 1: The state can be expressed as

x t = xt + xt with xt = t-1 s=0 A t-1-s Bu s , xt = t-1 s=0 A t-1-s w s . (4)
Note that that xt = E θ [x t ] solves the deterministic dynamics xt+1 = Ax t + Bu t and xt has zero mean and is independent of the control. The two terms xt and xt depend linearly on the Bu s and the w s respectively. The data-generating distribution knowing the parameter θ can be computed using the probability chain rule with dynamics (1):

p(τ |θ) = ( 1 √ 2πσ 2 ) T exp - 1 2σ 2 T -1 t=0 x t+1 -Ax t -Bu t 2 2 . (5)
We define the log-likelihood (up to a constant):

(τ, θ) = - 1 2σ 2 T -1 t=0 x t+1 -Ax t -Bu t 2 2 = - 1 2σ 2 Y -Zθ 2 F , (6)
where we have noted Y = (y 0 . . . y T -1 ) ∈ R T ×d and Z = (z 0 . . . z T -1 ) ∈ R T ×q the observations and the covariates associated with the parameter θ. If θ = (A B), then

y t = x t+1 , z t = x t u t . If θ = A, then y t = x t+1 -Bu t and z t = x t . We also note U = (u 0 . . . u T -1 ) ∈ R T ×m , X = (x 0 . . . x T -1 ) ∈ R T ×d and W = (w 0 . . . w T -1
) ∈ R T ×d the matrix representations of the input, the state and the noise. We define the moment matrix M t = t-1 s=0 z s z s and the Gramians of the system at time t:

Γ t (τ ; θ) = 1 t E θ [M t ]. Note that Z Z = M T

A. Ordinary least squares

Given the linear structure of the dynamics, a natural estimator for the matrix θ is the least squares estimator. The method of least squares provides us with a formula for the mean squared error with respect to the ground truth, which can be used as a measure of the quality of a control.

Proposition 2 (Ordinary least squares estimator): Given inputs U and noise W , the ordinary least squares (OLS) estimator associated with the resulting trajectory τ is

θ(τ ) = (Z Z) -1 Z Y . (7)
and its difference from θ is given by

θ(τ ) -θ = (Z Z) -1 Z W = Z + W, (8)
where Z + denotes the pseudo-inverse of Z. Noting θ t the least squares estimator obtained from the trajectory τ t = (x 0,t , u 0:t-1 ), we recall the recursive update formula

θ t+1 = M -1 t+1 M t θ t + z t y t . (9
) Definition 2 (OLS mean squared error): For a given trajectory τ generated with parameters θ and noise W , the Euclidean mean squared error (MSE) is

θLS -θ 2 F = (Z Z) -1 Z W 2 2 = tr Z(Z Z) -2 Z W W . ( 10
)
If the noise W and the covariates Z were independent, the expected error would reduce to the A-optimal design objective E[tr(Z Z) -1 ]. It is not the case in our framework since Z is generated with W .

B. Classical optimal design

The correlation between Z and W makes the derivation of a simple expression for the expectation of (10) challenging. In this section, we show how a more tractable objective can be computed by applying the theory of optimal design of experiments [START_REF] Fedorov | Theory of Optimal Experiments[END_REF], [START_REF] David | Experimental design: review and comment[END_REF]. In the classical theory of optimal design, the informativeness of an experiment is measured by the size of the expected Fisher information.

Definition 3 (Fisher information matrix): Let (τ, θ) = log p(τ |θ) be the log-likelihood of the data distribution knowing the parameter θ. The Fisher information matrix is defined as

I(θ) = -E θ ∂ 2 (τ, θ) ∂θ 2 ∈ R qd×qd . ( 11
)
Proposition 3: For the LTI system (1),

I(θ) = T σ 2 diag(Γ T , . . . , Γ T ), (12)
the number of blocks being d.

Proof: The log-likelihood ( 6) can be separated into a sum over the θ j (d symmetric terms in θ j ). The quadratic term in θ j is Zθ j 2 2 = θ j Z Zθ j and the other terms are constant or linear. Differentiating twice and taking the expectation gives E θ [Z Z], which yields the desired result after division by -σ 2 .

Definition 4: In classical optimal design, the size of the information matrix is measured by some criterion Φ : S + n (R) → R + , which is a functional of its eigenvalues λ 1 , . . . , λ d ≥ 0. The quantity Φ(I) represents the amount of information brought by the experiment and should be maximized.

Example 4: Some of the usual optimal design criteria include A-optimality Φ A (I) = -tr(I -1 ) and Doptimality Φ D (I) = log det I. The criteria have mathematical properties such as homogeneity, monotonicity and concavity in the sense of Loewner ordering, which can be interpreted in terms of information theory: monotonicity means that a larger information matrix brings a greater amount of information, concavity means that information cannot be increased by interpolation between experiments. We refer to [START_REF] Pukelsheim | Optimal design of experiments[END_REF] for more details.

The classical optimal design theory leads to the following definition of the optimal design planning functional.

Definition 5 (Optimal design functional): Let Φ denote an optimal design criterion. Then the associated cost is defined as

F Φ (π; θ, t) = -Φ Γ t τ (π); θ .
(13) Remark 1: We note from equation (4) that Z is affine in U . Hence, M T = Z Z is quadratic in U , and maximizing [START_REF] Sarkar | Near optimal finite time identification of arbitrary linear dynamical systems[END_REF] efficiently is challenging even with concavity assumptions on Φ. In [START_REF] Andrew J Wagenmaker | Taskoptimal exploration in linear dynamical systems[END_REF], the policy is chosen to optimize [START_REF] Sarkar | Near optimal finite time identification of arbitrary linear dynamical systems[END_REF] with the Aoptimal criterion. The inputs are restricted to periodic signals and are optimized in the frequency domain.

III. ONLINE GREEDY IDENTIFICATION

A simple and natural approach for system identification consists in updating the policy at each time step t in a greedy manner: the input u t is chosen with energy γ 2 so as to maximize a one-step-ahead objective. Then, a new observation x t is collected and the process repeats. In the formalism of Section I-D, this corresponds to the schedule t i = i.

A. One-step-ahead objective

Following Section II-B, we adopt the optimal design functional F = F Φ . Upon choosing u t , the policy π t should maximize the design criterion Φ applied on the one-step-ahead, u t -dependent information matrix, the past trajectory x 0:t being fixed. The one-step-ahead information matrix is M t + E θs [z t z t ], because the next u t -dependent covariate is z t . Therefore, one-step-ahead planning yields the following optimization problem:

max u∈R m Φ M t + z(u)z(u) such that u 2 ≤ γ 2 , z(u) = x t u . (14)
Remark 2: With this greedy policy, the energy constraint imposed for the choice of one input ensures that the global power constraint ( 2) is met. The corresponding identification process is detailed in Algorithm 2. We will see in Section III-B that the D-optimality version of problem ( 14) can be solved accurately and at a low cost. Moreover, Algorithm 2 has the advantage of improving the knowledge of θ at each time step, so that the planning objective constantly corrected. In this way, the bias introduced by the uncertainty about θ is minimized, whereas a large bias could impair the identification when planning over larger time sequences. Algorithm 2 Greedy system identification inputs initial guess θ 0 , noise variance σ 2 , power γ 2 , time horizon T , design criterion Φ output final estimate θ T for 0

≤ t ≤ T -1 do u t ∈ argmax u 2 ≤γ 2 Φ Mt + z(u)z(u) play u t , observe x t+1 M t+1 = M t + z t z t θ t+1 = M -1 t+1 M t θ t + z t y t end for

B. Solving the one-step D-optimal design problem

We show that one-step-ahead D-optimal planning for online system identification is equivalent to a quadratic optimization problem that can be solved efficiently.

Proposition 4: For D-optimality, there exists a symmetric matrix Q ∈ R m×m and b ∈ R m such that the problem ( 14) is equivalent to

min u∈R d u Qu -2b u such that u 2 ≤ γ 2 . ( 15
)
Proof: By the matrix determinant lemma, we find that

log det M t + z(u)z(u) = log det M t + log 1 + z(u) M t -1 z(u) .
(16) Maximizing this quantity with respect to u is equivalent to maximizing z(u) M t -1 z(u). The matrix M t -1 is symmetric because M t is symmetric, and so are its diagonal submatrices. Given the affine dependence of z in u and the (possible) block structure of z and M t , z(u) M t -1 z(u) is of the form u Qu -2b u, up to a constant. We provide an explicit formula for Q and b in the case where θ = A in Section III-C We now characterize the minimizers of Problem [START_REF] Wagenmaker | Active learning for identification of linear dynamical systems[END_REF]. We focus on maximal energy greedy policies ( u t 2 2 = γ 2 ) and thus consider the equality constrained problem

min u∈R d u Qu -2b u such that u 2 = γ 2 . ( 17
)
Proposition 5: Let us denote u * a minimzer of ( 17), {α i } the eigenvalues of Q, and u i and b i the coordinates of u * and b in a corresponding orthonormal basis. Then there exists a nonzero scalar µ such that

u i = b i /(α i + µ) and i b i 2 (α i + µ) 2 = γ 2 . (18)
Proof: By the Lagrange multiplier theorem, there exists a nonzero scalar µ such that Qu *b = -µ u * , where µ can be scaled such that Q + µI d is nonsingular. Inverting the optimal condition and expanding the equality constraint yields the two conditions. It follows from Proposition 5 that our greedy planning problem [START_REF] Wagenmaker | Active learning for identification of linear dynamical systems[END_REF] can be solved at the cost of a scalar root-finding search and an eigenvalue decomposition. In [START_REF] Hager | Minimizing a quadratic over a sphere[END_REF], bounds are provided so as to efficiently initialize the root-finding method.

C. Known B

In the case where θ = A, the next u t -dependent covariate upon choosing u t is x t+1 so one-step-ahead planning takes the form

max u∈R m Φ M t+1 + σ 2 I d + x(u)x(u) such that u 2 ≤ γ 2 , x(u) = A t x t + B u. (19)
For D-optimality, we obtain a quadratic problem of the same form as [START_REF] Wagenmaker | Active learning for identification of linear dynamical systems[END_REF], with the following matrices:

Q = -B M t -1 B , b = B M t -1 A t x t . (20)
IV. GRADIENT-BASED PLANNING

In this section, we propose a gradient-based approach to planning.

A. Gradient-based optimal design

In the adaptive identification scheme of Algorithm 1, the cost functions (3) and ( 13) can be optimized by projected gradient descent. Gradients with respect to U can be derived analytically (see [START_REF] Goodwin | Dynamic System Identification: Experiment Design and Data Analysis[END_REF], section 6 for the derivation of an adjoint equation) or automatically in an automatic differentiation framework. We propose a gradient-based planning method that consists in performing gradient descent directly on U in functional [START_REF] Sarkar | Near optimal finite time identification of arbitrary linear dynamical systems[END_REF]. Note that we optimize the inputs directly in the time domain, whereas other approaches such as [START_REF] Andrew J Wagenmaker | Taskoptimal exploration in linear dynamical systems[END_REF] perform optimization in the frequency domain by restricting control to periodic inputs. We rescale U at each step to ensure that the power constraint ( 2) is met. The t i are chosen arbitrarily. The computational complexity of the algorithm is linear in T : each gradient step backpropagates through the planning time interval.

B. Gradient through the MSE oracle

Given the true parameters θ = (A B ), the optimal control for the MSE minimizes the MSE cost (3), as explained in Example 3. However, the dependance between Z and W makes this functional complicated to evaluate and to minimize with respect to the inputs, even when the true parameters θ are known. We propose a numerical method to minimize (3) using Monte-Carlo sampling. One can sample a batch of b noise matrices W 1 , . . . , W b ∼ N (0, σ 2 I), then approximate the gradient of (3) by

∇MSE(U ) 1 b b i=1 ∇ U tr Z(Z Z) -2 Z W i W i , (21)
and perform projected gradient descent. Although we have no convergence guarantees due to the complicated structure of the objective function, gradient descent does converge in practice, to a control that outperforms the adaptive controls.

V. PERFORMANCE STUDY

We compare our greedy algorithm to the gradient algorithms of Section IV and to the TOPLE algorithm of [START_REF] Andrew J Wagenmaker | Taskoptimal exploration in linear dynamical systems[END_REF].

A. Complexity analysis Definition 6 (Performance): The performance of the policy π is measured by the average error over the experiments on the true system: ε = MSE(π). We study the performance of our algorithms as a function of the number of observations T and the computational cost C. We also introduce the computational rate c = C/T . Algorithm 2 and the gradient identification algorithm have linear time complexity. Hence, we define c greedy and c gradient for a given number of gradient iterations. In practice, we find that c greedy c gradient , where c gradient is the computational rate needed for the gradient descent to converge. Furthermore, we expect the squared error to scale approximately as 1/T with the number of observations [START_REF] Boyd | Introduction to applied linear algebra: vectors, matrices, and least squares[END_REF] (see [START_REF] Andrew J Wagenmaker | Taskoptimal exploration in linear dynamical systems[END_REF] 13) with {t i } = {0, 10, T /2, T }. The results obtained are to be compared with those of the greedy algorithm, which has a fixed, small computational rate c.

Our diagrams are plotted on Fig. 1. They show that the greedy algorithm is preferable in a phase of low computation rate: C < c × T , as suggested by (22). The phase separation corresponds to a relatively high number of gradient steps. Indeed, the iso-performances along this line are almost vertical, meaning that the gradient descent has almost converged. Furthermore, the maximum performance gain of the gradient algorithm over the greedy algorithm is 10%.

B. Average estimation error

We now test the performances of our algorithms on random matrices, with the same settings as in the previous experiment. For each matrix A , we also compute an oracle optimal control using the Monte-Carlo approach of Section IV-B with a batch size of b = 100, and run a random input baseline (see Example 1), as well as the TOPLE algorithm of [START_REF] Andrew J Wagenmaker | Taskoptimal exploration in linear dynamical systems[END_REF].

The performances over time are shown in Figure 2. Both the gradient algorithm and the greedy algorithm closely approach the oracle. The former performs slightly better than the latter in average. However, the computational cost of the gradient algorithm is much larger, as shown in Table I. The values of c for the gradient-based algorithms are those found to achieve convergence in this setting (∼ 100 for the gradient-based optimal design, ∼ 1000 for TOPLE).

C. Identification of an aircraft system

We now study a more realistic setting from aeronautics: we apply system identification to an aircraft system. We use the numerical values issued in a NASA report [START_REF] Gupta | Application of optimal input synthesis to aircraft parameter identification[END_REF]. The lateral motion of a Lockheed Jet star is described by the slideslip and roll angles and the roll and yaw rates (β, φ, p, r) := x. The control variables are the aileron and rudder angles (δ a , δ r ) := u. The linear dynamics for an aircraft flying at 573.7 meters/sec at 6.096 meters are given by the following matrices obtained after discretization and normalization of the continuous-time system described in [START_REF] Gupta | Application of optimal input synthesis to aircraft parameter identification[END_REF]:

with σ = 1 and γ 4 deg. We apply our algorithms on this LTI system. Our results are summarized in Table II. The greedy algorithm estimates the system parameters as well as the gradient-based algorithms, but with substantially lighter computational cost. With small T , the estimate of A is too inaccurate for long-term planning to be effective.

It is more efficient to update the estimate and the policy frequently. We obtain similar results for the longitudinal system of a C-8 Buffalo aircraft [START_REF] Gupta | Application of optimal input synthesis to aircraft parameter identification[END_REF].

VI. CONCLUSION

In this work, we explore a setting for linear system identification with strong constraints on the number of interactions with the real system and on the computing resources used for planning and estimation. We introduce an online greedy algorithm requiring minimal computing resources and show empirically that in a regime where the number of interactions with the system is small, it can actually outperform more sophisticated gradient-based methods. Extending this approach to optimal control for the LQR is an interesting direction for future research.

Fig. 1 :

 1 Fig. 1: Experimental (T, C) diagram. Left Performance of the gradient algorithm, with variable T and C (varying number of gradient steps). Right Relative performance of the gradient algorithm compared with the greedy algorithm: negative means that the gradient performs better.

Fig. 2 :

 2 Fig. 2: Identification error for random A , averaged over 1000 samples.

TABLE I :

 I Average computational rate for the different algorithms.

		Random TOPLE [15] MSE Gradient Greedy
	c	1	100	50	2.36

TABLE II :

 II Frobenius error for A in the lateral system of the aircraft, T = 150. Our oracle algorithm reaches an error of 8.0 × 10 -2 . The computational time is expressed in an arbitrary unit.

		Random	TOPLE [15] MSE gradient Greedy
	Error 1.1 × 10 -1	8.6 × 10 -2	8.3 × 10 -2	8.2 × 10 -2
	Time	1	55.7	25	1.13

Our code is available at https://github.com/MB-29/ greedy-identification


Jessica Delacroix

Sophie Cazottes

Armelle Daniélou

Siegfried Fouvry

Jean-Yves Buffière

Jean-Yves Buffiere

Influence of microstructure on the fretting resistance of Al-Cu-Li alloys

Keywords: Fretting, Al-Cu-Li, fretting damage, X-ray tomography, crack

Influence of microstructure on the fretting resistance of Al-Cu-Li alloys Introduction

Confronted by the increasing use of composite materials in aircrafts, aluminium alloys have to be constantly improved. From a weight saving point of view, lithium addition offers interesting perspectives, as it reduces the density while improving at the same time mechanical properties like elastic stiffness and strength-damage tolerance compromise. Originally, Al-(Li,Cu) alloys were developed between the 60's and 90's but their commercial applications were limited due to technical problems, such as a very low corrosion resistance. Nowadays a new generation, with alloys containing a lower fraction of lithium, is being developed. A key issue in aircraft design is the use of rivets to join components which highly contribute to the risk of failure by fatigue. Higher fatigue stresses (induced by the presence of a hole) can become limiting in the case of riveted or bolted joints. Fundamentally, this problem can be considered from at least two points of view:

• First, through the influence of the rivet hole which acts as a stress raiser and therefore promotes fatigue crack initiation.

• Second, through the presence of a contact between the rivet and the fuselage material which can generate sub-surface damage through a fretting fatigue phenomenon. It is generally observed that fretting leads to premature crack nucleation compared to classic fatigue loading.

Although considerable research has been carried out on the initiation and growth of fatigue cracks from notches or holes in a vast range of Al alloys including early Al-(Li,Cu) alloys, the fatigue properties of the recently developed Al-Cu-Li alloy has been poorly investigated not to mention their fretting properties. Besides, very few studies have focused on the influence of microstructure on the fretting resistance of Al alloys [START_REF] Shinde | Fretting Fatigue Behavior in 7075-t6 Aluminum Alloy[END_REF][START_REF] Shinde | Quantitative Analysis of Fretting Fatigue Degradation in 7075-t6 Alumunum Alloy[END_REF][START_REF] Giummarra | Fretting Fatigue in 2xxx Series Aerospace Aluminium Alloys[END_REF]. This is mostly due to the fact that the characterisation of fretting cracks is generally destructive: sub contact cracking does not allow direct crack visualisation; instead the fretting test must be stopped and the samples cut in order to analyse the extent of damage. In this paper, the fretting resistance of two Al-Cu-Li alloys with different Cu/Li ratios and different heat treatments is studied.

Materials and methods

Materials

Two Al-Cu-Li 15 mm thick plates of alloys 2050 and 2196 were supplied by Constellium, Voreppe Research Centre (Constellium CRV). The chemical compositions of the alloys are detailed in Table 1. The following processing route has been used: casting, homogenization, hot rolling, solution heat treatment, quenching, stress relieving by controlled tension and artificial ageing. Two ageing conditions have been investigated: peak aged (T8) and under aged (UA -96h at 120 • C). The corresponding mechanical properties are given in Table 2.

%C u

%L i %M g %M n %A g %Z r %Z n %S i %F e 2050 3,2-3,9 0,7-1,3 0,2-0,6 0,2-0,5 0,2-0,7 0,06-0,14 < 0,25 < 0,08 < 0,1 2196 2,5-3,3 1,4-2,1 0,25-0,8 < 0,35 0,25-0,6 0,04-0,18 < 0,35 < 0,12 < 0,15 Table 1: Chemical composition of the studied alloys (wt%).

After heat treatment, a mixture of semi-coherent T 1 and coherent δ ′ precipitates has been observed by transmission electron microscopy [START_REF] Delacroix | Etude des mécanismes de fissuration en fatigue et/ou fretting d'alliages Al-Cu-Li[END_REF] in the 2196 alloy in the T8 condition while δ ′ are predominant in the UA condition. In the 2050 alloy a majority of T 1 precipitates has been found in the T8 condition while no significant precipitation could be observed in the UA condition. Both alloys exhibit a classical rolling texture with a low fraction of recrystallised grains in the 2050 alloy [START_REF] Delacroix | Etude des mécanismes de fissuration en fatigue et/ou fretting d'alliages Al-Cu-Li[END_REF]. Those grains have a classical pancake shape with a very large dimension along the rolling direction (L). The number of Fe-rich intermetallic particles per unit volume (as studied by X-ray tomography -see below for a short presentation of the technique) in the 2050 alloy is slightly larger that in the 2196 alloy but volume fractions are comparable (0.32% and 0.23 % respectively). The average size of those particles (radius of a sphere of equivalent volume) is slightly larger in the 2196 alloy (

Fretting tests

The experimental setup used for the fretting experiments is shown schematically in figure 1.

The flat Al specimen is maintained in contact with a cylindrical counter-body in titanium alloy thanks to the normal force P, which is kept constant during the test. A relative sinusoidal displacement δ is imposed, with a 10 Hz frequency, between the two bodies giving rise to a contact shear force of amplitude Q. Al-Cu-Li specimens have a controlled planar surface with a roughness Ra of 0. During the tests, P, Q and δ values were recorded. A constant number of cycles N of amplitude δ/tangential force Q is imposed on the sample; in the present case, N was fixed at 100 000 cycles. Several values of δ have been investigated. All the tests are carried out in partial slip conditions which are characterized by no relative movement between the cylinder and the plane at the centre of the contact and, simultaneously, a relative movement (slip) at the edges of the contact area. Depending on the contact loading conditions, cracks may or may not nucleate in these slip zones. As mentioned above, this cannot be detected by direct observation of the cycled sample, as cracking occurs below the contact surface. Therefore when the test is stopped (N =100 000), the sample is cut in two in the middle of the fretting scars. One of the new surfaces thus created is mechanically polished and the crack length measured by optical observation after Keller etching. The surface is further polished and etched twice (approximately 100 μm of material removed along the X direction between each observation) in order to evaluate the homogeneity of the crack length in volume. The crack length is defined as the projected length along the Y axis (see figure 1 for a schematic of the whole process and for a definition of the X, Y and Z axis). The location of fretting damage versus macroscopic directions of the rolled plate (L: rolling, LT: long transverse, ST: short transverse) is defined in figure 2. The 3D shape of the fretting cracks has been characterized by X-ray micro-tomography at the European Synchrotron Radiation Facility in Grenoble, France (beamline ID19). Given the voxel size used (0.7 μm) the whole fretting pad cannot be observed in tomography. Needle-like specimens with a 1mm 2 square section have been extracted at the surface of the fretting pads. Each needle contains a fretting crack in its central part and is set in tension with a dedicated device in order to open the crack and facilitate its observation as explained in Proudhon et al. [START_REF] Proudhon | Characterisation of fretting fatigue damage using synchrotron x-ray micro-tomography[END_REF]. The result of the tomographic observation is a 3D image of the fretting crack. Note that with this method only a restricted part of the fretting crack which has formed along the X direction can be observed.

Tomographic imaging of fretting cracks

Results and analysis.

Fretting damage

Figure 3 shows a plot of the average crack length measured below the contact as explained on figure 1 b. From this figure, by extrapolating the crack length to a zero tangential force, one can determine a shear stress threshold below which no crack initiation occurs. This threshold is found to be larger for the UA alloys (124 N/mm for the 2050 alloy and 113 N/mm for the 2196 alloy) than for the T8 alloys (96 N/mm for the 2050 alloy and 102 N/mm for the 2196 alloy). Besides the crack growth rate (given by the slopes of the curves on figure 3) is lower for the UA heat treatment.

3D observation of fretting cracks

The 3D characterisation of the fretting cracks shows that various small cracks initiate at different locations, and finally join to form one macroscopic crack as already observed in a 2024 alloy [START_REF] Proudhon | Characterisation of fretting fatigue damage using synchrotron x-ray micro-tomography[END_REF]. The local orientation of the crack during fretting propagation are different depending on the alloy temper (T8 or UA) for alloy 2050 (see Figure 4) : while in the T8 conditions the average orientation of the crack tend to stabilise (in a zone probably corresponding to one grain) at a depth of about 50 μm below the contact (Y/LT direction) it remained very "chaotic" for the UA condition. This qualitative observation which has been consistently made on the different samples observed in 3D could account for the better resistance of the UA treatment and could be a result of the microstructure induced by the heat treatment. Similar observations have been conducted during fatigue tests of samples with a central hole on the same materials/heat treatments : the UA microstructure induced a more tortuous crack path (with the crack following crystallographic directions) than for the T8 microstructure [START_REF] Delacroix | Etude des mécanismes de fissuration en fatigue et/ou fretting d'alliages Al-Cu-Li[END_REF]. In UA temper, the slip mode is probably more crystallographic in fatigue but also in fretting conditions.

Summary

For the experimental conditions investigated, the two alloys in the UA conditions exhibit a better resistance to fretting damage which is attributed to a more tortuous crack path observed in 3D below the contact. This higher tortuosity can be linked to a microstructure which promotes planar slip (shearable precipitates), as observed on larger fatigue samples [START_REF] Delacroix | Etude des mécanismes de fissuration en fatigue et/ou fretting d'alliages Al-Cu-Li[END_REF].

Figure 1 :

 1 Figure 1: a)Schematic representation of the fretting device for the cylinder (Ti)/plane (Al-Cu-Li) configuration. b)Experimental method to investigate cracking after the fretting test (X//ST, Y //LT, Z//L)

Figure 2 :

 2 Figure 2: Schematic drawing illustrating the orientation of the fretting cracks with respect to the sample rolling directions and test conditions (L-T configuration in this case). The rectangular gray shape shows the fretting scar. The double arrow indicates the direction along which the fretting pad is oscillating. The hashed lines show schematically the crack orientation below the contact.

Figure 3 :

 3 Figure 3: Comparison of the crack nucleation and propagation under fretting for the alloys 2050 (T8 and UA) and 2196 (T8 and UA).

Figure 4 :

 4 Figure 4: Example of 3D observations of the fretting cracks as obtained by X-ray tomography (Q=210N/mm) a) 2D projection of the crack (in black) in LT-ST plane (2050-T8 L-T) b) 3D rendering of the crack observed below the contact (2050-T8 L-T) c) and d) : 3D rendering of the crack normal in the XYZ axis system; the color indicates the evolution of the local orientation of the crack surface (c : 2050 UA L-T, d: 2050 T8 L-T)

Table 2 :

 2 3.1 μm) than in the 2050 (2.6 μm).

	Alliage Treatment TYS (MPa) UTS (MPa) ǫ% E(MPa) ν σ d (MPa)
	2196	T8	559	594	5,9	79000	0,305 140-150
		UA	428	520	11,1
	2050	T8	525	550	6	77000	0,305	150
		UA	360	463	18,9

Mechanical properties of the two alloys in UA and T8 conditions. σ d is the fatigue strength at 10 6 cycles for R =0, 1.

Acknowledgements

The PhD work of J. Delacroix was funded by Constellium.