Chiara Grazia

Milena Gennari

Gaia Maria

Giorgia Quaroni

Costantino Creton

Paola Minghetti

Francesco Cilurzo

Gaia Maria Giorgia

Gaia Maria Giorgia Quaroni

SEBS block copolymers as novel materials to design transdermal patches

Keywords:

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

Introduction

Transdermal patches are nowadays designed as drug in adhesive systems in which a drug loaded pressure sensitive adhesive is sandwiched between a protective liner and a backing layer or outer covering [START_REF] Cilurzo | Adhesive properties: a critical issue in transdermal patch development[END_REF]. The former is removed and discharged immediately before the application of the patch to the skin, the latter prevents the adhesion to the cloths and the loss of drug through the backing surface. The transdermal patches are usually produced by solvent casting or, alternatively, by hot-melt methods. Basically, a slurry is spread on the protective layer and then the dry/cold adhesive layer is coupled with backing layer.

Thermoplastic elastomers such as styrenic block copolymers are widely used in PSAs because of their unique structure, which offer advantages in processing. They consist of two polystyrene end-blocks and a linear aliphatic mid-block. These end-blocks and mid-blocks are completely incompatible and therefore the material is arranged into domains on a 20-50 nm level. Since each molecule of a tri-block polymer has styrene on each end, the styrenic domains comprise several molecules. This association of end-blocks acts as a multifunctional physical crosslinker effectively giving a flexible and elastic material.

The elasticity of styrene block copolymers is due to the thermodynamic incompatibility between the polystyrene end-blocks and the elastomer midblock, creating a two-phase structure consisting of polystyrene "domains" dispersed in an elastomeric matrix. The exact morphology of the two-phase structure and the mechanical properties depend on the molecular structure of both phases and in particular on their molecular weights [START_REF] Bates | Block copolymer thermodynamics: theory and experiment[END_REF][START_REF] Creton | Effect of the diblock content on the adhesive and deformation properties of PSAs based on styrenic block copolymers[END_REF][START_REF] Creton | Large-Strain mechanical behavior of model block copolymer adhesives[END_REF].

Considering these characteristics, styrene-based block copolymers are the perfect base for hot melt adhesives [START_REF] Derail | Rheological properties of hot-melt pressure-sensitive adhesive (HMPSAs) based on styrene-isoprene copolymers. Part 2: Innovative molecular design form predictive formation[END_REF][START_REF] Kim | Thermal properties and adhesion strength of amorphous poly(α-olefins) /styrene-ethylene-butylene copolymer/terpene hot-melt adhesive[END_REF]: they are strong, rubbery solids at ambient temperature, but become viscous liquids at high temperatures and this change of state is reversible.

Among these copolymers, styrene-isoprene-styrene (SIS) copolymers were widely studied for the design of pressure sensitive adhesives [START_REF] Minghetti | Measuring adhesive performance in transdermal delivery systems[END_REF] suitable for the preparation of both transdermal patches [START_REF] Ma | Design and evaluation of a monolithic drug-in-adhesive patch for testosterone based on styrene-isoprene-styrene block copolymer[END_REF] and medical devices [START_REF] Jin | Influence of hydrophilic polymers on functional properties and wound healing efficacy of hydrocolloid based wound dressings[END_REF]. The adhesive properties have been conferred by compounding the SIS with paraffin oil and an aliphatic resin [START_REF] Jagisch | Styrenic block copolymers[END_REF]. Moreover, because of their chemical structure, SIS copolymers need a large amount of antioxidants to improve their stability [START_REF] Chanda | Plastics Fundamentals, Properties, and Testing, Characteristics of Polymers[END_REF]. Conversely, styrene-block-(ethylene-co-butylene)-blockstyrene (SEBS) copolymers, being hydrogenated derivatives, are elastomers with outstanding stability to UV radiations and well-known biocompatibility [START_REF] Costa | Effect of butadiene/styrene ratio, block structure and carbon nanotube content on the mechanical and electrical properties of thermoplastic elastomers after UV ageing[END_REF][START_REF] Hou | Construction of 3D micropatterned surfaces with worm like and superhydrophilic PEG brushes to detect dysfunctional cells[END_REF]. Moreover, their compounding with plasticizer(s) and tackifier(s) results in a hot melt pressure sensitive adhesive with a very high creep resistance [START_REF] Kim | Thermal properties and adhesion strength of amorphous poly(α-olefins) /styrene-ethylene-butylene copolymer/terpene hot-melt adhesive[END_REF].

Taking into account all these considerations, SEBS copolymers appear as good potential candidates for the design of transdermal patches and/or medicated plasters. Nevertheless, the performances of the system obtained by the casting technique has not been well investigated and the ability of the compounded SEBS based pressure sensitive adhesive to release a loaded drug is unknown.

The aim of this work was to design a pressure sensitive adhesive made of SEBS in order to demonstrate the potentialities of such class of materials in the development of transdermal patches.

The adhesive properties were conferred using a low molecular weight, but high glass transition temperature hydrocarbon resin as tackifier and a liquid paraffin as plasticizer (*** [START_REF] Pagani | Self-adhesive matrix system comprising a styrene block copolymer[END_REF].

The PSAs were designed by using SEBS with different average molecular weights and, since tackifiers are crucial components for tuning the initial bonding of a PSA [START_REF] Nakajima | Rheology, composition, and peel-mechanism of block copolymer-tackifier-based pressure sensitive adhesives[END_REF], two resins, with different structure, glass transition temperature (T g ) and molecular weight were added.

The linear viscoelastic properties of the formulated materials were characterized with a rheometer and the adhesive properties were characterized with a specially designed probe tack instrument [START_REF] Lakrout | Direct observation of cavitation and fibrillation in a probe tack experiment on model acrylic pressure-sensitive-adhesives[END_REF] focusing on the debonding behavior from a steel surface under different operating conditions. The debonding of PSAs from a rigid surface often occurs with the formation of a fibrillar structure. Since the energy is only dissipated rather rapidly when the fibrils are detached from the surface, it is essential to characterize the elastic properties of these adhesives not only in the linear viscoelastic regime, but also in the large strain non-linear elastic domain [START_REF] Creton | Large-Strain mechanical behavior of model block copolymer adhesives[END_REF][START_REF] Deplace | Fine tuning the adhesive properties of a soft nanostructured adhesive with rheological measurements[END_REF]. We have therefore also performed tensile tests of adhesive films based on SEBS. The in vitro biopharmaceutical performances of patches were assessed using ibuprofen, as a model drug.

Materials and methods

Materials

Pharmaceutical grade SEBS are available on the market under the trade name of Europrene SOL TH®. Three types of SEBS, were kindly donated by Versalis SpA (San Donato Milanese, Italy): Europrene SOL TH® 2311 (triblock, 30% bound styrene, average molecular weight 45.61 kDa, M n 43,65 kDa, M w /M n = 1.04, SEBS-L), Europrene SOL TH® 2312 (triblock, 30% bound styrene, average molecular weight 64.23 kDa, M n 59,82 kDa, M w /M n = 1.07, SEBS-M), and Europrene SOL TH® 2315 (triblock, 32% bound styrene, average molecular weight 176.35 kDa, M n 155,87 kDa, M w /M n = 1.13, SEBS-H). In order to make the system adhesive it was necessary to add a tackifier. The hydrocarbon resins Regalite™ R1100 (ring and ball softening point = 100 °C, glass transition temperature = 50 °C, cloud point = 78 °C, molecular weight = 850 g/mol) and Eastotac™ H100W (ring and ball softening point = 100 °C, glass transition temperature = 41 °C, cloud point = 81 °C, molecular weight = 1000 g/mol) were donated from Eingemann&Veronelli Spa (Rho, Italy). Paraffin oil was obtained from Carlo Erba (Milan, Italy). In order to produce transdermal patches a polyester film was chosen as backing layer, while a polyester siliconized film Saint Gobain (thickness 120 µm) was used as a release liner. Sibuprofen (IB) was purchased from Dipharma Francis (Baranzate, Italy). Toluene was chosen as solvent and it was purchased by Sigma Aldrich (Milan, Italy). All the solvents were of analytical grade.

Preparation of the formulations

The SEBS polymers were dissolved in toluene, and the mixtures were continuously maintained under stirring with a magnetic bar at room temperature for 1 h. Then, the other components of the formulation were added, and the solution was continuously maintained under stirring with a magnetic bar at room temperature for 23 h. When the drug-loaded formulations were prepared, IB was added in the polymeric solution, before adding tackifier and oil. PSAs and patches were prepared only after 24 h in order to reduce air bubbles formed during the stirring stage. A pre-formulation study was performed to achieve the final compositions of the formulations, which are illustrated in Table 1.

Rheological properties in the linear regime

The rheological parameters were determined on a Discovery Hybrid Instrument HR-3 (TA Instruments), by using a parallel crosshatched plate geometry (diameter: 22 mm). In order to obtain equilibrated samples of about 2 mm of thickness, a special cup-shaped sample holder (6 × 5 cm) made of Teflon was used. The solution was then poured in the cup and the sample was dried in the sample holder. A 2-steps drying process was used: first, the adhesive matrices were dried under a glass cover for 24 h at room temperature and subsequently in an oven at 45 °C under vacuum for 24 h. After 7 days, necessary for matrix maturation, each sample was cut (diameter: 22 mm) by using a punch. Strain sweep measurements were performed at 6.28 rad/s (1 Hz) in order to define the linear viscoelastic region (LVR). Once the LVR was set, a frequency sweep deformation in the range 10 -2 -10 2 rad/s was applied to the sample. The strain value was in the LVR. Each sample was analyzed first at 25 °C, and then at 32 °C. Storage modulus (G′), loss modulus (G″) and phase angle (δ) were used to characterize the samples. Each experiment was performed twice to test data reproducibility.

Texture analysis

In order to obtain a dry matrix of 180-200 μm thickness, 3 mL of polymeric mixture were deposited on a standard glass slide (2.6 × 10 × 0.2 cm 3 ), previously cleaned and activated by a plasma technique. A 2-steps drying process was used: first, PSAs were dried under a glass cover for 24 h at room temperature and subsequently in an oven at 45 °C under vacuum for 24 h. The adhesive matrix final thickness was evaluated by a white light scattering technique with an optical profilometer (Microsurf 3D, Fogale nanotech). Probe tack experiments were performed on a custom-designed apparatus adapted on an MTS 810 hydraulic testing machine, allowing the simultaneous observation of the debonding process through a transparent glass substrate [START_REF] Lakrout | Direct observation of cavitation and fibrillation in a probe tack experiment on model acrylic pressure-sensitive-adhesives[END_REF]. A circular flat ended probe was brought into contact with the adhesive layer at a constant velocity (30 μm/s).

Once the probe and the sample came into contact, a compressive force of 30 N was applied. The probe was maintained in contact with the adhesive for 10 s. At the end of the contact time, the probe was separated from the adhesive by retracting it at a predefined speed, namely 10, 100 and 1000 µm/s. The probe was made of stainless steel with a diameter of 9.7 mm. The probe flatness was well-controlled by polishing it with several grades of abrasive paper. The entire probe tack set-up was enclosed in a controlled-temperature chamber to perform the experiments at room temperature (25 ± 0.5 °C) and at skin surface temperature (32 ± 0.5 °C). The whole force-distance curve was recorded. The detachment force and the elongation at break were measured and expressed in Newtons (N) and millimeters (mm), respectively. Since the results are influenced by the PSA thickness, the forcedistance curve was converted into a stress-strain curve. The nominal tensile stress (σ) and strain (ε) were calculated according to the following equations:

= σ F A / (1) = - ε h h h ( ) / t o 0
(2) F is the force registered during the detachment, A is the surface area of the probe that is in contact during the compression stage, as determined from the video streaming, h t is the adhesive elongation at time t and h 0 is the adhesive layer initial thickness. The Adhesion energy (W adh ) can then be evaluated as the integral of the σ-ε curve multiplied by the initial thickness.

To evaluate the fracture occurring during the debonding process all tests were filmed and recorded through a 45°mirror. For each formulation and for each experimental condition three tests were carried out.

Non-linear elastic properties

Tensile tests under large strain were carried out in a standard tensile testing machine (Instron model 5565), equipped with a 10 N load cell and a non-contact video extensometer (SVE), to measure the local strain near the center of the sample. Rectangular samples of 2 mm thick, 5 mm wide, and 20 mm long (length between the clamps) were cut with a die-cutter. They were prepared at the desired thickness in the same way as the samples used for the rheological tests. Tensile tests were performed on the same formulations tested at room temperature since rheological measurements showed that the behavior of the SEBS-sample doesn't change when increasing the temperature from 25 °C to 32 °C. The experiments were carried out using two different cross-head velocities (kept constant): 0.833 mm/s and 8.33 mm/s; these velocities were chosen in order to be approximately equivalent to the strain rate applied to the materials in the texture analysis. Two white spots were drawn on the sample to follow the displacement with the laser extensometer. Two tests were performed for each sample at a given strain rate to ensure reproducibility.

During the test, force and displacement from the cross-head were measured. In addition, the extensometer measures the displacement of the white spots. A comparison of the two nominal strains obtained from the cross-head and of the extensometer ensured that no slippage occurred during the experiments. The following parameters were determined: nominal stress σ N , calculated by dividing the force by the original cross-section area of the specimen and expressed in force per unit area (MPa), and λ, that is the stretch or extension ratio calculated by dividing the length of the specimen by the initial length of the specimen.

Patches preparation

Patches were prepared by casting, using a laboratory-coating unit Mathis LTE-S(M) (Mathis, CH), equipped with a doctor knife. The coating thickness was set at 250 µm in order to obtain a dry matrix of about 50 µm. The mixture was spread on the backing layer, dried at 40 °C for 20 min, and covered with the release liner. To evaluate the final matrix thickness a MI 1000 µm (ChemInstrument, USA) was used. Samples were sealed in a well-closed airtight container and stored at 25 ± 0.1 °C until use.

Inclined ball-tack test

The inclined ball tack test probes the ability of a patch to quickly adhere to a stainless-steel surface. In this test, balls characterized by different diameters are rolled on the adhesive surface of a patch placed onto an inclined ramp; the largest ball that stops on the patch determines the tack value. According to the "Inclined ball tack testing" reported in the Japanese Pharmacopeia XVII (JP), a plane having an inclination angle of 30°was used. Samples were cut in order to obtain specimens wider than 10 mm and longer than 70 mm. Each specimen was fixed in the prescribed position on the ramp with the adhesive layer side up. The upper and the lower ends of the specimen were covered with clean sheets of paper, leaving 50 mm of adhesive free at the central position. The balls must be able to roll down without slipping. Several hard steel balls (from no. 2 to no. 32, JP XVII) were used. The higher the ball number, the larger the diameter and, therefore, the greater the weight. Before running, each ball was cleaned by rinsing it with acetone and wiping it dry with absorbent cleaning material followed by air drying for at least 10 min. Finally, each ball was rolled over the inclined plane from the top of the ramp onto the adhesive tape. Specimens were changed for each run. The number (No.) of the largest ball that stops on the adhesive represents the value of the inclined ball tack test. The assay was performed in triplicate.

Shear adhesion

Shear adhesion tests were performed by using an 8 Bank Oven Shear HT8 Instrument (ChemInstruments, Ichemco, Cuggiono, Italy), in agreement with an internal method described by [START_REF] Cilurzo | On the characterization of medicated plasters containing NSAIDs according to novel indications of USP and EMA: adhesive property and in vitro skin permeation studies[END_REF]. The adhesive sample was pressed onto the stainless-steel test panel with a bonding area of 2.5 × 2.5 cm 2 by a double pass of a 2.04 kg roller. A mass of 500 g was hung at the end of each sample. To perform tests at 32 °C a controlled-temperature chamber was used. Each result, namely the time required to completely detach the sample from the test panel, is expressed as the mean ± standard deviation of six determinations for each formulation.

Peel adhesion 180°test

Peel adhesion measures the patch adherence when peeled at 180°a ngle to a standard steel panel. Peel strength was measured on a tensile machine equipped with a 50 N cell (Instron 5965, ITW Test and Measurement Italia S.r.L., Trezzano sul Naviglio, Italy) at room temperature. Tests were performed according to an internal method [START_REF] Cilurzo | On the characterization of medicated plasters containing NSAIDs according to novel indications of USP and EMA: adhesive property and in vitro skin permeation studies[END_REF].

For each formulation, samples were run using a stainless-steel panel, at a peel angle of 180°and peel speed of 300 mm/min. The average force was calculated as the arithmetical mean of the values of the linear portion of the curve and the results are expressed as the mean ± standard deviation of six determinations for each formulation. The peel values are expressed in cN per centimeter width of the adhesive patch under test.

Cold flow

Cold flow represents the migration of the adhesive matrix outside the edge of the backing layer during the storage and involves drug leakage from the edge of membrane-controlled patch. It occurs if the matrix flows like a very viscous liquid between the backing layer and the release liner during the storage [START_REF] Quaroni | Tuning the rheological properties of an ammonium methacrylate copolymer for the design of adhesives suitable for transdermal patches[END_REF]. Circular samples (matrix and backing layer) were cut by using a punch (diameter: 32 mm), maintaining the whole release liner. After three months of storage in a sealed container impervious to water and humidity at 40 ± 1 °C, samples were evaluated in terms of migration of the adhesive matrix on the release liner. The dimensional changes were measured using a graph paper and cold flow was considered negligible if the matrix was not visually detectable outside the backing layer.

The analyses were performed in triplicate.

Drug content

To determine IB amount loaded in the patches, a specimen of 2.54 cm 2 , after release liner removing, was first weighed and then dissolved in 50 mL of methanol. To support the complete solubilization of the adhesive matrix and the active principle, the sample was sonicated for 1 h and then left at rest overnight. The solution was filtered through a 0.45 µm polypropylene filter (VWR International, Milan, Italy), diluted with mobile phase (1:9 v/v) and assayed by HPLC, according to the method reported below. Each value represents the average of three determinations for each formulation.

Dissolution test

The IB release rate was studied by using an apparatus SR8 PLUS Dissolution test station (Hanson Research, CA, USA) according to the "Dissolution test for transdermal patches" of European Pharmacopoeia 9.2, 2017.

A sample of 8.04 cm 2 was cut and placed flat on the disk with the release liner surface facing up. The backing layer was attached on the disk by using a cyanoacrylate adhesive and the release liner was removed. Each vessel was filled with 300 mL of PBS buffer (pH 7.4), the water bath temperature was maintained at 32 ± 0.5 °C and the paddle speed was set at 25 rpm. Samples (5 mL) were collected at predetermined intervals and assayed by HPLC, according to the methods reported in paragraph 2.13. An equal volume of fresh medium was immediately added to maintain the dissolution volume.

The release rate constant was calculated according to Higuchi's equation as follows:

= ∞ M M kt / t 0.5 (3)
where M t is the amount of drug released at time t, M ∞ is the drug loading in the matrix and k is the release rate constant expressed as h 0.5 . Each value represents the average of three measurements.

In vitro skin permeation study

Since pig ear skin shows similar histological and physiological characteristics and close permeability properties with respect to human skin [START_REF] Engesland | In vitro models to estimate drug permeation through the compromised stratum corneum barrier[END_REF][START_REF] Campani | Nanocarriers to enhance the accumulation of vitamin K1 into the Skin[END_REF], it was selected to study IB-permeation profiles from SEBS-patches. This skin used was obtained from the ears of different pigs, which were kindly provided by a local slaughterhouse (Milan, Italy). The pig ear skin was separated from the underlying cartilage, cut into squares of about 3.5 cm 2 and sealed in evacuated plastic bags and frozen at -20 °C. Before the experiments, the samples were equilibrated at room temperature for 1 h.

The integrity of all tissue samples was assessed by measuring their electrical resistance (voltage: 100 mV, frequency: 100 Hz; Agilent 4263B LCR Meter, Microlease, Cernusco sul Naviglio, Italy) [START_REF] Gennari | A glimpse in critical attributes to design cutaneous fil forming system based on ammonium methacrylate[END_REF].

The permeation experiments were performed using modified Franz cells under occlusive conditions. They have a diffusion area of 0.636 cm 2 and a receptor compartment volume of approximately 5 mL. This compartment was filled with physiological solution. A patch sample (2.54 cm 2 ) was applied onto pig-ear with slight pressure. The patch-skin system was carefully mounted on the lower part of the Franz cell with the patch facing upward. The upper and lower parts of the vertical Franz cell were sealed with Parafilm® and fastened together by means of a clamp. The system was kept at 37 ± 1 °C by means of a circulating water bath so that the skin surface temperature was at 32 ± 1 °C and the receiver medium was continuously maintained under stirring with a magnetic bar. At predetermined interval times (1, 3, 5, 7 and 24 h) 200 µL samples were withdrawn from the receptor compartment and an equivalent volume of fresh receiver medium was added. Sink conditions were maintained throughout the experiment. Samples were analyzed by HPLC according to the method described below.

The cumulative amount permeated through the pig ear skin per unit of area was calculated from the concentration of IB in the receiving medium and plotted as a function of time. The main calculated permeation parameters were the sum of the amount of drug permeated through the skin (Q 24h ), expressed as µg/cm 2 , and the steady flux (J), expressed as µg/cm 2 h and determined as the slope of the linear portion of the plot. The results of each permeation experiments are expressed as the mean ± standard deviation of three replicates for each formulation.

Drug assay

The IB concentrations were quantified by HPLC analysis (Agilent HP 1100, Chemstation, Hewlett Packard, Santa Monica, USA). 20 μL sample was injected at 25 °C on a HyperClone™ 5 μm BDS C18 130, 150 mm × 4.6 mm (Phenomenex, Torrance, CA) column. The composition of the mobile phase was acetonitrile/water pH 2.6 (60/40, % v/v). The flow rate was 1.5 mL/min. The wavelength was set at 230 nm. IB concentration was determined from a standard calibration curves in the 0.1-50 μg/mL range.

Statistical analysis

The performances of the patches in terms of drug release rates and adhesive properties were compared by analysis of the variance followed by Turkey post-analyses (Daniel's XL Toolbox 6.70). The level of significance was taken as p < 0.05.

Results and discussion

Pressure sensitive adhesive characterization

Texture experiments were performed not only to evaluate the instant adhesion properties of the PSAs, but also their debonding mechanisms. The compression stress (0.4 MPa) applied during the experiment was selected after performing different tests at different contact stresses. It was chosen to obtain a full contact between the adhesive layer and the probe surface, while avoiding a probe indentation higher than 15 µm, which can influence the results. The failure mechanisms were studied by changing the debonding velocity and two typical debonding mechanisms were identified.

At the lower debonding velocities (10 and 100 µm/s), all the formulations showed a debonding behavior typical of a viscous liquid: many digitations, intended as air fingers that propagate from the surface toward the centre of the adhesive, were observed [START_REF] Nase | Dynamic evolution of fingering patterns in a lifted Hele-Shaw cell[END_REF], because at these experimental conditions the adhesives had more time to relax and reorganize during the detachment process (Fig. 1). The shape of stress-strain curves and debonding mechanisms did not change significantly for different SEBS M w . The maximum value of strain (ε max ) and adhesion energy (W adh ) increased for SEBS-H, although no correlation was evident for the other two polymers (Fig. 1).

Moreover, the adhesive behavior during the debonding phase, as well as the shape of the stress-strain curve and the debonding mechanisms, were independent of the tackifier molecular weight (Fig. 1).

At the highest debonding velocity (1000 µm/s), the viscoelastic adhesion, characterized by a sharp increase of the force to a maximum value, followed by a decrease with a plateau (associated with a fibrillation process), was observed for all the formulations (Fig. 2, red line).

An increase of the temperature did not affect the detachment response, but decreased both stress and maximum strain, and therefore also the adhesion energy.

Interestingly, in each experimental condition, all the formulations detached adhesively without leaving obvious residues on the metal probe after the debonding process.

The observed stress-strain curve of SEBS-H at the highest debonding velocity shows a higher maximum strain compared to the other formulations, due to the formation of fibrils. When fibrillation occurs during the detachment process and the debonding is eventually adhesive, this implies that the adhesive interactions between probe and adhesive are high, allowing therefore more extensive bulk deformation of the adhesive layer, and that the adhesive material in the fibrils must experience some strain hardening to cause adhesive detachment [START_REF] Deplace | Fine tuning the adhesive properties of a soft nanostructured adhesive with rheological measurements[END_REF].

To confirm these hypotheses, tensile tests were performed in order to characterize the elastic properties in the large strain non-linear elastic domain and to show whether strain hardening is indeed observed in elongation [START_REF] Creton | Large-Strain mechanical behavior of model block copolymer adhesives[END_REF]. A typical extension curve can be divided in three regions: low (corresponding to Young's modulus), medium (corresponding to the softening) and high strain (corresponding to the strain hardening). Formulations L-R, M-R and H-R showed the typical nonlinear behavior of crosslinked rubbery gels (Fig. 3), due to the structure of the matrices: they are composed of styrene domains and soft deformable isoprene regions swollen with resin and plasticizer [START_REF] Creton | Effect of the diblock content on the adhesive and deformation properties of PSAs based on styrenic block copolymers[END_REF]. All the formulations, independent of the molecular weight of SEBS, exhibited a pronounced softening at intermediate strains, which is an indication of a notable viscoelastic behavior. The formulation based on SEBS-L did not display any strain hardening at large strain before breaking and failed at a relatively low extension of λ = 3 (Fig. 3).

The formulations based on SEBS-M and SEBS-H on the other hand were much more extensible and showed a softening stage followed by hardening. In particular, the SEBS-H is significantly stiffer with a stress at break of more than 300 kPa as opposed to 80 and 14 kPa for SEBS-M and SEBS-L respectively.

Placebo patches characterization

The thickness of both placebo and drug-loaded matrices was in the 49-55 μm range. Considering the accuracy of the instrument, all the matrices can be considered similar.

In terms of adhesive properties, the shear adhesion measures the ability of a PSA to adhere to a standard steel panel under constant stress [START_REF] Cilurzo | Adhesive properties: a critical issue in transdermal patch development[END_REF]. All the recorded values felt in an acceptable range since they were higher than values reported in literature for transdermal patches [START_REF] Minghetti | Evaluation of adhesive properties of transdermal therapeutic systems containing nitroglycerin[END_REF]. No significant differences were observed in the shear adhesion values of formulations based on SEBS-L and SEBS-M (Table 2). On the contrary, as expected, formulations H-R and H-E showed higher values than the corresponding formulations based on SEBS with lower molecular weights.

The absence of differences among SESB-L and SEBS-M can be explained considering that the patches M-R and M-E detached adhesively (i.e., the adherent plate was clean after the detachment): as a consequence, the shear adhesion results cannot be considered as a true measure of the internal strength of these formulations [START_REF] Cilurzo | Adhesive properties: a critical issue in transdermal patch development[END_REF]. On the contrary, for formulations L-R and L-E a matrix print on the steel panel test was observed after the detachment of the patches.

The results obtained from the rheological oscillatory tests in the linear regime performed to assess the PSA viscoelasticity can help to explain the different behavior among SEBS-H or SEBS-M and SEBS-L. The rheological measurements showed that the materials made from SEBS-L behaved as viscoelastic fluids with G′ and G″ varying markedly with frequency, while those prepared by SEBS-M and SEBS-H behave like moderately viscoelastic gels. (Fig. 4).

This means that formulations L-R and L-E possess the appropriate balance between cohesion and adhesion for this particular application. In other words, they behaved as viscoelastic fluids, guaranteeing a reasonable resistance to shear, and as viscous liquids, able to dissipate energy when peeled off the substrate [START_REF] Deplace | Fine tuning the adhesive properties of a soft nanostructured adhesive with rheological measurements[END_REF]. On the basis of these considerations, the detachment behavior of formulations L-R and L-E can be attributed to a polymer molecular disentanglement within the adhesive, resulting in a cohesive failure. On the contrary, the more elastic materials (based on SEBS-M and SEBS-H), under a shear stress, separate at the interface leaving no traces of matrix residues on the steel panel test.

The good cohesive properties of the prepared adhesive matrices are further supported by the lack of cold flow after storage at 40 °C over 3 months, in agreement with the elastic properties of these materials both in the linear viscoelastic region and in the large strain non-linear elastic domain. Indeed, it was demonstrated in a previous work that patches present cold flow when the polymeric matrices show a more viscous character [START_REF] Quaroni | Tuning the rheological properties of an ammonium methacrylate copolymer for the design of adhesives suitable for transdermal patches[END_REF]. Moving the attention toward the tackifiers, an influence of their molecular weight can be noted even if this effect resulted statistically significant only in the formulations made with SEBS-H (p < 0.05). In particular, as expected, the higher the molecular weight of the hydrogenated aliphatic resin used as tackifier, the higher the shear adhesion values. Again, all the formulations satisfied Dahlquist's criterion (Table 3), according to which a good PSA has to show an elastic modulus determined at the bonding frequencies (6.28 rad/s) lower than 0.1 MPa [START_REF] Dahlquist | Treatise on adhesion and adhesives[END_REF].

The peel adhesion represents the force required to peel away a patch from a substrate [START_REF] Cilurzo | Adhesive properties: a critical issue in transdermal patch development[END_REF]. The low peel values reported in Table 2 can be considered satisfactory, since it is well known that high values could hurt and cause skin damage upon removal. Because the debonding mechanisms are complex and depend on both the values of G′ and G″ and the strain hardening properties in large strain, it is not easy to correlate the peel force to the formulation in a simple way. The L-R formulations have a higher modulus at high strain rate and little strain hardening while M-R and H-R are softer (more penetration of the ball) and have some strain hardening (more stable fibrils and higher peel force).

The inclined rolling ball test provided information on the superficial properties of the adhesive. Indeed, differences in the diameter of the balls that stop on the adhesive layer were evident changing SEBS-molecular weight (Table 2) and a correlation among tack and shear adhesion was highlighted: the higher the number of the ball the higher the shear adhesion. The tackifier resin type did not influence the stickiness of the matrices, in accordance with the texture experiments.

In summary, the designed formulations having a low modulus, and therefore satisfying the so-called Dahlquist criterion, appeared suitable to design PSAs. However, their adhesive properties cannot be predicted considering only the linear viscoelastic regimen since they are soft materials. Indeed, the linear viscoelastic properties fail at predicting the cohesive properties of a PSA, which can be studied in the viscoelastic regime. By simple tensile tests it's possible to characterize the ability of the formed fibrils to be stretched before the final debonding. The softening, which occurs at intermediate strains, and the hardening, which occurs at large strains, can be used to predict the mode of failure and the energy of adhesion.

All together the obtained data suggested that all the formulations presented suitable adhesive properties to be used in the design of (trans)dermal patches. First, the shear adhesion values fell within the range which correspond to that of medicated plasters already available on the market [START_REF] Cilurzo | On the characterization of medicated plasters containing NSAIDs according to novel indications of USP and EMA: adhesive property and in vitro skin permeation studies[END_REF] that in many cases, when tested with the same method used in this work, showed an adhesive failure. Secondly, the peel adhesion values resulted lower than those registered for other formulations based on PSA, but at least the double than plaster based on hydrogels. Thirdly, the stickiness of the patch was sufficiently low to assure an accurate application on the skin [START_REF] Cilurzo | Adhesive properties: a critical issue in transdermal patch development[END_REF].

Drug loaded patches performances

In order to confirm the feasibility to design transdermal patches made from SEBS, patches loaded with IB were prepared. This active ingredient was selected since it is well recognized that its loading in a polymeric matrix significantly affects the adhesive properties [START_REF] Cilurzo | On the characterization of medicated plasters containing NSAIDs according to novel indications of USP and EMA: adhesive property and in vitro skin permeation studies[END_REF][START_REF] Cilurzo | Polymethacrylates as crystallization inhibitors in monolayer transdermal patches containing ibuprofen[END_REF][START_REF] Michaelis | Plasticization and antiplasticization of an acrylic pressure sensitive adhesive by ibuprofen and their effect on the adhesion properties[END_REF]. The thickness of the patches was homogenous and in the 49-55 μm range.

An effective reservoir function of these nanostructured systems was highlighted. As depicted in Fig. 5 the structure of the adhesive is not modified by the addition of IB and no-crystal are detected. Indeed, in the case of drugs that are poorly soluble and inclined to crystallize, such as IB, generally the amount that can be loaded in a patch is limited. As a matter of fact, a previous work showed that no more than 3% IB could be loaded in a polydimethylsiloxane PSA matrix without observing the growth of drug crystals after few days [START_REF] Cilurzo | Polymethacrylates as crystallization inhibitors in monolayer transdermal patches containing ibuprofen[END_REF]. Instead, the loaded IB amount in the SEBS based matrices was higher and it didn't require auxiliary excipients (such as crystallization inhibitors), suggesting the ability of the styrenic domains of these copolymers to work as drug reservoirs.

The loading of this large amount of IB in the adhesive matrix significantly influenced the adhesive properties of the patches. Indeed, the rolling ball tack values increased in all cases, while the shear adhesion values significantly decreased, suggesting that this small molecule can effectively act as plasticizer also in SEBS based formulations (Table 4). However, all the patches showed the same failure of the corresponding placebo formulation. In the case of the peel adhesion the effect of the drug loading appeared unpredictable, since the values increased for the formulations prepared with SEBS-L, remained substantially unchanged in the case of SEBS-M and significantly decreased for SEBS-H formulations. These modifications may be due to a reduction in the number of entanglements between the polymer chains, which determined an increase in the PSA fluidity, as highlighted by the rheological properties of the formulations. Literature data reported that for chemically cross-linked adhesive matrices the effect of the active ingredient was more evident in the formulations having greater G′ values [START_REF] Wang | Rheological characterization of cataplasm bases composed of cross-linked partially neutralized polyacrylate hydrogel[END_REF]. Accordingly, the reduction of the shear adhesion values due to the IB loading for SEBS-H formulations having the lowest G′ value resulted more evident (about 20 folds) than for SEBS-M formulations (about 10 folds) and SEBS-L (about a half). The opposite trend was found in the case of rolling ball tack test where the plasticizing effect of IB, producing a softer matrix, increased the tack values. Furthermore, the rolling ball tack test also showed the influence of the tackifier resin. This kind of excipients are added to PSAs to modify the viscoelastic properties (decreasing G′ values in the low frequency region) so that the adhesive performances in bonding are improved. According to such effect, the formulation based on SEBS-L containing Regalite™, which, having lower G′ value at 0.05 rad/s appears softer than the corresponding formulation containing Eastotac™, was more sensitive to the plasticization effect exerted by IB. Moreover, taking in consideration formulations L-R_IB and L-E_IB, a reduction of shear adhesion values resulted much more evident when Eastotac™ H100W was used, suggesting that in the softer matrices also the M w of aliphatic tackifier could play a key role in determining the shear adhesion performances of the patches.

Drug content (Table 5) satisfied the Ph. Eur. assay for the uniformity of dosage units.

IB release was completed within 7 h and it followed Higuchi pattern independently of the SEBS and tackifier molecular weight (R 2 > 0.992). The IB release constants (Table 5), calculated according to the Higuchi model, highlighted as the IB constant release rate was significantly faster by using SEBS-L as polymer (p < 0.05), but it was not dependent on the tackifier suggesting that the softening effect, verified in the study of adhesive properties of drug loaded patches, did not influence the drug diffusivity.

These data were in agreement with the rheological pattern: the higher the liquid character of the formulation, the faster the drug release rate. Moreover, comparing these results with the drug release data obtained with a commercially available patch loaded with IB, namely Ibupas®, the drug release rate from formulations L-R_IB, L-E_IB and H-E_IB resulted significantly different (p < 0.05). In particular, formulations L-R_IB and L-E_IB exhibited faster IB release rate if compared to Ibupas®, while in the case of H-E_IB it was slower.

Aiming to deepen the information on the biopharmaceutical performances, in vitro permeation studies by using Franz cells and pig ear skin as membrane were performed on the patches loaded with IB. Drug permeability from the commercial reference product (Ibupas®) was also evaluated.

The permeated amount after 24 h (Q 24h ) and flux (J) of IB from the formulated patches through the pig ear skin are summarized in Table 5.

The amount of IB permeated was influenced by polymer molecular weight, but not by the tackifier (Fig. 6): in particular, the lower the polymer molecular weight, the higher the IB permeated amount. Surprisingly, the IB diffusion through the skin from the formulations L-R_IB and L-E_IB was not limited by the polymeric matrix, if compared to the commercial reference product (Fig. 6). These data well correlated with the results on the drug release from the matrix.

Conclusions

The tested formulations satisfied the requirements for designing a PSA suitable for the development of transdermal patches, since they exhibited adhesive properties appropriate for an in vivo use and showed satisfactory biopharmaceutical properties. In particular, SEBSlow molecular weight is the polymer most worthy of consideration to design transdermal patches because of its favorable viscoelastic behavior and its capability to not limit in vitro IB permeability, if compared to the commercial reference product. The resins used to confer suitable tackiness to the copolymer did not significantly influence the rheological behavior and adhesive properties of placebo SEBS-matrices. On the contrary, concerning the drug loaded patches, not only the polymer and the resin molecular weight, but also the loaded drug affected the overall technological and in vitro biopharmaceutical properties of the final patch. A plasticizing effect exerted by the drug was evident considering the adhesive properties of the matrices, and it was more evident with formulations based on SEBS-M and SEBS-H (in other words the more elastic materials). However, no sign of cold flow was noticed, confirming the optimal cohesive properties of all these materials.

Finally, the effective reservoir function of the nanostructured styrenic domains of SEBS was confirmed. Indeed, 10% w/w of IB was loaded in these patches, avoiding the drug crystallization, without the addition of auxiliary agents.

Table 5

Drug content and main parameters calculated for in vitro drug release and in vitro skin permeation experiments: constant drug release rate (k), steady flux (J) and cumulative drug amount permeated at the end of the experiments (Q 24h ).

Form. Code

Drug content (µg/cm 2 ) k (h 0.5 ) J (µg/cm 2 h) Q 24h (µg/cm

Fig. 1 .

 1 Fig. 1. Formulations containing (a) Regalite™ R1000 and (b) Eastotac™ H100W tested at 25 °C and 10 µm/s. Red line: formulation based on SEBS-L, green line: formulation based on SEBS-M, blue line: formulation based on SEBS-H. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3 .

 3 Fig. 3. Effect of SEBS-molecular weight on the stress-lambda curves in the nonlinear tensile experiments. Formulations were tested at 25 °C and 5 mm/s as cross-head velocity. (a: formulation L-R, b: formulation M-R, c: formulation H-R).

Fig. 4 .

 4 Fig. 4. Evolution of G′ and G″ with frequency. Tests were performed at 25 °C (red line: L-R, green line: M-R, blue line: H-R). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5 .

 5 Fig. 5. TEM of L-R and L-R_IB patches.

Fig. 6 .

 6 Fig.6. In vitro permeation profiles of IB through ear-pig skin from the formulated transdermal patches and from the commercial reference product (Ibupas®).

Table 1

 1 Placebo and drug-loaded formulation compositions (%, w/w).

	Form. Code	SEBS-L	SEBS-M	SEBS-H	Regalite™ H1100	Eastotac™ H100W	Paraffin oil	IB
	L-R	10	-	-	40	-	50	-
	L-E	10	-	-	-	40	50	-
	M-R	-	10	-	40	-	50	-
	M-E	-	10	-	-	40	50	-
	H-R	-	-	10	40	-	50	-
	H-E	-	-	10	-	40	50	-
	L-R_IB	9	-	-	36	-	45	10
	L-E_IB	9	-	-	-	36	45	10
	M-R_IB	-	9	-	36	-	45	10
	M-E_IB	-	9	-	-	36	45	10
	H-R_IB	-	-	9	3 6	-	45	10
	H-E_IB	-	-	9	-	36	45	10

Table 2

 2 Adhesive properties of placebo patches: shear adhesion, peel adhesion and tack (ball no.) evaluated by inclined rolling ball test.

	Form. Code	Shear adhesion (min)	Peel adhesion (cN/cm)	Ball No.
	L-R	61.7 ± 14.6	37.8 ± 9.3	5
	L-E	57.6 ± 5.2	38.0 ± 9.0	5
	M-R	57.2 ± 4.1	26.4 ± 2.6	8
	M-E	38.4 ± 24.1	25.8 ± 3.3	8
	H-R	101.3 ± 8.2	48.0 ± 9.7	12
	H-E	277.5 ± 41.2	48.6 ± 10.3	12

Table 3

 3 Elastic modulus (G′) of placebo formulations as a function of frequency.

	Form	G′ (kPa)
		0.05 rad/s	6.28 rad/s	100 rad/s
	L-R	2.71	8.03	11.58
	L-E	7.93	11.54	15.18
	M-R	6.39	7.15	9.28
	M-E	6.16	6.79	9.05
	H-R	6.07	6.72	5.21

Table 4

 4 Adhesive properties of drug-loaded patches: shear adhesion, peel adhesion and tack (ball no.) evaluated by inclined rolling ball test.

	Form. Code	Shear adhesion (min)	Peel adhesion (cN/cm)	Ball No.
	L-R_IB	20.8 ± 0.5	75.8 ± 9.6	12
	L-E_IB	30.3 ± 5.3	76.0 ± 11.7	10
	M-R_IB	5.6 ± 1.0	42.3 ± 2.9	12
	1M-E_IB	5.9 ± 2.3	45.0 ± 3.8	10
	H-R_IB	5.5 ± 0.1	36.4 ± 10.0	15
	H-E_IB	5.8 ± 0.2	37.0 ± 10.1	15

Thus, the overall data suggested that SEBS copolymers are suitable materials to prepare drug in-adhesive patches by the casting technique.

CRediT authorship contribution statement

Chiara Grazia Milena Gennari: Conceptualization, Investigation, Formal analysis, Writing -original draft, Visualization. Gaia Maria Giorgia Quaroni: Conceptualization, Investigation, Formal analysis, Writing -original draft, Visualization. Costantino Creton: Formal analysis, Writing -review & editing, Supervision. Paola Minghetti: Conceptualization, Supervision. Francesco Cilurzo: Conceptualization, Methodology, Formal analysis, Writing -review & editing, Visualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Nelly Barbot
email: nelly.barbot@irisa.fr

Laurent Miclet
email: laurent.miclet@gmail.com

Henri Prade
email: prade@irit.fr

Analogy Between Concepts (Extended Abstract) *

Keywords:

Analogical proportions are statements of the form "x is to y as z is to t", where x, y, z, t are items of the same nature, or not. In this paper, we more particularly consider "relational proportions" of the form "object A has the same relationship with attribute a as object B with attribute b". We provide a formal definition for relational proportions, and investigate how they can be extracted from a formal context, in the setting of formal concept analysis.

Introduction

Analogical reasoning has raised the interest of psychologists and computer scientists for a long time; see, e.g., [Dedre Gentner and Kokinov, 2001;[START_REF] Prade | Computational Approaches to Analogical Reasoning: Current Trends[END_REF]. We focus here on analogical proportions which are statements of the form "x is to y as z is to t", expressing an analogical parallel between pairs (x, y) and (z, t). A statement such as "Carlsen is to chess as Mozart is to music" introduces Carlsen as a precocious virtuoso of chess, a quality that Mozart is well known to have concerning music. It relates two types of items, here people and activities. It is an example of what we call relational proportions which are statements of the form "object A has the same relationship with attribute a as object B with attribute b". This can be viewed as a special case of analogical proportions. In case where x, y, z, t are items which can be represented in terms of the same set of features, a formal definition has been proposed for analogical proportions in the setting of Boolean logic and then extended using multiple-valued logic for handling numerical features [START_REF] Miclet | Handling analogical proportions in classical logic and fuzzy logics settings[END_REF][START_REF] Prade | From analogical proportion to logical proportions[END_REF][START_REF] Dubois | [END_REF], by stating that "x differs from y as z differs from t and y differs from x as t differs from z". The nature of Relational Proportions (RP for short) suggests to handle them in the setting of formal concept analysis. This leads us to the question of defining analogical proportions between formal concepts. For more details and proofs, the reader is referred to [Barbot et al., 2019].

Definition 1 (Axioms of Analogical Proportion). An analogical proportion (AP) on a set X is a quaternary relation on X, i.e., a subset of X 4 . An element of this subset, written (x : y :: z : t), which reads "x is to y as z is to t", must obey the following axioms (studied in [Lepage, 2003]): 1. Reflexivity of 'as': (x : y :: x : y) 2. Symmetry of 'as': (x : y :: z : t) ⇔ (z : t :: x : y) 3. Exchange of means: (x : y :: z : t) ⇔ (x : z :: y : t) Then, thanks to (2) and (3), it can be easily seen that (x : y :: z : t) ⇔ (t : y :: z : x) should also hold (exchange of the extremes). According to the first two axioms, four other formulations are equivalent to the canonical form (x : y :: z : t). Finally, the eight equivalent forms of an analogical proportion are: (x : y :: z : t), (z : t :: x : y), (y : x :: t : z), (t : z :: y : x), (z : x :: t : y), (t : y :: z : x), (x : z :: y : t) and (y : t :: x : z). A fourth (optional) axiom, called determinism, insists on the uniqueness of the solution t = y of the equation in t: (x : y :: x : t).

With respect to this axiomatic definition of AP, [START_REF] Stroppa | Nicolas Stroppa and Franc ¸ois Yvon. Du quatrième de proportion comme principe inductif : une proposition et son application à l'apprentissage de la morphologie[END_REF] have given another definition, based on the notion of factorization when the set of objects is a commutative semigroup. From these previous works, [START_REF] Miclet | [END_REF] have derived the following definitions in the lattice framework.

Definition 2. A 4-tuple (x, y, z, t) of a lattice (L, ∨, ∧, ≤) 4 is a Factorial Analogical Proportion (FAP) (x : y :: z : t) iff x = (x ∧ y) ∨ (x ∧ z) x = (x ∨ y) ∧ (x ∨ z) y = (x ∧ y) ∨ (y ∧ t) y = (x ∨ y) ∧ (y ∨ t) z = (z ∧ t) ∨ (x ∧ z) z = (z ∨ t) ∧ (x ∨ z) t = (z ∧ t) ∨ (y ∧ t) t = (z ∨ t) ∧ (y ∨ t) Definition 3. A 4-tuple (x, y, z, t) of (L, ∨, ∧, ≤) 4 is a Weak Analogical Proportion (WAP) when x ∧ t = y ∧ z and x ∨ t = y ∨ z. It is denoted x : y WAP :: z : t. a 1 a 2 a 3 a 4 a 5 o 1 × × o 2 × × o 3 × × o 4 × × × Figure 1: The formal context R.
In the case of a distributive lattice (e.g., a Boolean lattice), this alternative definition is equivalent to the FAP. But, in general, a FAP is a WAP and the converse is false, which explains the use of adjective "weak" [START_REF] Miclet | [END_REF]. Example 1. Let Σ be a finite set associated with the Boolean lattice (2 Σ , ∪, ∩, ¬, ⊆). When saying that "x is to y as z is to t" where x, y, z, t ⊆ Σ, we express that x differs from y in the same way as z differs from t. For example, if x = {a, b, e} and y = {b, c, e}, we see that to transform x into y, we have to remove a and add c. Now, if z = {a, d, e}, we can construct t with the same operations, to get t = {c, d, e}. More formally, with this view, we should have x \ y = z \ t and y \ x = t \ z (with x \ y = x ∩ ¬y). This is equivalent to x ∩ t = y ∩ z and x ∪ t = y ∪ z. This relation linking x, y, z, t is clearly symmetrical, and satisfies the exchange of the means. Hence it is a correct definition of the AP in the Boolean setting [START_REF] Miclet | Handling analogical proportions in classical logic and fuzzy logics settings[END_REF].

We give here a simple example of FAP in a lattice. Proposition 1. Let y and z be two elements of a lattice, the proportion y : y ∨ z :: y ∧ z : z is a FAP. We call it a Canonical Analogical Proportion (CAP).

Analogical Proportions in FCA

In order to derive more specifically the AP notion in a Formal Concept Analysis (FCA) framework, we first recall some basic elements of FCA, before studying the relations between several kinds of AP and their characterization in FCA.

FCA starts with a binary relation R defined between a set O of objects and a set A of attributes (or properties). The tuple The set of all formal concepts is equipped with a partial order (denoted ≤) defined as:

(O, A, R) is called a formal context.
(o 1 , a 1 ) ≤ (o 2 , a 2 ) iff o 1 ⊆ o 2 (or, equivalently, a 2 ⊆ a 1 ). Then it is structured as a lattice, called the concept lattice of R. Example 2. The concept lattice of context R displayed in Figure 1 is shown in Figure 2.
Since a concept x is associated to a set of objects o x and a set of attributes a x , the objective of this section is to relate the AP definitions with these sets, and to study the links between APs in a concept lattice and APs on object or attribute sets. Proposition 2. Let x, y, z and t be 4 concepts, one has:

(x : y WAP :: z : t) iff a x ∩a t = a y ∩a z and o x ∩o t = o y ∩o z . ∅ {a 1 , a 2 , a 3 , a 4 , a 5 } {o 1 } {a 3 , a 4 } {o 2 } {a 1 , a 3 } {o 3 } {a 2 , a 4 } {o 4 } {a 1 , a 2 , a 5 } {o 1 , o 2 } {a 3 } {o 1 , o 3 } {a 4 } {o 2 , o 4 } {a 1 } {o 3 , o 4 } {a 2 } {o 1 , o 2 , o 3 , o 4 } ∅ x y z t z t x y ⊥ Figure 2:
The formal concept lattice of R. We give now a proposition which leads us to a corollary in which is defined yet another analogical proportion between formal concepts, the strongest of all. Proposition 4. Let x, y, z and t be 4 concepts, if (a x ∪ a t = a y ∪ a z and o x ∪ o t = o y ∪ o z ) then the FAP x : y :: z : t holds. Corollary 1. Let x, y, z and t be 4 concepts, the following 2 conjunctions are equivalent:

a x ∪ a t =
⊥ : z (which is a CAP), but o y ∪ o z = o ∪ o ⊥ and a y ∪ a z = a ∪ a ⊥ .
4 Formal Concepts and Relational Proportion

From a RP to Concepts in AP

We study here if we can deduce from a relational proportion "A is the B of a", or "A is to a as B is to b", formal concepts in WAP and an analogical complex from this knowledge.

As an example, we have found in a web magazine the following proportion "Massimiliano Alajmo is the Mozart of Italian cooking". The background knowledge allowing to understand this relational proportion is the following: music and Italian cooking are disciplines practiced by humans,

a 1 a 2 a 3 o 1 × × o 2 × × a 1 a 2 a 3 a 4 o 1 × × o 2 × × o 3 × × a 4 × × Figure 3: Constructing a formal context from a RP
such disciplines can be practiced with different levels of ability, Mozart is a musician and Mozart is a genius in music discipline. Since the quality "to be a genius" is not possessed by everybody, there must exist many "ordinary gifted" musicians. Then, the background knowledge can be expressed by the formal context on the left side of Figure 3. where o 1 stands for Mozart, o 2 for one of "ordinary gifted" musicians, a 1 is the attribute "practices music", a 2 "is a genius" and a 3 "has an ordinary ability". Now, when the new data "Alajmo is the Mozart of Italian cooking" is introduced, the knowledge extends as follows: Alajmo practices Italian cooking, and he has something in common with Mozart that is not Italian cooking. The relational proportion is a reduced form of "Alajmo is to Italian cooking as Mozart is to music". Since Mozart has only the other attribute "Genius", Alajmo must have it. Moreover, since cooking is a discipline practiced by humans, there must exist some ordinary gifted Italian cook. At last, we must introduce the notion of non-genius in our universe. If we do not, we implicitly suppose that everybody is a genius for some activity. The knowledge is now expressed by the formal context on the right side of Figure 3 where o 3 stands for Alajmo, o 4 an ordinary gifted Italian cook and a 4 Italian cooking. This context is called the analogical context. Considering the associated concept lattice, the closest analogical proportion to "Alajmo is the Mozart of Italian cooking" is ({o 3 }, {a 2 , a 4 }) : ({o 4 }, {a 3 , a 4 }) WAP :: ({o 1 }, {a 1 , a 2 }) : ({o 2 }, {a 1 , a 3 }) which translates into "Mozart is to some ordinary musician as Alajmo is to some ordinary cook".

More formally, from the relational proportion "o 1 is the o 2 of a", we can derive an analogical context as above. It is composed of objects o 1 and o 2 , described by four attributes: a is possessed by o 1 and not by o 2 , ã is possessed by o 2 and not by o 1 , b is possessed both by o 1 and o 2 and b is some attribute not possessed by o 1 nor o 2 . Secondly we complete the context with two objects o 3 and o 4 that are the complements of o 2 and o 1 with respect to the four attributes. The result is the analogical context where a 1 = b, a 2 = a, a 3 = ã and a 4 = b.

Analogical Complex

In the previous subsection, it turns out that the analogical context is an interesting pattern, from which we can construct relational proportions. A more general definition of this pattern, named analogical complex, has been given in [START_REF] Miclet | From formal concepts to analogical complexes[END_REF]. We present it here through an example.

Let us consider the context in Fig. 4, called SmallZoo, extracted from the Zoo data base [Lichman, 2013]. The context of Fig. 5 is a subcontext of SmallZoo with special characteristics described in Fig. 6. Any subcontext with such characteristics is called 'analogical complex'. In the left part of SmallZoo hair feathers eggs milk airborne aquatic predator toothed Fig. 6, a '×' between a subset of objects (say, o 1 ) and a subset of attributes (say, a 4 ) means that every object of o 1 is in relation with every attribute of a 4 , and a 'blank' between a subset of objects (say, o 1 ) and a subset of attributes (say, a 1 ) means that no object of o 1 is in relation with no attribute of a 1 . In practice, the right table is sufficient to define an analogical complex, since the left one is the same for all of them. Thus, an analogical complex is defined by the 8 sets o 1 , ..., o 4 , a 1 , ..., a 4 , and it is said to be complete when none of these sets are empty. From the analogical complex structure, we derive a formal definition of a relational proportion.

a 0 a 1 a 2 a 3 a 4 a 5 a 6 a 7 o 0 aardvark × × × × o 1 chicken × × × o 2 crow × × × × o 3 dolphin × × × × o 4 duck × × × × o 5 fruitbat × × × × o 6 kiwi × × × o 7 mink × × × × × o 8 penguin × × × × o 9 platypus × × × × × Figure 4: The context SmallZoo a 1 a 2 a 3 a 4 a 5 a 0 a 3 a 7 a 1 a 2 a 4 o 1 o 1 × × × o 2 × × × o 2 o 5 × × × × o 3 o 8 × × × o 4 o 7 × × × ×
Definition 4. Let (o 1,4 , a 1,4 ) be a complete analogical complex in a formal context, the following sets are said to be in the formal relational proportion (RP) (o 1 is to a 3 as o 2 is to a 2 ), and we write (o 1 ˜a3 ˜˜o 2 ˜a2 ).

Comments. The reduced form of the RP would be (o 1 is the o 2 of a 3 ). Besides, from the same complex, we can extract the RPs o 1 ˜a4 ˜˜o 3 ˜a1 , o 2 ˜a4 ˜˜o 4 ˜a1 and o 3 ˜a3 ˜˜o 4 ˜a2 . Since the operator ˜˜is commutative, it gives a total of 8, but permuting the extremes and the means in a RP may lead to awkward phrasings. Example 3. The complex described in Fig. 6 implies all attributes but a 6 (predator) and objects o 1 and o 2 (chicken and crow), o 5 (fruitbat), o 8 (penguin) and o 7 (mink). From this context, the RP in reduced form "a fruitbat is the mink of airborne animals" can be derived for instance, meaning that fruitbat and mink have hair, are toothed and produce milk, but that the mink is aquatic at the contrary of the fruitbat. Of course, the interest of such phrases has to be taken in context: the SmallZoo data base is supposed to be the only knowledge.

a 1 a 2 a 3 a 4 o 1 × × o 2 × × o 3 × × o 4 × × o 1,4 a 1,4 {o 1 , o 2 } {a 5 } {o 5 } {a 0 , a 3 , a 7 } {o 8 } {a 1 , a 2 } {o 7 } {a 4 }

WAP and Analogical Complex

We explore now the links between WAP between concepts and complete analogical complex, and then the formal RPs. First, we are interested in defining non degenerated WAPs, called complete and bi-complete, forbidding inclusion between two of its concepts. It is a key notion for building WAP between concepts with a sound cognitive interpretation.

Definition 5. Let us consider (x : y WAP :: z : t), this WAP is complete through attributes when (a x ∩ a y ) \ a ∩ , (a x ∩ a z ) \ a ∩ , (a y ∩ a t ) \ a ∩ and (a z ∩ a t ) \ a ∩ are nonempty where a ∩ = a x ∩ a y ∩ a z ∩ a t . A WAP complete through objects is similarly defined using subsets of objects.

A WAP is bi-complete when

:: (y ∧ t) : (z ∧ t)) is complete through attributes.
In order to derive RPs from an AP between concepts, we consider a complete WAP through attributes (a similar reasoning can be done from a complete WAP through objects) and introduce a process to extract an analogical complex.

Due to the completeness, sets a 1 = (a z ∩ a t ) \ a ∩ , a 2 = (a y ∩a t )\a ∩ , a 3 = (a x ∩a z )\a ∩ and a 4 = (a x ∩a y )\a ∩ are nonempty. We also define o 1 = o x the set of objects proper to x (that appear in o x but not in the objects of y, z and t) and similarly o 2 = o y , o 3 = o z and o 4 = o t .

By construction, every o ∈ o 1 is in relation with every a ∈ a 3 ∪ a 4 . It is also the case between o 2 and a 2 ∪ a 4 , o 3 and a 1 ∪ a 3 , o 4 and a 1 ∪ a 2 . For all the other combinations, for instance o 1 and a 1 , for any o ∈ o 1 , there exists a ∈ a 1 such that o / ∈ a ↓ . However, these properties do not guarantee that the subcontext (o 1,4 , a 1,4 ) is an analogical schema, even if it is a closed schema: it can exist an object o ∈ o i in relation with an attribute a ∈ a j , where (i, j) ∈ {(1, 1), (1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 4)}. In such a case, either a or o is removed and this postprocessing permits to obtain an analogical schema. But this schema is not necessarily a complex, since the associated subcontext may be not maximal. Then a second postprocessing maximises the schema into complex, adding new attributes and/or objects chosen among those which do not appear in a

x ∪ . . . ∪ a t nor o x ∪ . . . ∪ o t .
This method may lead to several complexes, according to the choices in both postprocessings. This set of complexes is a sub-lattice of the lattice of complexes and in case of a bicomplete WAP as input, one of the output lattices is entirely made of complete analogical complexes [Barbot et al., 2019]. After removing a 2 , the table in Fig. 8 is an analogical schema and we can check that it is maximal in SmallZoo. Note that if we had chosen to remove o 9 , the postprocessings would have produced the analogical complex detailed in Fig. 5. For example, from the complete analogical complex described above, we can derive the following RP: "the chicken and the crow are to the feathers as the fruitbat is to the hair, the milk and the teeth". It makes sense when considering that all these animals share the attribute "airborne".

a 5 a 0 a 3 a 1 a 2 a 4 o 1 × × × o 2 × × × o 5 × × × o 8 × × × o 7 × × × o 9 × × × × Figure 7: Subcontext of SmallZoo (example 4) a 1 a 2 a 3 a 4 a 5 a 0 a 3 a 1 a 4 o 1 o 1 × × o 2 × × o 2 o 5 × × × o 3 o 8 × × o 4 o 7 × × × o 9 × × ×

Conclusion

We have shown how relational proportions can be identified in a formal context. Relational proportions offer a basis for concise forms of explanations. Indeed, if B has some wellknown features, the proportion "object A is to attribute a as object B is to attribute b" provides an argument for stating that "object A is the B of a", when A possesses these well-known features also, as in "Carlsen is the Mozart of chess". It is worth pointing out that two cognitive capabilities, namely conceptual categorization and analogical reasoning can be handled together in the setting of formal concept analysis. This short presentation has left aside the algorithmic side (based on the identification of formal complexes), which is discussed in the long version of [Barbot et al., 2019].

  The notation (o, a) ∈ R or oRa means that object o has attribute a. We denote o ↑ = {a ∈ A | (o, a) ∈ R} the attribute set of object o and a ↓ = {o ∈ O | (o, a) ∈ R} the object set having attribute a. Similarly, for any subset o of objects, o ↑ is defined as {a ∈ A | a ↓ ⊇ o}. Then a formal concept is defined as a pair (o, a), such that a ↓ = o and o ↑ = a. One calls o the extension of the concept and a its intension.

Figure 5 :

 5 Figure 5: An analogical complex extracted from SmallZoo

Figure 6 :

 6 Figure 6: Notation of an analogical complex

Figure 8 :

 8 Figure 8: Analogical schema derived from Fig. 7

  Comments. The converse is false. Let us consider the formal context R in Fig.2. We have x : y : t due to Proposition 2. However, the Boolean APs a x : a y :: a z : a t and o x : o y :: o z : o t are both false. The WAP between concepts is less restrictive than the AP between sets of attributes: in a WAP, objects are allowed to possess attributes which are not shared by any other object concerned in the WAP. In this lattice, x, y, z and t are in WAP, but a x : a y :: a z : a t and o x : o y :: o z : o t are both false. Besides, x , y , z and t are in WAP and o

Proposition 3. Let x, y, z and t be 4 concepts, if (a x : a y :: a z : a t or o x : o y :: o z : o t ) then x : y WAP :: z : t. WAP :: z x : o y :: o z : o t is true, but a x : a y :: a z : a t and the FAP x : y :: z : t are both false.

  a y ∪ a z and o x ∪ o t = o y ∪ o z a Comments. From Corollary 1, a x : a y :: a z : a t and o x : o y :: o z : o t imply the FAP x : y :: z : t. However, the reciprocal is false. Let us consider the concept lattice displayed in Figure2: we have the FAP y :::

	This characterizes a particular case of FAP between concepts
	that we call a Strong Analogical Proportion (SAP). It is de-
	noted x : y

x : a y :: a z : a t and o x : o y :: o z : o t SAP :: z : t.

  • if it is complete through attributes and ((x ∨ y) : (x ∨ is complete through objects.• or if it is complete through objects and ((x ∧ y) : (x ∧

	z)
	z)	WAP

WAP

:: (y ∨ t) : (z ∨ t))

Acknowledgements

The authors acknowledge a partial support of ANR-11-LABX-0040-CIMI (project ISIPA).