Frederic Tewes

Carsten Ehrhardt

Anne Marie Healy
email: healyam@tcd.ie

Superparamagnetic iron oxide nanoparticles (SPIONs

Introduction

Pharmacotherapy of lung diseases often involves direct delivery of active pharmaceutical ingredients (APIs) by pulmonary inhalation. However, despite the progress in aerosol delivery to the lung, administration systems are still unable to effectively deliver the dose to the optimal areas of deposition within the respiratory tract. Optimising the deposition pattern of API within the most suitable part of the airways should increase the efficacy of the treatment and reduce side effects. This should be beneficial for treating localised lung diseases, such as respiratory infection and lung cancer, i.e. by targeting foci of bacterial infection or tumour nodules. Superparamagnetic iron oxide nanoparticles (SPIONs), such as nanoparticles of maghemite (-Fe 2 O 3 ) or magnetite (Fe 3 O 4 ) offer attractive magnetic properties. However, SPIONs dispersions are unstable at physiological pH and surface modifications are required to increase their stability in aqueous media. SPION coated dispersions are one of the few FDA approved nanoparticles for use as MRI contrast agents [START_REF] Yildirimer | Toxicological considerations of clinically applicable nanoparticles[END_REF]. Lately, SPIONs were also envisaged as sensitive devices for magnetic drug targeting after intravenous nanoparticles administration, in combination with a target-directed magnetic gradient field [START_REF] Chourpa | Molecular composition of iron oxide nanoparticles, precursors for magnetic drug targeting, as characterized by confocal Raman microspectroscopy[END_REF][START_REF] Munnier | On the interaction of doxorubicin with oleate ions: Fluorescence spectroscopy and liquid-liquid extraction study[END_REF][START_REF] Pouponneau | Co-encapsulation of magnetic nanoparticles and doxorubicin into biodegradable microcarriers for deep tissue targeting by vascular MRI navigation[END_REF][START_REF] Sapet | Magnetofection: Magnetically assisted & targeted nucleic acids delivery[END_REF][START_REF] Coates | Guiding Aerosol Deposition in the Lung[END_REF]. This concept has also been applied in an attempt to target dry nanoparticle aerosols within the lung using magnetically driven deposition [START_REF] Ally | Magnetic targeting of aerosol particles for cancer therapy[END_REF][START_REF] Upadhyay | Magnetised thermo responsive lipid vehicles for targeted and controlled lung drug delivery[END_REF]. However, it has been demonstrated that, even with optimised magnet design, the resulting magnetic forces would not be sufficient to efficiently guide individual SPIONs in a dry-powder aerosol because of their small magnetic moment [START_REF] Dames | Targeted delivery of magnetic aerosol droplets to the lung[END_REF]. In contrast, when a multitude of SPIONs are assembled in an liquid aerosol droplet as a nanomagnetosol, the magnetic moment of the assembly increased, which resulted in aerosols which were guidable by medically compatible magnetic fields [START_REF] Dames | Targeted delivery of magnetic aerosol droplets to the lung[END_REF]. Also, recently, inert SPIONs added to the nebuliser solution were used to guide the aerosol to the affected region of the lung by means of a strong external magnetic field. Various therapeutic agents have been administrated by this technique [START_REF] Laurent | Nucleic acid delivery using magnetic nanoparticles: The Magnetofection™ technology[END_REF].

However, the small size of nanoparticles can lead to particle-particle aggregation, making their physical handling difficult in liquid and dry powder forms [START_REF] Tsapis | Trojan particles: Large porous carriers of nanoparticles for drug delivery[END_REF]. The delivery of API to the lung may be achieved by using dry powder inhalers (DPIs), metered dose inhalers or nebulisers. In the solid form, as in a DPI, pharmaceutical ingredients are usual more stable than in liquid form. Therefore, with the same concept of increasing the magnetic moment, targeted aerosol delivery to a specific area of the lung could also be achieved with Trojan microparticles loaded with SPIONs. Trojan particles are microparticles composed of nanoparticles and additional components used to maintain the nanoparticles together. Once in the body, the microparticles disaggregate and release the nanoparticles. Trojan particles were previously formulated and reported as being an efficient means of administering nanoparticles to the lung by inhalation [START_REF] Tsapis | Trojan particles: Large porous carriers of nanoparticles for drug delivery[END_REF].

These nano-in-microparticle systems should allow for higher aerosolisation and delivery efficiency than nanoparticles and permit the focalization of the drug reservoir in the targeted area.

To reach the deep lung alveolar region, particles require a 1-5 µm aerodynamic diameter range. This aerodynamic diameter range corresponds to spherical particles of unit density having a 1 -5 µm geometric diameter range. Iron oxide density is 4.9 g/cm 3 and 5.2 g/cm 3 for maghemite and magnetite, respectively. Therefore, the best way to nebulise microparticles composed of iron oxide is to develop porous hybrid particles with a reduced density. In fact, to optimise the efficacy of dry powder inhalation, porous particles with low apparent density were developed [START_REF] Edwards | Large porous particles for pulmonary drug delivery[END_REF]. For instance, tobramycin powder was produced using the emulsion-based PulmoSphere technology, producing highly dispersible porous particles [START_REF] Geller | Development of an inhaled dry-powder formulation of tobramycin using PulmoSphere technology[END_REF]. Excipient-free nanoporous microparticles (NPMPs) [START_REF] Healy | Characterisation of excipientfree nanoporous microparticles (NPMPs) of bendroflumethiazide[END_REF][START_REF] Nolan | Excipient-free nanoporous microparticles of budesonide for pulmonary delivery[END_REF][START_REF] Nolan | Particle engineering of materials for oral inhalation by dry powder inhalers. II -Sodium cromoglicate[END_REF][START_REF] Tewes | Steroid/mucokinetic hybrid nanoporous microparticles for pulmonary drug delivery[END_REF] prepared by a spray drying process had improved in vitro deposition properties compare to non-porous microparticles. Trojan large porous particle composed of polymeric nanoparticles were also formulated by spray drying and exhibited much better flow and aerosolisation properties than the nanoparticles from which they were prepared [START_REF] Tsapis | Trojan particles: Large porous carriers of nanoparticles for drug delivery[END_REF]. Therefore, the aim of this study was to formulate SPIONs-loaded Trojan porous microparticles for aerosol lung magnetic targeting after administration using a dry powder inhaler.

1-Materials and methods 2.1 Materials

Hydroxypropyl-β-cyclodextrin (HPCD) with an average degree of substitution of 0.65 (Encapsin™ HPB) was purchased from Janssen Biotech, Olen, Belgium. Linear PEG 10kDa (PEG), alkanes with 99.8% purity (hexane, heptane, octane, nonane, decane and undecane), maghemite (Fe 2 O 3 ) nanoparticles (SPIONs) with a mean diameter of 50 nm, magnesium stearate (MgST) and ammonium carbonate ((NH 4 ) 2 CO 3 ) were all purchased from Sigma-Aldrich, (Dublin, Ireland).

Methods

Spray drying

Various suspensions containing SPIONs were spray dried using a B-290 Mini spray dryer (Büchi, Flawil, Switzerland) set in the closed cycle mode with a 2-fluid nozzle. The liquid phase of the suspensions was composed of butyl acetate/methanol/water mixture with a volume ratio 5:5:1., as used previously [START_REF] Tewes | Evaluation of HPβCD-PEG microparticles for salmon calcitonin administration via pulmonary delivery[END_REF]. The composition of the suspensions is described in Table 1. The spray dryer was operated as follows: Inlet temperature was 65 o C; feeding pump was set at 30%; spraying N 2 nozzle flow rate was 15 L/min; N 2 flowing at 670 NL/h was used as the drying gas. These conditions resulted in an outlet temperature ranging from 36 to 39°C.

25P- 75H- 5F 25P- 75H- 30F 50P- 50H- 50F 50P- 50H- 50F- CO 3 100P- 50F 100P- 50F- CO 3 100H- 50F 50P- 50H- 50F-ST 50P- 50H- 50F- CO 3 -ST

Scanning electron microscopy (SEM)

SEM micrographs of samples were taken using a Tescan Mira XMU (Tescan s.r.o., Czech Republic) electron microscope. The samples were fixed on aluminium stubs and coated with a 10 nm-thick gold film. Primary electrons were accelerated under a voltage of 5 kV. Images were formed from the collection of secondary electrons.

Powder X-ray diffraction (XRD)

XRD measurements were conducted on samples placed in a low background silicon holder, using a Rigaku Miniflex II desktop X-ray diffractometer (Rigaku, Tokyo, Japan).

The samples were scanned over a range of 5 -40º 2θ at a step size of 0.05º/s as previously described [START_REF] Tewes | Development and characterisation of soluble polymeric particles for pulmonary peptide delivery[END_REF].

Particle size distribution analysis

The geometric particle size distributions (PSD) were determined by laser diffraction using a Mastersizer 2000 (Malvern Instruments, Worcestershire, UK) with the Scirocco 2000 dry powder feeder to disperse the particles as described previously [START_REF] Tewes | Development and characterisation of soluble polymeric particles for pulmonary peptide delivery[END_REF]. The dispersive air pressure used was 3 bar and vibration feed rate was set to 50%. Data were analysed based on the equivalent volume median diameter, D 50 , and the span of the PSD.

Calculation was performed using Mie theory and refractive index part of 2 and absorption part of 1 as optical particle properties (n=2).

Particle true density

The true density of the materials was measured using an Accupyc 1330 Pycnometer (Micromeritics®) with helium (99.995% purity) to determine the volume of accurately weighed samples. Samples were dried prior to measurement for 24 h in a Gallenkamp vacuum oven operating at 600 mbar and 25°C (n=2).

Surface free energy measurement

Measurements were performed at 0% RH or 40% RH and 30 • C, (n = 3) using an inverse gas chromatography (iGC) instrument (SMS Ltd., London, UK). Powders were packed into a silanized glass column (300mm x 3mm), and then pre-treated for 1 h at 30 • C and 0% RH. Then, 250 L of the probe vapour-helium mixture was injected into the helium flow. All injections of probe vapours were performed at 0.03% v/v of the saturated probe vapour. A flame ionization detector was used to monitor the probes' elution. In acid-base theory, the total surface free energy of a solid (γ s T ) has 2 main components: a dispersive contribution (γ s d ) and specific or acid-base contribution (γ s AB ) which are independent and additive. In order to calculate γ s d of the particles, alkane probes with a known dispersive contribution (γ p d ) and a nil specific contribution (γ p AB ) were used. Methane was used as inert reference. At this low % of saturation (0.03% v/v), iGC was used in infinite dilution conditions and γ s d was calculated using the method developed by Schultz et al. [START_REF] Schultz | The role of the interface in carbon-fiber epoxy composites[END_REF].

Aerodynamic particle diameter analysis

The aerodynamic diameter (AD) distribution of the particles was measured using a Next Generation Impactor (NGI) as previously described [START_REF] Tewes | Development and characterisation of soluble polymeric particles for pulmonary peptide delivery[END_REF]. The flow rate was adjusted to get a pressure drop of 4 kPa in the powder inhaler (Handihaler ® , Boeringher Ingelheim, Ingelheim, Germany) and the time of aspiration was adjusted to obtain 4 L. The inhaler was filled with gelatin n o 3 capsule loaded with 20±2 mg of powder (n = 3). After inhaler actuation, particle deposition on the NGI was determined by the SPIONs assay as described below. The amount of particles recovered on each stage expressed as a percentage of the emitted recovered dose was considered as the fine particle fraction (FPF). The mass median aerodynamic diameter (MMAD) and FPF were calculated as previously described [START_REF] Tewes | Development and characterisation of soluble polymeric particles for pulmonary peptide delivery[END_REF]. Additional experiments were performed in the presence of a neodymium iron boron magnet (e-Magnets UK, Hertfordshire UK) of 20 mm of diameter and 20 mm of length having a maximum energy product (BH max ) of 413.8 kJ/m 3 placed on the bottom of the stage 2 of the NGI.

SPIONs concentration assay

SPIONs assay was performed by turbidity measurements at 510 nm. SPIONs were dispersed in 2% w/w poly (vinyl alcohol) (10kDa) solution containing 0.1M of NaOH using a sonicator bath. In the case of formulations containing MgST, particles were dispersed in a 1/1 (v/v) mixture of ethanol and an aqueous solution composed of 2% w/w PVA (10kDa) and 0.1M of NaOH. In these conditions, stable suspensions were obtained.

Calibration curves were constructed with standard suspensions composed of SPIONs dispersed in the same media as the formulations and having concentrations ranging from 0.001 to 0.05 mg/mL.

Statistical data analysis

Data were statistically evaluated by a two-way ANOVA using Excel ® software (Microsoft). Significance level was <0.05.

2-Results and discussion

Powder XRD patterns recorded for the powders made of PEG, HPCD and SPIONs (Fig. 1) had curved baselines and diffraction peaks at 30.3 and 35.6 2θ degrees corresponding to maghemite iron oxide [START_REF] Sotiriou | A novel platform for pulmonary and cardiovascular toxicological characterization of inhaled engineered nanomaterials[END_REF]. The intensity of the diffraction peaks increased with an increasing amount of SPIONs incorporated in the formulation. PEG residual crystallinity was also observed in formulations containing more than 33 wt% of PEG from the presence of the two major diffraction peaks at 19.2 and 23.3 2θ degrees [START_REF] Tewes | Evaluation of HPβCD-PEG microparticles for salmon calcitonin administration via pulmonary delivery[END_REF][START_REF] Tewes | Development and characterisation of soluble polymeric particles for pulmonary peptide delivery[END_REF]. The absence of diffraction peaks corresponding to the HPCD, suggested that this excipient was in the XRD amorphous state.

For spray dried systems comprising PEG, HPCD and SPIONs, SEM pictures showed individual spherical microparticles surrounded by SPIONs nanoparticles (Figure 2A-B-D). This morphology was obtained due to the high Peclet number of the SPIONs relative to the other excipients, resulting in accumulation on the surface of the microparticles [START_REF] Tsapis | Trojan particles: Large porous carriers of nanoparticles for drug delivery[END_REF][START_REF] Vehring | Pharmaceutical particle engineering via spray drying[END_REF]. Without PEG in the formulation, heterogeneous blends of free SPIONs and HPCD microparticles were obtained (data not shown). This failure to form the Trojan microparticles may be as a result of the van der Waals forces between the SPIONs accumulated on the microparticles being too low to enable them be retained on the surface [START_REF] Tsapis | Trojan particles: Large porous carriers of nanoparticles for drug delivery[END_REF] and could also be due to the weak adhesion between HPCD and SPIONs.

Volume weighted particle size distributions of these particles (100H-50F) showed a large proportion of nanoparticles (Fig. 3F). In the absence of HPCD, large particle aggregates were produced and the presence of SPIONs was not visible (Figure 2C). These aggregates were thought to result from the low and broad melting temperature of the PEG, leading to the formation in the spray dryer of partly solidified sticky particles [START_REF] Tewes | Evaluation of HPβCD-PEG microparticles for salmon calcitonin administration via pulmonary delivery[END_REF][START_REF] Tewes | Development and characterisation of soluble polymeric particles for pulmonary peptide delivery[END_REF]. The absence of visible SPIONs suggests that these soft particles embed SPIONs in their bulk, before complete solidification. These particles (100P-50F) had the highest geometric median diameter, as measured by laser diffraction, compared to the other formulations containing PEG and HPCD (Fig. 3 -curve A), resulting also in the lowest specific surface area (Table 2). Due to their large size, these particles were considered to be not suitable for pulmonary administration. These particles presented the lowest dispersive surface free energy (γ s d ) values (36 ± 1 mJ/m 2 ), similar to values found previously for PEG alone (37.7 ± 4 mJ/m 2 ) [START_REF] Tewes | Evaluation of HPβCD-PEG microparticles for salmon calcitonin administration via pulmonary delivery[END_REF], confirming that their surfaces were mainly composed of PEG molecules. The use of PEG allowed SPIONs to stick together to form the Trojan microparticles, but produced large aggregates in the absence of HPCD. Also, an appropriate ratio between PEG and HPCD was necessary to obtain isolated and spherical Trojan microparticles.

Particle deposition on the NGI impactor stages was dependent on the particle type. For SPIONs alone, only 30% of the nanoparticles emitted out of the gelatin capsule were deposited beyond the first impactor stage, which has a cut-off aerodynamic diameter of 8.9 µm (Fig. 5A). According to equation 1, these nanoparticles with a geometric diameter of 50 nm and having a true density of 4.9 g/cm 3 should have an aerodynamic diameter of 110 nm and deposit mainly on the filter stage of the impactor. The large deposition on stage 1 is presumably due to the agglomeration of the nanoparticles in non-inhalable large clusters. In fact, SPIONs had a high γ s d (147.5 ± 42 mJ/m 2 , Table 1), favouring their aggregation. The increase in PEG concentration in the formulation reduced γ s d (Table 2), decreasing the ability of the particles to aggregate. Besides preventing the particles from aggregating, the PEG should also play a role in preventing nonspecific and irreversible adsorption of foreign protein onto the particles, reducing opsonisation and particle phagocytosis.

In order to reach the pulmonary alveoli, particles must have an aerodynamic equivalent diameter (d a ) in the 1-5 µm range [START_REF] Edwards | Large porous particles for pulmonary drug delivery[END_REF]. The aerodynamic diameter is the diameter of a sphere of unit density, which reaches the same velocity in the air stream as the particle analysed, which can be nonspherical and have a different density [23,[START_REF] Decarlo | Particle Morphology and Density Characterization by Combined Mobility and Aerodynamic Diameter Measurements[END_REF]. It is linked to the volume-equivalent geometric diameter (d g ) by the particle shape and density, as described by the following equation [23,[START_REF] Decarlo | Particle Morphology and Density Characterization by Combined Mobility and Aerodynamic Diameter Measurements[END_REF]:

            . 0 p g a d d
Eq. 1

Where ρ 0 is the standard particle density (1g/cc), χ is the particle shape factor that is 1 for a sphere and ρ p is the apparent particle density. ρ p is equal to the mass of a particle divided by its apparent volume, i.e. the total volume of the particle, excluding open pores, but including closed pores [START_REF] Decarlo | Particle Morphology and Density Characterization by Combined Mobility and Aerodynamic Diameter Measurements[END_REF], which can be less than the material density (true density) if the particle is porous. Aerodynamic diameter decreases with increasing χ. For an irregular particle χ is always greater than 1 [START_REF] Decarlo | Particle Morphology and Density Characterization by Combined Mobility and Aerodynamic Diameter Measurements[END_REF]. Also, in order to decrease d a of large or dense particles below 5µm, several studies focused on enhancing the particle's porosity to increase χ and decrease ρ p [START_REF] Tsapis | Trojan particles: Large porous carriers of nanoparticles for drug delivery[END_REF][START_REF] Nolan | Excipient-free nanoporous microparticles of budesonide for pulmonary delivery[END_REF][START_REF] Nolan | Particle engineering of materials for oral inhalation by dry powder inhalers. II -Sodium cromoglicate[END_REF][START_REF] Tewes | Steroid/mucokinetic hybrid nanoporous microparticles for pulmonary drug delivery[END_REF].

To increase the porosity of the particles, ammonium carbonate (NH 4 ) 2 CO 3 was added to the formulation. In fact, (NH 4 ) 2 CO 3 is commonly used as a blowing agent, [START_REF] Edwards | Large porous particles for pulmonary drug delivery[END_REF][START_REF] Traini | In Vitro Investigation of Drug Particulates Interactions and Aerosol Performance of Pressurised Metered Dose Inhalers[END_REF] poreforming agent [START_REF] Edwards | Large porous particles for pulmonary drug delivery[END_REF] or process enhancer [START_REF] Healy | Characterisation of excipientfree nanoporous microparticles (NPMPs) of bendroflumethiazide[END_REF][START_REF] Nolan | Excipient-free nanoporous microparticles of budesonide for pulmonary delivery[END_REF]. This compound decomposes at 60°C and produces gases during spray drying, thus it is able to create porous or hollow particles. The addition of (NH 4 ) 2 CO 3 to the PEG-SPIONs mixture did not change the morphology of the particle aggregates (Figure 4A). It would appear that the partially solidified PEG did not allow the formation of void cavities in the particles, which are usually made by material solidification around the gas bubbles produced by the (NH 4 ) 2 CO 3 decomposition on spray drying. However, the addition of ammonium carbonate to the 50P-50H-50F formulation produced individual spherical and hollow microparticles (Figure 4B). These microparticles had a median geometric diameter of 3 µm (Fig. 3) and SPIONs were observable on their surface, but appeared more entrapped than when processing was undertaken in the absence of ammonium carbonate (Figure 2D). The deeper penetration of the SPIONs within the microparticles may allow for the avoidance of SPIONs desorption from the surface of the microparticles due to the mechanical stress produced during the dry powder inhalation. The large pores in these microparticles should enhance their aerodynamic properties, and be favourable for alveolar particle deposition.

The formulation of SPIONs-loaded microparticles made of PEG and HPCD with (NH 4 ) 2 CO 3 (50P-50H-50F-CO 3 ) decreased the amount of particles collected on stage 1 of the impactor from 49.8±9.0% (for SPIONs alone) to 35.1±6.4%, and increased the amount collected on the other stages, with a gradual decrease in the amount deposited with the decrease in the stage cut-off diameter (Fig. 5B). The application of a magnetic field on stage 2 of the impactor changed the particle deposition profile. The percentage of particles on stage 1, 3 and 4 decreased and the percentage on stage 2 significantly increased, 2.5-fold compared to the percentage obtained without magnetic field. Thus, this type of particles was sensitive to the magnetic field; however, their aerodynamic properties were not appropriate for targeting the deep lung area. The aerodynamic properties of the 50P-50H-50F-CO 3 microparticles showed a large measured MMAD value (10.2 ± 2.0 µm) compared to the d a (3.66 µm) calculated with equation 1 using the median volume-weighted geometric diameter D 50 , considering the particles to be spherical and using the particles' true density (Table 2). This difference can be attributed to the aggregation of the microparticles. Therefore, in order to reduce particle aggregation, magnesium stearate (MgST) was added to the formulation.

MgST is used in marketed DPI products (Seebri ® Breezhaler ® , Novartis; Foradil ® Certihaler ® , Novartis), and is commonly used to reduce the surface free energy [START_REF] Swaminathan | Measurement of the surface energy of lubricated pharmaceutical powders by inverse gas chromatography[END_REF] and agglomeration of particles [START_REF] Zhou | Improving aerosolization of drug powders by reducing powder intrinsic cohesion via a mechanical dry coating approach[END_REF]. The carboxylate group of MgST may be coordinated to the iron atom on the SPIONs surface via four different structures, as was previously observed with oleic acid [START_REF] Okassa | Optimization of iron oxide nanoparticles encapsulation within poly(d,l-lactide-co-glycolide) sub-micron particles[END_REF]. This would make the SPIONs surface less polar and less adhesive and would change the SPIONs behaviour. However, it was found that the dispersive part of the surface free energy ( s d ) of the microparticles was not changed by the addition of MgST. This component of the surface free energy of the Trojan microparticle was already low due to the presence of PEG at the surface. The addition of MgST to the formulations changed the particle morphology. SEM micrographs (Fig. 4C-D) showed individual microparticles with surfaces that appeared more porous compared to particles formulated without MgST (Fig. 4B). This increase in porosity induced an increase in the specific surface area, as measured by nitrogen adsorption (Table 2). Also, rough or irregular particles may have very low effective van der Waals adhesion forces [START_REF] Walton | Review of Adhesion Fundamentals for Micron-Scale Particles[END_REF], facilitating particle aerosolisation.

The addition of MgST into the formulation significantly decreased particle deposition on stage 1 of the NGI, from 49.8 ± 9.0% for nanoparticles alone to 1.8 ± 0.3% of the recovered dose for 50P-50H-50F-ST and increased the amount of particles recovered on the other stages (Figure 5C). For the 50P-50H-50F-ST particles the main deposition occurred on stages 2 and 3 (22.9 ± 3.4 and 20.9 ± 3.4 %, respectively), having a cut-off of 4.46 and 2.82 µm, respectively, with a gradual decrease in the amount of powder recovered on the other lower cut-off stages. These particles have a MMAD of 3.4 ± 1 µm which was higher than the aerodynamic diameter (2.0 µm) calculated using equation 1. In the presence of magnets on stage 2, the amount of particles recovered on this stage increased from 22.9 ± 3.4 to 32.6 ± 4.9% and decreased on the following stages.

The addition of (NH 4 ) 2 CO 3 and MgST to the formulation improved further the aerodynamic properties of the Trojan particles, leading to the main deposition being centred on stage 4 (23.4 ± 3.5%) which has a cut-off of 1.6 µm (Figure 5D). In the in vivo situation, it would be expected that these particles would be spread out within the whole lung. In the presence of a magnetic field on stage 2, the amount of particles deposited at this stage significantly increased 4-fold from 4.8 ± 0.7% to 19.5 ± 3.3%. These Trojan particles appeared more sensitive to the magnetic field than particles formulated without (NH 4 ) 2 CO 3 and MgST and the deposition on the NGI stage was significantly altered in the presence of the magnet (Figure 5D).

These particles may be useful for treating localised lung disease, by targeting foci of bacterial infection or tumour nodules. SPIONs released from the Trojan microparticles should be eliminated from the lungs via mechanisms such as mucociliary clearance and macrophage phagocytosis [START_REF] Buxton | Nanomedicine for the management of lung and blood diseases[END_REF]. Even though it has already been used in inhalation in humans, the inhalation of SPIONs may raise some toxicological concerns [START_REF] Moller | Mucociliary and longterm particle clearance in airways of patients with immotile cilia[END_REF]. A previous toxicological study [START_REF] Zhu | Particokinetics and extrapulmonary translocation of intratracheally instilled ferric oxide nanoparticles in rats and the potential health risk assessment[END_REF] showed that intratracheally instilled Fe 2 O 3 nanoparticles of 22 nm diameter could pass through the alveolar-capillary barrier into the systemic circulation at a clearance rate of 3.06 mg/day. The authors of this study suggested that this absorption was probably due to macrophages clearance function overloading, potentially resulting in lung cumulative toxicity of the nanoparticles.

PEG used in the formulations discussed here could be useful to prevent early stage particle phagocytosis by alveolar macrophage, by forming a swelling/viscous crown around the drug loaded microparticles to temporarily repulse macrophages [START_REF] El-Sherbiny | Controlled release pulmonary administration of curcumin using swellable biocompatible microparticles[END_REF][START_REF] El-Sherbiny | Swellable microparticles as carriers for sustained pulmonary drug delivery[END_REF]. Thus, the use of PEG in the current formulations could decrease the macrophage overloading by slowing down the rate at which the particles are phagocytosed. After complete solubilisation, PEG of 10 kDa should diffuse from the lung into the blood circulation and be eliminated by renal excretion, as its molecular weight is lower than 30kDa [START_REF] Fee | Size comparison between proteins PEGylated with branched and linear poly(ethylene glycol) molecules[END_REF][START_REF] Harris | Effect of pegylation on pharmaceuticals[END_REF].

Other study performed on rats exposed to magnetite particles with a MMAD of 1.3 μm for 13-week of inhalation showed no mortality, consistent changes in body weights, or systemic toxicity. Elevations of neutrophils in bronchoalveolar lavage appeared to be the most sensitive endpoint of the study [START_REF] Pauluhn | Subchronic inhalation toxicity of iron oxide (magnetite, Fe3O4) in rats: pulmonary toxicity is determined by the particle kinetics typical of poorly soluble particles[END_REF]. Particle size appears to be determinant in SPIONs toxicity. For example, ultra small superparamagnetic particle of iron oxide having a diameter of 5 nm were not toxic to human monocyte-macrophages in vitro and did not activate them to produce pro-inflammatory cytokines or superoxide anions [START_REF] Müller | Effect of ultrasmall superparamagnetic iron oxide nanoparticles (Ferumoxtran-10) on human monocyte-macrophages in vitro[END_REF].

Nevertheless, we suggest that this approach of lung drug targeting by an external magnetic field would be acceptable only in the case of a clear benefit such as in the case of anticancer drug delivery [START_REF] Rudolph | Magnetic aerosol targeting of nanoparticles to cancer: nanomagnetosols[END_REF] and if used at a low frequency.

3-Conclusion

This study demonstrates the feasibility of formulating SPIONs-loaded microparticles which may be useful in the treatment of localised lung disease, such as foci of bacterial infection or tumour nodules. The Trojan microparticles formulated were aerosolised using a dry powder inhaler to produce inhalable particles. These particles were sensitive enough to the magnetic field produced by a commercial magnet to induce a significant change of their distribution on a cascade impactor.

Figure 1 :

 1 Figure 1: Powder X-ray diffractograms of hybrid SPION loaded microparticles

Figure 2 :

 2 Figure 2: SEM micrographs of trojan hybrid SPION loaded microparticles: A) 25P-75H-5F , B) 25P-75H-30F , C) 100P-50F, D) 50P-50H-50F.

Figure 3 :

 3 Figure 3: Geometric particles size distribution of hybrid SPION loaded microparticles. A: 100P-50F; B: 100P-50F-CO3; C: 50P-50H-50F; D: 25P-75H-30F; E: 25P-75H-5F, F: 100HP-50F.

Figure 4 :Figure 5 :

 45 Figure 4: SEM micrographs of hybrid SPION loaded microparticles formulated with (NH 4 ) 2 CO 3 : A) 100P-50F-CO 3 , B) 50P-50H-50F-CO 3 . SEM micrographs of hybrid SPION loaded microparticles formulated with Mg-ST: C) 50P-50H-50F-ST, D) 50P-50H-50F-ST-CO 3 .

Table 1 : Concentration (g/L) of materials in the spray dried solutions and formulation code.

 1

Table 2 : Physicochemical properties of the Trojan particles. * Data from supplier.

 2

			25P-75H-5F	25P-75H-30F	50P-50H-50F	100P-50F	50P-50H-50F-CO3	50P-50H-50F-ST	50P-50H-50F-CO3-ST	SPION
	True density (g/cm 3 ) 1.39 ± 0.01 1.64 ± 0.01 1.68 ± 0.01 1.69 ± 0.01 1.76 ± 0.01 1.66 ± 0.01 1.71 ± 0.01 2.70 ± 0.01
	Specific surface area (m 2 /g)	4.5 ± 0.2	4.8 ± 0.2 3.9 ± 0.1 1.4 ± 0.2	3.2 ± 0.1	6.8 ± 0.4	4.6 ± 0.3	46.6*
	geometric median diameter (µm)	1.9 ± 0.2	2.0 ± 0.4 2.2 ± 0.4 7.2 ± 0.7	2.8 ± 0.3	1.54 ± 0.2	2.15 ± 0.3	0.05*
	γ s	d (mJ/m 2 )	119 ± 2	149 ± 27	43 ± 5	36 ± 1	39 ± 2
	MMAD (µm)					10.2 ± 2.0	3.4 ± 1.1	2.2 ± 0.8	9.3 ± 3.1
		GSD					2.4 ± 1.1	2.4 ± 0.5	2.1 ± 0.4	4.3 ± 1.8

4-Acknowledgement

The authors acknowledge funding by a Strategic Research Cluster grant (07/SRC/B1154) under the National Development Plan co-funded by EU Structural Funds and Science Foundation Ireland.


Jonathan Chappelon
email: jonathan.chappelon@um2.fr.

Kolja Knauer
email: kolja.knauer@googlemail.com

Pedro Luis

Jorge Montejano

Ramírez Luis

Alfonsín

Luis Pedro

Jonathan Chappelon ⋆

Luis Pedro Montejano
email: lpmontejano@gmail.com

Jorge Luis

Ram Írez
email: jramirez@um2.fr

Alfons Ín

Michel Las Vergnas

Connected covering numbers

Keywords: 2010 Mathematics Subject Classification. 05B05, 5E105 Covering design, Turán-system, uniform oriented matroid

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Introduction

Let n, k, r be positive integers such that n k r 1. A (n, k, r)-covering is a family B of k-subsets of {1, . . . , n}, called blocks, such that each r-subset of {1, . . . , n} is contained in at least one of the blocks. The number of blocks is the covering's size. The minimum size of such a covering is called the covering number and is denoted by C(n, k, r). Given a (n, k, r)-covering B, its graph G(B) has B as vertices and two vertices are joined if they have one r-subset in common. We say that a (n, k, r)-covering is connected if the graph G(B) is connected. The minimum size of a connected (n, k, r)-covering is called the connected covering number and is denoted by CC(n, k, r).

The graph corresponding to a connected [START_REF] Hamidoune | Directed switching games on graphs and matroids[END_REF][START_REF] Forge | Disconnected coverings for oriented matroids via simultaneous mutations[END_REF], 3)-covering can be nicely illustrated as shown in Figure 1. In this paper, we mainly pay our attention to coverings when k = r + 1 and thus, we will denote C(n, r + 1, r) (resp. CC(n, r + 1, r)) by C(n, r) (resp. by CC(n, r)) for short. The original motivation to study CC(n, r) comes from the following question posed by Las Vergnas.

Question 1.1. Let U r,n be the rank r uniform matroid on n elements. What is the smallest number s(n, r) of circuits of U r,n , that uniquely determines all orientations of U r,n ? That is, whenever two uniform oriented matroids coincide on these circuits they must be equal.

In [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF], Forge and Ramírez Alfonsín introduced the notion of connected coverings and proved that [START_REF] Applegate | On asymmetric coverings and covering numbers[END_REF] s(n, r) CC(n, r).

The latter was then used to improve the best upper bound, s(n, r)

n-1 r , known at that time due to Hamidoune and Las Vergnas [START_REF] Hamidoune | Directed switching games on graphs and matroids[END_REF]; see also [START_REF] Forge | Disconnected coverings for oriented matroids via simultaneous mutations[END_REF] for related results. It turns out that s(n, r) is also closely related to C(n, r). Indeed, by using results in [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF][START_REF] Forge | Disconnected coverings for oriented matroids via simultaneous mutations[END_REF] it can be proved that [START_REF] Björner | Oriented matroids[END_REF] C(n, r) s(n, r).

A proof (needing some oriented matroid notions and thus lying slightly out of scope of this paper) of a more general version of the above inequality can be found in [START_REF] Knauer | How many circuits determine an oriented matroid?[END_REF].

Covering designs have been the subject of an enormous amount of research papers (see [START_REF] Gordon | New constructions for covering designs[END_REF] for many upper bounds and [START_REF] Sidorenko | What we know and what we do not know about Turán numbers[END_REF] for a survey in the dual setting of Turán-systems).

Although the construction of block design is often very elusive and the proof of their existence is sometimes tough, here, we will be able to present explicit constructions yielding exact values and bounds for,C(n, r) and CC(n, r) for infinitely many cases. The study of C(n, r) and CC(n, r) seems to be interesting not only for Design Theory but also, in view of Equations ( 2) and ( 1), for the implications on the behavior of s(n, r) in Oriented Matroid Theory. This relationship was already remarked in [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF]Theorem 4.1] where it was proved that CC(n, r) 2C(n, r). The latter can be slightly improved as follows Many interesting variants of Question 1.1 can be investigated. For instance, for nonuniform (oriented) matroids (graphic, representable, etc.) and by varying the notion of what determine means (up to orientations, bijections, etc.). These (and other) variants are treated in another paper (see [START_REF] Knauer | How many circuits determine an oriented matroid?[END_REF]).

This paper is organized as follows. In the next section, we recall some basic definitions and results concerning (connected) coverings and its connection with Turán systems needed for the rest of the paper. In Section 3, we investigate connected covering numbers when the value r is either small or close to n. Among other results, we give the exact value for CC(n, 2) (Theorem 3.2), for CC(n, 3) for n 12 (Theorem 3.3) and for CC(n, n -3) (Theorem 3.6). A famous conjecture of Turán and its connection with our results is also discussed. In Section 4, we present a general upper bound for CC(n, r) (Theorem 4.8)

allowing us to improve the best known upper bound for s(n, r). We end the paper by discussing some asymptotic results in Section 5.

Basic results

Let n, m, p be positive integers such that n m p. A (n, m, p)-Turán-system is a family D of p-subsets of {1, . . . , n}, called blocks, such that each m-subset of {1, . . . , n} contains at least one of the blocks. The number of blocks is the size of the Turán-system. The minimum size of such a covering is called the Turán Number and is denoted by T(n, m, p). Given a (n, m, p)-Turán-system D, with 0 2pm p, its graph G(D) has as vertices D and two vertices are joined if they have one 2pm-subset in common. We say that a (n, m, p)-Turán-system with 0 2p

-m p is connected if G(D) is connected.
The minimum size of a connected (n, m, p)-Turán-system is the connected Turán Number and is denoted by CT(n, m, p). By applying set complement to blocks, it can be obtained that

(4) C(n, k, r) = T(n, n -r, n -k). Moreover, if 0 n -2k + r n -k then (5) CC(n, k, r) = CT(n, n -r, n -k).
Note that the precondition for ( 5) is fulfilled if k = r + 1.

Most of the papers on coverings consider n large compared with k and r, while for Turán numbers it has frequently been considered n large compared with m and p, and often focusing on the quantity lim n→∞ T(n, m, p)/ n p for fixed m and p. Thus, for Turán-type problems, the value C(n, k, r) has usually been studied in the case when k and r are not too far from n. Forge and Ramírez Alfonsín [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF] proved that [START_REF] Gordon | New constructions for covering designs[END_REF] CC(n, r) n r -1 r =: CC * 1 (n, r).

Moreover, Sidorenko [START_REF] Sidorenko | The method of quadratic forms in a combinatorial problem of Turán[END_REF] proved that T(n, r + 1, r)

n-r n-r+1 ( n r )
r . Together with (4), we obtain that

(7) CC(n, r) C(n, r) = T(n, n -r, n -r -1) r + 1 r + 2 n r+1 n -r -1 =: CC * 2 (n, r).
Combining ( 6) and ( 7), together with a straight forward computation we have (8) CC(n, r) max{CC * 1 (n, r), CC * 2 (n, r)}, where the maximum is attained by the second term if and only if r 2 3 (n -1). The following recursive lower bound for covering numbers was obtained by Schönheim [START_REF] Schönheim | On coverings[END_REF] and, independently, by Katona, Nemetz and Simonovits [START_REF] Katona | On a problem of Turán in the theory of graphs[END_REF] (9)

C(n, r) n r + 1 C(n -1, r -1)
which can be iterated yielding to

(10) C(n, r) n r + 1 n -1 r . . . n -r + 1 2 . . . =: L(n, r).
Forge and Ramírez Alfonsín [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF]Theorem 4.2] proved that CC(n, r)

n-1 i=r+1 C(i, r -1). In this proof, they used the following recursive upper bound that will be useful for us later, [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF] CC(n, r) CC(n -1, r) + C(n -1, r -1).

Results for small and large r

In this section, we investigate connected covering numbers for small and large r, that is, when r is very close to either 1 or n. Let us start with the following observations. All these values coincide with the corresponding covering numbers except in the case r = 1, where C(n, 1) = ⌈ n 2 ⌉. 3.1. Results when r is small. For ordinary covering numbers, Fort and Hedlund [START_REF] Fort | Minimal coverings of pairs by triples[END_REF] have shown that C(n, 2) := ⌈ n 3 ⌈ n 2 ⌉⌉ that coincides with the lower bounds given in [START_REF] Knauer | How many circuits determine an oriented matroid?[END_REF] when the case r = 2. We also have the precise value for the connected case when r = 2. Theorem 3.2. Let n be a positive integer with n 3. Then, we have

CC(n, 2) = n 2 -1 2 .
Proof. Note that the claimed value coincides with the lower bound CC * 1 (n, 2). This lower bound comes from the fact that every connected covering has a construction sequence, where every new triangle shares at least one edge with an already constructed triangle. We present a construction sequence where indeed every new triangle (except possibly the last one) shares exactly one edge with the already constructed ones. Therefore, we attain the lower bound. Part of the construction is shown in Figure 2. We start presenting the black triangles from left to right. Then we present all triangles of the form (2i -1, 2i, j) for 1 i n 2 and j 2i + 3. (These are not depicted in the figure.) Now we present the gray triangles from left to right. A gray triangle of the form (2i, 2i + 1, 2i + 4) is connected to the already presented ones via (2i -1, 2i, 2i + 4). Note (as in the figure) the last triangle may indeed share two edges of already presented triangles, depending on the parity of n. This accounts for the ceiling in the formula. It is easy to check that all edges are covered.

The precise value of C(n, 3) remains unknown only for finitely many n, see [START_REF] Mills | On the covering of triples by quadruples[END_REF][START_REF] Mills | A covering of triples by quadruples[END_REF][START_REF] Ji | An improvement on covering triples by quadruples[END_REF]. The situation for connected coverings is worse. Proof. Note that the claimed value coincides with the lower bound CC * 1 (n, 3). For n 6 this is already checked in [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF]. Figure 1 proves CC(7, 3) 12 = CC * 1 [START_REF] Hamidoune | Directed switching games on graphs and matroids[END_REF][START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF]. By using [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF], C(7, 2) = 7, and CC(7, 3) = 12, we obtain that CC [START_REF] Ji | An improvement on covering triples by quadruples[END_REF][START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF] 19 = CC * 1 [START_REF] Ji | An improvement on covering triples by quadruples[END_REF][START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF]. Figure 3 proves CC(9, 3) 28 = CC * 1 (9, 3). From equation ( 11) and the fact that CC(9, 3) = 28 and C(9, 2) = 12, we conclude that CC(10, 3) 40 = CC * 1 (10, 3). Now, Figure 4 proves that CC(11, 3) 55 = CC * 1 [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF][START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF]. Finally, to construct a connected covering witnessing CC * 1 (12, 3) we delete the block {2, 4, 6, 8} from the covering in Figure 4. One can check that this still leaves a covering B, whose graph now has three components. Now, we take the following (disconnected) (11, 3, 2)-covering: We add to each of these block the element 12 and thus together with B obtain a (12, 4, 3)covering B ′ . To see that B ′ is connected, note that each of the blocks containing 12 is connected to a block from B. Moreover, the blocks {1, 4, 6, 12}, {2, 4, 6, 12}, {4, 6, 8, 12} form a triangle and each of them has a neighbor in a different component of G(B). Thus, G(B ′ ) is connected and B ′ has 73 blocks which coincides with CC * 1 [START_REF] Kuzjurin | Explicit constructions of Rödl's asymptotically good packings and coverings[END_REF][START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF].

CC(n, n -3) = n 2 2 + n 2 2 + 1.
Proof. The parameter C(n, n -3) = T(n, 3, 2) was determined already by Mantel in 1907 [START_REF] Mantel | Problem 28[END_REF] and is

⌈ n 2 ⌉ 2 + ⌊ n 2 ⌋
2 . Turán proved that the unique minimal configuration of sets of size 2 hitting all sets of size 3 of an n-set are the edges of two vertex-disjoint complete graphs K ⌈ n 2 ⌉ and K ⌊ n 2 ⌋ , see [START_REF] Turán | Eine Extremalaufgabe aus der Graphentheorie[END_REF]. Now, by ( 4) and ( 5), the covering corresponding to the Turán-system is connected if and only if the graph whose edges correspond to the blocks of the Turán-system is connected. Thus, since the unique optimal construction by Turán is not connected but can be made connected by adding a single edge connecting the two complete graphs, this is optimal with respect to connectivity. Therefore, CC(n, n-3) = T(n, 3, 2)+1, giving the result. Proposition 3.7. Let n = 5, 6, 8, 9 be a positive integer with n 4. Then, we have

CC(n, n -4)      m(m -1)(2m -1) if n = 3m, m 2 (2m -1) if n = 3m + 1, m 2 (2m + 1)
if n = 3m + 2.

If n = 5, 6, 9 the value of CC(n, n -4) is one larger than claimed in the formula. Further, CC(8, 4) ∈ {20, 21}, i.e., it remains open if the above formula has to be increased by one or not in order to give the precise value.

Proof. We will show that a Turán-system D verifying the claimed bounds due to Kostochka [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF] is connected. Indeed the construction of [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF] is a parametrized family of Turán-systems, each of whose members attains the claimed bound. Our construction results from picking special parameters: Assume that n 12 and n is divisible by 3. Split [n] into three sets A 1 , A 2 , A 3 of equal size. Pick special elements x i , y i ∈ A i and denote

B i := A i \ {x i , y i } for i = 0, 1, 2.
The blocks of D consist of 3-element sets {a, b, c} of the following forms:

L i : a, b, c ∈ A i , T 1 i : a = x i and b, c ∈ A i+1 , T 2 i : a = y i and b, c ∈ B i-1 ∪ {x i+1 , y i+1 }, T 3 i : a ∈ B i and b, c ∈ B i-1 ∪ {x i+1 , y i-1
} where i = 0, 1, 2, and addition of indices is understood modulo 3. Let us now show that D is connected. Clearly, all blocks in a given A i are connected and all 2-element subsets in each A i are covered by a block in this A i . Thus, it suffices to verify that there are two 2-element sets {e, f } ⊆ A 0 and {e ′ , f ′ } ⊆ A 2 which can be connected by a sequence of blocks of D, because then any block in A 0 containing {e, f } is connected to any block in A 2 containing {e ′ , f ′ }. The connectivity of D then follows by the symmetry of the construction. Let {e, f } ⊆ B 0 . Take {e, f, y 1 } ∈ T 2 1 , then {e, y 1 , y 2 } ∈ T 2 1 , and then {e, f ′ , y 2 } ∈ T 3 0 , where f ′ ∈ B 2 , i.e, {y 2 , f ′ } ⊆ B 2 . Now, following [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF] deleting any element of such a system yields a Turán-system D ′ of the claimed size for n ′ = n -1. We can just delete any x i , since these are not used for connectivity. Following [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF], two elements can be removed from D to obtain a Turánsystem D ′′ of the claimed size for n ′′ = n -2, if the set formed by these two elements belongs to exactly n 3 -1 blocks. This is the case for {x i , x i+1 }, which belongs to exactly n 3 -1 blocks from T 1 i . Again, this preserves connectivity. We are left with the cases n 9. In [START_REF] Sidorenko | The method of quadratic forms in a combinatorial problem of Turán[END_REF] it is shown that the Turán-systems of the claimed size for n = 9 are exactly the members of the family constructed in [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF]. There are exactly two such systems: In both cases [START_REF] Katona | On a problem of Turán in the theory of graphs[END_REF] is split into three sets A 1 , A 2 , A 3 of size 3. In the first system we pick a x i ∈ A i and denote A i \ {x i } by B i . The blocks then are the 3-element sets {a, b, c} of the following forms:

L i : a, b, c ∈ A i , T 1 i : a = x i and b, c ∈ A i+1 , T 2 i : a ∈ B i and b, c ∈ B i-1 ∪ {x i+1 }.
The second system coincides with an instance of a construction due to Turán [START_REF] Turán | Research problems[END_REF]. It consists of the following 3-element sets:

L i : a, b, c ∈ A i , T 1 i : a ∈ A i and b, c ∈ A i+1 .
It is easy to check that both systems are not connected. On the other hand, the second one can be made connected adding a single block taking one element from each A i . This proves the claim for n = 9. Further, removing any vertex not contained in the added block, one obtains a connected Turán-system for n = 8 with 21 blocks. While there are Turán-systems showing T(8, 4, 3) = 20 we do not know if there is any such connected system. See Figure 1 for proving our statement for n = 7, Theorem 3.2 for n = 6, and Remark 3.1 for n = 4, 5.

A famous conjecture of Turán [START_REF] Turán | Research problems[END_REF] states that the bounds in Proposition 3.7 are best possible for C(n, n -4). By combining (1) and Proposition 3.7, for n 10 we have

(12) C(n, n -4) CC(n, n -4)      m(m -1)(2m -1) if n = 3m, m 2 (2m -1) if n = 3m + 1, m 2 (2m + 1) if n = 3m + 2.
Turán's conjecture has been verified for all n 13 by [START_REF] Sidorenko | The method of quadratic forms in a combinatorial problem of Turán[END_REF] and so, by [START_REF] Kuzjurin | Explicit constructions of Rödl's asymptotically good packings and coverings[END_REF], the connected covering number can also be determined for these same values.

Towards proving Turán's conjecture, it would be of interest to investigate the following.

Question 3.8. Is it true that one of the inequalities in ( 12) is actually an equality ?

Bounds and precise values for all CC(n, r) with n 14 are given in Table 1. All the exact values previously given in [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF] for the same range have been improved by using our above results. 9)) a -Lower bounds for C(n, r) (from [START_REF] Applegate | On asymmetric coverings and covering numbers[END_REF]) Question 3.9. Is the sequence (CC(n, i)) 0 i n-1 unimodal for every n ? or perhaps logarithmically concave 1 ?

A general upper bound

Let n and r be positive integers such that n r + 1 3. Forge and Ramírez Alfonsín [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF] obtained the following general upper bound [START_REF] Mantel | Problem 28[END_REF] S(n, r) :=

⌊ n-r+1 2 ⌋ i=1 n -2i r -1 + n -r 2 CC(n, r).
Let us notice that the upper bounds obtained by applying the recursive equation [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF], that were used in Table 1, are better than the one given by [START_REF] Mantel | Problem 28[END_REF]. Moreover, by iterating [START_REF] Kostochka | A class of constructions for Turán's (3, 4)-problem[END_REF] it can be obtained ( 14) CC(n, r)

n-1 i=r C(i, r -1).
Although [START_REF] Mills | On the covering of triples by quadruples[END_REF] might be used to get an explicit upper bound for s(n, r), it is not clear how good it would be since that would depend on the known exact values and the upper bounds of C(n, r) used in the recurrence (and thus intrinsically difficult to compute). On a i-1 a i+1 holds for every a i with 1 i n -1). Notice that a log-concave sequence is unimodal.

the contrary, in [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF] Equation ( 13) was used to give the best known (to our knowledge) explicit upper bound for s(n, r). In this section, we will construct a connected (n, r + 1, r)-covering giving an upper bound for CC(n, r) better than S(n, r) and so, yielding a better upper bound for s(n, r) than that given in [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF]. Theorem 4.1. Let n and r be positive integers such that n r + 1 3. Then CC(n, r) N(n, r), where

(15) N(n, r) := ⌈ n-r 2 ⌉-1 i=0 (n -r -2i) r -2 + 2i r -2 + n -r 2 -1 + δ 0 C(n -2, r -2),
and δ 0 is the parity function of nr, that is,

δ 0 = 0 if n -r is odd, 1 otherwise.
Proof. From this point on, for any positive integer s, we will denote [s] := {1, . . . , s} and by [s] t the set of all t-subsets of [s]. Moreover, for any subset of integers {b 1 , . . . , b s }, we may suppose that b i < b j for all integers i and j such that 1 i < j s.

Case 1. Suppose that nr is odd and let m such that nr = 2m + 1. We will construct a connected (r

+ 2m + 1, r + 1, r)-covering of size m + m i=0 r -2 + 2i r -2 (2m + 1 -2i).
We consider a particular (r+2m+1, r+1, r)-covering, which is constituted by a large number of blocks but whose associated graph has a small number of connected components.

For any i ∈ {0, . . . , m}, let N i be the following subset of (r + 1)-subsets of [r + 2m + 1]:

N i :=    {b 1 , . . . , b r+1 } {b 1 , . . . , b r-2 } ∈ [r+2i-2] r-2 b r-1 = r + 2i -1, b r = r + 2i b r+1 ∈ {r + 2i + 1, . . . , r + 2m + 1}    . Claim 4.2. The set m i=0 N i is a (r + 2m + 1, r + 1, r)-covering. Let b = {b 1 , . . . , b r } ∈ [r+2m+1] r . If b r-1 = r -1 + 2i for some i ∈ {0, . . . , m}, then b ⊂ B for some B ∈ N i . The same occurs if b r-1 = r + 2i. Claim 4.3. The graph G (N i ) is connected, for any i ∈ {0, . . . , m}. Let B = {b 1 , . . . , b r+1 } and C = {c 1 , . . . , c r+1 } in N i . Clearly, B is adjacent to {b 1 , . . . , b r , c r+1 } in G(N i ). Since b r-1 = c r-1 , b r = c r and {d 1 , . . . , d r-2 , c r-1 , c r , c r+1 } ∈ N i for all {d 1 , . . . , d r-2 } ⊂ [r -2 + 2i],
then there exists a path from B to C. Claim 4.4. There exists a (r + 1)-subset

C i such that G(N i ∪ C i ∪ N i+1 ) is connected for any i ∈ {0, . . . , m -1}. Let B i = {1, . . . , r -2, r -1 + 2i, r + 2i, r + 1 + 2i} ∈ N i and B i+1 = {1, . . . , r -2, r + 1 + 2i, r + 2 + 2i, r + 3 + 2i} ∈ N i+1 . Then, the (r + 1)-subset C i = {1, . . . , r -2, r + 2i, r + 1 + 2i, r + 2 + 2i} is adjacent to B i and B i+1 in G(N i ∪ C i ∪ N i+1
N i ) m-1 i=0 C i is a connected (r+2m+ 1, r + 1, r)-covering. Finally, since |N i | = r-2+2i r-2
(2m + 1 -2i) for any i ∈ {0, . . . , m}, the theorem holds in this case.

Case 2. Suppose nr is even and let m be such that nr = 2m. We are going to construct a (r + 2m, r + 1, r)-connected covering of size

m -1 + C(r -2 + 2m, r -2) + m-1 i=0 r -2 + 2i r -2 (2m -2i).
As already defined in Case 1, we consider the collection N i of (r + 1)-subsets of [r + 2m] defined by . We distinguish two cases on the value of It can be checked that N 0 ∪ N 1 is a (7, 5, 4)-covering and G(N 0 ) and G(N 1 ) are connected. Now, by taking C 0 = 12456, it follows that G(N 0 ∪ C 0 ∪ N 1 ) is connected.

N i := [r+2i-2]
b r-2 . First, if b r-2 < r + 2i -1, then {b 1 , . . . , b r-2 } ∈ [r+2i-2]

Hence, (

m i=0 N i ) ∪ ( m-2 i=0 C i ) is a connected (r + 2m, r + 1, r)-covering. Since |N i | = r-2+2i r-2 (
We may now show that S(n, r) > N(n, r). For this we need first the following Theorem and Proposition. Theorem 4.7. Let r and n be positive integers such that n r + 1 3. Then,

S(n, r) = N(n, r) + ⌊ n-r 2 ⌋-1 i=0 n -r 2 -i r -2 + 2i r -3 + δ 0 (1 -C(n -2, r -2)) ,
where δ 0 is the parity function of nr.

Proof. By induction on n > r. From ( 13) and ( 15), the identity is verified for n = r + 1 and n = r + 2. Suppose now that the identity is verified for a certain value of n and let D be the difference

D := (S(n + 2, r) -N(n + 2, r)) -(S(n, r) -N(n, r)) .
Then S(n + 2, r) = N(n + 2, r) + (S(n, r) -N(n, r)) + D.

By using [START_REF] Mantel | Problem 28[END_REF], we obtain

S(n+2, r)-S(n, r) = ⌊ n-r+1 2 ⌋+1 i=1 n + 2 -2i r -1 + n -r 2 + 1 - ⌊ n-r+1 2 ⌋ i=1 n -2i r -1 - n -r 2 = n r -1 + 1.
By using (15), we have

N(n+2, r)-N(n, r) = ⌈ n-r 2 ⌉ i=0 (n + 2 -r -2i) r -2 + 2i r -2 + n -r 2 + δ 0 C(n, r -2) - ⌈ n-r 2 ⌉-1 i=0 (n -r -2i) r -2 + 2i r -2 - n -r 2 + 1 -δ 0 C(n -2, r -2) = ⌈ n-r 2 ⌉-1 i=0 2 r -2 + 2i r -2 + n + 2 -r -2 n -r 2 r -2 + 2 n-r 2 r -2 +δ 0 (C(n, r -2) -C(n -2, r -2)) + 1.
Moreover, for nr odd, it follows that

N(n + 2, r) -N(n, r) = ⌈ n-r 2 ⌉ i=0 2 r -2 + 2i r -2 + (δ 0 -1) n -1 r -2 +δ 0 (C(n, r -2) -C(n -2, r -2)) + 1. Therefore D = n r -1 - ⌈ n-r 2 ⌉ i=0 2 r -2 + 2i r -2 +(1-δ 0 ) n -1 r -2 +δ 0 (C(n -2, r -2) -C(n, r -2)) . From the identity r-2+2i r-2 = r-1+2i r-2 -r-2+2i r-3
, we obtain that

⌈ n-r 2 ⌉ i=0 2 r -2 + 2i r -2 = ⌈ n-r 2 ⌉ i=0 r -2 + 2i r -2 + ⌈ n-r 2 ⌉ i=0 r -1 + 2i r -2 - ⌈ n-r 2 ⌉ i=0 r -2 + 2i r -3 = r-1+2⌈ n-r 2 ⌉ i=r-2 i r -2 - ⌈ n-r 2 ⌉ i=0 r -2 + 2i r -3 = r + 2 n-r 2 r -1 - ⌈ n-r 2 ⌉ i=0 r -2 + 2i r -3 .
Thus,

D = n r -1 - r + 2 n-r 2 r -1 + (1 -δ 0 ) n -1 r -2 + ⌈ n-r 2 ⌉ i=0 r -2 + 2i r -3 +δ 0 (C(n -2, r -2) -C(n, r -2)) .
If nr is even, then δ 0 = 1 and

n r -1 - r + 2 n-r 2 r -1 + (1 -δ 0 ) n -1 r -2 = n r -1 - n r -1 = 0.
Either, if nr is odd, then δ 0 = 0 and

n r -1 - r + 2 n-r 2 r -1 + (1 -δ 0 ) n -1 r -2 = n r -1 - n + 1 r -1 + n -1 r -2 = - n r -2 + n -1 r -2 = - n -1 r -3 .

Asymptotics

In [START_REF] Rödl | On a packing and covering problem[END_REF] Rödl uses the probabilistic method to show the existence of asymptotically good coverings. Restricted to our case this means that C(n, r)

n r → 1 r + 1
as n → ∞.

Since CC(n, r) 2C(n, r) (see [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF]) we immediately obtain:

CC(n, r) n r → a 2 r + 1
as n → ∞.

In [START_REF] Forge | Connected coverings and an application to oriented matroids[END_REF] it was shown that S(n, r)

n r → 1 2
as n → ∞ and since by Theorem 4.8 the difference N(n, r) -S(n, r) is in O(n r-1 ) we have the same asymptotic behavior for N(n, r).

It is however still a topic of research to find explicit constructions witnessing the bound of Rödl, see [START_REF] Kuzjurin | Explicit constructions of Rödl's asymptotically good packings and coverings[END_REF].

( 3 )

 3 CC(n, r) 2C(n, r) -1, since the graph G associated to a covering with C(n, r) blocks (and thus with |V (G)| = C(n, r)) can be made connected by adding at most C(n, r) -1 extra vertices (blocks), obtaining a graph corresponding to a (n, r+1, r)-connected covering with at most 2C(n, r)-1 blocks.

Remarks 3

 3 .1. a) CC(n, 0) = 1 since any 1-element set contains the empty set. b) CC(n, 1) = n -1 by taking the edges of a spanning tree of K n . c) CC(n, n -2) = n -1 by taking all but one (n -1)-sets. d) CC(n, n -1) = 1 by taking the entire set.

Figure 2 .

 2 Figure 2. Part of the construction proving CC(n, 2) CC * 1 (n, 2).

Figure 3 .Theorem 3 . 3 .

 333 Figure 3. An example proving CC(9, 3) 28. The circle-vertices are a covering.

  3, 11}, {1, 4, 6}, {1, 2, 8}, {1, 5, 9}, {1, 7, 10}, {3, 4, 9}, {2, 3, 10}, {3, 5, 6}, {3, 7, 8}, {2, 4, 6}, {4, 5, 7}, {4, 10, 11}, {4, 6, 8}, {2, 5, 11}, {2, 7, 9}, {5, 8, 10}, {6, 7, 11}, {8, 9, 11}, {6, 9, 10}    .

Figure 4 .

 4 Figure 4. An example proving CC(11, 3) 55. The circle-vertices are a covering.

r- 2 × 2 .

 22 {r + 2i -1} × {r + 2i} × {r + 2i + 1, . . . , r + 2m} for any i ∈ {0, . . . , m -1}. Let C be a (r + 2m -2, r -1, r -2)-covering of size C(r + 2m -2, r -2) and consider the set N m := {B ∪ {r + 2m -1, r + 2m} | B ∈ C}. Then, one can check that m i=0 N i is a (r + 2m, r + 1, r)-covering. Similarly as in the proofs of Claims 4.3 and 4.4, it follows that G(N i ) is connected for any i ∈ {0, . . . , m -1} and there exists a (r + 1)-subsetC i such that G(N i ∪ C i ∪ N i+1 ) is connected for any i ∈ {0, . . . , m -2}. Claim 4.5. . For any B ∈ N m , there exist i ∈ {0, . . . , m -1} and C ∈ N i such that B is adjacent to C in the graph G(N i ∪ N m ). Let B = {b 1 , . . . , b r-1 , r+2m-1, r+2m} ∈ N m . If b r-1 = r+2i-1 for some i ∈ {0, . . . , m-1}, then {b 1 , . . . , b r-2 } ∈ [r+2i-2] r-Let C = {b 1 , . . . , b r-2 , r + 2i -1, r + 2i,r + 2m}, by definition C ∈ N i and moreover, since {b 1 , . . . , b r-2 , r + 2i -1, r + 2m} ⊂ B and {b 1 , . . . , b r-2 , r + 2i -1, r + 2m} ⊂ C, we deduce that B and C are adjacent in the graph G(N i ∪ N m ). Either, if b r-1 = r + 2i for some i ∈ {0, . . . , m -1}, we have that {b 1 , . . . , b r-2 } ∈ [r+2i-1] r-2

r- 2 .

 2 Consider now C = {b 1 , . . . , b r-2 , r + 2i -1, r + 2i, r + 2m}. As above, since {b 1 , . . . , b r-2 , r + 2i, r + 2m} ⊂ B and {b 1 , . . . , b r-2 , r + 2i, r + 2m} ⊂ C, we deduce that B and C are adjacent in the graph G(N i ∪ N m ). Finally, suppose that b r-2 = r + 2i -1 and let α ∈ [r + 2i -2] \ {b 1 , . . . , b r-3 } and C = {b 1 , . . . , b r-3 , r + 2i -1, r + 2i, r + 2m} ∪ {α} ∈ [r+2m] r+1 . Since {b 1 , . . . , b r-3 , r + 2i -1, r + 2i, r + 2m} ⊂ B and {b 1 , . . . , b r-3 , r + 2i -1, r + 2i, r + 2m} ⊂ C, we deduce that B and C are adjacent in the graph G(N i ∪ N m ). This concludes the proof of Claim 4.5.

  2m -2i) for any i ∈ {0, . . . , m -1} and |N m | = C(n -2, r -2), the theorem holds.Let us illustrate the construction given in the above theorem.

Example 4 . 6 .

 46 N(7, 4) = 10. We consider N 0 = {12345, 12346, 12347} and N 1 = {12567, 13567, 14567, 23567, 24567, 34567}.

- 3 + 3 + 3 + 3 . 2 r- 1

 333321 δ 0 (C(n -2, r -2) -C(n, r -2)) = ⌊ n-r 2 ⌋ i=0 r -2 + 2i r -3 + δ 0 (C(n -2, r -2) -C(n, r -2)) .Now, with the induction hypothesis, we obtainS(n+2, r)-N(n+2, r) = (S(n, r) -N(n, r)) δ 0 (C(n -2, r -2) -C(n, rδ 0 (1 -C(n, r -2)) .Theorem 4.8. Let r and n be positive integers such that nr is an even number. Then,S(n, r) N(n, r) + Proof. It is known [6, page 7] that C(n, r)n-+C(n-2, r). By applying this inequality repeatedly we have C(n -2, r -2)

Table 1

 1 [20, 21 u ] [32 l , 35 r ] [53 l , 59 r ] [83 l , 89 r ] [124 l , 136 r ] [179 l , 193 r ] [250 l , 271 r ] , 61 r ] [96 a , 111 r ] [159 l , 177 r ] [258 l , 290 r ] [401 l , 447 r ]

	r \ n 1 2 3 4 5	6	7	8	9	10	11	12	13	14
	0	1 1 1 1 1	1	1	1	1	1	1	1	1	1
	1	1 2 3 4	5	6	7	8	9	10	11	12	13
	2	1 3 5 e,t 7 e	10 e	14 e	18 e	22 e	27 e	33 e	39 e	45 e
	3	1 4 7 p,t 12 p,u	19 p	28 p	40 p	55 p	73 p	[95 l , 97 r ]	[121 l , 123 r ]
	4 10 t 5 1 5 1 6 [51 l 6 13 t 31 u 1 7 17 t 45 u	[84 a , 95 r ] [165 a , 195 r ] [286 l , 327 r ] [501 l , 572 r ]
	7				1	8	21 t	63 u	[126 a , 147 r ] [269 a , 323 r ] [491 l , 587 r ]
	8					1	9	26 t	84 u	[185 a , 210 r ] [419 a , 505 r ]
	9						1	10	31 t	112 u	[259 s , 297 r ]
	10							1	11	37 t	[143 s , 144 u ]
	11								1	12	43 t
	12									1	13
	13										1

led us to consider the following.

Table 1 .

 1 Bounds and values of CC(n, r) for n 14.

Key of Table 1 :

r -Upper bound for CC(n, r)(from Equation (

11

)) e -Exact values for CC(n, 2) (Theorem 3.2) t -Exact values for CC(n, n -3) (Theorem 3.6) l -Lower bound CC * 1 (n, r) p -Some exact values for CC(n, 3) (Theorem 3.3) u -Upper bound for CC(n, n -4) (Proposition 3.7) s -Lower bound for C(n, r) (from Equation (

  ). This concludes the proof of Claim 3. By Claims 4.2, 4.3 and 4.4, we obtain that ( m i=0

Acknowledgments

Much of this work in particular for the construction of the connected covering designs in Figures 1,3, 4 strongly benefited from the La Jolla Covering Repository (http://www. ccrwest.org/cover.html) maintained by Dan Gordon.

The second and forth authors were supported by the ANR TEOMATRO grant ANR-10-BLAN 0207. The third author was supported by CONACYT.