Besim Ben-Nissan
email: b.ben-nissan@uts.edu.au

Innocent Macha

Sophie Cazalbou

Andy H Choi

Keywords: antibiotics, biofilm, drug delivery, hydroxyapatite, nanocoatings, nanocomposites, nanolam inates, osteomyelitis, thin films

Calcium phosphate nanocoatings and nanocomposites, part 2: thin films for slow drug delivery and osteomyelitis

During the last two decades although many calcium phosphate based nanomaterials have been proposed for both drug delivery, and bone regeneration, their coating applications have been somehow slow due to the problems related to their complicated synthesis methods. ln order to control the efficiency of local drug delivery of a biomaterial the critical pore sizes as well as good control of the chemical composition is pertinent. A variety of calcium phosphate based nanocoated composite drug delivery systems are currently being investigated. This review aims to give an update into the advancements of calcium phosphate nanocoatings and thin film nanolaminates. ln particular recent research on PLA/hydroxyapatite composite thin films and coatings into the slow drug delivery for the possible treatment of osteomyelitis is covered.

During the last hundred years due to its simi lari ty to hard tissue, the material of choice for the enhancement of bioactivity has been calcium phosphates [START_REF] Albee | Studies in bone growth: triple CaP as a stimulus to osteogenesis[END_REF][START_REF] Hulbert | Potential of ceramic materials as permanently implantable skeletal prostheses[END_REF][START_REF] Choi | Calcium phosphate nanocoatings and nanocomposites, part I: recent developments and advancements in tissue engineering and bioimaging[END_REF][START_REF] Choi | Nano-biomaterials coatings in dentistry[END_REF][START_REF] Choi | Advances in calcium phosphate nanocoatings and nanocomposites[END_REF][START_REF] Choi | Sol-gel production of bioactive nanocoatings for medical applications. Part II: current research and development[END_REF]. A number of bio degradable nanocoated biomaterials based on calcium phosphates are currently being examined for applications for slow drug delivery and dispersion of pharmaceuticals and minerais to the targeted area. In these new devices, which comprises a polymeric or ceramic matrix and porous inorganic par ticulate matter, factors such as the chemical composition as well as the critical pore size are the main factors that influence the disso lution rates. Additive particulate sizes which can be incorporated to these films can be from a few nanometers up to microns depen dent on the required thickness and the fonc tion of the films. They are usually designed to contain nano and meso pores for drug loading.

Classical approach of the systemic drug delivery has in the past generated a num-10.2217/nnm. [START_REF] Mukesh | Methotrexate loaded self stabilized calcium phosphate nanoparticles: a novel inorganic carrier for intracellular drug delivery[END_REF].220 ber of problems that can possibly be solved by local or targeted delivery. These include infl uence of the dissolving drugs to the whole body rather than pin point loca.l delivery. Moreover, aside from reduction of toxicity to healthy cells within the whole environment, targeting these drugs directly to required locations have the possibility of improving drug efficacy and efficiency, resulting in a significant cost savings for the health care system.

During the last decade a number of research groups have been describing the pro duction of nove! nanocoatings and thin film nanolaminates with hydroxyapatite (HAp) and other calcium phosphates for clinical applications. The nano to micro particles of oxide or mixed ceramics, and calcium phosphates are the inorganic components and natural or synthetic polymers such as collagen, chitosan and biodegradable poly mers such as polylactic acid (PLA) are the matrix materials in these new composites.

These nanocomposites can be synthesized by mixing the inorganic particles physically through heat or by solvent introduction into an already existing polymeric matrix material. Evaporation of the solvent aids the formation of the nanocomposite or the thin films.

The development of tissue engineering in the past has been related directly to the types of scaffolding materials used. At present, a number of synthetic bone graft biomaterials such as calcium phosphates are available as options to autogenous bone for augmentation, repair or substitution [START_REF] Choi | Calcium phosphate nanocoatings and nanocomposites, part I: recent developments and advancements in tissue engineering and bioimaging[END_REF]. New advancements associated with calcium phosphate scaffolds and their improvements in microstructure and surface properties have created new opportunities for bone regenerative technologies. These recent developments also make the calcium phosphate scaffolds to be thought of as being at least biologically constructive instead of as only osteoconductive scaffolds specifically with the addition of biologic materials such as bone morphogenic proteins and stem cells [7-9].

Evidence has shown that porous calcium phosphate has a direct influence on the proliferation and differentiation of human mesenchymal stem cells (MSCs). Tissue engineering along with new bioactive molecules enhanced the possible applications of calcium phosphate as scaffolds able to guide the behavior of these cells and the efficiency of bone regeneration as well as being carriers of these cells [START_REF] Ben-Nissan | Sol-gel production of bioactive nanocoatings for medical applications. Part I: an introduction[END_REF][START_REF] Gibson | Calcium phosphate as scaffolds for mesenchymal stem cell[END_REF].

In this review, we aim to introduce the recent developments of nanocoatings and thin films and composites containing calcium phosphate-based nanoparticles currently being investigated for the delivery of pharmaceutical substances and their use in medicine and specifically for the treatment of osteomyelitis.

Surface modifications & liposomes for drug delivery applications

In bioceramics, the critical pore size and interconnectivity can be altered in order to control the ease of delivery and dispersion of a material to the targeted area. Targeting usually achieved with appropriate functionalizing of the surfaces. The pores may range from a few nanometers to micron sizes dependent on the pharmaceuticals used or areas intended to be delivered such as long bones. Delivery systems based on calcium phosphate which is similar at least chemically to the hard tissues, have the potential to increase drug efficacy while at the same time minimizing toxicity to nondiseased cells. Nano drug-delivery systems, embedded within a matrix or not, also have the exceptional attribute of being capable of delivering and controlling dissolution with high precision due to their high surface areas. It is not surprising that the number of research papers covering drug, gene and mineral deliv-ery of nanoparticles, nanocoatings and composites published during the last decade is very high .

The appropriate dissolution rates and their control within the human body is the main concern for drug carriers containing nanoparticles and nano thin films [START_REF] Paul | Fatty acid conjugated calcium phosphate nanoparticles for protein delivery[END_REF]. The use of calcium phosphate as a delivery system also broadens its effectiveness as a result of their capacity to locally deliver minerals as well as calcium and phosphate, other active ions and biogenic materials such as bone morphogenetic proteins (BMPs) and MSCs if required to be used in the successful treatment of bone diseases. In addition the surface modification approach it can also be used to achieve enhancements of stability and long range solubility control of nanocoatings and thin films in aqueous media, as well introduction of new material properties and functions. In principle through the use of a wide range of biological, chemical and/or physical surface modifications methods, the surfaces of nanostructured materials such as nanocoatings can be altered and functionalized to assist us in slow drug delivery. In the quest for the surface modifications of nanostructured materials, approaches such as macro micro and nanocoating have emerged as the leading strategies resulting in better functionalization of the surfaces of materials and for better osseointegration in the long term.

The biological modification of surfaces of nanocoatings is at times essential for the functionality of the devices. Biospecific molecules can be incorporated into the nanocoatings or thin films by using physical or chemical methods, thus presenting biospecific sites for the further immobilization of ligands specific to these molecules. The immobilizations of specific ligands such as antibody-antigen and receptor-ligand can be carried out using biologically specific reactions [34]. Current research work in these areas is very promising.

It is well known that different biomedical applications require different functions and properties of materials. As a result, techniques available to modify nanostructured materials or thin films can vary in order to meet the demands of various biomedical systems. In spite of the advantages offered by nanocoatings and nanoparticle containing composite thin films, such as their small surface pore sizes and loading efficiency, a number of issues such as control of the appropriate drug release rates, restricted their use clinical applications.

The targeting ability and efficacy of any drug delivery system are sometimes hindered by the rapid dissolution of the carrier system within the human body. A good example is their side effects in chemotherapy drug delivery for the cancer patients. The long circulation time within the blood is the primary concern for drug carriers of both local and systemic delivery. For this reason, a number of investigations have been carried out to examine ways in which 'long-circulatingtime' carriers can be designed and engineered. Among these, the surface modification of thin films and nanocoatings with a variety of polymeric macromolecules or nonionic surfactant has been demonstrated to be the most effective for maintaining the presence of drug delivery particles in the blood for prolonged periods [START_REF] Xu | Modification of nanostructured materials for biomedical applications[END_REF].

The use of surface modification is used in gene therapy in an effort to obtain controlled delivery of small interfering RNA and plasmid DNA (pDNA) particularly in an acidic pH environment [START_REF] Zhou | Lipid-coated nano-calciumphosphate (LNCP) for gene delivery[END_REF][START_REF] Liu | An efficient calcium phosphate nanoparticle-based nonviral vector for gene delivery[END_REF][START_REF] Pittella | Pancreatic cancer therapy by systemic administration of VEGF siRNA contained in calcium phosphate/charge-conversion polymer hybrid nanoparticles[END_REF][START_REF] Li | Biodegradable calcium phosphate nanoparticle with lipid coating for systemic siRNA delivery[END_REF]. The use of cationic liposomes as transfection vectors has become an ideal choice and most widely employed in the transfer of pDNA due to their weak immunogenicity and low toxicity [START_REF] Olton | Nanostructured calcium phosphates (NanoCaPs) for non-viral gene delivery: influence of the synthesis parameters on transfection efficiency[END_REF]. A study by Zhou et al. [START_REF] Zhou | Lipid-coated nano-calciumphosphate (LNCP) for gene delivery[END_REF] has suggested that coating calcium phosphate with liposomes could provide consistently efficient and satisfactory delivery of pDNA. Using mammalian cell culture, their findings showed the application of a lipid coating resulted in a tenfold increase in the transfection of pDNA compared with uncoated calcium phosphate [25].

Considered as one of the most clinically recognized thin film nanoscale systems, liposomes consist of a single layer or multiple concentric lipid bilayers that encapsulate an aqueous compartment are currently utilized in the delivery of antifungal drugs, vaccines and genes [START_REF] Zhou | Lipid-coated nano-calciumphosphate (LNCP) for gene delivery[END_REF][START_REF] Xu | Encapsulation and release of a hydrophobic drug from hydroxyapatite coated liposomes[END_REF][START_REF] Anada | Synthesis of calcium phosphate-binding liposome for drug delivery[END_REF][START_REF] Zhu | Liposome combined porous β-TCP scaffold: preparation, characterization, and antibiofilm activity[END_REF][START_REF] Huang | Liposomes-coated hydroxyapatite and tricalcium phosphate implanted in the mandibular bony defect of miniature swine[END_REF][START_REF] Wang | Bisphosphonatederivatized liposomes to control drug release from collagen/ hydroxyapatite scaffolds[END_REF][START_REF] Kostarelos | Liposome-nanoparticle hybrids for multimodal diagnostic and therapeutic applications[END_REF]. The exceptional clinical profile of liposome coatings in comparison to other delivery systems is based on their reduced toxicity, biodegradability and capacity for size, and surface manipulations [START_REF] Lewis | The development of liposome encapsulated calcium phosphates for bone regeneration[END_REF]. An improvement in the biocompatibility of liposomes as well as an increase in nanoparticle hydrophilicity and stability in plasma can be achieved through the encapsulation of nanomaterials such as calcium phosphate within liposomes (Figure 1). In Lewis work multilayered liposomes were produced with the incorporation of nano-HAp and other minerals such as strontium, magnesium and zinc [START_REF] Lewis | The development of liposome encapsulated calcium phosphates for bone regeneration[END_REF]. Figure 1 shows a transmission electron microscopy (TEM) image of multilayered liposomes containing nano-HAp particles produced in a calcifying buffer. The figure was obtained in FEI Morgagni 268D transmission electron microscope (Eindhoven, the Netherlands) at 80 kV [START_REF] Lewis | The development of liposome encapsulated calcium phosphates for bone regeneration[END_REF]. The work showed an excellent encapsulation that can help to control drug delivery rates in medical applications such as chemotherapy drug delivery for oncology patients. This observed ease of coating and release delay ability is one of the strong reasons, calcium phosphate based nanoparticle containing thin films and liposome coatings are ideal candidates for drug delivery and bone regeneration systems [START_REF] Lewis | The development of liposome encapsulated calcium phosphates for bone regeneration[END_REF][START_REF] Ibara | Osteoconductivity and biodegradability of collagen scaffold coated with nano-β-TCP and fibroblast growth factor 2[END_REF][START_REF] Ben-Nissan | Natural bioceramics: from coral to bone and beyond[END_REF][START_REF] Milev | Morphological stability of plate-like hydroxyapatite[END_REF]. In addition, combinatory therapy-modalities can be accomplished by utilizing the ability of liposome coatings to carry hydrophobic and hydrophilic moieties as well as their capacity to incorporate therapeutic and diagnostic agents into a single liposome-delivery system [42].

Huang et al. [START_REF] Huang | Liposomes-coated hydroxyapatite and tricalcium phosphate implanted in the mandibular bony defect of miniature swine[END_REF] have suggested that the nucleation process for new bone formation could be improved by the presence of negatively charged liposome coatings.

In their experiments carried out in miniature swine, artificial bony defects on one side where implanted with liposome-coated tri calcium phosphate while defects on the other side served as controls. They reported that at three weeks postimplantation, dense connective tissues surrounded the implant material and new bone formation was visible after 6 weeks.

Using a different strategy, Wang et al. explored the possibility of producing collagen-calcium phosphate scaffolds with the incorporation of liposome thin films for the controlled release of bioactive molecules in bone regeneration and repair [40]. They suggested that bisphosphonate (BP) functionalized liposome encapsulation could be isolated within mineral-containing scaffolds can be better drug delivery system that localize their drug cargo to the directed area. The liposome encapsulation used consisted of cholesterol, distearoylphosphodioline, distearoylphosphoethanolaminepoly(ethylene glycol). Based on their observations, the encapsulation of BP within liposomes displayed a strong affinity to the scaffolds and the drugs entrapped within the BP-liposomes showed a slower release rate from the scaffolds as compared with drugs that were un-encapsulated or encapsulated in polyethylene glycol-liposomes. In a similar study by Chou et al. it was shown that this liposome coating thin films reduced the drug release rates in both BP and antibiotic additions [START_REF] Chou | Antibiotic delivery potential of nano and micro porous marine structures derived β-TCP spheres for medical applications[END_REF][START_REF] Choi | Current perspectives: calcium phosphate nanocoatings and nanocomposite coatings in dentistry[END_REF].

A study by Xu et al. [START_REF] Xu | Encapsulation and release of a hydrophobic drug from hydroxyapatite coated liposomes[END_REF] explored the possibility of synthesizing a multifunctional thin film drug carrier with sustained drug release capability provided by the inner core liposome and osteoconductivity for bone cells supported by the HAp outer layer. The liposomes were produced from 1,2-dimyristoyl-sn-glycero-3-phosphate and 1,2-dimyristoyl-sn-glycero-3-phosphocholine, which is then loaded with the lipophilic drug indomethacin. The release profile of indomethacin was measured at two different pH levels (4 and 7.4). As expected by coating the liposome with HAp, they reported a reduction in release rate of indomethacin in comparison to uncoated liposomes. They also reported that without these coatings, the rate of drug release occurred more rapidly at pH 7.4 rather than at pH 4.

It has been reported that the management of possible postoperative infections from bone grafts and prostheses as well as the treatment of bone diseases such as bone metastases will benefit gready if there is a delivery system which has a high affinity toward bone tissues thereby maximizing its therapeutic effect on bone-related diseases [START_REF] Xu | Encapsulation and release of a hydrophobic drug from hydroxyapatite coated liposomes[END_REF][START_REF] Huang | Liposomes-coated hydroxyapatite and tricalcium phosphate implanted in the mandibular bony defect of miniature swine[END_REF]. Using this approach, Anada et al. [START_REF] Anada | Synthesis of calcium phosphate-binding liposome for drug delivery[END_REF] attempted to develop a calcium phos phate-binding liposome coating for a bone targeting drug delivery system by synthesizing an amphipathic molecule bearing a BP head group to recognize and bind to HAp. Liposomes loaded with the drug doxoru bicin adsorbed onto the surfaces ofHAp were observed to significandy reduce the number of viable human osteosarcoma MG63 cells. Based on these observa tions, they suggested that the system can be excellent coated carriers for anticancer drugs as they specifically target bone tissue [START_REF] Xu | Encapsulation and release of a hydrophobic drug from hydroxyapatite coated liposomes[END_REF].

Infection & osteomyelitis

lt is widely accepted in the medical community that wound contamination as well as postoperative infec tions following implantation or during surgical inter vention in orthopedics and maxillofacial surgery can result in serious clinical problems and could jeopardize the osseointegration process. For these reasons, antibi otics either administrated orally or intravenously are often provided as prophylactics.

By far the most frequent complications related to the use of implantable medical devices such as orthopedic or dental prostheses and endotracheal tubes are bacte rial infections. Pseudonomas aeruginosa is regarded as an opportunistic pathogen causing indwelling device related infections especially catheters. P. aeruginosa infection is leading cause of morbidity and mortality in cystic fibrosis patients. On the other hand Staphy lococcus aureus infection causes serious infectious complications such as severe sepsis, septic-thrombosis and/or severe deep-seated infections (endocarditis, osteomyelitis and other metastatic infections). One of the basic hospital and surgical intervention acquired biofilm infections are those associated with S. aureus and Staphylococcus epidermidis mains, including meth icillin-resistant S. aureus (MRSA). A thorough and comprehensive understanding of the molecular bases of biofilm formation and their adhesion may help us to fight biofilm infections.

Osteoarticular infection is frequendy caused by coagulase-negative Staphylococci as main aetiologic agents in lace infections as well as Streptococci, Entero cocci and anaerobes. The number of infections is esti mated to be around 0.5-2.0% of cases and it increases continuously due to the rise in the need for implants.

The European Center for Disease Prevention and Controll (ECDPAC) reported that approximately 4,100,000 patients are estimated to acquire infections in the European Union every year (47]. The number of deaths occurring as a direct consequence of these infections is estimated to be at least 37,000 patients and these infections are thought to contribute to an additional 110,000 deaths each year. ln the USA, it was estimated that approximately 1,700,000 patients acquired infections for the year 2002. lt has been esti mated that 5% of patients undergoing clean surgical procedures and up to 20% of patients having intra abdominal surgical procedures develop a surgical site infection. Such infections result in 3.7 million excess hospital days and more than US$1.6-3 billion in excess hospital costs per year. ln order to mitigate this problem different strate gies have been proposed on either preventing and/or controlling bacterial infections. Modification or devel opment of thin film or nanocoated multilayer devices with surface properties that have an effect against microbial adhesion or viabili ty seems to be a promis ing approach for the prevention of device-related infec tions. Another strategy is to modify the surface of medical devices biologically, chemically or physically to render the surface free of microbial adhesion. Mul tifunctional thin films or nanocoatings can facilitate this new approach. ln clinical applications an ideal implant coatings must provide surgeons with several benefits such as providing primarily anchorage with appropriate bioac tivi ty involving osteoconduction and if possible, osteo induction. The nanocoating or thin films used should also provide antimicrobial property to prevent implants from developing acute or postoperative infections.

As stated earlier the development of bone infec tion is based on the formation of a bacterial biofilm where the bacteria differentiate from planktonic into sessile forms that protect some bacterial cells that can be released from the biofilm after antibiotic treatment has ceased.

Bacteria adhesion is a complex phenomenon affected by many factors, including properties of surface materials, some characteristics of bacteria itself and the environment where the adhesion takes, such as the presence of serum protein or bactericidal substances [START_REF] An | Concise review of mechanisms of bacterial adhesion to biomaterial surfaces[END_REF]. Some of the proposed theory and model of adhesion seem to be limited because they consider physical interaction between the surface and bacteria and neglecting biological aspects of adhesion in which specific bacterial structure responsible for adhesive activities called adhesins that control cell to cell or cell to abiotic surface adhesion. Bacteria may have different adhesives for different surfaces (different acceptor). The ionic strength and pH of the medium in which the adhesion takes place influence the charge of the cell wall and of the substrate (in terms of surface chemistry, charge and hydrophobicity) and therefore affects their interaction.

The antibiotic dosage required to act on bacteria in biofilm conditions can be many folds higher in concentration of the drug required for treatment of planktonic cells. Consequently, it is difficult to completely eradicate active infection by means of 'systemic antibiotics' which in doses active in biofilm can be toxic for a patient.

In orthopedics, osteomyelitis mainly occurs in tissues within the infected area, and consequently targeted delivery of antimicrobial agent locally is a more appropriate form of treatment. One of the most effective ways of achieving targeted delivery of antibiotics is to use a carrier device that can be placed into the body at a specific site such as coating on an implant, which will release slowly the correct antibiotic dosage. The major challenge associated with the use of antibiotic is ensuring retention of antibiotic release and activity for a prolonged period of time post operation [START_REF] Kargupta | Coatings and surface modifications imparting antimicrobial activity to orthopedic implants[END_REF].

Treatment of osteomyelitis can be with immobilization and antibiotic therapy with a number of drugs including flucloxacillin, gentamicin, tobermycin or vancomycin, and fusidic acid. Surgical drainage and removal of damaged bone, as its presence prevents healing, sequestrum may be possible but recurrence is common. Currently, commercial products in the form of pellets composed of acrylic polymers or ceramics such as calcium sulfate or bioactive glass are available as slow drug delivery devices. However, due to the problems related to their dissolution rates, shape, sizes and chemical composition, they are either nonresorbable that requires second surgery to remove or quickly resorbed by the body's natural physiological process, thus limiting their effectiveness. In addition, past investigations have shown that some antibiotics have been reported to be ototoxic and nephrotoxic at high dosage. For most controlled release systems, the loaded dosages are usually high, and therefore the systemic exposure of antibiotic in blood and urine is the major safety concern.

The concept of pathophysiology of osteomyelitis which has been widely accepted is the infected bone becomes devascularized and the resulting sequestered portions of necrotic cortical bone harbor bacteria. This sequestered, necrotic, infected bone is responsible for the chronicity of osteomyelitis. Pus and granulation tissue then surrounds the infected fragments and through increased intraosseous pressure, bacterial toxins and enzymes, further contribute to devascularization of the surrounding bone. The granulation tissue that surrounds the infected area as the infection becomes chronic is replaced by relatively avascular fibrous tissue, and stimulation to form new reactive bone referred to as involucrum takes place within the surrounding tissues and in the periosteum permeative mesenchymal cells that wraps around the sequestered, necrotic, infected bone. Antibiotics and antibodies must cross this involucrum and relatively avascular fibrous tissue to reach the microorganisms once the fibrous and bony encapsulation takes place. Consequently, the very effective effort by the body to quarantine the host from the infection also isolates the microorganism from the defenses of the host. The infection becomes chronic and cannot be eradicated when this pathological standoff occurs. This condition is the basis for intervention surgery and the excision of necrotic sequestra. Antibiotic therapy completes the treatment of chronic osteomyelitis by eradicating the microorganisms once they are no longer isolated [50].

The quest for a more effective means of delivering antibiotics without the complications related to longterm intravenous access and the toxicity of systemic antibiotics has been ongoing. Hence, the most accurate method of assessing diseases is provided by models that utilize eradication of infection as a criterion for success, the histological findings of new bone formation, inflammation, sequestration and intraosseous bacteria, combined with cultures.

In the past, ceramics as well as other materials incorporated to the thin films and bulk composites have been suggested as potential candidates to be used as biodegradable drug delivery systems. However, manipulating these materials into an appropriate shape with adequate microporosity in order to be fitted into bone defects of different size and form was found to be difficult. Recently, it has been demonstrated that marine shells, foraminifera and corals incorporated into thin films and nanocoatings with specific microspherical carriers offer desired functions for the delivery ofBP (paminodrate) and an antibiotic (gentamicin) [START_REF] Ben-Nissan | Marine materials in drug delivery and tissue engineering: from natural role models, to bone regeneration and repair and slow delivery of therapeutic drugs, proteins and genes[END_REF][START_REF] Chou | Coral exoskeletons as a precursor material for the development of calcium phosphate drug delivery system for bone tissue engineering[END_REF][START_REF] Ben-Nissan | Marine structures as templates for biomaterials[END_REF][START_REF] Chou | Effect of biomimetic zinccontaining tricalcium phosphate (Zn-TCP) on the growth and osteogenic differentiation of mesenchymal stem cells[END_REF][START_REF] Chou | Targeting and dissolution characteristics of bone forming and antibacterial drugs by harnessing the structure of micro-spherical shells from coral beach sand[END_REF]. This has been possible by virtue of its nano and mesoporous structure and architec ture of the foraminifera shells (Figure 2A) which are difficult if not impossible to produce with our cur rent manufacturing methods [START_REF] Ben-Nissan | Natural bioceramics: from coral to bone and beyond[END_REF][START_REF] Milev | Morphological stability of plate-like hydroxyapatite[END_REF]. Foraminifera and coral in addition to their unique interconnected porous structure are made of calcium carbonates that can be easily converted to bone such like phosphate structures [START_REF] Ben-Nissan | Natural bioceramics: from coral to bone and beyond[END_REF][START_REF] Milev | Morphological stability of plate-like hydroxyapatite[END_REF].

Bone repai r drug delivery systems & osteomyelitis

In early 1900s, a calcium phosphate compound (tri calcium phosphate) was first successfully used in bone repaie followed by the first clinical study [!). It took more than six decades until porous calcium phosphate scaffolds were proposed for the treatment of bony defects [START_REF] Hulbert | Potential of ceramic materials as permanently implantable skeletal prostheses[END_REF]. The reason behind the research and devel opment of calcium phosphate based biomaterials for bone repaie, augmentation and substitution was due to the similarity in composition between biological and synthetic apatites [START_REF] Chou | Targeting and dissolution characteristics of bone forming and antibacterial drugs by harnessing the structure of micro-spherical shells from coral beach sand[END_REF].

For bone repaie and slow drug delivery systems, the most appropriate materials are the calcium phos phate based bone substitutes [START_REF] Choi | Calcium phosphate nanocoatings and nanocomposites, part I: recent developments and advancements in tissue engineering and bioimaging[END_REF]. They offer easy pro duction, drug carrying capability and supply both calcium and phosphates during dissolution. They can be easily incorporated within thin films and nanocoat ings [55.56). There have been a large number of stud ies carried out on both experimental and commercial calcium phosphate based drug carriers during the last decade . Due to their wide areas of applicabili ry , the delivery of antibiotics has become a major focus in the prevention and treatment against infection during or postoperative surgical interventions.

Biodegradable composite thin films

Biodegradable polymer thin films loaded with gen tamicin have been synthesized to act as a 'composite coatings' for metallic implants and fracture fixation devices in an attempt to prevent implant-associated infections [48.57-59). Due to their tendency to uptake and release pharmaceuticals and minerais as well as the capability to degrade over time, the use of biodegrad able polymer thin films is advantageous. In addition, by controlling the pore sizes and interconnectivi ry of these particles, the rates of drug release could be tailored to suit the treatment.

In our experimental work [START_REF] Ben-Nissan | Marine structures as templates for biomaterials[END_REF][START_REF] Chou | Effect of biomimetic zinccontaining tricalcium phosphate (Zn-TCP) on the growth and osteogenic differentiation of mesenchymal stem cells[END_REF][START_REF] Chou | Targeting and dissolution characteristics of bone forming and antibacterial drugs by harnessing the structure of micro-spherical shells from coral beach sand[END_REF][START_REF] Legeros | Calcium phosphate-based osteoinductive materials[END_REF][START_REF] Macha | Marine structure derived calcium phosphate-polymer biocomposites for local antibiotic delivery[END_REF][START_REF] Macha | Development and dissolution studies of bisphosphonate (clodronate)containing hydroxyapatite-polylactic acid biocomposites for slow drug delivery[END_REF][START_REF] Macha | Improvement of elongation in nanosurface modified bioglass/PLA thin film composites[END_REF], hydrothermally converted coralline HAp particles containing nano and mesopores were loaded with medically active sub stances, which cover both the surfaces and intercon nected pores of the particles that ranged from a few hundred nanometers to micron sizes (Figure 3).

The influence of HAp particles within PLA matrix on the relea:se of gentamicin as well as its release kinet ics have been previously examined [START_REF] Macha | Improvement of elongation in nanosurface modified bioglass/PLA thin film composites[END_REF]. lt was discov ered that the release kinetics of gentamicin appeared to obey "Power law Korsmeyer Peppas' mode! with mostly diffusional process through a number of different drug transport mechanisms. Statistical analysis revealed a very significant difference on the release of gentamicin between gentamicin-containing PLA (PLAGM) and gentamicin-containing HAp microspheres within PLA matrix (PLAHApGM).

These thin film composite coatings, such as PLA HApGM displays slower release rates than PLA matrix alone. As stated earlier HAp and other calcium phos phates are also the source of Ca 2 • for the regeneration and repaie of diseased bone tissue. lt was also reported that the release profiles, exhibited an early burst stage and then a steady state release rate with significant antimicrobial activity against S. aureus (SHIO0O) even at high concentration of bacteria. The devices also indicated significant ability to control the growth of bacterial even after 4 weeks of drug release. It was sug gested that clinical release profiles can be easily tuned from drug-HAp physicochemical interactions and deg radation kinetics of polymer matrix. lt was concluded that the developed systems could be applied to prevent microbial adhesion to medical implant surfaces and to treat infections mainly caused by S. aureus in surgery. lt was reported that the degradation of the poly meric network towards the end of the initial release will 'favor' the dissolution of any drug residue. Accord ingly, this phenomenon (diffusion and degradation) should take place at the conclusion of surface drug release (when dissolution of drug have created second ary porosity inside the polymeric network and resul tant 'fragility' of it) and/or dependent on factors such has the surface area, dissolution rate and the molecular weight of the PLA film used.

For the coral converted to hydroxyapatite material (coralline HAp) structure contains meso-and nano pores and during the initial drug loading period gen tamicin penetrates the pores of the 'HAp-coral' and coats its surfaces and its porous network. The gentami cin contained in these porosities will be released gradu ally. Hence, the process of polymeric matrix thin film composites containing drug carrying particulate mat ter can be thought to proceed in three stages that can be named as 'progressive dissolution process'.

In the first stage (stage 1), the initial release (burst) of gentamicin can be regarded as the direct dissolution of surface bound drugs in water or the physiologie environ ment due to the exposed outer surface area and concen tration gradient. lt was shown in that the initial burst took approximately 1 week for gentamicin. At this stage it can be assumed that the release is purely governed by diffusion of drugs from polymeric matrix surface [START_REF] Severino | Polymers for drug delivery systems formulations[END_REF][START_REF] Liechty | Polymers for drug delivery systems[END_REF].

In the second stage the dissolution is driven by the internai diffusion of drugs impregnated within the matrix possibly in the porous part of the matrix generated during preparation. For gentamicin release from PLAHApGM samples, this stage is preceded by 'lag phase' which occurs between 1 and 2 h of release. The presence of HAp loaded with gentamicin could in many ways hinder (or slow down) the release of gentamicin through these micropores. This stage is relatively much slower compared with the previous one due to the drug transporting through mainly from the surface areas and large pores of the particles. This stage proceeds with release from narrow pores of the nano and mesopores of the particles to the matrix and then to the environment. There is no degradation of particles at this stage but dissolution of the drugs only.

The third stage involves slow degradation of the polymeric matrix combined with dissolution and deterioration of the particles (Figures 4 &2B) and slow release of the added drugs or the minerais into the environment. This is terminal release phase or stage for loaded devices. At this stage, there is pore growth due to both mass loss by polymer degradation and pore coalescence (micropores coalescing (or joining) to form larger pores). In recent dissolution studies of BPs, for the device containing HAp (PLAHApBP), the slower dissolution rates observed were reported to be due to the strong bonding of BP to apatite (HAp) as well as encapsulation within the particles. It was ear lier reported that for BP containing calcium phosphate particles have strong affinity to the nanocrystalline apatites with adsorption phenomena that occurs at the surface of apatite crystals [START_REF] Pascaud | Interaction between a bisphosphonate, tiludronate and nanocrystalline apatite: in vitro viability and proliferation of HOP and HBMSC cells[END_REF].

Antibacterial efficacy & biofilm formation

Published work states that poor diffusion and penetra tion of antibiotic through the biofilm, contribute to A number of reasons on the microbial resistance to antimicrobial agents have been postulated. An increase in the deple tion of oxygen and nutrients resulting into slow growth of bacteria, adaptive stress responses and formation of persister cells are hypothesized to constitute a multi layered defense. In recent research efforts the focus is directed toward disabling biofilm resistance, which may enhance the ability of existing antibiotics to treat infections involving biofilms [67). lt has been reported that in most cases, biofi lm can be prevented aggres sively by antibiotic in their early stages and can also be treated by chronic suppressive therapy. Mah et al. [START_REF] Mah | A genetic basis for Pseudomonas aeruginosa biofilm antibiotic resistance[END_REF] suggested that the use of traditional antibiotics com bined with drug that interferes with biofilm-specific resistance would be the right approach to render biofi lms more susceptible to treatment. A recent study by our group has suggested that effec tive drug delivery devices with adequate slow release rates can be produced using converted coralline HAp particles loaded with medically active drugs and sub stances embedded within a PLA thin film matrix [START_REF] An | Concise review of mechanisms of bacterial adhesion to biomaterial surfaces[END_REF][START_REF] Macha | Development and dissolution studies of bisphosphonate (clodronate)containing hydroxyapatite-polylactic acid biocomposites for slow drug delivery[END_REF][START_REF] Macha | Improvement of elongation in nanosurface modified bioglass/PLA thin film composites[END_REF]. To investigate the antibiotic delivery efficiency of the composite thin films produced and observe the biofi lm formation, we carried out a number of tests.

In this current work, we prepared PLA films and film composites based on the previously published work and loaded with gentamicin. Schematic repre sentation of the process is given in Figure 5. The films were eut inro circular shape and glued on the sterilized cylindrical coupons. They were sterilized using UV for 40 min. Tr ytpic Soy Broth (TSB) for S. aureus and Mueller Hinton II BR cation adjusted media (MHB) for P. aeru g onas were prepared by thoroughly dissolv ing 30 g of TSB powder (Bacto TM TSB (BD) and 22 g of MHB powder in I I of polished 18 MQ (MilliQ, Millipore, Victoria, Australia) water, respectively. The solutions were autoclaved to sterilize (121 °C, liquid cycle) and stored at room temperature.

Staphylococcus aureus

(ATCC 25923) and P. aeruginosa (ATCC 15692) were used for this study and were cultured in a shaking incubator, 250 rpm at 37 °C under anaerobic and aerobic conditions, respec tively. For static biofilm formation, cells were grown in broth medium overnight, and diluted I: 100 into TSB and MHB media, respectively. These cell suspensions were inoculated into a 12-well plate containing tripli cate of samples glued on coupons. The plate was sealed with sterile breathable film (Aeraseal; Excel Scientific, CA, USA) and statically grown at 37 °C, 5% C0 2 for 24 h. Biofilm samples were washed with PBS, stained using SYT09 Green fluorescent Nucleic Acid Stain (Life Te chnologies Corp, CA, USA) and fixed with 4% paraformaldehyde.

The morphological changes of the biofilms were anal yz ed using confocal laser scanning microscopy (CLSM; Nikon AI, Tokyo, Japan), using oil-immer sion lens (70 Objective lens and numerical aperture of 1.4 with Z-series images taken in 1.0 mm slices) with NIS Elements Confocal software. A total of eight images were acquired randomly f r om each specimen.

In biofilm study, four biofilm image features calcu lated by COMSTAT [START_REF] Heydorn | Quantification of biofilm structures by the novel computer program COMSTAT[END_REF] were chosen to characterize biofi lm development by S. aureus and P. aeruginosa on PLA thin film composites. These variables, biomass, average chickness, roughness coefficient and surface to biovolume ratio were selected for interpretation of biological and physical characceristics of biofilm [START_REF] Stigter | Incorporation of tobramycin into biomimetic hydroxyapatite coating on titanium[END_REF] on these surfaces. Biomass represents the overall vol ume of the biofilm, and also provides an estimate of the biomass in the biofilm, average thickness provides a measure of the spatial size of the biofilm, roughness represents a measure of biofilm heterogenei ty and sur face to biovolume ratio provided us how large a por tion of the biofilm is exposed to the nutrients flow. Recorded CLSM images were reconstructed using IMARIS imaging system (Bitplane AG, Zurich, Swit zerland) for biofilm structural quantification in com puter statiscics software COMSTAT and presented as 3D structures.

The preliminary resulcs obcained from 5 days experiments for S. aureus (SHIO00), and Pseudomonas aeroginosa on PLA thin film samples are both strains ac day 5 had grown into micro colonies, reflecced by their low surface to volume ratio compared with bio film at day 1. On the other hand, microscopy images of S. aureus on the films showed large and high micro colonies on PLA, PLAHAp and PLAHApGM samples while on PLAGM high single cells and small cell dus cers were observed (Figure 6). Confocal images on the composites without and wich gentamycin shows the intensity of the biofilm formation (Figure 6A &C) and the effect of the antibiotics after 24 h (Figure 6B &D). le can be suggested that cells were able to attach on the surface of PLAGM samples but the drug released from the surface suppresses the ability of bacteria to make biofilm. Ir could also be envisaged chat there is possibility under flow-biofilm-growth conditions chat more drugs would be released from the surface and may suppress the attachment of bacteria on the surfaces. Being high-level antimicrobial resistant in Gram-negacive, P. aeroginosa displays similar structure characceriscics on the surface of PLA thin films.

Pseudomonas aeroginosa showed a stronger cendency to form microcolonies on the surface of PLA films chan S. aureus, which was indicated by higher rough ness coefficients. The higher surface to biovolume ratio of P. aeroginosa is an indication of its fiat growth on the surface compared with S. aureus. This is also consistent with the lower average chickness of the biofilm it forms on the surface. le was also observed chat S. aureus grows faster chan P. aeroginosa on PLA surfaces indicated by lower surface to volume ratio, higher average thickness and hence higher biomass in the biofilm.

The effect of antibiocic in the films (PLAGM and PLAHApGM) against biofilm after 24 h seemed to be minimal for S. aureus and no at ail to P. aerogi nosa. This is possibly due the fact that starie conditions may influence low release rate of drugs f r om the device and also 24 h is not enough cime to realize polymer degradation and significant release of antibiotic. In summary , ail of the PLA and PLA/HAp genta micin loaded chin films under experimental conditions exhibiced significant abili ty to prevent bacterial growth even at high concentration of baccerial. The prolonged ability to release drug from these films was tested by subjecting films for 4 weeks under the same experimen tal conditions. Ir was observed that even after releasing drugs for 4 weeks, films were still able to release enough gentamicin to prevent microbial accivicies.

Conclusion & future perspective

At the moment, we are gradually discovering new ways of reproducing structures that are commonly found in nature with desirable properties. The use of biological microstructures is one versatile approach for the repro duction of inorganic structures with identical features. This is achieved by using techniques in biomineral inspired self-repairing materials chemistry. Multifunc tional and multilayered nano coatings and assembly will assise us in this endeavor.

Slow drug delivery and tissue engineering are two common tools chat could assise to help us to solve a number of bone related deficiencies and repaie. Cur rently, the demand is clear for becter tissue engineering scaffolds that possess more natural bioresponsive envi ronments favorable in guiding the natural processes of regeneration. To meet this biological challenge, nove! design and synthesis steps must be incorporated into the new generation scaffolds. We believe that there needs to be a new thinking in tissue scaffold systems that are responsive in which the synthesized bioma trix evolves in real-time to meet the requirements and optimize for the adaptive growth and regeneration of human tissues, while delivering the right biogene tic materials, pharmaceuticals and minerais. Con sequently, the environments of the scaffolds will be further adjusted as cells proliferate and differentiate. 3D printing methods are currently used for a number of inorganic and metallic materials and stem cells and other biologie structures are incorporated to these new generation 3D materials for organ regeneration and repa1r.

During the last decade atomic force microscopy was used as a powerful platform in nanomedicine for studying and controlling the forces involved in ce!! adhesion and biofilm formation. Atomic force microscopy is well regarded in measuring the small interaction forces between a sharp probe and the sur face of a sample. A relatively more recent equipment the single-cell force spectroscopy (SCFS), a single cell can be attached on the probe in order to measure the interaction forces between the cell probe and a solid substrate or another cells. SCFS-based techniques have recently established as an important tool for understanding how microbial pathogens attach to surfaces and form biofilms [START_REF] Dufrene | Understanding forces in biofilms[END_REF]. This can be a turning point in our fight against biofilms and infections. SCFS assays will allow us to clarify the specific and nonspecific forces driving cell adhesion on a singlecell basis. Future work in this area will open new avenues for the development of new tools, devices, methods and multifunctional coatings capable of detecting and destroying biofilms.

Nevertheless, nanosynthesis based on biological principles of design and assembly is still in its infancy. Better understanding of biofilm formation and the cell forces is pertinent. The use of bio-inspired nanofabrication techniques including multifunctional multilayer coatings in nanoscale thicknesses for slow drug delivery and tissue repair is a unique approach. This has enormous potential to improve scaffold or lab-on-chip designs with the capability to diagnose, identify, stop biofilm formation, self-repair, micro-evolve and osteointegrate fully. No writing assistance was utilized in the production of this manuscript.

Executive summary

Background

• Currently, researchers are focusing on the fabrication of new nanocoatings, nanomaterials, nanolaminates and thin film nanocomposites that are appropriate for applications such as drug delivery and tissue engineering. • Innovative approaches driven by nanotechnology are now available for the production of synthetic bone-like calcium phosphate nanomaterials.

Surface modifications and liposomes for drug delivery applications

• The appropriate circulation time in addition to the dissolution rates within the human body is the main issue for drug carriers containing nanoparticles. • The surfaces of nanostructured materials such as nanocoatings can be modified and functionalized with several types of reagents via various biological, chemical and/or physical methods. • The surface modification approach can be used to achieve functional improvements by design, better osteointegration and stability of nanomaterials and nanocoatings in aqueous media.

Infection & osteomyelitis

• Treatment of osteomyelitis can be with immobilization and antibiotic therapy with a number of drugs including gentamicin. • Modification or development of thin film or nanocoated multilayer devices with surface properties that have an effect against microbial adhesion or viability seems to be a promising approach for the prevention of device-related infections. • Bacteria adhesion is a complex phenomenon affected by factors such as the properties of surface materials and characteristics of the bacteria.

Bone repair drug delivery systems & osteomyelitis

• The delivery of antibiotics has become a major focus in research for the use in the treatment of bone infections or as prevention against infection during surgical interventions. • Although not fully satisfactory, the implantation of antibiotic-loaded PMMA microspheres and calcium sulfate based microspheres and powders into the infection site are the approach currently being used in both orthopedics and in maxillofacial surgery. • Biodegradable calcium phosphate drug delivery system would be better suited to this endeavor as it can also introduce calcium and phosphate ions and range of minerals to assist bone growth.

Biodegradable composite thin films

• The use of biodegradable polymer thin films is advantageous in the fight against bacterial infection.

• This is due to their tendency to uptake and release pharmaceuticals and minerals as well as the capability to degrade over time.

Figure 1 .

 1 Figure 1. Transmission electron microscopy image of multilamellar lipsomes containing nano hydroxyapatite particles and minerais in a calcifying buffer (scale 500 nm). Reproduced with permission from [42].

Figure 2 .

 2 Figure 2. (A) Foraminifera hydroxyapatite microsphere. (B) Enlarged degraded surface of Foraminifera hydroxyapatite structure within simulated physiological environment.

Figure 3 .

 3 Figure 3. (A) Hydrothermal converted coralline hydroxyapatite surface. (B) Gentamicin-coated coralline hydroxyapatite surface.

1 fFigure 4 .

 14 Figure 4. Gentamicin drug delivery device produced from polylactic acid-coralline hydroxyapatite thin film composite. (A) The scanning electron microscopy image of the coralline HAp surface used for drug delivery (only the struts are used as drug delivery vehicles shown in a circled area), and (B) drug release profile of only the gentamicin within PLA (PLAGM) and HAp coralline particles loaded with gentamicin embedded within a PLA matrix (PLHApGM) showing lower amounts of drugs released as well as a longer release rate up to 8 weeks. GM: Gentamicin; HAp: Hydroxyapatite; PLA: Polylactic acid. the persistence of biofi lm infections especially those associated with implanted devices [63-66].A number of reasons on the microbial resistance to antimicrobial agents have been postulated. An increase in the deple tion of oxygen and nutrients resulting into slow growth of bacteria, adaptive stress responses and formation of persister cells are hypothesized to constitute a multi layered defense. In recent research efforts the focus is directed toward disabling biofilm resistance, which may enhance the ability of existing antibiotics to treat infections involving biofilms [67). lt has been reported that in most cases, biofi lm can be prevented aggres sively by antibiotic in their early stages and can also be treated by chronic suppressive therapy. Mah et al.[START_REF] Mah | A genetic basis for Pseudomonas aeruginosa biofilm antibiotic resistance[END_REF] suggested that the use of traditional antibiotics com bined with drug that interferes with biofilm-specific resistance would be the right approach to render biofi lms more susceptible to treatment.A recent study by our group has suggested that effec tive drug delivery devices with adequate slow release rates can be produced using converted coralline HAp particles loaded with medically active drugs and sub stances embedded within a PLA thin film matrix[48,58- 59). To investigate the antibiotic delivery efficiency of the composite thin films produced and observe the biofi lm formation, we carried out a number of tests.In this current work, we prepared PLA films and film composites based on the previously published work and loaded with gentamicin. Schematic repre sentation of the process is given in Figure5. The films

HApFigure 5 .

 5 Figure 5. Schem atic representation of the drug loading and release method used in this current work. GM: Gent amycin; HAp: Hydroxyapatite; PBS: Phosphate-buffered saline; PLA: Polylactic acid.

Figure 6 .

 6 Figure 6. Confocal microscopy images showing 24 h biofilm growth of Staphylococcus aureus. Biofilm growth on (A) PLAHAp, (B) PLAHApGM films and their intensity map reflecting biofilm thickness in (C &D), respectively, where (B & D) reflects the effectiveness of the gentamicin-loaded composites. GM: Gentamycin; HAp: Hydroxyapatite; PLA: Polylactic acid .

  sity of Technology Sydney for the partial financial support of one of the authors during his PhD studies. The authors have no other relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript apart from those disclosed.

Acknowledgements

The authors would like to give special thanks to B Milthorpe, Faculty of Science UTS and R Cavaliere of iThree Institute, UTS for their assistance on biofilm work and for their continued support in search of the most appropriate biomaterial.

Financial & competing interests disclosure

The authors would like to acknowledge and thank the Australian Academy of science-Horizon2020 grant and the Univer-


Étienne Birmelé
email: etienne.birmele@parisdescartes.fr

Fabien De Montgolfier

Léo Planche
email: leoplanche@gmail.com

On the Minimum Eccentricity Isometric Cycle Problem

Keywords: isometric cycle, eccentricity, k-domination, k-covering

In this paper we investigate the Minimum Eccentricity Isometric Cycle (MEIC) problem. Given a graph, this problem consists in finding an isometric cycle with smallest possible eccentricity k. We show that this problem is NP-Hard and we propose a 3-approximation algorithm running in O(n(m + kn)) time.

Introduction

For both graph classification purposes and applications, it is an important issue to summarize a graph into a more simple object such as a tree or a path, or in the case of this article, a cycle. Different constructions and metrics offer such a characterization, for example tree-decompositions and tree-width [START_REF] Robertson | Graph minors. I. Excluding a forest[END_REF]. Another approach, on which we focus in this article, is to study the problem in terms of domination.

In the path case, the problem consists in finding a path such that every vertex in the graph belongs to or has a neighbor in the path. Several graphs classes were defined in terms of dominating paths. Graphs containing a dominating pair, that is vertices such that every path linking them is dominating are studied in [START_REF] Deogun | Dominating pair graphs[END_REF]. Graphs such that short dominating paths are present in all induced subgraphs are characterized in [START_REF] Bacsó | Characterization of graphs dominated by induced paths[END_REF]. Linear-time algorithms to find dominating paths or dominating vertex pairs were also developed for AT-free graphs [START_REF] Corneil | A linear time algorithm to compute a dominating path in an AT-free graph[END_REF][START_REF] Corneil | Linear time algorithms for dominating pairs in asteroidal triple-free graphs[END_REF]. Dominating paths do not exist however in every graph. A natural extension of the notion of domination is the notion of k-coverage (also called k-domination). For a given integer k, a path k-covers the graph if every vertex is at distance at most k from the path. The smallest k such that a path k-covers the graph is then a metric as desired.

Another formulation for this metric is the notion of eccentricity. Given a graph G, x ∈ V (G) and S ⊂ V (G), we note d(x, S) = min s∈S d G (x, s). The eccentricity of S, denoted ecc(S), is the smallest k such that for any x ∈ V (G), d G (x, S) ≤ k.

The minimum-eccentricity shortest path (MESP) problem was introduced by Dragan and Leitert [START_REF] Dragan | On the minimum eccentricity shortest path problem[END_REF]. The name is transparent as it consists in finding a shortest path with minimal eccentricity. The study of this problem was originally motivated by its link to the embedding with low distortion of graphs into the line. Dragan and Leitert have shown that this problem is NP-complete, and designed some approximation algorithms, and established its relationship with the Line Embedding problem [START_REF] Dragan | On the minimum eccentricity shortest path problem[END_REF]. The MESP problem is also related with graphs problems arising from biological data set [START_REF] Birmelé | Decomposing a graph into shortest paths with bounded eccentricity[END_REF].

A problem related to the MESP problem consists in finding an isometric cycle with minimal eccentricity, as defined in [START_REF] Birmelé | Decomposing a graph into shortest paths with bounded eccentricity[END_REF]. A cycle is isometric if it preserves distances:

Definition 1.1 [Isometric cycle] Let G be a graph and C a cycle of G. d G denotes the distance in G and d C the distance in G[C].
C is an isometric cycle if and only if, for every two vertices u, v of C we have:

d G (u, v) = d C (u, v).
In other words, one of the two paths linking u and v in the cycle is a shortest path in the graph. Note that an isometric cycle is necessarily an induced cycle. The problem we are interested in consists in finding an isometric cycle with minimum eccentricity. Definition 1.2 [Minimum Eccentricity Isometric Cycle Problem (MEIC)] Given a graph G, find an isometric cycle C such that, for every isometric cycle U , ecc(C) ≤ ecc(U ).

A longest isometric cycle may be computed in O(n 4.752 log(n)) time [START_REF] Lokshtanov | Finding the longest isometric cycle in a graph[END_REF], which gives a polynomial-time 3-approximation for the MEIC problem [START_REF] Birmelé | Decomposing a graph into shortest paths with bounded eccentricity[END_REF]. It was also shown that a graph admitting an isometric cycle with low eccentricity admits a cycle embedding of low distortion [START_REF] Birmelé | Decomposing a graph into shortest paths with bounded eccentricity[END_REF].

In this paper, we first show that the MEIC problem is NP-complete. To do so, we propose a reduction of MESP to MEIC. Then we propose a O(n(m + kn))-time 3-approximation algorithm. This is faster than the O(n 4.752 log(n)) time algorithm proposed in [START_REF] Birmelé | Decomposing a graph into shortest paths with bounded eccentricity[END_REF] but with the additional constraint that the size of the isometric cycle of minimal excentricity has to be at least 32 times its eccentricity (see Theorem 4.6). Notice that finding a minimum-eccentricity isometric cycle is useful only in "donut-shaped" graphs, implying the cycle is long and has short eccentricity.

Definitions and Notations

Through this paper we consider finite connected undirected graphs. For a graph G = (V, E), we use n = |V | and m = |E| to denote the cardinality of the vertex set and the edge set of G. A shortest path between two vertices u and v is a path whose length is minimal among all u, v-paths. Its length (counting edges) is the distance d(u, v). When the precision is needed, we denote d G (u, v) the distance between two vertices u and v in the graph G. Depending on the context, we consider a path either as a sequence, or as a set of vertices. The distance d(v, S) between a vertex v and a set S is the smallest distance between v and a vertex from S. The eccentricity ecc(S) of a set S is the largest distance between S and any vertex of G. The ball of center v and radius r, noted B(v, r), is the set of vertices at distance at most r from v. Given two sets of vertices A and B, we note A \ B the set of vertices of A which are not in B.

NP-completeness

We propose a polynomial reduction of MESP to MEIC.

Let f be the function which associates to a graph with vertices V (G) = {v 1 , ...v n } a family of graphs f (G) = {H (i,j)∈{1...n} 2 } such that for every (i, j) ∈ {1...n} 2 , H (i,j) is the graph G to which a path P i,j is added of size 2n between v i and v j .

Lemma 2.1 Given a graph G, the shortest path of minimal eccentricity of G is of eccentricity k if and only if, for any graph of f (G), its isometric cycle of minimal eccentricity is of eccentricity at least k.

Proof. Let us show that for every graph H i,j in f (G), any isometric cycle of minimal eccentricity contains P i,j and a shortest path of minimal eccentricity in G between v i and v j .

Claim 1 : Any isometric cycle of minimal eccentricity in H i,j contains P i,j .

Consider any cycle C of G. It does not contain any vertex of P i,j other than v i and v j . The path P i,j being of length 2n, the eccentricity of the cycle C in H i,j is of size at least n. Consider now any cycle in H i,j containing P i,j and a shortest path between v i and v j . This cycle is isometric and of eccentricity at most n -2.

Claim 2 : Any isometric cycle of minimal eccentricity in H i,j contains P i,j and a shortest path of minimal eccentricity in G between v i and v j .

We have already established that such a cycle C contains P i,j and a path Q in G between v i and v j . Furthermore, Q or P i,j has to be a shortest path as C is isometric. The distance d G (v i , v j ) being at most n -1 and P i,j being of size 2n, Q is thus a shortest path linking v i and v j . Moreover, as every shortest path between any vertex of P i,j and any vertex of G contains either v i or v j , ecc(C) = ecc G (Q), where ecc G stands for the eccentricity in the graph G. C being of minimal eccentricity among the isometric cycles, Q is of minimal eccentricity among the shortest paths between v i and v j . 2

Lemma 2.1 implies that the solution of MESP in G may be computed by solving MEIC in f (G), and the transformation f is clearly polynomial. MESP being an NP-complete problem [START_REF] Dragan | On the minimum eccentricity shortest path problem[END_REF], it follows that : Theorem 2.2 The problem MEIC is NP-complete.

Preliminary results

We define for every cycle C an arbitrary cyclic orientation and for every u, v vertices of the cycle, we note C uv and C vu the two paths in C linking u and v. In order to make proofs clearer, we define the operator ' as follows :

Definition 3.1 Let G be a graph with an isometric cycle C k-dominating G. For every vertex x in G, x denotes a vertex of C at distance at most k from x, randomly chosen if several choices are possible.

The following lemma is a crucial preliminary result, which will allow us later to build 3k-dominating cycles if the graph contains an isometric cycle k-dominating it. Lemma 3.2 Let G be a graph with an isometric cycle C k-dominating G, and u and v be any two vertices in G. Every path between u and v 2k-dominates either C u ,v or C v ,u .

Proof.

Notations used in the proof are illustrated in Figure 1.

Let P be a path between u and v. Suppose that P does not 2k-dominate some vertex b on the path C u ,v and consider any vertex a in C v ,u . We have u (resp. v ) in the path In this section we develop a new approximation algorithm for the MEIC problem, with a better complexity compared to [START_REF] Birmelé | Decomposing a graph into shortest paths with bounded eccentricity[END_REF] (Theorem 4) but requiring a graph with a longer isometric cycle. Consider a graph with an isometric cycle of eccentricity k. Algorithm 1 (F irstCycle) first computes a cycle U 0 of eccentricity at most 3k but non necessarily isometric. Its size is then iteratively reduced while keeping an eccentricity of at most 3k. The algorithm ends when the last computed cycle is isometric.

Proof.

Notations used in the proof are illustrated in Figure 1.

If F irstCycle(G) returns a cycle U 0 , it is the union of two paths P and P . Let m be a vertex in the middle of P (randomly chosen among the two possible ones if -P-is odd) and assume w.l.o.g. that m is in C a b .

Claim 1: P is of length at least 12k. Let a be a vertex of C furthest from a .

ecc(a) ≥ d(a, a ) ≥ d(a , a ) -k ≥ C 2 -k ≥ 12k + 3
As b is a vertex of G furthest from a, we have the result. The other inequality is a direct result of Lemma 4.2.

) ≥ |P | 2 -2k d(m , b ) ≥ d(b, m) -d(b, b ) -d(m, m ) ≥ |P | 2 -2k so that d(m, y) ≥ |P | 2 -3k ≥ |P | 4 + |P | 4 -3k > |P | 4
C b a is therefore at distance at least |P | 4 of m. Furthermore, d(a, m) = |P | 2 ≥ |P | 4 + 3k d(b, m) = |P | 2 ≥ |P | 4 + 3k
Claim 5: U 0 does not contain the same vertex twice Assume that a vertex x appears more than once in U 0 . As both P and P are shortest path, they each contain x at most (and exactly) once. If x is in P c,m then it is more distant from a than c and it should have been selected at line 8 instead of c. This leads to a contradiction. The same contradiction appears if we assume x in P m,d . 2 The function F irstCycle creates a cycle U 0 , not necessarily isometric, 3k-dominating the graph. Starting with U 0 , the size of the cycle is iteratively decreased by the N extCycle procedure described by Algorithm 2, while keeping the 3k-domination property.

NextCycle

Input: A graph G and a cycle U i Output:

A cycle U i+1 2 for a ∈ U i do 3 If S = {b | d U i (a, b) = |U i | 2 and d G (a, b) < d U i (a, b)} = ∅ then 4 Compute Q a shortest path between a and b 5 Compute the eccentricity of Q ∪ U i a,b and Q ∪ U i b,a 6
Return the set with the lowest eccentricity Proof.

7 Return U i
Notations used in the proof are ilustrated with figure 2. Let u and v be two vertices of C such that u and v both belong to P and such that P 2k-dominates C u v (such a pair exists as we may always pick u = v ). Assume that there exists a vertex w not in C u v and such that w is in P . By Lemma 3.2, P 2k-dominates either C u w or C w u . In the first case, C u v is a subset of C u ,w . In the second case, P 2k-dominates C w u and C u v hence C w v . As v is in C u w and u is in C w v , the sizes of C u w and C w v are strictly larger than the one of C u v . The length of C being finite, there exists a pair of vertices verifying the property and such that the considered arc is maximal. 2 Let z be the vertex of C a b at distance 4k + 1 of w . U is 3k-dominating G by hypothesis, therefore z is at distance at most 3k of a vertex y of U . The fact that d(w , z) > 4k then implies that y is not in C a ,w . We now have to study three different sub-cases, corresponding to y ∈ U a,w , y ∈ U w,b and y ∈ U b,a , but the arguments are very similar. Let U i be a cycle such that:

• U i 2k-dominates C. • |C| -6k -2 ≤ |U i | ≤ |C| + 4k
Then U i+1 = N extCycle(U i ) has the same properties and

|U i+1 | < |U i |.
Proof. The inequalities concerning the length are straightforward given Lemma 4.2 and the fact that U i+1 is obtained by replacing an arc of U i with a path of smaller length.

To prove the 2k-domination, consider the path Q linking the vertices a and b of U i as selected by the algorithm. By Lemma 3.

Fig. 1 .

 1 Fig. 1. Notations used in Lemma 3.2 (left) and in Lemma 4.1 (right)

1 ) then 7 Let P be a shortest path between a and b in G 8 9Algorithm 1 :

 1781 FirstCycle Input: A graph G Output: A cycle U 0 2 Let a be any vertex of G 3 Compute b a vertex furthest from a 4 Compute P a shortest path from a to b 5 Let m be a vertex in the middle of P 6 If a and b are connected in G = G \ B(m, |P | 4 Let c be the vertex of P a,m ∪ P furthest from a Let d be the vertex of P m,b ∪ P furthest from b 10 Return P c,d ∪ P c,d 11 Return an error "MEIC too short with respect to k" Pseudocode of the algorithm F irstCycle Lemma 4.1 Let G be a graph with a k-dominating isometric cycle C. If C is of size at least 26k + 6 then F irstCycle(G) returns a cycle U 0 2k-dominating C. Furthermore U 0 is of size at most |C| + 4k and at least |C| -2k -2.

Claim 2 :

 2 B(m, |P | 4 ) does not disconnect a from b. Consider any vertex y in C b a , as m is in C a b , d(m, y) ≥ d(m , y) -d(m, m ) ≥ min(d(m , a ), d(m , b )) -k But d(m , a ) ≥ d(a, m) -d(a, a ) -d(m, m

It follows that the paths of length k from a to a and b to b are at distance greater than |P | 4 + 4 from

 44 2k of m. The vertices a and b are thus connected in G . This claim implies that F irstCycle(G) returns a cycle.Claim 3: U 0 2k-dominates C. Without loss of generality, assume that m is in C c d . Let us first show that P cm 2k-dominates C c m . To do so, assume that it is not the case. Then, by Lemma 3.2, P cm 2k-dominates C m c . The vertex m being in C c d , it follows that d is in C m c and that there exists a vertex x in P cm at distance at most 2k from d . It follows that,|P cm | = d(c, x) + d(x, m) |P cm | ≥ d(c, d) -3k + d(d, m) -3k d(c, m) ≥ d(c, m) + d(m, d) + d(d, m) -6k 6k ≥ 2d(d, m)As d is a vertex of P , it is at distance more than |P | fact that P is of size larger than 12k, hence P cm 2k-dominates C c m . Similarly, we show that P md 2k-dominates C m d , it follows that P 2k-dominates C c d . Lemma 3.2 moreover implies that P 2k-dominates either C c d or C d c . But, as |P | 4 is greater than 3k, m cannot be 2k-dominated by a vertex in G . As m is on C c d , P 2k-dominates C d c . It follows that U 0 2k-dominates C. Claim 4: |C| -6k -2 ≤ |U 0 | ≤ |C| + 4k |P | ≤ |C a b | + 2k as P is a shortest path between a and b. Similarly, as P is a shortest path in G , |P | ≤ |C b a | + 2k. Consequently, |P | + |P | ≤ |C| + 4k.

Lemma 4 . 2 2 ,

 422 Let G be a graph with an isometric cycle C k-dominating G. Let a and b be two vertices, possibly a = b, linked by a path P 2k-dominating C a ,b . Then, |P | ≥ |C a b | -6k -2.In particular, if U is a cycle 2k-dominating C, U is of size at least |C| -6k -2.Proof. Consider any two vertices a and b and a path P linking them and 2k-dominating C a ,b . Let m be the middle of C a ,b and z a vertex of P which is 2k-dominating m. C a ,m being of length at most |C| |C a ,m | ≤ d(a , m) + 1. Similarly, |C m,b | ≤ d(m, b ) + 1. It follows that |C a b | ≤ d(a , m) + d(m, b ) + 2 ≤ (k + |P a,z | + 2k) + (k + |P z,b | + 2k) + 2 ≤ |P a,b | + 6k + 2 The affirmation concerning the cycle follows by choosing a = b. 2

Algorithm 2 :Fig. 2 .

 22 Fig. 2. Illustration of Lemma 4.3

Lemma 4 . 4

 44 Assume that |C| ≥ 32k + 7. Let U be a cycle of size at most |C| + 4k which 3k-dominates G and let a and b be two vertices of U . Then, C a b and C b a are both included in either I(U a,b ), or in I(U b,a ). Proof. Assume that this is not the case. By symmetry, assume that C a b is neither included in I(U a,b ), nor in I(U b,a ). Let w 1 be the extremity of I(U a,b ) such that C a w 1 ⊂ C a b and w 2 the extremity of I(U b,a ) such that C a w 2 ⊂ C a b . By symmetry, assume that C a w 2 ⊂ C a w 1 and let us note w = w 1 . There exists w ∈ U a,b such that d(w, w ) = k. Case 1: d(w , b ) ≤ 4k + 1 In this case, |C b ,w | ≥ |C| -4k -2. By definition of w , C b ,w ⊂ I(U w,b ) hence is 2k-dominated by U w,b . The Lemma 4.2 then implies that |U w,b | ≥ |C b ,w | -6k -2 ≥ |C| -10k -4. By the definition of w , we also have that C a ,w ⊂ I(U a,w ). Furthermore C b ,a ⊂ I(U b,a ) by Lemma 3.2 as, by hypothesis, I(U b,a ) does not contain C a ,b . Therefore, C b ,w ⊂ I(U b,w ) and by the Lemma 4.2, |U b,w | ≥ |C b ,w | -6k -2 ≥ |C| -10k -4. Finally, |U | = |U b,w | + |U w,b | ≥ 2|C| -20k -8. As |U | ≤ |C| + 4k, it implies that |C| ≤ 24k + 8, which contradicts our hypothesis.

Case 2 :

 2 d(w , b ) ≥ 4k + 1

Fig. 3 .

 3 Fig. 3. Notations used in the proof of Lemma 4.4, in case 1 on the left and case 2 on the right

  2, one can assume w.l.o.g. that I(Q) contains C a ,b and therefore 2k-dominates it. Furthermore, Lemma 4.4 implies that C b ,a is included in I(U i a,b ) or I(U i b,a ). By symmetry , assume that it is included in I(U i a,b ). Then, I(Q ∪ U i a,b ) = C, and it follows that Q ∪ U i a,b 2k-dominates C and is of eccentricity at most 3k. The property is verified ifU i+1 = Q ∪ U i a,b . If U i+1 = Q ∪ U i b,a , Q ∪ U i b,a also 3k-dominates G as it is of a lower eccentricity. Lemma 4.4 then implies that either U i b,a or Q 2k-dominates C b a , and therefore Q ∪ U i b,a 2k-dominates C. 2

ApproximationCycle

Input: A graph G Output: We know by Lemma 4.5 that the length of the cycles of the sequence U i strictly decreases and is at least |C|-6k-2. As |U 0 | ≤ |C|+4k, it follows that the sequence converges after at most one execution of F irstCycle and at most 10k + 2 executions of N extCycle. Not forgetting the O(nm)-time distance precomputation, we have : Theorem 4.6 Let a graph with an isometric cycle of eccentricity k and of size ≥ 32k + 7, an isometric cycle of eccentricity at most 3k may be computed in O(n(m + kn)) time.


C Quilliet

Depressions at the surface of an elastic spherical shell submitted to external pressure

Keywords: 46.70.De (Beams, plates and shells), 46.32.+x (Static buckling and instability), 89.75.Kd (Patterns)

Depressions at the surface of an elastic spherical shell

submitted to external pressure.

Introduction

Buckling, or sudden change of the shape of objects under constraint, has been a problem addressed for a long time, as it is of utmost practical importance (architecture, designing of tough containers or pipes etc) [1][2][3] . More recently, buckling was investigated at much smaller scales, mainly due to the improvement of observation and manipulation techniques at microscopic, or even nanoscopic level [START_REF] Komura | Good review of recent advances[END_REF] . This brings back into the spotlight the simplest symmetry that provides a closed vessel, the spherical one. In spite of important theoretical advances motivated by air and spatial navigation 5 , the spherical symmetry had been somewhat neglected compared to other symmetries (axisymmetrical, cylindrical etc) because less directly encountered in large scale systems where gravity plays a role. At smaller scales, when gravity becomes negligible, whole spheres are more commonplace: they are encountered in colloids, biological objects, and in most of the phenomena linked to surface tension. Flows, encapsulation processes, evaporation/dissolution phenomena in complex fluids may lead to buckling processes, generating non-trivial shapes. Motivated by recent experimental works [6][7][8] , we focus in this paper on thin spherical shells of an elastic and homogeneous material submitted to an isotropic constraint, such as an external pressure. The onset of buckling of hollow spheres submitted to an external pressure has been derived in earlier times [START_REF] Landau | Theory of elasticity[END_REF][START_REF] Pogorelov | Bendings of surfaces and stability of shells[END_REF] , recent theoretical work dealing mainly with the nonideality of materials 11 . For what concerns postbuckling, no analytical predictions concerning post-buckling shapes of spheres under isotropic external pressure are available, by lack of a complete buckling analysis.

Indeed, there is a lively field in biophysics aimed at understanding shapes of biological objects with simple physical ingredients. Since early work by Helfrich and others where bending elasticity and spontaneous curvature 12 (or area-difference elasticity 13 , which can be modelized in the same way [START_REF] Mukhopadhyay | 15 Among many papers dealing with this subject : R. Lipowsky[END_REF] ) were the main shape tuners 15 , some models now include additional stretching elasticity, which brings them quite close to what is discussed hereafter. To our knowledge, these are nevertheless inaccurate to interprete the buckling of homogeneous shells, as they precisely take into account inhomogeneities characteristic of the biological objects they aim to describe. For the closest models, these inhomogeneities are either the different origin (lipidic membrane and cytoskeleton) of bending and stretching elasticity [START_REF] Mukhopadhyay | 15 Among many papers dealing with this subject : R. Lipowsky[END_REF]16 , which allows bending/stretching ratios forbidden in a model of thin homogeneous plate (cf §2), or defects due to the discrete protein structure of a shell virus 17 .

(a) (b) In this paper concerning homogeneous objects, we propose a heuristic calculation concerning the optimum number of depressions that can be hold by a buckled thin spherical shell, in order to understand the apparently divergent observations of references 6 and 7 . The two papers reported the observation of strongly buckled objects, originally porous hollow spherical shells filled with a solvent, that buckle when the solvent evaporates. Buckling of a porous shell due to evaporation of an inner solvent is acknowledged as being of a capillary origin, and macroscopically (i.e. at scales larger than the pore size) equivalent to the effect of an isotropic external pressure 18 . These experiments therefore constitute a direct illustration of the postbuckling of hollow spheres under external pressure, for which no theoretical predictions exist. Surprisingly, conformations taken by the shells qualitatively differ between the two references 6 and 7 : as shown on Fig. 1a, there is occurrence of either a single and quite deep depression 6 , or several depressions distributed over the sphere's surface, leading to a coarsely cubic shape 7 . Single depression was also obtained by osmotic pressure (fig. 1b) 8 . In this paper we will determine whether this discrepancy can be interpreted through the equilibrium configurations of an elastic model, or if some drying artefacts should be invoked.

Elasticity

We will use elasticity theory results related to thin shells submitted to external constraints in order to get insight into the number of depressions expected for an equilibrium conformation. For a thin spherical shell of homogeneous elastic material (Young modulus E 3D , sphere radius R, thickness d), deformation occurs by bidimensional stretching (stretch modulus E 2D = E 3D d) or bending (bending modulus κ ~ E 3D d 3 ). The latter is related to the mean curvature, and we do not take into account energy variations linked to the Gaussian curvature, as its integral on a closed surface depends only on the topology (Gauss-Bonnet theorem). It has been shown [START_REF] Landau | Theory of elasticity[END_REF] that the ratio between bending and bidimensional stretch energies scales as :

U b /U s ~ (d/R) 2
where R is the radius of the spherical shell, and d its thickness. Hence for sufficiently thin shells, deformations occur mainly by bending, which is much less energetic than stretching. In this framework, a depression obviously corresponds to the inversion of a spherical cap [START_REF] Landau | Theory of elasticity[END_REF][START_REF] Pogorelov | Bendings of surfaces and stability of shells[END_REF] , which avoids stretching energy outside 0.5 mm the circular ridge that joins the undeformed part and the inverted cap, defined by its halfangle α (Fig. 2). For shells of zero spontaneous curvature (the case with nonzero spontaneous curvature will be treated in the paragraph 3), bending energy does not depend on the sign of the curvature and hence is not modified in the inverted cap.

Figure 2: Depression formed by inversion of a spherical cap. The circular ridge that allows a continuous jonction between the undeformed spherical part and the inverted cap has a lateral extension δ ~ (Rd) 1/2 (where d is the thickness of the shell). The aperture of the depression is defined by the half-angle α that extrapolates (dotted line) the ridge thickness down to zero.

Energy modifications then concentrate in the circular ridge, which lateral extension is imposed by minimization as δ ~ (Rd) 1/2 ; the relevant curvature radius being δ/tan α [START_REF] Pogorelov | Bendings of surfaces and stability of shells[END_REF]19 . As the total surface bended in the ridge has a width δ and a length 2πR sin α , the energy of a single depression writes:

U 1 = 2 π κ (d/R) -1/2 sin α tg 2 α ( 1
)
This expression diverges when α approaches π /2: thin shell theory breaks down when the radius of curvature becomes of order d. However, as will appear obvious later, other considerations different from energetic ones prevail in this limit, and looking for a more accurate expression is unnecessary within the current work.

The volume variation ∆V due to cap inversion is twice the volume of the spherical cap, hence the volume variation relative to the undeformed sphere volume:

(∆V/V sphere ) 1 depression = (1 -cos α ) 2 (2 + cos α ) / 2 (2)
In the case of N similar depressions, we have:

U N = N U 1 (3a)
and ∆V/V sphere = N (∆V/V sphere ) 1 depression (3b)

We therefore have the expressions of both the elastic energy and the volume variation corresponding to N similar depressions formed by inversion of spherical caps of half-angle α.

As the relative volume variation is the key parameter to appreciate the deformation intensity, it would be interesting to eliminate α in order to get the elastic energy U N as a fonction of ∆V/V sphere , and then to discuss the relative stability of the conformations with different numbers N of depressions ("states").

Explicit expression for small depressions: For small inverted caps ( α << 1), system (3) simplifies to:

U N ~ 2 π N κ (d/R) -1/2 α 3 and ∆V/V sphere ~ N α 4 / 8
Which, for a given relative volume variation ∆V/V sphere leads to: U N ~ 8 13/12 π N 1/4 κ (d/R) -1/2 (∆V/V sphere ) 3/4 (4)

For a given shell (E, d and R fixed), the N 1/4 dependence with ∆V/V sphere clearly leads to N=1 at equilibrium. The conformation with a single depression is favored in this regime, which corresponds to small deformations (∆V/V sphere << 1).

Implicit dependence for larger depressions:

For larger values of α , a parametric plot of U N as a fonction of ∆V/V sphere shows that for increasing ∆V/V sphere , the lowest energy state changes from N=1 to 2 to 3 etc (Fig. 3). However, system (3) is not sufficient α δ R to describe the conformations adopted at high deformations, since equation (3b) is valid only as long as depressions do not interpenetrate each other. There is of course a maximum size (i.e. maximum α ) above which it becomes impossible to find an arrangement of N similar depressions that avoids interpenetration between two of them. For the sake of simplicity, we will neglect the thickness of the circular ridge, which corresponds to the case d<<R. Then the problem becomes purely geometrical: how many similar spherical caps can be inverted without interpenetration at the surface of a sphere? ) as a function of ∆V/V sphere , for different numbers N of similar depressions (inverted spherical caps). The curves are traced up to the maximum value (∆V/V sphere ) max,N above which there would be an interpenetration between depressions. Black, green, pink, blue, red and yellow curves correspond respectively from N=1 up to 6. Note the (∆V/V sphere ) 3/4 behavior of eq. ( 4) for small ∆V/V sphere , that corresponds to small values of α . Insert: u N /(∆V/V sphere ) as a function of ∆V/V sphere (same scale in abscissa), displayed to enhance the difference between curves without modifying the ∆V/V sphere cross-over values.

A necessary condition is that the spherical caps themselves do not overlap before inversion. This amounts to the search for the maximum surface of a sphere that can be covered with N similar spherical caps, which is another formulation of the so-called Tammes' problem: maximize the minimum point-topoint distance for a set of N points placed on a sphere 20 . The solution of the Tammes' problem is much less obvious than its formulation; depending on N, analytical or numerical solutions exist [START_REF] Conway | Sphere packing, Lattices and Groups[END_REF][START_REF] Schütte | [END_REF][23] , that for our purpose allow to calculate (∆V/V sphere ) max for N ≥ 5 (Table 1 Values for N=5 to 8 correspond to the Tammes' problem and are given in reference [START_REF] Schütte | [END_REF] , N=9 in reference 27 , N=10 to 15 in reference 23 . For N higher than the values displayed in this table, the asymptotic formula 28 for the upper limit:

α max,N (rad) = (2 π / 3 ) 1/2 N -1/2
shows that it keeps decreasing; higher values of N are then useless for our purpose. Third column: the corresponding volume variation for the N inverted caps.

For up to N = 4, the necessary condition is not sufficient. This is obvious for N=2: the caps corresponding to the solutions of the Tammes' problem are two hemispheres ( α=90°), which cannot be reverted both simultaneously. Interpenetration of two opposite depressions begins when they contact at the center of the sphere, which corresponds to a maximum value of the depression half-angle α: α max,2 = 60°. We will repeat a likewise operation in

u N
∆V / V sphere what follows: keeping the centers of the spherical caps of the Tammes' problem solutions at their positions, then decrease α down to a value for which there is no interpenetration. For N=3, the caps that solves the Tammes' problem are centered on the vertices of an equilateral equatorial triangle 24 ; a little trigonometry show that interpenetration is avoided when α reduces to α max,3 with: cos α max,3 = 1/ 3 (Fig. 4). In a similar way, one can show for N=4 that cos α max,4 = 1 / (2 sin α T /2), where α T = 109°i s the angle between two center/vertex directions in the tetrahedron. Hence the halfangle of 54.59° (Tammes' problem solution for N=4, with tetraedric symmetry) corresponds to 52.24° for the non-interpenetrating inverted caps. For higher values of N (N>4), the polyhedron holding the bases of the spherical caps present only right or obtuse angles between adjacent faces: there is therefore no risk of interpenetration as long as the caps bases do not overlap -and the Tammes' condition is sufficient. With these values of α max,N , the maximum relative volume variation (∆V/V sphere ) max,N can be calculated for each N (Table I). For all N, ∆V/V sphere has no physical meaning above (∆V/V sphere ) max,N (as shown before, the mathematical solution provided by equations (3a) and (3b) corresponds to interpenetration of the inverted caps beyond this limit) : hence the energy U N of the state with N depressions is plotted and considered only up to (∆V/V sphere ) max,N in figure 3. We also restrict to values of N from 1 to 6, because higher values correspond to energies superior to U 6 for physical values of ∆V/V sphere (i.e. ; smaller than (∆V/V sphere ) max,N ).

Looking for the number N of depressions that corresponds to the lowest energy at every relative volume variation provides the phase diagram displayed on Fig. 5: at first, the optimum N increases with ∆V/V sphere due to energetic considerations, from N=1 up to a quite stable cubic organization of the depressions (N=6). The cap half-angle α keeps a value around 40° during this evolution. Then steric factors favorize a tetraedric-related conformation (N=4), followed by the final single depression state. It is interesting to note that the state N=1 for high relative volume variations is made compulsory by the geometry: this explains a posteriori why there is no point in refining the energy calculation in this limit. In observations by Gao et al 8 , which correspond obviously to a zero spontaneous curvature case as the capsule is composed of symmetrical polyelectrolyte layers, single depressions were observed for angles up to 46°f or osmotic pressures of 22 kPa, and beyond 76° for 44.5 kPa (increase of osmotic pressure leads to a loss of inner volume), as displayed in Fig. 1b. This is not far from the upper and lower limits for N=1, which suggests that it could be worthwhile exploring intermediate values of the osmotic pressure in order to check the presence of configurations presenting a higher number of depressions. In a similar but thinner system (d/R = 0.018), a transition from N=1 towards a higher number of depressions was observed when increasing the osmotic pressure 25 , but a further increase leads to a zoology of tortured shapes for which we do not have a model. With localized occurrence of important curvature, coalescence of distinct depressions can be prevented by high energy barriers. This would lead to quenching of the system in metastable conformations, which is not rare in buckling problems.

In the present model of zero spontaneous curvature, a single parameter (the total relative volume variation) happens to drive the conformations; this is worthwhile to α max,3 be commented. As previously stated, we did not consider the extension of the circular ridge that continuously links the inverted cap to the spherical undeformed part. Taking it into account would (i) increase for each depression the volume variation by a term scaling in δ 2 R, which means a correction in N(d/R) for ∆V/V sphere (ii) lower the maximum surface compactness of depressions, by adding an excluded corona around the caps (of width or order δ), which amounts to decrease d max,N and then (∆V/V sphere ) max,N for each N. If the prefactor of d/R in the volume correction varies slowly with α , the global effect of (i) will be a displacement of the transitions towards smaller values of ∆V/V sphere , proportionnally to d/R. Besides, by affecting (∆V/V sphere ) max,N , (ii) will cause an additional displacement of the "steric" transitions (i.e. transitions towards smaller values of N). Another consequence is a concommittant lowering of the range of α for the corresponding conformations, decreasing in particular the minimum value of α that can be obtained for the capsule. Both effects vanish for the thinnest shells, which is likely to be the case in aforementioned references.

Shells of nonzero spontaneous curvature

In this section, the reference state of the shell surface has a (algebric) spontaneous curvature c 0 (surface being oriented towards the outside: for example, an initially unstrained case corresponds to c 0 = +1/R). Curvature energy per unit surface then writes κ(c-c 0 ) 2 /2 instead of κc 2 /2, where c is the local mean curvature. The elastic energy of the ridge 19 can then be expressed as: U ridge = 2 π κ δR sin α ( (δ/tg α ) -1 -c 0 ) 2 . In the region enclosed by the ridge, changing the curvature of a spherical cap of surface 2 π R 2 (1 -cos α ) from (+1/R) to (-1/R) costs an energy: U cap = 2 π κ R 2 (1 -cos α ) 4 c 0 / R. The total elastic energy of a single depression then writes:

U 1,λ = 2πκ (d/R) -1/2 [sin α (tg α -λ) 2 + 4(1 -cos α ) λ]
(5) where λ = c 0 R (d/R) 1/2 . Equation (5) shows that the relative volume variation is not anymore the only parameter that drives the conformations: the relative thickness d/R and the relative spontaneous curvature c 0 R also play a qualitatively similar role.

For various values of λ, we used this U 1,λ like U 1 in §2, in order to trace U N,λ as a function of ∆V/V sphere for different numbers of depressions N, and then to determine which N has the lowest energy. Results are summarized in figure 6, that displays the phase diagram of the optimal N according to λ and ∆V/V sphere . In particular, the angle of a small single depression (corresponding to the upper limit of the N=1 region for low relative volume variations) varies from 44° for λ = 0 to 60° for λ = -1.2 and to 66° for λ = 1.2. For comparison with experimental results, we can remark that due to in situ formation, the elastic shell composed of aggregated polysterene colloids of reference 7 should be unstrained, hence c 0 R=1 and λ = (d/R) 1/2 = 0.3 in this case. The phase diagram shows that one can find a range of ∆V/V sphere presenting a N=6 state for this value. For what concerns the capsules of reference 6,26 , it is hard to propose an a priori value for λ. The templated spherical shells these capsules were obtained from must have been strainless after their synthesis, but subsequent modification of inner and outer solvant is likely to have caused a nonzero spontaneous curvature due to asymetric surface chargeing.

Interestingly enough, the steric boundaries of the phase diagram are not affected by λ, hence the observation of capsules with a single depression of half-angle ~90° is consistent with the position of the upper N=1 zone, whatever the value of λ.

Conclusion

We showed that a relatively simple model of elastic deformations at equilibrium, and mainly by curvature, is sufficient to make the first predictions concerning postbuckling conformations of a spherical shell submitted to an external pressure. In this model, the parameters that drive the transitions between different states (i.e. different number of depressions formed by inversion of a spherical cap) are the total relative volume variation and the adimensional curvature parameter λ taking into account both the thickness of the shell and spontaneous curvature (rescaled by the initial sphere radius). The buckled sphere exhibits a single depression for both small and important relative volume variations, and several depressions (up to 6, which leads to a cubic symmetry) for intermediate ones. This model proposes a phase diagram that is likely to explain the conformations observed in experiments: "capsules" with a nearly hemispheric single depression for important variations of the inner volume, observed by Zoldesi et al 6,26 , other N=1 states observed by Gao et al 8 , or coarsely cubic shapes for weaker variations observed by Tsapis et al 7 . For more accurate predictions, a more sophisticated model considering the geometry of the ridge is required. It should reveal, at higher orders, an extra influence of the relative shell thickness d/R on the boundaries of the phase diagram.

Figure 1 :

 1 Figure 1: Different shapes obtained after evaporation of the solvent contained in a spherical porous shell. (a) Left: silica/silicon « capsule » (N=1 ; d/R=0.17) observed by Zoldesi et al 6 (reproduced with author and editor permission). Right: more polyhedral shape (N~6) observed by Tsapis et al in shells made from aggregation of colloïdal particles at the surface of an evaporating droplet of colloidal suspension 7 ; d/R=0.09 (reproduced with permission by the author). (b) Deformation of polyelectrolyte capsules (d/R = 0.026) in solution through osmotic pressure (left: 22 kPa ; right: 44.5 kPa) 8 .

Figure 3 :

 3 Figure 3: Reduced energy u N = U N / (2 π κ (d/R) - 1/2

Figure 4 :

 4 Figure 4: Equatorial section of the N = 3 conformation: α max,3 = 54,74°. The solution to Tammes' problem would correspond to slightly larger spherical caps ( α = 60°) that could not revert without interpenetration.

Figure 5 :

 5 Figure 5: Black (right axis) : equilibrium values of the number of similar depressions N at the surface of the sphere, as a function of the relative volume variation ∆V/V sphere . The five first transitions occur at ∆V/V sphere = 0.106, 0.185, 0.274, 0.341 and 0.418. The following "steric" transitions occur at values given in Table I for N=6 and N=4. Grey (left axis) : in degrees, half-angle α of the depressions.

Figure 6 :

 6 Figure 6: Phase diagram displaying the equilibrium value of N (number of similar depressions) for shells of nonzero spontaneous curvature c0. Vertical axis : relative volume variation ∆V/V sphere . Horizontal axis : adimensional curvature parameter λ = c0R (d/R) 1/2 .

Table I :

 I ).

	N	α max (deg)	maximum ∆V/V sphere
	1	90	1
	2	60	0.625
	3	54.7	0.691
	4	52.2	0.785
	5	45	0.581
	6	45	0.697
	7	38.9	0.480
	8	37.4	0.474
	9	35.3	0.427
	10	33.1	0.373
	11	31.7	0.350
	12	31.7	0.382
	13	28.6	0.277
	14	27.8	0.270
	15	26.8	0.251

Maximum values α N,max of the half-angle α that defines a spherical cap at the surface of a sphere, over which one cannot invert N similar caps without interpenetration. Values of α max for cases N=1 to 4 are specifically treated in the text.

Acknowledgements : The author thanks E. B. Saff, R. Ferreol and J.-P. Raven for interesting suggestions, F. Graner and J.-P. R. for careful reading of the manuscript and B. Houchmandzadeh for his introduction to Scilab. Financial support was granted from D. G. A. (Direction Générale de l'Armement) and Transregio Template Program.