A note on the Hill-Ogden generalised strains.

This brief contribution provides an overview of the Hill-Ogden generalised strain tensors, and some considerations on their representation in generalised (curvilinear) coordinates, in a fully covariant formalism that is adaptable to a more general theory on Riemannian manifolds. These strains may be naturally defined with covariant components or naturally defined with contravariant components. Each of these two macro-families is best suited to a specific geometrical context.

Dynamic Deformation Calculation of Articular Cartilage and Cells Using Resonance-Driven Laser Scanning Microscopy.

The deformation of articular cartilage and its cells at the micro-scale during dynamic activities such as gait has high mechanoregulatory importance. Measuring the cellular geometries during such dynamics has been limited by the rate of microscopic image acquisition. The introduction of resonating mirrors for image rasterization (resonant scanning), rather than the conventional servo control (galvano scanning), has significantly improved the scanning rate by more than 100×. However, the high scanning rate comes at the cost of image quality, thereby posing challenges in image processing. Here, resonance-driven 3-D laser microscopy is used to observe the transient, micro-scale deformation of articular cartilage and its cells under osmotic challenge conditions. Custom image segmentation and deformable registration software were implemented for analysis of the resonance-scanned microscopy data. The software exhibited robust and accurate performance on the osmotic swelling measurements, as well as quantitative validation testing. The resonance-scanning protocol and developed analysis software allow for simultaneous strain calculation of both the local tissue and cells, and are thus a valuable tool for real-time probing of the cell-matrix interactions that are highly relevant in the fields of orthopedic biomechanics, cell mechanobiology, and functional tissue engineering.

Mobility decisions, economic dynamics and epidemic.

We propose a model, which nests a susceptible-infected-recovered-deceased (SIRD) epidemic model into a dynamic macroeconomic equilibrium framework with agents' mobility. The latter affect both their income and their probability of infecting and being infected. Strategic complementarities among individual mobility choices drive the evolution of aggregate economic activity, while infection externalities caused by individual mobility affect disease diffusion. The continuum of rational forward-looking agents coordinates on the Nash equilibrium of a discrete time, finite-state, infinite-horizon Mean Field Game. We prove the existence of an equilibrium and provide a recursive construction method for the search of an equilibrium(a), which also guides our numerical investigations. We calibrate the model by using Italian experience on COVID-19 epidemic and we discuss policy implications.

Optimal vaccination in a SIRS epidemic model.

We propose and solve an optimal vaccination problem within a deterministic compartmental model of SIRS type: the immunized population can become susceptible again, e.g. because of a not complete immunization power of the vaccine. A social planner thus aims at reducing the number of susceptible individuals via a vaccination campaign, while minimizing the social and economic costs related to the infectious disease. As a theoretical contribution, we provide a technical non-smooth verification theorem, guaranteeing that a semiconcave viscosity solution to the Hamilton-Jacobi-Bellman equation identifies with the minimal cost function, provided that the closed-loop equation admits a solution. Conditions under which the closed-loop equation is well-posed are then derived by borrowing results from the theory of Regular Lagrangian Flows. From the applied point of view, we provide a numerical implementation of the model in a case study with quadratic instantaneous costs. Amongst other conclusions, we observe that in the long-run the optimal vaccination policy is able to keep the percentage of infected to zero, at least when the natural reproduction number and the reinfection rate are small.

Taming the spread of an epidemic by lockdown policies.

We study the problem of a policymaker who aims at taming the spread of an epidemic while minimizing its associated social costs. The main feature of our model lies in the fact that the disease's transmission rate is a diffusive stochastic process whose trend can be adjusted via costly confinement policies. We provide a complete theoretical analysis, as well as numerical experiments illustrating the structure of the optimal lockdown policy. In all our experiments the latter is characterized by three distinct periods: the epidemic is first let to freely evolve, then vigorously tamed, and finally a less stringent containment should be adopted. Moreover, the optimal containment policy is such that the product "reproduction number × percentage of susceptible" is kept after a certain date strictly below the critical level of one, although the reproduction number is let to oscillate above one in the last more relaxed phase of lockdown. Finally, an increase in the fluctuations of the transmission rate is shown to give rise to an earlier beginning of the optimal lockdown policy, which is also diluted over a longer period of time.

Anisotropic Diffusivity Tensor in Articular Cartilage: Effective Medium Approach.

Due to the avascular nature of articular cartilage, molecular transport occurs via interstitial fluid flow as well as via diffusion. Diffusion in cartilage has been studied experimentally, but no mathematical models have been developed to interpret the experimental results and the observed isotropy or anisotropy in the different cartilage zones. Here, we propose a model for the determination of the diffusivity tensor of uncharged macromolecules in articular cartilage, accounting for the inhomogeneity and anisotropy arising from fiber arrangement, volumetric fraction, and radius. We study a representative element of volume (REV) comprising a fiber surrounded by fluid-saturated proteoglycan matrix. The REV permeability tensor is evaluated using a previously developed model, while the REV diffusivity tensor is obtained by incorporating the hydrodynamic effect and the steric effect of the fiber-reinforced matrix. Both effects are represented by anisotropic second-order tensors. The overall diffusivity tensor is obtained as the averaging integral of the REV diffusivity, weighted by the probability distribution of fiber orientation. The model's predictions of the trend of the magnitude of the diffusivity of spheroidal macromolecules as a function of molecular radius agree with published experimental results. For large linear macromolecules, the model underestimates the diffusivity magnitude (i.e., the equivalent isotropic diffusivity). The model correctly predicts the anisotropic behavior for linear macromolecules, although it underestimates the numerical value of the diffusivity anisotropy ratio of large linear macromolecules in the superficial zone, and overestimates it in the deep zone. In summary, this model constitutes a first step toward understanding the relation between diffusivity and permeability in articular cartilage.

Non-linear model for compression tests on articular cartilage.

Hydrated soft tissues, such as articular cartilage, are often modeled as biphasic systems with individually incompressible solid and fluid phases, and biphasic models are employed to fit experimental data in order to determine the mechanical and hydraulic properties of the tissues. Two of the most common experimental setups are confined and unconfined compression. Analytical solutions exist for the unconfined case with the linear, isotropic, homogeneous model of articular cartilage, and for the confined case with the non-linear, isotropic, homogeneous model. The aim of this contribution is to provide an easily implementable numerical tool to determine a solution to the governing differential equations of (homogeneous and isotropic) unconfined and (inhomogeneous and isotropic) confined compression under large deformations. The large-deformation governing equations are reduced to equivalent diffusive equations, which are then solved by means of finite difference (FD) methods. The solution strategy proposed here could be used to generate benchmark tests for validating complex user-defined material models within finite element (FE) implementations, and for determining the tissue's mechanical and hydraulic properties from experimental data.

Microscale investigation of the anisotropic swelling of cartilage tissue and cells in response to hypo-osmotic challenges.

Tissue swelling represents an early sign of osteoarthritis, reflecting osmolarity changes from iso- to hypo-osmotic in the diseased joints. Increased tissue hydration may drive cell swelling. The opposing cartilages in a joint may swell differently, thereby predisposing the more swollen cartilage and cells to mechanical injuries. However, our understanding of the tissue-cell interdependence in osmotically loaded joints is limited as tissue and cell swellings have been studied separately. Here, we measured tissue and cell responses of opposing patellar (PAT) and femoral groove (FG) cartilages in lapine knees exposed to an extreme hypo-osmotic challenge. We found that the tissue matrix and most cells swelled during the hypo-osmotic challenge, but to a different extent (tissue: <3%, cells: 11%-15%). Swelling-induced tissue strains were anisotropic, showing 2%-4% stretch and 1%-2% compression along the first and third principal directions, respectively. These strains were amplified by 5-8 times in the cells. Interestingly, the first principal strains of tissue and cells occurred in different directions (60-61° for tissue vs. 8-13° for cells), suggesting different mechanisms causing volume expansion in the tissue and the cells. Instead of the continuous swelling observed in the tissue matrix, >88% of cells underwent regulatory volume decrease to return to their pre-osmotic challenge volumes. Cell shapes changed in the early phase of swelling but stayed constant thereafter. Kinematic changes to tissue and cells were larger for PAT cartilage than for FG cartilage. We conclude that the swelling-induced deformation of tissue and cells is anisotropic. Cells actively restored volume independent of the surrounding tissues and seemed to prioritize volume restoration over shape restoration. Our findings shed light on tissue-cell interdependence in changing osmotic environments that is crucial for cell mechano-transduction in swollen/diseased tissues.

The Protective Function of Directed Asymmetry in the Pericellular Matrix Enveloping Chondrocytes.

The specialized pericellular matrix (PCM) surrounding chondrocytes within articular cartilage is critical to the tissue's health and longevity. Growing evidence suggests that PCM alterations are ubiquitous across all trajectories of osteoarthritis, a crippling and prevalent joint disease. The PCM geometry is of particular interest as it influences the cellular mechanical environment. Observations of asymmetrical PCM thickness have been reported, but a quantified characterization is lacking. To this end, a novel microscopy protocol was developed and applied to acquire images of the PCM surrounding live cells. Morphometric analysis indicated a statistical bias towards thicker PCM on the inferior cellular surface. The mechanical effects of this bias were investigated with multiscale modelling, which revealed potentially damaging, high tensile strains in the direction perpendicular to the membrane and localized on the inferior surface. These strains varied substantially between PCM asymmetry cases. Simulations with a thicker inferior PCM, representative of the observed geometry, resulted in strain magnitudes approximately half of those calculated for a symmetric geometry, and a third of those with a thin inferior PCM. This strain attenuation suggests that synthesis of a thicker inferior PCM may be a protective adaptation. PCM asymmetry may thus be important in cartilage development, pathology, and engineering.

Development of stimulus-sensitive electrospun membranes based on novel biodegradable segmented polyurethane as triggered delivery system for doxorubicin.

In this work, redox-sensitive polyurethane urea (PUU) based electrospun membranes have been exploited to chemically tether a pH-sensitive doxorubicin derivative achieved by linking a lipoyl hydrazide to the drug via a hydrazone linkage. First, the lipoyl-hydrazone-doxorubicin derivative labelled as LA-Hy-Doxo has been synthesized and characterized. Then, the molecule has been tethered, via a thiol-disulfide exchange reaction, to the redox-sensitive PUU (PolyCEGS) electrospun membrane. The redox-sensitive PolyCEGS PUU has been produced by using PCL-PEG-PCL polyol and glutathione-tetramethyl ester (GSSG-OMe)4 as a chain extender. The LA-Hy-Doxo tethered electrospun membrane has showed a dually controlled release triggered by acidic and reducing conditions, producing a significant cytotoxic effect in human breast cancer cell lines (MCF-7) which has validated the system for the post-surgical treatment of solid tumors to contrast recurrence.

Photothermal nanofibrillar membrane based on hyaluronic acid and graphene oxide to treat Staphylococcus aureus and Pseudomonas aeruginosa infected wounds.

Here we reported the fabrication of an electrospun membrane based on a hyaluronic acid derivative (HA-EDA) to be used as a bandage for the potential treatment of chronic wounds. The membrane, loaded with graphene oxide (GO) and ciprofloxacin, showed photothermal properties and light-triggered drug release when irradiated with a near-infrared (NIR) laser beam. Free amino groups of HA-EDA derivative allowed autocrosslinking of the electrospun membrane; thus, a substantial enhancement in the hydrolytic resistance of the patch was obtained. In vitro antibacterial activity studies performed on Staphylococcus aureus and Pseudomonas aeruginosa revealed that such electrospun membranes, due to the synergistic effect of the antibiotic and NIR-mediated hyperthermia, reduced the viability of both pathogens. Specific in vitro experiment demonstrated also that is possible to disrupt, through laser irradiation, the biofilms formed onto the membrane.

Composite Hydrogels of Alkyl Functionalized Gellan Gum Derivative and Hydroxyapatite/Tricalcium Phosphate Nanoparticles as Injectable Scaffolds for bone Regeneration.

An alkyl functionalized gellan gum derivative is here used to produce hydrogels containing hydroxyapatite and tricalcium phosphate nanoparticles as injectable nanostructured scaffolds for bone regeneration. The amphiphilic nature of the polysaccharide derivative along with its thermotropic behavior and ionotropic crosslinking features make possible to produce injectable bone mimetic scaffolds that can be used to release viable cells and osteoinductive biomolecules. The influence of different nanoparticles concentration on the rheological and physicochemical properties of the injectable systems is studied. It is found that the presence of inorganic nanoparticles reinforces the 3D hydrated polymeric networks without influencing their injectability but improving the physicochemical properties of ionotropic crosslinked hydrogels produced with two different curing media. Preliminary cytocompatibility tests performed with murine preosteoblast cells revealed that gellan gum based hydrogels can safely encapsulate viable cells. Loading and release experiments for dexamethasone and stromal cell-derived factor-1 demonstrate the drug delivery features of the obtained injectable systems.

Collagen fibres determine the crack morphology in articular cartilage.

Cracks in articular cartilage compromise tissue integrity and mechanical properties and lead to chondral lesions if untreated. An understanding of the mechanics of cracked cartilage may help in the prevention of cartilage deterioration and the development of tissue-engineered substitutes. The degeneration of cartilage in the presence of cracks may depend on the ultrastructure and composition of the tissue, which changes with aging, disease and habitual loading. It is unknown if the structural and compositional differences between immature and mature cartilage affect the mechanics of cartilage cracks, possibly predisposing one to a greater risk of degeneration than the other. We used a fibre-reinforced poro-viscoelastic swelling material model that accounts for large deformations and tension-compression non-linearity, and the finite element method to investigate the role of cartilage structure and composition on crack morphology and tissue mechanics. We demonstrate that the crack morphology predicted by our theoretical model agrees well with the histo-morphometric images of young and mature cracked cartilages under indentation loading. We also determined that the crack morphology was primarily dependent on collagen fibre orientation which differs as a function of cartilage depth and tissue maturity. The arcade-like collagen fibre orientation, first discussed by Benninghoff in his classical 1925 paper, appears to be beneficial for slowing the progression of tissue cracks by 'sealing' the crack and partially preserving fluid pressure during loading. Preservation of the natural load distribution between solid and fluid constituents of cartilage may be a key factor in slowing or preventing the propagation of tissue cracks and associated tissue matrix damage. STATEMENT OF SIGNIFICANCE: Cracks in articular cartilage can be detrimental to joint health if not treated, but it is not clear how they propagate and lead to tissue degradation. We used an advanced numerical model to determine the role of cartilage structure and composition on crack morphology under loading. Based on the structure and composition found in immature and mature cartilages, our model successfully predicts the crack morphology in these cartilages and determines that collagen fibre as the major determinant of crack morphology. The arcade-like Benninghoff collagen fibre orientation appears to be crucial in 'sealing' the tissue crack and preserves normal fluid-solid load distribution in cartilage. Inclusion of the arcade-like fibre orientation in tissue-engineered construct may help improve its integration within the host tissue.

An asymmetric electrospun membrane for the controlled release of ciprofloxacin and FGF-2: Evaluation of antimicrobial and chemoattractant properties.

Here, an asymmetric double-layer membrane has been designed and fabricated by electrospinning as a tool for a potential wound healing application. A hydrophobic layer has been produced by using a polyurethane-polycaprolactone (PU-PCL) copolymer and loaded with the antibacterial ciprofloxacin whereas an ion responsive hydrophilic layer has been produced by using an octyl derivative of gellan gum (GG-C8) and polyvinyl alcohol (PVA) and loaded with the growth factor FGF-2. This study investigated how the properties of this asymmetric membrane loaded with actives, were influenced by the ionotropic crosslinking of the hydrophilic layer. In particular, the treatment in DPBS and the crosslinking in CaCl2 0.1 or 1 M of the hydrophilic layer affected the release profile of the bioactive molecules allowing to modulate both the antimicrobial effect, as assayed by logarithmic reduction of the Staphylococcus aureus viable count, and the chemoattractant properties on NIH 3 T3 cell line, as assayed by scratch test and cell chemoattraction assay.

Chondrocyte Deformations Under Mild Dynamic Loading Conditions.

Dynamic deformation of chondrocytes are associated with cell mechanotransduction and thus may offer a new understanding of the mechanobiology of articular cartilage. Despite extensive research on chondrocyte deformations for static conditions, work for dynamic conditions remains rare. However, it is these dynamic conditions that articular cartilage in joints are exposed to everyday, and that seem to promote biological signaling in chondrocytes. Therefore, the objective of this study was to develop an experimental technique to determine the in situ deformations of chondrocytes when the cartilage is dynamically compressed. We hypothesized that dynamic deformations of chondrocytes vastly differ from those observed under steady-state static strain conditions. Real-time chondrocyte geometry was reconstructed at 10, 15, and 20% compression during ramp compressions with 20% ultimate strain, applied at a strain rate of 0.2% s-1, followed by stress relaxation. Dynamic compressive chondrocyte deformations were non-linear as a function of nominal strain, with large deformations in the early and small deformations in the late part of compression. Early compression (up to about 10%) was associated with chondrocyte volume loss, while late compression (> ~ 10%) was associated with cell deformation but minimal volume loss. Force continued to decrease for 5 min in the stress-relaxation phase, while chondrocyte shape/volume remained unaltered after the first minute of stress-relaxation.

Hyaluronan alkyl derivatives-based electrospun membranes for potential guided bone regeneration: Fabrication, characterization and in vitro osteoinductive properties.

The aim of the work was to determine the effects of the chemical functionalization of hyaluronic acid (HA) with pendant aliphatic tails at different lengths and free amino groups in terms of chemical reactivity, degradation rate, drug-eluting features, and surface properties when processed as electrospun membranes (EM) evaluating the osteoinductive potential for a possible application as guided bone regeneration (GBR). To this end, a series of HA derivatives with different aliphatic tails (DD-Cx mol% ≈ 12.0 mol%) and decreasing derivatization of free amino groups (DDEDA mol% from 70.0 to 30.0 mol%) were first synthesized, namely Hn. Then dexamethasone-loaded Hn EM, i.e. HnX were prepared from aqueous polymeric solutions with polyvinyl alcohol (PVA), as a non-ionogenic linear flexible polymeric carrier, and the multifunctional 2-hydroxypropyl- cyclodextrin (HPCD) which acted as a rheological modifier, a stabilizer of Taylor's cone, and a solubilizing agent. A comprehensive characterization of the membranes was carried out through ATR-IR, XRD, and WCA measurements. According to the in vitro hydrolytic and enzymatic degradation and drug release in different aqueous media for two months, the insertion of alkyl pendant grafts and the crosslinking process provided tuneable additional resistance to the whole membrane suitably for the final application of the membranes. Cell culture showed the cytocompatibility and cell proliferation until 7 days. Osteogenic differentiation and mineralization of pre-osteoblastic MC3T3 cells occurred for most of membranes after 35 days as valued by measuring ALP activity (50 nmol 4-np/h/nf DNA) and the deposition of calcium (120-140 µg ml-1).

Nano-structured myelin: new nanovesicles for targeted delivery to white matter and microglia, from brain-to-brain.

Neurodegenerative diseases affect millions of people worldwide and the presence of various physiological barriers limits the accessibility to the brain and reduces the efficacy of various therapies. Moreover, new carriers having targeting properties to specific brain regions and cells are needed in order to improve therapies for the brain disorder treatment. In this study, for the first time, Myelin nanoVesicles (hereafter defined MyVes) from brain-extracted myelin were produced. The MyVes have an average diameter of 100-150 â€‹nm, negative zeta potential, spheroidal morphology, and contain lipids and the key proteins of the myelin sheath. Furthermore, they exhibit good cytocompatibility. The MyVes were able to target the white matter and interact mainly with the microglia cells. The preliminary results here presented allow us to suppose the employment of MyVes as potential carrier to target the white matter and microglia in order to counteract white matter microglia-related diseases.

Effect of cracks on the local deformations of articular cartilage.

Despite significant evidence regarding the increased risk of cartilage degeneration due to traumatic injuries to joints, there is still a lack of understanding of the mechanisms underlying osteoarthritis development following a joint injury. Injuries in knee cartilage are often characterized by lesions or tears. In addition to acute traumatic joint injuries, microscale damages, which may form because of wear, are thought to be a contributing factor in the development of osteoarthritis. While the overall function of a joint may not be affected by the presence of microcracks, we hypothesized that strain magnification in the vicinity of microcracks might be significant. We tested this hypothesis by creating partial cuts in articular cartilages and measuring the strain within 20 µm from the edge of these cuts. Measurements were made in the superficial and middle zones of articular cartilage extract samples. We found that local strain in the vicinity of cuts is magnified by a factor of 1.2-1.6 compared to strains in intact regions for nominal compressions exceeding 5%. For nominal compressions of less than 5%, no strain magnification was detected in the vicinity of the cracks. We concluded that articular cartilage cracks magnify local strains by damaging the structural integrity and decreasing the fluid pressure in the matrix.

Effect of structural distortions on articular cartilage permeability under large deformations.

The permeability of articular cartilage has a key role in load support and lubrication in diarthrodial joints. The microstructural rearrangement and consequent alteration in permeability caused by the large deformations undergone by cartilage have been previously modelled with a multi-scale approach. At the microscopic scale, the tissue is regarded as a homogeneous fluid-filled proteoglycan matrix reinforced by collagen fibres. A material point is described by a representative element of volume (REV), comprising a collagen fibre surrounded by a jacket of fluid-saturated proteoglycan matrix. At the macroscopic scale, the statistical orientation of the fibres is accounted for via averaging of the REV over all possible directions. The previous models accounted for volumetric deformation and fibre reorientation, but did not consider the cross-sectional distortion of the REV, which changes the widths of the fluid channels in different directions. We account for REV cross-sectional distortion and demonstrate its effects by simulating confined compression tests for the superficial, middle and deep zones of articular cartilage. The proposed model captures published experimental results that were not reproduced correctly by the previous models, and shows that each factor (volumetric deformation, fibre reorientation, REV cross-sectional distortion) can be dominant, depending on fibre orientation and amount of compression, implying that all three factors should be accounted for when modelling cartilage permeability.

Gellan gum-based delivery systems of therapeutic agents and cells.

The purpose of this review is to make a summary of high quality research trends using gellan gum (GG) as a polymeric constituent for the design of innovative drug delivery systems and devices for biomedical applications, such as cell therapy and regenerative medicine. The use of gellan gum is described both in its native form and as chemically functionalized derivatives or physically mixed with natural or synthetic materials. Starting from a systematic study of recent research works, the main properties of the native polysaccharide have been highlighted and therefore some improvements have been focused thanks to the design of chemically functionalized derivatives and the use of composite materials.