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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.

Inverse Dynamics is used to calculate the forces and moments in the joints of multibody systems investigated in fields such as Biomechanics or Robotics. In a didactic spirit, this paper begins with an overview of the derivations of the kinematical and dynamical equations of rigid bodies from the point of view of modern Continuum Mechanics. Then, it introduces a matrix formulation for the solution of Inverse Dynamics problems and, finally, reports a simple two-dimensional example of application to a problem in Biomechanics.

In this paper we propose a theoretical model including a susceptible-infected-recovered-dead (SIRD) model of epidemic in a dynamic macroeconomic general equilibrium framework with agents' mobility. The latter affect both their income (and consumption) and their probability of infecting and of 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. Rational expectations of forward looking agents on the dynamics of aggregate mobility and epidemic determine individual mobility decisions. The model allows to evaluate alternative scenarios of mobility restrictions, especially policies dependent on the state of epidemic. We prove the existence of an equilibrium and provide a recursive construction method for finding equilibrium(a), which also guides our numerical investigations. We calibrate the model by using Italian experience on COVID-19 epidemic in the period February 2020 - May 2021. We discuss how our economic SIRD (ESIRD) model produces a substantially different dynamics of economy and epidemic with respect to a SIRD model with constant agents' mobility. Finally, by numerical explorations we illustrate how the model can be used to design an efficient policy of state-of-epidemic-dependent mobility restrictions, which mitigates the epidemic peaks stressing health system, and allows for trading-off the economic losses due to reduced mobility with the lower death rate due to the lower spread of epidemic.

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.

Under given prestress conditions, solitary waves in fluid-filled elastic tubes are confined to a rather narrow set of combinations of the background fluid velocity and the wave speed. This set, which we call the domain of existence, is bounded by several curves that represent various physical and mathematical restrictions. Remarkably, these restrictions can be cast as purely algebraic conditions to be imposed upon the governing system of differential equations. Paramount among the physical restrictions are the avoidance of wrinkles and the self-impenetrability of the wave profile. In particular, the existence of a critical wave speed of impending wrinkling, independent of the background fluid velocity, is established rigorously.

In the original paper, we erroneously used the total potential energy (Helmholtz free energy of the material plus potential energy of the external forces) in the entropy inequality.

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.

An alkyl functionalized gellan gum derivative was 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 made 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 was studied. We found that the presence of inorganic nanoparticles reinforces the three-dimensional 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. This article is protected by copyright. All rights reserved.

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 \emph{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., 17 pages, 4 figures

We propose a model, which nests a susceptible-infected-recovered-deceased (SIRD) model of epidemic 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.

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 syn- thesized 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

In this contribution, we propose a multiplicative decomposition of the deformation gradient corresponding to the imagined procedure that Eshelby (Proc R Soc Ser A 241:376–396, 1957) used to investigate the theory of inclusions in the case of infinitesimal deformations. The proposed multiplicative decomposition is inspired by classical multiplicative decompositions reported in the literature and encompasses, as particular cases, other decompositions proposed for Eshelby’s inclusion problem. The linearisation of the proposed multiplicative decomposition coincides with the additive decomposition of the infinitesimal strain in Eshelby’s original procedure.

The role played by magnetoelastic effects on the properties exhibited by magnetic domain walls propagating along the major axis of a thin magnetostrictive nanostrip, coupled mechanically with a thick piezoelectric actuator, is theoretically investigated. The magnetostrictive layer is assumed to be a linear elastic material belonging to the cubic crystal classes $$\bar{4}$$ 3m, 432 and m $$\bar{3}$$ m and to undergo isochoric magnetostrictive deformations. The analysis is carried out in the framework of the extended Landau–Lifshitz–Gilbert equation, which allows to describe, at the mesoscale, the spatio-temporal evolution of the local magnetization vector driven by magnetic fields and electric currents, in the presence of magnetoelastic and magnetocrystalline anisotropy fields. Through the traveling-wave transformation, the explicit expression of the key features involved in both steady and precessional regimes is provided and a qualitative comparison with data from the literature is also presented.

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.

The non-homogeneous, anisotropic material properties, and triphasic nature of articular cartilage enables diarthrodial joints to withstand large and complex physiological loading conditions. To develop biomaterials that provide similar functional properties as those found in articular cartilage, it is vital to have knowledge of the strain distributions in cartilage for a large range of loading conditions. Applied stress vs. strain properties of articular cartilage have been measured primarily for static conditions, but the dynamic properties are thought to be more relevant for joint function and cartilage biosynthesis. Furthermore, the dynamic stress-strain properties are expected to vary significantly from those obtained for static, steady-state conditions. Here, we present a method for the determination of axial strain fields throughout the depth of articular cartilage for static loading conditions and dynamic conditions performed at different loading rates. For the conditions tested here, the strain distributions throughout the cartilage depth were more uniform for the dynamic than the static loading conditions, and more uniform for high compared to low strain rates.

Starting from the observation that the growth of a body breaks the time translation symmetry of the body’s dynamics, we determine a scalar field, called internal time, that defines an indicator of the intrinsic time scale of the growth-related body’s structural evolution. By recasting the theory of growth for monophasic media within a variational framework, we obtain the internal time as the solution of a partial differential equation descending from Noether’s Theorem. We do this by considering two approaches, one formulated in terms of internal variables and one adopting the concept of augmented kinematics.

This study focusses on the remodelling of biological tissues in the framework of the theory of material uniformity. A constitutive evolution model is introduced, including fibre recruitment and reorientation, and subjected to the entropy inequality, which enforces the Second Principle of Thermodynamics. The model is applied to a numerical example describing a pressurised fibre-reinforced cylinder, roughly representing an artery, and is able to capture the major characteristics of remodelling in arteries, as reported in the literature.