Salvatore Federico

Nonlinear elasticity of biological tissues with statistical fibre orientation

The elastic strain energy potential for nonlinear fibre-reinforced materials is customarily obtained by superposition of the potentials of the matrix and of each family of fibres. Composites with statistically oriented fibres, such as biological tissues, can be seen as being reinforced by a continuous infinity of fibre families, the orientation of which can be represented by means of a probability density function defined on the unit sphere (i.e. the solid angle). In this case, the superposition procedure gives rise to an integral form of the elastic potential such that the deformation features in the integral, which therefore cannot be calculated a priori. As a consequence, an analytical use of this potential is impossible. In this paper, we implemented this integral form of the elastic potential into a numerical procedure that evaluates the potential, the stress and the elasticity tensor at each deformation step. The numerical integration over the unit sphere is performed by means of the method of spherical designs, in which the result of the integral is approximated by a suitable sum over a discrete subset of the unit sphere. As an example of application, we modelled the collagen fibre distribution in articular cartilage, and used it in simulating displacement-controlled tests: the unconfined compression of a cylindrical sample and the contact problem in the hip joint.

Towards an analytical model of soft biological tissues.

In the past years, soft-tissue modelling research has seen substantial developments, a significant part of which can be ascribed to the refinement of numerical techniques, such as Finite Element analysis. A large class of physico-mechanical properties can be effectively simulated and predictions can be made for a variety of phenomena. However, there is still much that can be conceptually explored by means of fundamental theoretical analysis. In the past few years, driven by our interest in articular cartilage mechanics, we have developed theoretical microstructural models for linear elasticity and permeability that accounted for the presence and arrangement of collagen fibres in cartilage. In this paper, we investigate analytically the non-linear elasticity of soft tissues with collagen fibres arranged according to a given distribution of orientation, a problem that, aside from the case of fibres aligned in a finite number of distinct directions, has been treated exclusively numerically in the literature. We show that, for the case of a tissue with complex fibre arrangement, such as articular cartilage, the theoretical framework commonly used leads to an integral expression of the elastic strain energy potential. The present model is a first attempt in the development of a unified analytical microstructural model for non-linear elasticity and permeability of hydrated, fibre-reinforced soft tissues.

Effect of Fluid Boundary Conditions on Joint Contact Mechanics and Applications to the Modeling of Osteoarthritic Joints

The long-term goal of our research is to understand the mechanism of osteoarthritis (OA) initiation and progress through experimental and theoretical approaches. In previous theoretical models, joint contact mechanics was implemented without consideration of the fluid boundary conditions and with constant permeability. The primary purpose of this study was to investigate the effect of fluid boundary conditions at the articular surfaces on the contact mechanics, in terms of load sharing and fluid flow properties using variable permeability. The tested conditions included totally sealed surfaces, open surfaces, and open surfaces with variable permeability. While the sealed surface model failed to predict relaxation times and load sharing properly, the class of open surface models (open surfaces with constant permeability, and surfaces with variable permeability) gave good agreement with experiments, in terms of relaxation time and load sharing between the solid and the fluid phase. In particular, the variable permeability model was judged to be the most realistic of the three models, from a biological and physical point of view. This model was then used to simulate joint contact in the early and late stages of OA. In the early stages of OA, the model predicted a decrease in peak contact pressure and an increase in contact area, while in the late stages of OA, peak pressures were increased and contact areas were decreased compared to normal. These findings agree well with experimental observations.

Remodelling in statistically oriented fibre-reinforced materials and biological tissues

We present a mathematical model of structural reorganisation in a fibre-reinforced composite material in which the fibres are oriented statistically, i.e. obey a probability distribution of orientation. Such a composite material exemplifies a biological tissue (e.g. articular cartilage or a blood vessel) whose soft matrix is reinforced by collagen fibres. The structural reorganisation of the composite takes place as fibres reorient, in response to mechanical stimuli, in order to optimise the stress distribution in the tissue. Our mathematical model is based on the Principle of Virtual Powers and the study of dissipation. Besides incompressibility, our main hypothesis is that the composite is characterised by a probability density distribution that measures the probability of finding a family of fibres aligned along a given direction at a fixed material point. Under this assumption, we describe the reorientation of fibres as the evolution of the most probable direction along which the fibres are aligned. To test our theory, we compare our simulations of a benchmark problem with selected results taken from the literature.

Mechanical behaviour of in-situ chondrocytes subjected to different loading rates: a finite element study

Experimental findings indicate that in-situ chondrocytes die readily following impact loading, but remain essentially unaffected at low (non-impact) strain rates. This study was aimed at identifying possible causes for cell death in impact loading by quantifying chondrocyte mechanics when cartilage was subjected to a 5% nominal tissue strain at different strain rates. Multi-scale modelling techniques were used to simulate cartilage tissue and the corresponding chondrocytes residing in the tissue. Chondrocytes were modelled by accounting for the cell membrane, pericellular matrix and pericellular capsule. The results suggest that cell deformations, cell fluid pressures and fluid flow velocity through cells are highest at the highest (impact) strain rate, but they do not reach damaging levels. Tangential strain rates of the cell membrane were highest at the highest strain rate and were observed primarily in superficial tissue cells. Since cell death following impact loading occurs primarily in superficial zone cells, we speculate that cell death in impact loading is caused by the high tangential strain rates in the membrane of superficial zone cells causing membrane rupture and loss of cell content and integrity.

Volumetric-Distortional Decomposition of Deformation and Elasticity Tensor

The deformation gradient admits a multiplicative decomposition into a purely volumetric component and a purely distortional component. For a hyperelastic material, based on this decomposition, the elastic strain energy potential, the stress, and the elasticity tensor can be expressed in general as a function of both the volumetric deformation and the distortional deformation. However, the volumetric—distortional decomposition of deformation has often been employed in a fully decoupled form of the elastic strain energy potential, which is expressed as the sum of a term depending solely on the volumetric deformation and a term depending solely on the distortional deformation. This work has three main objectives. First, to derive the elasticity tensor in the general (non-decoupled) case, in its material, spatial, and linear forms; this is achieved by extensive use of fourth-order tensor algebra, and in particular of the properties of the so-called spherical operator, which is largely used, but very seldom given the dignity of being assigned a symbol and a name, in the literature. Second, to show that a fully decoupled potential gives rise to an elasticity tensor which may be inconsistent with its linearized counterpart, as some components of the linear elasticity tensor in general do not have a corresponding term in nonlinear decoupled elasticity tensor. Third, to obtain the conditions under which a purely hydrostatic stress causes a purely volumetric deformation, by means of the developed theory; the results show that this condition is satisfied if and only if the elastic potential is fully decoupled. While the whole approach is completely independent of the material symmetry, the cases of isotropy and transverse isotropy are shown as an example.

The linear elasticity tensor of incompressible materials

With a universally accepted abuse of terminology, materials having much larger stiffness for volumetric than for shear deformations are called incompressible. This work proposes two approaches for the evaluation of the correct form of the linear elasticity tensor of so-called incompressible materials, both stemming from non-linear theory. In the approach of strict incompressibility, one imposes the kinematical constraint of isochoric deformation. In the approach of quasi-incompressibility, which is often employed to enforce incompressibility in numerical applications such as the Finite Element Method, one instead assumes a decoupled form of the elastic potential (or strain energy), which is written as the sum of a function of the volumetric deformation only and a function of the distortional deformation only, and then imposes that the bulk modulus be much larger than all other moduli. The conditions which the elasticity tensor has to obey for both strict incompressibility and quasi-incompressibility have been derived, regardless of the material symmetry. The representation of the linear elasticity tensor for the quasi-incompressible case differs from that of the strictly incompressible case by one parameter, which can be conveniently chosen to be the bulk modulus. Some important symmetries have been studied in detail, showing that the linear elasticity tensors for the cases of isotropy, transverse isotropy and orthotropy are characterised by one, three and six independent parameters, respectively, for the case of strict incompressibility, and two, four and seven independent parameters, respectively, for the case of quasi-incompressibility, as opposed to the two, five and nine parameters, respectively, of the general compressible case.

Extracellular matrix integrity affects the mechanical behaviour of in-situ chondrocytes under compression

Cartilage lesions change the microenvironment of cells and may accelerate cartilage degradation through catabolic responses from chondrocytes. In this study, we investigated the effects of structural integrity of the extracellular matrix (ECM) on chondrocytes by comparing the mechanics of cells surrounded by an intact ECM with cells close to a cartilage lesion using experimental and numerical methods. Experimentally, 15% nominal compression was applied to bovine cartilage tissues using a light-transmissible compression system. Target cells in the intact ECM and near lesions were imaged by dual-photon microscopy. Changes in cell morphology (N(cell)=32 for both ECM conditions) were quantified. A two-scale (tissue level and cell level) Finite Element (FE) model was also developed. A 15% nominal compression was applied to a non-linear, biphasic tissue model with the corresponding cell level models studied at different radial locations from the centre of the sample in the transient phase and at steady state. We studied the Green-Lagrange strains in the tissue and cells. Experimental and theoretical results indicated that cells near lesions deform less axially than chondrocytes in the intact ECM at steady state. However, cells near lesions experienced large tensile strains in the principal height direction, which are likely associated with non-uniform tissue radial bulging. Previous experiments showed that tensile strains of high magnitude cause an up-regulation of digestive enzyme gene expressions. Therefore, we propose that cartilage degradation near tissue lesions may be due to the large tensile strains in the principal height direction applied to cells, thus leading to an up-regulation of catabolic factors.

Mechanical model of the breast for the prediction of deformation during imaging

To predict changes in the shape of the breast in different imaging devices, a Coupled Eulerian-Lagrangian (CEL) mechanical model is developed. The CEL method allows for a more adequate representation of the very large deformations experienced by the soft and incompressible tissues of the breast. The mechanical response of the tissues is based on advanced mathematical formulations and experimental data from the literature. Realistic geometries generated from Magnetic Resonance (MR) images are used as study cases. Furthermore, specific boundary conditions are applied to the model to predict the shape of the breast and the location of the internal tissues in a prototype microwave breast imaging system, where an immersion medium is used. The accuracy of the model was assessed by comparing the numerical results with a laser scan of the same subject in the microwave breast imaging system.

A depth-dependent model of the pericellular microenvironment of chondrocytes in articular cartilage

Experimental studies suggest that the magnitude of chondrocyte deformation is much smaller than expected based on the material properties of extracellular matrix (ECM) and cells, and that this result could be explained by a structural unit, the chondron, that is thought to protect chondrocytes from large deformations in situ. We extended an existing numerical model of chondrocyte, ECM and pericellular matrix (PCM) to include depth-dependent structural information. Our results suggest that superficial zone chondrocytes, which lack a pericellular capsule (PC), are relatively stiff, and therefore are protected from excessive deformations, whereas middle and deep zone chondrocytes are softer but are protected by the PC that limits cell deformations in these regions. We conclude that cell deformations sensitively depend on the immediate structural environment of the PCM in a depth-dependent manner, and that the functional stiffness of chondrocytes in situ is much larger than experiments on isolated cells would suggest.