Jean-François Ganghoffer

A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization

Cosserat models of cancellous bone are constructed, relying on micromechanical approaches in order to investigate microstructure-related scale effects on the macroscopic properties of bone. The derivation of the effective mechanical properties of cancellous bone considered as a cellular solid modeled as two-dimensional lattices of articulated beams is presently investigated. The cell walls of the bone microstructure are modeled as Timoshenko thick beams. The asymptotic homogenization technique is involved to get closed form expressions of the equivalent properties versus the geometrical and mechanical microparameters, accounting for the effects of bending, axial, and transverse shear deformations. Considering lattice microrotations as additional degrees of freedom at both the microscopic and macroscopic scales, an anisotropic micropolar equivalent continuum model is constructed, the effective mechanical properties of which are identified. The effective elastic moduli of various periodic cell structures are computed in situations of low and high effective densities to assess the impact of the transverse shear deformation. The stress distribution in a cracked bone sample is computed based on the effective micropolar model, highlighting the regularizing effect of the Cosserat continuum in comparison to a classical elasticity continuum model.

A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure.

A 3D anisotropic micropolar continuum model of vertebral trabecular bone is presently developed accounting for the influence of microstructure-related scale effects on the macroscopic effective properties. Vertebral trabecular bone is modeled as a cellular material with an idealized periodic structure made of open 3D cells. The micromechanical approach relies on the discrete homogenization technique considering lattice microrotations as additional degrees of freedom at the microscale. The effective elastic properties of 3D lattices made of articulated beams taking into account axial, transverse shearing, flexural, and torsional deformations of the cell struts are derived as closed form expressions of the geometrical and mechanical microparameters. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the same microparameters. A finite element model of the local architecture of the trabeculae gives values of the effective moduli that are in satisfactory agreement with the homogenized moduli.

A multilayer braided scaffold for Anterior Cruciate Ligament : mechanical modeling at the fiber scale

An adapted scaffold for Anterior Cruciate Ligament (ACL) tissue engineering must match biological, morphological and biomechanical requirements. Computer-aided tissue engineering consists of finding the most appropriate scaffold regarding a specific application by using numerical tools. In the present study, the biomechanical behavior of a new multilayer braided scaffold adapted to computer-aided tissue engineering is computed by using a dedicated Finite Element (FE) code. Among different copoly(lactic acid-co-(ε-caprolactone)) (PLCL) fibers tested in the present study, PLCL fibers with a lactic acid/ε-caprolactone ratio of 85/15 were selected as a constitutive material for the scaffold considering its strength and deformability. The mechanical behavior of these fibers was utilized as material input in a Finite Element (FE) code which considers contact/friction interactions between fibers within a large deformation framework. An initial geometry issued from the braiding process was then computed and was found to be representative of the actual scaffold geometry. Comparisons between simulated tensile tests and experimental data show that the method enables to predict the tensile response of the multilayer braided scaffold as a function of different process parameters. As a result, the present approach constitutes a valuable tool in order to determine the configuration which best fits the biomechanical requirements needed to restore the knee function during the rehabilitation period. The developed approach also allows the mechanical stimuli due to external loading to be quantified, and will be used to perform further mechanobiological analyses of the scaffold under dynamic culture.

Finite element simulation of interactions between pelvic organs: Predictive model of the prostate motion in the context of radiotherapy

The setting up of predictive models of the pelvic organ motion and deformation may prove an efficient tool in the framework of prostate cancer radiotherapy, in order to deliver doses more accurately and efficiently to the clinical target volume (CTV). A finite element (FE) model of the prostate, rectum and bladder motion has been developed, investigating more specifically the influence of the rectum and bladder repletions on the gland motion. The required organ geometries are obtained after processing the computed tomography (CT) images, using specific softwares. Due to their structural characteristics, a 3D shell discretization is adopted for the rectum and the bladder, whereas a volume discretization is adopted for the prostate. As for the mechanical behavior modelling, first order Ogden hyperelastic constitutive laws for both the rectum and bladder are identified. The prostate is comparatively considered as more rigid and is accordingly modelled as an elastic tissue undergoing small strains. A FE model is then created, accounting for boundary and contact conditions, internal and applied loadings being selected as close as possible to available anatomic data. The order of magnitude of the prostate motion predicted by the FE simulations is similar to the measurements done on a deceased person, accounting for the delineation errors, with a relative error around 8%. Differences are essentially due to uncertainties in the constitutive parameters, pointing towards the need for the setting up of direct measurement of the organs mechanical behavior.

Identification of couple-stress moduli of vertebral trabecular bone based on the 3D internal architectures.

The purpose of this paper is to develop a homogeneous, orthotropic couple-stress continuum model as a substitute of the 3D periodic heterogeneous cellular solid model of vertebral trabecular bone. Vertebral trabecular bone is modeled as a porous material with an idealized periodic structure made of 3D open cubic cells, which is effectively orthotropic. The chosen architecture is based on studies of samples taken from the central part of vertebral bodies. The effective properties are obtained based on the response of the representative volume element under prescribed boundary conditions. Mixed boundary conditions comprising both traction and displacement boundary conditions are applied on the structure boundaries. In this contribution, the effective mechanical constants of the effective couple-stress continuum are deduced by an equivalent strain energy method. The characteristic lengths for bending and torsion are identified from the resulting homogenized orthotropic moduli. We conduct this study computationally using a finite element approach. Vertebral trabecular bone is modeled either as a cellular solid or as a two-phase material consisting of bone tissue (stiff phase) forming a trabecular network, and a surrounding soft tissue referring to the bone marrow present in the pores. Both the bone tissue forming the network and the pores are assumed to be homogeneous linear elastic, and isotropic media. The scale effects on the predicted couple stress moduli of these networks are investigated by varying the size of the bone specimens over which the boundary conditions are applied. The analysis using mixed boundary conditions gives results that are independent of unit cell size when computing the first couple stress tensor, while it is dependent on the cell size as to the second couple stress tensor moduli. This study provides overall guidance on how the size of the trabecular specimen influence couple stresses elastic moduli of cellular materials, with focus on bones. The developed approach is quite general and applicable to any heterogeneous cellular and composite materials.

A thermodynamic approach with internal variables using Lagrange formalism. Part I: General framework

Abstract We present some reflections on the application of the Lagrangian formalism for continuous media locally uniform subjected to internal irreversible evolutions. The Lagrangian density, defined as the time derivative of a non-equilibrium thermodynamic potential, [Thermodynamics of Relaxation Processes using Internal variables within a Lagrange-formalism. P. Germain’s Anniversary Volume 2000. Contiuum Thermomechanics: the Art and Science of Modeling Matter’s Behaviour, 2000], contains all the symmetry properties of the system. The generalised Lagrange co-ordinates correspond to the state and internal variables of the time derivative of the generalised Gibbs potential. The latter being used within the framework of the De Donder’s method, must also account for the memory effect of the physical medium. This first part is devoted to the thermodynamic framework called the distribution of non-linear relaxations approach (DNLR) developed by C. Cunat on the basis of the generalised Gibbs’ relation.

Mechanical behaviour of a fibrous scaffold for ligament tissue engineering: Finite elements analysis vs. X-ray tomography imaging

The use of biodegradable scaffolds seeded with cells in order to regenerate functional tissue-engineered substitutes offers interesting alternative to common medical approaches for ligament repair. Particularly, finite element (FE) method enables the ability to predict and optimise both the macroscopic behaviour of these scaffolds and the local mechanic signals that control the cell activity. In this study, we investigate the ability of a dedicated FE code to predict the geometrical evolution of a new braided and biodegradable polymer scaffold for ligament tissue engineering by comparing scaffold geometries issued from FE simulations and from X-ray tomographic imaging during a tensile test. Moreover, we compare two types of FE simulations the initial geometries of which are issued either from X-ray imaging or from a computed idealised configuration. We report that the dedicated FE simulations from an idealised reference configuration can be reasonably used in the future to predict the global and local mechanical behaviour of the braided scaffold. A valuable and original dialog between the fields of experimental and numerical characterisation of such fibrous media is thus achieved. In the future, this approach should enable to improve accurate characterisation of local and global behaviour of tissue-engineering scaffolds.

Computation of the effective mechanical response of biological networks accounting for large configuration changes

The asymptotic homogenization technique is involved to derive the effective elastic response of biological membranes viewed as repetitive beam networks. Thereby, a systematic methodology is established, allowing the prediction of the overall mechanical properties of biological membranes in the nonlinear regime, reflecting the influence of the geometrical and mechanical micro-parameters of the network structure on the overall response of the equivalent continuum. Biomembranes networks are classified based on nodal connectivity, so that we analyze in this work 3, 4 and 6-connectivity networks, which are representative of most biological networks. The individual filaments of the network are described as undulated beams prone to entropic elasticity, with tensile moduli determined from their persistence length. The effective micropolar continuum evaluated as a continuum substitute of the biological network has a kinematics reflecting the discrete network deformation modes, involving a nodal displacement and a microrotation. The statics involves the classical Cauchy stress and internal moments encapsulated into couple stresses, which develop internal work in duality to microcurvatures reflecting local network undulations. The relative ratio of the characteristic bending length of the effective micropolar continuum to the unit cell size determines the relevant choice of the equivalent medium. In most cases, the Cauchy continuum is sufficient to model biomembranes. The peptidoglycan network may exhibit a re-entrant hexagonal configuration due to thermal or pressure fluctuations, for which micropolar effects become important. The homogenized responses are in good agreement with FE simulations performed over the whole network. The predictive nature of the employed homogenization technique allows the identification of a strain energy density of a hyperelastic model, for the purpose of performing structural calculations of the shape evolutions of biomembranes.

Construction of Micropolar Continua from the Homogenization of Repetitive Planar Lattices

The derivation of the effective mechanical properties of planar lattices made of articulated bars is herewith investigated, relying on the asymptotic homogenization technique to get closed form expressions of the equivalent properties versus the geometrical and mechanical microparameters. Considering lattice microrotations as additional degrees of freedom at both scales, micropolar equivalent continua are constructed from discrete lattices made of a repetitive unit cell, from an extension of the asymptotic homogenization technique. We will show that it is necessary to solve on two different orders a linear system of equations giving the kinematic variables, at both the first and second order. The effective strain and effective curvature appear respectively as the first and second order strain variables. In the case of a centrosymmetric unit cell, there is no coupling between couple stresses and strains nor between stress and curvature. The unknown kinematic variables are determined by solving the translational and rotational equilibrium for the whole lattice. This in turn leads to the expression of the stress vector and couple stress vector, allowing to construct the Cauchy stress and couple stress tensors. The homogenized behavior of the tetragonal and hexagonal lattices is determined in terms of homogenized micropolar moduli.

Computer-Aided Tissue Engineering: Application to the Case of Anterior Cruciate Ligament Repair

Tissue engineering has the potential to overcome the limitations associated with current reconstructions strategies of the Anterior Cruciate Ligament (ACL). However, the design of a scaffold satisfying the key requirements associated with ACL tissue engineering is a challenging task. In order to avoid a costly trial-and-error approach, computer-based methods have been widely used in the case of various applications such as bone or cartilage. These methods can help to define the best scaffold and culture conditions for a given list of criteria, and may also enable to predict the ultimate evolution of the scaffold and to better understand some mechanobiological principles. Some of these methods are reviewed in the current chapter, and are applied for the first time in the case of ACL tissue engineering. The morphological and mechanical properties of a new scaffold based on copoly(lactic acid-co-(\(\upvarepsilon \)-caprolactone)) (PLCL) fibers arranged into a multilayer braided structure will be assessed using dedicated numerical tools. Preliminary biological assessments are also presented, and some conclusions concerning the suitability of the scaffold and the interest of CATE in this case will be drawn.