Multi-scale biomechanical study of transport phenomena in the intervertebral disc

Intervertebral disc (IVD) degeneration is primarily involved in back pain, a morbidity that strongly affects the quality of life of individuals nowadays. Lumbar IVDs undergo stressful mechanical loads while being the largest avascular tissues in our body: Mechanical principles alone cannot unravel the intricate phenomena that occur at the cellular scale which are fundamental for the IVD regeneration. The present work aimed at coupling biomechanical and relevant molecular transport processes for disc cells to provide a mechanobiological finite element framework for a deeper understanding of degenerative processes and the planning of regenerative strategies. Given the importance of fluid flow within the IVD, the influence of poroelastic parameters such as permeabilities and solid-phase stiffness of the IVD subtissues was explored. A continuum porohyperelastic material model was then implemented. The angles of collagen fibers embedded in the annulus fibrosus (AF) were calibrated. The osmotic pressure of the central nucleus pulposus (NP) was also taken into account. In a parallel study of the human vertebral bone, microporomechanics was used together with experimental ultrasonic tests to characterize the stiffness of the solid matrix, and to provide estimates of poroelastic coefficients. Fluid dynamics analyses and microtomographic images were combined to understand the fluid exchanges at the bone-IVD interface. The porohyperelastic model of a lumbar IVD with poroelastic vertebral layers was coupled with a IVD transport model of three solutes - oxygen, lactate and glucose - interrelated to reproduce the glycolytic IVD metabolism. With such coupling it was possible to study the effect of deformations, fluid contents, solid-phase stiffness, permeabilities, pH, cell densities of IVD subtissues and NP osmotic pressure on the solute transport. Moreover, cell death governed by glucose deprivation and lactate accumulation was included to explore the mechanical effect on cell viability. Results showed that the stiffness of the AF had the most remarkable role on the poroelastic behavior of the IVD. The permeability of the thin cartilage endplate and the NP stiffness were also relevant. The porohyperelastic model was shown to reproduce the local AF mechanics, provided the fiber angles were calibrated regionally. Such back-calculation led to absolute values of fibers angles and to a global IVD poromechanical behavior in agreement with experiments in literature. The inclusion of osmotic pressure in the NP also led to stress values under confined compression comparable to those measured in healthy and degenerated NP specimens. For the solid bone matrix, axial and transverse stiffness coefficients found experimentally in the present work agreed with universal mass density-elasticity relationships, and combined with continuum microporomechanics provided poroelastic coefficients for undrained and drained cases. The effective permeability of the vertebral bony endplate calculated with fluid dynamics was highly correlated with the porosity measured in microtomographic images. The coupling of transport and porohyperelastic models revealed a mechanical effect acting under large volume changes and high compliance, favored by healthy rather than degenerated IVD properties. Such effect was attributed to strain-dependent diffusivities and diffusion distances and was shown to be beneficial for IVD cells due to the load-dependent increases of glucose levels. Cell density, NP osmotic pressure and porosity were the most important parameters affecting the coupled mechano-transport of metabolites. This novel study highlights the restoration of both cellular and mechanical factors and has a great potential impact for novel designs of treatments focused on tissue regeneration. It also provides methodological features that could be implemented in clinical image-based tools and improve the multiscale understanding of the human spine mechanobiology

Investigating the post-yield behavior of mineralized bone fibril arrays using a 3D non-linear finite element unit-cell model.

In this study, we propose a 3D non-linear finite element (FE) unit-cell model to investigate the post-yield behavior of mineralized collagen fibril arrays (FAY). We then compare the predictions of the model with recent micro-tensile and micropillar compression tests in both axial and transverse directions. The unit cell consists of mineralized collagen fibrils (MCFs) embedded in an extrafibrillar matrix (EFM), and the FE mesh is equipped with cohesive interactions and a custom plasticity model. The simulation results confirm that MCF plays a dominant role in load bearing prior to yielding under axial tensile loading. Damage was initiated via debonding in shear and progressive sliding at the MCF/EFM interface, and resulted in MCF pull-out until brittle failure. In transverse tensile loading, EFM carried most of the load in pre-yield deformation, and then mixed normal/shear debonding between MCF and EFM began to form, which eventually produced brittle delamination of the two phases. The loading/unloading FE analysis in compression along both axial and transverse directions demonstrated perfect plasticity without any reduction in elastic modulus, i.e., damage due to the interfaces as seen in micropillar compression. Beyond the brittle and ductile nature of the stress-strain curves, in tensile and compressive loading, the simulated post-yield behavior and failure mechanism are in good quantitative agreement with the experimental observations. Our rather simple but efficient unit-cell FE model can reproduce qualitatively and quantitatively the mechanical behavior of bone ECM under tensile and compressive loading along the two main orientations. The model's integration into higher length scales may be useful in describing the macroscopic post-yield and failure behavior of trabecular and cortical bone in greater detail

A new framework for characterization of poroelastic materials using indentation

To characterize a poroelastic material, typically an indenter is pressed onto the surface of the material with a ramp of a finite approach velocity followed by a hold where the indenter displacement is kept constant This leads to deformation of the porous matrix, pressurization of the interstitial fluid and relaxation due to redistribution of fluid through the pores. In most studies the poroelastic properties, including elastic modulus, Poisson ratio and poroelastic diffusion coefficient, are extracted by assuming an instantaneous step indentation. However, exerting step like indentation is not experimentally possible and usually a ramp indentation with a finite approach velocity is applied. Moreover, the poroelastic relaxation time highly depends on the approach velocity in addition to the poroelastic diffusion coefficient and the contact area. Here, we extensively studied the effect of indentation velocity using finite element simulations which has enabled the formulation of a new framework based on a master curve that incorporates the finite rise time. To verify our novel framework, the poroelastic properties of two types of hydrogels were extracted experimentally using indentation tests at both macro and micro scales. Our new framework that is based on consideration of finite approach velocity is experimentally easy to implement and provides more accurate estimation of poroelastic properties

Particle-based modeling of the mechanical behavior of porous fluid-saturated viscoelastic solids

In the paper, we developed a macroscopic discrete element model of permeable fluid-saturated materials with solid skeleton characterized by viscoelastic rheological properties. The Biot's linear model of poroelasticity was used as a mathematical basis for describing the mechanical interrelation between the solid skeleton and interstitial fluid. Using this model, we numerically studied the dependences of the effective Young's modulus and strength of fluid-saturated viscoelastic materials on the loading rate, sample size and the mechanical parameters, which determine the relaxation time of the solid-phase skeleton and the time scale of redistribution of fluid in the pore space. We revealed two dimensionless control parameters that determine the dynamic values of the effective mechanical characteristics of the samples under compression loading. We obtained the general relations that describe the above-mentioned dependences in terms of the two proposed control parameters. These relations have a logistic nature and are described by sigmoid functions. The importance of the proposed empirical expressions is determined by the possibility of their application for predicting the mechanical response of fluid-saturated materials of different nature (bone tissue, rocks, porous materials with polymeric skeleton, including elastomers, etc.) under dynamic loading

Multiscale characterisation of cortical bone tissue

Multiscale analysis has become an attractive technique to predict the behaviour of materials whose microstructure strongly changes spatially or among samples, with that microstructure controlling the local constitutive behaviour. This is the case, for example, of most biological tissues-such as bone. Multiscale approaches not only allow, not only to better characterise the local behaviour, but also to predict the field-variable distributions (e.g., strains, stresses) at both scales (macro and micro) simultaneously. However, multiscale analysis usually lacks sufficient experimental feedback to demonstrate its validity. In this paper an experimental and numerical micromechanics analysis is developed with application to cortical bone. Displacement and strain fields are obtained across the microstructure by means of digital image correlation (DIC). The other mechanical variables are computed following the micromechanics theory. Special emphasis is given to the differences found in the different field variables between the micro- and macro-structures, which points out the need for this multiscale approach in cortical bone tissue. The obtained results are used to establish the basis of a multiscale methodology with application to the analysis of bone tissue mechanics at different spatial scales

Numerical model of bone remodeling sensitive to loading frequency through a poroelastic behavior and internal fluid movements

International audienceIn this article, a phenomenological numerical model of bone remodeling is proposed. This model is based on the poroelasticity theory in order to take into account the effects of fluid movements in bone adaptation. Moreover, the proposed remodeling law stands from the classical 'Stanford' law, enriched in order to take into account the loading frequency, through fluid movements. This coupling is materialized by a quadratic function of Darcy velocity. The numerical model is carried out, using a finite element method, and calibrated using experimental results at macroscopic level, from the literature. First results concern cyclic loadings on a mouse ulna, at different frequencies between 1 Hz and 30 Hz, for a force amplitude of 1.5 N and 2 N. Experimental results exhibit a sensitivity to the loading frequency, with privileged frequency for bone remodeling between 5 Hz and 10 Hz, for the force amplitude of 2 N. For the force amplitude of 1.5 N, no privileged frequencies for bone remodeling are highlighted. This tendency is reproduced by the proposed numerical computations. The model is identified on a single case (one frequency and one force amplitude) and validated on the other ones. The second experimental validation deals with a different loading regime: An internal fluid pressure at 20 Hz on a turkey ulna. The same framework is applied, and the numerical and experimental data are still matching in terms of gain in bone mass density

Fluid-driven deformation of a soft granular material

Compressing a porous, fluid-filled material will drive the interstitial fluid out of the pore space, as when squeezing water out of a kitchen sponge. Inversely, injecting fluid into a porous material can deform the solid structure, as when fracturing a shale for natural gas recovery. These poromechanical interactions play an important role in geological and biological systems across a wide range of scales, from the propagation of magma through the Earth's mantle to the transport of fluid through living cells and tissues. The theory of poroelasticity has been largely successful in modeling poromechanical behavior in relatively simple systems, but this continuum theory is fundamentally limited by our understanding of the pore-scale interactions between the fluid and the solid, and these problems are notoriously difficult to study in a laboratory setting. Here, we present a high-resolution measurement of injection-driven poromechanical deformation in a system with granular microsctructure: We inject fluid into a dense, confined monolayer of soft particles and use particle tracking to reveal the dynamics of the multi-scale deformation field. We find that a continuum model based on poroelasticity theory captures certain macroscopic features of the deformation, but the particle-scale deformation field exhibits dramatic departures from smooth, continuum behavior. We observe particle-scale rearrangement and hysteresis, as well as petal-like mesoscale structures that are connected to material failure through spiral shear banding