Claire Morin

A multiscale poromicromechanical approach to wave propagation and attenuation in bone

Ultrasonics is an important diagnostic tool for bone diseases, as it allows for non-invasive assessment of bone tissue quality through mass density-elasticity relationships. The latter are, however, quite complex for fluid-filled porous media, which motivates us to develop a rigorous multiscale poromicrodynamics approach valid across the great variety of different bone tissues. Multiscale momentum and mass balance, as well as kinematics of a hierarchical double porous medium, together with Darcys law for fluid flow and micro-poro-elasticity for the solid phase of bone, give access to the so-called dispersion relation, linking the complex wave numbers to corresponding wave frequencies. Experimentally validated results show that 2.25 MHz acoustical signals transmit healthy cortical bone (exhibiting a low vascular porosity) only in the form of fast waves, agreeing very well with experimental data, while both fast and slow waves transmit highly osteoporotic as well as trabecular bone (exhibiting a large vascular porosity). While velocities and wavelengths of both fast and slow waves, as well as attenuation lengths of slow waves, are always monotonously increasing with the permeability of the bone sample, the attenuation length of fast waves shows a minimum when considered as function of the permeability.

Modeling Tensile-Compressive Asymmetry for Superelastic Shape Memory Alloys.

In this article, the Zaki-Moumni (ZM) model for shape memory alloys is extended to account for tensile-compressive asymmetry over a wide temperature range. To this avail, a mathematical framework recently developed by Raniecki and Mróz is utilized to define new yield functions that are sign-sensitive. With respect to the original ZM model, the modifications are essentially made to the expressions of the Helmholtz free energy and of the internal constraints. The model is shown to properly simulate the asymmetric behavior of shape memory alloys both for martensite orientation and pseudoelasticity.

Direct Numerical Determination of the Asymptotic Cyclic Behavior of Pseudoelastic Shape Memory Structures

The design of shape memory alloys (SMAs) structures against fatigue requires the computation of the stabilized mechanical state. The classical computation method, based on a plasticity-like algorithm, requires a step-by-step calculation, leading to prohibitive computation time to reach this stabilized state. To overcome this issue, we propose to extend the direct cyclic method (DCM), for elastoplastic structures, for use with the Zaki-Moumni (ZM) model for SMAs. DCM is a large time increment method in which a periodicity condition is enforced on the state variables. Comparison with the classical incremental approach shows considerable reduction in computation time.

Mineralization-driven bone tissue evolution follows from fluid-to-solid phase transformations in closed thermodynamic systems

The fundamental mechanisms that govern bone mineralization have been fairly well evidenced by means of experimental research. However, rules for the evolution of the volume and composition of the bone tissue compartments (such as the mineralized collagen fibrils and the extrafibrillar space in between) have not been provided yet. As an original contribution to this open question, we here test whether mineralizing bone tissue can be represented as a thermodynamically closed system, where crystals precipitate from an ionic solution, while the masses of the fibrillar and extrafibrillar bone tissue compartments are preserved. When translating, based on various experimental and theoretical findings, this mass conservation proposition into diffraction-mass density relations, the latter are remarkably well confirmed by independent experimental data from various sources. Resulting shrinkage and composition rules are deemed beneficial for further progress in bone materials science and biomedical engineering.

A comprehensive study of layer-specific morphological changes in the microstructure of carotid arteries under uniaxial load

The load bearing properties of large blood vessels are principally conferred by collagen and elastin networks and their microstructural organization plays an important role in the outcomes of various arterial pathologies. In particular, these fibrous networks are able to rearrange and reorient spatially during mechanical deformations. In this study, we investigate for the first time whether these well-known morphological rearrangements are the same across the whole thickness of blood vessels, and subsequently if the underlying mechanisms that govern these rearrangements can be predicted using affine kinematics. To this aim, we submitted rabbit carotid samples to uniaxial load in three distinct deformation directions, while recording live images of the 3D microstructure using multiphoton microscopy. Our results show that the observed realignment of collagen and elastin in the media layer, along with elastin of the adventitia layer, remained limited to small angles that can be predicted by affine kinematics. We show also that collagen bundles of fibers in the adventitia layer behaved in significantly different fashion. They showed a remarkable capacity to realign in the direction of the load, whatever the loading direction. Measured reorientation angles of the fibers were significantly higher than affine predictions. This remarkable property of collagen bundles in the adventitia was never observed before, it shows that the medium surrounding collagen in the adventitia undergoes complex deformations challenging traditional hyperelastic models based on mixture theories. STATEMENT OF SIGNIFICANCE The biomechanical properties of arteries are conferred by the rearrangement under load of the collagen and elastin fibers making up the arterial microstructure. Their kinematics under deformation is not yet characterized for all fiber networks. In this respect we have submitted samples of arterial tissue to uniaxial tension, simultaneously to confocal imaging of their microstructure. Our method allowed identifying for the first time the remarkable ability of adventitial collagen fibers to reorient in the direction of the load, achieving reorientation rotations that exceeded those predicted by affine kinematics, while all other networks followed the affine kinematics. Our results highlight new properties of the microstructure, which might play a role in the outcomes of vascular pathologies like aneurysms.

A simple 1D model with thermomechanical coupling for superelastic SMAs

This paper presents an outline for a new uniaxial model for shape memory alloys that accounts for thermomechanical coupling. The coupling provides an explanation of the dependence of SMA behavior on the loading rate. 1D simulations are carried in Matlab using simple finite-difference discretization of the mechanical and thermal equations.

The concept of frozen elastic energy as a consequence of changes in microstructure morphology

Constitutive modelling of soft biological tissues has been the topic of abundant literature. These biological tissues, made of variously oriented and crimped fibers embedded in a soft matrix, exhibit a highly nonlinear anisotropic behavior with the ability to sustain large reversible strains. The existing constitutive models are mainly phenomenological hyperelastic models developed at the macroscopic scale (Holzapfel & Gasser 2000). Experimental mechanical tests performed on soft tissues and coupled to confocal microscopic imaging (Schrauwen et al. 2012) reveal that this nonlinear behavior originates in geometrical changes in the microstructure, such as progressive decrimping and re-alignement of the fibers along the load direction. This confirms the growing need to understand the relationship between phenomena taking place in the microstructure and macroscopic mechanical response; subsequently driving forward multi-scale approaches (Morin & Hellmich 2014). We here propose to model the reorientation of the fibers within the matrix through extension of the framework of continuum micromechanics (Zaoui 2002) and Eshelby’s inclusion problems (Eshelby 1957). We investigate the ability of the proposed model to capture, through microstructure morphology changes, the non-linear mechanical response of soft tissues, the possible path dependence of their response to multiaxial loading, and a remaining frozen elastic energy after complete unloading of the tissue.

Kinematics of collagen fibers in carotid arteries under tension-inflation loading

Biomechanics of the extracellular matrix in arteries determines their macroscopic mechanical behavior. In particular, the distribution of collagen fibers and bundles plays a significant role. Experimental data showed that, in most arterial walls, there are preferred fiber directions. However, the realignment of collagen fibers during tissue deformation is still controversial: whilst authors claim that fibers should undergo affine deformations, others showed the contrary. In order to have an insight about this important question of affine deformations at the microscopic scale, we measured the realignment of collagen fibers in the adventitia layer of carotid arteries using multiphoton microscopy combined with an unprecedented Fourier based method. We compared the realignment for two types of macroscopic loading applied on arterial segments: axial tension under constant pressure (scenario 1) and inflation under constant axial length (scenario 2). Results showed that, although the tissue underwent macroscopic stretches beyond 1.5 in the circumferential direction, fiber directions remained unchanged during scenario 2 loading. Conversely, fibers strongly realigned along the axis direction for scenario 1 loading. In both cases, the motion of collagen fibers did not satisfy affine deformations, with a significant difference between both cases: affine predictions strongly under-estimated fiber reorientations in uniaxial tension and over-estimated fiber reorientations during inflation at constant length. Finally, we explained this specific kinematics of collagen fibers by the complex tension-compression interactions between very stiff collagen fibers and compliant surrounding proteins. A tensegrity representation of the extracellular matrix in the adventitia taking into account these interactions was proposed to model the motion of collagen fibers during tissue deformation.

Time Integration and Assessment of a Model for Shape Memory Alloys Considering Multiaxial Nonproportional Loading Cases

The paper presents a numerical implementation of the ZM model for shape memory alloys that fully accounts for non-proportional loading and its influence on martensite reorientation and phase transformation. Derivation of the time-discrete implicit integration algorithm is provided. The algorithm is used for finite element simulations using Abaqus, in which the model is implemented by means of a user material subroutine. The simulations are shown to agree with experimental and numerical simulation data taken from the literature.Copyright

Review of “Universal” Rules Governing Bone Composition, Organization, and Elasticity Across Organizational Hierarchies

“Universal” organizational patterns in bone are reviewed and presented, in terms of mathematically expressed rules concerning the composition and elasticity of a large variety of tissues. Firstly, experimental data sets gained from dehydration-demineralization tests, dehydration-deorganification tests, and dehydration-ashing tests are thoroughly analyzed. On this basis, bilinear relations can be identified, between the mass density of the extracellular bone matrix on the one hand, and the apparent mass densities of its basic constituents (water, hydroxyapatite, and organic matter), on the other hand. Secondly, the question as to how hydroxyapatite is distributed in bone tissue is addressed. To that end, mass and volume measurements gained from wet, dehydrated, and demineralized tissue samples, as well as optical densities provided by transmission electron microscopy, are studied, confirming a rule on how the mineral is partitioned between fibrillar and extrafibrillar spaces in the ultrastructure of bone. Thirdly, a swelling rule for hydrating collagen is validated through processing of experimental data from X-ray diffraction, vacuum drying, and mass measurements, quantifying the change of the bone tissue composition upon hydration. And fourthly, application of the mass conservation law to extracellular bone matrix considered as closed thermodynamic system, allows for studying the change of bone tissue composition during mineralization. Finally, these compositional rules, which are shown to be “universally” valid throughout the vertebrate kingdom, enter a micromechanical homogenization scheme for upscaling the experimentally accessible elastic properties of the elementary mechanical building blocks of bone (hydroxyapatite minerals, type I collagen, and water with non-collageneous organics) to the macroscopic scale of cortical and trabecular bone.