A paradigm shift is taking place in medicine from using synthetic implants and tissue grafts to a tissue engineering approach that uses degradable porous material scaffolds integrated with biological cells or molecules to regenerate tissues. This new paradigm requires scaffolds that balance temporary mechanical function with mass transport to aid biological delivery and tissue regeneration. Little is known quantitatively about this balance as early scaffolds were not fabricated with precise porous architecture. Recent advances in both computational topology design (CTD) and solid free-form fabrication (SFF) have made it possible to create scaffolds with controlled architecture. This paper reviews the integration of CTD with SFF to build designer tissue-engineering scaffolds. It also details the mechanical properties and tissue regeneration achieved using designer scaffolds. Finally, future directions are suggested for using designer scaffolds with in vivo experimentation to optimize tissue-engineering treatments, and coupling designer scaffolds with cell printing to create designer material/biofactor hybrids.

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Porous scaffold design for tissue engineering

In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined. The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

https://hal.archives-ouvertes.fr/hal-01297725/document

A new constitutive framework for arterial wall mechanics and a comparative study of material models

Built upon the two original books by Mike Crisfield and their own lecture notes, renowned scientist Rene de Borst and his team offer a thoroughly updated yet condensed edition that retains and builds upon the excellent reputation and appeal amongst students and engineers alike for which Crisfield's first edition is acclaimed. Together with numerous additions and updates, the new authors have retained the core content of the original publication, while bringing an improved focus on new developments and ideas. This edition offers the latest insights in non-linear finite element technology, including non-linear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity. The authors' integrated and consistent style and unrivalled engineering approach assures this book's unique position within the computational mechanics literature.

Non-Linear Finite Element Analysis of Solids and Structures: de Borst/Non-Linear Finite Element Analysis of Solids and Structures

A method of approach to the correlation of theory and experiment for incompressible isotropic elastic solids under finite strain was developed in a previous paper (Ogden 1972) Here, the results of that work are extended to incorporate the effects of compressibility (under isothermal conditions) The strain-energy function constructed for incompressible materials is augmented by a function of the density ratio with the result that experimental data on the compressibility of rubberlike materials are adequately accounted for At the same time the good fit of the strain-energy function arising in the incompressibility theory to the data in simple tension, pure shear and equibiaxial tension is maintained in the compressible theory without any change in the values of the material constants A full discussion of inequalities which may reasonably be imposed upon the material parameters occurring in the compressible theory is included

Large Deformation Isotropic Elasticity—On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

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Non-Linear Elastic Deformations

Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379–4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2005.0073

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

When a rubber test piece is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved on the initial loading. This stress softening phenomenon is referred to as the Mullins effect . In this paper a simple phenomenological model is proposed to account for the Mullins effect observed in filled rubber elastomers. The model is based on the theory of incompressible isotropic elasticity amended by the incorporation of a single continuous parameter, interpreted as a damage parameter. This parameter controls the material properties in the sense that it enables the material response to be governed by a strain–energy function on unloading and subsequent submaximal loading different from that on the primary (initial) loading path from the virgin state. For this reason the model is referred to as pseudo-elastic } and a primary loading-unloading cycle involves energy dissipation. The dissipation is measured by a damage function which depends only on the damage parameter and on the point of the primary loading path from which unloading begins. A specific form of this function with two adjustable material constants, coupled with standard forms of the (incompressible, isotropic) strain–energy function, is used to illustrate the qualitative features of the Mullins effect in both simple tension and pure shear. For simple tension the model is then specialized further in order to fit Mullins effect data. It is emphasized that the model developed here is applicable to multiaxial states of stress and strain, not just the specific uniaxial tests highlighted.

Ray W. Ogden

Papers

Non-Linear Elastic Deformations

A new constitutive framework for arterial wall mechanics and a comparative study of material models

Large Deformation Isotropic Elasticity—On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

A pseudo-elastic model for the Mullins effect in filled rubber