Micromechanical analysis of volumetric growth in the context of open systems thermodynamics and configurational mechanics. Application to tumor growth

We adopt in this paper the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the expansion (resp. contraction) of initially small tissue elements distributed within a host surrounding matrix, due to the interfacial motion of their boundary. The interface motion is controlled by the availability of nutrients and mechanical driving forces resulting from the internal stresses that built in during the growth. A general extremum principle of the zero potential for open systems witnessing a change of their mass due to the diffusion of nutrients is constructed, considering the framework of open systems thermodynamics. We postulate that the shape of the tissue element evolves in such a way as to minimize the zero potential among all possible admissible shapes of the growing tissue elements. The resulting driving force for the motion of the interface sets a surface growth models at the scale of the growing tissue elements, and is conjugated to a driving force identified as the interfacial jump of the normal component of an energy momentum tensor, in line with Hadamard’s structure theorem. The balance laws associated with volumetric growth at the mesoscopic level result as the averaging of surface growth mechanisms occurring at the microscopic scale of the growing tissue elements. The average kinematics has been formulated in terms of the effective growth velocity gradient and elastic rate of deformation tensor, both functions of time. This formalism is exemplified by the simulation of the avascular growth of multicell spheroids in the presence of diffusion of nutrients, showing the respective influence of mechanical and chemical driving forces in relation to generation of internal stresses.     

BIOMECHANICAL MODELING OF SURFACE WRINKLING OF SOFT TISSUES WITH GROWTH-DEPENDENT MECHANICAL PROPERTIES

This paper explores growth induced morphological instabilities in biological soft materials.In view of that the growth of a living tissue not only changes its geometry but also can alter its mechanical properties,we suggest a refined volumetric growth model incorporating the effects of growth on the mechanical properties of materials.Analogy between this volumetric growth model and the conventional thermal stress model is addressed for both small and finite deformation problems,which brings great ease for the finite element analysis based on the suggested model.Examples of growth induced surface wrinkling behavior in soft composites,including coreshell soft cylinders and three-layered soft tissues,are explored.The results and discussions foresee possible applications of the model in understanding the correlation between the morphogenesis and growth of soft biological tissues(e.g.skins and tumors),as well as in evaluating the deformation and surface instability behavior of soft artificial materials induced by swelling/shrinkage.     

Sur le remodelage des tissus osseux anisotropesOn the remodelling of anisotropic bone tissue

Growth (change of relaxed lengths) and remodelling (change of mechanical properties) are both involved in the morphogenesis of biological tissues. To model them is of paramount import for progressing both in scientific understanding and health technologies. We model bone tissue as a microstructured continuum, whose mechanical properties at the macroscopic scale are described by a linear, anisotropic elastic response that evolves in time. Our kinematics is rich enough to allow for the microstructural evolution, as well as for the interplay between stress, growth and remodelling. This is a unified approach to the mechanics of growth and remodelling, in which all balance laws derive from one virtual-power principle. As a first application, we study the problem of stiffness remodelling due to planar rotation of the microstructure, excluding bulk growth and all physiological response to mechanical stimuli (passive remodelling). To cite this article: A. DiCarlo et al., C. R. Mecanique 334 (2006).     

A constitutive model for shape memory alloys accounting for thermomechanical coupling

This paper presents a generalized Zaki-Moumni (ZM) model for shape memory alloys (SMAs) [cf. Zaki, W., Moumni, Z., 2007a. A three-dimensional model of the thermomechanical behavior of shape memory alloys. J. Mech. Phys. Solids 55, 2455-2490 accounting for thermomechanical coupling. To this end, the expression of the Helmholtz free energy is modified in order to derive the heat equation in accordance with the principles of thermodynamics. An algorithm is proposed to implement the coupled ZM model into a finite element code, which is then used to solve a thermomechanical boundary value problem involving a superelastic SMA structure. The model is validated against experimental data available in the literature. Strain rate dependence of the mechanical pseudoelastic response is taken into account with good qualitative as well as quantitative accuracy in the case of moderate strain rates and for mechanical results in the case of high strain rates. However, only qualitative agreement is achieved for thermal results at high strain rates. It is shown that this discrepancy is mainly due to localization effects which are note taken into account in our model. Analyzing the influence of the heat sources on the material response shows that the mechanical hysteresis is mainly due to intrinsic dissipation, whereas the thermal response is governed by latent heat. In addition, the variation of the area of the hysteresis loop with respect to the strain rate is discussed. It is found that this variation is not monotonic and reaches a maximum value for a certain value of strain rate.     

Mathematical modeling of anisotropic avascular tumor growth

Cancer represents one the most challenging problems in medicine and biology nowadays, and is being actively addressed by many researchers from different areas of knowledge. The increasing development of sophisticated mathematical models and computer-based procedures has had a positive impact on our understanding of cancer-related mechanisms and the design of anticancer treatment strategies. However, further investigation and experimentation are still required to completely elucidate the tumor-associated mechanical responses, as well as the effect of mechanical forces on the net tumor growth. In this work we develop a theoretical framework in the context of continuum mechanics to investigate the anisotropic growth of avascular tumor spheroids. To that end, a specific anisotropic growth deformation tensor is considered, which also describes an isotropic growth law as a particular case. Mixtures theory and the notion of multiple natural configurations are then used to formulate a mathematical model of avascular tumor growth. More precisely, mass, momentum balance and nutrients diffusion equations are derived, where solid tumors are assumed as hyperelastic and compressible materials. Moreover, mechanical interactions with a rigid extracellular matrix (ECM) are considered, and the mechanical modulation of growing tumors in a rigid surrounding tissue is investigated by means of numerical simulations. Finally, the model results are compared with experimental data previously reported in the literature.     

Modeling of growth and remodeling in soft biological tissues with multiple constituents

Among the various important characteristics of biological tissues is their ability to grow and remodel. It is well-known that one of the primary triggers behind the growth and remodeling process is changes in the mechanical environment, for instance changes in stress, strain, etc. These mechanisms of mechanotransduction are the driving force behind many changes in structure and function including growth and remodeling. The purpose of this article is to formulate better constitutive equations for the stress in tissues with multiple constituents undergoing growth and remodeling. This is a very complex problem and is of tremendous importance. Here, we do the modeling from a mechanics point of view, utilizing the theory of natural configurations coupled with population dynamics to accurately model the production and removal of the different constituents that comprise the tissue. This is accomplished by deriving a generalized McKendrick equation for growth and remodeling and has the advantage of directly including the age distribution of constituents into the model. The population distribution function is then used to determine the stress in the tissue.     

On a new interpretation of the classical Maxwell model

Recently, [Rao, I.J., Rajagopal, K.R., 2007. Status of the K-BKZ model within the framework of materials with multiple natural configurations. Journal of Non-Newtonian Fluid Mechanics, 141, 79–84] showed that the K-BKZ Model is a special sub-class of models based on a thermodynamic framework that takes into account the fact that bodies are capable of existing stress free in multiple configurations with special choices being made for the way in which the body stores energy and the way it dissipates energy. They also showed that several generalizations of the K-BKZ model are possible. In this short note we show that two distinct methods of storing energy and dissipating energy lead to the classical Maxwell model. That is, in addition to the classical choice for the storage of energy and rate of dissipation (the usual spring dashpot analogy) a more complicated choice also leads to the same model. This result is rather important as it shows that a variety of means for storing and dissipating energy can lead to the same mechanical response, when one restricts oneself to purely mechanical considerations.     

Energy release rates for delamination of composite beams

This paper deals with the formulation of a mechanical/numerical model for analyzing the delamination effects of layered composite beams. The laminate is modelled through a multiple-beam model and interfacial constitutive laws are obtained by introducing interlaminar bilateral and unilateral springs. Delamination growth is described by employing the classical energy release rate criterion. A path-following procedure with delamination growth control is presented for the numerical analysis of the given model. Numerical results on delamination buckling and growth in compressed beams are given and comparisons with simplified theories are established.     

A mathematical model of cortical bone remodeling at cellular level under mechanical stimulus

A bone cell population dynamics model for cortical bone remodeling under mechanical stimulus is developed in this paper. The external experiments extracted from the literature which have not been used in the creation of the model are used to test the validity of the model. Not only can the model compare reasonably well with these experimental results such as the increase percentage of final values of bone mineral content (BMC) and bone fracture energy (BFE) among different loading schemes (which proves the validity of the model), but also predict the realtime development pattern of BMC and BFE, as well as the dynamics of osteoblasts (OBA), osteoclasts (OCA), nitric oxide (NO) and prostaglandin E₂ (PGE₂) for each loading scheme, which can hardly be monitored through experiment. In conclusion, the model is the first of its kind that is able to provide an insight into the quantitative mechanism of bone remodeling at cellular level by which bone cells are activated by mechanical stimulus in order to start resorption/formation of bone mass. More importantly, this model has laid a solid foundation based on which future work such as systemic control theory analysis of bone remodeling under mechanical stimulus can be investigated. The to-be identified control mechanism will help to develop effective drugs and combined nonpharmacological therapies to combat bone loss pathologies. Also this deeper understanding of how mechanical forces quantitatively interact with skeletal tissue is essential for the generation of bone tissue for tissue replacement purposes in tissue engineering.     

Medical image-based simulation of abdominal aortic aneurysm growth

Advances in theoretical modeling of biological tissue growth and remodeling (G&R) and computational biomechanics have been helpful to capture salient features of vascular remodeling during the progression of vascular diseases. Nevertheless, application of such advances to individualized diagnosis and clinical treatment of diseases such as abdominal aortic aneurysm (AAA) remains challenging. As a step toward that goal, in this paper, we present a computational framework necessary towards patient-specific modeling of AAA growth. Prior to AAA simulations, using an inverse optimization method, initial material parameters are identified for a healthy aorta such that a homeostatic condition is satisfied for the given medical image-based geometrical model under physiological conditions. Various shapes of AAAs are then computationally created by inducing elastin degradation with different spatio-temporal distributions. The simulation results emphasize the role of extent of elastin damage, geometric complexity of an enlarged AAA, and sensitivity of stress-mediated collagen turnover on the wall stress distribution and the rate of expansion. The results also show that the distributions of stress and local expansion initially correspond to the extent of elastin damage, but change via stress-mediated tissue G&R depending on the aneurysm shape. Finally, we suggest that the current framework can be utilized along with medical images from an individual patient to predict the AAA shape and mechanical properties in the near future via an inverse scheme.     

考虑应变率效应的混凝土随机损伤本构模型研究

混凝土材料组分复杂且具有随机分布的特点,其受力力学行为不可避免地存在非线性和随机性.同时,在动力荷载作用下,混凝土材料具有不可忽视的率敏感性.为了综合反映混凝土受力力学行为中的非线性、随机性与率敏感性,本文从对材料的纳-微观裂纹扩展分析入手,引入速率过程理论描述纳观裂纹的扩展速率,并研究了对应的能量耗散过程.在此基础上通过裂纹层级模型将纳观分析推演到微观尺度,建立了微观能量耗散的基本表达式.进而与微-细观随机断裂模型相结合,形成了混凝土纳-微-细观随机损伤本构模型.同时,基于速率相关势垒的分析,揭示了动力强度的提高源自加载速率和原子键断裂速率的竞争机制.据此,假定微裂纹间相互作用与应变率比值的相关关系以建立微弹簧能量耗散速率与应变率的联系,实现了从静力本构模型向动力本构模型的扩展.数值算例表明,建议模型能够同时反映混凝土材料力学行为中的非线性、随机性和率敏感性.最后通过与相关试验结果的对比,验证了建议模型的正确性.     

A dynamic cellular vertex model of growing epithelial tissues

Intercellular interactions play a significant role in a wide range of biological functions and processes at both the cellular and tissue scales, for example, embryogenesis, organogenesis, and cancer invasion. In this paper, a dynamic cellular vertex model is presented to study the morphome-chanics of a growing epithelial monolayer. The regulating role of stresses in soft tissue growth is revealed. It is found that the cells originating from the same parent cell in the monolayer can orchestrate into clustering patterns as the tis-sue grows. Collective cell migration exhibits a feature of spatial correlation across multiple cells. Dynamic intercel-lular interactions can engender a variety of distinct tissue behaviors in a social context. Uniform cell proliferation may render high and heterogeneous residual compressive stresses, while stress-regulated proliferation can effectively release the stresses, reducing the stress heterogeneity in the tissue. The results highlight the critical role of mechanical factors in the growth and morphogenesis of epithelial tissues and help understand the development and invasion of epithelial tumors.     

X-ray scattering measurements of particle orientation in a sheared polymer/clay dispersion

We report steady and transient measurements of particle orientation in a clay dispersion subjected to shear flow. An organically modified clay is dispersed in a Newtonian polymer matrix at a volume fraction of 0.02, using methods previously reported by Mobuchon et al. (Rheol Acta 46: 1045, 2007). In accord with prior studies, mechanical rheometry shows yield stress-like behavior in steady shear, while time dependent growth of modulus is observed following flow cessation. Measurements of flow-induced orientation in the flow-gradient plane of simple shear flow using small-angle and wide-angle X-ray scattering (SAXS and WAXS) are reported. Both SAXS and WAXS reveal increasing particle orientation as shear rate is increased. Partial relaxation of nanoparticle orientation upon flow cessation is well correlated with time-dependent changes in complex modulus. SAXS and WAXS data provide qualitatively similar results; however, some quantitative differences are attributed to differences in the length scales probed by these techniques.     

Numerical solution of the oxygen diffusion problem in cylindrically shaped sections of tissue

A mathematical model is presented which describes the diffusion of oxygen in absorbing tissue, and numerical solution of its partial differential equation is obtained by the finite difference equations. The diffusion with absorption model is associated with the process of a moving boundary which marks the furthest penetration of oxygen in the absorbing cylindrically shaped sections of tissue and also allows for an initial distribution of oxygen through the absorbing tissue. Copyright © 2007 John Wiley & Sons, Ltd.     

A predictive mechano-biological model of the bone-implant healing

The quality of the fixation orthopaedic implant to its surrounding bone determines its clinical longevity. Up to 20% of hip replacement operations are currently revisions for aseptic loosening. While this fixation quality is determined primarily by the bone and tissue anchoring the implant, conditions influencing bone growth in the early post-operative period include the surgical technique and coupled mechanical and biochemical factors. The aim of the study was to propose an original mechano-biological formulation of the healing process of periprosthetic tissue. The multiphasic porous model involved the solid osseous matrix, the extracellular fluid phase, the osteoblastic cellular phase responsible from the bone formation and the growth factor phase promoting the cellular activity. To derive the non-linear convective-diffuse governing equations, mass balance was associated to cell active haptotactic and chemotactic migration, growth factor diffusion, cell proliferation (logistic law) and bone formation (reactive medium). The in-vivo application concerned a canine axisymmetric implant which was stable and mechanically unloaded. Predictive numerical results were compared to ex-vivo data from a histologic study. The generic healing pattern involving two main oscillations of the radial bone formation was well predicted. In the future, the model could assist in evaluating the role of growth factor concentrations and their temporal delivering as far as the role of pertinent sources such as bioactive coating or additional biomaterials.     

Hydrocephalic cerebrospinal fluid flowing rotationally with pulsatile boundaries: A mathematical simulation of the thermodynamical approach

To study the kinematics of flow rate and ventricular dilatation, an analytical perturbation approach of hydrocephalus has been devised. This research provides a comprehensive investigation of the characteristics of cerebrospinal fluid (CSF) flow and pressure in a hydrocephalic patient. The influence of hydrocephalic CSF, flowing rotationally with realistic dynamical characteristics on pulsatile boundaries of subarachnoid space, was demonstrated using a nonlinear controlling system of CSF. An analytical perturbation method of hydrocephalus has been developed to investigate the biomechanics of fluid flow rate and the ventricular enlargement. In this paper presents a detailed analysis of CSF flow and pressure dynamics in a hydrocephalic patient. It was elaborated with a nonlinear governing model of CSF to show the influence of hydrocephalic CSF, flowing rotationally with realistic dynamical behaviors on pulsatile boundaries of subarachnoid space. In accordance with the suggested model, the elasticity factor changes depending on how much a porous layer, in this case the brain parenchyma, is stretched. It was improved to include the relaxation of internal mechanical stresses for various perturbation orders, modelling the potential plasticity of brain tissue. The initial geometry that was utilised to create the framework of CSF with pathological disease hydrocephalus and indeed the output of simulations using this model were compared to the actual progression of ventricular dimensions and shapes in patients. According to this observation, the non - linear and elastic mechanical phenomena incorporated into the current model are probably true. Further modelling of ventricular dilation at a normal pressure may benefit from the existence of a valid model whose parameters approximate genuine mechanical characteristics of the cerebral cortex.     

基于PVDF薄膜传感器的猪肝动态压缩力学性能测试

瞬态冲击载荷作用下肝脏的力学响应是损伤生物力学的重要研究内容.本文提出了一种可用于软组织动态压缩力学特性测试的改进SHPB(分离式霍普金森压杆,Split Hopkinson Pressure Bar)方法.该方法采用PVDF(聚偏氟乙烯,Polyvinylidene Fluor)压电薄膜传感器测量实验过程中试件两端面...