A theoretical model for surface bone remodeling under electromagnetic loads

A theoretical model for surface bone remodeling under electromagnetic loads is proposed in this paper. In the model, surface bone remodeling is assumed to be related to growth factors. Growth factors in latent form in osteocytes are released to the bone fluid after the osteocytes are absorbed by osteoclasts, and then regulate the bone formation process. At the same time, environmental loadings can influence the generation of growth factors. This paper shows how surface bone remodeling is triggered under the influence of growth factors. Based on this hypothesis, a computational model is established that simulates the bone coupling remodeling process, including internal and surface bone remodeling. The effects of various loadings, including electrical and magnetic loadings, are simulated and compared. The interactions between internal and surface bone remodeling are investigated via the numerical method. The results indicate that an electromagnetic field can strongly influence the bone remodeling process and that the remodeling process will be altered after surface bone remodeling is triggered, compared to the sole effect of the internal remodeling process.     

骨组织生长重建的力学生物学机制研究进展

生物力学已被证实是骨组织生长、重建及成形当中一个十分重要的因素. 骨组织的损 伤修复过程本质上是细胞的生物学过程和应力作用下的生长过程. 这虽然肯定了生物力学在 骨组织生长、重建过程中的重要地位, 但是, 人们对生物力学因素如何诱导骨生长、 重建的力学生物学机制仍不甚了解. 而骨组织工程需要更为科学完善的细胞生物学机制来研究和探 索骨组织的构建过程. 本文概述了国内外生物力学与骨组织生长重建的宏微观理论, 主要讨 论了骨组织结构及功能形成过程中的力学生物学相关问题.     

骨疲劳损伤研究

和所有人造的结构材料一样，骨在重复载荷作用下也会发生疲劳 断裂.近年来,在骨疲劳损伤领域的研究已取得较大发展.本文综述了骨 疲劳损伤的一些共同性研究课题,重点包括骨疲劳损伤的特性,基于骨疲 劳微裂纹的损伤评估模型及骨疲劳行为的实验研究结果等.最后提出要 更好理解骨疲劳损伤,修复和再生行为仍然有大量的研究工作需要开展.     

Effect of magnetic field on poroelastic bone model for internal remodeling

This paper studies the effects of the magnetic field and the porosity on a poroelastic bone model for internal remodeling. The solution of the internal bone remodeling process induced by a magnetic field is presented. The bone is treated as a poroelastic material by Biot’s formulation. Based on the theory of small strain adaptive elasticity, a theoretical approach for the internal remodeling is proposed. The components of the stresses, the displacements, and the rate of internal remodeling are obtained in analytical forms, and the numerical results are represented graphically. The results indicate that the effects of the magnetic field and the porosity on the rate of internal remodeling in bone are very pronounced.     

Damage-Bone Remodelling Theory: An Extension to Anisotropic Damage

Everyday activities damage bone, and this damage is normally repaired in a continuous remodelling process. When an imbalance in this remodelling process occurs, bones may become more susceptible to fracture. This paper is concerned with a derivation of: (a) the general rule for bone remodelling in interaction with anisotropic damage and (b) the associated thermodynamic restrictions. The procedure of Coleman and Noll, utilizing the Clausius–Duhem inequality is used and is provided as basis for a consistent formulation of bone remodelling involving a chemical reaction and mass transfer between two constituents in presence of microcracks. It is hoped that this work provides a rational basis for the practical use of such theoretical results.     

A fully anisotropic hierarchical hybrid cellular automaton algorithm to simulate bone remodeling

Computational models of the bone remodeling process have been utilized to further our understanding of the adaptation of bone architecture to changes in its mechanical environment. The hierarchical hybrid cellular automata (HHCA) algorithm is a multi-scale approach for the simulation of the adaptation of bone. Currently, this remodeling algorithm utilizes the apparent material properties of the trabecular architecture. The objective of this work is to increase the fidelity of the HHCA algorithm by incorporating the local anisotropic properties of these structures. Preliminary analyses display improved efficiency and a more consistent material distribution when incorporating anisotropic properties into the HHCA methodology.     

Parametric study of control mechanism of cortical bone remodeling under mechanical stimulus

The control mechanism of mechanical bone remodeling at cellular level was investigated by means of an extensive parametric study on a theoretical model described in this paper. From a perspective of control mechanism, it was found that there are several control mechanisms working simultaneously in bone remodeling which is a complex process. Typically, an extensive parametric study was carried out for investigating model parameter space related to cell differentiation and apoptosis which can describe the fundamental cell lineage behaviors. After analyzing all the combinations of 728 permutations in six model parameters, we have identified a small number of parameter combinations that can lead to physiologically realistic responses which are similar to theoretically idealized physiological responses. The results presented in the work enhanced our understanding on mechanical bone remodeling and the identified control mechanisms can help researchers to develop combined pharmacological-mechanical therapies to treat bone loss diseases such as osteoporosis.     

含损伤演化的TM耦合数值模型及其应用研究

从岩石材料的细观结构层次出发，应用损伤力学和热弹性理论，对热力耦合作用下岩石破裂 过程中热－应力相互作用关系进行了分析. 初步建立了细观岩石热－力(TM)耦合数值模型， 探讨了TM耦合作用下岩石材料的细观结构损伤及其诱发的材料力学性能演化机制，把岩石 热固耦合问题的研究从应力状态分析深入到损伤、破坏过程分析之中. 运用该数值模型对某 硬岩实验室开展的原位尺度实验中的废料处理井间柱稳定性进行了模拟分析，其应力场、岩 石剥离破坏形态及破坏诱发的AE特性等均与实验监测结果表现出了较好的一致性，证 明了该数值模型的合理性和有效性.     

Bone remodeling II: small strain adaptive elasticity

The general theory of adaptive elastic materials proposed in [1] as a model for the physiological process of bone remodeling is specialized to the case of small strains in isothermal processes. Boundary value problems in the context of this specialized theory are specified and solved. It is shown that remodeling will not occur in a long bone such as a femur as a result of a pure torsion about the long axis of the bone. Criteria for biologically normal remodeling are established for steady stress and steady strain.

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Zusammenfassung Die allgemeine Theorie über adaptives elastisches Material, wie sie in Referenz [1] als ein Modell für den physiologischen Prozess der Knochenrekonstruktion vorgeschlagen wird ist auf den Fall kleiner Dehnungen und isothermer Prozesse beschränkt. Randwertprobleme im Rahmen dieser Theorie werden angegeben und gelöst. Es wird gezeigt, das in langen knochen wie dem Femur, keine Rekonstruktion erfolgt als folge einer Torsionsbelastung um die Langsachse des knochens. Kriterien für eine biologisch normale Rekonstruktion unter konstanten spannungen und Dehnungen werden angegeben.

     

Identification of a constitutive law for trabecular bone samples under remodeling in the framework of irreversible thermodynamics

We construct in the present paper constitutive models for bone remodeling based on micromechanical analyses at the scale of a representative unit cell (RUC) including a porous trabecular microstructure. The time evolution of the microstructure is simulated as a surface remodeling process by relating the surface growth remodeling velocity to a surface driving force incorporating a (surface) Eshelby tensor. Adopting the framework of irreversible thermodynamics, a 2D constitutive model based on the setting up of the free energy density and a dissipation potential is identified from FE simulations performed over a unit cell representative of the trabecular architecture obtained from real bone microstructures. The static and evolutive effective properties of bone at the scale of the RUC are obtained by combining a methodology for the evaluation of the average kinematic and static variables over a prototype unit cell and numerical simulations with controlled imposed first gradient rates. The formulated effective growth constitutive law at the scale of the homogenized set of trabeculae within the RUC is of viscoplastic type and relates the average growth strain rate to the homogenized stress tensor. The postulated model includes a power law function of an effective stress chosen to depend on the first and second stress invariants. The model coefficients are calibrated from a set of virtual testing performed over the RUC subjected to a sequence of loadings. Numerical simulations show that overall bone growth does not show any growth kinematic hardening. The obtained results quantify the strength and importance of different types of external loads (uniaxial tension, simple shear, and biaxial loading) on the overall remodeling process and the development of elastic deformations within the RUC.     

基于应变能准则优化模型的骨骼重建数值模拟

将骨骼重建的适应性弹性理论及参考应变能理论与结构优化及有限元方法结合,建立了基于应变能准则优化模型的骨骼重建数值模拟方法,研究骨骼内部重建的机理和规律。以单元应变能密度为刺激源,由内部材料的分布变化来模拟骨重建的过程和规律。通过对股骨头重建的数值模拟,取得了与临床实验相符的结果,也证实了骨结构形态是对力学环境的最佳适应,定量地反映了力学刺激对骨骼重建的影响,得到了符合骨骼重建规律的结论。     

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.     

The influence of different loads on the remodeling process of a bone and bioresorbable material mixture with voids

A model of a mixture of bone tissue and bioresorbable material with voids was used to numerically analyze the physiological balance between the processes of bone growth and resorption and artificial material resorption in a plate-like sample. The adopted model was derived from a theory for the behavior of porous solids in which the matrix material is linearly elastic and the interstices are void of material. The specimen—constituted by a region of bone living tissue and one of bioresorbable material—was acted by different in-plane loading conditions, namely pure bending and shear. Ranges of load magnitudes were identified within which physiological states become possible. Furthermore, the consequences of applying different loading conditions are examined at the end of the remodeling process. In particular, maximum value of bone and material mass densities, and extensions of the zones where bone is reconstructed were identified and compared in the two different load conditions. From the practical view point, during surgery planning and later rehabilitation, some choice of the following parameters is given: porosity of the graft, material characteristics of the graft, and adjustment of initial mixture tissue/bioresorbable material and later, during healing and remodeling, optimal loading conditions.     

反演在骨生长方程参数识别中的应用

基于BFGS算法,根据自行设计的动物实验,得到在不同应力环境中,快速生长期大鼠股骨生 长与重建过程中骨密度实测数据,采用反演数值方法,获取了骨生长方程中随时间变化的 生物参数B和K. 通过正演验证,表明所建立的基于BFGS算法的参数识别方法具有较好的 稳定性和较高的识别精度,能够反演出比较切合实际的生物参数数值. 研究表明,反问 题方法是解决生命现象不可测性和未知性问题的有效手段,把反演方法应用到骨生长与重建 等生命现象的研究中,可确定、修正预设的数学模式,为数值量化骨适应生物模型的建立提 供了一条可行的途径.     

Surface remodeling of bones induced by intra-medullary nailing

The paper is devoted to a study on the surface remodeling of bones. Anisotropy and piezoelectricity of bone tissue (as per previous experimental studies) are incorporated in the analysis. Surface remodeling induced by intra-medullary nailing is of primary concern in the study.     

Bone remodeling I: theory of adaptive elasticity

A thermomechanical continuum theory involving a chemical reaction and mass transfer between two constituents is developed here as a model for bone remodeling. Bone remodeling is a collective term for the continual processes of growth, reinforcement and resorbtion which occur in living bone. The resulting theory describes an elastic material which adapts its structure to applied loading.

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Zusammenfassung Eine Thermo-mechanische kontinuum Theorie als Modell für die Knochenrekonstrucktion wird entwickelt, die eine chemische Reaktion und einen Massentransport zwischen zwei Komponenten behandelt. Knochenrekonstruktion ist ein Sammelbegriff für die kontinuierlichen Prozesse des Wachsens, der Verstärkung und des Abbaus wie sie im lebenden Knochen auftreten. Die Theorie beschreibt ein elastisches Material, das sich in der Form der Belastung anpasst.

     

基于Wolff法则连续体拓扑优化方法中的参数分析

Wolff法则是指骨骼在外部荷载变化时,骨骼内部小梁骨保持沿主应力方向分布以更好抵抗外部荷载。基于Wolff法则的连续体拓扑优化方法是模仿骨骼重建规律的一种新的连续体优化方法。基本思想是将待优化的结构看成是遵从Wolff法则的“骨骼”,仿照骨骼重建过程,连续体的拓扑优化过程即为“骨骼”重建过程。该方法中利用用于描述材料微结构几何及弹性性质的构造张量作为设计变量,采用参考应变区间确定构造张量特征主值的更新规律。本文通过对仿生过程中各因素的分析,解释了优化模型中各参数的物理意义。通过数值分析,给出了参数选取规律以保证算法稳定和快速收敛,从而使得本文优化方法更具实际应用价值。     

A fluid flow model in the lacunar-canalicular system under the pressure gradient and electrical field driven loads

The lacunar-canalicular system (LCS) is acknowledged to directly participate in bone tissue remodeling. The fluid flow in the LCS is synergic driven by the pressure gradient and electric field loads due to the electro-mechanical properties of bone. In this paper, an idealized annulus Maxwell fluid flow model in bone canaliculus is established, and the analytical solutions of the fluid velocity, the fluid shear stress, and the fluid flow rate are obtained. The results of the fluid flow under pressure gradient driven (PGD), electric field driven (EFD), and pressure-electricity synergic driven (P-ESD) patterns are compared and discussed. The effects of the diameter of canaliculi and osteocyte processes are evaluated. The results show that the P-ESD pattern can combine the regulatory advantages of single PGD and EFD patterns, and the osteocyte process surface can feel a relatively uniform shear stress distribution. As the bone canalicular inner radius increases, the produced shear stress under the PGD or P-ESD pattern increases slightly but changes little under the EFD pattern. The increase in the viscosity makes the flow slow down but does not affect the fluid shear stress (FSS) on the canalicular inner wall and osteocyte process surface. The increase in the high-valent ions does not affect the flow velocity and the flow rate, but the FSS on the canalicular inner wall and osteocyte process surface increases linearly. In this study, the results show that the shear stress sensed by the osteocyte process under the P-ESD pattern can be regulated by changing the pressure gradient and the intensity of electric field, as well as the parameters of the annulus fluid and the canaliculus size, which is helpful for the osteocyte mechanical responses. The established model provides a basis for the study of the mechanisms of electro-mechanical signals stimulating bone tissue (cells) growth.

     

Aging in the cortical bone: a constitutive law and its application

It is well known that the material behavior of human cortical bone changes from ductile to more brittle due to aging. This process is accompanied by a decrease of the maximum specific deformation energy. Numerous mechanical tests of specimens have shown a relationship between the mechanical behavior, age and microstructure, especially the porosity, mineralization and fraction of the secondary osteonal area. But up to now, this relationship is not explicitly considered in a constitutive law. Measured stress–strain curves, taken from the literature, from one-dimensional mechanical experiments in tension (McCalden et al. in J Bone Joint Surg Am 75(8):1193–1205, 1993) have been characterized by Young’s modulus, elastic, plastic and fracture energy, fracture stress and strain. The specimens have been harvested from the femora of 46 deceased individuals. Based on this data, we set up a system of equations taking into account the microstructure of the bone material by analogy to common procedures in fracture and damage mechanics. Solving this system for all measured experimental data leads to the determination of the independent damage parameters for each individual person. It turned out that some characteristic mechanical values and one independent damage parameter are statistically significant dependent on age and microstructure. We receive a constitutive law, which describes the mechanical behavior up to fracture by measurable parameters for the microstructure and the individual age and gender only. In turn, we calculate the individual tolerable load for bending, using a nonlinear stress–strain curve, and postulate an age-dependent fracture load for healthy bone by means of the statistical regression. Deviation from the standard is an indication for a bone disease in particular for osteoporosis.