A theoretical model of the effect of continuum damage on a bone adaptation model

Connecting mechanics and bone cell activities in the bone remodeling process: an integrated finite element modeling

Bone adaptation occurs as a response to external loadings and involves bone resorption by osteoclasts followed by the formation of new bone by osteoblasts. It is directly triggered by the transduction phase by osteocytes embedded within the bone matrix. The bone remodeling process is governed by the interactions between osteoblasts and osteoclasts through the expression of several autocrine and paracrine factors that control bone cell populations and their relative rate of differentiation and proliferation. A review of the literature shows that despite the progress in bone remodeling simulation using the finite element (FE) method, there is still a lack of predictive models that explicitly consider the interaction between osteoblasts and osteoclasts combined with the mechanical response of bone. The current study attempts to develop an FE model to describe the bone remodeling process, taking into consideration the activities of osteoclasts and osteoblasts. The mechanical behavior of bone is described by taking into account the bone material fatigue damage accumulation and mineralization. A coupled strain-damage stimulus function is proposed, which controls the level of autocrine and paracrine factors. The cellular behavior is based on Komarova et al.'s (2003) dynamic law, which describes the autocrine and paracrine interactions between osteoblasts and osteoclasts and computes cell population dynamics and changes in bone mass at a discrete site of bone remodeling. Therefore, when an external mechanical stress is applied, bone formation and resorption is governed by cells dynamic rather than adaptive elasticity approaches. The proposed FE model has been implemented in the FE code Abaqus (UMAT routine). An example of human proximal femur is investigated using the model developed. The model was able to predict final human proximal femur adaptation similar to the patterns observed in a human proximal femur. The results obtained reveal complex spatio-temporal bone adaptation. The proposed FEM model gives insight into how bone cells adapt their architecture to the mechanical and biological environment.

Using smooth particle hydrodynamics to investigate femoral cortical bone remodelling at the Haversian level

In the neck of the femur, about 70% of the strength is contributed by the cortical bone, which is the most highly stressed part of the structure and is the site where failure is almost certainly initiated. A better understanding of cortical bone remodelling mechanisms can help discern changes at this anatomical site, which are essential if an understanding of the mechanisms by which hips weaken and become vulnerable to fracture is to be gained. The aims of this study were to (i) examine a hypothesis that low strain fields arise because of subject-specific Haversian canal distributions causing bone resorption and reduced bone integrity and (ii) introduce the use of a meshless particle-based computational modelling approach SPH to capture bone remodelling features at the level of the Haversian canals. We show that bone remodelling initiated by strain at the Haversian level is highly influenced by the subject-specific pore distribution, bone density, loading and osteocyte density. SPH is shown to be effective at capturing the intricate bone pore shapes that evolved over time.

On a new law of bone remodeling based on damage elasticity: a thermodynamic approach

Background

Bone tissue is the main element of the human skeleton and is a dynamic tissue that is continuously renewed by bone-resorbing osteoclasts and bone-forming osteoblasts.

The bone is also capable of repairing itself and adapting its structure to changes in its load environment through the process of bone remodeling.

Therefore, this phenomenon has been gaining increasing interest in the last years and many laws have been developed in order to simulate this process.

Results

In this paper, we develop a new law of bone remodeling in the context of damaged elastic by applying the thermodynamic approach in the case of small perturbations.

The model is solved numerically by a finite difference method in the one-dimensional bone structure of a n-unit elements model.

Conclusion

In addition, several numerical simulations are presented that confirm the accuracy and effectiveness of the model.

Indentation experiments and simulation of ovine bone using a viscoelastic-plastic damage model

Indentation methods have been widely used to study bone at the micro- and nanoscales. It has been shown that bone exhibits viscoelastic behavior with permanent deformation during indentation. At the same time, damage due to microcracks is induced due to the stresses beneath the indenter tip. In this work, a simplified viscoelastic-plastic damage model was developed to more closely simulate indentation creep data, and the effect of the model parameters on the indentation curve was investigated. Experimentally, baseline and 2-year postovariectomized (OVX-2) ovine (sheep) bone samples were prepared and indented. The damage model was then applied via finite element analysis to simulate the bone indentation data. The mechanical properties of yielding, viscosity, and damage parameter were obtained from the simulations. The results suggest that damage develops more quickly for OVX-2 samples under the same indentation load conditions as the baseline data.

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

In 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 is based on 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.

Trabecular bone remodelling simulated by a stochastic exchange of discrete bone packets from the surface

Human bone is constantly renewed through life via the process of bone remodelling, in which individual packets of bone are removed by osteoclasts and replaced by osteoblasts. Remodelling is mechanically controlled, where osteocytes embedded within the bone matrix are thought to act as mechanical sensors. In this computational work, a stochastic model for bone remodelling is used in which the renewal of bone material occurs by exchange of discrete bone packets. We tested different hypotheses of how the mechanical stimulus for bone remodelling is integrated by osteocytes and sent to actor cells on the bone's surface. A collective (summed) signal from multiple osteocytes as opposed to an individual (maximal) signal from a single osteocyte was found to lead to lower inner porosity and surface roughness of the simulated bone structure. This observation can be interpreted in that collective osteocyte signalling provides an effective surface tension to the remodelling process. Furthermore, the material heterogeneity due to remodelling was studied on a network of trabeculae. As the model is discrete, the age of individual bone packets can be monitored with time. The simulation results were compared with experimental data coming from quantitative back scattered electron imaging by transforming the information about the age of the bone packet into a mineral content. Discrepancies with experiments indicate that osteoclasts preferentially resorb low mineralized, i.e. young, bone at the bone's surface.

A NUMERICAL TECHNIQUE TO EVALUATE THE FLEXURAL STIFFNESS OF LONG BONES AFFECTED BY CRACKS AND POROSITY

Bone maintains its structure through a constant process of resorption and formation, in a process called bone remodeling. An imbalance in this process caused by disease, abnormal mechanical demands, or fatigue may predispose bone to fracture injuries. Increase in bone resorption can increase the number of surface cracks and structural porosity of the bone and thus change its stiffness properties. In this study, a computational technique is proposed to investigate the stiffness properties in long bones based on dynamic responses. As the first attempt, defects such as porosity and cracks are detected based on changes in stiffness properties of the sample. The least square algorithm and the finite element method are used as tools in this study. The Wilson-θ numerical method is employed to generate artificially experimental results for acceleration vectors. The data obtained from the artificial experiment is later employed to the proposed computational investigation model as raw data.

Characterization of indentation response and stiffness reduction of bone using a continuum damage model

Indentation tests can be used to characterize the mechanical properties of bone at small load/length scales offering the possibility of utilizing very small test specimens, which can be excised using minimally-invasive procedures. In addition, the need for mechanical property data from bone may be a requirement for fundamental multi-scale experiments, changes in nano- and micro-mechanical properties (e.g., as affected by changes in bone mineral density) due to drug therapies, and/or the development of computational models. Load vs. indentation depth data, however, is more complex than those obtained from typical macro-scale experiments, primarily due to the mixed state of stress, and thus interpretation of the data and extraction of mechanical properties is more challenging. Previous studies have shown that cortical bone exhibits a visco-elastic response combined with permanent deformation during indentation tests, and that the load vs. indentation depth response can be simulated using a visco-elastic/plastic material model. The model successfully captures the loading and creep displacement behavior, however, it does not adequately reproduce the unloading response near the end of the unloading cycle, where a pronounced decrease in contact stiffness is observed. It is proposed that the stiffness reduction observed in bone results from an increase in damage; therefore, a plastic-damage model was investigated and shown capable of simulating a typical bone indentation response through an axisymmetric finite element simulation. The plastic-damage model was able to reproduce the full indentation response, especially the reduced stiffness behavior exhibited during the latter stages of unloading. The results suggest that the plastic-damage model is suitable for describing the complex indentation response of bone and may provide further insight into the relationship between model parameters and mechanical/physical properties.

On the role of bone damage in calcium homeostasis

Bone serves as the reservoir of some minerals including calcium. If calcium is needed anywhere in the body, it can be removed from the bone matrix by resorption and put back into the blood flow. During bone remodelling the resorbed tissue is replaced by osteoid which gets mineralized very slowly. Then, calcium homeostasis is controlled by bone remodelling, among other processes: the more intense is the remodelling activity, the lower is the mineral content of bone matrix. Bone remodelling is initiated by the presence of microstructural damage. Some experimental evidences show that the fatigue properties of bone are degraded and more microdamage is accumulated due to the external load as the mineral content increases. That damage initiates bone remodelling and the mineral content is so reduced. Therefore, this process prevents the mineral content of bone matrix to reach very high (non-physiological) values. A bone remodelling model has been used to simulate this regulatory process. In this model, damage is an initiation factor for bone remodelling and is estimated through a fatigue algorithm, depending on the macroscopic strain level. Mineral content depends on bone remodelling and mineralization rate. Finally, the bone fatigue properties are defined as dependent on the mineral content, closing the interconnection between damage and mineral content. The remodelling model was applied to a simplified example consisting of a bar under tension with an initially heterogeneous mineral distribution. Considering the fatigue properties as dependent on the mineral content, the mineral distribution tends to be homogeneous with an ash fraction within the physiological range. If such dependance is not considered and fatigue properties are assumed constant, the homogenization is not always achieved and the mineral content may rise up to high non-physiological values. Thus, the interconnection between mineral content and fatigue properties is essential for the maintenance of bone's structural integrity as well as for the calcium homeostasis.

A topology optimization based model of bone adaptation

A novel topology optimization model based on homogenization methods was developed for predicting bone density distribution and anisotropy, assuming the bone structure to be a self-optimizing biological material which maximizes its own structural stiffness. The feasibility and efficiency of this method were tested on a 2D model for a proximal femur under single and multiple loading conditions. The main aim was to compute homogenized optimal designs using an optimal laminated microstructure. The computational results showed that high bone density levels are distributed along the diaphysis and form arching struts within the femoral head. The pattern of bone density distribution and the anisotropic bone behavior predicted by the model in the multiple load case were both in good agreement with the structural architecture and bone density distribution occurring in natural femora. This approach provides a novel means of understanding the remodeling processes involved in fracture repair and the treatment of bone diseases.

A bone remodelling model coupling microdamage growth and repair by 3D BMU-activity

Bone as most of living tissues is able, during its entire lifetime, to adapt its internal microstructure and subsequently its associated mechanical properties to its specific mechanical and physiological environment in a process commonly known as bone remodelling. Bone is therefore continuously renewed and micro-damage, accumulated by fatigue or creep, is removed minimizing the risk of fracture. Nevertheless, bone is not always able to repair itself completely. Actually, if bone repairing function is slower than micro-damage accumulation, a type of bone fracture, usually known as "stress fracture", can finally evolve. In this paper, we propose a bone remodelling continuous model able to simulate micro-damage growth and repair in a coupled way and able therefore to predict the occurrence of "stress fractures". The biological bone remodelling process is modelled in terms of equations that describe the activity of basic multicellular units. The predicted results show a good correspondence with experimental and clinical data. For example, in disuse, bone porosity increases until an equilibrium situation is achieved. In overloading, bone porosity decreases unless the damage rate is so high that causes resorption or "stress fracture".

Finite Element Prediction of Proximal Femoral Fracture Patterns Under Different Loads

The main purpose of this work is to discuss the ability of finite element analyses, together with an appropriate anisotropic fracture criterion, to predict the ultimate load and type of fracture in bones and more specifically in the proximal femur. We show here that the use of a three-dimensional anisotropic criterion provides better results than other well-known isotropic criteria. The criterion parameters and the anisotropic elastic properties were defined in terms of the bone tissue microstructure, quantified by the apparent density and the so-called “fabric tensor”, whose spatial distributions were obtained by means of an anisotropic remodeling model able to capture the main features of the internal structure of long bones. In order to check the validity of the results obtained, they have been compared with those of an experimental work that analyzes different types of fractures induced in the proximal femur by a static overload.

Buckling of adaptive elastic bone-plate: theoretical and numerical investigation

During day-to-day activities, many bones in the axial and appendicular skeleton are subjected to repetitive, cyclic loading that often results directly in an increased risk of bone fracture. In clinical orthopedics, trabecular fatigue fractures are observed as compressive stress fractures in the proximal femur, vertebrae, calcaneus and tibia, that are often preceded by buckling and bending of microstructural elements (Müller et al. in J Biomechanics 31:150 1998; Gibson in J Biomechanics 18:317-328 1985; Gibson and Ashby in Cellular solids 1997; Lotz et al. in Osteoporos Int 5:252-261 1995; Carter and Hayes in Science 194:1174-1176 1976). However, the relative importance of bone density and architecture in the etiology of these fractures are poorly understood and consequently not investigated from a biomechanical point of view. In the present contribution, an attempt is made to formulate a bone-plate buckling theory using Cowin's concepts of adaptive elasticity (Cowin and Hegedus in J Elast 6:313-325 1976; Hegedus and Cowin J Elast 6:337-352 1976). In particular, the buckling problem of a Kirchhoff-Love bone plate is investigated numerically by using the finite difference method and an iterative solving approach (Chen in Comput Methods Appl Mech Eng 167:91-99 1998; Hildebland in Introduction to numerical analysis 1974; Richtmyer and Morton in Difference methods for initial-value problems 1967).

Computer simulation of an adaptive damage-bone remodeling law applied to three unit-bone bars structure

It is well admitted that the mechanical loading plays an important role in the growth and maintenance of our skeleton, and that microdamage (i.e.: microcracks) occurs naturally when the bone is overloaded during day-to-day activities. It is also argued, from experimental and theoretical viewpoint, that the cells which built and rebuilt the skeleton are sensitive for both strain and microdamage. The recent damage-bone remodeling theory is employed here to study the mechanical response of the three unit-bone bars that simulate bone trabeculae in the form of truss. It is shown that under constant load, such a structure exhibit inhomogeneous strain and it's response to external applied load depends strongly upon the manner in which the microdamage is distributed.

Skeletal biology, functional asymmetry and the origins of "Handedness"

The past decade has witnessed the (re)emergence of a debate as to whether handedness is apomorphic within hominins. There are both qualitative and quantitative arguments, some which draw non-human primates into the handed sphere and others which exclude them. Ultimate questions concern origins of structural asymmetry of both brain and body and lateralized behaviours with implications for tool use and language. Lateralization is thus an important realm of phylogenetic study, and archaeologists and psychologists alike have sought to identify handedness within material culture. However, hand preference for tool manufacture and use among extant non-human primates, such as Cebus and Pan, suggests that the archaeological record may well be mute regarding the origins of laterality. In this paper, an argument is put forward positioning skeletal biology as a viable approach to the handedness origins issue. Behaviour is a mediator of the complementary processes of geometric modelling (change in size and shape) and histological remodelling (disuse osteopenia; microfracture repair); therefore, directional asymmetry in the pattern of skeletal modelling and remodelling is a putative signal of lateralized activity.

Damaged-bone adaptation under steady homogeneous stress

In this work an extension of the adaptive-elasticity theory is proposed in order to include the contribution of bone microdamage as a stimulus. Some aspects of damaged-bone tissue adaptation, brought about by a change of the daily loading history, are investigated. In particular, under the assumption of a small strain approximation and isothermal conditions, the solution of the remodeling rate equation for steady homogeneous stress is discussed and the damage effect upon the remodeling time constant is shown. The result is both theoretical and numerical, based on a recent theory of internal damaged-bone remodeling (Ramtani, S., and Zidi, M., 1999, "Damaged-Bone Remodeling Theory: Thermodynamical Approach, " Mechanics Research Communications, Vol. 26, pp. 701-708. Ramtani, S., and Zidi, M., 2001, "A Theoretical Model of the Effect of Continum Damage on a Bone Adaption Model," Journal of Biomechanics, Vol. 34, pp. 471-479) and motivated by the works of Cowin, S. C., and Hegedus, D. M., 1976, "Bone Remodeling I: Theory and Adaptive Elasticity," Journal of Elasticity, Vol. 6, pp. 471-479 and Hegedus, D. H., and Cowin, S. C., 1976, "Bone Remodeling II: Small Strain Adaptive Elasticity," Journal of Elasticity, Vol. 6, pp. 337-352.