Can deterministic mechanical size effects contribute to fracture and microdamage accumulation in trabecular bone?

Failure of bone under monotonic and cyclic loading is related to the bone mineral density, the quality of the bone matrix, and the evolution of microcracks. The theory of linear elastic fracture mechanics has commonly been applied to describe fracture in bone. Evidence is presented that bone failure can be described through a non-linear theory of fracture. Thereby, deterministic size effects are introduced. Concepts of a non-linear theory are applied to discern how the interaction among bone matrix constituents (collagen and mineral), microcrack characteristics, and trabecular architecture can create distinctively differences in the fracture resistance at the bone tissue level. The non-linear model is applied to interpret pre-clinical data concerning the effects of anti-osteoporotic agents on bone properties. The results show that bisphosphonate (BP) treatments that suppress bone remodeling will change trabecular bone in ways such that the size of the failure process zone relative to the trabecular thickness is reduced. Selective estrogen receptor modulators (SERMs) that suppress bone remodeling will change trabecular bone in ways such that the size of the failure process zone relative to the trabecular thickness is increased. The consequences of these changes are reflected in bone mechanical response and predictions are consistent with experimental observations in the animal model which show that BP treatment is associated with more brittle fracture and microcracks without altering the average length of the cracks, whereas SERM treatments lead to a more ductile fracture and mainly increase crack length with a smaller increase in microcrack density. The model suggests that BPs may be more effective in cases in which bone mass is very low, whereas SERMS may be more effective when milder osteoporotic symptoms are present.     

A study of the effect of non-linearities in the equation of bone remodeling

In this paper, we introduced a high-order non-linear equation of bone remodeling to combine with FEM by introducing two non-linearities, i.e. the remodeling coefficient B(t) and the order of non-linear remodeling equation. The influence of each non-linearity was tested based on its mechanical and physiological implications discussed. We use two finite element models to investigate the influences of non-linearities in this equation: a plate subjected to a ramp load, and a 2D model of the cross-section of a vertebra. By importing the idea of topology optimization in engineering, their external shapes and internal density distributions were simulated from unfixed configurations. To a certain extent, the high-order non-linear equation of bone remodeling we suggested here can control the remodeling processes of bones in different stages of growth or at different anatomic sites more effectively, and make it more consistent with physiological reality, i.e. express the remodeling characteristic that bone's best morphology is adapted to its mechanical environment. Furthermore, it is likely to describe the process of bone growth and evolution.     

Stability analysis and finite element simulation of bone remodeling model

Bone remodeling is widely viewed as a dynamic process--maintaining bone structure through a balance between the opposed activities of osteoblast and osteoclast cells--in which the stability problem is often pointed out. By an analytical approach, we present a bone remodeling model applied to n unit-elements in order to analyze the stationary states and the condition of their stability. In addition, this theory has been simulated in a computer model using the Finite Element Method (FEM) to show a relationship between the bone remodeling process and the stability analysis.     

Phenomenological model of bone remodeling cycle containing osteocyte regulation loop

Biological parameters, such as bone resorption and formation constants, are important variables to achieve optimised hard tissue scaffolds design. To help to understand the modelling process that occurs when a scaffold is implanted it is vital to understand the rather complex bone remodeling process prevalent in native bone. One approach to developing a mathematical model that predicts osteoactivity both in scaffolds, as well as in bone in vivo, is based on a bio-cybernetic vision of basic multicellular unit (BMU) action -. In the case of the model presented in this paper, an additional loop of regulation based on osteocyte activity has been added. This approach has resulted in a four-dimensional system, which shows steady-quasi-cyclic behaviour using a particular range of constants with real biological meaning. The initial findings suggesting that the basic steady-state appears as a torus in multidimensional phase space have been discussed. The existence of this surface in the osteoclasts-osteoblasts-osteocytes-bone subspace indicates that there is a first integral for this dynamic system. Biological and physical interpretation of this integral as a conservative value has been proposed. It is possible to draw an analogy between this conservative value, as a kind of substrate-energy regenerative potential of the bone remodeling system with a molecular nature, to the classical physical value (energy). There are clear indications that there is recovering potential within the BMU that results in a steady operating genetically predominated bone remodeling process. This recovering potential is directed against both mechanical and biomechanical damage to the bone. The current model has credibility when compared to the normal bone remodeling process. However, additional work is required to study a wider range of constants.     

Functional adaptation in long bones: establishing in vivo values for surface remodeling rate coefficients

In this paper we describe a computational means, based on beam theory, for application of the theory of adaptive elasticity to examples of real bone geometries. The results of the animal experiments were taken from the literature, and each documented the temporal evolution of a change in bone shape after a significant change in the mechanical loading environment of the bone. For each of these studies, we establish preliminary estimates of the in vivo values of the surface remodeling rate coefficients--the key parameters in the theory of surface remodeling. Our preliminary parameter estimates are established by comparison of published animal experimental results with surface remodeling theory predictions generated by the computational method.     

Theoretical and numerical study of a bone remodeling model: The effect of osteocyte cells distribution

It is well argued that osteocytes are mechanosensory cells and are involved in the regulation of bone remodeling. In previous works, the predictions from a simulation model have suggested that both the influencing distance of osteocytes and the magnitude of the mechanical loads determine the thickness of trabeculae whereas the number of osteocytes primarily affects the rate of bone remodeling. The question that remains not completely answered is: for the same number of osteocytes, what is the effect of different distributions on the remodeling process? Based on a particular regulatory bone remodeling model, the question is addressed, in part, by performing a stability analysis in connection with numerical simulations. The results allow us to demonstrate that, on one hand, we cannot reach a conclusion about the stability of the model for a nonuniform osteocyte distribution. This implies that there is no relationship between the different parameters conveying the stability of the model. On the other hand, we show that the osteocyte cell distribution has a significant influence on the bone morphology, which seems to be confirmed by simulations with real data obtained from rat tibia.     

A study of the viscoelastic effect in a bone remodeling model

The present paper addresses the following question can a simple regulatory bone remodeling model predict effects of viscosity on the trabecular morphology? For that, we propose an extension of a previous bone remodeling model by taking into account the viscosity properties of the tissue. Zener’s law is used to describe the mechanical behavior of the bone and a specific law of the apparent bone density rate is proposed. Based on stability analysis, numerical simulations are then performed to investigate the viscosity role on simulations of the bone remodeling process. We show that the viscous contribution affects the evolution of the apparent bone density, by slowing down the adaptation process, which seems to be confirmed by simulations with real data obtained from rat tibia.     

The effect of direct and indirect force transmission on peri-implant bone stress – a contact finite element analysis

In almost all finite element (FE) studies in dentistry, virtual forces are applied directly to dentures. The purpose of this study was to develop a FE model with non-linear contact simulation using an antagonist as force transmitter and to compare this with a similar model that uses direct force transmission. Furthermore, five contact situations were created in order to examine their influence on the peri-implant bone stresses, which are relevant to the survival rate of implants. It was found that the peri-implant bone stresses were strongly influenced by the kind of force transmission and contact number.     

A Femur-Implant Model for the Prediction of Bone Remodeling Behavior Induced by Cementless Stem

Bone remodeling simulation is an effective tool for the prediction of long-term effect of implant on the bone tissue, as well as the selection of an appropriate implant in terms of architecture and material. In this paper, a finite element model of proximal femur was developed to simulate the structures of internal trabecular and cortical bones by incorporating quantitative bone functional adaptation theory with finite element analysis. Cementless stems made of titanium, two types of Functionally Graded Material (FGM) and flexible ‘iso-elastic’ material as comparison were implanted in the structure of proximal femur respectively to simulate the bone remodeling behaviors of host bone. The distributions of bone density, von Mises stress, and interface shear stress were obtained. All the prosthetic stems had effects on the bone remodeling behaviors of proximal femur, but the degrees of stress shielding were different. The amount of bone loss caused by titanium implant was in agreement with the clinical observation. The FGM stems caused less bone loss than that of the titanium stem, in which FGM I stem (titanium richer at the top to more HAP/Col towards the bottom) could relieve stress shielding effectively, and the interface shear stresses were more evenly distributed in the model with FGM I stem in comparison with those in the models with FGM II (titanium and bioglass) and titanium stems. The numerical simulations in the present study provided theoretical basis for FGM as an appropriate material of femoral implant from a biomechanical point of view. The next steps are to fabricate FGM stem and to conduct animal experiments to investigate the effects of FGM stem on the remodeling behaviors using animal model.     

Mandibular bone remodeling induced by dental implant

The ability to assess the effects of an implant on bone remodeling is of particular importance to prosthesis placement planning and associated treatment assurance. Prediction of on-going bone responses will enable us to improve the performance of a restoration. Although the bone remodeling for long bones had been extensively studied, there have been relatively few reports for dental scenarios despite its increasing significance with more and more dental implant placements. This paper aimed to develop a systematic protocol to assess mandibular bone remodeling induced by dental implantation, which extends the remodeling algorithms established for the long bones into dental settings. In this study, a 3D model for a segment of a human mandible was generated from in vivo CT scan images, together with a titanium implant embedded to the mandible. The results examined the changes in bone density and stiffness as a result of bone remodeling over a period of 48 months. Resonance frequency analysis was also performed to relate natural frequencies to bone remodeling. The density contours are qualitatively compared with clinical follow-up X-ray images, thereby providing validity for the bone remodeling algorithm presented in dental bone analysis.     

Bone tissue engineering: the role of interstitial fluid flow

It is well established that vascularization is required for effective bone healing. This implies that blood flow and interstitial fluid (ISF) flow are required for healing and maintenance of bone. The fact that changes in bone blood flow and ISF flow are associated with changes in bone remodeling and formation support this theory. ISF flow in bone results from transcortical pressure gradients produced by vascular and hydrostatic pressure, and mechanical loading. Conditions observed to alter flow rates include increases in venous pressure in hypertension, fluid shifts occurring in bedrest and microgravity, increases in vascularization during the injury-healing response, and mechanical compression and bending of bone during exercise. These conditions also induce changes in bone remodeling. Previously, we hypothesized that interstitial fluid flow in bone, and in particular fluid shear stress, serves to mediate signal transduction in mechanical loading- and injury-induced remodeling. In addition, we proposed that a lack or decrease of ISF flow results in the bone loss observed in disuse and microgravity. The purpose of this article is to review ISF flow in bone and its role in osteogenesis.     

Electro-mechanical behavior of wet bone--Part I: Theory

The remodeling properties of bones due to various stimuli have been of substantial interest to the scientists. Examination of electro-mechanical properties of bone and their relation to remodeling and osteogenesis have been investigated mainly by experimental means. In this study, by using continuum physics, it is shown that the remodeling of bones can be formulated theoretically in terms of electrical and mechanical effects. The interactions among the constituents of bone (bone matrix, bone salts, electrolytes and hydrogen ions) and effects of various stimuli (mechanical, electrical and chemical) on the remodeling mechanism of bone tissue are interpreted with this model. Moreover, the stimulation of osteogenesis by electrical means is predicted.     

Evaluation of bone remodeling around single dental implants of different lengths: a mechanobiological numerical simulation and validation using clinical data

Algorithmic models have been proposed to explain adaptive behavior of bone to loading; however, these models have not been applied to explain the biomechanics of short dental implants. Purpose of present study was to simulate bone remodeling around single implants of different lengths using mechanoregulatory tissue differentiation model derived from the Stanford theory, using finite elements analysis (FEA) and to validate the theoretical prediction with the clinical findings of crestal bone loss. Loading cycles were applied on 7-, 10-, or 13-mm-long dental implants to simulate daily mastication and bone remodeling was assessed by changes in the strain energy density of bone after a 3, 6, and 12 months of function. Moreover, clinical findings of marginal bone loss in 45 patients rehabilitated with same implant designs used in the simulation (n = 15) were computed to validate the theoretical results. FEA analysis showed that although the bone density values reduced over time in the cortical bone for all groups, bone remodeling was independent of implant length. Clinical data showed a similar pattern of bone resorption compared with the data generated from mathematical analyses, independent of implant length. The results of this study showed that the mechanoregulatory tissue model could be employed in monitoring the morphological changes in bone that is subjected to biomechanical loads. In addition, the implant length did not influence the bone remodeling around single dental implants during the first year of loading.     

On the optimal shape of hip implants

The success of a total hip arthroplasty is strongly related to the initial stability of the femoral component and to the stress shielding effect. In fact, for cementless stems, initial stability is essential to promote bone ingrowth into the stem coating. An inefficient primary stability is also a cause of thigh pain. In addition, the bone adaptation after the surgery can lead to an excessive bone loss and, consequently, can compromise the success of the implant. These factors depend on prosthesis design, namely on material, interface conditions and shape. Although, surgeons use stems with very different geometries, new computational tools using structural optimization methods have been used to achieve a better design in order to improve initial stability and therefore, the implant durability. In this work, a multi-criteria shape optimization process is developed to study the relationship between implants performance and geometry. The multi-criteria objective function takes into account the initial stability of the femoral stem and the effect of stress shielding on bone adaptation after the surgery. Then, the optimized stems are tested using a concurrent model for bone remodeling and osseointegration to evaluate long-term performance. Additionally, the sensitivity to misalignments is analyzed, since femoral stems are often placed in varus or valgus position. Results show that the different criteria are contradictory resulting in different characteristics for the hip stem. However, the multi-criteria formulation leads to compromise solutions, with a combination of the geometric characteristics obtained for each criterion separately.     

Stability analysis of a continuum-based constrained mixture model for vascular growth and remodeling

A stabilizing criterion is derived for equations governing vascular growth and remodeling. We start from the integral state equations of the continuum-based constrained mixture theory of vascular growth and remodeling and obtain a system of time-delayed differential equations describing vascular growth. By employing an exponential form of the constituent survival function, the delayed differential equations can be reduced to a nonlinear ODE system. We demonstrate the degeneracy of the linearized system about the homeostatic state, which is a fundamental cause of the neutral stability observations reported in prior studies. Due to this degeneracy, stability conclusions for the original nonlinear system cannot be directly inferred. To resolve this problem, a sub-system is constructed by recognizing a linear relation between two states. Subsequently, Lyapunov’s indirect method is used to connect stability properties between the linearized system and the original nonlinear system, to rigorously establish the neutral stability properties of the original system. In particular, this analysis leads to a stability criterion for vascular expansion in terms of growth and remodeling kinetic parameters, geometric quantities and material properties. Numerical simulations were conducted to evaluate the theoretical stability criterion under broader conditions, as well as study the influence of key parameters and physical factors on growth properties. The theoretical results are also compared with prior numerical and experimental findings in the literature.     

Theoretical solutions for internal bone remodeling of diaphyseal shafts using adaptive elasticity theory

In this paper we considered the problem of internal bone remodeling induced by casting a broken femur. It is shown that, after the removal of the plastercast, the healing bone is either becoming more porous and less stiff or it is osteoporotic. The result is theoretical, based on a theory of internal bone remodeling proposed by Hegedus and Cowin (Journal of Elasticity 6, 1976, 337-352).     

A semi-empirical cell dynamics model for bone turnover under external stimulus

The normal periodic turnover of bone is referred to as remodeling. In remodeling, old or damaged bone is removed during a 'resorption' phase and new bone is formed in its place during a 'formation' phase in a sequence of events known as coupling. Resorption is preceded by an 'activation' phase in which the signal to remodel is initiated and transmitted. Remodeling is known to involve the interaction of external stimuli, bone cells, calcium and phosphate ions, and several proteins, hormones, molecules, and factors. In this study, a semi-empirical cell dynamics model for bone remodeling under external stimulus that accounts for the interaction between bone mass, bone fluid calcium, bone calcium, and all three major bone cell types, is presented. The model is formulated to mimic biological coupling by solving separately and sequentially systems of ODEs for the activation, resorption, and formation phases of bone remodeling. The charateristic time for resorption (20 days) and the amount of resorption (~0.5%) are fixed for all simulations, but the formation time at turnover is an output of the model. The model was used to investigate the effects of different types of strain stimuli on bone turnover under bone fluid calcium balance and imbalance conditions. For bone fluid calcium balance, the model predicts complete turnover after 130 days of formation under constant 1000 microstrain stimulus; after 47 days of formation under constant 2000 microstrain stimulus; after 173 days of formation under strain-free conditions, and after 80 days of formation under monotonic increasing strain stimulus from 1000 to 2000 microstrain. For bone fluid calcium imbalance, the model predicts that complete turnover occurs after 261 days of formation under constant 1000 microstrain stimulus and that turnover never occurs under strain-free conditions. These predictions were not impacted by mean dynamic input strain stimuli of 1000 and 2000 microstrain at 1 Hz and 1000 microstrain amplitude. The formation phase of remodeling lasts longer than the resorption phase, increased strain stimulus accelerates bone turnover, while absence of strain significantly delays or prevents it, and formation time for turnover under monotonic increasing strain conditions is intermediate to those for constant strain stimuli at the minimum and maximum monotonic strain levels. These results are consistent with the biology, and with Frost's mechanostat theory.     

A mechanistic model for internal bone remodeling exhibits different dynamic responses in disuse and overload

Bone is a dynamic tissue which, through the process of bone remodeling in the mature skeleton, renews itself during normal function and adapts to mechanical loads. It is, therefore, important to understand the effect of remodeling on the mechanical function of bone, as well as the effect of the inherent time lag in the remodeling process. In this study, we develop a constitutive model for bone remodeling which includes a number of relevant mechanical and biological processes and use this model to address differences in the remodeling behavior as a volume element of bone is placed in disuse or overload. The remodeling parameters exhibited damped oscillatory behavior as the element was placed in disuse, with the amplitude of the oscillations increasing as the severity of disuse increased. In overload situations, the remodeling parameters exhibited critically sensitive behavior for loads beyond a threshold value. These results bear some correspondence to experimental findings, suggesting that the model may be useful when examining the importance of transient responses for bone in disuse, and for investigating the role fatigue damage removal plays in preventing or causing stress fractures. In addition, the constitutive algorithm is currently being employed in finite element simulations of bone adaptation to predict important features of the internal structure of the normal femur, as well as to study bone diseases and their treatment.     

The effect of stress concentration on bone remodeling: theoretical predictions

Theoretical predictions of internal bone remodeling around an elliptical hole are studied. The internal remodeling theory due to Cowin and Hegedus is employed. The bone is modeled as an initially homogeneous adaptive elastic plate with an elliptical hole under a superposed steady compressive load. It is shown that there will exist a final inhomogeneous remodeling distribution around the hole that will disappear away from the hole. The remodeling is such that the compressive stress concentration causes the bone structure to evolve to one of greater density and stiffer elastic coefficients. The speed of remodeling around the hole and its variation with respect to distance is investigated and discussed. It is shown that the rate of bone reinforcement in the area of compressive stress concentration is much higher than the rate of bone resorption in the area of existing tensile stress. Special cases of a circular hole and vertical and horizontal slots are studied and discussed.     

Periosteal and endosteal control of bone remodeling under torsional loading.

The shape changes that occur in the mid-diaphysis of a long bone due to adaptive remodeling induced by increasing or decreasing the axial and/or torsional loading of the bone are investigated using a simple model. In this model the mid-diaphysis of a long bone is represented as a hollow thick-walled right-circular cylinder, and different optimal strategies for bone remodeling are considered. It is shown that if such a thick-walled right-circular cylinder capable of surface remodeling is subjected to an axial compressive load and a twisting torque, then the remodeling patterns depend on whether the periosteal surface or the endosteal surface controls the limits of the remodeling process. It is shown that the effect of increasing the torque is always opposite to the effect of increasing the compressive load. Thus, similar remodeling patterns are obtained by increasing one type of loading and decreasing the other. Aside from the restriction of idealized cylindrical geometry, the only assumptions made are that the bone tissue is linearly elastic and that there exists a finite range of remodeling equilibrium stresses. Only those loading situations which maintain the bone in remodeling equilibrium are considered in this work. It follows that the results presented are independent of the specific type of rule governing the temporal evolution of the bone shape, since any such rule applies only in situations where there is active remodeling and, hence, no remodeling equilibrium.