Remodeling Equilibrium Research Articles

BONE ORTHOTROPIC REMODELING AS A THERMODYNAMICALLY-DRIVEN EVOLUTION

In this paper, we present and discuss a model of bone remodeling set up in the framework of the theory of generalized continuum mechanics which was first introduced by DiCarlo et al. [Sur le remodelage des tissus osseux anisotropes, Comptes Rendus Mécanique 334(11):651–661, 2006]. Bone is described as an orthotropic body experiencing remodeling as a rotation of its microstructure. Thus, the complete kinematic description of a material point is provided by its position in space and a rotation tensor describing the orientation of its microstructure. Material motion is driven by energetic considerations, namely by the application of the Clausius–Duhem inequality to the microstructured material. Within this framework of orthotropic remodeling, some key features of the remodeling equilibrium configurations are deduced in the case of homogeneous strain or stress loading conditions. First, it is shown that remodeling equilibrium configurations correspond to energy extrema. Second, stability of the remodeling equilibrium configurations is assessed in terms of the local convexity of the strain and complementary energy functionals hence recovering some classical energy theorems. Eventually, it is shown that the remodeling equilibrium configurations are not only highly dependent on the loading conditions, but also on the material properties.

Lights and shadows of NSAIDs in bone healing: the role of prostaglandins in bone metabolism.

In this review, we discuss the current data about the anatomy and function of bone tissue with particular regard to influence of prostaglandins. Bone tissue dynamics are characterized by a constant remodeling process that involves all bone tissue cells. The communication between bone component cells and other organs is necessary for bone remodeling equilibrium and confirms the dynamic character of bone tissue. Remodeling is also a vital element of healing processes and in adapting bone tissue to stress responses. Therefore, in our review we present the role and significance of bone cells and signaling pathways enabling maintenance of bone homeostasis and remodeling process stability. Cyclooxygenase (COX) is a crucial enzyme in the production of prostaglandins and thromboxane. We focus on the role of COX isoenzymes with highlighting their connection with bone formation, resorption and repair. Prostaglandins are known as arachidonic acid metabolites acting through specific membrane receptors and play an important role in the regulation of osteoblast and osteoclast functions. Prostaglandin PGE2 with its four defined receptors (EP1R, EP2R, EP3R and EP4R) is crucial to maintain balanced bone turnover. Their stimulatory or inhibitory effects appear to depend on different structure-activity relations and signaling pathways. We have described the role of these receptors in bone metabolism and healing. We conclude that the activity of prostaglandins in bone tissue is defined by maintaining bone remodeling balance and its reactions to humoral mediators and mechanical stress. Most data confirm that among prostaglandins, PGE2 takes part in all processes of trauma response, including homeostasis, inflammation and healing, and plays a key role in bone physiology.

Capturing microscopic features of bone remodeling into a macroscopic model based on biological rationales of bone adaptation.

To understand Wolff's law, bone adaptation by remodeling at the cellular and tissue levels has been discussed extensively through experimental and simulation studies. For the clinical application of a bone remodeling simulation, it is significant to establish a macroscopic model that incorporates clarified microscopic mechanisms. In this study, we proposed novel macroscopic models based on the microscopic mechanism of osteocytic mechanosensing, in which the flow of fluid in the lacuno-canalicular porosity generated by fluid pressure gradients plays an important role, and theoretically evaluated the proposed models, taking biological rationales of bone adaptation into account. The proposed models were categorized into two groups according to whether the remodeling equilibrium state was defined globally or locally, i.e., the global or local uniformity models. Each remodeling stimulus in the proposed models was quantitatively evaluated through image-based finite element analyses of a swine cancellous bone, according to two introduced criteria associated with the trabecular volume and orientation at remodeling equilibrium based on biological rationales. The evaluation suggested that nonuniformity of the mean stress gradient in the local uniformity model, one of the proposed stimuli, has high validity. Furthermore, the adaptive potential of each stimulus was discussed based on spatial distribution of a remodeling stimulus on the trabecular surface. The theoretical consideration of a remodeling stimulus based on biological rationales of bone adaptation would contribute to the establishment of a clinically applicable and reliable simulation model of bone remodeling.

Analyses of RANK and RANKL in the Post-GWAS Context: Functional Evidence of Vitamin D Stimulation Through a RANKL Distal Region

Over the past decade, many genome-wide association studies (GWAs) and meta-analyses have identified genes and regions involved in osteoporotic phenotypes. Nevertheless, the large majority of these results were not tested at any functional level. GWA-associated single-nucleotide polymorphisms (SNPs) near candidate genes such as RANK and RANKL suggest that these SNPs and/or other variants nearby may be involved in bone phenotype determination. This study focuses on SNPs along these two genes, which encode proteins with a well-established role in the bone remodeling equilibrium. Thirty-three SNPs, chosen for their location in evolutionary conserved regions or replicated from previous studies, were genotyped in the BARCOS cohort of 1061 postmenopausal women and tested for association with osteoporotic phenotypes. SNP rs9594738, which lies 184 kb upstream of the RANKL gene, was the only SNP found to be associated with a bone phenotype (dominant model: beta coefficient = -0.034, p = 1.5 × 10(-4) , for lumbar spine bone mineral density). Functional experiments exploring a distal region (DR) of 831 bp that harbors this SNP in a centered position (nt 470) demonstrated its capacity to inhibit the RANKL promoter in reporter gene assays. Remarkably, this DR inhibition was significantly reduced in the presence of vitamin D. In conclusion, the GWA-associated SNP rs9594738 lies in a region involved in transcription regulation through which vitamin D could be regulating RANKL expression and bone mineral density.

Stiffness Design of Continuum Structures by a Bionics Topology Optimization Method

A heuristic approach is presented to solve continuum topology optimization problems with specified constraints, e.g., structural volume constraint and/or displacement constraint(s). The essentials of the present approach are summarized as follows. First, the structure is regarded as a piece of bone and the topology optimization process is viewed as bone remodeling process. Second, a second-rank positive and definite fabric tensor is introduced to express the microstructure and anisotropy of a material point in the design domain. The eigenpairs of the fabric tensor are the design variables of the material point. Third, Wolff’s law, which states that bone microstructure and local stiffness tend to align with the stress principal directions to adapt to its mechanical environment, is used to renew the eigenvectors of the fabric tensor. To update the eigenvalues, an interval of reference strain, which is similar to the concept of dead zone in bone remodeling theory, is suggested. The idea is that, when any one of the absolute values of the principal strains of a material point is out of the current reference interval, the fabric tensor will be changed. On the contrary, if all of the absolute values of the principal strains are in the current reference interval, the fabric tensor remains constant and the material point is in a state of remodeling equilibrium. Finally, the update rule of the reference strain interval is established. When the length of the interval equals zero, the strain energy density in the final structure distributes uniformly. Simultaneously, the volume and the displacement field of the final structure are determined uniquely. Therefore, the update of the reference interval depends on the ratio(s) between the current constraint value(s) and their critical value(s). Parameters, e.g., finite element mesh the initial material and the increments of the eigenvalues of fabric tensors, are studied to reveal their influences on the convergent behavior. Numerical results demonstrate the validity of the method developed.

Characterization of oxidized low-density lipoprotein-induced hormesis-like effects in osteoblastic cells

Epidemiological studies indicate that patients suffering from atherosclerosis are predisposed to develop osteoporosis. Atherogenic determinants such as oxidized low-density lipoprotein (oxLDL) particles have been shown both to stimulate the proliferation and promote apoptosis of bone-forming osteoblasts. Given such opposite responses, we characterized the oxLDL-induced hormesis-like effects in osteoblasts. Biphasic 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) reductive activity responses were induced by oxLDL where low concentrations (10-50 microg/ml) increased and high concentrations (from 150 microg/ml) reduced the MTT activity. Cell proliferation stimulation by oxLDL partially accounted for the increased MTT activity. No alteration of mitochondria mass was noticed, whereas low concentrations of oxLDL induced mitochondria hyperpolarization and increased the cellular levels of reactive oxygen species (ROS). The oxLDL-induced MTT activity was not related to intracellular ROS levels. OxLDL increased NAD(P)H-associated cellular fluorescence and flavoenzyme inhibitor diphenyleneiodonium reduced basal and oxLDL-induced MTT activity, suggesting an enhancement of NAD(P)H-dependent cellular reduction potential. Low concentrations of oxLDL reduced cellular thiol content and increased metallothionein expression, suggesting the induction of compensatory mechanisms for the maintenance of cell redox state. These concentrations of oxLDL reduced osteoblast alkaline phosphatase activity and cell migration. Our results indicate that oxLDL particles cause hormesis-like response with the stimulation of both proliferation and cellular NAD(P)H-dependent reduction potential by low concentrations, whereas high concentrations lead to reduction of MTT activity associated with the cell death. Given the effects of low concentrations of oxLDL on osteoblast functions, oxLDL may contribute to the impairment of bone remodeling equilibrium.

647 荷重条件が骨梁のリモデリング刺激量に与える影響

Distribution function of mechanical quantities at single trabecular level as remodeling stimuli was obtained by using digital image-based finite element models of normal vertebral body of rat. When a linear rate equation was assumed, integration of strain energy density (SED), von Mises equivalent stress, and local stress nonuniformity were more candidate stimuli of trabecular bone remodeling near remodeling equilibrium than SED. The sensitivity of distribution function of stress nonuniformity to change in physiological loading condition was the smallest in the four mechanical quantities.

Pathophysiological and clinical importance of insulin-like growth factor-I with respect to bone metabolism.

The modern concept of causality of diseases emphasizes the study of natural defense functions of the organism and possibilities of influencing them, which will lead to effective prevention of these diseases. A great deal of information has been obtained on the system growth hormone (GH)/insulin-like growth factor (IGF)-I, which is of quite fundamental importance for the integrity of the organism. A dysbalance of the system may be the cause of diseases of the neonatal period, as well as diseases associated with aging. In old age, the synthesis of the crucial peptide system, IGF-I, declines as well as the sensitivity of tissues to this hormone. At the same time the changes in the expression of IGF-binding proteins (IGFBP) occur. Systemic growth factors are present in measurable concentrations in the circulation, they are, however, taken up or synthesized by some tissues, where they act as local cellular regulators. IGF-I is produced by many tissues, including bones under the control of estrogens, growth hormone and the parathyroid hormone. A decline of bone IGF-I in the cortical portion of bones is one of the many mechanisms leading to the development of involutional osteoporosis. Correlation studies, which have provided evidence of a relationship between the IGF system and the building of peak bone mass and its subsequent loss contributed to the understanding of the pathogenesis of this disease. It may be foreseen that the results of intervention studies focused on the effects of the recombinant IGF-I will also influence therapeutic and preventive approaches. Modern antiresorption pharmacotherapy stabilizes or enhances bone density and reduces the risk of fractures. The addition of effective anabolics might increase the effectiveness of treatment by shifting the remodeling equilibrium in favor of formative processes. Because both recombinant GH and IGF-I have certain therapeutic limitations, it is considered to utilize substances which either stimulate endogenous IGF-I production directly in the bone or modulate synthesis and distribution of binding proteins for the peptide. Further new findings related to physiology and pathophysiology of this peptide will contribute to designing new strategies in the prevention of osteoporosis and other serious diseases of old age, such as diabetes, neoplasias or cardiovascular diseases.

How morphology predicts mechanical properties of trabecular structures depends on intra-specimen trabecular thickness variations

Two observations underlie this work. First, that the architecture of trabecular bone can accurately predict the mechanical stiffness characteristics of bone specimens when considering the combination of volume fraction and fabric, which is a measure of architectural anisotropy. Second, that the same morphological measures could not accurately predict the mechanical properties of porous structures in general. We hypothesize that this discrepancy can be explained by the special nature of trabecular bone as a structure in remodeling equilibrium relative to the external loads. We tested this hypothesis using a generic model of trabecular bone. Five series of 153 different architectures were created with this model. Each architecture was subjected to morphological analysis, and four different fabric measures were calculated to evaluate their effectiveness in characterizing the architecture. Relationships were determined relating morphology to the elastic constants. The quality of these relationships was tested by correlating the predicted elastic constants with those determined from finite element analysis. We found that the four fabric measures used could estimate the mechanical properties almost equally well. So the suggestion that fabric measures based on trabecular bone volume better represent the architecture than mean intercept length could not be affirmed. We conclude that for structures with equally sized elliptical voids the mechanical properties can be predicted well only if trabecular thickness variations within each structure are limited. These structures closely resemble previously developed models of trabecular bone. Furthermore, they are stiff in the principal fabric direction, hence, according to Cowin (J. Biomech. Eng. (108) (1986) 83), they are in remodeling equilibrium. These structures are also stiff over a large range of loading orientations, hence, are relatively insensitive to deviations in direction of loading.

A model of bone adaptation as an optimization process

The internal bone adaptation of the proximal femur is considered. A three-dimensional finite element model of the proximal femur is used. The bone remodeling in this work is numerically described by an evolutionary remodeling scheme with anisotropic material parameters and time-dependent loading. The memory of past loading is included in the model to account for the delay in the bone response from the load changes. The remodeling rate equation is derived from the structural optimization task of maximizing the stiffness in each time step. Additional information can be extracted from the optimization process; the remodeling equilibrium parameter where no apposition or resorption takes place, is defined as the element optimality conditions and the optimal design is used as an initial design for the onset of the remodeling simulation. Two examples of bone adaptation resulting from load changes are given, and the irreversible nature of the model is illustrated.

Three-Dimensional Lattice Continuum Model of Cancellous Bone for Structural and Remodeling Simulation.

In this article, we discuss a mechanical model for structural and remodeling analysis / simulation of cancellous bone, that takes into account tissue microstructure and residual stress. A three-dimensional lattice continuum is used as a structural model of cancellous bone with trabecular architecture, and its mechanical behavior is investigated concerning the dependence of structural parameters on the apparent mechanical properties of the tissue. Assuming the local uniform stress state to be an optimal stress state realized at the remodeling equilibrium, a remodeling rate equation is proposed to express the stress regulation process at the microstructural level for the three-dimensional lattice continuum. In terms of the lattice continuum, a vertebral body is modeled based on quantitative measurements of the trabecular architecture of the cancellous bone, and a remodeling simulation is conducted under the conditions of repetitive bending with compression. By comparison of the obtained distributions of the residual stress and the volume fraction with the experimental observations, the validity of the proposed model in predicting the adaptive remodeling of cancellous bone using a three-dimensional lattice continuum is demonstrated.

Loading Mode Interactions in Simulations of Long Bone Cross-Sectional Adaptation

Although many bone adaptation theories have been formulated to address both trabecular and cortical adaptation, most applications have focused on trabecular adaptation. Thus far, no thorough investigations of the influence of different types of loading on predicted patterns of long bone cross-sectional adaptation have been reported. In the current study, we present a new model for long bone cross-sectional adaptation that incorporates axial, bending and torsional loading components. We found that bending moments have a strong potential to modulate cross-sectional geometry, but can produce unforseen (and unrealistic) geometric instabilities. Torsional moments have the ability to suppress these instabilities, suggesting that torsion may play a more significant role in guiding long bone development than previously recognized. Our results also call into question the concept of strict “remodeling equilibrium,” suggesting that long bones do not necessarily approach a state of uniform mechanical stimulation. This modeling approach provides an additional perspective on experimental studies, and may lead to a greater understanding of the interaction between mechanics and biology in long bone adaptation.

An Analysis and Comparison of Convergence and Uniqueness of Time-Independent Bone Adaptation Models

Several stimuli are proposed in the bone remodeling theory. It is not clear, if a unique solution exists and if the result is convergent using a certain stimulus. In this study, the strain stimulus, strain energy stimulus and the von Mises stress stimulus for bone remodeling are compared and applied to a square plate model using the finite element method. In the plane stress state, the remodeling equilibrium equations are transformed into functions of only the principal strains and the graphs of these functions are drawn in a diagram using the principal strains as the variables of two coordinate axes. The equation of the sum of principal strain squared equal to a constant is a circle in the diagram. The remodeling equilibrium equation of the strain stimulus is a quadrangle fitting into the circle, the remodeling equilibrium equation of the strain energy stimulus is an ellipse and the remodeling equilibrium equation of the von Mises stress stimulus is also an ellipse close to the principal strains circle when we take the same constants in the above equations. Using the finite element method, two models are performed with the uniform initial elastic properties and with the semi-random initial distribution of the elastic properties. The principal strains as the final finite element results converge within 2% of the objective constant for all the different stimuli. The obtained Young's moduli of two models as the adaptation object are different but in equilibrium, i.e. the equilibrium solution of adaptation model is not unique. The principal strains can not be used to examine the uniqueness of solution, since two different solutions can have the same results of principal strains. Using a certain stimulus, certain initial properties and a certain iterative equation, the solution is unique in equilibrium. The results using the model in this study show also that the same results can be obtained using any of the three stimuli when a proper constant in each remodeling equilibrium equation is chosen.

Bone maintenance and remodeling: a control system based on fatigue damage.

This paper explores the relationship between damage, repair, and remodeling in compact bone. A model of microcrack growth is developed that takes account of recent findings on the behaviour of small fatigue cracks in other materials. This is combined with a simple model of a repair process, envisaged as a constant rate of decrease in crack length. The system that results is capable of achieving a stable and precise control of crack length without the need to measure it. This is very useful because it implies that bone does not require the complexities of crack-measuring transducers or active decision-making processes. A simple explanation is suggested for the presence of a "lazy zone" of remodeling equilibrium strains, and the limits of this zone are quantified. The model is developed through a necessarily simplified geometry and loading scheme but can be extended to provide a general solution applicable to in vivo conditions.

Mechanical Remodeling of Bone Structure Considering Residual Stress

Mechanical remodeling of bone is a kind of adaptation and is performed to regulate the stress and/or strain in the tissue in response to the changing mechanical environment due to tissue growth and atrophy. We propose a phenomenological model of mechanical remodeling of bone structure considering residual stress. The lattice continuum model is used to represent the bone structure, such as the trabecular structure of cancellous bones. The basic idea in the previous report, which concerns mechanical remodeling that takes into account the residual stress, is extended to the continuum with the internal tissue structure. A remodeling rate equation of the tissue structure is expressed so as to result in an equistress state at the remodeling equilib-rium as an optimality of the bone structure in the steady state. A case study of a long bone under bending moment reveals basic features of the proposed model of the stress regulation process. Remodeling simulation for the vertebral body under repetitive bending with compression demonstrates apparent density and residual stress distribu-tions that coincide with the experimental observations.

On the sufficiency conditions for the stability of bone remodeling equilibrium

In this technical note a sufficiency condition is established for the stability of a strain-energy-based bone remodeling theory in the special case of a beam loaded by an axial force and a bending moment. In a previous report the same condition was shown to be a necessary condition for stability in the same situation. The remodeling scheme is one characterized by a remodeling stimulus equal to the strain energy density divided by the bulk or apparent density raised to an exponent m as well as an elastic modulus proportional to bulk or apparent density raised to an exponent n. In order for a remodeling scheme to be stable for an elastic beam loaded by an axial force and a bending moment, it is established that the condition that m must be greater than n is not only necessary, but also sufficient.

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.

Bone Loss in the Distal Anterior Femur After Total Knee Arthroplasty

Bone loss in the distal anterior femur in asymptomatic total knee arthroplasty (TKA) patients has been noted roentgenographically and during revision surgery. A retrospective roentgenographic review of 147 TKA cases was carried out to document bone loss. The influence that the mode of fixation (porous coated and cemented) and the implant design have on bone loss was examined. The time of onset and the progression of bone loss were studied. Bone loss occurred in the distal anterior femur in the majority of cases reviewed (68%). The prevalence of bone loss was independent of the mode of fixation and the implant design. By qualitative observation, roentgenographically detectable bone loss occurred within the first postoperative year and did not progress further. Previously three-dimensional finite element analysis demonstrated that the replacement of the bearing surface of the femur with a stiff metallic implant reduces the stress in the distal anterior femur by at least one order of magnitude. It is therefore speculated that the observed bone loss results from stress shielding. The apparent lack of progression may reflect the development of a new remodeling equilibrium under the altered stress conditions. The bone loss in the distal anterior femur described has not been implicated as a source of failure. However, since the bone strength in the femoral region is compromised as it becomes osteopenic, bone failure may occur with longer periods of cyclic loading. Furthermore, as a result of bone loss, revision arthroplasty may be more difficult.

Structural Adaptation of Bones

Living bone tissue, like many other connective tissues, is a structural material that adapts its form and microstructure to changing environmental loading conditions. Bone tissue adapts not only its shape, but also its density and the details of its microstructure including its anisotropy. The anisotropy of bone is adapted in both its degree or strength and in the orientation of its principal axes of symmetry. These adaptive features of bone tissue are often referred to as aspects of Wolff’s law, although, strictly speaking, the term “Wolff’s law” applies only to the structural adaptation of spongy or trabecular bone. In this paper the composition, microstructure, mechanical properties and structurally adaptive features of bone are briefly reviewed. An algebraic formulation of Wolff’s law at remodeling equilibrium is described, and the nature of an evolutionary Wolff’s law is sketched.

Wolff's law of trabecular architecture at remodeling equilibrium.

An elastic constitutive relation for cancellous bone tissue is developed. This relationship involves the stress tensor T, the strain tensor E and the fabric tensor H for cancellous bone. The fabric tensor is a symmetric second rank tensor that is a quantitative stereological measure of the microstructural arrangement of trabeculae and pores in the cancellous bone tissue. The constitutive relation obtained is part of an algebraic formulation of Wolff's law of trabecular architecture in remodeling equilibrium. In particular, with the general constitutive relationship between T, H and E, the statement of Wolff's law at remodeling equilibrium is simply the requirement of the commutativity of the matrix multiplication of the stress tensor and the fabric tensor at remodeling equilibrium, T*H* = H*T*. The asterisk on the stress and fabric tensor indicates their values in remodeling equilibrium. It is shown that the constitutive relation also requires that E*H* = H*E*. Thus, the principal axes of the stress, strain and fabric tensors all coincide at remodeling equilibrium.