Large-scale microstructural simulation of load-adaptive bone remodeling in whole human vertebrae

Parameter reduction, sensitivity studies, and correlation analyses applied to a mechanobiologically regulated bone cell population model of the bone metabolism

When striving for reconstructing and predicting bone remodeling processes by means of mathematical models, cell population models have become a popular option. From a conceptual point of view, these models are able to take into account an arbitrary amount of regulatory mechanisms driving the development of bone cells and their activities. However, in most cases, the models include a large number of parameters; and most of those parameters cannot be measured, which certainly compromises the credibility of cell population models. Here, new insights are presented as to the potential improvement of this unsatisfactory situation. In particular, a previously published bone remodeling model was considered, and based on combination and merging of the original parameters, the total number of parameters could be reduced from 28 to 18, without impairing the model's versatility and significance. Furthermore, a comprehensive number of one- and two-variable sensitivity studies were performed, pointing out which parameters (alone and in combination with other parameters) influence the model predictions significantly - for that purpose, the mean squared relative error (MSRE) between simulations based on the original parameters and based on varied parameters was considered as failure measure. It has turned out that the model is significantly more sensitive to parameters which can be considered as phenomenological (such as differentiation, proliferation, and apoptosis rates) than to parameters which are directly related to specific processes (such as dissociation rate constants, and maximum concentrations of the involved factors). Using common correlation measures (such as Pearson, Spearman, and partial ranked correlation coefficients), correlation studies revealed that the correlations between most parameters and the MSRE are weak, while a few parameters exhibited moderate correlations. In conclusion, the results shown in this paper provide valuable insights concerning the design of new experiments allowing for measurement of the parameters which are most influential in the context of bone remodeling simulation.

Bone Mechanoregulation Allows Subject-Specific Load Estimation Based on Time-Lapsed Micro-CT and HR-pQCT in Vivo

Patients at high risk of fracture due to metabolic diseases frequently undergo long-term antiresorptive therapy. However, in some patients, treatment is unsuccessful in preventing fractures or causes severe adverse health outcomes. Understanding load-driven bone remodelling, i.e., mechanoregulation, is critical to understand which patients are at risk for progressive bone degeneration and may enable better patient selection or adaptive therapeutic intervention strategies. Bone microarchitecture assessment using high-resolution peripheral quantitative computed tomography (HR-pQCT) combined with computed mechanical loads has successfully been used to investigate bone mechanoregulation at the trabecular level. To obtain the required mechanical loads that induce local variances in mechanical strain and cause bone remodelling, estimation of physiological loading is essential. Current models homogenise strain patterns throughout the bone to estimate load distribution in vivo, assuming that the bone structure is in biomechanical homoeostasis. Yet, this assumption may be flawed for investigating alterations in bone mechanoregulation. By further utilising available spatiotemporal information of time-lapsed bone imaging studies, we developed a mechanoregulation-based load estimation (MR) algorithm. MR calculates organ-scale loads by scaling and superimposing a set of predefined independent unit loads to optimise measured bone formation in high-, quiescence in medium-, and resorption in low-strain regions. We benchmarked our algorithm against a previously published load history (LH) algorithm using synthetic data, micro-CT images of murine vertebrae under defined experimental in vivo loadings, and HR-pQCT images from seven patients. Our algorithm consistently outperformed LH in all three datasets. In silico-generated time evolutions of distal radius geometries (n = 5) indicated significantly higher sensitivity, specificity, and accuracy for MR than LH (p < 0.01). This increased performance led to substantially better discrimination between physiological and extra-physiological loading in mice (n = 8). Moreover, a significantly (p < 0.01) higher association between remodelling events and computed local mechanical signals was found using MR [correct classification rate (CCR) = 0.42] than LH (CCR = 0.38) to estimate human distal radius loading. Future applications of MR may enable clinicians to link subtle changes in bone strength to changes in day-to-day loading, identifying weak spots in the bone microstructure for local intervention and personalised treatment approaches.

Maintaining Bone Health in the Lumbar Spine: Routine Activities Alone Are Not Enough

Public health organisations typically recommend a minimum amount of moderate intensity activities such as walking or cycling for two and a half hours a week, combined with some more demanding physical activity on at least 2 days a week to maintain a healthy musculoskeletal condition. For populations at risk of bone loss in the lumbar spine, these guidelines are particularly relevant. However, an understanding of how these different activities are influential in maintaining vertebral bone health is lacking. A predictive structural finite element modelling approach using a strain-driven algorithm was developed to study mechanical stimulus and bone adaptation in the lumbar spine under various physiological loading conditions. These loading conditions were obtained with a previously developed full-body musculoskeletal model for a range of daily living activities representative of a healthy lifestyle. Activities of interest for the simulations include moderate intensity activities involving limited spine movements in all directions such as, walking, stair ascent and descent, sitting down and standing up, and more demanding activities with large spine movements during reaching and lifting tasks. For a combination of moderate and more demanding activities, the finite element model predicted a trabecular and cortical bone architecture representative of a healthy vertebra. When more demanding activities were removed from the simulations, areas at risk of bone degradation were observed at all lumbar levels in the anterior part of the vertebral body, the transverse processes and the spinous process. Moderate intensity activities alone were found to be insufficient in providing a mechanical stimulus to prevent bone degradation. More demanding physical activities are essential to maintain bone health in the lumbar spine.

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.

Theoretical concept of cortical to cancellous bone transformation

Microstructures of cortical and cancellous bones are altered continually by load-adaptive remodeling; in addition, their cellular mechanisms are similar despite the remarkably different porosities. The cortico-cancellous transitional zone is a site of vigorous remodeling, and intracortical remodeling cavitates the inner cortex to promote its trabecularization, which is considered the main cause of bone loss because of aging. Therefore, to prevent and treat age-related cortical bone loss effectively, it is indispensable to gain an integrated understanding of the cortical to the cancellous bone transformation via remodeling. We propose a novel theoretical concept to account for the transformation of dense cortical bone to porous cancellous bone. We develop a mathematical model of cortical and cancellous bone remodeling based on the concept that bone porosity is determined by the balance between the load-bearing function of mineralized bone and the material-transporting function of bone marrow. Remodeling simulations using this mathematical model enable the reproduction of the microstructures of cortical and cancellous bones simultaneously. Furthermore, current remodeling simulations have the potential to replicate cortical-to-cancellous bone transformation based on changes in the local balance between bone formation and resorption. We anticipate that the proposed mathematical model of cortical and cancellous bone remodeling will contribute to highlighting the essential features of cortical bone loss due to trabecularization of the cortex and help predict its spatial and temporal behavior during aging.

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A novel theoretical concept to account for cortical-to-cancellous bone transformation is proposed.

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A remodeling model to reproduce cortical and cancellous bone microstructures is developed.

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The remodeling simulation replicates cortical-to-cancellous bone transformation.

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The proposed method is valuable in clinical applications such as in predicting age-related cortical bone loss.

A mathematical multiscale model of bone remodeling, accounting for pore space-specific mechanosensation

While bone tissue is a hierarchically organized material, mathematical formulations of bone remodeling are often defined on the level of a millimeter-sized representative volume element (RVE), "smeared" over all types of bone microstructures seen at lower observation scales. Thus, there is no explicit consideration of the fact that the biological cells and biochemical factors driving bone remodeling are actually located in differently sized pore spaces: active osteoblasts and osteoclasts can be found in the vascular pores, whereas the lacunar pores host osteocytes - bone cells originating from former osteoblasts which were then "buried" in newly deposited extracellular bone matrix. We here propose a mathematical description which considers size and shape of the pore spaces where the biological and biochemical events take place. In particular, a previously published systems biology formulation, accounting for biochemical regulatory mechanisms such as the rank-rankl-opg pathway, is cast into a multiscale framework coupled to a poromicromechanical model. The latter gives access to the vascular and lacunar pore pressures arising from macroscopic loading. Extensive experimental data on the biological consequences of this loading strongly suggest that the aforementioned pore pressures, together with the loading frequency, are essential drivers of bone remodeling. The novel approach presented here allows for satisfactory simulation of the evolution of bone tissue under various loading conditions, and for different species; including scenarios such as mechanical dis- and overuse of murine and human bone, or in osteocyte-free bone.

Computational modeling of long-term effects of prophylactic vertebroplasty on bone adaptation

Cement augmentation in vertebrae (vertebroplasty) is usually used to restore mechanical strength after spinal fracture but could also be used as a prophylactic treatment. So far, the mechanical competence has been determined immediately post-treatment, without considering long-term effects of bone adaptation. In this work, we investigated such long-term effects of vertebroplasty on the stiffness of the augmented bone by means of computational simulation of bone adaptation. Using micro-finite element analysis, we determined sites of increased mechanical stress (stress raisers) and stress shielding and, based on the simulations, regions with increased or decreased bone loss due to augmentation. Cement volumes connecting the end plates led to increased stress shielding and bone loss. The increased stiffness due to the augmentation, however, remained constant over the simulation time of 30 years. If the intervention was performed at an earlier time point, it did lead to more bone loss, but again, it did not affect long-term stability as this loss was compensated by bone gains in other areas. In particular, around the augmentation cement, bone structures were preserved, suggesting a long-term integration of the cement in the augmented bone. We conclude that, from a biomechanical perspective, the impact of vertebroplasty on the bone at the microstructural level is less detrimental than previously thought.

Bone remodeling as a spatial evolutionary game

Bone remodeling is a complex process involving cell-cell interactions, biochemical signaling and mechanical stimuli. Early models of the biological aspects of remodeling were non-spatial and focused on the local dynamics at a fixed location in the bone. Several spatial extensions of these models have been proposed, but they generally suffer from two limitations: first, they are not amenable to analysis and are computationally expensive, and second, they neglect the role played by bone-embedded osteocytes. To address these issues, we developed a novel model of spatial remodeling based on the principles of evolutionary game theory. The analytically tractable framework describes the spatial interactions between zones of bone resorption, bone formation and quiescent bone, and explicitly accounts for regulation of remodeling by bone-embedded, mechanotransducing osteocytes. Using tools from the theory of interacting particle systems we systematically classified the different dynamic regimes of the spatial model and identified regions of parameter space that allow for global coexistence of resorption, formation and quiescence, as observed in physiological remodeling. In coexistence scenarios, three-dimensional simulations revealed the emergence of sponge-like bone clusters. Comparison between spatial and non-spatial dynamics revealed substantial differences and suggested a stabilizing role of space. Our findings emphasize the importance of accounting for spatial structure and bone-embedded osteocytes when modeling the process of bone remodeling. Thanks to the lattice-based framework, the proposed model can easily be coupled to a mechanical model of bone loading.

Load-adaptive bone remodeling simulations reveal osteoporotic microstructural and mechanical changes in whole human vertebrae

Osteoporosis is a major medical burden and its impact is expected to increase in our aging society. It is associated with low bone density and microstructural deterioration. Treatments are available, but the critical factor is to define individuals at risk from osteoporotic fractures. Computational simulations investigating not only changes in net bone tissue volume, but also changes in its microstructure where osteoporotic deterioration occur might help to better predict the risk of fractures. In this study, bone remodeling simulations with a mechanical feedback loop were used to predict microstructural changes due to osteoporosis and their impact on bone fragility from 50 to 80 years of age. Starting from homeostatic bone remodeling of a group of seven, mixed sex whole vertebrae, five mechanostat models mimicking different biological alterations associated with osteoporosis were developed, leading to imbalanced bone formation and resorption with a total net loss of bone tissue. A model with reduced bone formation rate and cell sensitivity led to the best match of morphometric indices compared to literature data and was chosen to predict postmenopausal osteoporotic bone loss in the whole group. Thirty years of osteoporotic bone loss were predicted with changes in morphometric indices in agreement with experimental measurements, and only showing major deviations in trabecular number and trabecular separation. In particular, although being optimized to match to the morphometric indices alone, the predicted bone loss revealed realistic changes on the organ level and on biomechanical competence. While the osteoporotic bone was able to maintain the mechanical stability to a great extent, higher fragility towards error loads was found for the osteoporotic bones.

Voxel size dependency, reproducibility and sensitivity of an in vivo bone loading estimation algorithm

A bone loading estimation algorithm was previously developed that provides in vivo loading conditions required for in vivo bone remodelling simulations. The algorithm derives a bone's loading history from its microstructure as assessed by high-resolution (HR) computed tomography (CT). This reverse engineering approach showed accurate and realistic results based on micro-CT and HR-peripheral quantitative CT images. However, its voxel size dependency, reproducibility and sensitivity still need to be investigated, which is the purpose of this study. Voxel size dependency was tested on cadaveric distal radii with micro-CT images scanned at 25 µm and downscaled to 50, 61, 75, 82, 100, 125 and 150 µm. Reproducibility was calculated with repeated in vitro as well as in vivo HR-pQCT measurements at 82 µm. Sensitivity was defined using HR-pQCT images from women with fracture versus non-fracture, and low versus high bone volume fraction, expecting similar and different loading histories, respectively. Our results indicate that the algorithm is voxel size independent within an average (maximum) error of 8.2% (32.9%) at 61 µm, but that the dependency increases considerably at voxel sizes bigger than 82 µm. In vitro and in vivo reproducibility are up to 4.5% and 10.2%, respectively, which is comparable to other in vitro studies and slightly higher than in other in vivo studies. Subjects with different bone volume fraction were clearly distinguished but not subjects with and without fracture. This is in agreement with bone adapting to customary loading but not to fall loads. We conclude that the in vivo bone loading estimation algorithm provides reproducible, sensitive and fairly voxel size independent results at up to 82 µm, but that smaller voxel sizes would be advantageous.

Computational modelling of bone augmentation in the spine

Computational models are gaining importance not only for basic science, but also for the analysis of clinical interventions and to support clinicians prior to intervention. Vertebroplasty has been used to stabilise compression fractures in the spine for years, yet there are still diverging ideas on the ideal deposition location, volume, and augmentation material. In particular, little is known about the long-term effects of the intervention on the surrounding biological tissue. This review aims to investigate computational efforts made in the field of vertebroplasty, from the augmentation procedure to strength prediction and long-term in silico bone biology in augmented human vertebrae. While there is ample work on simulating the augmentation procedure and strength prediction, simulations predicting long-term effects are lacking. Recent developments in bone remodelling simulations have the potential to show adaptation to cement augmentation and, thus, close this gap.