ABSTRACT Diversity in the body shapes and sizes of dinosaurs was foundational to their widespread success during the Mesozoic era. The ability to quantify body size and form reliably is therefore critical to the study of dinosaur biology and evolution. Body mass estimates for any given fossil animal are, in theory, most informative when derived from volumetric models that account for the three‐dimensional shapes of the entire body. In addition to providing estimates of total body mass, volumetric approaches can be used to determine the inertial properties of specific body segments and the overall distribution of mass throughout the body, each of which are essential for the modelling and interpretation of form–function relationships and their associations with ecology. However, the determination of body volumes in fossil taxa is often subjective, and may be sensitive to varied artistic inference. This highlights the need for an approach to body mass estimation in which body segment volumes are systematically constrained by quantitative scaling relationships between the hard tissues that fossilise and the soft tissues only observable in extant taxa. To this end, we used recently published skeletal to soft tissue volumetric scaling factors derived from CT data of extant sauropsids to estimate body segment mass properties from skeletal models of 52 non‐avian dinosaurs representing the majority of major clades and body plans. The body masses estimated by this study range from less than 200 g in the tiny avialan Yixianornis to over 60 tonnes in the giant sauropod Patagotitan , which is currently the largest dinosaur known from mostly complete skeletal remains. From our models, we infer that many previous reconstructions of soft tissue envelopes may be too small, and that many dinosaurs were therefore heavier than previous estimates. Our models generally overlap with the range of body mass estimates derived from limb bone shaft dimensions, but with considerable quantitative variability among major clades. This suggests that different taxa either differed in skeletal to soft tissue volume ratios, or that their limb bone dimensions varied relative to body mass, perhaps related to differences in locomotor dynamics and postural evolution. Our models also allowed us to investigate variation in mass distribution and body proportions across different dinosaurs from a perspective grounded in extant anatomical data, framing long‐standing hypotheses about their form, function, and behaviour in a quantitative context. For example, reconstructed disparity in whole‐body centres of mass reflects a broad array of postures in different dinosaur clades, while the lack of strong positive allometry in the dimensions of the weight‐bearing limb segments relative to total body mass corroborates previous studies suggesting an overall decrease in dinosaur locomotor performance as body size increased.
Guidelines for assessment of bone microstructure in rodents using micro–computed tomography
Abstract Use of high-resolution micro–computed tomography (µCT) imaging to assess trabecular and cortical bone morphology has grown immensely. There are several commercially available µCT systems, each with different approaches to image acquisition, evaluation, and reporting of outcomes. This lack of consistency makes it difficult to interpret reported results and to compare findings across different studies. This article addresses this critical need for standardized terminology and consistent reporting of parameters related to image acquisition and analysis, and key outcome assessments, particularly with respect to ex vivo analysis of rodent specimens. Thus the guidelines herein provide recommendations regarding (1) standardized terminology and units, (2) information to be included in describing the methods for a given experiment, and (3) a minimal set of outcome variables that should be reported. Whereas the specific research objective will determine the experimental design, these guidelines are intended to ensure accurate and consistent reporting of µCT-derived bone morphometry and density measurements. In particular, the methods section for papers that present µCT-based outcomes must include details of the following scan aspects: (1) image acquisition, including the scanning medium, X-ray tube potential, and voxel size, as well as clear descriptions of the size and location of the volume of interest and the method used to delineate trabecular and cortical bone regions, and (2) image processing, including the algorithms used for image filtration and the approach used for image segmentation. Morphometric analyses should be based on 3D algorithms that do not rely on assumptions about the underlying structure whenever possible. When reporting µCT results, the minimal set of variables that should be used to describe trabecular bone morphometry includes bone volume fraction and trabecular number, thickness, and separation. The minimal set of variables that should be used to describe cortical bone morphometry includes total cross-sectional area, cortical bone area, cortical bone area fraction, and cortical thickness. Other variables also may be appropriate depending on the research question and technical quality of the scan. Standard nomenclature, outlined in this article, should be followed for reporting of results. © 2010 American Society for Bone and Mineral Research
A multiscale mechanobiological model of bone remodelling predicts site-specific bone loss in the femur during osteoporosis and mechanical disuse
We propose a multiscale mechanobiological model of bone remodelling to investigate the site-specific evolution of bone volume fraction across the midshaft of a femur. The model includes hormonal regulation and biochemical coupling of bone cell populations, the influence of the microstructure on bone turnover rate, and mechanical adaptation of the tissue. Both microscopic and tissue-scale stress/strain states of the tissue are calculated from macroscopic loads by a combination of beam theory and micromechanical homogenisation. This model is applied to simulate the spatio-temporal evolution of a human midshaft femur scan subjected to two deregulating circumstances: (i) osteoporosis and (ii) mechanical disuse. Both simulated deregulations led to endocortical bone loss, cortical wall thinning and expansion of the medullary cavity, in accordance with experimental findings. Our model suggests that these observations are attributable to a large extent to the influence of the microstructure on bone turnover rate. Mechanical adaptation is found to help preserve intracortical bone matrix near the periosteum. Moreover, it leads to non-uniform cortical wall thickness due to the asymmetry of macroscopic loads introduced by the bending moment. The effect of mechanical adaptation near the endosteum can be greatly affected by whether the mechanical stimulus includes stress concentration effects or not.
<i>In silico</i> biology of bone modelling and remodelling: adaptation
Modelling and remodelling are the processes by which bone adapts its shape and internal structure to external influences. However, the cellular mechanisms triggering osteoclastic resorption and osteoblastic formation are still unknown. In order to investigate current biological theories, in silico models can be applied. In the past, most of these models were based on the continuum assumption, but some questions related to bone adaptation can be addressed better by models incorporating the trabecular microstructure. In this paper, existing simulation models are reviewed and one of the microstructural models is extended to test the hypothesis that bone adaptation can be simulated without particular knowledge of the local strain distribution in the bone. Validation using an experimental murine loading model showed that this is possible. Furthermore, the experimental model revealed that bone formation cannot be attributed only to an increase in trabecular thickness but also to structural reorganization including the growth of new trabeculae. How these new trabeculae arise is still an unresolved issue and might be better addressed by incorporating other levels of hierarchy, especially the cellular level. The cellular level sheds light on the activity and interplay between the different cell types, leading to the effective change in the whole bone. For this reason, hierarchical multi-scale simulations might help in the future to better understand the biomathematical laws behind bone adaptation.
A novel mathematical model of bone remodelling cycles for trabecular bone at the cellular level
After an initial phase of growth and development, bone undergoes a continuous cycle of repair, renewal and optimisation by a process called remodelling. This paper describes a novel mathematical model of the trabecular bone remodelling cycle. It is essentially formulated to simulate a remodelling event at a fixed position in the bone, integrating bone removal by osteoclasts and formation by osteoblasts. The model is developed to construct the variation in bone thickness at a particular point during the remodelling event, derived from standard bone histomorphometric analyses. The novelties of the approach are the adoption of a predator–prey model to describe the dynamic interaction between osteoclasts and osteoblasts, using a genetic algorithm–based solution; quantitative reconstruction of the bone remodelling cycle; and the introduction of a feedback mechanism in the bone formation activity to co-regulate bone thickness. The application of the model is first demonstrated by using experimental data recorded for normal (healthy) bone remodelling to predict the temporal variation in the number of osteoblasts and osteoclasts. The simulated histomorphometric data and remodelling cycle characteristics compare well with the specified input data. Sensitivity studies then reveal how variations in the model’s parameters affect its output; it is hoped that these parameters can be linked to specific biochemical factors in the future. Two sample pathological conditions, hypothyroidism and primary hyperparathyroidism, are examined to demonstrate how the model could be applied more broadly, and, for the first time, the osteoblast and osteoclast populations are predicted for these conditions. Further data are required to fully validate the model’s predictive capacity, but this work shows it has potential, especially in the modelling of pathological conditions and the optimisation of the treatment of those conditions.
Towards a new spatial representation of bone remodeling
Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed. There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes. We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur in vitro and in vivo. By employing the level set approach it is possible to develop computational and mathematical representations of the spatialbehavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as in vitro and in vivo. This makes it possible to perform in silica experiments more closely resembling experimental observations.
Standardized nomenclature, symbols, and units for bone histomorphometry: A 2012 update of the report of the ASBMR Histomorphometry Nomenclature Committee
Before publication of the original version of this report in 1987, practitioners of bone histomorphometry communicated with each other in a variety of arca
Integrated Bone Tissue Physiology: Anatomy and Physiology
<i>In silico</i> models of bone remodeling from macro to nano—from organ to cell
Abstract Computational modeling is a tool through which researchers can achieve a greater understanding of the mechanisms governing biological systems. In the field of bone biology, a plethora of models exist which attempt to replicate and investigate bone's dynamic behavior at different scales. At organ level, models are continuum based and describe the variation of bone's apparent density as a function of both biological and external mechanical stimuli. At tissue level, models include bone microarchitecture and more descriptive parameters such as trabecular thickness, osteoclast resorption depth, and activation frequency. Finally, at cell level, models employ partial differential equations to describe complex cellular interactions in the temporal domain. Although informative, these models exist in isolation. Consequently, their interpretation is limited. In this review, we present an overview of the organ‐, tissue‐, and cell‐level models and assess their ability to reflect bone's metabolic processes reliably. Existing interscale synergies are then presented along with a computational framework which could be exploited to achieve a fully integrated, multiscale modeling approach. WIREs Syst Biol Med 2011 3 241–251 DOI: 10.1002/wsbm.115 This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models
A review of mathematical modeling of bone remodeling from a systems biology perspective
Bone remodeling is an essential, delicately balanced physiological process of coordinated activity of bone cells that remove and deposit new bone tissue in the adult skeleton. Due to the complex nature of this process, many mathematical models of bone remodeling have been developed. Each of these models has unique features, but they have underlying patterns. In this review, the authors highlight the important aspects frequently found in mathematical models for bone remodeling and discuss how and why these aspects are included when considering the physiology of the bone basic multicellular unit, which is the term used for the collection of cells responsible for bone remodeling. The review also emphasizes the view of bone remodeling from a systems biology perspective. Understanding the systemic mechanisms involved in remodeling will help provide information on bone pathology associated with aging, endocrine disorders, cancers, and inflammatory conditions and enhance systems pharmacology. Furthermore, some features of the bone remodeling cycle and interactions with other organ systems that have not yet been modeled mathematically are discussed as promising future directions in the field.
A new biological bone remodeling in silico model combined with advanced discretization methods
Abstract Bone remodeling remains a highly researched topic investigated by many strands of science. The main purpose of this work is formulating a new computational framework for biological simulation, extending the version of the bone remodeling model previously proposed by Komarova. Thus, considering only the biological aspect of the remodeling process, the action of osteoclasts and osteoblasts is taken into account as well as its impact on bone mass. It is conducted a spatiotemporal analysis of a remodeling cycle obtaining a dynamic behavior of bone cells very similar to the biological process already described in the literature. The numerical example used is based on bone images obtained with scanning electron microscopy. During simulation, it is possible to observe the variation of bone's architecture through isomaps. These maps are obtained through the combination of biological bone remodeling models with three distinct numerical techniques—finite element method (FEM), radial point interpolation method (RPIM), and natural neighbor radial point interpolation method (NNRPIM). A study combining these numerical techniques allows to compare their performance. Ultimately, this work supports the inclusion of meshless methods due to their smoother results and its easiness to be combined with medical images from CT scans and MRI.
Automated Scale Reduction of Nonlinear <span style="font-variant:small-caps;">QSP</span> Models With an Illustrative Application to a Bone Biology System
Integrating quantitative systems pharmacology ( QSP ) into pharmacokinetics/pharmacodynamics ( PKPD ) has resulted in models that are highly complex and often not amenable to further exploration via estimation or design. Because QSP models are usually depicted using nonlinear differential equations it is not straightforward to apply some model reduction techniques, such as proper lumping. In this study, we explore the combined use of linearization and proper lumping as a general method to simplification of a nonlinear QSP model. We illustrate this with a bone biology model and the reduced model was then applied to describe bone mineral density ( BMD ) changes due to denosumab dosing. The methodologies used in this study can be applied to other multiscale models for developing a mechanism‐based structural model for future analyses.
A review of recent developments in mathematical modeling of bone remodeling
In this article, we summarize the developments in the mathematical modeling of the mechanics of bone and related biological phenomena. We will devote special attention to the results of the last 10–15 years, although we will cover some relevant classical work to better frame the more recent researches. We will propose a division of the literature based on the main aim of the model (mechanical/biomathematical) and the type of biological phenomena considered (stimulus, growth, cell population dynamics). Finally, we will suggest some possible directions for future investigations.
A review of mathematical modeling of bone remodeling from a systems biology perspective
Bone remodeling is an essential physiological process in the adult skeleton. Due to the complex nature of this process, many mathematical models of bone remodeling have been developed. Each of these models has unique features, but they have underlying patterns. In this review, the authors highlight the important aspects frequently found in mathematical models for bone remodeling and discuss how and why these aspects are included when considering the physiology of the bone basic multicellular unit, which is the term used for the collection of cells responsible for bone remodeling. The review also emphasizes the view of bone remodeling from a systems biology perspective. Understanding the systemic mechanisms involved in remodeling will help provide information on bone pathology associated with aging, endocrine disorders, cancers, and inflammatory conditions and enhance systems pharmacology. Furthermore, some features of the bone remodeling cycle and interactions with other organ systems that have not yet been modeled mathematically are discussed as promising future directions in the field.
Growth, metabolic partitioning, and the size of microorganisms
Population growth rate is a fundamental ecological and evolutionary characteristic of living organisms, but individuals must balance the metabolism devoted to biosynthesis and reproduction against the maintenance of existing structure and other functionality. Here we present a mathematical model that relates metabolic partitioning to the form of growth. The model captures the observed growth trajectory of single cells and individuals for a variety of species and taxa spanning prokaryotes, unicellular eukaryotes, and small multicellular eukaryotes. Our analysis suggests that the per-unit costs of biosynthesis and maintenance are conserved across prokaryotes and eukaryotes. However, the relative metabolic expenditure on growth and maintenance of whole organisms clearly differentiates taxa: prokaryotes spend an increasing fraction of their entire metabolism on growth with increasing cell size, whereas eukaryotes devote a diminishing fraction. These differences allow us to predict the minimum and maximum size for each taxonomic group, anticipating observed evolutionary life-history transitions. The framework provides energetic insights into taxonomic tradeoffs related to growth and metabolism and constrains traits that are important for size-structured modeling of microbial communities and their ecological and biogeochemical effects.
Mechanobiological osteocyte feedback drives mechanostat regulation of bone in a multiscale computational model
Significant progress has been made to identify the cells and signaling molecules involved in the mechanobiological regulation of bone remodeling. It is now well accepted that osteocytes act as mechanosensory cells in bone expressing several signaling molecules such as nitric oxide (NO) and sclerostin (Scl) which are able to control bone remodeling responses. In this paper, we present a comprehensive multiscale computational model of bone remodeling which incorporates biochemical osteocyte feedback. The mechanostat theory is quantitatively incorporated into the model using mechanical feedback to control expression levels of NO and Scl. The catabolic signaling pathway RANK–RANKL–OPG is co-regulated via (continuous) PTH and NO, while the anabolic Wnt signaling pathway is described via competitive binding reactions between Wnt, Scl and the Wnt receptors LRP5/6. Using this novel model of bone remodeling, we investigate the effects of changes in the mechanical loading and hormonal environment on bone balance. Our numerical simulations show that we can calibrate the mechanostat anabolic and catabolic regulatory mechanisms so that they are mutually exclusive. This is consistent with previous models that use a Wolff-type law to regulate bone resorption and formation separately. Furthermore, mechanical feedback provides an effective mechanism to obtain physiological bone loss responses due to mechanical disuse and/or osteoporosis.
A mechano-chemo-biological model for bone remodeling with a new mechano-chemo-transduction approach
Bone remodeling is a fundamental biological process that develops in bone tissue along its whole lifetime. It refers to a continuous bone transformation with new bone formation and old bone resorption that changes the internal microstructure and composition of the tissue. The main objectives of bone remodeling are: repair of the internal microcracks; adaptation of the macroscopic stiffness and strength to the actual changing mechanical demands; and control of the calcium homeostasis. Understanding this process and predicting its evolution is critical to reduce the effects of long-term disuse as happens during periods of reduced mobility. It is also important in the design of bone implants to avoid long-term stress shielding. Many mathematical models have been proposed from the earliest purely phenomenological to the latest that include biological knowledge. However, there still exists a lack of connection between the mechanical driving force and the biochemical and cell processes it triggers. Here, and following previous works that model independently the mechanobiological and biochemical processes in bone remodeling, we present a more complete model, useful for both cortical and trabecular bone, that uses a new mechanotransduction approach based on the effect of strains onto the bonding–unbonding rate of RANK/RANKL/OPG receptor–ligand reactions. We compare the results of this model with previous ones, showing a good agreement in similar conditions. We also apply it to realistic situations such as a femoral bone after implantation of a hip prosthesis, getting similar results to the clinical ones in the final bone density distribution. Finally, we extend this approach to the anisotropic case, getting not only the mean density, but also the directional homogenization of the microstructure. This biochemical approach permits, not only to predict the bone evolution under changes in the mechanical loads, but also, to consider anabolic and catabolic drugs to control bone density, such as those used in osteoporosis.
