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Keywords:

  • mathematical modeling;
  • in silico simulation;
  • bone remodeling;
  • drug testing;
  • osteoporosis

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. BACKGROUND ON CONCEPTUAL AND MATHEMATICAL MODELING IN BONE RESEARCH
  5. TOWARD THE FUTURE WITH IN SILICO MODELING
  6. CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES

Contemporary, computer-based mathematical modeling techniques make it possible to represent complex biological mechanisms in a manner that permits hypothesis testing in silico. This perspective shows how such approaches might be applied to bone remodeling and therapeutic research.

Currently, the dominant conceptual model applied in bone research involves the dynamic balance between the continual build-up and breakdown of bone matrix by two cell types, the osteoblasts and osteoclasts, acting together as a coordinated, remodeling unit. This conceptualization has served extraordinarily well as a focal point for understanding how mutations, chemical mediators, and mechanical force, as well as external influences (e.g., drugs, diet) affect bone structure and function. However, the need remains to better understand and predict the consequences of manipulating any single factor, or combination of factors, within the context of this complex system's multiple interacting pathways. Mathematical models are a natural extension of conceptual models, providing dynamic, quantitative descriptions of the relationships among interacting components. This formalization creates the ability to simulate the natural behavior of a system, as well as its modulation by therapeutic or dietetic interventions. A number of mathematical models have been developed to study complex bone functions, but most include only a limited set of biological components needed to address a few specific questions. However, it is possible to develop larger, multiscale models that capture the dynamic interactions of many biological components and relate them to important physiological or pathological outcomes that allow broader study. Examples of such models include Entelos' PhysioLab platforms. These models simulate the dynamic, quantitative interactions among a biological system's biochemicals, cells, tissues, and organs and how they give rise to key physiologic and pathophysiologic outcomes. We propose that a similar predictive, dynamical, multiscale mathematical model of bone remodeling and metabolism would provide a better understanding of the mechanisms governing these phenomena as well as serve as an in silico platform for testing pharmaceutical and clinical interventions on metabolic bone disease.


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. BACKGROUND ON CONCEPTUAL AND MATHEMATICAL MODELING IN BONE RESEARCH
  5. TOWARD THE FUTURE WITH IN SILICO MODELING
  6. CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES

AWIDE RANGE of bone-related disorders, including arthritis, osteoporosis, osteolytic bone disease, and bone fractures, can significantly reduce an individual's ability to lead an active and healthy life. The U.S. Centers for Disease Control and Prevention estimate that 59.4 million Americans will suffer from arthritis by 2020,(1) and the National Osteoporosis Foundation estimates that 50% of women and 25% of men >50 years of age will have an osteoporosis-related fracture in their lifetime.(2) These numbers highlight the urgent need to establish a comprehensive and accurate understanding of bone biology and pathophysiology and create novel research approaches to expedite the development of more effective therapeutic strategies.

A number of the fundamental characteristics of bone biology and physiology significantly complicate the task of elucidating the complex biological mechanisms that underlie multiple bone diseases and pathologies. Bone function is simultaneously modulated by the hormonal, cytokine and immune systems, and is exquisitely sensitive to variations in mechanical force. Bone function is thus ultimately the consequence of multiple cell-cell interactions operating in both a coordinated and hierarchical fashion. In addition, changes in bone structure and function occur over time scales ranging from days to decades, but how the faster or slower changes contribute individually or in combination to bone diseases that develop over months to years is unclear. Finally, the heterogeneity of the human population and the influences of diverse environmental factors add to the challenge of understanding and treating human bone diseases.

Tractable research approaches that allow the investigation and manipulation of specific pathways in the context of the integrated bone system would prove invaluable in the study of bone physiology. In vivo animal and clinical approaches to study the mechanisms of bone physiology maintain the whole system context but are poorly amenable to control and/or measurement of numerous important variables. Alternatively, the relative simplicity of in vitro experimental systems allows greater control and observability, but the influence of the important physiological context is sacrificed. We propose that large-scale mathematical models of bone biology should be used to complement and bridge these traditional experimental approaches.

BACKGROUND ON CONCEPTUAL AND MATHEMATICAL MODELING IN BONE RESEARCH

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. BACKGROUND ON CONCEPTUAL AND MATHEMATICAL MODELING IN BONE RESEARCH
  5. TOWARD THE FUTURE WITH IN SILICO MODELING
  6. CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES

The use of conceptual and mathematical models has already proven valuable in bone research. Over the years, a relatively simple conceptual model of the dynamic balance between the continual build-up and breakdown of bone matrix by osteoblasts (OBs) and osteoclasts (OCs) has served extraordinarily well as a focal point for understanding bone turnover and metabolism. In particular, this model addresses the complex behaviors arising from the interaction of these two key cell types acting together (“coupled”) as a coordinated remodeling unit (Fig. 1). One of the first clear proponents of this concept was Dr Harold Frost, who termed this interacting functional unit the “basic multicellular unit (BMU) of remodeling.”(3) The BMU continues to be used to account for the dynamics of bone turnover as it responds to pathology, the aging process (notably in osteoporosis), and variations in mechanical load. Another concept of Frost's is the “mechanostat,” which incorporates environmental influences by viewing a bone's response to mechanical force as a threshold-sensitive, feedback loop, modified by genetic and environmental factors, that ultimately changes bone structure by altering OB and OC dynamics in the BMU (Fig. 2).(4) The mechanostat idea wedded what was perhaps the first conceptualization of bone remodeling as a physiological process responsive to mechanical forces (Wolff's Law)(5) to the more contemporary notion of coordinated OB and OC activity (the BMU).

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Figure FIG. 1.. Formation-resorption coupling cycle (conceptual model). The functional coupling between OC resorptive activity and the bone-forming role of OBs is shown in this drawing prepared by one of the authors (AJK) >20 years ago. This conceptual model incorporates precursor cells into the coupling process and shows sites in the cycle where the cells would be sensitive to putative regulatory agents. The model also suggests places where therapeutic agents might be effective in altering bone formation and/or resorption.

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Figure FIG. 2.. Illustration of the “mechanostat” conceptual model as proposed by Frost.(3,4) (A) As shown by the dotted line, the mechanostat consists, collectively, of the bone, the “minimum effective strain” (MES) mechanism and the growth, modeling, and remodeling systems. Applied strain to the MES above or below a load threshold value induces modeling or remodeling, respectively. (B) The additional influences of mechanical usage (MU) and numerous regulatory factors on the mechanostat are depicted in this second illustration. The MU captures the consequences of physical motion on bone compensatory responses. The influences of currently unknown factors are indicated by X. Integration of these regulatory factors by the mechanostat determines whether, to what extent, and where the bone modeling or remodeling will take place.

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Whereas these conceptualizations have proven valuable in addressing some of the complications in bone biology, they do not address time-dependent and quantitative aspects of biological function. The ability to explicitly and quantitatively account for the dynamics and magnitude of various functions clearly differentiates mathematical models from static conceptual diagrams.

Mathematical models have already been used to study various aspects of bone biology, remodeling, and pathophysiology. For example, building on the mechanostat concept described above, Turner(6) created a model of bone mass regulation based on the ability of bone cells to adapt to transient signals in their environment. He showed that this model could reconcile many phenomena observed on how bone mass responds to changes in strain, hormones, and other stimuli better than models based on fixed setpoints. A thematically similar model by van der Linden et al.(7) was used to resolve apparent bone level strains into tissue level strains in trabecular bone specimens. Computer simulations of several bone loss scenarios in this model showed that small amounts of bone loss at highly strained locations could severely compromise the mechanical properties of the bone, whereas large losses at low strain locations had only minor effects. Furthermore, these simulations have shown that the character of the loss is as important as the quantity of loss. Loss by thinning of trabeculae had little effect on stiffness, whereas the same amount of loss through resorption cavities caused a marked decrease in stiffness, yielding high strain peaks at the bottom of the cavities. This finding showed that a reduction in the number and size of resorption cavities with antiresorptive drug treatment can result in large reductions in fracture risk with only small increases in bone mass. In another study, Hernandez et al.(8) used a computer model of trabecular bone remodeling to study the relative influences of peak BMD, onset time of menopause, and age-related bone loss on the development of osteoporosis. By varying these three parameters by ±10%, the model predicted that the onset of osteoporosis could be delayed longest by increasing peak BMD, which they argued suggests that intervention should focus on this aspect of the disease rather than the others.

Some models have now represented the BMU concept directly. A model from Huiskes et al.(9) examined the properties of cross-regulation between osteocytes, OBs, and OCs and the role of mechanical strain in maintenance and adaptation of a small region of trabecular bone. This work showed that cell coupling by feedback from mechanical load transfer can account for the emergence and maintenance of trabecular architecture as an optimal mechanical structure and its ability to adapt to varying external loads. In contrast, the model of Lemaire et al.(10) describes a representative BMU consisting of OCs and OBs and a few regulatory molecules that shows good agreement with many known aspects of bone function. An interesting model prediction that awaits experimental verification is that increasing the preosteoblastic population by increasing the differentiation of mesenchymal progenitors or modulating apoptosis would be an effective means of therapeutic stimulation of bone formation.

TOWARD THE FUTURE WITH IN SILICO MODELING

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. BACKGROUND ON CONCEPTUAL AND MATHEMATICAL MODELING IN BONE RESEARCH
  5. TOWARD THE FUTURE WITH IN SILICO MODELING
  6. CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES

The modeling examples cited above have helped elucidate some of the complex phenomena governing bone biology. However, these models were limited to relatively few variables and were designed to address a specific set of biological questions. While very useful in their specific applications, such models do not have the flexibility to address a wider set of questions based on a broad range of pathway interactions. Today, advances in scientific knowledge, as well as in mathematical modeling and computational infrastructure, enable the research community to develop large-scale mathematical models that more broadly include the functions and interactions of multiple biological components, integrate systemic and local effects, and span multiple time scales and organizational levels (biochemical, cell, tissue, and organ). Within such a framework, it is possible to simulate and study how modulating specific molecules or cellular functions affects key physiological and pathological outcomes within an integrated system context. Thus, such models complement in vivo and in vitro studies by providing a system framework for all of the components while allowing manipulation and subsequent observation of the system variables represented.

Large-scale, quantitative, predictive models enable investigators to test competing hypotheses about the mechanisms underlying system functions and compare the relative contributions of multiple pathways, cell functions, and mediators to health and disease states. This approach enables one to identify important knowledge gaps within the system and provides a means for assessing the impact of these gaps on both the physiologic outcomes and the potential efficacy of therapeutic agents.

The Entelos Rheumatoid Arthritis (RA) PhysioLab platform provides an example of this approach. This large-scale, mechanism-based, dynamic model of a prototypical human rheumatoid joint is designed to study potential therapeutic strategies at the discovery, preclinical, and clinical levels. The RA PhysioLab model focuses on the dynamics of chronic RA at the cartilage-pannus junction, representing the functions of different cell types present and their many interactions (Fig. 3). It was constructed based on publicly available data describing cell and tissue functions and interactions. Using several hundred ordinary differential equations to describe cell populations and interactions, the model calculates how the various functions contribute to synovial tissue inflammation and their impact on cartilage degradation and bone erosion. The functions represented occur on time scales from minutes to months, and the model simulates chronic disease over a period of years.

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Figure FIG. 3.. Schematic representation of the area of the inflamed joint represented in the RA Physiolab platform. The insets on the right show the cellular components and some of the mechanisms represented in the model. The top inset (synovial hyperplasia) represents the cells and their relationships modeled in the synovial tissue compartment. The arrows show the cross-talk between these cells through cell-cell interactions and production of soluble mediators. The middle inset (cartilage degradation) represents the modeled cartilage compartment, comprised of chondrocytes and the cartilage matrix. The model captures the contribution of both the synovial tissue and cartilage to the production of matrix metalloproteinases (MMPs) that can destroy the cartilage matrix. The arrows show that the soluble mediators produced in the synovial tissue compartment can influence the behaviors of the cells in the cartilage, and inversely, the enzymes produced by chondrocytes will contribute to the cartilage matrix degradation along with enzymes from the synovial tissue. The bottom inset (bone erosion) shows the functions involved in OC differentiation and activation that are included. Rather than modeling the OC population dynamics, the model tracks the mediators that regulate OC differentiation and activation, shown by the arrows. EC, endothelial cells; FLS, fibroblast-like synoviocytes.

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Simulation of therapeutics in this model allows the prediction and interpretation of complex dynamics resulting from the interplay of various mediators and cells. For example, bone erosion is influenced by both OC differentiation and activation, which are in turn regulated both positively and negatively by numerous mediators, as shown by the bone erosion module of the RA PhysioLab platform (Fig. 4). Furthermore, individual therapeutic modalities frequently have discrepant effects on different biologic functions; for instance, steroid effects on bone erosion are both positive (e.g., the potentiation of RANKL-induced OC differentiation) and negative (e.g., the inhibition of TNF-α-induced OC differentiation; Fig. 4). Mathematical models such as this quantitatively integrate these complex, often competing, effects into a unified, interpretable context, allowing one to predict the impact of a specific therapeutic intervention.

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Figure FIG. 4.. Calculation of the bone erosion index in the RA Physiolab platform. The bone erosion index calculates the net erosive nature of the inflamed synovial tissue in RA by tracking three main functions relevant to bone erosion: OC differentiation, OC activation, and the number of OC precursors in the synovial tissue. This figure shows the mediators synthesized in the synovial tissue that influence the activation and/or differentiation of OC precursors into fully active and resorptive cells. White- and black-tipped arrows indicate inhibitory and stimulatory effects, respectively. Specific equations (not shown) are embedded within the nodes and arrows in this diagram.

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Model predictions of the dynamic human in vivo response can guide preclinical and clinical studies. Experimental observation of tissue effects of therapies, especially on a short time scale, is often impractical in clinical trials and limited in animal models. On the other hand, assessing the long-term efficacy in the clinic is challenging and very costly. Thus, a priori estimates of the dynamic response to treatment derived from models like the RA PhysioLab platform can help researchers determine when to expect improvement, and therefore how long to run a clinical trial. Furthermore, these simulations can reveal potential transient effects that might confound a result, which can be taken into account when planning a clinical trial. For example, simulation of methylprednisolone treatment in the RA PhysioLab platform predicts a biphasic response, with an increase in bone erosion shortly after treatment begins but a net reduction later (2 months and beyond; Fig. 5). This longer-term effect is consistent with the known effects of glucocorticoid treatment on bone erosion.(11) Further in silico analysis of the effects of methylprednisolone reveals that the biphasic response is due, at least in part, to a rapid increase in OC differentiation followed by a subsequent slow decrease in OC activation and precursor (monocyte) recruitment (Fig. 4). In this manner, using the model to identify the key pathways driving therapeutic effects can guide the design of more directed treatments that maximize efficacy while minimizing adverse effects (e.g., glucocorticoid-induced osteoporosis(12)).

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Figure FIG. 5.. Simulated effect of methylprednisolone treatment on normalized bone erosion index over 12 months of treatment in chronic RA. This graph shows the effect of 5 mg/day methylprednisolone treatment on bone erosion index, normalized to the value for the untreated patient at each time-point, calculated in the RA PhysioLab platform. Values <1 indicate improvement in bone erosion index relative to the untreated patient at that time-point, whereas values >1 correspond to worsening.

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Based on the recent advances in the field of bone biology, a model of bone metabolism analogous to the RA PhysioLab platform can be imagined. Such a large-scale model could bring together into a single platform multiple elements such as hormonal regulation, biomechanical influences, and the dynamics of the BMU. The breadth and depth of the detailed model (i.e., its scope) would be based on the specific research questions it is meant to address.

For illustration, one can consider the biological elements and functions needed to represent bone state and function related to effective treatment of osteoporosis. Research areas of interest might include the role of estrogens and other hormones in the maintenance of healthy homeostasis, the effects of hormone replacement therapy in menopausal women, the regulatory differences between males and females and how these are affected by menopause and aging, and the heterogeneity in the human population. In addition, such a platform might focus on the effects of lifestyle and metabolic factors on osteoporosis, enabling exploration of the influences of exercise and diet on bone loss. Because it has been shown that cortical and spongy bone respond differently to treatments, representing their divergent regulation within the same model would provide a mechanism for evaluating the effectiveness of new and existing therapies on both bone types.

With the goal of studying osteoporosis, the behaviors that a model might be required to reproduce include healthy homeostatic regulation of bone turnover and calcium balance and the dynamic changes associated with the progressive loss of bone mass in aging individuals. To do this, a model design might focus on the biology of a representative volume of bone in contact with a blood compartment. An essential part of the bone compartment would include OC and OB population dynamics (e.g., proliferation, apoptosis, activation, differentiation, precursor sources), anabolism and catabolism of bone matrix, and modulation of these processes by hormones, growth factors, cytokines, exercise, mechanical forces, and other factors. Additional elements to consider would be OC population dynamics and the influx of other cell types that might be precursors to the OCs and OBs. The blood compartment would serve as a source and/or sink for dietary elements such as Ca2+, endogenous factors such as hormones, and therapeutic drugs. It is important to note that although such a large-scale model would be developed around specific research areas, the resulting platform would allow exploration of multiple complex and intertwining dynamics, bringing a higher level of understanding to guide research in these areas. Finally, to extend the use of such a platform, it is possible to envision embedding such a model within a 3-D model of bone mass, thus relating the “microscopic” biology to the macroscopic bone mechanics.

In the development and use of such large-scale mathematical models, one must deal with uncertainties regarding which biological components and functions contribute to a system-level behavior (e.g., the identity of OC precursors), as well as quantitative aspects of their contributions (e.g., rates, magnitudes). These uncertainties can be addressed in two ways; first, by using a top-down or behavior-driven approach to modeling, and second, by using the model as a hypothesis testing platform. The top-down approach begins by identifying the system level behaviors that a model should reproduce, including dynamics and quantitative aspects. These behaviors guide the selection of biological components that seem to be involved in the system and have properties or functions that might contribute to system behavior. This perspective contrasts with reductionist approaches, including bottom-up modeling, that focus first on identifying “all of the parts.” Such reductionism does not take advantage of the guiding information available from system level behaviors. Second, developing a model is also a hypothesis testing opportunity. For instance, if integration of component functions does not fully reproduce system dynamics, the researcher can hypothesize additional functionalities to achieve the desired behavior based on related biological systems and an understanding of dynamical systems. These hypotheses are represented mathematically, and the model's function is reevaluated. A good match with known system dynamics indicates that the hypotheses are plausible, and appropriate experiments can be designed to verify them. Whereas any model developed now will not be able to simulate every conceivable condition encountered in bone remodeling, even a basic platform like that imagined above could represent more dimensions of bone remodeling than models currently available. As knowledge of the biology evolves, a model can be reconfigured to include new data, expanding the model's scope and applicability for testing new hypotheses on bone remodeling. In these ways, models should be viewed as continual “works in progress” that integrate current knowledge, help make explicit key knowledge gaps, and provide a means to test in an analytic manner multiple hypotheses about system mechanisms and the importance of biological variation.

Optimally, the development and use of large-scale models should be integrated with experimental work, where biosimulation is used to quickly test multiple hypotheses, and the empirical experiments are used to confirm in silico predictions or collect data to fill key knowledge gaps. These new data can be incorporated into the model to expand its knowledge framework and to drive both in silico and empirical research.

Possible challenges to using computational modeling more extensively exist, but examination of related endeavors shows that they can be overcome. A frequent concern is whether there is sufficient and appropriate data to build sound models in this area. Based on prior modeling experience, we believe there is ample data. For example, the extensive literature on bone biology and related disorders includes data analogous to that used to develop the RA PhysioLab model discussed above. Whereas obtaining research funding for nontraditional approaches is always a concern, NIH and NSF both have multiple initiatives that support biological systems modeling that can be found through a search on their web sites. Furthermore, the FDA espouses an integral role for computational modeling to help reverse increasingly long and expensive pharmaceutical development cycles.(13)

Another challenge is to establish productive interdisciplinary teams of biologists and mathematical modeling experts (e.g., engineers, applied mathematicians). To address this, we suggest tapping the experience of investigators in fields where such collaborations are common, such as neurobiology and genetics, modern bioengineering and systems biology, and industry modeling groups in the pharmaceutical field. Computational infrastructure should not be perceived as limiting, because the requisite computer power for running large simulations is available and inexpensive, and software for modeling and simulation continues to be improved and made commercially available. Thus, the motivation, funding, and infrastructure for the development of large-scale mathematical models of biological systems are present and growing.

CONCLUSIONS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. BACKGROUND ON CONCEPTUAL AND MATHEMATICAL MODELING IN BONE RESEARCH
  5. TOWARD THE FUTURE WITH IN SILICO MODELING
  6. CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES

Predictive biosimulation using models that integrate data in a dynamic, quantitative framework offers a powerful approach to the study of bone biology. The integration of diverse elements, including the regulation of OC and OB differentiation, cell population dynamics, and hormonal and biomechanical regulation of bone biology in one dynamic, mechanism-based model could contribute to a broader understanding of bone metabolism in physiological and pathophysiological states. Such models can also be used to identify efficacious therapeutic interventions.

Acknowledgements

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. BACKGROUND ON CONCEPTUAL AND MATHEMATICAL MODELING IN BONE RESEARCH
  5. TOWARD THE FUTURE WITH IN SILICO MODELING
  6. CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES

The authors acknowledge the researchers in addition to Defranoux and Young who developed the RA PhysioLab platform, Herbert Struemper, Saroja Ramanujan, and Kalle Soderstrom, and thank Kalle Soderstrom for helpful discussions about this manuscript. AJK contribution was supported, in part, by NIH Grant R01 AG18717.

REFERENCES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. BACKGROUND ON CONCEPTUAL AND MATHEMATICAL MODELING IN BONE RESEARCH
  5. TOWARD THE FUTURE WITH IN SILICO MODELING
  6. CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES
  • 1
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