Peter Pivonka

Mathematical modeling in bone biology: from intracellular signaling to tissue mechanics.

Although conceptual and experimental models are historically well incorporated in bone biology studies, mathematical modeling has been much less-frequently utilized. This review aims to introduce mathematical modeling to readers who are not familiar with the concept underlying this methodology, to outline how mathematical models can help to improve current understanding of bone biology and to discuss examples where mathematical modeling was used to provide new insights into important questions of bone biology.

Spatio-temporal structure of cell distribution in cortical Bone Multicellular Units: A mathematical model

Bone remodelling maintains the functionality of skeletal tissue by locally coordinating bone-resorbing cells (osteoclasts) and bone-forming cells (osteoblasts) in the form of Bone Multicellular Units (BMUs). Understanding the emergence of such structured units out of the complex network of biochemical interactions between bone cells is essential to extend our fundamental knowledge of normal bone physiology and its disorders. To this end, we propose a spatio-temporal continuum model that integrates some of the most important interaction pathways currently known to exist between cells of the osteoblastic and osteoclastic lineage. This mathematical model allows us to test the significance and completeness of these pathways based on their ability to reproduce the spatio-temporal dynamics of individual BMUs. We show that under suitable conditions, the experimentally observed structured cell distribution of cortical BMUs is retrieved. The proposed model admits travelling-wave-like solutions for the cell densities with tightly organised profiles, corresponding to the progression of a single remodelling BMU. The shapes of these spatial profiles within the travelling structure can be linked to the intrinsic parameters of the model such as differentiation and apoptosis rates for bone cells. In addition to the cell distribution, the spatial distribution of regulatory factors can also be calculated. This provides new insights on how different regulatory factors exert their action on bone cells leading to cellular spatial and temporal segregation, and functional coordination.

The influence of bone surface availability in bone remodelling—A mathematical model including coupled geometrical and biomechanical regulations of bone cells

Bone is a biomaterial undergoing continuous renewal. The renewal process is known as bone remodelling and is operated by bone-resorbing cells (osteoclasts) and bone-forming cells (osteoblasts). An important function of bone remodelling is the repair of microcracks accumulating in the bone matrix due to mechanical loading. Cell–cell communication between cells of the osteoclastic lineage and cells of the osteoblastic lineage is thought to couple resorption and formation so as to preserve bone integrity and achieve homeostatic bone renewal. Both biochemical and biomechanical regulatory mechanisms have been identified in this coupling. Many bone pathologies are associated with an alteration of bone cell interactions and a consequent disruption of bone homeostasis. In osteoporosis, for example, this disruption leads to long-term bone loss and increased fragility, and can ultimately result in fractures. Here we focus on an additional and poorly understood potential regulatory mechanism of bone cells, that involves the morphology of the microstructure of bone. Bone cells can only remove and replace bone at a bone surface. However, the microscopic availability of bone surface depends in turn on the ever-changing bone microstructure. The importance of this geometrical dependence is unknown and difficult to quantify experimentally. Therefore, we develop a sophisticated mathematical model of bone cell interactions that takes into account biochemical, biomechanical and geometrical regulations. We then investigate numerically the influence of bone surface availability in bone remodelling within a representative bone tissue sample. Biochemical regulations included in the model involve signalling molecules of the receptor–activator nuclear factor κB pathway (rank–rankl–opg), macrophage colony-stimulating factor (mcsf), transforming growth factor β(tgfβ), and parathyroid hormone (pth). For the biomechanical regulation of bone cells, a multiscale homogenisation scheme is used to determine the microscopic strains generated at the level of the extravascular matrix hosting the osteocytes by macroscopic loading. The interdependence between the bone cells’ activity, which modifies the bone microstructure, and changes in the microscopic bone surface availability, which in turn influences bone cell development and activity, is implemented using a remarkable experimental relationship between bone specific surface and bone porosity. Our model suggests that geometrical regulation of the activation of new remodelling events could have a significant effect on bone porosity and bone stiffness in osteoporosis. On the other hand, geometrical regulation of late stages of osteoblast and osteoclast differentiation seems less significant. We conclude that the development of osteoporosis is probably accelerated by this geometrical regulation in cortical bone, but probably slowed down in trabecular bone.

Solute transport in cartilage undergoing cyclic deformation

There are no blood vessels in cartilage to transport nutrients and growth factors to chondrocytes dispersed throughout the cartilage matrix. Insulin-like growth factor-I (IGF-I) is a large molecule with an important role in cartilage growth and metabolism, however, it first must reach the chondrocytes to exert its effect. While diffusion of IGF-I through cartilage is possible, it has been speculated that cyclic loading can enhance the rate of solute transport within cartilage. To better understand this process, here a one-dimensional axisymmetric mathematical model is developed to examine the transport of solutes through a cylindrical plug of cartilage undergoing cyclic axial deformation in the range of 10− 3–1 Hz. This study has revealed the role of timescales in interpreting transport results in cartilage. It is shown that dynamic strains can either enhance or inhibit IGF-I transport at small timescales ( < 20 min after onset of loading), depending on loading frequency. However, on longer timescales it is found that dynamic loading has negligible effect on IGF-I transport. Most importantly, in all cases examined the steady state IGF-I concentration did not exceed the fixed boundary value, in contrast to the predictions of Mauk et al. (2003).

Femoral shaft strains during daily activities: Implications for atypical femoral fractures.

BACKGROUND Atypical femoral fractures are low-energy fractures initiating in the lateral femoral shaft. We hypothesized that atypical femoral fracture onset is associated with daily femoral strain patterns. We examined femoral shaft strains during daily activities. METHODS We analyzed earlier calculations of femoral strain during walking, sitting and rising from a chair, stair ascent, stair descent, stepping up, and squatting based on anatomically consistent musculoskeletal and finite-element models from a single donor and motion recordings from a body-matched volunteer. Femoral strains in the femoral shaft were extracted for the different activities and compared. The dependency between femoral strains in the lateral shaft and kinetic parameters was studied using multi-parametric linear regression analysis. FINDINGS Tensile strain in the lateral femoral shaft varied from 327 με (squatting) to 2004 με (walking). Walking and stair descent imposed tensile loading on the lateral shaft, whereas the other activities mainly imposed tensile loads on the anterior shaft. The multi-parametric linear regression showed a moderately strong correlation between tensile strains in the lateral shaft and the motion kinetic (joint moments and ground reaction force) in the proximal (R(2)=0.60) and the distal shaft (R(2)=0.46). INTERPRETATION Bone regions subjected to tensile strains are associated with atypical femoral fractures. Walking is the daily activity that induces the highest tensile strain in the lateral femoral shaft. The kinetics of motion explains 46%-50% of the tensile strain variation in the lateral shaft, whereas the unexplained part is likely to be attributed to the way joint moments are decomposed into muscle forces.

Modelling the anabolic response of bone using a cell population model.

To maintain bone mass during bone remodelling, coupling is required between bone resorption and bone formation. This coordination is achieved by a network of autocrine and paracrine signalling molecules between cells of the osteoclastic lineage and cells of the osteoblastic lineage. Mathematical modelling of signalling between cells of both lineages can assist in the interpretation of experimental data, clarify signalling interactions and help develop a deeper understanding of complex bone diseases. Several mathematical models of bone cell interactions have been developed, some including RANK-RANKL-OPG signalling between cells and systemic parathyroid hormone PTH. However, to our knowledge these models do not currently include key aspects of some more recent biological evidence for anabolic responses. In this paper, we further develop a mathematical model of bone cell interactions by Pivonka et al. (2008) to include the proliferation of precursor osteoblasts into the model. This inclusion is important to be able to account for Wnt signalling, believed to play an important role in the anabolic responses of bone. We show that an increased rate of differentiation to precursor cells or an increased rate of proliferation of precursor osteoblasts themselves both result in increased bone mass. However, modelling these different processes separately enables the new model to represent recent experimental discoveries such as the role of Wnt signalling in bone biology and the recruitment of osteoblast progenitor cells by transforming growth factor β. Finally, we illustrate the power of the new models capabilities by applying the model to prostate cancer metastasis to bone. In the bone microenvironment, prostate cancer cells are believed to release some of the same signalling molecules used to coordinate bone remodelling (i.e.,Wnt and PTHrP), enabling the cancer cells to disrupt normal signalling and coordination between bone cells. This disruption can lead to either bone gain or bone loss. We demonstrate that the new computational model developed here is capable of capturing some key observations made on the evolution of the bone mass due to metastasis of prostate cancer to the bone microenvironment.

Role of mathematical modeling in bone fracture healing

Bone fracture healing is a complex physiological process commonly described by a four-phase model consisting of an inflammatory phase, two repair phases with soft callus formation followed by hard callus formation, and a remodeling phase, or more recently by an anabolic/catabolic model. Data from humans and animal models have demonstrated crucial environmental conditions for optimal fracture healing, including the mechanical environment, blood supply and availability of mesenchymal stem cells. Fracture healing spans multiple length and time scales, making it difficult to know precisely which factors and/or phases to manipulate in order to obtain optimal fracture-repair outcomes. Deformations resulting from physiological loading or fracture fixation at the organ scale are sensed at the cellular scale by cells inside the fracture callus. These deformations together with autocrine and paracrine signals determine cellular differentiation, proliferation and migration. The local repair activities lead to new bone formation and stabilization of the fracture. Although experimental data are available at different spatial and temporal scales, it is not clear how these data can be linked to provide a holistic view of fracture healing. Mathematical modeling is a powerful tool to quantify conceptual models and to establish the missing links between experimental data obtained at different scales. The objective of this review is to introduce mathematical modeling to readers who are not familiar with this methodology and to demonstrate that once validated, such models can be used for hypothesis testing and to assist in clinical treatment as will be shown for the example of atrophic nonunions.

A Multiscale Framework Based on the Physiome Markup Languages for Exploring the Initiation of Osteoarthritis at the Bone–Cartilage Interface

The initiation of osteoarthritis (OA) has been linked to the onset and progression of pathologic mechanisms at the cartilage-bone interface. Most importantly, this degenerative disease involves cross-talk between the cartilage and subchondral bone environments, so an informative model should contain the complete complex. In order to evaluate this process, we have developed a multiscale model using the open-source ontologies developed for the Physiome Project with cartilage and bone descriptions at the cellular, micro, and macro levels. In this way, we can effectively model the influence of whole body loadings at the macro level and the influence of bone organization and architecture at the micro level, and have cell level processes that determine bone and cartilage remodeling. Cell information is then passed up the spatial scales to modify micro architecture and provide a macro spatial characterization of cartilage inflammation. We evaluate the framework by linking a common knee injury (anterior cruciate ligament deficiency) to proinflammatory mediators as a possible pathway to initiate OA. This framework provides a “virtual bone-cartilage” tool for evaluating hypotheses, treatment effects, and disease onset to inform and strengthen clinical studies.

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.

Poromicromechanics reveals that physiological bone strains induce osteocyte-stimulating lacunar pressure

Mechanical loads which are macroscopically acting onto bony organs, are known to influence the activities of biological cells located in the pore spaces of bone, in particular so the signaling and production processes mediated by osteocytes. The exact mechanisms by which osteocytes are actually able to “feel” the mechanical loading and changes thereof, has been the subject of numerous studies, and, while several hypotheses have been brought forth over time, this topic has remained a matter of debate. Relaxation times reported in a recent experimental study of Gardinier et al. (Bone 46(4):1075–1081, 2010) strongly suggest that the lacunar pores are likely to experience, during typical physiological load cycles, not only fluid transport, but also undrained conditions. The latter entail the buildup of lacunar pore pressures, which we here quantify by means of a thorough multiscale modeling approach. In particular, the proposed model is based on classical poroelasticity theory, and able to account for multiple pore spaces. First, the model reveals distinct nonlinear dependencies of the resulting lacunar (and vascular) pore pressures on the underlying bone composition, highlighting the importance of a rigorous multiscale approach for appropriate computation of the aforementioned pore pressures. Then, the derived equations are evaluated for macroscopic (uniaxial as well as hydrostatic) mechanical loading of physiological magnitude. The resulting model-predicted pore pressures agree very well with the pressures that have been revealed, by means of in vitro studies, to be of adequate magnitude for modulating the responses of biological cells, including osteocytes. This underlines that osteocytes may respond to many types of loading stimuli at the same time, in particular so to fluid flow and hydrostatic pressure.