Pascal R. Buenzli

Quantifying the osteocyte network in the human skeleton.

Osteocytes form an extensive cellular network throughout the hard tissue matrix of the skeleton, which is known to regulate skeletal structure. However due to limitations in imaging techniques, the magnitude and complexity of this network remain undefined. We have used data from recent papers obtained by new imaging techniques, in order to estimate absolute and relative quantities of the human osteocyte network and form a more complete understanding of the extent and nature of this network. We estimate that the total number of osteocytes within the average adult human skeleton is ~42 billion and that the total number of osteocyte dendritic projections from these cells is ~3.7 trillion. Based on prior measurements of canalicular density and a mathematical model of osteocyte dendritic process branching, we calculate that these cells form a total of 23 trillion connections with each other and with bone surface cells. We estimate the total length of all osteocytic processes connected end-to-end to be 175,000 km. Furthermore, we calculate that the total surface area of the lacuno-canalicular system is 215 m(2). However, the residing osteocytes leave only enough space for 24 mL of extracellular fluid. Calculations based on measurements in lactation-induced murine osteocytic osteolysis indicate a potential total loss of ~16,000 mm(3) (16 mL) of bone by this process in the human skeleton. Finally, based on the average speed of remodelling in the adult, we calculate that 9.1 million osteocytes are replenished throughout the skeleton on a daily basis, indicating the dynamic nature of the osteocyte network. We conclude that the osteocyte network is a highly complex communication network, and is much more vast than commonly appreciated. It is at the same order of magnitude as current estimates of the size of the neural network in the brain, even though the formation of the branched network differs between neurons and osteocytes. Furthermore, continual replenishment of large numbers of osteocytes in the process of remodelling allows therapeutic changes to the continually renewed osteoblast population to be rapidly incorporated into the skeleton.

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.

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.

The Casimir force at high temperature

The standard expression of the high-temperature Casimir force between perfect conductors is obtained by imposing macroscopic boundary conditions on the electromagnetic field at metallic interfaces. This force is twice larger than that computed in microscopic classical models allowing for charge fluctuations inside the conductors. We present a direct computation of the force between two quantum plasma slabs in the framework of non-relativistic quantum electrodynamics including quantum and thermal fluctuations of both matter and field. In the semi-classical regime, the asymptotic force at large slab separation is identical to that found in the above purely classical models, which is therefore the right result. We conclude that when calculating the Casimir force at non-zero temperature, fluctuations inside the conductors cannot be ignored. Aspects of this subject are treated in a companion letter by Jancovici B. and Samaj L., Casimir force between two ideal-conductor walls revisited (Europhys. Lett., 72 (2005) 35).

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.

Bone refilling in cortical basic multicellular units: insights into tetracycline double labelling from a computational model

Bone remodelling is carried out by ‘bone multicellular units’ (

Microscopic origin of universality in Casimir forces

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