Modelling the anabolic response of bone using a cell population model
Highlights
► Precursor osteoblasts proliferation is a potent mechanism for anabolic response. ► Osteoblast differentiation and proliferation concur for proper bone formation control. ► Action of Wnt and pth in bone remodelling may explain a variety of bone responses. ► Wnt/Dkk1 production by prostate cancer can drive the type of metastatic bone lesions.
Introduction
Bone is a dynamic living tissue which continuously undergoes remodelling to ensure mineral homeostasis and to repair micro damage (Parfitt, 2002, Martin et al., 1998). The two main bone cell types executing bone remodelling are osteoclasts which resorb the mineralised bone matrix and osteoblasts which deposit osteoid (which subsequently becomes mineralised) (Martin et al., 1998). The third cell type involved in bone remodelling are osteocytes (i.e., terminally differentiated cells derived from mature osteoblasts that have been trapped in the mineralised bone matrix Bonewald and Johnson, 2008). The entire ensemble of bone cells contributing to bone remodelling is referred to as the basic multicellular unit (bmu) (Parfitt, 1994, Parfitt, 1983).
Within the bmu, pre-osteoblasts, which express rankl have been hypothesised to control the differentiation of osteoclasts from hematopoietic progenitors (Ma et al., 2001, Martin, 2004, Roodman, 1999, Gori et al., 2000). The bone resorption phase is subsequently followed by bone formation, driven in part by factors produced by the osteoclast that stimulate osteoblastogenesis (Roodman, 1999). This coupling between resorption and formation phase in bmus is required to maintain bone mass. Many bone pathologies, such as osteoporosis, Paget's disease and cancer metastasis to bone, are associated with the dysregulation of this coupling process leading to abnormal bone loss or bone gain. Mathematical modelling can be employed to interpret experimental data, clarify signalling interactions, investigate therapeutic interventions, and to generally better understand bone remodelling from a systems perspective (Pivonka and Komarova, 2010).
Bone remodelling has been represented mathematically in a variety of ways including bone cell population models (ODEs) (Komarova et al., 2003, Lemaire et al., 2004, Pivonka et al., 2008), continuum models (PDEs) (Ryser et al., 2009, Buenzli et al., 2011, Ji et al., in press) and discrete cell models (van Oers et al., 2008, Buenzli et al., 2012). The bone cell population model by Lemaire et al. (2004) proposes an interesting approach based on fundamental chemical reaction principles such as material balance and mass action kinetics. This model incorporates some of the most important bone biology known at that time. Extensions to include further components of bone biology can be formulated using the same framework. We have used this framework to include new knowledge in bone biology in our bone cell population model (Pivonka et al., 2008) (such as the expression of rankl and opg by osteoblasts of various maturities) (Pivonka et al., 2008), and to introduce a spatial variation in cell numbers to represent a single basic multicellular unit (Buenzli et al., 2011). We have also applied the model by Pivonka et al. (2008) to examine possible therapeutic interventions to restore bone mass following dysregulation of the rank–rankl–opg signalling system (Pivonka et al., 2010), and studied osteolytic lesions in multiple myeloma (Wang et al., 2011).
However, while the model by Pivonka et al. (2008) does some things well, it does not capture the anabolic effects of precursor osteoblast proliferation. Recent experimental evidence suggests that wnt signalling is a critically important regulator of bone remodelling—wnt signalling plays an important role in normal bone homeostasis under mechanical loading, and excessive wnt signalling is responsible for some osteopetrotic (excess) bone states (Henriksen et al., 2009, Jilka, 2007). In addition, recent clinical evidence demonstrates that administration of intermittent pth is an effective anabolic intervention (Jilka, 2007, Hodsman et al., 2005). The exact molecular mechanisms leading to anabolic responses under intermittent pth administration are incompletely understood and probably multifactorial, involving differential regulations of osteoblast differentiation, proliferation and apoptosis (Jilka, 2007). While we do not model intermittent pth administration in this paper, it is important to include these three cellular behaviours regulating the number of osteoblasts for future investigations. In this paper, we thus further develop the model by Pivonka et al. (2008) by introducing the proliferation of osteoblasts in a way such that the new model is consistent with the original model and can incorporate osteoblast proliferation through wnt signalling or via other signalling systems. We then explore the effect of parameter changes in the new model on net bone balance, and see that the new model is capable of effectively representing osteopetrotic bone disease states arising from disruption of normal osteoblastic proliferation.
Finally we illustrate the capabilities of the new model in a complex bone disease that arises when prostate cancer cells metastasise to the bone microenvironment. This disease is characterised by a variable phenotype that often initially involves net bone loss (Clarke and Fleisch, 2008, Hall et al., 2006a, Roudier et al., 2008), and finally net bone gain (coupled with focal bone loss). We show that the new model developed here can model bone gain and bone loss via secretion of signalling molecules such as wnt, psa and pthrp by the prostate cancer cells.
Section snippets
Background
A recent review by Khosla et al. highlights the importance of osteoblast development in the regulation of bone remodelling and the potential for therapeutic interventions that target the osteoblastic lineage (Khosla et al., 2008). Osteoblasts are mesenchymal cells derived from the mesoderm. Sequential expression of several molecules (such as runx2 and osx), driven by signal transduction pathways, facilitates the differentiation of the progenitor cell into a proliferating pre-osteoblast, then
Description of the model
For simplicity, osteoblast progenitors and pre-osteoblasts are pooled into a single proliferative cell type in our model, which we call pre-osteoblast and denote by . Three stages of osteoblast development and two stages of osteoclast development are included in the cell population model.
Osteoclasts. Pre-osteoclasts () represent circulating cells of haematopoietic origin. Pre-osteoclasts are assumed to mature into active osteoclasts () upon activation of their rank receptor by the
Properties of the model
The steady-state cell densities represented by the model correspond to physiological cell densities (averaged at the tissue level) of a normal, healthy adult whose skeleton undergoes remodelling. While a baseline of mesenchymal stem cells and hematopoetic stem cells is implicitly assumed, a bone remodelling event is not necessarily induced. Indeed, the system of ODEs (3), (4), (5) governing the evolution of , , and always admits vanishing bone cell densities as a solution,
Application to prostate cancer metastasis
Many bone pathologies are due to an altered bone balance and an altered bone turnover rate during remodelling. Bone imbalance is associated with under-refilling (bone loss) or over-refilling (bone gain) in bmus. Bone turnover rate is associated with the number of active bmus and indicates how fast bone may be lost, gained, and/or turned over. Our computational model represents bone remodelling at the tissue scale, where bmu quantities are spatially averaged. At this scale, bone imbalance and
Conclusions
Recent experimental evidence suggests that osteoblast proliferation plays an important role in the regulation of bone remodelling. In this paper, we have developed a novel computational model of bone cell interactions that includes osteoblast proliferation. This model takes into account a catabolic regulatory mechanism of bone remodelling, mediated by the rank–rankl–opg pathway, and a new anabolic regulatory mechanism of bone remodelling, driven by osteoblast proliferation. From our numerical
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