Modelling the anabolic response of bone using a cell population model

Abstract

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 $\mathrm{\beta }$. Finally, we illustrate the power of the new model's 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.

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.