Analytic approach to explore dynamical osteoporotic bone turnover
Abstract
Background
The dynamics of the osteoporotic bone turnover is studied in this paper with the aid of stability analysis of the associated mathematical model. Osteoporosis, which is a common bone disorder, is studied in this papper in detail with an emphasis on the relative threshold values. We examine the expository signaling among the bone cells named osteoclast and osteoblast. Main functioning of osteoblasts is bone formation, whereas osteoclasts are bone removal cells.
Methods
Mathematical framework for osteoporotic bone turnover comprising of the communication between osteoclasts and osteoblasts has been presented to exhibit the conditions for stability in bone turnover.
Results
The percentage ratios of the population of osteoblasts/osteoclasts have been determined via numerical simulations. The remedial upshots of targeting osteoporotic cells participating in such process are examined.
Conclusions
From our analysis we have conclude that the role of external agents in treating the diseased bone can be better interpreted with the aid of a theoretical model.
Keywords
Bone remodeling Osteoporosis Stability analysis Numerical simulations1 Introduction
The bone formation and resorption is a continuous process and is known that collagen strands and lifeless bone minerals cover about 60–70 percent of the area of a bone at a local site, whereas the remaining proportion carries water. An infant body contains 270 bones, whereas an adult skeleton is made of 260 bones, which happens due to the continuous process of bone resorption and formation. There are two forms of bones, called cortical and trabecular (also known as cancellous). A cortical bone is an external layer of bone, which is hard and thick, whereas a spongy and soft structure lying inside the bone is called a trabecular bone. The main functioning of the bone includes the support to body and provides the base to marrow. There are numerous kinds of bone cells, but mainly two cells, named osteoclast (C) and osteoblast (B), take part in the functioning and maintenance of a bone. The main activity of osteoclast includes the resorption of mineralized tissues, whereas osteoblasts are boneforming cells. Osteocytes are bone cells lying at the deep rooted area of a bone.
A bone guarantees redesigning for the duration of whole life in spatially and transiently discrete destinations in an organized procedure including resorption, trailed by formation of a new bone [19]. Bone remodeling is accountable for development and mechanically instigated adaptation of bone and entails the processes of bone formation and resorption, however, globally correlated, takes place separately at anatomical different sites. Such a tightly correlated event needs harmonized activities of multicellular components to confirm that bone removal and making take place successively at the identical anatomical place to preserve the bone volume.
The two cells mainly responsible for bone remodeling process are osteoclasts (C) and osteoblasts (B). The key role of both kinds of cells is quite opposite; osteoclasts are boneresorbing cells, whereas osteoblasts function as boneforming cells. Osteoclasts are lethally differentiated myeloid multinucleated osteoclast precursors, which are exclusively transformed to eradicate mineralized bone milieu. Precursors of osteoclasts flow in the blood, and bone marrow and adult osteoclasts are produced from synthesis of the precursors. Such an evolution takes place after the signaling through receptor activators of NFκβ and its ligand. During bone resorption, osteoclasts append to the bone and develop tight seals with the underlying bone matrix in roughly circular extensions of their cytoplasm, and within these sealed zones, they form ruffled border membranes. By this sealing and secretory means the bone matrix is concurrently degraded, and bone minerals are dissolved while protecting neighboring cells from the harmful effects of HCl. Osteoblasts are mononucleated cellular units derived from osteoblast precursors. They are responsible for manufacturing of the major proteins of bone and type 1 collagen like osteocalcin and osteopontin, which compile the organic milieu of bone. These cells are also accountable for the mineralization of bone and fabricate enzyme alkaline phosphates. Recent investigations exhibit the role of osteoblasts as an imperative component in governing osteoclastogenesis. Another type of bone cells namely osteocytes, a latent state of osteoblasts, also participate in bone maintenance and mineralization. Many therapies of different bone disorders affect through osteocytes.
Various growth factors give paracrine coupling amongst C and B and additionally autocrine cycles for positive and negative input control of every cell kind. Among the most essential components, there are the proresorptive cytokine receptor activator of nuclear factor β ligand (RANKL) and its decoy receptor osteoprotegerin (OPG), which are communicated by B and corresponding control of C [17, 24]. Complex formation of OPG and RANK with RANKL give anabolic and catabolic effects on bones, respectively. Several other elements like transforming growth factorβ (TGFβ) and parathyroid hormone (PTH) also incorporate in such phenomena.
Multiple and dispensable activities of TGFβ family incorporate monitoring of various characteristics and outcomes of cell functions, including growth, prolifiration, and dispersal, in all organisms of the human body. TGFβ plays a vital role in administrating the perpetuation of bone metabolic homeostasis [18]. TGFβ is discharged and enacted by resorbing osteoclasts and gives an anabolic effect on a bone by directly influencing the osteoblasts [3, 7]. Although TGFβ directly takes part in the emergence of osteoclasts during the latent state of osteoblasts, it hinders bone removal by diminishing the expression of RANKL on osteoblasts [23].
PTH employs typical actions on bone metabolism by triggering its receptor on target cells. Its main functioning includes calcium regulation. PTH has dual effects on bone turnover process. Constant exposure to PTH gives a negative feedback, whereas a sporadic contact to low quantity of PTH is coupled with positive effects. A number of studies have been conducted to investigate the effect of PTH drug in dysregulated bone remodeling and showed anabolic effects on bones.
Sclerostin, encrypted by SOST gene, is an extravasated glycoprotein expressed by osteocytes [4, 28, 29]. It is a Cterminal cysteine knotlike (CTCK) domain and has resemblance to the DAN family of bone morphogenetic protein (BMP) antagonists. Animal™ models have been found potentially valuable for investigating sclerostin for the reason that they have high conservancy over vertebrates (the percentages of the sequences of amino acids measured in the mouse and rate are 88 and 89, respectively, which are quite similar to human sequences) [4]. Binding of sclerostin to its coreceptors expressed on the surface of osteoblasts constrains the Wntβcatenin signaling [14] and, consequently, suppresses differentiation, production, and undertakings of osteoblastic cells [2]. Experimental study on SOST knockout mice showed high levels of bone mass phenotype [15], whereas the results were rather contrasting, i.e., a lowmass phenotype was observed in a transgenic mice with an overexpression of SOST gene [12]. Therefore, osteoblast movement is selfsynchronized by a negative feedback setup.
Poroelasticity is the study based on the relation among deformation and fluid flow in a fluid drenched malleable medium [6]. The main activities of bone fluid include the transportation of nutrients to cells engrossed in medium and moving waste of cells away. It also provides minerals to bone tissue for relocation. A mechanosensory system of bone and bone fluid are strongly interconnected. A mechanical stress plays a vital role in supplying the bone fluid. A mechanical strain supports the bone deformation process, which in turn produces the fluid flow [20, 30]. Bone cells also get affected through such a mechanism of mechanical strain; it helps in maintaining the bone remodeling process. Many bone diseases like osteoporosis have been examined by applying a continuous mechanical stress; such a stress gave anabolic effects on bone formation, whereas catabolic affects bone resorption [8]. A disparity among RANKL/OPG signaling equipoise gives rise to several bone pathological processes, mostly associated with ageing, called osteopenia, and with more extremity, osteoporosis. Osteoporosis is a bonerelated disorder regarded as skeletal infirmity with low bone mineral density (BMD), which subsequently causes recurrent microharms and impulsive cracks; it is a persistent syndrome demanding longterm cure. Middleaged women and elderly people are the prime targets of osteoporosis, and currently its influence is intensely rising socially and economically; WHO has declared it as the second foremost healthcare issue. In a regular bone remodeling process, RANKL/OPG is prudently balanced; the rise in RANKL plays a vital part in supporting resorption via osteoclast development, activities, and persistence. As the human body gets older, the density of the osteons expands, and the cortical sponginess and architectural damages of bone also rise due to the execution of a large number of remodeling cycles. This scenario initiates a vicious cycle where microharms occur often, consequently weakening bone configuration and increasing proportion of impulsive fractures [31]. Furthermore, latest analyses on bone remodeling propose that plasma levels OPG and RANKL are contrariwise linked to BMD and take part in the growth of osteoporosis in postmenopausal women [9].
Recently, many researchers have developed several mathematical models of bone remodeling to study the deep dynamics of intracellular and intercellular mechanisms of bone. Some of the models allow just the estimation of cell population dynamics and alteration in bone density; the first attempt in this regard was by Komarova et al. [11], whereas some of them deal with several bone diseases including the tight coupling between bone cells and their potential therapies based on their efficacy, first introduced by Lemaire et al. [13]. Komarova et al. [11] model depicts that the dynamical behavior of bone remodeling strictly depends on the regulation of bone resorbing cells by autocrine factors, and the osteoclast–osteoblast communication produces a complex nonlinear structure, which cannot be figured out from the analysis of each cell class alone. Many researchers extended the model of Komarova et al. afterward. Ayati et al. [1] developed a model to study myeloma bone disorder and demonstrated how therapeutic strategies might be examined through such computational systems. ChenCharpentier and Diakite [5] introduced delays in the model of Komarova et al. [11] and checked the stability and bifurcation. Ryser et al. [25] developed a mathematical model based on partial differential equations with time delays to simulate the dynamics of bone multicellular units. Their system illustrates the RANKL/OPG signaling pathway along with osteoclast–osteoblast communication. Lemaire et al. [13] proposed a mathematical framework comprising of cellular control and biochemical feedbacks systems responsible for the modulation of bone remodeling. Their model also incorporates simulation of the growth of several metabolic disorders such as osteoporosis, estrogen dearth, or vitamin D deficiency and determines possible treatments based on their effectiveness. Since RANK/RANKL/OPG signaling pathway is a main signaling cascade synchronizing bone turnover, practical insinuation of particular RANKL/OPG expression profiles on bone density were profoundly investigated by Pivonka et al. [21]. They developed a model comprising of the dynamics of bone cellular units, based on the work of Lemaire et al. [13] integrating the RANK/RANKL/OPG pathway along with the modulating effect of TGFÎ^{2} on bone cells. Pivonka et al. [22] for the first time introduced a PK model of drug denosumab along with a PD model of bone remodeling and analyzed the effect of denosumab on osteoporosis through this PK/PD model.
2 Mathematical models
2.1 Osteoporotic model
 1.
The number of osteoclasts (osteoblasts) includes the populations of osteoclasts (osteoblasts) precursors and osteoclasts (osteoblasts).
 2.
The death rate of bone resorbing and boneforming cells is proportional to the present populations of that particular cell.
 3.
As the ageing factor (\(a_{\mathrm{ageing}}\)) plays a crucial role in the development of osteoporosis and weakens the bones by destroying the bone cells, this effect has been included in bone removal and forming bone cells.
 4.
No external effects have been included in the population system.
Parametric values
Parameter values  

Parameter  Description  Value 
\(a_{1}\)  Cdifferentiation rate  3/day 
\(a_{2}\)  Bdifferentiation rate  4/day 
\(b_{1}\)  Capoptosis rate  0.2/day 
\(b_{2}\)  Bapoptosis rate  0.02/day 
\(\delta_{1}\)  paracrine effect of C onto B  0 
\(\delta_{2}\)  paracrine effect of B onto C  1 
\(s_{1}\)  bone removal rate  0.0748/day 
\(s_{2}\)  bone formation rate  0.0006395/day 
\(a_{\mathrm{ageing}}\)  ageing factor  2 
\(\delta_{OL}\)  RANKL effect  0.1 
C  steady level of C  (Control, Osteoporosis) = (1.16, 1.78) 
B  steady level of B  (Control, Osteoporosis) = (231.72, 177.91) 
\(C_{0}\)  initial value of C  (Control, Osteoporosis) = (11.16, 11.78) 
\(B_{0}\)  initial value of B  (Control, Osteoporosis) = (231.72, 177.91) 
\(Y_{0}\)  initial value of Y  (Control, Osteoporosis) = (1, 1) 
\(d_{1}\)  proportionality constant  0.00005/day 

\(\delta_{11}\), net effect of osteoclasts autocrine factors,

\(\delta_{21}\), osteoblastsreleased paracrine factors influencing osteoclasts,

\(\delta_{12}\), osteoclastsreleased paracrine factors influencing osteoblasts,

\(\delta_{22}\), net effect of osteoblasts autocrine factors.

\(C^{\delta_{11}}\propto C\) and \(B^{\delta_{22}}\propto B\)

\(\delta_{1}=\delta_{21}\) and \(\delta_{2}=\delta_{12}\), where \(\delta _{i}\), \(i=1,2\), represent the paracrine effects produced by RANKRANKLOPG signalling pathway and other factors including PTH and TGFβ. In addition, the effect of \(\delta_{1}\) is catabolic as it represses the concentration of osteoclasts, whereas \(\delta_{2}\) stimulates the production of osteoblasts and so produces an anabolic effect on osteoblasts population.
2.2 Theoretical investigation of the model
Theorem 1
A unique positive oscillatory solution of nonlinear system (4) exists subjected to positive initial conditions and the property\((\delta_{1}+\delta_{OL})\delta_{2}\leq 0\).
Proof
2.3 External agents influencing the osteoporotic bone turnover
2.3.1 Stability analysis of the model
Theorem
The nonlinear system (6) with the initial condition\((C_{0},B_{0})\), provided that\(C,B>0\)and\((\delta_{1}+\delta_{OL})\delta_{2}\geq0\), exhibits a positive solution, which shows affinity toward the stable equilibrium solution\(( C,B)\).
Proof
3 Numerical simulations
The inspiration behind this scheme is that we can now solve the normalized boundary value problem, although nonlinear, quite easily, using both analytic and numerical schemes. The most widely employed numerical method for the boundary value problems is the collocation method. The advantage of this method is that it reduces the nthorder differential equation(s) into n firstorder differential equations, thus reducing the computational cost on a large domain with small step size and a range of parameters. We have simplified the three systems using the generalized collocation method (GCM). Collocation methods are basically implicit Runge–Kutta quadrature techniques.
3.1 Conclusions and future work
In this paper, complex dynamics of bone diseases are studied with the aid of numerical experiments. The stability analysis is of great significance in the field of computational biology. It helps to validate the parametric values, the correct selection of thresholds values, which are useful during diseases diagnosis and therapeutics. In this paper, we have considered these factors with the aid of an extended model, where the external agent and its role as a therapeutic strategy are examined. In our future work, we aim to extend this study by interfacing it with clinical trials.
Notes
Acknowledgements
Not applicable.
Availability of data and materials
This paper contains no any studies with human participants or animals performed by any of the authors. All the data related to the current study were provided along with this paper.
Authors’ contributions
SJ conceived the paper and prepared materials and methods. MY did the analytic analysis. YB did numerical computations. AS did results and discussion. AbS did the literature review. All authors equally contributed in the final version of the manuscript. All authors read and approved the final manuscript.
Funding
Funding was received via project NRPU 5420 and NRPU 4250.
Competing interests
The authors declare that there is no conflict of interests.
References
 1.Ayati, B.P., Edwards, C.M., Webb, G.F., Wikswo, J.P.: A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease. Biol. Direct 5(1), 28 (2010) CrossRefGoogle Scholar
 2.Baron, R., Rawadi, G.: Targeting the wnt/βcatenin pathway to regulate bone formation in the adult skeleton. Endocrinology 148(6), 2635–2643 (2007) CrossRefGoogle Scholar
 3.Bonewald, L.F., Dallas, S.L.: Role of active and latent transforming growth factor β in bone formation. J. Cell. Biochem. 55(3), 350–357 (1994) CrossRefGoogle Scholar
 4.Brunkow, M.E., Gardner, J.C., Van Ness, J., Paeper, B.W., Kovacevich, B.R., Proll, S., Skonier, J.E., Zhao, L., Sabo, P.J., Fu, Y.H., et al.: Bone dysplasia sclerosteosis results from loss of the sost gene product, a novel cystine knotcontaining protein. Am. J. Hum. Genet. 68(3), 577–589 (2001) CrossRefGoogle Scholar
 5.ChenCharpentier, B.M., Diakite, I.: A mathematical model of bone remodeling with delays. J. Comput. Appl. Math. 291, 76–84 (2016) MathSciNetCrossRefGoogle Scholar
 6.Cowin, S.C.: Mechanosensory Mechanisms in Bone. Lecture Notes – ABIOMED (2001) Google Scholar
 7.Erlebacher, A., Filvaroff, E.H., Ye, J.Q., Derynck, R.: Osteoblastic responses to TGFβ during bone remodeling. Mol. Biol. Cell 9(7), 1903–1918 (1998) CrossRefGoogle Scholar
 8.Haba, Y., Lindner, T., Fritsche, A., Schiebenhöfer, A.K., Souffrant, R., Kluess, D., Skripitz, R., Mittelmeier, W., Bader, R.: Relationship between mechanical properties and bone mineral density of human femoral bone retrieved from patients with osteoarthritis. Open Orthop. J. 6, 458 (2012) CrossRefGoogle Scholar
 9.Jabbar, S., Drury, J., Fordham, J.N., Datta, H.K., Francis, R.M., Tuck, S.P.: Osteoprotegerin, rankl and bone turnover in postmenopausal osteoporosis. J. Clin. Pathol. 64(4), 354–357 (2011) CrossRefGoogle Scholar
 10.Jerez, S., Chen, B.: Stability analysis of a Komarova type model for the interactions of osteoblast and osteoclast cells during bone remodeling. Math. Biosci. 264, 29–37 (2015) MathSciNetCrossRefGoogle Scholar
 11.Komarova, S.V., Smith, R.J., Dixon, S.J., Sims, S.M., Wahl, L.M.: Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling. Bone 33(2), 206–215 (2003) CrossRefGoogle Scholar
 12.Kramer, I., Loots, G.G., Studer, A., Keller, H., Kneissel, M.: Parathyroid hormone (PTH)induced bone gain is blunted in sost overexpressing and deficient mice. J. Bone Miner. Res. 25(2), 178–189 (2010) CrossRefGoogle Scholar
 13.Lemaire, V., Tobin, F.L., Greller, L.D., Cho, C.R., Suva, L.J.: Modeling the interactions between osteoblast and osteoclast activities in bone remodeling. J. Theor. Biol. 229(3), 293–309 (2004) MathSciNetCrossRefGoogle Scholar
 14.Li, X., Zhang, Y., Kang, H., Liu, W., Liu, P., Zhang, J., Harris, S.E., Wu, D.: Sclerostin binds to LRP5/6 and antagonizes canonical wnt signaling. J. Biol. Chem. 280(20), 19883–19887 (2005) CrossRefGoogle Scholar
 15.Lin, C., Jiang, X., Dai, Z., Guo, X., Weng, T., Wang, J., Li, Y., Feng, G., Gao, X., He, L.: Sclerostin mediates bone response to mechanical unloading through antagonizing wnt/βcatenin signaling. J. Bone Miner. Res. 24(10), 1651–1661 (2009) CrossRefGoogle Scholar
 16.Liò, P., Paoletti, N., Ali Moni, M., Atwell, K., Merelli, E., Viceconti, M.: Modelling osteomyelitis. BMC Bioinform. 13(14), S12 (2012) CrossRefGoogle Scholar
 17.Hofbauer, L.C., Heufelder, A.E.: Role of receptor activator of nuclear factorκB ligand and osteoprotegerin in bone cell biology. J. Mol. Med. 79(5–6), 243–253 (2001) CrossRefGoogle Scholar
 18.łukasz, A., Poniatowski, Wojdasiewicz, P., Gasik, R., Szukiewicz, D.: Transforming growth factor beta family: insight into the role of growth factors in regulation of fracture healing biology and potential clinical applications. Mediators of Inflammation, 2015 (2015) Google Scholar
 19.Parfitt, A.M.: Osteonal and hemiosteonal remodeling: the spatial and temporal framework for signal traffic in adult human bone. J. Cell. Biochem. 55(3), 273–286 (1994) CrossRefGoogle Scholar
 20.Piert, M., Zittel, T.T., Becker, G.A., Jahn, M., Stahlschmidt, A., Maier, G., Machulla, H.J., Bares, R.: Assessment of porcine bone metabolism by dynamic [18F] fluoride ion PET: correlation with bone histomorphometry. J. Nucl. Med. 42(7), 1091–1100 (2001) Google Scholar
 21.Pivonka, P., Zimak, J., Smith, D.W., Gardiner, B.S., Dunstan, C.R., Sims, N.A., Martin, T.J., Mundy, G.R.: Model structure and control of bone remodeling: a theoretical study. Bone 43(2), 249–263 (2008) CrossRefGoogle Scholar
 22.Pivonka, P., Zimak, J., Smith, D.W., Gardiner, B.S., Dunstan, C.R., Sims, N.A., Martin, T.J., Mundy, G.R.: Theoretical investigation of the role of the RANK–RANKL–OPG system in bone remodeling. J. Theor. Biol. 262(2), 306–316 (2010) CrossRefGoogle Scholar
 23.Quinn, J.M.W., Itoh, K., Udagawa, N., Häusler, K., Yasuda, H., Shima, N., Mizuno, A., Higashio, K., Takahashi, N., Suda, T., et al.: Transforming growth factor β affects osteoclast differentiation via direct and indirect actions. J. Bone Miner. Res. 16(10), 1787–1794 (2001) CrossRefGoogle Scholar
 24.Raisz, L.G.: Physiology and pathophysiology of bone remodeling. Clin. Chem. 45(8), 1353–1358 (1999) Google Scholar
 25.Ryser, M.D., Komarova, S.V., Nigam, N.: The cellular dynamics of bone remodeling: a mathematical model. SIAM J. Appl. Math. 70(6), 1899–1921 (2010) MathSciNetCrossRefGoogle Scholar
 26.Tunc, C.: Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations. Appl. Comput. Math. 10(3), 449–462 (2011) MathSciNetzbMATHGoogle Scholar
 27.Tunç, C., Tunç, O.: A note on certain qualitative properties of a second order linear differential system. Appl. Math. Inf. Sci. 9(2), 953 (2015) MathSciNetzbMATHGoogle Scholar
 28.Van Bezooijen, R.L., Roelen, B.A.J., Visser, A., Van Der Weepals, L., De Wilt, E., Karperien, M., Hamersma, H., Papapoulos, S.E., Ten Dijke, P., Löwik, C.W.G.M.: Sclerostin is an osteocyteexpressed negative regulator of bone formation, but not a classical BMP antagonist. J. Exp. Med. 199(6), 805–814 (2004) CrossRefGoogle Scholar
 29.Veverka, V., Henry, A.J., Slocombe, P.M., Ventom, A., Mulloy, B., Muskett, F.W., Muzylak, M., Greenslade, K., Moore, A., Zhang, L., et al.: Characterization of the structural features and interactions of sclerostin molecular insight into a key regulator of wntmediated bone formation. J. Biol. Chem. 284(16), 10890–10900 (2009) CrossRefGoogle Scholar
 30.Weinbaum, S., Cowin, S.C., Zeng, Y.: A model for the excitation of osteocytes by mechanical loadinginduced bone fluid shear stresses. J. Biomech. 27(3), 339–360 (1994) CrossRefGoogle Scholar
 31.Whitfield, J.F.: Growing Bone. Landes Bioscience (2007) Google Scholar
Copyright information
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.