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An impulsive mathematical model of bone formation and resorption: effects of parathyroid hormone, calcitonin and impulsive estrogen supplement
- Inthira Chaiya^{1} and
- Chontita Rattanakul^{1, 2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1206-2
© The Author(s) 2017
- Received: 10 January 2017
- Accepted: 12 May 2017
- Published: 30 May 2017
Abstract
In this paper, we propose an impulsive mathematical model of bone formation and resorption accounting for the number of active osteoclastic cells, bone resorbing cells, and the number of active osteoblastic cells, bone forming cells, based on the effects of parathyroid hormone and calcitonin with impulsive estrogen supplement. The model is then analyzed theoretically in terms of its stability, permanence and oscillatory behavior. The conditions on the model parameters, for which the desirable behaviors of the solution of the system can be expected, are derived. Numerical simulations are also carried out in order to support our theoretical predictions. The results indicate that the frequency and dosage of the estrogen supplements are important since the behavior of the solution of the system depends on the frequency and dosage of the estrogen supplements.
Keywords
- bone remodeling process
- calcitonin
- parathyroid hormone
- estrogen
- impulsive treatment
- mathematical model
1 Introduction
Bone is a highly organized tissue. Apart from attending to its own structural integrity, the skeleton must also respond to systemic needs for adequate amounts of calcium in the extracellular fluid. In order to maintain normal functions of bone and calcium homeostasis, bone remodeling process is needed to be taken into account. In bone remodeling process, osteoclasts act as bone resorbing cells, while osteoblasts act as bone forming cells. The process begins with the appearance of active osteoclasts on a previously inactive surface of bone, after which a lacuna will be excavated. Subsequently, osteoblasts then refill the resorption cavity and become inactive osteoblasts [1–3]. Serious health conditions such as osteoporosis may occur when the bone remodeling process is imbalanced (e.g. there is excessively deep resorption space produced by osteoclasts, or the replenishment of the resorption space by the activation of osteoblasts is incomplete). For a better understanding of bone remodeling process, the basic knowledge about osteoblasts, osteoclasts and the involving hormones such as parathyroid hormone (PTH) and calcitonin (CT) is necessary.
Osteoblasts derived from mesenchymal cells are responsible for bone formation. The stromal stem cells proliferate preosteoblast precursors, and then preosteoblasts precursors proliferate preosteoblasts. After that, preosteoblasts differentiate into osteoblasts, and then osteoblasts become osteocytes or resting osteoblasts [4]. Note that the differentiation and activation of osteoclasts require direct physical contact with osteoblasts that produce at least two indispensable cytokines [1]. On the other hand, osteoclasts derived from the hemopoietic cells are responsible for bone resorption. The hemopoietic stem cells proliferate preosteoclast precursor cells, and then preosteoclasts precursors differentiate into preosteoclasts. After that, preosteoclasts differentiate into osteoclasts [4].
PTH and CT are principal hormones involved in bone formation and resorption. PTH is secreted from the parathyroid glands in response to the low calcium level in blood. It increases calcium concentration in blood by various direct and indirect actions on bone, kidney and intestine [1]. On bone, PTH enhances bone resorption by stimulating the activation of osteoclasts indirectly through osteoblasts since osteoclasts do not possess PTH receptors while osteoblast precursors possess them. Moreover, PTH has both stimulating and inhibiting effects on the differentiation and activation of osteoblasts [1]. PTH stimulates the differentiation of mesenchymal stem cells to preosteoblasts, but it inhibits the differentiation of osteoblasts in mature cells through the process of down-regulation of PTH receptors on osteoblasts [5–7]. In contrast to PTH, CT produced by the parafollicular cells of the thyroid gland inhibits bone resorption by acting on osteoclasts resulting in the decrease of calcium level in blood [1, 2].
As discussed above, an excessively deep resorption space produced by osteoclasts or an incomplete replenishment of the resorption space by the activation of osteoblasts can result in bone remodeling imbalance. If remodeling imbalance exists after the completion of a remodeling cycle, the degree of bone loss will be exacerbated and that leads to osteoporosis [8]. Osteoporosis is a bone metabolic disease which is characterized by low bone mass, the structural deterioration of bone and an increased risk of fracture. It occurs most frequently in postmenopausal women [3]. An increase in the activation frequency of new bone remodeling units and an increase in remodeling imbalance, especially resulting from the increase of osteoclastic formation, are observed when estrogen is deficient [8–12]. In osteoporosis patients, estrogen replacement therapy has been widely used to prevent menopausal bone loss and reduce the risk of fracture [8–12]. Kanatani et al. [10] and Riggs et al. [12] reported that the presence of estrogen results in the decrease of bone resorption by inhibiting the activity of osteoclasts. The works of Prestwood et al. [9] and Albright et al. [11] indicated the decrease in the values of biochemical markers of bone turnover due to the short-term estrogen treatment. However, there are some serious risks and side effects from the estrogen replacement therapy such as breast cancer and heart disease [8, 13]. High doses of estrogen results in weight loss in rats, and an increase in tumor formation was observed in aging rats with long-term treatment of estrogen [14]. Hence, the appropriate dose and duration of estrogen treatment are necessary and need to be investigated in details.
Mathematical models of bone remodeling process were proposed and analyzed theoretically and numerically [15–20]. However, the model that incorporates the effects of PTH, CT and the impulsive treatments of estrogen has not been proposed and analyzed yet. In the next section, we will develop a system of impulsive differential equations to study the effects of PTH, CT and the impulsive estrogen replacement therapy on bone remodeling process.
2 Model development
Here, the rate of change of PTH concentration in blood is represented by (1a). The first term on the right-hand side accounts for the secretion rate of PTH from the parathyroid glands which is inhibited by the increase in the calcium levels indicated by the increase in the number of active osteoclasts [1].
The rate of change of CT concentration in blood is represented by (1b). The elevation of calcium levels indicated by the increase in the number of active osteoclasts stimulates the secretion of CT from the thyroid gland, whereas the increase in the level of PTH suppresses the secretion of CT as reported in [34]. Hence, the secretion rate of CT is then assumed to be represented by the first term on the right-hand side of (1b).
The rate of change of the number of active osteoclasts is represented by (1c). The first term on the right-hand side accounts for the stimulating effect of PTH and the inhibiting effect of CT on the reproduction of active osteoclasts which requires the cell to cell interaction of osteoclasts and osteoblasts as indicated in [20].
The rate of change of the number of active osteoblasts is represented by (1d). The first and the second terms on the right-hand side account for the stimulating effect and the inhibiting effect of PTH on the reproduction and differentiation of active osteoblasts as mentioned in [20], respectively.
The last terms of (1a)-(1d) account for the removal rates of PTH, CT, active osteoclasts and active osteoblasts, respectively.
The inhibition effect of estrogen treatment on the number of active osteoclasts is represented by (1e), while the stimulating effect of estrogen treatment on the number of active osteoblasts is represented by (1f).
3 Preliminaries
Definition 1
- (a)
V is continuous in \((nT,(n+1)T]\times R_{+}^{2}\to{R_{+}}\) and for each \(S\in R_{+}^{2},n\in{Z_{+}}\), \(\lim_{(t,Y)\to(nT^{+},S)} V(t,Y)=V(nT^{+},S)\) exists, and
- (b)
V is locally Lipschitzian in S.
Definition 2
In what follows, we assume that the solution of (4a)-(4d), \(S(t)=(z(t),w(t))\), is a piecewise continuous function. That is, \(S(t):{R_{+}}\to{R}_{+}^{2}\), \(S(t)\) is continuous on \((nT,(n+1)T],n\in{Z_{+}}\) and \(\lim_{t\to nT^{+}}S(t)=S(nT^{+})\) exists. Then the global existence and uniqueness of solution to (4a)-(4d) is guaranteed by the smoothness properties of F (see [37] for more details).
Since \(\frac{dz}{dt}=0\) whenever \(z(t)=0, t\ne nT,\frac {dw}{dt}>0\) whenever \(w(t)=0, t\ne nT\) and \(z(n{{T}^{+}})=(1-\rho )z(nT), 0<\rho<1\), \(w(n{{T}^{+}})=w(n{{T}})+\mu, \mu>0\), we have the following lemma.
Lemma 1
Suppose \(S(t)=(z(t),w(t))\) is a solution of (4a)-(4d) with \(S({{0}^{+}})\ge0\). Then \(S(t)\ge0\) for all \(t\ge0\).
Lemma 2
Proof
We let \(v(t)=z(t)+w(t)\), \({M_{1}}=\sup z{f_{1}}(z)=\frac{a_{1}}{b_{1}}\) and \({M_{2}}=\sup{f_{1}}(z)=\frac{a_{1}}{{b_{1}}{k_{1}}}\).
4 Stability when there is no active osteoclast
Lemma 3
System (6)-(8) has a positive periodic solution \(\tilde{w}(t)\), and for every solution \(w(t)\) of (6)-(8), we have \(w(t)\to\tilde{w}(t)\) as \(t\to\infty \).
Theorem 1
Proof
5 Permanence of the system
Definition 3
Theorem 2
Proof
Suppose that \(S(t)=(z(t),w(t))\) is a solution of system (4a)-(4d) with \(z(0^{+})>0\) and \(w(0^{+})>0\). Since (5) holds, Lemma 2 implies that there is a constant \(M>0\) such that, for sufficiently large t, \(z(t)\le M\) and \(w(t)\le M\).
Step 1. We will prove by contradiction that there exists \(t_{1}\) such that \(z(t_{1})\ge{m_{3}}\).
The comparison theorem [37] then implies that \(w(t)\geq P(t)\).
Step 2. If \(z(t)\ge{m_{3}}\) for all \(t>{t_{1}}\), then the proof is complete. Otherwise, \(z(t)<{m_{3}}\) for some \(t>{t_{1}}\). Letting \(t^{*}=\inf_{t>t_{1}}\{z(t)< m_{3}\}\). There are two possible subcases.
Case 1: \({t^{*}}={n_{1}}T\) for some \({n_{1}}\in{Z_{+}}\). This means \(z(t)\ge {m_{3}}\) for \(t\in(t_{1},t^{*}]\) and, by the continuity of \(z(t)\), we have \(z(t^{*})={m_{3}}\).
Case 2: \(t^{*}\ne nT\) for all \(n\in{Z_{+}}\). This means \(z(t)\ge{m_{3}}\) for \(t\in[t_{1},t^{*})\) and \(z(t^{*})={m_{3}}\). Suppose \(t^{*} \in ({n'_{1}}T,({n'_{1}}+1)T)\) for some \({n'_{1}}\in{Z_{+}}\). There are two possible subcases.
For \(t>\underline{t}\), we can continue in the same way since \(z(\underline{t})\ge{m_{3}}\). Since \({m_{2}}<{\bar{m}_{2}}<{m_{3}}\), we have \(z(t)\ge{m_{2}}\) for \(t\ge{t_{1}}\). The proof is complete. □
6 Existence and stability of positive periodic solution
7 Numerical simulations
In this section, numerical simulations are carried out in order to support our theoretical predictions.
8 Conclusion
We have developed an impulsive system of bone remodeling process in order to investigate the effect of estrogen supplements. The model is then analyzed theoretically so that the conditions on the system parameters, for which the desirable behaviors of the solution of the system can be expected, are derived. We can see that the frequency of estrogen treatments \(\frac{1}{T}\), the dosages of estrogen supplements indicated by μ and ρ play important roles in controlling the number of active osteoclasts and active osteoblasts, the different frequencies and dosages of estrogen supplements might lead to the undesirable behavior of the system. For example, the solution of the system might be unbounded, which means that the number of active osteoclasts, bone resorbing cells, might increase to very high levels or the number of active osteoblasts, bone forming cells, decreases to very low levels resulting in the net bone resorption instead of bone formation, which is the expected outcome of the estrogen supplements.
Moreover, a condition needed to guarantee the existence of positive oscillation in the number of active osteoblasts and active osteoclasts resembling clinical observation stated in Theorem 3 is \(T>T_{\mathrm{max}}\), where \(T_{\mathrm{max}}\) depends on μ and ρ. Hence, if the dosage of estrogen supplement reflected by μ and ρ is fixed, an appropriate frequency of estrogen supplement \(\frac{1}{T}\) can be chosen so that \(T>T_{\mathrm{max}}\) as required by Theorem 3 to guarantee the desirable levels of active osteoblasts and active osteoclasts. Therefore, the dosage and the frequency of estrogen supplement are the keys to the effectiveness of estrogen supplement in osteoporosis patients.
Declarations
Acknowledgements
This work was supported by the Royal Golden Jubilee Ph.D. Program (contract number PHD53K0191), Centre of Excellence in Mathematics, Commission on Higher Education, Thailand and Mahidol University, Thailand.
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.
Authors’ Affiliations
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