Dynamical Modelling of Bone Formation and Resorption under Impulsive Estrogen Supplement: Effects of Parathyroid Hormone and Prolactin

Osteoporosis, a bone metabolic disease, is one of the major diseases occurring in aging population especially in postmenopausal women. A system of impulsive differential equations is developed in this paper in order to investigate the effects of parathyroid hormone and prolactin on bone-forming cells, namely, osteoblasts, and bone-resorbing cells, namely, osteoclasts, under the impulsive estrogen supplement. %e theoretical analysis of the developed model is carried out so that we obtain the conditions on the system parameters in which the stability and permanence of the model can occur. Computer simulations are also provided to illustrate the theoretical predictions.


Introduction
Osteoporosis is a disease which shows no symptoms until there is a bone fracture incidence [1]. It is characterized by an impairment of bone mass, measured by a Bone Mineral Density (BMD) test as the T-score is less than or equal to 2.5, and microarchitectural deterioration of bone tissue [2][3][4][5].
is disease causes a major public health issue for all races and both sexes [1]. Since BMD decreases as the population ages, osteoporosis occurs prevalently in menopause woman and elderly men between 75 and 80 years of age [6]. More than 200 million people worldwide were approximated to be osteoporotic patients with hip fractures [7]. In Europe and the United States, a report revealed that 30% of women are diagnosed with osteoporosis [8]. It was reported in ailand that 19.8% of postmenopausal women are prevalently osteoporotic [9]. Furthermore, approximately 40% of women who experience menopause and 30% of men might have an osteoporosis-related fracture in the remaining lifetime [10].
Bone is a living tissue [11]. ere is a process in its place to resorb old bone and form new bone to maintain its strength and health, called bone-remodeling process, and such a process occurs throughout a person's life [10,12,13]. In osteoporosis patients, the loss of bone mass occurs because the removal of older bone is more than the replacement of new bone which is an imbalance between bone resorption and bone formation [10]. ere are two families of cells mainly involved in the process of bone remodeling, namely, osteoclastic cells, breaking down the bone cells, and osteoblastic cells, forming the bone tissue [12,13]. Boneremodeling process is composed of four phases sequentially, which are activation, resorption, reversal, and formation [12]. In this process, osteoblast lineage cells play a crucial role in the activation stage by taking action on blood cell precursors or hematopoietic cells to establish osteoclastic resorption of bones. Under a layer of lining cells, the removal of bone is done by osteoclasts in the resorption step. e mineral here is dissolved, and the bone matrix is broken down. In the reversal phase, the surface of the resorbed bone creates a thin layer of protein to make it ready for the following stage. Finally, in the formation step, osteoblasts acting in successive waves start to form up new bones by laying down multiple layers of matrix in an orderly arrangement. en, some of these osteoblasts stay inside the bone and become osteocytes, remaining in contact to adjacent osteocytes and osteoblasts on the bone surface. is process is successfully finished in 3-4 months, while the first three stages take 2 to 3 weeks to finish [12]. ere are many factors involved in bone-remodeling process such as parathyroid hormone (PTH), prolactin (PRL), calcitonin, and estrogen.
PTH plays an important role in the bone-remodeling process [14]. It is released from the parathyroid gland, and both stimulating and inhibiting effects have been reported on the development of osteoblasts from its progenitors [14][15][16][17][18]. On the one hand, PTH indirectly enhances the development of osteoclasts from its progenitors by operating through osteoblasts since osteoclasts lack PTH receptors while preosteoblast precursors and preosteoblasts possess them [14][15][16][17][18]. On the other hand, PRL is a polypeptide hormone synthesized in and secreted from specialized cells of the anterior pituitary gland, the lactotrophs [19]. According to [20], PRL receptors have been reported on osteoblasts, and hence, PRL then has effects on boneremodeling process as well. PRL enhances bone resorption by increasing receptor activator of NF-ligand (RANKL) and decreasing osteoprotegerin (OPG) expressions by osteoblastic cells [21].
Bone mass reduces significantly due to the declining secretion of estrogen during and after the menopause years in women [22]. Estrogen deficiency in postmenopausal women leads to accelerating resorption of bone by osteoclasts, and osteoporosis will then occur [23]. e direct and indirect studies indicated that estrogen also plays a key role on skeletal health in men [24]. Several hormones play an important role in the process of bone remodeling and also in the treatments of osteoporotic patients. Estrogen replacement therapy has a beneficial effect on postmenopausal women in preventing the loss of bone as shown in clinical studies [25,26]. In a long-term study of following up postmenopausal women who take estrogen replacement therapy, in vivo study results demonstrated that replacement estrogen therapy can prevent or slow down osteoporosis in postmenopausal patients compared to an unsupplemented control group [27]. A recent research using cancellous rats reported that decreased estrogen levels led to bone changes which affect flow of interstitial around osteocytes [28]. Many studies have shown that estrogen impacts positively in improving bone mineral density, and taking lower doses of estrogen is shown to be effective and cause fewer side effects. At the cellular level, estrogen increases the level of OPG which is able to bind with RANKL to block the differentiation, the activity, and the survival of osteoclasts.
Since bone is an alive and dynamic tissue [11], many researchers have been trying to describe the process of bone remodeling inside our bodies using several types of mathematical models. Chudtong et al. [29] introduced an impulsive system to describe bone-remodeling process involving PTH supplements by extending Rattanakul et al.'s model in [30]. Rattanakul et al. [31] constructed a mathematical model as a system of nonlinear differential equations to investigate bone formation and resorption process based on the effect of calcitonin. Motivated by [31], Panitsupakamon and Rattanakul [32] modified the model proposed in [31] by incorporating the time delay observed in the bone-remodeling process. e influence of estrogen supplement is also considered by adding a term to the dynamics of the active osteoclast population in the modified model. Chaiya and Rattanakul [33] proposed an impulsive mathematical model of bone-remodeling process incorporating the effects of prolactin and the impulsive control strategies of parathyroid hormone supplement on osteoblasts and osteoclasts. e dosage of parathyroid hormone can be added appropriately to the system to ensure the suitable levels of active osteoblasts and active osteoclasts.
ey also proposed another model accounting for the effects of PTH and calcitonin on bone-remodeling process together with the effects of impulsive treatments of estrogen in [34]. However, a mathematical model of bone resorption and bone formation consisting of osteoblastic cells, osteoclastic cells, parathyroid hormone, and prolactin with the effect of impulsive treatment of estrogen has never been established.

Model Development
In this section, we propose the following impulsive differential equation model to investigate the dynamics of boneforming cells and bone-resorbing cells based on the effects of PTH, PRL, and estrogen supplements: with where In what follows, x(t), y(t), z(t), and w(t) account for the level of PTH in blood at time t, the level of PRL in blood at time t, the number of active osteoclasts at time t, and the number of active osteoblasts at time t, respectively. T accounts for the period between each impulsive estrogen treatment, n ∈ Z + � 1, 2, 3, . . . { }, ρ accounts for the inhibiting effect of estrogen treatment on osteoclasts, 0 < ρ < 1, and μ accounts for the stimulating effect on osteoblasts of estrogen treatment, μ > 0. In addition, all parameters in (1) and (2) are assumed to be positive.
PTH secretion from the parathyroid gland is controlled by the calcium level in blood. Once the number of active osteoclasts increases, blood calcium levels then increase due to the increase in bone resorption resulting in the decrease of PTH secretion in order to maintain calcium level within the normal range [35,36]. Hence, we then assumed that the rate of change of the concentration of PTH in blood at time t denoted by dx/dt in equation (1) varies inversely with the number of active osteoclasts as shown in the first term on the right-hand side of the first equation in equation (1). According to [37], an increase in the level of plasma PRL also enhances the release of PTH from the parathyroid gland and hence the second term on the right-hand side denotes the stimulating effect of PRL on PTH. e last term stands for the removal rate of PTH from the system. However, PRL stimulates osteoclastic differentiation by decreasing OPG and increasing RANKL expressions by osteoblastic cells [21]. RANKL then binds to RANK, its receptor on preosteoclast, and stimulates the differentiation of osteoclasts. To balance the bone-remodeling cycle, when the number of osteoclasts increases, the secretion of PRL is then decreased. In the second equation of (1), the rate of change of the level of PRL in blood denoted by dy/dt is then assumed to vary inversely with the number of active osteoclasts as shown in the first term on the right-hand side. On the other hand, the increase in the number of osteoclasts leads to the increase in the calcium level in blood and then the secretion of PTH from the parathyroid gland, as well as the secretion of PRL from the anterior pituitary gland, which also enhances the secretion of PTH will be decreased in order to maintain the calcium level in blood within the normal range [35,36]. en, the second term on the righthand side stands for the stimulating effect of PTH on the release of PRL from the anterior pituitary gland. e last term stands for the removal rate of PRL from the system.
Osteoclasts lack PTH and PRL receptors, whereas osteoblasts possess them. erefore, the effects of PTH and PRL on the differentiation of osteoclasts are both indirect effects. However, it has been reported in [17] that the differentiation of active osteoclasts also requires the production of osteoclast differentiation factor (ODF) and its receptor on osteoclasts as well and hence the differentiation of active osteoclasts requires the presence of both osteoblastic and osteoclastic cells. Even though PTH has the stimulating effects on the differentiation of active osteoclasts as reported in [18,38], it has also been observed clinically in [17] that PTH inhibits the differentiation of active osteoclasts when the level of PTH increases further as well. On the other hand, PRL enhances bone resorption by decreasing OPG and increasing RANKL expressions by osteoblastic cells [21]. RANKL then binds to RANK, its receptor on preosteoclast, and stimulates the differentiation of osteoclasts [21]. us, the rate of change of the number of active osteoclasts denoted by dz/dt is then assumed to depend on the stimulating and inhibiting effects of PTH and PRL as presented on the first term on the right-hand side of the third equation in (1). e last term stands for the removal rate of active osteoclasts.
Osteoblasts possess both PTH receptors and PRL receptors, and hence, the effects of PTH and PRL on the differentiation of osteoblasts are both direct effects. PTH prevents a suicidal process called apoptosis of osteoblasts [39,40]. In addition, both stimulating and inhibiting effects of PTH on osteoblastic differentiation process have been clinically observed depending on the differentiation stages [14]. As for the stimulating effect of PTH on the differentiation of osteoblasts, we assumed that the rate of change of the number of active osteoblasts denoted by dw/dt varies directly with the level of PTH as presented on the first term on the right-hand side of the fourth equation of (1), whereas the inhibiting effect of PTH on the differentiation of osteoblasts is presented on the second term on the right-hand side. On the other hand, it has been reported in [41] that the osteoblast number (DNA content) is declined significantly due to PRL, resulting in a decrease in osteoblast proliferation, and hence, the third term on the right-hand side is assumed to represent the inhibiting effect of PRL on osteoblasts. e last term represents the removal rate of active osteoblasts. e first equation of (2) accounts for the inhibitory effect of impulsive treatment with estrogen supplement on the number of active osteoclasts. e second equation accounts for the simulative effect of impulsive treatment with estrogen supplement on the number of active osteoblasts.
Several clinical observations indicate that the dynamics of PTH and PRL are so much faster than the dynamics of osteoclasts and osteoblasts [35,36,42,43]. In what follows, PTH and PRL are then assumed to equilibrate rapidly to their equilibrium for which dx/dt � 0 and dy/dt � 0, respectively. at is, and where us, the reduced system of (1) and (2) can be obtained as follows: International Journal of Mathematics and Mathematical Sciences with

Stability Analysis
Now, we let where Let us denote the map defined by the right-hand side of (5) and exists and is finite for each S ∈ R 2 + , n ∈ Z + , and V being locally Lipschitzian in S, then the upper right derivative of U(t, S) with respect to (5) and (6) is defined as where F � (F 1 , F 2 ). e solution S(t) � (z(t), w(t)) to systems (5) and (6) is assumed to be a piecewise continuous function which means that S(t) is continuous on (nT, (n + 1)T] for n ∈ Z + and lim t⟶nT + S(t) � S(nT + ) exists. According to the smoothness properties of F [44], the solution to systems (5) and (6) exists and is unique.  (5) and (6), (5) and (6) and en, for sufficiently large t, there exists a positive con- For t ≠ nT, it follows that International Journal of Mathematics and Mathematical Sciences By applying Lemma 2.2 in [44], it can be derived that erefore, u(t) ≤ M as t ⟶ ∞ and u(t) is uniformly ultimately bounded, which means that when t is large enough with z(t) ≤ M and w(t) ≤ M for some positive constant M.

Lemma 3. System (14) has a positive periodic solution w(t).
In addition, the solution w(t) of (14) converges to w(t) as t ⟶ ∞.
e local stability of the solution (0, w(t)) may be determined by considering the behavior of small amplitude perturbations of the solution.
Here, we let It follows that International Journal of Mathematics and Mathematical Sciences where Φ(t) is the fundamental solution matrix, in which and Φ(0) is the identity matrix I. us, we obtain the fundamental solution matrix as follows: Note that the terms * and * * are not required in further analysis and their exact expressions are not necessary.
Linearization of (6) provides Next, the Floquet theory is then applied to guarantee the local stability of the solution (0, w(t)).
Let us consider e solution (0, w(t)) is locally asymptotically stable if the magnitudes of λ 1 and λ 2 and the eigenvalue of M are both less than 1.

Proof. Let S(t) � (z(t), w(t)) be any solution of
Consider the second equation of (5); we can see that It follows that for sufficiently large t, there exists ε > 0 such that erefore, we obtain when t is large enough. Next, we will show that there is a positive constant m 2 for which z(t) > m 2 . Firstly, for some m 3 > 0, we let Secondly, we will do the following two steps.

□
Step 1. We will show that z(t 1 ) ≥ m 3 for some t 1 > 0 by contradiction. Suppose that z(t) < m 3 for all positive values of t. From the second equation of (5) and (6), we can see that Let us consider the following comparison system: 6 International Journal of Mathematics and Mathematical Sciences A periodic solution of (29) can then be obtained as follows: with P(0 + ) ≡ μ/1 − e − CT + B/C. erefore, the positive solution of (29) is From the first equation of reduced system (5), we have According to the comparison theorem [45], we obtain w(t) ≥ P(t). It follows that for some T 1 > 0 and for ε 1 > 0 which is small enough. Hence, Suppose that n ∈ Z + and NT ≥ T 1 . en, by integration over (nT, (n + 1)T], n ≥ N, we obtain We can see that en, for a positive constant ε 1 which is sufficiently small, we have erefore, if (19) and (30) hold, then a small positive constant m 3 can be chosen so that ln η > 0, and hence η > 1. It follows that z((n + k)T) ≥ z(nT)η k ⟶ ∞ when k ⟶ ∞ which contradicts the boundedness of z(t). Hence, there exists t 1 > 0 such that z(t 1 ) ≥ m 3 .
Step 2. If z(t) ≥ m 3 for all t ≥ t 1 , then the proof is complete; otherwise, there exists t > t 1 such that z(t) < m 3 and we then let t * � inf t>t 1 z(t) < m 3 , so that the following two possible cases are obtained.

Case 1.
ere exists n 1 ∈ Z + , such that t * � n 1 T. It follows that for t ∈ (t 1 , t * ], z(t) ≥ m 3 , and we obtain z(t) � m 3 by the continuity of z(t). When t is large enough, z(t) and w(t) are both bounded above by a positive constant M and w(t) is also bounded below by a positive constant m 1 which imply that we can choose positive constants M ′ and m ′ for which z(t) < M ′ and m 1 ′ < w(t) < M ′ such that We also choose n 2 , n 3 ∈ Z + that satisfy the following conditions: and where Here, we let T ′ � n 2 T + n 3 T. We claim that there exists a constant t 2 ∈ (t * , t * + T ′ ] in which z(t 2 ) > m 3 . Otherwise, by considering (38) with P(t * + ) � w(t * + ), for t ∈ (nT, (n + 1)T] and n 1 ≤ n ≤ n 1 + n 2 + n 3 , we obtain International Journal of Mathematics and Mathematical Sciences (51) It follows that Similar to Step 1, we then obtain According to the first equation of (5), By integrating (54) over [t * , t * + n 2 T], it follows that erefore, we obtain which is a contradiction, and hence, z(t 2 ) > m 3 for some t 2 ∈ (t * , t * + T ′ ]. Next, we let t � inf t>t * z(t) > m 3 . It follows that z(t) < m 3 for t ∈ (t * , t), and we can obtain z(t) � m 3 by the continuity of z(t).
en, l ∈ Z + is chosen where l ≤ n 2 + n 3 and t * + lT ≥ t. Suppose t ∈ (t * + (l − 1)T, t * + lT]. From (54), we obtain Since η 1 is negative and l < n 2 + n 3 , we let so that z(t) ≥ m 2 for (t * , t). We can use t instead of t * , the proof can be proceeded in the same manner and consequently, we will obtain z(t) ≥ m 2 for sufficiently large t.

Periodic Solution: Existence and Stability
To investigate the possibility of a bifurcation of the nontrivial periodic solution to systems (5) and (6) near (0, w(t)), we first interchange the variables of z and w for convenience on computation. In what follows, let us consider the following system instead: with Δz(t) � μ, Let Note that According to Lakmeche and Ario [45], the following notations are used: International Journal of Mathematics and Mathematical Sciences zΦ 1 We can see that I > 0 if while J < 0 provided that (19) and hold where K 0 � a 7 f 1 ′ (0), K 1 � − (a 8 k 5 f 1 ′ (0)/(k 5 + f 1 (0)) 2 ) and K 2 � − (a 9 k 6 f 2 ′ (0)/(k 6 + f 2 (0)) 2 ). is leads to the following theorem according to Lakmeche and Ario [45].

Numerical Simulations
In this section, numerical simulations are performed to illustrate the theoretical predictions as proved in the previous section.
To illustrate the theoretical prediction in eorem 1, the parameters in systems (5) and (6) Figure 1. We can see that the solution of systems (5) and (6) converges asymptotically to the oscillating solution (0, w(t)) as time passes for which the active osteoblasts are vanished as predicted in eorem 1.
To illustrate the theoretical prediction in eorem 3, the parameters in systems (5) and (6)  e computer simulation is shown in Figure 3. We can see that the solution of systems (5) and (6) exhibits sustained oscillations as predicted in eorem 3.
As shown in Figure 4, we can see that the period between each estrogen administration T is significant. e smaller value of T yields the better result on the number of osteoclasts and osteoblasts.

Conclusion
In this paper, an impulsive mathematical model of the process of bone remodeling accounted for bone-resorbing cells, osteoclasts, and bone-forming cells, osteoblasts, is developed in order to investigate the effect of impulsive estrogen supplement. e effects of parathyroid hormone and prolactin are also taken into account. We then apply the Floquet theory and the comparison theorem to derive the conditions in which the periodic solution is locally asymptotically stable. Moreover, the permanence of the system is also investigated as well so that we arrived at the conditions for which the sustained oscillation of the solution is guaranteed. In addition, computer simulations are presented to illustrate the theoretical predictions. e results indicate that the dosage of estrogen supplement indicated by μ and ρ and the frequency of estrogen supplement indicated by 1/T play important roles in the treatment of osteoporosis patients. Even though the smaller value of T yields the better result on the number of osteoclasts and osteoblasts, the side effects of estrogen administration are also needed to take into account. erefore, the appropriate dosage and frequency of estrogen supplement that could control the number of bone-forming cells and bone-resorbing cells to lie within the desirable range might lead to an efficient treatment in osteoporosis patients.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.