Abstract—Calcium is essential for human. Apart from providing

skeletal strength, calcium also plays important role in a wide range ofbiological functions. In this paper, calcium homeostasis, themechanism that maintains the serum calcium level to be in the normalrange, is investigated mathematically. A mathematical model isformulated to incorporate the effects of parathyroid hormone, vitaminD and time delay on calcium homeostasis. The conditions on themodel parameters for which a periodic solution exists are thenderived by means of Hopf bifurcation theorem. Moreover, variouskinds of dynamic behavior of the model are also investigatednumerically.Keywords—Calcium homeostasis, parathyroid hormone, time

delay, vitamin D.

I. INTRODUCTION

VER 99% of total body of calcium is stored in bone [1]-[5]. Calcium is essential for many mechanisms in human

body. It plays a key role in the regulation of enzymaticactivities and fundamental cellular events including thecontraction of muscles, hormone secretion, cell division andblood clotting [1]-[5]. Serum calcium levels are regulated bythree main mechanisms which are bone turnover, intestinal

This work was supported by the Centre of Excellence in Mathematics,Commission on Higher Education, Thailand and the Royal Golden JubileePh.D. Program (contract number PHD53K0191).

I. Chaiya is with the Department of Mathematics, Faculty of Science,Mahidol University, Thailand and the Centre of Excellence in Mathematics,the Commission on Higher Education, Thailand (e-mail:g5438789@student.mahidol.ac.th).

C. Rattanakul is with the Department of Mathematics, Faculty of Science,Mahidol University, Thailand and the Centre of Excellence in Mathematics,the Commission on Higher Education, Thailand (corresponding author,phone: 662-201-5340; fax: 662-201-5343; e-mail: chontita.rat@mahidol.ac.th).

S. Rattanamongkonkul is with the Department of Mathematics, Faculty ofScience, Burapha University, Thailand and the Centre of Excellence inMathematics, the Commission on Higher Education, Thailand (e-mail:sahattay@buu.ac.th).

W. Panitsupakamon is with the Department of Mathematics, Faculty ofScience, Silpakorn University, Thailand and the Centre of Excellence inMathematics, the Commission on Higher Education, Thailand (e-mail:wannapa@su.ac.th).

S. Ruktamatakul is with the Department of Mathematics, Faculty ofLiberal Arts Science, Kasetsart University, Thailand and the Centre ofExcellence in Mathematics, the Commission on Higher Education, Thailand(e-mail: faasspr@nontri.ku.ac.th).

absorption and renal reabsorption [2]. Parathyroid hormone(PTH) and vitamin D are two major regulators that areresponsible for normal calcium homeostasis [1]-[5].

The increase in serum level of PTH leads to the increase inthe mobilization of calcium from bone matrix through thestimulating effect on the osteoclastic activity [1]. The initialphase can be seen within 1-2 hours and more pronouncedphase becomes evident after about 12 hours [1]. PTH alsoincreases the serum level of calcium by acting on the kidneythrough the promoting of the reabsorption of calcium in thedistal nephron and the promoting of the conversion of vitaminD into its active form which results in the increase in theintestinal uptake of calcium [1].

On the other hand, vitamin D in its active form mediates itsbiological effects by binding to vitamin D receptors (VDR)located on the target organs which are bone, intestine, kidneyand the parathyroid glands [1], [7]-[11]. Active vitamin Dincreases calcium uptake at intestine. The result can be seenwithin approximately 2 hours [1]. Due to the sequentialhydroxylation in liver and kidney, the longer duration isneeded when vitamin D is given [1]. Moreover, active vitaminD also acts on bone to increase both the number and activity ofosteoclasts [1], [7]-[11].

In the next section, we then develop a delay-differentialequations model to investigate the change in the serum level ofcalcium ion due to the change in the serum levels ofparathyroid hormone and vitamin D. The time delay observedclinically in the stimulating effects of parathyroid hormone andvitamin D on calcium release will also be incorporated in themodel.

II. MODEL FORMULATION

Let us denote the concentration of PTH above the basallevel in blood at time t by P t , the concentration of PTH

above the basal level in blood at time t by P t , theconcentration of the active form of vitamin D in blood at timet by D t , the concentration of the active form of vitamin D

in blood at time t by D t , and the concentration of

calcium in blood at time t by C t .

A delay-differential equations model of calciumhomeostasis: Effects of parathyroid hormone

and vitamin D

Inthira Chaiya, Chontita Rattanakul, Sahattaya Rattanamongkonkul, Wannapa Panitsupakamon,

and Sittipong Ruktamatakul

O

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Firstly, PTH secreted from the parathyroid glands is one ofthe major regulators of calcium homeostasis. Parathyroid cellsare unusual in the respect that hormone synthesis anddegradation are adjusted due to physiological demand forsecretion [1]. As much as 90% of the hormone synthesizedmay be destroyed within the chief cells, which degrade PTH atan accelerated rate when serum level of calcium is high [1]. Onthe other hand, the release of PTH is stimulated by the lowlevel of calcium in blood [1]-[6]. PTH increases the serumlevel of calcium by various direct and indirect actions on bone,intestine and kidney as discussed in the previous section inorder to maintain the normal range of calcium concentration inblood. Moreover, in the absence of PTH, the serum level ofcalcium decreases dramatically over a period of several hours[1]. In addition, the chief cells of the parathyroid glands arealso targets for the active vitamin D and response to it in anegative feedback manner [1]. The equation for the rate ofchange of PTH concentration above the basal level in blood isthen assumed to have the form

11

1 2 3 4

   ( )( )

udP vdT w w D w w C

P

(1)

where the parameters 1 1 2 3 4, , , ,u w w w w and 1v are assumed tobe positive. The first term on the right hand side represents thesecretion rate of PTH from the parathyroid glandscorresponding to the serum levels of active vitamin D andcalcium. The last term represents the removal rate of PTHfrom the system.

Secondly, vitamin D is another principal regulator ofcalcium homeostasis. The body itself produced vitamin Dwhen it is exposed to the sun. Vitamin D is then synthesized into an active form so that it can mediate its biological effects.Active vitamin D then binds to vitamin D receptors expressedon the target organs. It enhances calcium absorption in theintestine and increases calcium mobilization from bone [1],[7]-[11]. On the other hand, PTH also stimulates the synthesisof the active form of vitamin D [1], [7]-[11]. The equation forthe rate of change of serum level of active vitamin D is thenassumed to have the form

2 322

6 8

4 5

5 7

( )( )  ( )( )u u P u u DdD v

dT w w PD

w wD

C

(2)

where the parameters 2 3 4 5 5 6 7 8, , , , , , ,u u u u w w w w and 2v areassumed to be positive. The first term on the right hand siderepresents the synthesis rate of active vitamin D correspondingto the serum levels of PTH, calcium and active vitamin Ditself. The last term represents the removal rate of activevitamin D from the system.

Finally, the rate of change of serum level calcium is thenassumed to have the form

6 7 3   dC u P t u D t vd

CT

(3)

where the parameters 6 7,u u and 3v are assumed to bepositive. The first term on the right hand side represents thedelay effect in the increase of serum level of calcium due tothe increase in the concentration of PTH. The second termrepresents the delay effect in the increase of serum level ofcalcium due to the increase in the serum level of active

vitamin D. The last term represents the removal rate ofcalcium from the system.

III. MODEL ANALYSIS

In order to investigate the possibility of periodic dynamicsin our system of (1)-(3), we scale the variables and the

parameters in the model as follows:0 0 0

, , ,P D CX Y ZP D C

0 1 0 2 0 31 2 32

0 2 4 0 0 0 6 8 0 0 6 8 0 0

, , ,T u T u T uTt a a aT w w P D C w w P C w w P C

5 0 0 6 0 0 7 044 5 6 72 2

6 8 0 0 6 8 0 0 0 0

, , , ,u D T u P T u Dua a a aw w P C w w P C C C

3 5 711 2 3 4 1 1 02

2 0 4 0 6 0 8 0

, , , , ,w w wwk k k k d v Tw D w C w P w C

2 2 0 ,d v T 3 3 0d v T . The system (1)-(3) can then be written as

11

1 2

   ( )( )

adX d Xdt k Y k Z

(4)

4322

3 4

52( )( )  ( )( )

a a X a a Y YdY d Ydt k X k Z

(5)

6 7 3dZ a X t a Y t d Zdt

(6)

Assuming that , ,S S SX Y Z is a non washout steady state ofthe system (4)-(6). Letting Sx X X , Sy Y Y ,

Sz Z Z , we will be led to the following linearized systemof (4)-(6)

S

x xy J yz z

(7)

where SJ is the corresponding Jacobian matrix evaluated at

, ,S S SX Y Z , namely

1 1 1

6 7 3

S

d d A d BJ C D E

a e a e d

(8)

where

1

2

23 2 3 3 2

22 3 3

2 4 52

4 5

2 4

,

,

2,

2,

S

S

S

S

S S S

S S

S S

S S

S S

XAk Y

XBk Z

a X a X a k d YC

a a X k X

d Y a a YD d

a a Y Y

E d Y k Z

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The characteristic equation of SJ can then be written as

3 2 0F a b d c e e (9)

where

1 3

1 3 1 3 1

6 1 7

1 3 1 3

7 1 6 1

,,

,,

a d d Eb d d d d D d ACc a d B a Ed d d D d d ACE a d BC E a d AE BD

According to the Hopf bifurcation theorem, it is necessarythat (9) has a pair of purely imaginary complex roots i for some value of so that a periodic solution exists. In orderthat such a pair can be found, one must have 0F i , thatis,

3 2 0ii a i b i d c i e e (10)

Equating real and imaginary parts on the left of (10) tozero, we obtain the following equations:

2 cos sina d e c (11) 3 cos sinb c e (12)

By squaring both sides of (11) and (12), and then adding,we obtain

0 (13)where

6 2 4 2 2 2 2 22 2a b b ad c d e

Letting 2 , (13) can be written as

3 2 0U V W (14)

where 2 2 2 2 22 , 2 ,U a b V b ad c W d e .

Therefore, if (14) has a positive real solution 2 0 then (9) will have a pair of complex solutions, i .

According to the work of Ruan and Wei [12], for apolynomial in the form of (14), the following lemmas areobtained and so we state them without proofs.

Lemma 1(a) If 0W , then (14) has at least one positive root.

(b) If 0W , then the necessary condition for (14) to have apositive real root is that 2 3 0.U V

Lemma 2 If0W and 0 (15)

then (14) has a positive root if and only if

1 0 and 1 0 (16)

where 1 3U

.

Therefore, by the above lemmas, we assume that either0W or (15) and (16) hold so that (14) has positive roots.

Assuming that it has three positive roots denoted by 1 , 2and 3 . Then, (13) has three positive roots

, 1,2,3.k k k

Now, let 0 0 be the smallest of such for which¸.i Substituting k into (11)-(12) and solving for ,

we obtain

3

2 2 2

1 21 arcsin2

j k kk

k k k

ac e be cd jc e

(17)

where 1,2,3,k and 1,2,...j

Theorem 1 Suppose that

0, 0a d e and a b c d e (18)

(a) If 0W and 0 , then all roots of (9) have nonzeroreal parts for all 0.

(b) If either0W (19)

or 10, 0, 0W and 1 0 (20)

then all roots of (9) have negative real parts when 00, ,

where 0 1 3, 1

min , 0j jk kk j

(21)

with jk defined in (17).

Proof(a) By contradiction, if (9) has a root with zero real part forsome 0 which implies that (14) has a positive real root. ByLemma 1(b), the necessary condition for (9) to have a positivereal root is that 0 which contradicts the fact that 0 .Therefore, all roots of (9) have nonzero real parts for all

0 .

(b) For 0 , equation (9) is reduced to

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3 2 0a b c d e (22)

Since the conditions in (18) hold, the Routh-Hurwitz criterionthen implies that all roots of (9) have negative real parts andhence, all roots¸ of (9) have negative real parts at the

point 0 . From the continuity of , all roots of (9) willhave negative real parts for values of in some open intervalcontaining 0 . Therefore, all roots of (9) have negative realparts for positive values of 0, c for some 0.c

However, c is defined by (21) to be the minimum of all

the positive jk where j

k is defined as in (17). Hence,

0 is the minimum of such positive 's for which the realparts of some roots of (9) vanish, provided that (19) or (20)holds. Thus, 0c , which completes the proof.

Theorem 1 implies that if either (19) or (20) are satisfiedand (18) holds, the steady state , ,S S SX Y Z of our system of

(4)-(6) is stable for some values of 00, . At 0 ,

Re 0 by the definition of 0 and hence the stability

of the steady state , ,S S SX Y Z is lost at 0 . In order for aHopf bifurcation to occur, and hence a periodic solution of oursystem of (4)-(6) may be expected, we still need to show that

0

Re0

dd

which is done in the next theorem.

Theorem 2 Suppose that conditions (19) or (20) in Theorem 1hold, then i is a pair of purely imaginary roots of (9).Moreover,

0

Re0

dd

(23)

provided that 0 0 (24)

where0

20 0 0, .k

ProofThe first part of this theorem is an immediate consequence ofTheorem 1 and the definition of 0 . In order to prove that

0

Re0

dd

, let us consider (9),

3 2 0F a b d c e e Then,

23 2

0

dF da b c e e c e ed d

and hence,

1 23 2 0d a b cd c e e c e

Since 3 2c e e a b d , then

1 2

3 2

3 2d a b cd c ea b d

At 0 0, i and thus,

0

210 0

4 2 300 0 0 0

20 0

3 2b i ad id b i a d

cc i e

Therefore,

0

4 2 2 210 0

6 2 4 2 2 20 0 0

2

2 2 20

3 2 4 2Re

2 2

a b b addd a b b ad d

cc e

(13)

implies that

6 2 4 2 2 2 2 2 20 0 0 02 2a b b ad d c e

then,

0

4 2 2 2 210 0

2 2 20

20

2 2 20

3 2 2 2Re

0

a b b ad cdd c e

c e

Hence,0

1

Re 0dd

and the proof is complete. We thus

have the following result.

Theorem 3 If either (19) or (20) holds, then a periodicsolution occurs in our model equations (4)-(6) for a positivetime delay 0 given by (21) provided that (18) and (24) aresatisfied.

IV. NUMERICAL INVESTIGATION

A computer simulation of the system (4)-(6) is presented inFig. 1 and 2, with parametric values chosen to satisfy theconditions in Theorem 3. The solution trajectory projected

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onto the ,x y plane, ,x z plane and ,y z plane are asshown in Fig. 1a, 1b and 1c, respectively. The correspondingtime courses of the PTH concentration above the basal level,

the concentration of active vitamin D and the concentration ofcalcium are as shown in Fig. 2a, 2b and 2c, respectively,showing a periodic behavior as theoretically predicted.

Fig. 1 A computer simulation of the model systems (4)-(6) with 1 2 3 4 5 6 7 10.007, 0.1, 0.8, 0.5, 0.01, 0.02, 0.08, 0.08,a a a a a a a k

2 3 4 1 2 30.01, 3.9, 0.06, 0.07, 0.145, 0.1, =12, (0) 1,k k k d d d x (0) 1, (0) 1.y z (a) The solution trajectory projected onto the(x,y)-plane, (b) The solution trajectory projected onto the (x,z)-plane and (c) The solution trajectory projected onto the (y,z)-plane, respectively.

Fig. 2 A computer simulation of the model systems (4)-(6) with 1 2 3 4 5 6 7 10.007, 0.1, 0.8, 0.5, 0.01, 0.02, 0.08, 0.08,a a a a a a a k

2 3 4 1 2 30.01, 3.9, 0.06, 0.07, 0.145, 0.1, 12, (0) 1,k k k d d d x (0) 1, (0) 1.y z (a) The corresponding time courses of the PTH

concentration above the basal level (x), (b) the concentration of active vitamin D (y) and (c) the concentration of calcium (z), respectively.

a) b) c)

a) b) c)

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A computer simulation of the system (4)-(6) is presented inFig. 3 and 4. The solution trajectory projected onto the ,x y plane, ,x z plane and ,y z plane are as shownin Fig. 3a, 3b and 3c, respectively, showing a solutiontrajectory tends to a stable equilibrium solution. The

corresponding time courses of the PTH concentration abovethe basal level, the concentration of active vitamin D and theconcentration of calcium are as shown in Fig. 4a, 4b and 4c,respectively.

Fig. 3 A computer simulation of the model systems (4)-(6) with 1 2 3 4 5 6 7 10.007, 0.1, 0.8, 0.5, 0.01, 0.02, 0.08, 0.08,a a a a a a a k

2 3 4 1 2 30.01, 3.9, 0.06, 0.07, 0.145, 0.1, 5, (0) 1,k k k d d d x (0) 1, (0) 1.y z (a) The solution trajectory projected onto the(x,y)-plane, (b) The solution trajectory projected onto the (x,z)-plane and (c) The solution trajectory projected onto the (y,z)-plane, respectively.

Fig. 4 A computer simulation of the model systems (4)-(6) with 1 2 3 4 5 6 7 10.007, 0.1, 0.8, 0.5, 0.01, 0.02, 0.08, 0.08,a a a a a a a k

2 3 4 1 2 30.01, 3.9, 0.06, 0.07, 0.145, 0.1, 5, (0) 1,k k k d d d x (0) 1, (0) 1.y z (a) The corresponding time courses of the PTH

concentration above the basal level (x), (b) the concentration of active vitamin D (y) and (c) the concentration of calcium (z), respectively.

a) b) c)

a) b) c)

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A computer simulation of the system (4)-(6) is presented inFig. 5 and 6. The solution trajectory projected onto the ,x y plane, ,x z plane and ,y z plane are as shownin Fig. 5a, 5b and 5c, respectively, showing a chaotic behavior

exhibited by our model. The corresponding time courses of thePTH concentration above the basal level, the concentration ofactive vitamin D and the concentration of calcium are asshown in Fig. 6a, 6b and 6c, respectively.

Fig. 5 A computer simulation of the model systems (4)-(6) with 1 2 3 4 5 6 7 10.007, 0.1, 0.8, 0.5, 0.01, 0.02, 0.08, 0.08,a a a a a a a k

2 3 4 1 2 30.01, 3.9, 0.06, 0.07, 0.145, 0.1, 80, (0) 1,k k k d d d x (0) 1, (0) 1.y z (a) The solution trajectory projected onto the(x,y)-plane, (b) The solution trajectory projected onto the (x,z)-plane and (c) The solution trajectory projected onto the (y,z)-plane, respectively.

Fig. 6 A computer simulation of the model systems (4)-(6) with 1 2 3 4 5 6 7 10.007, 0.1, 0.8, 0.5, 0.01, 0.02, 0.08, 0.08,a a a a a a a k

2 3 4 1 2 30.01, 3.9, 0.06, 0.07, 0.145, 0.1, 80, (0) 1,k k k d d d x (0) 1, (0) 1.y z (a) The corresponding time courses of the PTH

concentration above the basal level (x), (b) the concentration of active vitamin D (y) and (c) the concentration of calcium (z), respectively.

a) b) c)

a) b) c)

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V. CONCLUSION

A delay-differential equations model accounted for theeffects of PTH, vitamin D and time delay is developed in orderto investigate the calcium homeostasis. Hopf bifurcationtheorem is then utilized in order to derive the conditions on themodel parameters that guarantee the existence of a periodicsolution. Numerical investigation is then carried out by usingRunge-Kutta method [13]-[16]. The results indicate that ourmodel can exhibit nonlinear behavior corresponding to thepulsatile patterns observed clinically in the serum levels ofparathyroid hormone, vitamin D and calcium [17]-[19].

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[14] M. Racila and J.M. Crolet, “Sinupros: Mathematical model of humancortical bone”, in Proc. 10th WSEAS Inter. Conf. on Mathematics andComputers in Biology and Chemistry, WSEAS Press, Prague, CzechRepublic, 2009, pp. 53.

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[16] A. Chirita, R. H. Ene, R.B. Nicolescu and R.I. Carstea, “A numericalsimulation of distributed-parameter systems”, in Proc. 9th WSEASInter. Conf. on Mathematical Methods and Computational Techniquesin Electrical Engineering, WSEAS Press, Arcachon, 2007, pp. 70.

[17] V. Tangpricha, P. Koutkia, S.M. Rieke, T.C. Chen, A.A. Perez, andM.F. Holick, “Fortification of orange juice with vitamin D: a novelapproach for enhancing vitamin D nutritional health”, Am. J. Clin.Nutr., vol.77, pp. 1478-1483, 2003.

[18] M.F. Holick, “Sunlight and vitamin D for bone health and preventionof autoimmune diseases, cancers, and cardiovascular disease”, Am. J.Clin. Nutr., vol.80 (suppl), pp. 1678S-1688S, 2004.

[19] K. N. Muse, S. C. Manolagas, L.J. Deftos, N. Alexander, and S.S.C.Yen, “Calcium-regulating hormones across the menstrual cycle”, J.Clin. Endocrinol. Metab., vol.62, no.2, pp.1313-1315, 1986.

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