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September  2012, 17(6): 2185-2200. doi: 10.3934/dcdsb.2012.17.2185

Dynamics of bone cell signaling and PTH treatments of osteoporosis

David S. Ross 1, , Christina Battista 1, , Antonio Cabal 2, and  Khamir Mehta 3,

1. 

School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, United States, United States
2. 

ESD - Modeling and Simulations, Merck and Co., West Point, PA 19486, United States
3. 

Applied Computer Science and Mathematics, Merck and Co., West Point, PA 19486, United States

Received  September 2011 Revised  February 2012 Published  May 2012

In this paper we analyze and generalize the dynamical system introduced by Lemaire and co-workers [14] as a model of cell signaling in bone remodeling. We show that for large classes of parameter values, including the physically-realistic baseline values, the system has a unique physically relevant equilibrium which is a global attractor. We generalize that model, minimally, to incorporate a mechanism by which parathyroid hormone (PTH) retards osteoblast apoptosis. We show that with this mechanism, which is a simplified version of that proposed by Bellido and co-workers [4], the model exhibits a well-known phenomenon that has puzzled researchers: the system responds catabolically to the continuous administration of PTH and ally to appropriately pulsed administration of PTH.
Keywords: global stability, Cell signaling, osteoporosis, Lyapunov, parathyroid hormone, PTH..
Mathematics Subject Classification: Primary: 34D23, 37C75; Secondary: 92C45, 92C5.
Citation: David S. Ross, Christina Battista, Antonio Cabal, Khamir Mehta. Dynamics of bone cell signaling and PTH treatments of osteoporosis. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2185-2200. doi: 10.3934/dcdsb.2012.17.2185
References:
[1]

E. Agyingi, D. S. Ross and K. Bathena, A model of the transmission dynamics of Leismaniasis, J. Bio. Systems, 19 (2011), 237-250. doi: 10.1142/S0218339011003841.

[2]

B. P. Ayati, C. M. Edwards, G. F. Webb and J. P. Wikswo, Mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease, Biology Direct, 5 (2010), 28.

[3]

C. Battista, "Mathematical Models of Bone Remodeling at the Cellular Level," M.S. Thesis, Rochester Institute of Technology, 2011.

[4]

T. Bellido, A. Ali, L. I. Plotkin, Q. Fu, I. Gubrij, P. K. Roberson, R. S. Weinstein, C. A. O'Brien, S. C. Manolagas and R. I. Jilka, Proteasomal degradation of Runx2 shortens parathyroid hormone-induced anti-apoptotic signaling in osteoblasts; A putatitve explanation for why intermittent administration is needed for bone anabolism, J. Bio. Chem., 278, (2003), 50259-50272.

[5]

F. Brohner and W. E. Stein, Calcium homeostasis, an old problem revisited, J. Nutr., 125 (1995), 1987S-1995S.

[6]

F. Cosman, N. E. Lane, M. A. Bolognese, J. R. Zanchetta, P. A. Garcia-Hernandez, K. Sees, J. A. Matriano, K. Gaumer and P. E. Daddona, Effect of transdermal teriparatide administration on bone mineral density in postmenopausal women, J. Clin. Endocrinol. Metab., 95 (2010), 151-158. doi: 10.1210/jc.2009-0358.

[7]

D. W. Dempster, F. Cosman, M. Parisien and V. Shen, Anabolic action of parathyroid hormone on bone, Endocr. Rev., 14 (1993), 690-709.

[8]

P. Ducy, T. Schinke and G. Karsenty, The osteoblast: A sophisticated fibroblast under central surveillance, Science, 289 (2000), 1501-1504. doi: 10.1126/science.289.5484.1501.

[9]

J. S. Finkelstein, Jason J. Wyland, B. Z. Leder, S. A. M. Burnett-Bowie, H. Lee, H. Jüppner and R. M. Neer, Effects of teriparatide retreatment in osteoporotic men and women, J. Clin. Endocrinol. Metab., 94 (2009), 2495-2501. doi: 10.1210/jc.2009-0154.

[10]

C. Y. Guo, W. E. G. Thomas, A. W. al-Dehaimi, A. M. A. Assiri and R. Eastell, Longitudinal changes in bone mineral density and bone turnover in postmenopausal women with primary hyperparathyroidism, J. Clin. Endocrinol. Metab., 81 (1996), 3487-3491. doi: 10.1210/jc.81.10.3487.

[11]

R. L. Jilka, R. S. Weinstein, T. Bellido, P. Roberson, A. M. Parfitt and S. C. Manolagas, Increased bone formation by prevention of osteoblast apoptosis with parathyroid hormone, J. Clin. Invest., 104 (1999), 439-446. doi: 10.1172/JCI6610.

[12]

S. V. Komarova, R. J. Smith, S. J. Dixon, S. M. Sims and L. M. Wahl, Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling, Bone, 33 (2003), 206-215. doi: 10.1016/S8756-3282(03)00157-1.

[13]

M. H. Kroll, Parathyroid hormone temporal effects on bone formation and resorption, Bull. Math. Biol., 62 (2000), 163-688. doi: 10.1006/bulm.1999.0146.

[14]

V. Lemaire, F. L. Tobin, L. D. Greller, C. R. Cho and L. J. Suva, Modeling the interactions between osteoblast and osteoclast activities in bone remodeling, J. Theo. Bio., 229 (2004), 293-309. doi: 10.1016/j.jtbi.2004.03.023.

[15]

S. C. Manolagas, Birth and death of bone cells: Basic regulatory mechanisms and implications for the pathogenesis and treatment of osteoporosis, Endocrine Reviews, 21 (2000), 115-137. doi: 10.1210/er.21.2.115.

[16]

R. M. Neer, C. D. Arnaud, J. R. Zanchetta, R. Prince, G. A. Gaich, J. Y. Reginster, A. B. Hodsman, E. F. Eriksen, S. Ish-Shalom, H. K. Genant, O. Wang and B. H. Mitlak, Effect of parathyroid hormone (1-34) on fractures and bone mineral density in postmenopausal women with osteoporosis, N. Engl. J. Med., 344 (2001), 1434-1441. doi: 10.1056/NEJM200105103441904.

[17]

R. O. C. Oreffoi, G. R. Mundyf, S. M. Seyedin and L. F. Bonewald, Activation of the bone-derived latent TGF-beta complex by isolated osteoclasts, Biochem. Biophys. Res. Comm., 158 (1989), 817-823. doi: 10.1016/0006-291X(89)92795-2.

[18]

M. Peterson and M. Riggs, A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling, Bone, 46 (2010), 49-63. doi: 10.1016/j.bone.2009.08.053.

[19]

K. E. S. Poole and J. Reeve, Parathyroid hormone-a bone anabolic and catabolic agent, Current Opinion in Pharmacology, 5 (2005), 612-617.

[20]

J. F. Raposo, L. G. Sobrinho and H. G. Ferriera, A minimal mathematical model of calcium homeostasis, J. Clinical Endocrin. Metab., 87 (2002), 4330-4340. doi: 10.1210/jc.2002-011870.

[21]

D. S. Ross, K. Mehta and A. Cabal, A mathematical model of bone biochemistry,, in preparation., (). 

[22]

E. Seeman and P. D. Delmas, Reconstructing the skeleton with intermittent parathyroid hormone, Trends Endocrinol. Metab., 12 (2001), 281-283. doi: 10.1016/S1043-2760(01)00460-X.

[23]

R. P. Shrestha, C. V. Hollot, S. R. Chipkin, C. P. Schmitt and Y. Chait, A mathematical model of parathyroid hormone response to acute changes in plasma ionized calcium concentration in humans, Mathematical Biosciences, 226 (2010), 46-57. doi: 10.1016/j.mbs.2010.04.001.

[24]

E. Slatopolsky, A. Brown and A. Dusso, Pathogenesis of secondary hyperparathyroidism, Kidney Int. Suppl., 73 (1999), S14-S19. doi: 10.1046/j.1523-1755.1999.07304.x.

[25]

R. V. Talmage, G. E. Lester, P. F. Hirsch and J. Musculoske, Parathyroid hormone and plasma calcium control: An editorial, J. Musculoskel. Neuron. Interact., 1 (2000), 121-126.

[26]

S. L. Teitelbaum, Bone resorption by osteoclasts, Science, 289 (2000), 1501-1504. doi: 10.1126/science.289.5484.1504.

[27]

H. Vaananen and T. Laitala-Leinonen, Osteoclast lineage and function, Arch. Biochem. Biophys., 473 (2008), 132-138. doi: 10.1016/j.abb.2008.03.037.

[28]

L. Xing and B. F. Boyce, Regulation of apoptosis in osteoclasts and osteoblastic cells, Biochemical and Biophysical Research Communications, 328 (2005), 709-720. doi: 10.1016/j.bbrc.2004.11.072.

[29]

M. Zumsande, D. Stiefs, S. Siegmund and T. Gross, General analysis of mathematical models for bone remodeling, Bone, 48 (2011), 910-917. doi: 10.1016/j.bone.2010.12.010.

show all references

References:
[1]

E. Agyingi, D. S. Ross and K. Bathena, A model of the transmission dynamics of Leismaniasis, J. Bio. Systems, 19 (2011), 237-250. doi: 10.1142/S0218339011003841.

[2]

B. P. Ayati, C. M. Edwards, G. F. Webb and J. P. Wikswo, Mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease, Biology Direct, 5 (2010), 28.

[3]

C. Battista, "Mathematical Models of Bone Remodeling at the Cellular Level," M.S. Thesis, Rochester Institute of Technology, 2011.

[4]

T. Bellido, A. Ali, L. I. Plotkin, Q. Fu, I. Gubrij, P. K. Roberson, R. S. Weinstein, C. A. O'Brien, S. C. Manolagas and R. I. Jilka, Proteasomal degradation of Runx2 shortens parathyroid hormone-induced anti-apoptotic signaling in osteoblasts; A putatitve explanation for why intermittent administration is needed for bone anabolism, J. Bio. Chem., 278, (2003), 50259-50272.

[5]

F. Brohner and W. E. Stein, Calcium homeostasis, an old problem revisited, J. Nutr., 125 (1995), 1987S-1995S.

[6]

F. Cosman, N. E. Lane, M. A. Bolognese, J. R. Zanchetta, P. A. Garcia-Hernandez, K. Sees, J. A. Matriano, K. Gaumer and P. E. Daddona, Effect of transdermal teriparatide administration on bone mineral density in postmenopausal women, J. Clin. Endocrinol. Metab., 95 (2010), 151-158. doi: 10.1210/jc.2009-0358.

[7]

D. W. Dempster, F. Cosman, M. Parisien and V. Shen, Anabolic action of parathyroid hormone on bone, Endocr. Rev., 14 (1993), 690-709.

[8]

P. Ducy, T. Schinke and G. Karsenty, The osteoblast: A sophisticated fibroblast under central surveillance, Science, 289 (2000), 1501-1504. doi: 10.1126/science.289.5484.1501.

[9]

J. S. Finkelstein, Jason J. Wyland, B. Z. Leder, S. A. M. Burnett-Bowie, H. Lee, H. Jüppner and R. M. Neer, Effects of teriparatide retreatment in osteoporotic men and women, J. Clin. Endocrinol. Metab., 94 (2009), 2495-2501. doi: 10.1210/jc.2009-0154.

[10]

C. Y. Guo, W. E. G. Thomas, A. W. al-Dehaimi, A. M. A. Assiri and R. Eastell, Longitudinal changes in bone mineral density and bone turnover in postmenopausal women with primary hyperparathyroidism, J. Clin. Endocrinol. Metab., 81 (1996), 3487-3491. doi: 10.1210/jc.81.10.3487.

[11]

R. L. Jilka, R. S. Weinstein, T. Bellido, P. Roberson, A. M. Parfitt and S. C. Manolagas, Increased bone formation by prevention of osteoblast apoptosis with parathyroid hormone, J. Clin. Invest., 104 (1999), 439-446. doi: 10.1172/JCI6610.

[12]

S. V. Komarova, R. J. Smith, S. J. Dixon, S. M. Sims and L. M. Wahl, Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling, Bone, 33 (2003), 206-215. doi: 10.1016/S8756-3282(03)00157-1.

[13]

M. H. Kroll, Parathyroid hormone temporal effects on bone formation and resorption, Bull. Math. Biol., 62 (2000), 163-688. doi: 10.1006/bulm.1999.0146.

[14]

V. Lemaire, F. L. Tobin, L. D. Greller, C. R. Cho and L. J. Suva, Modeling the interactions between osteoblast and osteoclast activities in bone remodeling, J. Theo. Bio., 229 (2004), 293-309. doi: 10.1016/j.jtbi.2004.03.023.

[15]

S. C. Manolagas, Birth and death of bone cells: Basic regulatory mechanisms and implications for the pathogenesis and treatment of osteoporosis, Endocrine Reviews, 21 (2000), 115-137. doi: 10.1210/er.21.2.115.

[16]

R. M. Neer, C. D. Arnaud, J. R. Zanchetta, R. Prince, G. A. Gaich, J. Y. Reginster, A. B. Hodsman, E. F. Eriksen, S. Ish-Shalom, H. K. Genant, O. Wang and B. H. Mitlak, Effect of parathyroid hormone (1-34) on fractures and bone mineral density in postmenopausal women with osteoporosis, N. Engl. J. Med., 344 (2001), 1434-1441. doi: 10.1056/NEJM200105103441904.

[17]

R. O. C. Oreffoi, G. R. Mundyf, S. M. Seyedin and L. F. Bonewald, Activation of the bone-derived latent TGF-beta complex by isolated osteoclasts, Biochem. Biophys. Res. Comm., 158 (1989), 817-823. doi: 10.1016/0006-291X(89)92795-2.

[18]

M. Peterson and M. Riggs, A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling, Bone, 46 (2010), 49-63. doi: 10.1016/j.bone.2009.08.053.

[19]

K. E. S. Poole and J. Reeve, Parathyroid hormone-a bone anabolic and catabolic agent, Current Opinion in Pharmacology, 5 (2005), 612-617.

[20]

J. F. Raposo, L. G. Sobrinho and H. G. Ferriera, A minimal mathematical model of calcium homeostasis, J. Clinical Endocrin. Metab., 87 (2002), 4330-4340. doi: 10.1210/jc.2002-011870.

[21]

D. S. Ross, K. Mehta and A. Cabal, A mathematical model of bone biochemistry,, in preparation., (). 

[22]

E. Seeman and P. D. Delmas, Reconstructing the skeleton with intermittent parathyroid hormone, Trends Endocrinol. Metab., 12 (2001), 281-283. doi: 10.1016/S1043-2760(01)00460-X.

[23]

R. P. Shrestha, C. V. Hollot, S. R. Chipkin, C. P. Schmitt and Y. Chait, A mathematical model of parathyroid hormone response to acute changes in plasma ionized calcium concentration in humans, Mathematical Biosciences, 226 (2010), 46-57. doi: 10.1016/j.mbs.2010.04.001.

[24]

E. Slatopolsky, A. Brown and A. Dusso, Pathogenesis of secondary hyperparathyroidism, Kidney Int. Suppl., 73 (1999), S14-S19. doi: 10.1046/j.1523-1755.1999.07304.x.

[25]

R. V. Talmage, G. E. Lester, P. F. Hirsch and J. Musculoske, Parathyroid hormone and plasma calcium control: An editorial, J. Musculoskel. Neuron. Interact., 1 (2000), 121-126.

[26]

S. L. Teitelbaum, Bone resorption by osteoclasts, Science, 289 (2000), 1501-1504. doi: 10.1126/science.289.5484.1504.

[27]

H. Vaananen and T. Laitala-Leinonen, Osteoclast lineage and function, Arch. Biochem. Biophys., 473 (2008), 132-138. doi: 10.1016/j.abb.2008.03.037.

[28]

L. Xing and B. F. Boyce, Regulation of apoptosis in osteoclasts and osteoblastic cells, Biochemical and Biophysical Research Communications, 328 (2005), 709-720. doi: 10.1016/j.bbrc.2004.11.072.

[29]

M. Zumsande, D. Stiefs, S. Siegmund and T. Gross, General analysis of mathematical models for bone remodeling, Bone, 48 (2011), 910-917. doi: 10.1016/j.bone.2010.12.010.

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