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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

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.
Citation: David S. Ross, Christina Battista, Antonio Cabal, Khamir Mehta. Dynamics of bone cell signaling and PTH treatments of osteoporosis. Discrete & 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.  doi: 10.1142/S0218339011003841.  Google Scholar

[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).   Google Scholar

[3]

C. Battista, "Mathematical Models of Bone Remodeling at the Cellular Level,", M.S. Thesis, (2011).   Google Scholar

[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.   Google Scholar

[5]

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

[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.  doi: 10.1210/jc.2009-0358.  Google Scholar

[7]

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

[8]

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

[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.  doi: 10.1210/jc.2009-0154.  Google Scholar

[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.  doi: 10.1210/jc.81.10.3487.  Google Scholar

[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.  doi: 10.1172/JCI6610.  Google Scholar

[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.  doi: 10.1016/S8756-3282(03)00157-1.  Google Scholar

[13]

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

[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.  doi: 10.1016/j.jtbi.2004.03.023.  Google Scholar

[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.  doi: 10.1210/er.21.2.115.  Google Scholar

[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.  doi: 10.1056/NEJM200105103441904.  Google Scholar

[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.  doi: 10.1016/0006-291X(89)92795-2.  Google Scholar

[18]

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

[19]

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

[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.  doi: 10.1210/jc.2002-011870.  Google Scholar

[21]

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

[22]

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

[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.  doi: 10.1016/j.mbs.2010.04.001.  Google Scholar

[24]

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

[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.   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

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.  doi: 10.1142/S0218339011003841.  Google Scholar

[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).   Google Scholar

[3]

C. Battista, "Mathematical Models of Bone Remodeling at the Cellular Level,", M.S. Thesis, (2011).   Google Scholar

[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.   Google Scholar

[5]

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

[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.  doi: 10.1210/jc.2009-0358.  Google Scholar

[7]

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

[8]

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

[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.  doi: 10.1210/jc.2009-0154.  Google Scholar

[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.  doi: 10.1210/jc.81.10.3487.  Google Scholar

[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.  doi: 10.1172/JCI6610.  Google Scholar

[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.  doi: 10.1016/S8756-3282(03)00157-1.  Google Scholar

[13]

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

[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.  doi: 10.1016/j.jtbi.2004.03.023.  Google Scholar

[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.  doi: 10.1210/er.21.2.115.  Google Scholar

[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.  doi: 10.1056/NEJM200105103441904.  Google Scholar

[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.  doi: 10.1016/0006-291X(89)92795-2.  Google Scholar

[18]

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

[19]

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

[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.  doi: 10.1210/jc.2002-011870.  Google Scholar

[21]

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

[22]

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

[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.  doi: 10.1016/j.mbs.2010.04.001.  Google Scholar

[24]

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

[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.   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

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