2012, 9(2): 281-295. doi: 10.3934/mbe.2012.9.281

Towards a new spatial representation of bone remodeling

1. 

Department of Mathematics/Program, in Applied Mathematical and Computational Sciences, University of Iowa, Iowa City, IA 52242-1419, United States, United States

2. 

Department of Orthopaedics and Rehabilitation, University of Iowa Hospitals and Clinics, University of Iowa, Iowa City, IA 52242, United States, United States

Received  October 2011 Revised  November 2011 Published  March 2012

Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed.
    There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes.
    We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur in vitro and in vivo.
    By employing the level set approach it is possible to develop computational and mathematical representations of the spatial behavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as in vitro and in vivo. This makes it possible to perform in silica experiments more closely resembling experimental observations.
Citation: Jason M. Graham, Bruce P. Ayati, Prem S. Ramakrishnan, James A. Martin. Towards a new spatial representation of bone remodeling. Mathematical Biosciences & Engineering, 2012, 9 (2) : 281-295. doi: 10.3934/mbe.2012.9.281
References:
[1]

T. Akchurin, T. Aissiou, N. Kemeny, E. Prosk, N. Nigam and S. V. Komarova, Complex dynamics of osteoclast formation and death in long-term cultures,, PLoS One, 3 (2008). Google Scholar

[2]

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

[3]

J. P. Bilezikian, L. G. Raisz and G. A. Rodan, "Principles of Bone Biology," Second edition,, Academic Press, (2002). Google Scholar

[4]

L. Geris, A. Gerisch, C. Maes, G. Carmeliet, R. Weiner, J. Vander Sloten and H. Van Oosterwyck, Mathematical modeling of fracture healing in mice: Comparison between experimental data and numerical simulation results,, Med. Biol. Eng. Comput., 44 (2006), 280. doi: 10.1007/s11517-006-0040-6. Google Scholar

[5]

L. Geris, A. Gerisch and R. C. Schugart, Mathematical modeling in wound healing, bone regeneration and tissue engineering,, Acta Biotheor, 58 (2010), 355. doi: 10.1007/s10441-010-9112-y. Google Scholar

[6]

L. Geris, A. Gerisch, J. Vander Sloten, R. Weiner and H. Van Oosterwyck, Angiogenesis in bone fracture healing: A bioregulatory model,, J. Theor. Bio., 25 (2008), 137. doi: 10.1016/j.jtbi.2007.11.008. Google Scholar

[7]

L. Geris, A. A. C. Reed, J. Vander Sloten, A. Hamish, R. W. Simpson and H. Van Oosterwyck, Occurrence and treatment of bone atrophic non-unions investigated by an integrative approach,, PLoS Comput. Bio., 6 (2010), 189. Google Scholar

[8]

L. Geris, J. Vander Sloten and H. Van oosterwyck, Connecting biology and mechanics in fracture healing: An integrated mathematical modeling framework for the study of nonunions,, Biomech. Model. Mechanobiol., 9 (2010), 713. doi: 10.1007/s10237-010-0208-8. Google Scholar

[9]

C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations," With separately available software, Frontiers in Applied Mathematics, 16,, Society for Industrial and Applied Mathematics (SIAM), (1995). Google Scholar

[10]

C. T. Kelley, "Solving Nonlinear Equations with Newton's Method,", Fundamentals of Algorithms, 1 (2003). Google Scholar

[11]

S. V. Komarova, Mathematical model of paracrine interactions between osteoclasts and osteoblasts predicts anabolic action of parathyroid hormone on bone,, J. Endocrinol., 146 (2005), 3589. doi: 10.1210/en.2004-1642. Google Scholar

[12]

S. V. Komarova, R. J. Smith, S. J. Dixon, S. M. Sims and L. H. Wahl, Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling,, Bone, 33 (2003), 225. doi: 10.1016/S8756-3282(03)00157-1. Google Scholar

[13]

M. J. Martin and J. C. Buckland-Wright, Sensitivity analysis of a novel mathematical model identifies factors determining bone resorption rates,, Bone, 35 (2004), 918. doi: 10.1016/j.bone.2004.06.010. Google Scholar

[14]

M. J. Martin and J. C. Buckland-Wright, A novel mathematical model identifies potential factors regulating bone apposition,, Calcif. Tissue Int., 77 (2005), 250. doi: 10.1007/s00223-005-0101-0. Google Scholar

[15]

I. M. Mitchell, The flexible, extensible and efficient toolbox of level set methods,, J. Sci. Comp., 35 (2008), 300. doi: 10.1007/s10915-007-9174-4. Google Scholar

[16]

S. Osher and R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces," Applied Mathematical Sciences, 153,, Springer-Verlag, (2003). Google Scholar

[17]

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. Google Scholar

[18]

S. Osher and C. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations,, SIAM J. Numer. Anal., 28 (1991), 907. doi: 10.1137/0728049. Google Scholar

[19]

A. M. Parfitt, Osteonal and hemi-osteonal remodeling: The spatial and temporal framework for signal traffic in adult human bone,, J. Cell. Biochem., 55 (1994), 273. Google Scholar

[20]

V. Peiffer, A. Gerisch, D. Vandepitte, H. Van Oosterwyck and L. Geris, A hybrid bioregulatory model of angiogenesis during bone fracture healing,, Biomech. Model. Mechanobiol., 10 (2011), 383. doi: 10.1007/s10237-010-0241-7. Google Scholar

[21]

D. Peng, B. Merriman, S. Osher, H. Zhao and M. Kang, A PDE-based fast local level set method,, J. Comput. Phys., 155 (1999), 410. doi: 10.1006/jcph.1999.6345. Google Scholar

[22]

P. Pivonka and S. V. Komarova, Mathematical modeling in bone biology: From intracellular signaling to tissue mechanics,, Bone, 47 (2010), 181. doi: 10.1016/j.bone.2010.04.601. Google Scholar

[23]

P. Pivonka, J. Zimak, D. W. Smith, B. S. Gardiner, C. R. Dunstan, N. A. Sims, T. J. Martin and G. R. Mundy, Model structure and control of bone remodeling: A theoretical study,, Bone, 43 (2008), 249. doi: 10.1016/j.bone.2008.03.025. Google Scholar

[24]

L. G. Raisz, Physiology and pathophysiology of bone remodeling,, Clinical Chemistry, 45 (1999), 1353. Google Scholar

[25]

A. G. Robling, A. B. Castillo and C. H. Turner, Biomechanical and molecular regualtion of bone remodeling,, Annu. Rev. Biomed. Eng., 8 (2006), 455. doi: 10.1146/annurev.bioeng.8.061505.095721. Google Scholar

[26]

M. D. Ryser, S. V. Komarova and N. Nigam, The cellular dynamics of bone remodeling: A mathematical model,, SIAM J. Appl. Math., 70 (2010), 1899. doi: 10.1137/090746094. Google Scholar

[27]

M. D. Ryser, N. Nigam and S. V. Komarova, Mathematical modeling of spatio-temporal dyanmics of a single bone multicellular unit,, J. Bone Miner. Res., 24 (2009), 860. doi: 10.1359/jbmr.081229. Google Scholar

[28]

D. Salac and W. Lu, A local semi-implicit level-set method for interface motion,, J. Sci. Comput., 35 (2008), 330. doi: 10.1007/s10915-008-9188-6. Google Scholar

[29]

J. A. Sethian, "Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science," Second edition, Cambridge Monographs on Applied and Computational Mathematics, 3,, Cambridge University Press, (1999). Google Scholar

[30]

B. Sumengen, A matlab toolbox implementing level set methods, 2004., Available from: \url{http://barissumengen.com/level_set_methods/}., (). Google Scholar

[31]

H. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion,, J. Comput. Phys., 127 (1996), 179. doi: 10.1006/jcph.1996.0167. Google Scholar

[32]

H. Zhao, B. Merriman, S. Osher and L. Wang, Capturing the behavior of bubbles and drops using the variational level set approach,, J. Comput. Phys., 143 (1998), 495. doi: 10.1006/jcph.1997.5810. Google Scholar

show all references

References:
[1]

T. Akchurin, T. Aissiou, N. Kemeny, E. Prosk, N. Nigam and S. V. Komarova, Complex dynamics of osteoclast formation and death in long-term cultures,, PLoS One, 3 (2008). Google Scholar

[2]

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

[3]

J. P. Bilezikian, L. G. Raisz and G. A. Rodan, "Principles of Bone Biology," Second edition,, Academic Press, (2002). Google Scholar

[4]

L. Geris, A. Gerisch, C. Maes, G. Carmeliet, R. Weiner, J. Vander Sloten and H. Van Oosterwyck, Mathematical modeling of fracture healing in mice: Comparison between experimental data and numerical simulation results,, Med. Biol. Eng. Comput., 44 (2006), 280. doi: 10.1007/s11517-006-0040-6. Google Scholar

[5]

L. Geris, A. Gerisch and R. C. Schugart, Mathematical modeling in wound healing, bone regeneration and tissue engineering,, Acta Biotheor, 58 (2010), 355. doi: 10.1007/s10441-010-9112-y. Google Scholar

[6]

L. Geris, A. Gerisch, J. Vander Sloten, R. Weiner and H. Van Oosterwyck, Angiogenesis in bone fracture healing: A bioregulatory model,, J. Theor. Bio., 25 (2008), 137. doi: 10.1016/j.jtbi.2007.11.008. Google Scholar

[7]

L. Geris, A. A. C. Reed, J. Vander Sloten, A. Hamish, R. W. Simpson and H. Van Oosterwyck, Occurrence and treatment of bone atrophic non-unions investigated by an integrative approach,, PLoS Comput. Bio., 6 (2010), 189. Google Scholar

[8]

L. Geris, J. Vander Sloten and H. Van oosterwyck, Connecting biology and mechanics in fracture healing: An integrated mathematical modeling framework for the study of nonunions,, Biomech. Model. Mechanobiol., 9 (2010), 713. doi: 10.1007/s10237-010-0208-8. Google Scholar

[9]

C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations," With separately available software, Frontiers in Applied Mathematics, 16,, Society for Industrial and Applied Mathematics (SIAM), (1995). Google Scholar

[10]

C. T. Kelley, "Solving Nonlinear Equations with Newton's Method,", Fundamentals of Algorithms, 1 (2003). Google Scholar

[11]

S. V. Komarova, Mathematical model of paracrine interactions between osteoclasts and osteoblasts predicts anabolic action of parathyroid hormone on bone,, J. Endocrinol., 146 (2005), 3589. doi: 10.1210/en.2004-1642. Google Scholar

[12]

S. V. Komarova, R. J. Smith, S. J. Dixon, S. M. Sims and L. H. Wahl, Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling,, Bone, 33 (2003), 225. doi: 10.1016/S8756-3282(03)00157-1. Google Scholar

[13]

M. J. Martin and J. C. Buckland-Wright, Sensitivity analysis of a novel mathematical model identifies factors determining bone resorption rates,, Bone, 35 (2004), 918. doi: 10.1016/j.bone.2004.06.010. Google Scholar

[14]

M. J. Martin and J. C. Buckland-Wright, A novel mathematical model identifies potential factors regulating bone apposition,, Calcif. Tissue Int., 77 (2005), 250. doi: 10.1007/s00223-005-0101-0. Google Scholar

[15]

I. M. Mitchell, The flexible, extensible and efficient toolbox of level set methods,, J. Sci. Comp., 35 (2008), 300. doi: 10.1007/s10915-007-9174-4. Google Scholar

[16]

S. Osher and R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces," Applied Mathematical Sciences, 153,, Springer-Verlag, (2003). Google Scholar

[17]

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. Google Scholar

[18]

S. Osher and C. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations,, SIAM J. Numer. Anal., 28 (1991), 907. doi: 10.1137/0728049. Google Scholar

[19]

A. M. Parfitt, Osteonal and hemi-osteonal remodeling: The spatial and temporal framework for signal traffic in adult human bone,, J. Cell. Biochem., 55 (1994), 273. Google Scholar

[20]

V. Peiffer, A. Gerisch, D. Vandepitte, H. Van Oosterwyck and L. Geris, A hybrid bioregulatory model of angiogenesis during bone fracture healing,, Biomech. Model. Mechanobiol., 10 (2011), 383. doi: 10.1007/s10237-010-0241-7. Google Scholar

[21]

D. Peng, B. Merriman, S. Osher, H. Zhao and M. Kang, A PDE-based fast local level set method,, J. Comput. Phys., 155 (1999), 410. doi: 10.1006/jcph.1999.6345. Google Scholar

[22]

P. Pivonka and S. V. Komarova, Mathematical modeling in bone biology: From intracellular signaling to tissue mechanics,, Bone, 47 (2010), 181. doi: 10.1016/j.bone.2010.04.601. Google Scholar

[23]

P. Pivonka, J. Zimak, D. W. Smith, B. S. Gardiner, C. R. Dunstan, N. A. Sims, T. J. Martin and G. R. Mundy, Model structure and control of bone remodeling: A theoretical study,, Bone, 43 (2008), 249. doi: 10.1016/j.bone.2008.03.025. Google Scholar

[24]

L. G. Raisz, Physiology and pathophysiology of bone remodeling,, Clinical Chemistry, 45 (1999), 1353. Google Scholar

[25]

A. G. Robling, A. B. Castillo and C. H. Turner, Biomechanical and molecular regualtion of bone remodeling,, Annu. Rev. Biomed. Eng., 8 (2006), 455. doi: 10.1146/annurev.bioeng.8.061505.095721. Google Scholar

[26]

M. D. Ryser, S. V. Komarova and N. Nigam, The cellular dynamics of bone remodeling: A mathematical model,, SIAM J. Appl. Math., 70 (2010), 1899. doi: 10.1137/090746094. Google Scholar

[27]

M. D. Ryser, N. Nigam and S. V. Komarova, Mathematical modeling of spatio-temporal dyanmics of a single bone multicellular unit,, J. Bone Miner. Res., 24 (2009), 860. doi: 10.1359/jbmr.081229. Google Scholar

[28]

D. Salac and W. Lu, A local semi-implicit level-set method for interface motion,, J. Sci. Comput., 35 (2008), 330. doi: 10.1007/s10915-008-9188-6. Google Scholar

[29]

J. A. Sethian, "Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science," Second edition, Cambridge Monographs on Applied and Computational Mathematics, 3,, Cambridge University Press, (1999). Google Scholar

[30]

B. Sumengen, A matlab toolbox implementing level set methods, 2004., Available from: \url{http://barissumengen.com/level_set_methods/}., (). Google Scholar

[31]

H. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion,, J. Comput. Phys., 127 (1996), 179. doi: 10.1006/jcph.1996.0167. Google Scholar

[32]

H. Zhao, B. Merriman, S. Osher and L. Wang, Capturing the behavior of bubbles and drops using the variational level set approach,, J. Comput. Phys., 143 (1998), 495. doi: 10.1006/jcph.1997.5810. Google Scholar

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