# American Institute of Mathematical Sciences

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). [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). [3] J. P. Bilezikian, L. G. Raisz and G. A. Rodan, "Principles of Bone Biology," Second edition, Academic Press, Boston, 2002. [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), 280-289. doi: 10.1007/s11517-006-0040-6. [5] L. Geris, A. Gerisch and R. C. Schugart, Mathematical modeling in wound healing, bone regeneration and tissue engineering, Acta Biotheor, 58 (2010), 355-367. doi: 10.1007/s10441-010-9112-y. [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), 137-158. doi: 10.1016/j.jtbi.2007.11.008. [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), 189-193. [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), 713-724. doi: 10.1007/s10237-010-0208-8. [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), Philadelphia, PA, 1995. [10] C. T. Kelley, "Solving Nonlinear Equations with Newton's Method," Fundamentals of Algorithms, 1, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. [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), 3589-3595. doi: 10.1210/en.2004-1642. [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), 225-234. doi: 10.1016/S8756-3282(03)00157-1. [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), 918-928. doi: 10.1016/j.bone.2004.06.010. [14] M. J. Martin and J. C. Buckland-Wright, A novel mathematical model identifies potential factors regulating bone apposition, Calcif. Tissue Int., 77 (2005), 250-260. doi: 10.1007/s00223-005-0101-0. [15] I. M. Mitchell, The flexible, extensible and efficient toolbox of level set methods, J. Sci. Comp., 35 (2008), 300-329. doi: 10.1007/s10915-007-9174-4. [16] S. Osher and R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces," Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003. [17] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [18] S. Osher and C. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907-922. doi: 10.1137/0728049. [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), 273-286. [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), 383-395. doi: 10.1007/s10237-010-0241-7. [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), 410-438. doi: 10.1006/jcph.1999.6345. [22] P. Pivonka and S. V. Komarova, Mathematical modeling in bone biology: From intracellular signaling to tissue mechanics, Bone, 47 (2010), 181-189. doi: 10.1016/j.bone.2010.04.601. [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), 249-263. doi: 10.1016/j.bone.2008.03.025. [24] L. G. Raisz, Physiology and pathophysiology of bone remodeling, Clinical Chemistry, 45 (1999), 1353-1358. [25] A. G. Robling, A. B. Castillo and C. H. Turner, Biomechanical and molecular regualtion of bone remodeling, Annu. Rev. Biomed. Eng., 8 (2006), 455-498. doi: 10.1146/annurev.bioeng.8.061505.095721. [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), 1899-1921. doi: 10.1137/090746094. [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), 860-870. doi: 10.1359/jbmr.081229. [28] D. Salac and W. Lu, A local semi-implicit level-set method for interface motion, J. Sci. Comput., 35 (2008), 330-349. doi: 10.1007/s10915-008-9188-6. [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, Cambridge, 1999. [30] B. Sumengen, A matlab toolbox implementing level set methods, 2004., Available from: \url{http://barissumengen.com/level_set_methods/}., (). [31] H. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), 179-195. doi: 10.1006/jcph.1996.0167. [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), 495-518. doi: 10.1006/jcph.1997.5810.

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). [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). [3] J. P. Bilezikian, L. G. Raisz and G. A. Rodan, "Principles of Bone Biology," Second edition, Academic Press, Boston, 2002. [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), 280-289. doi: 10.1007/s11517-006-0040-6. [5] L. Geris, A. Gerisch and R. C. Schugart, Mathematical modeling in wound healing, bone regeneration and tissue engineering, Acta Biotheor, 58 (2010), 355-367. doi: 10.1007/s10441-010-9112-y. [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), 137-158. doi: 10.1016/j.jtbi.2007.11.008. [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), 189-193. [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), 713-724. doi: 10.1007/s10237-010-0208-8. [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), Philadelphia, PA, 1995. [10] C. T. Kelley, "Solving Nonlinear Equations with Newton's Method," Fundamentals of Algorithms, 1, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. [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), 3589-3595. doi: 10.1210/en.2004-1642. [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), 225-234. doi: 10.1016/S8756-3282(03)00157-1. [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), 918-928. doi: 10.1016/j.bone.2004.06.010. [14] M. J. Martin and J. C. Buckland-Wright, A novel mathematical model identifies potential factors regulating bone apposition, Calcif. Tissue Int., 77 (2005), 250-260. doi: 10.1007/s00223-005-0101-0. [15] I. M. Mitchell, The flexible, extensible and efficient toolbox of level set methods, J. Sci. Comp., 35 (2008), 300-329. doi: 10.1007/s10915-007-9174-4. [16] S. Osher and R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces," Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003. [17] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [18] S. Osher and C. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907-922. doi: 10.1137/0728049. [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), 273-286. [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), 383-395. doi: 10.1007/s10237-010-0241-7. [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), 410-438. doi: 10.1006/jcph.1999.6345. [22] P. Pivonka and S. V. Komarova, Mathematical modeling in bone biology: From intracellular signaling to tissue mechanics, Bone, 47 (2010), 181-189. doi: 10.1016/j.bone.2010.04.601. [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), 249-263. doi: 10.1016/j.bone.2008.03.025. [24] L. G. Raisz, Physiology and pathophysiology of bone remodeling, Clinical Chemistry, 45 (1999), 1353-1358. [25] A. G. Robling, A. B. Castillo and C. H. Turner, Biomechanical and molecular regualtion of bone remodeling, Annu. Rev. Biomed. Eng., 8 (2006), 455-498. doi: 10.1146/annurev.bioeng.8.061505.095721. [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), 1899-1921. doi: 10.1137/090746094. [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), 860-870. doi: 10.1359/jbmr.081229. [28] D. Salac and W. Lu, A local semi-implicit level-set method for interface motion, J. Sci. Comput., 35 (2008), 330-349. doi: 10.1007/s10915-008-9188-6. [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, Cambridge, 1999. [30] B. Sumengen, A matlab toolbox implementing level set methods, 2004., Available from: \url{http://barissumengen.com/level_set_methods/}., (). [31] H. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), 179-195. doi: 10.1006/jcph.1996.0167. [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), 495-518. doi: 10.1006/jcph.1997.5810.
 [1] J. R. Fernández, R. Martínez, J. M. Viaño. Analysis of a bone remodeling model. Communications on Pure and Applied Analysis, 2009, 8 (1) : 255-274. doi: 10.3934/cpaa.2009.8.255 [2] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems and Imaging, 2021, 15 (2) : 315-338. doi: 10.3934/ipi.2020070 [3] Yoshikazu Giga, Hiroyoshi Mitake, Hung V. Tran. Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3983-3999. doi: 10.3934/dcdsb.2019228 [4] Esther Klann, Ronny Ramlau, Wolfgang Ring. A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. Inverse Problems and Imaging, 2011, 5 (1) : 137-166. doi: 10.3934/ipi.2011.5.137 [5] Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems and Imaging, 2021, 15 (3) : 387-413. doi: 10.3934/ipi.2020073 [6] Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047 [7] Bin Dong, Aichi Chien, Yu Mao, Jian Ye, Fernando Vinuela, Stanley Osher. Level set based brain aneurysm capturing in 3D. Inverse Problems and Imaging, 2010, 4 (2) : 241-255. doi: 10.3934/ipi.2010.4.241 [8] Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029 [9] Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic and Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237 [10] Grzegorz Dudziuk, Mirosław Lachowicz, Henryk Leszczyński, Zuzanna Szymańska. A simple model of collagen remodeling. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2205-2217. doi: 10.3934/dcdsb.2019091 [11] Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390 [12] Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems and Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459 [13] Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347 [14] Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems and Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479 [15] Jonas Lampart. A remark on the attainable set of the Schrödinger equation. Evolution Equations and Control Theory, 2021, 10 (3) : 461-469. doi: 10.3934/eect.2020075 [16] Ariosto Silva, Alexander R. A. Anderson, Robert Gatenby. A multiscale model of the bone marrow and hematopoiesis. Mathematical Biosciences & Engineering, 2011, 8 (2) : 643-658. doi: 10.3934/mbe.2011.8.643 [17] Avner Friedman, Kang-Ling Liao. The role of the cytokines IL-27 and IL-35 in cancer. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1203-1217. doi: 10.3934/mbe.2015.12.1203 [18] 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 [19] Urszula Ledzewicz, Heinz Schättler. Controlling a model for bone marrow dynamics in cancer chemotherapy. Mathematical Biosciences & Engineering, 2004, 1 (1) : 95-110. doi: 10.3934/mbe.2004.1.95 [20] Ana Isabel Muñoz, J. Ignacio Tello. On a mathematical model of bone marrow metastatic niche. Mathematical Biosciences & Engineering, 2017, 14 (1) : 289-304. doi: 10.3934/mbe.2017019

2018 Impact Factor: 1.313

## Metrics

• PDF downloads (25)
• HTML views (0)
• Cited by (7)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]