# 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
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