Modeling of Bone Remodeling and Bone Metastases

Marc D. Ryser, N. Nigam and S.V. Komarova

The process of bone remodeling is very important for the removal of fatigue damage and the renewal of old bone tissue. Responsible for the remodeling process are the so-called bone multicellular units (BMUs), complex entities consisting of several interacting cell types. Pathologies in the remodeling process are known to lead to diseases such as osteoporosis and osteoarthritis, and hence a precise understanding of the BMU functioning is very important for clinical purposes.

Since in vivo experiments on BMUs are very lengthy, difficult and expensive, mathematical modeling provides a promising tool in the quest for a better understanding of the underlying processes. Within the framework of this project, we developed a nonlinear PDE model, which captures the dynamics of osteoclasts (bone resorption) and osteoblasts (bone production), as well as the RANK/RANKL/OPG pathway [1;2]. More recently, we modified the model to investigate the impact of the RANK/RANKL/OPG pathway on bone metastases of breast and prostate cancers [3]. From a mathematical point of view, the nonlinear nature of the equations and the emergence of traveling waves pose interesting questions in their own right.

[1] Mathematical modeling of spatio-temporal dynamics of a single bone multicellular unit. M.D. Ryser, N. Nigam, S.V. Komarova; J Bone Min Res (2009) 24:860-70

[2] The cellular dynamics of bone remodeling: a mathematical model. M.D. Ryser, N. Nigam, S.V. Komarova; SIAM J. Appl. Math. Volume 70, Issue 6, pp. 1899-1921 (2010)

[3] Osteoprotegerin in bone metastases: mathematical solution to the puzzle. M.D. Ryser, S.V. Komarova, Y. Qu, Submitted to J Bone Min Res (2011)

Figure 1: BMU moving across the bone tissue. First, 10-20 osteoclasts resorb old or damaged tissue; then they recruit 1,000-2,000 osteoblasts which produce new bone matrix. The BMU moves at a speed of 20-40 microns/day and survives for up to 6 months.

Figure 2: Modeling the BMU evolution: RANKL (osteoclast stimulator), OPG (decoy receptor for RANKL) and osteoclast (OC) densities after 90 days.

Nonlinear Stochastic Partial Differential Equations in Higher Dimensions

Marc D. Ryser, P.F. Tupper, N. Nigam, M. Hairer, and H. Weber

The theory of linear and nonlinear parabolic SPDEs driven by additive white noise is well established in the one dimensional case. In addition, the convergence properties of space-time discretizations of such equations have been extensively studied. However, in dimensions two and higher, the nature of these equations is quite different: already the linearized versions admit merely distribution-valued solutions. Even though the higher-dimensional nonlinear equations are generally assumed to be ill-posed in the mathematics community, they are frequently used to model physical systems in the applied sciences. We investigated this discrepancy on the example of the white noise-driven Allen-Cahn equation in two dimensions. In silico experiments suggested that numerical discretizations converged to the zero-distribution with vanishing mesh size [4]. Using tools of stochastic quantization, we devised a rigorous proof for this limit result [5]. [4] On the well-posedness of the stochastic

[4] On the well-posedness of the stochastic Allen-Cahn equation in two dimensions. M.D. Ryser, P.F. Tupper, N. Nigam; Preprint on ArXiv.org (2011)

[5] M.D. Ryser, H. Weber, M. Hairer. In preparation (2011)

Figure 3: Numerical solution of the white noise-driven Allen-Cahn equation on the two-dimensional torus. By refining the grid (A to D), the underlying pattern of the deterministic equation (E) gets washed out: the noise becomes (relatively speaking) too strong, and we observe convergence to the zero-distribution.