Analysis of a bone remodeling model with myeloma disease arising in cellular dynamics

Abstract

In this work we study a bone remodeling model for the evolution of the myeloma disease. The biological problem is written as a coupled nonlinear system consisting of parabolic partial differential equations. They are written in terms of the concentrations of osteoblasts and osteoclasts, the density of the relative bone and the concentration of the tumor cells. Then, we deal with the numerical analysis of this variational problem, introducing a numerical approximation by using the finite element method and a hybrid combination of both implicit and explicit Euler schemes. We perform some a priori error estimates and show a few numerical simulations to demonstrate the accuracy of the approximation. Finally, we present the comparison with previous works and the behavior of the solution in two‐dimensional examples.

Spatial extension of a bone remodeling dynamics model and its finite element analysis

“…The ripples shown in Refs. 29,30 also appear in the results of this paper, decaying to the uniform solution as expected.…”

“…With this model, we focus on the dynamical relations between cells. In the one‐dimensional example the solution shows an ondulatory behavior also seen in previous published works 29,30 . Qualitatively, the decay to the steady‐state solution shown in the original work by Graham and colleagues 27 and seen in experiments also appears in this example.…”

Tracing the footsteps of autophagy in computational biology

“…PDE, unlike ODE, investigates spatial patterns of inherently heterogeneous systems with regionally varying fields. A literature study has revealed the application of these models in various fields, including bone remodeling [ 142 ], dengue [ 143 ] and cancer [ 144 ]. Studies complimented with PDE and ML approaches have also opened a new window for disease treatment [ 145 , 146 ].…”