Analysis of the uniqueness and stability of solutions to problems regarding the bone-remodeling process

• Published in: Medical engineering & physics Vol. 85; pp. 113 - 122
• Main Authors: Oening Dicati, Gabriela Wessling, Gubaua, José Eduardo, Pereira, Jucélio Tomás
• Format: Journal Article
• Language: English
• Published: England Elsevier Ltd 01-11-2020
• Subjects: Bone remodeling simulation; Lazy zone; Phenomenological model; Uniqueness of the solution; Phenomenological model; Bone remodeling simulation; Lazy zone; Uniqueness of the solution
• ISSN: 1350-4533; 1873-4030
• DOI: 10.1016/j.medengphy.2020.10.007

•A set of modifications that guarantee the uniqueness and stability of the solutions in phenomenological bone-remodeling simulations.•Checkerboard control using simple nodal average approach applied on the stress field.•The lazy zone is replaced by the lazy point concept similar to the classic fully stressed design concept.•Bone remodeling simulation performed in a 3D human femur geometry.•When all proposed modifications are used it is possible to observe the tendency toward a unique solution.
Simulation of the bone remodeling process is extremely important because it makes possible the structure forecast of one or several bones when anomalous situations, such as prosthesis installation, occur. Thus, it is necessary that the mathematical model to simulate the bone remodeling process be reliable; that is, the numerical solution must be stable regardless of initial density field for a phenomenological approach to model the process. For several models found in the literature, this characteristic of stability is not observed, largely due to the discontinuities present in the property values of the models (e.g., Young's modulus and Poisson's ratio). In addition, checkerboard formation and the lazy zone prevent the uniqueness of the solution. To correct these difficulties, this study proposes a set of modifications to guarantee the uniqueness and stability of the solutions, when a phenomenological approach is used. The proposed modifications are: (a) change the rate of remodeling curve in the lazy zone region and (b) create transition functions to guarantee the continuity of the expressions used to describe Young's modulus and Poisson's ratio. Moreover, the stress smoothing process controls the checkerboard formation. Numerical analysis is used to simulate the solution behavior from each proposed modification. The results show that, when all proposed modifications are applied to the three-dimensional models simulated here, it is possible to observe the tendency toward a unique solution.