Mathematical modelling of the degradation behaviour of biodegradable metals

Abstract

A mathematical model for the biodegradation of magnesium is developed in this study to inspect the corrosion behaviour of biodegradable implants. The aim of this study was to provide a suitable framework for the assessment of the corrosion rate of magnesium which includes the process of formation/dissolution of the protective film. The model is intended to aid the design of implants with suitable geometries. The level-set method is used to follow the changing geometry of the implants during the corrosion process. A system of partial differential equations is formulated based on the physical and chemical processes that occur at the implant-medium boundary in order to simulate the effect of the formation of a protective film on the degradation rate. The experimental data from the literature on the corrosion of a high-purity magnesium sample immersed in simulated body fluid is used to calibrate the model. The model is then used to predict the degradation behaviour of a porous orthopaedic implant. The model successfully reproduces the precipitation of the corrosion products on the magnesium surface and the effect on the degradation rate. It can be used to simulate the implant degradation and the formation of the corrosion products on the surface of biodegradable magnesium implants with complex geometries.

A simple model for long‐time degradation of magnesium under physiological conditions

“…The introduced mathematical model describes and even reproduces the complex degradation/corrosion models of Deslousis et al, Höche, Bajger et al, or Sun et al Comparing the parameters with the existing models their physicochemical background can be described like: rate constant: describes the dissolution and the effective Mg transport through the deposits toward interaction with the electrolyte, and its balancing. In this way it includes: porosity information (barrier properties) electrochemical kinetics qualitative impurity level information and/or information on secondary phases/alloying elements information on the strength of electrolyte interaction with the material layer growth parameter: a describes the affinity for the formation of a layer (deposit) and thus includes information on: chemical kinetics (incl.…”

“…The rate constant and also the layer growth parameter empirically describe the interaction between electrochemical reactions, degradation product compound formation, transport phenomena, chemical conversion, and the final degradation, while the offset parameter h 0 describes the initially accelerated degradation of the bare metallic surface. The introduced mathematical model describes and even reproduces the complex degradation/corrosion models of Deslousis et al, Höche, Bajger et al, or Sun et al [24][25][26][27] Comparing the parameters with the existing models their physicochemical background can be described like:…”

Continuum Modeling and Simulation in Bone Tissue Engineering

“…In this context, Grogan et al [115,116] presented physically based continuum models, based on reaction-diffusion equations, available to analyze the corrosion of magnesium metal stents. A similar mathematical modeling was shown in [117] but using the level-set strategy to simulate the dissolution of the biomaterial. Sanz-Herrera et al [118] developed a continuum model for the biodegradation of magnesium accounting for the dynamics and evolution of secondary species, such as pH or corrosion products.…”

“…Sci. 2019, 9, x FOR PEER REVIEW 9 of 16 similar mathematical modeling was shown in [117] but using the level-set strategy to simulate the dissolution of the biomaterial. Sanz-Herrera et al [118] developed a continuum model for the biodegradation of magnesium accounting for the dynamics and evolution of secondary species, such as pH or corrosion products.…”

Numerical Modelling of Effects of Biphasic Layers of Corrosion Products to the Degradation of Magnesium Metal In Vitro

“…There are numerous studies of similar problems for a range of metals that neglect the advective contribution to the corrosion dynamics, resulting in Stefan-like problems to describe corrosion in, for example, zirconium [ 20 ] and steel [ 21 ]. Magnesium corrosion has been the subject of a small number of modelling studies [ 22 , 23 , 24 ]. In [ 23 ], a simple two-phase bulk model of magnesium corrosion was proposed and parameter fitted to experimental data; however, the model is not explicit in the products of corrosion.…”