A phenomenological model for the degradation of biodegradable polymers Ying Wang, Jingzhe Pan * , Xiaoxiao Han, Csaba Sinka, Lifeng Ding Department of Engineering, University of Leicester, Leicester LE1 7RH, UK article info Article history: Received 26 February 2008 Accepted 24 April 2008 Available online 19 May 2008 Keywords: Biodegradable polymers Biodegradation Modelling Finite element analysis abstract This paper presents a phenomenological diffusion–reaction model for the biodegradation of bio- degradable polymers. The biodegradation process is modelled using a set of simplified reaction–diffusion equations. These partial differential equations are non-dimensionalised giving two normalised param- eters which control the interplay between the hydrolysis reaction and the monomer diffusion. The equations are firstly solved for simple cases of plates and pins. The numerical results are presented in the form of biodegradation maps which show the conditions where the biodegradation is controlled by auto- catalysed hydrolysis, non-catalysed hydrolysis, a combination of auto-catalysed and non-catalysed hy- drolyses, or a combination of hydrolysis and monomer diffusion, respectively. The degradation maps provide a clear guide for the design of biodegradable fixation devices used in orthopaedic surgeries. Finally the diffusion–reaction equations are solved using the finite element method for strip and square meshes, showing how the model can be used to assist the design of sophisticated fixation devices. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction Biodegradable polymers, typically a combination of polylactic acids (PLA) and polyglycolic acid (PGA) of various forms, are cur- rently being used in the human body as fixation devices in the form of screws, pins, meshes, etc., to protect the healing of fractured bones. The advantage of using biodegradable fixations instead of metallic ones is obvious – a device simply ‘‘disappears’’ after the bone heals, eliminating the need for secondary surgery and avoiding stress shielding, which weakens living bones. Dedicated commer- cial companies have been set up to manufacture such devices. Many case studies have been published by orthopaedic surgeons [1,2].A large amount of detailed studies, both in vitro and in vivo, have been published on the degradation mechanisms and the factors con- trolling the degradation rate of a range of biodegradable polymers [3]. It has been shown that the degradation behaviour of the bio- degradable implants is complicated. For example, a thicker plate consisted of PLA degrades faster than a thinner one made of the same polymer [4]. This is because the structure change of PLA is heterogeneous in an implant due to the autocatalytic nature of the hydrolysis reaction of PLA [5]. The choice of dimensions for a par- ticular device is therefore not straightforward – a thicker device gives higher initial strength comparing with a thinner one but it also degrades faster. This complicated behaviour makes it difficult to optimise the design of a biodegradable device. The typical degra- dation period of an implant can be several months; hence, the trial-and-error approach is problematic. A computer-aided design approach, as being practiced routinely in engineering, would be ideal to overcome the difficulty. However, there has been very limited amount of work in the literature on mathematical modelling of biodegradation. Most of the published papers are in the context of controlled drug release, almost all of which are for very simple geometries [6–10]. Furthermore the mathematical equations have often been presented in terms of specific reactions, giving an im- pression that they are only valid for specific polymer systems. The purpose of this paper is to demonstrate that it is possible to extract a general phenomenological model for the biodegradation from the existing experimental and modelling works, and that modern numerical techniques can be used to solve the corresponding equations for devices of both simple and complicated geometries. The computer model can be used to assist material selection and device design, and as a valuable tool for surgeons’ training. 2. Reaction–diffusion equations for biodegradation Many biodegradable polymers have been developed but mate- rials used for the orthopaedic fixation are mostly limited to poly- glycolic acid (PGA), polylactic acids (PLA) and their copolymers because of their well-established biocompatibility. Two different morphological forms of PLA, i.e. DLPLA and LPLA, are often used. PGA is the simplest linear, aliphatic polyester and highly crystalline. It is the most hydrophilic one among PGA, DLPLA and LPLA and fully biodegrades within 6–12 months. DLPLA is always amorphous and takes 12–16 months to biodegrade. LPLA is semicrystalline. It is the most hydrophobic one among the three and takes more than 24 months to biodegrade. By co-polymerising PGA, LPLA and DLPLA * Corresponding author. Tel.: þ44 116 223 1092; fax: þ44 116 252 2525. E-mail address: jp165@leicester.ac.uk (J. Pan). Contents lists available at ScienceDirect Biomaterials journal homepage: www.elsevier.com/locate/biomaterials 0142-9612/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.biomaterials.2008.04.042 Biomaterials 29 (2008) 3393–3401