Finite element analysis of degradation of biodegradable medical devices

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1 Page 1 of 10 Medical Biotechnology Finite element analysis of degradation of biodegradable medical devices X Han 1, J Pan 2 * Abstract Introduction This paper presents a set of case studies for calculating the degradation rates of medical devices made of biodegradable polymers using the finite element method. Case Study Firstly, a set of experimental data in the literature showing size effect of the degradation was analysed; it was shown that the finite element model is able to fit the experimental data fairly well. Then four different devices were analysed to demonstrate the applications of finite element analysis in device design. These include (a) a simple cube to demonstrate the three dimensional effect of device degradation, (b) a typical scaffold for tissue engineering, (c) a fixation screw for orthopaedic surgery and (d) a coronary stent. Conclusion The analysis shows how the design details can affect the degradation rate of the various implants/devices. Introduction Biodegradable polymers, especially linear aliphatic polymers, have found great attractions in a broad range of medical applications: they were firstly used to made sutures successfully in the 1970s 1, afterwards they drew great interests in fields of orthopaedic fixation devices, controlled drug release and scaffolds in tissue engineering. Biodegradable polymers gradually replace conventional *Corresponding author jp165@le.ac.uk 1 Wolfson school of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, LE11 3TU, UK 2 Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK biomaterials in many medical applications owing to their nature of degradation. Hydrolysis degradation turns polymers into smaller dissolvable molecules which are then eventually metabolised into carbon dioxide and water after they served their functions. Polyesters, such as polyglycolide (PGA), poly (L-lactide) (PLA) and their copolymers (PGA-co-PLA), are of the greatest interest because of their well-established biodegradability, biocompatibility and mechanical properties. Degradation of biodegradable devices is, however, a complicated chemical-physical process. It depends on the chemical structure of polymer, the shape and size of the device and the degradation environment. Degradation therefore ranges from weeks to years for different polymers and devices. Heterogeneous degradation was demonstrated by Li et al. 2 and Hurrell et al. 3 through a set of experiments. They showed that the core of samples degraded much faster than the surface. In particular, the experimental results obtained by Grizzi et al. 4 showed that a plate of 2 mm thick degrades faster than a film of 0.3 mm thick. This is known as size effect in PLA/PGA degradation. Grizzi et al. 4 suggested that the heterogeneous degradation and size effect are results of auto-catalytic hydrolysis reaction. Dissolved short chains produced by chain scissions have carboxylic end groups and hydroxyl end groups. The carboxylic end groups have a high degree of proton donor rate, thus significantly accelerates the hydrolysis rate. Diffusion of short chains therefore plays a critical role in controlling the overall degradation profile. Size effect and heterogeneous degradation make it difficult to transfer experimental experience from one device to another even if they are made of the same polymer. A mathematical framework has been developed by Pan and his co-workers 5 11, which captures the dominating mechanisms in degradation including hydrolysis, crystallisation and short chain diffusion. It has been shown that the model can fit a wide range of the experimental data and that the size effect and heterogeneous degradation can be predicted using the mathematical model 6,10. The purpose of this paper is to show that the mathematical model can be implemented in commercial finite element software and used for the design of medical implants of any sophisticated shapes. Firstly, the experimental data obtained by Grizzi et al. 4 were analysed which revealed a complicated mechanism for the reported size effect. Then, four different devices are analysed to demonstrate the application of the finite element in device design. These include (a) a simple cube to demonstrate the three dimensional effect of device degradation, (b) a typical scaffold for tissue engineering, (c) a fixation screw for orthopaedic surgery and (d) a coronary stent. The analysis demonstrated how the design variables can affect the degradation rate. The protocol of this study has been approved by the relevant ethical committee related to our institution in which it was performed. The mathematical model We focus on amorphous polymers in this study for simplicity although the mathematical model can handle semi-crystalline polymers without

2 Page 2 of 10 any difficulty 5,6. Following Han et al. 6,8, the rate equation for polymer chain scission during hydrolysis degradation can be written as drs Rs = Ce β dt 0 1 α Ce 0 k + k C n 1 2 e0 1 C m C ol e0 n (1) in which R s is the mole concentration of polymer chain scissions, C ol is the mole concentration of ester units of dissolvable short chains and C e0 is the initial mole concentration of ester units of the polymer chains. The reaction-diffusion equation for C ol can be written as dc dt ol R s drs = αβ + Ce0 dt 3 i= 1 x i β 1 D C ol. xi (2) In the above equations, k 1 and k 2 are rates for non-catalytic hydrolysis and auto-catalytic hydrolysis reactions respectively; a and b are related to the production rate of the short chains by chain scission, m is the average degree of polymerisation of the short chains and n is the exponent for acid disassociation which is usually taken as 0.5. D is the diffusion coefficient of the short chains in a degrading polymer which is given by ( ) D= D V 03. V ( Dpore Dpolymer ) 2 3 polymer pore pore (3) in which V pore is the porosity of the polymer caused by the loss of short chains and given by V pore ( ) Rol Col Col0 = Ce0 Rs Col Col = (4) β 0 α Ce0 Ce0 and D polymer and D pore are diffusion coefficients of the short chains in nondegraded polymer and liquid-filled pores respectively. C ol0 is the concentration of residual short chains that may exist in the polymer. The average molecular weight is calculated using M M n n0 Rs 1 Ce0 = α Rs α R s 1 + N dp0 Ce0 m Ce0, (5) in which M n0 is the initial numberaveraged molecular weight, N dp0 is the initial average degree of polymerisation of the polymer. N dp0 = M n0 /M 0 in which M 0 is the molecular weight of a single repeating unit of the polymer. Full details of the mathematical model and its experimental validation can be found in the work by Pan and his coworkers In the current paper, equations (1) and (2) are solved for different devices numerically using a commercial finite element package COMSOL Multiphysics. Convergence studies are carried out for each case in order to determine the appropriate element density and time step length. Analysis of experimental data obtained by Grizzi et al. Grizzi et al. 4 presented a set of experimental data to demonstrate size dependence in degradation of PLA. Among many other data, they showed molecular weight and mass loss as functions of time for plate samples of 2 mm in thickness and thin film samples of 0.3 mm in thickness made from a PLA50. Their results show that the thick samples degrade significantly faster than the thin ones. They proposed a schematic reaction-diffusion model to explain the size effect of the degradation. Wang et al. 10 and Han and Pan 8 analysed these data using actual reaction-diffusion models. However, in these previous models, the potential existence of a large amount of residual monomers in the samples β β was not considered. Here these data are re-analysed assuming different amounts of residual monomers in the samples. Grizzi et al. 4 did not report any data on residual monomers in their samples. However, their mass loss data show that the thin samples lost 5% of their weight at the very beginning of the degradation tests. It is not possible for polymer chain scission to produce this amount of short chains. The only logical explanation is that these samples must have 5% residual monomers before the test began which dissolved into the aqueous medium. Their thick samples on the other hand did not show the same level of mass loss. In this study, separate finite element models for their plate and film samples are built. We limit our analysis to the first 10 weeks of the degradation experiment because after which time the samples start to break up and the mathematical model becomes invalid. The following initial values and boundary conditions are used: (a) Initial conditions: R s = 0, C ol = C ol0, at t = 0 (6) (b) Boundary condition: C ol = 0, at the surface of the samples (7) Table 1 provides the full set of model parameters which are used in the finite element analysis. In the table, the values for initial concentration of ester units of the polymer, C e0, and molecular weight of a repeating unit of the polymers, M 0, are typical values for PLA. The initial molecular weight, M n0, is different for the thin and thick samples. The values for M n0 are directly taken from the paper by Grizzi et al. 4. It is widely recognised that short PLA/PGA chains under 8 repeating units are water soluble. Hence the average number of ester units for the short chains, m, is set as 4. It is assumed that the polymer degradation occurs through

3 Page 3 of 10 random chain scission 8, which leads to the values for a and b in the table. The values for C ol0, which represent the concentration of residual monomers in the samples, are set as 5% for the thin samples and 0% for the thick ones. The fast mass loss for the thin samples also indicates either a very large diffusion coefficient for the small chains or other mechanisms of weight loss such as direct dissolution off the sample surface. The values of k 1, k 2 (hydrolysis rate constants) and D polymer (diffusion coefficient of short chains in non-degraded polymer) are obtained by fitting the finite element predictions with the experimental data for molecular weight and mass loss as functions of time. Figures 1 and 2 present the best fitting achieved using the parameters in Table 1. The finite element analysis provides spatial distribution of the average molecular weight as a function of time. The average molecular weight shown in Figure 1 is the average value of the molecular weight over the entire volume of the samples. It can be seen that the finite element models can fit the experimental data fairly well. It is however important to highlight the role played by the residual monomers in the model prediction. Figure 3 shows the calculated average molecular weight comparing with the experimental data using the same set of parameters except that residual monomers were set as zero for both the film and plate samples. It can be observed that the fitting actually improved. However, the model for thin samples will no longer be able to predict any significant mass loss. Figure 4 shows the calculated mass loss as function of time for the thin and thick samples assuming zero residual monomers. It can be observed from Figure 4 that the mass loss is very small in the first ten weeks. Mass loss of the thick samples accelerates around week 5 which agrees with the trend shown by the experimental data. In the remaining demonstration case studies, parameters in Table 1 are used except that (a) the initial Figure 1: Normalised number average molecular weights of plate and film as functions of time. Discrete symbols are the experimental data by Grizzi et al. 4 and the lines are fitting of the finite element models. Figure 2: Mass loss of plate and film samples as functions of time. Discrete symbols are the experimental data by Grizzi et al. 4 and the lines are fitting of the finite element models. number average molecular weight is set as 30,000 g/mol, (b) C ol0 is set as zero to reflect perfect polymerisation and (c) a common diffusion coefficient of D polymer = m 2 /week is used. This is a very large diffusion coefficient which is chosen here to reflect fast dissolution of short chains into the aqueous medium in consistency with the study in the previous section. The demonstration can be made by setting the parameters at any other set of values. A - a three dimensional cube Figure 5 shows a mm cube which is the simplest possible device and used here to demonstrate the three dimensional effect of

4 Page 4 of 10 Table 1 C e0 = mol/m 3 M 0 = 72 g/mol M n0 = g/mol for plate Model parameters used for finite element analysis of data by Grizzi et al. 4 device degradation. Governing equations (1) and (2) are solved numerically using the PDE interfaces module from COMSOL Multiphysics. A total number of 2321 elements are used as shown in figure 5(a) which has degrees of freedom in the finite element model. This mesh density was tested to be fine enough for obtaining a converged result. The boundary condition is that C ol = 0 at all the boundaries. Figure 6 shows the distribution of average molecular weight over the cross- section indicated by Figure 5b at three different times. The molecular weight is normalised by its initial value. It can be observed that the degradation k 1 = /week k 2 = (m 3 /mol) 0.5 /week D polymer = m 2 /week M n0 = g/mol for film D pore = D polymer 1000 m = 4 α = 28, β = 2 C ol0 = 5% C e0 for plate C ol0 = 0 C e0 for film Figure 3: Repeated calculation of Figure 1 assuming zero residual monomers for both film and plate samples. Discrete symbols are the experimental data by Grizzi et al. 4 and the lines are fitting of the finite element models. starts to become heterogeneous by week 3. At week 5, M n in the centre of the cube is reduced to 45% of the initial value while on the surface M n is about 60 70% of the initial value. A shell-like structure emerges around week 5 when the molecular weight averaged over the entire volume of the cube is about 50% of the initial value. This centre-surface differentiation has been widely observed for devices made of PLA and PGA 2 4,12,13. B tissue engineering scaffolds In bone tissue engineering, PLA, PGA and their copolymers are widely used to make scaffolds for tissue generation because of their manufacturability to highly porous forms. The microstructure of a scaffold plays an important role in nutrient transport and waste diffusion. Pore size and their distribution can be designed for different applications. The purpose here is to demonstrate how to use the finite element analysis to evaluate the degradation rate in the design of a scaffold. Figure 7 shows three cubes each containing a spherical pore of different sizes. Each case can be regarded as a representative unit of either a uniform scaffold or a different part of a same scaffold that has non-uniform porosity. Case B-I has a unit size of 0.6 mm and pore size of 650 μm in diameter which correspond to a porosity of 64.85%. Case B-II has a unit size of mm and pore size of 100 μm in diameter which correspond to exactly the same porosity as that of Case B-I. The representative units of cases B-I and B-II appear identical in Figure 7 but their absolute sizes are different. Case B-III has a unit size of 0.5 mm and pore size of 650 μm which correspond to a porosity of 89.59%. The inner surface of the pore, highlighted in green in the figure, is where the scaffold is exposed to the aqueous medium. The boundary condition at the pores surface is set as C ol = 0. The other surfaces of the representative units are actually local symmetry planes. It is assumed that the short chains do not diffuse across these planes. Figure 8 shows the calculated distribution of average molecular weight at week 10 for the three representative units. Figure 9 shows the overall average of the molecular weights over the entire volume of the units as functions of time. Comparing case B-I and B-II which share the same porosity, it can be observed from the figures that both their overall averages and the spatial distributions of the average molecular weight are significantly different. This is because the absolute size of case B-I is larger than B-II. Acids accumulation in B-I

5 Page 5 of 10 Figure 4: Calculated mass loss for thin and thick samples assuming zero residual monomers. Figure 5: Finite element model for a mm cube. The red cross-section on the right figure indicates where molecular weight distribution is presented in Figure 6. leads to a faster degradation and spatial differentiation of its degradation. The typical wall thickness for case B-I is mm while that of case B-II is mm. Because of its small size, case B-II degrades almost uniformly. Comparing case B-I and B-III which share the same pore size, it can be observed from the figures that B-III degrades slightly slower and less heterogeneously than B-I. This is because the absolute size, hence its typical wall thickness, of B-III is smaller than B-I. These case studies demonstrate that pore size and porosity of a scaffold act together to control its degradation rate. C internal bone fixation screws Biodegradable internal fixation devices such as screws, pins and plates are already in clinical applications such as orthopaedic surgeries. The advantage of using biodegradable fixation devices instead of metallic ones is obvious the device simply disappears after the bone heals. Load can slowly shift from a degrading protection device to the healing bone in the remodelling process which ensures complete healing of the

6 Page 6 of 10 Figure 6: Distribution of normalised average molecular weight (normalised by initial value) over the cross-section indicated by Figure 5b at (a) t = 3 weeks, (b) t = 5 weeks and (c) t = 7 weeks. Figure 7: Three representative units of scaffods. Figure 8: Distribution of normalised average molecular weight (normalised by the initial value) at week 10 calculated using finite element models for the three cases. bone. The degradation rate of the fixation devices has to be carefully controlled in order not to endanger the healing bones. In this section we demonstrate how the finite element analysis can help in the design of a biodegradable fixation screw as shown in Figure 10. There are several geometric variables for the fixation screw. Here we focus on the diameter of the inside hollow tube. In surgeries, the hollow tube is used to guide the screw into an intended position. Figure 10 shows two screws of different inside diameters with all the other dimensions being identical. The screws have a major diameter of 6 mm, minor d iameter of 4.75 mm, length of 10 mm, pitch of 2 mm and shred angle of 60o. The diameter of the hollow tube is 3 mm for case C-I and 4 mm for case C-II. Figure 11 shows the calculated distributions of average molecular weight over the vertical cross- section of the screws at week 5 while Figure 12 shows the average molecular weights averaged over the entire screw as functions of time. In the finite element model, both the inside and outside surfaces of the screws are treated as boundaries with the aqueous medium. A clear size effect can be observed from Figures 11 and 12. The screw of C-I degrades faster and less uniform than C-II because it has a larger wall thickness. Focusing on case C-I shown in Fig 11, it is interesting to note the dark blue spots at the core of the shred. This is the area of low average molecular weight caused by acid accumulation. The identical pattern of degradation was observed by Schwach and Vert14 in their in-vivo experiment. D coronary stents Biodegradable coronary stents are currently under large scale patent studies. These stents are made of polylactide and designed to restore blood flow by opening a narrowed artery and providing support while the opened area heals. The stent combines scaffolding and drug release for the artery the narrowing can be treated with resolution of the patient s symptoms and the released drug attenuates the response of injured tissue that is caused by the high pressure deployment of the stent. Once the stent is no longer required it slowly dissolves over a period of 2 years through pathways in the Krebs cycle to carbon dioxide and water. A permanent implant is

7 Page 7 of 10 Figure 9: Molecular weights normalised by the initial value and averaged over unit volume as functions of time for the three cases shown in Figure 8. Figure 10: Biodegradable internal fixation screws of two different hollow tubes. not left behind allowing the artery to be more functionally normal. Here, we demonstrate how the finite element analysis can be used to understand the degradation process of a biodegradable stent. Figure 13 shows a representative unit of a stent that is modelled in the finite element analysis. Although a stent may look complicated in its 3-dimensional shape, a large part of it is made of straight columns which degrade in a 2- dimensional pattern. For these columns, the size of their cross section is the characteristic diffusion distance for the small chains and hence Figure 11: Distributions of average molecular weight (normalised by initial value) over the vertical cross-sections at week 5. decides the degradation rate. The junctions of the columns however degrade in a full three dimensional pattern. Because of their relatively large size, they also degrade faster than the columns. In the finite element model, all the surfaces are treated as boundary with the aqueous medium except for the cross- sections of the columns marked by C. The cross-sections marked by C are symmetry planes and it is assumed that the short chains do not diffuse across them. Figure 14 shows the distributions of average molecular weight

8 Page 8 of 10 Figure 12: Molecular weight normalised by the initial value and averaged over the volume of the screw as functions of time for the two cases shown in Figure 10. Figure 13: A representative unit of a stent that is analysed in the finite element analysis. at different times for (a) the crosssection of a column marked by A-A in Figure 13 and (b) the cross-section of a junction marked by B-B in Figure 13 respectively. The typical size of the cross-section of the columns is mm. This is rather small and means that the columns degrade more or less uniformly. Figure 15 compares the molecular weight as functions of time for two points located at the centre of the column and the centre of the junction respectively. It is clear that the core of the junction degrades faster than the rate of the device. If the junction fails, the stent would collapse and lose its ability to function as a stent. It is therefore important to design the junctions carefully so that premature failure can be avoided. Discussion The degradation of biodegradable polymeric devices can be modelled using a reaction-diffusion model. The mathematical equations can be solved for biodegradable implants

9 Page 9 of 10 Figure 14: Distribution of molecular weight at different times of degradation for (a) the cross-section of a column marked by A in Figure 13 and (b) the cross-section of a junction marked by B-B in Figure 13. Figure 15: Comparison of molecular weights as functions of time at two different points located at the centre of a column and centre of a junction, respectively. and devices of any sophisticated shapes and sizes giving the average molecular weight at any location and any time. Although the demonstration in the current paper is for devices made of amorphous polymers, the mathematical equations have already been developed for devices made of semi-crystalline polymers 5,6 and can be solved similarly using the finite element method. The current paper focuses on the degradation rate of the various devices. In device design, the change in mechanical properties, such as the Young s modulus and strength of the polymers, can also be important considerations especially for internal fixation devices. Once the relation between the mechanical properties and the average molecular weight is established, it is then straightforward to set up a finite element model for any devices that link the degradation analysis with a simultaneous stress analysis to predict its mechanical performance 11. Conclusion The interplay between bone remodelling and device degradation leads to load transfer between a degrading device and the healing bone. This is however a research topic still under development. It should also be pointed out that currently our

10 Page 10 of 10 understanding on the interaction between the surrounding tissues and a degrading device is not sufficient enough to be included in the mathematical model. A perfect sink condition for the short chains diffusion has been assumed in the finite element analysis. Further work is required to understand how different applications affect this boundary condition. Acknowledgements All the finite element analysis performed in this paper were done using COMSOL Multiphysics software, please see the software details below. COMSOL Multiphysics v. 4.3b. COMSOL AB, Stockholm, Sweden. References 1. Reed AM, Gilding DK. Biodegradable polymers for use in surgery poly(glycolic)/ poly(iactic acid) homo and copolymers: 2. In vitro degradation. Polymer April;22: Li SM, Garreau H, Vert M. Structureproperty relationships in the case of the degradation of massive aliphatic poly(a-hydroxy acids) in aqueous media Part 1: Poly(DL-lactic acid). J Mat Sci. 1990;1(3): Hurrell S, Cameron RE. The effect of initial polymer morphology on the degradation and drug release from polyglycolide. Biomaterials. 2002;23: Grizzi I, Garreau S, Li S, Vert M. Hydrolytic degradation of devices based on poly(dl-lactic acid) size-dependence. Biomaterials Mar;16: Gleadall A, Pan J, Atkinson H. A simplified theory of crystallisation induced by polymer chain scissions for biodegradable polyesters. Polymer Degradation and Stability. 2012;97: Han X, Pan J. A model for simultaneous crystallisation and biodegradation of biodegradable polymers. Biomaterials Jan;30: Han X, Pan J, Buchanan F, Weir N, Farrar D. Analysis of degradation data of poly(llactide co-l,d-lactide) and poly(l-lactide) obtained at elevated and physiological temperatures using mathematical models. Acta Biomater Oct;6: Han X, Pan J. Polymer chain scission, oligomer production and diffusion: a two-scale model for degradation of bioresorbable polyesters. Acta Biomater Feb;7: Pan J, Han X, Niu W, Cameron RE. A model for biodegradation of composite materials made of polyesters and tricalcium phosphates. Biomaterials. 2011;32: Wang Y, Pan J, Han X, Sinka C, Ding L. A phenomenological model for the degradation of biodegradable polymers. Biomaterials. 2008;29: Wang Y, Han X, Pan J, Sinka C. An entropy spring model for the Young s modulus change of biodegradable polymers during biodegradation. J Mech Behav Biomed Mater. 2010;3: Hurrell S, Cameron R. Polyglycolide: degradation and drug release. Part I: Changes in morphology during degradation. J Mater Sci Mat Med Sep;12: Hurrell S, Cameron R. The effect of buffer concentration, ph and bufferions on the degradation and drug release frompolyglycolide. Polymer International. 2003;52: Schwach G, Vert M. In vitro and in vivo degradation of lactic acid-based interference screws used in cruciate ligament reconstruction. Int J Biol Macromol Jun Jul;25: