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For citation purposes: Han X, Pan J. Finite element analysis of degradation of biodegradable medical devices. OA Biotechnology 2013 Jul 01;2(3):22. Com

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AbstractIntroductionThis paper presents a set of case stud-ies for calculating the degradation rates of medical devices made of bio-degradable polymers using the finite element method.Case StudyFirstly, 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 applica-tions 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.ConclusionThe analysis shows how the design details can affect the degradation rate of the various implants/devices.

IntroductionBiodegradable polymers, especially linear aliphatic polymers, have found great attractions in a broad range of medical applications: they were firstly used to made sutures success-fully in the 1970s1, afterwards they drew great interests in fields of ortho-paedic fixation devices, controlled drug release and scaffolds in tissue engineering. Biodegradable poly-mers gradually replace conventional

biomaterials in many medical appli-cations owing to their nature of deg-radation. Hydrolysis degradation turns polymers into smaller dissolva-ble molecules which are then eventu-ally metabolised into carbon dioxide and water after they served their functions. Polyesters, such as polyg-lycolide (PGA), poly (L-lactide) (PLA) and their copolymers (PGA-co-PLA), are of the greatest interest because of their well-established biodegradabil-ity, biocompatibility and mechanical properties. Degradation of biode-gradable devices is, however, a com-plicated chemical-physical process. It depends on the chemical structure of polymer, the shape and size of the device and the degradation environ-ment. Degradation therefore ranges from weeks to years for different pol-ymers 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 heter-ogeneous degradation and size effect are results of auto-catalytic hydroly-sis reaction. Dissolved short chains produced by chain scissions have car-boxylic 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 degrada-tion profile. Size effect and heteroge-neous degradation make it difficult to transfer experimental experience

from one device to another even if they are made of the same poly-mer. A mathematical framework has been developed by Pan and his co- workers5–11, which captures the dominating mechanisms in degra-dation including hydrolysis, crystal-lisation and short chain diffusion. It has been shown that the model can fit a wide range of the experimen-tal data and that the size effect and heterogeneous degradation can be predicted using the mathematical model6,10. 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 demon-strate the three dimensional effect of device degradation, (b) a typical scaf-fold for tissue engineering, (c) a fixa-tion screw for orthopaedic surgery and (d) a coronary stent. The analysis demonstrated how the design vari-ables can affect the degradation rate.

Case studyThe protocol of this study has been approved by the relevant ethical com-mittee related to our institution in which it was performed.

The mathematical modelWe focus on amorphous polymers in this study for simplicity although the mathematical model can handle semi-crystalline polymers without

Finite element analysis of degradation of biodegradable medical devices

X Han1, J Pan2*

*Corresponding authorEmail: [email protected] Wolfson school of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, LE11 3TU, UK

2 Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK

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For citation purposes: Han X, Pan J. Finite element analysis of degradation of biodegradable medical devices. OA Biotechnology 2013 Jul 01;2(3):22.

any difficulty5,6. Following Han et al.6,8, the rate equation for polymer chain scission during hydrolysis degrada-tion can be written as

dRdt

CRC

k k Cm

CC

se

s

e

en ol

e

= −

+

0

0

1 2 0

0

1

1

αβ

n

(1)

in which Rs is the mole concentra-tion of polymer chain scissions, Col is the mole concentration of ester units of dissolvable short chains and Ce0 is the initial mole concentration of ester units of the polymer chains. The reac-tion-diffusion equation for Col can be written as

dCdt

RC

dRdt

xDCx

ol s

e

s

i

ol

ii

=

+

∂∂

∂∂

−

=∑

αββ

0

1

1

3

(2)

In the above equations, k1 and k2 are rates for non-catalytic hydrolysis and auto-catalytic hydrolysis reac-tions 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 expo-nent 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 V

D D

polymer pore pore

pore polymer

= + −( )−( )

1 3 0 32 3

. .

(3)

in which Vpore is the porosity of the polymer caused by the loss of short chains and given by

VR C C

C

R

C

C CC

poreol ol ol

e

s

e

ol ol

e

=− −( )

=

−

−

0

0

0

0

0

αβ

(4)

and Dpolymer and Dpore are diffusion coefficients of the short chains in non-degraded polymer and liquid-filled pores respectively. Col0 is the concen-tration of residual short chains that may exist in the polymer. The average molecular weight is calculated using

MM

RC

NRC m

RC

n

n

s

e

dps

e

s

e

0

0

0

0 0

1

1

=−

+ −

α

α

β

β,

(5)

in which Mn0 is the initial number-averaged molecular weight, Ndp0 is the initial average degree of polymerisa-tion of the polymer. Ndp0 = Mn0/M0 in which M0 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 coworkers5–11. 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 experi-mental 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 sam-ples 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 pro-posed a schematic reaction-diffusion model to explain the size effect of the degradation. Wang et al.10 and Han and Pan8 analysed these data using actual reaction-diffusion models. However, in these previous models, the poten-tial existence of a large amount of residual monomers in the samples

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 scis-sion 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 aque-ous medium. Their thick samples on the other hand did not show the same level of mass loss.

In this study, separate finite ele-ment models for their plate and film samples are built. We limit our analy-sis to the first 10 weeks of the deg-radation 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:Rs = 0, Col = Col0, at t = 0 (6)

(b) Boundary condition: Col = 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, Ce0, and molecular weight of a repeating unit of the polymers, M0, are typical values for PLA. The initial molecular weight, Mn0, is different for the thin and thick samples. The val-ues for Mn0 are directly taken from the paper by Grizzi et al.4. It is widely rec-ognised 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

.

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random chain scission8, which leads to the values for a and b in the table. The values for Col0, which represent the concentration of residual mono-mers 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 k1, k2 (hydrolysis rate constants) and Dpolymer (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 pro-vides spatial distribution of the aver-age 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 mod-els can fit the experimental data fairly well. It is however important to high-light the role played by the residual monomers in the model prediction. Figure 3 shows the calculated aver-age 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 pre-dict any significant mass loss. Figure 4 shows the calculated mass loss as func-tion of time for the thin and thick sam-ples assuming zero residual mono-mers. 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

number average molecular weight is set as 30,000 g/mol, (b) Col0 is set as zero to reflect perfect polymerisa-tion and (c) a common diffusion coef-ficient of Dpolymer = 5×10–8 m2/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 consist-ency with the study in the previous

section. The demonstration can be made by setting the parameters at any other set of values.

Case study A - a three dimensional cubeFigure 5 shows a 1 × 1 × 1 mm cube which is the simplest possible ‘device’ and used here to demon-strate the three dimensional effect of

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.

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device degradation. Governing equa-tions (1) and (2) are solved numeri-cally using the PDE interfaces mod-ule from COMSOL Multiphysics®. A total number of 2321 elements are used as shown in figure 5(a) which has 18285 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 Col = 0 at all the boundaries. Figure 6 shows the distribution of average molecu-lar weight over the cross- section indicated by Figure 5b at three dif-ferent times. The molecular weight is normalised by its initial value. It can be observed that the degradation

starts to become heterogeneous by week 3. At week 5, Mn in the centre of the cube is reduced to 45% of the initial value while on the surface Mn 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 differen-tiation has been widely observed for devices made of PLA and PGA2–4,12,13.

Case study B – tissue engineering scaffoldsIn bone tissue engineering, PLA, PGA and their copolymers are widely used

to make scaffolds for tissue genera-tion because of their manufactur-ability 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 pur-pose 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 spheri-cal 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 0.0923 mm and pore size of 100 μm in diameter which correspond to exactly the same porosity as that of Case B-I. The rep-resentative 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 sur-face 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 sur-face is set as Col = 0. The other ‘sur-faces’ 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 dis-tribution of average molecular weight at week 10 for the three representa-tive units. Figure 9 shows the over-all 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 aver-ages and the spatial distributions of the average molecular weight are sig-nificantly different. This is because the absolute size of case B-I is larger than B-II. Acids accumulation in B-I

Table 1 Model parameters used for finite element analysis of data by Grizzi et al.4

Ce0 = 17300 mol/m3 k1= 1.4×10-4/week

M0 = 72 g/mol k2 = 0.0019(m3/mol)0.5/week

Mn0 = 20000 g/mol for plate Dpolymer = 8×10-7m2/week

Mn0 = 34000 g/mol for film Dpore = Dpolymer×1000

m = 4 Col0 = 5% Ce0 for plate

α = 28, β = 2 Col0 = 0 Ce0 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.

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leads to a faster degradation and spa-tial differentiation of its degradation. The typical wall thickness for case B-I is 0.175 mm while that of case B-II is 0.02691 mm. Because of its small size, case B-II degrades almost uni-formly. 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 stud-ies demonstrate that pore size and porosity of a scaffold act together to control its degradation rate.

Case study C – internal bone fixation screwsBiodegradable internal fixa-tion devices such as screws, pins

and plates are already in clini-cal applications such as ortho-paedic surgeries. The advantage of using biodegradable fixation devices instead of metallic ones is obvious – the device simply disap-pears after the bone heals. Load can slowly shift from a degrading pro-tection device to the healing bone in the remodelling process which ensures complete healing of the

Figure 4: Calculated mass loss for thin and thick samples assuming zero residual monomers.

Figure 5: Finite element model for a 1 × 1 × 1 mm cube. The red cross-section on the right figure indicates where molecular weight distribution is presented in Figure 6.

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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 ele-ment analysis can help in the design

of a biodegradable fixation screw as shown in Figure 10. There are sev-eral geometric variables for the fix-ation 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 diameter 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 molecu-lar weight over the vertical cross- section of the screws at week 5 while Figure 12 shows the average molecu-lar 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 uni-form 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 pat-tern of degradation was observed by Schwach and Vert14 in their in-vivo experiment.

Case study D – coronary stentsBiodegradable 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

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.

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not left behind allowing the artery to be more functionally normal. Here, we demonstrate how the finite ele-ment analysis can be used to under-stand 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 col-umns, the size of their cross section is the characteristic diffusion dis-tance for the small chains and hence

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 ele-ment 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 sym-metry planes and it is assumed that the short chains do not diffuse across them.

Figure 14 shows the distribu-tions of average molecular weight

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 hol-low tubes.

Figure 11: Distributions of average molecular weight (normalised by initial value) over the vertical cross-sections at week 5.

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at different times for (a) the cross-section 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 0.175 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 col-lapse 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.

DiscussionThe degradation of biodegradable polymeric devices can be modelled using a reaction-diffusion model. The mathematical equations can be solved for biodegradable implants

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.

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and devices of any sophisticated shapes and sizes giving the aver-age molecular weight at any loca-tion and any time. Although the demonstration in the current paper is for devices made of amorphous polymers, the mathematical equa-tions have already been developed for devices made of semi-crystal-line polymers5,6 and can be solved similarly using the finite element method. The current paper focuses

on the degradation rate of the vari-ous 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 molecu-lar 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 pre-dict its mechanical performance11.

ConclusionThe interplay between bone remod-elling and device degradation leads to load transfer between a degrad-ing device and the healing bone. This is however a research topic still under development. It should also be pointed out that currently our

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.

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understanding on the interaction between the surrounding tissues and a degrading device is not suf-ficient enough to be included in the mathematical model. A perfect sink condition for the short chains diffu-sion has been assumed in the finite element analysis. Further work is required to understand how differ-ent applications affect this boundary condition.

AcknowledgementsAll the finite element analysis per-formed in this paper were done using COMSOL Multiphysics® software, please see the software details below.

COMSOL Multiphysics® v. 4.3b. www.comsol.com. COMSOL AB, Stockholm, Sweden.

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