Modelling the evolving ductility of biodegradable polymers

Transcription

1 Modelling the evolving ductility of biodegradable polymers Aoife Hill, Sophie Bezela, Reyhaneh N. Shirazi, William Ronan Biomechanics Research Centre, Biomedical Engineering, NUI Galway Aoife Hill Modelling biodegradable polymers 1

2 Overview Background Evolving mechanical properties: Ø Stiffness Ø Ductility Existing models: Ø Phenomenologically based degradation model Ø Physically based scission model Updated model Results Aoife Hill Modelling biodegradable polymers 2

3 Research objective o Develop a modelling framework for biodegradable polymers; o For use in the medical device industry, specifically cardiovascular stents [1] In-silico testing: o Reduces costs o Speeds up development of new devices o Reduces reliance on animal trials [1] Shine et al., Aoife Hill Modelling biodegradable polymers 3

4 Coronary Heart Disease o Arteries narrow o Blood flow is reduced o Plaque builds up on artery wall [2] Treatment: o Stent is inserted in artery o Blood flow is improved [3] ~30% occurrence of in-stent restenosis for bare metal stents [4] [2] COR Medical Group, Inc., [3] labroots. [4] Cassesse et al., Aoife Hill Modelling biodegradable polymers 4

5 How do the mechanical properties evolve? Mechanical properties [5] Time, degradation [5] Bezela, Unpublished ME thesis, NUI Galway, Aoife Hill Modelling biodegradable polymers 5

6 Amorphous biodegradable Biodegradable polymers polymers Autocatalysis: Trapped degradation products Accelerates end scission process [6] [6] Wang, PhD Thesis, University of Leicester, Aoife Hill Modelling biodegradable polymers 6

7 Molecular weight o Molecular weight does not have a single value o Different chains have different lengths/molecular weights o Average molecular weight is calculated o! " = % &' & ' & o ( ) is the number of chains of weight! ) # of molecules Low molecular weight! " Molecular weight Sample GPC curve High molecular weight Peak heights normalised Aoife Hill Modelling biodegradable polymers 7

8 Predicting evolving stiffness Freely jointed chain: o Behaves as entropic spring ) = 1040 o Good model for rubber-like elasticity ) = ) = 347. ) <. ) %6(7 $ %&'()* = $ %&'()* 1 Young s modulus:! = 3$ %&'()* +, $ %&'()* chains per unit volume + Boltzmann s constant, absolute temperature [9] Wang et al., JMBBM, Modifications [9]: 1. No-rise rule incorporated: $ %&'()* never increases 2. Molecular weight threshold introduced:. ) <. ) %6(7 : $ %&'()* = $ %&'()* 1 Aoife Hill Modelling biodegradable polymers 8

9 Evolution of failure strain Freely jointed chain: Engineering strain:! " = $ " $ & $ & = ' ' $ " = ' $ + True strain, / " : ' ( ) ( ) = ' ( ) 1 / " = ln 1 +! " = ( ) ln ' $ + $ & = ' $ + [10] / " 3 4 = ( ) ln(6 7) ' - number of polymer units / 9 " = ( ln ' 3 4?3@ABC 4 ) :;<=7> [10] Ward et al., Wiley, Aoife Hill Modelling biodegradable polymers 9

10 Existing experimental data (a) [7] (b) [5]! " Time, degradation (days) (a) Poly(lactic-co-glycolic) acid (b) Poly(lactic) acid (c) Polypropylene Time, degradation (days) (c) [8]! " (%) [7] Shirazi et al., Acta Biomaterialia, 10: , [5] Bezela, Unpublished ME thesis, NUI Galway, [8] Fayolle et al., Polymer, 45: , # $ (&' ()* +, ) Aoife Hill Modelling biodegradable polymers 10

11 Phenomenological degradation model Wang et al., 2008: Change in ester concentration Change in monomer concentration!" (!$ = (& '" ( + & * " ( " +.- # ) Non-catalytic degradation mechanism Autocatalytic degradation mechanism!" #!$ = & '" ( + & * " ( " +.- # +. 0." # Diffusion term Ignoring diffusion and location, 4" ( 4$ = (& '" ( + & * " ( " # +.- ) 4" # 4$ = & '" ( + & * " ( " # +.- Aoife Hill Modelling biodegradable polymers 11

12 Exploring various reaction rates!" #!$ = (( )" # + ( + " # ", -./ )!",!$ = ( )" # + ( + " # ", -./ Solutions easily obtained in MATLAB using the ode45 solver. [( ) ] = day 8), [( + ] = m ; mol 8) -./ day 8) ( ) = ( + (L), ( ) (R) ( + = Aoife Hill Modelling biodegradable polymers 12

13 Degradation model results As monomers are too small to detect, [11]:! "! "# = % & % &# ' ( = day /(, ' 1 = m 5 mol /( #.8 day /( ode45 Experimental [7] # of molecules Molecular weight Sample GPC curve 9 1 = [7] Shirazi et al., Acta Biomaterialia, 10: , [11] Shirazi et al., JMBBM, 54:48-59, Aoife Hill Modelling biodegradable polymers 13

14 Implementing scission model Based on Shirazi et al., Generate representative polymer chain distribution. 2. Store initial chain lengths in an array. 3. Perform randomly assigned sequences of chain and end scissions (ratio fixed). 4. Track! " for each initial chain individually as scissions occur. If! " <! " $%&' : does not contribute to mechanical properties. 5. Obtain information about evolving properties. # of molecules Molecular weight Chain scission x x End scission ( e4! " 5e4 ( ) e6 2e6 # Scissions Aoife Hill Modelling biodegradable polymers 14

15 Scission model results Monomers included Monomers excluded 8e4 # $ 4e4 R1000:E1 R10:E1 R1:E1 R1:E10 R1:E1000!! " 0 0 1e6 2e6 # Scissions e6 # Scissions e6 Aoife Hill Modelling biodegradable polymers 15

16 Comparing models Degradation model: Scission model:!" #!$ = (( )" # + ( + " # ", - )!",!$ = ( )" # + ( + " # ", - o o o Generate chain distribution Perform scissions Track evolving / - for chain groups Advantages: o Time dependent o Can include spatial dependence. Useful for 3D applications, e.g. FEM Disadvantages: o Provides no information on evolving mechanical properties Advantages: o Provides information on evolving mechanical properties Disadvantages: o Difficult to relate to time o Random:End held constant Aoife Hill Modelling biodegradable polymers 16

17 Comparing sample solutions Degradation model: Scission model: 8e4! " 4e e6 2e6 # Scissions Don t obtain information about evolving mechanical properties Do obtain information about evolving mechanical properties - #, % & Aoife Hill Modelling biodegradable polymers 17

18 Combining models How many random and end scissions occur in a time step?!" #!$ = (( )" # + ( + " # ", -./ )!",!$ = ( )" # + ( + " # ", -./ Hydrolysis: Autocatalysis: Trapped degradation products Causes end and chain/random scissions Accelerates end scission process [6] Non-catalytic degradation: ( ) " # Autocatalytic degradation: ( + " # ", -./ " # - concentration of esters ", - concentration of monomers Calculate at each time step: 12! = ( ) " # + ( + " # ", -./ =!",!$ 892!7: = ( ) " # [6] Wang, PhD Thesis, University of Leicester, Aoife Hill Modelling biodegradable polymers 18

19 Predictions for molecular weight D 5 = day I5, \\\\\\\. 6 = D 6 = mol I5 #.B day I5 % &, combined model % &, deg. model Experimental! " [7]! ", monomers incl.! ", monomers excl. Model needs to be recalibrated Move from phenomenological approach towards physically based calculation for! "! "! "# = % & % &# = % &! " =! () ( = % &!+,-.!-// ) ( #1h-34/! "# = 5# 6!! = 5!!, % &# = 10;! " = 1 = % &! " = < 6!! = 4.5!!, % & = 9;! " = 0.9 = % &! " = = 3!!, % & = 9;! " B 0.9 = % & [7] Shirazi et al., Acta Biomaterialia, 10: , Aoife Hill Modelling biodegradable polymers 20

20 Evolving mechanical properties Stiffness: Monomers incl. Monomers excl. Experimental [7] $% $% : 1 = Time (days)!" # & ' = day /', & 1 = m 5 mol /' 8.9 day /' [7] Shirazi et al., Acta Biomaterialia, 10: , Aoife Hill Modelling biodegradable polymers 21

21 Evolving mechanical properties Ductility: Monomers incl. Monomers excl , Time (days) :; <! " = day *",!, = m 0 mol *" 3.4 day *" Aoife Hill Modelling biodegradable polymers 22

22 Further work Degradation model: Scission model: Failure criteria:! " # $ = & ' ln(+,) <C D <Q = (B &C D + B ' C D C F, ) <C F <Q = B &C D + B ' C D C F, :;<=>?@@?A;@ = B & C D + B ' C D C F G.I = JK L JM NO;<AP=>?@@?A;@ = B & C D o o o o!. " = & ln / # $ 5# 6789 $ ' 0123,4 Add to experimental data for amorphous polymers Calibrate the model Link to continuum finite element models Aid in designing medical devices with suitable degradation rates and mechanical integrity Aoife Hill Modelling biodegradable polymers 23

23 Acknowledgements MechBio Research Group: o Dr William Ronan o Rosa Shine o Bobby Huxford o Ruth Tarpey o Ali Reza Abaei Ardebili Funding: College of Engineering and Informatics Postgraduate Research Scholarship Aoife Hill Modelling biodegradable polymers 24