High-performance simulation of biodegradation behavior of magnesium-based biomaterials Mojtaba Barzegari Liesbet Geris Biomechanics Section, Department of Mechanical Engineering, KU Leuven, Leuven, Belgium
Biodegradable Metals • Mg, Zn, and Fe • Great mechanical properties • Biocompatibility and contribution in metabolism • Potential applications: • Cardiovascular stents • Orthopedic implants 2
An Example in Orthopedics • Osteoarthritis • Hip Osteoarthritis • Total hip replacement (https://www.arthritis-health.com/types/osteoarthritis/videos) 3
So What Is The Problem? • Problem: Tuning the degradation of the implant to the regeneration of the new bone • Can be solved by: Building a mathematical framework for the assessment of biodegradation (Source: 3D Systems Inc.) 4
Model Workflow Underlying Science Mathematical Models Computational Models Partial differential equations Finite difference method Chemistry of biodegradation Reaction-Diffusion-Convection Finite element method Physics of perfusion Biology of tissue growth Level set method Scientific computing libraries Open source solvers 5
Chemistry of Biodegradation Medium H 2 H 2 H 2 H 2 O H 2 O H 2 O H 2 O H 2 O H 2 O ՜ Mg 2+ + 2𝑓 − Mg Cl − OH − OH − OH − OH − OH − OH − 2H 2 O + 2𝑓 − ՜ H 2 + 2OH − Cl − 𝑓 − 𝑓 − 𝑓 − 𝑓 − 𝑓 − 𝑓 − Cl − Mg 2+ + 2OH − ՜ Mg OH 2 Mg OH 2 Mg 2+ Cl − Mg OH 2 + 2Cl − ՜ Mg Cl 2 + 2OH − Mg 2+ ՜ Mg 2+ + 2Cl − + 2OH − Cl − Mg Cl − Mg 2+ 6
Constructing Mathematical Model The model captures: 1. The chemistry of dissolution of metallic implant 2. Formation of a protective film 3. Effect of ions in the medium (1) (2) (3) H 2 H 2 O Mg 2+ OH − Cl − Mg 2+ 𝑓 − Mg 2+ OH − Mg OH 2 Mg OH 2 Mg Mg Mg 7
Mathematical Representation Chemical reactions Concentration notations 𝑙 1 Mg 2+ + H 2 + 2OH − ՜ 𝑙 1 Mg OH 2 + H 2 Mg 2+ ⇉ Mg Cl − ⇉ Cl Mg + 2H 2 O ՜ 𝑙 2 Mg 2+ + 2Cl − + 2OH − Mg OH 2 + 2Cl − ՜ Mg OH 2 ⇉ [Film] Derived PDEs Film 𝜖 Mg 𝑓 𝛼 Mg −𝑙 1 Mg +𝑙 2 Film Cl 2 = 𝛼. 𝐸 Mg 1 − Film max 𝜖𝑢 𝜖 Film Film −𝑙 2 Film Cl 2 = 𝑙 1 Mg 1 − 𝜖𝑢 Film max 𝜖 Cl 𝑓 𝛼 Cl = 𝛼. 𝐸 Cl Film max = 𝜍 Mg OH 2 × (1 − 𝜗) 𝜖𝑢 8
Identifying Moving Biodegradation Interface • Different approaches: 1. Interface tracking methods 2. Interface fitting methods 3. Interface capturing methods 1) 2) 3) 9
Interface Capturing using Implicit Interfaces • Implicit interfaces can be defined by an implicit distance function 𝜚 𝑦 = 𝑦 2 − 2 𝜚 𝑦, 𝑧 = 𝑦 2 + 𝑧 2 − 𝑠 2 1D implicit function 2D implicit function 𝜚 = 0 interface 𝜚 > 0 outside 𝜚 < 0 inside 10
Level Set Method • A PDE to capture the moving implicit surface • 𝜚 = 𝜚(𝑦, 𝑧, 𝑨, 𝑢) 𝜖𝜚 𝜖𝑢 + 𝑊. 𝛼𝜚 + v 𝛼𝜚 = 𝑐𝜆 𝛼𝜚 External velocity field Normal direction motion Curvature−dependent term 11
Level Set Method for Biodegradation Medium v 𝜖𝜚 𝜖𝑢 + v 𝛼𝜚 = 0 Level set: Scaffold v 𝐊 𝑦, 𝑢 − 𝑑 sol − 𝑑 sat v(𝑦, 𝑢) . 𝑜 = 0 Rankine-Hugoniot: 𝑓 𝛼 𝐸 Mg 𝑜 Mg − [Mg] sol −[Mg] sat v = 0 Mg scaffold: Mg 2+ 𝑑 sol Mg 2+ 𝑓 𝛼 𝐸 Mg 𝑜 Mg 𝜖𝜚 𝜖𝑢 − 𝛼𝜚 = 0 𝑑 sat PDE to solve: 𝑑 [Mg] sol −[Mg] sat Mg 12
Constructing Computational Model • Not feasible to implement models in commercial software packages • Discretizing PDE equations, numerical computation o Finite difference method (temporal terms) o Finite element method (spatial terms) 𝜖[Mg] 𝜖[Mg] 𝜖[Mg] Film Film Film 𝑓 𝛼[Mg] − 𝑙 1 Mg 𝑓 𝛼[Mg] − 𝑙 1 Mg 𝑓 𝛼[Mg] − 𝑙 1 Mg + 𝑙 2 Film Cl 2 + 𝑙 2 Film Cl 2 + 𝑙 2 Film Cl 2 = 𝛼. 𝐸 Mg = 𝛼. 𝐸 Mg = 𝛼. 𝐸 Mg 1 − 1 − 1 − 𝜖𝑢 𝜖𝑢 𝜖𝑢 Film max Film max Film max Implicit backward Euler 1 st order Lagrange polynomial as the basis function 13
Computational Mesh • Euler Mesh • High accuracy in the interface • Refining the mesh adaptively 14
An Example of 3D Mesh 15
Implementing Computational Model • Mesh generation (SALOME, TetGen) A typical 3D simulation: #Elements ~= 800,000 • Weak form implementation (FreeFem++) #DOF ~= 500,000 • Parallelization • Message Passing Interface (MPICH) • High-performance Domain Decomposition (HPDDM) • High-performance solvers (MUMPS, PETSc) 16
Verification of the code Verifying the correct behavior of: • Mass transfer and ion release • Level set surface tracking 17
Convergence Studies Formed Hydrogen Gas 12 • Mesh and time step sensitivity Elements: ~13,000 • Crucial for in-house codes 10 8 Time to Simulate 5 Days (Hour) 40 6 35 30 4 25 Elements: 20 ~600,000 15 2 10 5 0 0 0 1 2 3 4 5 0 100,000 200,000 300,000 400,000 500,000 600,000 Time (day) Number of Elements 18
Benchmarking parallelization • Serial and parallel code should produce the same result • Evaluating scale-up Time to simulate every 40 time steps Run time of each time step 28 30 200 Time (second) 25 Time (minute) 150 19 20 100 15 Assembly time 50 8 10 Solver time 5 0 1 2 6 0 MPI Cores Serial code Paralle code #1 Paralle code #2 (DOF: 381,205, Elements: 2,233,524, MPI Cores: 4) (DOF: 44,663, MPI Cores: 4) 19
Calibration of the Model • Different approaches to calibrate models • Mass loss • Formed hydrogen gas • Obtaining reaction rates and diffusion coefficients 𝜖[Mg] Film 𝑓 𝛼[Mg] − 𝑙 1 Mg + 𝑙 2 Film Cl 2 = 𝛼. 𝐸 Mg 1 − 𝜖𝑢 Film max 𝜖 Film Film − 𝑙 2 𝐺 Cl 2 = 𝑙 1 Mg 1 − 𝜖𝑢 Film max 𝜖[Cl] 𝑓 𝛼[Cl] = 𝛼. 𝐸 Cl 𝜖𝑢 20
Experimental Data and Model Calibration (Abidin et al., Corrosion Science, 2013) 21
Model Parameters Estimation • Each simulation takes ~8 hours to run • Using a Bayesian optimization algorithm • Cost function is the RMSE of difference in experimental data and model output 22
Application for Porous Scaffolds • Sample mesh based on a CT image of a porous Mg scaffold 23
2D Mg Scaffold – Film Formation 24
2D Mg Scaffold – Film Formation 25
2D Mg Scaffold – Film Formation 26
3D Porous Scaffold Degradation 27
Model Validation • Validation is currently taking place • We use another experimental setup to validate the model • Models will be extended to capture pH changes, and that will be used for model validation (Mei et al., Corrosion Science, 2019) 28
Conclusion • A quantitative mathematical model to assess the degradation behavior of biodegradable metallic implants in-silico • Once fully validated, the model will be an important tool to find the right design and properties of the magnesium-based implants 29
Thank you for your attention This research is financially supported by the PROSPEROS project, funded by the Interreg VA Flanders - The Netherlands program
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