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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

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High-performance simulation of biodegradation behavior of magnesium-based biomaterials

Mojtaba Barzegari Liesbet Geris

Biomechanics Section, Department of Mechanical Engineering, KU Leuven, Leuven, Belgium

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  • Mg, Zn, and Fe
  • Great mechanical properties
  • Biocompatibility and contribution in metabolism
  • Potential applications:
  • Cardiovascular stents
  • Orthopedic implants

2

Biodegradable Metals

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3

An Example in Orthopedics

(https://www.arthritis-health.com/types/osteoarthritis/videos)

  • Osteoarthritis
  • Hip Osteoarthritis
  • Total hip replacement
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  • 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

4

So What Is The Problem?

(Source: 3D Systems Inc.)

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5

Model Workflow

Underlying Science Mathematical Models Computational Models Finite difference method Finite element method Scientific computing libraries Open source solvers Partial differential equations Reaction-Diffusion-Convection Level set method Chemistry of biodegradation Physics of perfusion Biology of tissue growth

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6

Chemistry of Biodegradation

՜ Mg2+ + 2π‘“βˆ’ Mg 2H2O + 2π‘“βˆ’ ՜ H2 + 2OHβˆ’ Mg2+ + 2OHβˆ’ ՜ Mg OH 2 Mg OH 2 + 2Clβˆ’Υœ Mg Cl 2 + 2OHβˆ’ Mg

Medium

Mg2+ Mg2+ Mg2+

H2O π‘“βˆ’ π‘“βˆ’ π‘“βˆ’ π‘“βˆ’ π‘“βˆ’ π‘“βˆ’ OHβˆ’ OHβˆ’ OHβˆ’ OHβˆ’ OHβˆ’ OHβˆ’ H2 H2 H2

Clβˆ’ Clβˆ’ Clβˆ’ ՜ Mg2+ + 2Clβˆ’ + 2OHβˆ’ Clβˆ’ Clβˆ’ Clβˆ’ Mg OH 2

H2O H2O H2O H2O H2O

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The model captures:

  • 1. The chemistry of dissolution of metallic implant
  • 2. Formation of a protective film
  • 3. Effect of ions in the medium

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Constructing Mathematical Model

Mg

Mg2+ π‘“βˆ’ OHβˆ’ H2 Clβˆ’ H2O

Mg Mg OH 2

Mg2+

Mg Mg OH 2

(1) (2) (3)

OHβˆ’ Mg2+

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Derived PDEs

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Mathematical Representation

Mg + 2H2O ՜

𝑙1 Mg2+ + H2 + 2OHβˆ’ ՜ 𝑙1 Mg OH 2 + H2

Concentration notations

Mg2+ ⇉ Mg Clβˆ’ ⇉ Cl Mg OH 2 ⇉ [Film]

Chemical reactions

Mg OH 2 + 2Clβˆ’ ՜

𝑙2 Mg2+ + 2Clβˆ’ + 2OHβˆ’

πœ– Mg πœ–π‘’ = 𝛼. 𝐸Mg

𝑓 𝛼 Mg βˆ’π‘™1 Mg

+𝑙2 Film Cl 2 πœ– Cl πœ–π‘’ = 𝛼. 𝐸Cl

𝑓 𝛼 Cl

πœ– Film πœ–π‘’ = 𝑙1 Mg βˆ’π‘™2 Film Cl 2 1 βˆ’ Film Film max 1 βˆ’ Film Film max

Film max = 𝜍Mg OH 2 Γ— (1 βˆ’ πœ—)

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  • Different approaches:
  • 1. Interface tracking methods
  • 2. Interface fitting methods
  • 3. Interface capturing methods

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Identifying Moving Biodegradation Interface

1) 2) 3)

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𝜚 = 0 interface

  • Implicit interfaces can be defined by an implicit distance function

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Interface Capturing using Implicit Interfaces

1D implicit function 𝜚 𝑦 = 𝑦2 βˆ’ 2

𝜚 < 0 inside 𝜚 > 0

  • utside

2D implicit function 𝜚 𝑦, 𝑧 = 𝑦2 + 𝑧2 βˆ’ 𝑠2

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  • A PDE to capture the moving implicit surface
  • 𝜚 = 𝜚(𝑦, 𝑧, 𝑨, 𝑒)

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Level Set Method

πœ–πœš πœ–π‘’ + π‘Š. π›Όπœš

External velocity field

+ v π›Όπœš

Normal direction motion

= π‘πœ† π›Όπœš

Curvatureβˆ’dependent term

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Level Set Method for Biodegradation

πœ–πœš πœ–π‘’ + v π›Όπœš = 0

Scaffold Medium v v

Mg

Mg2+

𝑑 𝑑sol 𝑑sat

Mg2+

Level set: Rankine-Hugoniot:

𝐊 𝑦, 𝑒 βˆ’ 𝑑sol βˆ’ 𝑑sat v(𝑦, 𝑒) . π‘œ = 0

Mg scaffold:

𝐸Mg

𝑓 𝛼 π‘œ Mg βˆ’ [Mg]solβˆ’[Mg]sat v = 0

πœ–πœš πœ–π‘’ βˆ’ 𝐸Mg

𝑓 𝛼 π‘œ Mg

[Mg]solβˆ’[Mg]sat π›Όπœš = 0

PDE to solve:

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  • Not feasible to implement models in commercial software packages
  • Discretizing PDE equations, numerical computation
  • Finite difference method (temporal terms)
  • Finite element method (spatial terms)

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Constructing Computational Model

πœ–[Mg] πœ–π‘’ = 𝛼. 𝐸Mg

𝑓 𝛼[Mg] βˆ’ 𝑙1 Mg

1 βˆ’ Film Film max + 𝑙2 Film Cl 2 πœ–[Mg] πœ–π‘’ = 𝛼. 𝐸Mg

𝑓 𝛼[Mg] βˆ’ 𝑙1 Mg

1 βˆ’ Film Film max + 𝑙2 Film Cl 2 πœ–[Mg] πœ–π‘’ = 𝛼. 𝐸Mg

𝑓 𝛼[Mg] βˆ’ 𝑙1 Mg

1 βˆ’ Film Film max + 𝑙2 Film Cl 2 Implicit backward Euler 1st order Lagrange polynomial as the basis function

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  • Euler Mesh
  • High accuracy in the interface
  • Refining the mesh adaptively

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Computational Mesh

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An Example of 3D Mesh

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  • Mesh generation (SALOME, TetGen)
  • Weak form implementation (FreeFem++)
  • Parallelization
  • Message Passing Interface (MPICH)
  • High-performance Domain Decomposition (HPDDM)
  • High-performance solvers (MUMPS, PETSc)

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Implementing Computational Model

A typical 3D simulation: #Elements ~= 800,000 #DOF ~= 500,000

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Verifying the correct behavior of:

  • Mass transfer and ion release
  • Level set surface tracking

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Verification of the code

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  • Mesh and time step sensitivity
  • Crucial for in-house codes

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Convergence Studies

2 4 6 8 10 12 1 2 3 4 5 Time (day)

Formed Hydrogen Gas

Elements: ~13,000 Elements: ~600,000

5 10 15 20 25 30 35 40 100,000 200,000 300,000 400,000 500,000 600,000 Number of Elements

Time to Simulate 5 Days (Hour)

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  • Serial and parallel code should produce the same result
  • Evaluating scale-up

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Benchmarking parallelization

28 19 8 5 10 15 20 25 30 Serial code Paralle code #1 Paralle code #2 Time (minute)

Time to simulate every 40 time steps

50 100 150 200 1 2 6 Time (second) MPI Cores

Run time of each time step

Assembly time Solver time (DOF: 44,663, MPI Cores: 4) (DOF: 381,205, Elements: 2,233,524, MPI Cores: 4)

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  • Different approaches to calibrate models
  • Mass loss
  • Formed hydrogen gas
  • Obtaining reaction rates and diffusion coefficients

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Calibration of the Model

πœ–[Mg] πœ–π‘’ = 𝛼. 𝐸Mg

𝑓 𝛼[Mg] βˆ’ 𝑙1 Mg

1 βˆ’ Film Film max + 𝑙2 Film Cl 2 πœ– Film πœ–π‘’ = 𝑙1 Mg 1 βˆ’ Film Film max βˆ’ 𝑙2𝐺 Cl 2 πœ–[Cl] πœ–π‘’ = 𝛼. 𝐸Cl

𝑓 𝛼[Cl]

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Experimental Data and Model Calibration

(Abidin et al., Corrosion Science, 2013)

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  • 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

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Model Parameters Estimation

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Application for Porous Scaffolds

  • Sample mesh

based on a CT image of a porous Mg scaffold

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2D Mg Scaffold – Film Formation

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2D Mg Scaffold – Film Formation

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2D Mg Scaffold – Film Formation

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3D Porous Scaffold Degradation

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  • 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

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Model Validation

(Mei et al., Corrosion Science, 2019)

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  • 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

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Conclusion

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Thank you for your attention

This research is financially supported by the PROSPEROS project, funded by the Interreg VA Flanders - The Netherlands program