Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Micro-Macro Coupling for Optimizing Scaffold Mediated Bone Regeneration

Patrick Dondl, Oliver Suchan

TL;DR

The paper introduces a micro-macro scaffold bone regeneration framework that embeds microscale effects through periodic homogenization and leverages $FE^2$ and $FE-FFT$ strategies to couple diffusion, mechanics, and cell dynamics. It formulates geometry-aware microscopic constitutive laws, computes homogenized coefficients, and then optimizes scaffold density under PDE constraints using adjoint derivatives, incorporating correctors to ensure accurate homogenized stimuli. Key findings show that accounting for microstructure materially affects regeneration predictions and that stress-driven density patterns dominate optimization outcomes, with the fully homogenized model (eds) producing the strongest regeneration signals. The approach enables precision scaffold design by linking bone density, scaffold geometry, and mechanical environment, potentially benefiting patient-specific therapies albeit at high computational cost.

Abstract

This work presents a framework for modeling three-dimensional scaffold-mediated bone regeneration and the associated optimization problem. By incorporating microstructure into the model through periodic homogenization, we capture the effects of microscale fluctuations on the bone growth process. Numerical results and optimized scaffold designs that explicitly account for the microstructure are presented, demonstrating the potential of this approach for improving scaffold performance.

Micro-Macro Coupling for Optimizing Scaffold Mediated Bone Regeneration

TL;DR

The paper introduces a micro-macro scaffold bone regeneration framework that embeds microscale effects through periodic homogenization and leverages and strategies to couple diffusion, mechanics, and cell dynamics. It formulates geometry-aware microscopic constitutive laws, computes homogenized coefficients, and then optimizes scaffold density under PDE constraints using adjoint derivatives, incorporating correctors to ensure accurate homogenized stimuli. Key findings show that accounting for microstructure materially affects regeneration predictions and that stress-driven density patterns dominate optimization outcomes, with the fully homogenized model (eds) producing the strongest regeneration signals. The approach enables precision scaffold design by linking bone density, scaffold geometry, and mechanical environment, potentially benefiting patient-specific therapies albeit at high computational cost.

Abstract

This work presents a framework for modeling three-dimensional scaffold-mediated bone regeneration and the associated optimization problem. By incorporating microstructure into the model through periodic homogenization, we capture the effects of microscale fluctuations on the bone growth process. Numerical results and optimized scaffold designs that explicitly account for the microstructure are presented, demonstrating the potential of this approach for improving scaffold performance.

Paper Structure

This paper contains 9 sections, 3 theorems, 35 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Micro-Macro Model
  3. Micro-Macro Coupling
  4. Implementation Details
  5. Optimization
  6. Simulations
  7. Cell Evolution
  8. Optimized Scaffolding Structure
  9. Conclusion

Key Result

Lemma 2.1

Let $f \in L^p(\Omega; L^\infty_\#(Y))$ for some $p \in [1, \infty)$. Then $f_\varepsilon(x) \coloneqq f(x, x/\varepsilon) \in L^p(\Omega)$ with

Figures (10)

  • Figure 1: Visualization of the micro-macro-coupling approach where we have a spatially varying bone density (color gradient within defect site) and a constant scaffold density. For each combination of scaffold and bone density one obtains a specific microstructure for which one has to solve the microscopic problem. Above the defect site three different Gyroid microstructures are depicted with gradually increasing bone volume fraction but constant scaffold volume fraction.
  • Figure 2: Interaction of cells. The mechanical stimulation given through linear elasticity, depending on this stimulus the cells can either differentiate, proliferate or die (apoptosis). Cells with a double circle are also able to migrate within the bone defect.
  • Figure 3: The two base geometries that will be used throughout this work.
  • Figure 4: Illustration of how the bone (orange) grows only on the surface of the scaffolding structure (black) and thus forming the bone-scaffold composite on the microscopical level.
  • Figure 5: Comparison of the cell evolutions for the different modes n, ed and eds for a simulation time of 140d.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Lemma 2.1: Weak limits for rapidly oscillating periodic functions, see WeakOscIntroHomog
  • Theorem 2.2: Corrector Result, IntroHomog
  • Theorem 2.3: Homogenized Stimulus
  • proof
  • Remark 2.4