A multiscale optimization framework for bone remodeling: Integrating material and structural adaptations across hierarchical scales
Data files
Oct 15, 2025 version files 98.99 MB
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bone_multitopopt.zip
36.89 MB
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BoneHealthAssessment.zip
62.10 MB
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README.md
9.37 KB
Abstract
Bone exhibits a hierarchical organization across multiple length scales, integrating functional properties through adaptive remodeling mechanisms. In this article, we present a concurrent material-structure optimization framework that identifies optimal macroscale bone density and microstructural configurations, including collagen and hydroxyapatite distribution and lacunae orientation, across the length scales in bone’s hierarchical organization. Our framework formulates a compliance minimization problem with coupled material and structure optimization sub-problems and leverages a continuum micromechanics-based homogenization approach to efficiently capture bone’s hierarchical material behavior. This enables computationally tractable optimization independent of the number of hierarchical scales, addressing key limitations of conventional remodeling approaches. We apply the framework to a human proximal femur under realistic musculoskeletal loading conditions and demonstrate its ability to capture self-optimizing mechanisms consistent with physiological adaptation. While not intended as a clinical diagnostic tool at this stage, the model provides a physics-based rationale for estimating microstructural distributions of bone constituents and highlights deviations that may inform future assessments of bone quality. These findings offer a foundation for targeted therapeutic strategies, personalized diagnostics, and regenerative medicine applications.
File: bone_multitopopt.zip
Overview
This dataset contains computational code and results from bone multi-scale optimization simulations for bone materials to carry out bone remodeling in Python.
The research investigates a concurrent material–structure optimization framework that identifies optimal macroscale bone density and microstructural configurations, including collagen and hydroxyapatite distribution and lacunae orientation, across the hierarchical length scales of bone.
Repository Structure
bone_multitopopt/
├── src/ # Main source code
│ ├── __init__.py
│ ├── bone_assembly_util.py # Finite element assembly utilities
│ ├── bone_element_util.py # Element-level estimation operations
│ ├── bone_material_optim_util.py # Material-level optimization algorithms
│ ├── bone_micro_util.py # Continuum micromechanics-based homogenization utilities
│ ├── bone_struct_optim_util.py # Structure-level topology optimization routine
│ ├── fem_util.py # Finite Element Method (FEM) core module
│ ├── input_file.py # Input parameter handling
│ ├── tensor_util.py # Tensor operations
│ ├── visual_SIMP.py # SIMP visualization
│ └── visualization.py # General plotting utilities
├── test/ # Comprehensive test suite
│ ├── __init__.py
│ ├── test_FEM_solver.py # FEM solver validation tests
│ ├── test_bone_assembly.py # Assembly utility tests
│ ├── test_bone_element.py # Element operation tests
│ ├── test_bone_material_optim.py # Material optimization tests
│ ├── test_bone_micro.py # Microstructure tests
│ ├── test_fem_util.py # FEM utility tests
│ └── test_input_file.py # Input handling tests
├── Examples/
│ └── Examples_paper/
│ └── Multiscale_bone/ # Paper reproduction case study
│ ├── input_file.py # Multiscale simulation configuration
│ ├── main.py # Main script for multiscale optimization
│ ├── Combined_BC.mat # Boundary condition data (CT scan-based)
│ ├── squared_mesh_data.mat # Mesh geometry data (CT scan-based)
│ └── pictures/ # Simulation result visualizations (collagen, HA,
density, orientation)
│ └── bone_sub_theta_all-itr/ # Iteration-wise lacunae orientation
visualization data
│ ├── theta_0.png # Lacunae orientation vector field at iteration 0
│ ├── theta_5.png # Lacunae orientation vector field at iteration 5
│ ├── theta_10.png # Lacunae orientation vector field at iteration 10
│ ├── theta_15.png # Lacunae orientation vector field at iteration 15
│ ├── theta_20.png # Lacunae orientation vector field at iteration 20
│ ├── theta_25.png # Lacunae orientation vector field at iteration 25
│ ├── theta_30.png # Lacunae orientation vector field at iteration 30
│ ├── theta_final.png # Final converged lacunae orientation vector field
│ ├── bone_sub_theta_all-itr.xdmf # Descriptor file linking iteration-wise
orientation data for visualization
│ └── bone_sub_theta_all-itr.h5 # HDF5 data file containing all
iteration-wise orientation field values
├── images/ # Supporting figures for conceptual and graphical illustrations
│ ├── bone_opt.png # Conceptual sketch of the proposed material and structure
optimization problem for a proximal femur
├── setup.py — A standard Python packaging script used to define the project structure. It allows the src directory to be recognized as an installable Python package using setup tools
Software Requirements
Essential Software:
- Python 3.7+ with dependencies: NumPy, SciPy, Matplotlib, Jupyter
To install all dependencies:
python setup.py install
File-Specific Instructions:
- Run Python scripts:
python main.py - Launch Jupyter notebook:
jupyter notebook bone_multitopopt.ipynb - View PNG images: any standard image viewer
Simulation Results Visualization: - ParaView — Free visualization of
.xdmfresult files
(Each .xdmf file is accompanied by a corresponding .h5 (HDF5) file that stores the actual numerical data (mesh, fields, and solution variables). The .xdmf file serves as a lightweight descriptor that references the .h5 data for efficient visualization in ParaView.)
Data Files Description
Optimization Results — Examples/Examples_paper/Multiscale_bone/pictures/
Files:
f_HA_itr_(iteration).png— Optimized hydroxyapatite (mineral) fraction per iterationf_col_itr_(iteration).png— Optimized collagen fraction per iterationrho_itr_(iteration).png— Optimized density distribution per iteration- bone_sub_theta_all-itr/
theta_(iteration).png— Optimized microstructural orientation angle per iteration vertical_slice_mirrored_70.png— 70th vertical slice of the femur extracted from the CT dataset, represented by Hounsfield Unit (HU) map
Model Parameters:
- Design domain: Segmented bone geometry (
inp.Bone()) - Total elements: Computed from mesh (
n_ele = elem.shape[0]) - Estimated total volume: 5860.3385 (arbitrary unit³)
- Element volume:
vol_el = vol_dom / n_ele - Initial element density: ρ = 0.75 (uniform initialization)
- Target volume fraction: 0.65 (material utilization ratio)
- Filter radius: 2 (element units)
- Optimization iterations: 30
- Move limit (density update step): 0.02
- Damping coefficient: 0.5 (stabilization)
- Tolerance (micro-iteration convergence): 1×10⁻²
- Maximum micro-iterations: 7
- Initial microstructural parameters:
f_HA_bar = 0.55— initial hydroxyapatite fractionf_col_bar = 0.45 × 0.9— initial collagen fractionθ_lac = 0— initial microstructural orientation
File: BoneHealthAssessment.zip
Overview
This clinical dataset contains CT scans for quantitative bone health assessment in two female patients of different ages, demonstrating the application of the bone multi-topology optimization framework to patient-specific data.
BoneHealthAssessment/
├── Pat064_58F/ # Patient 064: 58-year-old female
│ ├── Pat064_CT_Data.nrrd # 3D CT scan volume
│ ├── Pat064_Slice2D.nrrd # Extracted 2D slice
│ ├── bone_material-optimization2D-HA.xdmf # Optimized mineral density
│ ├── bone_material-optimization2D-rho.xdmf # Optimized macro density
│ └── Pat064_Slice2D.png # 2D slice visualization
└── Pat095_82F/ # Patient 095: 82-year-old female
├── Pat095_CT_Data.nrrd # 3D CT scan volume
├── Pat095_Slice2D.nrrd # Extracted 2D slice
├── bone_material-optimization2D-HA.xdmf # Optimized mineral density
├── bone_material-optimization2D-rho.xdmf # Optimized macro density
└── Pat095_Slice2D.png # 2D slice visualization
Software Requirements
Medical Imaging Tools:
Simulation Results Visualization:
- ParaView — Free visualization of
.xdmfresult files
(Each .xdmf file is accompanied by a corresponding .h5 (HDF5) file that stores the actual numerical data (mesh, fields, and solution variables). The .xdmf file serves as a lightweight descriptor that references the .h5 data for efficient visualization in ParaView.)
Citation
If you use this dataset or code, please cite the associated research and this repository.
License
This repository and its accompanying datasets are released under the Creative Commons Zero (CC0 1.0 Universal Public Domain Dedication).
By applying CC0, the author waives all rights to the data and associated materials to the extent permitted by law, allowing unrestricted reuse, distribution, and modification without any requirement for attribution.
All code components within this repository are likewise made openly available for academic and non-commercial use under the same CC0 waiver, ensuring full transparency and reproducibility of the presented research.