# A two-scale approach for trabecular bone microstructure modeling based on computational homogenization procedure

## Abstract

In this paper the numerical implementation of two-scale modelling of bone microstructure is presented. The study is a part of long-term project on bone remodelling which drives bone microstructure change based directly on trabeculae surface energy. The proposed approach is based on a first-order computational homogenization technique. The coincidence of macro- and micro-model kinematics is done with the use of uniform displacement and traction boundary conditions. The computational homogenization procedure is driven by a self-prepared manager which is coded in Python. The computation on real bone structure (a piece of female Wistar rat bone) is performed as well.

## Keywords

Two-scale modelling Computational homogenization Bone microstructure modelling## 1 Introduction

Bone tissue is a heterogeneous material which can be classified into two basic types, cortical and cancellous. The cortical bone is dense and creates hard outer layer for cancellous bone which is geometrically complex three-dimensional structure, a mixture of trabeculae in a stochastic form of small beams, struts or rods and voids (see Fig. 1).

The size reduction of FE model in the case of material which is heterogeneous at a certain scale can be done using a homogenization approach [1, 13, 31, 32]. There are a few different, well-known methods of homogenization of a heterogeneous material. The simplest one bases on a rule of mixtures. In this case, the homogenized properties are calculated as an average over the particular properties of ingredients, which are weighed with their volume fractions. This approach can take into account only one microstructural characteristics. The other method assumes an effective medium approximation [9, 17, 26] which can be used only for structures which have regular and patterned geometry. In this method the equivalent material properties are calculated based on an analytical solution of a boundary value problem. Thus, it cannot take into consideration the non-regular microstructure of cancellous bone. The next homogenization method bases on the mathematical asymptotic homogenization theory [3]. It makes it possible to obtain both: a homogenized properties and local stress and strain values. However, this homogenization method can be used only in cases of very simple microscopic geometries. Today, a group of methods called unit cell [34] are very common for complex microstructure. The unit cell method helps to obtain the information on material state on a local microstructural level and simultaneously on homogenized global material properties. These macroscopic properties are calculated using closed-form phenomenological constitutive equations based on the averaged state (stress–strain) from the analysis of a representative cell of microstructure subjected to a corresponding load scheme. From the viewpoint of the problem discussed, the key weakness of the unit cell method is the requirement of a representative cell of microstructure. In the case of cancellous bone as microstructure it is impossible to define for each macro level material point corresponding representative volume element (RVE). This limitation is of course crucial in the same sense for all homogenization methods discussed above. To sum up, the typical homogenization methods cannot be used directly for simulation of cancellous bone microstructure.

In the context of the discussed problem, the useful alternative for the classical homogenization methods can be a hierarchical approach [5, 14, 23]. This method bases on two-scale approaches and uses the micro- and the macro-scale to describe global bone characteristics. In the global scale, bone is modeled as a continuum material which is characterized by homogenized properties. In the micro scale the bone tissue is considered to be a representative cell of trabeculae of a cancellous bone. This representative cell is assumed to be periodic and has predefined geometric topology [5, 23].

The two-scale modelling approach proposed in this paper determines the macroscopic constitutive behavior based directly on detail geometry of cancellous bone microstructure.

## 2 Computational homogenization

The homogenization of heterogeneous materials with the use of multi-scale technique is a common method today [12, 22, 33, 35, 36, 37]. The main idea of this method, also called global-local analysis, bases on the direct calculation of the local macroscopic constitutive response from the corresponding microstructure boundary value problem. The foundation of the computational homogenization method is consistent with the other concepts of homogenization techniques. It can be described as a classical four-steps homogenization scheme proposed by Suquet [33]. In the first step, a microstructural RVE is defined. Its constitutive behaviour is assumed to be known. In the second step, microscopic boundary conditions from the macroscopic input variables are formulated and applied to the RVE (so-called macro-to-micro transition). The third step is the calculation of the homogenized macroscopic characteristics based on the analysis of the microstructural RVE (so called micro-to-macro transition). In the last step, the relation between the macroscopic characteristics and inputs is calculated.

The computational homogenization technique has a few crucial advantages. First of all, it does not require any explicitly formulated constitutive equations on a macro-scale level. The macroscopic constitutive behaviour is directly obtained from the corresponding micro-scale boundary value problem analysis [35]. The second advantage is the possibility of taking into consideration the evolution of the microstructure on the macroscopic level analysis and coupling this evolution with results of the macro-scale analysis. Last but not least, there is the advantage that this method can be still effectively used even if the requirement of RVE is not exactly fulfilled for all micro-scale levels analyses.

### 2.1 Basics

### 2.2 Models

In the presented approach the finite element method is used to realize computational homogenization procedure on macro- and micro-levels. Both macro- and micro-models are created in Abaqus/Standard code. In this section the macro- and micro-models are described.

## 3 Benchmark problems

Because the developed two-scale modelling implementation is based on homogenization technique, the homogenization of a reference microstructure, i.e. one material, continuous cube, is done as a verification in the first place (test A1–A5). Next, this two-scale procedure is verified for bone-like discontinuous material (B6–D12). The proposed approach was implemented in Abaqus FEA code (Dassault Systèmes) driven by self-prepared manager codded in Python.

This reference microstructure is a cube of edge length equal to 10 mm. The structure is divided into 125,000 8-node linear continuum elements. The linear, isotropic material properties are assigned to the structures with Young’s modulus and Poisson’s ratio equal 20e9 Pa and 0.3, respectively. The mechanical parameters of the structures correspond to bone tissue mechanical characteristics [38]. For the reference microstructure, a macro-model and a set of micro-models have been created.

The discontinuity is generated by adding ten spherical holes. The coordinates of the centers and radiuses are randomly generated. The obtained test structures for the three layout of holes are shown in Fig. 5.

Tests models specification

Test id | Macro-model discretization | Reference microstructure | Total void percentage (%) | |
---|---|---|---|---|

Element number | Node number | |||

A1 | \(1\times 1\times 1\) | 125,000 | 132,652 | 0.0 |

A2 | \(2\times 2\times 2\) | |||

A3 | \(5\times 5\times 5\) | |||

A4 | \(7\times 7\times 7\) | |||

A5 | \(9\times 9\times 9\) | |||

B6 | \(5\times 5\times 5\) | 115,572 | 125,212 | 7.5 |

C7 | \(1\times 1\times 1\) | 93.524 | 104,559 | 25.2 |

C8 | \(2\times 2\times 2\) | |||

C9 | \(5\times 5\times 5\) | |||

C10 | \(7\times 7\times 7\) | |||

C11 | \(9\times 9\times 9\) | |||

D12 | \(5\times 5\times 5\) | 69,614 | 81,097 | 44.3 |

Verification of homogenization—the global deformation of tested specimens for one-scale and two-scale analyses along with percentage error

Test id | Two-scale approach | one-scale approach | Error | |||||||
---|---|---|---|---|---|---|---|---|---|---|

L1 (mm) | L2 (mm) | L3 (mm) | L1 (mm) | L2 (mm) | L3 (mm) | L1 (%) | L2 (%) | L3 (%) | mean (%) | |

A1 | \(-\)3.72e\(-\)9 | 1.30e\(-\)8 | 7.80e\(-\)10 | \(-\)4.70e\(-\)9 | 1.93e\(-\)8 | 8.95e\(-\)10 | 20.9 | 32.7 | 12.8 | 22.1 |

A2 | \(-\)4.50e\(-\)9 | 1.58e\(-\)8 | 7.83e\(-\)10 | \(-\)4.70e\(-\)9 | 1.93e\(-\)8 | 8.95e\(-\)10 | 4.2 | 18.1 | 12.5 | 11.6 |

A3 | \(-\)4.68e\(-\)9 | 1.89e\(-\)8 | 8.86e\(-\)10 | \(-\)4.70e\(-\)9 | 1.93e\(-\)8 | 8.95e\(-\)10 | 0.3 | 2.4 | 1.0 | 3.7 |

A4 | \(-\)4.68e\(-\)9 | 1.90e\(-\)8 | 8.88e\(-\)10 | \(-\)4.70e\(-\)9 | 1.93e\(-\)8 | 8.95e\(-\)10 | 0.4 | 1.8 | 0.8 | 1.1 |

A5 | \(-\)4.71e\(-\)9 | 1.93e\(-\)8 | 8.98e\(-\)10 | \(-\)4.70e\(-\)9 | 1.93e\(-\)8 | 8.95e\(-\)10 | 0.2 | 0.4 | 0.4 | 0.3 |

B6 | \(-\)5.35e\(-\)9 | 2.16e\(-\)8 | 9.74e\(-\)10 | \(-\)5.48e\(-\)9 | 2.28e\(-\)8 | 1.01e\(-\)9 | 2.4 | 5.3 | 3.2 | 3.6 |

C7 | \(-\)1.06e\(-\)8 | 3.11e\(-\)8 | 1.87e\(-\)9 | \(-\)1.63e\(-\)8 | 5.74e\(-\)8 | 4.12e\(-\)9 | 34.8 | 45.7 | 54.6 | 45.0 |

C8 | \(-\)1.21e\(-\)8 | 3.49e\(-\)8 | 2.06e\(-\)9 | \(-\)1.63e\(-\)8 | 5.74e\(-\)8 | 4.12e\(-\)9 | 26.0 | 39.2 | 50.2 | 38.5 |

C9 | \(-\)1.61e\(-\)8 | 5.22e\(-\)8 | 3.59e\(-\)9 | \(-\)1.63e\(-\)8 | 5.74e\(-\)8 | 4.12e\(-\)9 | 0.8 | 9.1 | 12.9 | 7.6 |

C10 | \(-\)1.54e\(-\)8 | 5.24e\(-\)8 | 3.65e\(-\)9 | \(-\)1.63e\(-\)8 | 5.74e\(-\)8 | 4.12e\(-\)9 | 5.6 | 8.6 | 11.4 | 8.5 |

C11 | \(-\)1.58e\(-\)8 | 5.46e\(-\)8 | 3.84e\(-\)9 | \(-\)1.63e\(-\)8 | 5.74e\(-\)8 | 4.12e\(-\)9 | 3.0 | 4.8 | 6.8 | 4.9 |

D12 | \(-\)7.60e\(-\)9 | 3.19e\(-\)8 | 1.57e\(-\)9 | \(-\)8.00e\(-\)9 | 3.51e\(-\)8 | 1.72e\(-\)9 | 4.9 | 9.1 | 8.7 | 7.6 |

The crucial aspect of the verification of the homogenization procedure is the mesh convergence evaluation done with the tests A1–A5 for completely filled cube and C7–C11 for porous microstructure. In both cases the mesh refinement reduces the mean error to 0.3 and 4.9 % for tests A5 and C11, respectively. However, the error below 10 % can be achieved already for the macro-model which has at least five elements per each edge of the cube (tests A3 and C9). The character of the mesh convergence curve and small values of the error for fine meshes enable us to conclude that global responses of the one-scale and two-scale models are comparable. It confirms that macro-to-micro and micro-to-macro transitions are done in a proper way and the error due to homogenization procedure in the case o f the two-scale model is acceptable.

The tests B, C and D illustrate the influence of material discontinuity which is a crucial aspect from the viewpoint of bone microstructure modelling. The results obtained are qualitatively the same and quantitatively almost the same (Table 2). Since these three tests can represent three stages of evolution of the cancellous bone microstructure, the verification clearly shows that the presented procedure can be used in the case of evaluating bone-like material.

## 4 Real case study: Wistar rat trabecular bone

The global deformation of rat bone model in direction of applied load for one-scale and two-scale analyses

Load case | Two-scale approach | One-scale approach | Error (%) |
---|---|---|---|

compression (L1) | 5.52271e\(-\)3 | 5.2218e\(-\)3 | 5.7 |

shear (L2) | 2.67052e\(-\)2 | 2.5564e\(-\)2 | 4.7 |

## 5 Conclusions

The paper deals with the concept of procedure which, using a two-scale modelling, allows for a fruitful analysis of bone considering its porous, non-homogenous, non-continuous, stochastic form. The global displacement (equivalently strain energy) comparison for test problems, as well as real rat bone geometry, show that proposed modelling approach can be used to simulate bone microstructure with satisfactory accuracy. It can be done not only in an efficient way but also with an acceptable calculation cost. The analysis of a piece of rat bone shows that presented two-scale approach makes it possible to reduce significantly the size of the problem from system of questions with more then 36 million unknowns to set of 2,774 systems of questions with no more then 27,000 unknowns each one. The solving of these set of systems of questions in the two-scale approach is perfectly linearly scalable. The two-scale model can be analyzed with the use of the regular PC workstation instead of the Beowulf cluster (8 CPUs vs 20 CPUs) and nonetheless the time of calculation is four times faster. Moreover, the two-scale technique makes it possible to calculate mechanical bone behavior for large models and to keep the detailed information about cancellous bone microstructure simultaneously. This technique opens a door to the effective simulations of bone remodelling on the level of particular trabeculae as a common practice in biomechanical studies.

The proposed approach has several general significant advantages as well. Firstly, there are no assumptions concerning micro-scale geometry. Secondly, the macroscopic behavior of other heterogeneous materials, not only bone-like, can be described without the explicitly formulated constitutive relations on the macro-scale level. Next, from the viewpoint of engineering application, any arbitrary non-linear and time dependent material behavior can be taken into consideration on the micro-scale level. At the same time this method can be used in the cases of complex loading paths, large deformations and large rotations on the macro-scale level. Next, the hierarchical approach significantly reduces the cost and time of calculation in the case of large models of complex structures. Finally, the averaged stress can be calculated to estimate stress level for homogenized material.

The disadvantage of the method should also be mentioned. In the fully coupled micro-macro technique the computational cost of all nested boundary value problems analyses can be very large. However, this problem can be successfully solved with the use of parallel computing. In the proposed approach the parallelization ratio is close to one.

## Notes

### Acknowledgments

The support of the Ministry of Science and Higher Education under Grants R13 0020/06, N518 328835 and Polish National Science Centre under Grant DEC-2011/01/B/ST8/06925 is highly acknowledged.

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