Institute of Fundamental Technological Research

In previous works, the authors introduced a multiscale optimization method to maximize the stiffness of elastic solids with biomimetic cancellous microstructures described by a finite set of parameters. Although effective, the procedure is computationally expensive when solving large-scale problems using per-element non-linear constraints to impose local bounds on the solid volume fraction. This work improves the computational performance of the method by exploring two strategies to completely dispense with nonlinear local constraints: to bound the microparameters so the microsctructures are always within the solid fraction of trabecular bone, and to map the microparameters onto an auxiliary set of parameters that are linearly bounded. As a side effect, the design spaces are reduced. Such reductions are assessed in terms of the bulk and shear moduli and elastic symmetries, which are compared to those of natural bone. Performances of the two strategies are assessed by solving a series of benchmark problems and studying the stiffness of a hip prosthesis. The strategy based on the isoparametric mapping achieves the best results, performing up to 2000 times faster while marginally reducing the design space. Thus, the isoparametric mapping approach makes the multiscale design method a suitable tool for solving large-scale problems of practical interest.

Multiscale optimization , Trabecular bone , Parameterized microstructures , Computational performance , Large-scale problems

Cancellous bone is a highly porous, heterogeneous, and anisotropic material which can be found at the epiphyses of long bones and in the vertebral bodies. The hierarchical architecture makes cancellous bone a prime example of a lightweight natural material that combines strength with toughness. Better understanding the mechanics of cancellous bone is of interest for the diagnosis of bone diseases, the evaluation of the risk of fracture, and for the design of artificial bones and bone scaffolds for tissue engineering. A multiscale optimization method to maximize the stiffness of artificial bones using biomimetic cellular microstructures described by a finite set of geometrical micro-parameters is presented here. The most outstanding characteristics of its implementation are the use of: an interior point optimization algorithm, a precalculated response surface methodology for the evaluation of the elastic tensor of the microstructure as an analytical function of the micro-parameters, and the adjoint method for the computation of the sensitivity of the macroscopic mechanical response to the variation of the micro-parameters. The performance and effectiveness of the tool are evaluated by solving a problem that consists in finding the optimal distribution of the microstructures for a proximal end of a femur subjected to physiological loads. Two strategies for the specification of the solid volume fraction constraints are assessed. The results are compared with data of a computed tomography study of an actual human bone. The model successfully predicts the main features of the spatial arrangement of the trabecular and cortical microstructures of the natural bone.

multiscale optimization, cancellous bone, bone scaffolds, parameterized microstructures

Natural (or biological) materials usually achieve outstanding mechanical performances. In particular, cancellous bone presents a high stiffness/strength toweight ratio, so its structure inspires the development of novel ultra-light cellularmaterials. Amultiscale method for the design of elastic solids with a cancellous bone parameterized biomimetic microstructure is introduced in this work. The method combines a finite element model to evaluate the stiffness of the body at the macroscale with a gradient-based nonlinear constrained optimization solver to obtain the optimal values of themicroparameters andmicrostructure orientation over the body domain. The most salient features of the implementation are an offline response surface methodology for the evaluation of the microstructure elastic tensor in terms of the microparameters, an adjoint method for the computation of the sensitivity of the macroscopic stiffness to the microparameters, a quasi-Newton approximation for the evaluation of the Hessian matrix of the nonlinear optimizer, and a distanceweighted filter of the microparameters to remediate checkerboard effects. The settings of the above features, the optimizer termination options, and the initial values of the microparameters are investigated for the best performance of the method. The effectiveness of the method is demonstrated for several examples, whose results are compared with the reference solutions calculated using a SIMP method. The method shows to be effective; it attains results coherent with SIMP approaches in terms of stiffness and spatial material distribution. The good performance of themultiscalemethod is attributed to the capability of the parameterized mimeticmicrostructure to attain bulk and shear moduli that are close to the Hashin-Shtrikman upper bounds over the complete solid volume fraction range.

multiscale optimization, cancellous bone, parameterized microstructure, interior-point optimizer, biomimetic materials

Bone tissue mechanical properties and trabecular microarchitecture are the main factors that determine the biomechanical properties of cancellous bone. Artificial cancellous microstructures, typically described by a reduced number of geometrical parameters, can be designed to obtain a mechanical behavior mimicking that of natural bone. In this work, we assess the ability of the parameterized microstructure introduced by Kowalczyk (Comput Methods Biomech Biomed Eng 9:135–147, 2006. doi:10.1080/10255840600751473) to mimic the elastic response of cancellous bone. Artificial microstructures are compared with actual bone samples in terms of elasticity matrices and their symmetry classes. The capability of the parameterized microstructure to combine the dominant isotropic, hexagonal, tetragonal and orthorhombic symmetry classes in the proportions present in the cancellous bone is shown. Based on this finding, two optimization approaches are devised to find the geometrical parameters of the artificial microstructure that better mimics the elastic response of a target natural bone specimen: a Sequential Quadratic Programming algorithm that minimizes the norm of the difference between the elasticity matrices, and a Pattern Search algorithm that minimizes the difference between the symmetry class decompositions. The pattern search approach is found to produce the best results. The performance of the method is demonstrated via analyses for 146 bone samples.

Cancellous bone, Parameterized microstructure, Elastic properties, Homogenization, Symmetry classes, Optimization

The Fast Fourier Transform–based method, originally introduced by Moulinec and Suquet in 1994 has gained popularity for computing homogenized properties of composites. In this work, the method is used for the computational homogenization of the elastic properties of cancellous bone. To the authors' knowledge, this is the first study where the Fast Fourier Transform scheme is applied to bone mechanics. The performance of the method is analyzed for artificial and natural bone samples of 2 species: bovine femoral heads and implanted femurs of Hokkaido rats. Model geometries are constructed using data from X‐ray tomographies, and the bone tissue elastic properties are measured using microindentation and nanoindentation tests. Computed results are in excellent agreement with those available in the literature. The study shows the suitability of the method to accurately estimate the fully anisotropic elastic response of cancellous bone. Guidelines are provided for the construction of the models and the setting of the algorithm.

accelerated FFT method, cancellous bone, homogenized elastic properties

Natural (or biological) materials generally achieve excellent mechanical performance. In particular, trabecular bone has a high stiffness / strength / weight ratio, which is why its structure inspires the development of new ultralight cellular materials. This work introduces a multiscale method for the design of elastic solids with a parametrized biomimetic microstructure of trabecular bone. The method combines a finite element model to evaluate the stiffness of the body at the macroscale with a non-linear optimizer to obtain the optimal values ​​of the microparameters and the orientation of the microstructure on the body domain. The most outstanding characteristics of its implementation are the use of: (i) an interior point type optimization algorithm, (ii) a precalculated response surface methodology for the evaluation of the elastic tensor of the microstructure as a function of the microparameters, ( iii) the attached method for calculating the sensitivity of the macroscopic mechanical response to the variation of the microparameters, and (iv) a weighted filter on the microparameters to correct for checkerboard effects. The performance and effectiveness of the tool are evaluated by solving a problem that consists of finding the optimal distribution of microstructures for the proximal end of a femur subjected to physiological loads. Three strategies are evaluated for the specification of the solid volume fraction constraints. The results are compared with data from a CT scan of a real human bone. The model successfully predicts the main features of the spatial arrangement of the trabecular and cortical microstructures of natural bone.

multiscale optimization, trabecular bone, parametric miscrostructure

multiscale structural optimization, biomimetic microstructures, finite element analysis