Multiscale methodology for bone remodelling simulation using coupled finite element and neural network computation

The aim of this paper is to develop a multiscale hierarchical hybrid model based on finite element analysis and neural network computation to link mesoscopic scale (trabecular network level) and macroscopic (whole bone level) to simulate the process of bone remodelling. As whole bone simulation, including the 3D reconstruction of trabecular level bone, is time consuming, finite element calculation is only performed at the macroscopic level, whilst trained neural networks are employed as numerical substitutes for the finite element code needed for the mesoscale prediction. The bone mechanical properties are updated at the macroscopic scale depending on the morphological and mechanical adaptation at the mesoscopic scale computed by the trained neural network. The digital image-based modelling technique using μ-CT and voxel finite element analysis is used to capture volume elements representativeof 2 mm³ at the mesoscale level of the femoral head. The input data for the artificial neural network are a set of bone material parameters, boundary conditions and the applied stress. The output data are the updated bone properties and some trabecular bone factors. The current approach is the first model, to our knowledge, that incorporates both finite element analysis and neural network computation to rapidly simulate multilevel bone adaptation.     

Derivation of the mesoscopic elasticity tensor of cortical bone from quantitative impedance images at the micron scale

This paper addresses the relationships between the microscopic properties of bone and its elasticity at the millimetre scale, or mesoscale. A method is proposed to estimate the mesoscale properties of cortical bone based on a spatial distribution of acoustic properties at the microscopic scale obtained with scanning acoustic microscopy. The procedure to compute the mesoscopic stiffness tensor involves (i) the segmentation of the pores to obtain a realistic model of the porosity; (ii) the construction of a field of anisotropic elastic coefficients at the microscopic scale which reflects the heterogeneity of the bone matrix; (iii) finite element computations of mesoscopic homogenized properties. The computed mesoscopic properties compare well with available experimental data. It appears that the tissue anisotropy at the microscopic level has a major effect on the mesoscopic anisotropy and that assuming the pores filled with an incompressible fluid or, alternatively, empty, leads to significantly different mesoscopic properties.     

A generic 3-dimensional system to mimic trabecular bone surface adaptation

The paper presents the structure optimizing system based on surface remodelling. The grounds for algorithm formulation are given by the bone remodelling phenomenon leading to optimization of trabecular network in the bone. The assumptions, algorithms and limitations of the own mesh generator Cosmoprojector are described. Unlike other approaches, the system is able to mimic real bone evolution including tissue consolidation and separation. The article presents a closed system consisting of finite element mesh generation, decision criteria for structure adaptation and finite element analysis in parallel environment. It also provides some computation results obtained by using specially designed software.     

A generic 3-dimensional system to mimic trabecular bone surface adaptation

The paper presents the structure optimizing system based on surface remodelling. The grounds for algorithm formulation are given by the bone remodelling phenomenon leading to optimization of trabecular network in the bone. The assumptions, algorithms and limitations of the own mesh generator Cosmoprojector are described. Unlike other approaches, the system is able to mimic real bone evolution including tissue consolidation and separation. The article presents a closed system consisting of finite element mesh generation, decision criteria for structure adaptation and finite element analysis in parallel environment. It also provides some computation results obtained by using specially designed software.     

Experimental and numerical identification of cortical bone permeability

Bone is a complex system, and could be modeled as a poroelastic media. The aim of this paper is to identify the macroscopic value of the cortical bone permeability coefficient. A simple experimental method was designed in order to determine the permeability coefficient. Two bone samples taken from different ox femurs were filled with water, to place them under internal pressure. The measurements gave both the fluid flow through the lateral surfaces and the internal pressure. The originality of this work is the coupling between an experimental process and a structural computation performed with a finite element method. The mean cortical bone permeability coefficient identified was about k=1.1x10(-13)m(2). This value tends to confirm other values found in the literature, obtained by different methods and often at macroscopic scale. It confirms also the domination of vascular permeability (Haversian and Volkmann's canals).     

A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties

At its highest level of microstructural organization—the mesoscale or millimeter scale—cortical bone exhibits a heterogeneous distribution of pores (Haversian canals, resorption cavities). Multi-scale mechanical models rely on the definition of a representative volume element (RVE). Analytical homogenization techniques are usually based on an idealized RVE microstructure, while finite element homogenization using high-resolution images is based on a realistic RVE of finite size. The objective of this paper was to quantify the size and content of possible cortical bone mesoscale RVEs. RVE size was defined as the minimum size: (1) for which the apparent (homogenized) stiffness tensor becomes independent of the applied boundary conditions or (2) for which the variance of elastic properties for a set of microstructure realizations is sufficiently small. The field of elastic coefficients and microstructure in RVEs was derived from one acoustic microscopy image of a human femur cortical bone sample with an overall porosity of 8.5%. The homogenized properties of RVEs were computed with a finite element technique. It was found that the size of the RVE representative of the overall tissue is about 1.5 mm. Smaller RVEs (~0.5 mm) can also be considered to estimate local mesoscopic properties that strongly depend on the local pores volume fraction. This result provides a sound basis for the application of homogenization techniques to model the heterogeneity of cortical microstructures. An application of the findings to estimate elastic properties in the case of a porosity gradient is briefly presented.     

Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone

We compare theoretical predictions of the effective elastic moduli of cortical bone at both the meso- and macroscales. We consider the efficacy of three alternative approaches: the method of asymptotic homogenization, the Mori-Tanaka scheme and the Hashin-Rosen bounds. The methods concur for specific engineering moduli such as the axial Young's modulus but can vary for others. In a past study, the effect of porosity alone on mesoscopic properties of cortical bone was considered, taking the matrix to be isotropic. Here, we consider the additional influence of the transverse isotropy of the matrix. We make the point that micromechanical approaches can be used in two alternative ways to predict either the macroscopic (size of cortical bone sample) or mesoscopic (in between micro- and macroscales) effective moduli, depending upon the choice of representative volume element size. It is widely accepted that the mesoscale behaviour is an important aspect of the mechanical behaviour of bone but models incorporating its effect have started to appear only relatively recently. Before this only macroscopic behaviour was addressed. Comparisons are drawn with experimental data and simulations from the literature for macroscale predictions with particularly good agreement in the case of dry bone. Finally, we show how predictions of the effective mesoscopic elastic moduli can be made which retain dependence on the well-known porosity gradient across the thickness of cortical bone.     

Neural adaptive control for vibration suppression in composite fin-tip of aircraft

In this paper, we present a neural adaptive control scheme for active vibration suppression of a composite aircraft fin tip. The mathematical model of a composite aircraft fin tip is derived using the finite element approach. The finite element model is updated experimentally to reflect the natural frequencies and mode shapes very accurately. Piezo-electric actuators and sensors are placed at optimal locations such that the vibration suppression is a maximum. Model-reference direct adaptive neural network control scheme is proposed to force the vibration level within the minimum acceptable limit. In this scheme, Gaussian neural network with linear filters is used to approximate the inverse dynamics of the system and the parameters of the neural controller are estimated using Lyapunov based update law. In order to reduce the computational burden, which is critical for real-time applications, the number of hidden neurons is also estimated in the proposed scheme. The global asymptotic stability of the overall system is ensured using the principles of Lyapunov approach. Simulation studies are carried-out using sinusoidal force functions of varying frequency. Experimental results show that the proposed neural adaptive control scheme is capable of providing significant vibration suppression in the multiple bending modes of interest. The performance of the proposed scheme is better than the H(infinity) control scheme.     

Optimization of fed-batch bioreactor using neural network model

An algorithm using feedforward neural network model for determining optimal substrate feeding policies for fed-batch fermentation process is presented in this work. The algorithm involves developing the neural network model of the process using the sampled data. The trained neural network model in turn is used for optimization purposes. The advantages of this technique is that optimization can be achieved without detailed kinetic model of the process and the computation of gradient of objective function with respect to control variables is straightforward. The application of the technique is demonstrated with two examples, namely, production of secreted protein and invertase. The simulation results show that the discrete-time dynamics of fed-batch bioreactor can be satisfactorily approximated using a feedforward sigmoidal neural network. The optimal policies obtained with the neural network model agree reasonably well with the previously reported results.     

Micromechanical modelling of cortical bone

Cortical bone is a heterogeneous material with a complex hierarchical microstructure. In this work, unit cell finite element models were developed to investigate the effect of microstructural morphology on the macroscopic properties of cortical bone. The effect of lacunar and vascular porosities, percentage of osteonal bone and orientation of the Haversian system on the macroscopic elastic moduli and Poisson's ratios was investigated. The results presented provide relationships for applying more locally accurate material properties to larger scale and whole bone models of varying porosity. Analysis of the effect of the orientation of the Haversian system showed that its effects should not be neglected in larger scale models. This study also provides insight into how microstructural features effect local distributions and cause a strain magnification effect. Limitations in applying the unit cell methodology approach to bone are also discussed.     

Modeling of bone at a single lamella level

This paper focuses on the ultrastructure of bone at a single lamella level. At this scale, collagen fibrils reinforced with apatite crystals are aligned preferentially to form a lamella. At the next structural level, such lamella are stacked in different orientations to form either osteons in cortical bone or trabecular pockets in trabecular bone. We use a finite element model, which treats small strain elasticity of a spatially random network of collagen fibrils, and compute anisotropic effective stiffness tensors and deformations of such a single lamella as a function of fibril volume fractions (or porosities), prescribed microgeometries, and fibril geometric and elastic properties.     

可渗透管网络非定常跨壁传输问题的解耦算法

提出了可渗透管网络非定常跨壁传输问题的解耦算法,这是一种有限元法与初值问题Ｃａｕｃｈｙ方法相结合的半解析算法。在一维管流的假设下,通过引入插值函数,得到用管外变量表示的管内变量的解析解,使最终导出的有限元方程中只包管外变量,从而减少了联立方程的数目,大大节省了计算时间,本方法特别适用于大型网络的数值计算。     

Application of neural networks for the prediction of cartilage stress in a musculoskeletal system

Traditional finite element (FE) analysis is computationally demanding. The computational time becomes prohibitively long when multiple loading and boundary conditions need to be considered such as in musculoskeletal movement simulations involving multiple joints and muscles. Presented in this study is an innovative approach that takes advantage of the computational efficiency of both the dynamic multibody (MB) method and neural network (NN) analysis. A NN model that captures the behavior of musculoskeletal tissue subjected to known loading situations is built, trained, and validated based on both MB and FE simulation data. It is found that nonlinear, dynamic NNs yield better predictions over their linear, static counterparts. The developed NN model is then capable of predicting stress values at regions of interest within the musculoskeletal system in only a fraction of the time required by FE simulation.     

Micromechanical model of a surrogate for collagenous soft tissues: development, validation and analysis of mesoscale size effects

Aligned, collagenous tissues such as tendons and ligaments are composed primarily of water and type I collagen, organized hierarchically into nanoscale fibrils, microscale fibers and mesoscale fascicles. Force transfer across scales is complex and poorly understood. Since innervation, the vasculature, damage mechanisms and mechanotransduction occur at the microscale and mesoscale, understanding multiscale interactions is of high importance. This study used a physical model in combination with a computational model to isolate and examine the mechanisms of force transfer between scales. A collagen-based surrogate served as the physical model. The surrogate consisted of extruded collagen fibers embedded within a collagen gel matrix. A micromechanical finite element model of the surrogate was validated using tensile test data that were recorded using a custom tensile testing device mounted on a confocal microscope. Results demonstrated that the experimentally measured macroscale strain was not representative of the microscale strain, which was highly inhomogeneous. The micromechanical model, in combination with a macroscopic continuum model, revealed that the microscale inhomogeneity resulted from size effects in the presence of a constrained boundary. A sensitivity study indicated that significant scale effects would be present over a range of physiologically relevant inter-fiber spacing values and matrix material properties. The results indicate that the traditional continuum assumption is not valid for describing the macroscale behavior of the surrogate and that boundary-induced size effects are present.     

Neural network predicts sequence of TP53 gene based on DNA chip

We have trained an artificial neural network to predict the sequence of the human TP53 tumor suppressor gene based on a p53 GeneChip. The trained neural network uses as input the fluorescence intensities of DNA hybridized to oligonucleotides on the surface of the chip and makes between zero and four errors in the predicted 1300 bp sequence when tested on wild-type TP53 sequence. AVAILABILITY: The trained neural network is available for academic use by contacting steen@cbs.dtu.dk     

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

The discrete nature of trabecular bone microarchitecture affects implant stability

Small endosseous implants, such as screws, are important components of modern orthopedics and dentistry. Hence they have to reliably fulfill a variety of requirements, which makes the development of such implants challenging. Finite element analysis is a widely used computational tool used to analyze and optimize implant stability in bone. For these purposes, bone is generally modeled as a continuum material. However, bone failure and bone adaptation processes are occurring at the discrete level of individual trabeculae; hence the assessment of stresses and strains at this level is relevant. Therefore, the aim of the present study was to investigate how peri-implant strain distribution and load transfer between implant and bone are affected by the continuum assumption. We performed a computational study in which cancellous screws were inserted in continuum and discrete models of trabecular bone; axial loading was simulated. We found strong differences in bone-implant stiffness between the discrete and continuum bone model. They depended on bone density and applied boundary conditions. Furthermore, load transfer from the screw to the surrounding bone differed strongly between the continuum and discrete models, especially for low-density bone. Based on our findings we conclude that continuum bone models are of limited use for finite element analysis of peri-implant mechanical loading in trabecular bone when a precise quantification of peri-implant stresses and strains is required. Therefore, for the assessment and improvement of trabecular bone implants, finite element models which accurately represent trabecular microarchitecture should be used.     

Effect of including damage at the tissue level in the nonlinear homogenisation of trabecular bone

Being able to predict bone fracture or implant stability needs a proper constitutive model of trabecular bone at the macroscale in multiaxial, non-monotonic loading modes. Its macroscopic damage behaviour has been investigated experimentally in the past, mostly with the restriction of uniaxial cyclic loading experiments for different samples, which does not allow for the investigation of several load cases in the same sample as damage in one direction may affect the behaviour in other directions. Homogenised finite element models of whole bones have the potential to assess complicated scenarios and thus improve clinical predictions. The aim of this study is to use a homogenisation-based multiscale procedure to upscale the damage behaviour of bone from an assumed solid phase constitutive law and investigate its multiaxial behaviour for the first time. Twelve cubic specimens were each submitted to nine proportional strain histories by using a parallel code developed in-house. Evolution of post-elastic properties for trabecular bone was assessed for a small range of macroscopic plastic strains in these nine load cases. Damage evolution was found to be non-isotropic, and both damage and hardening were found to depend on the loading mode (tensile, compression or shear); both were characterised by linear laws with relatively high coefficients of determination. It is expected that the knowledge of the macroscopic behaviour of trabecular bone gained in this study will help in creating more precise continuum FE models of whole bones that improve clinical predictions.     

Implementing Gaussian process inference with neural networks

Gaussian processes compare favourably with backpropagation neural networks as a tool for regression, and Bayesian neural networks have Gaussian process behaviour when the number of hidden neurons tends to infinity. We describe a simple recurrent neural network with connection weights trained by one-shot Hebbian learning. This network amounts to a dynamical system which relaxes to a stable state in which it generates predictions identical to those of Gaussian process regression. In effect an infinite number of hidden units in a feed-forward architecture can be replaced by a merely finite number, together with recurrent connections.