Multi-level hp-finite cell method for embedded interface problems with application in biomechanics.

This work presents a numerical discretization technique for solving 3-dimensional material interface problems involving complex geometry without conforming mesh generation. The finite cell method (FCM), which is a high-order fictitious domain approach, is used for the numerical approximation of the solution without a boundary-conforming mesh. Weak discontinuities at material interfaces are resolved by using separate FCM meshes for each material sub-domain and weakly enforcing the interface conditions between the different meshes. Additionally, a recently developed hierarchical hp-refinement scheme is used to locally refine the FCM meshes to resolve singularities and local solution features at the interfaces. Thereby, higher convergence rates are achievable for nonsmooth problems. A series of numerical experiments with 2- and 3-dimensional benchmark problems is presented, showing that the proposed hp-refinement scheme in conjunction with the weak enforcement of the interface conditions leads to a significant improvement of the convergence rates, even in the presence of singularities. Finally, the proposed technique is applied to simulate a vertebra-implant model. The application showcases the method's potential as an accurate simulation tool for biomechanical problems involving complex geometry, and it demonstrates its flexibility in dealing with different types of geometric description.

An Integrated Design, Material, and Fabrication Platform for Engineering Biomechanically and Biologically Functional Soft Tissues.

We present a design rationale for stretchable soft network composites for engineering tissues that predominantly function under high tensile loads. The convergence of 3D-printed fibers selected from a design library and biodegradable interpenetrating polymer networks (IPNs) result in biomimetic tissue engineered constructs (bTECs) with fully tunable properties that can match specific tissue requirements. We present our technology platform using an exemplary soft network composite model that is characterized to be flexible, yet ∼125 times stronger (E = 3.19 MPa) and ∼100 times tougher (WExt = ∼2000 kJ m-3) than its hydrogel counterpart.

Biofabricated soft network composites for cartilage tissue engineering.

Articular cartilage from a material science point of view is a soft network composite that plays a critical role in load-bearing joints during dynamic loading. Its composite structure, consisting of a collagen fiber network and a hydrated proteoglycan matrix, gives rise to the complex mechanical properties of the tissue including viscoelasticity and stress relaxation. Melt electrospinning writing allows the design and fabrication of medical grade polycaprolactone (mPCL) fibrous networks for the reinforcement of soft hydrogel matrices for cartilage tissue engineering. However, these fiber-reinforced constructs underperformed under dynamic and prolonged loading conditions, suggesting that more targeted design approaches and material selection are required to fully exploit the potential of fibers as reinforcing agents for cartilage tissue engineering. In the present study, we emulated the proteoglycan matrix of articular cartilage by using highly negatively charged star-shaped poly(ethylene glycol)/heparin hydrogel (sPEG/Hep) as the soft matrix. These soft hydrogels combined with mPCL melt electrospun fibrous networks exhibited mechanical anisotropy, nonlinearity, viscoelasticity and morphology analogous to those of their native counterpart, and provided a suitable microenvironment for in vitro human chondrocyte culture and neocartilage formation. In addition, a numerical model using the p-version of the finite element method (p-FEM) was developed in order to gain further insights into the deformation mechanisms of the constructs in silico, as well as to predict compressive moduli. To our knowledge, this is the first study presenting cartilage tissue-engineered constructs that capture the overall transient, equilibrium and dynamic biomechanical properties of human articular cartilage.

Uncertainty quantification for personalized analyses of human proximal femurs.

Computational models for the personalized analysis of human femurs contain uncertainties in bone material properties and loads, which affect the simulation results. To quantify the influence we developed a probabilistic framework based on polynomial chaos (PC) that propagates stochastic input variables through any computational model. We considered a stochastic E-ρ relationship and a stochastic hip contact force, representing realistic variability of experimental data. Their influence on the prediction of principal strains (ϵ1 and ϵ3) was quantified for one human proximal femur, including sensitivity and reliability analysis. Large variabilities in the principal strain predictions were found in the cortical shell of the femoral neck, with coefficients of variation of ≈40%. Between 60 and 80% of the variance in ϵ1 and ϵ3 are attributable to the uncertainty in the E-ρ relationship, while ≈10% are caused by the load magnitude and 5-30% by the load direction. Principal strain directions were unaffected by material and loading uncertainties. The antero-superior and medial inferior sides of the neck exhibited the largest probabilities for tensile and compression failure, however all were very small (pf<0.001). In summary, uncertainty quantification with PC has been demonstrated to efficiently and accurately describe the influence of very different stochastic inputs, which increases the credibility and explanatory power of personalized analyses of human proximal femurs.

Stochastic description of the peak hip contact force during walking free and going upstairs.

Uncertainty quantification for the response of a patient specific femur is mandatory when advocating finite element (FE) models in clinical applications. Reliable stochastic descriptions of physiological hip contact forces are an essential prerequisite for such an endeavor. We therefore analyze the in-vivo available data of seven individuals from HIP98 and OrthoLoad with the objective of characterizing the variability of the peak hip contact force magnitude and two corresponding spatial angles (in sagittal and frontal plane) during walking free and going upstairs. Regression analyses with linear mixed-effects models were performed resulting in six normal random variables, one for each force component and activity. Importantly, the statistical analysis accounts for the fact that same individuals performed both activities. The mean of the peak force magnitude was found to be linearly dependent on the body weight with an additional, activity-specific intercept and all variances were dominated by the inter-patient variability. No distinct correlation was found between the two angles and the force magnitude. The proposed stochastic description of the peak hip contact force during walking free and going upstairs contributes towards future uncertainty quantification of patient-specific FE models.

Prediction of the mechanical response of the femur with uncertain elastic properties.

A mandatory requirement for any reliable prediction of the mechanical response of bones, based on quantitative computer tomography, is an accurate relationship between material properties (usually Young's modulus E) and bone density ρ. Many such E-ρ relationships are available based on different experiments on femur specimens with a large spread due to uncertainties. The first goal of this study is to pool and analyze the relevant available experimental data and develop a stochasticE-ρ relationship. This analysis highlights that there is no experimental data available to cover the entire density range of the human femur and that some "popular" E-ρ relationships are based on data that contains extreme scatter, while others are based on a very limited amount of information. The second goal is to use the newly developed stochastic E-ρ relationship in high-order finite element analyses (FEAs) for the computation of strains and displacements in two human proximal femurs, mimicking in vitro experiments. When compared with the experimental observations, the FEA predictions using the median of the stochastic E-ρ relationship follow the underlying distribution of the stochastic E-ρ relationship. Thus, most deviations of the FEA predictions from experimental observations can possibly be explained by uncertain elastic properties of the femur.

The finite cell method for bone simulations: verification and validation.

Standard methods for predicting bone's mechanical response from quantitative computer tomography (qCT) scans are mainly based on classical h-version finite element methods (FEMs). Due to the low-order polynomial approximation, the need for segmentation and the simplified approach to assign a constant material property to each element in h-FE models, these often compromise the accuracy and efficiency of h-FE solutions. Herein, a non-standard method, the finite cell method (FCM), is proposed for predicting the mechanical response of the human femur. The FCM is free of the above limitations associated with h-FEMs and is orders of magnitude more efficient, allowing its use in the setting of computational steering. This non-standard method applies a fictitious domain approach to simplify the modeling of a complex bone geometry obtained directly from a qCT scan and takes into consideration easily the heterogeneous material distribution of the various bone regions of the femur. The fundamental principles and properties of the FCM are briefly described in relation to bone analysis, providing a theoretical basis for the comparison with the p-FEM as a reference analysis and simulation method of high quality. Both p-FEM and FCM results are validated by comparison with an in vitro experiment on a fresh-frozen femur.

Lattice Boltzmann simulations of binary fluid flow through porous media.

The lattice Boltzmann equation is often advocated as a simulation tool that is particularly effective for complex fluids such as multiphase and multicomponent flows through porous media. We construct a three-dimensional 19 velocity lattice Boltzmann model for immiscible binary fluids with variable viscosities and density ratio based on the model proposed by Gunstensen. The model is tested for the following binary fluid flow problems: a stationary planar interface among two fluids; channel flow of immiscible binary fluids; the Laplace problem; and a rising bubble. The results agree well with semi-analytic results in a range of the Eötvös, Morton and Reynolds number. We also present preliminary simulation results for two large-scale realistic applications: the flow of an air-water mixture in a waste-water batch reactor and the saturation hysteresis effect in soil flow. We discuss some limitations of the lattice Boltzmann method in the simulation of realistic and difficult multiphase problems.