An in silico study on the influence of extracellular matrix mechanics on vasculogenesis.

BACKGROUND AND OBJECTIVES: Blood vessels form a network of capillaries throughout the body that perform essential functions for life. Vasculogenesis, i.e. the formation of new blood vessels, is regulated by many factors, biochemical ones being among the most important. However, others such as the biomechanical influence on shape, organization and structure of vessel networks require further investigation. In this paper, we develop a 3D agent-based mechanobiological model of vasculogenesis with the aim of analyzing how the mechanics of the extracellular matrix (ECM) affects vasculogenesis. METHODS: For this purpose, we consider a growing domain composed of different cells: tip cells, which are the driving cells located at the end of the vessels and stalk cells, which are found in the interior of the vascular network. ECM is considered as particles (agents) that surround the growth of the vascular network. Depending on the cell type, different sets of forces are considered, such as chemotactic, mechanical, random and viscoelastic forces among others. RESULTS: The growth of the network is iteratively analyzed and updated at each time step based on a mechanically-driven proliferation rule. The influence of different biomechanical factors, such as ECM stiffness or viscoelasticity are explored through in silico simulations. A number of indicators are defined along the algorithm, like number of cells, branches, tortuosity and anisotropy, in order to compare topological differences of the vascular network during vasculogenesis under different ECM conditions. The obtained results are qualitatively compared with other related works in the literature. CONCLUSIONS: The present study sheds some light and partially explain, from an in silico perspective, the role of ECM mechanics on vasculogenesis. The main conclusions of this work are: (i) increased stiffness increases proliferation, (ii) the network tends to migrate towards stiffer areas, and (iii) increased viscoelasticity decreases proliferation.

Advanced in silico validation framework for three-dimensional traction force microscopy and application to an in vitro model of sprouting angiogenesis.

In the last decade, cellular forces in three-dimensional hydrogels that mimic the extracellular matrix have been calculated by means of Traction Force Microscopy (TFM). However, characterizing the accuracy limits of a traction recovery method is critical to avoid obscuring physiological information due to traction recovery errors. So far, 3D TFM algorithms have only been validated using simplified cell geometries, bypassing image processing steps or arbitrarily simulating focal adhesions. Moreover, it is still uncertain which of the two common traction recovery methods, i.e., forward and inverse, is more robust against the inherent challenges of 3D TFM. In this work, we established an advanced in silico validation framework that is applicable to any 3D TFM experimental setup and that can be used to correctly couple the experimental and computational aspects of 3D TFM. Advancements relate to the simultaneous incorporation of complex cell geometries, simulation of microscopy images of varying bead densities and different focal adhesion sizes and distributions. By measuring the traction recovery error with respect to ground truth solutions, we found that while highest traction recovery errors occur for cases with sparse and small focal adhesions, our implementation of the inverse method improves two-fold the accuracy with respect to the forward method (average error of 23% vs. 50%). This advantage was further supported by recovering cellular tractions around angiogenic sprouts in an in vitro model of angiogenesis. The inverse method recovered higher traction peaks and a clearer pulling pattern at the sprout protrusion tips than the forward method. STATEMENT OF SIGNIFICANCE: Biomaterial performance is often studied by quantifying cell-matrix mechanical interactions by means of Traction Force Microscopy (TFM). However, 3D TFM algorithms are often validated in simplified scenarios, which do not allow to fully assess errors that could obscure physiological information. Here, we established an advanced in silico validation framework that mimics real TFM experimental conditions and that characterizes the expected errors of a 3D TFM workflow. We apply this framework to demonstrate the enhanced accuracy of a novel inverse traction recovery method that is illustrated in the context of an in vitro model of sprouting angiogenesis. Together, our study shows the importance of a proper traction recovery method to minimise errors and the need for an advanced framework to assess those errors.

Data-Driven Computational Simulation in Bone Mechanics.

The data-driven approach was formally introduced in the field of computational mechanics just a few years ago, but it has gained increasing interest and application as disruptive technology in many other fields of physics and engineering. Although the fundamental bases of the method have been already settled, there are still many challenges to solve, which are often inherently linked to the problem at hand. In this paper, the data-driven methodology is applied to a particular problem in tissue biomechanics, a context where this approach is particularly suitable due to the difficulty in establishing accurate and general constitutive models, due to the intrinsic intra and inter-individual variability of the microstructure and associated mechanical properties of biological tissues. The problem addressed here corresponds to the characterization and mechanical simulation of a piece of cortical bone tissue. Cortical horse bone tissue was mechanically tested using a biaxial machine. The displacement field was obtained by means of digital image correlation and then transformed into strains by approximating the displacement derivatives in the bone virtual geometric image. These results, together with the approximated stress state, assumed as uniform in the small pieces tested, were used as input in the flowchart of the data-driven methodology to solve several numerical examples, which were compared with the corresponding classical model-based fitted solution. From these results, we conclude that the data-driven methodology is a useful tool to directly simulate problems of biomechanical interest without the imposition (model-free) of complex spatial and individually-varying constitutive laws. The presented data-driven approach recovers the natural spatial variation of the solution, resulting from the complex structure of bone tissue, i.e. heterogeneity, microstructural hierarchy and multifactorial architecture, making it possible to add the intrinsic stochasticity of biological tissues into the data set and into the numerical approach.

Inverse method based on 3D nonlinear physically constrained minimisation in the framework of traction force microscopy.

Traction force microscopy is a methodology that enables to estimate cellular forces from the measurement of the displacement field of an extracellular matrix (ECM)-mimicking hydrogel that a cell is mechanically interacting with. In this paper, a new inverse and physically-consistent methodology is developed and implemented in the context of 3D nonlinear elasticity. The proposed method searches for a displacement field that approximates the measured one, through the imposition of fulfillment of equilibrium with real and known forces acting in the hydrogel. The overall mathematical formulation leads to a constrained optimisation problem that is treated through a Lagrange operator and that is solved numerically by means of a nonlinear finite element framework. In order to illustrate the potential and enhanced accuracy of the proposed inverse method, it is applied to a total of 5 different real cases of cells cultured in a 3D hydrogel that is considered to behave as a nonlinear elastic material. Different error indicators are defined in order to compare ground truth simulated displacements and tractions to the ones recovered by the new inverse as well as by the forward method. Results indicate that the evaluation of displacement gradients leads to errors, in terms of recovered tractions, that are more than three times lower (on average) for the inverse method compared to the forward method. They highlight the enhanced accuracy of the developed methodology and the importance of appropriate inverse methods that impose physical constraints to traction and stress recovery in the context of traction force microscopy.

In silico dynamic characterization of the femur: Physiological versus mechanical boundary conditions.

It is established that bone tissue adapts and responds to mechanical loading. Several studies have suggested an existence of positive influence of vibration on the bone mass maintenance. Thus, some bone regeneration therapies are based on vibration of bone tissue under circumstances of disease to stimulate its formation. Frequency of loading should be properly selected and therefore a correct characterization of the dynamic properties of this tissue may be critical for the success of such orthopedic techniques. On the other hand, many studies implement vibration techniques with in silico models. Numerical results are exclusively dependent on properties of bone tissue, i.e. geometry, density distribution and stiffness, as well as boundary conditions. In the present study, the influence of boundary conditions and material properties on the dynamic characteristics of bone tissue was explored in a human femur. Bone shape and density were directly reconstructed from computer tomographies, whereas natural frequencies and modes of vibration were obtained for different boundary conditions including physiological and mechanical ones. Results of this study show the moderate effect of material properties compared to the much substantial effect of boundary conditions. A factor of 2 in the natural frequency was obtained depending on imposed boundary conditions, highlighting the importance in the selection of appropriate conditions in the analysis of the bone organ.

Model of dissolution in the framework of tissue engineering and drug delivery.

Dissolution phenomena are ubiquitously present in biomaterials in many different fields. Despite the advantages of simulation-based design of biomaterials in medical applications, additional efforts are needed to derive reliable models which describe the process of dissolution. A phenomenologically based model, available for simulation of dissolution in biomaterials, is introduced in this paper. The model turns into a set of reaction-diffusion equations implemented in a finite element numerical framework. First, a parametric analysis is conducted in order to explore the role of model parameters on the overall dissolution process. Then, the model is calibrated and validated versus a straightforward but rigorous experimental setup. Results show that the mathematical model macroscopically reproduces the main physicochemical phenomena that take place in the tests, corroborating its usefulness for design of biomaterials in the tissue engineering and drug delivery research areas.

In silico design of magnesium implants: Macroscopic modeling.

Magnesium-based biomedical implants offer many advantages versus traditional ones although some challenges are still present. In this context, mathematical modeling and computational simulation may be a useful and complementary tool to evaluate in silico the performance of magnesium biomaterials under different conditions. In this paper, a phenomenologically-based model to simulate magnesium corrosion is developed. The model describes the physico-chemical interactions and evolution of species present in this phenomenon. A set of 7 species is considered in the model, which allows to simulate hydrogen release, pH evolution, corrosion products formation as well as degradation of magnesium. The model is developed under the continuum media theory and is implemented in a finite element framework. In the results section, the effect of model parameters on outcomes is firstly explored. Second, model results are qualitative validated versus two examples of application found in the literature. Two main conclusions are derived from this work: (i) the model captures well the experimental trends and allows to analyze the main variables present in magnesium corrosion, (ii) even though further validation is needed the model may be a useful standard in the design of degradable metal implants.

A novel method for visualising and quantifying through-plane skin layer deformations.

Skin is a multilayer composite and exhibits highly non-linear, viscoelastic, anisotropic material properties. In many consumer product and medical applications (e.g. during shaving, needle insertion, patient re-positioning), large tissue displacements and deformations are involved; consequently large local strains in the skin tissue can occur. Here, we present a novel imaging-based method to study skin deformations and the mechanics of interacting skin layers of full-thickness skin. Shear experiments and real-time video recording were combined with digital image correlation and strain field analysis to visualise and quantify skin layer deformations during dynamic mechanical testing. A global shear strain of 10% was applied to airbrush-patterned porcine skin (thickness: 1.2-1.6mm) using a rotational rheometer. The recordings were analysed with ARAMIS image correlation software, and local skin displacement, strain and stiffness profiles through the skin layers determined. The results of this pilot study revealed inhomogeneous skin deformation, characterised by a gradual transition from a low (2.0-5.0%; epidermis) to high (10-22%; dermis) shear strain regime. Shear moduli ranged from 20 to 130kPa. The herein presented method will be used for more extended studies on viable human skin, and is considered a valuable foundation for further development of constitutive models which can be used in advanced finite element analyses of skin.

Scaffold microarchitecture determines internal bone directional growth structure: a numerical study.

A number of successful results have been reported in bone tissue engineering, although the routine clinical practice has not been reached so far. One of the reasons is the poor understanding of the role of each scaffold design parameter in its functional performance, which yields an uncertain outcome of each clinical application. Specifically, the role of internal scaffold microarchitectural shape on the regeneration rate and distribution of newly formed bone is still unknown. This work is focused on the in-silico determination of the role of scaffold microstructural anisotropy in bone tissue regeneration. A multiscale approach of the problem is established distinguishing between macroscopic region domain (bone organ and scaffold) and microscopic domain (scaffold microstructure). Results show that, once the scaffold microstructure is properly interconnected and the porosity is sufficiently high, similar rates of bone regeneration are found. However, the main conclusion of the work is that initial scaffold microstructural anisotropy has important consequences since it determines the spatial distribution of the newly formed tissue.

A mathematical approach to bone tissue engineering.

Tissue engineering is becoming consolidated in the biomedical field as one of the most promising strategies in tissue repair and regenerative medicine. Within this discipline, bone tissue engineering involves the use of cell-loaded porous biomaterials, i.e. bioscaffolds, to promote bone tissue regeneration in bone defects or diseases such as osteoporosis, although it has not yet been incorporated into daily clinical practice. The overall success of a particular bone tissue engineering application depends strongly on scaffold design parameters, which do away with long and expensive clinical protocols. Computer simulation is a useful tool that may reduce animal experiments and help to identify optimal patient-specific designs after concise model validation. In this paper, we present a novel mathematical approach to bone regeneration within scaffolds, based on a multiscale framework. Results are presented over an actual scaffold microstructure, showing the potential of computer simulation, and how it can aid in the task of making bone tissue engineering a reality in clinical practice.

On scaffold designing for bone regeneration: A computational multiscale approach.

Scaffold design for bone tissue engineering applications involves many parameters that directly influence the rate of bone tissue regeneration onto its microstructural surface. To improve scaffold functionality, increasing interest is being focused on in vitro and in vivo research in order to obtain the optimal scaffold design for a specific application. However, the evaluation of the effect of each specific scaffold parameter on tissue regeneration using these techniques requires costly protocols and long-term experiments. In this paper, we elucidate the effect of some scaffold parameters on bone tissue regeneration by means of a mathematically based approach. By virtue of in silico experiments, factors such as scaffold stiffness, porosity, resorption kinetics, pore size and pre-seeding are analyzed in a specific bone tissue application found in the literature. The model predicts the in vivo rate of bone formation within the scaffold, the scaffold degradation and the interaction between the implanted scaffold and the surrounding bone. Results show an increasing rate of bone regeneration with increasing scaffold stiffness, scaffold mean pore size and pre-seeding, whereas the collapse of the scaffold occurs for a faster biomaterial resorption kinetics. Requiring further experimental validation, the model can be useful for the assessment of scaffold design and for the analysis of scaffold parameters in tissue regeneration.

A mathematical model for bone tissue regeneration inside a specific type of scaffold.

Bone tissue regeneration using scaffolds is receiving an increasing interest in orthopedic surgery and tissue engineering applications. In this study, we present the geometrical characterization of a specific family of scaffolds based on a face cubic centered (FCC) arrangement of empty pores leading to analytical formulae of porosity and specific surface. The effective behavior of those scaffolds, in terms of mechanical properties and permeability, is evaluated through the asymptotic homogenization theory applied to a representative volume element identified with the unit cell FCC. Bone growth into the scaffold is estimated by means of a phenomenological model that considers a macroscopic effective stress as the mechanical stimulus that regulates bone formation. Cell migration within the scaffold is modeled as a diffusion process based on Fick's law which allows us to estimate the cell invasion into the scaffold microstructure. The proposed model considers that bone growth velocity is proportional to the concentration of cells and regulated by the mechanical stimulus. This model allows us to explore what happens within the scaffold, the surrounding bone and their interaction. The mathematical model has been numerically implemented and qualitatively compared with previous experimental results found in the literature for a scaffold implanted in the femoral condyle of a rabbit. Specifically, the model predicts around 19 and 23% of bone regeneration for non-grafted and grafted scaffolds, respectively, both with an initial porosity of 76%.