Mechanical interaction of angiogenic microvessels with the extracellular matrix.

Angiogenesis is the process by which new blood vessels sprout from existing blood vessels, enabling new vascular elements to be added to an existing vasculature. This review discusses our investigations into the role of cell-matrix mechanics in the mechanical regulation of angiogenesis. The experimental aspects of the research are based on in vitro experiments using an organ culture model of sprouting angiogenesis with the goal of developing new treatments and techniques to either promote or inhibit angiogenic outgrowth, depending on the application. Computational simulations were performed to simulate angiogenic growth coupled to matrix deformation, and live two-photon microscopy was used to obtain insight into the dynamic mechanical interaction between angiogenic neovessels and the extracellular matrix. In these studies, we characterized how angiogenic neovessels remodel the extracellular matrix (ECM) and how properties of the matrix such as density and boundary conditions influence vascular growth and alignment. Angiogenic neovessels extensively deform and remodel the matrix through a combination of applied traction, proteolytic activity, and generation of new cell-matrix adhesions. The angiogenic phenotype within endothelial cells is promoted by ECM deformation and remodeling. Sensitivity analysis using our finite element model of angiogenesis suggests that cell-generated traction during growth is the most important parameter controlling the deformation of the matrix and, therefore, angiogenic growth and remodeling. Live two-photon imaging has also revealed numerous neovessel behaviors during angiogenesis that are poorly understood such as episodic growth/regression, neovessel colocation, and anastomosis. Our research demonstrates that the topology of a resulting vascular network can be manipulated directly by modifying the mechanical interaction between angiogenic neovessels and the matrix.

Computational modeling of multicellular constructs with the material point method.

Computational modeling of the mechanics of cells and multicellular constructs with standard numerical discretization techniques such as the finite element (FE) method is complicated by the complex geometry, material properties and boundary conditions that are associated with such systems. The objectives of this research were to apply the material point method (MPM), a meshless method, to the modeling of vascularized constructs by adapting the algorithm to accurately handle quasi-static, large deformation mechanics, and to apply the modified MPM algorithm to large-scale simulations using a discretization that was obtained directly from volumetric confocal image data. The standard implicit time integration algorithm for MPM was modified to allow the background computational grid to remain fixed with respect to the spatial distribution of material points during the analysis. This algorithm was used to simulate the 3D mechanics of a vascularized scaffold under tension, consisting of growing microvascular fragments embedded in a collagen gel, by discretizing the construct with over 13.6 million material points. Baseline 3D simulations demonstrated that the modified MPM algorithm was both more accurate and more robust than the standard MPM algorithm. Scaling studies demonstrated the ability of the parallel code to scale to 200 processors. Optimal discretization was established for the simulations of the mechanics of vascularized scaffolds by examining stress distributions and reaction forces. Sensitivity studies demonstrated that the reaction force during simulated extension was highly sensitive to the modulus of the microvessels, despite the fact that they comprised only 10.4% of the volume of the total sample. In contrast, the reaction force was relatively insensitive to the effective Poisson's ratio of the entire sample. These results suggest that the MPM simulations could form the basis for estimating the modulus of the embedded microvessels through a parameter estimation scheme. Because of the generality and robustness of the modified MPM algorithm, the relative ease of generating spatial discretizations from volumetric image data, and the ability of the parallel computational implementation to scale to large processor counts, it is anticipated that this modeling approach may be extended to many other applications, including the analysis of other multicellular constructs and investigations of cell mechanics.

A coupled model of neovessel growth and matrix mechanics describes and predicts angiogenesis in vitro.

During angiogenesis, sprouting microvessels interact with the extracellular matrix (ECM) by degrading and reorganizing the matrix, applying traction forces, and producing deformation. Morphometric features of the resulting microvascular network are affected by the interaction between the matrix and angiogenic microvessels. The objective of this study was to develop a continuous-discrete modeling approach to simulate mechanical interactions between growing neovessels and the deformation of the matrix in vitro. This was accomplished by coupling an existing angiogenesis growth model which uses properties of the ECM to regulate angiogenic growth with the nonlinear finite element software FEBio (www.febio.org). FEBio solves for the deformation and remodeling of the matrix caused by active stress generated by neovessel sprouts, and this deformation was used to update the ECM into the current configuration. After mesh resolution and parameter sensitivity studies, the model was used to accurately predict vascular alignment for various matrix boundary conditions. Alignment primarily arises passively as microvessels convect with the deformation of the matrix, but active alignment along collagen fibrils plays a role as well. Predictions of alignment were most sensitive to the range over which active stresses were applied and the viscoelastic time constant in the material model. The computational framework provides a flexible platform for interpreting in vitro investigations of vessel-matrix interactions, predicting new experiments, and simulating conditions that are outside current experimental capabilities.

Extracellular matrix density regulates the rate of neovessel growth and branching in sprouting angiogenesis.

Angiogenesis is regulated by the local microenvironment, including the mechanical interactions between neovessel sprouts and the extracellular matrix (ECM). However, the mechanisms controlling the relationship of mechanical and biophysical properties of the ECM to neovessel growth during sprouting angiogenesis are just beginning to be understood. In this research, we characterized the relationship between matrix density and microvascular topology in an in vitro 3D organ culture model of sprouting angiogenesis. We used these results to design and calibrate a computational growth model to demonstrate how changes in individual neovessel behavior produce the changes in vascular topology that were observed experimentally. Vascularized gels with higher collagen densities produced neovasculatures with shorter vessel lengths, less branch points, and reduced network interconnectivity. The computational model was able to predict these experimental results by scaling the rates of neovessel growth and branching according to local matrix density. As a final demonstration of utility of the modeling framework, we used our growth model to predict several scenarios of practical interest that could not be investigated experimentally using the organ culture model. Increasing the density of the ECM significantly reduced angiogenesis and network formation within a 3D organ culture model of angiogenesis. Increasing the density of the matrix increases the stiffness of the ECM, changing how neovessels are able to deform and remodel their surroundings. The computational framework outlined in this study was capable of predicting this observed experimental behavior by adjusting neovessel growth rate and branching probability according to local ECM density, demonstrating that altering the stiffness of the ECM via increasing matrix density affects neovessel behavior, thereby regulated vascular topology during angiogenesis.

A computational model of in vitro angiogenesis based on extracellular matrix fibre orientation.

Recent interest in the process of vascularisation within the biomedical community has motivated numerous new research efforts focusing on the process of angiogenesis. Although the role of chemical factors during angiogenesis has been well documented, the role of mechanical factors, such as the interaction between angiogenic vessels and the extracellular matrix, remains poorly understood. In vitro methods for studying angiogenesis exist; however, measurements available using such techniques often suffer from limited spatial and temporal resolutions. For this reason, computational models have been extensively employed to investigate various aspects of angiogenesis. This paper outlines the formulation and validation of a simple and robust computational model developed to accurately simulate angiogenesis based on length, branching and orientation morphometrics collected from vascularised tissue constructs. Microvessels were represented as a series of connected line segments. The morphology of the vessels was determined by a linear combination of the collagen fibre orientation, the vessel density gradient and a random walk component. Excellent agreement was observed between computational and experimental morphometric data over time. Computational predictions of microvessel orientation within an anisotropic matrix correlated well with experimental data. The accuracy of this modelling approach makes it a valuable platform for investigating the role of mechanical interactions during angiogenesis.

Simulation of soft tissue failure using the material point method.

Understanding the factors that control the extent of tissue damage as a result of material failure in soft tissues may provide means to improve diagnosis and treatment of soft tissue injuries. The objective of this research was to develop and test a computational framework for the study of the failure of anisotropic soft tissues subjected to finite deformation. An anisotropic constitutive model incorporating strain-based failure criteria was implemented in an existing computational solid mechanics software based on the material point method (MPM), a quasi-meshless particle method for simulations in computational mechanics. The constitutive model and the strain-based failure formulations were tested using simulations of simple shear and tensile mechanical tests. The model was then applied to investigate a scenario of a penetrating injury: a low-speed projectile was released through a myocardial material slab. Sensitivity studies were performed to establish the necessary grid resolution and time-step size. Results of the simple shear and tensile test simulations demonstrated the correct implementation of the constitutive model and the influence of both fiber family and matrix failure on predictions of overall tissue failure. The slab penetration simulations produced physically realistic wound tracts, exhibiting diameter increase from entrance to exit. Simulations examining the effect of bullet initial velocity showed that the anisotropy influenced the shape and size of the exit wound more at lower velocities. Furthermore, the size and taper of the wound cavity was smaller for the higher bullet velocity. It was concluded that these effects were due to the amount of momentum transfer. The results demonstrate the feasibility of using MPM and the associated failure model for large-scale numerical simulations of soft tissue failure.