Fig 1.
FEBio + AngioFE Graphical Summary.
Presented here is a graphical summary of the simulation framework. The physical parameters of the model were generated and prescribed in FEBio/FEBioStudio. Other user-defined parameters associated with vessel growth and traction behaviors were included to configure AngioFE. Model initialization: Parent microvessel fragments were superimposed on the finite element mesh. Degrees of freedom (DOF) such as collagen EFDs and density were mapped to the finite element mesh. Vessel orientation: The direction each vessel grew, ψnew, was determined by 1) interpolating EFDs from the finite element integration points to the vessel tip, 2) pseudo-deforming the local EFD, 3) sampling the EFD for a single contact guidance direction θ, and 4) determining the balance between persistence along the previous vessel orientation ψ and contact guidance by θ. Vessel extension: The function ν(ρ,FA) scaled the vessel extension rate. This function decreased inversely with collagen density and increased directly with collagen anisotropy. After new vessels grew, branches were added along the existing vessels. Finally, cellular tractions were applied at the tips of the new vessels. Mechanics & kinematics: Cell tractions were sent to FEBio, which then solved the equations of motion. Vessel positions were then updated by AngioFE. Model I/O: The updated vessel positions and finite element degrees of freedom were saved after each time step.
Fig 2.
Generation of collagen gels with varying anisotropy.
Collagen gels were aligned to low, moderate, and high anisotropy and imaged via second harmonic generation (SHG) in a prior study [1]. Image data was used to extract ODFs and fit EFDs for each level of alignment.
Fig 3.
Comparison of experimental and simulated angiogenesis.
A. Z projections of experimental (confocal) and simulated microvascular networks grown in low, medium, or high anisotropy collagen from our previous study [1]. Qualitative agreement was visible for all cases of low and medium anisotropy but vector field simulations visually differed from experiments and EFD simulations at high anisotropy. Depth of field = 200 μm. B. Averaged microvessel ODFs for each experimental or simulated case were projected onto the XY plane to simplify comparison since there was little growth in the Z direction. Microvessel orientations from EFD simulations were in good agreement with experimental microvascular ODFs at all three levels of anisotropy. In contrast, microvascular ODFs from vector field simulations diverged from the experimental data for the cases of medium and high anisotropy.
Fig 4.
Visualizations and verification of EFD pseudo-deformation.
A. Visualization of pseudo-deformed collagen fibril ODFs (glyphs colored by GFA) during large-scale uniaxial tension in biphasic materials. Cutouts highlight scaling and rotation at the top-right model corner and along the center row of elements (quarter-symmetry view from the edge to the center). B-C. Comparison between ODFs undergoing “true” deformation (Ω) and ODFs undergoing pseudo-deformation (Ωp) for tension/compression (stretch ratio λ ∈ [0.5, 1.5]) and simple shear (shear ratio κ ∈ [0, 0.5]) of a single element. Differences in GFA were less than 1e-3 for all cases, which indicated good agreement between Ω and Ωp. Heat maps of the 3D ODFs were generated for the test cases with the highest strain to demonstrate the agreement in ODF magnitude and orientation between Ω and Ωp.
Fig 5.
Predictions of microvascular growth in response to spatial anisotropy gradients.
Simulations were performed with microvessels seeded on the proximal (left) end of a rectilinear domain. Growth was simulated across an isotropic matrix (baseline) or a matrix characterized by a positive or negative anisotropy gradient. Local matrix alignment is indicated by ellipsoidal glyphs above each representative image with the color indicating the anisotropy (blue: low; red: high). Microvessels in the baseline model failed to reach the distal region after 12 days. Anisotropy gradients resulted in increased vascularization of the middle and distal regions. A negative anisotropy gradient resulted in the most vascularization in the middle region, although there was no difference in vascularization of the distal region between gradient cases (1 way ANOVA with Sidakholm post hoc. *: p<0.05 w.r.t baseline; @: p<0.05 post hoc pairwise comparison).
Fig 6.
Tumor associated collagen signatures (TACS) differentially facilitate neovessel recruitment.
Microvessels were simulated to originate in the stroma of a tumor periphery (inset, orange). Microvessels grew within the periphery toward the tumor (inset, magenta), separated by a structural interface (inset, yellow). Alignment and density of the interface was varied to mimic various TACS. The interface comprised an isotropic collagen ODF (circle), or aligned ODFs (ellipses) running perpendicular or along interface. The interface density was either low or high. Representative Z projections of the interface and tumor region are presented at the bottom. High interface density reduced the length of microvessels that crossed into the field (TACS-1 interfaces). Interface alignment along the tumor (TACS-2) deflected vessels or trapped them within the interface, while alignment radiating from the tumor facilitated vascular invasion (TACS-3). Fibril alignment in TACS-3 nullified the effects of increased matrix density. 1 way ANOVA. ***: p<0.001 w.r.t. baseline.
Fig 7.
Pseudo-deformation concept map & Monte-Carlo EFD sampling visualization.
A. EFD deformation & pseudo-deformation concept map. Vectors originating from the center of a unit sphere ΩI were mapped by P0 to the undeformed EFD Ω0. After deformation by F, the fibrils of Ω0 were stretched and rotated, yielding the deformed ODF Ω. The net mapping from ΩI to Ω was given by Fn. Polar decomposition of Fn yielded Fn = VnRn. An EFD Ωp that closely approximated Ω was found by applying the pseudo-deformation Fp = VnI to ΩI. B. Visualizations of EFD Monte-Carlo sampling are provided for the cases of transverse isotropy (semiprincipal axis lengths β1 = 5, β2 = β3 = 1) and orthotropy (semiprincipal axis lengths β1 = 5, β2 = 3, β3 = 1). Sampled directions were mapped onto the surface of a unit sphere. The ODF was generated by summing points on the surface of the unit sphere that corresponded to faces on an icosahedron.