Figure 1.
(A) SEM image of collagen fibers from a normal mouse mammary gland. (B) Representative SHG images of in vitro 2 mg/ml collagen gel under 3% strain (B) and 30% strain (C). (D) Schematic illustration of a bead-and-spring collagen fiber network model. Black lines represent collagen fibers, and red lines represent crosslinkers. Black dots represent beads, which have elastic connection with other beads. Constructed fiber network model with 1 mg/ml density and 2× total number of fibers [N] crosslinker density, random network (E) and prealigned network (F).
Figure 2.
Shear simulation test using elastic fiber network model.
(A) The simulation box is 200 µm (length) ×200 µm (width) ×300 µm (height). The bottom 50 µm and top 50 µm of the box are anchored region. The beads in the bottom anchored region are fixed and the beads in the top anchored region are deformed to y-axis. (B) Snapshot images for initial and quasi-equilibrium state of 0.1 shear strain with 2 mg/ml collagen density, 8N crosslinker density, 400 KPa crosslinker strength, and random fiber network. The shear strain step size is 0.01 (2 µm) and total ten shear strains are applied to the simulation box. (C) Maximum force of ten shear strain test from 0.01 strain to 0.1 strain, assuming that the fiber network reaches the quasi-equilibrium state when the maximum force decreases the below of 10−5 of the maximum value at each deformed state.
Table 1.
Parameters for elastic collagen fiber network simulations.
Figure 3.
We simulated 8 different crosslinker densities (2, 4, …, 16N, 2N increment), 16 different crosslinker strengths (50, 100, …, 800 KPa, 50 KPa increment), and 4 different collagen densities (1, 2, 3, 4 mg/ml) for random fiber networks, which is total 512 different parameter sets. Shear stress - shear strain curves for ten strains using a 0.01 strain step size are shown in various crosslinker densities with fixed 400 KPa crosslinker strength (A) of 2 mg/ml collagen density case, various crosslinker strengths with fixed 8N crosslinker density (B) of 2 mg/ml collagen density case, and various collagen densities with fixed 400 KPa crosslinker strength and fixed 8N crosslinker density (C). Five independent runs were conducted for each parameter set. Only four curves for each varied parameter are shown for the better visualization. (D) Shear modulus surface plot for four different collagen densities, 8 different crosslinker densities and 16 different crosslinker strength. Each modulus value was calculated from the regression line slope of the stress-strain curve.
Figure 4.
Finding the best-fit crosslinker parameter values.
(A) The sum of squared residuals (SSR) between the shear elastic moduli data from Stein et al. [30] and our simulations (figure 3D), for four different collagen densities. (B) The spline interpolation of (A) using the half of the prior parameter intervals provides smoother surface of the sum of squared residuals. (C) Intersection lines of experimental data plane with simulation surfaces of figure 3D. (D) Zoomed-in contour plot of the 7th spline interpolation of the SSR around the minimum value to the two times of the minimum value. Five points were selected for comparison: P1 (290.23 KPa for crosslinker strength, 15.19N for crosslinker density), P2 (400 KPa, 13.27N), P3 (634.38 KPa, 11.28N), P4 (700 KPa, 10.89N), and P5 (775.39 KPa, 10.58N). (E) Validation for the best-fit crosslinker parameter values. We have compared spline interpolated SSR estimation values with calculated SSR value using simulation results for these 5 selected crosslinker parameter points. 30 independent simulations were run to calculate SSR values. The P3 crosslinker parameter values were chosen as the best-fit value because both spline estimated SSR and calculated SSR using simulation are the lowest value.
Figure 5.
Validation of the best-fit crosslinker parameter values.
(A) Shear modulus of simulation results (Sim) using the best-fit crosslinker parameter values and elastic modulus (G′) in shear experiments (Exp) from Stein et al. [30]. 5 independent runs were simulated for seven different collagen densities (1, 1.5, …, 4 mg/ml using a 0.5 mg/ml increment). (B) Tensile modulus of various strain rate experiments, experiments from Provenzano et al. [41], Roeder et al. [40], Riching et al. [24], Lopez-Garcia et al. [42], predicted values (Pre) using a power-law fitting from Lopez-Garcia et al. [42], and simulation results using the best-fit crosslinker parameter values. Inset figure is magnified view of our experimental data of 2 mg/ml collagen gels at very slow train rate of 0.046/min.
Figure 6.
Stress-strain curves from small strain toe region to medium strain linear region.
(A) Schematic stress-strain curve to illustrate toe, linear, plastic, and failure region. A collagen fiber network is soft at the small deformation state, but stiff at the large deformation state because realigning fibers through crosslinkers play a pivotal role in the strain stiffening. Realignment illustration of fiber network model for shear test (A1: zero strain, A2: small strain, A3: medium strain) and tensile test (A4: zero strain, A5: small strain, A6: medium strain). Black solid lines represent collagen fibers and red solid lines are crosslinkers. Black dots represent beads which can have elastic connection with other beads. Light blue solid arrows represent force vectors, and a light blue dot represents anchored fixed beads for shear test. Simulated stress-strain curves of shear test (B) and tensile test (C) for two different collagen densities: 1, 4 mg/ml and two different network geometries: prealigned network and random network using the best-fit crosslinker parameter values. The errorbars are standard deviation from the mean in 5 independent simulations. We simulated up to 0.5 strain with a 0.01 strain step size. (D) Poisson's ratio of tensile tests in (C) for both random and prealigned networks.
Figure 7.
Simulation of a local deformation test using the calibrated collagen model of 2 mg/ml.
(A) A cubic test box (20 µm×20 µm×20 µm) is located at the center of the simulation box (300 µm×300 µm×300 µm). All beads in the test box are anchored and displaced by 60 µm in the z-direction (black arrow) with a 2 µm displacement step size for 30 steps. Beads in the outer layer of the simulation box (within 50 µm of all the box sides) are anchored. All fiber-beads are initially at equilibrium before the test box is displaced. Average force value was calculated at the quasi-equilibrium state after each displacement step. Average force value of all beads in the test box (B), anchored layer (C), and internal box (D) over 30 displacement steps. Force vectors at the quasi-equilibrium state of 60 µm displacement in the test box (E), anchored layer (F), and internal box (G). Each colorbar shows force scale in the figure. Force histogram at the quasi-equilibrium state of 60 µm displacement in the test box (H), anchored layer (I), and internal box (J). Inset images of figure I and J are magnified views to illustrate the tails of distribution at larger force values.