Fig 1.
Schematic representation of the coupling between ABMs and FEMs.
All measures are expressed in mm. Red boxes identify the regions of interest for strain distribution prediction and are identical to Borgiani et al. (2019) [43] to facilitate the comparison between the models. PS = periosteal, I = intracortical, ES = endosteal. Purple and green arrows represent displacement boundary conditions applied to the cortices along x and y, respectively.
Table 1.
ECM material properties and cell activities parameters.
Fig 2.
Algorithm implemented to simulate vessel growth.
P1 = probability of tip EC migration in the direction of the previous iteration; P2 = probability of tip EC migration in a random direction; P3 = probability of tip EC migration following the strain-based rules (P3 = 1-P2-P1); EPmax_abs = absolute maximum principal strain; θvessel = vessel growth direction; θEPmax_abs = absolute maximum principal strain direction; θ⊥EPmax_abs = perpendicular to the absolute maximum principal strain direction.
Fig 3.
Predictions of mechanical strains, vessel patterning and density, and comparison with experimental data.
On the top, the healing region on the 7th day post-osteotomy under rigid (top row) and semirigid (bottom row) fixation conditions; (A) predicted strain distribution; (B) principal strain directions; (C) in silico predictions of vessels pattern; (D) ex vivo vessels pattern (Emcn, Endomucin stained); (E) regions of interest (ROIs); (F-G) scatter plot of experimental vs. predicted vessel density in ROIs under rigid vs. semirigid fixation conditions. Circles represent experimental samples; crosses represent the in silico realizations; solid lines indicate the average experimental vessel density; dashed lines indicate the average in silico vessel density. Asterisks indicate a significant difference: *P value < 0.05, **P value < 0.01.
Fig 4.
Qualitative and quantitative comparison of the predicted vessel orientation with experimental data in the different ROIs.
A) In silico vs. ex vivo comparison between the zoomed ROIs (from the top, ROI 1, ROI 2, ROI 3) for each fixator type (rigid on the left, semirigid on the right); (B) Vessel orientation analysis results for the rigid (top) and semirigid (bottom) fixators. The colours represent the ROIs, the patterns represent the preferred direction: 90°, 0°, other. Asterisks indicate a significant difference: *P value < 0.05, **P value < 0.01.
Fig 5.
Predicted OVSCs organization under rigid fixation and comparison with experimental data.
(A) Alignment of newly formed collagen fibres (white arrows) in the proximity of the cortex (second harmonic signal, two-photon microscopy) during the initial healing phase. The regions of interest considered in the analysis are identified with dashed lines; (B) Percentage distribution of OVSCs orientation for the rigid fixator within the gap and along the periosteum. On the top right, the colour code for cell orientation is reported; (C) Predicted OVSCs self-organization at day 7 for the rigid fixator. On the top right, the colour code for cell orientation is reported.
Fig 6.
Predicted vessel organization in the baseline model vs. upon endothelial cells mechano-response inhibition.
(A) Predicted vessel distribution for the baseline (left) and EC_MR_KO (right). The zoomed images of the osteotomy gap are reported below; (B) vessel orientation analysis for EC_MR_KO and baseline within the gap. Lines indicate a significant difference in the pairwise comparisons between angle bins in the EC_MR_KO model (p<0.05); (C) Box plots of vessel lengths for the baseline and EC_MR_KO within the whole healing region. The average vessel length is significantly different between the two groups (p<0.01); (D) The table shows the average number of self-intersecting vessels for the baseline and EC_MR_KO within the whole healing region. EC_MR_KO = inhibition of tip ECs mechano-response.
Fig 7.
Predicted vessel organization in the baseline model vs. upon stromal cell traction forces inhibition.
(A) Predicted vessel distribution for the baseline (left) and OVSCs_TF_KO (right); (B) vessel orientation analysis for the baseline vs. OVSCs_TF_KO model within the ROIs. OVSCs_TF_KO = inhibition of OVSCs traction forces.
Fig 8.
Predicted vessel and cell organization after unloading the cortices with the baseline model vs. upon stromal cell traction forces inhibition.
(A) Distribution of the absolute maximum principal strains for the unloaded baseline model with a zoom into the gap; (B) Directions of the absolute maximum principal strains for the unloaded baseline model with a zoom into the gap; (C) Predicted vessel distribution for the unloaded baseline model (on the left) and for the unloaded OVSC_TF_KO model (on the right) and zoomed vessel organization in ROI 1; (D) vessel orientation analysis for the unloaded baseline and OVSC_TF_KO models within ROI1. Empty circles indicate a significant difference (p<0.05) between the baseline and OVSC_TF_KO model under unloading. Lines indicate a significant difference in the pairwise comparisons between angle bins in the same model (p<0.05); (E) Predicted OVSCs distribution for the unloaded baseline model and zoomed OVSCs organization within the gap; (F) percentage distribution of OVSCs orientations for the unloaded baseline model within the gap. OVSC_TF_KO = inhibition of OVSCs traction forces.