Fig 1.
Parametric finite element model of the functionally graded scaffold utilized in the study.
CAD model (A-B) and finite element mesh (C-D) of the scaffold (A, C) and granulation tissue (B, D). Circular pores with variable radius A have been modelled. The nodes of the bottom surface of the model were clamped (E-G) while those of the upper surface were tied to a rigid plate (represented in blue). Three different loading conditions were hypothesized: a compression force (E); a shear force
(F); a mixed compression-shear force
(G). The pore pressure ppore on the outer surfaces of the granulation tissue was set equal to zero to simulate the free exudation of fluid.
Fig 2.
Porosity distribution laws analyzed in the study.
(A) constant; (B) linear; (C) bi-linear; (D) tri-linear. The specific coefficients Ai (i = 1, 2, 3, 4) of these laws were determined via the optimization algorithm.
Table 1.
Porosity distribution laws implemented in the study.
Fig 3.
Schematic of the algorithm implemented in Matlab environment to optimize the porosity distribution law in functionally graded scaffolds.
Fig 4.
Computed values of A and BO% in the case of compression loading.
Pore radius A (A, C, E) (vs. location y) and percentages of the scaffold volume occupied by bone BO% (B, D, F) predicted by the optimization algorithm in the case of compression loading FV for different scaffold Young’s moduli and after implementing different porosity distribution laws. The schematic figure shown on the top indicates the loading condition to which the diagrams refer. All the values of BO% reported in the diagrams refer to the optimal configuration, i.e. the configuration for which Ω reaches its minimum value.
Fig 5.
Computed values of A and BO% in the case of shear loading.
Pore radius A (A, C, E) (vs. location y) and percentages of the scaffold volume occupied by bone BO% (B, D, F) predicted by the optimization algorithm in the case of shear loading FH for different scaffold Young’s moduli and after implementing different porosity distribution laws. The schematic figure shown on the top indicates the loading condition to which the diagrams refer. All the values of BO% reported in the diagrams refer to the optimal configuration, i.e. the configuration for which Ω reaches its minimum value.
Fig 6.
Computed values of A and BO% in the case of mixed load.
Pore radius A (a, c, e) (vs. location y) and percentages of the scaffold volume occupied by bone BO% (b, d, f) predicted by the optimization algorithm in the case of mixed load FM for different scaffold Young’s modulus values and after implementing different porosity distribution laws. The schematic figure shown on the top indicates the loading condition to which the diagrams refer. All the values of BO% reported in the diagrams refer to the optimal configuration, i.e. the configuration for which Ω reaches its minimum value.
Fig 7.
Computed values of PVPD for different loading conditions.
Percent Variation of the Pore Dimension (PVPD) for the compression FV (B), the shear FH (C) and the mixed FM (D) load and for all the hypothesized scaffold Young’s modulus values. (A) reference schematic utilized to calculate the parameter PVPD. Note: AH and AL are the highest and lowest value of A that can be located in correspondence of any value of y and not necessarily, as reported in the figure, of the furthest values y = 0 μm and y = h = 3822 μm.
Fig 8.
Computed values of iBO% for compression (A), shear (B) and mixed (C) load.
Fig 9.
3D view of the best geometrical configurations (tri-linear porosity distribution) predicted by the optimization algorithm for the shear loading condition.
Fig 10.
Patterns of bone predicted in the case of: (i) compression loading; (ii) scaffold Young’s modulus E = 1000 MPa; (iii) porosity distribution law: constant.
Elements in gray are representative of the regions within the scaffold pores where the algorithm predicts bone formation. Interestingly, the predicted bony tissue patterns appear consistent with those of new tissue formed in three-dimensional matrix channels observed in an in vitro study [39]. Bone formation starts from the pore walls and propagates towards the pore center.