Finite Element-Derived Surrogate Models of Locked Plate Fracture Fixation Biomechanics

Abstract

Internal fixation of bone fractures using plates and screws involves many choices-implant type, material, sizes, and geometric configuration-made by the surgeon. These decisions can be important for providing adequate stability to promote healing and prevent implant mechanical failure. The purpose of this study was to develop mathematical models of the relationships between fracture fixation construct parameters and resulting 3D biomechanics, based on parametric computer simulations. Finite element models of hundreds of different locked plate fixation constructs for midshaft diaphyseal fractures were systematically assembled using custom algorithms, and axial, torsional, and bending loadings were simulated. Multivariate regression was used to fit response surface polynomial equations relating fixation design parameters to outputs including maximum implant stresses, axial and shear strain at the fracture site, and construct stiffness. Surrogate models with as little as three regressors showed good fitting (R ² = 0.62-0.97). Inner working length was the strongest predictor of maximum plate and screw stresses, and a variety of quadratic and interaction terms influenced resulting biomechanics. The framework presented in this study can be applied to additional types of bone fractures to provide clinicians and implant designers with clinical insight, surgical optimization, and a comprehensive mathematical description of biomechanics.

Keywords: Finite element; Fracture fixation; Locking plate and screw; Response surface.

FIGURE 1.

Modularized finite element modeling used to generate large numbers of fracture fixation design configurations. (a) bone with or without screw hole; (b) locking plate parts; (c) locking screw; (d) an example of fracture fixation design achieved by automated assembly of bone, plate, and screw parts, in this case with 5 mm fracture gap, 14 hole plate, and screws positioned at holes 1, 3, and 6; (e) design variables of fracture fixation designs which served as regressors in the response surface statistical models; (f) Eight points (red dots) at the fracture gap used to compute strains in this region.

FIGURE 2.

Examples of FEA results for fracture fixation with 26 cm plate and axial loading. Left and right columns show 0.5 and 1.5 cm fracture gap, respectively, and each row shows a different screw configuration. Von Mises stresses are displayed for the plate. White circles indicate screw locations.

FIGURE 3.

Scatter plots of surrogate models. (Left column) Scatter plots show example fits between R²-based selection (5%) surrogate statistical model- predicted values and FE model ‘observed’ values that were used to fit the surrogate model (45° line indicates perfect fitting). (Right column) Plots of leverage vs. R-Student for the same models.

FIGURE 4.

Response surfaces based on R²-based selection (5%) surrogate models reported in Table 3. Red dots are the FEA results used to fit the surrogate models. (a): in axial loading σ_{plate_max} response surface as a function of L_inner and E_implant. (b): in axial loading ε_axial response surface as a function of d_gap and L_inner. (c): in torsion loading ε_shear response surface as a function of d_gap and L_inner. In (a), L_outer was constant (26.5 cm), and E_implant and was N_screws constant (200 GPa and 2 respectively) in (d).

FIGURE 5.

Comparison of response surfaces between statistical models based on large data set and DOE-based data set (also see Table S-5). Response surface plots similar to those presented in Fig. 4 are presented, with individual data points resulting from FE analyses also visible. In the right-sided column of images, response surfaces derived from both the large and DOE-based data set are superimposed on one another for comparison. In order to enable 3D plotting of the surrogate models that included more than two independent variables, additional variables were set constant: In response surface of σ_{plate_max}, L_outer was constant (26.5 cm), and E_implant was constant (200 GPa) in response surface of ε_shear.