Methods

Finite element models (Fig.1) were developed from pre-operative CT scans of the tibiae of 11 female patients with osteoarthritis (age: 58–77 years). We sought to compare two loading conditions from Smith et al.;^[1] these corresponded to a mechanically aligned knee and a knee with 4° of varus. Consequently, we virtually implanted each model with a two-peg cementless baseplate following two tibial alignment strategies: mechanical alignment (i.e., perpendicular to the tibial mechanical axis) and 2° tibial varus alignment (the femoral resection accounts for additional 2° varus). The baseplate was modeled as solid titanium (E=114.3 GPa; v=0.33). The pegs and a 1.2 mm layer on the bone-contact surface were modeled as 3D-printed porous titanium (E=1.1 GPa; v=0.3). Bone material properties were non-homogeneous, determined from the CT scans using relationships specific to the proximal tibia.^[2,4] The bone-implant interface was modelled as frictional with friction coefficients for solid and porous titanium of 0.6 and 1.1, respectively. The tibia was fixed 77 mm distal to the resection. For mechanical alignment, instrumented TKR loads previously measured in vivo^[5] were applied to the top of the baseplate throughout level gait in 2% intervals (Fig.1a). For varus alignment, the varus/valgus moment was modified to match the ratio of medial-lateral force distribution from Smith et al.^[1] (Fig.1b).

Methods

We developed finite element (FE) models of the three TER mechanisms. To reduce computational cost, we did not include the humeral and ulnar stems. Materials were linear-elastic for the metallic components (E_Ti6Al4V=114.3 GPa, E_CoCr=210 GPa, v=0.33) and linear elastic-plastic for the polyethylene components (E=618 MPa, v=0.46; S_Y=22 MPa; S_U=230.6 MPa; ε_U=1.5 mm/mm). The models were meshed with linear tetrahedral elements of sizes 0.4–0.6 mm. We assumed a friction coefficient of 0.02 between metal and polyethylene. In all simulations, the ulnar component was fixed and the humeral component loaded. We computed the constraint profiles in full extension by simulating each mechanism from 8° varus to 8° valgus and from 8° internal to 8° external rotation. All other degrees-of-freedom except for flexion extension were unconstrained. Then, we identified the instant during feeding that generated the highest moments at the elbow[2], and we applied the joint forces and moments to each TER to evaluate the stresses in the polyethylene. To validate the FE results, we experimentally evaluated the constraint of the design with highest polyethylene stresses in pure internal-external rotation and compared the results against those from a FE model that reproduced the experimental setup (Fig.1-a).

Methods

The 2D stem geometry (initially Profemur^®TL) was parameterized with 8 variables. Limits were established to keep tapered stem shape, avoid intersecting the cortexes and assure proper cortical contact. A functional gradation of the stem's material properties was generated by prescribing the values of the elastic modulus (E) on a 53 points grid. Values for E were between 2 GPa (highly porous meta-material made of Ti6Al4V) and 110 GPa (solid Ti6Al4V). The stem neck and a 1.5 mm layer around the stem were kept solid.

Two contradictory objective functions were considered: 1) a function of the total bone loss, accounting for the bone losses due to the resection for the implant insertion and due to stress shielding; 2) a function of the interfacial shear stresses, accounting for their uniformity and value. This multi-objective optimization problem was solved using genetic algorithms for stair climbing load case^[2], with 30090 stem design evaluations for a total of 50 generations (iterations).

Two representative optimized stem designs were selected to undergo a second step of tailoring their porous meta-material for obtaining the desired material properties distribution. Simple-cubic unit cell was considered at the mesoscale of the porous meta-material, with a fixed unit-cell length of 1.5 mm. The strut diameter at each point of the grid was optimized to match the prescribed E using a previously developed model of porous meta-materials that includes the manufacturing irregularities^[3].

Methods

The 2D models were generated by the intersection of the 3D model with the stem symmetry plane. Three approaches were considered to assure 3D-2D correspondence: 1) consider variable thickness for the cortical elements so that their transversal area moment of inertia equals the cross-sectional area moment of inertia from the 3D model (model WOSP); 2) include an additional side-plate with variable thickness to match the area moment of inertia from the 3D model, and consider constant thickness for the cortical bone elements (model SPCT); 3) include the side-plate but consider variable thickness for the cortical bone elements, derived from the 3D model (model SPVT). In all cases, the cancellous bone and stem elements had variable thickness computed so that their transversal area moment of inertia was equal to the cross-sectional area moment of inertia measured in the 3D model. This was done at different levels (Fig.1), providing a thickness distribution for the 2D elements. FE analyses were carried out for the static loading condition simulating stair climbing[4]. All materials were defined as linear isotropic and homogeneous. The post-operative situation where bone ingrowth is achieved was considered, resulting in bonded contact between the bone and the implant. The comparison between the 2D and 3D models was done based on three physical quantities: the Von Mises stresses (σ_VM); the strain energy density (U) and the interfacial shear stress (t) along the stem-bone interface.

Methodology

The 2D-geometry of a femur model (Sawbones®) with an implanted Profemur-TL stem (Wright Medical Technology Inc.) was used for FE simulations. The stem geometry was parameterized using a set of 8 variables (Figure 1-a). To optimize the stem's material properties, a grid was generated with equally spaced points for a total of 96 points (Figure 1-b).

Purely elastic materials were used for the stem and the bone. Two bone qualities were considered: good (Ecortical=20 GPa, Etrabecular=1.5 GPa) and medium (Ecortical=15 GPa, Etrabecular=1 GPa). Poisson ratio was fixed to v=0.3. Loading corresponded to stair climbing. Hip contact force along with abductors, vastus lateralis and vastus medialis muscles were considered5 for a bodyweight of 847 N.

The resorbed bone mass fraction was evaluated from the differences in strain energy densities between the intact bone and the implanted bone2. The displacement of the load point on the femoral head was computed.

The optimization problem was formulated as the minimization of the resorbed bone mass fraction and the head displacement. It was solved using a genetic algorithm.

Methodology

A coronal cut was performed on a femur model (Sawbones®) with an implanted Profemur®TL (Wright Medical Inc.) stem to obtain the 2D-geometry for FE simulations.

The central part of the FGI stem was made porous, the neck and inferior tip were solid. Ti6Al4V elastic material was assumed (E=120 GPa, v=0.3). Three bone qualities were considered for the optimization: poor (E=6GPa; v=0.3); good (E=12GPa; v=0.3); excellent (E=30GPa; v=0.3).

The structure of bone evolves to maintain a reasonable level of the strains. Similarly in the proposed algorithm, the strut sections of the porous material evolve to keep stresses (proportional to strains) at a reasonable level. Starting with a very small strut section, resulting in an almost zero-rigidity stem, strut sections are increased or decreased as a function of the stresses they support. This is done incrementally, until force values corresponding to normal walking of an 80 kg person (1867 N)⁴ are reached. Force direction was vertical and no action of the abductors was considered, to analyze the worst case scenario. The optimized FGI microstructure is defined by the strut diameter distributions. Since the distance between struts remain constant, variations in strut diameters result in variations in density.

Optimized FGI porous structure was compared for the three bone qualities considered and with a solid stem in terms of bone stresses.

EXPERIMENTAL METHODS

Two computational models of the porous structure were developed: a 3D solid model, considered as the reference for comparison, and a beam model as a simplified and computationally inexpensive approximation (Fig. 1). CATIA V5R20 (3D modelling) and ANSYS V13 (simulations) were used.

Isotropic elastic material model was used. Strut diameter (ϕ_b) was set to 450 μm, pore diameter (ϕ_p) was varied between 600 and 5000 μm, and pore number (n_p) between 2 and 9. Structures sizes varied from 2.1 × 2.1 × 2.1 mm³ to 49.05 × 49.05 × 49.05 mm³. Apparent elastic modulus (E_ap) and its difference between both models (error) were analyzed for the different values of ϕ_p and slenderness ratio (S_R). In addition, the influence of loading direction was analyzed with the beam model for cubic and diamond cell geometries. E_ap variations were compared.