Three different approaches to deformable nails modeling have been compared (Figure 2); in each of them, the nail element has been studied as a three-dimensional slender beam, made of biocompatible stainless steel AISI 316 LVM (E = 2 × 10¹¹ MPa, ρ = 8,000 Kg/m³ and υ = 0.3). The first method consists of a discretization of the nail through a combination of lumped elements with inertial and stiffness properties. Each element is composed of two rigid spheres, placed at the ends of a massless beam, which produces forces and moments linearly dependent on relative displacements and velocities of the beam's endpoints (MSC Adams, 2000; MSC SimCompanion, 2017c). Inertial properties of the nail are attributed to the rigid spheres in accordance with the lumped mass method; all these elements are arranged in series to reproduce the actual nail geometry. Therefore, the mass is equal to half the mass of the beam connecting two spheres for the first and the last spheres, whereas the mass is twice this value for internal bodies. On the other side, stiffness properties of the beams have been computed from a beam with a circular cross section identical to the nail cross section. The number of rigid bodies for the discretization has been determined imposing that at least two spheres were engaged into the ring's hole (7 mm long), during each step of the sliding phase; as a result, the nail has been discretized by 41 spheres.

The second approach required creating and importing a finite element model of the nail to define a “flexible body.” The geometry has been replicated extruding a circular section, meshed by 100 4-node shell elements; this section was extruded along a spline coincident with the centreline and the final model reproduced the nail's volume through 18600 20-node solid elements (4th order shape functions). The finite element model was imported into Adams as a MNF (Modal Neutral File, Zanetti et al., 2017). As it is based on the Craig-Bampton theory, boundary DOF must be defined (Ansys Inc., 2013; MSC SimCompanion, 2017b); therefore, the nodes placed at the center of the upper and lower nail sections have been selected as interface points.

The last method made use of the so called FE Parts, that are a modeling tool for flexible bodies provided by the multibody software (MSC SimCompanion, 2017a). This tool allows generating both straight and curved 3D beams: a spline was defined that was a guide curve for the centreline, along with the shape and the size of the beam section and material properties. The final model was a beam, meshed by a set of about 2570 tetrahedric solid elements (they are based on a non-linear finite element formulation, used in flexible multibody dynamics to describe large deformation of moving bodies, but no further details are disclosed by the software producer).

The ring nut is a key element, governing the contact forces driving the device performance.

The direct importation into the model of the 3D ring geometry as a rigid body leads to the difficult calculations of contact forces between two coaxial cylinders (Webex, 2012). As a first step, the full ring geometry has been therefore segmented into smaller elements in order to reduce the contact detection area (see differently colored volumes in Figure 3). In addition, a “conceptual” model of the ring nut has been created in order to further simplify numerical simulations and reduce computation time. This model simulates the central cylindrical part of the nut and is reported in Figure 3. The advantage of this solution compared to the segmented ring model is that localized contacts take the place of distributed contacts and the number of contacts per nail is reduced. This simplified model has been validated comparing its results to the results obtained using the original geometry (section Contact Parameters Tuning).

The fractured bone has been modeled as a solid body, through the importation of the 3D femur geometry into the simulation environment (Zanetti and Bignardi, 2009). Only the cortical bone has been modeled in order to obtain a conservative estimate: the trabecular bone is very thin inside the medullary canal and it could be easily fractured by the nails; a fracture along a plane inclined of 30° [type 32-A2, according to AO classification (Femoral shaft - Diagnosis - AO Surgery Reference)¹ has been simulated. The initial relative position of the fractured bone parts has been governed through a massless element, oriented as the fracture plane, with linear stiffness and damping properties (MSC SimCompanion, 2017c).

One end of all nails has been fully constrained by a fixed joint, in order to simulate the stub (Figure 1). The ring slides along the vertical direction (y axis in Figure 2) by means of a translational joint, following a law of motion defined by a quintic polynomial (“STEP5” function). The displacement law has been imposed so that the ring covers 33 cm within 5 s, both throughout the upward and the downward motion. The two bone's fragments have been completely constrained by two fixed joints applied at their center of mass; these boundary conditions allow removing bone displacements and rotations during the first two phases of simulation, approximating actual conditions in surgery.

All contact forces have been simulated, making use of IMPACT formulation in Adams.

This formulation includes an elastic and a damping component, defined through four parameters (K, e, C_max, and d_max), as detailed in the following:

where g is a parameter that represents the penetration between the involved geometries, dg/dt is the penetration velocity evaluated at the contact point, e is the elastic force exponent, d_max is the penetration depth and C_max is the damping coefficient. The STEP function imposes a gradual variation of the damping coefficient, according to the following cubic polynomial law:

where x₀ and x₁ are limit input values of the STEP function; h₀ and h₁ are the initial and final function values; a = h₀ - h₁ and Δ = (x- x₀)/(x₁ - x₀).

All these parameters have been initially estimated taking into consideration the geometry and materials involved in the model; subsequently, they have been tuned examining results of a set of simulation tests: the “ideal” set of parameters should allow performing simulations in reasonable times, avoiding excessive penetration among objects.

All simulations have been performed with frictionless contacts between the nails and the medullary canal; this is a conservative hypothesis, since friction would have a stabilizing action, limiting movements of the nails inside the bone during the loading phase.

A simple finite element model of a single nail has been created (by Ansys v. 17) as a benchmark to check MB model results. Nail's centreline has been replicated through a spline, meshed by means of 50 beam elements. All six DOF of the base node (“fixed node” in Figure 4) have been constrained, and sequential analyses have been performed, where, node by node, a z displacement equal to the node's z coordinate itself (Figure 4) has been imposed (that is the examined node has been brought to the vertical axis) and the respective constraining force has been measured.

As a result of this analysis, the force required to bring each node to the zero z coordinate was assessed and it has been plotted in Figure 4; the force magnitude initially grows, it reaches a maximum value on node 20th (about 5 N), and finally it decreases.

Comparisons among the three nail models have been performed analyzing the interaction of a single nail with the ring. At this stage, contact parameters have been set equal to their default values. The discretized nail model has required implementing 41·8 contacts (since the spheres are 41 and the segments composing each nail guide in the ring are 8), whereas 8 contact couples were sufficient for the Flexible Body and the FE Part models. Each model has been validated analyzing the pattern of z force component at the nail's constraint (Figure 5), which had to replicate the force behavior reported in Figure 4. Obtained results are shown in the Figure 5.

The contact force for the lumped parameters model of the nail exhibits many peaks, corresponding to spheres coming in and out of contact within the ring hole. The blue curve in Figure 5B can be obtained with a filtering operation (a second order, low pass Butterworth filter, with a cut-off frequency of about 1 Hz has been applied to the signal), and its pattern is similar to the one depicted in Figure 4, however absolute values are quite different.

In Figure 5C, the sudden change of the force amplitude at 2.5 s time instant, along with the consequent vibrations are a clear sign that the Flexible Body model is not able to keep the nail in contact with the ring: there is a sudden compenetration and since that time instant the contact boundary is over.

With reference to the FE Part model, the curve is smooth and its pattern and amplitude are comparable to those obtained from FE model; again, compenetration takes place as in the Flexible Body simulation, however it occurs at the final phase of the ring movement.

Simulation times of the three formulations are 3.4, 6.5, and 5.3 min respectively. On the basis of these results and of further similar tests, the Flexible Body model has been discarded, because of the higher computational times and because the contact management is more critical. FE Part formulation has proved to be sufficiently robust to obtain an accurate contact force curve even when using default contact parameters values, leaving a wide margin for their tuning. Having considered this aspect and the lower number of required contacts, the FE Part model has been established to be the best approach, in spite of its higher simulation time compared to the lumped parameters model.

A gradual tuning process of contact parameters has been performed; as described above, the only contact between a single nail and the ring nut has been modeled as a first step; after, a second nail has been added as well as the contact between two adjacent FE Parts; finally, the complete system has been considered. Table 1 shows the best set of parameters obtained for each model, and images illustrate the respective configurations at the final stage of ring nut movement.

The tuning process has demonstrated that “optimal” contact parameters may change in relation to the number of elements (and consequently the expected contact force), even with reference to the same contact couple (i.e., nail-to-ring or nail-to-nail). Once parameters for the six nails-real ring model have been established, the “simplified” ring has been validated. Figure 6 shows the magnitude of the force at the bounded end of one of the nails, for the actual and the simplified ring: there is a good match between the two curves, the maximum difference between peak values reaching 4%.

The single nail model has been validated comparing the ring-to-nail force with the force computed from the FE model (Figure 7). The comparison has showed that these patterns are very close, with a deviation of the MB peak value (5.16 N) equal to about 3% compared to the FE peak value (5.30 N). Besides, the location of this peak value along the nail (abscissa in Figure 7) is the same.

The simplified ring has allowed a further optimization of contact parameters in order to limit compenetration phenomena (Table 2); this was made possible by the reduction of the model complexity: with the original nut model, parameters reported in Table 2 would have led to unacceptable simulation times. In fact, using the internal solver setting and the real ring geometry model, the simulation calculates around one half of the ring path in 2 h and, from that point on, it cannot proceed due to high penetration phenomena between the nails and the ring. In the same conditions, the simulation with the simplified ring geometry is completed in 50 min (simulations have been performed on laptop equipped with a i5 processor and 12 GB RAM).

The two bone's fragments have been constrained by two fixed joints, and standard contact parameters have been set, to study the contact between the nails and the inner wall of the medullary cavity. When the device is fully open into the femur (Figure 8), the nails adapt their shape to the bone cavity.

The most relevant forces are those between the nails and the distal femur fragment. These forces are shown in Figure 9; they are different for each nail, because the geometry of the canal is not axial-symmetric.

The values of forces at the end of the opening phase are produced not only by the action of the distributed contact along the medullary canal, but also by the forces that nails exert on one another (Figure 8). This observation explains why the final force values of curves reported in Figure 9 stand between 7 and 12 N, while both FE and MB single nail models output was equal to 4 N (Figure 7).

After the implantation of the Marchetti-Vicenzi nail, the fractured femur has been loaded: the proximal fragment has been subjected to a growing quasi-static horizontal force [ranging from 0 to 200 N with a quintic polynomial step law (Terzini et al., 2017), see contoured image in Figure 10]. The forces between the six nails and the proximal bone fragment throughout the loading phase are depicted in Figure 10: the nails start contacting the proximal bone at time 10 s, and from that moment on, contact forces rise.

The final contact force ranges between 15 and 118 N; these different values are due to the fact that nails come into contact with different portions of the proximal fragment.

Figure 11 shows the relative rotation between the bone fragments vs. the external load (see the arrow in contoured image in Figure 11). The curve here reported proves the device capability to stiffen the broken femur toward bending loads: if the device would not be present, the slope of this curve would be theoretically infinite. The video provided in the Supplementary Material section shows the full simulation.

These data allow to reach a conservative estimate since they have been obtained without taking into account the friction between nails and bone, the contact force and friction between the two bone fragments and without modeling the porous spongy component; therefore the actual femoral rotation is expected to be lower. According to these results, the nail action is relevant, however a moderate (200 N) lateral force applied on a standing subject (fixed foot) would be able to significantly dislocate the fracture. This could be the reason why the performance of this fixation device has been proved to be poor on the femoral bone (Anastopoulos et al., 2001; Madan et al., 2003), where acting forces are higher compared to upper extremity bones, on average.

The authors have here chosen to follow a “global” approach, considering the nail stiffening action: this action is strictly related to relative displacement at the fracture site since the larger is this action the lesser relative displacements are. The reason why more localized measurements (such as the above cited relative displacements) have not been here considered is the respective dependency on boundary conditions, such as the geometry of the fracture rim (i.e., transverse or oblique) and on friction between contacting surfaces. This would hamper obtaining general results on the performance of a given bone-nail system. Indeed, a patient-specific planning would require performing such evaluations since healing is strictly related to displacements at the fracture site.

However, it should be stressed that there is still a margin to improve. Figure 12 reports the net force at the distal fragment vs. the ring position. With reference to the opening phase, this force reaches a peak value before the ring has been completely withdrawn. As a consequence, the most stable configuration of the fractured bone—nail system is not the completely open configuration.

The model described in the previous sections was based on the hypothesis of an ideal assembly of the nails; that is nails centrelines have been placed at exactly 60° steps. A further model has been developed where an angular position error ranging between 0° and 10° has been randomly assigned to each nail. Figure 13 clearly outlines differences between FE Part contact curves in the “ideal” nails system or in the “realistic” one. Differences in contact forces can reach 50%, therefore the employment of a probabilistic design is strongly advocated for this sort of devices.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.