To help with comprehending the development of NBA, we briefly review the gait cycle (i.e. the ambulatory phase of walking). There are five phases of ambulation; 1) initial contact, 2) foot flat 3) midstance 4) heel lift and 5) toe off. The gait pattern, such as step width, step length, etc., is complex and unique to everyone. The midstance is the single limb support phase of gait cycle where the foot assumes a support and overall stability role. In an instantaneous midstance motion, the leg becomes rigid and is ready for propulsion while the center of mass is momentarily above the ankle joint. The complete sole of the foot is weight bearing as this single limb supports the entire body weight [2, 3]. In this orientation, the contact force between the femoral condyles and the tibial plateau is in compression, and the external knee adductor moment is resisted by a combination of muscle and ligament forces [21]. Therefore, the single rigid limb midstance phase provides an instant condition to apply a pseudo-static model following the minimum energy principal for the development and analysis of the NBA algorithm. Instead of a traditional MA approach, we apply structural engineering principles by the following steps: 1) identify for the hip, knee, and ankle (HKA) contact boundaries; 2) modeling of the femur and tibia as truss structures; 3) establish the neutral boundary axis (NB axis).
Defining contact boundaries (boundary conditions)
Figure 1a shows a coronal HKA radiograph in the midstance of gait for the right leg (not a traditional two-legged standing XR). It is noted that the method could be more practically applied using a traditional two-legged standing XR, CT or MRI. The figure illustrates that the contact boundaries are defined for HKA on the radiograph in the approximate midstance phase in a coronal view. The hip contact boundary HB is defined as the contact bounds (h1, h2) of the superior surface femoral head onto the inferior surface of the acetabulum in pelvis. The knee boundary KB is defined as the contact bounds (k1, k2) in the medial and lateral femoral condyles on the tibial plateau. Finally, the ankle boundary AB is defined as the contact bounds (a1, a2) on the medial and lateral talus corners.
Modelling of truss structure for femur and tibia
As a structure, the femur and tibia are bones with dense cortical bone and supporting internal trabecular network. It is believed that the trabecular bone is aligned within the stress lines of the femur, especially, at the femoral calcar [22, 23]. It is recognized that the femur and tibia are more than shells of cortical bone, as the trabecula provide an internal truss for the bone, mechanically providing strength at the lowest weight possible. For this reason, we infer the femur and tibia can be modeled as a truss structure [20].
An iterative algorithm of structural layout patterns of truss is developed. This approach includes Ptolemy’s theorem [10] and Delaunay and Voronoi tessellations [9, 11]. Ptolemy’s theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. Ptolemy’s theorem is applied to constrain the HKA boundary points. Further, the structural layout patterns are a truss as Delaunay and Voronoi tessellations and are applied for the truss connectivity of the HKA boundaries (Fig. 1b). Figure 1c exhibits the construction of femoral and tibial quadrilaterals as trusses with diagonals. The intersecting points of the femoral and tibial diagonals are defined as P1 and P2, where the points P1 and P2 represent the center points of femoral and tibial quadrilaterals, respectively.
Establishing neutral boundary Axis
Once the femoral and tibial trusses are constructed, the straight-line L, connecting the points of P1 to P2, is defined as “Neutral Boundary Axis” in Fig. 1d. Although an individual’s gait cycle is unique, he or she prefers to move in an optimal way of walking. This implies that the NB axis should follow the direction of gravity according to the minimum energy principal [2] in the midstance phase. Consequently, aligning the NB axis in the direction of gravity is an important factor in determining the overall leg limb alignment of the native knee, while the NB axis being situated within the HKA boundaries for the single limb support is important to prevent Varus/Valgus (V/V) instability. These conditions assure structural stability and knee balance in the midstance phase during ambulation.
Defining joint line (initial condition)
Figure 1d exhibits the joint line J (the femoral axis of initial knee flexing) and is defined as the line perpendicular to the NB axis (which is in the direction gravity). This leads to inevitable conclusion that the joint line is parallel to the ground [13, 14], which is assumed to be one shared feature for everyone. Despite the complex dynamic knee motion, the joint line being parallel to the ground is of significance as the initial condition in the knee structure when ready for a transition toward propulsion in gait cycle.