Control Variables

There are two control variables that specify the slack length for each ligament bundle. One control variable specifies the slack length of one margin fiber, and the other specifies the slack length of the other margin fiber (Fig. 4). The given slack lengths are used to calculate the strain at the margin fibers of the ligament bundle, and these strains are linearly interpolated to define the slack lengths of the remaining 23 fibers in the ligament bundle.

Note

A ligament bundle’s two given slack lengths are not directly interpolated to assign slack length to the remaining 23 fibers. This is because the femoral and tibial fiber insertion points likely do not lie in the same plane. Additionally, the imaged insertion-to-insertion length may vary across the ligament bundle due to wrapping.

control variable

Fig. 4 An example of (a) a ligament mesh with 25 fibers from a finite element model. (b) A simplified representation of that same model showing 5 fibers. (inset) The magenta points show the insertions of the fibers at the margins of the ligament bundle insertion, and the insertion-to-insertion length of the first (\(L_0^{im}\)) and second (\(L_1^{im}\)) fiber at the margins of the insertion in the imaged joint position.

The given slack lengths for a ligament bundle’s margin fibers, \(x_0\) and \(x_1\) and the fiber’s imaged insertion-to-insertion length, \(L_0^{im}\) and \(L_1^{im}\), are used to calculate the strain (\(\epsilon^*\)) needed to be applied to the fiber’s imaged insertion-to-insertion length to achieve the given slack lengths (\(x_0\), \(x_1\)).

(1)\[\epsilon^*_i = \frac{x_i - L_i^{im}}{L_i^{im}}\]

Note

\(\epsilon^*_i\) is the strain that is applied to the ligament relative to the fibers images insertion-to-insertion length. This is not the prestrain carried by the fiber, which is relative to the slack length, \(\epsilon = \frac{L^{im} - L^{slack}}{L^{slack}}\)

The strains for the two margin fibers (\(\epsilon^*_0\) , \(\epsilon^*_1\)) are linearly interpolated across the ligament bundle. Each fiber’s slack length is calculated using the interpolated strain value and imaged insertion-to-insertion length (1).

Insertion-to-Insertion Length

The insertion-to-insertion length of a ligament fiber is defined as the distance between the fiber’s insertion points, including wrapping around the femur, tibia, meniscus, and cartilage. This length is defined from the results of the “reference simulation” described in Cleveland State University’s Model Development documentation.

In short, the ligament geometry was originally defined as springs that spanned between insertion points without any bone, cartilage, or meniscus wrapping. The reference simulation enforced ligament wrapping while maintaining the imaged joint position for the ligament insertion points, bones, cartilage, and menisci. After wrapping was enforced, the positions of the ligament nodes were extracted from the reference simulation results. These coordinates are used to determine each ligament fiber’s insertion-to-insertion length when the joint is in the imaged position.