Objective Function

For each objective function evaluation, the specified control variables are applied to the knee model and a subset of the experimental test cases are simulated. The knee model’s joint kinematics [GS83] are measured from the results of the simulation. The objective function is the sum of the squared residual between the model and experimentally measured joint kinematics.

(3)\[\begin{split}\begin{split} f(\bar{x}) &= \sum_{j=1}^4\sum_{k=1}^6w_k\Big(M_{jk}(\bar{x}) - E_{jk}\Big)^2 \\ h(x_i) &\geq 0. \text{ for } i=1 \ldots 22 \end{split}\end{split}\]

where \(\bar{x}\) is the set of 22 control variables, \(x_i\) is the \(i^{th}\) control variable, and \(h(x_i)\) is an inequality constraint that is applied to every control variable, for a total of 22 inequality constraints. The experimental test case is represented with \(j\), \(k\) represents the degree of kinematic freedom, and \(w_k=[0, 1, 0, 0, 2, 2]\) is the weight that is applied to each kinematic degree of freedom (Table 2).

Table 2 The weights and the corresponding kinematic degree of freedom.
i Weight DOF
\(i=1\) 0 Medial tibial translation
\(i=2\) 1 Anterior tibial translation
\(i=3\) 0 Superior tibial translation
\(i=4\) 0 Flexion
\(i=5\) 2 Varus
\(i=6\) 2 Internal tibial rotation

The weighting factor, \(w_k\), is used to equate tibial translation to tibial rotations. A weighting factor of 2 was arbitrarily selected to equate \(5^o\) of rotational error to 10 mm of anterior tibial translational error.

The weighting factor is also used to exclude kinematic degrees of freedom from the objective function. Flexion is excluded because the experimental flexion angles are prescribed in the finite element model. Medial and superior tibial translation are excluded from the objective function, similar to previous studies [BH96] [BCF+12] [HCA+16]. The same weights are applied to the kinematics, regardless of the experimental test that is being evaluated. This means that the same kinematic degrees of freedom are included in the objective function, regardless of the experimental load case.

[BCF+12]Mark A. Baldwin, Chadd W. Clary, Clare K. Fitzpatrick, James S. Deacy, Lorin P. Maletsky, and Paul J. Rullkoetter. Dynamic finite element knee simulation for evaluation of knee replacement mechanics. Journal of Biomechanics, 45(3):474–483, February 2012. URL: http://www.sciencedirect.com/science/article/pii/S0021929011007469, doi:10.1016/j.jbiomech.2011.11.052.
[BH96]L. Blankevoort and R. Huiskes. Validation of a three-dimensional model of the knee. Journal of Biomechanics, 29(7):955–961, July 1996. URL: http://www.sciencedirect.com/science/article/pii/0021929095001492, doi:10.1016/0021-9290(95)00149-2.
[GS83]E. S. Grood and W. J. Suntay. A Joint Coordinate System for the Clinical Description of Three-Dimensional Motions: Application to the Knee. Journal of Biomechanical Engineering, 105(2):136–144, May 1983. URL: http://dx.doi.org/10.1115/1.3138397, doi:10.1115/1.3138397.
[HCA+16]Michael D. Harris, Adam J. Cyr, Azhar A. Ali, Clare K. Fitzpatrick, Paul J. Rullkoetter, Lorin P. Maletsky, and Kevin B. Shelburne. A Combined Experimental and Computational Approach to Subject-Specific Analysis of Knee Joint Laxity. Journal of Biomechanical Engineering, 138(8):081004–081004, June 2016. URL: http://dx.doi.org/10.1115/1.4033882, doi:10.1115/1.4033882.