Model Calibration Summary¶
This section summarizes the steps involved in calibrating a finite element knee model. The summary includes links to other chapters where specific details are provided.
These steps apply to the DU data set. The documentation for the OpenKnee(s) data set can be found in a separate document.
Knee Model¶
The knee model that was developed during the Model Development phase is used for model calibration. Note that a simplified model will be used for calibration, where removal of various anatomy will increase computational efficiency. Details are described in the Tibiofemoral Model section.
Additionally, a mistake was found in the plACL insertion (specifically the file dat/DU02/mriData/Segmentation/plAclLine.fcsv, which is located in the compressed directory titled DU02ModelDevelopment_IntermediateOutputs.tar.gz). The mistake likely occurred in 3D Slicer where the insertion point may have been moved when rotating the model in the 3D viewing window. Fortunately, the file is in a local data repository, and original version of the file was restored (before the insertion point was accidentally moved). For the included ligaments, the insertions are the same as are reported in the Model Development outputs, except for one of the the plACL tibial insertion points. This is described in the Model Development Deviation section.
Optimization Scheme¶
Optimization Test Cases¶
There is data for several laxity style test cases available across multiple flexion angles. A subset of these data is used in the calibration procedure. Each objective function evaluation will include \(0^\circ\) and \(90^\circ\) flexion and the following test cases: anterior drawer, posterior drawer, varus moment, valgus moment. The maximum loads from each test case will be included in the objective function, which will result in evaluation of 8 total loading points. See the Test Cases and Test Case Step sections for more information.
Control Variables¶
The knee model is composed of 11 ligament bundles, and the slack length for each bundle is a parameter in the optimization. Specifically, each ligament bundle has two slack length parameters, and these parameters are used to define the slack length of every fiber in the ligament bundle. More details can be found in the Control Variables section.
Forward Kinematics Model Evaluation¶
A forward kinematic simulation was used to facilitate definition of the initial guess values as well as bounds for the control variables. The purpose of this simulation is to determine the maximum length of each ligament bundle throughout the experimental data used for calibration. The kinematics from the calibration test cases are used as inputs to a kinematically driven finite element model, and the lengths of the fibers at the margins of each ligament bundle are recorded. More details can be found in the Forward Kinematics Simulation section.
Control Variable Bounds¶
Upper and lower bounds for each control variable are used in the optimization. The lower bounds are set to prevent the optimization algorithm from evaluating slack lengths that may cause instability in the knee model. The upper bounds are used to limit the size of the parameter space. These bounds are defined using the values recorded from the forward kinematics model evaluation. The upper bound for each control variable is defined as the maximum length of the corresponding ligament fiber in the forward kinematics simulation results. The lower bounds are defined as 60% of the upper bound. More details can be found in the Control Variable Bounds section.
Initial Guess¶
Each control variable’s initial guess value is defined as 90% of the corresponding variable’s upper bound (i.e. the maximum length throughout the kinematically controlled simulation). This approach is uniformly applied to every control variable in an attempt to avoid influencing the optimization solution with a judicious selection of initial guess values. More information can be found in the Initial Guess section.
Optimization Algorithm¶
The gradient based “SLSQP” algorithm that is available in the SCIPY optimize toolbox will be utilized. This particular approach allows use of inequality constraints and bounds on the control variables. Details are described in optimization scheme section
Objective Function¶
The objective function calculates the weighted sum of squared residual between model and experimentally measured kinematics during the simulated test cases. The objective function will evaluate the residual in anterior-posterior tibial displacement, varus-valgus rotation, and internal-external tibial rotation. More details can be found in the Objective Function section. Information on the specific steps used in the Abaqus simulation can be found in the Simulation Steps section.
Optimization Constraints¶
Inequality constraints are used to enforce that each controlled ligament fiber experiences at least a nominal force of 0.1 N at one point through the simulated calibration test cases.
The constraints are used to prevent the optimization algorithm from effectively removing control variables from the model. A control variable can be effectively removed by specifying a large slack length where the ligament fibers are slack throughout the entire knee model evaluation. In this case the slack fibers will not influence the objective function, and this can result in the optimization algorithm “forgetting” the ligament when it otherwise should not. More details can be found in the Constraints section.