Initial Deviations

The only deviation that is needed from the Model Calibration phase is in the processing of the given DU kinematics. The original Model Calibration documentation misinterpreted the reported translations as joint translations rather than clinical translations [GS83]. The model calibration workflow was setup to measure the joint translations from the knee model, and to avoid changing that workflow, the experimental data will be converted from clinical translations to joint translations. Any data that is reported will specify whether clinical translations or joint translations are used.

Experimental Data Processing - DU

The following process is used to convert the given clinical translations to joint translations. This uses definitions and equations from [GS83].

\[\begin{split}\begin{split} q_1 &= S_1 + S_3 cos(\beta) \\ q_2 &= S_2 \\ q_3 &= -S_3 - S_1 cos(\beta) \end{split}\end{split}\]

Where \(q_1\), \(q_2\), \(q_3\) are the reported lateral, anterior, and superior tibial translation values, and \(\beta\) is the angle between the tibia’s z-axis and the femur’s x-axis.

\[\begin{split}\beta = \begin{cases} \frac{\pi}{2} - varus & \text{right knee} \\ varus + \frac{\pi}{2} & \text{left knee} \end{cases}\end{split}\]

Note

The definition of \(\beta\) varies from that reported by [GS83] in Table 2. The definition of \(\beta\) follows the convention shown in Figure 1 [GS83].

Solving the above equation for \(S_1\) and \(S_3\).

\[\begin{split}\begin{split} S_1 &= q_1 - S_3 cos(\beta) \\ S_3 &= -\frac{q_3 + q_1 cos(\beta)}{1 - cos(\beta)^2} \end{split}\end{split}\]

The calculated joint translations (\(S_1, S_2, S_3\)) will be used in model calibration. Following this conversion, the experimental data will be processed in the same manner as described in the Process Experimental Data - DU section of the Model Calibration documentation. The relevant section is reproduced below, and changes were made for clarity with the above calculations. Note that the load definitions were not changed in the Model Benchmarking phase.

Kinematics and Kinetics Adjustment 1 - CSU/WSU convention

The joint kinematics and kinetics were converted to the convention used by the CSU/WSU lab. This step was not strictly necessary, however. It was implemented to keep the data consistent with the modeling workflow used in the CSU/WSU lab, which also allowed retention of our current documentation of the description of motion. The CSU/WSU convention describes motions as:

  • Medial tibial translation

  • Anterior tibial translation

  • Superior tibial translation

  • Flexion

  • Varus

  • Internal tibial rotation.

To convert to the CSU/WSU convention, the sign of the kinematics for three motions were changed (Table 1):
  1. TF ML (mm) (\(S_1\) above) kinematics are multiplied by -1 to convert to medial tibial translation.

  2. TF VV (deg) kinematics are multiplied by -1 to convert to varus rotation.

  3. TF IE (deg) kinematics are multiplied by -1 to convert to internal tibial rotation.

To convert to the CSU/WSU convention, the sign of the kinetics for three loads were changed (Table 1):
  1. Force TF ML (N) kinetics are multiplied by -1 to convert to medial tibial drawer force.

  2. Torque TF VV (Nmm) kinetics are multiplied by -1 to convert to varus torque.

  3. Torque TF IE (Nmm) kinetics are multiplied by -1 to convert to internal tibial rotation torque.

Table 1 The heading of the reported kinematic and kinetic degrees of freedom, the descriptions of the positive reported values, and the value that is multiplied by the experimental data to convert to the CSU conventions.

Reported

Reported Positive Direction

CSU/WSU convention

TF ML (mm) (\(S_1\) above)

Lateral tibial translation

-1

TF AP (mm) (\(S_2\) above)

Anterior tibial translation

1

TF SI (mm) (\(S_3\) above)

Superior tibial translation

1

TF FE (deg)

Flexion

1

TF VV (deg)

Valgus

-1

TF IE (deg)

External tibial rotation

-1

Force TF ML (N)

Lateral tibial drawer force

-1

Force TF AP (N)

Anterior tibial drawer force

1

Force TF SI (N)

Distraction force

1

Torque TF FE (Nmm)

Flexion torque

1

Torque TF VV (Nmm)

Valgus torque

-1

Torque TF IE (Nmm)

External tibial rotation torque

-1

Note

There was no conversion needed for right or left knees because the kinematics were measured using the connector elements defined in the Model Development documentation. These connector elements compose the joint coordinate system ([GS83]), and their definition in developing the model takes into account whether the specimen is a right or left knee.

Kinetics Adjustment 2 - Right or Left Knee

The signs of the loads may be changed depending on whether a right or left knee specimen was simulated. This is because the loads are applied with respect to the fixed tibial coordinate system, and the positive directions of this coordinate system may not match the experimental data’s loading directions.

For example, based on the CSU/WSU convention, the positive x direction points to the right. The reported load that corresponds to this direction is Lateral Drawer. For a right knee, the Lateral Drawer direction and the tibia’s x-axis point in the same direction. Conversely, for a left knee the Lateral Drawer direction and the tibia’s x-axis point in opposite directions. The sign of the reported data was changed because the loads were applied with respect to the tibia’s fixed coordinate system.

These adjustments were applied in the part of the Abaqus .inp file that was used to define the test cases step. An example of these adjustments can be seen in the code block below.

Below is a description of how the experimental loads were adjusted for each degree of freedom.
  • After the previous adjustment (Table 2), the data is described as medial tibial drawer. For a right knee, the data is multiplied by -1 (Table 2) to change the load to lateral tibial drawer.

  • The data is reported as anterior tibial drawer, and not adjusted in the previous step (Table 2). The anterior direction is in the positive y direction relative to the fixed tibial coordinate system for left and right knees, so this value is not changed for right or left knees (Table 2).

  • The data is reported as distraction, and not adjusted in the previous step (Table 2). A load that acts in the inferior direction relative to the fixed tibial coordinate system will cause joint distraction regardless of a right or left knee. The positive direction for the tibia’s z-axis points in the superior direction, therefore the sign for both right and left knees is changed to make the distraction load act in the inferior direction (Table 2).

  • The flexion torque from the previous step is multiplied by -1 (Table 2). The medial-lateral axis points to the right (for both a right and left knee), and a negative torque about the tibia’s “flexion axis” will cause knee flexion.

  • The data is reported as varus torque, and not adjusted in the previous step (Table 2). The y-axis of tibia’s fixed coordinate system points in the anterior direction. For a right knee, a positive torque around the tibia’s y-axis causes varus, so the sign is not changed for a right knee. However the sign is changed for a left knee because a positive torque about the y-axis of a left tibia causes valgus.

  • After the previous adjustment (Table 2), the data is described as internal tibial rotation torque. A positive rotation about the positive z-direction of a right knee causes internal tibial rotation, so the sign is not changed for a right knee. However the sign is changed for a left knee because a positive rotation about the positive z-direction of a left knee causes external tibial rotation.

Table 2 The degree of freedom at the node that the loads are applied to, and the corresponding load description, and the relevant sign convention changes. The reported description is from the given .tdms files that contain experimental data under the State.JCS Load heading.

Axis

Reported

CSU/WSU convention

Right knee

Left knee

\(x_1\)

Force TF ML (N)

-1

-1

1

\(x_2\)

Force TF AP (N)

1

1

1

\(x_3\)

Force TF SI (N)

1

-1

-1

\(x_4\)

Torque TF FE (Nmm)

-1

-1

-1

\(x_5\)

Torque TF VV (Nmm)

1

1

-1

\(x_6\)

Torque TF IE (Nmm)

-1

1

-1

Below is an example code showing how loads are specified in the Abaqus input file for a simulation of a right knee:

** Medial tibial drawer force, -1. for right knee and 1. for left knee
*CLOAD, amplitude=medialTibialDrawerForce, follower, op=new
JointCoordSys.1, 1, -1.
** Anterior tibial drawer force, 1. for right and left knee
*CLOAD, amplitude=anteriorTibialDrawerForce, follower, op=new
JointCoordSys.1, 2, 1.
** Distraction force, -1. for right and left knee
*CLOAD, amplitude=distractionForce, follower, op=new
JointCoordSys.1, 3, -1.
** Flexion torque, -1 for a right and left knee
*CLOAD, amplitude=flexionTorque, follower, op=new
JointCoordSys.1, 4, -1.
** Varus torque, 1. for right knee and -1. for left knee
*CLOAD, amplitude=varusTorque, follower, op=new
JointCoordSys.1, 5, 1.
** Internal tibial rotation torque, 1. for right knee and -1. for left knee
*CLOAD, amplitude=internalTibialRotationTorque, follower, op=new
JointCoordSys.1, 6, 1.

Where each amplitude is the magnitude of the corresponding load, where the adjustments from adjustment 1 are reflected. The node JointCoordSys.1 is the node that was coincident with the origin of the tibia’s fixed coordinate system. The integer following JointCoordSys.1 indicates the degree of freedom that the load was applied to, and this corresponds with \(x_i\) in Table 2. The final integer is multiplied by the specified amplitude. This was used to adjust for a right or left knee according to Table 2.

GS83(1,2,3,4,5)

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.