Coordinate Systems

This section describes the methods used to define the coordinate systems of the femur, tibia, and patella. These coordinate systems are used to describe tibiofemoral and patellofemoral kinematics.

This describes how the coordinate system is defined using the OpenKnee(s) data set, which consists only of MR images. These methods consist of manually defining osseous landmarks in the MR images, and using those landmarks to define the fixed femoral, tibial, and patellar coordinate systems.

These methods use 3D Slicer (https://www.slicer.org/) to visualize the MR images and place fiducials at the desired locations.

Femur

Six total fiducial points are used to define the femur’s fixed coordinate system. These methods are similar those described in [grood_joint_1983]. The MR images do not include the femoral head, so the femur’s SI axis will be initially defined with points around the femoral shaft. This temporary SI axis is used to define the AP axis. After the AP axis is defined, the SI axis will be rotated 5 degrees about the AP axis to define the “mechanical” femur SI axis. The “mechanical” SI axis is taken as the femur’s fixed SI axis.

Fiducial locations:
  • \(F_1\) = Medial femoral epicondyle: The most medial point on the medial femoral condyle (using the axial view in the MR images).
  • \(F_2\) = Lateral femoral epicondyle: The most lateral point on the lateral femoral condyle (using the axial view in the MR images).
  • \(F_3\) = Proximal femoral shaft (anterior): The anterior point on the femoral shaft in the most proximal slice (using the axial view in the MR images).
  • \(F_4\) = Proximal femoral shaft (posterior): The posterior point on the femoral shaft in the most proximal slice (using the axial view in the MR images).
  • \(F_5\) = Proximal femoral shaft (medial): The medial point on the femoral shaft in the most proximal slice (using the axial view in the MR images).
  • \(F_6\) = Proximal femoral shaft (lateral): The lateral point on the femoral shaft in the most proximal slice (using the axial view in the MR images).

Note

\(F_3\), \(F_4\), \(F_5\), and \(F_6\) should be defined using the same slice in the MR images.

Origin: the midpoint between the medial (\(F_1\)) and lateral (\(F_2\)) epicondyles.

\[P_{OriginF} = \frac{F_1 + F_2}{2}\]

The femur’s temporary superior-inferior axis (\(z_{SI\_temp}\)) is defined with the line connecting the midpoint between the medial (\(F_1\)) and lateral (\(F_2\)) epicondyles, and the average of the points digitized around the proximal femoral shaft (\(F_3\)-\(F_6\)).

\[\begin{split}\begin{split} P_{proximal} =& \frac{F_3 + F_4 + F_5 + F_6}{4} \\ \bar{z}_{SI\_temp} =& \frac{P_{proximal} - P_{OriginF}}{||P_{proximal} - P_{OriginF}||} \end{split}\end{split}\]

The femur’s anatomic anterior-posterior (\(y_{AP}\)) axis is defined as the normal to the plane defined with the proximal femoral shaft center (\(P_{Proximal}\)) and the medial (\(F_1\)) and lateral (\(F_2\)) epicondyles. The \(y_{AP}\) axis points anteriorly, and is defined as

\[y_{AP} = \bar{z}_{SI\_temp} \times \bar{CD}\]

Where \(\bar{CD}\) is a unit vector that points to the right, and is parallel to a line that connects the medial (\(F_1\)) and lateral (\(F_2\))

Note

For a right knee:

\[\bar{CD} = \frac{F_2 - F_1}{||F_2 - F_1||}\]

For a left knee:

\[\bar{CD} = \frac{F_1 - F_2}{||F_1 - F_2||}\]

To determine the femur’s fixed \(z_{SI}\) axis, \(z_{SI\_temp}\) will be rotated 5 degrees about \(y_{AP}\) (using the method described in section 9.2 found here). Recognizing that \(z_{SI\_temp}\) is already a unit vector that is perpendicular to the unit vector \(y_{AP}\),

\[\begin{split}\begin{split} \bar{w} =& \bar{y}_{AP} \times \bar{z}_{SI\_temp} \\ \bar{z}_{SI} =& \bar{z}_{SI\_temp}cos\theta + \bar{w}sin\theta \end{split}\end{split}\]

Where \(\theta=5\) degrees for a left knee, and \(\theta=-5\) degrees for a right knee.

The anatomic medial-lateral (\(x_{ML}\)) axis points to the right, and is defined with as the cross product between the \(y_{AP}\) and \(z_{SI}\) axes.

\[x_{AP} = y_{AP} \times z_{SI}\]

Tibia

Six total fiducial points are used to define the tibia’s fixed coordinate system. These methods are similar those described in [grood_joint_1983].

Fiducial anatomy:
  • \(T_1\) = Medial tibial plateau
  • \(T_2\) = Lateral tibial plateau
  • \(T_3\) = Distal tibial shaft (anterior): The anterior point on the tibial shaft in the most distal slice (using the axial view in the MR images).
  • \(T_4\) = Distal tibial shaft (posterior): The posterior point on the tibial shaft in the most distal slice (using the axial view in the MR images).
  • \(T_5\) = Distal tibial shaft (medial): The medial point on the tibial shaft in the most distal slice (using the axial view in the MR images).
  • \(T_6\) = Distal tibial shaft (lateral): The lateral point on the tibial shaft in the most distal slice (using the axial view in the MR images).

Note

\(T_3\), \(T_4\), \(T_5\), and \(T_6\) should be defined using the same slice in the MR images.

Origin: defined as the midpoint between the medial (\(T_1\)) and lateral (\(T_2\)) tibial plateau.

\[P_{OriginT} = \frac{T_1 + T_2}{2}\]

The tibial superior-inferior axis (\(z_{SI}\)) is defined as the line between the midpoint of the medial (\(T_1\)) and lateral (\(T_2\)) tibial plateau, and the average of the digitized medial and lateral malleolus (\(T_3\)-\(T_6\)).

The tibial anterior-posterior axis (\(y_{AP}\)) points anteriorly, and is defined as

\[y_{AP} = z_{SI} \times \bar{AB}\]

Where \(\bar{AB}\) is a unit vector line that points to the right, and is parallel to the line that passes through the medial (\(T_1\)) and lateral (\(T_2\)) tibial plateau.

Note

For a right knee:

\[\bar{AB} = \frac{T_2 - T_1}{||T_2 - T_1||}\]

For a left knee:

\[\bar{AB} = \frac{T_1 - T_2}{||T_1 - T_2||}\]

The medial-lateral (\(x_{ML}\)) axis points to the right, and is defined with as the cross product between the \(y_{AP}\) and \(z_{SI}\) axes.

\[x_{AP} = y_{AP} \times z_{SI}\]

Patella

Four fiducial points are used to define the patella’s fixed coordinate system.

Digitized anatomy:
  • \(P_1\) = Most medial point
  • \(P_2\) = Most lateral point
  • \(P_3\) = Most superior point
  • \(P_4\) = Most inferior point

Origin: defined as the midpoint between the medial (\(P_1\)) and lateral (\(P_2\)) points.

\[P_{OriginP} = \frac{P_1 + P_2}{2}\]

The medial-lateral (\(x_{ML}\)) points to the right. For a right knee \(x_{ML}\) is defined as the line passing through the lateral point (\(P_2\)) and the origin of the patella’s coordinate system (\(P_{OriginP}\). For a left knee \(x_{ML}\) is defined as the line passing through the medial point (\(P_1\)) and the origin of the patella’s coordinate system (\(P_{OriginP}\).

\[\begin{split}\begin{split} x_{ML-right} =& \frac{P_2 - P_{OriginP}}{||P_2 - P_{OriginP}||} \\ x_{ML-left} =& \frac{P_{OriginP} - P_1}{||P_{OriginP} - P_1||} \end{split}\end{split}\]

The anterior-posterior (\(y_{AP}\)) points anteriorly, and is defined as

\[y_{AP} = \bar{EF} \times x_{ML}\]

where \(\bar{EF}\) is a unit vector that points superiorly and is parallel with a line that passes through the inferior (\(P_4\)) and superior (\(P_3\)) points.

\[\bar{EF} = \frac{P_3 - P_4}{||P_3 - P_4||}\]

The superior-inferior (\(z_{SI}\)) axis points superiorly, and is defined as the cross product between the medial-lateral axis (\(x_{ML}\)), and the anterior-posterior axis (\(y_{AP}\)).

\[z_{SI} = x_{ML} \times y_{AP}\]