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 DU data set, which consists MR and CT images, as well as points that were digitized around the femur, tibia, and patella during testing. The digitized points include individual landmarks, and sets of points that were digitized around the osseous and articulating surfaces.

These methods define the digitized landmarks with respect to the MR image’s coordinate system by using a registration procedure (described here: DU Registration). The landmarks in the MR image coordinate system are used to define the fixed coordinate systems for the femur, tibia, and patella.

Femur (DU)

Five digitized landmarks around the distal femur and several digitized points around the femoral head are reported. Four of the distal points and the center of the femoral head are used to define the femur’s fixed coordinate system. The center of the femoral head is calculated from the points digitized around the femoral head. This coordinate system is similar to that described by [GS83].

Fiducial locations:
  • \(F_1\) = Medial femoral epicondyle
  • \(F_2\) = Lateral femoral epicondyle
  • \(F_3\) = distal point between condyles
  • \(F_4\) = Posterior medial femoral condyle
  • \(F_5\) = Posterior lateral femoral conldyle
Calculated location:
  • \(F_6\) = Femoral head center, calcualted by fitting a sphere to the points dititized around the femoral head

Origin: defined as \(F_3\), the distal point between condyles.

The femoral superior-inferior axis (\(z_{SI}\)) points superiorly, and is defined as the unit vector that is parallel with the line that passes through the center of the femoral head (\(F_5\)) and the mid point between the medial and femoral condyles (\(F_3\)).

\[z_{SI} = \frac{F_6 - F_3}{||F_6 - F_3||}\]

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

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

Where \(\bar{EF}\) is a unit vector line that points to the right, and is parallel to the line that passes through the posterior medial (\(F_4\)) and posterior lateral (\(F_5\)) femoral condyles.

Note

For a right knee:

\[\bar{EF} = \frac{F_5 - F_4}{||F_5 - F_4||}\]

For a left knee:

\[\bar{EF} = \frac{F_4 - F_5}{||F_4 - F_5||}\]

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.

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

Tibia (DU)

Five digitized landmarks are reported, and four of these points are used to define the tibia’s fixed coordinate system. This coordinate system is similar to that described by [GS83].

Fiducial locations:
  • \(T_1\) = Medial tibial plateau dwell point
  • \(T_2\) = Lateral tibial plateau dwell point
  • \(T_3\) = Proximal tip of the medial spine of the tibial eminence (Not used in coordinate system definition)
  • \(T_4\) = Center of the tibial eminence
  • \(T_5\) = Center of the distal intramedullary canal

Origin: defined as \(T_4\), the center of the tibial eminence.

The tibial superior-inferior axis (\(z_{SI}\)) points superiorly, and is defined as the unit vector that is parallel with the line that passes through the center of the distal intramedullary canal (\(T_5\)) and the center of the tibial eminence (\(T_4\)).

\[z_{SI} = \frac{T_4 - T_5}{||T_4 - T_5||}\]

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 dwell points.

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 (DU)

Five fiducial points are used to define the patella’s fixed coordinate system. The article [ASC+16] does not state the coordinate system used for the patella, so the patella’s coordinate system is defined using the same procedure as the OpenKnee(s) data.

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 (\(y_{AP}\)).

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

[ASC+16]Azhar A. Ali, Sami S. Shalhoub, Adam J. Cyr, Clare K. Fitzpatrick, Lorin P. Maletsky, Paul J. Rullkoetter, and Kevin B. Shelburne. Validation of predicted patellofemoral mechanics in a finite element model of the healthy and cruciate-deficient knee. Journal of Biomechanics, 49(2):302–309, January 2016. URL: http://www.sciencedirect.com/science/article/pii/S0021929015007265, doi:10.1016/j.jbiomech.2015.12.020.
[GS83](1, 2) 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.