Coordinate Systems

The way that the femoral and tibial coordinate systems were calculated was changed, however workflow to define registration between the experimental and MR image coordinate systems was not changed. It was verified that the previous workflow (documented under the Tibiofemoral Model > Registration section in the Model Calibration phase) for defining the digitized points around the registration fiducials yielded the same results as the given data. Therefore, the previous workflow was used instead of importing point coordinates from the following files:

  • data-MB-oks003/RECALIBRATION-Probed_Points_Files/Fem_RM_lateral.xyz

  • data-MB-oks003/RECALIBRATION-Probed_Points_Files/Fem_RM_medial.xyz

  • data-MB-oks003/RECALIBRATION-Probed_Points_Files/Fem_RM_posterior.xyz

  • data-MB-oks003/RECALIBRATION-Probed_Points_Files/Tib_RM_lateral.xyz

  • data-MB-oks003/RECALIBRATION-Probed_Points_Files/Tib_RM_medial.xyz

  • data-MB-oks003/RECALIBRATION-Probed_Points_Files/Tib_RM_posterior.xyz

Fixed Femoral Coordinate System

The femur’s fixed coordinate system was calculated using the digitized landmark coordinates reported in the file data-MB-oks003/RECALIBRATION-Probed_Points_Files/Fem_AL.xyz. The methods described on pg. 5 of data-MB-oks003/DESCRIPTION-DataRepresentation_OpenKnees.docx were used to define the femur’s fixed coordinate system in the optotrak sensor’s global coordinate system (\(T_{Sensor1\_Fem}\)).

\[\begin{split}T_{Sensor1\_FEM} = \begin{bmatrix} F_x^0 & F_y^0 & F_z^0 & O_F^0 \\ F_x^1 & F_y^1 & F_z^1 & O_F^1 \\ F_x^2 & F_y^2 & F_z^2 & O_F^2 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

Where \(F_x^i\), \(F_y^i\), \(F_z^i\), and \(O_F^i\) are defined on pg. 5 in data-MB-oks003/DESCRIPTION-DataRepresentation_OpenKnees.docx. The superscripts in the above equation indicate the vector’s index.

Next the transform from the MR image’s global coordinate system to the femur’s fixed coordinate system in the MR image (\(T_{Im\_Sensor1}\)) was defined. \(T_{Im\_Sensor1}\) (which is determined using methods described in the Tibiofemoral Model > Registration section in the Model Calibration phase) is multiplied by \(T_{Sensor1\_Fem}\)

\[T_{Im\_Fem} = T_{Im\_Sensor1}T_{Sensor1\_Fem}\]

\(T_{Im\_Fem}\) can be used to define the femur’s fixed coordinate system in the knee model using the SimVitro convention, where the positive direction on the anterior-posterior axis points posteriorly. To match the convention used by the CSU lab, the fixed femoral coordinate system defined by \(T_{Im\_Fem}\) was rotated 180 degrees about the fixed femur’s z-axis. This defined the positive direction along the anterior-posterior axis as pointing anteriorly.

\[\begin{split}S_{Im\_Fem} = T_{Im\_Fem}*\begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

The last row in \(S_{Im\_Fem}\) is neglected, and the columns are used to define the axes of the fixed femoral coordinate system and its origin.

\[\begin{split}\begin{split} x_{axis} &= [S_{Im\_Fem}(0,0), S_{Im\_Fem}(1,0), S_{Im\_Fem}(2,0)]\\ y_{axis} &= [S_{Im\_Fem}(0,1), S_{Im\_Fem}(1,1), S_{Im\_Fem}(2,1)]\\ z_{axis} &= [S_{Im\_Fem}(0,2), S_{Im\_Fem}(1,2), S_{Im\_Fem}(2,2)]\\ origin &= [S_{Im\_Fem}(0,3), S_{Im\_Fem}(1,3), S_{Im\_Fem}(2,3)]\\ \end{split}\end{split}\]

Fixed Tibial Coordinate System

The tibia’s fixed coordinate system was calculated using the digitized landmark coordinates reported in the file data-MB-oks003/RECALIBRATION-Probed_Points_Files/Fem_AL.xyz. The methods described on pg. 4-5 of data-MB-oks003/DESCRIPTION-DataRepresentation_OpenKnees.docx were used to define the tibia’s fixed coordinate system in the optotrak sensor’s global coordinate system (\(T_{Sensor2\_Tib}\)).

\[\begin{split}T_{Sensor2\_Tib} = \begin{bmatrix} T_x^0 & T_y^0 & T_z^0 & O_T^0 \\ T_x^1 & T_y^1 & T_z^1 & O_T^1 \\ T_x^2 & T_y^2 & T_z^2 & O_T^2 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

Where \(T_x^i\), \(T_y^i\), \(T_z^i\), and \(O_T^i\) are defined on pg. 4-5 in data-MB-oks003/DESCRIPTION-DataRepresentation_OpenKnees.docx. The superscripts in the above equation indicate the vector’s index.

Next the transform from the MR image’s global coordinate system to the tibia’s fixed coordinate system in the MR image (\(T_{Im\_Sensor2}\)) was defined. \(T_{Im\_Sensor2}\) (which is determined using methods described in the Tibiofemoral Model > Registration section in the Model Calibration phase) is multiplied by \(T_{Sensor2\_Tib}\)

\[T_{Im\_Tib} = T_{Im\_Sensor2}T_{Sensor2\_Tib}\]

\(T_{Im\_Tib}\) can be used to define the tibia’s fixed coordinate system in the knee model using the SimVitro convention, where the positive direction on the anterior-posterior axis points posteriorly. To match the convention used by the CSU lab, the fixed tibial coordinate system defined by \(T_{Im\_Tib}\) is rotated 180 degrees about the fixed tibia’s z-axis. This defines the positive direction along the anterior-posterior axis as pointing anteriorly.

\[\begin{split}S_{Im\_Tib} = T_{Im\_RB2}*\begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

The last row in \(S_{Im\_Tib}\) is neglected, and the columns are used to define the axes of the fixed tibial coordinate system and its origin.

\[\begin{split}\begin{split} x_{axis} &= [S_{Im\_Tib}(0,0), S_{Im\_Tib}(1,0), S_{Im\_Tib}(2,0)]\\ y_{axis} &= [S_{Im\_Tib}(0,1), S_{Im\_Tib}(1,1), S_{Im\_Tib}(2,1)]\\ z_{axis} &= [S_{Im\_Tib}(0,2), S_{Im\_Tib}(1,2), S_{Im\_Tib}(2,2)]\\ origin &= [S_{Im\_Tib}(0,3), S_{Im\_Tib}(1,3), S_{Im\_Tib}(2,3)]\\ \end{split}\end{split}\]