Tibiofemoral Coordinate System - Connectors

The fixed femoral and tibial Coordinate Systems are used to define the connector elements that make up the tibiofemoral coordinate system in the FE model.

Nodes

Four nodes are used to define the three connector elements that compuse the tibiofemoral coordinate system.

  • Node 1: The origin of the tibia’s fixed coordinate system
  • Node 2: The origin of the femur’s fixed coordinate system
  • Node 3: The point where the AP axis intersects the SI axis
  • Node 4: The point there the AP axis intersects the ML axis
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Fig. 46 The nodes that are used to define the connector elements.

Referring to Fig. 46, and using the line equation (\(p_1 = p_0 + t\bar{v}\)), the points that lie on the AP axis can be written as (noting that \(\bar{v_1}\), \(\bar{v_2}\), and \(\bar{v_3}\) are unit vectors that are parallel with the \(x_{ML}\), \(y_{AP}\) and \(z_{SI}\) axes respectively)

\[\begin{split}\begin{split} P_4 =& P_2 + t_2\bar{v_2} \\ P_3 =& P_4 + t_3\bar{v_3} \\ =& P_2 + t_2\bar{v_2} + t_3\bar{v_3} \\ \end{split}\end{split}\]

Recognizing that \(P_3\) can also be written as \(P_3 = P_1 + t_1\bar{v_1}\), the following expression can be written

\[\begin{split}\begin{split} P_3 =& P_1 + t_1\bar{v_1} \\ =& P_2 + t_2\bar{v_2} + t_3\bar{v_3} \\ P_1 + t_1\bar{v_1} =& P_2 + t_2\bar{v_2} + t_3\bar{v_3} \\ \end{split}\end{split}\]

Rearranging the above equation,

\[P_1 - P_2 = -t_1\bar{v_1} + t_2\bar{v_2} + t_3\bar{v_3}\]

Recognizing that the only unknown values in the above equation are \(t_1\), \(t_2\), and \(t_3\), the above equation can be expressed in matrix form

\[\begin{split}\begin{bmatrix} -v_{11} & v_{21} & v_{31} \\ -v_{12} & v_{22} & v_{32} \\ -v_{13} & v_{23} & v_{33} \end{bmatrix} \begin{bmatrix} t_1 \\ t_2 \\ t_3 \end{bmatrix} = \begin{bmatrix} P_{11} - P_{21} \\ P_{12} - P_{22} \\ P_{13} - P_{23} \end{bmatrix}\end{split}\]

where \(v_{ij}\) refers to the j index in unit vector i, and \(P_{kl}\) refers to the l index in point number k.

The values for \(t_i\) can be solved for in the above equation, and

\[\begin{split}\begin{split} P_3 =& P_1 + t_1\bar{v_1} \\ P_4 =& P_2 + t_1\bar{v_2} \\ \end{split}\end{split}\]

Elements

The nodes are used to define the elements in the FE model. These elements are used to apply or measure kinematics, and apply external loads. The order that the nodes are defined in the Abaqus input file defines the positive direction of the element. A minimum of three elements are needed, however three additional elements are added to keep consistency in kinematic and load descriptions between right and left knees. For example, translation along the \(y_{AP}\) axis has the same sign between left and right knees, however the same rotation about the \(y_{AP}\) (varus) has an opposite sign between left and right knees.

The specific ordering of the nodes will change between right and left knees, but the elements will contain the following nodes:

  • Element 1: node 2 and node 4 - medial tibial translation and force
  • Element 2: node 4 and node 3 - anterior tibial translation and force
  • Element 3: node 1 and node 3 - superior tibial translation and force
  • Element 4: node 2 and node 4 - extension rotation and moment
  • Element 5: node 4 and node 3 - varus rotation and moment
  • Element 6: node 1 and node 3 - internal tibial rotation and moment

Ties

Some of the nodes are tied to the femur and tibia geometry

  • Node 1: Tied to the tibia
    • The node that is at the tibia’s fixed coordinate system origin is rigidly tied to the tibia bone geometry
  • Node 2: Tied to the femur
    • The node that is at the femur’s fixed coordinate system origin is rigidly tied to the femur bone geometry