TableOfContents

Target Outcome

The overall goal of this specification is to provide coordinate systems for each registration marker sets on femur, tibia, and patella with the intention to allow transformation between joint testing and anatomical imaging coordinate systems. In this regard, the goals are

Prerequisities

Infrastructure

Previous Protocols

Procedures

Obtaining Location of Registration Marker Centers

The registration markers are spherical objects. The boundaries of these spheres will be digitized from magnetic resonance imaging and during joint testing using a digitizing probe, see Previous Protocols above. More than 10 points are expected to be acquired for each registration marker for a given experimentation modality. The location of a given registration marker will be obtained from a sphere fit to the locations of these points. A Python script will allow the following input/output relationship in the coordinate system of measured points.

input

coordinates of n locations on a given registration marker (n > 10)

output

center and radius of sphere (x, y, z, r) that best represents the spherical distribution of n locations

Transformation Matrices

Femur

It is assumed that all the registration marker locations described below are obtained following the protocol described above. Variables to establish the transformation matrix between femur and joint testing coordinate systems are:

The transformation matrix between femur and joint testing coordinate system can be obtained as:

\begin{displaymath}
\begin{array}{l}
o_F = p_1^O \\
u_F = \frac {p_2^O - o_F } {||p_2^O - o_F||} \\
v_F = u_F \times \frac {p_3^0 - o_F} {||p_3^O - o_F||} \\
w_F = u_F \times v_F \\
T_{OF} = \left [ \begin{array}{cccc} u_F & v_F & w_F & o_F \\ 0 & 0 & 0 & 1 \end{array} \right]
\end{array}
\end{displaymath}

Variables to establish the transformation matrix between femur and imaging coordinate systems are:

The transformation matrix between femur and imaging coordinate system can be obtained by following the steps already outlined above.

Tibia

It is assumed that all the registration marker locations described below are obtained following the protocol described above. Variables to establish the transformation matrix between tibia and joint testing coordinate systems are:

The transformation matrix between tibia and joint testing coordinate system can be obtained through similar calculations conducted for femur above. Variables to establish the transformation matrix between tibia and imaging coordinate systems are:

The transformation matrix between tibia and imaging coordinate system can be obtained by following the steps outlined for the femur, see above.

Patella

The following information should be available from CAD drawing of the patella registration marker assembly:

The location of divot coordinates should be measured during specimen preparation:

Singular value decomposition, e.g. Söderkvist and Wedin (1993), using divot coordinates measured in two coordinate systems can provide the following transformation matrix:

Using center of patella registration markers with this transformation matrix one can obtain:

The transformation matrix between patella and joint testing coordinate system can be obtained by following the steps outlined for the femur, see above.

Variables to establish the transformation matrix between patella and imaging coordinate systems are:

The transformation matrix between patella and imaging coordinate system can be obtained by following the steps outlined for the femur, see above.

References

Söderkvist I, Wedin PA. Determining the movements of the skeleton using well-configured markers. J Biomech. 1993 Dec;26(12):1473-7. [http://www.ncbi.nlm.nih.gov/pubmed/8308052 PubMed]