Automatic Quantification of Congruity from Knee MRI - Computational Biomechanics for Medicine - page 50

2.3 Contact Area Estimation

The second step in the quantification framework is estimating the region where the
inferior surface of femoral compartment and the superior surface of the tibial
compartment are in contact, which we defined as the CA. The surfaces of the tibial
and femoral cartilages in the CA (tibial and femoral contact surfaces) are estimated
by employing the Euclidean binary distance transform ( DT ). In this, a kd-tree
implementation was used for faster computation. The tibial contact surface is
located by estimating the tibial cartilage voxels that are less than the voxel width
( vw ) from the femoral cartilage, which is denoted as TibProx . The femoral contact
surface is located by estimating the femoral cartilage voxels that are less than a
voxel width from tibial cartilage that is denoted as FemProx . Let t be a voxel
location in TibProx and f be a voxel location in FemProx . So, t ( t ! [ t x , t y , t z ]) and f
( f ! [ f x , f y , f z ]) are the 3D scan coordinates in the segmentation in sagittal, coronal,
and axial direction, respectively. The estimation of TibProx and FemProx is
expressed mathematically as:

TibProx ¼ft 2 tibjDTðt ; femÞ< vwg

FemProx ¼ff 2 femjDTðf ; tibÞ< vwg;

where fem is the femoral cartilage.
Therefore, ideally, in the CA, any voxel location of the TibProx could be
expressed as a function TibProx(FemProx(t)) , where tib is the tibial cartilage, i.e.,
for every location on the TibProx , there is either one or more corresponding
locations on the FemProx at a voxel width apart and vice versa. See Fig. 2 ato
visualize the location of CA in a healthy MTF joint.

2.4 Congruity Computation

The local congruence of the tibiofemoral joint is calculated by computing the first
and second order features on TibProx and FemProx at every voxel location and
associating them. If TibProx and FemProx are locally congruent, then the first and
second order features will match well.

2.4.1 First Order Features

Let n t be the normal vector at t in the TibProx and n f be the normal vector at the
corresponding voxel location f in the FemProx . Mathematically, these normal
vectors are compu te d from first order partial derivatives of the level set representa-
tion . Therefore, n t ¼ rf t

jrf t j

f , where ∇f t is the gradient j ∇ f t j and is the gradient

magnitude. Similarly, for n f .