Fig 3. Additional procedures required for some erosions in our algorithm.

A. This step may be necessary if the erosion VOI overflows beyond the concavity. a, b) These images represent overflow of the VOI (white arrow) on 2D (a) and 3D (b) images. c, d) The appropriate VOI is created by adding shrinking operations at the step in Fig 1E. B. These steps may be necessary if the erosion VOI connects with bone marrow spaces. a) The primary VOI is connected with the bone marrow spaces. b) The shrinking operation is executed iteratively until the continuity is completely interrupted. c) The dilation operation is executed the same number of times until a return to the original shape is achieved. d) Only the VOI of the erosion is extracted. C. Adjustment of rotation in the axial view may be necessary if the rotation is markedly different in longitudinal analysis. a, c) These images corresponding to the image in Fig 1D are created from the same image. A different rotation causes a slight difference in filling concavities. b, d) The two VOIs of erosions are slightly different due to differences in rotation.