We present a new semi-supervised method for segmenting multiple interrelated object boundaries with spherical
topology in volumetric images. The core of our method is a novel graph-theoretic algorithm that simultaneously
detects multiple surfaces under smoothness, distance, and elasticity constraints. The algorithm computes the
global optimum of an objective function that incorporates boundary, regional and surface elasticity information.
A single straight line drawn by the user in a cross-sectional slice is the sole user input, which roughly indicates
the extent of the object. We employ a multi-seeded Dijkstra-based range competition algorithm to pre-segment
the object on two orthogonal multiplanar reformatted (MPR) planes that pass through the input line. Based
on the 2D pre-segmentation results, we estimate the object and background intensity histograms, and employ
an adaptive mean-shift mode-seeking process on the object histogram to automatically determine the number of
surface layers to be segmented. The final multiple-surface segmentation is performed in an ellipsoidal coordinate
frame constructed by an automated ellipsoid fitting procedure. We apply our method to the segmentation of
liver lesions with necrosis or calcification, and various other tumors in CT images. For liver tumor segmentation,
our method can simultaneously delineate both tumor and necrosis boundaries. This capability is unprecedented
and is valuable for cancer diagnosis, treatment planning, and evaluation.