Analysis of spatial patterns in tumor imaging data: A method for assessment of tubularity

The spatial distribution of PET-imaged tracer uptake in a tumor mass can be an important prognostic indicator of patient survival. Our own experience with PET FDG imaging of sarcoma has found that the characterization of spatial heterogeneity by the degree of conformity to an ellipsoidal uptake pattern substantially enhances the ability to differentiate high and low risk patient groups. The present work is motivated by a generalization of this approach in which heterogeneity is evaluated in terms of the degree of conformity of the observed uptake pattern to a tubular form. Unlike the ellipsoid, the tubular structure is allowed to have a curvilinear central spine. Also while the uptake contours transverse to the spine are described by an angular function, the functional form is general and not the simple trigonometric form of the ellipse. The tubular model is formulated as a voxel-level description of the observed uptake in a user-defined region of interest. A method based on a regularization criterion with a weighted least squares data-fit and a Laplacian penalty term is developed for estimation. However, because the mapping from the set of high-dimensional parameters that define the tubular form to the measured uptake is non-linear, the regularization criterion has complex non-quadratic elements both in the data-fit and in the penalty. This complicates the analysis. We present an algorithm based on B-spline representation of key functional components in the tubular model and an adaptation of the Gauss-Newton including Levenberg-Marquart line searches for optimization. A key technical aspect of the regularization approach is that it allows for simplified 1-dimensional control over the simultaneous smoothness of a number of functional parameters. Indeed it is feasible to also implement a generalized cross-validation scheme for data-adaptive choice of model smoothness. We present results using real and simulated data.