Cross-Validated Spline Methods for the Estimation of Three-Dimensional Tumor Size Distributions from Observations on Two-Dimensional Cross Sections

Abstract

We study the problem of estimating the distribution of the three-dimensional radiuses of a collection of spheres, given measurements of the two-dimensional radiuses of a sample of planar cross sections. This problem arises in the estimation of the tumor size distribution of spherical microtumors induced in mouse livers following injection of a carcinogen. We first convert this problem to a form suitable for the application of cross-validated spline methods for the solution of ill-posed integral equations given noisy data. Then we develop special numerical techniques that will allow the spline methods to be accurately applied to integral equations like those associated with the present problem. We apply the resulting method to some mouse-liver data. The subject mouse liver has been completely dissected, allowing a rare comparison of the estimate with the “truth.” The statistical properties of the estimate are explored via Monte Carlo methods. The interplay between statistical and numerical analytic methods for problems like this are explored and the use of eigensequence plots for studying “ill posedness” is described.

Key Words:

• Random spheres model
• Stereology
• Cross-validated splines
• Tumor size distribution
• Ill-posed problem