Functional principal component analyses of biomedical images as outcome measures

Medical imaging data are often valuable in evaluating disease and therapeutic effects. However, in formal assessment of treatment efficacy, it is usual to discard most of the rich information within the image, instead relying on simple summary measures. This reflects the absence of satisfactory statistical tools for the description and analysis of variability between images. We present extended techniques of functional data analysis applied to distributions of variable values extracted from specified regions within images, which are used to produce displays of 'principal densities' that allow interpretation of principal modes of variation in terms of features in the distributions of the voxel values. These techniques are especially relevant in circumstances where the spatial distribution of variables within the specified region is not of interest. Tumours, for example, are disorganized in nature and may change shape rapidly so it is not possible, even in principle, to create a 1–1 correspondence between images before and post treatment. The techniques that are introduced here, however, enable us to distinguish differences between pretreatment and post-treatment densities. These methods are essentially exploratory; hence we develop a permutation test providing more formal assessment of differences of treatment, which assesses the changes within dose group. Extensions to multivariate images of two or more variables are also illustrated and we show that the methodology makes bivariate functional data just as easy to handle as univariate data.