One application of positron emission tomography (PET), a nuclear imaging technique, in neuroscience involves in vivo estimation of the density of various proteins (often, neuroreceptors) in the brain. PET scanning begins with the injection of a radiolabeled tracer that binds preferentially to the target protein; tracer molecules are then continuously delivered to the brain via the bloodstream. By detecting the radioactive decay of the tracer over time, dynamic PET data are constructed to reflect the concentration of the target protein in the brain at each time. The fundamental problem in the analysis of dynamic PET data involves estimating the impulse response function (IRF), which is necessary for describing the binding behavior of the injected radiotracer. Virtually all existing methods have three common aspects: summarizing the entire IRF with a single scalar measure; modeling each subject separately; and the imposition of parametric restrictions on the IRF. In contrast, we propose a functional data analytic approach that regards each subject's IRF as the basic analysis unit, models multiple subjects simultaneously, and estimates the IRF nonparametrically. We pose our model as a linear mixed effect model in which population level fixed effects and subject-specific random effects are expanded using a B-spline basis. Shrinkage and roughness penalties are incorporated in the model to enforce identifiability and smoothness of the estimated curves, respectively, while monotonicity and non-negativity constraints impose biological information on estimates. We illustrate this approach by applying it to clinical PET data with subjects belonging to three diagnosic groups. We explore differences among groups by means of pointwise confidence intervals of the estimated mean curves based on bootstrap samples.
Keywords: Bootstrap; Constrained estimation; Function-on-scalar regression; Nonparametric; Splines.