Spheroids—three-dimensional aggregates of cells grown from a cancer cell line—represent a model of living tissue for chemotherapy investigation. Distribution of chemotherapeutics in spheroid sections was determined using the matrix-assisted laser desorption/ionization mass spectrometry imaging (MALDI MSI). Proliferating or apoptotic cells were immunohistochemically labeled and visualized by laser scanning confocal fluorescence microscopy (LSCM). Drug efficacy was evaluated by comparing coregistered MALDI MSI and LSCM data of drug-treated spheroids with LSCM only data of untreated control spheroids. We developed a fiducial-based workflow for coregistration of low-resolution MALDI MS with high-resolution LSCM images. To allow comparison of drug and cell distribution between the drug-treated and untreated spheroids of different shapes or diameters, we introduced a common diffusion-related coordinate, the distance from the spheroid boundary. In a procedure referred to as “peeling”, we correlated average drug distribution at a certain distance with the average reduction in the affected cells between the untreated and the treated spheroids. This novel approach makes it possible to differentiate between peripheral cells that died due to therapy and the innermost cells which died naturally. Two novel algorithms—for MALDI MS image denoising and for weighting of MALDI MSI and LSCM data by the presence of cell nuclei—are also presented.

In this paper we focus on the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a larger class of Poisson point processes.

We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino–Schoen/Chrusciel–Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ‘Morawetz vector field’ to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino–Schoen/Chrusciel–Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows larger classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino–Schoen/Chrusciel–Delay space-times.

The focus of our attention is the limit distribution of the sum of independent and identically distributed random vectors from which all the extreme summands are removed. The problem is rather trivial if the summands are ordered by their norms. It is of much more interest when the vertices of the convex hull generated by the vectors are taken as the extremes.

The calculation of probabilities on pedigrees of arbitrary complexity is discussed for a basic model of transmission and penetrance (encompassing Mendelian inheritance, and certain environmental influences).

The structure of pedigrees, and the types of loops occurring, is discussed. Some results in graph theory are obtained and, using these, a recurrence relation derived for certain probabilities. The recursive procedure enables the successive peeling off of certain members of the pedigree, and the condensation of the information on those individuals into a function on a subset of those remaining. The underlying theory is set out, and examples given of the utilization of the resulting algorithm.