Stochastic models for objects and images in oncology and virology: application to PI3K-Akt-mTOR signaling and COVID-19 disease

Purpose: The goal of this research is to develop innovative methods of acquiring simultaneous multidimensional molecular images of several different physiological random processes (PRPs) that might all be active in a particular disease such as COVID-19. Approach: Our study is part of an ongoing effort at the University of Arizona to derive biologically accurate yet mathematically tractable models of the objects of interest in molecular imaging and of the images they produce. In both cases, the models are fully stochastic, in the sense that they provide ways to estimate any estimable property of the object or image. The mathematical tool we use for images is the characteristic function, which can be calculated if the multivariate probability density function for the image data is known. For objects, which are functions of continuous variables rather than discrete pixels or voxels, the characteristic function becomes infinite dimensional, and we refer to it as the characteristic functional. Results: Several innovative mathematical results are derived, in particular for simultaneous imaging of multiple PRPs. Then the application of these methods to cancers that disrupt the mammalian target of rapamycin signaling pathway and to COVID-19 are discussed qualitatively. One reason for choosing these two problems is that they both involve lipid rafts. Conclusions: We found that it was necessary to employ a new algorithm for energy estimation to do simultaneous single-photon emission computerized tomography imaging of a large number of different tracers. With this caveat, however, we expect to be able to acquire and analyze an unprecedented amount of molecular imaging data for an individual COVID patient.