Many different physiological processes affect the growth of malignant lesions and their response to therapy. Each of these processes is spatially and genetically heterogeneous; dynamically evolving in time; controlled by many other physiological processes, and intrinsically random and unpredictable. The objective of this paper is to show that all of these properties of cancer physiology can be treated in a unified, mathematically rigorous way via the theory of random processes. We treat each physiological process as a random function of position and time within a tumor, defining the joint statistics of such functions via the infinite-dimensional characteristic functional. The theory is illustrated by analyzing several models of drug delivery and response of a tumor to therapy. To apply the methodology to precision cancer therapy, we use maximum-likelihood estimation with Emission Computed Tomography (ECT) data to estimate unknown patient-specific physiological parameters, ultimately demonstrating how to predict the probability of tumor control for an individual patient undergoing a proposed therapeutic regimen.

We consider the geometrical optics problem of finding a system of two reflectors that transform a spherical wavefront into a beam of parallel rays with prescribed intensity distribution. Using techniques from optimal transportation theory, it has been shown before that this problem is equivalent to an infinite dimensional linear programming (LP) problem. We investigate techniques for constructing the two reflectors numerically. A straightforward discretization of this problem has the disadvantage that the number of constraints increases rapidly with the mesh size. So with this technique only very coarse meshes are practical. To address this well-known issue we propose an iterative solution scheme. In each step an LP problem is solved. Information from the previous iteration step is used to reduce the number of constraints necessary. As a proof of concept we apply our proposed scheme to solve a problem with synthetic data. We give evidence that the scheme converges. We also show that it allows for much finer meshes than a simple discretization scheme. There exists a growing literature for the application of optimal transportation theory to other beam shaping problems. Our proposed scheme is easy to adapt for these problems as well.

In this paper, we propose Wasserstein Isometric Mapping (Wassmap), a parameter-free nonlinear dimensionality reduction technique that provides solutions to some drawbacks in existing global nonlinear dimensionality reduction algorithms in imaging applications. Wassmap represents images via probability measures in Wasserstein space, then uses pairwise quadratic Wasserstein distances between the associated measures to produce a low-dimensional, approximately isometric embedding. We show that the algorithm is able to exactly recover parameters of some image manifolds including those generated by translations or dilations of a fixed generating measure. Additionally, we show that a discrete version of the algorithm retrieves parameters from manifolds generated from discrete measures by providing a theoretical bridge to transfer recovery results from functional data to discrete data. Testing of the proposed algorithms on various image data manifolds show that Wassmap yields good embeddings compared with other global techniques.