Long time existence of solutions to an elastic flow of networks

The L2-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods

Existence and uniqueness of the motion by curvature of regular networks

We prove existence and uniqueness of the motion by curvature of networks with triple junctions in Rd when the initial datum is of class Wp2-2/p and the unit tangent vectors to the concurring curves form angles of 120 degrees. Moreover, we investigate the regularisation effect due to the parabolic nature of the system. An application of the well-posedness is a new proof and a generalisation of the long-time behaviour result derived by Mantegazza et al. in 2004. Our study is motivated by an open question proposed in the 2016 survey from Mantegazza et al.: does there exist a unique solution of the motion by curvature of networks with initial datum being a regular network of class C 2? We give a positive answer

Riparian ecosystems in human cancers

Intratumoral evolution produces extensive genetic heterogeneity in clinical cancers. This is generally attributed to an increased mutation rate that continually produces new genetically defined clonal lineages. Equally important are the interactions between the heritable traits of cancer cells and their microenvironment that produces natural selection favoring some clonal ‘species’ over others. That is, while mutations produce the heritable variation, environmental selection and cellular adaptation govern the strategies (and genotypes) that can proliferate within the tumor ecosystem. Here we ask: What are the dominant evolutionary forces in the cancer ecosystem? We propose that the tumor vascular network is a common and primary cause of intratumoral heterogeneity. Specifically, variations in blood flow result in variability in substrate, such as oxygen, and metabolites, such as acid, that serve as critical, but predictable, environmental selection forces. We examine the evolutionary and ecological consequences of variable blood flow by drawing an analogy to riparian habitats within desert landscapes. We propose that the phenotypic properties of cancer cells will exhibit predictable spatial variation within tumor phenotypes as a result of proximity to blood flow. Just as rivers in the desert create an abrupt shift from the lush, mesic riparian vegetation along the banks to sparser, xeric and dry-adapted plant species in the adjacent drylands, we expect blood vessels within tumors to promote similarly distinct communities of cancer cells that change abruptly with distance from the blood vessel. We propose vascular density and blood flow within a tumor as a primary evolutionary force governing variations in the phenotypic properties of cancer cells thus providing a unifying ecological framework for understanding intratumoral heterogeneity