Research into mechanisms of hematogenous metastasis has largely become genetic in focus, attempting to understand the molecular basis of `seed-soil relationships. Preceeding this biological mechanism is the physical process of dissemination of circul
ating tumour cells (CTCs). We utilize a `filter-flow paradigm to show that assumptions about CTC dynamics strongly affect metastatic efficiency: without data on CTC dynamics, any attempt to predict metastatic spread in individual patients is impossible.
Mathematical modeling in cancer has been growing in popularity and impact since its inception in 1932. The first theoretical mathematical modeling in cancer research was focused on understanding tumor growth laws and has grown to include the competit
ion between healthy and normal tissue, carcinogenesis, therapy and metastasis. It is the latter topic, metastasis, on which we will focus this short review, specifically discussing various computational and mathematical models of different portions of the metastatic process, including: the emergence of the metastatic phenotype, the timing and size distribution of metastases, the factors that influence the dormancy of micrometastases and patterns of spread from a given primary tumor.
The evolutionary process has been modelled in many ways using both stochastic and deterministic models. We develop an algebraic model of evolution in a population of asexually reproducing organisms in which we represent a stochastic walk in phenotype
space, constrained to the edges of an underlying graph representing the genotype, with a time-homogeneous Markov Chain. We show its equivalence to a more standard, explicit stochastic model and show the algebraic models superiority in computational efficiency. Because of this increase in efficiency, we offer the ability to simulate the evolution of much larger populations in more realistic genotype spaces. Further, we show how the algebraic properties of the Markov Chain model can give insight into the evolutionary process and allow for analysis using familiar linear algebraic methods.
This paper presents improved approximation algorithms for the problem of multiprocessor scheduling under uncertainty, or SUU, in which the execution of each job may fail probabilistically. This problem is motivated by the increasing use of distribute
d computing to handle large, computationally intensive tasks. In the SUU problem we are given n unit-length jobs and m machines, a directed acyclic graph G of precedence constraints among jobs, and unrelated failure probabilities q_{ij} for each job j when executed on machine i for a single timestep. Our goal is to find a schedule that minimizes the expected makespan, which is the expected time at which all jobs complete. Lin and Rajaraman gave the first approximations for this NP-hard problem for the special cases of independent jobs, precedence constraints forming disjoint chains, and precedence constraints forming trees. In this paper, we present asymptotically better approximation algorithms. In particular, we give an O(loglog min(m,n))-approximation for independent jobs (improving on the previously best O(log n)-approximation). We also give an O(log(n+m) loglog min(m,n))-approximation algorithm for precedence constraints that form disjoint chains (improving on the previously best O(log(n)log(m)log(n+m)/loglog(n+m))-approximation by a (log n/loglog n)^2 factor when n = poly(m). Our algorithm for precedence constraints forming chains can also be used as a component for precedence constraints forming trees, yielding a similar improvement over the previously best algorithms for trees.
