title: Entropy, complexity, and Markov diagrams for random walk cancer models.
abstract: The notion of entropy is used to compare the complexity associated with 12 common cancers based on metastatic tumor distribution autopsy data. We characterize power-law distributions, entropy, and Kullback-Liebler divergence associated with each primary cancer as compared with data for all cancer types aggregated. We then correlate entropy values with other measures of complexity associated with Markov chain dynamical systems models of progression. The Markov transition matrix associated with each cancer is associated with a directed graph model where nodes are anatomical locations where a metastatic tumor could develop, and edge weightings are transition probabilities of progression from site to site. The steady-state distribution corresponds to the autopsy data distribution. Entropy correlates well with the overall complexity of the reduced directed graph structure for each cancer and with a measure of systemic interconnectedness of the graph, called graph conductance. The models suggest that grouping cancers according to their entropy values, with skin, breast, kidney, and lung cancers being prototypical high entropy cancers, stomach, uterine, pancreatic and ovarian being mid-level entropy cancers, and colorectal, cervical, bladder, and prostate cancers being prototypical low entropy cancers, provides a potentially useful framework for viewing metastatic cancer in terms of predictability, complexity, and metastatic potential.
authors.affiliation: Viterbi School of Engineering, Department of Mathematics, and Norris Comprehensive Cancer Center, University of Southern California, Los Angeles, CA 90089-1191, USA.
authors.forename: Paul K
authors.initials: PK
authors.lastname: Newton
authors.affiliation: Viterbi School of Engineering, Department of Mathematics, and Norris Comprehensive Cancer Center, University of Southern California, Los Angeles, CA 90089-1191, USA.
authors.forename: Jeremy
authors.initials: J
authors.lastname: Mason
authors.affiliation: Viterbi School of Engineering, Department of Mathematics, and Norris Comprehensive Cancer Center, University of Southern California, Los Angeles, CA 90089-1191, USA.
authors.forename: Brian
authors.initials: B
authors.lastname: Hurt
authors.affiliation: Scripps Clinic Medical Group, 10666 N. Torrey Pines Rd. MC 211C, La Jolla CA 92037.
authors.forename: Kelly
authors.initials: K
authors.lastname: Bethel
authors.affiliation: UCSD Moores Cancer Center, 3855 Health Sciences Drive, La Jolla, CA, 92093.
authors.forename: Lyudmila
authors.initials: L
authors.lastname: Bazhenova
authors.affiliation: Keck School of Medicine, University of Southern California, 1441 Eastlake Avenue Suite 3440, Los Angeles, CA 90033.
authors.forename: Jorge
authors.initials: J
authors.lastname: Nieva
authors.affiliation: Department of Biological Sciences, University of Southern California, 3430 S. Vermont Ave, Suite 105, Los Angeles CA 90089-3301.
authors.forename: Peter
authors.initials: P
authors.lastname: Kuhn
mesh_terms: D019277:Entropy; D006801:Humans; D008390:Markov Chains; D008954:Models, Biological; D009369:Neoplasms / Q000378:metabolism*