  Cancers are complex adaptive diseases regulated by the nonlinear feedback
systems between genetic instabilities, environmental signals, cellular protein
flows, and gene regulatory networks. Understanding the cybernetics of cancer
requires the integration of information dynamics across multidimensional
spatiotemporal scales, including genetic, transcriptional, metabolic,
proteomic, epigenetic, and multi-cellular networks. However, the time-series
analysis of these complex networks remains vastly absent in cancer research.
With longitudinal screening and time-series analysis of cellular dynamics,
universally observed causal patterns pertaining to dynamical systems, may
self-organize in the signaling or gene expression state-space of cancer
triggering processes. A class of these patterns, strange attractors, may be
mathematical biomarkers of cancer progression. The emergence of intracellular
chaos and chaotic cell population dynamics remains a new paradigm in systems
oncology. As such, chaotic and complex dynamics are discussed as mathematical
hallmarks of cancer cell fate dynamics herein. Given the assumption that
time-resolved single-cell datasets are made available, a survey of
interdisciplinary tools and algorithms from complexity theory, are hereby
reviewed to investigate critical phenomena and chaotic dynamics in cancer
ecosystems. To conclude, the perspective cultivates an intuition for
computational systems oncology in terms of nonlinear dynamics, information
theory, inverse problems and complexity. We highlight the limitations we see in
the area of statistical machine learning but the opportunity at combining it
with the symbolic computational power offered by the mathematical tools
explored.
