A Review of Mathematical and Computational Methods in Cancer Dynamics

Abstract

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 medicine. 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.

Mathematical modeling of regulatory networks of intracellular processes – Aims and selected methods

“…Each year, new models appear, describing systems that previously have not been dealt with. However, most efforts concentrate on pathways involved in determination of cell fate [64] , cell cycle [3] , response to stress [63] , [89] , carcinogenesis [125] , immune system responses [29] , [76] , or, most recently, micro-RNA-mediated regulation of intracellular processes [61] , [92] , [104] .…”

Bistability and chaotic behaviors in a 4D cancer oncolytic Virotherapy mathematical model: Pspice and FPGA implementations

“…Various mathematical approaches allow the analysis of biological systems. For example, to infer gene expression patterns together with the cell fate trajectory in cancer therapies, and with the objectives to dimensionally reduce the patterns' spaces, pseudotemporal ordering methods are useful [54][55][56][57]. In the dynamical systems point of view, there are some weakness to construct cancer networks, because of lack of enough time series measurements on some cell expressions such as protein oscillations, gene dynamics etc...…”

A Monte Carlo simulation for moving light source in intracavity PDT

“…By comparison, it is a critical part of the study of radiation oncology. While there is a vast literature on x-ray radiation [7][8][9][10] and several commercial software programs to aid physicians in this treatment, there is little similar information for PDT researchers and physicians. In fact, to date there is no commercial software for *gene@simphotek.com, beesonk@simphotek.com; phone 1 973 621-2340; simphotek.com PDT that encompasses all of the three main areas; light propagation, photo-physics, and oxygen.…”