A Stochastic Markov Chain Model to Describe Lung Cancer Growth and Metastasis

Department of Aerospace & Mechanical Engineering and Department of Mathematics, University of Southern California, Los Angeles, California, United States of America.
PLoS ONE (Impact Factor: 3.23). 04/2012; 7(4):e34637. DOI: 10.1371/journal.pone.0034637
Source: PubMed


A stochastic Markov chain model for metastatic progression is developed for primary lung cancer based on a network construction of metastatic sites with dynamics modeled as an ensemble of random walkers on the network. We calculate a transition matrix, with entries (transition probabilities) interpreted as random variables, and use it to construct a circular bi-directional network of primary and metastatic locations based on postmortem tissue analysis of 3827 autopsies on untreated patients documenting all primary tumor locations and metastatic sites from this population. The resulting 50 potential metastatic sites are connected by directed edges with distributed weightings, where the site connections and weightings are obtained by calculating the entries of an ensemble of transition matrices so that the steady-state distribution obtained from the long-time limit of the Markov chain dynamical system corresponds to the ensemble metastatic distribution obtained from the autopsy data set. We condition our search for a transition matrix on an initial distribution of metastatic tumors obtained from the data set. Through an iterative numerical search procedure, we adjust the entries of a sequence of approximations until a transition matrix with the correct steady-state is found (up to a numerical threshold). Since this constrained linear optimization problem is underdetermined, we characterize the statistical variance of the ensemble of transition matrices calculated using the means and variances of their singular value distributions as a diagnostic tool. We interpret the ensemble averaged transition probabilities as (approximately) normally distributed random variables. The model allows us to simulate and quantify disease progression pathways and timescales of progression from the lung position to other sites and we highlight several key findings based on the model.

Download full-text


Available from: Paul K Newton,
  • Source
    • "While we learn about the population-level propensity and temporal dynamics of spread from the models of Chen [45] and Newton [46] [47], what is lacking is a framework by which these models could be applied to an individual patient. "
    [Show abstract] [Hide abstract]
    ABSTRACT: 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 competition 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.
  • [Show abstract] [Hide abstract]
    ABSTRACT: In breast cancer, mortality is driven by the metastatic process, whereby some cancer cells leave their primary site of origin and travel to distant vital organs. Despite improved screening and therapies to treat breast cancers, metastasis continues to undermine these advances. The pervasive albatross of metastasis necessitates improved prevention and treatment of metastasis. To this end, clinicians routinely employ post-operative or adjuvant therapy to decrease the risk of future metastasis and improve the chance for cure. This article evaluates the limitations of breast cancer therapies within the context of growth curves, and in doing so, provides new insight into the metastatic process as well as more effective means for therapeutic delivery. Two critical developments evolve from this mathematical analysis: first, the use of dose dense chemotherapy to improve survival among breast cancer patients; and second, the theory of self-seeding, which fundamentally changes our understanding of metastasis and the trajectory of drug development.
    Journal of Mammary Gland Biology and Neoplasia 09/2012; 17(3). DOI:10.1007/s10911-012-9267-z · 4.53 Impact Factor
  • Source

    10/2012; 4(5):444-5. DOI:10.3978/j.issn.2072-1439.2012.08.13
Show more