Abstract
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.
Highlights
The identification of circulating tumor cells (CTCs) in the human circulatory system dates back to Ashworth’s 1869 paper [1] in which he identified and pointed out the potential significance of cells similar to those found in the primary tumor of a deceased cancer victim
There has been sporadic focus on CTCs as a key diagnostic tool in the fight against cancer, based mostly on the so-called ‘seed and soil’ hypothesis [2,3,4] of cancer metastasis, in which the CTCs play the role of seeds which detach from the primary tumor, disperse through the bloodstream, and get trapped at various distant sites, if conditions are favorable, extravasate, form metastases, and subsequently colonize
The computational model we develop and discuss in this paper is an ensemble based Markov chain/random walk model of disease progression in which we use a stochastic transition matrix with entries that are normally distributed
Summary
The identification of circulating tumor cells (CTCs) in the human circulatory system dates back to Ashworth’s 1869 paper [1] in which he identified and pointed out the potential significance of cells similar to those found in the primary tumor of a deceased cancer victim. The metastatic sites offer the soil for potential subsequent growth of secondary tumors. Paget’s 1889 seed-and-soil hypothesis [3] asserts that the development of secondary tumors is not due to chance alone, but depends on detailed interactions, or cross-talk, between select cancer cells and specific organ microenvironments. While the seed-and-soil hypothesis remains a bedrock theory in cancer research, it has been significantly refined over the years to incorporate our current level of understanding on how the ability for a tumor cell to mestastasize depends on its complex interactions with the homeostatic factors that promote tumor cell growth, cell survival, angiogenisis, invasion, and metastastasis [2]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
