Jeremy Mason

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

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.

Spatiotemporal progression of metastatic breast cancer: a Markov chain model highlighting the role of early metastatic sites

Background:Cancer cell migration patterns are critical for understanding metastases and clinical evolution. Breast cancer spreads from one organ system to another via hematogenous and lymphatic routes. Although patterns of spread may superficially seem random and unpredictable, we explored the possibility that this is not the case.Aims:Develop a Markov based model of breast cancer progression that has predictive capability.Methods:On the basis of a longitudinal data set of 446 breast cancer patients, we created a Markov chain model of metastasis that describes the probabilities of metastasis occurring at a given anatomic site together with the probability of spread to additional sites. Progression is modeled as a random walk on a directed graph, where nodes represent anatomical sites where tumors can develop.Results:We quantify how survival depends on the location of the first metastatic site for different patient subcategories. In addition, we classify metastatic sites as “sponges” or “spreaders” with implications regarding anatomical pathway prediction and long-term survival. As metastatic tumors to the bone (main spreader) are most prominent, we focus in more detail on differences between groups of patients who form subsequent metastases to the lung as compared with the liver.Conclusions:We have found that spatiotemporal patterns of metastatic spread in breast cancer are neither random nor unpredictable. Furthermore, the novel concept of classifying organ sites as sponges or spreaders may motivate experiments seeking a biological basis for these phenomena and allow us to quantify the potential consequences of therapeutic targeting of sites in the oligometastatic setting and shed light on organotropic aspects of the disease.

Entropy, complexity, and Markov diagrams for random walk cancer models

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.

Adrenal metastases in lung cancer: clinical implications of a mathematical model.

Adrenal gland metastases are common in lung cancer. It is well recognized that aggressive treatment of solitary adrenal metastases leads to improved outcomes but the exact nature of adrenal deposits is not well understood. Controversy exists as to the routing of cancer cells to the adrenal gland with some believing that this transmission is lymphatic, in contrast to the more generally accepted theory of hematogenous spread. Recently published mathematical modeling of cancer progression strongly supports the lymphatic theory. With that in mind, we performed a literature review to look for biological plausibility of simulation results and believe that evidence supports the contention that metastases to the adrenal gland can be routed by means of lymphatic channels. This could explain improved survival for patients in whom solitary adrenal metastases are managed aggressively with surgical or radiation modalities. We are calling for clinical trials prospectively testing this hypothesis.

Markov chain models of cancer metastasis

We describe the use of Markov chain models for the purpose of quantitative forecasting of metastatic cancer progression. Each site (node) in the Markov network (directed graph) is an organ site where a secondary tumor could develop with some probability. The Markov matrix is an N x N matrix where each entry represents a transition probability of the disease progressing from one site to another during the course of the disease. The initial state-vector has a 1 at the position corresponding to the primary tumor, and 0s elsewhere (no initial metastases). The spread of the disease to other sites (metastases) is modeled as a directed random walk on the Markov network, moving from site to site with the estimated transition probabilities obtained from longitudinal data. The stochastic model produces probabilistic predictions of the likelihood of each metastatic pathway and corresponding time sequences obtained from computer Monte Carlo simulations. The main challenge is to empirically estimate the N^2 transition probabilities in the Markov matrix using appropriate longitudinal data.

Abstract 4532: Adrenal metastases in lung cancer: Clinical implications of a mathematical model

Adrenal gland metastases are common in lung cancer. It is well recognized that aggressive treatment of solitary adrenal metastases leads to improved outcomes but the exact nature of adrenal deposits is not well understood. Controversy exists as to the routing of cancer cells to the adrenal gland with some believing that this transmission is lymphatic, in contrast to the more generally accepted theory of hematogenous spread. Using an autopsy dataset of 3827 untreated cancer patients, we use the metastatic distribution of common primary cancer types to create Markov models of progression. The anatomical sites of spread in the body represent states in the model, while the transition probability between states represents the probability of metastatic spread from one site to another. We then use the Markov models to run Monte Carlo simulations of random walkers representing circulating tumor cells traveling within the body to simulate metastatic spread. We calculate mean first passage times (MFPT) to each site as a representative of time to metastasis formation. Analysis of 6 common cancer types (bladder [n=120 autopsies; 289 metastases], breast [n=432; 2235], colorectal [n=161; 420], lung [n=560; 859], ovarian [n=418; 806], and prostate [n=62; 212]) from the dataset showed distinct metastatic distributions across the populations. MFPT calculations to anatomical sites in the 6 analyzed primary cancer types indicated model progression times to the adrenal gland similar to the regional and distal lymph nodes in only lung cancer. The times associated with adrenal gland in the other primary cancers were similar to other metastatic sites of hematogenous spread. The Markov models created strongly support the lymphatic theory for metastatic spread to the adrenal glands. After performing a literature review to look for the biological plausibility of the simulated results, we believe evidence supports this theory and validates the models. This could explain improved survival for patients in whom solitary adrenal metastases are managed aggressively with surgical or radiation modalities. In order to further validate this theory, we are calling for clinical trials prospectively testing this hypothesis. Citation Format: Jeremy M. Mason, AnneMarie Ciccarella, Lori Marx-Rubiner, Lyudmila Bazhenova, Paul K. Newton, Kelly Bethel, Jorge J. Nieva, Peter Kuhn. Adrenal metastases in lung cancer: Clinical implications of a mathematical model [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2017; 2017 Apr 1-5; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2017;77(13 Suppl):Abstract nr 4532. doi:10.1158/1538-7445.AM2017-4532

Abstract 2711: Spatiotemporal progression patterns in metastatic lung cancer treated with bevacizumab

Proceedings: AACR 107th Annual Meeting 2016; April 16-20, 2016; New Orleans, LA We describe a Markov chain mathematical model of lung cancer progression based on a longitudinal dataset of 722 lung cancer patients. Specifically, we investigate the patterns of metastatic spread in lung cancer and how the VEGF inhibitor, bevacizumab, alters these metastatic patterns. Stochastic models were used to simulate metastatic spread by means of random walk processes on directed graphs. We created spatiotemporal diagrams of cancer progression to analyze the differences between patients treated and not treated with bevacizumab. Patients with squamous lung cancer or brain metastases at baseline were ineligible for bevacizumab and excluded from analysis. Our results demonstrated that not only does bevacizumab extend survival, but it also alters patterns of metastatic spread. Patients treated with bevacizumab were characterized by greater heterogeneity in their pathways of metastatic progression, compared to those patients not treated with bevacizumab, which showed more homogeneity in their pathways. Furthermore, we quantify this spread and characterize metastatic sites as ‘spreaders’ and ‘sponges’ based on their probability of spreading the disease. Citation Format: Jeremy M. Mason, Paul K. Newton, Gino K. In, Sonia Lin, Peter Kuhn, Jorge J. Nieva. Spatiotemporal progression patterns in metastatic lung cancer treated with bevacizumab. [abstract]. In: Proceedings of the 107th Annual Meeting of the American Association for Cancer Research; 2016 Apr 16-20; New Orleans, LA. Philadelphia (PA): AACR; Cancer Res 2016;76(14 Suppl):Abstract nr 2711.