Increasing spacing between vessels slows tumour growth and creates areas of necrosis.

<p>Tumours are grown in equally sized, regularly vascularised domains with decreasing vessel density from boxes 1 to 5 (from top left: Θ = 0.0072, 0.0041, 0.0033, 0.0025, 0.0013). All plots show the automaton state at the final time point at time <i>t</i> ≈ 190 days. Only the smallest vessel density (0.0013 in this figure) entirely constrains growth. Growth rate over the first ≈190 days is summarized in the lower right panel.</p

Homogeneous and heterogeneous vessel patterns with same density have very different carrying capacity and cellular oxygen distributions.

<p>We plot the equilibrium cellular-oxygen distributions and spatial oxygen distributions from the minimum and maximum cellularity examples from two representative families (24 and 54 vessels per domain) of simulations. We see nearly 20% changes in carrying capacity in favour of the more homogeneous distributions in both cases, and while the second and third moments of the distributions of oxygen distribution change in the same direction, the magnitude of the changes are highly varied from the low to high density cases.</p

Varying the vascular density with regular spacing affects the carrying capacity and cellular-oxygen distribution in normal tissue.

<p>We plot healthy tissue growth and maintenance as we vary the vessel density (decreasing density from top to bottom Θ = 0.0018, 0.0024 and 0.0032). We plot cellular distributions (left) with associated spatial oxygen concentration (middle) and non-spatial distribution of cells versus oxygen concentration (right). These plots represent the system at dynamic equilibrium, in which cell death and birth is balanced across the tissue. The mean of the cellular-oxygen distribution decreases with vessel density (0.26, 0.18 and 0.16, top to bottom) while the standard deviation and skewness stay approximately constant (std = 0.09, 0.1 and 0.1, skewness = 3.18, 3.23 and 3.32). Domains are of size 100 × 100.</p

Surviving cells versus vessel density for all simulations.

<p>We plot the number of surviving cells after 2Gy of simulated radiation in each simulation as calculated using <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004712#pcbi.1004712.e005" target="_blank">Eq (5)</a> modified by the OER from Eqs (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004712#pcbi.1004712.e006" target="_blank">6</a>) and (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004712#pcbi.1004712.e007" target="_blank">7</a>) versus the number of vessels in each case for each of the 500 simulations with constant vessel number, but random placement, on domain size 73 × 73 at dynamic equilibrium. The edges of the boxes represent the 25th and 75th percentiles, the whiskers extend to the most extreme data points not considered outliers. Outliers are defined as any simulation outside approximately 2.7 standard deviations, and they are plotted as red crosses. Inset we plot the standard deviation for each family of simulations versus the vessel number.</p

Schematic of discrete time updating algorithm for the HCA.

<p>At each cellular automaton update, each cell in the domain undergoes a series of fate decisions based on intrinsic cell parameters and microenvironmental cues.</p

Varying vascular density affects the carrying capacity of normal tissue and the cell-oxygen concentration distribution.

<p>We show the results of normal tissue growth and maintenance as we increase the number of randomly seeded vessels from 18 (top) to 24 (middle) to 32 (bottom) on a fixed domain (100 × 100). Cells (left) and oxygen concentration (centre) are visualized. We plot the average distribution of healthy cells by oxygen concentration (right) over ten runs of the simulation with different vascular distributions but constant number of vessels. Every simulation ends in a dynamic equilibrium.</p

Ripley’s <i>L</i> function versus surviving cells after radiation.

<p>We present six scatter plots showing the relationship in each of the 500 simulations represented in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004712#pcbi.1004712.g005" target="_blank">Fig 5</a> for a given initial vessel density between cell number surviving after 2 Gy of radiation (<i>x</i>-axis) and the mean of Ripley’s <i>L</i> function (<i>y</i>-axis). We find that there is a positive correlation in the low vessel densities, and a negative correlation in the high vessel densities. All correlations are significant (<i>p</i> ≪ 0.05, see Fig D in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004712#pcbi.1004712.s002" target="_blank">S1 Text</a>).</p

Cell signaling heterogeneity is modulated by both cell-intrinsic and -extrinsic mechanisms: An integrated approach to understanding targeted therapy

<div><p>During the last decade, our understanding of cancer cell signaling networks has significantly improved, leading to the development of various targeted therapies that have elicited profound but, unfortunately, short-lived responses. This is, in part, due to the fact that these targeted therapies ignore context and average out heterogeneity. Here, we present a mathematical framework that addresses the impact of signaling heterogeneity on targeted therapy outcomes. We employ a simplified oncogenic rat sarcoma (<i>RAS</i>)-driven mitogen-activated protein kinase (MAPK) and phosphoinositide 3-kinase-protein kinase B (PI3K-AKT) signaling pathway in lung cancer as an experimental model system and develop a network model of the pathway. We measure how inhibition of the pathway modulates protein phosphorylation as well as cell viability under different microenvironmental conditions. Training the model on this data using Monte Carlo simulation results in a suite of in silico cells whose relative protein activities and cell viability match experimental observation. The calibrated model predicts distributional responses to kinase inhibitors and suggests drug resistance mechanisms that can be exploited in drug combination strategies. The suggested combination strategies are validated using in vitro experimental data. The validated in silico cells are further interrogated through an unsupervised clustering analysis and then integrated into a mathematical model of tumor growth in a homogeneous and resource-limited microenvironment. We assess posttreatment heterogeneity and predict vast differences across treatments with similar efficacy, further emphasizing that heterogeneity should modulate treatment strategies. The signaling model is also integrated into a hybrid cellular automata (HCA) model of tumor growth in a spatially heterogeneous microenvironment. As a proof of concept, we simulate tumor responses to targeted therapies in a spatially segregated tissue structure containing tumor and stroma (derived from patient tissue) and predict complex cell signaling responses that suggest a novel combination treatment strategy.</p></div

Microenvironmental Variables Must Influence Intrinsic Phenotypic Parameters of Cancer Stem Cells to Affect Tumourigenicity

<div><p>Since the discovery of tumour initiating cells (TICs) in solid tumours, studies focussing on their role in cancer initiation and progression have abounded. The biological interrogation of these cells continues to yield volumes of information on their pro-tumourigenic behaviour, but actionable generalised conclusions have been scarce. Further, new information suggesting a dependence of tumour composition and growth on the microenvironment has yet to be studied theoretically. To address this point, we created a hybrid, discrete/continuous computational cellular automaton model of a generalised stem-cell driven tissue with a simple microenvironment. Using the model we explored the phenotypic traits inherent to the tumour initiating cells and the effect of the microenvironment on tissue growth. We identify the regions in phenotype parameter space where TICs are able to cause a disruption in homeostasis, leading to tissue overgrowth and tumour maintenance. As our parameters and model are non-specific, they could apply to any tissue TIC and do not assume specific genetic mutations. Targeting these phenotypic traits could represent a generalizable therapeutic strategy across cancer types. Further, we find that the microenvironmental variable does not strongly affect the outcomes, suggesting a need for direct feedback from the microenvironment onto stem-cell behaviour in future modelling endeavours.</p></div

Model prediction and validation.

<p>(A) Histograms of relative cell viabilities (log2 scale) of all in silico cells after treatments of EGFRi, METi, <i>RAS_m</i>i, AKTi, RAFi, MEKi, and ERKi. First, a bimodal distribution EGFRi-treated in silico cells (one mode: −1.25; second mode: −0.5). Second, a skewed distribution after METi (a skew toward 0; 0: no change). Third, a uniform distribution in response to <i>RAS_m</i>i (almost uniform distribution from −1.5 to 1.5). Fourth, a slight bimodal distribution after AKTi. Fifth, a distributional response after RAFi. Sixth, a normal distribution in response to MEKi. Seventh, a normal distribution after ERKi. (B) Histograms of relative cell viabilities of all in silico cells in log2 scale. First: EGFRi only (blue), EGFRi/ERKi (orange), and EGFRi/MEKi (yellow). Second: METi only (blue), METi/ERKi (orange), and METi/MEKi (yellow). Third: <i>RAS_m</i>i only (blue), <i>RAS_m</i>i/RAFi (green), <i>RAS_m</i>i/ERKi (orange), and METi/MEKi (yellow). Fourth: AKTi only (blue), AKTi/RAFi (green), AKTi/ERKi (orange), and AKTi/MEKi (yellow). Fifth: RAFi only (blue), RAFi/ERKi (orange), and RAFi/MEKi (yellow). (C) Validation. Model predicted relative cell viabilities (red bars) and experimental data (gray bars) after 10 different treatments. First: EGFRi, EGFRi/MEKi, and EGFRi/ERKi. Second: METi, METi/MEKi, METi/ERKi. Third: AKTi, AKTi/RAFi, AKTi/MEKi, AKTi/ERKi. The numerical data used in Fig 3 are included in the second sheet <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.2002930#pbio.2002930.s013" target="_blank">S1 Data</a>. AKT (PKB), protein kinase B; DMSO, Dimethyl sulfoxide (control); EGFR, epidermal growth factor receptor; ERK, extracellular receptor kinase; MEK, mitogen-activated protein kinase kinase; MET (c-MET), tyrosine-protein kinase Met or hepatocyte growth factor receptor (HGFR); RAF, rapidly accelerated fibrosarcoma; RAS, rat sarcoma.</p