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

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

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

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

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

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

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

The three qualitatively different tissue scale phenotypes plotted as cell numbers over time for the example simulations in figure 4.

<p>The black trace, representing the unsustainable simulation, grows quickly though never expands its stem population and then outstrips the available oxygen and collapses. The blue trace, representing the homeostatic simulation, reaches a critical size and then maintains a steady birth-death balance. The red trace, representing the tumorigenic simulation, settles into an effectively linear trace on this log-log plot, suggesting power law growth.</p

Computational model description.

<p>(A) The model includes three different cell types: stem, progenitor and differentiated. All cell types interact with the microenvironment in the form of oxygen tension. (B) The behaviour of each cell type is captured by a flowchart. The last segment with discontinuous arrows represents behaviour that is specific to the stem cells. (C) The cells are represented as agents inhabiting points in a grid in a 2D space with 500×500 grid points. Stem cells are represented as red points, progenitor as green and fully differentiated as blue. The vasculature is represented as oxygen source points in black.</p