Fig 1.
PhysiCell and multiple time scales.
PhysiCell uses BioFVM to update the microenvironment at the short green tick marks, corresponding to Δtdiff. It updates cell mechanics (including cell position) less frequently at the medium black tick marks (Δtmech), and it runs the cell volume and cycle/death models least frequently at the long red tick marks (Δtcell). Note that the time steps shown are for illustrative purpose; the default step sizes are given in Time steps.
Fig 2.
Hanging drop spheroid (HDS) simulations with deterministic necrosis (left) and stochastic necrosis (right), plotted at 4, 8, and 16 days.
Videos are available at S1 and S2 Videos. Legend: Ki67+ cells are green before mitosis (K1) and magenta afterwards (K2). Pale blue cells are Ki67- (Q), dead cells are red (apoptotic) and brown (necrotic), and nuclei are dark blue. Bottom: Hanging drop spheroid experiment (HCC827 non-small cell lung carcinoma) showing a similar necrotic core microstructure. PhysiCell is the first simulation to predict this structure arising from cell-scale mechanical interactions. Image courtesy Mumenthaler lab, Lawrence J. Ellison Center for Transformative Medicine, University of Southern California.
Fig 3.
Left: The deterministic and stochastic necrosis models both give approximately linear growth (left), but the HDS with deterministic necrosis model grows faster (∼ 5% difference in diameter at day 18). Right: The HDS with stochastic necrosis has fewer cells than the deterministic model (∼ 26% difference in cell count at day 18), due to its delay in necrosis. The difference in cell count is larger than the difference in tumor diameter because most of the difference lies in the number of necrotic cells, and necrotic cells are smaller than viable cells.
Fig 4.
HDS computational cost scaling.
Left: Wall-time vs. cell count for the stochastic (red) and deterministic (blue) necrosis necrosis models on a single HPC compute node. Both models show approximately linear cost scaling with the number of cell agents. Right: Wall time vs. cell count for stochastic necrosis model on the desktop workstation (orange) and the single HPC node (green).
Fig 5.
Ductal carcinoma in situ (DCIS) simulations with deterministic necrosis (left) and stochastic necrosis (right), plotted at 10 and 30 days (multiple views).
Videos are available at S3 and S4 Videos. The legend is the same as Fig 2.
Fig 6.
The deterministic and stochastic necrosis models both result in linear DCIS growth at approximately 1 cm/year (left), even while their cell counts differ by 21% by the end of the simulations (right).
Fig 7.
Director cells (green) release a chemoattractant c1 to guide worker cells (red). Cargo cells (blue) release a separate chemoattractant c2. Unadhered worker cells chemotax towards ∇c2, test for contact with cargo cells, form adhesive bonds, and then pull them towards the directors by following ∇c1. If c1 exceeds a threshold, the worker cells release the cargo and return to seek more cargo cells, repeating the cycle. Bar: 200 μm. A video is available at S5 Video.
Fig 8.
Anti-cancer “biorobots” example.
By modifying the worker cells in the previous example (Fig 7) to move up hypoxic gradients (along −∇pO2) and drop their cargo in hypoxic zones, we can deliver cargo to a growing tumor. In this example, the cargo cells secrete a therapeutic that induces apoptosis in nearby tumor cells, leading to partial tumor regression. Bar: 200 μm. A video is available at S6 Video.
Fig 9.
Each cell has an independent expression of a mutant “oncoprotein” p (dimensionless, bounded in [0, 2]), which scales the oxygen-based rate of cell cycle entry. Blue cells have least p, and yellow cells have most. Initially, the population has normally distributed p with mean 1, standard deviation 0.3, and a “salt and pepper” mixed spatial arrangement. The proliferative advantage for cells with higher p leads to selection and enrichment of the most yellow cells. Stochastic effects lead to emergence of fast-growing foci and a loss of tumor symmetry. Bar: 200 μm. A video is available at S7 Video.
Fig 10.
In this 3-D example, each tumor cell secretes an immunostimulatory factor, and its immunogenicity is modeled as proportional to its mutant oncoprotein expression. (See the previous example in Fig 9.) After 14 days, red immune cells perform a biased random walk towards the immunostimulatory factor, test for contact with cells, form adhesions, and attempt to induce apoptosis for cells with greater immunogenicity. The immune cells successfully attack the tumor initially, leading to partial regression; apoptotic cells are cyan. But strong homing towards gradients of the immunostimulatory factor causes immune cells to “pass” some cells at the outer edge, leading to tumor regrowth. Eventually, immune cells leave the necrotic regions and press their attack on the tumor. This highlights the importance of stochasticity in immune cell movement in mixing with the tumor cells for a more successful immune response. A video is available at S8 Video.