Cellular Automata Simulation of Prostate Cancer: Understanding Macrophage-Driven Tumor Dynamics

Introduction: Prostate cancer (PC) remains the second leading cause of cancer-related deaths among men, particularly affecting those over the age of 65 [1]. Despite improvements in early detection and treatment of localized disease, metastatic PC continues to pose a significant challenge due partly to the complex interactions within the tumor microenvironment (TME) and the acquisition of drug resistance. The immune system plays a dynamic role within the TME, with functional arms that could be tumoricidal and tumor-supportive. Macrophages, particularly anti-inflammatory M2-like macrophages, makeup 50% of the TME and thus significantly influence clinical outcomes, leading to disease progression and immunosuppression [2]. M1 macrophages, however, promote a pro-inflammatory microenvironment that would be more immunosupportive and, consequently, more tumoricidal. Modeling the interactions between the TME, M1/M2 ratio, and tumor progression can provide beneficial information about disease progression and be beneficial in simulating therapeutic regimens proposed for PC. Cellular automata (CA) models follow a set of governing rules that can exhibit global behavior that emerge from cell:cell interactions and is an ideal candidate for modeling biological systems, such as macrophage polarization within the TME. Our model builds on the foundational model established by Qi et al. by including the functionality of M1 and M2 macrophages within the governing rules to provide deeper insights into the mechanisms by which macrophages influence PC progression and treatment response.

Materials and

Methods: The model, adapted from Qi et al., incorporates pro-inflammatory M1 and anti-inflammatory M2 macrophages alongside cancerous and dead cells, represented by M1, M2, C, and D, respectively, to focus on the dynamics between cancer cells and macrophage polarization. This modification allowed for a more detailed exploration of macrophage-mediated immunomodulation within the PC TME. We utilized a grid-based CA model, initialized on a 501x501 lattice where each point represents a cell. The grid was sequentially updated to reflect tumor progression and cellular interactions based on predefined rules for cell proliferation, cytotoxic effects, cell death, and macrophage polarization. These interactions are quantified using reaction constants where k1-5 represents cancer cell proliferation, M1 binding rate, M1 effector rate, dead cell dissolution rate, and M1-M2 polarization rate, respectively, influenced by the surrounding density where once the tumor reaches a critical density, the cancer cells are more likely to proliferate on the edge of the tumor. Simulation parameters were adjusted to reflect human physiological criteria, where derived rates for cell growth and macrophage dynamics from current literature were used. The model monitors changes in cell populations and interactions over 150 days, with each grid iteration representing one day to evaluate the evolving dynamics within the TME. The model was developed by inputting governing rules and critical information from Qi et al. into ChatGTP to generate corresponding MATLAB code that would support the visualization and analysis of cellular interactions and was compared to the Gompertz model of tumor growth.

Results, Conclusions, and Discussions: The model demonstrated that metastatic tumors grew more extensively, supporting our hypothesis that M2 macrophages would enhance the model’s fidelity to real-world tumor progression (Fig 1A). The balance between M2 macrophages and cancer cells remained stable throughout the simulation across disease states, where M1 macrophages were notably depleted prior to the end of the simulation (Fig1 1B, and D). The metastatic simulation showed a Gompertzian growth pattern, which is consistent with established literature (Fig 1C). Sensitivity analysis indicated that the final tumor size was sensitive to variations in the cancer cell proliferation rate, k1, with significant differences observed across a range of tested values (Fig1 E and F).
Our study supports the use of CA models to simulate the complex interplay between PC cells and macrophages within the TME. Incorporating the interplay between pro-inflammatory M1 and anti-inflammatory macrophages within the model allows us to exhibit the immunomodulatory roles of these cells more accurately. The model successfully simulated tumor behaviors consistent with physiological observations, such as the necrotic core formation and a Gompertzian growth pattern observed in the metastatic simulation. However, in the localized and hormone-resistant simulation runs, we did not observe the same overall growth; instead, we observed the number of cancer cells to decrease, and the final iteration did not yield an expected tumor. For both of these conditions, the proliferation rate of PC cells and the PC cell death rate were similar; thus, the PC cells proliferated at the same rate as cell death, causing the expected tumor structure to break down and a deviation from Gompertzian growth. The rapid depletion of M1 macrophages within one week of simulation time may be linked to the functionality of M2 macrophages, where the M2 macrophages successfully create an immunosuppressive environment, and M1 macrophages are less likely to infiltrate the tumor. Our model provides valuable insights into the interactions within the PC TME, such as immunomodulation within the TME, by modifying parameter values to assess key players in macrophage-mediated immunomodulation.

Acknowledgements (Optional): Reference
1. Siegel, R.L. et al. (2022) 'Cancer statistics, 2022', CA: A Cancer Journal for Clinicians, 72(1), pp. 7–33. doi:10.3322/caac.21708.
2. Poh, A.R. and Ernst, M. (2018) “Targeting macrophages in cancer: From bench to bedside,” Frontiers in Oncology, 8. https://doi.org/10.3389/fonc.2018.00049.
3. Qi, A.-S. et al. (1993) 'A cellular automaton model of cancerous growth', Journal of Theoretical Biology, 161(1), pp. 1–12. doi:10.1006/jtbi.1993.1035.