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Lenia & artificial life

How cool is the name 'Artificial Life'? It literally means bringing life to a computer, letting you test your craziest ideas without any of the real-world side effects! We used one such system, Lenia[1], to explore tumor growth and study interactions within the tumor-immune environment. Lenia (from the Latin lenis, meaning “smooth”) is a cellular automata framework that operates with continuous space and time, inspired by Lenia Mathematical Lifeforms—a continuous generalization of John Conway’s Game of Life.

Before going into the technical details, it’s important to reflect on why we created this model. Artificial life systems are designed to capture essential biological features such as self-organization (morphogenesis), self-regulation (homeostasis), self-direction (motility), and self-replication (reproduction). They also model processes like growth, response to environmental stimuli, evolvability, and adaptation—emergent properties critical in living systems.

These same characteristics—cellular replication, growth, motility, evolvability, and adaptation—are not only hallmarks of biological life but also key drivers in the development of complex, multi-factorial diseases such as cancer. By modeling these processes, Lenia allows us to simulate the intricate behaviors of cancer cells, providing deeper understanding into how tumors grow, evolve, and respond to treatment. Let’s explore some examples using Lenia.

Logistic Growth with an Allee effect

In our recent paper[2], we first imagine a small group of tumor cells in the early stages of growth. At this point, there may not be enough cells to release sufficient growth factors, evade the immune system, or establish a proper blood supply (angiogenesis). As a result, the tumor grows slowly, or it might even shrink. However, once the tumor reaches a critical size, there are enough cells to start processes like angiogenesis, which brings nutrients to the tumor. This increased communication between cells helps the tumor grow more rapidly. Thus, there’s a threshold, called the Allee threshold, that, once crossed, speeds up tumor growth.

In tumor growth dynamics literature, a logistic growth model (Equation 1) is often used. This model creates an "S-curve," where the tumor initially grows like it would under exponential growth when it's far from its maximum size (carrying capacity). But as it gets closer to the carrying capacity, growth slows down due to resource limitations.

$$ \begin{equation} \frac{dN}{dt}=γN(1- \frac{N}{k}) \\ \end{equation} $$ To account for the possibility that tumors might go extinct early on—due to the Allee effect—we can introduce the Allee threshold into the model (Equation 2). Figure 1 compares how tumor growth behaves under the basic logistic model, a small Allee threshold, and a large one. Figures 1(A-C) show that having a higher initial number of tumor cells can help avoid early extinction.

$$ \begin{equation} \frac{dN}{dt}=γN(1- \frac{N}{k})\frac{(N-L)}{k} \end{equation} $$

“Short-range interaction kernels are more robust to Allee effects.”

These models don’t fully capture the heterogeneity and interactions within the tumor microenvironment. For instance, when considering the Allee effect, we might want to explore how this threshold interacts with spatial dynamics in tumors. More precisely, the interaction kernel of a tumor accounts for a spectrum of spatial interactions, ranging from local (cell-scale) to global (tumor-scale). We’re interested in understanding how this range of interactions can affect tumor dynamics across different Allee thresholds.

The pattern we observe in Video 1 suggests that a well-mixed model may overestimate the role of the Allee effect, and therefore, the likelihood of tumor extinction. We observed how the size of the interaction kernel, which governs the spatial interaction, can impact tumor dynamics under the Allee effect.

Spatial Interactions and Heterogeneity in Tumor-Immune Dynamics

While the logistic and Allee models provide a foundation for understanding tumor growth, they don’t fully capture the spatial heterogeneity within the tumor microenvironment. Lenia offers a unique advantage by supporting multi-interaction kernels in the system. In the context of tumor-immune interactions, different types of interactions—such as tumor-tumor, tumor-immune, immune-tumor, and immune-immune—can each influence the dynamics in different ways. These interaction kernels represent the spatial dynamics and distances over which cells interact. To describe tumor-immune dynamics in this more complex setting, we turn to a predator-prey model described in Equations (3) and (4).

Tumor cell growth dynamics: $$ \begin{equation} \dot{x}=rx(x-L)(1-x)-bxy \end{equation} $$
Immune cell dynamics: $$ \begin{equation} \dot{y}=gbxy-dy \end{equation} $$

Where `r` is the growth rate of tumor, `L` is the Allee threshold, `b` is the killing rate of tumor cells by immune response and immune cells expand at rate `g` and die at rate `d`.

We consider four interaction kernels (Figure 2) to capture all the dynamics in the tumor-immune system. The tumor-immune kernel (K12) represents immune sensitivity, where a short-range K12 (small R12) leads to a higher immune kill rate per unit area. To visualize this, think of the tumor-immune kernel as the average distance an immune cell travels to kill a tumor. For example, a large tumor-immune kernel could represent a tumor with high PD-L1 expression, where immune cells need to travel farther before they can act effectively.

Next, we define the immune-tumor kernel (K21) as spatial immune specificity, where a short-range K21 (small R21) leads to localized recruitment of immune cells near tumor cells (this is different from T-cell antigen recognition specificity). To understand this better, consider the immune-tumor kernel as the average distance at which immune cells are recruited by the tumor. An immunologically quiet tumor would have a large, diffuse kernel with undirected recruitment, whereas a highly antigenic tumor would have a small kernel with strong, localized immune cell recruitment.

All the figures and simulations presented in this blog post were implemented using our Artificial Life system, Lenia. Lenia offers a wide range of features essential for cancer modeling, including cellular replication, growth, motility, evolvability, and adaptation. As demonstrated throughout this blog, Lenia is designed to replicate a variety of models commonly used in cancer research, such as:

1. Classical Analytical Models: Simulating ordinary differential equations widely applied in cancer research.
2. Stochastic Agent-Based Models: Supporting stochastic simulations of cancer cell behavior.
3. Evolutionary Game Theory: Modeling interactions between multiple cell types competing on Darwinian fitness landscapes.
4. Cell Migration: Incorporating generalized cell migration models like chemotaxis, which are prevalent in mathematical modeling of cancer.

Mini-tutorial for Lenia

If you'd like to use Lenia in your project or recreate the simulations shown in this blog, you can easily download and implement it via github.com/mathonco/Lenia-in-hal, built in HAL[3].

Users can define growth functions and customize kernel sizes and interaction functions for spatial modeling. Implementing a model in Lenia involves two key steps:

1. Defining the growth function, (G(u))
2. Defining the interaction kernel, (K(r))

For example, to implement logistic growth with an Allee effect, you can override the G(u) function as follows:

// Growth function
@Override
public double G(double u) {
    return gamma * u * (u - L) * (C - u);
}

Here, gamma, L, and C are constants. This function defines the growth rate at each lattice location based on u, the density potential, which is calculated as the weighted sum of densities in a cell's neighborhood as defined by the interaction kernel. You can define the neighborhood kernel like this:

@Override
public double K(double r) {
    if (r <= Rstar) {
        return 1.0;
    } else {
        return 0.0;
    }
}

In this example, all cells within a radius of Rstar are equally weighted. If the neighborhood is empty, then u = 0; if the neighborhood is at full capacity, then u = 1. Lenia provides a powerful and flexible framework to explore and model the complexities of cancer dynamics, enabling researchers to test hypotheses and create custom simulations with ease.

References

1. Chan, B.W.C., 2018. Lenia-biology of artificial life. arXiv preprint arXiv:1812.05433.
2. Marzban, S., Srivastava, S., Kartika, S., Bravo, R., Safriel, R., Zarski, A., Anderson, A.R., Chung, C.H., Amelio, A.L. and West, J., 2024. Spatial interactions modulate tumor growth and immune infiltration. NPJ Systems Biology and Applications, 10(1), p.106.
3. Bravo, R.R., Baratchart, E., West, J., Schenck, R.O., Miller, A.K., Gallaher, J., Gatenbee, C.D., Basanta, D., Robertson-Tessi, M. and Anderson, A.R., 2020. Hybrid Automata Library: A flexible platform for hybrid modeling with real-time visualization. PLoS computational biology, 16(3), p.e1007635.