Non‑isothermal Allen–Cahn tumour growth model

A reproducible, high–performance simulator for the non‑isothermal Allen–Cahn tumour growth model. This package implements a three‑variable partial differential equation system coupling tumour cell concentration, temperature and nutrient dynamics. The continuous model is derived in Gatti et. al. and describes the interplay of tumour proliferation, apoptosis, nutrient consumption and heat conduction. The equations are implemented in JAX and solved with either the exponax pseudo‑spectral framework (if available) or a fallback finite–difference scheme.

Simulation Progression and Interpretation

To illustrate the dynamical behaviour of the non-isothermal Allen–Cahn tumour model, we present simulation snapshots at three different time horizons:

• 200 steps (early stage)
• 1000 steps (intermediate stage)
• 10000 steps (late stage / quasi-steady state)

These correspond to increasing physical time and reveal how the coupled fields ($\varphi$) (tumour), ($\theta$) ( temperature), and ($\sigma$) (nutrient) co-evolve.

Early Stage (≈ 200 steps)

At early times, the tumour is still close to its initial condition (typically a small localized region).

• The tumour interface is sharp but beginning to diffuse due to the Allen–Cahn term

$$ \Delta \varphi - F'(\varphi) $$

• Temperature ($\theta$) starts to increase locally where tumour growth occurs because of the source term ( $\varphi_t^2$).
• Nutrient ($\sigma$) remains nearly uniform, with only slight depletion near the tumour. $\varphi_t^2$).

Interpretation: This regime is dominated by interface relaxation and initial proliferation. The system has not yet developed strong nonlinear feedback.

---

Intermediate Stage (≈ 1000 steps)

At intermediate times, nonlinear coupling becomes significant.

• The tumour region expands and smooths, with curvature-driven motion of the interface.
• Temperature shows localized hotspots correlated with active tumour growth.
• Nutrient depletion becomes clearly visible, especially inside the tumour bulk.

Interpretation: This stage reflects active tumour growth under resource constraints. The coupling term

$$ (P\sigma - A)h(\varphi) $$

starts to regulate growth: regions with lower nutrient slow down or stabilize.

---

Late Stage / Quasi-Steady State (≈ 10000 steps)

At long times, the system approaches a steady or metastable configuration.

• The tumour profile stabilizes into a diffuse but stationary structure.
• Temperature equilibrates through diffusion ( \nabla \cdot (\kappa(\theta)\nabla\theta) ).
• Nutrient reaches a balance between consumption and supply:

$$ -C\sigma h(\varphi) + B(\sigma_B - \sigma) \approx 0 $$

Interpretation: The system reaches a dynamic equilibrium where:

• tumour growth is balanced by apoptosis and nutrient limitation,
• temperature production is balanced by diffusion,
• nutrient consumption is balanced by vascular supply.

This corresponds to a homeostatic tumour state in the model.

---

Key Observations

• The model exhibits strong nonlinear coupling between tumour growth, metabolism (nutrient), and thermodynamics ( temperature).
• Tumour evolution is not purely diffusive; it is regulated by resource availability and internal feedback.
• Long-term behaviour depends critically on parameters ($P, A, C, B, \sigma_B$). Adjusting these can lead to different growth patterns, steady states or even tumour regression.

---

Reproducing These Results

You can reproduce similar trajectories using:

python examples/basic_3d_simulation.py

Mathematical model

The simulator solves the simplified non-dimensional system of coupled PDEs (equations(3.3) in the reference). Under homogeneous Neumann boundary conditions, the model reads

φ_t = Δφ − F′(φ) + θ + (P σ − A) h(φ),
θ_t = ∇·(κ(θ)∇θ) + (φ_t)^2 − θ φ_t,
σ_t = Δσ − C σ h(φ) + B (σ_B − σ),

where φ is the tumour cell concentration, θ the absolute temperature, and σ the nutrient concentration. The terms on the right–hand side correspond to diffusion, a double-well potential (F), thermal coupling and reaction/source mechanisms. The governing equations come from the derivation in the reference; in particular the authors show that, after non-dimensionalisation and setting (c_V=β=ε=1), the model reduces to the above system.

Constitutive functions

The simulator uses the following constitutive relations, taken directly from the physical model:

• Double-well potential: The bulk free energy is (F(φ)=φ^2(1−φ)^2). Its derivative appears in the Allen–Cahn equation and is given by (F′(φ)=4φ^3−6φ^2+2φ).
• Activation function h: The tumour regulates proliferation and apoptosis through a bounded, monotonic increasing function h. Biologically, one expects (h(0)=0) and (|h(r)|+|h′(r)|≤C); in practice we implement a smooth ramp that approximates (\max(0,φ)). These properties ensure that the source terms remain bounded and reflect the underlying biology.
• Heat conductivity: The heat conduction coefficient is (κ(θ)=1+|θ|^q) for some exponent (q≥2). By default the simulator uses q=2 but this can be changed via the parameter dataclass.

The positive constants P, A, C, B and σ_B respectively denote the tumour proliferation rate, the apoptosis rate, the nutrient consumption rate, the blood–tissue transfer rate and the nutrient concentration in the vasculature. When (P σ − A) > 0 the tumour grows; otherwise it decays. The nutrient equation combines consumption C σ h(φ) with supply B(σ_B−σ). Homogeneous Neumann boundary conditions (zero normal fluxes) are imposed for all fields.

Package structure

The repository is organised as follows:

• src/acthermal_tumor/core.py – constitutive functions, discrete differential operators and the right‑hand side of the PDE system.
• src/acthermal_tumor/solver.py – a high‑level ThermalTumorSimulator class that builds the computational grid, initializes state variables and advances the solution using a fourth–order Runge–Kutta scheme. The solver uses exponax to build spectral Laplacians when the library is available; otherwise it falls back to a finite–difference implementation of the Laplacian and divergence operators with Neumann boundary conditions.
• src/acthermal_tumor/parameters.py – a dataclass encapsulating all physical parameters and numerical settings.
• src/acthermal_tumor/utils.py – helper routines for generating initial conditions and plotting simulation output.
• tests/ – a small pytest suite that checks basic stability (e.g. no NaN values after a few time steps).
• docs/ – introductory documentation and usage examples.

Installation

The package uses PEP517 build isolation via hatchling. To install the development version locally, clone this repository and run:

pip install .[dev]

This will install the simulator along with its optional test dependencies. You can then run the test suite with pytest.

Quick start

The following example sets up a two‑dimensional grid, initializes a small circular tumour and runs the simulator for a few time steps. See docs/index.md for a detailed tutorial.

import jax.numpy as jnp
from acthermal_tumor.parameters import Parameters
from acthermal_tumor.solver import ThermalTumorSimulator
from acthermal_tumor.utils import generate_initial_conditions, plot_state

# Define domain and parameters
shape = (64, 64)  # 2D grid
params = Parameters(
    P=2.0,
    A=1.0,
    C=1.0,
    B=0.5,
    sigma_B=1.0,
    q=2,
    dt=1e-3,
    dx=1.0 / 64
)

# Generate initial conditions: tumour in the centre, uniform temperature and nutrient
phi0, theta0, sigma0 = generate_initial_conditions(shape, radius_fraction=0.1)

# Create and run simulator
sim = ThermalTumorSimulator(shape=shape, params=params)
state = sim.initialize_state(phi0, theta0, sigma0)
for _ in range(100):
    state = sim.step(state)

# Plot the final state
plot_state(state)

Contributing

Contributions that improve accuracy, extend functionality or add additional time steppers are welcome. Please open an issue or pull request on the project repository.