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A stochastic agent-based extension of the GSM2 model for particle therapy: cell-cycle dynamics, dose-rate dependence, and fractionation effects

This paper introduces a stochastic agent-based framework that extends the GSM2 model to simulate single-cell dynamics, including cell-cycle progression and DNA repair, within 3D tumour spheroids, successfully reproducing complex radiobiological phenomena such as dose-rate dependence and fractionation effects without relying on empirical correction factors.

Original authors: Francesco G. Cordoni, Marco Battestini, Marta Missiaggia

Published 2026-05-01

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CC BY 4.0

Original authors: Francesco G. Cordoni, Marco Battestini, Marta Missiaggia

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tumor not as a solid, uniform block of bad tissue, but as a bustling, chaotic city made of millions of individual citizens (cells). Some of these citizens are sleeping in a quiet neighborhood (the hypoxic core), while others are bustling in the busy downtown (the active outer rim). Some are young and energetic, while others are older and more cautious.

This paper introduces a new, highly detailed "city simulator" for particle therapy (a type of cancer treatment using protons or carbon ions). Instead of treating the tumor as a single, average blob, this model tracks every single "citizen" individually, watching how they react to radiation, how they repair themselves, and how they move around.

Here is a breakdown of what the researchers built and what they found, using simple analogies:

1. The Simulator: A "Next-Event" City Watch

Most old models are like a time-lapse video that checks the city every hour to see what changed. This new model is like a super-advanced security system that only wakes up when something actually happens.

The Agents: Every cell is an autonomous "agent" with its own internal state. It knows its own DNA damage, its current "mood" (cell cycle phase), and how much oxygen it has.
The Clock: The simulation doesn't tick forward in fixed seconds. Instead, it jumps from one event to the next. If a particle hits a cell, the clock jumps to that moment. If a cell repairs a wound, the clock jumps to that moment. This makes the simulation incredibly fast and efficient, even for huge cities (tumors) with thousands of cells.

2. The Radiation: A Rainstorm of Particles

The researchers simulated two types of "rain" falling on this city:

Protons (1H): Like a steady, light drizzle. It causes damage, but the city can often patch it up.
Carbon Ions (12C): Like a heavy, destructive hailstorm. These particles carry more energy and cause complex, clustered damage that is much harder to fix.

The model tracks every single "drop" (particle) as it hits the city. It doesn't just say "10% of the city is damaged"; it counts exactly how many broken windows (DNA lesions) each specific house has.

3. The City's Reaction: Repair, Rest, and Rebuilding

When the radiation hits, the cells react in three main ways, which the model captures perfectly:

Repair: If a cell gets a small scratch (sublethal damage), it tries to fix it. If it fixes it in time, it survives.
The "Redistribution" Dance: Cells have different jobs at different times. Some are in a "busy work" phase (G2/M) where they are very fragile, while others are in a "resting" phase (S phase) where they are tougher. The model shows that if you hit the city slowly, the cells have time to shift their schedules. They might move from the fragile phase to the tough phase before the next "drop" hits, making the city harder to kill.
The Oxygen Factor: The center of the tumor is often starved of oxygen (hypoxic), making those cells like "ghosts" that are hard to hit with standard radiation. The model accounts for this, showing that heavy carbon ions are better at hitting these "ghosts" than protons are.

4. Key Discoveries from the Simulation

The "Slow Rain" Effect (Dose Rate)
The researchers tested what happens if you deliver the radiation very quickly (a flash) versus very slowly (a drizzle over a long time).

The Finding: When you drizzle radiation slowly, the cells have time to fix their wounds between drops. This makes the treatment less effective.
The Twist: This "slow rain" effect disappears when using Carbon Ions. Because Carbon Ions cause such massive, complex damage (like a hailstone smashing a window), the cells can't fix it fast enough, no matter how slow the rain falls. The model proved this without needing to add any "magic numbers" or guesswork; it just emerged naturally from the physics of the simulation.

The "Split Dose" Puzzle
The team simulated giving the radiation in two doses, separated by a break (like a lunch break).

The Finding: The time between the two doses matters a lot.
- If you wait a little bit, the cells fix their wounds (good for the cells, bad for the cure).
- If you wait a bit longer, the cells shuffle their schedules and move into the "tough" phase (bad for the cure).
- If you wait even longer, the surviving cells start multiplying to fill the empty spots (repopulation), making the tumor grow back.
The Result: The model showed a "wave" pattern in survival rates. It went up, then down, then up again, depending on exactly how long the break was. This matches real-world observations but explains why it happens by watching the individual cells dance through their life cycles.

5. Why This Matters (According to the Paper)

The authors state that this is the first time a single model has successfully combined:

Physics: Tracking individual particles hitting the cell.
Biology: Tracking DNA repair and cell cycles.
Space: Watching cells move and die in a 3D tumor shape.

They didn't just predict the outcome; they showed the mechanism. They proved that complex biological behaviors (like why slow radiation is less effective, or why splitting doses has a weird effect) are the natural result of cells trying to repair themselves while the radiation keeps hitting them.

In short: The paper presents a digital twin of a tumor that lets scientists watch, in slow motion, how every single cell reacts to particle therapy. It confirms that heavy ions (Carbon) are superior for tough, oxygen-starved tumors and explains the complex rules of how timing affects the treatment's success, all without using any "cheat codes" or empirical guesses.

1. Problem Statement

Radiobiological modeling faces a critical challenge in linking microscopic energy deposition from ionizing radiation to emergent biological outcomes, particularly in particle therapy. Current dominant frameworks, such as the Linear-Quadratic (LQ) model and its extensions (e.g., Microdosimetric Kinetic Model, MKM), rely on population averages and deterministic assumptions. They fail to explicitly capture:

Stochastic variability at the single-cell level (e.g., random lesion induction and repair).
Cell-cycle dynamics, including phase-dependent radiosensitivity (G1, S, G2/M) and redistribution.
Spatial heterogeneity, such as oxygen gradients (hypoxia) and cell migration within tumors.
Complex irradiation scenarios, such as dose-rate effects and split-dose fractionation, often requiring empirical correction factors (e.g., the Lea–Catcheside factor) rather than deriving them from first principles.

The authors aim to bridge the gap between physical track-structure simulations and tissue-level outcomes by developing a unified, stochastic, single-cell framework that resolves these dynamics without empirical ad-hoc adjustments.

2. Methodology

The authors propose a Stochastic Agent-Based Model (ABM) that extends the Generalized Stochastic Microdosimetric Model (GSM2).

Core Architecture

Agent Definition: Each cell is an autonomous agent on a 3D lattice. Its internal state includes DNA lesion counts (sublethal $X$ and lethal $Y$ ), cell-cycle phase ( $G_1, S, G_2, M, G_0$ ), and local oxygenation.
Stochastic Dynamics: The system evolves via a Continuous-Time Markov Chain (CTMC) driven by a next-event, event-driven algorithm. This avoids fixed time steps, improving computational efficiency.
Competing Events: The global scheduler manages four competing event categories:
1. Radiation Physics: Stochastic arrival of individual particles based on prescribed dose rates.
2. Biological Response: DNA damage induction, repair, misrepair, and conversion (Gillespie's Stochastic Simulation Algorithm).
3. Cell-Cycle Progression: Phase transitions and division (modeled via Gamma-distributed durations).
4. Migration: Random walk on the lattice based on available space.

Key Sub-Models

GSM2 Integration: Damage induction and repair rates ( $r, a, b$ ) are derived directly from GSM2 master equations, calibrated to reproduce phase-specific LQ parameters ( $\alpha, \beta$ ) from experimental literature.
Oxygenation: A quasi-static, radially symmetric profile models oxygen diffusion in tumor spheroids, modulating lesion yield via the Oxygen Enhancement Ratio (OER).
Energy Deposition: Uses the Kiefer–Chatterjee Amorphous Track (AT) model to calculate energy deposition in sub-nuclear domains, accounting for Linear Energy Transfer (LET).
Simulation Environment: Implemented in Julia, simulating 3D tumor spheroids irradiated with protons ( $^1$ H) and carbon ions ( $^{12}$ C) across various energies and dose rates ( $10^{-5}$ to $10^{-2}$ Gy s $^{-1}$ ).

3. Key Contributions

Unified Stochastic Framework: The first model to integrate GSM2 microdosimetry with explicit single-cell agent dynamics, capturing repair, reassortment, and repopulation simultaneously.
Emergent Dose-Rate Effects: The model naturally reproduces dose-rate dependence (including the attenuation of the quadratic $\beta$ term at low dose rates) and the inverse dose rate effect in split-dose irradiation without using empirical correction factors like the Lea–Catcheside factor.
High-LET vs. Low-LET Differentiation: Explicitly demonstrates how high-LET carbon ions mitigate dose-rate dependence due to the dominance of single-track lethal events, whereas low-LET protons show strong sensitivity to temporal dose structure.
Computational Efficiency: The event-driven algorithm allows for the simulation of large cell populations (thousands of cells) with full single-cell resolution, scaling approximately linearly with system size.

4. Key Results

Spatiotemporal Dynamics: Simulations of 3D spheroids show that carbon ions cause more rapid and complete depletion of the proliferating rim compared to protons at the same energy per nucleon, consistent with higher biological effectiveness.
Cell-Cycle Redistribution: The model captures the "reassortment" effect. Following irradiation, cells redistribute through phases, leading to oscillatory survival probabilities in split-dose scenarios.
- Observation: Survival probability exhibits a non-monotonic dependence on inter-fraction time: an initial increase (repair), a second increase (redistribution to radioresistant S-phase), and a subsequent decrease (re-entry into sensitive G2/M).
Dose-Rate Dependence:
- The linear parameter ( $\alpha$ ) remains constant across dose rates.
- The quadratic parameter ( $\beta$ ) decreases monotonically as dose rate decreases, approaching zero at very low rates.
- This effect is negligible for high-LET carbon ions but pronounced for protons.
Stochastic Variability: The model generates distributions of outcomes rather than single deterministic curves, quantifying the uncertainty arising from stochastic damage induction and repair at the single-cell level.

5. Significance and Future Outlook

Mechanistic Insight: By removing empirical corrections, the model provides a mechanistic explanation for classical radiobiological phenomena (the "4 Rs": Repair, Reassortment, Repopulation, Reoxygenation).
Clinical Relevance: The framework offers a pathway to predict Tumor Control Probability (TCP) and Normal Tissue Complication Probability (NTCP) by aggregating single-cell outcomes, potentially improving treatment planning for particle therapy.
Limitations & Future Work:
- Current cell fate is binary (survive/die); future versions will include apoptosis, senescence, and mitotic catastrophe.
- Oxygenation is currently static; dynamic coupling with vascular networks is planned.
- Quantitative validation against experimental spheroid data is the next critical step.

In conclusion, this work represents a significant advancement in radiobiological modeling, moving from population-averaged phenomenology to a rigorous, stochastic, single-cell description that is essential for optimizing particle therapy in heterogeneous and hypoxic tumor environments.