Manchester Inhomogeneous Radiation Chemistry (MIRaCLE)

• MIRaCLE is a hybrid computational framework that simulates the spatio-temporal evolution of radiolytic species generated by high-energy electron interactions in water.
• It combines stochastic sampling of inelastic collisions with a deterministic spectral PDE solver to achieve near–Monte Carlo accuracy with significant performance gains.
• Its one-shot computation models bridge microscopic track-structure details with macroscopic chemical kinetics, facilitating applications in radiotherapy and nuclear safety.

Manchester Inhomogeneous Radiation Chemistry by Linear Expansions (MIRaCLE) is a hybrid computational framework that simulates the spatio-temporal evolution of radiolytic species generated by high-energy electron interactions in water. Developed as an efficient alternative to conventional Monte Carlo and continuum reaction–diffusion models, MIRaCLE stochastically determines initial event locations and then deterministically solves the subsequent chemical evolution using a spectral Partial Differential Equation (PDE) approach. This platform enables “one-shot” computation of time-dependent concentrations and radiolytic yields (“G-values”) at dose rates and timescales previously inaccessible to standard methodologies, delivering near–Monte Carlo accuracy with drastic performance gains, and providing a tractable bridge between microscopic track-structure models and macroscopic chemical kinetics scenarios (Perkins et al., 5 Jan 2026).

1. Theoretical Framework

MIRaCLE’s core methodology comprises two sequential stages:

1. Stochastic sampling of inelastic electron collisions: The method places $N$ electron collision “vertices” $\{\mathbf R_j\}_{j=1}^N$ within the simulation volume, reflecting the spatial distribution of radiolysis events based on physical track-structure.
2. Continuum mapping via linear expansions: Each discrete event is converted to a smooth concentration “reaction packet” by expressing the initial concentration field of chemical species $i$ as

$$c_i(\mathbf x,t_0) = \sum_{j=1}^N a_{ij}\,\phi_i(\mathbf x-\mathbf R_j)$$

where $\phi_i(\mathbf r)$ is a normalized basis function (typically Gaussian or gamma-shaped) and $a_{ij}$ are coefficients representing the local yield per event.

For common radiolytic species, this representation utilizes established packet shapes:

• Gaussian packets (Models 2–4):

$$\phi_i(\mathbf r) = \frac{1}{(2\pi \sigma_i^2)^{3/2}}\exp\left(-\frac{|\mathbf r|^2}{2\sigma_i^2}\right)$$

with $\sigma_i$ species-dependent,

• Regularized gamma distributions (Model 3, solvated electron per Kreipl et al.), with explicit cutoff and scale parameters.

The coefficient $a_{ij}= M_i/N$ assigns equal particle weight to all vertices generating species $i$.

2. Reaction–Diffusion Dynamics

Following initialization, MIRaCLE evolves the concentration profiles $c_i(\mathbf x,t)$ deterministically by numerically integrating the coupled reaction–diffusion equations: $$\frac{\partial c_i}{\partial t} = D_i\nabla^2 c_i + \sum_{j,k} s_{i,jk}\,k_{jk}\,c_j\,c_k - \sum_{\ell}k_{i\ell}c_i c_\ell$$ where $D_i$ is the diffusion coefficient, $k_{jk}$ are bimolecular rate constants (sourced from the standard water-radiolysis network), and $s_{i,jk}$ quantifies species production per reaction.

Representative equations for solvated electrons ($c_e$), hydroxyl radicals ($c_{HO}$), and hydrogen atoms ($c_H$) are: $$\begin{aligned} \partial_t c_e &= D_e\nabla^2c_e - 2k_{ee}c_e^2 - k_{eHO}c_e c_{HO} - k_{eH}c_e c_H - \dots \ \partial_t c_{HO} &= D_{HO}\nabla^2c_{HO} - 2k_{HO\,HO}c_{HO}^2 - k_{eHO}c_e c_{HO} - k_{H\,HO}c_H c_{HO} + \dots \end{aligned}$$

A spectral PDE solver, employing truncated Fourier/Chebyshev bases (typically up to 100 modes per spatial direction), advances this system with adaptive time-stepping.

3. Correction for Continuum Self-Interaction

A key artifact in continuum modeling is unphysical self-interaction—radicals within the same reaction packet inappropriately reacting with themselves, inflating bimolecular yields. MIRaCLE introduces an effective homodimerisation rate correction: $$k^{\rm eff}_{AA} = k_{AA} \frac{n_A}{n_A + J_{0,A}(t_{\rm eff})}$$ where $n_A = N_A / V$ (density of packets), $J_{0,A}(t)$ is the self-overlap integral for a single packet, and $t_{\rm eff}$ is identified when the diffusion length equates to mean packet separation. Beyond this time, spatial mixing negates origin memory. This correction robustly restores long-time G-values without excessive dimer formation.

4. Computational Workflow

MIRaCLE’s calculation pipeline is explicitly structured, with the principal workflow steps as follows:

Input: E_dep, geometry, chemistry (D_i, k_jk), mode_limit Output: c_i(x, t), G_i(t) 1. Compute total yields: for each i, M_i = (E_dep / 100 eV) * G_i^init 2. Estimate vertices: N_V = E_dep / (I * eta) 3. Sample collision vertices (track-structure) 4. Assign reaction packets: a_ij = M_i / N_V for card(i, j) 5. Expand initial conditions: c_i(x, t_0) = sum_j a_ij * phi_i(x - R_j) 6. Spectral PDE evolution with k_AA^eff 7. Post-process G_i(t) by integration

Where $I\approx 79.9$ eV (mean ionization energy), $\eta\approx0.848$ (fraction not relaxing to water), and the rest of the parameters are tabulated per species.

Intermediate results such as spatial maps $c_i(\mathbf x,t)$ and time-dependent radiolytic yields $G_i(t)$ (molecules per 100 eV) are generated on the fly via integration: $$G_i(t) = \frac{N_A \int c_i(\mathbf x,t)\,d^3x}{E_{\rm dep}/(100\,{\rm eV})}$$ where $N_A$ is Avogadro’s number.

5. Performance and Benchmarking

MIRaCLE achieves high-precision simulation fidelity within minimal computational resources—reported “one-shot” models (no trajectory averaging) yield relative errors $\chi \lesssim 0.07$ compared to Geant4-DNA Monte Carlo standards. For example, using an M2 MacBook Pro, Model 3 requires approximately 480 s for initial condition assembly and 3480 s for evolution to 100 μs. Spectral-convergence tests show error $\chi$ scales inversely with the number of mode truncations per direction, and CPU cost for initial conditions grows linearly with dose, whereas subsequent PDE evolution is nearly dose-independent.

A comparison highlights MIRaCLE’s efficiency: for a 10,000 Gy case, standard stochastic Monte Carlo would require $\sim5$ CPU-years, yet MIRaCLE delivers results for 1 ps–100 μs in approximately half a day on a single core.

6. Applicability, Limitations, and Future Directions

MIRaCLE is suited to a range of domains:

• Precision radiotherapy, including ultra-high dose-rate (“FLASH”) regimes,
• Water chemistry and safety in nuclear technologies,
• Environmental processing (e.g., waste irradiation),
• Electron microscopy of liquids (resolving beam-induced chemistry artifacts).

Limitations include:

• Vertex positions are Poisson-distributed; real spur clustering and heterogeneity are not yet modeled.
• Packet width and shape parameters are drawn from existing literature—not ab initio or dynamically updated.
• Correction term $k^{\rm eff}$ is applied at a single “mixing” time, neglecting details of short-time radical kinetics.
• Interfaces and highly heterogeneous geometries are not currently supported.

Planned advancements include dynamic, color-partitioned correction algorithms to suppress early-time self-overlap, ab initio packet shape derivation via TDDFT, extension to high-LET ions and complex three-dimensional geometries (e.g., tissue, nanoparticle), and GPU-accelerated spectral solvers. A plausible implication is expanded applicability to direct ab initio radiolysis modeling and integration with experimental microdosimetry.

7. Significance in Radiation Chemistry Modeling

MIRaCLE redefines the simulation of radiation-induced chemical processes by foundationally merging stochastic event architecture with deterministic continuum evolution. This approach enables rapid, accurate estimation of radiolytic yields and spatio-temporal species distributions, efficiently mimicking the trajectory average of Monte Carlo without extensive computational expense. Consequently, MIRaCLE establishes itself as a versatile platform, potentially facilitating theory-experiment feedback and operational studies across emerging applications in medicine, nuclear safety, materials, and environmental sciences (Perkins et al., 5 Jan 2026).