A neural network embedding the linear-quadratic model for improved prediction of cellular response to ion beam exposure

Abstract

Predicting biological responses to ionizing radiation is challenging due to the complex, multi-scale mechanisms involved. Traditional machine learning (ML) models that fit experimental data on cell survival can be sensitive to experimental uncertainties and lack theoretical consistency. This study developed and validated a neural network (NN) framework that integrates established radiobiological constraints directly into its architecture and training process. The objective was to predict cell surviving fractions (SF) and key radiosensitivity parameters for various cell types and radiation qualities in a manner that remains consistent with known dose-response behaviour. A multi-branch feedforward NN was developed to simultaneously predict SF and derived parameters including the linear-quadratic model (LQM) parameters α and β, the mean inactivation dose (MID), the dose at 10% survival (D₁₀), and the relative biological effectiveness at 10% survival (RBE₁₀). The model architecture explicitly embeds the LQM formula and the calculations of radiosensitivity parameters within dedicated custom layers. A custom loss function penalizes deviations from known ranges and balances multiple prediction objectives. Trained and validated using the Particle Irradiation Data Ensemble (PIDE) via ten-fold cross-validation, the NN achieved high prediction accuracy. Predicted SF values were in excellent agreement with experimental data (R² ≈ 0.95). The model also robustly predicted D₁₀ and MID values (R² ≈ 0.80) and RBE₁₀ value (R² ≈ 0.75). Predictions of LQM’s α parameter were reasonably correlated (R² ≈ 0.65), whereas β parameter predictions exhibited greater uncertainty. The integration of the LQM formalism into the NN architecture yielded a model that produces consistent predictions parameterized by mechanistically relevant variables across diverse radiation qualities. Constraining the model with established radiobiological constraints improved its ability to link input features to cellular outcomes, ensuring accurate dose-response modeling and valid extrapolations.

Introduction

Understanding the biological effects of various types of radiation on living cells is crucial for predicting radiation-induced damage. The cellular response to radiation exposure is a complex process involving multiple biological mechanisms, including the induction of DNA damage, repair pathways, and modulation of cell cycle progression, all of which collectively determine cell fate¹. The linear-quadratic model (LQM) encapsulates these interactions through two mechanistically distinct components, grounded in quantitative biological descriptions rather than empirical polynomial curve fitting: α, representing single-track lethal events, and β, reflecting lethal events formed by the interaction of mis-repaired sub-lethal lesions from separate tracks that can lead to chromosomal aberrations^2–5. Parameters α and β can exhibit considerable variability depending on cell type, radiation quality, and experimental conditions, necessitating more sophisticated approaches to comprehensively predict and interpret radiobiological outcomes.

In recent years, the advent of large-scale databases and advanced computational techniques has created new opportunities for analyzing and modeling radiobiological data. The Particle Irradiation Data Ensemble (PIDE) database^6,7, curated by the Gesellschaft für Schwerionenforschung (GSI), is a comprehensive collection of experimental data on cell survival after exposure to different types of ionizing radiation. These resources enable systematic exploration of the relationships between physical radiation parameters and biological outcomes across diverse experimental conditions.

Data-driven approaches have shown considerable promise in predicting biological responses to ionizing radiation. Papakonstantinou et al.⁸ and Zanni et al.⁹ employed a random forest (RF)-based ML model, utilizing the data on DNA double-strand breaks (DSBs) calculated by the fast Monte Carlo damage simulation software (MCDS) to predict the α and β parameters. Their findings indicated reliable predictions for the α parameter, which was mainly influenced by LET and cell line features, whereas the β parameter was more challenging to model due to its small magnitude and susceptibility to experimental noise.

Debreceni et al.¹⁰ utilized the PIDE database in an RF-based ML model for predicting cell surviving fractions based on the physical features of heavy ions. The RF model, incorporating dose and LET as input, outperformed both the linear-quadratic model and local regression models, achieving higher prediction accuracy. Cordoni et al.¹¹ developed Artificial iNtelligence bAsed model for (radiation-induced) cell KIlliNg prediction (ANAKIN), an ML model for predicting cell surviving fraction. ANAKIN utilized the RF¹² and XGBoost¹³ algorithms in model training with the PIDE database including cell surviving fractions calculated from the LQM parameters. ANAKIN predictions of RBE were comparable to those of the microdosimetric kinetic model and the local effect model (LEM) version III¹⁴.

Despite these successes, standard ML techniques often encounter limitations. They can be highly sensitive to experimental uncertainties, potentially misinterpreting them as meaningful structural patterns. This may result in predictions that lack biological plausibility (such as non-monotonic survival curves). Furthermore, these techniques struggle to make reliable estimates or clear interpretations beyond the training data, as purely data-driven parameters can lack inherent radiobiological meaning¹⁵. In addition, in multi-output regression scenarios, standard RF methodology optimizes for each target variable in isolation. Consequently, this independent optimization approach may not adequately capture the inherent interdependencies that might exist between the different output variables¹⁶.

The strong performance of traditional ML models such as RF in predictive modeling tasks is well-documented and often outperform more complex models^17,18. However, their inherent structure, while powerful, can present limitations when customization is desired. Consequently, an RF can obtain higher R² for individual target variables such as SF and a directly predicted RBE₁₀. However, the RBE₁₀ value recalculated from the RF-predicted SF curve may be inconsistent and markedly less accurate.

The integration of photon irradiation data into ion data-based ML models used to predict SF enables the models to implicitly extract patterns induced by photons, thereby improving ion prediction accuracy^8,9,11. However, this may introduce a limitation such that ion-related target variable predictions become dependent on the corresponding photon-related target variables and their inclusion becomes mandatory. Tree-based methods do not differentiate between features utilized for prediction and those for pure support. As a result, supporting features may inadvertently act as shortcuts, potentially leading to biased or inflated model performance¹⁹.

Neural networks (NNs), by contrast, offer distinct advantages in terms of flexibility and customizability, allowing for the explicit incorporation of domain-specific constraints directly into the model architecture^20,21. In addition, supporting features can be utilized with constrained influence on model training, ensuring that the relationships among features are respected. This approach enhances the consistency and interpretability of predictions, while reducing the risk of inadvertently exploiting shortcuts that could compromise the robustness of the predictions. Tian and Lühr²² utilized a NN model to validate the concept of ion type-independent beam quality Q for RBE prediction, demonstrating robust performance in clinically relevant dose ranges (2–4 Gy). However, the developed NN application was confined to interpolation within data-supported regions.

In this paper, we developed a NN framework that integrates established radiobiological constraints. This integration is achieved through two primary mechanisms: (1) a special model architecture featuring custom layers that embed fundamental dose-response formulas including the LQM for SF and the derivation of radiosensitivity parameters (MID, D₁₀, RBE₁₀); and (2) a custom loss function designed to enforce constraints during the training process. Trained and validated using experimental data from the PIDE, the model simultaneously predicts multiple target variables: SF, LQM parameters α and β, MID, D₁₀, and RBE₁₀ for diverse cell types and radiation conditions.

Methods

Data collection and pre-processing

The primary source of data used for model training was PIDE^6,7. PIDE included 1,118 experiments, each of which reported key physical and biological parameters such as radiation type (X-ray or ion species represented by ion mass number), dose, LET values (keV/µm), cell type (identification), cell origin (human or rodent), cell class (tumor or normal), DNA content (e.g., genome size), and cell cycle phase during irradiation, and the measured cell surviving fractions at various doses along with the LQM α and β values fitted to the corresponding experiment survival curves.

This study focused primarily on the ion exposure data. The PIDE data of SF from ion exposure were collected for training. The SF values on photon exposure used for pairwise calculation of RBE₁₀ values were taken from the LQM curves reconstructed with the α and β parameters that PIDE provides.

Following initial preparation, data were processed through multiple steps as follows:

• Standardization of survival data: SF data from ion beam exposures were standardized such that the SF values were 1.0 at 0 Gy. This was achieved by adding data points at (dose, SF) = (0, 1) if such points were not already present in the dataset, and by rescaling experimental SF values to ensure that data points of (dose, SF) lie within (0, 1]. All data points were divided by the largest SF value for experiments containing SF values exceeding 1.0. This method was preferred over capping values at 1.0, as it preserves the relative shape and gradient of the dose-response curves. However, we acknowledge that this may introduce a systematic shift if the fluctuation is isolated rather than methodological.

• Integration of radiosensitivity parameters: The radiosensitivity parameters including the mean inactivation dose (MID), the dose at 10% survival (D₁₀) from X-ray (D₁₀,_x-ray) and ion (D₁₀,_ion) exposures, and the relative biological effectiveness at 10% survival (RBE₁₀) were added to the dataset. D₁₀ values were determined by analytically solving the LQM equation for D at 10% survival. The LQM equation was defined by the given α and β values from the PIDE database. RBE₁₀ was calculated as the ratio of D₁₀ from photon exposure to that from ion exposure. MID was computed by applying the trapezoidal rule to integrate the SF curve, reconstructed using the LQM equation, at 0.1 Gy intervals over the dose range from 0 to 15 Gy.

• Data exclusion criteria: Experiments with negative LQM coefficients or from exposure to ions of LET above 500 keV/µm were excluded. Experiments with LET above 500 keV/µm, representing about 12% of the PIDE data, covered sparsely distributed LET values over a wide range up to 16,500 keV/µm. Experiments consisting of fewer than two independent observations or recording responses on fewer than five different doses were excluded.

The output parameters comprised SF, α, β, D₁₀, and MID. RBE₁₀ (≡ D₁₀,_x-ray/D₁₀_,ion) was calculated within the custom loss function to be utilized during the NN optimization. After pre-processing, the final dataset comprised 5,166 individual SF measurements, derived from 629 unique experiments. An experiment is defined by a unique combination of the features (‘cell type’, ‘cell class’, ‘cell origin’, ‘cell cycle phase’, ‘DNA content, ‘ion mass number’, ‘irradiation condition’, ‘LET’) and a series of SF measurements at varying doses. The final dataset was structured such that each data instance represented a single dose point consisting of (1) input features (biological and physical conditions, and dose), (2) target reference values including the specific SF at that dose, and (3) the radiosensitivity parameters (D₁₀, MID, and RBE₁₀). Radiosensitivity parameters were repeated across all data instances within a unique experiment. This structure ensures that the model learns to predict consistent curve parameters regardless of the specific input doses. Furthermore, this repetition implicitly weights the influence of these parameters by the number of SF data, ensuring that experiments defined by higher data density, potentially better characterized, are prioritized during model training. A statistical summary of the processed PIDE data is presented in Table 1.

Table 1.

Summary of pre-processed datasets from the PIDE database for training and testing the NN model.

	Parameter	Details	Number of unique categories/range
Biological features	Cell type	Included cell types are in footnote*	102
	Cell class	Tumor, Normal ​	2
	Cell origin	Human, Rodent	2
	Cell-cycle phase	Asynchronous, G0, G0/G1, G1, G1/G0, G1/S, M, S, Sl, Sm	7
	DNA content (10⁹ bp)	5.4, 5.6, 6.0	3
Physical features	Irradiation condition (beam modulation)	Mono-energetic beam, Spread-out Bragg Peak (SOBP) ​	2
	Ions (ion mass number)	¹H, ²H, ³He, 4He, 6Li, ⁷Li, ¹⁰B, ¹¹B, ¹²C, ¹⁴N, ¹⁶O, ²⁰Ne, ²⁸Si, ⁴⁰Ar, ⁵⁶Fe.	15
	LET (keV/µm)	Dose‑ or track‑mean linear energy transfer	(0.52–493) keV/µm
	D (Gy)	Absorbed dose from experimental SF curves	(0–10.0) Gy
Radiobiological response parameters	SF	Surviving fraction after ion irradiation	(10⁻⁵ – 1.00)
	α (Gy⁻¹)	Linear LQ coefficient, ion fit	(0.0–3.61) Gy⁻¹
	β (Gy⁻²)	Quadratic LQ coefficient, ion fit	(0.0–1.07) Gy⁻2
	D₁₀ (Gy)	Dose for 10% survival, ions	(0.64–9.49) Gy
	MID (Gy)	Mean inactivation dose, ions	(0.28–4.57) Gy
	RBE10	Relative biological effectiveness of ions at 10% survival	(0.45–5.91)

* Included cell types: 92-1, A-172, A-172mp53, A-172neo, A-549, AA, AA8, AG01522, AG1522, AG1522B, Aprt, B16, Becker, C32TG, C3H10T1/2, CGL1, CHO, CHO-10B, CHO-K1, CRL-1500, Caski, ChangHL, Colo679, DLD1, EUE, H1299-wtp53, H1299tp53, H1299tp53-null, H1437, H4, H460, HCT116, HE, HE20, HEK, HEP2, HF19, HFIB15, HFIB2, HFIB30, HFL-I, HFL-III, HL-60, HMF, HMV-I, HMV-II, HSF, HSG, HTh7, HeLa, HeLaS3, IEC-6, IGR, KNS-60, KNS-89, KS-1, L5178Y, LC-1sq, LN229, LS-174T, M/10, M10, M3-1, MEF, MOLT4, Marcus, MeWo, NB1RGB, OCUB-M, ONS-76, ONS76, PDV, PDVC57, PS1, RAT-1, SASmp53, SCC25, SCC61, SF126, SHE, SK-MG-1, SL3-147, SQ20B, T1, T98G, TK1, U-251MG, U-251MGKO, U-343MG, U-87, U-87MG, V3, V79, irs1, irs1SF, irs2, xrs5, AT1OS, B16-F0, H1299wtp53, SW-1573, and clone431.

Feature engineering

The following feature engineering steps were performed to prepare the dataset for effective modeling:

• Categorical encoding: Non-numeric features (‘cell type’, ‘cell class’, ‘cell origin’, ‘cell cycle phase’, and ‘irradiation condition’) were transformed into numerical representations by assigning a unique integer to each distinct category. These integers served as indices for the embedding layers (described in Sect. 2.3), mapping discrete categories to continuous vector representations. This approach enables the model to learn relationships between categories without imposing unintended ordinal assumptions^23–25.

• Numerical feature scaling: Numerical features (‘DNA content’, ‘ion mass number’, ‘LET’) were normalized such that their maximum values equalled one. This standardized the range of these features, preventing variables with larger magnitudes from dominating outcomes due to differing scales²⁶.

Model architecture and training

The model architecture utilized a multi-branch feedforward neural network implemented using the TensorFlow Keras library²⁷. Keras provides a user-friendly high-level application programming interface (API) for building and training deep learning models.

The primary design objective was to achieve constrained predictions of SF that are consistent with the LQM formalism across diverse cell lines and irradiation conditions. The model was optimized to minimize the prediction error for SF_pred compared to the reference SF_ref, while also minimizing errors in derived radiosensitivity parameters (MID, D₁₀, and RBE₁₀) (see Sect. 2.4 for the detailed loss function).

The network processes the input features (‘cell type’, ‘cell class’, ‘cell origin’, ‘cell cycle phase’, ‘DNA content’, ‘irradiation condition’, ‘ion mass number’, ‘LET’) to predict the α and β values. These parameters, along with the input ‘dose’, are fed into custom layers that calculate the final target variables SF, MID, D₁₀, and RBE₁₀. This architecture ensures that the predicted α and β parameters and the resulting derived SF(D), D₁₀, MID, and RBE₁₀ encapsulate information of the entire survival curve and that the model can map input features to the entire dose-response curve anchored by the LQM, rather than merely fitting individual SF data points. Consequently, this reduces model sensitivity to experimental noise and the need for a large training dataset²¹.

Figure 1 depicts the overall architecture of the model, illustrating the different processing blocks from input parameters to predicted outputs. The network starts with processing the input categorical features (‘cell type’, ‘cell class’, ‘cell origin’, ‘cell cycle’, and ‘irradiation condition’) and the numerical features (’DNA content’, ‘ion mass number’, ‘LET’, and ‘dose’). Categorical features are processed through embedding layers transforming each discrete category into a continuous vector representation. The embedding dimension was set equal to the number of unique categories for each respective feature to enable distinct representation of the unique traits. These representations are optimized via backpropagation during training, allowing the model to capture latent relationships among categories (e.g., grouping similar cell types)²⁵.

Fig. 1.

Block diagram of the multi-branch neural network, showing the flow from data embedding and processing through separate and combined branches to α and β estimation, and ultimately the calculation of the target variables (SF, MID, and D₁₀) from ion-beam exposure. The diagram details the calculation of the MSE_Total, computed as the weighted sum of individual loss components (MSE_k), where k denotes the specific target variable index (SF, D₁₀, MID, RBE₁₀). The weighting factors (w_k) are the weights for specific loss components. The bold red dashed line indicates the backpropagation process used to update the network weights and biases based on MSE_Total.

Following initial processing and embedding, features are channeled into distinct branches reflecting biological properties (‘cell type’, ‘cell class’, ‘cell origin’, ‘cell cycle phase’, ‘DNA content’) only (Bio network), physical properties (‘irradiation condition’, ‘ion mass number’, ‘LET’) only (Phys network), and both properties (Phys-Bio network). This multi-branch approach, consisting of distinct learnings, allows for the development of specialized representations^28,29. These three branches are processed separately through hidden layers before being combined and fed into a final shared pathway for predictions. Each processing pathway consisted of three fully connected hidden layers with 128 neurons per layer, utilizing the ReLU activation function []. The outputs of these three branches were concatenated to form a vector of size 384, which served as the input for the final shared pathway. The final shared pathway subsequently passes through another three fully connected hidden layers (128 neurons each) before connecting to preliminary output layers for  and These output layers employ a scaled sigmoid activation function [] to constrain the predicted and values within a reasonable range from zero to a general maximum value exceeding their respective maxima in the reference data (​∈ [0.0, 4.0 Gy⁻¹], β ∈ ​[0.0, 1.1 Gy⁻²]).

These were set to be general limits that could potentially apply to irradiation experiments beyond the scope of the training data. These limits were rarely exceeded in the full PIDE database. The few outliers mostly involved negative β terms that produce upward-bending survival curves, which is biologically implausible and likely caused by noisy data. The largest parameter values of (​, ​) in the training datasets were (3.61 Gy⁻¹, 1.07 Gy⁻²), respectively.

The predicted LQM parameters ( and ​) together with the input dose are then passed to the following custom layers dedicated to deriving the final target variables data:

• SF calculation layer calculates the SF using the LQM formula: ;

• D₁₀ calculation layer calculates D₁₀ by analytically solving the LQM equation for D when SF = 0.1; and

• MID calculation layer calculates MID​ by numerically integrating the predicted survival curve applying the trapezoidal rule to integrate the SF curve, reconstructed using the LQM equation, at 0.1 Gy intervals over the dose range from 0 to 15 Gy.

Custom loss function

A composite custom loss function was implemented to guide the NN training, enabling simultaneous optimization of SF predictions and the derived radiosensitivity metrics. This loss function integrates multiple mean squared error (MSE) terms and boundary penalty terms.

The loss function comprises weighted MSE terms for the primary target variables: SF, D₁₀, and MID. An MSE term for RBE₁₀ (calculated within the loss function as (D₁₀,_x-ray/D₁₀_,ion) was also included. The reference photon dose (D₁₀,_x-ray) is utilized exclusively within the custom loss function for RBE₁₀ calculation. This approach contrasts with tree-based models, which treat all provided variables as potential split features without distinguishing between predictive and auxiliary inputs. Boundary penalty term () is incorporated for each of D₁₀, MID, and RBE₁₀ to enforce biological plausibility and prevent extrapolation into unphysical regimes. The penalty, calculated using Rectified Linear Unit (ReLU) functions, activates selectively when prediction (ŷ) falls below the minimum or exceeds the maximum values observed in the training data. The target variables α and β are optimized indirectly via the dependent variables (SF, MID, D₁₀, and RBE₁₀). Prior work^8–10 and our initial testing indicated large uncertainty in β estimation. Therefore, we excluded direct MSE terms on α and β to avoid simultaneous supervision of correlated quantities and to mitigate noise propagation into the network optimization, while preserving their role in shaping the predicted dose-response curve.

Each component of the loss function (MSE term and boundary penalty) is scaled by its own weight () calculated as the inverse variance of the corresponding target variable in the training dataset ( where represents the variance of the target variable in the training data. This scheme accounts for the varying magnitudes and physical units of different target variables by standardizing the contribution of each term to the total loss³⁰. This ensures that the model optimization is not dominated by target variables with larger magnitudes. Table 2 summarizes the loss components and their associated weights in the custom loss function.

Table 2.

Loss components considered to guide NN training. The loss value () is calculated between predicted (ŷ) and reference (y) values. The total loss () is computed as the sum of component weighted loss values (), where k and n refer to the specific loss component and the number of training data points, respectively.

Loss component	Loss weight	Weighted loss term
[]

The Adam optimizer³¹ was utilized to optimize the network parameters (weights and biases), including the embedding layer weights, throughout the training process. The NN internal learnable parameters, referred to as its weights and biases, are iteratively adjusted to minimize the overall loss () during training. At each training iteration, the custom loss function evaluates the model predictions against the reference values to compute as the weighted sum of individual loss components (Table 2). Gradients of with respect to all network weights and biases are computed using the standard backpropagation algorithm^27,32. The optimizer uses these gradients to update the network weights and biases aiming to reduce prediction error in subsequent iterations.

Model training and evaluation

The NN model, incorporating the custom architecture and loss function detailed in Sect. 2.3 and 2.4, was trained and evaluated using a 10-fold cross-validation strategy on the prepared dataset as described in Sect. 2.1. This process involved training ten separate models: each model was trained on nine training data folds (90% of the data) and evaluated on the remaining data fold (10% of the data). Each data fold served as a test set exactly once. Integer indexing of categorical features (categorical encoding) was performed at the beginning of full training to ensure the same index mapping was applied consistently across training and test datasets. These integer indices were then fed into the embedding layers, whose vector representations were learned during the training process. The model training and evaluation proceeded as follows.

Data partitioning

The entire dataset was divided into ten data folds, ensuring that each fold contains mutually exclusive random sets of experiments and that individual SF measurements from the same experiment belong to the same fold. Fold sizes remained highly balanced, ranging from 516 to 517 data points. This experiment-based stratification prevented data leakage between training and testing sets and provided robust assessment of model performance on entirely new experimental conditions.

Independent model training

Numerical features in the training data were normalized (as described in Sect. 2.2). Normalization was then applied to the test-fold. The normalized training data were further randomly divided into one training subset (about half the data in nine folds) and the other validation subset. The validation subset was used to monitor model performance during training and to stop training before overfitting the model to the training data.

Ten separate models were trained with a unique combination of nine data folds. For each model, the weights of loss components () were set to the inverse variance of the respective target variables, and the model was trained in two main cycles:

1. First cycle: The model was trained using the training subset for up to 800 cycle iterations (epochs). After each epoch, the custom loss function computed the validation loss () on the validation subset and network weights were updated via backpropagation as detailed in Sect. 2.4. Throughout the 800 epochs, the network configuration (weights, biases) that resulted in the smallest validation loss were monitored and saved. This cycle allowed for the selection of an optimal starting point for the second cycle.

2. Second cycle: The model was further trained using both training and validation subsets of the nine training data folds for an additional 100 epochs, starting with the best network configuration identified in the first cycle. The trained model was used to make predictions on its corresponding test data fold.

Cross validation and statistical robustness

Following the training of all ten models, predictions of target variables from the ten mutually exclusive test datasets were aggregated to form a complete set of predictions for the entire dataset.

The statistical significance and stability of the predictions were assessed by repeating the entire ten-fold cross-validation process (Subsections 2.5.1 to 2.5.3) five times using different random seeds for data partitioning and weight initialization. The predicted values for each individual data point were averaged across these five independent runs, and the performance metrics were determined based on these mean predictions.

The coefficient of determination (R²) was calculated for each target variable by comparing the predictions against their corresponding reference values from experiments as follows:

where y_i is the experimental reference value, ŷ_i is the predicted value, and ȳ is the mean of the reference values.

Analysis of feature importance

For each of the ten models across the five runs, feature permutation analysis was performed to assess the relative importance of input features in predicting SF and radiosensitivity parameters including D₁₀, MID, and RBE₁₀. Feature importance was assessed by independently permuting the values of each input feature and measuring the resulting reduction in model performance¹².

To evaluate each input feature j, its values in the test dataset were randomly shuffled times, while keeping all other features unchanged. The importance score () was calculated as the mean reduction in the performance metric () resulting from the permutation compared to the baseline performance () obtained with undisturbed test dataset:

where represents the score obtained in the k^th permutation of feature j.

Results and discussion

Model performance

The NN model demonstrated consistent prediction performance for SF and radiosensitivity parameters across five independent training runs. SF predictions were highly correlated with experimental data (R²0.954). Prediction performance for SF was stable across different types of ions, with R² of approximately 0.94, 0.95, and 0.96 for protons, carbon ions, and other ion types, respectively (Fig. 2a). Prediction performance for RBE₁₀ varied depending on the ion type (R² = 0.48, 0.67, and 0.73 for proton, carbon, and other ions, respectively), resulting in an overall correlation of R²= 0.75 (Fig. 2b).

Fig. 2.

Evaluation of the NN model performance with PIDE as reference data: in prediction of (a) SF and (b) RBE₁₀, across different ion types; (c) compared to the LEM version IV in RBE₁₀ prediction; (d) compared to the LEM in RBE₁₀​ prediction as a function of LET across different ion types; and (e) RBE₁₀ predictions as a function of LET for individual ion types. Data presented in (c) and (d) are restricted to the data points (n= 320) corresponding to those reported by Pfuhl et al.¹⁵.

The prediction performance of the NN model for RBE₁₀ was compared with the LEM version IV¹⁵ against PIDE reference data. Figure 2c is a comprehensive view of the RBE₁₀ predictions across different ion beams specified by a broad range of LET values. The NN model shows a better performance in RBE₁₀ prediction. Figure 2d shows the trend of RBE₁₀ predicted by the NN and LEM IV models, as a function of LET across the range of up to 500 keV/µm, in comparison with that of PIDE data. The RBE₁₀ predictions of both models followed the trend observed with the PIDE dataset peaking at comparable LET values around 150 keV/µm. RBE₁₀ predictions by the NN model were more closely aligned with experimental PIDE values compared to LEM estimates. Comparisons with LEM were limited to the data categories reported by Pfuhl et al.¹⁵.

LEM IV underestimates RBE at low LET for all ion species and tends to overestimate RBE for light ions such as protons and helium at high LET, while showing good agreement for heavier ions at high LET¹⁵. Additionally, experimental LET–RBE₁₀ curves exhibit ion-specific behaviour, with distinct peak heights and curve shapes for each ion³³. Data fitting might be optimized with separate models for individual ion types, however, this would substantially reduce both the data range and the number of training samples available for each model. This study adopted a unified modelling approach by pooling all ions together therefore provides a more comprehensive training space, enabling the model to capture common patterns across radiation qualities while still accommodating ion-specific behaviours.

The NN model accommodates these complexities by learning the underlying nonlinear dependencies directly from experimental data. Table 3 summarizes the performance of the NN model across all ions. Model predictions closely followed the reference RBE₁₀ values, reproducing the expected trend that RBE₁₀ approaches peak values at LET of around 150 keV/µm and that the peak values are higher with lighter ions (e.g., ⁴He) than with heavier ions (e.g., ¹²C, ²⁰Ne) (Table 3; Fig. 2e). The variation in RBE₁₀ across different ions at similar LET values (Fig. 2e), together with the close agreement between the predicted and reference mean RBE₁₀ values for each ion (Table 3), confirms that the model captures the ion-specific trends rather than a common single trend.

Table 3.

Performance of the NN model in RBE₁₀ prediction with PIDE as reference data. Results are presented for the aggregate dataset (Overall) and stratified by ion type.

Ion	N*	Max. RBE₁₀		Mean RBE₁₀		R²
		Reference	Prediction	Reference	Prediction
Overall	629	5.91	5.71	2.12	2.09	0.75
¹H	139	3.49	2.92	1.36	1.33	0.48
²H	15	3.81	3.48	2.11	1.83	0.65
³He	32	4.00	4.70	2.56	2.74	0.89
4He	57	5.91	5.71	2.65	2.71	0.77
6Li	2	4.71	4.73	4.42	4.43	-
⁷Li	4	4.02	2.99	2.73	2.12	0.95
¹⁰B	2	3.25	2.86	2.60	2.55	-
¹¹B	3	4.15	3.81	3.19	3.17	0.30
¹²C	249	5.05	4.31	2.16	2.12	0.67
¹⁴N	7	5.55	3.61	3.19	2.68	0.16
¹⁶O	7	2.89	2.87	2.21	2.15	0.95
²⁰Ne	76	3.91	3.40	2.48	2.43	0.74
²⁸Si	9	2.69	2.62	2.21	2.24	0.70
⁴⁰Ar	16	3.01	3.09	2.33	2.38	0.48
56Fe	11	3.09	2.66	2.1	2.01	0.23

*Number of experimental data sets.

Strong correlations were observed in predictions of D₁₀ values (R² = 0.79) and MID values (R² = 0.78) (Fig. 3a and b). Predicting α and β​ parameters using the NN model was challenging, even though the model successfully captured the overall dose-response trend. The α predictions showed a moderate correlation (R² = 0.65) with reference values (Fig. 3c), whereas the performance in β​ prediction was significantly weak (R² = 0.01) (Fig. 3d). These results are consistent with those reported in previous studies^8,9,11.

Fig. 3.

Performance of NN model with PIDE as reference data in prediction of: (a) D₁₀, (b) MID, (c) α, and (d) β across different ion types.

The robust prediction of RBE₁₀ (R² = 0.75) (Fig. 2b), observed despite poor β prediction and a slight underestimation of high α values (Fig. 3c), is attributed to the stabilizing effect of multi-objective optimization. Multiple (α, β) pairs can generate statistically similar survival curves over a sampled dose interval⁶. Optimizing the model depending on MSE terms of derived quantities (SF, D₁₀, MID) and excluding those for (α, β) parameters, the model prioritizes the functional shape of the survival curve over the specific numerical values of α and β. Thus, the resulting dose-response trend, characterized by the predicted pair (α, β), remains consistent even when predictions of individual parameters exhibit variation. The quality of the predicted α and β was further verified by calculating the continuous RBE across dose levels; the model maintained reliable accuracy in RBE prediction across the survival curve (R² ranging from 0.70 to 0.75, at SF of 0.9 to 0.1).

Feature importance

Feature permutation analysis was performed to assess the relative importance of input features in predicting SF and radiosensitivity parameters including D₁₀, MID, and RBE₁₀. Feature importance was assessed by independently permuting the values of each input feature and measuring the resulting reduction in model performance¹².

Dose was by far the dominant predictor for SF. For intrinsic radiosensitivity parameters (D₁₀, MID​, and RBE₁₀​), LET and cell line specificity were the most important predictive features. This is consistent with the radiobiological understanding that radiation quality (expressed in terms of LET) and cellular properties are key determinants of response. D₁₀​, MID, and RBE₁₀ are determined by the SF curve, which shows the trend of SF as a function of dose. These results confirm that the network prioritizes fundamental physical determinants (e.g., dose for SF, LET for RBE) rather than exploiting statistical shortcuts or confounding variables in the training data.

The ion type, expressed in ion mass number, played a secondary role in D₁₀, MID, and RBE₁₀ prediction​. Other biological features including cell cycle phase, cell origin, and DNA content and physical feature of irradiation condition were of relatively low importance, and cell class, whether normal or tumoral, appeared to have little predictive impact. LET and cell-line specificity, features that have a major impact on D₁₀, MID, and RBE₁₀ prediction, have a moderate impact on SF prediction.

Fig. 4.

Permutation feature importance scores for SF, D₁₀, MID, and RBE₁₀. Error bars indicate the standard errors of feature importance across the five repetitions of 10-fold cross-validation.

Discussion

The performance of the developed NN framework (R² = 0.95 and 0.75 for SF and RBE₁₀, respectively) is comparable to prior data-driven approaches. For example, the mean absolute percentage error (MAPE) for RBE₁₀ predictions was 15% in this study compared to 26% using the ANAKIN model¹¹. The results in this study are in general agreement with prior data-driven approaches that ML models can significantly improve prediction power for cell survival and RBE compared to purely mechanistic models ^10,11,15.

The primary distinction in this study lies in the architectural enforcement of biological constraints by explicitly embedding the LQM within the architectural layers of the NN. This enforced the prediction of monotonic dose-response curves governed by non-negative radiosensitivity parameters (α and β), thereby preventing invalid extrapolations often observed in unconstrained ML models. Furthermore, interpretability is inherent to this architecture as the network learns to map biological and physical features directly to the mechanistic LQM parameters instead of learning shortcuts to isolated target variables.

Despite these advantages, the study is subject to certain limitations. First, the prediction capability of the model is intrinsically bound by the quality and diversity of the PIDE training data. The inherent heterogeneity and imbalance within the PIDE database, such as the scarcity of data for certain ion species compared to protons and carbon ions, may bias the learning process, potentially resulting in lower prediction accuracy for under-represented exposure conditions. Second, the model assumes the validity of the LQM across all conditions. While being the key for obtaining consistent predictions, this may limit the ability to capture complex phenomena such as low-dose hypersensitivity or saturation effects at very high doses that deviate from the standard LQM.

Conclusions

This study successfully developed and validated a NN framework that directly incorporates LQM within its architecture and training process. The model demonstrated high prediction accuracy for SF across diverse radiation qualities and cell types, achieving an R² value of approximately 0.96. Furthermore, it provided robust predictions for key radiosensitivity parameters such as D₁₀​, MID, and RBE₁₀​ and reasonable estimates for the LQM parameter α. Incorporating these radiobiological constraints through custom layers embedding LQM and a customized loss function ensured that model predictions remained consistent with radiobiological models.

Author contributions

A.A. developed computational models and analyzed the data. E.K. organized research and interpreted the results. A.A and E.K wrote the manuscript.

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.