Enhancing X-Ray and Gamma-Ray Detector Calibration via AI-Driven Digital Twins: Predicting Extracorporeal Photon Emission from OpenDose Specific Absorbed Fraction Datasets Using Uncertainty-Aware Transformer Ensembles

Abstract

Patient-specific dosimetry and quantitative external counting for in vivo monitoring of radiopharmaceutical therapies require detector calibration coefficients (k) that are currently obtained through computationally intensive Monte Carlo (MC) simulations. We present a digital-twin framework that learns the mapping from organ-level Specific Absorbed Fractions (SAFs) to clinical k values for X-ray and gamma-ray detectors, trained on 4.47 million SAF entries from the OpenDose collaboration covering the ICRP-110 adult male (AM) and adult female (AF) reference phantoms, 91 photon energies (5–10,000 keV), and 336 source organs. An uncertainty-aware ensemble combining Histogram Gradient Boosting, ExtraTrees, Random Forest, Quantile-HGBT, and a Feature-Tokenizer Transformer (414 K parameters) was stacked via ridge regression and calibrated using Conformalized Quantile Regression. The ensemble achieved a test mean absolute error of 0.220 in log10-SAF space (R² = 0.921) with an empirical 95% prediction interval coverage of 95.0%. Validation against 10,487 independent published S-values (EMDOSE) yielded a Pearson correlation of 0.990 (log space). The framework reduces k-coefficient computation time from hours of MC simulation to milliseconds, supporting real-time detector calibration for theranostic workflows.

1. Introduction

Internal dosimetry in nuclear medicine relies on the Medical Internal Radiation Dose (MIRD) formalism, in which the absorbed dose to a target region is the product of the time-integrated activity in a source region and an S-value that depends on radionuclide decay characteristics and on the Specific Absorbed Fraction (SAF) of the irradiated anatomy [1,2]. The development of voxel-based reference phantoms, most notably the ICRP Publication 110 adult male (AM) and adult female (AF) computational phantoms [3], and of the associated reference anatomical and physiological values [4] has enabled highly standardised computation of SAFs for radiation protection and nuclear medicine. Building on this work, the ICRP Publication 133 framework tabulated SAFs for a wide range of photon and electron energies and source–target combinations [5].

More recently, the OpenDose collaboration has released independently cross-verified SAF datasets produced by six Monte Carlo (MC) codes (EGSnrc, FLUKA, GATE, Geant4, MCNP and PENELOPE) covering the ICRP-110 AM and AF phantoms [6]. The OpenDose dataset, openly available with statistical uncertainties, is now a community reference, and the associated OpenDose3D extension for 3D Slicer supports patient-specific dosimetry [7]. Olguin and co-workers complemented these resources with the EMDOSE library of S-values for clinically relevant therapeutic radionuclides, also derived from the ICRP-110 phantoms [8].

In parallel, peptide-receptor and α-emitter radionuclide therapies (e.g., [¹⁷⁷Lu]Lu-DOTATATE, [²²³Ra]RaCl₂, [¹³¹I]NaI) have moved towards personalised, image-based dosimetry [9,10]. The dominant computational obstacle is the Monte Carlo simulation needed to compute either dose-point kernels, voxel S-values or detector calibration coefficients [10,11]. Machine-learning (ML) surrogates have therefore attracted growing attention as fast, scalable alternatives. Lee et al. [12] introduced Deep-Dose, a convolutional architecture for voxel dose estimation; Akhavanallaf et al. [13] trained a deep network to predict whole-body voxel dose for [¹⁸F]FDG; Li et al. [14] proposed DblurDoseNet for resolution-compensated voxel dosimetry; Mansouri et al. [15] applied transformer architectures to [¹⁷⁷Lu]Lu-DOTATATE; Scarinci et al. [16] used computational intelligence to predict dose-point kernels; Eleftheriadis et al. [17] developed a fast voxel-level dose estimator for [^99mTc]; and Pastor-Serrano and Perkó [18] reported millisecond-scale deep-learning proton-dose calculation at near-MC accuracy. Brosch-Lenz et al. [19] documented the substantial inter-workflow variability of therapy-relevant endpoints in [¹⁷⁷Lu]Lu-DOTATATE dosimetry [9,15].

On the detection side, semiconductor X-ray and γ-ray detectors, Cadmium Telluride (CdTe), Cadmium Zinc Telluride (CdZnTe) and high-purity germanium (HPGe), provide the spectroscopic backbone of whole-body counters, hand-held surveymeters and emerging intraoperative γ-cameras [20,21,22]. Calibrating such detectors for organ-localised activity quantification requires either physical anthropomorphic phantoms or extensive MC simulations that propagate the source spectrum through the phantom and detector response [23]. A practical bottleneck is that calibration coefficients depend on photon energy, source organ, phantom and detector geometry, leading to hundreds of MC runs per clinical scenario [1].

From a methodological perspective, recent advances in tabular deep learning offer a promising path. The Feature-Tokenizer Transformer (FT-Transformer) of Gorishniy et al. [24], built on the original transformer architecture [25], has been shown to be competitive with gradient-boosted trees on heterogeneous tabular data while supporting principled uncertainty estimation when combined with deep ensembles [26] or Monte Carlo dropout [27]. Conformalized Quantile Regression (CQR) [28] provides a distribution-free, finite-sample-valid mechanism for calibrating prediction intervals, complementing the well-established gradient boosting [29,30] and tree ensemble [31] baselines. Bayesian and ensemble methods for predictive uncertainty have been reviewed in [32] and form the basis of much current dosimetry machine learning [15,19].

Despite this progress, several gaps remain. First, no published ML surrogate has been trained on the cross-verified OpenDose SAF database (https://www.opendose.org, accessed 10 May 2026 [6]) which uniquely eliminates single-MC-code bias [1]. Second, existing dosimetry ML studies focus almost exclusively on dose maps inside the patient body; the inverse problem, predicting the extracorporeal photon emission spectrum reaching an external detector, and from there the detector calibration coefficient, has not been Conformalized Quantile Regression (CQR) is well established in the statistics literature [28], its application to high-dimensional internal-dosimetry surrogates has not been reported. The present work addresses these three gaps by introducing an uncertainty-aware digital twin that: (i) ingests the full OpenDose AM/AF SAF dataset (5.14 million entries; 91 photon energies; 338 source organs); (ii) stacks five complementary tabular learners, including an FT-Transformer trained on GPU, within a ridge meta-learner with CQR-calibrated 95% prediction intervals; (iii) propagates the predicted SAFs to clinically relevant detector calibration coefficients for four representative spectroscopy chains (HPGe whole-body counter, CdTe γ-camera, NaI(Tl) surveymeter, CdZnTe spectrometer) Panel colours are used only to group conceptually related elements (blue: input data and tree-based learners; green: phantom and physics inputs; orange: external validation data; red: the deep learning model; yellow: the final ensemble and its outputs) and carry no quantitative meaning.; and (iv) is independently validated against 10,487 published S-values from the EMDOSE library [8]. The framework is openly released under an open-source licence and is designed to be deployed as the calibration back-end for the workflows envisioned by the COST Action CA24131-ENRICH consortium (https://www.cost.eu/actions/CA24131/, accessed 10 May 2026) [8,28].

2. Materials and Methods

An overview of the computational workflow is shown in Figure 1. The pipeline ingests the openly available OpenDose Monte Carlo SAF dataset, performs phantom-aware feature engineering, trains a heterogeneous ensemble of five base learners, fuses their predictions through a ridge stacker, and calibrates the resulting 95% prediction intervals via Conformalized Quantile Regression. Downstream modules translate the predicted SAFs into clinical S-values and detector calibration coefficients (k) and validate them against the independent EMDOSE library [8]. Each stage is checkpointed and re-entrant to support unreliable GPU runtimes.

2.1. Data Sources

Three openly available datasets were combined. (i) The OpenDose Monte Carlo SAF dataset for the ICRP Publication 110 reference phantoms [3,6] was retrieved as an optimised parquet bundle (84 MB, SHA-256 archived in the Supplementary Materials). OpenDose data is produced independently by six Monte Carlo codes within the collaboration, EGSnrc, FLUKA, GATE, Geant4, MCNP and PENELOPE, and each SAF value is cross-verified by an automated discrepancy-detection workflow, eliminating single-code physics-model bias [6]. (ii) Organ volumes and masses for both phantoms were extracted from the OpenDose adult reference supplement data (v1.2) [3,6], calibrated to the reference anatomical values of ICRP Publication 89 [4]. (iii) For external validation, the EMDOSE S-value library [8] was retrieved as 24 JSON files spanning [¹³¹I], [¹⁷⁷Lu] and [²²³Ra], the three radionuclides for which alpha, beta and gamma S-values are tabulated for both AM and AF phantoms. Full data-provenance details, including download URLs, file hashes and parsing scripts, are provided in Supplementary Note S1. Principal photon emissions of the therapeutic radiopharmaceuticals used in the external validation of this work are as follows: [¹³¹I] emits gamma rays at 364.5 keV (81.5% intensity), 636.99 keV (7.1%), 284.3 keV (6.1%) and 80.2 keV (2.6%); [¹⁷⁷Lu] emits gamma rays at 112.9 keV (6.2%) and 208.4 keV (10.4%); [²²³Ra] and its short-lived daughters emit gamma rays predominantly at 269.5 keV (13.7%), 154.2 keV (5.6%), 351.1 keV (13.0%) and 401.8 keV (6.6%) [1].

2.2. Computational Environment

All experiments were executed in Google Colaboratory (Google LLC, Mountain View, CA, USA; paid Pro tier) using a single NVIDIA Tesla T4 GPU (NVIDIA Corporation, Santa Clara, CA, USA; 16 GB VRAM) for the FT-Transformer and CPU-only in Google Colaboratory (paid Pro tier) (51 GB RAM) for the gradient-boosting and tree-ensemble baselines. The software environment was Python 3.10.12, NumPy 2.0.2, pandas 2.2.2, scikit-learn 1.5.2 [31], LightGBM 4.5 [30], XGBoost 2.1 [29], and PyTorch 2.10.0 with CUDA 12.8 [33]. An auxiliary FT-Transformer reference implementation was adapted from the RTDL library [24]. Random seeds were fixed at 42 throughout for reproducibility. Each pipeline stage was checkpointed (joblib + parquet) and mirrored to Google Drive to survive runtime disconnections. Total wall-clock time on the cited hardware was 38 min; a full per-stage breakdown is given in Supplementary Table S1.

2.3. Dataset Construction and Pre-Processing

After parsing the OpenDose archives, the SAF database contained 5,136,768 entries (AM: 2,583,672; AF: 2,553,096) over 91 photon energy bins from 5.0 keV to 10 MeV. 169 AM source organs and 168 AF source organs target a unified 173-organ vocabulary (union of AM and AF region nomenclature) [3,6]. Entries with strictly zero SAF (forbidden by geometry or below MC statistical sensitivity) were removed, yielding 4,474,288 non-zero training samples. The target variable was the natural logarithm of the SAF (in kg⁻¹), used throughout to symmetrise the dynamic range that spans more than ten orders of magnitude. Phantom anatomy (AM: 73.0 kg, voxel volume 36.5 mm³; AF: 60.0 kg, voxel volume 15.25 mm³) is summarised in

Table 1. Dataset composition. SAF: Specific Absorbed Fraction; AM/AF: ICRP-110 Adult Male/Adult Female reference phantoms; EMDOSE: independent S-value library used for external validation [8]. Voxel sizes and region counts are taken from the OpenDose phantom metadata; AM has fewer voxels but a larger voxel volume than AF, leading to comparable total tissue mass but different Monte Carlo statistical sensitivities.

Item	Value
Phantoms	2 (ICRP-110 AM, AF) [3]
AM total mass/height	73.0 kg/1.76 m
AM voxel size/count	36.53 mm3/8,193,532 voxels
AM number of regions	169
AF total mass/height	60.0 kg/1.63 m
AF voxel size/count	15.25 mm3/14,255,124 voxels
AF number of regions	168
Source organs (AM/AF)	168/167
Target organ vocabulary	173
Photon energies	91 (logarithmic spacing)
Energy range	5.0–10,000 keV
Total SAF rows	5,136,768
Non-zero SAFs (training)	4,474,288
Features per sample	9
Mass coverage (AM/AF direct)	169/169/168/168
External validation pairs (EMDOSE) [8]	10,487
Monte Carlo codes (OpenDose) [6]	EGSnrc, FLUKA, GATE, Geant4, MCNP, PENELOPE

. As a sanity check, the AM liver self-SAF at 100 keV computed from the parsed parquet was 0.1005 kg⁻¹, consistent with the value reported by Bolch et al. [5]. The AM voxel volume (36.53 mm³) is larger than the AF voxel volume (15.25 mm³) because the ICRP-110 male phantom was segmented at coarser axial resolution (8.0 mm slice thickness) than the female phantom (4.84 mm slice thickness) [3]; this reflects the historical origin of the two datasets and leads to correspondingly different Monte Carlo statistical sensitivities per voxel that OpenDose accounts for through its per-entry uncertainty tables. A complete uncertainty budget covering the MC statistical, feature-tokenisation, model epistemic, aleatoric and CQR-calibration contributions is provided in Supplementary Table S7 and Supplementary Figure S3.

2.4. Feature Engineering

Each training sample was represented by nine features: (i) phantom index (categorical, AM/AF); (ii–iii) integer indices of source and target organ over the unified vocabulary; (iv) the natural log of the photon energy in MeV; (v–vi) the log10 of the source-organ and target-organ masses in grams; (vii–viii) a binary self-irradiation flag (source = target) and a binary phantom-sex flag; and (ix) the OpenDose statistical uncertainty when reported. Continuous features were standardised to zero mean and unit variance; categorical indices were left as integers for tree-based models and passed through learned embeddings of dimension 32 for the FT-Transformer. The full feature-engineering script is reproduced in Supplementary Note S2.

2.5. Train/Test Split and Validation Strategies

Five complementary validation schemes were employed (Figure 1d). (1) Random 70/15/15 train/validation/test split (3,132,001/671,143/671,144 samples) provided an upper bound on in-distribution performance. (2) Leave-one-source-organ-out (LOSO) was performed for five anatomically distinct organs: Liver, Brain, Thyroid, Spleen and Pancreas, to estimate generalisation to unseen source geometries. (3) Leave-one-energy-out (LOEN) was performed at 50, 300 and 2000 keV to probe extrapolation across the photon-energy axis. (4) Cross-phantom transfer (AM → AF and AF → AM) examined sex-specific anatomical transfer. (5) External validation used 10,487 S-value pairs from the EMDOSE library [8]. Numerical performance for each scheme is summarised in

Table 2. Cross-validation performance of base learners and the stacked ensemble across all five validation strategies. MAE and RMSE are in log10(SAF) space. PI95 cov. = empirical 95% prediction-interval coverage. The cross-phantom and leave-one-out schemes use the full Stack + CQR ensemble (per-fold breakdowns are given in Supplementary Tables S4 and S5). LOSO: leave-one-source-organ-out (organs: Liver, Brain, Thyroid, Spleen, Pancreas); LOEN: leave-one-energy-out (energies: 50, 300, 2000 keV).

Validation Scheme	Model	MAE	R2	RMSE	PI95 cov.
Random 70/15/15	HGBT	0.846	0.529	1.419	–
Random 70/15/15	ExtraTrees	0.309	0.766	1.001	–
Random 70/15/15	Random Forest	0.202	0.862	0.767	–
Random 70/15/15	Quantile-HGBT	0.851	0.378	1.631	93.9%
Random 70/15/15	FT-Transformer [24]	0.304	0.896	0.668	–
Random 70/15/15	Stack + CQR [28]	0.220	0.921	0.581	95.0%
Cross-phantom AM → AF	Stack + CQR	1.011	0.394	–	–
Cross-phantom AF → AM	Stack + CQR	0.977	0.424	–	–
LOSO (median over 5 organs)	Stack + CQR	0.917	0.346	–	–
LOEN (median over 3 energies)	Stack + CQR	0.721	0.519	–	–

(Section 3.1) and discussed in detail in Section 3.2.

2.6. Base Learners

Five complementary models were trained as base learners: a Histogram Gradient Boosting (HGBT) regressor, an ExtraTrees (ET) ensemble, a Random Forest (RF), a Quantile-HGBT producing raw predictive intervals at q = 0.025, 0.500, 0.975 [30,31], and a deep Feature-Tokenizer Transformer (FT-Transformer, hereafter FT) [24,25] with 414,337 parameters trained on a Tesla T4 GPU for 35 epochs. Three bootstrap repetitions of the FT-Transformer with different random seeds were used to estimate epistemic uncertainty [26,27]. All hyperparameter values, including tree depths, learning rates, embedding dimensions and optimiser settings, were chosen from preliminary experiments on 5% of the data and are tabulated in full in Supplementary Table S2; no hyperparameter optimisation was performed on the test set [24,26,28,31].

2.7. Stacked Ensemble and Conformalized Quantile Regression

Base-learner predictions on the held-out validation set were combined through a ridge-regression meta-learner (regularisation strength α = 10⁻³, fitted to validation set targets). The ridge coefficients (HGBT: 0.095; ET: −0.112; RF: 0.512; Quantile-HGBT median: −0.104; FT-Transformer: 0.592) reflect partial multicollinearity among predictors, negative weights act as residual-correction terms rather than as primary predictors, and the dominance of RF and FT-Transformer in the ensemble. Empirical 95% prediction intervals were then calibrated by Conformalized Quantile Regression [28]: on the validation set we computed the conformity score $E_i=\max \left({\widehat{q}}_{\left\{0.025\right\}\left(x_i\right)}- y_i, y_i- {\widehat{q}}_{\left\{0.975\right\}\left(x_i\right)}\right)$, and the (1 − α)-quantile of the $\left|E_i\right|$ set was used to symmetrically widen the test-set quantile predictions. This procedure provides finite-sample-valid marginal coverage [28] without distributional assumptions.

2.8. Detector Calibration Coefficient Propagation

Predicted SAFs were converted to clinical detector calibration coefficients k (in counts per second per becquerel, cps Bq⁻¹, tabulated for practical readability as mcps MBq⁻¹) by convolving the SAF-derived organ photon escape spectrum with a parametric model of the detector intrinsic efficiency ε(E). Four representative spectroscopy chains were evaluated [20,21,22,23]: an HPGe whole-body counter (2.5 in × 2.5 in ≈ 6.35 cm diameter, 6.35 cm thickness planar crystal; density ρ = 5.32 g cm⁻³, effective atomic number Z_eff = 32; ε(E) ∝ E^−0.5 for E > 100 keV), a CdTe portable γ-camera (10 mm × 10 mm × 2 mm side/thickness pixel; ρ = 5.85 g cm⁻³, Z_eff = 50; parallel-hole collimator, ε(E) calibrated to manufacturer data sheets [20,21]), a NaI(Tl) surveymeter (76 mm diameter × 76 mm height cylindrical; ρ = 3.67 g cm⁻³, Z_eff = 50; well-known ε(E) curve from [22]), and a CdZnTe spectrometer (15 mm side × 15 mm side × 7.5 mm thickness; ρ = 5.78 g cm⁻³, Z_eff = 49; ε(E) from [21]). For each phantom × source-organ × radionuclide × detector combination (n = 120 in total), the predicted k coefficient was compared with the value obtained when ground-truth OpenDose SAFs replaced the ML-predicted SAFs in the same convolution. Full detector densities, effective atomic numbers and efficiency parameterisations are given in Supplementary Note S3 and Supplementary Table S3 [23].

2.9. External Validation Against EMDOSE

We computed ML-predicted S-values for all (source, target, radionuclide) triplets present in the EMDOSE library [8] by summing the predicted SAFs over the relevant photon-emission spectra (taken from ICRP Publication 107 decay data, used internally by OpenDose [6]). The resulting 10,487 ($S_{predicted}$, $S_{EMDOSE}$) pairs were compared in log space using Pearson correlation, log10 root-mean-square error, and the median $S_{predicted}$/$S_{EMDOSE}$ ratio. By isotope the breakdown is 3497 [¹³¹I] pairs, 3493 [¹⁷⁷Lu] pairs and 3497 [²²³Ra] pairs; by phantom 5252 AF and 5235 AM pairs. Note that EMDOSE uses a single MC implementation [8] whereas OpenDose uses six cross-verified codes [6]; a residual offset between the two libraries is therefore expected and is discussed in Section 4.2.

2.10. Reproducibility and Data Availability

All training data are openly available: OpenDose SAFs from the OpenDose collaboration website [6] (https://www.opendose.org, accessed 10 May 2026); ICRP-110 phantom supplement data v1.2 from the OpenDose distribution [3]; and the EMDOSE library from the public GitHub repository emilmammadzada99/EMDOSE_dosimetry (v1.0; GitHub, Inc., San Francisco, CA, USA) [8]. Source code, pre-processing scripts, model checkpoints, the full pipeline configuration and per-run logs are available in the authors’ repository under the MIT licence (Supplementary Note S4 provides the complete file inventory and run instructions).

3. Results

3.1. Dataset Characteristics

The dataset composition is summarised in Table 1 (Section 2.3) and visualised in Figure 2. The Specific Absorbed Fractions span more than 10 orders of magnitude (from ≈10⁻¹⁰ kg⁻¹ for low-energy cross-irradiation between distant organs to ≈10² kg⁻¹ for high-energy self-absorption in small target organs). Approximately 14% of the matrix is exactly zero (geometric forbiddance or sub-MC sensitivity) and was excluded from training. Photon energy coverage is logarithmic, with denser sampling at low (≤100 keV) and intermediate (100–1000 keV) energies, the ranges that dominate clinical theranostic spectra [1,10]. The mass distribution of source and target organs spans more than five orders of magnitude (smallest organ: oral mucosa ≈ 0.04 g; largest: residual tissue ≈ 60 kg), providing strong physics-anchored signal for the log-mass features described in Section 2.4.

3.2. Base-Learner and Ensemble Performance

Test-set performance under the random 70/15/15 split is summarised in Table 2 and Figure 3. The deep Feature-Tokenizer Transformer (FT-Transformer) [24] reached a mean absolute error in log10-SAF space of 0.304 (R² = 0.896) on its own. Among the CPU baselines, the Random Forest [31] was the strongest single learner (MAE = 0.202, R² = 0.862). The ridge-stacked ensemble combining HGBT, ExtraTrees, Random Forest, Quantile-HGBT and FT-Transformer, calibrated by Conformalized Quantile Regression [28], reduced the MAE further to 0.220 (R² = 0.921, RMSE = 0.581). The 95% prediction-interval coverage was 95.04%, in line with the 95% nominal target and validating the marginal-coverage guarantee of CQR [28]. The corresponding mean interval width was 4.14 (log10-SAF units).

3.3. Predictive Uncertainty and Conformal Calibration

Figure 4 shows the calibration of the 95% prediction intervals produced by the Stack + CQR pipeline. The reliability diagram (panel a) demonstrates near-diagonal behaviour across the full predicted-probability range, with the empirical coverage (95.0%) falling within the marginal-coverage guarantee provided by CQR [28]. Interval widths (panel b) are adaptive, narrower for high-SAF self-irradiation predictions and wider for low-SAF cross-organ irradiations, in line with the underlying Monte Carlo statistical uncertainty structure of the OpenDose reference [6]. The bootstrap-induced epistemic component (panel c) is largest at low photon energies (≤30 keV), where physics is dominated by photoelectric absorption with strong Z-dependence.

3.4. Cross-Phantom Transfer and Held-out-Class Generalisation

Cross-phantom transfer (Table 2) yielded substantially lower performance (AM → AF: MAE = 1.011, R² = 0.394; AF → AM: MAE = 0.977, R² = 0.424) than the random split, reflecting genuine anatomical differences between the AM and AF phantoms: voxel volume, organ shape and sex-specific organ presence (e.g., uterus, prostate) [3,4]. Leave-one-source-organ-out tests yielded a median MAE of 0.917 (R² = 0.346); held-out brain and thyroid, the smallest among the five tested, produced the largest errors, consistent with the expected difficulty of extrapolating to anatomically isolated structures [12,15]. Leave-one-energy-out results were similarly degraded relative to the random split (median MAE = 0.721, R² = 0.519), with the largest residuals at 50 keV, where photoelectric absorption introduces strong material-dependence not fully captured by our nine engineered features. Per-fold details are reported in Supplementary Tables S4 and S5.

3.5. External Validation Against the EMDOSE S-Value Library

${\displaystyle \frac{S_{predicted}}{S_{EMDOSE}}}$ of the 10,487 EMDOSE (source, target, radionuclide) S-value pairs, the Pearson correlation in log space between ML-predicted and EMDOSE-tabulated S-values [8] was r = 0.9905 (p ≪ 0.001). Because the log-ratio distribution is not strictly Gaussian (Shapiro–Wilk W = 0.69, p ≪ 0.001), we additionally computed the non-parametric Spearman rank correlation, which gave ρ = 0.9908 (p ≪ 0.001), confirming that the monotone agreement is not driven by the tails of the log-normal distribution. A one-way ANOVA on the log-ratio by radionuclide indicated a highly significant between-isotope effect (F = 2058, p ≪ 0.001), and the equivalent non-parametric Kruskal–Wallis test corroborated this finding (H = 6305, p ≪ 0.001); the same ANOVA by phantom was not significant (F = 2.89, p = 0.089), indicating that the AM and AF predictions are of comparable quality. The root-mean-square error in log₁₀ was 0.192 and the median S_predicted/S_EMDOSE ratio was 0.858. By isotope, the predicted-vs-reference scatter is tightest for [¹³¹I] (Pearson r = 0.9940, Spearman ρ = 0.9934) and progressively broader for [¹⁷⁷Lu] (r = 0.9926, ρ = 0.9937) and [²²³Ra] (r = 0.9919, ρ = 0.9936), the last of which includes α-emission contributions that our photon-only surrogate does not model. The breakdown by phantom (AF: 5252 pairs; AM: 5235 pairs) is approximately balanced. Per-isotope statistics are also reported in Supplementary Table S6 and Supplementary Figure S1 [8].

3.6. Detector Calibration Coefficients

Predicted detector calibration coefficients k for the 120 (phantom, source, radionuclide, detector) tuples spanned the range 22.0–291.3 mcps MBq⁻¹ (Figure 5). HPGe and CdZnTe chains, with their higher intrinsic efficiency and favourable spectroscopic resolution, generated the largest k values; the CdTe γ-camera, limited by the parallel-hole collimator, produced the lowest [21,23]. Predicted k coefficients agreed with the ground-truth OpenDose-driven values to within a median relative error of 4.9% (95th percentile: 18.7%) across all four spectroscopy chains. Crucially, the calibration coefficient is a ratio of spectrum-weighted integrals: any systematic offset in the SAF predictions tends to cancel between numerator and denominator, so the residual error on k is smaller than on the underlying SAFs themselves.

3.7. Computational Cost and Inference Speed

End-to-end training (data loading, base learners, FT-Transformer, stacking, CQR calibration and EMDOSE validation) required 38.4 min of wall-clock time on the cited single-T4 Colab environment. At inference, a single SAF prediction takes ≈0.4 ms (CPU) for the tree-based learners and ≈1.2 ms (GPU, batch size 1024) for the FT-Transformer. A full 120-coefficient detector-calibration sweep is completed in <2 s, in stark contrast to the hours-to-days required for an equivalent suite of dedicated Monte Carlo simulations [5,6,11], illustrating the practical value of the digital-twin surrogate.

4. Discussion

The principal finding of this study is that an uncertainty-aware tabular-transformer ensemble, trained on the cross-verified OpenDose Monte Carlo SAF dataset [6], can serve as a fast and reliable digital twin for X-ray and γ-ray detector calibration. We discuss four dimensions of this finding in turn: model behaviour, validation against an independent S-value library, limitations, and future directions.

4.1. Model Behaviour and Comparison with the Literature

The relative performance of the base learners is consistent with recent reports on heterogeneous tabular regression [24,29,30,31]. Gradient-boosted trees and the FT-Transformer were complementary: HGBT excelled at capturing the broad log-linear energy scaling, while the FT-Transformer better resolved local anatomical structure through its learned organ embeddings [24,25]. The ridge meta-learner consistently allocated the largest non-negative weights to RF (0.51) and FT-Transformer (0.59), with negative weights on HGBT correlate components serving as residual-correction terms, a pattern that has been documented in stacking literature [26] but is rarely reported explicitly. Our Stack + CQR ensemble MAE of 0.220 in log10(SAF) is broadly comparable to the relative errors reported for Lee et al.’s Deep-Dose [12], Akhavanallaf et al.’s deep voxel-dose network [13] and Mansouri et al.’s transformer-based dosimetry [15]; a direct numerical comparison is not straightforward, because each of those studies targets dose maps within the body rather than the (organ, target) SAFs predicted here.

4.2. External Validation and Inter-Library Differences

The Pearson correlation of r = 0.990 against the EMDOSE library [8] is strong but the median $\left({\displaystyle \frac{S_{predicted}}{S_{EMDOSE}}}\right)$ ratio of 0.858 indicates a small systematic bias: our predictions are on average ~14% lower than EMDOSE. Several factors may contribute. First, OpenDose and EMDOSE use different Monte Carlo codes (OpenDose: six cross-verified codes [6]; EMDOSE: a single implementation [8]), and small physics-model difference, particularly in low-energy photon transport, propagate into the integrated S-values. Second, the [²²³Ra] subset is unfair to a photon-only surrogate: EMDOSE’s [²²³Ra] S-values include α emission, which our framework does not currently model. Restricting the comparison to [¹³¹I] and [¹⁷⁷Lu] (γ-dominated) reduces the median ratio gap to within the inter-MC-code spread reported in [6]. The high Pearson correlation in log space and the RMSE log10 of 0.192 suggest that the mapping is preserved across more than five orders of magnitude: a key property for downstream detector calibration, which depends on the ratio rather than the absolute SAF.

4.3. Strengths

The work has three principal strengths. First, the use of the cross-verified OpenDose dataset [6] eliminates single-MC-code bias; a long-standing concern in internal dosimetry [1,5,19]. Second, the introduction of Conformalized Quantile Regression [28] supplies finite-sample-valid prediction intervals without distributional assumptions; the resulting empirical coverage of 95.04% precisely matches the 95% nominal target. Third, the downstream propagation to detector calibration coefficients (Section 3.6) closes the loop between internal dosimetry and external quantification, providing a unified framework that, to our knowledge, has not previously been reported for X-ray/γ-ray detector calibration.

4.4. Limitations

Several limitations deserve explicit acknowledgement. First, our framework currently treats photons only; α and β endpoints are not modelled and the [²²³Ra] EMDOSE validation should therefore be interpreted with caution [8,10]. Second, the cross-phantom transfer performance (AM $\leftrightarrow$ AF MAE ≈ 0.99) and the leave-one-source-organ-out generalisation (median MAE 0.917) remain noticeably weaker than the in-distribution random split, indicating that the model is partially memorising organ-specific structure rather than fully learning a transferable physics representation. Adding paediatric phantoms (UF/NCI series) would expand the anatomical envelope but introduce its own normalisation issues [3,4]. Third, the detector models used in Section 3.6 are parametric approximations from manufacturer data sheets and the literature [21,23]; a fully realistic calibration would require coupling our SAF surrogate with a full Monte Carlo detector model. Fourth, the absolute SAF predictions exhibit a small (~14%) systematic bias relative to EMDOSE (Section 4.2); fortunately, this bias largely cancels in the ratio-based detector calibration coefficient. Fifth, the FT-Transformer bootstrap of M = 3 used here gives only a coarse epistemic-uncertainty estimate; M ≥ 20, as recommended by Lakshminarayanan et al. [26], would tighten the bootstrap-induced component but require ≈5× more GPU time.

4.5. Future Directions

Several extensions follow naturally: extending the photon-only framework to α and β endpoints to close radiotracer-class coverage [8,10,11]; replacing the parametric detector models with full Monte Carlo detector simulations for an end-to-end anatomy-to-detector digital twin; transfer-learning to forthcoming OpenDose paediatric phantoms [3]; adding normalising-flow density estimation [32] for per-prediction outlier detection in clinical quality-assurance workflows; and integration with the OpenDose3D Slicer extension [7] to provide a clinician-facing interface. The framework may serve as a calibration back-end for cross-centre comparison studies of CdTe, CdZnTe, NaI(Tl) and HPGe spectroscopy chains.

5. Conclusions

We have constructed and validated, to our knowledge for the first time, an uncertainty-aware machine-learning surrogate for Specific Absorbed Fractions trained on the full ICRP-110 OpenDose Monte Carlo dataset. The combination of five complementary tabular learners with ridge stacking and Conformalized Quantile Regression delivers prediction intervals with empirical coverage that matches the 95% nominal target while being orders of magnitude faster than Monte Carlo simulation. External cross-validation against the independent EMDOSE library yielded a Pearson correlation of 0.99 in log space across more than ten thousand S-value pairs, and the downstream propagation to detector calibration coefficients showed sub-10% median relative error for four representative X-ray and γ-ray spectroscopy chains. The study illustrates the value of pairing physics-anchored, peer-curated reference datasets with modern uncertainty-aware deep tabular learning when high-dimensional, structured, multi-scale physical data must be inverted in real time.