Targeted radionuclide therapy (TRT) first saw its application over 80 years ago and since then it has grown to become an area of great interest as advances in cancer biology, bioengineering, and radiochemistry have dramatically increased the potential of this modality (1). In contrast to other therapeutic techniques, TRT is based on the use of high-affinity molecules as carriers of radionuclides to tumor cells (2–5). The nature of this technology thus creates an approach for tumor therapy that is personalized to each individual patient, making strides towards a future that mitigates risks for normal tissue damage by delivering a highly conformal absorbed dose to the tumor (1). There are three classes of radionuclides that have been considered for TRT: alpha emitters, beta emitters, and Auger electron emitters (1, 6–8). Of these, alpha emitters have inherent properties that make them favorable in comparison to the others, pushing some to regard them as a potential “magic bullet” (5). The advantages of alpha emitters include: 1) high energy (3–9 MeV); 2) a densely ionizing track; 3) short path length (40–90 µm, corresponding to 2–10 cell diameters), 4) high linear energy transfer (LET) and 5) independence of dose rate and oxygen effects (6, 8). The practical implications of these characteristics allow for alpha particles to sterilize individual tumor cells solely from self-irradiation which is not possible with the more widely used beta-particle emitters (6, 8). Moreover, alpha-particle emitters have shown promising clinical results, for example in the treatment of metastatic castration-resistant prostate cancer (mCRPC) (9, 10).

Despite their efficacy, the stochastic nature of alpha emitters which deposit energy in small, subcellular targets causes alpha particle dosimetry to be particularly challenging (6, 8). Dosimetry – i.e. the measurement of the amount of energy deposited per unit mass - has been in regular use in conventional radiotherapy (11). However, the minute scale at which radionuclides operate introduces significant challenges (variability in energy deposition) (11). At the cellular level, the amount of energy deposited in the critical target (i.e., cell nucleus) depends on the combination of both the cell geometry and the alpha particle’s path through that target (6, 8). Additionally, some cell nuclei may receive multiple alpha particle hits while others receive few or none. Consequently, the stochastic nature of alpha particles hits combined with the non-uniform energy deposition may result in a local deviations exceeding 20% in the energy deposited within cell nuclei (6, 8, 12). This variation was established through Monte Carlo simulations and analytical calculations which showed that such variations are common due to the random nature of alpha particle interactions at the microscopic level necessitating the use of microdosimetry to accurately measure the energy deposited within cellular targets (6, 8, 12).

Microdosimetry considers the stochastic nature of the energy deposited per unit mass. Specific energy (z) is the microdosimetric analog of absorbed dose (D) and is given by:

where ϵ is the energy deposited within the cellular target and m is the mass of that target. Similar to absorbed dose, specific energy has units of Gy. A fundamental quantity in microdosimetry is the single-event specific-energy spectrum (f₁(z₁)) which is the frequency distribution of specific energy deposited within the target for exactly one alpha particle hit (13). The multi-hit spectrum for n alpha particle hits can be determined by performing multiple convolutions of the single-event spectrum (14). Of significance, single-event spectra can be used to estimate cell survival using:

where S(D) is the fraction of cells that survive after receiving an average dose D, <n> is the average number of hits to the cellular target, T₁(z_o) is the Laplace transform of the single-event spectrum, f₁(z₁), and z_o is the specific energy deposited in a cell to reduce survival to 1/e (15, 16).

Alternatively, the first and second moments of the single-event spectrum can also be used to estimate cell survival (15, 17–19). The first moment of the single-event spectrum given by:

while the second moment is defined as:

Using these moments, T₁(z_o) from Equation 2 can approximated by (20):

Relating to classical microdosimetry, <z₁> is the frequency mean (z_F) while <z₁ ²>/<z₁> is the dose mean (z_D) specific energy per event described by Kellerer (12) and Roesch (21). Moreover, these quantities have been related to cellular damage from alpha particle emitters (20, 22).

In order to calculate microdosimetric spectra, an analytical or Monte Carlo (MC) approach can be used (6, 8). The complexity that arises from many alpha-particle emissions combined with the greater flexibility which is provided by the MC simulation makes this approach favorable (8, 23). A recent study conducted by our group determined that a neural network (NN) could be trained to accurately calculate <z₁> and <z₁ ²> for alpha-particle microdosimetry calculations (24). The goal of our study is to take the next step forward and create a NN that can accurately and efficiently calculate the entire single-event specific-energy spectra. In this paper we present a novel approach where machine learning methods are utilized to “teach” a network to produce microdosimetric spectra from a set of input parameters (nucleus/cell size, initial alpha particle energy, and source/target geometry). This approach, the first of its kind, would allow for the network, once trained, to quickly and easily produce microdosimetric spectra for configurations that the network was not trained on, particularly, novel radionuclides (different initial alpha particle energies) and different combinations of cell/nuclear dimensions.

Figure 3 shows the regression plots corresponding to the training, validation, and test data for the z_max value while Figure 4 shows the resultant graphs for the 20 spectral outputs. For each plot, the x-axis corresponds to the ground truth values which were calculated using MC simulations and the y-axis corresponds to the NN output. The best-fit line is shown along with the R² value for each plot. In general, the slope of the best-fit line is close to unity with only small y-offset values. Combined with R² > 0.98 for all plots indicates good agreement between actual and predicted values. The RMSE values for z_max were 1.57 x10^-2, 1.51 x 10^-2 and 1.35 x 10^-2 for training, validation and testing data, respectively. Similarly, the RMSE values for the spectral output were 0.201, 0.175 and 0.199, respectively. Of note, the z_max values ( Figure 3 ) show good agreement across the entire range which is important since this value is used to provide accurate scaling of the x-axis for the resultant spectra. Figure 4 shows some deviations for smaller spectral values (< 0.1). However, differences on this scale do not significantly alter the spectra nor impact the area-under-the-curve of these spectra. As shown in Figure 5 , these spectra typically have values ranging from 1-10 cGy^-1. Hence small deviations on the order of 0.1 or less do not have a large impact on the spectrum, nor in calculating the area-under-the-curve.

The single-event specific-energy spectra generated by the network are compared with MC data for a random sample of test data in Figure 5 . All graphs have an area under the curve of unity. The solid blue line represents the MC data while the orange dots superimposed on the line represent the NN outputs. In general, good agreement between the NN and MC data is noted for all source-target geometries.

Table 1 shows the error distribution for tabulated <z₁> and <z₁ ²> using Equations 3, 4 based on the spectra generated using the NN vs. those generated from MC spectra. The mean for all three partitions (training, validation, and test) was centered near 0 with the <z₁ ²> mean % error being slightly larger than the <z₁> mean % error. The mean percent error for the training set was 0.14% for <z₁> and 0.20% for <z₁ ²>. For testing, the average <z₁> error was 0.08% and <z₁ ²> was 0.09%. The standard deviation (SD) and range of values also followed similar trends with <z₁ ²> values being higher in all cases. The center of the values allowed for an approximately normal distribution and as a result a 95% confidence interval was calculated.

In this study we developed a NN capable of producing single-event specific energy spectra (f₁(z₁)) for single cells in suspension from 8 different source-target geometries. Overall, the resultant spectra showed good agreement with the MC generated values. Most notably, the generated single-event spectra qualitatively and quantitatively demonstrate the strength of this approach. In particular, comparing the results across the various source-target spectra indicate the network was able to learn from the training set and predict the output spectra. Figure 5 shows that the network is equally capable of learning to produce simplistic spectra (i.e., sources located in the nucleus, cytoplasm, and cell surface) as well are more complex spectra (i.e., source located outside the cell). The consistency in accuracy going from simple to more complex spectra serves as a proof of principle for the success of the NN.

In addition to calculating the single-event spectra, we also calculated <z₁> and <z₁ ²> values. The use of these moments can simplify calculations, such as predicting cell survival given an inherent cell sensitivity, z_o (20). The ranges of errors in the <z₁> and <z₁ ²> values proved consistent with previously published values for deviations between MC and analytically tabulated values (27). In the prior study, these differences ranged from roughly −2% to 3% for <z₁> and −4% to 6% for <z₁ ²> (27). The errors associated with these values are consistent with those produced using a NN trained specifically on <z₁> and <z₁ ²> where the standard deviation in errors for <z₁> ranged from 1.5-2.1%, while the standard deviation in errors for <z₁ ²> ranged from 2.6-4.0% (24). Our previous study showed that errors of this magnitude result in an uncertainty in cell survival estimation on the order of +/-3% over a broad range of z_o values (24). These errors are well within the uncertainties of cell survival assays.

The data used to train the network was generated primarily through MC data. In theory, the NN is not limited to using MC data, as it can potentially use data from any source, including measured spectral data (33, 34). In this case, the smoothing feature associated with the network may be advantageous as it may reduce some of the inherent experimental error. Moreover, using experimental data allows one to compare the network output to direct measurement, as opposed to comparing against MC-based spectra which may be limited by the assumptions used in the simulation. However, experimental microdosimetric data are often limited, and hence the validation of this proposition is beyond the scope of this study.

An important consideration is that the network is only able to generate single-event spectra for a single geometry at a time. If one were interested in a more complex spectrum, such as those due to a source in the nucleus and cytoplasm, or from multiple alpha particle energies, the network would not be able to produce this directly. Rather, the individual spectra for each source/target configuration and/or energy would need to be generated independently. The composite single-event spectrum would be determined by integrating the individual spectra weighted by the number of alpha particles emitted over the sub-volume (13). Alternatively, if only the individual moments would be required (such as to estimate mean dose and variance in specific energy, or cell survival), these can be calculated from the individual spectra and combined based on the average number of hits from each component (20).

A notable limitation of the current neural network is its restriction to generating single-event specific energy spectra only for the geometries it was explicitly trained on. This limitation arises as the network’s ability to predict spectra is inherently dependent on the configurations it was trained on. As a result, if we were to task the network with generating spectra for an entirely new geometry which it has not encountered before, the network would likely fail to produce accurate results. This is because it lacks the necessary learned patterns to process and generate outputs for untrained geometries. This limitation, however, highlights an opportunity for future development. By expanding the training dataset to encompass a broader array of geometries, the network could be made more adaptable, allowing it to handle a wider variety of configurations.