Framework to calculate ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb radiation dose and biological effect in metastatic castration‐resistant prostate cancer treatment

Abstract

Background

In the treatment of metastatic castration‐resistant prostate cancer (mCRPC), the standard radionuclide ¹⁷⁷Lu (β⁻ emitter) is being challenged by alternatives, particularly the α‐emitter ²²⁵Ac and Auger electron emitter ¹⁶¹Tb, due to their superior radiobiological properties. These include higher linear energy transfer (LET) and shorter ranges, which enhance localized cell killing while minimizing off‐target effects. While these radionuclides induce DNA damage both in source‐cells and neighboring cells (crossfire effect), their distinct radiation profiles provide critical metrics to compare their therapeutic efficacy. By quantifying these differences, especially in micrometastatic settings, the optimal radionuclide for specific clinical scenarios could be selected.

Purpose

This Study aims to develop a comprehensive pipeline based on Monte Carlo (MC) simulations to compare the therapeutic efficacy of ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb in the treatment of mCRPC based on clinically administered activity (7.4 GBq for ¹⁷⁷Lu, ¹⁶¹Tb, and 7 MBq for ²²⁵Ac) in a multi‐cell model. Besides evaluating the absorbed dose at the cellular level, the Biological Effect Cell Kernel (BECK) method is proposed to compare the radiobiological effect of radionuclides, accounting for the crossfire effect using 3D convolution.

Methods

A 2 µm resolution cell model was constructed with a 20 µm cell, an 8 µm nucleus diameter, and a 26 µm center‐to‐center distance. This configuration resulted in a cellular fraction of 0.24 mL/g, in agreement with that estimated from six prostate cancer patients using CT Perfusion. The prostate time‐integrated activity (TIA) in the model was estimated from a patient based on a dynamic 300 MBq ¹⁸F‐DCFPyL PET scan after scaling to account for the higher administered activity in therapy. The TOPAS‐nBio MC tool was used to calculate the absorbed dose and the DNA breaks in the cell model. To account for the crossfire effect, we created the BECK, an isotropic 3D kernel, demonstrating the DNA breaks in the source‐containing cell, and those induced in its neighboring cells. The BECK was convolved with the 3D TIA maps of the cell model to obtain the DNA break maps.

Results

The cellular absorbed dose was higher than the macro‐scale dose based on SPECT‐derived TIA, by 31.3%, 15.7%, and 39.8% for ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb, respectively. At the single‐cell level, ²²⁵Ac induced markedly higher DNA breaks per source – 48 double‐strand breaks (DSBs), and 32 complex DSBs, compared to ¹⁷⁷Lu – 0.022 and 0.017, and ¹⁶¹Tb – 0.083 and 0.073, respectively. Crossfire effects were dominant for ²⁵⁵Ac and ¹⁷⁷Lu at ∼75% and less pronounced for ¹⁶¹Tb at ∼41%. The maximum ranges at which 99.99% of the total DNA breaks were observed were approximately 85 µm for ²²⁵Ac, 150 µm for ¹⁷⁷Lu, and 110 µm for ¹⁶¹Tb.

Conclusions

DPKs and BECKs of ^{2 2}⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb were developed to quantify cellular‐level dose distributions and biological efficacy, revealing micrometer‐scale heterogeneity accentuated by short‐range emissions (α, CEs, AEs). Results demonstrate ¹⁶¹Tb's optimal performance for micrometastases—surpassing ¹⁷⁷Lu's efficacy with lower toxicity than ^{2 2}⁵Ac—while relative biological effectiveness predicts activity requirements: ^{2 2}⁵Ac << ¹⁶¹Tb < ¹⁷⁷Lu.

1. INTRODUCTION

Targeted radionuclide therapy (TRT) or radiopharmaceutical therapy (RPT) uses specialized ligands to bind to specific biomarkers, allowing for precise cancer cell elimination at the cellular level. ¹ In TRT, radionuclides emitting energetic particles are conjugated to ligands to induce cytotoxicity by delivering radiation in close proximity to the tumor cells. ² TRT provides several advantages over other existing modalities. By targeting cells that only express a particular biomarker, healthy cells can be potentially preserved, diminishing the side effects of the treatment compared to other non‐targeted modalities such as external beam radiotherapy. ² , ³ , ⁴ In contrast to chemotherapy, TRT enables selective delivery of radiation doses to tumor cells while minimizing toxicity to healthy tissues. By exploiting general cytotoxicity due to DNA injury, TRTs are less susceptible to compensatory biologic pathways compared to molecularly targeted drugs like kinase inhibitors. Finally, TRTs generally have non‐overlapping toxicity profiles compared to other systemic therapies. ⁴ , ⁵ Recent advances in molecular biology have paved the way for targeted therapies with radionuclides using β and α particles, and Auger electrons. These therapies are increasingly being used to treat neuroendocrine tumors (NETs) and metastatic castration‐resistant prostate cancer (mCRPC), with ongoing research exploring their potential for other cancers. ¹ , ⁶

While its expression on normal cells is limited, prostate‐specific membrane antigen (PSMA) is highly expressed on the cell's surface of CRPC mCRPC, acting as a receptor for PSMA ligands. ¹⁷⁷Lu‐PSMA‐617 was first introduced in 2015 as a promising treatment modality for mCRPC, and later, with published data from the phase III VISION trial, it became the favorable approach for battling mCRPC. ⁷ , ⁸ While studies show a remarkable response after ¹⁷⁷Lu‐PSMA therapy in mCRPC, ⁹ , ¹⁰ ongoing research continues to explore various aspects of TRT in mCRPC, including overall survival, dosimetry, toxicity, alternative radionuclides, and its combination with other treatment modalities. ¹¹ , ¹² , ¹³ , ¹⁴ , ¹⁵

Internal dosimetry of TRT can be conducted either pre‐ or post‐treatment. Studies have demonstrated a correlation between the measured standardized uptake value (SUV) from either pre‐ or post‐treatment imaging and the absorbed dose in lesions and normal organs. ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ However, the correlation cannot be generalized to different imaging protocols and individual patient characteristics. From a dosimetric perspective, the tumor and tissue absorbed dose from internally distributed radionuclides is directly correlated to the time‐integrated activity (TIA) in the body, which can be estimated from serial uptake data acquired with either sequential planar or SPECT/CT imaging studies over several days. ²⁰ , ²¹ Because of the limited resolution of SPECT and PET imaging devices and the heterogeneous nature of radiopharmaceutical distributions at the cellular level, calculated TIA from such techniques and hence absorbed dose may be less accurate. ²² Furthermore, the micrometer range of alpha particles and the nanometer to micrometer range of Auger electrons necessitates a microdosimetric approach to assess their effects on cells more accurately. Such an approach enables a more precise evaluation of the localized energy deposition within small volumes, to understand the biological impact of radiation.

The current radionuclide of choice with the Food and Drug Administration (FDA) approval, ¹⁷⁷Lu, emits beta particles. Even with multiple treatment cycles, the therapeutic efficacy of its beta particles is reported to be limited in some cases. ²³ Alpha emitters, such as ²²⁵Ac offer potential advantages in the treatment of mCRPC due to the high linear energy transfer (LET) of various alpha particles in its decay chain and short penetration range. It has been reported that patients who developed radio‐resistance to ¹⁷⁷Lu‐PSMA therapy can still demonstrate positive responses to ²²⁵Ac. ²⁴ Although there is an increasing interest in ²²⁵Ac for TRT, limited accessibility and its low production rate are still drawbacks. ²⁵ , ²⁶ The newly emerged radionuclide ¹⁶¹Tb, shares therapeutic characteristics with ¹⁷⁷Lu, and additionally emits a high yield of conversion and Auger electrons (CEs/AEs). These high LET ultra‐short range emissions enable target elimination of residual cancer cells in micrometastases through a potent self‐irradiation effect. ²⁷ , ²⁸ , ²⁹ TRT induces a wide variety of DNA lesions, including single‐strand breaks (SSBs), double‐strand breaks (DSBs), and other complex configurations of breaks. It has been reported that DSBs and complex DSBs are the dominant mechanisms for radiation‐induced cell killing and also adverse events, ³⁰ , ³¹ making DNA molecules primary targets for cancer treatment. Therefore, the evaluation of DNA breaks in addition to the absorbed dose is important to determine the biological effects of TRT.

Given the ongoing debate regarding the optimal radionuclide for TRT of mCRPC—particularly among ¹⁷⁷Lu, ²²⁵Ac, and ¹⁶¹Tb—in terms of therapeutic efficacy, adverse events, and their suitability for treating micrometastases, ²⁴ , ³² , ³³ this study aims to develop a comprehensive pipeline for comparing the therapeutic efficiency of the three radionuclides in the treatment of mCRPC. Track‐structure Monte Carlo simulations were used to model interactions in the micrometer (DNA) scale to estimate the absorbed dose and early DNA breaks. These simulations are based on clinically used activities and a clinically grounded cellular model, offering a realistic framework for assessing the potential impact of each radionuclide in a clinical context.

2. METHODS

2.1. Micro scale cell model

A 2 µm resolution cell model was constructed using 16 µm cubic cells, and 4 µm cubic nuclei, that have approximately the same volume as the 20 µm diameter spherical cells. Cubic cells were used to facilitate 3D convolution calculations, as further discussed in subsequent sections. The cellular fraction of the cell model was estimated from the prostate CT Perfusion (CTP) studies of six patients in our IGPC‐02 clinical trial database (NCT04009174), a non‐randomized, prospective trial of men who were scheduled for radical prostatectomy for prostate cancer treatment. These patients underwent a series of pre‐operative multi‐modality imaging studies, including MRI, CTP, and a 22‐minute dynamic PET study with an injection of 300 MBq of ¹⁸F‐DCFPyL. CTP is a dynamic imaging technique that monitors the passage of a contrast agent through tissue. CTP functional maps provide information on various tissue characteristics, including blood flow (BF), blood volume (BV), and the distribution volume (V_d ) of the contrast agent. ³⁴ As CT contrast agents are hydrophilic, they distribute primarily within the intercellular space rather than diffusing into the cells. The parameter, V_d , represents the volume (mL) in which the contrast agent distributes per 100 grams of tissue, allowing for an estimation of the cellular fraction in mL/g as 1.0 – $V_d$. Using a prototype CTP software (GE Healthcare), prostate cell volume was measured to be 0.24 ± 0.031 mL/g in six patients. ³⁵ , ³⁶

Cells were positioned with a 26 µm center‐to‐center distance to correspond to a cellular fraction of 0.24 mL/g from CTP measurement. The dimension of the models was selected to ensure that at least three cells at the center of each model would experience the full crossfire effect associated with the radionuclide radiation range. Specifically, these dimensions were 296 µm, long enough for ²²⁵Ac and approximately 2 mm for both ¹⁷⁷Lu and ¹⁶¹Tb cell models. These models were utilized to calculate the 3D absorbed dose distribution, and 3D DNA break maps based on the clinically administered radiopharmaceutical activity.

2.2. Time‐integrated activity (TIA)

2.2.1. Cell model TIA

One patient from the six used for the CTP calculation was randomly selected for dose calculations and DNA break simulations. The prostate activity‐time curve (TAC) of the patient was used to generate the time‐integrated activity (TIA). The TIA was normalized from the diagnostic ¹⁸F‐DCFPyL dose (300 MBq) to the clinically administered activity per treatment cycle for each radionuclide, specifically 7.4 GBq/cycle for both ¹⁷⁷Lu and ¹⁶¹Tb, and 100 KBq/Kg per cycle for ²²⁵Ac (equating to 7 MBq for a 70 Kg patient). ⁸ , ³³ , ³⁷ The TIA for each cell in the model was derived from the maximum TIA value within the prostate tumor and the cellular fraction determined by the CTP. It was assumed that the TIA was uniformly distributed within the cytoplasm, with no activity present at the cell nucleus and membrane or the intercellular space. Details on the calculation of the patient's TIA and TIA per cell in the model are provided in Appendix 1.

2.2.2. SPECT uniform TIA

As stated, the limited resolution of SPECT images results in a blurred version of the simulated TIA in the µm scale. Consequently, the total TIA from each model was uniformly distributed across all 2 µm voxels instead of only being at the cell's cytoplasm, to represent the averaged TIA measured with SPECT.

2.3. 3D Dose distributions

Dose‐point kernels (DPKs) for the ²²⁵Ac chain, ¹⁷⁷Lu, and ¹⁶¹Tb were simulated at 2 µm resolution using the Tools for Particle Simulation (TOPAS) toolkit, ³⁸ , ³⁹ a wrapper around the GEANT4 Monte Carlo code. DPK represents the spatial distribution of absorbed dose around a point source in a homogeneous infinite medium, typically water. For these DPK simulations, physics modules “g4em‐Livermore”, “g4decay”, and “g4radioactivedecay” were used. To compute the 3D absorbed dose, the DPKs were convolved with the corresponding 2 µm resolution TIA map for each model as well as the SPECT uniform TIA, using the following equations:

$$D\left(x,y,z\right)=TIA\left(x,y,z\right)\otimes DPK\left(x,y,z\right)$$

$$\begin{array}{cc}\hfill D\left(x,y,z\right)& =\underset{-\infty }{\overset{\infty }{\int }}d\chi \underset{-\infty }{\overset{\infty }{\int }}d\psi \underset{-\infty }{\overset{\infty }{\int }}d\omega TIA\left(\chi ,\psi ,\omega \right)\hfill \\ & DPK\left(x-\chi ,y-\psi ,z-\omega \right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill D\left(x,y,z\right)& =\sum _{i=-\infty }^{\infty }\sum _{j=-\infty }^{\infty }\sum _{k=-\infty }^{\infty }TIA\left({\chi }_i,{\psi }_j,{\omega }_k\right)\hfill \\ & DPK\left(x-{\chi }_i,y-{\psi }_{j,}z-{\omega }_k\right)\mathrm{\Delta }{\chi }_i\mathrm{\Delta }{\psi }_j\mathrm{\Delta }{\omega }_k\hfill \end{array}$$

where $D(x,y,z)$ is the 3D absorbed dose at the voxel $(x,y,z)$ contributed by point sources at locations ${\chi }_i,{\psi }_{j,}{\omega }_k$.

2.4. Single‐ and double‐strand DNA breaks (SSB, DSB)

To estimate the number of direct SSBs, DSBs, and complex DSBs, track‐structure simulations were performed using the Density‐Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm as adopted by GEANT4‐DNA and implemented into TOPAS‐nBio extension. ⁴⁰ The algorithm does not directly account for the DNA geometry, instead, the interactions of source particles and secondary particles with the medium are tracked. This allows the calculation of energy deposited at each interaction site. The number of interactions within the nucleus resulting in DNA damage is specified by two parameters in the TOPAS parameter file. First, considering chromosomes are randomly distributed in the nucleus, the parameter (SampleHitsWithProbability) defines the probability that an interaction occurs with DNA, which was set to 0.16 as implemented by Francis et al. ⁴¹ The second parameter defines the lower and upper energy thresholds for sampling SSBs (Upper/LowerEnergyForSamplingSSB). The probability of inducing SSBs is 0 for energy deposits below 5 eV and increases linearly to 1 at 37.5 eV and above.

SSB occurs when the damage is situated on one of the DNA strands. In the DBSCAN, both strands have an equal probability of sustaining SSBs. The algorithm examines all SSB locations and if two SSBs occur on opposite strands within 10 base pairs (bp) (∼3.2 nm), a DSB is formed. Furthermore, if three or more SSBs are located within a 10 bp radius, with at least one SSB on the opposite strand, a complex DSB is formed.

For simulations, cells were modeled as two concentric spheres representing the cell and the nucleus with 20 and 8 µm diameters, respectively, and the “G4_WATER” medium. The radioactive sources per cell or TIA were calculated as described in Appendix 1, and were randomly distributed within the cytoplasm with uniform probability. The track‐structure physics module “g4em‐dna_opt4” was used to track electrons down to low energies, and the resulting DNA breaks were scored by the DBSCAN. Due to the computational expense of the “g4em‐dna_opt4”, it was applied exclusively to the nucleus region, and the “g4em‐livermore” physics module was assigned to other components. Figure 1 shows a TOPAS‐nBio simulation of ²²⁵Ac of self‐irradiation and crossfire induced DNA breaks between two cells (26 µm apart). One cell contains uniformly distributed cytoplasmic ²²⁵Ac sources, while the other has none. The right panel highlights dense ionization clusters from alpha tracks in the nucleus, simulated using the track‐structure physics module “g4em‐dna_opt4” (right).

FIGURE 1.

A demonstration of the TOPAS nBio MC simulation and crossfire effect with two cells separated by 26 µm (left), and the dense ionization around the ²²⁵Ac alpha particle track inside the nucleus (right). Alpha, beta (electron), and gamma tracks are shown using red, green, and yellow trajectories, respectively. The figure was generated by the TOPAS‐nBio Qt system.

2.5. Biologic effect cell kernel (BECK)

When radioactive sources are inside a cell, neighboring cells are affected, with the impact depending on the radiation type and energy. To have a more accurate way of estimating the DNA breaks, a pipeline was developed to investigate the crossfire effect on neighboring cells. First, the distance of each adjacent cell from the central cell was calculated in the model. For each unique distance, ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb sources were introduced into the cytoplasm of the central cell, and DNA breaks were estimated in the non‐radioactive neighboring cells at those unique distances. The simulations were run with 10,000 histories, and the number of sources was set to 20 for ²²⁵Ac and 2,000 for ¹⁷⁷Lu and ¹⁶¹Tb, respectively, to achieve good statistics. Collected data was converted into an isotropic 3D Biologic Effect Cell Kernel (BECK), representing the number of DNA breaks in the central cell (self‐irradiation) as well as surrounding cells (crossfire) caused by radiation from a single source. Finally, a 3D DNA breaks map was generated by convolving the BECK with the cellular TIA map of the model. If the TIA is unity throughout the cell model, then convolving this unity TIA with the BECK would give the total number of DNA breaks for a cell from self‐irradiation as well as crossfire from adjacent cells.

2.6. Relative biological effectiveness (RBE) for TRT

For external beam radiation therapy, Relative Biological Effectiveness (RBE) is a parameter that quantifies the effectiveness of different types of ionization radiation to achieve a specific biological effect and is defined as:

$$RBE=\frac{D_{Reference}}{D_{Test}}$$

where $D_{Reference}$ and $D_{Test}$ are the doses for the reference and test radiation to achieve the specific biological effect. For cancer treatment, the relevant biological effect is the number of lethal lesions created. In this investigation, we used complex DSBs as the biological effect of interest, as they are mostly the lethal lesions that lead to cell death. ⁴² , ⁴³ Also, instead of normalizing to the same dose, for TRT, it is appropriate to normalize to per radionuclide source. Given these considerations, we define RBE for TRT as:

$$\begin{array}{cc}& RB{E}_{TRT}=\hfill \\ & \frac{ComplexDSBsperradionuclidesourc{e}_{Test}}{ComplexDSBsperradionuclidesourc{e}_{Reference}}\hfill \end{array}$$

For the reference radionuclide, ⁶⁰Co was used as its radiation is commonly used in RBE investigations of external beam radiotherapy. We created complex DSB BECKs of our cell model for ⁶⁰Co, ¹⁷⁷Lu, ¹⁶¹Tb, and ²²⁵Ac sources, and then the total number of complex DSBs from each radionuclide as described in Section 2.5. These numbers were then used to calculate $RB{E}_{TRT}$ as shown in the equation above.

3. RESULTS

3.1. ²²⁵Ac decay chain, ¹⁷⁷Lu, and ¹⁶¹Tb DPKs

Figure 2 illustrates the DPKs for the ²²⁵Ac chain, ¹⁷⁷Lu, and ¹⁶¹Tb with a spatial resolution of 2 µm, plotted on a semi‐logarithmic scale. The maximum dose at the source location was 17.396 Gy/Bq.s for the ²²⁵Ac chain, while ¹⁷⁷Lu and ¹⁶¹Tb exhibited corresponding values of 0.0378 and 0.23 Gy/Bq.s, respectively.

FIGURE 2.

²²⁵Ac decay chain, ¹⁷⁷Lu, and ¹⁶¹Tb Dose‐Point Kernels (DPKs) with a 2 µm spatial resolution.

3.2. Cell model vs. uniform SPECT absorbed dose

Figure 3 presents the cell model's dose profiles in comparison to the uniform SPECT dose for the ²²⁵Ac chain (A), ¹⁷⁷Lu (B), and ¹⁶¹Tb (C), focusing on the center slice of the model. These profiles were generated by convolving the DPK with the heterogeneous TIA of the model, and the uniform SPECT TIA.

FIGURE 3.

Cell model absorbed dose profiles compared to the uniform SPECT dose for a: ²²⁵Ac chain, b: ¹⁷⁷Lu, and c: ¹⁶¹Tb.

For each radionuclide, the maximum dose was in the cytoplasm as expected. The ²²⁵Ac chain maximum dose was approximately 31.36% higher than its uniform SPECT dose for the same total TIA. The dose decreased within the nucleus and intercellular regions with a peak‐to‐trough ratio of approximately 1.35 for this radionuclide. The discrepancies between the maximum cellular dose and the SPECT uniform dose for ¹⁷⁷Lu and ¹⁶¹Tb were 15.73% and 39.81%, while the peak‐to‐trough ratios were 1.15 and 1.65, respectively.

Table 1 presents the radiation dose (Gy) delivered to the nucleus of the central cell for each radionuclide, calculated using the cell model's TIA. Specifically, it includes: (1) the self‐irradiation dose (from radionuclides within the central cell itself), (2) the total dose (combining self‐irradiation and crossfire effects from neighboring cells, computed via spatial convolution), and (3) the relative contribution of crossfire (expressed as a percentage of the total dose). These metrics collectively quantify how crossfire propagation enhances radiation delivery beyond localized self‐irradiation.

TABLE 1.

For each radionuclide, the table lists nuclear doses (Gy) to the central cell from self‐irradiation alone, total dose (self + crossfire via convolution), and crossfire's percentage contribution—All derived from time‐integrated activity.

	Nucleus dose at the central cell (without crossfire)	Nucleus dose at the central cell (with crossfire)
225Ac	0.058	0.22 (74%)
177Lu	0.144	1.2 (88%)
161Tb	0.496	1.6 (69%)

3.3. DNA breaks from Monte Carlo simulations

A linear relationship was observed between the number of radioactive sources (TIA) and the resulting DNA breaks. This linearity was also confirmed when examining the neighboring cells at varying distances from the source‐containing cell, further demonstrating that DNA damage scales proportionally with the number of sources, regardless of spatial distance. These linearity relationships suggest that DNA damages can be modeled with the BECK and estimated by convolution in direct analogy to estimating absorbed doses with DPKs. For brevity, we have only presented the ²²⁵Ac chain results. Each data point was simulated 10 times, and the averaged results are shown in Figure 4.

FIGURE 4.

The linear relationship between the number of DNA lesions versus the number of ²²⁵Ac sources in the central cell (self‐irradiation) (column a), and in neighboring cells at 26 µm and 58.14 µm away from the central cell (columns b and c, respectively). Error bars represent the standard deviation (SD), calculated from 10 measurements per data point. The solid line represents the fit to the data, with an R² value of 0.99 for all plots.

Figure 5 illustrates single‐strand breaks (SSBs) for each isotope, presented on a semi‐logarithmic scale up to 200 µm from the source‐containing cell. The results were scaled to represent DNA breaks from a single source, allowing for a standardized comparison. For self‐irradiation, ²²⁵Ac source induced 233 SSBs/source through its chain. In comparison, ¹⁷⁷Lu and ¹⁶¹Tb resulted in 0.6 and 2.2 SSBs/source, respectively. In the first adjacent cell located 26 µm away, SSBs/source were reduced to 16, 0.04, and 0.08 for the ²²⁵Ac chain, ¹⁷⁷Lu, and ¹⁶¹Tb respectively, which decreased further at larger distances.

FIGURE 5.

SSBs scored by the DBSCAN algorithm for ²²⁵Ac chain, ¹⁷⁷Lu, and ¹⁶¹Tb, are plotted on a semi‐logarithmic scale for both the central cell containing the sources and surrounding non‐radioactive cells at varying distances in the cell model. Error bars, which in most cases are not visible, represent the standard deviation (SD), calculated from 10 measurements per data point.

Figure 6 illustrates DSBs and complex DSBs on a semi‐logarithmic scale. In the central cell, the number of DSBs/source induced by ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb was approximately 48, 0.022, and 0.083 respectively. Correspondingly, the number of complex DSBs was 32, 0.019, and 0.073 per source, respectively. In the first neighboring cell located 26 µm from the central cell, approximately 5, 0.0013, and 0.003 DSBs, and 3, 0.0012, and 0.0025 complex DSBs/source were recorded for the ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb, respectively.

FIGURE 6.

DSBs and Complex DSBs scored by the DBSCAN algorithm for ²²⁵Ac chain, ¹⁷⁷Lu, and ¹⁶¹Tb are plotted on a semi‐logarithmic scale for both the central cell containing the sources and surrounding non‐radioactive cells at varying distances in the cell model. Error bars, which in most cases are not visible, represent the standard deviation (SD), calculated from 10 measurements per data point.

Figures 5 and 6 show that the number of DNA breaks exhibited a sharp decrease at ∼85 µm corresponding to the Bragg peak of ²¹³Po, a daughter with the highest alpha energy in the ²²⁵Ac chain.

Figure 7 shows a cross‐section of the 3D isotropic BECK for the decay chain of ²²⁵Ac. Due to space limitations, only ²²⁵Ac data was presented. For the convolution model of DNA breaks to be valid, two conditions must be met: linearity and spatial invariance. Linearity, demonstrated in Figure 4, and spatial invariance, inherent to a homogeneous medium, ensure that the convolution of BECK and TIA maps accurately evaluates DNA breaks from self‐irradiation while accounting for the crossfire effect. Figure 7 column B shows the result of the convolution of ²²⁵Ac BECK with the model's TIA. The highest DNA breaks of 450, 140, and 74 for SSBs, DSBs, and complex DSBs, respectively occurred in the central cluster of cells, in agreement with the maximum 85 µm range of alpha particles in the chain. The comparison between the number of DNA breaks in the central cell from self‐irradiation, as well as the total breaks including the crossfire effect, as given by the convolved maps, is summarized in Table 2. The contribution of crossfire DNA breaks increased in the order ¹⁶¹Tb, ²²⁵Ac, to ¹⁷⁷Lu.

FIGURE 7.

A cross‐section of the isotropic 3D biologic effect cell kernel (BECK) for ²²⁵Ac (column a), and the cell model's DNA lesion map obtained by convolving the 3D BECK with the 3D cell model's TIA map (column b).

TABLE 2.

Comparison between the single‐cell DNA breaks without crossfire versus the model's total DNA break map accounting for the crossfire from neighboring cells. The DNA breaks were calculated for the administered activity of 7 MBq for ²²⁵Ac and 7.4 GBq for ¹⁶¹Tb and ¹⁷⁷Lu. DNA breaks from crossfire are shown as percentages in parentheses.

	DNA breaks in the central cell (without crossfire)			Cell model DNA break map a (with crossfire)
	SSBs	DSBs	Complex DSBs	SSBs	DSBs	Complex DSBs
225Ac chain	174	36	24	450 (61%)	140 (74%)	74 (68%)
177Lu	497	17	15	1998 (75%)	69 (75%)	60 (75%)
161Tb	1736	63	55	2953 (41%)	109 (42%)	98 (44%)

^a BECK convolved with the TIA map based on the clinically administered activity.

Figure 8 illustrates the radionuclide ranges as a function of the percentage of total induced complex DSBs. For the source‐containing cell, ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb recorded 26%, 25%, and 60% of the total complex DSBs, respectively. The maximum ranges at which 99.99% of the total recorded complex DSBs were observed were approximately 85 µm for ²²⁵Ac, 150 µm for ¹⁷⁷Lu, and 110 µm for ¹⁶¹Tb. Similar behavior was also observed for the DSBs (figure not shown).

FIGURE 8.

²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb ranges as a function of complex DSBs.

Figure 9 compares the complex DSBs induced in various cell layers in our cell model, where only the central 11 × 11 × 11 cells had activity distributed uniformly in the cytoplasm to simulate a micrometastasis of volume 0.26×0.26×0.26 mm^3, which had accumulated the targeted radionuclide. Each marker on the profiles shows one layer of cells around the central cell in 3D, and a horizontal reference line is drawn where 90% of total complex DSBs were observed. We obtained these profiles by creating a 3D TIA map in which we put activity in the cytoplasm of 11 cells (one at the center and 5 cells at each side), while other cells beyond these 11 cells had zero activity. This TIA map was then convolved with BECK to generate a 3D complex DSBs map. As shown, ¹⁷⁷Lu was capable of inducing > 90% of total complex DSBs to three layers of cells away from the center, while ²²⁵Ac and ¹⁶¹Tb deliver the same percentage of total complex DSBs to two layers of cells before decreasing below 90%. Additionally, ¹⁷⁷Lu has a wider profile, while ²²⁵Ac and ¹⁶¹Tb tend to decrease with sharper slopes after two layers of cells. DSBs also show the same trend (figure not shown).

FIGURE 9.

Comparison of %complex DSBs profiles for ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb for different cell layers away from the central cell.

3.4. Relative biological effectiveness (RBE)

$RB{E}_{TRT}$ for each radionuclide in our cell model was calculated as described in Sections 2.5 and 2.6. Table 3 summarizes the RBE_TRT values compared to the LET of each radionuclide.

TABLE 3.

The TRT relative biological effectiveness (RBE_TRT) of ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb.

Radionuclide	LET(KeV/µm)	RBE
177Lu	0.2–0.5 (β−) 29	1.1
161Tb	0.2–0.5 (β−) 29 4–26 (CE & AE) 29 , 32	1.8
225Ac	80–100 (α) 29	1434

4. DISCUSSION

A comprehensive pipeline was developed to evaluate the therapeutic efficacy of the alpha emitter ²²⁵Ac, the beta emitter ¹⁷⁷Lu, and the Auger emitter ¹⁶¹Tb for the treatment of mCRPC based on clinically administered activities. Alpha particles and Auger electrons have a high LET, producing a large number of ion pairs along their short path, which induces clustered DSBs and leads to cell death even from a small exposure. ⁴⁴ Alpha particles in the ²²⁵Ac chain have ranges between 40 to 85 µm in soft tissue, making them suitable for the treatment of metastatic lesions while preserving surrounding healthy tissues. In contrast, beta‐emitting radionuclides typically generate less than 20 ion pairs per micrometer, primarily resulting in more SSBs with lower lethality than DSBs. ⁴⁴ While studies have demonstrated notable success utilizing ²²⁵Ac TRT in some patients with resistance to ¹⁷⁷Lu‐PSMA therapy, some relevant side effects were reported, particularly in cases where it was administered without adjunctive therapies. ⁴⁵ , ⁴⁶

¹⁶¹Tb is a new emerging radionuclide offering similar characteristics to ¹⁷⁷Lu including half‐life (¹⁷⁷Lu: 6.644 d vs. ¹⁶¹Tb: 6.69 d), mean β‐particles energies (¹⁷⁷Lu: 148.8 KeV|79.44%, 112.3 KeV|8.89% vs. ¹⁶¹Tb: 157.4 KeV|65%, 137 KeV|25.7%), and the total Q‐value (¹⁷⁷Lu: 496.9 KeV vs. 161 Tb: 596 KeV). Additionally, ¹⁶¹Tb emits a significant proportion of conversion and Auger electrons (CEs & AEs), which are considered ultra‐low‐range high‐LET particles with an increased chance of inducing clustered DSBs in micrometastases. A phase I/II clinician trial, VIOLET, was designed to investigate the safety and efficacy of ¹⁶¹Tb‐PSMA‐I&T for the treatment of mCRPC as well as the maximum tolerated dose (MTD). ³³

4.1. Absorbed dose

An analysis of absorbed dose per individual cell, accounting for the crossfire effect from neighboring cells, reveals that based on administered activity, ¹⁶¹Tb had the highest peak‐to‐trough ratio and the greatest discrepancy between the micro‐scale dose and the simulated uniform SPECT dose. This spatially heterogeneous absorbed dose results from the CEs and AEs depositing their energies at less than a few hundred nanometers to several micrometers. ^{47 225}Ac exhibited the next highest discrepancy, attributed to the high‐energy alpha particles within its decay chain. Conversely, the ¹⁷⁷Lu dose showed the smallest discrepancy between the micro‐scale dose and the uniform SPECT‐derived dose, due to its low‐LET beta particles, which were further smoothed by gamma photons from its decay. Thus, the absorbed dose calculated from the SPECT‐derived TIAs, might not be as accurate as the micro‐scale dose due to its limited resolution. As summarized in Table 1, the relative contribution of crossfire effects to total absorbed dose varied significantly among radionuclides due to their distinct emission profiles. ¹⁷⁷Lu demonstrated the highest crossfire contribution (88%), attributable to the long penetration range of its β⁻ particles and accompanying γ emissions. ^{2 2}⁵Ac with its decay chain and ¹⁶¹Tb exhibit reduced crossfire effects (74% and 69%, respectively) compared to ¹⁷⁷Lu (88%). This phenomenon stems from their high‐LET emissions (α particles for ^{2 2}⁵Ac; CEs/AEs for ¹⁶¹Tb), which deposit >90% of their energy within the source cell or immediate micro‐environment. Such localized energy deposition renders both radionuclides ideal for micrometastases targeting, where confined dose delivery maximizes tumor control while sparing healthy tissues. In contrast, ¹⁷⁷Lu's long‐range β⁻ particles and accompanying γ‐rays enable broader dose distribution, making it better suited for macro‐metastases treatment.

Besides self‐irradiation, the DPK, from reciprocity considerations, also characterizes both the bystander and crossfire irradiation. Provided linear superimposition applies, then the absorbed dose can be estimated as the 3D convolution of the TIA of the cell model and the DPK. In this study, we assumed activity is uniformly distributed in the cell's cytoplasm while none in the intercellular space and nucleus to represent one extreme of heterogeneous source distribution. This distribution resulted in peaks (cytoplasm, self‐irradiation) and troughs in the absorbed dose profiles: the shallow troughs in nuclei (from bystander effect) and the deeper troughs in intercellular spaces with no sources. Notably, the detectable dose in the source‐free intercellular regions arises mainly from the crossfire effect—radiation spillover from neighboring cells. Similarly, the BECK can be interpreted in the same way as DPK in characterizing DNA breaks from self‐irradiation, bystander, and crossfire effects.

4.2. DNA breaks

The DBSCAN algorithm implemented in TOPAS‐nBio focuses exclusively on physical interactions, thereby quantifying direct DNA damages occurring in the early stages of irradiation, excluding those induced by subsequent chemical processes like radiolysis and free radicals. The decay chain of ²²⁵Ac includes the emission of five high‐energy alpha particles: ²²⁵Ac: 5.83 MeV (100%), ²²¹Fr: 6.34 MeV (100%), ²¹⁷At: 7.06 MeV (100%), ²¹³Bi: 5.87 MeV (2.09%) and ²¹³Po: 8.3 MeV (97.92%) inducing an average of 233 SSBs, 49 DSBs, and 32 complex DSBs per ²²⁵Ac source. These values are substantially higher than those observed with the β‐emitter ¹⁷⁷Lu which yields an average of 0.6 SSBs, 0.02 DSBs, and 0.019 complex DSBs per source disintegration within a single cell (i.e., no crossfire effect). This discrepancy arises from the difference in energy deposition per unit length, which is approximately 80 keV/µm for a 5.9 MeV alpha particle, compared to 0.2–0.5 keV/µm for 100–500 keV electrons. ²⁹ The decay of a ¹⁶¹ Tb source leads to an average of 2.2 SSBs, 0.083 DSBs, and 0.073 complex DSBs within a cell. This is approximately four times higher than the DNA damage levels of ¹⁷⁷Lu. This superiority over ¹⁷⁷Lu is mainly due to low energy CE and Auger electrons (mostly < 50 KeV) with high linear energy transfers (LET) of (4‐26 KeV/µm), inducing highly localized energy depositions at subcellular scales. ²⁷

DSBs and complex DSBs are recognized as the most critical type of DNA damage influencing radiobiological effects. ⁴⁸ , ⁴⁹ Among the investigated radionuclides, ²²⁵Ac exhibited the highest levels of DSBs and complex DSBs within a range of up to approximately 80–85 µm, aligning with the Bragg peak ranges of alpha particles in the ²²⁵Ac decay chain. ²⁹ As presented in Figure 6, it was observed that DSBs and complex DSBs in the first neighboring cell positioned at 26 µm away from the central cell, decreased by approximately 90% for ²²⁵Ac, while ¹⁷⁷Lu and ¹⁶¹Tb showed a decrease of approximately 94% and 97% at the first adjacent cell respectively. This general reduction in DNA damage with distance is primarily due to the inverse‐square law effect associated with isotropic emissions and the attenuation of radiation within the tissue medium. Focusing on the DSBs and complex DSBs, Table 2 shows that the ¹⁶¹Tb has the lowest crossfire effect (∼ 42%–44%) to the neighboring cells, while that for ²²⁵Ac is ∼74% for DSBs and ∼68% for complex DSBs, which is roughly similar to the ∼75% contribution of crossfire for ¹⁷⁷Lu. To ensure that the crossfire contribution remains independent of the specific cell model configuration, a second cell model was constructed with a 16 µm cell diameter, 6 µm nucleus diameter, and 20 µm center‐to‐center distance, resulting in a cellular fraction of 0.27 mL/g. The BECK was recalculated for this new configuration, revealing a comparable crossfire contribution with an average variation of ±5% for each radionuclide. These findings indicate that the crossfire effect is an intrinsic property of the radionuclide rather than the cell model itself. However, the absolute number of DNA breaks varied between models, primarily due to differences in nucleus size and total cellular fraction.

4.3. Radiobiology—lethal lesions vs. DNA breaks

As outlined in Appendix 2, the modified linear quadratic model (MLQM) enables calculation of the average number of radiation‐induced lethal lesions by accounting for dose rate, repair mechanisms, and tumor repopulation. While TOPAS‐nBio/DBSCAN currently lacks the capability to incorporate these modifying effects in its Monte Carlo simulations of lethal lesion generation, comparing MLQM predictions with TOPAS‐nBio/DBSCAN DNA break data remains valuable for interpreting simulation results. For our cell model exposed to a clinically administered activity of ¹⁷⁷Lu (7.4 GBq, delivering 1.3 Gy), the MLQM predicted 0.34 lethal lesions using parameters from Table B1. Experimental studies indicate that while most DSBs are efficiently repaired and non‐lethal, ⁵⁰ , ⁵¹ repairing complex DSBs may be a challenging task for the cellular DNA repair mechanism; a small fraction of these lesions may be misrepaired, and a small portion (0.7 – 1.2%) may become lethal lesions that ultimately contribute to cell death. ⁴² , ⁴³ At clinically administered activity of 7.4 GBq, TOPAS‐nBio/DBSCAN simulated 60 complex DSBs for ¹⁷⁷Lu (Table 2), suggesting 0.42–0.72 lethal lesions via misrepair—a range that aligns reasonably well with the MLQM prediction of 0.34, despite the latter's parameters not being ¹⁷⁷Lu‐specific. This consistency supports the hypothesis that complex DSBs, rather than simple DSBs, are the primary contributors to lethal lesions.

4.4. Comparison of ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb in TRT of mCRPC

The evidence presented in Section 4.2 strongly supports the hypothesis that complex DSBs serve as the primary determinant of lethal lesions, validating our definition of RBE_TRT based on complex DSBs in Section 2.6. As demonstrated in Tables 2 and 3, ²²⁵Ac exhibits an RBE_TRT approximately 1,000 times greater than ¹⁷⁷Lu and ¹⁶¹Tb due to its significantly higher yield of complex DSBs. In contrast, ¹⁷⁷Lu and ¹⁶¹Tb show nearly identical RBE_TRT values. These findings suggest that for mCRPC treatment, the administered activity of ²²⁵Ac could be reduced by three orders of magnitude compared to ¹⁷⁷Lu, while ¹⁶¹Tb would require similar activity levels to ¹⁷⁷Lu—predictions that agree with several Phase I/II clinical trials. ³³ , ³⁷ , ⁵²

Besides RBE, a further evaluation of these radionuclides is to compare self‐irradiation and crossfire contributions. Table 2 shows that the crossfire‐to‐self‐irradiation ratios for complex DSB induction differ significantly: 2.1 for ^{2 2}⁵Ac, 3.0 for ¹⁷⁷Lu, and 0.8 for ¹⁶¹Tb. These ratios must be interpreted alongside their corresponding crossfire ranges (Figure 8): 85 µm (^{2 2}⁵Ac), 150 µm (¹⁷⁷Lu), and 110 µm (¹⁶¹Tb). Critically, the biological impact depends on how these factors combine:

• A low crossfire ratio (¹⁶¹Tb's 0.8) can be counterbalanced by its intermediate range (110 µm), sustaining complex DSB production over distance.

• A high ratio (^{2 2}⁵Ac's 2.1) compensates for its short range (85 µm), achieving similar complex DSB yields within confined volumes.

• The combination of the highest ratio (¹⁷⁷Lu's 3.0) and longest range (150 µm) maximizes crossfire‐driven complex DSBs over extended distances.

This interplay is visualized in Figure 9 using a micrometastasis model (0.26 mm edge length). Despite their differing ratios and ranges, ^{2 2}⁵Ac and ¹⁶¹Tb produce nearly identical spatially confined complex DSB profiles (>95% within target boundaries), while ¹⁷⁷Lu's emissions (β⁻/γ‐rays) generate lesions extending to 0.52 mm, double the target size. This precision confirms ^{2 2}⁵Ac/¹⁶¹Tb's suitability for micrometastases (minimizing off‐target damage) ⁵³ , ⁵⁴ , ⁵⁵ , ⁵⁶ and ¹⁷⁷Lu's advantage for macro‐metastases.

Lastly, when evaluating the radionuclides for TRT application, normal tissue toxicity must be considered alongside treatment efficacy for micro‐versus macro‐metastases. A recent meta‐analysis ⁴⁶ demonstrated that ^{2 2}⁵Ac‐PSMA exhibits higher rates of toxicity (e.g., anemia, xerostomia) compared to ¹⁷⁷Lu‐PSMA—a difference attributable to their distinct linear energy transfer (LET) profiles (80‐100 keV/µm for ²²⁵Ac α‐particles vs. 0.2‐0.5 keV/µm for ¹⁷⁷Lu β⁻ emissions; Table 3). By extension, ¹⁶¹Tb's intermediate LET (4‐26 keV/µm from CEs/AEs) would predictably yield toxicity rates between these extremes: lower than ^{2 2}⁵Ac but higher than ¹⁷⁷Lu. Thus, while ^{2 2}⁵Ac and ¹⁶¹Tb show comparable efficacy against micrometastases, ¹⁶¹Tb's better safety profile (reduced normal tissue complications) makes it the more optimal choice for such applications.

4.5. Validation and translating the workflow into clinical settings

In this study, the predictions were made using the previously validated DBSCAN algorithm in conjunction with patient‐derived prostate tumor cellular fractions and clinically relevant administered activities for each radionuclide, thereby closely approximating clinical conditions. The microscale absorbed dose was computed by the Monte Carlo simulation and convolution, and—while it can be cross‐validated using various imaging modalities, micro‐/nano‐dosimeters, or chemical dosimeters such as Fricke gel dosimeters, ⁵⁷ —Monte Carlo remains the gold standard and the most accurate method for such scenarios. ⁵⁸ , ⁵⁹ , ⁶⁰

The DNA break predictions can also be further investigated using methods such as γ‐H2AX immunofluorescence, ⁶¹ Pulsed‐Field Gel Electrophoresis (PFGE), ⁶² or other techniques. For example, subcutaneous prostate tumors can be induced in mice using PC‐3 or LNCaP cell lines, with cellular fraction quantified by CT perfusion. Following radiolabeled‐PSMA injection, multiple time‐point SPECT scans can be obtained to calculate the TIA in the tumor. DNA break maps can be calculated by convolving the BECK with the TIA map of the model. This experiment's results can be compared to the γ‐H2AX staining performed on the same tumor for experimental validation.

In clinical practice, this workflow could be adapted for patient care as follows: The tumor's cellular fraction would be derived from CT perfusion imaging, while its TIA would be calculated using serial SPECT/CT scans. These parameters would then enable calculation of absorbed doses to both the tumor and organs at risk. The resulting dose estimates could subsequently be applied to established radiobiological models—specifically, the linear‐quadratic model—to predict two key outcomes: tumor control probability and normal tissue complication probability. These predictive metrics would then guide personalized adjustments to administered activity for subsequent treatment cycles while providing prognostic insights into therapeutic efficacy.

5. CONCLUSION

In this study, we have generated the DPKs and BECKs for the ²²⁵Ac chain, ¹⁷⁷Lu, and ¹⁶¹Tb as a tool to evaluate the absorbed dose and biological efficacy of these radionuclides in a cell model taking into account the crossfire effect. Using the generated DPKs, we showed that the heterogeneous distribution of radioligand at the cellular micrometer level leads to corresponding nonuniform absorbed dose with peaks and troughs that are accentuated by radiations that have short ranges like alpha particles, CEs, and AEs. Our findings indicate that ¹⁶¹Tb, similar to ²²⁵Ac, demonstrates enhanced efficacy for the treatment of micrometastases, making it a promising candidate for TRT. RBE_TRT predicts the following activity requirements for mCRPC treatment: ²²⁵Ac (lowest) ≪ ¹⁶¹Tb < ¹⁷⁷Lu (highest). For micrometastases, compared to ¹⁷⁷Lu‐PSMA therapy, ¹⁶¹Tb is more effective while exhibiting lower cytotoxicity than ²²⁵Ac‐PSMA, due to its reduced crossfire effect and intermediate range and LET, hence, lesser impact on neighboring healthy cells in the tumor environment.

CONFLICT OF INTEREST STATEMENT

Lee TY licenses CT Perfusion to GE Healthcare (Waukesha, Wisconsin, USA) and Neusoft Medical (Shenyang, China). Other authors do not have a conflict of interest.

APPENDIX A. THE TIME‐INTEGRATED ACTIVITY (TIA) CALCULATION FROM THE DYNAMIC PET SCAN

Patients in our clinical trial on the utility of ¹⁸F‐DCFPyL in diagnosing prostate cancer (NCT04009174) underwent a 22‐minute dynamic PET scan with an injection of 300 MBq of ¹⁸F‐DCFPyL. The tracer circulated throughout the body and became bound to tumor cells. If the tumor and blood activity were in equilibrium, the distribution volume (DV) is the ratio of tissue ($Q(\infty ))$ and blood ($C_a(\infty ))$ activity:

$$DV=\frac{Q\left(\infty \right)}{C_a\left(\infty \right)}$$ (A1)

Assuming that Equation (A1) is true for all times, therefore:

$$DV=\frac{Q\left(t\right)}{C_a\left(t\right)}\forall t$$ (A2)

and:

$$Q\left(t\right)=DV·C_a\left(t\right)\forall t$$ (A3)

The TIA can be calculated by integrating both sides of Equation (A3) over time:

$$\underset{0}{\overset{\infty }{\int }}Q\left(t\right)dt=DV·\underset{0}{\overset{\infty }{\int }}{C}_a\left(t\right)dt$$ (A4)

where ${\int }_0^{\infty }Q(t)dt$ is the tumor TIA from 300MBq of ¹⁸F‐DCFPyL. The arterial TIA measured from one of the internal femoral arteries was extrapolated to background activity using a sum of three exponentials. The distribution volume of the prostate was determined by Logan Graphical Analysis. ⁶³ Although Equation (A4) was derived under the simplifying assumption that Equation (A2) holds prior to tumor‐blood equilibrium, pharmacokinetic modeling demonstrates that this assumption is unnecessary. For conciseness, the full derivation is omitted here.

The tumor TIA as calculated from Equation (A4) was scaled to the clinically administered activities in therapy (7.4 GBq of ¹⁷⁷Lu and ¹⁶¹Tb‐PSMA and 7 MBq of ²²⁵Ac‐PSMA).

Calculation of the number of sources (TIA) per cell based on the clinically administered therapeutic activity

The example below is provided for the 7.4 GBq administered activity of ¹⁷⁷Lu‐PSMA. The same steps were repeated for other radionuclides:

The voxel tumor TIA map was calculated for a patient using Equation (A4), and the maximum TIA was determined to be 4.49E+10 Bq. s/mL.

The PET voxel has a volume of 0.02 mL (2.73 × 2.73 × 2.73 mm³). Therefore, the maximum TIA in a voxel was:

$$VoxelTIA\left(Bq.s\right)=voxelvolume\left(mL\right)\times TIA\left(\frac{Bq.s}{mL}\right)$$ (A5)

Based on the 0.24 mL/g cellular fraction determined from CT Perfusion, the cell model has a 26 µm center‐to‐center distance. The number of cells in a PET voxel was:

$$Numberofcellsinavoxel={\left[\frac{Voxelsidelength}{centertocenterdistance}\right]}^3$$ (A6)

The number of sources (TIA) per cell, can be calculated by dividing Equation (A5) by Equation (A6), assuming uniform activity throughout the tumor.

APPENDIX B. MODIFIED LINEAR‐QUADRATIC MODEL ACCOUNTING FOR DOSE RATE, REPAIR AND REPOPULATION

In the simplified linear‐quadratic model, the survival fraction (SF) of a cell population is defined by ⁶⁴

$$SF=e^{-\alpha D-\beta D^2}=e^{-\lambda }$$ (B1)

where D is the radiation dose, α is the linear sensitivity coefficient, β is the quadratic sensitivity coefficient, and λ is the average number of lethal lesions produced by D.

To account for the dose rate and DNA repair, and the repopulation, the modified SF equation would be ⁶⁵

$$\lambda =\alpha D+G\beta D^2-\gamma \left(T_t-T_k\right)$$ (B2)

where γ is the repopulation rate, T_t is the treatment time, T_k is the kick‐off time for repopulation, and G is the unitless Lea‐Catcheside factor described as ⁶⁶

$$G=\frac{2}{\mu -\kappa }{\left(\frac{\kappa }{1-e^{-\kappa T_t}}\right)}^2\left(\frac{e^{-\left(\mu +\kappa \right)T_t}-1}{\mu +\kappa }-\frac{e^{-2\kappa T_t}-1}{2\kappa }\right)$$ (B3)

where μ is the repair rate constant, which accounts for the dose rate and the LET of radiation, and $\kappa$ is the decay rate constant of the radionuclide.

The following parameters are used in this study to calculate the $\lambda$ and the G factor:

TABLE B1.

Parameters used in this study for the calculation of lethal lesions and repair (G) factor.

Parameter	Description	Value
α	Linear sensitivity coefficient	0.217 Gy−1 67
α/$\beta$	Alpha/Beta ratio	3 Gy 67
κ	Decay rate constant	0.103 days−1
μ	Repair rate	11.09 days−1 68
T 1/2	Physical half‐life	6.67 days
T rep	Repair half‐life	1.5 hours 68
Tt	Treatment time	33.5 days
Tk	Tumor kick‐off time	56 days 68
γ = ln (2)/𝑇𝑝	Repopulation rate	0.0027 days−1 68

Ghaseminejad S, De Sarno D, Bauman G, Lee T‐Y. Framework to calculate ²²⁵Ac, ¹⁷⁷Lu, and ¹⁶¹Tb radiation dose and biological effect in metastatic castration‐resistant prostate cancer treatment. Med Phys. 2025;52:e18035. 10.1002/mp.18035