TOPAS a tool to evaluate the impact of cell geometry and radionuclide on alpha particle therapy

Due to the increasing clinical application of alpha particles, accurate assessment of their dosimetry at the cellular scale should be strongly advocated. Although observations of the impact of cell and nuclear geometry have been previously reported, this effect has not been fully quantified. Additionally, alpha particle dosimetry presents several challenges and most conventional methodologies have poor resolution and are limited to average parameters across populations of cells. Meaningful dosimetry studies with alpha particles require detailed information on the geometry of the target at a subcellular scale. Methods. The impact of cellular geometry was evaluated for 3 different scenarios, a spherical cell with a concentric nucleus, a spherical cell with an eccentric nucleus and a model of a cell attached to a flask, consisting of a hemispherical oblate ellipsoid, all exposed to 1,700 211At radionuclide decays. We also evaluated the cross-irradiation of alpha particles as function of distance to a source cell. Finally, a nanodosimetric analysis of absorbed dose to the nucleus of a cell exposed to 1 Gy of different alpha emitting radionuclides was performed. Results. Simulated data shows the dosimetry of self-absorbed-dose strongly depends on activity localization in the source cell, but that activity localization within the source cell did not significantly affect the cross absorbed dose even when cells are in direct contact with each other. Additionally, nanodosimetric analysis failed to show any significant differences in the energy deposition profile between different alpha particle emitters. Conclusions. The collected data allows a better understanding of the dosimetry of alpha particles emitters at the sub-cellular scale. Dosimetric variations between different cellular configurations can generate complications and confounding factors for the translation of dosimetric outcomes into clinical settings, but effects of different radionuclides are generally similar.


Introduction
Alpha particle-based radionuclides have emerged as an attractive therapeutic option with increased clinical interest since the approval of 223 RaCl2 in 2013, after the ALSYMPCA trial [1], as the first alpha emitting radionuclide for the treatment of metastatic castration resistant prostate cancer (mCRPC). This has stimulated clinical interest in other alpha emitting radionuclides, such as astatine-211 ( 211 At), actinium-225 ( 225 Ac) and thorium-227 ( 227 Th) in different clinical settings.
The effectiveness of alpha particles comes from their high LET (60 to 230 keV μm −1 ), resulting in a dense ionization pattern, that culminates in collections of multiple or complex DSB in close proximity to each other [2]. Despite the increasing interest in alpha particles, their dosimetry presents several challenges and accurate assessments of the dosimetry should be strongly advocated [3,4]. Meaningful dosimetry studies with alpha particles require detailed information on the geometry of the target on a subcellular scale. Therefore, studying the effects of alpha particles at the sub-cellular level is of interest to determine the suitability of a given radionuclide for targeted radiotherapy., also known as radiopharmaceutical therapy.
The impact of cellular geometry was first identified in 1979 by Lloyd et al who showed that from in vitro studies flattened cells can withstand more particle hits due to a variation in the nuclear cross section between spherical and attached cells [5]. Later, Raju et al showed that flattened cells needed a higher number of nuclear traversals to be inactivated, due a decrease in the energy deposition per traversal in a thinner nucleus [6].
The effects of cell shape on self-absorbed dose for different radionuclides and monoenergetic electron and alpha particles were previously studied in detail by Goddu et al, which has since become a a reference for dosimetry among the medical physics community [7]. Recently, Falzone et al reported a significant effect of nucleus eccentricity for Auger electron dosimetry [8].
Similarly, Arnaud et al showed major discrepancies between a complex real cell model and a spherical model when exposed to Auger electrons [9]. While it is known that Auger electrons have much smaller ranges than alpha particles tracks, cell specific radiosensitivity and variations in absorbed dose in cell lines with different cellular dimensions have been experimentally shown following exposure to 223 RaCl2 [10]. Furthermore, Raghavan et al shown significant dosimetric variations among 12 Auger emitters and 2 alpha emitters in a model for the treatment of primary brain cancer [11]. It has been previously demonstrated, by theoretical models, that nucleus size and nuclear geometry can have an impact on the calculated absorbed dose [12].
It is also important to note some previous seminal studies which focus on microdosimetry of single cells exposed to alpha particle emitters such as the ones developed by Goddu et al and Stinchmob et al [7,[13][14][15]. In addition, the user-friendly software MIRDcell V2.1 allows a better understanding of the impact of radionuclide choice, activity distribution, cell dimensions and intercellular distances [16]. However, MIRDcell is limited on the number of geometries available, such as 3D clusters in the shape of spheres, ellipsoids, rods and, cones, where all cells and nucleus in the clusters are modelled only as a concentric sphere with the same dimensions. In contrast, TOPAS give the user total freedom to define the desired geometry from a very simple sphere to very complex systems.
The aim of this study was to evaluate the effect of different cellular geometries, cell packing density and energy distribution on the dosimetry of several alpha emitting radionuclides that have clinical potential with the TOPAS Monte Carlo toolkit. Here, we seek to determine whether TOPAS is a valuable alternative to other available software to perform high LET dosimetry at the subcellular scale and enable the investigation of the impact of different cell geometries which are not feasible in other tools. To test this, a Monte Carlo model that simulates alpha particle interactions was applied to 3 different cell geometries. The model takes into account the stochastic nature of alpha particles for a range of clinically relevant macroscopic absorbed doses.

TOPAS
Dosimetry values were calculated with the TOPAS Monte Carlo modelling tool (version 3.0.1). TOol for PArticle Simulation (TOPAS) is a publicly available MC simulation platform supporting user-friendly dosimetry simulation for research and clinical physicists [17]. TOPAS is a top-level user-friendly software layer over Geant4, which includes the standard Geant4 toolkit, plus additional codes to access Geant4 functionality [18]. TOPAS provides a list of default physics models list that includes: 'g4em-standard_opt4' 'g4h-phy_QGSP_BIC_HP' 'g4decay' 'g4ion-binarycascade' '4h-elastic_HP' and 'g4stopping'. More information about the TOPAS physic list can be found at https:// topas.readthedocs.io/en/latest/.
In this study simulations were carried out with 1700 histories, selected as TOPAS isotropic sources with origin and orientation randomly selected inside of the chosen compartment, in 20 independent runs. Emitted alpha particle energies were obtained from the Bureau International des Poids et Mesures ' Table  of Radionuclides', as presented in Supplementary table S1 (available online at stacks.iop.org/BPEX/7/ 035008/mmedia). For each radionuclide, it was assumed that daughters remained in equilibrium with their parent where the complete decay chain was taken into account with the respective emission yield as presented in table S1. Emitted electrons and photons were not included in the calculations [19].

Influence of cell geometry
The effect of different cell geometries and nuclear positions when exposed to alpha particle irradiation was simulated for 3 different cell models. The deposited dose and number of particle hits at a microscopic level for each individual cellular component were recorded with TOPAS.
The initial model is a spherical cell, consisting of 2 homogeneous spheres of liquid water (r=1 g cm 3 -), representing the cell and its nucleus, immersed in an infinite water medium (figures 1(A)-(B)). Cell R C ( )and nucleus R N ( ) radii values used were 10 μm and 5 μm, respectively, representative of cellular dimensions that have been previously used [8]. In the eccentric nucleus model, the nucleus has been translocated 5 μm towards the membrane, such that they touched at one point. Lastly, a model of a cell attached to a flask (figure 1(C)) consists of 1 hemispherical oblate ellipsoid (a C =10 μm, b C =14.14 μm, c C =14.14 μm) of liquid water representing the cell, sitting on a plastic plane with water medium distributed above the plastic plane, and 1 oblate ellipsoid (a N =2.5 μm, b N =7.07 μm, c N =7.07 μm) located inside of the cell, representing the nucleus. The cell parameters are representative values only chosen to simulate dimensions of an average cell and maintain constant volumes between the different simulations.
To evaluate the potential effect eccentricity and sub-cellular geometry could have on dosimetric outcomes, 1700 alpha emitting radionuclides ( 211 At) per cell were distributed (1) uniformly inside the nucleus, (2) uniformly inside the cell cytoplasm but outside the nucleus, (3) exclusively at the cell membrane, and (4) uniformly in the surrounding medium (30 μm around the cell). This number of radionuclides was derived by Chouin et al from the activity and number of radiolabelled antibodies per cell as a function of time after injection of 6 MBq of 211 At-MX35 within micrometastases from murine models of ovarian carcinoma [20]. It should be noted that, for radionuclides distributed in the medium, the total cross-dose will depend on the activity per unit volume in the medium, and thus the size of the volume simulated in the vicinity of the cell. However, as this work focuses on ratios of dose due to changes in cell geometry, a relatively small volume can be considered without loss of accuracy.
This model took as target compartments: (1) nucleus, (2) cytoplasm and (3) the whole cell. For each scenario, 20 independent runs were simulated to obtain distributions of absorbed doses and decrease uncertainties in their mean values. Based on the TOPAS scorers DoseToMedium, SurfaceTrackCount and EnergyDeposit it was possible to calculate, for each target, the absorbed dose, number of hits and energy deposited, respectively. Here, a hit is defined as when an alpha particle enters a target (cell or nucleus) and stop within or traverses through the target volume and exits, and the absorbed dose was calculated as the sum of energy deposited in the compartment divided by the mass of the compartment.

Influence of cell geometry
Being aware of the typical nonuniform irradiation profile of alpha-particles, we evaluated the crossirradiation to neighbouring cells in a concentric model (1 source cell, 14 unlabelled) as a function of the distance to the source cell. These simulations began with the 14 cells placed in a face-centered cubic (fcc) lattice in direct contact with the source cell. This mimics one of the highest cell packing density without overlaps between different cells (fraction of occupied space is 0.74). These simulations were then repeated with increasing membrane to membrane separation from the source cell to neighbouring cells of 0 μm, 5 μm, 10 μm and 20 μm, mimicking a decrease in cellular distribution density (figure 2). 1700 decay events of 211 At were distributed in different compartments of the central source cell. Again, the distribution of hits, the absorbed dose and energy deposited were recorded in the cytoplasm, nucleus and whole cell, for each cell in the simulation.

Physical comparison between different alpha emitting radionuclides
To compare the dose deposition of a range of alpha emitting radionuclides with that of 211 At, a nanodosimetric analysis of absorbed dose to the cell nucleus was performed. The nucleus of the concentric cell model (figure 1 A) was used, divided into voxels of 1000 nm 3 , and energy deposited in each voxel was recorded. For each simulation, radionuclide decays were randomly distributed in the cellular cytoplasm to generate a cell absorbed dose of 1 Gy, taking into account only the alpha decays of the complete decay chain of each radionuclide (Supplementary table S2). Each scenario and radionuclide was repeated 20 times to obtain a distribution of hits and absorbed doses in the voxels, and decrease uncertainties in their mean values.

Statistical Analysis
Data were analysed and presented using GraphPad Prism version 7.01 and MATLAB R2017b. Statistical significance was determined using two-way analysis of variance (ANOVA). P values of <0.05 were considered statistically significant 3. Results

Influence of cell geometry
In figure 3, the absorbed dose to the target and number of hits in each target were calculated for concentric, eccentric and attached cell configurations. Also, the absorbed dose ratio / D D , ecc conc defined as the ratio of the absorbed dose to the target in an eccentric cell compared with a concentric cell, and the dose ratio attached conc defined as the ratio of the absorbed dose to the target in attached cell compared with a concentric cell, were also quantified. Finally, the hit ratio / H H ecc conc is defined as the ratio of the number of hits in an eccentric cell compared with a concentric cell was assessed. Figure 3 presents the ratio of the absorbed dose to different targets in an eccentric model compared with a concentric model, together with the hit ratio for uniformly distributed activity of 211 At in different source compartments. The evaluation of cellular geometry showed an increased absorbed dose to the nucleus for the eccentric model when the activity is uniformly distributed on the cellular membrane or in the surrounding medium. In contrast, the dose to the nucleus decreases, by 25%, when the activity is distributed at the cytoplasm, as seen in figure 3. Additionally, table 1 shows detailed information on the effect nucleus eccentricity on the energy deposition and hit probability per emitted particle at the nucleus. Moreover, it is important to note that the range of absorbed doses for different hits can be significantly below or above the average dose reported in table 1 (Supplementary figure 2). When emitted from the cell nucleus, each decay will result in a hit to the nucleus (100%). From the cytoplasm, only 13.2% and 10.5% of alpha particles will hit the nucleus for the concentric and eccentric models respectively, while the remaining fraction deposits energy elsewhere. From the cell membrane, the probability of nucleus hits is 6.9% and 11.1% for the concentric and eccentric models respectively. Finally, only 1.4% and 1.6% of the alpha particles will directly intersect the nucleus when emitted from the medium. This last scenario is also dependent on the relative volume of the medium to the cell -here our simulation was carried out considering the distribution of the activity in a volume of a sphere with a radius of 30 μm around the cell.
We also compared self-absorbed dose to the nucleus obtained with TOPAS with two widely accepted scientific methodologies: (1) MIRDcell V2.1 [16] and (2) S-values from the work of Goddu et al [7]. Variations between TOPAS and those models are in all simulated scenarios equal or below 10%, which may be attributed to a more detailed energy list for the decay scheme of 211 At than the average energies used in this work, and as expected these variations decrease as the distance between the source compartment and the target is increased (table 2).
In the concentric models, the rank order of energy deposited per hit in the nucleus of all investigated configurations was as follows: Medium>Mem-brane>Cytoplasm>Nucleus. This is because alpha particles incident from greater distances arrive with lower energies and typically higher LETs. However, the probability of hit on the nucleus in both models follows the reverse order. In contrast, in the eccentric models the rank order of energy deposited per hit was as follows: Medium>Cytoplasm>Mem-brane>Nucleus. This is because the nucleus is now much closer to the membrane than large portions of the cytoplasm, reducing the difference between activity deposited in these two components. Figure 4 presents a bar plot of the ratio of the absorbed dose to target for the attached cell model compared with a concentric model, for uniformly distributed activity of 211 At in different source compartments. The absorbed dose in the attached cell is, as expected, the same as the absorbed dose in the concentric model when both are exposed to the same activity uniformly distributed in the surrounding medium. ( / D D Attached conc > 0.95). However, when activity is distributed inside the cell, that is in the nucleus or cytoplasm, the absorbed dose in the cell and its compartments in the in attached model, is reduced in comparison with the concentric model It is important to state that variables such as, cell and nucleus size will substantially influence the absorbed dose to the nucleus. It has been previously demonstrated that cells with larger cytoplasmatic radii and nuclear to cytoplasm volume ratios have greater alpha particle radiation sensitivity [21]. However, a detailed exploration of such variables are outside the focus of this work, Figure 5 presents the relationship between the average absorbed dose to cells and distance to the source cell. It is possible to observe that the absorbed dose to the nucleus and hits to the cellular and nuclear membranes decreases with distance and that the subcellular localization of the activity in the source cell does not influence the absorbed cross absorbed-dose for any of the evaluated distances (0-20 μm). Additionally, a similar trend of absorbed dose in the cytoplasm and the whole cell is seen as function of the distance to the source cell (Supplementary figure 3).

Influence of cell packing density
The absorbed dose and the number of hits obtained for cross-irradiation at different distances from the source in each cell are shown in figure 6. In this scenario the absorbed dose to the nucleus of the source cell is approximately 33 Gy. The absorbed cross absorbed-dose radiation follows an approximately Poisson distribution with a mean of 4.02 Gy and Standard Deviation (SD) of 1.23 Gy, and a coefficient of  (figure 6 B). Once again, these values are in good agreement with MIRDcell simulations where cross-dose from cells exposed to the same activity at equivalent distances will have an absorbed dose of 6.78 Gy when cells are in direct contact with the source cell and 1.19 Gy when cell membranes are 20 μm away from the membrane of the source cell. As mentioned before these differences in absorbed dose can be attributed to the more detailed decay scheme of MIRDcell software. Interestingly, changing from the default TOPAS physics models to a modular list which uses the low-energy 'g4em-penelope' instead of the standard physics, which is optimized for higher energy particles, increases the absorbed dose to the nucleus when cells are in direct contact in our simulations to a value in closer agreement with MIRDcell (5.46 Gy TOPAS versus 6.78 Gy MIRDcell). Replacing these physics lists with the very low-energy Geant4-DNA physics lists may further improve agreement, however this comes with a substantial cost in computational power.
Similar variations to the number of hits to the nucleus of cells exposed to a uniform distribution of alpha sources have been also reported by Humm, where such variation follows a Poisson distribution and the average hit number is dependent on the radionuclide activity at the target to [22].
In these figures cell-to-cell fluctuations in absorbed dose and hits to the nucleus is also evident. Despite all 14 cells being located at the same distance from the source cell, there is a nonuniform profile of the absorbed doses and hits on the cell membrane and nucleus. Moreover, it is clear that different hits to the nucleus can deposit different energies. For example, cells 7, 8 and 9 in figure 6(D) have the same number of hits to the nucleus, but different absorbed doses.

Physical comparison between different alpha emitting radionuclides
For the simulated conditions 99.9% of the voxels within the nucleus see 0 deposited energy, and this is independent of the chosen radionuclide. Figure 7(A) shows the fractional probability distribution for energy deposited in voxels (1000 nm 3 ) with absorbed energy different from zero for different alpha emitters, when activity is randomly distributed in the cellular cytoplasm, this is, voxels with zero absorbed energy were not taken into account. Detailed analysis of the energy distribution on a subcellular scale fails to show any substantial differences between different radionuclides when only the energy and number of alpha particles emitted per radionuclide are considered.
In addition, these analyses show that it is highly improbable for a voxel to be hit by more than one alpha particle at clinically relevant activities.
Additionally, figures 7(B) and (C) display a comparison of the ratio of the absorbed dose to the nucleus among different alpha emitting radionuclides and cell models as previously described. Once again, all alpha emitting radionuclides have the same behavior, with absorbed dose to the nucleus in the eccentric model around 25% and 50% higher than in the concentric model when activity is distributed in the surrounding medium and cellular membrane respectively, and always slightly lower when distributed in the cytoplasm. A similar trend is observed between the attached and concentric model when the activity is distributed in the membrane, medium or in cytoplasm, but there are also variations in this absorbed dose ratio when activity is distributed in the nucleus, the absorbed dose in the attached model is around 10% lower than in the concentric model when activity is distributed in the nucleus. These variations are significant between models (P<0.0001 2-way ANOVA) and marginally statistically significant between radionuclides (P=0.033 2-way ANOVA). However, it should be noted that although it is statistically significant within these simulations, the magnitude of the differences between radionuclides is on the order of a few percent, likely smaller than differences in uptake and distribution between the isotopes. It is also important to highlight that to obtain absorbed dose to the nucleus of 1 Gy, will require different numbers of decays due the different decay schemes of each radionuclide in study. For radionuclides that only emits a single alpha particle ( 211 At, 212 Bi and 213 Bi) per decay, 1 Gy is obtained with 80 to 90 decays of the radionuclide in the cytoplasm. However, this number is smaller for radionuclides with multiple alpha particle emissions, such as 19 for 223 Ra and 225 Ac and 14 for 227 Th (Supplementary Table S2).

Discussion
Despite the remarkable advances in alpha particle dosimetry and imaging, there is still much debate around its accuracy at a cellular scale and how it can be translated to clinical settings. In this work, the dosimetry of alpha particles at a sub-cellular level in clinically relevant scenarios was investigated using computational modelling in different cell geometries.
Self-absorbed dose to the nucleus simulated by TOPAS are in good agreement with the values obtained with two other well accepted approaches (MIRDcell and S-values from Goddu et al), with variations equal or less than 10%. These discrepancies can be attributed to the different energies used in this study (5.87 MeV -42% branching ratio and 7.45%-58% branching ratio) compared with the more detailed decay scheme of MIRDcell software and the monoenergetic alpha particle used in Goddu et al work (7.0 MeV). These variations between our data and other models tend to decrease as the distance between the source compartment and the target compartment increases, as the S-values for different compartments coalesce with distance.
The effects of cell packing density were also evaluated. The evaluation of cellular geometry showed an increased absorbed dose to the nucleus for the eccentric model when the activity is uniformly distributed on the cellular membrane or in the surrounding medium. In contrast, the dose to the nucleus decreases, when the activity is distributed at the cytoplasm. Regardless of the targeting strategy adopted, the cellular geometry can influence the mean absorbed dose to the nucleus and thus the biological effect of a therapeutic agent [8]. Similar conclusions were previously demonstrated by Nettleton et al, who noted differences between S values, that is, the specific absorbed dose per cumulated activity (Gy/Bq.s) in spherical and ellipsoid cell geometries [23].
Interestingly, a similar trend is observed for the comparison of the absorbed dose between the attached model with the concentric model. An increased absorbed dose to the nucleus is observed for alpha particles emitted from the cellular membrane or surrounding medium and a decreased effect when alpha particles are emitted inside of the cell when compared with a spherical cell, with variations up to 17% (Two-way ANOVA, p=0.004 compared with a constant ratio of 1) ( figure 4).
This phenomenon was first reported by Lloyd et al who showed that in in vitro studies flattened cells can withstand more particle hits due to a variation in the nuclear cross section between spherical and attached cells [5]. Later, Raju et al in seminal work with alpha particles showed that the mean number of alpha particle transversals required to induce a lethal lesion was not constant among all cells, and flattened cells needed a higher number of nuclear hit transversals to be inactivated, due a decrease on the energy deposition per transversal across a thinner nucleus [6,24]. Recently, Arnaud et al showed major discrepancies between a complex real cell model and a spherical model when exposed to Auger electrons, highlighting the importance of modelling geometries as close as possible to the real scenarios [9].
Recently 6% variations in absorbed doses to the whole cell were reported [10] for different cell lines exposed to the exactly same irradiations conditions due to geometric variations in these cell lines. These experimental observations correlate well with the variations found between our three different cell models in silico. Additionally, variations between different cellular configurations, specifically between the spherical cell model and the cell attached to a flask model, a so called geometrical effect can generate complications and confounding factors for the translation of dosimetric evaluations to clinical settings.
Furthermore, absorbed dose is not only dependent on the target cell geometry, but deposition sites of the radionuclide within the cell are another important factor to consider during therapeutic applications, as described by Roeske and Thomas [25] and Goddu et al [26]. These dosimetric variations resulting from the impact of intracellular localization are easily available in MIRDcell software [16].
Dose deposition by alpha particles has variations along their tracks, as the LET of alpha particles in the first few microns of their track is lower than when near the Bragg peak. That is, for comparable paths across the cell nucleus, alpha particles that are emitted from the cell membrane or from the medium will deposit more energy than those emitted from the nucleus. Additionally, the average path lengths traversed in the nucleus are generally shorter for alpha particles originating in the nucleus than when originating from the cell surface or medium [25]. This is accounted in analytic dosimetric models as well as for the different Monte Carlo codes available. As it is observed in our results, where the absorbed dose to the nucleus from an alpha particle localized in the cell nucleus (8.26 0.3  cGy) is less than the absorbed dose when the alpha particle is distributed outside the nucleus (15.±0.1 cGy) (cytoplasm, membrane or medium).
However, to infer the real effectiveness of localized alpha therapy, it is necessary to consider the probability of a hit per emitted particle for different source compartments (table 1). Absorbed dose variations and Charlton showed very little dependence on the activity distribution on the fraction of cells surviving in a 3D spheroid when uniformly exposed to 211 At. But in contrast, the cellular distribution density and size of the cluster had a large contribution to the fraction of cells surviving [27].
This can be attributed to the short path length of alpha particles there is a diminished cross absorbed dose, increasing the need to effectively target all cells in the tumour to achieve a uniform distribution of the absorbed dose in the target. Moreover, data presented here shows that activity localization within the source cell does not significantly affect the cross absorbeddose even when cells are in direct contact with each other, figure 5. This is in good agreement with the work of Goddu et al, that cross absorbed-dose plays a diminished role in a cluster of cells if only 10% or less cells are labelled, without significant differences for different cellular compartments (cytoplasm or nucleus) [26].
This, nonuniform distribution of the radiopharmaceutical has been demonstrated in pre-clinical trials with alpha emitting radionuclides, meaning this may be an important consideration for future optimisation [28,29].
Here different cross-irradiation scenarios were simulated to understand the behavior of absorbed dose distribution among cells exposed to identical source cells. Despite cells being exposed to identical conditions there is a significant variation in the absorbed dose per cell, not only because different cells will experience different number of hits with some cells having multiple hits while other might receive no hits at all, but also because not every particle hit will deposit the same energy, as previously explained.
In the scenario of 20 μm separation between the membrane of the source cell and the unlabelled cells the average absorbed dose is 1.23 Gy with a coefficient of variation of 61%, and in this situation ∼3% of cells do not absorb any dose. Humm showed in their microdosimetric model that in a highly nonuniform radioactivity distribution scenario, such as when radioactivity is retained in the capillary, only cells near them will be directedly affected by the alpha particles, and a significant proportion of cells, those away from the capillary receive no hits, regardless of the specific activity used [22]. Similar results have also been previously reported [27,[30][31][32]. These reports indicate that alpha emitter exposures are likely to lead to highly heterogeneous absorbed dose distributions with a highly localized absorbed dose around targeted cells. But this means that its efficacy in such heterogeneous activity distribution scenarios may be dependent on a second order mechanism such as the bystander effect or tumour anti-vascular therapy, not taken into consideration in these simulations [33,34].
There are several alpha emitting radionuclides with potential for clinical application and researchers within the field are trying to progress toward a more specific and effective targeted alpha therapy with 225 Ac-PSMA as an alternative to 223 Ra [35]. A comparative study among different alpha emitting radionuclides in identical conditions is timely and of interest to the community. In a dosimetric analysis at the microscopic level, it was observed that when different radionuclides are used with the same initial spatial activity distribution, there was no significant difference in how they deposited energy on the subnucleus scale ( figure 7). These results suggest that the exact energy distribution of generated alpha particles does not significantly impact on biological effects, in contrast to recently published observations on Auger emitters [8].
A closer look at the ratios of the absorbed dose to the nucleus between the three different cell geometry models for activity distributed in different compartments within the cell showed that the cell geometry effect was radionuclide independent, that is, variations follows the same trends no matter the chosen radionuclide. While the differences between radionuclide were statistically significant, their magnitude was on the order of a few percent, likely smaller than differences in up-take and distribution which are not included in this model. This analysis was based on the assumption that primordial alpha emitting radionuclide and their progeny will decay in the same compartment and that there is no diffusion of these radionuclides between compartments between successive decays. This is a reasonable estimate as animal studies have demonstrated little redistribution of 223 Ra daughters away from the original tissue compartment [36,37]. Moreover, in an interesting study by Howell and colleagues, a good correlation was observed between experimental data and an empirical model to calculate possible biological effects of 223 Ra and its daughters, without redistribution of 223 Ra daughters [38]. However, in some particular scenarios, the detachment of daughters from the targeting carrier can occur due to the recoil energy upon alpha decay having more than sufficient energy to destroy any chemical bond. This would allow the daughter nuclides to diffuse away, leading to energy deposition elsewhere in the surrounding tissues, which is an important aspect that should be taken into account in future studies. Another important aspect to take into account is the physical half-life of the daughter products, as long half-lives will allow these new free radionuclides to diffuse away from the target and therefore contribute to radiation toxicity. As no significant differences are detected between different alpha particle emitters when only physical aspects of the emitted alpha particles are taken into account, the selection of the ideal alpha emitting radionuclide should be based on chemical and biological properties for targeting and conjugation, availability and physical half-life. Different radionuclides have different decays schemes and consequently different numbers of alpha particle emissions. Radionuclides with multiple alpha particle emissions will need fewer atoms to be delivered to the target than radionuclides with a single alpha particle emission.
The physical half-life of the radionuclide as well as the pharmacokinetics of the radionuclide (biological uptake at the target and clearance half-times) plays a vital role in achieving an effective accumulated absorbed dose per unit time, as different half-lives imply different doses rates for the same accumulated activity at the target, the possible impact of physical half-life and pharmacokinetics are further debated by Raghavan et al [11]. For instance, a shorter half-life could achieve larger doses per unit of time when paired with good pharmacokinetics. This is critically important if the surviving cells in the irradiated volume are continuously proliferating [39,40]. Additionally, physical half-life has a major importance in the industrial logistics of production and distribution, where a very short physical half-life implies a significant waste of activity during transportation.

Conclusions
This study evaluates the effect of cellular geometry on self-absorbed-dose and cross-dose with alpha irradiation from different radionuclides using the same conditions. Our simulations take into consideration the stochastic nature of alpha particles at typical clinical doses using the TOPAS Monte Carlo toolkit. The presented data allowed a better understanding of the dosimetry of alpha particles at the sub-cellular scale. Simulated data obtained in this study are in good agreement with values from two more widely accepted approaches, namely MIRDcell and S-values from Goddu et al, and helps to support a strong dependence of activity localization in the source cell to selfabsorbed-dose dosimetry. However, activity localization within the source cell does not significantly affect the cross absorbed-dose even when cells are in direct contact with each other.
Our results correlate with previous reports from alpha particle therapy suggesting a heterogeneous absorbed dose distribution within the target and a highly localized dose deposition. Moreover, we report no significant differences in the energy deposition profile between different alpha particle emitters, as even for very different numbers of decays required between radionuclides to have an absorbed dose to the cell of 1 Gy. As the clinical implementation of TAT is increasing, this type of analyses may be useful in interpreting future clinical results.