Impact of cell geometry, cellular uptake region, and tumour morphology on ²²⁵Ac and ¹⁷⁷Lu dose distributions in prostate cancer

Abstract

Background

Radiopharmaceutical therapy with ²²⁵Ac- and ¹⁷⁷Lu-PSMA has shown promising results for the treatment of prostate cancer. However, the distinct physical properties of alpha and beta radiation elicit varying cellular responses, which could be influenced by factors such as tumour morphology. In this study, we use simulations to examine how cell geometry, region of pharmaceutical uptake within the cell to model different internalization fractions, and the presence of tumour hypoxia and necrosis impact nucleus absorbed doses and dose heterogeneity with ²²⁵Ac and ¹⁷⁷Lu. We also develop nucleus absorbed dose kernels for application to autoradiography images.

Methods

We used the GATE Monte Carlo software to simulate three geometries of LNCaP prostate cancer cells (spherical, cubic, and ovoid) with activity of ²²⁵Ac or ¹⁷⁷Lu internalized in the cytoplasm or bound to the extracellular membrane. Nucleus S-values were calculated for each geometry, source region, and isotope. The cell models were used to create nucleus absorbed dose kernels for each source region describing the dose to each nucleus in a cell layer, which were applied to simulated tumours composed of normoxic, hypoxic, or necrotic cancer cells to obtain dose rate maps. Absorbed doses within the tumours and dose heterogeneity were analyzed for each tumour morphology and isotope. Cell geometry made a minimal impact on S-values to the nucleus, however internalization resulted in higher nucleus doses. Applying the kernels to the simulated tumour maps showed that doses to each cell type varied between ²²⁵Ac and ¹⁷⁷Lu depending on tumour morphology. Dose heterogeneity within tumours was slightly higher with ²²⁵Ac, however the tumour morphology made a larger impact on dose heterogeneity compared to the choice of isotope, with hypoxic and necrotic tumours having very heterogeneous dose distributions.

Conclusions

Cell geometry simplifications may still allow robust results in simulation studies. Furthermore, the morphology of the tumour itself may make a larger impact on treatment response compared to other variables such as ratio of internalization. Finally, nucleus absorbed dose kernels were created that could enable microdosimetric studies with autoradiography.

Introduction

Radiopharmaceutical therapy (RPT), which involves the administration of cancer cell binding pharmaceuticals labelled with alpha, beta minus, or Auger-Meitner electron emitting radionuclides, has shown promising results in treating prostate cancer due to its ability to target cancer cells while minimizing toxicity to healthy tissues [1]. Lutetium-177 (¹⁷⁷Lu) labelled prostate-specific membrane antigen (PSMA-617) ([¹⁷⁷Lu]Lu-PSMA-617), was approved by the U.S. Food and Drug Administration (FDA) as Pluvicto™ in March 2022 [2] due to the therapeutic efficacy of ¹⁷⁷Lu’s beta emissions and the high PSMA expression in metastatic castration resistant prostate cancer (mCRPC) [1]. PSMA binding radiopharmaceuticals labelled with alpha emitting radionuclides such as actinium-225 (²²⁵Ac) have also been shown to be effective with minimal toxicity and may result in superior treatment of mCRPC compared to beta emitters due to the high linear energy transfer (LET) of alpha particles [3].

¹⁷⁷Lu emits beta minus particles and internal conversion electrons with mean and maximum ranges of 0.28 mm and 2.0 mm in tissue respectively [4, 5]. Conversely, the ²²⁵Ac decay chain emits four alpha particles with tissue ranges of 47–85 µm [5]. After binding to the prostate cancer cell surface, PSMA binding radiopharmaceuticals either remain bound to the cell membrane or are internalized within the cell [6]. For very short-ranged particles, the amount of the radiopharmaceutical internalized compared to surface bound may impact the total energy absorbed by the cell nucleus.

There has been an increasing interest in the utilization of simulation-based models of irradiated cells [7] to gain an understanding of what is occurring on the sub-cellular level. These models often assume cancer cells are spherical for computational simplicity, however real prostate cancer cells exhibit irregularities in shape and size which introduce a substantial challenge to model [7]. Several studies have assessed the validity of using spherical cell models for absorbed dose calculations [7,8,9], but findings remain conflicting and it still needs to be determined if spherical models are appropriate for absorbed dose calculations.

Furthermore, work is being done to determine the impact of radiopharmaceutical internalization [1, 10,11,12] and many simulation studies separate the radionuclide source locations into internalized versus externally bound configurations. However, most simulation-based studies do not assess the impact of differing internalization ratios [13, 14]. For example, a study may find higher absorbed nucleus doses when the radiopharmaceutical is internalized within the cell, but this omits the impact of if the ratio of membrane bound activity is much higher than the amount internalized. There are many new PSMA binding radiopharmaceuticals under development [15], likely all with different ratios of externally bound to internalized activity.

Beyond the cell morphology, also of importance is the morphology of the tumour itself. Solid tumours are composed of normoxic, hypoxic, and necrotic regions, corresponding to normal, low, and nearly zero oxygen levels respectively [16]. Hypoxic and necrotic regions may increase absorbed dose heterogeneity within tumours due to decreased blood vessel density and vascularization [17], which make it challenging for the radiopharmaceutical to reach these regions; the presence of hypoxia is known to compromise treatment efficacy and encourage cancer progression [16, 18].

These three factors (cell geometry, the ratio of the radiopharmaceutical membrane bound versus internalized, and the tumour morphology) could impact absorbed doses to prostate cancer cells and ultimately treatment efficacy. Because of the different ranges of ¹⁷⁷Lu and ²²⁵Ac particulate emissions, ¹⁷⁷Lu and ²²⁵Ac labelled pharmaceuticals may behave differently under different conditions (i.e. different tumour morphologies and cell geometries). To these ends, the goal of this work is firstly to determine the impact of cell geometry, source region (cytoplasm vs. membrane), and ratio of internalization on absorbed doses to simulated prostate cancer cell nuclei and within multicellular prostate cancer tumour models. We then assess the impact of hypoxia and necrosis in the tumour models on the absorbed dose distributions. We use Monte Carlo simulations for all tests, and perform all tests with both ¹⁷⁷Lu and ²²⁵Ac and compare the results between the two radioisotopes.

Methods

Using Geant4 Application for Tomographic Emission (GATE), we first simulated three different single cell models with activity in the cytoplasm (to represent internalized activity) or membrane (to represent surface bound activity) of each model to determine the impact of cell geometry and source location on nucleus absorbed doses as an indicator of deoxyribonucleic acid (DNA) damage. We expanded two of the single cell models to a cell layer because of the extra-cellular range of some particulate emissions, and created nucleus absorbed dose kernels that can be combined in different ratios of cytoplasm to membrane uptake to model radiopharmaceuticals with different uptake patterns. The kernels were applied to 2D multicellular prostate cancer tumour maps (with varying amounts of normoxic, hypoxic, and necrotic regions) to obtain absorbed dose rate maps within the tumours. The impact of cell geometry, internalization ratio, and tumour morphology on the absorbed doses and absorbed dose distributions within the tumours was assessed for both ¹⁷⁷Lu and ²²⁵Ac.

Simulation validation

To validate our simulations, we used MIRDcell V3.10, a multicellular dosimetry tool [19]. MIRDcell analytically calculates the absorbed doses from beta and alpha particles using the continuous slowing down approximation in simple spherical cellular geometries. In MIRDcell, we created two spherical cells with 14 µm diameters directly adjacent to each other, each with a 4 µm radius nucleus within a 7 µm radius cytoplasm. Homogeneous activity of ²²⁵Ac or ¹⁷⁷Lu was placed in the cytoplasm of one cell and the self-dose per decay to the nucleus and the cross-dose per decay to the neighbor nucleus was calculated. In GATE, two identical water density cells were defined, and the same absorbed dose values were scored in the nuclei to compare against MIRDcell. Details of the GATE simulations are described below.

Monte Carlo simulations

The Monte Carlo simulation software GATE version 9.0, based on Geant4 version 10.6.2, was used for all simulations. For simulations of ²²⁵Ac, the Geant4-DNA emDNAphysics physics list was used which allows the simulation of physical interactions of charged particles (e.g. beta and alpha particles) down to very low energies (7.4 eV) in liquid water [20]. The daughter emissions of ²²⁵Ac were included in the simulations. For ¹⁷⁷Lu, the emlivermore physics list was used which allows the simulation of internal conversion electron and beta particle interactions with matter down to about 250 eV. While the emlivermore list enabled us to obtain accurate results with ¹⁷⁷Lu in less time than the emDNAphysics list, it was not satisfactory for ²²⁵Ac due to not simulating alpha particles [21]. For both ²²⁵Ac and ¹⁷⁷Lu, the GATE “ion source” was used, which simulates the full photon spectrum and all charged particle emissions of the simulated radionuclide and any daughter radionuclides.

All simulations were split into 100 jobs which were run in semi-parallel. No cuts (cut-off values below which particles are no longer tracked, which are converted to energies by GATE) were used for ²²⁵Ac as the emDNAphysics list already simulates sufficiently low energies, while cuts of 0.1 µm were used for simulations of ¹⁷⁷Lu. The source and geometry definitions for the cell geometry comparisons and nucleus absorbed dose kernels are found below in the respective sections.

Cell geometry comparison

To determine the effect of cell geometry on absorbed doses to the nucleus, we modelled three cell geometries: a spherical, ovoid, and cubic cell. All models had a nucleus, nuclear membrane, cytoplasm, and cell membrane which were water density, and the cells were surrounded by water. The total cell and nucleus volumes were approximately identical in all models (1287 µm³ for the total cell and 377 µm³ for the nucleus) and were based on cross-sectional areas of human lymph node carcinoma of the prostate (LNCaP) cells derived from human prostate cancer cells [22].

The three different geometries were modelled as such:

• The spherical model consisted of concentric spheres with radii of 4.483 µm and 6.74 µm representing the nucleus and cytoplasm respectively.

• The cubic model had a nucleus and nuclear membrane identical to the spherical model while the cytoplasm was a 10.87 µm length cube.

• The ovoid model had an ellipsoidal nucleus with dimensions of 7.7 µm × 5.4 µm × 17.4 µm and a cytoplasm with dimensions of 12.0 µm × 8.0 µm × 25.6 µm.

In all models, the nuclear and cellular membranes were 5 nm thick and enveloped the nucleus or cytoplasm respectively. Figure 1 shows each cell model.

Fig. 1

The three cell geometries analyzed in this work, the spherical (left), cubic (middle), and ovoid (right) cell. Images were generated using visualization options in GATE

The GATE DoseActor tool was attached to each nucleus to score the deposited energy and ultimately calculate the absorbed radiation dose within each voxel. The DoseActor outputs 3D images of the deposited energy (“edep”) in units of MeV and the associated uncertainty. The absorbed dose was calculated from the deposited energy outputs as described below in Sect. "S-value calculation". For each single cell geometry, the DoseActor had a voxel size of 0.2 µm × 0.2 µm × 0.2 µm. 1 × 10⁹ and 1 × 10⁷ primaries were simulated for ¹⁷⁷Lu and ²²⁵Ac respectively. Simulations with the emDNAphysics list are more time intensive (taking approximately 6 days to simulate 1 × 10⁷ primaries of ²²⁵Ac, compared to approximately 30 min to simulate 1 × 10⁹ primaries of ¹⁷⁷Lu with the emlivermore list) and 1 × 10⁷ primaries was the maximum amount that was feasible with ²²⁵Ac.

Nucleus absorbed dose kernels

While human tissue consists of layers or clusters of cells, simulating thousands of cells with GATE is computationally intensive and time consuming. To best replicate this biological environment with our simulation constraints, we created a 2D multi-purpose nucleus absorbed dose kernel which describes the absorbed dose deposited in each cell nucleus in a layer of cells when activity is present in the cell at the center of the grid. This kernel can be convolved with simulated or real tissue activity images to move from activity values to absorbed dose rate values, enabling dosimetry and analysis of tissue heterogeneity. We created kernels for both ¹⁷⁷Lu and ²²⁵Ac, both source locations, and with both spherical and cubic cell geometries to investigate the effect of cell clustering (i.e. the presence of gaps in between the cells) on kernel values.

To create the kernels, cells were replicated in the positive X and Y directions every cell length (in this case 13.498 µm or 10.875 µm for the spherical and cubic cells respectively) using a Bash script. This created a layer of cells with the cell containing activity placed at the left most bottom quadrant (see Fig. 2). To reduce the simulation memory, each of the replicated cells only contained a nucleus and a cytoplasm; however 10 µm was added to the replicated cytoplasm to account for the missing membranes and maintain consistent cellular spacing. The original source cell had all four cell regions (nucleus, nuclear membrane, cytoplasm, and cellular membrane). The areas in between the cells were water density.

Fig. 2

The ²²⁵Ac cell distribution for the cubic cell kernel modelled in GATE. The cell with the green nucleus is the source cell. The ¹⁷⁷Lu cubic kernel was identical, however was 56 × 56 cells instead of 10 × 10

For each radionuclide and geometry (spherical and cubic), two kernels were created (see Fig. 3): i) when activity was placed homogenously in the cytoplasm of the central cell to simulate radiopharmaceutical internalization, and ii) when activity was placed on the cell membrane to simulate a radiopharmaceutical that is cell surface bound. The GATE DoseActor was attached to each nucleus in the cellular grid and was 1D, meaning the “edep” output file ascribed a singular value corresponding to the total energy deposited in each cell nucleus (compared to the single cell simulations where the DoseActor was 3D, resulting in a 3D matrix as output). The pixel size was one cell size (see Table 1) to enable convolution with cellular activity maps. Despite this, because the DoseActor is attached to the nucleus it only describes the energy deposited in the nucleus. 1 × 10⁸ and 1 × 10⁶ primaries were simulated for ¹⁷⁷Lu and ²²⁵Ac respectively.

Fig. 3

A 2D, 5 cell × 5 cell subset of the input for the simulation of the spherical absorbed dose kernels (not to scale) with one full cell geometry in the bottom left quadrant. All other cells contain only the cell nucleus and extranuclear region. The white gaps are water density and the black lines represent the pixel boundaries

Table 1 The final dimensions and lengths of all kernels post-processing

Using Python, the GATE DoseActor “edep” outputs for each cell nucleus were combined to create a 2D one-quarter kernel of the deposited energy with the source activity in the bottom left corner. This quarter kernel was transformed appropriately to create a full kernel, with one of the center rows and columns removed to obtain a kernel describing the deposited energy in each cell nucleus from activity in the source region of the center cell. The kernel was converted to units of S-value (absorbed dose per decay) as described in Sect. "S-value calculation". The final sizes of each kernel are presented in Table 1. While simulating only a one-quarter kernel and extrapolating to a full kernel omits the contribution from scattered radiation off of cells outside the quarter region, we believe this effect would be minimal.

While the maximum range of beta particles from ¹⁷⁷Lu is around 2.0 mm in tissue, the uncertainty in S-value of the kernel at 2.0 mm from the center cell was very high, given the very small pixel sizes of these kernels. Because the X₉₀ (the radius of a sphere in which the beta particles have released 90% of their energy) of ¹⁷⁷Lu in water is approximately 600 µm [23], we did not encompass the full range of beta particles from ¹⁷⁷Lu (from the kernel center to the end in each direction) as that was not computationally feasible.

To assess the impact of different ratios of internalized versus extracellular bound activity, we combined the kernels in three different ratios of cytoplasm to membrane activity: 3:1, 1:1, and 1:3. This was done using the following equation:

$${K}_{final}=\frac{c \cdot {K}_{cytoplasm}+ m\cdot {K}_{membrane}}{c+m}$$

(1)

where c and m are the ratios of activity in the cytoplasm to membrane respectively, and K_cytoplasm and K_membrane are the cytoplasm and membrane kernels respectively. The ratios of 3:1, 1:1, and 1:3 were chosen to represent the difference between radiopharmaceuticals with cytoplasm dominant uptake, membrane dominant uptake, or equal uptake. However, these kernels can be summed in any ratio to match the internalization pattern of a specific radiopharmaceutical that can be measured with internalization assays.

S-value calculation

S-values were calculated according to the MIRD schema, which defines the S-value as the mean absorbed dose to a target region per radioactive decay in a source region [24]. The “edep” matrices obtained from the GATE DoseActor were used to calculate all S-values. For the single cell simulations, the deposited energy in each voxel of the matrix was summed, converted from MeV to Joules, and divided by the nucleus mass and number of simulated primaries (also known as decays or histories) to obtain the total S-value in the cell nucleus from the source region in units of Gy/Bq/s as shown in Eq. 2. The same calculation was performed for each pixel of the extended nucleus absorbed dose kernels, however no summation was necessary as each pixel already represented the total energy deposited in each nucleus.

$${S}_{value}= \frac{\sum_{i=1}^{n}{E}_{i}}{{M}_{nucleus}\cdot {N}_{primaries}}$$

(2)

where E_i is the energy deposited in the ith voxel, n is the total number of voxels in the cell nucleus, M_nucleus is the mass of the nucleus, and N_primaries is the number of primaries simulated with GATE.

2D tumour maps

2D multicellular tumour maps containing normal (healthy) cells, blood vessel cells, normoxic cancer cells, hypoxic cancer cells, and necrotic cancer cells were used to study the impact of hypoxia and necrosis on the absorbed dose distributions. These maps were created by Ahn et al., and a description of their creation and validation is described in the cited work [25]. Three distinct tumour morphologies were used in this study, composing mostly normoxic cancer cells, hypoxic cancer cells, or necrotic cancer cells. Two sizes of each morphology were studied, corresponding to approximately 5.5 mm and 10 mm in diameter.

To convert the tumour grids to activity maps, we assumed that the activity uptake within each cell depended on its oxygenation level, as the amount of oxygen was assumed to correlate to blood vessel availability and the ability for the radiopharmaceutical to reach the cell. According to Ahn et al., cells in the tumour model became hypoxic or necrotic when they had oxygen levels of 0.08 to 0.5% and < 0.08% respectively [25] and we assumed a value of 6% O₂ for normoxic cells based on typical tissue oxygen levels [18, 26]. Based on these numbers, we assumed 91.2%, 7.6%, or 1.2% of the activity of each radionuclide went to each normoxic, hypoxic, and necrotic cell respectively. This was implemented by assigning 91.2 Bq, 7.6 Bq, or 1.2 Bq to each pixel of the tumour map depending on the cancer cell in that pixel, and then normalizing each pixel by the tumour’s total activity (obtained by summing the activity in each pixel) to ensure a total activity of 1 Bq per tumour. This enabled absorbed dose rate maps to be normalized in units of Gy/Bq/s. No activity was assumed to accumulate in the normal cells or blood vessel cells, under the assumption that there would be no uptake in off-target cells.

The summed 3:1, 1:1, and 1:3 cytoplasm to membrane ratio nucleus S-value kernels were convolved with the tumour activity maps to produce a 2D image of the absorbed dose rate per unit activity (DR/A) within the tumours in units of Gy/Bq/s. To perform the convolution, we used the ndimage.convolve function available in the Python SciPy package [27].

Data analysis

For each tumour morphology and size, the mean DR/A to each cell type (normal cells, blood vessel cells, and the three cancer cell types) was calculated. Dose-volume histograms (DVHs) for each cancer cell type and tumour morphology were created as a measure of absorbed dose heterogeneity within the tumours.

To assess the differences between the three different internalization ratios and the two different cell geometries used in the kernels on the tumour DR/A maps, a statistical analysis of variance (ANOVA) test was done on the mean DR/A to each cell type for each morphology using the scipy.stats function by SciPy in Python [27]. Two sets of ANOVA tests were done: the first compared the kernel geometries (the mean DR/A across all cells after the tumours were convolved with the kernels of different geometries, while the internalization ratio was kept the same) and the second compared the internalization ratios (the mean DR/A across all cells after the tumours were convolved with the kernels of different internalization ratios, while the geometry was kept the same). These tests were evaluated for both ¹⁷⁷Lu and ²²⁵Ac.

Results

Simulation validation

Table 2 summarizes the results of our simulation validation with MIRDcell. We observed that differences in self-doses for ¹⁷⁷Lu and ²²⁵Ac were within 1% and 4%, respectively. Differences in cross-doses were within 5% and 3%, respectively.

Table 2 The self-dose and cross-dose S-values from MIRDcell and our GATE simulations in units of mGy/Bq/s

Cell geometry comparison

Table 3 shows the S-values from the source region (cytoplasm or cell membrane) to the cell nucleus for all combinations of cell models (spherical, cubic, and ovoid) and radionuclides (¹⁷⁷Lu and ²²⁵Ac). We also include the S-value ratio between ²²⁵Ac and ¹⁷⁷Lu and the percent difference in the S-value to the same regions between each cell geometry. A difference in cell geometry caused percent differences in S-value ranging from 0.8% to 13.4%. The ovoid cell model had the largest difference in nucleus absorbed doses from the other two cell models, likely because of the variation in nucleus shape (ovoid vs. spherical). Nucleus doses were consistently highest within spherical cells and lowest within ovoid cells, although differences were relatively small.

Table 3 The S-values in units of Gy/Bq/s for ¹⁷⁷Lu and ²²⁵Ac from the source (column) to the target nucleus (row) for the different cell geometries

S-values were 39% to 46% higher for ¹⁷⁷Lu and 35% to 42% higher for ²²⁵Ac when the radionuclide was internalized in the cytoplasm compared to when bound on the cell membrane. S-values from ²²⁵Ac were 127–138 times higher than those from ¹⁷⁷Lu, with the ratio increasing when the radionuclide was membrane bound.

Nucleus absorbed dose kernels

The 2D distributions and profiles of the nucleus absorbed dose kernels are shown in Fig. 4. Visually, there is very little difference between the spherical and cubic cell kernels. The kernels show that ²²⁵Ac deposits its energy much more rapidly than ¹⁷⁷Lu, with the deposited energy decreasing by six orders of magnitude after 100 µm compared to four orders of magnitude after 300 µm for ¹⁷⁷Lu. Plots of the percent uncertainty for each kernel can be found in the supplementary material.

Fig. 4

a All ²²⁵Ac kernels. b All ¹⁷⁷Lu kernels. c Profile view through the center of the kernels intended to show the differences between the spherical and cubic kernels. One half of the symmetrical kernel is shown. The spherical kernels are solid lines and the cubic kernels are dashed lines. d Profile view through the center of the cytoplasm source kernels intended to show the differences between ¹⁷⁷Lu and ²²⁵Ac. Each pixel represents one cell

Tumour images

Figure 5 shows the multicellular tumour models and the respective absorbed dose rate per unit activity (DR/A) maps after convolution with the ¹⁷⁷Lu or ²²⁵Ac nucleus kernels. Table 4 shows the number of cells of each cell type present in each tumour morphology. For simplicity, all the following images and tables shown are the 3:1 cytoplasm to membrane radiopharmaceutical internalization ratio and created using the spherical cell kernels if not otherwise stated.

Fig. 5

The top row shows the multicellular tumour models, while the subsequent rows show DR/A images in units of nGy/Bq/s after applying the ¹⁷⁷Lu (middle) and ²²⁵Ac (bottom) spherical nucleus kernels (with a 3:1 cytoplasm to membrane internalization ratios)

Table 4 The number of cells of each type in each tumour morphology

Figure 6 shows the mean DR/A to each cell type for all tumour morphologies and the ratio of DR/A values between ²²⁵Ac and ¹⁷⁷Lu. The mean DR/A to a given cell type varies amongst the different morphologies but stays relatively consistent between the small and large tumours of a given morphology. DR/A values from ²²⁵Ac are around 100–150 times higher than those from ¹⁷⁷Lu depending on cell type, with the ratio being on average lowest in normal cells and highest in blood vessel cells.

Fig. 6

Mean DR/A values (nGy/Bq/s) to all cell types for each tumour type (3:1 cytoplasm to membrane radiopharmaceutical internalization ratio and a: spherical and b: cubic cell kernel) for ¹⁷⁷Lu (left column) and ²²⁵Ac (middle column). The right column shows the ratio of the DR/A from ²²⁵Ac to ¹⁷⁷Lu for each cell type and tumour morphology

Table 5 shows the mean DR/A to each cell type averaged after convolution with each nucleus absorbed dose kernel (the spherical and cubic kernels and the 3:1, 1:1, and 1:3 internalization ratio kernels for each geometry) and the associated standard deviation. This is presented for each tumour morphology. The tumour morphology makes a visible difference in the DR/A to each cell type. In contrast, the standard deviation never exceeds 4% of the mean, indicating that the variation in DR/A between each kernel is small; the choice of nucleus absorbed dose kernel (in other words the choice of cell geometry and internalization ratio) makes a small impact on the absorbed dose rates.

Table 5 The mean DR/A and associated standard deviation to each cell type and each tumour morphology from all the nucleus absorbed dose kernels (the spherical and cubic kernels and the 3:1, 1:1, and 1:3 internalization ratio kernels for each geometry) for ¹⁷⁷Lu and ²²⁵Ac for all tumour morphologies

The mean DR/A to all cancer cells for each morphology is also presented in Table 5. Smaller tumours had higher DR/A values. The DR/A from ²²⁵Ac was 138.6 and 138.4 times higher than ¹⁷⁷Lu for the small and large normoxic tumours respectively, 139.8 and 139.6 times higher than ¹⁷⁷Lu for the small and large hypoxic tumours respectively, and 141.3 and 141.1 times higher than ¹⁷⁷Lu for the small and large necrotic tumours respectively. Despite there being variation in the mean DR/A ratio to each cell type from ²²⁵Ac and ¹⁷⁷Lu (see Figs. 6(a) and 6(b)), the ratio of the total tumour DR/A was relatively consistent amongst all tumours.

Figure 7 shows DVHs for each tumour morphology, cancer cell type, and radionuclide. The heterogeneity of the absorbed dose distributions was dependent on the tumour morphology and size, with larger tumours exhibiting more heterogeneity. Normoxic tumours had relatively homogeneous dose distributions, but the necrotic and hypoxic tumours exhibited very heterogeneous absorbed dose distributions. Most notably, only 10–20% of the necrotic tumour volume received the maximum absorbed dose. In these tumours, most of the absorbed dose was delivered to regions densely populated with normoxic cells, which were predominantly on the tumour periphery with up to 100 times less dose delivered to the central regions in some cases (see Fig. 5). It should be noted that tumours in a patient may not have the same cell patterns observed here and could have significantly different cell distributions. The cell geometry and internalization ratio made almost no difference on the tumour heterogeneity, in contrast to the tumour morphology. Absorbed dose distributions from ¹⁷⁷Lu were more homogeneous than ²²⁵Ac as expected due to the larger range of beta particles, although the difference was relatively small.

Fig. 7

DVHs for each tumour morphology and cancer cell type for the spherical cell geometry and 3:1 membrane to cytoplasm internalization ratio as a percentage of the total absorbed dose

Statistical ANOVA tests

All tests resulted in a failure to reject the null hypothesis, so we cannot say that differences found in the mean DR/A between the different cell geometries or internalization ratios are statistically significant. Tables of the P-values and F statistics can be found in the supplementary material. Between both ¹⁷⁷Lu and ²²⁵Ac, the minimum P-value when comparing the cell geometries was 0.9096 (F statistic 0.0135). When comparing the internalization ratios the minimum P-value was 0.991 (F statistic 0.0089). We did observe a downward trend in absorbed dose with decreased cytoplasm to membrane uptake and using the cubic instead of spherical kernels (for example, the mean absorbed doses to cancer cells in the small normoxic tumour after convolution with the 3:1, 1:1, and 1:3 ²²⁵Ac spherical kernels was 1147, 1114, and 1082 nGy/Bq/s respectively, and these values were 1093, 1077, and 1061 nGy/Bq/s after convolution with the same cubic kernels) but the difference was not high enough to be statistically significant.

Discussion

We simulated three different prostate cancer cell geometries with activities of ¹⁷⁷Lu or ²²⁵Ac internalized within the cytoplasm or bound to the cell membrane and compared the differences in S-values to the cell nuclei from each source region. We then simulated cell layers with activity in the cytoplasm or membrane of the bottom left cell and scored the deposited energy in the nucleus of each cell in the layer, which we used to create 2D absorbed dose kernels representing the S-value to each cell nucleus in the layer from activity in each source location of the center cell. The kernels for each source location can be summed in any cytoplasm to membrane uptake ratio to represent the characteristics of a radiopharmaceutical of choice. We applied these kernels to simulated multicellular tumour models of various morphologies with differing levels of cell hypoxia and necrosis to obtain maps of absorbed dose rate per decay. We evaluated absorbed dose rate heterogeneity by creating DVHs, found mean absorbed dose rates per unit activity to each cell type in the tumour models, and performed a statistical analysis to compare the results of different cell geometries and internalization ratios on our results.

We found that cell geometry made a relatively minimal impact on nucleus S-values (< 14% difference between any geometry). This aligns with previous research on alpha and Auger-Meitner emitters; in particular Bastiaannet et al. [7] obtained similar results (a difference of 16.4% based on cell geometry) when comparing spherical cells with irregularly shaped cells segmented using real breast cancer cells and Auger-Meitner emitting radionuclides [7, 8]. However, as Auger-Meitner electrons have different properties to beta and alpha particles these works cannot be directly compared with ours. Regardless, this result contrasts with another study which found a difference in S-value of up to 40% with different cell geometries using ¹⁷⁷Lu [9]. Nucleus S-values may change with cell volume or nucleus size [28], which was not explored in the scope of this study. S-values may also be more dependent on the clustering behaviour of the cells compared to the geometry as cells tend to change their shapes when closely packed leaving very little gaps between cells [7]. However, we found no statistically significant difference between the spherical and cubic cell kernels when convolved with the tumour models despite the clustering differences between spheres and cubes (cubic cells leave no gaps between the cells).

We also found that S-values were higher when the radiopharmaceutical was internalized in the cytoplasm compared to bound to the cell membrane which has been verified by others [9, 13]. This would suggest that using radiopharmaceuticals with a larger internalization ratio would result in higher nucleus doses and better cancer cell eradication. This has been a common thought for decades, as it has traditionally been thought that RPT with receptor agonists (which activate receptors) are superior to antagonists (which bind to receptors but do not activate them) as agonists are internalized after receptor binding [11]. However, recent research has shown using antagonists may result in better RPT outcomes despite the reduced internalization, seemingly due to greater cell binding [1, 11, 12], which highlights the importance of the ratio of internalized to extracellular bound activity.

Despite obtaining higher nucleus S-values from internalized activity, we found no statistically significant difference in the mean DR/A to tumour cells based on the internalization ratio. We did notice a downward trend in absorbed doses when the ratio of internalized to extracellular bound activity was lower, but it was not statistically significant. It should be noted that these results assume that the nucleus absorbed dose is the only effect impacting cell survival. Cytoplasm irradiation has also been shown to influence cell survival and DNA synthesis due to both direct and indirect radiation effects (e.g. the bystander effect [29]) although the specific mechanisms behind this are not fully understood. Furthermore, radiopharmaceuticals may have uptake in the cell nucleus or other regions not studied here, which could be especially of impact for radionuclides like Ac that have short-range particle emissions.

A potentially unexpected observation was the fast reduction in the ¹⁷⁷Lu absorbed dose per decay with distance seen in the nucleus absorbed dose kernels. Typically, ¹⁷⁷Lu absorbed dose kernels are thought to exhibit a gradual decline in absorbed dose with distance. Most kernels feature considerably larger pixel sizes (ranging from 0.1 mm up to 5 mm) as these values are more relevant for clinical dosimetry [30, 31]. As our pixel sizes are approximately 0.01 mm², we can capture information in the 0–100 µm range that kernels with larger pixel sizes omit. Despite the maximum beta particle range of ¹⁷⁷Lu being relatively large (2.0 mm), a substantial portion of the absorbed dose may be localized very close to the source which is not considered with large pixel size kernels. This is less surprising considering the combined mean range of internal conversion and beta particles from ¹⁷⁷Lu is 0.28 mm, considerably smaller than the 2.0 mm maximum [4]. We must note that these kernels do not conform to the standard absorbed dose kernel framework but rather describe the absorbed dose distribution to cell nuclei within a cell layer; however, we believe the distribution is relatively similar to a standard absorbed dose kernel.

Upon applying our kernels to the tumour models, we found heterogeneous absorbed dose distributions across non-normoxic tumour morphologies. This aligns with research finding suboptimal absorbed doses to tissues situated in poorly vascularized regions, such as hypoxic and necrotic tumour microenvironments [32]. Our findings indicate that low oxygen tissues, such as those experiencing hypoxia, may receive suboptimal treatment under the assumption that heterogeneous radiopharmaceutical uptake corresponds to subtumour oxygen levels. However, it must be determined whether lower absorbed doses are seen in these regions due to lower radiopharmaceutical uptake like we assumed here or other factors. The tumour morphology and size had a much larger impact on the DR/A and the absorbed dose heterogeneity than the cell geometry and amount of internalization. There was no statistically significant difference between the tumour cell DR/A values when the spherical versus cubic kernels or the different internalization ratio kernels were applied. This is positive, as future studies at this scale may not be dependent on the cell morphology allowing robust results even if simplifications on the cellular level are made.

²²⁵Ac has been shown to be more therapeutically effective than ¹⁷⁷Lu for many reasons, including the higher LET of alpha particles [3, 33]. One group also found a higher uptake of [²²⁵Ac]Ac-PSMA-617 per unit activity in prostate cancer cells compared to [¹⁷⁷Lu]LuPSMA-617 [34]. We found mean DR/A values within tumours were 138 to 140 times higher with ²²⁵Ac compared to ¹⁷⁷Lu, and dose distributions within tumours from ²²⁵Ac were more heterogeneous than those from ¹⁷⁷Lu, although the difference was relatively small (see Fig. 7). This difference would likely increase using the microdosimetry formalism for alpha particles instead of the MIRD formalism like we used here. As low administered activities of ²²⁵Ac are used in RPT, there are often very few alpha particles traversing each nucleus and some nuclei are not hit at all [35], increasing absorbed dose heterogeneity. In the single cell models, the ²²⁵Ac to ¹⁷⁷Lu absorbed dose ratio was higher when the activity was surface bound instead of internalized, likely because the impact of the different particle ranges is more important at a farther distance from the cell nucleus.

Although this work is simulation-based, it opens the door to the generation of absorbed dose rate images using autoradiography of alpha and beta emitters. For example, quantitative alpha or beta particle images of tumours and healthy tissue cross-sections can be obtained using imaging devices such as the ionizing-radiation Quantum Imaging Detector (iQID) [36]. Convolution of our kernels with these autoradiography images could allow the determination of absorbed dose distributions and the study of tissue heterogeneity. This work utilized simulated tumour cross-sections as a substitution for tissue cross-sections, but future studies should move towards autoradiography to obtain more realistic results. To enable and facilitate this research, we will be uploading and sharing our kernels on Zenodo.

We recognize that this study has some limitations. Firstly, due to the very small pixel sizes in our kernels, more simulated decays than are computationally feasible are required to obtain low statistical error throughout the entire kernel. Statistical errors become greater than 10% after approximately 60 µm and 350 µm from the kernel center for ²²⁵Ac and ¹⁷⁷Lu respectively. Unfortunately, increasing the number of primaries in the kernels required more time and computational memory than we had access to. As the absorbed dose distribution is arguably more important near the radiation source, we feel this does not impact our results. Secondly, we used the MIRD formalism of absorbed dose, as opposed to microdosimetric techniques, for ²²⁵Ac calculations. This formalism assumes that many statistically independent radiation events in a single location are required to induce a biological effect [35]. For particles with very high LET such as alpha particles, a single event may be significant enough to render the mean absorbed dose misleading, and therefore this formalism may be less accurate for describing radiation damage at the cellular level [35]. Finally, we assumed that the radiopharmaceutical uptake in each cell was dependent on the oxygenation in that region. While studies have correlated radiopharmaceutical uptake with oxygen concentration [37], this is likely radiopharmaceutical dependent and the exact mechanisms of this are not entirely known at this time.

Conclusion

We demonstrated that cell geometry has a relatively small impact on absorbed radiation doses to prostate cancer cell nuclei for both ²²⁵Ac and ¹⁷⁷Lu, and tumour morphology plays a much more important role in absorbed doses and absorbed dose heterogeneity within tumours than cell geometry and the amount of radiopharmaceutical internalized versus membrane bound. We also generated absorbed dose kernels that enable microdosimetric studies with autoradiography and can be tailored to the internalization ratio of a radiopharmaceutical of interest, which showed that for ¹⁷⁷Lu a large portion of the absorbed dose may be localized closer to the source than previous kernels have described. Application of our kernels to tumour models of varying morphologies revealed that poorly vascularized tumour regions may receive suboptimal absorbed doses for both ²²⁵Ac and ¹⁷⁷Lu.