Visual Abstract
Abstract
Radiopharmaceutical cocktails have been developed over the years to treat cancer. Cocktails of agents are attractive because 1 radiopharmaceutical is unlikely to have the desired therapeutic effect because of nonuniform uptake by the targeted cells. Therefore, multiple radiopharmaceuticals targeting different receptors on a cell is warranted. However, past implementations in vivo have not met with convincing results because of the absence of optimization strategies. Here we present artificial intelligence (AI) tools housed in a new version of our software platform, MIRDcell V4, that optimize a cocktail of radiopharmaceuticals by minimizing the total disintegrations needed to achieve a given surviving fraction (SF) of tumor cells. Methods: AI tools are developed within MIRDcell V4 using an optimizer based on the sequential least-squares programming algorithm. The algorithm determines the molar activities for each drug in the cocktail that minimize the total disintegrations required to achieve a specified SF. Tools are provided for populations of cells that do not cross-irradiate (e.g., circulating or disseminated tumor cells) and for multicellular clusters (e.g., micrometastases). The tools were tested using model data, flow cytometry data for suspensions of single cells labeled with fluorochrome-labeled antibodies, and 3-dimensional spatiotemporal kinetics in spheroids for fluorochrome-loaded liposomes. Results: Experimental binding distributions of 4 211At-antibodies were considered for treating suspensions of MDA-MB-231 human breast cancer cells. A 2-drug combination reduced the number of 211At decays required by a factor of 1.6 relative to the best single antibody. In another study, 2 radiopharmaceuticals radiolabeled with 195mPt were each distributed lognormally in a hypothetical multicellular cluster. Here, the 2-drug combination required 1.7-fold fewer decays than did either drug alone. Finally, 2 225Ac-labeled drugs that provide different radial distributions within a spheroid require about one half of the disintegrations required by the best single agent. Conclusion: The MIRDcell AI tools determine optimized drug combinations and corresponding molar activities needed to achieve a given SF. This approach could be used to analyze a sample of cells obtained from cell culture, animal, or patient to predict the best combination of drugs for maximum therapeutic effect with the least total disintegrations.
- radiation physics
- radiation therapy planning
- radiobiology/dosimetry
- radionuclide therapy
- metastases
- radiopharmaceutical therapy
Combinations of drugs carrying radionuclides have been studied as an attempt to overcome the nonuniformity of absorbed dose distribution within a population of tumor cells (1). A cocktail of antibodies (Abs) can target different receptors of a cell population, making the complementary labeling of the tumor cells with the different Abs more effective in achieving homogeneity in labeling (2–4). Studies with cocktails of radiolabeled Abs have been conducted with varying degrees of success in terms of reducing tumor growth and lengthening survival time (5–11). Furthermore, liposomes with tumor-penetrating properties have also been studied as carriers of radionuclides with moderate success in controlling tumor growth (12,13). Combinations of Abs and liposomes carrying radionuclides have also provided success in retarding tumor growth when compared with either of them being used alone (14,15). Although the mixtures of radiopharmaceutical cocktails can be effective in reducing tumor growth, few studies have been done to optimize the mixture to minimize the absorbed dose to the normal tissues while achieving the desired therapeutic effect. Therefore, there is a need for methodologies that optimize the constituents of a radiopharmaceutical cocktail, namely the radionuclides, pharmaceutical targeting agents, and molar activities of the agents, in a manner that can lower administered activities without compromising the therapeutic goal.
Investigations, both computational simulation and experimental, have been conducted to study the effects of single-agent and multiagent mixtures of radiopharmaceuticals and radiopharmaceuticals carrying multiple radionuclides. Peer-Firozjaei et al. (16) conducted a simulation study to estimate the optimal range for the radii of spheric tumors for different ratios of 90Y/177Lu in a cocktail to maximize the absorbed dose uniformity within the tumors. Pourhabib et al. (17) have shown the benefits of [188/186Re]Re-hydroxyethylidene diphosphonate, a radiopharmaceutical carrying 2 radionuclides, in treating bone metastases. The clinical trial results from Alavi et al. (18) show the successful applicability of the [153Sm]Sm- and [177Lu]Lu-ethylenediaminetetramethylene phosphonate cocktail for bone pain palliation in patients with skeletal metastases. Madsen et al. (19) have presented a theoretic model illustrating an increased tumor dose delivery with combined [131I]I-metaiodobenzylguanidine and [90Y]Y-DOTATOC for neuroendocrine tumors.
Although radiopharmaceutical cocktails targeting different receptors in tumor cells have been investigated, their success in vivo has been limited to a reduced tumor growth rate. One of the contributors that prevents eliminating cancer is the lack of optimization strategies to estimate the amount of each radiopharmaceutical in the cocktail that is needed to achieve the desired therapeutic goal with minimum absorbed dose to the normal tissues. Strategies in the articles cited above were empiric in nature. Hobbs et al. have developed an optimization methodology, at the organ level, based on tumor biologically effective dose within the 3D-RD software framework (Rapid Dosimetry) (20). However, to the best of our knowledge, optimization at the cellular level to treat circulating tumor cells, disseminated tumor cells, and micrometastases has not been undertaken. Optimizing mixtures of radiopharmaceuticals for treating microscopic disease needs to account for the nonuniform binding characteristics of the vehicles to the cancer cells and the characteristics of the radiation emitted by the radionuclides that are being delivered to arrive at optimal molar activities for each radiopharmaceutical in the cocktail.
In this product of the MIRD committee, we present an artificial intelligence (AI) feature that has been added to the MIRDcell software platform (21) to optimize cocktails of radiopharmaceuticals that achieve a predetermined therapeutic goal with the minimum number of disintegrations from all radiopharmaceuticals. The optimization algorithm analyzes the spatial distributions of each radiopharmaceutical and calculates the molar activities of each radiopharmaceutical in the cocktail to achieve the desired surviving fraction (SF) of cells while minimizing the total disintegrations in a user-defined region. Tools are provided to conduct these analyses for a suspension of cells that represent circulating and disseminated tumor cells or for a spheric multicellular cluster that represents a spheroid or micrometastasis. We also present examples illustrating the utility of the AI tools. These AI tools are a step forward toward developing curative cancer therapies that minimize absorbed dose to normal tissues.
MATERIALS AND METHODS
The MIRDcell AI uses an optimization engine based on the sequential least-squares programming (SLSQP) algorithm which was originally written in Fortran (22–24). We developed a JAVA programming version of the SLSQP package that consists of the optimizer and a wrapper that implements the optimizer. The optimizer is the core of the package in which the actual optimization takes place. The algorithm implements the method of Han and Powell with a Broyden–Fletcher–Goldfarb–Shanno algorithm of the b-matrix and L1-test function within the step-length algorithm (22,23). According to notes in the SLSQP package, it was first implemented by Dieter Kraft in 1981, revised to Fortran 77 in March 1987, revised to Microsoft Fortran in March 1989, in-line coded by Hesse in April 1989, revised to Fortran/2 version 1.04 in February 1991, tested with Salford Fortran 77/386 compiler version 2.40 in protected mode in March 1991, copyrighted in 1991 by Dieter Kraft, and released under a Berkeley Software Distribution license. This version was translated to JAVA by Jianchao Wang in May 2021. The java wrapper slsqp4j is an open-source API release under a Berkeley Software Distribution license. MIRDcell V4 uses a slightly modified slsqp4j. MIRDcell V4 can be downloaded at https://mirdcell.njms.rutgers.edu/ for Windows, Mac, and Linux operating systems.
1-Dimensional (1D) Planning AI Tool
The 1D Planning AI tool can be used to find the optimum combination from a given set of radiopharmaceuticals to achieve a specified therapeutic effect on a population of cells that are sufficiently far apart such that there is no cross-absorbed dose from neighboring cells, such as a suspension of cells in culture, circulating tumor cells, or disseminated tumor cells. Only the self-absorbed dose (absorbed dose from disintegrations within the cell) is used to calculate the survival probability for each cell. The therapeutic effect, specified as a SF of the cells, is achieved while minimizing the number of total disintegrations occurring among all of the cells.
Each cell has a different number of receptors for each drug that targets a unique receptor (Fig. 1). Given the nonuniform distributions of these different receptor sites, the molar activity of each radiopharmaceutical needs to be optimized to realize the desired therapeutic effect while minimizing the total number of disintegrations.
Nonuniform distribution of radiopharmaceuticals (drugs) among cells.
The data to be uploaded to MIRDcell are the molecules per cell for each radiopharmaceutical drug in each cell, Mkj, where the indices k and j represent the kth cell and jth radiopharmaceutical drug, respectively. Such data can be obtained with quantitative flow cytometry when fluorescence can be used (1,25,26). Alternatively, the fluorescence intensity units per cell (FIU/cell) for each drug on each cell can be uploaded along with specifying L, the calibration coefficient for molecules per cell per FIU/cell. When the matrix product of this distribution is calculated with a column vector parameterizing the product of molar activity, qj, of the jth radiopharmaceutical, and the time-integrated activity coefficient, , divided by the Avogadro number, NA, this results in a vector of total disintegrations,
, in each cell, k.
Eq. 1
In Equation 1, n is the number of cells in the population and m is the number of radiopharmaceuticals used. This matrix product is calculated for each combination of the set of radiopharmaceuticals used. For example, if there are 4 radiopharmaceuticals (drug 1, drug 2, drug 3, drug 4), the matrix product given in Equation 1 is calculated for each combination (i.e., drug 1, drug 2, drug 3, drug 4, drug 1–drug 2, drug 1–drug 3,…, drug 1–drug 2–drug 3–drug 4).
The optimizing algorithm in the 1D Planning AI tool works by minimizing the total number of disintegrations in the entire cell population that are required to achieve a specified target SF, subject to several constraints. The optimization process is performed for each combination of drugs. The optimization algorithm can be summarized as follows:Eq. 2
Eq. 3
Eq. 4where in constraint 2, the subscripts 1, 2, etc. denote a subset of the drug combination in question. As an example, consider the 3-drug combination of drug 1–drug 3–drug 4. Constraint 2 ensures that total disintegrations resulting from the 3-drug combination would remain less than or equal to the minimum of total disintegrations from each of the 1- and 2-drug combinations of drugs 1, 3, and 4 (i.e., minimum of
,
,
,
,
, and
. The survival probability of each cell, Pk, is calculated in MIRDcell using a noninteracting linear quadratic (LQ) model (21). The contributions from source regions of the cell (nucleus, cytoplasm, and cell surface) are accounted for by tallying the absorbed doses to the nucleus of each cell from each radiation type. It is important to note that the “1D Planning” tab uses the self-absorbed dose from each radiation type to calculate the survival probability of a cell from that radiation type. The product of the probabilities for each radiation type gives the net probability of surviving the radiation insult from all radiation types. As an example, when the cell nucleus is considered as the source and target and the radiation type is designated by the ICODE, the probability that the kth cell survives is given by Equation 5. The ICODEs for several radiation types are given in the MIRD: Radionuclide Data and Decay Schemes monograph (27). Following the notation used in MIRD Pamphlet No. 27 (21),
Eq. 5where
Eq. 6
In Equation 6, fN, fCy, and fCS are the fractions of cell activity in the nucleus, cytoplasm, and cell surface, respectively. The quantity is the time-integrated activity in the kth cell, and the S coefficients,
,
, and
, are the self-absorbed doses to the cell nucleus per decay in the nucleus, cytoplasm, and cell surface, respectively, from the radiation type designated by ICODE (21). When all radiation types (i.e., ICODEs) are considered, within the noninteracting LQ model (21),
can be written as
Eq. 7
Once the optimization is completed, the optimized number of total disintegrations from the drug j is divided by its corresponding time-integrated activity coefficient to obtain the optimized total activity, . The optimized molar activity of drug j,
, is given by
Eq. 8where
is the total number of moles of the radionuclide drug j that is summed over all n cells.
3-Dimensional (3D) Planning AI Tool
The 3D Planning AI tool optimizes the formulation of radiopharmaceuticals for treating spheric multicellular clusters (i.e., spheroids, micrometastases). The main algorithm used in the 3D optimizer takes a similar approach as the 1D version; however, the cross-dose is included and there is a difference in the manner in which the total disintegrations are calculated. Unlike the 1D optimizer, the 3D version supports a variety of activity distributions (uniform, normal, lognormal, exponential, linear, and polynomial) in a spheric multicellular cluster geometry. These activity distributions are built into MIRDcell and can be selected by the user. In addition, experimental radial distributions of disintegrations per cell can be uploaded to the 3D optimizer. One of the main differences in 3D is that it allows the user to specify a region within which the disintegrations are minimized, whereas in 1D, the total disintegrations among all the cells are always minimized. This option in 3D becomes helpful when analyzing spheric clusters bathed in a radioactive medium in which the cross-irradiation from disintegrations in the medium is considered. In that situation, the user can instruct the algorithm to minimize the disintegrations in the medium while achieving a user-specified target SF in the multicellular cluster. The thickness of the shell corresponding to the medium should be approximately equal to the longest range of the particle radiations emitted by the radionuclides that are dosimetrically and radiobiologically significant. In this case, only the disintegrations in the user-specified region are considered for constraint 2 (Eq. 4).
Built-in Activity Distributions
Built-in activity distributions include uniform (random), lognormal (random), normal (random), linear (radial), and exponential (radial). Here, the user specifies the average number of molecules per cell for each drug in the cocktail. A mean activity per cell of 1 mBq is initially assumed, and the number of disintegrations in the cell are calculated by multiplying the mean activity per cell by the time-integrated activity coefficient. This results in a distribution of disintegrations among cells. The self-absorbed doses to a target region within the cell from the disintegrations in source regions within the cell are calculated as described above using the corresponding S coefficients. In the 3D optimizer, both self- and cross-absorbed doses from each radiation type (ICODE) and source region are used in calculating the survival probability of a given cell with the assumption of a noninteracting LQ model. In the noninteracting LQ model, the survival probabilities that are calculated from the self-doses are multiplied by the probabilities calculated from the cross-doses to evaluate the probability of survival of a given cell. Similar to the 1D optimizer, the 3D version works by minimizing the objective function given by Equation 2 subject to the 2 constraints given by Equations 3 and 4. One key difference in 3D is that only the number of disintegrations in a user-specified region is minimized in the objective function. Another difference in 3D is that, at each iteration of the optimizer, the absorbed dose distributions (i.e., for each ICODE from each source region) resulting from a given drug j is scaled by a factor, Xj, until the resulting SF from all drugs is matched with the specified target SF within a specified accuracy. The initial distribution of disintegrations from a drug j is then multiplied by the final scaling factor Xj, to obtain the optimized distribution of disintegrations for that drug. This is given in Equation 9.
Eq. 9
The optimized activity distribution of the drug j is obtained by dividing the optimized disintegrations by the corresponding time-integrated activity coefficient. The total activity from the drug j, , in the cluster is obtained by summing the optimized activity over all of the cells. The 3D optimizer requires the average number of drug molecules per cell,
, for each drug as one of the inputs. This information is used to calculate the optimized molar activity,
, of the radiopharmaceutical (Eq. 10).
Eq. 10where NA is the Avogadro number.
Uploaded Radial Distributions of Disintegrations per Cell
The 3D AI tool accepts a .csv file containing radial distributions of disintegrations per cell for 1 or more drugs. The first column contains the radial positions, and the latter columns contain the disintegrations per cell. The multicellular cluster is divided into concentric shells, and the cells within each assigned concentric shell are assigned the same activity per cell determined from the uploaded radial distributions. The absorbed doses and the surviving probability of each cell are calculated in the same manner as described in the section “Built-in Activity Distributions.” The optimization process is similar to what is given in that same section and minimizes the disintegrations in the user-specified region (Eq. 2) subject to the 2 constraints given by Equations 3 and 4. However, the optimized molar activity is not computed in the same way as is with built-in distributions described in that section. When an experimental distribution is uploaded, the optimizer requires the user to input the molar activity for each drug that caused the said distributions of disintegrations. At the termination of the 3D optimizer, the initial molar activity of a given drug j () is scaled by the scaling factor Xj (described in the section “Built-in Activity Distributions”) to obtain the optimized molar activity of that drug.
Eq. 11
MIRDcell-Ã: A Software Tool to Calculate Radial Distributions of Disintegrations from Experimental Spatiotemporal Data
The “3D Planning” tab in MIRDcell requires the uploaded experimental data to be radial distributions of disintegrations per cell in a spheric multicellular cluster (e.g., spheroid or micrometastasis). However, when the experimental data for the drugs are spatiotemporal distributions of activity or fluorescence intensity, the data need to be converted to a radial distribution of disintegrations per cell before being uploaded to the “3D Planning” tab. To accomplish this preprocessing of experimental data, a software tool named MIRDcell-Ã, written in Python, has been developed. Its graphical user interface was designed using the software package Qt Designer (The Qt Company) (28) and is incorporated with an in-house Python script. This software tool is provided as a separate downloadable application with this MIRDcell V4 release.
MIRDcell-Ã accepts an Excel file (.xlsx file) containing a spatiotemporal distribution of drug concentration (in µM) as the raw data file. The spatial distributions at each time point should be along the radial direction of a spheric geometry. The other input parameters for MIRDcell-Ã that are required from the user are given in Supplemental Figure 1 (supplemental materials are available at http://jnm.snmjournals.org). The fractional drug concentration inside the spheroid relative to the outside (medium) of the spheroid is used to obtain the activity concentration at each radial position within the spheroid (29). This is achieved by multiplying the outer (medium) activity concentration by the corresponding fractional drug concentration. After correcting for physical decay of the radionuclide being used in the drug, the time-integrated activity concentration is calculated at each radial position of the spheroid. MIRDcell-Ã supports 2 main integration methods: a biexponential fit and variations of the trapezoidal method. These 2 methods are discussed in detail in (29). A pure trapezoidal integration or a hybrid method combining the trapezoidal either with the physical decay or with an exponential fit to the clearance data can be selected. At each radial position, the time-integrated activity concentration is converted to disintegrations per cell, and a file containing a radial distribution of disintegrations per cell is saved in .csv format.
Validation
Numerous validation tests were performed during the development of the AI optimizer. Examples of the validations conducted include the following: (1) Optimized cocktails of radiopharmaceuticals were generated with the “Multi-Drug” < “1D Planning” or “Multi-Drug” < “3D Planning” tab to arrive at optimized molar activities for a given target SF. Then we recreated the same scenario in the “Multi-Drug” < “1D Suspension” or “Multi-Drug” < “3-D Cluster” tabs, entered the optimized molar activities, and ensured that the target SF was obtained. (2) It was ensured that the order of radiopharmaceuticals does not affect the result. (3) A cocktail of 4 identical radiopharmaceuticals was created in “Multi-Drug” < “3D Planning” tabs, and it was shown that the various combinations have equal contributions from each constituent. (4) The “Multi-Drug” < “Results” < “Surviving Fraction” tab was created to visualize the checks that are conducted to ensure that the traditional Monte Carlo cell survival analyses used in the “1D Suspension” and “3D Cluster” tabs yield the user-specified target SF in the “1D Planning” and “3D Planning” tabs. (5) Optimizer exit modes were written to the “Output” tab and checked.
RESULTS
The 1D and 3D AI tools were tested with both created data and experimental data to illustrate realistic scenarios that highlight the benefit of the tool when used for designing radiopharmaceutical cocktails. The following examples demonstrate the utility of the MIRDcell V4 optimizer algorithm.
Cell Suspension Treated with a Cocktail of 4 211At-Labeled Antibodies
In this example, the data from a study conducted by Pasternack et al. (1) to assess the relative advantage of Ab cocktails targeting cultured MDA-MB-231 human breast cancer cells are used to illustrate the applicability of the AI tool for planning treatments of circulating and disseminated tumor cells. The 4 Abs used in the Pasternack study were Ab1 (APC anti-EGFR (AY13)), Ab2 (AF-488 anti-CD-44 (691534)), Ab3 (Pacific Blue anti-CD-73 (AD2)), and Ab4 (PE anti-CD-44 (BJ18)). The experimental data consisted of the number of molecules of each Ab on each of 298,000 cells after treatment with a cocktail of Abs with each Ab at a concentration of 1 μg mL−1. The “Multi-Drug” < “1D Planning” tab of MIRDcell V4 was used to create 4 drugs, and the radii of the cell and its nucleus were set to 5 and 3 μm, respectively. The radiation type was selected as At-211 + daughters, and each Ab was distributed on the cell surface. The “Source Radiation” was selected as 211At + daughters, which emit 5.87 and 7.45 MeV α-particles with yields of 0.42 and 0.58 per decay of 211At, respectively. The other parameters used in the optimization are given in Supplemental Figure 2. The distributions of drug molecules on cells for each drug combination are shown in Supplemental Figure 3. The AI tool was run with a target SF of 0.0001 to simulate tumor eradication.
The graphical output of the AI tool from the MIRDcell V4 “Multi-Drug” < “Results” < “Decays for Equieffect” tab is shown in Figure 2. Ab4 required the most 211At disintegrations when used alone, and Ab3 required the least. Mixing Ab1 and Ab2 afforded little or no benefit relative to using either alone in terms of reducing the total 211At disintegrations required. In contrast, mixing Ab2 and Ab3 had a considerable effect on the total 211At disintegrations needed; it reduced the required disintegrations by a factor of 1.6 relative to the best single drug (i.e., Ab3). A similar but less prominent effect can be seen between Ab3 and Ab4. In the example provided here, little or no benefit is seen from adding 3 or more drugs to the mixture. However, it is important to note that this is specific to this example and could change if the drugs had different distributions in the cell population that favored a more complex mixture. Note that the comparative responses to each single Ab do not dictate how they will behave in a cocktail of 2 Abs. One should not expect that 2 Abs that have similar outcomes alone will have equal contributions when used as a cocktail. That would only occur if they were identically distributed (i.e., each cell has an equal amount of each Ab, but amounts differ on a cell-to-cell basis). A key feature of the MIRDcell AI tool is that it determines the optimized total number of disintegrations required and the corresponding molar activities for each drug of the different combinations. These are provided in the MIRDcell V4 “Output” tab (see supplemental materials) and are summarized in Table 1. This can be used to guide the radiochemical preparation of the optimal cocktail.
Total disintegrations required to achieve target SF of 0.0001 for each drug combination. Bar for Ab4 (2.66 × 109 Bq) has been cut off to provide this zoomed-in version focusing on disintegrations up to ∼5 × 108 Bq.
Optimized Results for Each Combination of 211At-Labeled Abs
Multicellular Cluster with 2 Lognormally Distributed 195mPt-Radiopharmaceuticals
In this example, the “Multi-Drug” < “3D Planning” tab of the MIRDcell V4 software was used to create 2 radiopharmaceuticals (drugs) with the “Source Radiation” being 195mPt, a radionuclide that emits an average of 36 Auger electrons per decay (30,31). The drugs were each distributed lognormally in a multicellular cluster with a spheric geometry (radius, 200 μm). An average number of 100,000 drug molecules per cell was set for both drugs, and the activity for each was distributed in the nucleus of the cell. The other parameters used in the simulation are given in Supplemental Figure 4. The optimizer was set to run to achieve a target SF equal to 0.001 while minimizing the total number of disintegrations within the cluster.
Figure 3 illustrates the optimized results of the MIRDcell AI tool in terms of the total number of disintegrations required by each combination to achieve the target SF. It is clear from Figure 3 that the combination of drug 1 and drug 2 achieved the required SF with approximately half of the total disintegrations used by any individual drug. Furthermore, it eases constraints on radiochemistry because the SF can be achieved with reduced molar activities for each of the drugs, as given in Table 2.
Total number of disintegrations required to achieve target SF of 0.001 for each combination of 2 195mPt-radiopharmaceuticals that are independently distributed lognormally among cells comprising spheric multicellular cluster with radius of 200 μm. Details regarding parameter settings are provided in Supplemental Figure 4.
Optimized Results for Each 195mPt–Drug Combination
Multicellular Cluster with 2 Radially Distributed 225Ac-Radiopharmaceuticals
This example illustrates the utility of the optimizer when using 225Ac delivered by 2 radiopharmaceuticals that provide different radial distributions within a tumor of spheric geometry. The “Multi-Drug” < “3D Planning” tab of the MIRDcell V4 software was used to create 2 drugs transporting 225Ac within a multicellular sphere with a radius of 190 μm (Supplemental Fig. 5). The “Source Radiation” was selected as 225Ac + daughters, with the principal radiation being 5 α-particles and 2 β-particles. The experimental radial distribution of molecules per cell of Stras et al. was used for one of the drugs (32). These data were for liposomes with properties that allowed deeper penetration into the spheroid. The spatiotemporal distribution of disintegrations per cell was calculated for 225Ac-liposomes by Katugampola et al. using the Stras et al. data (33). The outer region of the spheroid had lower disintegrations per cell values due to a low liposome concentration closer to the edge of the spheroid (see Supplemental Fig. 5); this is discussed in more detail in Katugampola et al. (29). The second drug was distributed exponentially from the outer edge of the spheroid to a depth of 30 μm using a built-in feature of MIRDcell. The other parameters used in the MIRDcell AI are given in Supplemental Figure 6. A target SF of 0.001 was set, and the optimization was run to determine the best combination of the 2 drugs that would result in the minimum number of disintegrations within the cluster to achieve the required SF.
Figure 4 illustrates the optimized total disintegrations required within the spheroid to achieve the target SF for each combination of the 2 drugs. Drug 2 alone, which had an exponential distribution to a depth of 30 μm from the edge from the spheroid and required the largest number of disintegrations to achieve the required target SF. Drug 1 alone, namely tumor-penetrating liposomes (Supplemental Fig. 5), required about 42,500 disintegrations, which was about 1,500 times smaller than that required by drug 2 alone. However, when both drugs are combined, the optimizer predicts that the same SF could be achieved by roughly half the disintegrations required by drug 1 alone. The order of entry of the drugs had no impact on the results. This is an important result because, even though drug 1 is quite effective on its own, the required disintegrations could be reduced almost by a factor of 2 by adding the less effective drug 2 to the mixture. This could potentially halve the absorbed dose to normal tissues.
Total disintegrations of 225Ac required to achieve target SF of 0.001 in multicellular cluster with radius of 190 μm for each possible combination of 2 225Ac-labeled drugs. Drug 1 is 225Ac-liposomes; drug 2 is exponential radial distribution of 225Ac with penetration depth of 30 μm. In this zoomed-in version, blue bar for drug 2 alone extends beyond top of figure. Drug 1 has 46,640 decays; drug 2 has 68,390,000 decays; drug 1 (17,844) with drug 2 (6,668) has 24,451 decays. 1 decay = 1 Bq s.
The reduction in the required number of disintegrations, when the drug mixture was used with the optimized molar activity ratios, was mainly due to the more uniform absorbed dose distribution within the spheroid (Fig. 5A). The drop in absorbed dose to the cells from the 225Ac-liposomes (drug 1) near the edge of the spheroid (magenta color in Fig. 5A) is compensated by the absorbed dose from drug 2 (blue color in Fig. 5A). The optimizer adjusts the molar activities of each drug to minimize the total disintegrations required to achieve the predefined target SF for the given spatial distributions of the 2 drugs. Figure 5B shows the dose–volume histogram for the spheroid, and it is seen that, with the optimized results, 100% of the cells in the spheroid received at least 4.5 Gy.
(A) Radial distribution of absorbed doses to cells within spheroid from optimized cocktail of 225Ac-liposomes (drug 1, magenta) and 225Ac-labeled drug with exponential radial distribution (drug 2, blue). (B) Dose–volume histogram resulting from optimization of 2 drugs. As per Figure 4, total decay is 24,451 Bq.
DISCUSSION
The main objective of this work is to present the MIRDcell V4 AI tool that optimizes the therapeutic efficacy of a cocktail of radiopharmaceuticals within our established MIRDcell platform. The features and capabilities of the optimizer are limited to some extent by those in MIRDcell V3. Among the limitations is that MIRDcell V4 permits only a single type of cell and models the cell and cell nucleus as concentric spheres. Nevertheless, Goddu et al. (34) have shown that the effect of the shape of the cell is minimal on the absorbed dose provided that the range of the particle is not very close to the cellular dimensions (35). Therefore, the results for each of the 3 examples presented (211At on the cell surface,195Pt in the cell nucleus, and 225Ac on the cell surface) should not be affected substantially by assuming spheric cell geometry. MIRDcell presently only accounts for effects due to direct energy depositions by different radiation types, and bystander effects are not considered (36–38). Other assumptions and limitations of MIRDcell are discussed in detail in Vaziri et al. (39) and Katugampola et al. (21). Despite these limitations, some studies have shown a good agreement between SFs predicted by MIRDcell and experimental measurements (21,29).
Additional limitations are introduced into the AI 3D Planning in MIRDcell V4. Only a single multicellular cluster geometry (spheric) and size are used in the optimization process. Diffusion kinetics within the tumor are not explicitly considered in the optimization process (40); however, an experimentally measured 3D spatiotemporal activity distribution can be analyzed with the companion software tool, MIRDcell-Ã (section “MIRDcell-Ã: A Software Tool to Calculate Radial Distributions of Disintegrations from Experimental Spatiotemporal Data”), that integrates time-varying radiopharmaceutical kinetics including diffusion effects. The MIRDcell-Ã output can be uploaded to the 3D Planning tool. It is important to note that for the 1D AI tool, in which an experimental distribution of fluorescence or carrier molecules per cell is uploaded to the “1D Planning” tab, the time variation of those quantities are not explicitly modeled in the optimization process. In addition, MIRDcell V4 optimizer does not specifically consider the absorbed dose delivered to normal tissues. The user should be aware that specific radiobiologic and organizational characteristics of the tissues of the organs at risk will influence the probability of harming them. The optimal cocktail formulation would be that which provides the highest tumor control probability with the lowest normal-tissue complication probability. Nevertheless, the 3D Planning tool does permit one to minimize the decays in the optimization region to achieve the desired SF in the target SF region. The optimization region could be considered a normal-tissue region if desired.
The optimizer is designed to handle a wide range of input values: fluorescence, molecules per cell, decays per cell, and built-in activity distributions. However, if one drug is significantly more effective at killing cells than another or if the target SF is set to a value that corresponds to killing all or nearly all of the cells in the multicellular cluster (e.g., 0.0001 target SF in a cluster with 10,000 cells), there is a possibility that some combinations involving those drugs might cause the SLSQP optimization algorithm to fail. This may be caused by the algorithm failing to converge or having a matrix singularity when optimizing. Accordingly, the exit modes of the SLSQP are written to the “Output” tab along with their definitions as a guide to the user. These should always be checked after each run—exit mode of 0 means that the optimization terminated successfully, which instills confidence in the result. The most common failure is an exit mode of 8, “positive directional derivative for linesearch”. This can often be overcome by changing the target SF to a higher value. Results with nonzero exit modes should be considered suspect. It is the user’s responsibility to check the exit modes after every run as an important quality control check.
The optimizer minimizes the disintegrations within a user-specified region and accepts only 2 constraints. However, most of the practical scenarios (e.g., unavailability of large quantities of 1 drug, desire to limit 1 drug over another drug, etc.) can be addressed using the upper and lower molar activity limits of the optimizer. The molar activity limits are especially useful when constraints imposed by physical half-life and radiochemistry restrict the maximum molar activity that can be achieved for a given radiopharmaceutical. Furthermore, the AI optimization process uses MIRDcell-calculated cellular absorbed doses and then minimizes the disintegrations required to achieve the specified SF. Although, as in MIRDcell V3, a complex set of LQ parameters are used to account for responses to different radiation types and cellular target←source combinations, MIRDcell V4 does not use an equieffective dose or biologically effective dose to account for the effects of dose rate. Nevertheless, users can provide LQ parameters derived from experimental studies with external beams of low-linear-energy-transfer γ-rays (41) and α-particles (42) delivered at time-dependent dose rates that match the dose-rate profile of the radiopharmaceutical to achieve the same end.
It should be noted that the MIRDcell AI tool can also be used to plan treatments with a cocktail of unlabeled primary drugs (e.g., Abs) followed by administration of a radiolabeled secondary (43) or a cocktail of radiolabeled drug-specific secondaries. For example, consider the former case using the Pasternack et al. (1) cell culture data shown in Figure 2 and Table 1. When a cocktail of radiolabeled Abs is used directly at concentrations of 1 μg mL−1 each, the best 2-drug combination includes Ab2 and Ab3 with molar activities of 4.2 × 106 and 1.4 × 107 GBq mol−1, respectively. If a 2-step process is used, one could first treat the cells with 1 and 3.3 μg mL−1 of unlabeled Ab2 and Ab3 (i.e., the ratio of the molar activities for the 1 μg mL−1 case), then wash the cells to remove Ab2 and Ab3 from the medium, and then treat the washed cells with a radiolabeled molecule that attaches to Ab2 and Ab3. If linear uptake is assumed, administering these concentrations would increase the Ab3 molecules per cell by a factor of 3.3, thereby maintaining the same ratio of decays from Ab2 and Ab3 provided that the radiolabeled molecule binds to them with equal probability. An analogous strategy could be used to target tumor cells in vivo.
Although the utility and the effectiveness of the AI tools are demonstrated in this work, rigorous in vivo and in vitro studies with cocktails of different radiopharmaceuticals are needed to further validate the predictions. These should include studies with α-, β-, and Auger electron–emitting radionuclides and targeting molecules that achieve different subcellular distributions and penetration distances into spheroids or micrometastases. These studies should include cocktails that use a single radionuclide but also a combination of radionuclides (e.g., α- and β-emitter, Auger electron, and β-emitter, etc.). MIRDcell V4.14 presently accommodates combinations of radionuclides for a single user-specified tissue geometry. Ultimately, optimizations will need to consider a variety of tissue geometries within a given analysis. Furthermore, the time and computational costs of the SLSQP algorithm in the JAVA programming environment can lead to long run times. A MIRDcell V4 optimization involving 4 drugs with 300,000 cells typically takes about 20 h for completion on an Apple Mac computer with an M1 Max chip and 64 GB memory because of the code’s limited use of parallel processing. Future versions would benefit from full parallelization of the code and perhaps judicious curation of the data to reduce the size of the dataset without affecting the result significantly.
Finally, it should be noted that the ability to upload measured distribution data to MIRDcell provides a unique opportunity to optimize radiopharmaceutical cocktails on the basis of data from biopsies of circulating tumor cells, disseminated tumor cells, and tumors. However, analyses of such data should be done for experimental purposes only and not to design treatments for humans. MIRDcell has not been approved by the U.S. Food and Drug Administration. The optimization algorithms in MIRDcell V4 do not consider explicitly the absorbed doses received by normal tissues and their respective radiobiologic characteristics (i.e., LQ model parameters). A more complete optimization process will require consideration of the biologic responses in organs at risk, which are not necessarily the critical tissues that surround the tumor tissue and do not respond radiobiologically in the same way as the tumors.
CONCLUSION
An AI tool that can optimize a cocktail of radiopharmaceuticals has been developed and implemented within the MIRDcell platform. Coupled with other functionalities of MIRDcell, the AI tool can predict the optimum combination of radiopharmaceuticals required to achieve a predefined therapeutic goal (SF) with minimum total disintegrations along with the corresponding molar activities of each radiopharmaceutical in the cocktail, either in a 3D tumor geometry or in a suspension of cells. Importantly, formulating cocktails using this approach can reduce the required molar activities of the constituents, thereby easing molar activity constraints often encountered in radiochemistry.
DISCLOSURE
This work was supported in part by NIH 1R01CA245139. MIRDcell V2 and V3 are patented under US 8,874,380 B2; US 9,623,262 B2; US 9,804,167 B2; and US 10,295,543. In addition, the V4 USPO application 18/476,134 is pending. No other potential conflict of interest relevant to this article was reported.
KEY POINTS
QUESTION: Can patient-specific cocktails of radiopharmaceuticals be formulated to eliminate cancer cell populations with fewer total radionuclide decays than a single radiopharmaceutical?
PERTINENT FINDINGS: AI tools are developed within MIRDcell V4 to determine the molar activities for each drug that minimize the total disintegrations required for a cocktail of radiopharmaceuticals to achieve the desired killing of a population of cancer cells. Examples representing both circulating tumor cells and micrometastases demonstrate that radiopharmaceutical cocktails can be formulated that perform substantially better than any one of the radiopharmaceuticals alone.
IMPLICATIONS FOR PATIENT CARE: The long-term goal is to use patient biopsies to determine the optimal drug combination and use that information to create a patient-specific radiopharmaceutical cocktail to maximize the therapeutic effect for that patient using the minimum number of disintegrations.
ACKNOWLEDGMENTS
This work was done in collaboration with the Society of Nuclear Medicine and Molecular Imaging MIRD Committee: Rachel Marie Barbee, Alejandro Bertolet, Wesley E. Bolch, Yuni K. Dewaraja, William D. Erwin, Valentina Ferri, Roger W. Howell, Oleksandra V. Ivashchenko, Adam L. Kesner, Richard Laforest, Todd E. Peterson, Joseph G. Rajendran, George Sgouros, Carlos F. Uribe and Pat B. Zanzonico (Chair). Special thanks are given to the NJMS Rutgers IT team, who created and support the MIRDcell website and maintain the server. Thanks to Adam Kesner for supporting the distribution of MIRDcell via the MIRDsoft platform. Finally, thanks to Caroline Bolch for designing the MIRDcell and MIRDcell-Ã icons and logos.
Footnotes
Published online Oct. 24, 2024.
- © 2024 by the Society of Nuclear Medicine and Molecular Imaging.
REFERENCES
- Received for publication December 13, 2023.
- Accepted for publication September 18, 2024.